A thermogravimetric study of oxygen
diffusion in YBa2Cu3O7-δ
María Dolores Vázquez-Navarro
A dissertation submitted to the University of Cambridge for the
Degree of Doctor of Philosophy
Corpus Christi College
Cambridge August 1998
To my parents
Recte vives Licini, neque altum
semper urgendo neque, dum procellas
cactus horrescis, nimiun pretendo
litus iniquum
Auream quisquis mediocritatem
diligit tuts caret obsoleti
sordibus tecti, caret invidenda
sobrius aula
Saepis ventis agitatur ingens
pinus et celsae graviore casu
decidunt turres feriuntque summos
fulgura montes
Sperta infestis, metuit secundis
alterum sortem bene preparatum pectus,
informes hieles reducir Iupiter,
idem summovet.
Non si male nunc, et olim sic erit:
quondam cithara tacentem suscitat
Musam neque semper arcum
tendit Apollo
Rebus anguistis animosus atque fortis
appare; sapienter idem
contrahes vento nimium secundo
turgida vela.
Horace
Vivirás mejor, Licinio, no instando siempre
a la alta mar ni, por evitar prudente
las tormentas, acercándote demasiado a la
peligrosa orilla.
Quien ama la dorada medianía
carece seguro de la miseria de un
techo impropio; carece de un sobrio
palacio envidiable.
Con más frecuencia es zarandeado por los vientos el
pino ingente, las elevadas torres se precipitan
con más dura caída y los rayos hieren
los montes más altos.
Espera en la dificultad, y teme en la
prosperidad la suerte contraria con el pecho
bien dispuesto. Júpiter trae los espantosos inviernos;
también los remueve.
No porque ahora vaya mal será así siempre:
a veces la cítara motiva a la callada musa,
y no siempre Apolo
tiende su arco.
En la circunstancia adversa muéstrate animoso
y fuerte: de igual modo sabiamente
reducirás las hinchadas velas a
un viento favorable.
You will take a better course, Licinius,
if you do not always thrust over the deep sea,
or hug the dangerous coast too close,
shivering at the prospect of squalls.
Whoever loves the Golden Mean
is safe (no squalor for him in a filthy garret),
and temperate (for him no mansion
that men will envy).
The hughe pine is more cruelly tossed
by the winds, the loftiest towers
have the heaviest fall and lightning strikes
the tops of mountains.
The heart well prepared hopes in adversity
for a change in fortune, and fears in prosperity.
Jupiter brings back ugly winters
and Jupiter
removes them. If all goes badly now, some day
it will not be so. Sometimes Apollo rouses
the silent Muse with his lyre. He does not always
stretch his bow.
In difficult straits show spirit
and courage, and when the wind
is strong at your back, be wise
and shorten the bulging sail.
iSUMMARY
YBa2Cu3O7-δ (YBCO) was one of the first high temperature superconductors
discovered, and its superconducting properties are strongly dependent on oxygen
stoichiometry. A large amount of work has been done on the variation of
stoichiometry and its effect on the superconducting properties of the material.
However, in spite of all the work done, the results published in the literature are very
scattered.
This thesis presents a thermogravimetric study of oxygen diffusion in YBCO under
isothermal and non-isothermal conditions and tries to reconcile the data available
based on the results obtained and taking into account the factors that may have
affected the data presented by other groups, such as the effects of the microstructure
and the different diffusion coefficients measured with the techniques used. An
Arrhenius expression for the chemical diffusion of oxygen has been calculated from
the analysis of isothermal oxygenation data, and it has been corroborated by a study of
the non-isothermal experiments carried out. This work includes the development of a
macroscopic model for oxygen diffusion in YBCO based on the diffusion coefficient
calculated from experimental data. The model is used to simulate for the first time
oxygenations under both isothermal and non-isothermal conditions.
The study of non-isothermal oxygenations has led directly to the design of novel
cooling procedures that can be introduced at the end of the processing stage of YBCO
samples, producing highly oxygenated specimens in shorter times than for
conventional isothermal and ramped oxygenation procedures.
The final section of this dissertation presents a study of the Direct Current Zoning
effect. The generation of a mobile hot zone in a polycrystalline YBCO bar when
passing a current across it is directly linked to the diffusion of oxygen ions in the
material. A mechanism for the motion of the zone along the sample has been
suggested. A computer model has been developed to reproduce this process taking
into account the motion of ions due to chemical diffusion and the potential difference
established. The results from this model have corroborated the mechanism suggested
and give for the first time the opportunity to study this phenomenon in more detail.
ii
PREFACE
This thesis describes the work carried out in the Department of Materials Science and
Metallurgy of the University of Cambridge between October 1994 and July 1998
under the supervision of Prof. J.E. Evetts.
The work described is original except where reference is made to the work of others.
No part of this work has been submitted for any other degree at this or another
university. The work described has not been done in collaboration with others and it
does not exceed 60,000 words in length.
I would like to thank the Engineering and Physical Sciences Research Council, E.A.
Technology, the Interdisciplinary Research Centre (IRC) in Superconductivity, and
Corpus Christi College, Cambridge, for financial support during this research. I am
also grateful to Prof. C.J. Humphreys, Prof. A.H. Windle and Prof. Y. Liang for the
provision of laboratory facilities in the Department of Materials Science and
Metallurgy and at the IRC in Superconductivity.
Some of the work described in this dissertation has been published as follows:
“Microwave assisted oxygenation of melt-processed bulk YBa2Cu3O7-δ ceramics”.
A.T. Rowley, R. Wroe, D. Vázquez-Navarro, Wai Lo and D.A. Cardwell, J. Mat. Sci.,
Vol. 32 (1997) pp. 4541-4547.
“Analysis and Modelling of Oxygen Diffusion in YBa2Cu3O7-δ under Non-isothermal
Conditions”. M.D. Vázquez-Navarro, A. Kuršumoviç, C.Chen and J.E. Evetts, Inst.
Phys. Conf. Ser. No. 158 (paper presented at the European Conference on Applied
Superconductivity, The Netherlands, 30 June-3 July, 1997) pp. 1037-1040.
Mª Dolores Vázquez-Navarro
August 1998
iii
ACKNOWLEDGEMENTS
I am very grateful to all that have helped me during my career as a research student. I
would like to thank my supervisor, Dr. (now Prof.) Jan Evetts for his support,
encouragement and patience during these years. I am extremely grateful to Dr.
Lindsay Greer, for allowing me to use his Thermal Analysis System, and to Dr. Ilan
Blech and Dr. José Ruiz for their help with the development of the theoretical models
to the DCZ effect and macroscopic oxygen diffusion (respectively). I would also like
to thank Dr. Chen for kindly supplying the single crystals used. I am very grateful to
Dr. Ahmed Kuršumovic for arriving just at the right time to give me the breath of
encouragement and positiveness towards my work I needed at the time. No matter
what happens, it is always good to have a smile ready for everyone!. Thanks a lot too
to Andrew Moss, Dave Nicol , and Bryan Barber and Carol Best for their help when I
had to use the X-rays, SEM and needed to print just that little bit of the photograph
(but twenty of them, please!).
I once was told that we spend most of our lives working (not counting the time we are
asleep, of course); in the case of a PhD student this working life seems to be much
longer than anybody else’s. I would therefore like to thank everyone in the lab for
making it a nicer place where to spend all those hours: Gavin (incredibly patient with
my computer-related questions), Wilfred (thanks for all the support and for your help
in making the ‘box’… I am sure it will be useful to coming generations!), Richard
(being positive and enthusiastic all the time), Bee-San (ah, those long talks during the
long weekends!), Peter (you will always get the highest score!), Kim (go slowly, go
slowly!), Andrés (thanks for answering all my physics questions and for the morning
chats about almost everything! - and apologies to the rest of the office for the loud
Spanish natter). To you all and the rest of the lab (Mark, Zoe, Bartek... the ones that
are still here and those that have already left), “heaps of thanks” for having had the
patience to share all this time with a (very often) stressed Dolores who used to go to
Coffee and never went to Tea.
Thanks to Margaret for always being there and for giving me a room in 25 Cranmer
Rd., as I had the chance of sharing the house with Jon (best ‘uncle’ ever!), Alison :-),
Meera (thanks for all the de-stress lunches and so much more!) and Emma. There
iv
won’t be a place like no. 25. And to everyone in Leckhampton, especially the
‘Venezuelan contingent’: Juan Carlos, Deisy, Steve, Alena, Ernesto, Nadia and
Thomas for introducing me to Latin American music, going to all those dance-till-
you-drop parties and memorable punting trips to Grantchester (plus other random
celebrations).
I am most grateful to my London connection: Pablo, Javier (Molero), Javier
(Madurga), Dalia, Tatiana, Pete & Co. for being such good listeners and preserving
my sanity by giving me the chance of running away from Cambridge to good ol’
London every now and then.
A very special ‘thanks’ to Miguel, for his support and advice during these years,
always being at the other end of the phone/E-mail line to listen to whatever I had to
say. I wouldn’t have been able to get this far on my own. (Apologies for the long
coach journeys to and fro!).
And finally, I would like to thank my parents and sister for all they have done for me
all these years, and without whose love, encouragement and understanding I would
not be writing these lines. You are next, Marga!.
Chapter 1  Introduction to YBa2Cu3O7-δ
4
Chapter 1
Introduction to YBa2Cu3O7-δ
1.1 Introduction
This chapter presents a brief introduction to the subject of superconductivity: its
discovery and the search for a room temperature superconducting material. The
phenomenon of superconductivity is briefly outlined and the chapter ends with a
description of the dependence of the YBa2Cu3O7-δ chemical structure on its oxygen
stoichiometry.
1.2 Discovery
In the early decades of this century, after having successfully liquefied helium in 1909,
the Dutch physicist Kamerlingh Onnes conducted a study of the electrical properties
of metals at low temperatures. He observed that when mercury was cooled to
temperatures in the range of the boiling point of liquid helium, its resistivity suddenly
decreased to such a small value that he was unable to measure it.
Fig. 1.1 Relative variation in resistance ρ/ρRT vs. temperature (K) similar to that observed by K. Onnes
in Hg. This plot corresponds to the change in resistance with temperature of Tl2Ba2Ca2Cu3O10 (from
Cyrot and Pavuna (1992))
Chapter 1  Introduction to YBa2Cu3O7-δ
5
Onnes observed that below 4 K, mercury passed into a new state with electrical
properties unlike those previously known. This new state was called the
“superconducting state”. Superconducting materials offer no resistance to the flow of
electrical currents through them as long as their temperature is below a certain value
known as critical temperature: Tc. The discovery of superconductivity in mercury was
the starting point of the quest for a room temperature superconductor. This material
would allow the transport of energy without any losses, increasing the efficiency of
this process to 100 %.
During the following years it was found that more than 20 metallic elements exhibited
superconductivity at low temperatures. In 1913 Pb was found to be superconducting at
7.2 K and in 1930 Nb was discovered to have a critical temperature of 9.2 K, the
highest of all pure metals.
The study to find new materials with higher Tc’s continued with alloys and
compounds, which have higher critical temperatures than elemental metals. Nb
compounds have become very important, as they can be used in a large number of
small and large scale applications (e.g. magnetic field sensors and superconducting
magnets respectively). Nb3Ge exhibits the highest critical temperature of the group,
with Tc = 23.3 K. Further work on this field led to the discovery of superconductivity
in the Chevrel phases. The highest critical temperature in the series was 15 K,
measured in PbMo6S8. Organic superconducting materials were also discovered in the
early 1980’s. However, their critical temperatures are much lower than those of any of
the inorganic compounds mentioned so far, the highest being 10 K.
From 1973 to 1986 the 23 K figure of Nb3Ge remained at the top of the list of critical
temperatures. Oxide ceramics, such as the perovskites SrTiO3 and BaBiO3 could be
made to superconduct at low temperatures by reduction or by doping, but their Tc was
lower than that of Nb3Ge. In 1986, however, Bednorz and Müller (1986) synthesised
La2-xBaxCuO4 (LBCO). The crystal structure of this compound is based on that of the
first superconducting oxide, the perovskite SrTiO3. This compound was the first of
what are known today as high temperature superconductors, with a Tc = 35 K. After
this encouraging discovery an increasing amount of work was dedicated to studying
and developing new perovskite-like structures and in 1987, Wu and co-workers (Wu
Chapter 1  Introduction to YBa2Cu3O7-δ
6
et al. (1987)) and Zhao et al. (1987) found that substituting the La with Y resulted in a
compound, YBa2Cu3O7-δ (YBCO), with a Tc of 93 K, almost 40 degrees higher than
LSCO. This was a revolutionary discovery because finally, the material discovered
had a Tc above the boiling point of liquid nitrogen (77 K). This meant that the
applications for this superconducting compound would not require such a complex
cooling system as with the other materials that needed liquid He for their operation. In
the same year another superconducting compound, Bi-Sr-Ca-Cu-O (BSCCO) was
discovered. Of the two superconducting phases of this compound, Bi2Sr2Ca2Cu3O10+δ,
(2223) has the highest critical temperature: 110 K. Shortly after another compound
with the same crystal structure, Tl2Ba2Ca2Cu3O10, was found to superconduct at 125K.
In 1993 a mercury compound with a similar structure was discovered,
HgBa2Ca2Cu3O8-x, and its critical temperature is the highest observed up to date: 133
K. If the material is subjected to high pressure the Tc can be increased to 164 K. The
discovery of high temperature superconductors seems to have reached a temporary
standstill due to the increased complexity of the crystal structures of the compounds
discovered. This structure also limits the extent of their industrial applications, as
these materials are brittle and difficult to form and process for most large scale
applications.
1.3 Structure and stoichiometry of YBa2Cu3O7-δ
Most high Tc superconducting oxides are cuprate compounds with a characteristic
layered structure. The unit cell of superconducting cuprates of the RBa2Cu3O7-δ type
(R = rare earth) is developed from that of a tetragonal perovskite repeated along the c
axis, and it consists of a sequence of copper-oxygen layers. This section presents a
review of the structure of YBa2Cu3O7-δ as an introduction to the work presented in this
thesis.
The perovskite structure has the general formula ABO3. It was first found from the
mineral perovskite CaTiO3 and it consists of Ti4+ ions contained in octahedral cages of
O2- ions leaving a large cuboctahedral hole where the larger Ca2+ ion sits (see Fig.
1.2). Changes in temperature alter the coordination of the Ti ions with its
Chapter 1  Introduction to YBa2Cu3O7-δ
7
neighbouring oxygen ions, and this modifies the crystal structure of the material: from
tetragonal at low temperatures to orthorhombic and orthogonal at high temperatures.
Ca
Ti
O
Fig. 1.2 The perovskite structure
The crystal structure of YBCO consists of three perovskite-like unit cells stacked one
above the other. The central cell has an Y atom sandwiched between two corrugated
CuO2 planes. Above and below these CuO2 planes there is a BaO2 layer on top of
which lies a Cu-O basal plane with a variable oxygen content. The whole structure is
symmetrical with respect to the Y ion that can be substituted by a number of rare
earths (except Pr and Yb) without losing its superconducting properties.
a) b)
O(1)
O(4)
O(5)
O(2)
O(3)
Cu(2)
Cu(1)
Ba
Y
Basal plane
Cu-O chains
Basal
plane
CuO2 planes
Fig. 1.3 a) Structure and model unit of layered YBa2Cu3O7, and, b) YBa2Cu3O7-δ..When the oxygen
content is <7, O(1) and O(5) sites are partially filled
Chapter 1  Introduction to YBa2Cu3O7-δ
8
The stoichiometry of YBa2Cu3O7-δ varies with temperature and partial oxygen
pressure, the value of δ ranging from 0 to 0.85 (see Fig. 1.5). As oxygen is added to or
removed from the structure the oxygen ions rearrange themselves in the Cu-O basal
plane with a predominant tendency to form Cu-O chains (Poulsen et al. (1991) and
Jorgensen et al. (1990)). This change in oxygen content modifies the crystal structure
of YBCO and its superconducting properties. The decrease in oxygen has a marked
effect on the critical temperature, Tc (Fig. 1.4). Table 1.1 shows that there is some
disagreement in the published values linking the critical temperature of YBCO to its
characteristic oxygen content. Some authors observe a peak in the critical temperature
for oxygen contents of 6.97, while others have measured a constant value of Tc in this
region. It appears that the measurement of properties related to the oxygen
stoichiometry in YBCO is always associated to a certain amount of uncertainty in the
values calculated. This is probably due to the errors associated to the measurement of
the oxygen content of the material.
Fig. 1.4 Schematic variation of critical temperature with oxygen content (from Cyrot and Pavuna
(1992))
Chapter 1  Introduction to YBa2Cu3O7-δ
9
Oxygen content Tc (K) (1) Tc (K) (2) Tc (K) (3)
6.99 94 93 94
6.98 93 94 93
6.97 92 93 92
6.96 90 93 90
6.91 84.3 91 90
6.87 63 86.5 89
6.7 60 66.5 82.9
6.62 57.1 61 68.6
6.54 50 58 60
6.45 47.1 54.2 57.6
6.4 31 47.9 57.5
Table 1.1 Variation of critical temperature with oxygen content from different authors: (1) Jacobson et
al. (1987), (2) Poulsen et al. (1989), (3) Cava et al. (1987)
1.3.1 Orthorhombic to Tetragonal Transition
The stoichiometry of YBCO is the most important factor to consider in order to
optimise its superconducting properties, and a great deal of work has been dedicated
to the study of the variation in oxygen content of this material. It has been found that
the equilibrium stoichiometry depends both on temperature and partial oxygen
pressure (Gallagher (1987), Kishio et al. (1987), etc.). Fig. 1.5 shows data by
Gallagher that represents the variation in oxygen content in YBCO with respect to
temperature as well as partial oxygen pressure.
Fig. 1.5 Variation in oxygen content with temperature and oxygen pressure (from Gallagher (1987))
Chapter 1  Introduction to YBa2Cu3O7-δ
10
If we take a sample of YBCO at equilibrium (δ = 0), the oxygen ions in the Cu-O
basal plane are ordered on the O(1) sites only (the O(5) sites remain empty), forming
one-dimensional linear Cu-O chains along the b axis. The crystal structure has
orthorhombic (Pmmm) symmetry (a ≠ b). If the temperature of the sample is
increased, oxygen is lost from the structure (see Fig. 1.5). This decrease in oxygen
content results in a disorder of the ordered orthorhombic structure; oxygen ions move
into the adjacent and vacant O(5) sites until the loss of oxygen is so great that the
oxygen-oxygen interactions are not strong enough to maintain the orthorhombic
structure and the material undergoes a structural transformation to a tetragonal
symmetry (P4/mmm) (see Fig. 1.6).
a)  
O(5)
Tetragonal
O(1)
Orthorhombic
Cu(1)
b)
Filled oxygen site
Cu site
Empty oxygen site
Fig. 1.6 a) Occupancy of oxygen ions in lattice sites (from Jorgensen et al. (1987-a)), and b)
orthorhombic and tetragonal basal planes
Neutron diffraction studies carried out by Jorgensen et al. (1987-a) show the change in
the occupancy of the O(1) (1/2, 0, 0) and O(5) (0, 1/2, 0) sites in the crystal structure
as it loses or gains oxygen (see Fig. 1.6 a)). Strauven et al. (1988) measured changes
in oxygen occupancy from 0.85 to 0.25 for (1/2, 0, 0) and from 0 to 0.25 for (0, 1/2, 0)
as fully oxygenated samples were heated in air from room temperature to 720 °C,
agreeing with the results presented by Jorgensen and co-workers. The difference in
oxygen content, 0.35, evolves from the material into the surrounding atmosphere. This
transition from orthorhombic to tetragonal crystal structure has been found to be a
second order order-disorder transformation (Freitas and Plaskett (1987)) and it
depends on partial oxygen pressure as well as on the kinetics of oxygen diffusion in
the material (Shi (1989)).
Chapter 1  Introduction to YBa2Cu3O7-δ
11
The results of data published by the different authors on this subject are fairly
consistent. The transition has been found to occur at temperatures ranging from 676 to
700 °C in pure oxygen (Specht et al. (1988), Jorgensen et al. (1987-b)) to
approximately ≤ 520 °C at 0.09 atm O2 (O’Bryan and Gallagher (1987)).
The value of the oxygen content at which the orthorhombic to tetragonal transition
occurs has not been observed to depend on partial oxygen pressure, and the values
reported for the oxygen content at the transition vary from 6.5 (Freitas and Plaskett
(1987)) to 6.68 (Kishio et al. (1988)) (see Fig. 1.7 b)). The scatter in the data is very
likely to be the result of differences in sample preparation or poorly controlled
experimental conditions. This transition is a direct consequence of the oxygen content
and its ordering in the crystal structure.
450
500
550
600
650
700
750
0.010.1110100
oxygen pressure (atm)
Te
m
pe
ra
tu
re
 (C
)
Jorgensen Freitas
Eatough Krebs
O’Bryan LaGraff
Average Specht
a)
6
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7
0.010.1110100
oxygen pressure (atm)
O
x
yg
en
 
co
n
te
nt
 
 
 
Jorgensen Specht
Krebs O’Bryan
Gallagher Average
b)
Fig. 1.7 a) O-T transition temperature values, and b) O-T oxygen contents at different oxygen
pressures collected from literature
Chapter 1  Introduction to YBa2Cu3O7-δ
12
The O-T transition has been found to be irreversible in inert atmospheres, this is a
direct consequence of its relation to oxygen content. On heating, the orthorhombic
material loses oxygen to the surrounding atmosphere. If this atmosphere is an inert
gas, such as N2 or He, oxygen cannot be absorbed back from it when the temperature
decreases, so the sample remains tetragonal and no O-T transformation is observed.
The authors mentioned so far have also measured the variation in the values of the
lattice parameters with oxygen content and partial oxygen pressure. a and b have been
observed to become shorter with decreasing oxygen content until they become
equivalent (tetragonal crystal structure); c, on the other hand, is observed to increase
slowly with decreasing oxygen content (Jorgensen et al. (1987-b), Schuller et al.
(1987), LaGraff et al. (1990)).
3.8
3.84
3.88
3.92
3.96
4
6.46.56.66.76.86.97
Oxygen content 
La
tti
ce
 
pa
ra
m
et
er
s 
(A
)
c /3
a, b
a
b
Fig 1.8 Change in lattice parameters of YBCO with oxygen content (average of data collected from
literature)
For an oxygen content of 6.91, the crystal lattice of YBCO is orthorhombic with
lattice parameters: a = 0.382 nm, b = 0.388 nm and c = 1.168 nm, and the values for
the tetragonal structure (for an oxygen content equal to 6.62) are: a = b = 0.387 nm
and c = 1.172 nm (Datta (1992)).
1.3.2 Ortho I and Ortho II phases
It has been mentioned in section 1.2.1 that the critical superconducting temperature,
Tc, varies with oxygen content. The generally accepted view is that superconductivity
Chapter 1  Introduction to YBa2Cu3O7-δ
13
is intimately related to the CuO2 planes, and the variation of Tc is the result of a
charge redistribution in the structure that increases the hole content in these planes.
This is directly related to the oxygen content of the material.
Cava et al. (1990) observed a non-linear decrease in the electronic charge with
decreasing oxygen content in the copper of the CuO2 plane; this corresponds exactly
to the variation of Tc with oxygen content. LaGraff et al. (1990) observed in their
resistance measurements of YBCO that as the oxygen content decreased so did the
carrier concentration (electron holes). This agrees with theoretical work by Zaanen et
al. (1988). However, Veal et al. (1990) observed that after annealing an orthorhombic
sample at low temperature (≈ 150 °C) the Tc increased; they point out that this is due
to a rearrangement of the oxygen ions in the basal plane. Veal and co-workers suggest
that when the Cu-O chains are ordered, there is an increase in the in-plane carrier
concentration that alters the Tc. The preference of the oxygen ions to align themselves
into chains maximises the number of 2-coordinated monovalent coppers. Each Cu
acquires an electron from the CuO2 planes, maximising the hole concentration and
increasing the critical transition temperature. In this process oxygen ions that are in
normally vacant chain sites (O(5)) move to normally chain occupied sites (O(1)). High
resolution electron microscopy and electron diffraction studies (Beyers et al. (1989))
support the observations made by Veal and co-workers; these show the tendency for
oxygen to order in 2-dimensional chain structures with all the O(1) sites in a chain
along the b axis being either full or empty and with no oxygens on the O(5) sites on
the a axis.
The variation of the critical temperature due to a rearrangement of the oxygen ions in
the structure but not as a consequence of a change in stoichiometry stresses the
importance of the study of the different crystallographic structures present in YBCO.
X-ray diffraction, neutron diffraction, high resolution electron microscopy and
theoretical studies have been carried out for this purpose. The most important
orthorhombic structures in YBCO are known as Ortho I and Ortho II (OI and OII), and
they have been found to correspond to the two plateaux present in the variation of Tc
with composition at 93 K and 58 K (see Fig. 1.9 a)).
Chapter 1  Introduction to YBa2Cu3O7-δ
14
The OI (93 K) phase is characteristic of a fully oxgyenated material and it consists of a
sequence of full Cu-O chains (parallel to the b axis). As the oxygen in the structure
decreases many kinds of short range ordered full-empty chain arrays are observed until
at the 58 K plateau, the OII phase dominates. This structure has alternating full and
empty Cu-O chains (see Fig. 1.9 b)). Poulsen et al. (1991) suggests that the 58 K
plateau is a consequence of the dynamic coexistence of domains (or clusters) of the OI
and OII phases.
O(5)
O(1)
Ortho I Ortho II
Two empty
O(5) chains
b)
Fig 1.9 a) Variation of Tc with oxygen content (Poulsen et al. (1991)), and b) Ortho I and Ortho II
structures
The formation of the orthorhombic structure due to the incorporation of oxygen to the
tetragonal lattice results in an increase in the length of the b lattice parameter; this
gives rise to a crystal lattice distortion generating stresses in the crystal structure (see
Figs. 1.10 a) and b)) that are relieved by the formation of twins along the 110 planes.
In the ideal compound YBa2Cu3O6, the basal plane is free from oxygen ions and the
Chapter 1  Introduction to YBa2Cu3O7-δ
15
structure is tetragonal, with a lattice parameter equal to 3.82 Å (a = b). However, as
oxygen is incorporated to the lattice, it occupies either the O(1) or O(5) sites (see
chapters 1 and 3), increasing the length of the a and b parameters to 3.88 Å. This
difference in length is accommodated by the material with the formation of twins, as
shown in Fig. 1.10.
Twins can be observed by simple optical microscopy (see Fig. 1.11). This optical
examination was used by Beyers et al. (1989) to follow the O-T transition in YBCO.
They found that at oxygen contents as low as 6.28 it was possible to see the formation
of microdomains with a tweed microstructure, and that macroscopic twins were
clearly present for oxygen contents between 6.5 and 6.65. The presence of these twins
makes of optical examination a quick way of checking whether the material has been
oxygenated.
Cu
Cu
O
 a ~ 3.82 Å
 b ~ 3.88 Å
 a ~ 3.82 Å
 b ~ 3.82 Å
Cu
Cu Cu
Cu
Cu
Cu
O
Tetragonal Orthorhombic
a)
b) 
Fig. 1.10 a) Lattice parameters of deoxygenated and oxygenated basal plane showing schematic of
lattice deformation, and b) diagram showing how the lattice parameter difference is accommodated in
a twin boundary in YBa2Cu3O7
Chapter 1  Introduction to YBa2Cu3O7-δ
16
0.15 mm
Fig. 1.11 Twinned orthorhombic grains
This section has reviewed the changes in the crystal structure of YBCO that result
from the variation of its oxygen stoichiometry. This compound has been observed to
undergo a second order order-disorder transition from an orthorhombic structure (high
oxygen content) to a tetragonal structure (low oxygen content) at an oxygen content of
approximately 6.5. The transition temperature is observed to depend on partial oxygen
pressure. Within the range of oxygen contents at which YBCO is superconducting, a
number of orthorhombic structures have been identified (OI and OII among others –
see chapter 3) either by experimental methods or theoretical calculations. The results
present in the literature related to this orthorhombic to tetragonal transition and the
arrangement of oxygen ions in the crystal structure are fairly consistent.
1.4 Superconductivity
Soon after the discovery of superconductivity in mercury it was found that
temperature was not the only limiting factor for the transition from normal to
superconducting state. Onnes discovered that electrical currents could flow through
the superconductor without resistance as long as the current per unit cross-sectional
area did not exceed a certain value known as critical current, Ic. He also observed that
if the superconductor was placed in a magnetic field, the magnitude of this field had a
critical value, Bc, above which the superconductor would also revert to its normal
state.
Chapter 1  Introduction to YBa2Cu3O7-δ
17
1.4.1 Meissner effect
Meissner and Ochsenfeld (1933) observed that when a metal becomes a
superconductor when it is cooled in the presence of a magnetic field, it expels the
magnetic flux from its interior. This phenomenon is known as the Meissner effect. It
occurs because when the material is in its superconducting state, the presence of a
magnetic field sets up screening electrical currents on the surface of the
superconductor producing a magnetic field equal in strength but opposite in direction
to the externally applied one. These screening currents make the magnetic field inside
the sample exactly zero. The expulsion of the magnetic field occurs independently of
the sequence used in cooling or applying the magnetic field, which means that the
superconducting state is a true thermodynamic state.
a) 
b) 
Fig. 1.12 Schematic of Meissner effect showing that the magnetic field inside the superconductor will
be equal to 0 independently of the path followed to reach the superconducting state (from: Rose-Innes
(1988))
Chapter 1  Introduction to YBa2Cu3O7-δ
18
1.4.2 Penetration depth
From the discovery of the Meissner effect, F. and H. London (1950) studied the
relationship between the magnetic field and electrical currents in superconductors, and
predicted that the applied magnetic field did penetrate in a small region near the
surface in spite of the opposing screening currents. They suggested that the applied
field would penetrate in the superconductor over a characteristic length (λ). This
length is known as the penetration depth; the magnetic field decreases as exp(-x/λ)
where x is the distance from the sample surface.
Fig. 1.13 Schematic decay of the magnetic field in a superconductor (from: Cyrot and Pavuna (1992))
1.4.3 Type I and Type II superconductors
The type of superconductor described in the previous sections is known as Type I;
when the magnetic field exceeds a critical value, Bc, it penetrates inside the sample
eliminating its superconducting properties. The ability of these materials to exclude
magnetic fields makes them the perfect materials to use in ultrasensitive magnetic
instruments. However, the current densities carried by these materials are very low,
which makes them useless for many large scale applications (e.g. wires for
electromagnets). Most metallic elements are Type I superconductors.
In 1957 Abrikosov showed that certain superconductors exhibited a different
behaviour in the presence of a magnetic field. These materials allow the magnetic
field to penetrate in groups of quantised magnetic flux lines called vortices, each
vortex containing a single quantum of magnetic flux Φo = h/2e = 2 × 10-15 Wb. The
vortices arrange themselves into a regular lattice (the lowest possible energy state) and
the material is said to be in the mixed state. These are called Type II superconductors
Chapter 1  Introduction to YBa2Cu3O7-δ
19
and they enter the normal state when the magnetic field applied is so large that no
more vortices can be packed into the sample. Transition elements like Nb and V are
Type II superconductors, as well as alloys and copper oxides.
Type II superconductors can carry much larger currents than Type I, which increases
enormously their field of application (e.g. large permanent magnets, wires and tapes).
1.5 Conclusions
This chapter has reviewed the history of superconducting materials and it has given an
insight into the complex structure of YBa2Cu3O7-δ, one of the first high temperature
superconductors discovered. The change in the stoichiometry of YBCO modifies its
crystal structure and its electrical properties, as the material changes from being a
semiconductor at low oxygen contents to becoming a superconductor at high oxygen
contents.
The superconducting properties of YBCO are beyond the scope of this work and
therefore have been reviewed briefly. The following chapters are going to deal in more
detail with the study of diffusion and the optimisation of the oxygen stoichiometry in
this superconducting material.
Chapter 2 Theory of diffusion
20
Chapter 2
Theory of diffusion
2.1 Introduction
There are a large number of processes that involve the transport of a physical quantity
via a generalised flux from one part of the system to another under the action of a
force which often takes the form of a gradient of another physical quantity. These are
known as transport phenomena.
The following table summarises a number of these phenfomena:
Phenomenon Gradient Flux Law
Heat transfer
κ = thermal conductivity Temperature, T Energy, (1/2)mv2
Fourier,
jz = -κ (∂T/∂z)
Chemical diffusion
D = diffusion coefficient Concentration, C Mass, m
Fick,
jz = -D (∂C/∂z)
Viscosity
η = viscosity coefficient Velocity, v Momentum, mv
Newton,
jz = -η (∂v/∂z)
Electrical current
σ = electrical conductivity
Potential
difference, V Current, I
Ohm,
jz = -σ (∂V/∂z)
Table 2.1 Transport phenomena
Of all these processes, this study is going to concentrate on diffusion. This
phenomenon is very important because it is fundamental to many physical processes,
such as phase transformations, solid state reactions, sintering, etc.. Diffusion controls
for instance the achievement of a uniform concentration within a single phase where
there is an initial concentration gradient1 or a chemical potential gradient (see
                                                
1
 There are exceptions to this situation. In the case of diffusion of interstitial C in Fe, for example,
diffusion occurs in the direction of increasing C concentration (spinodal decomposition). This is known
as "up-hill" diffusion and it increases the concentration gradient.
Chapter 2 Theory of diffusion
21
equations (2.17)-(2.20)). This results in a reduction of the total free energy of the
system.
The diffusion process can be analysed by looking at the atomic mechanism through
which it takes place or by considering a macroscopic diffusion coefficient that makes
no reference to the atomic nature of matter and therefore does not specify the atomic
processes taking place. The following sections are going to consider these two aspects,
starting with the overall study of diffusion and following with the atomic mechanisms.
The various diffusion coefficients that can be measured and the relationship between
them will also be discussed. The chapter will finish with the derivation of a number of
solutions to the diffusion equation that will be used in the study of the oxygen
diffusion process in YBCO (chapter 5).
2.2 Definitions and fundamental relationships
The starting point for the study of the diffusion process is its definition by means of a
flux equation. Let us consider a volume of unit cross sectional area; the rate of transfer
of a diffusing substance through this section of unit area is proportional to the
concentration gradient measured normal to the section, i.e.



∂
∂
−=
x
CDj , (2.1)
where j is the rate of mass transfer per cross-sectional area, C the concentration of
diffusing species, x the distance travelled by the species in the direction perpendicular
to the cross-sectional area considered and D is the diffusion coefficient. This equation
is known as Fick’s first law. The negative sign indicates that diffusion occurs in the
direction opposite to that of increasing concentration. This equation describes the
effect of a concentration or chemical potential gradient of one type of particle present
in the absence of an electric field and it is only valid for an isotropic system with a
                                                                                                                                           
In the case of fast ionic conductors, diffusion can also occur in the absence of a concentration gradient.
There is a direct link between the chemical diffusion and the ionic conductivity of the diffusing
substance (Nernst-Einstein equation), and it is going to be seen in chapter 6 how the presence of an
electric field can modify the concentration profile within a sample of a fast ionic conducting material
such as YBa2Cu3O7-δ.
Chapter 2 Theory of diffusion
22
concentration gradient in the x direction. If the flux equation described the flow in all
three crystallographic directions, it would be expressed:




∂
∂
+
∂
∂
+
∂
∂
−=
z
CD
y
CD
x
CDj zyx . (2.2)
However, to simplify the study and analysis of the diffusion process, from now on the
system in which diffusion occurs is going to be considered isotropic (Dx = Dy = Dz)
unless stated otherwise. In the case of diffusion in the YBCO system considered in
this thesis, tracer diffusion measurements (Rothman et al. (1991)) have shown that
diffusion in the ab plane is approximately 106 times faster than along the c direction.
This large difference in the diffusion values justifies the assumption that oxygen
diffusion in YBCO occurs in a two-dimensional system (see chapters 3 and 5). For
this introductory analysis of the diffusion process, diffusion is going to be considered
to occur only along the x axis.
x x+∆x
jx+∆x
jx
+
_
Fig. 2.1 Schematic of diffusion process in one dimension (along the x-axis)
Fick’s first law can be derived as follows: let us take two adjacent planes where
diffusion is occurring by nearest neighbour jumps of one type of atoms between those
planes (Fig. 2.1). In this case the diffusion coefficient is taken to be independent of
concentration. The number of atoms per unit area of the plane at x is Nx and that of the
plane at x+∆x is Nx+∆x.
We must introduce a new quantity: the mean jump frequency of the diffusing atoms,
Γ; this is the average number of times an atom changes lattice sites per second.
Chapter 2 Theory of diffusion
23
The flux of atoms moving from x to x + ∆x and from x + ∆x to x is:
xx Nj Γ= 2
1
, and xxxx Nj ∆+∆+ Γ= 2
1
, respectively. (2.3)
The net flux between both planes is therefore given by:
( )xxxxxx NNjjj −Γ=−= ∆+∆+ 2
1
. (2.4)
This can also be expressed in terms of number of atoms per unit volume:
( ) ( )
x
NN
xj xxx
∆
−∆Γ= ∆+
2
1
. (2.5)
Since Nx/(∆x) is equal to C, the concentration, substituting and taking limits we have:
( )
x
C
xj
∂
∂∆Γ−= 2
2
1
, or 
x
CDj
∂
∂
−= . (2.6)
∆x is equal to the lattice spacing, d, (the distance of separation between nearest
neighbours). It can be seen that in this case describing linear diffusion the value of the
diffusion coefficient can be expressed by: D = 1/2Γ(∆x)2. In cubic lattices diffusion is
isotropic and therefore equally likely to occur in any of the orthogonal directions, so
the above expression would change to:



∂
∂Γ−=
x
Cdj
x
2
6
1
, (2.7)
which is equivalent to:
x
CDj
∂
∂
−= ,      with 2
6
1 dD xΓ= . (2.8)
For f.c.c. systems, 2
12
1 dD Γ= and for b.c.c. systems: 2
24
1 dD Γ= .
The dimensions of the terms of equations (2.1) to (2.7) are the following:
x = [L], D = [L2T-1],
d = [L], Γ = [T-1] .
The concentration can be expressed in several ways, since the units of the flux must be
consistent, they depend on those given to the concentration. For example, if the
concentration is given in g/cm3 ([ML-3]), the flux will be given in: g/cm2s ([ML-2T-1]).
Chapter 2 Theory of diffusion
24
However, if the concentration is given in moles/cm3, the units of the flux will be
moles/cm2s, this is: C = [L-3] and j = [L-2T-1]. D is usually expressed in cm2/s.
Fick’s first law only applies if the system considered is in steady state, this is, the
concentration at every point is invariant: 0=
∂
∂
t
C
.
To deal with non-steady state flow in which C changes with time, let us consider an
element of volume of unit cross sectional area with a thickness equal to ∆x. If this
distance is very small, the flux at x can be expressed as:
x
j
xjj xx ∂
∂∆−= ∆+ . (2.9)
jx jx+∆x
∆x
a)
x+∆xx
C
x
b)
jx
jx+∆x
x+∆xx
x
c)
Fig. 2.2 a) Element of volume, b) assumed concentration profle C(x), and c) flux of material, j(x) in this
element of volume
The amount of material that enters the element per unit time, jx, is different from that
which leaves it, jx+∆x, therefore the concentration in the element changes so that the
net increase in matter in the element can be expressed by:
t
C
x
x
j
xjj xxx ∂
∂∆=
∂
∂∆−=− ∆+ . (2.10)
Fick’s first law is valid at any point in time, so Eq. (2.1) can be substituted into (2.10)
giving:
Chapter 2 Theory of diffusion
25
2
2
x
j
x
CD
xt
C
∂
∂
=


∂
∂
∂
∂
=
∂
∂
. (2.11)
This is known as Fick’s second law of diffusion. In three directions it becomes:
j
z
CD
zy
CD
yx
CD
xt
C
zyx ∇=


∂
∂
∂
∂
+



∂
∂
∂
∂
+


∂
∂
∂
∂
=
∂
∂
. (2.12)
2.3 Mechanisms of diffusion
The diffusion process must also be examined in the context of the mechanism by
which it takes place. This mechanism can involve interstitial atoms, vacancies or a
complex combination of these. The mobility of these particles can also be affected by
the imperfections of the lattice such as dislocations, grain boundaries, point defects,
etc.
The atoms that form the lattice oscillate around their equilibrium positions and
sometimes this oscillation is so big that they change sites contributing to the diffusion
process. This can occur in several ways that are going to be reviewed in the following
sections:
2.3.1 Interstitial mechanism
This is one of the simplest mechanisms (see Fig. 2.3 a)). Atoms that have dissolved in
a host lattice into interstitial positions move through the interstices of the lattice
without displacing permanently any of the matrix atoms.
This type of diffusion is controlled by the energy needed to distort the host lattice
enough so that the change from one site to another is possible. This mechanism is
common in alloys where the solute normally dissolves interstitially, such as C in α- or
γ-iron, or in matrices damaged with radiation, where atoms that are normally in the
lattice have shifted to interstitial positions.
Chapter 2 Theory of diffusion
26
a) b)
Fig. 2.3 Schematic diagrams of: a) interstitial diffusion, and b) vacancy diffusion
2.3.2 Vacancy mechanism
This is the most general and important of all mechanisms (see Fig. 2.3 b)). Diffusion
occurs via an exchange of position of one atom with an adjacent vacancy. Therefore, a
flux of diffusing atoms in any direction requires a vacancy flux of equal magnitude in
the opposite direction. This type of diffusion is controlled by the energy needed to
form a vacancy plus the energy required to move it within the lattice. This mechanism
predominates in close packed metals and substitutional solutes in alloys. It is also
found in ionic compounds and oxides.
This mechanism was confirmed by the experiments carried out by Kirkendall. The
setup used for these experiments consisted of an α-brass/copper couple with the
interface between both allows marked by Mo wires (Fig. 2.4).
+ +
+
+ +
+
+ +
+
jZn
jCu
Fig. 2.4 Kirkendall effect
It was observed that after annealing the sample the markers had shifted to the side
closer to the α-brass end showing that the zinc atoms diffuse out of the alloy faster
Chapter 2 Theory of diffusion
27
than the copper atoms diffuse in. This can be explained in terms of vacancies: they
seem to exchange places more readily with the zinc atoms and as a result of the loss of
Zn pores are created in the brass side while on the Cu side of the couple the material
increases volume due to the diffusion of Zn into the structure.
2.3.3 Interstitialcy and Crowdion mechanisms
These mechanisms occur in situations where the interstitial atom is so large that if
interstitial diffusion were to occur it would produce a very large distortion of the
lattice. The activation energy required for interstitial diffusion to occur in this case
would be too big, so either of these two mechanisms become active. The interstitialcy
mechanism (Fig. 2.5 a)) is a combination of an interstitial and a substitutional process.
An interstitial atom moves onto a regular lattice site by pushing the atoms on that site
into an interstitial position and therefore interstitialcy diffusion occurs instead.
a) b)
Fig. 2.5 Schematic illustration of: a) interstitialcy mechanism and b) Crowdion mechanism
Atoms may adopt a Crowdion configuration to accommodate for the extra atom
introduced in the lattice (usually as the result of radiation damage). In this case n+1
atoms occupy n lattice sites. This configuration looks like an edge dislocation because
the distortion is spread out along a line (Fig. 2.5 b)). The energy to move this array of
atoms is also smaller than if an interstitial mechanism took place.
2.4 Types of diffusion coefficient
Diffusion processes can be studied with various techniques and some of these measure
different types of diffusion coefficients. This section reviews the different coefficients
and the relation between them.
Chapter 2 Theory of diffusion
28
2.4.1 Self diffusion
The self diffusion coefficient, Dsi, of the species i in a material is the diffusivity due to
thermal excitation in a homogeneous solid in the absence of a chemical potential
gradient. The motion of the ions can be modelled by a random walk analysis (Le
Claire et al. (1975)) to give (in an isotropic cubic lattice – see section 2.2):
2
6
1
iisi rD Γ= (2.13)
where ri is the jump distance and Γi is the mean jump frequency of each particle. Γi
depends on the fraction of ions that have sufficient thermal energy to overcome the
activation barrier for hopping and the number of vacant sites available for atoms to
hop into.
This coefficient can be estimated using a technique known as internal friction. It
involves applying stress to a sample that will induce a rearrangement of the mobile
atoms or ions to accommodate the stress. The stress is then released and the solid
returns to its original atomic configuration. In both cases the diffusion of atoms is
thermally activated. The value of the diffusion coefficient of the mobile species can be
calculated measuring the relaxation time necessary for the diffusing species to return
to the equilibrium configuration.
2.4.2 Tracer diffusion
This occurs when a radioactive tracer diffuses in a chemically homogeneous solid. In a
completely homogeneous material the mean square displacement of the tracer atoms is
experimentally determined from the spatial distribution of the tracer particles (tracer
concentration profile). This can be carried out by taking sections of the material
perpendicularly to the principal diffusion axis and measuring the radioactivity of each
section.
There are cases in which radioactive isotopes are not available, such as when oxygen
diffusion is being studied. A stable isotope of the diffusing species is then chosen as a
tracer. The penetration of this tracer as a function of distance is observed after the
diffusion experiment using Secondary Ion Mass Spectrometry (SIMS). This technique
Chapter 2 Theory of diffusion
29
is very powerful because it can analyse very small samples and it can be used to
determine the possible anisotropy of the tracer diffusion coefficient, D*, as a function
of crystallographic direction.
It should be noted that when the radioactive tracer chosen is of the same element as
the matrix material (e.g. 57Fe in Fe or 22Na in Na), the process is also known as self
diffusion, although it is not the same as the one discussed in the previous section, as it
is discussed below:
2.4.3 Correlation factors
The self diffusion coefficient, Ds, is a measure of the mobility of diffusing particles in
a matrix. But this diffusion of particles can occur via different mechanisms that may
not be equivalent.
If the jumps of i in the lattice are random, the path followed by all i particles can be
described as a “random walk”. According to the random walk theory, the diffusion
coefficient an isotropic material is proportional to the mean square displacement of
the diffusing particles in time t (see section 2.2):
6
2x
Ds ∝ (2.14)
where 〈x〉 is the mean distance between jumps. This is known as the Einstein diffusion
equation.
Let us compare two cases: self diffusion of A in A (as described in section 2.4.1) and
the diffusion of a radioactive tracer, A* in A. The diffusion mechanism is via
vacancies in both cases.
If we consider the first situation, A particles will diffuse in the matrix of A with all the
diffusing jumps being equivalent to each other and the random walk approach can be
applied. This does not occur in the second case, where once particle A* has taken the
place of the next nearest vacancy the probability of the particle exchanging sites with
the same vacancy and cancelling the diffusional step is greater than the probability
that A* will move in any other direction with the help of another vacancy coming
along. This means that diffusion of A* in the A lattice is not a random walk. However,
Chapter 2 Theory of diffusion
30
diffusion of vacancies in the A lattice is a random walk, so A* will keep on jumping
back and forth, exchanging positions with the original vacancy until another vacancy
becomes its next nearest neighbour.
In this case the jumps of the tracer atoms are said to be correlated and the comparison
between self diffusion and tracer diffusion is expressed by:
*fDDs = (2.15)
where D* denotes the tracer diffusion coefficient. The correlation factor, f, accounts
for the non-random motion of tracers and it is a measure of the efficiency of atomic
jumping for long distance migration.
The value of the correlation factor is always going to be less than 1 because the
diffusion process is always less efficient than in a perfectly random situation. f is a
function of the displacement between jumps and the angle between them. Therefore,
its value changes depending on the type of crystal structure and on the diffusion
mechanism. Table 2.2 summarises the value of the correlation factor for vacancy
diffusion for different lattice structures.
Crystal structure f
Two dimensional diffusion:
− Honeycomb  0.5
− Square  0.46694
− Hexagonal  0.56006
 Three dimensional diffusion:  
− Diamond  0.5
− Simple cubic  0.65311
− BCC  0.72722
− FCC  0.78146
− HCP  0.78121 ( ⊥ to c)
  0.78146 ( // to c)
 Table 2.2 Values of the correlation factor for vacancy diffusion in various lattices (after J.R. Manning,
(1968))
 The correlation factor is also present when comparing different diffusion mechanisms,
e.g., when diffusion occurs via interstitials or vacancies. Solvent particles B diffusing
through interstitials in A will move with a random walk because the particle always
Chapter 2 Theory of diffusion
31
finds itself in the same geometrical situation as in the previous step. However, when
diffusion occurs via vacancies the possibility that the interaction between the solvent
particle, B, and the vacancy be different from that between the vacancies and the
atoms of the host lattice makes the jumps of B non-random and therefore a correlation
factor, f, must be included when comparing diffusion occurring by both mechanisms.
 2.4.4 Chemical diffusion
 When a concentration gradient is imposed on a non-stoichiometric compound, re-
equilibration of the diffusing substance may occur by a diffusional process involving
the propagation of a concentration gradient throughout the specimen to minimise the
concentration differences in the sample. There are some special cases, such as that of
spinodal decomposition, where the concentration differences are not minimised by the
diffusion process, but increased.
 The rate constant that describes the chemical diffusion process is known as chemical
diffusion coefficient, ~D , and it is the constant of proportionality in Fick’s second law
of diffusion. In a one-dimensional case this law can be expressed as:
 


∂
∂
∂
∂
=
∂
∂
x
CD
xt
C ~
, (2.16)
 where C is the concentration of the mobile species, t is the time and x is the diffusion
distance.
 The chemical diffusion coefficient, ~D , and the self diffusion coefficient, Ds are
related by a thermodynamic enhancement factor that can be derived as follows:
 In order to achieve equilibrium in a system the value of chemical potentials of the
different mobile particles of the system must be constant. Let us consider a system
where particles, i, and vacancies, v, are the only mobile species. Gradients in their
chemical potential, µ, will generate a flux expressed by:
 
x
L
x
Lj viviiii ∂
∂
−
∂
∂
−=
µµ
, and
 
x
L
x
Lj vvviviv ∂
∂
−
∂
∂
−=
µµ
, (2.17)
Chapter 2 Theory of diffusion
32
 where the L terms are the transport coefficients of i and v, and µx is the chemical
potential of particles (x = i) and vacancies (x = v) in the system.
 For the above expressions to be correct the sum of the fluxes present in the system
must be zero, i.e. 0=+ vi jj . Taking the off-diagonal terms to be so small that they
can be neglected and considering that all the vacancies are at equilibrium in the
system (the vacancies that flow through the material are created at one end (x = 0) and
annihilated at the other) leaves µv = 0 and ( 0=∂∂ xiµ ). We have that the flux equals:
 
x
C
D
x
Lj isiiiii ∂
∂
−=
∂
∂
−=
µ
. (2.18)
 If we compare Fick’s first law with this equation we have that the value of Lii is:
 
i
i
siii
C
DL
µ∂
∂
−= (2.19)
 The chemical potential can be expressed by: µ µi io ikT a= + ln , where µoi is the
chemical potential of i at T = 0 K and ai is its activity. The self diffusion coefficient
can be considered a measure of the rate at which atoms diffuse in an ideal solution,
therefore the activity of the diffusing particle can be expressed as its mole fraction in
the system, ai = ni, and the chemical potential becomes: i
o
ii nkT ln+= µµ
 Substituting in the flux equation for both cases we have:
 Chemical diffusion:
x
akTL
x
L
x
C
Dj iiiiiiiii ∂
∂
−=
∂
∂
−
∂
∂
−=
ln~ µ
, and 
 Self diffusion :
x
nkTL
x
L
x
C
Dj iiiiiiisii ∂
∂
−=
∂
∂
−
∂
∂
−=
lnµ
.          (2.20)
 Assuming that the transport coefficients are the same in both cases and defining the
activity coefficient as iii na=γ , we obtain:
 



∂
∂
+−=
∂
∂
−=
i
i
si
i
i
sii
n
D
n
a
DD
ln
ln1
ln
ln~ γ
. (2.21)
 Where 



∂
∂
+
i
i
nln
ln1 γ  is the thermodynamic enhancement factor. This factor takes into
account that chemical diffusion occurs in the presence of a concentration gradient
whereas this gradient is absent in the self diffusion process.
Chapter 2 Theory of diffusion
33
 Very few studies have been made of the value of this thermodynamic factor for
oxygen diffusion in YBCO. Faupel and Hehenkamp (1993) propose a relationship
between the thermodynamic factor, Φ, and oxygen content as shown in Fig. 2.6.
Chapter 3 will analyse these data in the context of the diffusion results present in the
literature.
 
 Fig. 2.6 Variation of thermodynamic factor with oxygen content for YBa2Cu3O6+x (from Faupel and
Hehenkamp (1993))
 The thermodynamic factor links the individual chemical diffusion coefficient with the
self diffusion coefficient by a concentration dependent activity coefficient, γ. This γ
describes the interaction of the diffusing substances with respect to the lattice. The
value of γ is constant at small concentrations of the diffusing substance (Henry’s law)
and equals one as the mole fraction, ni, tends to one (Raoult’s law). In these cases the
chemical diffusion coefficient is approximately equal to the self diffusion coefficient.
 When the mixing process decreases the net enthalpy of the system, there is an
increased tencency towards interdiffusion and in this case the thermodynamic factor is
greater than one. This results in a chemical diffusion coefficient faster than the self
diffusion coefficient. When mixing increases the net enthalpy the thermodynamic
factor is less than unity, and the rate at which mixing occurs is reduced. In this case
chemical diffusion occurs more slowly than self diffusion.
 2.4.5 Factors affecting diffusion
 The diffusion coefficient can be affected by a large number of factors, the most
important ones being composition, temperature and structure.
Chapter 2 Theory of diffusion
34
• Composition
 There are cases in which the diffusion coefficient changes with composition. This
makes the analysis of the diffusion process more complicated since there is one extra
variable to be taken into account. The Kirkendall effect mentioned in section 2.2.1 is a
good example of a concentration dependent diffusion system. Boltzmann–Matano
analysis can be used to calculate the diffusion coefficient for each concentration.
There is a good review of the study of concentration dependent diffusion in Crank
(1975). The diffusion coefficient depends on concentration because the value of the
binding energies of the vacancy to the diffusing ion may vary with the amount of
diffusing ions present in the material. The value of the diffusion coefficient may
decrease when the number of vacant sites is reduced. The change in D may be greater
than an order of magnitude over a very small concentration range (10-4 atomic
percent). An example of this concentration dependent diffusion is that of Fe ions
diffusing in Fe1-xO. It can be seen in Fig. 2.7 how at 802 °C the value of D decreases
with increasing Fe content.
 The diffusion coefficient considered in this thesis is assumed to be independent on
concentration, so the possible solutions to the diffusion equation and the analysis of
the process presented will only refer to concentration-independent diffusion. The
range of activation enthalpies for diffusion processes observed experimentally is
considerable. Values can vary from approximately 0.1 eV allowing interstitial motion
at low temperatures to 1-2 eV, characteristic of vacancy motion in oxides at higher
temperatures (hundreds of degrees).
Chapter 2 Theory of diffusion
35
 
 Fig. 2.7 Log D as a function of deviation from stoichiometry for cation self-diffusion: 59Fe in Fe1-xO, at
different temperatures (from N.L. Peterson (1973))
• Temperature
 From diffusion experiments the dependence of the diffusion coefficient on
temperature can be expressed by an Arrhenius type relationship:
 ( )kTADD eo −= exp (2.22)
 where Do and Ae are independent of temperature. The activation energy of the
diffusion process, Ae, is associated to the energy required for the formation of defects,
and that necessary for their motion. Do, the pre-exponential term, is proportional to the
entropy of formation and motion of these particles as well as to the bond strength and
the jump frequency of the diffusing particles:
 
( )( )kSSfrD
mfo +Γ= exp6
1 2
, (2.23)
 (2.24)
 Where r is the jump distance and Γ is the jump frequency of the diffusing particles; f
is the correlation factor already mentioned in section 2.4.3. If self diffusion occurs via
vacancies the activation energy is equal to the enthalpy of formation and motion of
these vacancies (bond strength), while in interstitial diffusion it is not necessary to
form a defect for diffusion to occur and the activation energy would be the value of
the enthalpy of motion of the diffusing particle through interstitials in the lattice.
Diffusion of weakly bonded ions (metals) will have lower activation energies than
those that are strongly bonded to the lattice (ionic solids).
Chapter 2 Theory of diffusion
36
 In the case of oxygen diffusion in YBCO, the activation energy is equal to the binding
energy of the oxygen ions to the lattice. The creation of vacancies in the ab plane
would not alter significantly the final value, since even in the case of a lattice that is
almost fully oxygenated there are always O(5) sites available for the motion of oxygen
ions. Even though this thesis concentrates on the macroscopic aspects of diffusion,
chapter 3 reviews some of the theoretical work carried out to study the ionic motion in
the YBCO lattice during the diffusion process. It has already been mentioned at the
beginning of section 2.2 that diffusion in YBCO can be considered a two-dimensional
process, as the motion of oxygen ions and vacancies in the ab plane occurs around 106
times faster than in the c direction (see also chapter 3).
• Structure
D increases with lattice irregularity. Due to a reduction in packing density in
dislocations and grain boundaries, atoms can diffuse along these paths more easily.
Analysis of grain boundary diffusion will be dealt with in more detail in the following
section. Since the grain boundary area to volume ratio of a polycrystalline solid is
small, grain boundary diffusion only becomes important in certain phenomena, like
sintering or oxidation. It seems that grain boundary diffusion has a more pronounced
effect on D at lower temperatures because its lower activation energy makes it less
sensitive to temperature changes in this range. As the temperature decreases the value
for the diffusion rate along grain boundaries decreases more slowly than that of
diffusion through the bulk.
2.5 Application to real geometries. Solutions
of the diffusion equation
The diffusion process is a very important phenomenon, but it can be very complicated
in the general case of an anisotropic, inhomogeneous, multicomponent system subject
to various driving forces. General solutions of the diffusion equation can be obtained
for a number of initial and boundary conditions provided the diffusion coefficient is
constant.
Chapter 2 Theory of diffusion
37
2.5.1 Thin film solution
Let us first consider a non-steady state situation where two solute-free infinite rods are
joined by the ends with a thin film of solute concentration Co between them (see Fig.
2.8). The equation that describes this situation is Fick’s second law of diffusion:



∂
∂
−
∂
∂
=
∂
∂
x
CD
xt
C ~
; (2.16)
by differentiation we can see that:
( )tDx
t
AC ~4exp 2−= (2.25)
is a solution of the above equation. The total amount of solute per unit cross section,
M, is equal to: ∫∞
∞−
= CdxM ; substituting and integrating we can calculate the value of
A: 
D
MA
~2 pi
= . so the variation in concentration along the rods can be expressed by:
( )tDx
D
MC ~4exp
~2
2
−=
pi
. (2.26)
0 1 2 3 4-1-2-3-4
t1
t3
t2
t1 < t2 < t3
C
x
Fig. 2.8 Time sequence of diffusion profiles displaying thin film diffusion into an infinite rod
2.5.2 Semi-infinite sources
There are cases in which the initial distribution is not just confined to a plane but it
occupies a finite region, as it happens in the case where a semi-infinite rod with a
constant solute concentration is attached to a solute-free rod, i.e.:
Chapter 2 Theory of diffusion
38
C = Co for x < 0, t = 0
C = 0
  
  for x > 0, t = 0
Co
xP
ξ
δξ
C
0
Fig. 2.9 Definition of the symbols used in this section. Case of diffusion from a semi-infinite source
Taking a small element of volume of thickness δξ and unit area, the concentration at P
at any point in time is:
( )tD
tD
CC o ~4exp
~2
2ξ
pi
δξ
−= . (2.27)
Doing a summation of all the δξ elements:
( ) ( )∫ −=∫ −= ∞∞ tDxoo dCdtD
tD
CC ~2
22 exp~4exp
~2
ηη
pi
ξξ
pi
δξ
pi
, (2.28)
where Dt2ξη = , putting ( ) ∫= x dxerf 0 2exp2 ηηpi , with ( ) ( )xerfxerf −=− , and
( ) 1=∞erf . Values of the error function can be found in many tabulations (Carslaw
and Jaeger (1959)). The concentration profile varies according to:






−=
Dt
x
erfCC o
2
1
2
. (2.29)
Co
C
0 x
Fig. 2.10 Diffusion profile from a semi-infinite source at time t > 0. Black solid line represents the final
solute distribution. Grey lines below the curve show the different error functions that describe the
process and add up according to equation (2.29) to give the solute distribution shown.
Chapter 2 Theory of diffusion
39
2.5.3 Finite systems. Complete homogenisation
Let us consider the case of a plane of finite thickness with a solute concentration Co
diffusing into two semi-infinite planes at each side. The boundary conditions are:
C = Co for 0 < x < l at t = 0,
C = 0 for x = 0, and C = 0 for x = l for t > 0
0 1 2 3 4-1-2-3-4
t1
t3
t2
t1 < t2 < t3 < t4
C/Co
x
t4
1.0
0.2
0.6
0.8
0.4
0.0
Fig. 2.11 Concentration-distance curves for diffusion from a finite source at C = 0 for different times
The final expression for the variation in concentration is:
( )
( ) ( )∑∞
=


 +



 +
−
+
=
0
2
22 12
sin12~exp
12
14
n
o
l
xn
l
tnD
n
CC pipi
pi
. (2.30)
2.5.4 Diffusion in a slab
In the case of the diffusion considered in this work, the situation can be approximated
to that of diffusion in or out of a plane sheet. The diffusing species is assumed to enter
through the x axis only, the initial concentration is uniform and the concentration at
the surfaces is constant.
If the region -l < x < l is initially at a uniform concentration Co, and the surfaces are
kept at t constant concentration C∞, the solution of the diffusion equation becomes:
( )
( )
( )
( )
( ) ( )∑∞
=
∞


 +



 +
−
+
−
−=
−
−
1
2
22
2
12
cos
4
12~
exp
12
141
n
n
o
o
l
xn
l
tnD
nCC
CC pipi
pi
. (2.31)
This solution gives the variation of the concentration profile with time t and distance
x. If Mt is the total amount of diffusing substance that has entered the sheet at time t
Chapter 2 Theory of diffusion
40
and M
∞
 is the corresponding quantity after infinite time, then the previous equation
can be simplified to:
( )
( )∑∞
=
∞



 +
−
+
−=
1
2
22
22 4
12~
exp
12
81
n
t
l
tnD
nM
M pi
pi
, (2.32)
which gives the weight change with time in the diffusion process.
z
x
y
-l
a)
0 l
0
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
x/l
(C
-
C
o
)/(
Ci
n
f-C
o
)
0.0050.02
0.1
0.2
0.6
0.4
1.0
1.5
b)
0.8
Fig. 2.12 a) Schematic of diffusion in a slab with concentration profile overimposed, and b)
concentration distributions for the same diffusion situation at different times (equation (2.31)).
Numbers next to the curves are the values of Dt/l2
2.5.5 Grain boundary diffusion
All the solutions given above describe volume diffusion in perfect solids. Real solids,
however, may have defects, dislocations or boundaries that affect the diffusion
process, as mentioned in 2.3.5. This section is going to cover the mathematical
solution to only one of these fast diffusion pathways: grain boundary diffusion. This
type of diffusion is present in polycrystalline materials and its effect is more important
at low temperatures. This phenomenon was first studied by Fisher (1951) and it has
been followed by a number of authors that have proposed solutions to more complex
situations (e.g.: Whipple, (1954), Suzuoka, (1964) and Kwok et al. (1984-a), (1984-
b)).
Let us consider a grain boundary of thickness l perpendicular to a constant solute
source of concentration Co. The rate of change in concentration for an element of grain
boundary is given by:
Chapter 2 Theory of diffusion
41
x
y
x
y
yy jly
j
dyjdy
y
jjjl
dylt
C 221 −
∂
∂
−=


−



∂
∂
−−=
∂
∂ (2.33)
l
dy
jy Cp
jx jx
jy+(∂j/∂y)dy
Fig. 2.13 Schematic of the concentration profile along a grain boundary and into the lattice in the case
of grain boundary diffusion
Let DGB be the diffusion coefficient at the grain boundary and that in the bulk be DL.
As long as DGB > DL and the flux along the x direction is perpendicular to the
boundary, the above equation can be solved taking the solution to the thin film case
(Eq. (2.26)) and calculating an initially unknown factor that takes into account the
effect of the fast diffusion on the boundary acting as a second source of solute that
diffuses into the bulk:
( ) 







−



−
=
tD
x
erf
DlDtD
yCC
LLGBL
o 2
1
2
exp 41
pi
. (2.34)
2.6 Methods of measurement
There are many approaches to measure the magnitude of a diffusion coefficient; these
depend on the system studied and the characteristics of the diffusion process.
Therefore, there are steady-state and transient methods, methods based on the analysis
of concentration-distance curves or observation the overall rate or uptake or loss of
diffusion substance by a specimen of known size and shape. The latter is going to be
considered here. These methods are going to be applied to the analysis of oxygen
diffusion data collected with thermogravimetry presented in chapter 5.
Chapter 2 Theory of diffusion
42
2.6.1 Sorption method. Constant D~
This experiment has been used to determine the diffusion coefficient of dyes in
cellulose sheet and of oxygen in muscle (Crank (1975)). The diffusion coefficient is
assumed constant and the sheet does not change volume during the diffusion process.
The experimental procedure used is thermogravimetry, where the weight change of the
sheet is monitored as it is held at a certain temperature and the gas is diffusing in. The
solution of the diffusion equation is equation (2.32):
( )
( )∑∞
=
∞



 +
−
+
−=
1
2
22
22 4
12~
exp
12
81
n
t
l
tnD
nM
M pi
pi
. (2.32)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 300 600 900 1200 1500 1800
Time (s)
Fr
ac
t. 
w
t. 
ch
an
ge
 
Fig. 2.14 Fractional weight change vs. t plot from experimental data of an oxygenation of YBa2Cu3O7-δ
powder at 400 ºC. See chapter 5 for detailed analysis of experimental results
From this equation there are two ways of calculating the diffusion coefficient in the
sheet. The first one considers the value of t/l2 for which 21=
∞
MM t , (t/l2)1/2. This
is well approximated by:
( ) 







−=
922
221
2
169
1
16
ln1 pipi
pi D
lt , (2.35)
the error being ≈ 0.001%. We then have:
( ) 212
0049.0~
lt
D = . (2.36)
Chapter 2 Theory of diffusion
43
Therefore, if the half-time of a sorption process is observed experimentally for a
system with a constant diffusion coefficient the value of D~  can be determined from
(2.36).
2.6.2 Initial rates of sorption and desorption
It is also possible to calculate an average diffusion coefficient from the initial gradient
of the sorption curve when the fractional weight change is plotted against the square
root of time. In the early stages of diffusion, for D~  constant and a sheet of thickness l,
we have:
21
22
~4




=
∞ l
tD
M
M t
pi
. (2.37)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 50 60 70 80 90 100
t^0.5/l (x 1000)
Fr
ac
t. 
w
t. 
ch
an
ge
 
4 D^0.5/(pi)^2
Fig. 2.15 Fractional weight change vs. t1/2/l plot from the initial data of an oxygenation of YBa2Cu3O7-δ
powder at 400 ºC (particle diameter: 5.2 µm). See chapter 5 for detailed analysis of experimental data
2.6.3 Final rates of sorption and desorption
When the previous type of study is not possible there is a chance to calculate the value
of the diffusion coefficient taking the later stages of the diffusion process. In this case
only the first term of equation (2.32) needs to be considered and therefore, for
diffusion in and out of a sheet of thickness l:
( )( )
2
2
~ln
l
D
dt
MMd t pi
−=
−
∞
. (2.38)
Chapter 2 Theory of diffusion
44
The variation of the value of ln(M∞ -Mt) with time gives a straight line from whose
slope the value for D~  can be calculated.
-6
-5
-4
-3
-2
-1
0 480 960 1440 1920 2400 2880 3360
Time (s)
ln
(M
-
M
t)
-D (pi / l)^2
Fig. 2.16 Fractional weight change vs. t curve for experimental data of an oxygenation at 400 ºC of
YBa2Cu3O7-δ powder (particle diameter: 5.2 µm). See chapter 5 for detailed analysis of experimental
data
2.7 Conclusions
The diffusion process can be studied from two different points of view. The first one
(microscopic) is concerned with with the study of the movement of the ions within the
crystal lattice. The second approach studies the diffusion process measuring the effect
it has on the physical properties of the sample (change in resistivity, weight changes,
etc.), without looking in detail at the movement of ions in the crystal structure. This is
the macroscopic point of view, and it is going to be the one used in this thesis.
This chapter has reviewed the study of the diffusion process starting with the
formulation of Fick’s laws of diffusion. This has been followed by a presentation of
the different mechanisms for diffusion (vacancy, interstitial, etc.) and the types of
diffusion coefficients that can be measured depending on the technique used. The
relationships between these coefficients have been discussed.
The chapter finishes with the application of Fick’s laws to analyse and study diffusion
data obtained from real experiments, taking into account the geometry of the sample,
setting the appropriate boundary conditions for each case and giving the mathematical
expressions of a number of diffusion situations. The solution to Fick’s second law for
Chapter 2 Theory of diffusion
45
diffusion in a slab discussed here is going to be used in the analysis of the
experimental results presented in chapter 5.
Chapter 3 Literature Review
46
Chapter 3
Literature review
3.1 Introduction
Chapter 1 has introduced the subject of oxygen non-stoichiometry of YBCO, one of
the parameters that determine its normal state and superconducting properties.
Knowledge of the oxygen content of the samples as well as that of the rate at which it
diffuses in and out of them has become a key issue in order to be able to optimise its
properties.
The manufacture of YBCO samples involves (see chapter 4) heating the precursor
powders so that they react and produce the required copper oxide with the right
stoichiometric ratio between the Y, Ba and Cu ions. Once YBCO is obtained,
polycrystalline samples are produced by sintering or melt-processing; single crystals
are grown using a variety of methods, most of them involving heating to high
temperatures. The oxygen content at these high temperatures is very low and in order
to obtain YBCO with good superconducting properties an oxygenation step has to be
included in the manufacturing process. This step involves heating the sample to
temperatures below 500 °C to increase the oxygen content of YBCO and restore its
superconducting properties. The annealing time during this step is determined not only
by the diffusion coefficient, but also by the microstructural features that affect the
diffusion process, such as porosity (interconnecting voids), grain size, twin density,
grain orientation, sample surface area, impurity content and initial oxygen distribution.
In order to study the kinetics of oxygen diffusion a number of techniques have been
used, but the results obtained differ considerably. One of the reasons is that sometimes
the diffusion coefficient measured depends on the technique used, and the necessary
correlation factors are not applied before comparisons are made. There has been a
large amount of data published in this subject, but as it is going to be seen in the
Chapter 3 Literature Review
47
following sections the variables affecting the diffusion process (especially in
polycrystalline samples) are numerous and, in many cases, difficult to assess or
control, giving rise to the wide range of values for oxygen diffusion shown.
3.2 Thermogravimetry (TG)
Thermogravimetric measurements can be used to study the chemical diffusion of a
species in a material (see chapter 4 for a more detailed description of the technique).
These measurements consist on placing a sample of the material in a balance in an
enclosed atmosphere; the conditions of temperature and/or partial gas pressure –
oxygen in this case – are changed so that a concentration gradient is established
between the sample and the atmosphere and the diffusing species is either absorbed or
desorbed from the sample, changing its weight. This change in weight is recorded
with respect to time and temperature and is a direct measurement of the chemical
diffusion process that is taking place.
Thermogravimetric analysis has been applied to the study of oxygen diffusion in
YBCO powder and polycrystalline samples of varying density. The use of single
crystals with this method has not been reported, perhaps due to the difficulty in
producing the large amount of crystals needed so that the data from the weight
measurements is outside the experimental error region inherent to the system used.
The scatter in the published data is quite large, as it is going to be seen in the
following paragraphs. A summary of the results published using this technique is
presented in Table 3.1. The difference between these results is thought to be due
mainly to the high sensitivity of this technique; the variation in the microstructural
features of the samples affects considerably the measurements taken (see chapter 5,
section 5.2.3). But it could also have other reasons that are discussed in the following
analysis:
The first pieces of work published with results from thermogravimetry concentrate on
the measurement of the equilibrium oxygen content of YBCO with temperature and
pressure. Gallagher (1987), Strobel et al. (1987) and Kishio et al. (1987) present data
characterising the non-stoichiometry of YBCO at different partial oxygen pressures.
Chapter 3 Literature Review
48
From their experiments, Gallagher and Strobel et al. (1987) have calculated the value
for the activation energy of the diffusion process taking place. Gallagher (1987)
obtained an activation energy equal to 1.03 - 1.15 eV while Strobel and co-workers
calculated a higher value: 1.5 - 1.65 eV. The range in partial pressures used by both
authors is the same, and so is the temperature range; since the activation energy
characteristic of a vacancy diffusion process (see section 3.7 in this chapter and
chapter 2) is approximately equal to 1 eV, the value of Strobel et al. (1987) is too
high. Errors in the measurement or contamination of the sample could be two of the
reasons for such a high activation energy.
Fig 3.1 Oxygen stoichiometry vs. partial oxygen pressure diagram (Kishio et al. (1989-b))
In the two following papers published, Kishio et al. (1989-a), (1989-b) present a more
complete version of their initial work (Kishio et al. (1987)). Comparing their
concentration vs. partial oxygen pressure results with those from Strobel et al. (1987),
it can be seen there is a difference of 0.05 in the oxygen content measured, which can
be attributed to experimental error. Kishio and co-workers study in more depth the
diffusion process, concluding that the diffusion coefficient is dependent on the oxygen
content of the specimen, contradicting their initial assumption of a concentration-
independent diffusion coefficient. They calculated an activation energy that changed
with concentration between 1.38 eV (for δ = 1) to 1.81 eV (for δ = 0). The value of the
pre-exponential coefficient changed in one order of magnitude with oxygen content.
They find that their data, once converted to self-diffusion data with the
Chapter 3 Literature Review
49
thermodynamic factor, agrees quite well with those from Ikuma and Akiyoshi (1988).
Kishio et al. (1989-b) pointed out that more work should be done to investigate the
effect of grain boundary diffusion in the final results obtained. The values of the
diffusion coefficient calculated by this group have a very high activation energy and
pre-exponential coefficient, which suggests that they could have used the dimensions
of the polycrystalline pellets as their diffusion distance, instead of the grain size.
Taking a longer distance increases the value of the diffusion coefficient, as it will be
seen in chapter 5 (Table 5.1). In terms of the mechanism for diffusion, Strobel et al.
(1987) only mention that it involves motion of oxygen ions from (1/2, 0, 0) and (0,
1/2, 0) sites in the ab plane. This mechanism is discarded by Kishio and co-workers
(Kishio et al. (1989-b)) on the grounds that the chemical diffusion coefficient is said
to be dependent on oxygen content, which is more typical of an interstitial
mechanism. However, the proposed motion of oxygen along interstitials is due just to
a change of reference point where the O(5) sites are not considered part of the crystal
structure of YBCO, becoming interstitial sites.
1.0E-18
1.0E-16
1.0E-14
1.0E-12
1.0E-10
1.0E-08
1.0E-06
1.0E-04
0.0009 0.0011 0.0013 0.0015 0.0017
1/T (1/K)
lo
g 
D
Krebs Saiz Kishio
Mozahev Tang Krauns
Umemura in- Haller Glowacki DSC
800 700 600 500 400 300
T (C)
Fig. 3.2 Collected chemical diffusion data
Another study of oxygen diffusion that is going to be included in this section even
though it was carried out using a different thermal analysis technique is that of
*ÆRZDFNL et al. (1988). They used differential scanning calorimetry to study the
absorption and desorption of oxygen in YBCO. The experiments were carried out
under non-isothermal conditions, unlike most of the thermogravimetry work presented
Chapter 3 Literature Review
50
here, and they measured the heat released during the absorption of oxygen while the
sample was heated at a certain rate in a pure oxygen atmosphere. These non-
isothermal measurements present the advantage of giving information about the
oxygenation process from very low temperatures (below 300 ºC), which is not
possible to do with other techniques. The activation energy of the absorption process
was calculated to range between 1.5 and 1.6 eV, agreeing with that calculated by Tu
and co-workers (1988), (1989) for out-diffusion of oxygen in their analysis of oxygen
diffusion with resistance measurements (see section 3.4). These values are slightly
higher than the rest of the activation energies found in the literature. This is justified
by the authors proposing that the process is limited by the formation of a highly
oxygenated layer that moves towards the centre of the sample (agreeing with
Furukawa et al. (1992)), and suggesting that out-diffusion is surface reaction limited
(see Tu et al. (1989) and LaGraff et al. (1991) in section 3.4).
It has been suggested by a number of authors that the value of the diffusion coefficient
depends on the oxygen content of the sample. However, the bulk of the results
presented in the literature describes the process with an Arrhenius equation. This
means that a plot of lnD vs. 1/T is a straight line. Any concentration dependence of the
diffusion coefficient would be reflected in this graph as a deviation from this straight
line; as the dependence observed is an increase in the value of the activation energy
with oxygen content, the lnD vs. 1/T plot would curve downwards to slower values of
D (an increase in the slope of the line) as temperature decreased (see Fig. 3.3 a)). The
arguments behind the increase in activation energy are that the relaxation curves of the
property measured (weight change, resistivity, etc.) show a fast variation at the
beginning of the measurement, when the oxygen content is low (Fig. 3.3 b)); this is
followed by an apparent slowing down in the change of this property at longer times,
when a large amount of oxygen has diffused in the sample. The diffusion study
presented in this work (see chapter 5) shows that this behaviour is characteristic of a
single diffusion process, independently of the amount of oxygen absorbed by the
specimen, ruling out the proposed concentration dependence.
Chapter 3 Literature Review
51
lnD
Standard Arrhenius relationship:
D = Doexp(-Ae/kT)
Concentration dependent D
1/Ta)
Ae1:D1
Ae2:D2
Fractional change
Timeb)
Fig. 3.3 a) Schematic of concentration-dependent and concentration-independent diffusion behaviour,
and b) schematic of relaxation measurement with two apparent activation energy regions
Going back to the rest of the diffusion work, Saiz and Moya (1988) studied the
influence of the microstructure on the diffusion coefficient and observed that the
absorption of oxygen was up to 50 times faster in tetragonal samples than in
orthorhombic samples. They also observed that for the polycrystalline samples used
(grains 35 µm long, different densities), grain boundary diffusion was not the rate
limiting factor for the diffusion process. The activation energies calculated for both
processes (from a standard isothermal weight gain trace) were: 0.12 eV for tetragonal
samples (the initial part of the curve) and 0.29 eV for orthorhombic samples (from the
final part of the curve). These activation energy values are extremely low and have not
been corroborated by any other work in the literature. It is difficult to explain the
reason for these low values; perhaps the choice of the diffusion distance was not
correct, or the samples were contaminated during the preparation process.
Chapter 3 Literature Review
52
Haller et al. (1988) studied diffusion in a very porous polycrystalline sample and,
assuming that the diffusion coefficient was independent of the oxygen content of the
sample, they calculated an activation energy for the process equal to 1.15 eV and a
pre-exponential term of 2.2 × 10-3 cm2/s. Using the calculated value of the diffusion
coefficient, they also calculated the jump frequency of the oxygen ions in the lattice
and they found it to be quite similar to that characteristic of a vacancy diffusion
mechanism: 10-13 s-1. They observed a slight variation in the values of the activation
energy with oxygen content, but it was considered to be within the experimental error
of the measurements taken.
From their oxygenation-deoxygenation cycles, Umemura et al. (1989) calculated two
expressions for in- and out-diffusion. The activation energies of each are rather low:
0.7 and 0.5 eV respectively. An important aspect to consider again is the choice of the
diffusion distance. Umemura and co-workers do not mention the grain size of their
polycrystalline samples (diffusion distance), necessary to calculate the value of the
diffusion coefficient. This leads to think that they have used the full dimensions of the
sample as their diffusion distance; this could explain the fast values obtained for the
pre-exponential factor and the activation energy. A peculiar behaviour in the weight
relaxation times for in-diffusion suggests possible contamination of the samples or
perhaps the onset of grain boundary diffusion.
The work presented by Lau et al. (1991) is contradicting, suggesting diffusion
coefficients two orders of magnitude different for the same temperature. They have
observed a non-Arrhenius dependence of the diffusion coefficient, also observed by
Krauns and Krebs (1993), contradicting most of the results published. Krauns and
Krebs (1993) carried out oxygenation studies in polycrystalline YBCO with a grain
size of 1 to 5 µm and an average density of about 75%. The values they calculated for
the activation energy from their experimental measurements change from 1.3 eV at
400 °C to 0.3 eV at 700 °C. They explain this change suggesting that the decrease in
oxygen content at high temperatures enhances the diffusion process due to the
increase in oxygen vacancies.
Even though the work carried out by Tang and Lo (1991) observed a concentration
dependence of the diffusion coefficient, they show in a lnD vs. 1/T plot that their
Chapter 3 Literature Review
53
diffusion values fall in a straight line, following an Arrhenius equation. This
contradicts the previous statement of a concentration dependent diffusion coefficient.
It has already been discussed (see Fig. 3.3 a)) that a concentration dependent diffusion
coefficient deviates from the straight-line Arrhenius behaviour depending on the
dependence on the oxygen content (upwards if the activation energy decreases and
downwards if it increases with the amount of oxygen present in the sample). Tang and
Lo (1991) worked with powder samples at temperatures around 500 °C and calculated
an activation energy of 1.3 eV for the oxygenation process. The calculations were
carried out based on a solution of the diffusion equation with the appropriate boundary
conditions and at 500 °C the diffusion coefficient was found to be of the order of 10-10
cm2/s. They suggest that the source of disagreement between their calculations and
other data published in literature is the presence of porosity in polycrystalline samples
and the effect of large angle grain boundaries in the diffusion process.
On the same lines as Saiz and Moya (1988), Mozahev and co-workers (Mozahev et
al., (1992), Mozahev and Chernyaev, (1994)) studied the effect of sample porosity on
the diffusion of oxygen in polycrystalline samples, obtaining a reasonable value of the
activation energy equal to 1.2 eV. They observed that for sintered materials of
densities lower ≈ 80-85 % of the theoretical density, the oxygenation process could be
described quite accurately with a model of a diffusion in a sphere. This means that
both porous polycrystals and powders would have the same oxygenating behaviour.
This could be due to the presence of a large number of open pores that facilitate the
supply or removal of oxygen to and from inner parts of the polycrystal. If this were
true, then the only factor to take into account in order to calculate the diffusion
coefficient would be the diffusing distance, this is, the size of the grains in
polycrystalline specimens and the powder particle diameter in powder samples. The
temperatures at which their studies were carried out cover a very wide range (100–
750°C), and the effect of grain boundaries does not appear to modify the diffusion
results in the lower temperature region.
Chapter 3 Literature Review
54
100% dense
O2 in
a)
Closed porosity
(94 % dense)
Open porosity
(85 % dense)
O2 in
b)
O2 inc)
c axis perpendicular
to sheet
Bulk D
Grain boundary  D
Fig. 3.4 Schematic of diffusion process in polycrystalline samples with different degrees of porosity
With respect to the effect of porosity on diffusion, three cases should be taken into
account, depending on the degree of porosity present in the sample studied. It can be
seen in Fig. 3.4 a) that if the sample is fully dense, the oxygen source is the
atmosphere surrounding the sample and the only way this oxygen diffuses to the
centre is via grain boundary diffusion and bulk diffusion. In this case the diffusion
distance considered should be the dimensions of the sample (if the temperature at
which this occurs is such that grain boundary diffusion effects can be ignored). The
opposite situation would be that of a specimen with open porosity, where the oxygen
diffuses in the sample not only from the surrounding atmosphere but also from the
interconnected pores. These are a continuous oxygen source, as they are connected to
the atmosphere outside the sample. In this case the diffusion process occurs more
readily because oxygen can reach inner regions of the specimen through the pores, and
Chapter 3 Literature Review
55
the diffusion distance considered should be that between interconnected pores (see
Fig. 3.4 c)).
In the case of samples with closed porosity, (see Fig. 3.4 b)) some oxygenation would
occur from the oxygen trapped in the closed pores into the material, but the amount
would be so small that it would not have any major effect on the overall diffusion
behaviour.
Furukawa et al. (1992) studied the oxygenation of YBCO powder at very low
temperatures and at two different oxygen contents (fully deoxygenated and with an
oxygen content equal to 6.44). They observed a slower oxygen uptake in the YBCO
samples with a higher oxygen content (6.44), which suggests that the diffusion process
is controlled by the speed at which the interface between the tetragonal and
orthorhombic phases moves towards the centre of the sample. They have modelled
this process using an “unreacted core model”, where the grains are spherical and, as
soon as oxygenation starts, an orthorhombic skin forms and grows towards the centre
of the particle. It seems that in this model the skin is fully orthorhombic at the highest
oxygen content possible for that temperature and all the way across, which is difficult
to believe since abrupt changes in concentration are not common in non-
stoichiometric materials. The variation in concentration occurs gradually: there would
be a decreasing concentration gradient towards the centre of the sample, where the
outer surface is oxygenated to the highest possible oxygen content at the temperature
the experiment is carried out and the oxygen content decreases gradually as the
distance to the centre decreases. The slow oxygen uptake of the oxygenated sample
could be due to surface contamination of the powder.
Fig. 3.5 Schematic of unreacted core model (after Furukawa et al. (1992))
Chapter 3 Literature Review
56
3.2.1 Summary
Most of the thermogravimetric analysis of oxygen diffusion has been carried out in
polycrystalline samples of varying density, although work on powdered YBCO has
also been reported.
The values obtained with thermogravimetric analysis present a wide range of diffusion
coefficients and activation energies. TGA gives a value for the chemical diffusion
coefficient, i.e., diffusion in the presence of a concentration gradient, so the initial
oxygen distribution in the sample as well as the microstructure can have a very
important effect in the final result obtained. All the experiments have been carried out
using polycrystalline samples of varying density and grain size, in most cases these
variables have either been ignored or not mentioned in their publications, which
makes it difficult to assess whether the result obtained is characteristic of that
particular sample and agrees with a general value for D or the analysis of the collected
data has been done incorrectly.
The use of polycrystalline samples for diffusion studies with thermogravimetry
introduces a large source of uncertainties in the measurements. These specimens
should be considered complex systems where a large number of factors can influence
the diffusion process. The degree of densification (open or closed porosity), the size of
the grains and the presence of grain boundaries should be taken into account before
comparing the results obtained to those from powder specimens or to polycrystalline
samples with different microstructural features.
3.3 Tracer diffusion experiments
Tracer diffusion studies to monitor oxygen diffusion in YBCO are carried out using
18O as the tracer since there is no radioactive isotope of oxygen. Diffusion
experiments are performed equilibrating the samples in 16O, annealing them in 18O,
measuring the 18O depth profile by SIMS and determining the value of D* from a fit of
the depth profile to the solution of the diffusion equation.
Chapter 3 Literature Review
57
These studies can be carried out using single crystal or polycrystalline samples. The
results obtained from tracer diffusion experiments are affected less by the different
microstructural features of the samples (e.g. in polycrystalline materials), as SIMS
depth profiles average the results over the different grain orientations of the specimen.
The use of single crystals is very useful to obtain data from the behaviour in different
orientations, although diffusion in the ab plane is limited by the small thickness of the
crystals. Table 3.2 presents a summary of the results discussed below:
Rothman et al. (1989), (1991) and Routbort et al. (1990) have carried out a complete
study of the phenomenon of oxygen tracer diffusion in YBCO. They have measured
18O diffusion in polycrystalline and single crystal samples, calculating an expression
for diffusion in polycrystals that is corroborated by most of the other works published
(Turrillas et al. (1989), Sabras et al. (1990)). They did not observe a dependence of
the tracer diffusion coefficient on partial oxygen pressure or oxygen content.
Fig 3.6 Penetration plot for YBCO polycrystal annealed for 1/2 h at 300 °C (after Rothman et al.
(1991))
Their studies of diffusion on single crystals have been set as a reference point by all
the diffusion work carried out later on as they were the first to measure the anisotropy
in the diffusion process in the ab plane and along the c axis. The difference was
calculated to be of the order of 104 – 106 cm2/s. This has also been observed by other
authors that conducted the same type of study on single crystals: both Bredikhin et al.
(1991) and Tsukui et al. (1991-a), (1991-b) measured the difference between D*ab and
D*c in a single crystal and found it to be of approximately five orders of magnitude.
Chapter 3 Literature Review
58
Comparing the single crystal data with that from the polycrystalline samples, they
observed that the diffusion coefficient of the latter was approximately the same as that
measured in single crystal samples in the ab direction. This corroborates the initial
assumption that tracer diffusion measurements are not affected by the microstructure
of the specimen, giving an average value of the process in the sample. Since diffusion
along the c axis occurs so slowly it is reasonable that the result from a polycrystalline
sample is similar to that of diffusion along ab in a single crystal, the fastest of the two
diffusion process.
To complement the diffusion study, Rothman and co-workers (Rothman et al. (1991)
and Routbort and Rothman (1994)) calculated the value of the tracer diffusion
coefficient along the a and b directions in single crystal samples, finding a difference
between them of one to two orders of magnitude. This is the only work that has
measured such difference; the rest of the studies just consider diffusion in the ab plane
and along the c axis. They propose a mechanism of diffusion based on organised
diffusion along O(5) sites. This is similar to the diffusion mechanisms suggested by
Ronay and Nordlander (1988).
There is a third piece of work (Tsukui et al. (1991-a), (1991-b)) that measured the
value of D* in single crystals with a stair-like shape; this configuration allowed them
to carry out measurements along the ab and c directions in the same crystal, which
they suggest gives more reliable results than if the studies had been carried out on
different samples for each direction. However, they calculated a difference between
the ab and c diffusion almost two orders of magnitude higher than that presented by
Rothman et al. (1990). This could be due to the presence of impurities or flux on the
surface of the crystal, as they report having observed the presence of flux in other
crystals used, which altered the diffusion results obtained.
Chapter 3 Literature Review
59
Fig 3.7 Isotopic penetration profiles along the ab and c directions for in-diffusion of 18O at 800 °C in
an YBCO single crystal (after Bredikhin et al. (1991))
Another study of D* in single crystals is presented by Bredikhin et al. (1991). They
carry out a systematic study at two different temperatures of the diffusion coefficient
along the ab and the c directions, and although the results for D* in the ab plane agree
with those of Rothman and co-workers and the rest of the groups, the activation
energy calculated for D*c is rather high (2.3 eV); this is understandable given the
difficulty for the oxygen ions to move through the crystal structure in the c direction.
The difference between the measured activation energies along the ab plane and the c
axis seems to show that at one point at very high temperatures both values will
coincide. However, this would occur beyond the melting point of YBCO. The
activation energy calculated for D*ab is also slightly higher than that reported by
Rothman and co-workers, although it is within the range energies that are expected for
this type of process: around 1 eV.
It can be seen in Fig. 3.8 (and from the diffusion data collected with different
measurements – see Tables 3.1 and 3.2) that the slopes of the tracer diffusion data are
similar to those obtained for other chemical diffusion studies (Figs 3.2 and 3.11). This
seems to indicate that the effect of the thermodynamic factor, Φ, mentioned in chapter
2 (Fig. 2.6) is not as large as expected from the data published by Faupel and
Hehenkamp (1993). Their results show a considerable increase in the value of the
thermodynamic factor with increasing oxygen content. This would mean that the
difference between chemical and tracer diffusion data would be bigger at low
temperatures, as in this region the equilibrium oxygen content of YBCO is higher. It
Chapter 3 Literature Review
60
can be seen, however, that this is not the case, as most of the activation energies
obtained for chemical and tracer diffusion processes are in the range of 0.9 – 1.2 eV.
This could be an indication of the presence of a second effect that compensated the
radical increase in Φ in this low temperature region. Salomons and de Fontaine (1990)
have carried out MonteCarlo calculations to study theoretically the oxygen diffusion
process and they have also presented the change in the thermodynamic factor with
oxygen content in the sample. Their calculations present some variations with respect
to the results of Faupel and Hehenkamp (1993), but the magnitude of the
thermodynamic factor is also high for high oxygen contents of YBCO. They also
observed a sharp increase in Φ around 6.5.
It has been suggested that the effects of a concentration dependence of the diffusion
coefficient could counteract those of the thermodynamic factor, but the variation in
the activation energy due to the change in oxygen content would have to be so large
that it would have not passed unnoticed by the groups studying chemical diffusion.
The dependence of the diffusion coefficient on concentration and how this can be
observed using different measurement techniques should be the subject of a more
detailed study so that these points can be clarified.
1.0E-18
1.0E-16
1.0E-14
1.0E-12
1.0E-10
1.0E-08
1.0E-06
1.0E-04
0.0009 0.0011 0.0013 0.0015 0.0017
1/T (1/K)
lo
g 
D
Ikuma and Akiyoshi Pxtl Rothman Pxtl
Turrillas Pxtl Bredikh ab
Tsukui ab Bredikh c
Tsukui c
800 700 600 500 400 300
T (C)
Fig. 3.8 Collected tracer diffusion data
Chapter 3 Literature Review
61
There is another group of papers that studies the diffusion of 18O in YBCO but do not
use SIMS to evaluate the tracer diffusion coefficient. These groups use
thermogravimetric analysis to calculate the speed at which oxygen is absorbed in the
16O saturated sample. This work has been included in the tracer diffusion section
although it does not fit in the standard tracer diffusion measurement group.
The first work published of this kind is that by Ikuma and Akiyoshi (1988), who
studied oxygen isotope substitution in polycrystalline YBCO. They calculated an
expression for this phenomenon with a very high activation energy (1.76 eV). This
results in a lower value of the tracer diffusion coefficient. Rothman et al. (1989)
suggest that the phenomenon measured by Ikuma and Akiyoshi may not represent
diffusion. Their different result could be due to the presence of impurities in the
polycrystal, as thermogravimetry is quite sensitive to microstructural changes. The
decrease in D with increasing oxygen deficiency observed by these authors disagrees
with the Arrhenius expression for the diffusion coefficient presented. According to
this expression, D* increases with increasing temperature and this is associated to a
loss in oxygen, since the equilibrium oxygen content of YBCO decreases with
increasing temperature (see Fig. 1.5). It is therefore not possible to observe a decrease
in D* with increasing oxygen deficiency as temperature increases, which indicates that
there must be a flaw in their measurements.
The second work has been published by Conder and co-workers (Conder et al. (1993),
Krüger et al. (1993)). They studied 18O substitution in the YBCO lattice using
thermogravimetry. They calculated the activation energies characteristic of the
different types of ion exchange in the crystal structure. These range between 0.71 eV
for substitution of 18O in chains to 1.88 eV for the substitution of 18O in planes, apex
positions and chains. In a later paper, where they relate their study of the oxygen
diffusion to the superconducting properties, Conder et al. (1994) give a slightly higher
value for the activation energy of the isotope exchange in the chains: 1.23 eV. This
agrees better with the rest of the data presented by other authors. They confirm the
temperature (and not pressure or concentration) dependence of the oxygen diffusion
coefficient in YBCO.
Chapter 3 Literature Review
62
All their analysis has been carried out using a highly empirical method that consists on
finding the best fit to the experimental data from a set of curves that are characteristic
of a number of mechanisms (fifteen different types). The curve that seems to fit their
results best is characteristic of diffusion controlled by the formation of an oxygen
saturated outer region and the speed at which the ions diffuse through this layer, which
agrees with the reaction core oxygenation process suggested by Furukawa et al.
(1992), but disagree with the results obtained in this work. These results (see chapter
5) do not support the formation of a highly oxygenated outer layer that hinders the
diffusion process.
3.3.1 Summary
Diffusion results from tracer diffusion measurements are in very good agreement with
the exception of the work presented by Ikuma and Akiyoshi (1988). This technique
presents the advantage of being independent of the microstructure of the sample used,
as the results shown in Fig. 3.8 have been obtained from polycrystallinea dn single
crystal specimens. A second advantage is the possibility of measureing the value of
oxyen diffusion along the c axis, although it is necessary to have thick single crystals,
which are not always easy to obtain.
However, its correlation with chemical diffusion through the thermodynamic factor,
Φ, is not very clear, as this factor has been calculated to increase to very high values at
high oxygen contents. This change in Φ with concentration would result in a non-
Arrhenius dependence of D* with temperature, disagreeing with all the experimental
results published in the literature. More work should be carried out in this field to
clarify the relationship between the tracer and chemical diffusion coefficients.
3.4 Resistance measurements
Resistance methods are based on the change in resistivity of YBCO with oxygen
content. The resistivity of a material is determined by the product of the carrier
concentration times their electronic mobility. It is known that holes in YBCO can be
charge carriers and the concentration of these holes varies with oxygen doping.
Chapter 3 Literature Review
63
Therefore, changes in resistivity can be related to oxygen stoichiometry and to the rate
of change of this stoichiometry with time and temperature. Chapter 6 deals in more
detail with the variation in the resistivity of YBCO with oxygen content in the high
temperature range (>300 °C).
The groups that have used resistance methods to measure the diffusion coefficient of
oxygen in YBCO have studied this process in single crystals and polycrystalline
specimens. As it has been seen in the previous sections, sometimes the work
published does not give enough information about the microstructural features of the
samples to be able to compare the data in the literature. The results of these
measurements have been summarised in Table 3.3.
Tu et al. (1988-a), (1988-b) and (1989) studied the change in resistance during
oxygenation and deoxygenation of polycrystalline samples. They carried out both
isothermal and non-isothermal experiments.
Fig 3.9 Isothermal resistivity measurements (after Tu et al. (1989))
The isothermal experiments showed an initial sharp decrease in the resistivity with a
gradually decrease in slope with time which they attribute to a concentration
dependent in-diffusion, as the process seems to occur much faster at the beginning of
the measurement than at the end. From the slopes of the initial and final sections of
the resistivity vs. time curve, they obtained two activation energies: 0.45 eV for δ =
0.38 and 1.1 eV at saturation (δ = 0). Tu et al. (1988-a) suggest that the difference in
activation energies is due to the mechanism of oxygen diffusion. In the case of the
lower activation energy, the oxide has a high concentration of oxygen empty sites in
Chapter 3 Literature Review
64
the CuO planes and the energy required is that of oxygen motion down the plane.
However, as the sample saturates with oxygen the diffusion process seems to slow
down due to the reduction in the number of excess vacancies in the lattice, which
means that for diffusion to occur these empty sites need to be created before oxygen
can move within the material. This explains the higher value of the activation energy
(Ae) at the end of the isothermal experiment; the value of Ae contains the energy for
oxygen motion (0.48 eV) plus the energy for the formation of a defect (0.6 eV), giving
the final calculated value of 1.1 eV. This apparent slowing down of the change of
resistivity with time has been suggested by other authors to be characteristic of a
concentration dependent diffusion coefficient (LaGraff et al. (1991), LaGraff and
Payne (1993-a)).
However, chapter 5 will show that the shape of these measurements is characteristic of
an Arrhenius behaviour controlled by a single diffusion coefficient. Although the
speed at which oxygen diffuses in the material appears to slow down at longer times,
what happens is that the sample is being saturated with the characteristic oxygen
content for that temperature (see Fig. 1.5 for characteristic oxygen contents vs.
temperature and Fig. 3.3 for variation from Arrhenius behaviour) and the amount of
oxygen diffusing in the sample is less. The reduction in the variation of the oxygen
stoichiometry results in a reduction of the change in resistivity, as it depends on the
amount of oxygen present in the sample.
In the same work, Tu et al. (1988-a) observed that out-diffusion did not show the
same concentration dependence reported for in-diffusion and calculated the activation
energy of this process to be 1.76 eV. This high value is suggested to be the energy
required to overcome the surface barrier to oxygen diffusion during deoxygenation. It
should be taken into account that a small increase in the oxygen content will create
high conductivity paths for the flow of current, causing the resistivity value to drop
dramatically. The sudden decrease in resistivity observed by Tu et al. (1989) is
therefore not an indication of a faster in-diffusion or of a surface controlled out-
diffusion as it is suggested in their paper. This linear increase in resistivity with
decreasing oxygen content observed has not been reported by other authors, which
suggests that this effect might be a consequence of the behaviour of their
Chapter 3 Literature Review
65
polycrystalline samples due to microstructural features or to the experimental method
used. The value of the activation energy for out-diffusion compares only to that
obtained by Kishio et al. (1989-b) and Ikuma and Akiyoshi (1988) by
thermogravimetric analysis. There are other groups that agree with a surface-reaction
limited out-diffusion, such as LaGraff et al. (1991); they suggest that oxygen
evolution from the sample is controlled by the recombination and desorption process
of oxygen ions. However, the activation energy calculated by LaGraff et al. (1991) for
out-diffusion is much lower than that calculated by Tu and co-workers: 1.2 eV.
1.0E-18
1.0E-16
1.0E-14
1.0E-12
1.0E-10
1.0E-08
1.0E-06
1.0E-04
0.0009 0.0011 0.0013 0.0015 0.0017
1/T (1/K)
lo
g 
D
Tu 
Elschner
LaGraff and Payne Pxtl.
800 700 600 500 400 300
T (C)
Fig. 3.10 Collected resistance data
From the non-isothermal studies of oxygen absorption, Tu et al. (1988-b), (1989) used
Ozawa’s thermal analysis (Ozawa (1970)) and calculated a value for the activation
energy equal to 1.35 ± 0.1 eV (and a value of Do = 0.035 cm2/s for δ = 0). This
activation energy is higher than that calculated from the data from the isothermal in-
diffusion experiments and is thought to be a consequence of the formation of a highly
oxygenated layer at the surface of the sample that hinders the in-diffusion process.
These non-isothermal oxygenation curves have a very characteristic shape (see Fig.
3.11): when heating in pure oxygen a deoxygenated sample, the resistivity increases
until around 300 °C, where a maximum is observed. This maximum is believed to
mark the onset of diffusion into the sample. If the temperature is increased further, the
Chapter 3 Literature Review
66
resistivity decreases as the oxygen content of the sample increases, but at temperatures
around 500 °C the resistivity reaches a minimum value, thought to correspond with
the point at which the sample has saturated with oxygen. From that point onwards the
resistivity increases linearly again. Similar experiments have been carried out in this
work using thermogravimetry giving the same characteristic curve, as it will be shown
and discussed in more detail in chapter 5.
Fig 3.11 Non-isothermal resistivity measurements (after Tu et al. (1989)). 1 and 2 correspond to the
variation in resistivity on heating and cooling at 3 °C/min in He, and 3 and 4 to the changes in
resistivity when heating (3) and cooling (4) at 1°C/min in O2
Ottaviani et al. (1989), in their isothermal oxygenation experiments carried out with
polycrystalline samples, have also calculated two different activation energies from
the initial and final stages of the variation of resistivity with time, obtaining values
equal to 0.4 eV for the beginning of the process (agreeing with Tu et al. (1988-b)) and
0.9 eV for the final part of the isothermal curve. Thermogravimetric measurements
carried out simultaneously did not register any weight change in the last stages of the
oxygenation. They suggest that the annealing at low temperatures in the presence of
oxygen promoted the rearrangement of oxygen atoms from the (½, 0, 0) site where
oxygen gives no additional carrier, to the (0, ½, 0) position where oxygen adds mobile
charge carriers to the system. This process is activated by the rearrangement of a small
number of oxygen atoms, and they only have to move to adjacent positions, which
explains why the oxygen content does not change appreciably. The small change in
position explains the low value of the activation energy calculated for this process.
Chapter 3 Literature Review
67
Work done by Veal et al. (1990-a) (1990-b) supports this proposed rearrangement of
oxygen within the sample without changing the oxygen content. They report an
increase in the Tc of samples quenched from 520 °C after leaving the samples at room
temperature for several days. This is attributed to an increase in the order of the
vacancies in the sample, i.e. to an increase in the orthorhombicity of the sample that is
not associated to any oxygen absorption (the temperatures used are too low for any
significant diffusion to take place during the timescale of the experiments). The results
presented by Ottaviani et al. (1989) should have been complemented with Tc
measurements to corroborate their suggested rearrangement of oxygen ions into
orthorhombic domains. This is a very interesting behaviour that should be studied in
more detail, as it offers the possibility of improving the superconducting properties of
a material with a given oxygen content by annealing at low temperatures (< 300 ºC).
There is also the possibility that their thermogravimetric system was not sensitive
enough to record the small weight changes taking place during the last part of the
diffusion process.
Elschner et al. (1992) studied the dependence of the microstructure on the different
mechanisms of oxygen transport, working with polycrystals with different density and
grain size. The samples used had densities equal to 85% and 100% of the theoretical
density. Of the latter type, two samples were made by conventional sintering but in
one of them some melting occurred, so the microstructure of the samples was different
(40µm grains in the first one and presence of wide cracks 30 µm apart in the second).
They did not observe any difference between the in- and out-diffusion processes (see
Fig. 3.12), disagreeing with Tu et al. (1988-a) and LaGraff et al. (1991). The
activation energy they calculated for the diffusion process was the same for all the
samples used, irrespective of their density and grain size. The value of obtained for the
activation energy was equal to 0.5 eV. The pre-exponential coefficients were different
for each of them, which marks the difference in diffusion coefficients between the
three samples.
Chapter 3 Literature Review
68
Fig 3.12 Time dependence of electric resistance with changing oxygen partial pressure at T = 750 °C
(after Elschner et al. (1992))
It was observed that the diffusion of the 100 % dense sample sintered without melting
remained constant at temperatures below 450 °C. Elschner et al. (1992) suggest that
above this temperature the diffusion process measured is that occurring in the bulk of
the samples, whereas below 450 °C grain boundary diffusion takes over and controls
the process. The activation energy observed for this grain boundary diffusion is equal
to 0 eV. If this type of diffusion had such a low activation energy it would also be
expected to be present in the other two samples, but this is not observed. This
irregularity in the diffusion measurements is perhaps due to the presence of secondary
phases or impurities in the specimen. The rate limiting factors suggested for each
sample are bulk diffusion in the case of the 85% dense specimen and grain boundary
diffusion (diffusion down cracks) for the highly dense specimens.
LaGraff and Payne (1993-a), (1993-b) and (1993-c) carried out experiments with
single crystals at high temperatures and polycrystals with different grain sizes and
porosity in a wider temperature range (see Fig. 3.13 a)). As it has been mentioned
before, they found in-diffusion to be much faster than out-diffusion and the activation
energy for the in-diffusion process in single crystals was calculated to be equal to 1.1
eV, whereas that for a 95% dense polycrystal was 0.4 eV. This value agrees with those
presented by Ottaviani et al. (1989) and Tu et al. (1988-a). The lower activation
energy was thought to be characteristic of grain boundary diffusion, while the higher
one (1.1 eV) was suggested to be that of bulk diffusion. However, it has been
mentioned above that the shape of the variation in resistivity with time during an
isothermal experiment can be characteristic of a diffusional process with a single
value for the diffusion coefficient, rather than of a combination of a fast and a slow
Chapter 3 Literature Review
69
diffusion mechanism. They suggest the possibility of an important contribution of
grain boundary diffusion in polycrystalline samples for temperatures above 350 °C, as
it is the point above which they observe a non-linear variation of the resistivity
increase in non-isothermal experiments (c.f. Tu et al. (1989)). Since this onset of non-
linear resistivity occurs at 500 °C for single crystals, it has been suggested that 350 to
500 °C is the range within which grain boundary diffusion is faster than bulk diffusion
and above 500 °C diffusion occurs mainly inside the grains.
a)      b) 
Fig 3.13 a) Resistance variation for oxygen in-diffusion at 708 °C in polycrystalline and single crystal
YBCO (after LaGraff et al. (1993-b)), and b) change in resistivity during in- and out-diffusion
measurements when a highly oxygenated layer and a highly deoxygenated layer are formed
(respectively) in the samples.
Erb et al. (1996) carried out resistivity studies on single crystals and did not find any
difference between in- and out-diffusion processes, agreeing with Elschner et al.
(1992). Neither did they observe any dependence on the oxygen concentration. The
activation energy calculated seemed slightly low (0.8 eV), which could be due to
impurity effects. They point out that the method used to measure the changes in
resistivity with time and temperature in the sample could affect the final results
obtained, as some techniques are affected by grain boundary and surface effects
whereas others only measure changes in the bulk of the sample.
3.4.1 Summary
The use of resistance methods to measure the oxygen diffusion coefficient presents a
number of problems: one of them is that a small change in the oxygen content of the
material can have a drastic effect on the value of the resistivity. LaGraff and Payne
Chapter 3 Literature Review
70
(1993-a) calculated the effects the formation of a highly oxygenated shell and a
deoxygenated shell would have on in- and out-diffusion respectively, and they found
that a deoxygenated region formed during out diffusion would have a lesser effect on
the change in resistivity than an equally thick oxygenated shell during in-diffusion
(Fig. 3.13 b)). This oxygenated layer would form around the perimeter of the grains
and in the necks formed between the grains during the sintering process. It would act
as a short-circuiting pathway yielding resistivities not representative of the bulk
material. The low activation energies calculated by Tu et al. (1988-a), (1988-b),
(1989), LaGraff and co-workers (1991), (1992), (1993-a,b,c) and Elschner et al.
(1992) could be due to the presence of this highly oxygenated region at the surface of
the samples.
With respect to the evaluation of the diffusion coefficient from the change in
resistivity of the specimen it should be noted that it is necessary to know the value of
the effective diffusion length. When using resistance methods the external dimensions
of the sample (in the case of single crystals) may not necessarily be the diffusion
distances, so any diffusion coefficient given should be taken as an estimate of the
upper limit of the value of this coefficient. Polycrystals with open porosity (density
lower than 94 % of the theoretical value) present more internal surface area for oxygen
exchange to occur. This means that the diffusion length will be shorter if the shell
forms. Therefore the results are not representative of bulk behaviour.
It seems there is not an agreement on whether in-diffusion is faster or slower than out-
diffusion. Some authors argue that out-diffusion is surface reaction limited and this is
why it is slower than in-diffusion (Tu and co-workers), while this same argument is
presented by other authors to justify their findings of slower in-diffusion than out-
diffusion (LaGraff and Payne (1992)). The formation of a highly oxygenated layer is
also another argument presented when trying to explain a slower in-diffusion process.
At the same time, there are groups who have observed that both processes occur at the
same rate (Elschner et al. (1992)). It seems that since diffusion occurs due to the
presence of a chemical gradient, the oxygenation of the surface should not hinder but
form part of the diffusion process. The argument against surface reaction limited out-
diffusion has already been covered at the beginning of this section.
Chapter 3 Literature Review
71
3.5 Internal friction methods
A number of authors have measured internal friction of oxygen ions in YBCO and
related it to the diffusion of oxygen in this material (see Table 3.4). This technique
measures the stress relaxation of individual atoms as they diffuse in a random walk
through the structure. The activation energy of this process is the potential energy
barrier that the oxygen atoms must overcome in the diffusion process. The
measurements are not affected by the microstructure of the sample (e.g. presence of
grain boundaries), as the distance considered (jump distance of an oxygen ion) is only
a few Å, much smaller than the size of the microstructural features. This technique
gives good results at low temperatures, where normal methods of studying diffusion
are inoperative because of the very slow diffusion rates.
Internal friction measurements give a peak that in this case appears at 210 °C at 1.18
Hz and shifts to 800 °C for a vibration frequency of 40 Hz. This peak is attributed to
be characteristic of hopping of oxygen ions between the O(1) and O(5) sites as the
result of the applied stress. Most of the results obtained using this technique agree in
that the activation energy for oxygen hopping is approximately equal to 1 eV, which is
the characteristic value for the migration of point defects (e.g. vacancies).
Two groups (Tallon and Staines (1990) and Xie et al. (1989)) observe a change in the
value of the activation energy with increasing temperature, but the final value of the
diffusion coefficient is observed by Tallon and Staines (1990) to increase with
temperature, while Xie et al. (1989) observed that D decreased as the temperature
increased. The change in activation energy was attributed to a change in the potential
energy of the O(1) and O(5) sites as the structure becomes tetragonal and the sites
become equivalent, but this does not explain why the deviations from linearity go in
opposite directions for each group.
Xie et al. (1989) suggest an explanation for the different in- and out-diffusion
processes: The diffusion current is proportional to the density of oxygen-defective
supercells and in the case of in-diffusion, oxygen atoms find that there are a large
number of these in the structure, whereas in the case of out-diffusion the material is
saturated with oxygen, which makes the process very slow since it restricts the
Chapter 3 Literature Review
72
diffusion process to the outer deoxygenated layer of the sample. This argument is
valid only for fully deoxygenated or fully oxygenated specimens. In the case of a
partially oxygenated sample there should be enough defective supercells for both
processes to occur at equivalent rates.
1.0E-18
1.0E-16
1.0E-14
1.0E-12
1.0E-10
1.0E-08
1.0E-06
1.0E-04
0.0009 0.0011 0.0013 0.0015 0.0017
1/T
lo
g 
D
Xie Tallon
Cost-Lo Cost-Hi
800 700 600 500 400 300
T (C)
Fig. 3.14 Collected internal friction data
In two consecutive papers, Zhang and co-workers (Zhang et al. (1990-a), (1990-b))
present the results of very low frequency studies of oxygen relaxation deconvoluting a
series of peaks from the initial ones obtained. Two of these deconvoluted peaks are
associated to the motion of oxygen in the structure. This group is the only one that
considers oxygen motion not only along the Cu-O chains, but in the CuO2 planes and
along the c axis. They give activation energies for these two types of displacement, for
the in-plane approximately equal to 1 eV and for motion perpendicular to the CuO2
planes and Cu-O chains equal to 1.35 - 1.5 eV. Further studies should be carried out to
calculate the value of the pre-exponential coefficient to be able to compare these
results with those obtained with other techniques.
Cost and Stanley (1991) have carried out an internal friction study over a wide range
of frequencies which allowed them to calculate the activation energy for the relaxation
of the motion of oxygen ions in the structure. The value of the activation energy was
found to vary between 0.97-1.4 eV and the pre-exponential factor was equal to
Chapter 3 Literature Review
73
1.4×10-4 cm2/s to 6.3 × 10-4 cm2/s; these values agree with those presented by Xie et
al. (1989).
The relationship between oxygen diffusion and internal friction, however, is not
completely certain. On one hand the activation energies agree fairly well, especially
considering the range of energies obtained with other methods. But on the other hand
the curved Arrhenius plots in different directions and the fact that the motion of
oxygen ions occurs in the absence of a concentration gradient indicate that more work
should be carried out to find the correlation between the internal friction results and
chemical diffusion data.
3.6 Electrochemical methods
Using solid-state electrochemical cells it is possible to obtain information about
diffusion and ion conduction processes. In the case of YBCO, oxygen ions can be
driven in or out of the sample either by establishing a partial pressure gradient across
the cell or applying a potential difference across it.
A typical galvanic cell may be represented as follows:
O2 (P’O2), Au | electrolyte | electrode | Au, O2 (P’’O2)
where: electrode = YBCO, electrolyte = fast ionic conductor, e.g. YSZ
The potential, E, of this oxygen concentration cell can be related to the difference in
partial pressures on either side of the electrolyte by the Nernst equation:
( )22 ’’’ln OO PP
zF
RTE −= (3.1)
where (P’O2) and (P’’O2), are the oxygen pressures, R is the gas constant, F the
Faraday constant and z is the number of units of charge carried.
A number of studies have carried out electrochemical measurements to calculate the
oxygen diffusion coefficient of YBCO and their results are quite scattered. This
method is dependent on the microstructure of the sample and on the type of interaction
at the surface of the electrolyte, as reported by MacManus et al. (1992). The
Chapter 3 Literature Review
74
dependence on the microstructural features of the sample may be the reason for the
scatter in the results published:
O’Sullivan and Chang (1988) studied diffusion in a polycrystalline sample at 550 ºC
obtaining a value of 5 × 10-8 cm2/s for in- and out-diffusion, while this same value for
the diffusion coefficient was obtained by Belzner et al. (1990) for a temperature of
800 ºC. In a preliminary paper, MacManus et al. (1989) studied the introduction of
oxygen in YBCO using an electrochemical cell. From decay curves of the current
passing across the cell for set potentials applied, they calculated a value of the
intrinsic (bulk) diffusion coefficient equal to 9 × 10-8 cm2/s for 600 ºC, which can be
compared with that reported by O’Sullivan and Chang (1988). Further experiments
using resistance measurements are reported in this paper, giving values for the
diffusion coefficient equal to 8 × 10-7 cm2/s for 600 ºC and 1 × 10-4 cm2/s for 800 ºC.
The activation energy obtained from these two values is equal to 1.9 eV. These values
are so high because the diffusion distance taken for the calculations was half the pellet
thickness rather than half the average grain size.
In a following paper, MacManus et al. (1990) developed a computational model to
reproduce the variation in oxygen content as the electrochemical conditions varied,
and in an AC impedance analysis of various polycrystalline samples, the same group
(MacManus et al. (1992)) calculated values for the activation energy of the in- and
out-diffusion processes that agreed with Tu et al. (1989) (0.4 and 1.76 eV
respectively).
AC impedance studies consist on applying a small alternating voltage across a
galvanic cell and measuring the current response. Depending on the frequency of the
alternating voltage it is possible to observe the different processes of ion transport in
the cell, as each of them has a characteristic time constant. This type of measurements
can differentiate between bulk diffusion, diffusion along grain boundaries and across
the electrode. MacManus and co-workers found that the interface between the
electrode and the electrolytes could modify the final result of the measurement. The
values cited above were characteristic of an electrochemical cell with YBCO paste
electrodes; however, if these electrodes were made of a polished polycrystalline
sample the activation energy was found to be 1.5 eV. This activation energy is lower
Chapter 3 Literature Review
75
than that calculated from the data in their earlier paper (MacManus et al. (1989)).
However, as it was mentioned when analysing resistance methods all these activation
energy values are too high compared with the typical values for a vacancy diffusion
process and with the values calculated in this work (see chapter 5).
The diffusion coefficient calculated by Scolnik et al. (1991), 10-11-10-12 cm2/s, is four
to five orders of magnitude faster than other values reported for oxygen diffusion at
room temperature (e.g. Mogilevsky et al. (1994): 10-18-10-19 cm2/s). The reason why
all the diffusion coefficients obtained by O’Sullivan and Chang (1988), MacManus et
al. (1989) and the latter one by Scolnik et al. (1991) are so high compared to the rest
is unknown. The need to use polycrystalline samples in this technique and the possible
effects of grain boundary diffusion and the effect of the contact between electrodes
and electrolytes could be the reason for these fast diffusion coefficients.
Patrakeev et al. (1993) point out that obtaining activation energies from an Arrhenius
plot is not very accurate since the material is never at its equilibrium oxygen content.
They carried out oxygen permeation measurements of YBCO polycrystalline samples
and obtained a concentration dependent diffusion coefficient with activation energies
ranging from 2.0 to 2.4 eV for concentrations varying between 6.35 and 6.6, which are
amongst the highest values for activation energy reported.
These results are summarised in Table 3.5. The influence of sample microstructure on
the results obtained using this technique and the effects of the contact between the
electrode and electrolyte shows that great care must be taken when carrying out
electrochemical measurements.
3.7 Theory
The study of oxygen diffusion from a theoretical point of view is important since it
contributes to the understanding of the superconducting properties of YBCO and their
dependence on oxygen ordering in the structure, as well as helping to clarify the
diffusion process.
Alario-Franco and Chaillout (1989) suggested a simple tunnelling mechanism by
which oxygen atoms move in the b direction along the O(5) sites. This agrees with the
Chapter 3 Literature Review
76
work presented by Ronay and Nordlander (1988), who, by using effective medium
theory found that there is no potential energy barrier for oxygen motion in the
[1/2,b,0] tunnel, whereas oxygen motion in the a and c directions requires overcoming
barriers of 1.7 and 1.6 eV.
Bakker et al. (1988), (1989) presented a thermodynamic model to describe the oxygen
content and ordering in YBCO, and a solution of the Ising model yielded a slight
dependence of the diffusion coefficient on partial pressure of oxygen as well as a
break at the orthorhombic-tetragonal transformation.
Fig 3.15 Possible routes for oxygen to move in the a and c directions (Ronay and Nordlander (1988))
Most of the studies agree in the tendency of oxygen atoms to align in chains, forming
ordered regions within the material. De Fontaine and co-workers (de Fontaine et al.
(1989), (1990-a), and (1990-b), Ceder et al. (1990), Asta et al. (1991) and Salomons
and de Fontaine (1990)) used a Monte Carlo simulation of the antisymmetric next-
nearest-neighbour Ising (ASYNNNI) model and a cluster variation method (CVM) to
study oxygen diffusion in YBCO and related the different arrangement of the oxygen
atoms in the α (along the a direction) and β (along the b direction) sublattices to the
oxygen content and the critical temperature, Tc, characteristic for each oxygen
concentration in YBCO and calculated a phase diagram relating the different
orthorhombic ordered structures to the 60 and 90 K plateaux observed in Tc. This
phase diagram is developed in more detail by Semenovskaya and Khatchaturyan
(1992), who were able to distinguish not only the OI and OII phases, but also OI , OIII
and OIV . There is no calculation of the diffusion coefficient in the work by the group
Chapter 3 Literature Review
77
of de Fontaine and co-workers although they predict a diffusion coefficient that
depends strongly on concentration, noting the anisotropy between the diffusion along
the a or b axis in the structure.
Fig. 3.16 Oxygen ordering phase diagram of YBCO (de Fontaine et al. (1990))
Ausloos aQG 3 NDOVNL  DOVR XVHG WKH ,VLQJ PRGHO DQG WKH\ REVHUYHG DQ
irregularity in the value of the diffusion coefficient at an oxygen content of 6.5, which
has already been mentioned by Bakker et al. (1988), (1989), as well as being noted by
Salomons and dH )RQWDLQH  $XVORRV DQG 3 NDOVNL  VXJJHVW WKDW WKH
diffusion process occurs along the Cu-O chains when the oxygen content is high
(O(1)-O(1) or O(5)-O(5) jumps) and it is only at low concentrations of oxygen that
O(1)-O(5) jumps occur.
Rothman and co-workers (Rothman et al. (1991)) propose an oxygen diffusion
mechanism in which oxygen can only be removed from or be added to ends of ordered
chains and move down tunnels within the crystal structure. They also suggest that the
same type of jump is responsible for diffusion in the orthorhombic and tetragonal
phases.
The study of the path that follows an oxygen atom in its diffusion through YBCO has
been modelled by Islam (1990). The calculations yielded a series of diffusional jumps
along the O(1)-O(4)-O(1) sites, which means that diffusion in the c direction would be
more favourable than that in the ab plane, contradicting all the work presented so far.
Chapter 3 Literature Review
78
A number of authors have calculated the rate of formation of oxygen vacancies at
different positions in the lattice (Jorgensen et al. (1987-a), Capponi et al. (1987)), and
the occupancy of the O(4), O(2) and O(3) sites is always much lower than that of the
O(1) and O(5) sites, which makes diffusion along the c direction or in the CuO2 planes
a very restricted event.
With respect to the activation energies calculated with these theoretical methods, the
scatter is not bigger than that present in other methods, varying from 0.8 eV for
tetragonal YBCO (Choi et al. (1990)) to 1.2 eV (Salomons and de Fontaine (1990)).
It can be seen from the work reviewed in this section that no universally accepted
theory of diffusion in YBCO is yet available.
3.8 Conclusions
The study of oxygen diffusion in YBCO has been carried out using a number of
methods, and in spite of the large amount of work done in this field, this review
chapter shows that the results presented vary considerably, in some cases up to five to
six orders of magnitude.
The presence of microstructural features in the samples, such as porosity, grain size
and orientation, twin density and presence of microcracks among others affect the
final results of the diffusion process obtained when using certain experimental
techniques (TGA, electrochemical methods, resistivity measurements). These effects
are more noticeable when using polycrystalline samples, but using this type of
samples is in some cases difficult to avoid, because the small size of single crystals
limits the techniques that can be used.
Other techniques that are less affected by these microstructural features such as tracer
diffusion and internal friction methods, however, measure a different type of diffusion
coefficient: a process occurring in the absence of a concentration gradient. In order to
be able to correlate these results to chemical diffusion processes it is necessary to
know the value of the factors that link both types of diffusion (e.g. the thermodynamic
factor – see chapter 2). The value of these factors is also subject of study but has not
Chapter 3 Literature Review
79
been clarified completely, which makes the equivalence between diffusion in the
absence and in the presence of a concentration gradient rather difficult.
A large number of studies are incomplete, only giving values of the activation energy
of the diffusion process. This is an indication of how difficult it is for the oxygen ions
to move in the lattice, it gives the slope of the line obtained when plotting the value of
D vs. 1/T, but it does not locate the diffusion values. It is necessary to know the pre-
exponential coefficient (the intercept at 1/T = 0 of the D vs. 1/T line) in order to be
able to compare diffusion coefficients with each other. There are cases when the
analysis of the data collected using the different techniques does not allow to calculate
the pre-exponential coefficient; this limits the diffusion studies to a comparison of the
activation energy values. This does not give enough to be able to use it for practical
purposes of designing oxygenation stages during processing or calculating
oxygenation times for different samples.
The ideal situation would be a study of the diffusion process in large single crystals;
however, this is not very practical because of the extremely long times required to
oxygenate them fully at low temperatures, when the equilibrium oxygen content is
high and the superconducting properties can be optimised.
It has been shown that diffusion studies are very complex due to the large number of
factors affecting the techniques used and because of the modifications to the diffusion
behaviour introduced by the microstructural features of the samples studied. The
following chapters show how a careful design of the experimental procedure can lead
to a fairly complete study of the diffusion process and how these results can then be
used to model the outcome of a theoretical heat treatment in terms of final oxygen
concentration of the sample and distribution of oxygen within it, which can be applied
to the optimisation of the oxygen content in different types of samples.
Chapter 4 Experimental procedure
80
Chapter 4
Diffusion model and experimental
procedure
4.1 Introduction
This chapter is divided in two sections: the first one contains a description of the
diffusion model developed to reproduce the oxygenation process in YBCO. This
model was written to complement the analysis of the oxygen diffusion data obtained
experimentally, providing a means to follow the evolution of the distribution of
oxygen within the sample as well as giving the chance to predict the outcome of
hypothetical experiments and optimise the oxygenation during the cooling stage in the
processing of YBCO samples (see chapter 5).
This is followed by a second section, where a brief account of the thermal analysis
techniques available is given. This review concentrates on thermogravimetry, the
technique chosen to study the oxygen diffusion process in YBCO. An overview of the
preparation of the samples used in this work is presented, and the experimental
procedure followed to carry out the diffusion study is summarised at the end of the
chapter.
4.2 Modelling of oxygen diffusion in
YBa2Cu3O7-δ
The aim of the present study of oxygen diffusion is to reconcile the widely scattered
data present in the literature on this topic (see chapter 3). The computer programme
written was developed to model the diffusion process theoretically. The diffusion
study has been complemented with theoretical predictions of the outcome of
oxygenating and deoxygenating experiments calculated with the model developed.
Chapter 4 Experimental procedure
81
These oxygenations would have taken otherwise an unreasonable length of time to
complete if they had been carried out experimentally (oxygenation of large single
crystals). The use of this theoretical diffusion model is not limited to the prediction of
final oxygen contents during isothermal experiments; it can also be applied to the
study of the evolution of the oxygen content in multi-stage heat treatments and in non-
isothermal oxygenations, where the temperature is continuously changing with time
(see chapter 5).
4.2.1 Development of the model
It has been shown at the beginning of chapter 2 that heat conduction and diffusion are
analogous transport phenomena. This model has been developed based on a finite
element programme developed to study the heat transfer in steel specimens under the
effects of a concentrated heat source (Ruiz et al. (1989), Rodríguez-Donoso et al.
(1995)).
The model was initially designed for powder particles. These were assumed to be
spheres of radius r. It has been mentioned in chapter 3 that diffusion in YBCO occurs
more readily in the ab plane than along the c direction (Rothman et al. (1989)), so the
sphere is divided in parallel sections a distance dh apart perpendicular to the c axis.
This means that the circular faces of the flat cylinders are parallel to the ab plane (see
Figs. 4.1 a) and b)). Each one of these cylinders is then divided into concentric rings
separated by a distance dr.
The model calculates the rate of transfer of oxygen through each one of the rings in
the discs with respect to the volume of each ring and integrates this value over the
volume of the sphere.
When the distance dr is small compared to the radius of the sphere, it is possible to
calculate the rate of transfer of material through a chosen element of the sample taking
Fick’s first law of diffusion



∂
∂
−=
x
CDj , (4.1)/(2.1)
and applying it as follows:
Chapter 4 Experimental procedure
82
dr
r
dh
b
a
a
c
a)
b)
jr js
R S
C0
C1
C2 C3
C
r
c)
Fig. 4.1 Schematic of: a) side view, and b) top view of the divisions of the spherical particle for the
model developed, and c) Fick’s first law of diffusion
In a short time step, dt, the net amount of oxygen accumulated in the element shown
in Fig. 4.1 c) is equal to the amount that flows in through R minus the amount that
flows out of the element through S, this is:
( )( )20122 CCCdhrdr
Ddtjjj rsrnet −−−=−= pi (4.2)
where 2pirrdh equals the cross sectional area of each ring of radius r, and dr and dh are
the width and height of each ring, as shown in Figs. 4.1 a) and b). The variation in
accumulation or loss of oxygen in the theoretical powder particle with diffusion time
can be calculated integrating the above expression over the whole volume of the
spherical particle.
It is assumed that the variation in concentration between the boundaries of two
adjacent elements is linear and that the outer ring in each of the cylindrical slices is
always at the equilibrium oxygen content for each temperature. Diffusion ocurs due to
the concentration gradient established within the sample and the change in the value
of the diffusion coefficient with temperature. The process is not considered to be
surface diffusion controlled and the effects of grain boundary diffusion or diffusion
between the powder particles in agglomerates are ignored.
Chapter 4 Experimental procedure
83
4.2.2 Optimisation of time increments and stability of
solution
The stability of the calculation depends on the relationship between the size of the
time steps, dt, and of the width of the rings, dr, in each cylindrical section.
It is necessary to find a compromise between the length of the time increments and the
stability of the calculations. Computing time is reduced when the size of the time
increments increases, but this decreases the accuracy of the calculations and the
stability of the model.
To guarantee the stability of the model, the size of the time increment is chosen
according to the dimensions of the distance elements; in this case the elements are
concentric rings. Going back to Fick’s first law of diffusion:



∂
∂
−=
x
CDj ,   (4.1)/(2.1)
let Q be the number of oxygen atoms that pass through a plane of area A perpendicular
to the oxygen concentration gradient in time dt. Substituting in the equation above:
( ) dt
dr
CCADdt
dr
dCADQ ii −−=−= +1 . (4.3)
If Ni and Ni+1 are the number of oxygen atoms in adjacent elements i and i+1, and
since C = N/V, where V is the volume of the element considered, the above expression
can be rewritten as:
( ) ( )iiii NNKNN
dr
dtDQ −=−−= ++ 112  , (4.4)
this is: 
2dr
dtDK = . (4.5)
The value of K needs to be chosen with care. If K is very large for a given value of dr,
it would be possible to reverse the oxygen concentration gradient locally. If this
happens the solution becomes unstable in the following time step.
Chapter 4 Experimental procedure
84
ii-1 i+1
Fig. 4.2 Schematic of cylindrical elements in disk
The instability is limited to a region consisting of two adjacent elements plus first
order neighbours within this time increment. In this model each element is surrounded
by 2 elements. The concentration of the given finite element after a test time
increment is given by 2KCo, where Co is the total amount of material transferred per
unit time through each step. The concentration of the first order neighbouring finite
element is given by (1-2K)Co; therefore, the critical value of K such that the gradient
does not reverse is given by:
1-2K = 2K, (4.6)
which implies that:  Kc = 1/2           (4.7)
The simulation is stable provided Kc is not exceeded. For a given diffusion coefficient,
the value of K is inversely proportional to the square of the distance step, dr (Eq.
(4.5)). This distance is determined by the number of concentric rings into which each
cylinder is divided, ka, and this number is chosen so that the inherent error introduced
in the integration of the weight gain over the whole area is small. The value of kr
chosen for powder particles has been 20 (so that dr = r/20), and that for single crystals
has been 30 (with dr = r/30). The time interval between iterations, dt, is then chosen
(see Eq. (4.5)) so that the value of K is set equal to or just below 0.5 (as shown above).
Separate checks have shown no reduction in the accuracy of the fractional weight
change results when the value of K was altered (K<Kc). Decreasing the value of K had
the only effect of increasing computing time.
Chapter 4 Experimental procedure
85
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 10
Step
O
x
yg
en
 s
to
ic
hi
o
m
et
ry t = 108 s
t = 54 s
t = 162 s
a)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 10
Step
O
xy
ge
n
 s
to
ic
hi
o
m
et
ry
b)
t = 54 s
t = 108 s
t = 162 s
Fig. 4.3 a) Initial concentration traces for diffusion at 500 °C in a 2.1 µm radius particle when K = 0.2
b) Red lines: Initial concentration traces for diffusion at 500 °C in a 2.1 µm radius particle when K =
0.6. Blue lines: Comparison with concentration traces shown in a)
4.2.3 Error due to the finite element method
Finite element error originates from the assumptions of the method (see 4.2.1). The
first assumption made was that the concentration profile between two elements is
linear, but in practice this profile is slightly curved. The second error is introduced
because the centre of the oxygen distribution in an element does not quite coincide
with its geometrical centre. The effect of diffusion between the agglomerated powder
particles was not taken into account, as the agreement between the model for diffusion
in a single particle and the experimental results is very good.
These errors are proportional to the finite element density. An increase in the number
of elements in the calculation reduces the error introduced by the calculation in each
step. The size of the finite elements was chosen as a compromise between the
computation time and the required accuracy of the calculation.
4.2.4 Flowchart of the computer algorithm for the
diffusion model
The computer algorithm was written in QBASIC. The diffusion and geometrical data
necessary to do the calculations are stored as constants in the programme. These are:
the pre-exponential factor of the diffusion coefficient, Do, the value of the activation
energy, q, and the radius of the particle, r. The particle is divided into kr cylindrical
sections and each cylinder into ka rings, which allows to calculate the height of the
Chapter 4 Experimental procedure
86
cylindrical sections, dh, and the distance between the rings, dr. This first step also
includes setting the appropriate time interval, dt, in order to avoid instability in the
program at high temperatures according to what has been discussed in the previous
section. A second group of geometrical factors is calculated for each cylindrical
section: the external radius, volume and external surface area.
The user then inputs the initial oxygen concentration of the sample, the number of
steps of the heat treatment, the initial and final temperature of each step and the time
spent either changing the temperature between steps or keeping constant the
temperature set at the beginning of the step.
Once the initial concentration of the particle is known, this value is set as the starting
point for every ring and cylindrical slice, C(kr,ka). This calculation is repeated as
many times as cylinders there are (kr).
Time starts counting at this point; after each time increment the new temperature and
value of the diffusion coefficient, D(T), are calculated. From the variation in the
equilibrium concentration profile with temperature, the program can calculate the new
oxygen concentration for each ring C(kr,ka) (Eq. (4.1)). This is followed by a series of
iterations to calculate the change in concentration for all the rings with time and
temperature. Once the concentration profile within the disc is calculated, the computer
integrates the area below the curve obtained to calculate the relative weight change for
that time step. This series of calculations is repeated until the whole heat treatment has
been completed and the process starts again for the next disc in the sphere.
Concentration profiles, times, temperatures and weight changes are exported to a file
that can be read in spreadsheet form allowing the comparison between theoretical
values and experimental data.
Chapter 4 Experimental procedure
87
Geometrical factors
Diffusion parameters
Input processing
parameters
Concentration,  C(kr,ka)
t = 0
T = T0
t = t + dti
T = T + rate dti
D(T)
Change in concentration
All rings done
(ka = na)?
Overall weight change
ti = tiTot?
All discs done
(kr = nr)?
Export Concentration, Weight
and Temperature data
END
For every ring in the discs
and for each disc:
Yes
Yes
Yes
No
No
No
For each ring
(ka = 1 to na)
For each disc
(kr = 1 to nr)
Fig. 4.4 Flowchart of the diffusion algorithm developed
Chapter 4 Experimental procedure
88
4.3 Experimental techniques
The work presented in this thesis has been carried out using thermogravimetry. This
section reviews this technique applied to the study the oxygen diffusion process in
YBCO.
4.3.1 Thermogravimetry (TG)
Thermogravimetry (TG) a thermal analysis technique (see Table 4.1) that consists in
the study of weight changes of a specimen as a function of temperature, time and/or
surrounding atmosphere. This technique is used to quantify changes in weight
resulting from chemical reactions, decompositions, Curie point transitions,
transformations due to the interaction of a sample with the surrounding atmosphere
(e.g. oxidation) and absorption or release of gases from a sample. The last application
makes this technique particularly suitable for the study of the oxygen diffusion process
in YBCO.
Property Technique Abbreviation
Mass Thermogravimetry*
Derivative thermogravimetry
TG
DTG
Temperature Differential thermal analysis DTA
Enthalpy Differential scanning calorimetry DSC
Dimensions Thermodilatometry
Mechanical properties Thermomechanical analysis
Dynamic mechanical analysis
TMA
DMA
*The term thermogravimetric analysis and its abbreviation TGA have not been accepted by
the ICTA Nomenclature Committee.
Table 4.1 Thermal analysis techniques
4.3.1.1 TG design
The thermogravimetric analyser used in these experiments is a Perkin-Elmer TGA 7.
This system operates between room temperature and 1000 °C. The sample size can
have a maximum diameter of approximately 4 mm (bulk samples) and a volume of
≈12 mm3 (powder specimens). The usual sample weights are 25-30 mg and according
Chapter 4 Experimental procedure
89
to the manufacturer, the sensitivity of the balance is of 1µg. The errors associated to
the temperature and weight deviations in the system are discussed in more detail in
chapter 5.
The standard thermogravimetric analyser consists of five elements: balance,
temperature controller, furnace, atmosphere controller and a recording system. As it
can be seen in Fig. 4.5, a sample pan (often made of Pt, Au or porcelain) is suspended
from a high precision balance. In order to perform the experiments the sample pan has
to be introduced in a furnace. A thermocouple is placed in the furnace very close to
the pan but not touching it not to interfere with the weight measurements. The balance
chamber and the furnace are placed in a hermetically sealed enclosure where the
atmosphere can be controlled. Two flowmeters were connected to the gas intake for
the system so that the atmosphere could be a single gas (e.g. O2, N2, Ar) or a
controlled mixture of gases.
-
-
-
-
-
-
-
-
Gas 1 Gas 2
Balance gas
supply
Sample gas
supply
Tare
Sample
Furnace
Gas exhaust to ambient
Controller /
outputThermocouple
Fig. 4.5 Schematic of TGA
4.3.1.2 General operation and calibration
Thermogravimetry is a very sensitive technique and there are a number of variables
that may alter the final result of the measurement and should therefore be taken into
account. Experimental conditions must be carefully controlled and accurate calibration
of the system is necessary.
Chapter 4 Experimental procedure
90
• Control of the experimental conditions
 One of the main problems in thermogravimetry is the effect buoyancy and convection
currents of hot gas within the furnace enclosure may have on the final weight
measurement. These effects are reduced by either altering the design of the furnace to
minimise the convection currents or by keeping a certain flow rate that is, according to
the manufacturer of the system, the optimum to minimise these effects. The gas flow
rates recommended by Perkin-Elmer for this system are: 20 to 35 cm3/min for the
sample gas supply and 40 to 60 cm3/min for the balance gas supply
 It should also be taken into account that it is not possible to know the exact
temperature of the sample without interfering in the weight measurement, so a general
assumption must be made that the specimen is at the same temperature as the
thermocouple, which, as it has been mentioned in the previous section, is located as
close to the sample pan as possible.
 With respect to the samples used, it is advised that the sample size is maintained
approximately constant in all experiments. It is necessary to know the properties of the
sample in advance and whether the reaction rate is going to be affected by the state of
aggregation (powder, polycrystalline) of the specimen. Such is the case of diffusion
experiments, where the use of polycrystalline samples modifies the results; this will be
covered in more detail in the following chapter.
• Calibration
Thermogravimetric analysers are calibrated for weight and temperature. There is no
accepted standard for weight and temperature calibration, and it is usual that the
manufacturer supplies weight and temperature standards to calibrate the instrument.
Ferromagnetic samples with different Curie points are used for the temperature
calibration of the TGA. These samples are placed in the balance and the total weight
of the sample is altered by placing a magnet outside the furnace. When the
temperature increases above the characteristic Curie point of the specimens, they lose
their ferromagnetic properties and are no longer attracted by the magnet. This results
in a sudden change in weight in the measurement recorded. The use of several
Chapter 4 Experimental procedure
91
ferromagnetic materials is advised in order to extend the calibration over as wide a
temperature range as determined by the experiments to be carried out.
With respect to the variation of temperature with time, it is very important to note that
the heating rate modifies the lag of the sample temperature behind the programmed
temperature. It is therefore necessary to quantify the magnitude of this temperature
difference as a function of heating rate. Curie point measurements can also be used for
this purpose. The difference between the thermocouple reading and the real Curie
temperature of the sample at different heating and cooling rates gives the value of the
temperature lag for each ramp rate. Results obtained on heating at several rates are
shown in Fig. 4.6. The lag observed for the Fe sample, with a Curie temperature of
780 °C is much higher than that for Ni or Perkalloy, with lower Curie points. The
temperatures at which the experiments are carried out in this work are in a lower
temperature range: 300 to 550 °C. Therefore, the lag in the temperature measurements
carrid out in this work will never be higher than 20 °C.
0
10
20
30
40
50
300 400 500 600 700 800
Temperature (C)
Te
m
pe
ra
tu
re
 la
g 
(C
)  alfa 30
alfa 50
alfa 90
Ni
 Tc = 355.5 C
Perkalloy
 Tc = 596 C
Fe
 Tc = 780 C
Fig. 4.6 Increase in thermal lag with heating rate for samples with different Curie temperatures.
4.3.1.3 TG data analysis
There are a number of ways of analysing the data collected in TGA measurements.
One of the methods involves comparing the curves obtained to a set of already defined
curves that have been formulated to describe the behaviours of transformations
controlled by different mechanisms, such as boundary controlled growth or first or
Chapter 4 Experimental procedure
92
second order reactions. However, this method is not appropriate for the series of
experiments carried out in this dissertation since it is not known what mechanism
controls the diffusion of oxygen in and out of YBa2Cu3O7-δ. The microstructure of the
sample used can modify the value of the oxygen diffusion. It has been seen from the
scatter in the data shown in chapter 3 that diffusion is a complex process regulated by
different factors that vary with temperature: in bulk polycrystalline samples, for
example, it is controlled by grain boundary diffusion at low temperatures, with the
contribution from bulk diffusion becoming more important as the temperature
increases. The point at which these two mechanisms overlap is not known, and
assuming that only one of the two controls diffusion would be incorrect. Chapter 5 is
going to present the possible ways of carrying out diffusion studies that are not
affected by the microstructure of the samples.
The second method of analysis that can be used involves comparing the experimental
data to the profile and weight gain obtained using the solution to Fick’s second law
that is formulated according to the conditions present in the experiments (see chapter
2 for the different boundary conditions and solutions to the diffusion equation). The
study of initial rates of sorption and desorption also gives an approximation of the
diffusion coefficient. This is the analysis used in this thesis and its results are
described in more detail in the following chapter.
4.4 Materials and processing
The material used for these experiments was YBa2Cu3O7-δ in powder, polycrystalline
and single crystal form.
4.4.1 Powder
The powder used in the experiments was supplied by Seattle Speciality Ceramics, Inc.
(SSC). The manufacturer states in the specifications that the stoichiometry of the
material is YBa2Cu3O7, which has been confirmed with X-ray diffraction analysis.
Chapter 4 Experimental procedure
93
The particle size of the powder was measured using a Malvern Instruments Particle
analyser SB.0C and it has been found that the average particle diameter is 5.2 µm, and
the range of particle sizes present in more that 50 % in weight oscillated between 3
and 9 µm in diameter (see Fig. 4.7).
0
2
4
6
8
10
12
14
16
0.1 1 10 100
Particle size (microns)
A
m
o
u
n
t (a
rb
itr
.
 
u
n
its
)   
 
5.23 
Fig. 4.7 Particle size distribution in SSC powder
Fabrication processes normally start with a homogeneous powder of a suitable particle
size. Powder is frequently made using a solid state reaction method that involves
calcining oxide and carbonate precursors (Y2O3, BaCO3, CuO) mixed in the right
proportions. After the precursor powder is calcined at temperatures near 900 °C in O2,
the resulting material may contain secondary phases and unreacted components. The
grinding and calcining steps are repeated a number of times in order to completely
react the components into a phase pure powder. The YBCO powder used in this
experiments (see Fig. 4.7) was made following a slightly different technique, in order
to eliminate the effect of repeated grinding and calcining on the quality of the sample.
This technique combines solution, spray drying and combustion synthesis to produce a
homogeneous precursor powder consisting of an intimate mixture of Y2O3, BaCO3
and CuO (see Fig. 4.8). These powder particles were subsequently calcined and
converted to phase pure YBa2Cu3O7.
Chapter 4 Experimental procedure
94
Fig 4.7 SEM micrograph of the YBa2Cu3O7-δ powder used in this thesis. Average poarticle size:
5.23µm
Solution of
mixed
metal ions
Atomised
solution
Spray dry
-
dehydration
Ignited
precursor
droplets
Y2O3
Ba(NO3)2
CuO
HNO3
Carbohydrate
Calcination
YBa2Cu3O7-δ
(Y2O3,
BaCO3,
CuO)
YBa2Cu3O7-δ
Fig 4.8 Powder processing
Other alternative techniques to produce YBCO powder have been reviewed
extensively in Bourdillon and Bourdillon (1994) and Shi (1995).
4.4.2 Polycrystalline samples
The SSC powder was pressed at 50-76 MPa for two to five minutes to make pellets
and slabs that were sintered in oxygen in a tube furnace to produce polycrystalline
samples of different final densities. Before sintering the pellets were 1 cm in diameter
and had a thickness of around 4 to 5 millimetres. The rectangular slabs were 5 cm
long and 3 cm wide. The thickness was usually around 5-6 mm. Densification takes
place during sintering and the final volume decreases 15-25 %.
Chapter 4 Experimental procedure
95
Sintering depends strongly on temperature, and the choice of sintering temperature is
determined by the phase equilibrium diagram of the Y-Ba-Cu-O system. It is
important to have in mind that YBCO melts incongruently (decomposes peritectically
into solid Y2BaCuO5 and an Y-poor liquid phase) at around 1000 °C (see Fig. 4.9 b)).
Lindemer et al. (1991) review work done studying the decomposition of YBCO and
on the dependence of phase stability on the partial oxygen pressure of oxygen, and
they present a phase stability region depending on partial oxygen pressure and
temperature shown in Fig. 4.9 a).
Fig 4.9 a) Temperature oxygen partial pressure stability regime for YBCO, b) YBCO-CuO section of
CuO-BaO-Y2O3 diagram at 105 Pa (after Lindemer et al. (1991))
There are certain processing methods that require the formation of a liquid phase to
favour the formation of textured grains (melt texturing or flux growth methods)
(Salama et al. (1994)). However, for this study it was necessary to produce a series of
polycrystalline specimens with different densities so that it is possible to evaluate the
effect of porosity and of the presence of grain boundaries in the diffusion process,
therefore the formation of liquid phase during the sintering process was avoided.
The heat treatments consisted in a series of ramps at 240 °C/min to 950, 940 and
930°C, holding the samples at that temperature for 10-12 hours in a 100 % oxygen
atmosphere and leaving the sample to cool in the furnace to room temperature. The
densities of the samples were measured with a pycnometer and they were found to
Chapter 4 Experimental procedure
96
range between 81 % and 94 % of the theoretical density. Samples with densities below
91 % of the theoretical density exhibit open porosity whereas above this value the
pores between the grains are closed.
240 °C/h
10-12 h
950
940
930
T (°C)
time
Furnace cool
Fig. 4.10 Schematic of sintering process in 1 atm O2
4.4.3 Single crystals
The single crystals were kindly supplied by Dr. Chen of the Clarendon Physics
Laboratory in Oxford. These crystals were grown by the flux growth method in a
BaZrO3 crucible with the following heat treatment: heated to 920 °C and cooled (in
air) to 600 °C at 100 °C/h, then from 600 to 400 °C cooled at 10 °C/h and to room
temperature again at 100 °C/h, and ranged in size between 250 µm to a few
millimetres.
   0.3 mm
Fig. 4.11 Single crystal
Chapter 4 Experimental procedure
97
4.5 Characterisation techniques
The characterisation techniques reviewed in this section were used regularly during
the course of the measurements to corroborate the results obtained using
thermogravimetry.
4.5.1 Microscopy
Optical microscopy was used to examine the microstructure of the polycrystalline
samples and single crystals. The best viewing conditions for the microstructure of
YBCO exist in reflected polarised light. In these conditions the high-oxygen
orthorhombic phase shows a characteristic twinned structure. This structure is a
consequence of the accommodation of the stress generated in the crystal structure
when the lattice parameters of the unit cell change during the oxygenation process (see
chapter 1).
The extent of the twinning gives an idea of the degree of oxygenation of the grains/
single crystals. The tetragonal regions have lower oxygen content and appear
untwinned.
Camscan S2 electron microscopes were used in examination of the shapes of powder
particles as well as fracture surfaces of polycrystalline pellets. All images were formed
at a beam intensity of 40 kV.
4.5.2 X-ray diffraction
The diffraction patterns of specimens were taken using a Philips PW1370 vertical
diffractometer. Powder samples were mounted on a silicon substrate to eliminate
background signals and polycrystalline samples were directly mounted on the sample
holder aligned to the correct position in the diffraction cell.
Continuous θ-2θ scanning was carried out over the 2θ range 10° < 2θ < 100°, and
plots of X-ray intensity against 2θ were obtained. Step scans were also carried out.
Scan rates of 0.02°/min and preset times of 2-4 seconds were employed and the data
was also plotted in intensity vs. 2θ angle graphs. The 2θ range used was the same as in
Chapter 4 Experimental procedure
98
the continuous scans (10° < 2θ < 100°). The long counting times over narrow angular
ranges allowed to measure the peak positions precisely. Peak data were used to
estimate the oxygen stoichiometry in YBCO, calculating the lattice parameters from
the characteristic peak positions.
4.5.3 Tc measurements
An AC magnetic induction method can be used to monitor the diamagnetic response
of a sample placed between two induction coils. This response is directly related to the
oxygen content of the sample. The equipment used for making these measurements
was a ‘Lakeshore Cryotronic’ AC Susceptometer. The sample under study was
initially cooled in zero field. Then an a.c. signal of 333.3 Hz frequency and 79.6 A/m
was applied to the coils, and both the in-phase and out-of-phase components of the
magnetic susceptibility were recorded on heating the samples from approximately 4 K
to 100 K at 3 K/min.
60
90
120
150
180
210
240
0 20 40 60 80 100
Temperature (K)
In
-
ph
as
e 
sig
n
al
 (m
icr
o
v
o
lts
)   
a)
70
80
90
100
110
120
0 20 40 60 80 100
Temperature (K)
In
-
ph
as
e 
sig
na
l (
mi
cro
v
o
lts
)   
b)
Fig. 4.13 AC susceptometer traces of powder oxygenated at: a) 400 ° C in pure oxygen and b) in a
20%O2-80%Ar mixture at 450 °C. Note how in spite of the different oxygen contents of the samples
(6.97 and 6.9 respectively) the Tc appears to have the same value for both, when, according to Poulsen
et al. (1991) a decrease in the Tc should have been observed in the sample oxygenated at the lower
partial oxygen pressure and higher oxygenating temperature (b)).
Measurements were carried out on a number of samples with a range of oxygen
contents, from fully oxygenated at 400 °C to fully deoxygenated in Ar at 900 °C. Fig.
4.12 shows the results obtained from a sample oxygenated in pure oxygen at 400 °C
and another powder sample oxygenated in 20 % O2 at 450 °C. Both samples appear to
have the same critical temperature. It has already been pointed out in chapter 1 (see
section 1.3, Table 1.1) that there is a certain degree of uncertainty in the literature
Chapter 4 Experimental procedure
99
concerning the relationship between oxygen content and critical temperature of YBCO
samples; therefore, it was decided not to use this technique to determine accurately the
oxygen contents of the specimens.
4.6 Experimental procedures
4.6.1 Deoxygenation of samples
In order to be able to study the oxygenation behaviour of YBCO it was necessary to
produce oxygen-poor samples before carrying out the oxygenating experiments.
Deoxygenation of powders, polycrystals and single crystals was done heating the
specimen in Ar at 50 ºC/min to either 750 or 800 ºC, and holding the specimen at that
temperature until no further weight change was observed. The samples were cooled to
room temperature at 50 ºC/min.
4.6.2 Isothermal experiments
The aim of these experiments was to be able to calculate a value for the diffusion
coefficient that could be used later on to analyse the rest of the data collected:
4.6.2.1 Powder samples
Due to the small particle size of the powders it was not possible to heat powder
samples to a very high temperature, as the sample reached the equilibrium oxygen
content before it was heated to the final temperature set. Therefore, the range of
temperatures had a maximum limit of 450 ºC. This is a consequence of the increase in
diffusion coefficient with temperature (these aspects are covered in more detail in
chapter 5).
The powder was heated at a ramp rate of approximately 90 ºC/min to 300, 350, 400
and 450 ºC and it was held at those temperatures until no further weight change was
observed. The sample was subsequently cooled at a rate of 90 ºC/min.
Chapter 4 Experimental procedure
100
Additional experiments at different partial oxygen pressures were also carried out. The
pressures chosen were 20%, 50%, and 80% O2 in Ar.
4.6.2.2 Single crystal samples
The size of the single crystals used ranged between 212 µm to ≈7 mm. Their larger
size (diffusion distance) compared to the powder specimens increased the time it took
to equilibrate the samples in the low temperature range. Therefore, these single
crystals were used to study the oxygenation behaviour at the temperatures that were
too high for powder samples. The heating and cooling procedure is the same as for the
powders, with higher equilibration temperatures: from 400 to 550 °C. Data from these
experiments are combined with those from powder samples to be able to study the
diffusion process over a wide temperature range. All of these isothermal runs were
carried out in a pure oxygen atmosphere.
4.6.2.3 Polycrystalline samples
The oxygenation experiments using polycrystalline samples were carried in the same
temperature range as the powder specimens (300 to 450 °C) in spite of their larger
particle size. The ramp rates during oxygenation (heating and cooling) were
90°C/min.
4.6.3 Non-Isothermal experiments
Non-isothermal experiments were carried out on powders, polycrystals and single
crystal samples. After having deoxygenated the specimens, they were heated up to
700°C at the following ramp rates: 1, 10, 30, 50 and 90 °C/min. The samples were
held at 700 °C until no weight change was observed and they were cooled at the same
rate as they were heated up. These experiments were also carried out in a pure oxygen
atmosphere.
Chapter 4 Experimental procedure
101
4.7 Conclusions
This chapter has presented a description of the diffusion model developed to become a
theoretical basis of the study of oxygen diffusion in YBCO. The model will be used to
predict the oxygen contents of the samples after oxygenation and the distribution of
these contents within the samples used. It will also provide with a means to optimise
oxygenation times during processing (see chapter 5).
Thermogravimetry, the experimental technique used in this thesis for the study of
oxygen diffusion has been presented. This is followed by a description of the YBCO
powder and single crystals specimens as well as the preparation method of the bulk
samples used in these experiments. The chapter concludes with a description of the
experimental methods followed in the course of the experiments, covering the
deoxygenation pre-treatment and the isothermal and non-isothermal oxygenations
carried out.
Chapter 5 Results and discussion
102
Chapter 5
Results and discussion
5.1 Introduction
This chapter presents the experiments performed in this diffusion study and discusses
the results obtained. Their analysis is used to clarify the current situation on the value
of the oxygen diffusion coefficient in YBa2Cu3O7-δ. It has already been shown
(chapter 3) that the scatter in values of D~  published is quite large. The aim of this
thesis is to study the reasons behind this scatter as well as to give a value for the
chemical diffusion coefficient that is not affected by compositional or microstructural
features.
The results presented were obtained from diffusion studies carried out with powder,
single crystal and polycrystalline samples. These are discussed for experiments done
under isothermal and non-isothermal conditions. The section that describes the
diffusion analysis done on powder samples includes a comparison of the experimental
results with those from the theoretical calculations obtained with the model of the
oxygenation process developed (see chapter 4).
The final section of this chapter is dedicated to further applications of the model for
hypothetical oxygenation experiments. These have been performed to optimise the
oxygenation step during the processing of YBCO.
5.2 Isothermal experiments
The purpose of this first set of experiments was to calculate a value for the diffusion
coefficient of YBCO. The expression for D obtained will be used later on to analyse
the rest of the data collected and it will become a reference value for future
applications.
Chapter 5 Results and discussion
103
The technique used in these experiments was thermogravimetry (see chapter 4 for
experimental procedure followed). Monitoring the rate of change of the sample weight
it is possible to calculate the value of the chemical diffusion coefficient of oxygen in
YBCO (see chapter 2).
The state of aggregation of the sample can alter the final result of the measurements,
so it is necessary to make the following considerations:
The use of polycrystalline samples introduces a large number of factors whose effects
on diffusion are unknown or difficult to measure, such as the effect of porosity (open
and closed), the presence of grain boundaries, the temperature region in which grain
boundary diffusion is dominant (short-circuit diffusion), the presence of a range of
grain sizes within the sample, etc.
These microstructural parameters could be eliminated if a more simple form of the
material was chosen: for example, a single crystal. Unfortunately, the large sizes of
these crystals increase diffusion times so much that the experimentation times are
prolonged beyond practicality. The duration of the experiments can be decreased
increasing the diffusion temperature, but it would be more interesting if, instead of
extrapolating high-temperature results to the low temperature regions, the experiments
could be carried out at these lower temperatures, closer to those at which the samples
would achieve a high level of oxygenation. It has already been mentioned in chapter 1
that highly oxygenated samples have good superconducting properties. The
distribution of this oxygen within the sample to optimise the superconducting
properties is very important, the aim being an evenly distribution of oxygen within the
material.
Using YBCO in powder form solves most of these problems as each powder particle
is a very small single crystal. This allows one to study of the diffusion process at low
temperatures with more reasonable oxygenation times.
These points explain the strategy followed in this diffusion study. The first part is
dedicated to the oxygenation of powder; this is followed by a study of the oxygenation
of small single crystals and polycrystalline samples of different densities.
Chapter 5 Results and discussion
104
The following sections are going to deal with the experiments performed and the
analysis of the results obtained in this study of oxygen diffusion in YBCO.
5.2.1 Powders
Section 4.3.1 describes the fabrication process of the YBCO powder used. Its final
particle size ranges between 2 and 10 µm in diameter (see Fig. 4.7).
In order to standardise the initial condition of the specimens used, all of them were
deoxygenated beforehand. The samples were heated in the thermogravimetric analyser
(TGA) in Ar up to temperatures between 700 and 800 °C, and maintained at that
temperature until no further weight change was observed. At the end of the
deoxygenation stage they were cooled at 90 °C/min to room temperature.
90 °C/min
Initial transient
Isothermal stage
90 °C/min
Temperature (°C)
Time
Fig. 5.1 Schematic of isothermal heat treatment
During the isothermal heat treatments the weight changes experienced by the powder
samples were recorded with respect to time and temperature. The samples were heated
in pure oxygen at 90 °C/min to the temperature chosen for the oxygenation study (Fig.
5.1), held at this temperature until the weight stabilised and cooled to room
temperature at the same rate of 90 ºC/min.
The particle size of the powder used limits the temperature range of the experiments
(D ≈ x2/t); for a same value of the diffusion coefficient, if the particle size is small, the
diffusion distance is short and oxygenation times are shorter. Since the value of the
diffusion coefficient increases with increasing temperature, if the oxygenating
Chapter 5 Results and discussion
105
temperature chosen is high, diffusion of oxygen will occur more readily. This process
could speed up so much that the amount of oxygen absorbed during the initial ramp to
the isothermal stage (heating transient) is so high that no information about D~  can be
drawn from the isothermal stage. Section 5.3 explains how this absorption of oxygen
during the heating stage can also yield information about D~  and be applied to
optimise the oxygenation stage during the processing of YBCO.
This section, however, deals with the analysis of the diffusion process under
isothermal conditions, so the aim is to be able to reduce the amount of oxygen
absorbed during the initial heating transient. It was observed that if the isothermal
temperature was higher than 450 °C, the amount of oxygen absorbed at constant
temperature was too small (and if the temperature was increased high enough, even
non-existent) to be able to carry out any diffusion analysis of the data collected. 450°C
was therefore taken to be the upper limit of the range of temperatures chosen.
The lower limit is again determined by the length of the oxygenation steps (given the
particle size of the sample), and it was taken to be 300 °C. The isothermal
experiments were done at 300, 350, 400 and 450 °C; weight gain of the powder
samples was recorded with time and when their weight stabilised, the samples were
cooled at 90 °C/min to room temperature. A typical weight gain trace is shown in Fig.
5.2.
30.6
30.7
30.8
30.9
31
31.1
31.2
0 24000 48000 72000 96000 120000
Time (s)
W
ei
gh
t (m
g)
Fig. 5.2 Oxygenation of powder at 300 °C (isothermal temperature reached after ≈180 s)
Chapter 5 Results and discussion
106
Once the weight variation with time was obtained, the next step was to calculate the
diffusion coefficient characteristic for each temperature. A number of methods have
been used and their results are going to be reviewed in the following section.
5.2.1.1 Diffusion in a slab and in a cylinder
It has already been described in chapter 2 how setting the appropriate boundary
conditions to Fick’s second law it is possible to obtain an expression that describes the
diffusion process in the system considered.
The first condition taken for this work is that the diffusion coefficient is independent
of concentration. Some groups have observed that the diffusion coefficient depends on
the oxygen content of the sample but there is no agreement between the authors in the
type of variation, as some say it increases with decreasing oxygen content while others
do not observe this behaviour (see chapter 3 for a review of the results published in
the literature). It has been considered that the variation of D~  with concentration will
never be as large as the existing scatter of seven orders of magnitude present in the
values of D~  itself, so for this study it has been assumed that D~  is independent of
concentration.
Diffusion in YBCO takes place preferentially in the ab plane, diffusion in the c
direction being approximately 106 times slower. Therefore, it was decided that the
initial theoretical study of the value of D~  could be characterised as a diffusion in a
plane sheet.
To calculate the solution to Fick’s second law in this case, the concentrations at the
surfaces of the sheet are considered to be constant and equal throughout the
experiment. The initial concentration of diffusing substance is constant within the
sheet, and it is assumed for simplicity that the size of the sheet does not change during
the absorption process (see chapter 2).
The solution of the diffusion equation with these boundary conditions is:
( )
( )∑∞
=
∞



 +
−
+
−=
1
2
22
22 4
12~
exp
12
81
n
t
l
tnD
nM
M pi
pi
, (5.1)/(2.32)
Chapter 5 Results and discussion
107
where Mt is the total amount of material absorbed at time t, M∞ is the material
absorbed after infinite time, and l is one half of the sheet thickness.
Substituting in (5.1) the known values of powder particle size, l, and time it takes to
equilibrate the oxygen content of the sample for each temperature it is possible to
generate a series of oxygenation profiles (fractional weight change, Mt /M∞, vs. t) that
vary with the value chosen for the diffusion coefficient, as it can be seen in Fig. 5.3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 900 1800 2700 3600 4500 5400
Time (s)
Fr
ac
t. 
w
t. 
ch
an
ge
 
 
 
 2e-11
1e-11
5e-12
1e-12
5e-11
1e-10
Fig. 5.3 Variation in weight gain during oxygenation with different diffusion coefficients for a particle
with a 5 µm radius. Numbers next to curves are the values of D for each case (cm2/s)
If instead of a flat sheet a cylindrical geometry was considered, this diffusion analysis
would be carried out following the same steps using the solution for diffusion in a
cylinder (Eq. (5.2)). This is going to be covered in more detail later in this section.
Comparing the theoretical and experimental curves and changing the value of the
diffusion coefficient of the theoretical curve, it is possible to find a value of D~  for
which the difference between both curves is minimum. Repeating this process for
each one of the experiments carried out at 300, 350, 400 and 450 °C we obtain a series
of values for the diffusion coefficient at each temperature.
Fig. 5.4 shows one of the comparisons made between the experimental data and the
result of the solution to the diffusion equation for a particle 5.2 µm wide. It can be
seen how the fit is very good. The graph includes two extra theoretical absorption
curves that show how the choice of the diffusion coefficient with this method does not
Chapter 5 Results and discussion
108
introduce a large error in the calculations. A small variation in the value of D~  has a
pronounced effect on the shape of the fractional weight change curve.
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100 120
Time (s) (x 1000)
Fr
ac
t. 
w
t. 
ch
an
ge
4e-13
2e-13
3e-13
Fig. 5.4 Comparison between weight gain traces of an experimental oxygenation of powder at 300 °C
(in blue) and the theoretical calculation of the same oxygenation using the solution to the case of
diffusion in a flat sheet (in red). Diffusion distance: 2.6 µm; D~ = 3.0 × 10-13 cm2/s. Two extra
theoretical lines with their respective values of the diffusion coefficients (in cm2/s) are included for
comparison (for the same diffusion distance)
It has already been described in chapter 4 that the diffusion model developed divides
the powder particles into disks to carry out the diffusion calculations. It was thought
necessary to assess the difference in the values of the diffusion coefficient when using
a cylindrical geometry instead of a flat sheet.
Fig. 5.5 shows the two different situations considered: in the case of the sheet,
diffusion occurs only in one direction, whereas in the case of a cylinder, the
concentration gradient moves towards the centre of the cylinder in all directions.
z
x
y
Fig 5.5 Diffusion in flat sheet and cylindrical geometries
Chapter 5 Results and discussion
109
The solution of the diffusion equation for this cylindrical geometry with the same
boundary conditions mentioned above, takes the form:
( )tD
aM
M
n
t 2
1
22
~41 α
α
−−= ∑∞
=∞
, (5.2)
where Mt and M∞ are defined as in the previous case, a is the radius of the cylinder
and αn are the roots of ( ) 0=no aJ α , where Jo(x) is the Bessel function of the first kind
of order zero. The values for these roots are tabulated in B.A.A.S., Mathematical
tables, Vol. X (1952).
Following the same procedure as with the solution for sorption in a flat sheet, a series
of curves is obtained, and these match the plots of experimental weight gain with
respect to time (Fig. 5.6). The overlap of both curves is again very good. The diffusion
coefficient obtained with the flat sheet solution is slightly faster than that obtained
with the cylindrical geometry difference. It can be seen in Fig. 5.5 that a flat sheet with
the same width as the radius of a cylinder can absorb more oxygen. This gives rise to a
faster diffusion coefficient so that equilibrium is achieved in the same length of time.
It is going to be seen in section 5.2.1.2 that the difference between both values is
within the limits of the error introduced due to the change in particle size.
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100 120 140
Time (s) (x 1000)
Fr
ac
t. 
w
t. 
ch
an
ge
Fig. 5.6 Comparison between experimental (blue) and theoretical cylindrical geometry (red) weight
gain when oxygenating at 300 °C (radius = 2.6 µm); D~  = 1.5 × 10-13 cm2/s. Dashed line shows the
solution for diffusion in a flat sheet (diffusion distance = 2.6 µm); D~  = 3 × 10-13 cm2/s
Chapter 5 Results and discussion
110
From the above comparisons between experimental and theoretical diffusion model
data it can be seen that the diffusion process is not surface reaction dependent. Neither
of the theoretical solutions chosen describe a surface reaction limited process and the
agreement with the experimental data is very good. Chemical diffusion occurs as a
result of the presence of a concentration gradient in the sample, therefore it does not
seem reasonable that the outer part of the sample oxygenates fully leaving the inside
deoxygenated and that after this has occurred, diffusion proceeds slowly as a highly
oxygenated front moving towards the centre of the specimen, as some authors suggest
(Furukawa et al. (1992), Ikuma and Akiyoshi (1988), for example).
The diffusion process occurs in the presence of a decreasing gradient of oxygen
content towards the centre of the sample. The value of the gradient will become
smaller as diffusion continues until the sample is fully oxygenated and the oxygen
content is uniform throughout. This will be seen more clearly in section 5.2.1.5 (see
Fig. 5.18 for an example of evolution of concentration profiles within samples).
It should be considered that to do these calculations it has been necessary to input not
only the value of D~ , but also the particle size (the diffusion distance). Since the
powder used contains a range of particle sizes, the value of D~  was calculated for a
number of them in order to find an average for the value of D~  that could describe the
diffusion process with the minimum error possible. Table 5.1 shows the results of the
calculations for these particle sizes. It can be seen how the difference between the
values of D~  is always less than an order of magnitude. The choice of the maximum
and minimum particle size is done according to the particle size distribution of the
powder (see Fig. 4.7). 50 % of the powder particles have radii between 4.5 µm and
1.5µm; therefore these have been chosen to be the upper and lower error limit for the
diffusion coefficient.
From the calculations of the average D~  calculated it is possible to obtain an
expression of the diffusion coefficient from the slope (the activation energy) and
intercept (pre-exponential factor) of a plot of the logarithm of these D~  values against
1/T. The Arrhenius expression for the calculated D~  is:
( )( )kTD /eV24.1exp033.0~ −= (5.3)
Chapter 5 Results and discussion
111
Temperature
(°C)
D~ 1.5 µm (cm2/s) D~ 2.6 µm (cm2/s) D~ 4.5µm (cm2/s) Average(cm2/s)
300 1.0×10-13 3.0×10-13 9.4×10-13 4.5×10-13
350 9.3×10-13 1.9×10-12 5.6×10-12 2.8×10-12
400 4.9×10-12 9.5×10-12 2.8×10-11 1.4×10-11
450 2.5×10-11 6.0×10-11 1.8×10-10 8.8×10-11
Table 5.1 Diffusion coefficients at different temperatures and particle sizes (diffusion distance)
obtained from the solution to the equation describing diffusion in a flat sheet
5.2.1.2 Estimation of errors in the calculations
The errors associated to all these calculations and measurements are going to be
reviewed in this section.
The first error is that of the experimental measurements. According to the
manufacturer, the thermogravimetric analyser TGA 7 carries an error in its
measurements of ± 1 µg. Taking a more conservative approach and considering the
error to be one order of magnitude bigger, i.e. ± 10 µg, the measurements of the
weight change during oxygenation of a 35 mg sample with an initial oxygen content of
6.2 would carry an error of ≈ 1%, which is reasonable in view of the errors that are
introduced during the theoretical calculations, as it will be seen in this section.
The next factor to consider is the effect of the initial heating rate in the oxygenation of
the sample. It is not possible to heat up the material instantaneously to the temperature
at which the isothermal experiment is going to take place (see Fig. 5.7). Therefore,
during the initial heating stage some oxygen diffuses in the sample, increasing its
weight before the isothermal stage begins.
Chapter 5 Results and discussion
112
0
0.2
0.4
0.6
0.8
1
0 3600 7200 10800 14400 18000
Time (s)
Fr
ac
t. 
w
t. 
ch
an
ge
0
50
100
150
200
250
300
350
400
Te
m
pe
ra
tu
re
 (C
)
Isothermal stage
a)
0
0.2
0.4
0.6
0.8
1
0 360 720 1080 1440 1800
Time (s)
Fr
a
ct
.
 
w
t. 
ch
a
n
ge
0
100
200
300
400
500
T
em
pe
ra
tu
re
 
(C
)
Isothermal stage
b)
Fig. 5.7 Comparison between oxygenation of powder at 10 ºC/min to: a) 350 °C and b) 450 °C with the
initial transient (temperature ramp to the required temperature)
The amount of oxygen that diffuses in during this transient depends on the heating rate
and the temperature of the isothermal step. More oxygen will diffuse in the sample
when slow heating rates are used, although the effect is smaller if the isothermal stage
temperature is low. The use of fast heating rates reduces the amount of oxygen
absorbed during this initial transient (see Fig. 5.8). The initial absorption of oxygen
would be eliminated if an infinitely fast heating rate could be used.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 300 600 900 1200 1500 1800
Time (s)
Fr
ac
t. 
w
t. 
ch
an
ge
0
100
200
300
400
500
Te
m
pe
ra
tu
re
 (C
)
a)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 120 240 360 480 600 720 840
Time (s)
Fr
ac
t. 
w
t. 
ch
an
ge
0
100
200
300
400
500
Te
m
pe
ra
tu
re
 (C
)
b)
Fig. 5.8 Variation in initial oxygen content when heating a sample with a 2.1 µm radius to 450 °C at:
a) 30 °C/min,, and b) 90 °C/min (theoretical calculation)
It must be pointed out that one of the assumptions of this work is the independence of
the oxygen content of the diffusion coefficient. This initial transient does not, in
principle, present any problems for the calculation of the value of D~ . However, the
ideal situation would be that where all the weight gain occurred during the isothermal
stage.
The heating rate used for all the isothermal experiments was 90 °C/min, the fastest
possible while still keeping the temperature lag in the TGA 7 to less than 20 °C. The
Chapter 5 Results and discussion
113
time spent heating the sample was so short that the maximum amount of oxygen
absorbed during the non-isothermal stage was never higher than 35% of the total
amount at 450 °C and much less at lower temperatures
When carrying out the theoretical calculations to overlap the theoretical weight gain
curve with the experimental curve, two aspects should be considered: the diffusion
distance taken and the choice of D~ . With respect to the diffusion distance, it has
already been mentioned that the powder samples have a range of particle sizes. The
value of D~  was calculated for a number of these sizes in order to know the amount of
error introduced in the calculations. This error is the largest of all calculated here. The
value of the diffusion coefficient given in this work is an average of the values of D~
calculated for a number of particle sizes (see Table 5.1). Fig. 5.9 shows its variation
with temperature compared to the range of values obtained from literature with the
error bars that result from the variation in particle size in the calculations. The line in
bold is the variation of the D~  calculated using a cylindrical diffusion model. It can be
seen how the values obtained are within the error region of the diffusion coefficient
calculated using a sheet geometry.
1.0E-17
1.0E-15
1.0E-13
1.0E-11
1.0E-09
1.0E-07
1.0E-05
0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
1000/T (K)
D
iff
u
sio
n
 
co
ef
fic
ien
t (c
m
^
2/
s) 
 
 
800 700 600 500 400 300
T (C)
lower limit of values 
from literature 
upper limit of  values 
from literature
Fig. 5.9 Calculated D~  values (with error bars) obtained with the solution of diffusion in a slab. Line
in bold shows D~  values obtained for the solution of diffusion in a cylinder. These values are shown
within the upper and lower limits of diffusion data collected from the literature
Chapter 5 Results and discussion
114
The second aspect to consider is the choice of D~  once the diffusion distance is set. It
has been shown in the previous section (Fig. 5.4) that small changes in the value of D~
alter considerably the shape of the weight gain trace with time. The error introduced
by the choice of D~  is much smaller than that introduced by the variation in particle
size.
5.2.1.3 Other methods used to calculate D~
Two other theoretical methods can be used to estimate the value of the diffusion
coefficient at each temperature; these concentrate respectively on the weight changes
observed at the beginning and at the end of the oxygenation process. These
calculations are included here as a confirmation of the previous results obtained.
• Initial rates of absorption
 It has been shown in section 5.2.1.2 that the influence of the initial transient on the
overall shape of the experimental weight gain curve compared to the theoretical case
with no initial transient is minimum. Therefore, it is possible to carry out an analysis
of the initial rates of absorption of oxygen to calculate the diffusion coefficient at each
temperature.
 As it was explained in chapter 2, for small times and in the case of constant D~ , it is
possible to simplify the expression of the relative weight gain with time (5.1) to:
 
21
221
~4




=
∞
l
tD
M
M T
pi
(5.4)
 The value of D~  can be calculated from the slope of the straight line obtained when
plotting the relative weight change of the sample vs. the square root of time.
 Fig. 5.10 shows the line and slope of the line obtained during an oxygenation carried
out at 350 °C after ramping to that temperature at 90 °C/min. According to the
manufacturer the average particle size of the powder is 5.2 µm in diameter, so the
value of the diffusion coefficient calculated with this diffusion distance from the slope
of the line is: 3.1 × 10-12 cm2/s; this agrees very well with the diffusion coefficient
value obtained using the flat sheet solution.
Chapter 5 Results and discussion
115
 
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150 200 250
t^0.5/l (x 1000)
Fr
ac
t. 
w
t. 
ch
an
ge
3.1e-12
 Fig. 5.10 Fractional weight change (Mt /M∞) vs. t1/2/l for oxygenation of powder at 350 °C (in blue.
Particle size: 2.6 µm radius). Line in red represents theoretical fractional weight change for
oxygenation at 350 °C using the solution to the case of diffusion in a flat sheet.
 The experimental line has a slightly different slope than the solution to the diffusion
equation. This can be attributed to the experimental error of the measurements.
However, the variation in values of D~  for the three different cases is within the error
calculated above that results from the variation in grain size (diffusion distance). The
simplification of the equation that includes a summation to a one-term expression also
reduces the accuracy of the result.
• Final rates of absorption
 In the same way as the analysis of the initial rates of absorption, if the diffusion is
independent of composition it is possible to calculate the value of the diffusion
coefficient from the final stages of the absorption of oxygen. The solution to the
diffusion equation (5.1) for a plane sheet becomes in this simplified case:
 ( )( ) 2
2~
ln
l
DMM
dt
d
t
pi
−=−
∞
(5.5)
 A plot of ln (M∞ - Mt) against t will be a straight line of slope equal to D~ pi2/l2, where l
is the diffusion distance.
Chapter 5 Results and discussion
116
 Fig 5.11 shows such a plot for data from an oxygenation of powder at 400 °C. The
value of the diffusion coefficient obtained for this case from the slope of the line is
6.4×10-12 cm2/s. This value is within the experimental error margin calculated in
5.2.1.2
 
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
0 480 960 1440 1920 2400
Time (s)
ln
(M
-
M
t)
6.4e-12
 Fig. 5.11 Plot of ln(M∞ - Mt) vs. t characteristic of an oxygenation of YBCO powder at 400 °C (particle
size: 2.6 µm radius). Value next to slope in graph is the calculated diffusion coefficient following the
analysis explained above.
 5.2.1.4 Comparison with literature
 Fig. 5.9 showed the position of the data calculated with the results from these
experiments with respect to all the range of data published in this subject. This section
is going to review the differences found with results published in the literature (see
chapter 3):
• Tracer diffusion (Fig. 5.12)
 Comparing the results presented above with data collected in tracer diffusion
experiments, it is interesting to see that even though most of these were carried out
using polycrystalline samples there are a number of them that agree with the results of
the chemical diffusion in powder samples obtained in this work.
 The results from Rothman et al. (1989), (1991), for example, lie quite close to our
results even though the samples were polycrystalline with a density close to 99% of
the theoretical density. Similar results obtained for single crystals by Tsukui et al.
(1991-b) in the ab plane are one order of magnitude higher than those of Rothman and
Chapter 5 Results and discussion
117
co-workers, but this variation could be a result of the difficulty in carrying out SIMS
measurements along the ab plane, as it is necessary to have a thick crystal which is not
always available. It is not known whether the samples achieved equilibrium during the
pre-oxygenation stage; this technique could be more sensitive to these initial
conditions than thermogravimetry (see section 5.1) and a lower saturation of oxygen
could favour a faster substitution with the isotope.
 However, their c axis results agree quite well with those obtained by Bredikhin et al.
(1991), whose ab data also agrees within one order of magnitude with that of
Rothman et al. and with that obtained from this chemical diffusion study. Tracer data
calculated by Turrillas et al. (1989) also agrees within the order of magnitude error
mentioned above.
 
1.0E-18
1.0E-16
1.0E-14
1.0E-12
1.0E-10
1.0E-08
1.0E-06
1.0E-04
0.0009 0.0011 0.0013 0.0015 0.0017
1/T (1/K)
lo
g 
D
Ikuma and Akiyoshi Pxtl Rothman Pxtl
Turrillas Pxtl Bredikh ab
Tsukui ab This work
Bredikh c Tsukui c
800 700 600 500 400 300
T (C)
 Fig. 5.12 Tracer diffusion data from literature compared to the chemical diffusion in powder result
 The tracer diffusion result by Ikuma and Akiyoshi (1988) differs markedly from the
rest of the tracer data in the literature. They used thermogravimetry to study the
diffusion of 18O in polycrystalline samples previously oxygenated in 16O. Their initial
oxygenation conditions were altered by a slow heating rate and it is possible that the
presence of impurities could have modified the final results of the measurements. The
activation energy calculated in their experiments is unreasonably high: 1.8 eV (the
activation energy for vacancy diffusion is usually around 1 eV; see chapter 2). They
suggest a surface reaction mechanism controlling the diffusion process, but the
Chapter 5 Results and discussion
118
formation of a surface layer that may hinder diffusion has not yet been observed in any
of the traces obtained with SIMS, which probably rules out this explanation. It has
been suggested by Rothman’s group that the measurements could be of something else
other than oxygen diffusion.
• Resistance measurements (Fig. 5.13)
 There is a lot of work published with results obtained from resistance measurements,
but most of these only calculate values of the activation energy of the diffusion
process, which limits the comparison to the value of Ae.
 In order to do a complete study of the diffusion process it is not only necessary to
know the value of the activation energy, but also that of the pre-exponential factor. In
this way the straight line with a slope of Ae is anchored at the intercept for 1/T = 0,
which allows one to know the values of D~  for the whole temperature range. It has
been observed that rearrangement of the oxygen atoms within the lattice can be
measured at low temperatures. This could mean that this technique can be used to
measure not only diffusion of oxygen into and out from the sample, but also its
distribution once it is inside the structure. This should be taken into account when
analysing the results from these resistance measurements.
 
1.0E-18
1.0E-16
1.0E-14
1.0E-12
1.0E-10
1.0E-08
1.0E-06
1.0E-04
0.0009 0.0011 0.0013 0.0015 0.0017
1/T (1/K)
lo
g 
D
Tu 
Elschner 
LaGraff and Payne 
This work
800 700 600 500 400 300
T (C)
 Fig. 5.13 Diffusion data from resistance measurements in literature compared to the result of chemical
diffusion in powder
Chapter 5 Results and discussion
119
 Only Tu et al. (1988-a), (1988-b) and LaGraff and co-workers (1993-a) give a full
Arrhenius expression for the diffusion of oxygen in YBCO. The values obtained by
Tu and co-workers were calculated from a dynamic (changing temperature)
measurement of the resistance during oxygenation and deoxygenation of
polycrystalline samples similar to the non-isothermal studies that are going to be
presented in section 5.3. It is interesting to see that their values lie quite close to those
obtained in this study of chemical diffusion in powder samples. The effects of porosity
and grain boundary diffusion would be expected to alter the final results giving a
faster diffusion coefficient than that obtained with thermogravimetry, but it can be
seen that for most of the temperature range the values obtained by Tu et al. are less
than one order of magnitude lower than those of chemical diffusion. This set of values
lies within the experimental error region calculated in section 5.2.1.2.
 LaGraff et al. (1993-a), (1993-b) studied out- and in-diffusion in polycrystalline and
single crystal samples under isothermal conditions and they have carried out a
calculation of the activation energy of oxygen diffusion in YBCO: the first value,
equal to 0.96 eV, is calculated from measurements of resistivity in single crystals, and
the second, 0.5 eV, from polycrystalline samples. The latter is suggested to be
characteristic of grain boundary and pore diffusion and the former, with a higher
value, is thought to be typical of bulk diffusion. The value of 0.96 eV is in good
agreement with the diffusion of oxygen via vacancies mentioned earlier on (~ 1 eV).
 The work by Elschner et al. (1992) concentrates on the studying the effect of
microstructure in the diffusion process. They differentiate between diffusion along
cracks, which has a null activation energy, and bulk diffusion, with an activation
energy of approximately 1 eV. They have calculated a value for intragrain diffusion of
10-10 cm2/s at 600 °C, although they do not give a mathematical expression for the
variation of D~
 
with temperature. This diffusion value appears to be slightly low
compared to the other results, but it may be a consequence of the effects of the
microstructure, most probably due to the presence of closed porosity that may slow the
diffusion process to the centre of the sample.
Chapter 5 Results and discussion
120
• Chemical diffusion (Fig. 5.14)
 The widest scatter in data appears to be in the results obtained with studies of
chemical diffusion using thermogravimetric analysis. This technique is very sensitive
to the alteration in microstructural features of the specimens, so it is thought that the
change in density and grain size of the samples could be the reason for the wide range
of values published. All the samples used were polycrystalline, with varying density
and particle size.
 As it is going to be seen in section 5.2.3, the study of diffusion in polycrystalline
samples is very complex. It requires taking into account the effect of grain boundaries,
and the presence of open or closed or open porosity. Grain boundaries provide a fast
path for the transport of oxygen content at low temperatures (short-circuit diffusion).
The amount of grain boundaries is determined by the degree of porosity and the grain
size of the sample.
 If the presence and effects of grain boundaries are ignored the analysis would be
limited to setting the diffusion distance. It has already been discussed in chapter 3 that
in the case of porous samples, the diffusion distance chosen should be the average
between open pores, whereas if the sample is fully dense it should be the size of the
specimen. The wrong choice of the sample size as diffusion distance can affect
considerably the result of the calculation of D~ .
 The data presented by Kishio et al. (1989-b) differs considerably in the value of the
activation energy with respect to any of the other results presented in the literature and
those calculated in this work (see Table 3.1 and Fig. 5.14). Their study of the variation
of oxygen content with partial oxygen pressure is very complete, and although it
agrees with other studies of the same type (Ikuma and Akiyoshi (1988)), their
calculation of the diffusion coefficient seems to disagree with most of the data. Their
polycrystalline samples had a density of 98 % of the theoretical density and a grain
size of 50 µm, the largest obtained compared to most of the other groups. It is possible
that the time spent at each pressure and temperature was not enough to equilibrate the
sample; this, together with the high density of the sample could have given rise to
calculations of oxygen diffusion in a material that had not achieved equilibrium at the
Chapter 5 Results and discussion
121
end of the oxygenation process, introducing a larger degree of error in the calculations
of the diffusion coefficient.
 Differential scanning calorimetry studies of oxygen diffusion in YBCO powder by
Glowacki et al. (1988) have also yielded a high value of the activation energy for this
process, between 1.5-1.6 eV, close to the results by Kishio et al. (1989-b). However,
the calculation of the pre-exponential factor, Do, located the series of diffusion values
quite close to the results obtained in this thesis using thermogravimetry, as well as
those published by Haller et al. (1989) and Mozahev and co-workers (1992), (1994)
(see Fig. 5.14). The slight difference in the results is probably a result of experimental
errors, such as the temperature lag of the differential scanning calorimeter with
heating rate. A re-evaluation of the activation energy once the temperature lag has
been estimated would yield a more reliable value for the activation energy of oxygen
absorption in YBCO. The use of differential scanning calorimetry to study oxygen
diffusion presents the advantage of providing information about the evolution of this
process from very low temperatures (if the heating rate is not faster than 50 ºC/min).
Low temperature diffusion data is difficult to obtain sometimes using other
techniques. More work should be carried out in this field to investigate the
possibilities of applying DSC to the study of oxygen diffusion.
 
1.0E-18
1.0E-16
1.0E-14
1.0E-12
1.0E-10
1.0E-08
1.0E-06
1.0E-04
0.0009 0.0011 0.0013 0.0015 0.0017
1/T (1/K)
lo
g 
D
Krebs Saiz
Kishio Mozahev
Tang Krauns
Umemura in- Haller
This work Glowacki 
800 700 600 500 400 300
T (C)
 Fig. 5.14 Chemical diffusion data from the literature compared to the results of chemical diffusion in
powder
Chapter 5 Results and discussion
122
 The work carried out by Mozahev and co-workers (1992), (1994) combined studies of
oxygenation of very fine powders with polycrystalline samples of varying density.
They studied diffusion in the powder and low density polycrystals using the solution
to Fick’s second law characteristic of diffusion in a sphere. The study of the higher
density samples was done using the solution to diffusion in a plane sheet, taking the
diffusing distance to be the size of the polycrystalline sample, instead of the grain size.
This could be the reason for the slightly higher values (a longer diffusion distance than
the real distance results in a faster diffusion coefficient), but they lie within the region
of experimental error of the diffusion values calculated in this thesis.
 The high values of the diffusion coefficient obtained by Umemura et al. (1989) could
again be explained due to a miscalculation of the diffusion distances in the
polycrystalline sample. Their choice of solution of the diffusion equation to be applied
was probably not very accurate, as it required to take into account the height and
diameter of the sample (which they assume to be a cylinder) suggesting that they
consider diffusion along the three crystallographic axes, when diffusion occurs
primarily in the ab plane. The values of the activation energy calculated are quite low
(0.7 and 0.5 eV), which could be characteristic of grain boundary diffusion, as
suggested by LaGraff et al. (1993-a). (1993-b) in their electrical resistance studies.
 Krauns and Krebs (1993) carried out two measurements of D~  for a polycrystalline
sample with open porosity. They found that the diffusion coefficient was strongly
dependent on composition, with the activation energy changing as much as 0.8 eV
between specimens. The activation energy that can be calculated from the two
diffusion values given is quite low (see table 3.1); this corroborates the analysis
presented at the beginning of this section concerning the form of the material for the
experiments. Polycrystals are very complex and the use of powder samples was
thought to introduce less variables in the study of the diffusion process.
 Tang and Lo (1991) also studied the oxygenation of oxygen in YBCO in a very porous
polycrystalline sample, and the only diffusion value mentioned agrees with the
diffusion coefficient calculated in this work. The data by Haller et al. (1988) also
agrees quite well with the results presented in section 5.2.1.1 obtained using powder
samples, although Haller and co-workers used polycrystalline specimens for their
Chapter 5 Results and discussion
123
studies. The porosity of their samples and perhaps the choice of a slightly large
diffusion distance yield lower values for the diffusion coefficient than those obtained
in the work presented in this thesis. However, the agreement is within the order of
magnitude error mentioned in 5.2.1.2.
 5.2.1.5 Modelling of the diffusion process under isothermal
conditions
 Once the expression for D~  was calculated it was possible to reproduce the
experimental results with the computational model developed described in chapter 4.
 The calculated values of the pre-exponential factor and activation energy for diffusion
are entered in the programme and the variation in weight of a powder particle with a
set grain size (average grain size of the powder sample) can be calculated for the
oxygenation treatment chosen.
 This first step taken was to test the validity of the model, setting a theoretical
calculation of the weight gain recorded in the isothermal runs shown in the previous
section. Fig. 5.15 shows the results obtained for two of these runs. The first one is an
oxygenation with a fast ramp rate to a low temperature (90 °C/min to 350 °C) and the
second is an oxygenation to a higher temperature with a slower ramp rate (10 °C/min
to 450 °C). This experimental result was not used in the calculation of D~ , but is
included to show that the model can reproduce any kind of experimental situation.
 
0
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b)
 Fig. 5.15 a) 90 °C/min to 350 °C, and b) 10 °C/min to 450 °C. Experimental data shown in blue and
model data in red. Particle radius: 2.1 µm
 The reproduction of the weight change with temperature (or time) is very useful, as it
can help in the design of experiments where it is necessary to know the diffusion time
Chapter 5 Results and discussion
124
for any type of oxygenation treatment as well as what weight change is expected. To
complement further the analysis of oxygenation, this model also calculates the change
in the oxygen distribution gradient within the sample as the oxygenation occurs. This
feature is very useful when it is necessary to know the distribution of oxygen of a
sample after a certain heat treatment.
 The choice of particle radius equal to 2.1 µm was made so that both isothermal and
non-isothermal theoretical studies agreed with the experimental results. This
difference in the diffusion distance could be a consequence of the slight disagreement
in shape between the real powder particles and the spherical powder particles assumed
in the computer model.
 Fig. 5.16 shows how the model can successfully reproduce oxygenation experiments
at low and at higher temperatures (90 °C/min to 350 °C, Fig. 5.16 a), and 90 °C/min to
450°C, Fig. 5.16 b), respectively).
 It can be seen how during the treatment at low temperature the amount of oxygen
absorbed during the heating transient is minimum, while when the temperature is
raised the distribution of oxygen in the specimen at the beginning of the isothermal
step is much higher (marked by red lines in the oxygen distribution).
 Fig. 5.16 c) shows an extreme case of very slow heating rate: 10 °C/min to 450 °C.
From its characteristic weight gain plot and oxygen distribution graph it can be seen
that the oxygen starts being absorbed at 260 °C and most of the oxygenation occurs
during the initial transient mentioned in the previous sections. By the time the
isothermal stage is reached (at 450 °C) almost 90 % of the total amount of oxygen has
been absorbed. The distribution is not yet uniform, as the equilibrium oxygen content
for that temperature in a pure oxygen atmosphere is 6.97, so the concentration gradient
through the sample continues decreasing with time until equilibrium is reached. The
same occurs when heating at a faster rate, although in this case most of the
oxygenation occurs during the isothermal stage (see Fig. 5.7 a)).
Chapter 5 Results and discussion
125
 
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etc…
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O
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ge
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 c
o
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te
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t  
 
 Fig. 5.16 Concentration profile within a sample during oxygenations at: a) 90 °C/min to 350 °C, b)
90°C/min to 450 °C (time between concentration lines: 30 s), and c) 10 °C/min to 450 °C (time between
concentration lines: 60 s). Red lines show oxygenation during transient stage. Black lines show
oxygenation during isothermal stage. Sample size: 2.1 µm radius
 It has already been mentioned that high, uniform oxygen contents are the aim of the
manufacturing and processing of YBCO samples. Using the model developed to
reproduce the heat treatment will allow manufacturers and researchers to find out the
true oxygenation of the sample, i.e. its amount and distribution, which will be of great
help when it comes to analysing the results of experiments whose value depends on
oxygen content.
Chapter 5 Results and discussion
126
• Use of initial and final rates of absorption
 The results obtained from theoretical oxygenations with the model developed can also
be analysed in the same way as the experimental data. This section shows that an
analysis of the initial and final rates of absorption of these theoretical oxygenations
yields very similar results to those presented in section 5.2.1.3 (Other methods used to
calculate D~ ).
 Fig 5.17 shows the straight line obtained in the study of the initial rate of absorption of
a theoretical computation of an oxygenation at 90 °C/min to 350 °C using the
oxygenation model developed. The particle radius used was 2.1 µm. The value of the
diffusion coefficient obtained from the slope of the line is 1.3 × 10-12 cm2/s. The red
line in the graph shows the fractional weight change obtained using the solution to the
case of diffusion in a flat sheet (no initial transient).
 It can be seen from Fig. 5.17 that the diffusion coefficient obtained from this analysis
of the theoretical data gives a value for this coefficient quite similar to that obtained
from the experimental oxygenation. The difference in the values could be due to small
inaccuracies due to shape variation between the real sample and the theoretical model
and perhaps due to the combination of the oxygenations of powder particles of
different sizes.
 
0
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0.5
0.6
0 50 100 150 200 250
t^0.5/l (x 1000)
Fr
ac
t. 
w
t. 
ch
an
ge 3.1e-12
1.3e-12
 Fig. 5.17 Fractional weight change (Mt /M∞) vs. t1/2/l for a theoretical oxygenation at 350 °C
calculated with the diffusion model developed (particle size: 2.1 µm radius) (in blue). Line in red
represents theoretical fractional weight change for oxygenation at 350 °C using the solution to the
case of diffusion in a flat sheet (see Fig. 5.10)
 Concerning the analysis of the final rates of absorption, Fig. 5.18 shows a plot of the
data from an oxygenation at 400 °C oxygenation reproduced with the diffusion model
Chapter 5 Results and discussion
127
developed. The diffusion coefficients obtained for this case obtained from the slope of
the line is equal to 8.0 × 10-12 cm2/s. This value is within the error margin calculated
for the diffusion measurements presented in section 5.2.1.3.
 
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0 480 960 1440 1920 2400
Time (s)
ln
 
(M
-
M
t)
8.0e-12
 Fig. 5.18 ln(M∞ - Mt) vs. t for a theoretical oxygenation of powder at 400 °C (ramp at 90 °C/min)
calculated with the diffusion model developed (particle size 2.1 µm radius). D~  = 8.0 × 10-12 cm2/s
 5.2.2 Oxygenation of polycrystalline and single crystal
samples
 In order to follow the natural extension of this oxygenation work, a preliminary study
of the behaviour of polycrystalline and single crystal specimens was carried out.
 It has already been pointed out that the oxygenation process in polycrystals is very
complex. The presence of grain boundaries, open and closed porosity and an unknown
range of particle sizes increases the difficulty in the analysis of the results obtained.
The oxygenation of polycrystalline samples is beyond the scope of this thesis and
these results are only included as preliminary steps towards the complete analysis of
the diffusion process in YBCO.
 Two polycrystalline samples were made with SSC powder following the process
described in section 4.4.2 of chapter 4. The densities of these specimens were
determined by pycnometry and were 85 and 94 % of the theoretical density
respectively. The average grain size of both samples was measured using optical
microscopy; it was found to be 12.5 µm for the 85 % dense polycrystal and 150 µm
for the 94 % dense sample.
Chapter 5 Results and discussion
128
 Following the same oxygenation study carried out with powder samples, the
polycrystalline specimens were deoxygenated in Ar at 750 °C and subsequently
oxygenated at different temperatures in the range 300 - 450 °C.
 Fig. 5.19 a) and b) show that the oxygenation traces obtained in the case of
polycrystalline specimens had a different shape in the initial stages of the process:
instead of an exponential behaviour (see oxygenation of powder samples in 5.2.1;
Figs. 5.2 and 5.6, for example), the increase in weight appears to be linear with time
until most of the oxygenation has occurred. This behaviour was observed at low
temperatures for both polycrystalline specimens.
 
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a)
 
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b)
 Fig. 5.19 a) Oxygenation at 300°C of 85 % dense polycrystal, and b) oxygenation at 300 °C of 94%
polycrystal
 As the temperature of the isothermal oxygenations increased, this linear increase in
weight with time became less pronounced, disappearing altogether at high
temperatures, as it can be seen in Fig. 5.20 a) and b).
 It is not known why this initial linear weight gain occurs. The fact that it disappears as
the oxygenation temperature increases suggests that it might be linked to grain
boundary diffusion. It is known that this type of diffusion has a low activation energy
(lower than bulk diffusion) at low temperatures. However, the effect of grain
boundaries is thought to enhance the diffusion process, and the linear increase in
weight with time appears to be slowing down the absorption of oxygen in the sample.
This effect should be studied in more detail.
 
Chapter 5 Results and discussion
129
 
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a)
 
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b)
 Fig. 5.20 Oxygenation of 94% polycrystal at: a) 350°C, and b) 400 °C
 The calculation of the diffusion coefficient for polycrystalline samples was carried out
in the same way as it was done with powder data: the solution to the diffusion
equation for the case of diffusion in a slab was fitted to the experimental fractional
weight change. Using the values of the grain sizes measured with optical microscopy,
the curves were matched to the final variation in weight change, irrespectively of the
shape of the initial part of the curve (Fig. 5.21). Due to the small grain size of the 85%
dense polycrystal the temperature range over which the isothermal experiments were
carried out was very limited (300 – 350 °C) and it was not possible to carry out a
calculation of the value of the diffusion coefficient without incurring in a large error.
The value of the diffusion coefficient in a polycrystalline sample was calculated using
data from the oxygenation of the 94 % dense polycrystal.
 
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a)
 
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Time (s) (x 10)
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b)
 Fig. 5.21 Oxygenation of 94 % polycrystal at: a) 300°C (D = 1.2 × 10-8 cm2/s), and b) 400 °C
(D=1.7×10-7 cm2/s). Blue line: experimental data; red line: solution to the diffusion equation.
(Diffusion distance: x = 0.05cm)
 The diffusion coefficients for the 94 % dense sample were calculated ignoring the
effect of grain boundary diffusion in this specimen with closed porosity. The diffusion
distance used was therefore one half of the dimensions of the specimen. This was
Chapter 5 Results and discussion
130
calculated to be approximately 0.5 mm. The values obtained for the diffusion
coefficients obtained were:
 Temperature (°C)  D 94 % densepolycrystal (cm2/s)
 300  1.2 × 10-8
 350  5.1 × 10-8
 400  1.7 × 10-7
 450  4.8 × 10-7
 Table 5.2 Values of the diffusion coefficient calculated for a 94 % dense polycrystal with an average
diffusion distance of 0.05 cm
 Oxygenation experiments carried out with single crystal samples at higher
temperatures than those used with powder samples also yielded very high values of
the oxygen diffusion coefficient (see Fig. 5.22). It is not known why diffusion occurs
faster in these single crystals (average size: 1×1×~0.05 mm) than in powder samples.
 It has been observed that ageing a powder sample for 2½ weeks in a high CO2
environment increased the diffusion coefficient above the error margin calculated in
section 5.2.1.2 (see Fig. 5.22). The presence of impurities in the single crystals
introduced during processing could be one of the reasons for the faster diffusion
behaviour observed.
 
1.0E-17
1.0E-16
1.0E-15
1.0E-14
1.0E-13
1.0E-12
1.0E-11
1.0E-10
1.0E-09
1.0E-08
1.0E-07
1.2 1.3 1.4 1.5 1.6 1.7 1.8
1000/T (1/K)
D
 (c
m
^
2/
s)  
This work 
Aged powder
Single crystal
 Fig 5.22 D vs. 1/T results for single crystal specimens (diffusion distance: 50 µm) and aged SSC
powder (radius: 2.6 µm) compared to the chemical diffusion results presented in section 5.2.1.2 (Fig.
5.9)
Chapter 5 Results and discussion
131
 5.2.3 Summary
 This section has presented a study of oxygen diffusion in YBCO using
thermogravimetry under isothermal conditions.
 Thermogravimetry is a very useful technique for the measurement of chemical
diffusion, as it records weight changes as a function of time and temperature in a
controlled atmosphere. The atmosphere is chosen so that a chemical gradient is
established across the sample and oxygen (in this case) is driven in or out of the
specimens.
 Using the solution to Fick’s second law for diffusion in a slab it has been possible to
calculate a value for the chemical diffusion of oxygen in YBCO comparing the
theoretical results obtained using this solution (see Eq. (5.1)) from this solution with
the experimental data.
 The value of D~  obtained has been compared to the results published in the literature,
suggesting possible reasons for the differences observed between these and the results
obtained in this thesis. The effect of sample microstructure (use of polycrystals, single
crystals or powder specimens) appears to be quite important. This justifies the choice
of YBCO powder for the thermogravimetric study presented here. The choice of this
form of material eliminates the effects of microstructural characteristics such as grain
boundaries, presence of secondary phases, etc. In addition, the size of the powder
particles is so small that it has allowed one to carry out the experiments at
temperatures close to those ideally used for the oxygenation of samples during
processing.
 The computer programme developed to study the diffusion process theoretically has
already been described in chapter 4. The diffusion parameters used are calculated from
the thermogravimetry experiments. The validity of the programme has been checked,
and section 5.2.1.5 shows that it has been possible to reproduce theoretically the
experimental results from isothermal oxygenations at a series of heating rates.
 Since it has been proven that it is possible to reproduce experimental data using the
diffusion programme developed, the effect of the initial heating rate on the
oxygenation results has been studied using only theoretical results. This has allowed
Chapter 5 Results and discussion
132
one to carry out a study of the diffusion process that would have otherwise taken a
long time and a large number of samples if it had been carried out in practice.
 A preliminary analysis of diffusion data for polycrystalline and single crystal samples
has been carried out, but the number of factors affecting the process, such as grain
boundary diffusion, porosity (for single crystals) and the presence of impurities in the
sample (single crystals) have been observed to alter the oxygenation behaviour. A full
study of these factors and their effect on the diffusion process should be carried out to
complete the study of oxygen diffusion in YBCO.
 5.3 Non-Isothermal experiments
 One of the limitations of the use of powders is that, as it has been mentioned before
(section 5.2.1), the maximum temperature at which the experiments can be carried out
is limited by their particle size. If the particle size is very small, the temperature range
will be reduced because the maximum temperature at which isothermal experiments
can be carried out is low. Should this temperature value be exceeded, the sample will
oxygenate to its equilibrium value during the initial heating transient. The same effect
would be observed if the heating rate was reduced to very slow values. Under these
circumstances full oxygenation of the sample occurs during the heating stage, and it is
not possible to extract any information about the diffusion process (the value of D~ )
with the methods described in the previous sections.
 However, it was thought that the analysis of non-isothermal experiments could yield
interesting information about the diffusion process, as well as providing one with a
novel approach to the oxygenation step of the samples during processing (this will be
discussed in more detail at the end of this chapter). This section presents the analysis
of the experimental results obtained from oxygenations of powder, polycrystalline and
single crystal samples under non-isothermal conditions.
 5.3.1 Powders. Effect of ramp rates
 Powder samples were deoxygenated following the heat treatment in Ar mentioned in
4.6.1. These samples were then heated in an oxygen atmosphere (1 atm) at different
Chapter 5 Results and discussion
133
rates: 1, 10, 30, 50 and 90 °C/min to 700 °C. They were held at 700 °C for a few
minutes and then cooled to room temperature at the same rate as they were heated up.
 The shape of the weight gain curve in these non-isothermal experiments is very
characteristic (Fig. 5.23):
 
0
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2.4
0 100 200 300 400 500 600 700
Temperature (C)
Fr
ac
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2
3
4
5
 Fig. 5.23 Characteristic weight gain during a non-isothermal oxygenation of YBCO powder at
1°C/min to 700 °C in a pure oxygen atmosphere and cooling to room temperature at the same rate
 During the first part of the experiment, the weight of the sample remains constant (1).
When the temperature increases above a certain value, Tonset, the weight of the sample
starts increasing (2). This weight gain is steady until a certain point (3), where a
maximum occurs. The temperature at which this maximum weight gain is observed is
going to be referred to as maximum temperature, Tmax. After this point, the weight of
the sample starts decreasing continuously with increasing temperature (4). When the
maximum temperature of the heat treatment is reached and the sample is cooled down,
the change in weight of the specimen follows the last part of the heating curve (4),
weight increasing with decreasing temperature. When the point of maximum weight is
reached (3) and the temperature is decreased further, the sample continues to gain
weight (5) slowly until it stabilises in the very low temperature range.
 These changes in weight can be directly related to the change in oxygen content of the
originally deoxygenated YBCO sample. The explanation is as follows:
 At low temperatures (1) the weight of the sample remained constant because the value
of the diffusion coefficient is too small for diffusion to take place. As temperature
Chapter 5 Results and discussion
134
increases, D~  increases too, and at a certain point, (2), the value of this coefficient has
increased so much that the oxygen present in the furnace chamber has enough time to
diffuse in the sample, faster as temperature increases. This diffusion of oxygen is
observed in an increase in weight of the sample.
 The weight gain continues until a certain temperature, Tmax, when a maximum is
reached (3). At this point the sample is saturated with oxygen. The equilibrium oxygen
content at any temperature above this point is lower than that at the maximum of the
weight curve. Therefore, as the temperature increases the sample starts losing oxygen
(4). During this part of the non-isothermal heating curve the sample is always at its
equilibrium oxygen content.
 During cooling of the sample in the same atmosphere, it will absorb oxygen at the
same rate as it lost it during the last part of the heating curve (4). As the temperature
decreases below the point at which the maximum in weight occurred during heating,
the weight gain becomes more sluggish until it stops altogether at low temperatures
(5).
 The temperatures at which the oxygenation process starts, Tonset, (point at which the
weight gain begins) and at which the maximum in weight is reached, are determined
by the diffusion coefficient and heating rate.
 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 100 200 300 400 500 600 700
Temperature (C)
Fr
ac
t. 
w
t. 
ch
an
ge
10C/min
30C/min
90C/min
 Fig. 5.24 Non-isothermal experiments: powder samples heated in 100 % O2 to 700 °C at 10, 30 and
90°C/min
Chapter 5 Results and discussion
135
 Fig. 5.24 shows that the values of the minima and maxima shift to higher temperatures
with increasing heating rates. This occurs because as the ramp rate increases the time
spent at each temperature is shorter, and the oxygen absorption during each time step
is smaller. The onset of absorption will then be observed at higher temperatures.
 It is possible to calculate the activation energy of the absorption process from the
dependence on the heating rate of the temperature at which diffusion starts and Tmax:
 The process taking place is that of oxygen diffusion inside the sample along the ab
planes. Therefore, there is a flux of oxygen that alters the concentration according to
the flux equation (in 1-D):
 
x
CDJ
∂
∂
−=
~ (5.6)/(2.1)
 D~  is the chemical diffusion coefficient of oxygen in the sample, and it can be
expressed with an Arrhenius equation:
 
( )kTADD eo /exp~ −= . (5.7)
 According to Fick’s second law, the variation in concentration with time can be
expressed as:
 2
2
~
x
CDJ
t
C
∂
∂
=∇=
∂
∂
, (5.8)
 provided the diffusion coefficient is independent of concentration. It is possible to
integrate this expression with respect to time in order to calculate the total change in
concentration:
 ∫∫ ∂∂=∇= tt dtxCDJdtC 0 2
2
0
~
. (5.9)
 Being a non-isothermal experiment, the temperature changes with time, and it is
possible to change the integral variable:
 ∫ ∂∂= max 2
2
~
T
RT
dT
dT
dt
x
CDC , (5.10)
Chapter 5 Results and discussion
136
 where Tmax is the temperature at which the maximum in concentration occurs.
Substituting the expression for the diffusion coefficient and rearranging terms the
integration becomes:
 



−








∂
∂
=


max
2
max
2
2
exp
kT
A
A
kT
x
CD
dt
dTC e
e
o (5.11)
 Taking logarithms of both sides, it is possible to see that the activation energy for
oxygenation can be calculated from the slope of a plot of ( )dtdTln  vs. 1/kTmax . It
would also be possible to plot ( )( )2
max
ln TdtdT  vs. 1/kTmax. Fig. 5.25 shows a plot of
the results from both expressions, whose slopes give activation energies equal to 1.4
eV and 1.2 eV respectively given a total concentration change, C, of 0.4. These results
agree with the activation energy calculated using the diffusion analysis of isothermal
diffusion presented in the previous section (5.2.1.2)
 
-1
0
1
2
3
4
5
6
0.0012 0.0013 0.0014 0.0015
1/Tmax (1/K)
ln
 
alf
a
-14
-13
-12
-11
-10
-9
-8
ln
 
(al
fa
/T
m
x
^
2) 
 
 
ln (alfa)
ln (alfa/Tmax^2)
Ae = 1.4 eV
Ae = 1.2 eV
500 400T (C)
 Fig. 5.25 Plots of ln(α/T2max) and ln (α) vs. 1/Tmax for non-isothermal oxygen absorption at several
heating rates
 The values obtained in this calculation have an inherent error associated to the
temperature lag of the furnace, which has been calculated to be around 20 °C at high
temperatures. The average activation energy obtained is: 1.3 ± 0.1 eV. Having into
account that this is a rough initial calculation using data collected under dynamic
(non-equilibrium) conditions, this value agrees (within the experimental error limits)
with that obtained using the methods described in sections 5.2.1.1 and 5.2.1.3.
Chapter 5 Results and discussion
137
 5.3.2 Comparison with results from model
 The computer model developed can reproduce quite accurately the experimental
results under non-isothermal conditions, as it can be seen in Fig. 5.26 a). The curves
obtained with this model reproduce slow and fast heating rate experiments.
 The difference in maximum height between the curve modelled with the program and
the experimental trace is attributed to the choice of a single particle size to calculate
the theoretical curve, whereas the powder consists of a range of particle sizes. Their
overall effect is a reduction in the height of the maximum weight. This can be clearly
seen in Fig. 5.26 b). This figure shows a series of curves obtained for different particle
sizes and their average for the same heat treatment. The maximum in the fractional
weight change decreases and shifts to higher temperatures because the slower relative
weight gain of bigger particles counteracts the effect of smaller particles and decreases
the maximum in the average fractional weight change profile.
Chapter 5 Results and discussion
138
 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 100 200 300 400 500 600 700 800
Temperature (C)
Fr
ac
t. 
w
t. 
ch
an
ge
a)
 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 100 200 300 400 500 600 700 800
Temperature (C)
Fr
ac
t. 
w
t. 
ch
an
ge
Exp. data
Ratio
0.3
0.6
1.2
2.4
3.6
b)
 Fig. 5.26 Comparison of non-isothermal experimental and model curves. a) 50 °C/min to 700 °C (in
red) for a particle with a radius of 2.1 µm (experimental result for the same heat treatment in blue),
and b) model curves at different particle sizes (in microns) and their average (in red) compared to the
experimental data (in blue).Heat treatment: 50 °C/min to 700 °C
 The advantage of the use of this model is that it allows one to follow the evolution of
the oxygenation process in the specimen while the temperature increases, as it
calculates the concentration profile inside the sample. The variation in the oxygen
distribution is shown in Fig. 5.27.
 At the beginning (1) there is no weight change; only an outer cylinder of the disc is
oxygenated at the equilibrium oxygen content for RT, representing the surface of the
sample. In (2) the diffusion coefficient increases and the sample starts absorbing
oxygen, decreasing the concentration gradient from the surface to the centre.
Chapter 5 Results and discussion
139
 
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16 18 20
Step
O
xy
ge
n
 s
to
ic
hi
om
et
ry
  
1
 
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16 18 20
Step
O
xy
ge
n
 s
to
ic
hi
om
et
ry
  
2
 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 100 200 300 400 500 600 700 800
Temperature (C)
Fr
ac
t. 
w
t. 
ch
an
ge
1
2
3
4
Non-equilibrium Equilibrium
 
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16 18 20
Step
O
x
yg
en
 s
to
ic
hi
o
m
et
ry
  
3
 
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16 18 20
Step
O
xy
ge
n
 s
to
ic
hi
om
et
ry
  
4
 Fig. 5.27 Concentration profiles at different points during the heating in oxygen of a sample under
non-isothermal conditions (50 °C/min to 700 °C). Time between lines 40 s. Particle radius: 2.1 µm
 During (1) and (2) the sample is not at its equilibrium oxygen content because the
ramp rate and the slow diffusion coefficient at these low temperatures the sample
cannot oxygenate completely. But when the temperature is sufficiently high (the
diffusion coefficient has become faster), the sample has absorbed enough oxygen to
have a uniform distribution within the sample, and it has the value of the equilibrium
oxygen content at that temperature (3). From this point onwards the distribution of
Chapter 5 Results and discussion
140
oxygen in the sample is at equilibrium and decreases uniformly until the maximum
temperature of the non-isothermal run is reached (4).
 With respect to the cooling part of this non-isothermal experiment, it is very
interesting to see that as the temperature decreases, the sample is at the equilibrium
oxygen content (with a uniform distribution) until the temperature is decreased to
Tmax. Below this point, the diffusion coefficient becomes slower and the time spent at
each temperature (determined by the cooling rate) is not long enough for full
oxygenation of the sample to occur. This means that at the end of the cooling cycle,
the sample has highly oxygenated edges and a centre with a lower oxygen content (see
Fig. 5.28).
 
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 12 14 16 18 20
Step
St
o
ic
hi
om
et
ry
 
 
Concentration 
profile at Tmx
Oxygen distribution at 25 C
Cooling
 Fig. 5.28 Theoretical calculation of the concentration profiles at different points during the cooling in
oxygen at 50 °C/min from 700 °C of a 2.1 µm radius sample under non-isothermal conditions. Time
between lines: 20 s
 5.3.3 Summary
 This section has analysed the characteristic shape of non-isothermal oxygenation of
YBCO powders. It has been shown how an analysis of this dynamic measurement can
also yield information about the diffusion process.
 The diffusion programme developed has successfully reproduced the experimental
data taking into account the range of particle sizes present in the powder. The
possibility of following the evolution of the oxygen concentration profile during the
heat treatment corroborates the transition from a non-equilibrium situation where the
Chapter 5 Results and discussion
141
diffusion coefficient is too slow to oxygenate the sample fully, to a high temperature
regime where the sample is always at its equilibrium oxygen content. This regime has
a very interesting application that is going to be described in the following section.
 5.4 Further applications of the model
 5.4.1 Quasi-equilibrium cooling
 It has been shown throughout this chapter that the model of the diffusion process
developed can be applied to the study of oxygenation under isothermal and non-
isothermal conditions. This model not only reproduces with accuracy the fractional
weight changes of specimens under these conditions but also provides extra
information, calculating the variation in the distribution of the oxygen content within
the sample during the oxygenation process. Knowledge of this oxygen distribution is
very important, as it can help predicting (or explaining) their superconducting
behaviour.
 This section is going to describe the development of a novel cooling procedure that
could be introduced in the processing of YBCO specimens to give optimum
superconducting properties by maximising their oxygen content in shorter
oxygenation times.
 It is known that in order to have a sample with good superconducting behaviour, its
oxygen content must be above 6.9, and if possible it should be uniformly distributed.
It has been shown in section 5.3.1 that during the cooling part of the non-isothermal
experiments, the region above Tmax corresponds to a uniform increase in oxygen
content, where the sample is always at the equilibrium content for each temperature.
The point at which this maximum occurs depends on the cooling rate, and it decreases
with slower cooling rates (Fig. 5.29).
Chapter 5 Results and discussion
142
 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 100 200 300 400 500 600 700 800
Temperature (C)
Fr
ac
t. 
w
t. 
ch
an
ge
90 C/min
50 C/min
30 C/min
 Fig. 5.29 Theoretical non-isothermal runs at 30, 50 and 90 °C/min showing the decrease in peak
temperature with slower heating rates for a particle size 2.1 µm in radius
 The cooling procedure developed here combines these two factors: on one hand, the
possibility to cool a sample to a certain temperature, Tmax, while its oxygen content is
uniformly distributed and at equilibrium, and on the other hand being able to decrease
this temperature, Tmax, (below which the sample is no longer at “equilibrium”) slowing
the cooling rate. The cooling process developed keeps the sample at its equilibrium
oxygen content while the sample is being cooled decreasing the cooling rate. The
change in oxygen content is only possible if the equilibrium condition is altered for a
short period of time, establishing a concentration gradient across the sample.
Therefore, this improved oxygenation process is going to be referred to as quasi-
equilibrium cooling.
 Two assumptions have been made in the development of this cooling procedure:
- The cooling rate of the sample is never faster than 90 °C/min,
- The material used for these preliminary theoretical experiments is powder with an
average particle size of 2.1 µm in radius.
The first step is to obtain an expression relating the maximum temperatures to the
cooling rates. There is an exponential relationship between them that can be calculated
empirically from the slope and intercept of the straight line obtained when plotting
log(cooling rates) vs. Tmax (Fig. 5.30). The expression obtained is:
Cooling rate = 8 × 10-7 exp(0.0338×Tmax) (5.12)
Chapter 5 Results and discussion
143
y = 8E-07e0.0337x
0.1
1
10
100
300 350 400 450 500 550 600
Peak temperature (C)
Co
o
lin
g 
ra
te
 
(C
/m
in
)  
90 C/min 
Fig. 5.30 Plot of cooling rates, α (log scale), against maximum temperature, Tmax characteristic of a
particle with a radius equal to 2.1 µm
Fig. 5.28 shows that the distribution of the oxygen absorbed during cooling is uniform
as long as the temperature is higher than Tmax (schematic (3) in Fig. 5.27). Below this
temperature, the absorption ceases to be uniform because the diffusion coefficient
becomes too slow; at these lower temperatures oxygen concentrates on the outer
regions of the particle. This sets a limit in the uniform distribution of oxygen within
the specimen as well as on the value of its oxygen content. The final distribution is not
going to be constant, as shown in Fig. 5.28. But it is guaranteed to be at least as high
as the equilibrium oxygen content for Tmax of the heating rate used. Oxygen will be
uniformly distributed in the sample up to this content. The values of the equilibrium
oxygen content of each Tmax considered are therefore going to be taken as the oxygen
contents obtained when applying the quasi-equilibrium procedure. This gives a
certain margin of error, as the oxygen content is always going to be somewhat higher
than the values given in the following tables and graphs.
Table 5.2 shows a series of cooling rates, their characteristic peak temperatures
obtained from the non-isothermal experiments carried out with powder samples and
the equilibrium oxygen contents for those temperatures (from Gallagher (1987)).
These data have been represented in Fig. 5.31 below.
Chapter 5 Results and discussion
144
α (°C/min) Tmax (°C) Oxygen content
0.1 350 6.95
0.5 400 6.95
1 410 6.95
10 490 6.91
30 515 6.89
50 530 6.87
90 550 6.85
Table 5.2 Cooling rates, associated maximum temperatures and their equilibrium oxygen content for
powder samples (experimental data)
0.1
1
10
100
350 400 450 500 550 600
Peak temperature (C)
Co
o
lin
g 
ra
te
 
(C
/m
in
)  
6.82
6.84
6.86
6.88
6.9
6.92
6.94
6.96
6.98
7
O
x
yg
en
 
co
n
te
n
t  
 
1
 2
3
4
Fig. 5.31 Plot of cooling rates vs. Tmax and equilibrium oxygen content vs. Tmax
Fig. 5.31 can be used to make the predictions mentioned above:
If the minimum oxygen content required for a sample is 6.92, taking this point in the
oxygen content vs. Tmax line, it is possible to see that the temperature at which 6.92 is
the equilibrium content is approximately 485 °C (line 1). This temperature is the Tmax
of a cooling rate that can be read from the α vs. Tmax line (line 2). The cooling rate for
which 480 °C is Tmax is ~9 °C/min, therefore, if this sample is cooled at 9 °C/min
from 700 °C to room temperature, its oxygen content will be at least 6.92.
If, on the other hand, the powder has been cooled at 90 °C/min, the peak temperature
characteristic of this rate is 550 °C (line 3), which means that the oxygen content of
Chapter 5 Results and discussion
145
the powder will be at least 6.87 (line 4). This amount will be uniformly distributed in
the powder particles.
Knowing the cooling rate it is possible to calculate the time it will take to decrease the
temperature from 700 °C to Tmax:
Tmax (°C) α (°C/min) Cooling time from 700 °C Oxygen content
350 0.1 58 h 20 min 6.98
400 0.4 12 h 30 min 6.97
415 1 4 h 75 min 6.96
480 9 24 min 26 s 6.92
518 25 7 min 17 s 6.90
530 45 3 min 45 s 6.89
550 90 1 min 40 s 6.87
Table 5.3 Cooling times from 700 °C to the Tmax characteristic of each cooling rate, α, for a 2.1 µm
radius particle
If the oxygen content required is high, the maximum temperature will be low; this
means that the powder should be cooled at a very slow rate. However, it can be seen
from Table 5.3 that there is no need to keep the cooling rate constant in order to cool
the specimen in the quasi-equilibrium conditions described. Fast rates can be used at
high temperatures and slower rates at low temperatures to obtain the same final
oxygen concentration. Provided the temperature is never below the characteristic Tmax
for each rate chosen the material will be at equilibrium, and it will be possible to
combine the cooling rates so that if the final oxygen content is 6.97. Taking these
temperature steps to be the as big (and different in size) as those in the table above, the
cooling procedure would be:
From 700   to 550 °C at 90 °C/min  1 min 40 s
From 550   to 530 °C at 45 °C/min 24 s
From 530   to 518 °C at 25 °C/min 29 s
From 518   to 480 °C at   9 °C/min   4 min 13 s
From 480   to 410 °C at    1 °C/min   1 h 10 min
From 410   to 400 °C at 0.4 °C/min          25 min
Total ~   1 h 51 min
Chapter 5 Results and discussion
146
The difference in total cooling time between the single-rate cooling to 400 ºC (at
0.4°C/min) to the step-cooling shown above is quite large: approximately 10 h.
This quasi-equilibrium cooling procedure can be improved setting the temperature
intervals to be of a certain size. This would allow the construction of a table with the
time it takes to cool down from one temperature to another. The reduction in
temperature could be set to occur in 20 °C intervals, for example, and the cooling rates
for each of them would be the characteristic rates of the lower temperatures in the
step. Table 5.4 shows an example of these quasi-equilibrium cooling times.
To obtain a powder sample with an oxygen content of 6.97, for example, the table
shows how long it would take to cool from 700 °C and the cooling rates required for
each one of the temperature intervals. As it has been shown above, the final
temperature of the cooling procedure should be 400 °C, and the cooling time would be
1 h. The column on the right hand side shows the time it would have taken to cool to
that temperature using only the characteristic cooling rate for 400 °C (0.42°C/min):
approximately 12 hours.
The data from this table can also be presented in the form of a graph showing the
equilibrium oxygen content and reduced cooling times (see Fig. 5.32). Extended times
(single cooling rate) are included for comparison. The overlap of these two curves at
high temperatures is due to the limitation in cooling rate mentioned at the beginning
of this section: rates cannot be faster than 90 °C/min, which means that cooling to
560°C will always be carried out at the same rate, and reduced and extended times
will coincide. Below this temperature, the curves separate showing the difference in
cooling times between both procedures (single cooling rate and quasi-equilibrium
cooling).
Chapter 5 Results and discussion
147
Oxygen
content
Peak T
(°C)
Cooling rate
(°C/min)
Reduced time
from 700 °C
Extended time from
700 °C
6.98 300 0.01 28 h 42 min 19 days 16 h 42 min
6.98 320 0.03 14 h 36 min 13 days 17 h
6.98 340 0.06 7 h 27 min 4 days 14 h 22 min
6.97 360 0.11 3 h 48 min 53 h
6.97 380 0.22 1 h 57 min 25 h 28 min
6.97 400 0.42 60 min 12 h 10 min
6.96 420 0.83 31 min 5 h 47 min
6.95 440 1.64 17 min 2 h 45 min
6.93 460 3.23 9 min 1 h 17 min
6.92 480 6.34 6 min 36 min
6.91 500 12.47 3 min 17 min
6.89 520 24.52 2 min 8 min
6.88 540 48.20 2 min 4 min
6.85 560 90 1 min 2 min
6.82 580 90 1 min 1 min
6.80 600 90 1 min 1 min
6.76 620 90 < 1 min 1 min
6.73 640 90 < 1 min < 1 min
6.70 660 90 < 1 min < 1 min
6.67 680 90 < 1 min < 1 min
6.64 700
Table 5.4 Comparison between reduced and extended times for quasi-equilibrium cooling from 700 °C
of powder with 1.2 µm radius particles
Fig. 5.32 shows the same example used to demonstrate the application of the quasi-
equilibrium cooling table: 6.97 is the equilibrium oxygen content at 400 °C, which
means that cooling from 700 °C to this temperature following the quasi-equilibrium
procedure it would take 1 h, while it would take approximately 12 h to achieve the
same oxygen concentration using a single cooling rate. The time needed to cool to
room temperature would be added to this value to give the total time of the cooling
step.
Chapter 5 Results and discussion
148
0.1
1
10
100
1000
10000
300 400 500 600 700 800
Temperature (C)
Ti
m
e 
(m
in
)  
6.5
6.6
6.7
6.8
6.9
7
O
x
yg
en
 
co
0n
te
n
t  
Reduced time
Extended 
time
60 min
12 h 10 min
Fig. 5.32 Combination plot showing equilibrium oxygen content, reduced and extended cooling times
vs. Tmax (diffusion distance: 2.1 µm)
The starting temperature can be altered to suit the processing route, and the
temperature intervals can be modified to accommodate to the programming options of
the furnace used. If the temperature steps were very small, the total cooling stage
would be reduced to shorter times, but this would not be very practical experimentally,
because furnaces have a limited number of heat treatment steps. However, no matter
what the size of the temperature steps is, the quasi-equilibrium cooling procedure
always provides an alternative oxygenation with shorter processing times.
5.4.2 Extrapolation to different particle sizes
The alternative cooling route presented in the previous section is valid for small
powder particles (2.1 µm in radius). However, the size of the specimens used for
practical applications often bigger than just a few microns (sometimes of the order of
tens of mm), so it would be ideal if these curves were available for a range of particle
sizes.
One way of calculating these quasi-equilibrium cooling procedures for a number of
particle sizes would be to carry out experimentally a series of non-isothermal
oxygenations and calculating the relationship between cooling rate and maximum
temperature from the peaks obtained. This presents a problem that has already been
referred to at the beginning of the chapter: as the size of the specimen increases (the
Chapter 5 Results and discussion
149
diffusion distance increases) diffusion times increase too, slowing the data gathering
process considerably.
The diffusion model developed can be used to calculate theoretically the non-
isothermal experiments required to establish the relationship between cooling rates
and maximum temperatures. This can be done either fixing the grain size and
recalculating the expression of cooling rate (α) vs. Tmax, as it has been shown in the
previous section, or extrapolating these values from a known group of particle sizes to
the required particle size.
In the first case, Fig. 5.33 shows that an increase in the particle size requires a
decrease in the cooling rates used. If the rates are too fast, the sample will only be
partially oxygenated at the end of the process. The relationship between cooling rates
and Tmax obtained for a 10 µm radius particle is:
Cooling rate = 2 × 10-5 exp (0.0219×Tmax) (5.13)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 100 200 300 400 500 600 700 800
Temperature (C)
Fr
ac
t. 
w
t. 
ch
an
ge 10C/min
30C/min 
1C/min
Fig. 5.33 Theoretical non-isothermal oxygenations of a 10 µm radius particle. Note the slower cooling
rates compared with those used to oxygenate a 2.1 µm radius particle (Fig. 5.29)
This reduction in ramp rates means that the oxygenation times are going to increase
considerably, but as it is shown in Table 5.5 and Fig. 5.34, the quasi-equilibrium
cooling procedure developed still offers a faster alternative way of oxygenating the
specimens.
Chapter 5 Results and discussion
150
Oxygen
content
Peak T
(°C)
Cooling rate
(°C/min)
Reduced time from
700 °C
Extended time from
700 °C
6.98 300 0.03 1 day 8 h 48 min 9 days 16 h 40 min
6.98 320 0.04 21 h 9 min 5 days 22 h 38 min
6.98 340 0.07 13 h 40 min 3 days 15 h 12 min
6.97 360 0.11 8 h 49 min 2 days 5 h 9 min
6.97 380 0.17 5 h 41 min 1 days 8 h 17 min
6.97 400 0.26 3 h 40 min 19 h 32 min
6.96 420 0.40 2 h 22 min 11 h 46 min
6.95 440 0.61 1 h 32 min 7 h 3 min
6.93 460 0.95 59 min 4 h 12 min
6.92 480 1.48 38 min 2 h 29 min
6.91 500 2.29 25 min 1 h 27 min
6.89 520 3.54 16 min 51 min
6.88 540 5.49 10 min 29 min
6.85 560 8.51 7 min 16 min
6.82 580 13.19 4 min 10 min
6.80 600 49.08 3 min 5 min
6.76 620 76.06 2 min 3 min
6.73 640 90  1 min 1 min
6.70 660 90  1 min  1 min
6.67 680 90 < 1 min < 1 min
6.64 700
Table 5.5 Comparison between reduced and extended times for quasi-equilibrium cooling from 700 °C
of powder with 10 µm radius particles. Note the increase in cooling time and decrease in ramp rates
for the same oxygen contents (and peak temperatures)
Going back to the case of oxygenation to 6.97 again, the time it would take to increase
the oxygen content to that value using the quasi-equilibrium cooling method would be
approximately 3 h 40 min, which is a much better alternative than the ~ 19½ hours it
would take using a single cooling rate (Table 5.5 and Fig. 5.34).
Chapter 5 Results and discussion
151
0.1
1
10
100
1000
10000
100000
300 400 500 600 700 800
Temperature (C)
Ti
m
e 
(m
in
)  
6.4
6.5
6.6
6.7
6.8
6.9
7
O
xg
ye
n
 c
o
n
te
n
t
3 h 40 min
19 h 32 min
Reduced times
Extended times
Fig. 5.34 Combination plot showing equilibrium oxygen content, reduced and extended cooling times
vs. Tmax for a particle with a 10 µm radius (note that the cooling rates are much slower in the case of a
2.1 µm particle (Fig. 5.32))
If instead of fixing the particle size the study was carried out setting the cooling rates
constant, the procedure would be the following:
In the first place, it is necessary to obtain the empirical relationship between particle
size and Tmax for a chosen cooling rate. This relationship has been found to be
logarithmic; taking the maximum temperatures and cooling rates characteristic for a
2.1 µm radius particle (Eq. (5.12)), those for a 10 µm particle (Eq. (5.13)) and similar
data for other different particle sizes, it has been found that grouping the theoretical
results by cooling rates and plotting Tmax vs. log(particle size) gives a series of parallel
lines with the same slope. Fig. 3.35, for example, shows the change in Tmax that
corresponds to a cooling rate of 0.1 °C/min.
An increase in the grain size results in an increase of the temperature below which the
sample is not at its equilibrium oxygen content. This means that the temperature at
which the cooling procedure starts may have to be increased to 800 or 900 °C.
Chapter 5 Results and discussion
152
300
400
500
600
700
800
900
1000
0.0001 0.001 0.01 0.1 1
Radius (cm)
Tm
ax
 
(C
)
Tmax = 66.182 ln(radius) + 908.68
Fig. 5.35 Empirical relationship between grain sizes (radii) and maximum temperatures for a cooling
rate of 0.1 °C/min
It has already been mentioned that the slope of the line relating Tmax and particle radius
is the same for any cooling rate (α) chosen, so in order to complete Fig. 3.35 for a
wider range of cooling rates, it is only necessary to shift the Tmax characteristic of the
0.1 °C/min rate so that it intersects at the different Tmax -α points already calculated
for one of the particle sizes (see Fig. 5.36).
200
300
400
500
600
700
800
900
1000
0.0001 0.001 0.01 0.1 1
Radius (cm)
Tm
ax
 
(C
)   
10 C/min 5 C/min
1 C/min 0.1 C/min
0.01 C/min 0.001 C/min
From Fig. 5.35
Reference points 
from Fig. 5.30
Fig. 5.36 Empirical relationships between grain sizes (radii) and maximum temperatures for a series of
cooling rates
The graph that results from this series of relationships can be used as a guide of the
temperature range and cooling rates that need to be used for a quasi-equilibrium
Chapter 5 Results and discussion
153
oxygenation of specimens of any size; this would allow the calculation of the
oxygenation times that would result from using the cooling rates shown.
If the specimen to oxygenate had the dimensions 1×1×1 mm3, the diffusion distance
would be equal to 0.5 mm. For a particle with a radius equal to 0.5 mm, the following
cooling procedure (from ~900 °C in pure oxygen) can be drawn up from the data
plotted in Fig. 5.36 (approximate oxygen contents characteristic of each temperature
are shown in brackets):
Cool: at 10 °C/min to 880 °C (6.41)
at 5 °C/min to 830 °C (6.46) 10 min
at 1 °C/min to 790 °C (6.50) 50 min
at 0.1 °C/min to 720 °C (6.59) 11 h 40 min
at 0.6 °C/h to 650 °C (6.70)   4 days 20 h 40 min
at 1.4 °C/day to 580 °C (6.82)    1 month    18 days  14 h              .
~1 month 23 days
This cooling procedure will yield a sample with an oxygen content equal to ~6.82.
However, this content is very low for the material to have good superconducting
properties, so it would be necessary to decrease the temperature further. Due to the
increased particle size and slower diffusion rates at low temperatures, continuing the
oxygenation to increase the oxygen content using the quasi-equilibrium cooling
procedure would require the use of slower cooling rates (of the order of degrees per
month), which would result in unreasonably long cooling times.
The application of this oxygenation procedure for larger particles at lower
temperatures appears to be very complex due to the slow coooling rates needed.
However, it should be pointed out that these large specimens are usually polycrystals,
and diffusion in these materials occurs not only through the bulk, but combines with
grain boundary diffusion in the lower temperature range. The effects of other
microstructural features, such as porosity or presence of impurities should also be
taken into account. This could result in a faster overall diffusion process that would
involve faster cooling rates and yield shorter oxygenation times.
Chapter 5 Results and discussion
154
5.4.3 Summary
This section describes the development of a dynamic cooling procedure that can be
introduced in the processing of YBCO to yield highly oxygenated samples with a
uniform distribution of oxygen. This procedure has been termed quasi-equilibrium
cooling and has been developed based on data collected experimentally during non-
isothermal oxygenations of powder samples at different rates.
Using the diffusion model developed it has been possible to obtain theoretical results
of non-isothermal oxygenations of larger particles that would have otherwise taken a
very long time to carry out and it has been possible to present an initial extrapolation
of this cooling procedure to bigger samples. The application of this quasi-equilibrium
cooling procedure to oxygenate large specimens is still in an initial stage. All the work
carried out in this thesis has been done using a bulk diffusion coefficient, but
polycrystalline samples have a different effective diffusion coefficient, incorporating
grain boundary diffusion at lower temperatures. More work should be carried out to
evaluate the use of this cooling procedure in the oxygenation of polycrystalline
samples.
5.5 Conclusions
This chapter has presented an analysis of the results obtained using thermogravimetry
to measure the oxygenation of YBCO powder samples. Fitting a solution to Fick’s
first law with the appropriate boundary conditions it has been possible to calculate the
pre-exponential coefficient, Do, and the value of the activation energy of this diffusion
process that can be described with an Arrhenius equation. The values obtained for the
variation of the oxygen diffusion coefficient with temperature have been compared to
the results published in the literature.
The diffusion model developed described in chapter 4 uses the values of Do and the
activation energy calculated from the experimental data. It has been shown that this
model can be successfully applied to the analysis and prediction of isothermal
oxygenations. The evolution of the oxygen concentration profile within the sample
can also be followed using the programme developed; this is of great interest as this
Chapter 5 Results and discussion
155
information could be applied to the analysis of the superconducting properties of
oxygenated specimens.
This isothermal study of powder samples is followed by a brief account of the work
done on polycrystalline specimens of different density. The results obtained show that
the diffusion process in polycrystals is very complex, as it has already been pointed
out in chapters 2 and 3. The analysis of the data collected with the solution to Fick’s
first law used for the study of oxygenation of powder specimens has only attempted to
calculate a rough estimate of the oxygen diffusion coefficient for polycrystalline
samples.
The final section of this chapter concentrates on the study of non-isothermal
oxygenations. It analyses the characteristic shape of non-isothermal oxygenation
traces; these curves can be divided into a non-equilibrium region during which the
sample absorbs oxygen at a rate too slow to achieve full oxygenation, and an
equilibrium region in which the specimen is at the equilibrium oxygen content for
each temperature. A preliminary analysis of the non-isothermal results has allowed
one to calculate a value for the activation energy of the oxygenation process that
agrees well with that calculated from isothermal oxygenations.
The characteristic behaviour of the non-isothermal oxygenation trace at high
temperatures has been used to develop a novel cooling procedure that consists in
decreasing the temperature of the specimen from the last high-temperature stage
during manufacturing at a series of cooling rates that ensure that the sample is at the
equilibrium oxygen content of each temperature. This procedure has been termed
quasi-equilibrium cooling.
It has been calculated that the quasi-equilibrium cooling procedure reduces the
duration of the oxygenation step; this difference increases with the size of the samples.
Therefore, the procedure could be implemented at the end of the processing of YBCO
superconducting samples instead of the isothermal oxygenation stage usually
employed. This would eliminate (or reduce the effects of) one of the problems linked
to the use of high temperature superconductors in industrial applications: the need of
long oxygenation stages that become increasingly long with sample size; this makes
Chapter 5 Results and discussion
156
the manufacturing process of YBCO for high temperature superconducting
applications not viable commercially.
The diffusion programme has been used to calculate the final distribution of oxygen in
the sample at the end of the quasi-equilibrium cooling. It has been possible to
extrapolate diffusion results to larger samples reducing the overall experimentation
time in this thesis and completing the design of novel cooling procedures for samples
of different sizes. These extrapolated values can be used as a guideline to design the
cooling/oxygenating procedure appropriate for the size of the sample used in each
case. This study should be developed further using a diffusion expression typical of
polycrystalline YBCO that took the effects of grain boundaries and porosity in the
diffusion process. This would allow the development of cooling procedures for
samples used in large-scale applications.
The study of isothermal and non-isothermal oxygenations of YBCO carried out using
thermogravimetry and the development of a diffusion model to reproduce these results
theoretically have allowed one to gain more information about the oxygen diffusion
process. Using the programme to reproduce oxygenation experiments theoretically it
has been possible to develop an oxygenating procedure that could reduce the length of
the processing of YBCO specimens for large scale applications.
Chapter 6 Direct Current Zoning effect
156
Chapter 6
Direct Current Zoning effect
6.1 Introduction
The previous chapters have presented a study of the diffusion of oxygen in
YBa2Cu3O7-δ (YBCO). It has already been pointed out that the properties of
superconducting oxides are strongly dependent on the oxygen non-stoichiometry and
its distribution over the whole sample. Because of this non-stoichiometry YBCO
behaves as a solid mixed conductor at high temperatures, showing predominantly
electronic conductivity but also a significant ionic mobility; its overall resistivity
increases steeply for T > 400 °C due to the loss of oxygen and the increase in the
scattering of carriers. Above this temperature the oxygen ion conductivity (similar to
that of fast ionic conductors at high temperatures, see Fig. 6.26) becomes significant
in the conduction process. The subject of this chapter is directly related to the mixed
conductor behaviour of YBCO and to the diffusion of oxygen ions in this material.
This chapter presents a brief review of the high temperature electrical properties of
YBCO together with a review of the work carried out to study the Direct Current
Zoning effect. This is followed by a presentation and analysis of the work done and
concludes with the description of the computer model developed to reproduce the
effect of Direct Current Zoning in an ideal sample.
6.2 Direct Current Zoning effect (DCZ)
It has been observed that when a DC current is applied to a polycrystalline bar of
YBCO (and other superconducting materials) a hot zone forms at a certain point along
the bar. The zone has been observed to drift towards the cathode. This phenomenon is
known as Direct Current Zoning (DCZ), and it is going to be reviewed in the
following sections.
Chapter 6 Direct Current Zoning effect
157
The formation of a hot zone when a DC current is applied across the sample is related
to the increase in mobility of the oxygen ions in the sample with temperature and the
resulting change in ionic conductivity as well as to the steep increase in the resistivity
of the sample with temperature (see Fig. 6.3). Inhomogeneities in composition, density
or cross sectional area of the sample are thought to be the nucleation points of the
zone, although the precise combination of factors that determine the onset of zone
formation is not at present understood. The hot zone moves towards the negative
electrode. This drift is related to the migration of oxygen ions towards the anode due
to the potential difference, and within and out of the sample due to the changes in
temperature. The fast ionic motion of the ions in the structure could introduce the use
of YBCO as an oxygen sensor and it can also help in the understanding of the
complex kinetics and thermodynamics during the processing of high temperature
superconductors. The following sections present the work carried out by other groups.
6.2.1 DCZ in YBCO
Apart from the work reported here, three other groups world-wide have studied this
phenomenon and the results published disagree in certain aspects. The materials
studied by the different groups were: YBCO (Osip’yan et al. (1988), Bobrov et al.
(1990) and Nefedov et al. (1992)), Bi-Pb-Sr-Ca-Cu-O (Yanmaz et al. (1992), (1993))
and GdBCO (Okamoto et al. (1994-a) (1994-b)). All of the work reported has been
carried out on untextured polycrystalline ceramic samples.
Osip’yan et al. (1988) applied a DC current to YBCO bars and observed the formation
of a hot zone drifting towards the negative electrode. This behaviour was associated
with a non-linear dependence of the resistivity of the material with temperature and
the evolution of oxygen during Joule heating. They pointed out that such a loss would
be accompanied by the formation of a high concentration of positively charged oxygen
vacancies whose mobility in an electric field would determine the drift of the zone. In
another report by the same group, Bobrov et al. (1990) propose a model for the
presence of the hot zone in YBCO. In this model the effects of heat release and heat
transfer are linked to the increase in resistance and the oxygen vacancy concentration
and diffusion through the material.
Chapter 6 Direct Current Zoning effect
158
However, the I-V characteristic of the formation of the zone and the relationship
between domain drift velocity and current do not agree with reports published later
and with the work presented in this report.
Bobrov et al. (1990) claim that an increase in current increases the speed at which the
zone moves. But it has been observed in this work that the zone becomes immobile
when the voltage is increased above a certain value. This results in an increase in the
resistance of the sample, which in turn raises the temperature of the zone up to the
point where catastrophic failure could occur, melting the sample.
Nefedov et al. (1992) reported the DC treatment on the YBCO superconducting
ceramics at 77K and observed that light impurities (Na, K, Cl) and oxygen migrate
towards the appropriate electrode, resulting in a change of the properties of the
material.
Fig. 6.1 Sequence photograph after applying a DC voltage of 7V to GdBa2Cu3O7-δ. The pictures are
taken at 2 min intervals. The movement reverses each time the polarity is reversed (after Okamoto et
al. (1994-b))
Okamoto et al. (1994-a), (1994-b) report the same DCZ effect for samples of GdBCO
when a DC voltage of 7 V is applied at room temperature. They studied the effects of
partial oxygen pressure in the generation and movement of the zone. They reported
that the current necessary to create and maintain the hot zone appeared to be very
sensitive to the oxygen pressure in the atmosphere, and once the zone was present it
was observed to increase size with increasing voltage (Fig. 6.2). This group observed
Chapter 6 Direct Current Zoning effect
159
that the total current after the appearance of the hot zone was independent of the
applied voltage, which contradicts the measurements presented in this chapter. They
report that the value of the current needed to create a zone increased with increasing
oxygen pressure and so did the drift velocity of the zone, agreeing with Bobrov et al.
(1990). They conclude that these variations in oxygen pressure affect the oxygen
migration in the hot zone.
Fig. 6.2 Voltage dependence of the spot size (after Okamoto et al. (1994-b))
Migration of the hot spot is explained by Okamoto and co-workers in the following
way:
“[...] The oxygen ions in the hot zone move easily due to the high temperature
and high concentration of oxygen vacancies. When oxygen ions move to the
positive potential side of the hot spot, the resistivity increases on the negative
side and decreases on the positive side. [...] Because the system tries to stay in
equilibrium, oxygen will diffuse into the sample at the positive side which will
further decrease the resistivity and the temperature. At the side of the negative
electrode oxygen will be released from the sample, hence the resistivity and
the temperature will increase. Consequently, the spot will move to the negative
electrode”.
This is a very good description of the phenomenon that is going to be explained in
more detail in section 6.4.
Chapter 6 Direct Current Zoning effect
160
6.2.2 DCZ in other materials: Bi-Pb-Sr-Ca-Cu-O
Yanmaz et al. (1992), (1993) report another application of this heating effect by
applying an electric field across the sample. They observed that an increase in 0.2 A
produced another zone in the sample once the first one appeared. The passing of the
zone was observed to alter the final microstructure of the specimen. The formation of
two zones in the bar has also been observed in the experiments carried out for this
dissertation (see section 6.4). This result does not agree with the I-V characteristic of
the DCZ effect shown by Okamoto et al. (1994-a), where the current was constant
once a zone formed, independently of the voltage applied.
Yanmaz and co-workers observed the modification of the microstructure of the zoned
samples, and so have the other authors like Bobrov et al. (1990), for example, who
reported a change in the structure of the sample from orthorhombic to tetragonal
where the zone was present and a change to orthorhombic again once the zone had
moved towards the negative electrode. They agreed with Bobrov et al. (1990) in that
the speed of the zone increased with increasing current.
The suggested explanation for the DCZ effect by Yanmaz et al. (1992), (1993) is not
very clear. They propose that the creation and movement of the zone could be due to
solid electrolysis, with O- ions migrating under an electric field gradient from the
negative to the positive electrode, thus increasing:
“[...] the O- ion concentration at the positive electrode with a consequent
increase in electrical resistance. This results in a “hot spot” and the creation of
a zone due to resistance heating. Because the material becomes hot, oxygen is
lost, the resistance falls and the zone then moves to another high resistance
region”.
This decrease in resistance with decreasing oxygen ion concentration contradicts the
reported change in the electrical properties of YBCO with increasing temperature: as
temperature increases, the oxygen content decreases and the resistivity of the material
increases, as it is going to be seen in section 6.3 (e.g. Gurvitch and Fiori (1987)).
Chapter 6 Direct Current Zoning effect
161
6.2.3 Effect of different atmospheres
Osip’yan et al. (1994-b) and Bobrov and co-workers studied the effects of different
atmospheres on the DCZ effect and found that the zone drift velocity increased when
the samples were studied in oxygen atmosphere, and the size of the zone and the drift
velocity decreased when using helium. In vacuum a hot zone appeared but no drift
was observed. This resulted in a continuous heating of the sample and led to
catastrophic failure by melting at the zone. This difference in behaviour when
changing the oxygen content of the atmosphere means that the movement of oxygen
ions is very important in this process but a full explanation to this phenomenon has
not been presented yet.
6.3. High temperature transport properties
6.3.1 Review
After the discovery of YBCO much of the experimental and theoretical work on the
electronic behaviour of this material concentrated on the range of temperatures around
and below Tc in order to explain the mechanism of superconductivity in this regime.
But interest has also developed in the study of normal state electronic properties of
high temperature superconductors with the hope that such properties might help in
understanding the mechanism of high Tc superconductivity.
Conductivity measurements provide information about the product of the mobility of
the change carriers and their concentration in the sample. However, in the work
published on the YBCO system there has been an accumulation of conflicting reports
concerning its structural and electrical properties. Most of the work concentrates in the
determination of data within very limited regimes which does not allow one to draw
general conclusions concerning the defect structure of the oxide phase or its electrical
properties. In spite of this there is a general agreement on the decrease of conductivity
with decreasing oxygen content.
Chapter 6 Direct Current Zoning effect
162
In order to be able to study the Direct Current Zoning process it is necessary to know
how the conductivity of YBCO changes at high temperatures. This section presents a
review of the work carried out in this field.
Grader et al. (1987) observed a variation in the slope in the plot of resistivity vs.
temperature in a region near the orthorhombic-tetragonal transition and associated this
change in resistivity to the phase change in the crystal structure. This has also been
suggested by Fiori et al. (1987) (Fig. 6.3), Fisher et al. (1987) and Davidson et al.
(1988).
Fig. 6.3 Temperature dependence of the resistivity of YBCO at various partial oxygen pressures (after
Fiori et al. (1987))
However, Park and Kim (1988) and Yamaguchi et al. (1988) did not link this change
in slope with the order-disorder transition. It has been corroborated by other authors
that it occurs due to a change in the charge transport behaviour from metallic to
semiconducting. The temperatures at which this change is observed vary from 427 ºC
in air (Munakata et al. (1987)), higher than 400 ºC (Park and Kim (1988)),
approximately 500 ºC in air (Brabers et al. (1988)) to 620 ºC in air and 680 ºC in O2
(Park et al. (1988)). The last two values seem too high compared with the rest.
This variation in conducting behaviour is directly related to the oxygen non-
stoichiometry of the sample; since its distribution and the measurement of oxygen
motion in the sample appears to be very dependent on microstructure (see chapter 3),
Chapter 6 Direct Current Zoning effect
163
some scatter in the values is expected. The conductivity of the specimen is determined
by its thermal history, its microstructure and the measuring technique used.
The change in slope in the conductivity has also been measured in terms of oxygen
content. The values at which this change in slope has been observed to occur range
from 6.3 (Leonidov et al. (1989)) and 6.5 (Yamaguchi et al. (1988)), to 6.8 (Yoo and
Lee (1994)). The result from Leonidov et al. (1989) seems to be too low for the
temperatures considered (around 400-500 ºC). In this temperature range the
equilibrium oxygen content is much higher, closer to the 6.8 given by Yoo and Lee
(1994).
The strong dependence on the non-stoichiometry could be another explanation for the
hysteresis observed in the conductivity (resistivity) values during heating and cooling
cycles of the sample observed by Yamaguchi et al. (1988) and Freitas and Plaskett
(1987).
Fig. 6.4 Resistivity of YBCO as a function of oxygen content (after Grader et al. (1987))
All the authors agree on a dependence of the resistivity on temperature and partial
oxygen pressure; this is a direct consequence of the relationship between the
stoichiometry and the transport properties. However, there are slight discrepancies as
to what this dependence is. Leonidov et al. (1989) suggest a dependence of σ ∝ pO21/6
or pO21/4, Yoo (1989) observed a σ ∝ pO21/2 between 400 and 700 °C, Davidson et al.
Chapter 6 Direct Current Zoning effect
164
(1988) also suggest a square root dependence on partial oxygen pressure and Ronay
(1987) and Chang et al. (1988) observed a square root dependence in the tetragonal
phase, and σ ∝ pO21/4 in the orthorhombic phase. This dependence is directly related
to the type of current carrier, agreed by all authors to be holes.
During the study of the electrical resistivity of YBCO a change in the transport
process from p to n is observed to occur when the partial oxygen pressure is reduced
to very low values (10-3-10-4 atm at 700 °C to 800 °C (Chang et al. (1988)) or if the
temperature is increased (877 °C, according to Yoo and Lee (1994)).
The transport model has been suggested to be small polaron hopping by most authors:
Yoo (1989), Su et al. (1988), Nowotny and Rekas (1990), Davidson et al. (1988).
Fisher et al. (1987) proposed a narrow band model, applicable for the tetragonal
region, and Ronay suggested a different method based on the law of mass action.
Fig. 6.5 summarises some of the results published for the conductivity of YBCO at
different temperatures. The measurements carried out by the different groups measure
the total conductivity of the material; this will be mainly due to electron motion at low
temperatures with an increasing contribution of the ionic conductivity as temperature
increases. Fig. 6.24 in section 6.4.3 shows the increasing value of the ionic
conductivity of YBCO calculated using the relationship between ionic conductivity
and the diffusion coefficient of the ions in the material.
With respect to the direct measurement of the ionic conductivity of YBCO, the
amount of work carried out is not very great and the reports give quite scattered
values: Carrillo-Cabrera et al. (1989) have measured a conductivity of 1 Ω-1cm-1 at
500 °C (in air), while Park et al. (1988) give a value of 1.6 ×10-6 Ω-1cm-1 for the same
temperature (in oxygen). There is a serious discrepancy between these two results. It
has already been mentioned that the ionic conductivity is proportional to the
concentration and mobility of the diffusing species. The stoichiometry of this material
in air and oxygen changes only by approximately 0.3 atoms per formula unit
(Gallagher (1987)), which leads one to think that the measurements must have been
affected by an extra factor that has given rise to such a difference in the values.
Chapter 6 Direct Current Zoning effect
165
The activation energies calculated for ionic conductivity also present a large variation:
from 1.23 eV and 1.5 eV (Vischjager et al. (1990)) to around 2.5 eV (Carrillo-Cabrera
et al. (1989)), and 2.46 in the tetragonal phase or 1.6 eV in the orthorhombic phase
(Yoo (1989)). These variations could be a consequence of the different methods used
to measure this conductivity (DC two–probe conductivity, AC impedance, four point
probe, etc.) and it is very likely that the microstructure of the polycrystalline sample
affects the final result in the same way as it affects the diffusion measurements carried
out. This could be due to an irregular distribution of the oxygen in the grains or the
presence of grain boundaries, pores or impurities that can interfere with the
measurement.
It will be seen in the next section (6.3.2) that the value of the ionic conductivity can be
linked to the diffusion coefficient (Eq. (6.16)) using the Nernst-Einstein equation (Eq.
(6.7)). It seems surprising that of all the authors that have studied the ionic
conductivity of YBCO, only Park et al. (1988) have completed their work by relating
their values of the ionic conductivity to the oxygen diffusion in the material. They
obtained a value for the diffusion coefficient equal to 9.5 ×10-8 cm2/s at 500 °C. This
value is two orders of magnitude slower than that calculated for oxygen diffusion in a
polycrystalline material presented in chapter 5 (section 5.2.2).
This brief review shows that in spite of all the work carried out for the study of the
high temperature transport properties of YBCO the situation still presents conflicting
points over the transport mechanism as well as the value of the electronic and ionic
conductivity of the material. The dependence of this behaviour on the non-
stoichiometry introduces variables in the study similar to those considered in the study
of oxygen diffusion. This suggests that more work should be conducted in this topic
so that conclusive values for the electronic and ionic conductivity are produced.
Chapter 6 Direct Current Zoning effect
166
0
50
100
150
200
250
300
350
400
450
400 500 600 700 800 900 1000
Temperature (C)
Co
n
du
ct
iv
ity
 
(1/
oh
m
 
cm
)   Su et al. Leonidov et al.Park et al. Park and Kim
Chang et al. Grader et al.
Fiori et al. average
Eq. (6.1) This work
Fig. 6.5 Literature data for the electronic conductivity of YBCO (pO2 = 1 atm)
 In order to be able to carry out the analysis of the Direct Current Zoning effect, it was
necessary to obtain an expression for the average of the electronic conductivity in
YBCO. The value of the conductivity that is going to be used in the calculations
presented in section 6.4.3 of this chapter was calculated from the average of the
literature values (see Fig. 6.5). It was assumed to vary exponentially with temperature
(in °C) and has the form:
σ = 1.8×104 exp(-8.0×10-3 T)      (6.1)
6.3.2 Nernst-Einstein equation. Correlation between
diffusion and conductivity
In order to be able to study the causes of the Direct Current Zoning effect, it is
necessary to look at the motion of both ions and electrons in a concentration or
activity gradient and their simultaneous transport in an electric field (drift). Chemical
diffusion occurs in a concentration gradient and drift is due to electric field effects, so
while diffusion occurs for all species, drift (or electromigration) affects only charged
species.
Chapter 6 Direct Current Zoning effect
167
The study of diffusion in a concentration gradient has already been discussed in
chapter 2. The net flux of particles in a concentration (or chemical potential) gradient
is expressed by Fick’s first law:
j = - D (∂C/∂x), (6.2)/(2.1)
when diffusion occurs only in the x direction. D is the diffusion coefficient and C is
the concentration at each point.
The relationship between the diffusion coefficient and the mobility can be derived as
follows:
The chemical potential of the diffusing particles takes the form:
µ = µ ° + RT ln C,            (6.3)
where µ° is the chemical potential in the standard state, C is the concentration of the
particles, R is the gas constant and T the absolute temperature. The relation is valid for
mobile ions as well as for defects (e.g. vacancies).
In the absence of an electric field the diffusive force experienced by each particle is:
F = - (1/NA) (∂µ/∂x) = -(RT/NAC) (∂C/∂x)P,T. (6.4)
where NA is Avogadro’s number. The ion mechanical mobility, B, can be defined as
the average drift velocity per unit force. Therefore the particle velocity, v, is equal to:
v = B
 
F. (6.5)
Since the flux of diffusing particles is equal to the net amount of particles moving per
unit time, the flux of particles in a concentration gradient can also be expressed by:
j = C v = - (BRT/NA) (∂C/∂x)P,T. (6.6)
Comparison with Fick’s first law of diffusion (Eq. (6.2)) gives:
D = BRT/NA = BkT. (6.7)
This expression shows the direct proportionality between diffusion coefficient and
mobility. It is known as the Nernst-Einstein relation (k is the Boltzmann constant).
If the particle was driven by the potential difference of an electric field of strength E =
- ∂φ/∂x, the force on each particle due only to the applied electric field would be:
Chapter 6 Direct Current Zoning effect
168
F = - ze (∂φ/∂x) = zeE, (6.8)
where z is the charge of the migrating species (2 in the case of oxygen ions). The
average velocity at which the ions move is:
v = BF = BzeE. (6.9)
Using the expression for the velocity of the ions and Eq. (6.6), the flux of ions would
be equal to:
j = C v = CBzeE. (6.10)
Substituting the Nernst-Einstein equation (Eq. (6.7)), the mobility may then be
replaced by the diffusion coefficient of the ions:
j = CDzeE/kT. (6.11)
Therefore, the flux of particles (the number of particles per unit area and time, Eq.
(6.10)) can be expressed by:
j = - D [(∂C/∂x) - zC (F/kT) (∂φ/∂x)], (6.12)
where the second term on the right hand side represents electromigration. This term
shows the importance of charge of the species, z, and of the value of the electric field
gradient, ∂φ/∂x. F is Faraday’s number, the charge of a mole of electrons.
The conductivity of a material is a result of the movement of all ions and all other
charge carriers (electrons or holes) in the material under the influence of an electric
field. As long as the electric field is not too strong we can assume that there is a linear
relationship between the total flux of the charged particles (the electrical current
density), J, and the electric field, E. Ohm’s law is then valid:
J = σ E; (6.13)
σ is the electrical conductivity (assumed in this derivation to be independent of the
direction in the crystal). The total conductivity, σ, is the sum of the partial
conductivities σi of the various types of particles, i, if the different charge carriers are
transported independently of one another.
J = ∑ Ji = ∑ σi E,      so:      σ = ∑ σI. (6.14)
Chapter 6 Direct Current Zoning effect
169
Mobility and conductivity are related in the following way:
Assuming that only one type of ion is mobile, the flux of charge carried by the flowing
ions, j, is defined as:
j = J/(z F) = σ E /(z F). (6.15)
Substituting equation (6.11) above, the relation between partial electrical conductivity,
the mechanical mobility and the diffusion coefficient of the moving ions is:
σ = DC(zF)2/RT, (6.16)
and in terms of mobility:     σ = CBz2eF.         (6.17)
To calculate roughly the value of the ionic conductivity in YBCO at a temperature of
500°C, it is sufficient to know the value of the diffusion coefficient at that
temperature. For this preliminary calculation, D is going to be taken as 1.2×10-6 cm2/s
(see chapter 5 for diffusion coefficient of polycrystals). The charge of the mobile
species, z, is –2 and the concentration of ions is: 8.7×10-3 moles/cm3 in air (the
equilibrium oxygen content at this temperature is ~ 6.88). Substituting these values in
equation (6.16) the oxygen ionic conductivity has a value of roughly 6×10-2 Ω-1cm-1.
This differs with the value calculated by Park et al. (1988) in four orders of magnitude
and with that calculated by Carrillo-Cabrera et al. (1989) in two orders of magnitude.
The dependence of these values on the non-stoichiometry of the sample and its
microstructure could be the reason for the difference observed.
Charge transport is also sometimes expressed in terms of the electronic mobility, b,
defined as the drift velocity of charge carriers per unit electric field:
b = v/E, (6.18)
so that from (6.9):
b = B z e, (6.19)
and: J = C z F v = n z e b E,         (6.20)
where n is the ion density (ions per unit volume) and:
Chapter 6 Direct Current Zoning effect
170
σ = n z e b,    (6.21)
following the convention for charge transport in semiconductors and metals.
6.4 Results and discussion
The previous sections have discussed the phenomenon of Direct Current Zoning and
the work that has been carried out in this topic. This has been followed by a study of
the research done on the electrical properties of YBCO at high temperatures (section
6.3.1). YBCO has a high oxygen ion mobility at high temperature. This is associated
to the increase in the ionic conductivity at these temperatures (above 600 °C this
conductivity becomes significant, as it is going to be seen in section 6.4.3), and makes
YBCO a candidate for fast ionic conductor applications, although this field has not
been developed yet.
This review of the electronic and ionic conductivity values published in the literature
has been followed by a derivation of the relationship between conductivity and
diffusion. This is going to be used in the study of the motion of the zone combining
the effects of ionic conductivity and chemical diffusion on the distribution of oxygen
ions within the specimen (see section 6.3.2).
This section is going to present and discuss the work carried out in the analysis of the
Direct Current Zoning effect.
The first section describes the experimental procedure. This is followed by a
discussion of the results obtained, pointing out the unique characteristics of the DCZ
effect. The drop in the power needed to maintain the zone present in the bar and the
stopping of the zone are discussed. The third section presents an analysis of the
processes that take place simultaneously during the DCZ effect: the change in the
temperature profile when the zone appears, the variation in electronic and ionic
conductivity with temperature, and the magnitude and direction of the ionic fluxes due
to the potential difference established as well as to the concentration gradient present.
This analysis has corroborated the description of the process presented by Okamoto
and co-workers (1994-b) (see section 6.2). Reasons for the stopping of the zone have
been suggested.
Chapter 6 Direct Current Zoning effect
171
This is followed by a description of the computer model developed to reproduce the
formation of the zone. This programme has carried out for the first time theoretical
calculations of the processes that take place in the Direct Current Zoning effect: from
the moment when a current is applied to the YBCO bar to the formation of the zone
and its motion. The results from this model have been compared to experimental data,
finding a very good agreement between them. The suggestions made in the previous
sections concerning the motion of the zone, the change in the temperature profile with
time and the variation in the oxygen concentration within the sample have been
corroborated by the model, which, in spite of being at an early stage of development,
has produced very good results that have contributed to the understanding of the
processes taking place in the Direct Current Zoning effect. This section is followed by
a summary of the work presented.
6.4.1 Sample preparation and experimental procedure
For the DCZ experiments the YBCO samples were prepared via a polymer processing
technique. Polymer processed sheets were produced by a technique known as
calendering. It consists of mixing YBCO powder with a binder (PVB), a plasticiser
and a solvent (cyclohexanone) in the correct amounts in a rolling mill (see Appendix
2A). Extensive mixing of the components during successive rolling steps results in a
rubbery paste that can be rolled into a tape of controlled thickness with a
homogeneous distribution of the material. The tape forms what is known as “green
body” and it is an unreacted mixture of the components. It can be pressed in a further
step so that proper compaction is ensured and it is then cut to shape. The resulting tape
produced was about 15 cm wide and approximately 1.5 mm thick. After pressing it
was cut in thin strips 1-2 mm wide.
These strips required further processing. The binder is eliminated from the mixture in
a burn-off step at around 500 °C. This is followed by a sintering step at high
temperatures (see Fig. 6.6). It is essential that the binder is burned off carefully to
avoid the presence of large pores in the microstructure. A typical binder burn-off
period for samples 1-2 mm thick consists of a slow heating (1 to 2 °C/min) to 500 °C.
After this first step, conventional processing of the material is carried out, sintering at
Chapter 6 Direct Current Zoning effect
172
temperatures around 900 °C for 10 hours. The atmosphere used during the burn-off
and sintering steps is air. In order to make highly oxygenated specimens, the strips
were cooled to 500 °C and held in pure oxygen (with a flow of 30 mm3/min) for
approximately 10 hours. This produces compact oxygenated polycrystalline samples.
900 1-2 h
sintering
50 ºC/min
1 h
burn-off
1-2 ºC/h
500
~ 10 h
oxygenation
Furnace
cool
Air Oxygen
(30 cm3/min)
Temperature (ºC)
Time
50 ºC/min
Fig. 6.6 Schematic of heat treatment
The ends of the samples were impregnated with silver paste and contact silver wire
was mounted to the sample with the same paint to reduce contact resistance. An
increasing voltage was applied using a Coutant LQT 200 power supply. The
experimental setup is shown in Fig. 6.7.
Power Supply
YBCO sample
A
V
Fig. 6.7 Schematic of setup
The change in current with increasing voltage is monitored with a 4-point
measurement as pictured above. Temperature measurements were carried out using a
K-type thermocouple.
Chapter 6 Direct Current Zoning effect
173
6.4.2 Results and discussion
An increase in the voltage by 0.1 V every minute resulted in a linear increase of the
current passing through the sample. When the voltage and current reached a certain
value, a decrease in the current was observed (see Fig. 6.8); this is characteristic of the
onset in the formation of a hot zone. This hot region becomes visible when its
temperature is above 550-600 °C, and the drop in current was not always associated to
the presence of a visible zone. If this was the case, it was necessary to increase the
voltage further to raise the temperature of the sample and make the zone visible.
0
0.5
1
1.5
2
2.5
3
0 1 2 3 4 5
Voltage (V)
Cu
rr
en
t (
A)
Zone formation
Fig. 6.8 I-V characteristic of one of the samples
It should be noted that once the zone appeared in the sample, the amount of power
required to keep the zone present in the bar was reduced considerably compared to the
power applied just before it occurred. This decrease in power could be as high as 25%.
This characteristic of the formation of the zone is going to be studied in more detail
later on in this section.
It was observed that holding the current and voltage constant after the zone occurred it
was possible to see that it moved towards the negative electrode at speeds that ranged
between 2 to 0.3 mm/min. Specimens with lower porosity exhibited zones that moved
more slowly; this suggests that the speed at which the zone moves along the sample is
strongly correlated to the degree of intergranular contact in polycrystalline samples
and to the variation in the diffusion process due to the microstructure of the sample.
Chapter 6 Direct Current Zoning effect
174
Figure 6.9 shows the initial stages of the formation of the zone and its motion along
the sample.
+ + + + + + + +
_ _ _ _ _ _ _ _
Fig. 6.9 Sequence of photographs while applying a DC voltage of 5.3 V to an YBCO bar. The pictures
are taken at 10min intervals. Note the change of colour of the zone at the formation indicating a
change in temperature. Distance between lines on the marker beside the sample: 5 mm
A change in the polarity was observed to reverse the direction of travel:
+ + + + _ _ _ _
_ _ _ _ + + + +
Fig. 6.10 Sequence of photographs showing the change in direction of the mobile zone when the
polarity is reversed. V = 5.2 V and I = 3.6 A. Time interval between pictures: 10 min. Distance
between lines on the marker beside the sample: 5 mm
An increase in the voltage was observed to increase the current, contrary to what was
reported by Okamoto et al. (1994-a), (1994-b), and Yanmaz et al. (1992), (1993). The
current was observed to decrease every time the increase in voltage stopped.
Increasing the voltage once the zone is present in the sample increases the overall
temperature of both, which, instead of increasing its travelling speed (Bobrov et al.
(1990)), slowed down to a halt. The continuous increase of the potential difference
across the sample and its associated increase in temperature always resulted in an
Chapter 6 Direct Current Zoning effect
175
increase in the width of the zone followed by catastrophic failure due to melting of the
sample at the zone (see Fig. 6.2).
The zone disappeared when the voltage was decreased. If the sample was allowed to
cool down and then V was increased again, the zone always appeared at the point
where it had first occurred unless some damage had been done to the sample while the
previous zone was present, such as partial melting. If this was the case, the zone was
always observed to appear at the damaged region.
When observed under the optical microscope, the areas along which the zone had
moved presented a tetragonal structure, while the “unzoned” areas showed golden-
coloured orthorhombic grains with a large number of twins in them, typical of the
orthorhombic structure (see Fig. 6.12). The “zoned” regions showed in some cases a
small degree of twinning in the grains, but the twins were less noticeable. This means
that the transformation from orthorhombic to tetragonal had not been completed in the
time the zone moved along the bar. This result agrees with the changes in the
microstructure of the samples due to the effects of the zone reported by Osip’yan et al.
(1988) and Bobrov and co-workers: an increase in the amount of tetragonal phase
present after the passage of a hot zone. To confirm this change, the samples were
analysed with XRD, and it can be seen in the graphs enclosed (Fig. 6.11) that the
orthorhombic sample shows very clearly the two peaks at 32 degrees, first a smaller
one and then a larger one, whereas the zoned area gives a diffraction pattern where the
small peak is just showing to the right of the larger peak, which is characteristic of the
tetragonal phase. But the difference between them is not very pronounced probably
due to the uncompleted phase change mentioned before.
Chapter 6 Direct Current Zoning effect
176
a)
b)
Fig. 6.11 XRD patterns of a) tetragonal region in DCZ sample, and b) orthorhombic area in DCZ
sample
Chapter 6 Direct Current Zoning effect
177
0.15 mm 0.06 mm
a)   b)
Fig. 6.12 Microstructures of: a) zoned area (tetragonal) and b) unzoned area (orthorhombic, showing
characteristic twins in the grains)
The absence of microstructural changes in terms of grain size change is attributed to
the high porosity of the samples, since the starting size of the grains was small and
they were not very strongly interconnected. There is one case, however, where the
microstructure showed some grain growth (Fig. 6.13): in this sample the zone
remained immobile for a long period of time after which it melted the bar. The area
next to where the zone melted the specimen shows considerable grain growth
compared to the grain size of the regions in the surroundings. The increase in
temperature of the zone and the time it remained immobile at this high temperature
allowed grain growth to occur.
0.3 mm
Fig. 6.13 Microstructure of zoned area where grain growth occurred
Chapter 6 Direct Current Zoning effect
178
Temperature measurements were carried out to follow the evolution of the
temperature profile along the bar (Fig. 6.14):
0
100
200
300
400
500
600
700
0 10 20 30 40 50
Distance along the sample (mm)
Te
m
pe
ra
tu
re
 (C
) 
0.3 V
0.9 V
1.5 V
1.8 V
2.1 V
2.4 V
3.3 V
3.4 V - 1
3.4 V - 2
Fig. 6.14 Temperature profile of the sample with increasing voltage. Zone became visible at 3.4 V
It is interesting to note that before the zone becomes visible the temperature profile
appears to indicate the nucleation point of the zone. A slight increase in temperature is
observed 10 mm away from the left hand end of the sample (the anode). This region
with higher temperature moves towards the cathode (the right hand side) while the
voltage is increased. Raising the current that passes down the specimen increases the
temperature and the irregularity becomes more pronounced until it becomes visible at
~ 3.4 V.
There is still no consensus on the mechanism leading to zone formation. It is normally
attributed to a fluctuation in the resistivity of the material that increases the
temperature so much that the heat cannot be dissipated fast enough and the
perturbation remains in the form of a zone. The change in the temperature can be
explained as follows:
The temperature of the sample increases uniformly with voltage due to the Joule
effect. The resistivity of the sample changes accordingly with the increase in
temperature (Fig. 6.5) and this change occurs uniformly along the sample. If a small
fluctuation (increase) of the resistivity occurs, the temperature increases at that point.
This fluctuation is thought to occur at a microstructural irregularity such as a crack or
Chapter 6 Direct Current Zoning effect
179
a region of high porosity in the sample. The reduction in cross sectional area increases
the heating due to the Joule effect at that point, giving rise to the formation of the
zone.
When the heat generated by this increase in resistivity is small, it is transferred to the
surroundings by convection, conduction and radiation and the sample returns to its
previous stable state. However, if the fluctuation is so large that the heat generated is
transferred to the surroundings with a net increase in the temperature, the resistivity
gradient formed becomes stable in the sample. The width of the zone generated due to
this resistivity gradient is controlled by the amount of heat dissipated away from the
sample by radiation and along the sample by conduction. The zone moves along the
sample due to the motion of oxygen ions driven across the zone towards the anode by
the potential difference established. These changes in the temperature profile are
shown schematically in Fig. 6.15.
Temperature
Distance along the bar
O2- flux
Cathode Anode
Irregularity forms
and grows
Zone moves
t5
t4
t3
t2
t1
Fig. 6.15 Zone generation process
Other possible reason for the formation of a zone is the presence of impurities or
variations in concentration in the material that increase the local current density and
local heating so that the temperature raises to such levels that a zone is generated. This
has been observed in some of the specimens that had impurities adhered to the
surface, probably from a slight contamination of the furnace. These impurities could
have the effect of hindering the dissipation of heat from the sample, hence creating a
hot region that would give rise to a high resistivity nucleus, triggering the formation of
the zone.
Chapter 6 Direct Current Zoning effect
180
Sometimes the zone was observed to pin at one point in the sample, probably where
there was high porosity, a crack or a high concentration of defects. These irregularities
could anchor the hot spot in the sample. Making a notch of considerable depth in the
bar, the zone was observed to occur at the centre of this notch and remain immobile at
that point. At this moment a second zone formed in the sample. This zone moved
along the sample whereas that at the notch remained stationary. Once the mobile zone
disappeared it was necessary to increase the voltage (and current) again to form
another mobile zone.
Figure 6.16 shows the second occasion when two zones were observed in a sample. In
this case the zone that had initially formed, remained immobile and fainted with time.
When the voltage was increased to make it reappear, two zones were observed to
form: one hotter than the other. The voltage had to be decreased so that the hotter zone
did not melt the specimen and at this moment the zone that was at a lower temperature
disappeared. The zone that formed in the second place began to move along the
sample as it has already been observed in Figs. 6.9 and 6.10.
+ + + + + +
_ _ _ _ _ _
Fig. 6.16 Sequence of photographs showing the formation of two zones in a sample (see text above).
Initial voltage applied: 4.6 V (3.8 A). Voltage was increased to 5.1 V (4.2 A) between the second and
third photograph.Time between photographs: ~ 10 min. Distance between lines on the marker beside
the sample: 5 mm
A possible explanation of the movement of the zone towards the negative electrode in
the DCZ effect is the following:
An increase in temperature increases the oxygen diffusion coefficient, favouring the
flow of oxygen within and out of the sample. This reduction in oxygen content
Chapter 6 Direct Current Zoning effect
181
increases the resistivity of the material, raising the temperature of the sample until at a
certain point a hot zone appears (see Fig. 6.17).
O2-
Zone shifts O
2-
Resistivity
Zone shifts
High resistivity
area
Fig. 6.17 Schematic of movement of zone towards the negative electrode and variation in resistivity
within the sample associated to the presence of a zone
The movement of this hot region towards the cathode occurs because the oxygen ions
are diffusing both out of the sample and within it as a result of the temperature
gradient established and the electric field applied. This movement of ions creates an
oxygen-rich area on the positive side of the zone and an oxygen-depleted area on the
negative side of the zone. The resistivity of YBCO is related to its stoichiometry (see
Fig. 6.4), oxygen-rich samples being less resistive than oxygen-poor samples.
Therefore, the oxygen-rich end has a lower resistivity (and lower temperature) than
the oxygen-depleted end, that will be at a higher temperature as a result of this
increase in resistivity.
The changes in resistivity along the sample in the DCZ effect give rise to the
temperature profile presented in Fig. 6.14. A more detailed study of the heat transfer
processes that are taking place during this effect can be used to explain the drop in
current when the zone appears. These calculations have been introduced in the
program developed to model the DCZ effect (see section 6.4.4) and are presented in
Appendix 2B.
It is thought that the drop in current once the zone has formed (see Fig. 6.8), this is,
the drop in power when the zone forms could be linked to the following variation in
the temperature profile:
Chapter 6 Direct Current Zoning effect
182
Before the zone occurs
After the zone occurs
Distance along
the sample
Temperature
Fig. 6.18 Possible variation in the temperature profile before and after the zone forms
In order to compensate for the local sharp increase in temperature at the zone, the rest
of the sample temperature decreases. This occurs to maintain the amount of power
supplied constant. However, the decrease in resistivity that compensates the local
increase at the zone results in an overall decrease in temperature and a decrease in the
power needed to keep the zone present in the sample. Unfortunately, this could not be
seen clearly from the measurements taken (Fig. 6.14) due to the low accuracy of the
temperature measurement. However, the results from the computer simulation
developed that are going to be discussed in section 6.4.4 confirm this suggestion (see
Fig. 6.31).
6.4.3 DCZ: Analysis
In order to characterise fully the phenomenon of Direct Current Zoning there are two
areas where further study is required:
1) Analysis of the current flow in a mixed ionic conductor, the mechanism by
which a zone originates and the reasons for this instability to occur.
2) Modelling of the steady state motion of a stable zone after it has formed.
The mechanism of nucleation of a zone in the bar is very complex, as it involves an
irreversible transition from an essentially uniform high dissipation state to a
heterogeneous state with lower overall dissipation. The experimental work carried out
in this section of the thesis has concentrated on the analysis of the movement of the
Chapter 6 Direct Current Zoning effect
183
zone. Once the study of this aspect of the Direct Current Zoning effect is completed, it
will be possible to incorporate to it the effects of the microstructure, being able to
carry out a full characterisation of the DCZ effect.
This section is going to study the ionic fluxes established due to the potential
difference applied across the bar as well as due to the concentration gradient present in
the sample (Joule effect). Their change in magnitude depending on the value of the
electric field, E, and the change in concentration are used to explain the observed
stopping of the zone. This is followed by an analysis of the changes in ionic and
electronic conductivity, the number of mobile ions due to each type of flux and their
direct link to the speed at which the zone moves. It will be seen at the end of this
section that the calculations carried out are in very good agreement with the
experimental results.
Therefore, let us consider the zone already present in an YBCO bar. Fig. 6.19 shows
the characteristic temperature profile and the variation in oxygen distribution in the
bar. This oxygen distribution is due to the increase in the value of the diffusion
coefficient, D, and the decrease in the value of the equilibrium oxygen content with
temperature (see Figs. 5.9 and 1.5, respectively). Oxygen ions will tend to diffuse into
or out of the sample until the equilibrium content for each temperature is reached.
The initial temperature and concentration profiles considered for these calculations are
those shown in Fig. 6.19. Looking at the oxygen motion due to chemical diffusion, the
ions move within the sample due to the gradient established and out of the sample due
to the high temperatures reached at the zone. The following paragraphs present a
preliminary calculation of the out-diffusion process to assess its contribution to the
DCZ effect.
Chapter 6 Direct Current Zoning effect
184
Oxygen content
6.95
6.4
Zone
Temperature (ºC)
~700
300
Fig. 6.19 Schematic of temperature profile and distribution of oxygen content in an YBCO bar with a
zone
Table 6.1 shows the values of oxygen diffusion in polycrystalline samples calculated
from a diffusion expression fitted to the results obtained in chapter 5 (section 5.2.2.
See values of the diffusion coefficients used in Table 6.3). If the bar is 2 mm thick and
oxygen ions diffuse from the centre of the sample, the time it will take one oxygen ion
to get from the centre to the surface can be calculated (D ≈ x2/t):
Temperature (ºC) Time
400 16.3 h
500 2.3 h
600 ~ 30 min
700 9.4 min
800 3.5 min
Table 6.1 Time it would take an oxygen ion to diffuse from the centre of an YBCO bar with a zone to
the surface of the bar (diffusion distance, x = 0.1 cm)
It has already been mentioned in this section that the zone was observed to move at
speeds up to 2 mm/min in the samples used. It can be seen in Table 6.1 that oxygen
out-diffusion from the centre of the specimen takes longer to occur than what it takes
the zone to move down the bar. Therefore, diffusion within the sample dominates the
process and only the oxygen atoms that are very close to the surface diffuse out of the
specimen in the timescale of the experiment.
Taking again the 2 mm/min at which the zone moves along the sample, and assuming
that the experiment has been carried out for one minute, the zone will have shifted
Chapter 6 Direct Current Zoning effect
185
2mm. Looking at an oxygen ion in the region where the material is at 800 °C, during
this period of time it will have moved 6.9×10-2 mm. This distance is one order of
magnitude larger than the size of the YBCO unit cell (along the a or b axes), but not
very long on a macroscopic scale. Therefore, during this minute, only the oxygen ions
that are within ~7×10-2 mm from the surface of the bar diffuse to the surrounding
atmosphere. Since the zone is moving continuously and the temperature distribution is
also changing, the ions will be diffusing at the fastest speeds (at the highest
temperatures) for shorter periods of time than that considered above, so it can be
assumed that the effect of out-diffusion is negligible compared to that due to the
electric field. Therefore, the diffusion of oxygen out of the sample is not going to be
considered in the following calculations.
Fig. 6.20 shows an element of the bar considered. The processes taking place can be
expressed mathematically with Eq. (6.12). This equation describes the flux of matter
in a specimen under the effect of a concentration gradient (-D∂C/∂x) and an electric
field (DzCF / kT (-∂φ/∂x)):
j = - D [(∂C/∂x) - zC (F/kT) (∂φ/∂x)]. (6.12)
The electric field drives the oxygen ions towards the anode (jElectrical) and, at the same
time, the change in the temperature distribution (see Fig. 6.19) gives rise to a
redistribution of the oxygen ions in the structure (jFick). These ions move out of the
sample and within it so that the equilibrium oxygen content is achieved. It has already
been shown that the phenomenon of out-diffusion can be ignored for these initial
calculations due to the small effect it would have on the final result. The magnitude of
the two fluxes (jElectrical, jFick) on the motion of oxygen ions within the sample is
discussed below:
Chapter 6 Direct Current Zoning effect
186
jElectrical
jFick
Decreasing
concentration
gradient
C
V
0 x-x Distance
Motion of O2- ions
towards the anode
0 x-x
Out-diffusion
jFick
jElectrical
jFick
jFick
jFick
jElectrical ≠ jFick
jElectrical
a)
b)
c)
Fig. 6.20 a) Schematic of the two types of ion flux taking place in the zone, b) shape of the
concentration gradient established across it, and c) potential difference at the zone
Fig. 6.20 shows the two processes that contribute to the motion of oxygen ions in the
sample: the driving force of the potential gradient present across the bar and the
concentration gradient that results from the non-uniform heating of the sample (see
also Eq. (6.12)). In order to evaluate the importance of each process, the current
generated by the ions moving due to the chemical concentration gradient has been
calculated, and it has been compared with that due to the presence of an electric field.
It has already been shown that the temperature profile varies continuously along the
sample, with a pronounced increase closer to the zone region (Figs. 6.14 and 6.19). In
order to perform the calculations that follow it is necessary to divide this continuous
distribution in steps of a certain length. For this initial calculation, the steps are taken
every 100 °C. Each of these temperature steps has a characteristic oxygen content
from which the value of ∂C/∂x can be calculated. For a 5 cm bar, the approximation of
Chapter 6 Direct Current Zoning effect
187
the temperature distribution used in the following calculations is shown in Fig. 6.21
a).
400
500
600
700
0.1
0.7
2.5
800
0.30.4
Length
(cm)
Temperature (°C)
a)
6.5
6.6
6.7
6.8
2.5
6.9
Length
(cm)
Concentration
b)
ZoneZone
6.4
1
7.0
Fig. 6.21 Schematics showing: a) variation in temperature and step temperature subdivision (in red) of
half a bar of YBCO where there is a zone present (at the centre of the bar), and b) variation in
concentration in the same half bar of YBCO. The line in red shows the concentration steps between
each temperature from which ∂C/∂x is calculated
If the sample has an initial oxygen content of 6.95 and the concentration profile
corresponds to the equilibrium oxygen content of YBCO for each temperature (see
Fig. 6.21 b)), then the change with temperature of the two fluxes expressed in terms of
current is shown in the table below (6.2):
T (°C)  Concentration dx (cm) IFick (A) I E (A)
400 0.95 1 0 4.8×10-5
500 0.88 0.7 6.6×10-6 6.0×10-4
600 0.70 0.4 1.3×10-4 4.2×10-3
700 0.53 0.3 5.4×10-4 2.1×10-2
800 0.40 0.1 3.4×10-3 8.5×10-2
Table 6.2 Variation of current due to chemical diffusion of ions in the concentration gradient
established as shown in Fig. 6.20 due to the formation of a zone in an YBCO bar compared with the
current due to the potential difference across the bar (see Table 6.4)
It can be seen that for the concentration gradients assumed the motion of oxygen ions
is dominated by the electric field applied, with values around one to two orders of
magnitude higher than that due to chemical diffusion (see Fig. 6.22).
Chapter 6 Direct Current Zoning effect
188
1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
400 500 600 700 800
Temperature (C)
Cu
rr
en
t (A
)
j Fick
j E
Fig. 6.22 Variation of currents flowing in the sample due to the voltage applied and chemical diffusion
when a zone is present in an YBCO bar. The flow of ions due to the electric field dominates oxygen
motion in the sample
This flux analysis can be used to explain why a moving zone stops at a certain point
along the bar (see section 6.4.2). There are two situations where the zone has been
observed to stop moving:
In the first case, the zone stops at a point where there is a defect in the sample, such as
a notch or perhaps a change in the microstructure (a region with higher porosity, for
example). This defect increases locally the resistivity of the bar and the zone stops at
that point. Some of these defects have also been observed to be the nucleation points
of immobile zones in a sample (section 6.4.2).
In the second case, the zone moves down the sample and it stops due to an increase in
the current that passes through the bar. This results in an increase of the overall
temperature of the bar and the zone. Considering this case in terms of ion fluxes, when
the current is increased the electrical and chemical diffusion fluxes could become
equal in magnitude and balance each other at the anode end of the zone (see schematic
of zone element in Fig. 6.23). This would result in a decrease in the number of oxygen
ions flowing, and the zone would stop. If, at any moment the accumulation of oxygen
due to the flux from the cathode end into the zone increased the content above the
equilibrium value, this oxygen would leave the sample through the out-diffusion,
although it has already been discussed that for this preliminary study out-diffusion
would be neglected in these calculations.
Chapter 6 Direct Current Zoning effect
189
jElectrical jElectrical
jFick jFick
jFick
jFick
jElectrical = jFick
anodecathode
Fig.6.23 Schematic of fluxes in an element of zone when the zone is stationary
The results obtained using the DCZ model developed that are going to be discussed in
section 6.4.4 show the change in concentration gradient within the sample due to these
two fluxes. The combined effect of the electrical and chemical diffusion (Fick) fluxes
at the cathode end gives rise to a smooth concentration gradient that decreases towards
the high temperature region in the zone. At the anode end, the drift of ions due to the
potential difference across the sample shifts them towards lower temperature regions.
Fick diffusion becomes more sluggish and this results in an accumulation of ions (see
Fig. 6.32). The zone stops when the concentration gradient created due to this higher
concentration of ions at the anode end is such that the driving force on the oxygen ions
due to the potential difference is not strong enough to overcome the concentration
difference.
The main effect in DCZ is the motion of oxygen ions in the sample through the grain
boundaries (faster) and the grains (more slowly) under the effects of the electric field
(electromigration) and the temperature (diffusion), as it is going to be discussed in the
next section.
As it has been mentioned in this section, The sample is assumed to be always at the
equilibrium oxygen content for each temperature. The conductivity of oxygen ions in
YBCO can be calculated using Eq. (6.16) provided the chemical diffusion coefficient
of the oxygen ions in the sample is known. This series of values is taken from the
measurements of diffusion in polycrystalline YBCO samples presented in chapter 5:
Chapter 6 Direct Current Zoning effect
190
Temperature
(°C)
Diffusion coefficient
(cm2/s)
Ionic conductivity
(Ω-1cm-1) (1)
Electronic conductivity
(Ω-1cm-1) (2)
400 1.7×10-7 1.1×10-2 757.9
500 1.2×10-6 6.0×10-2 341
    600 (3) 5.4×10-6 1.9×10-1 153.6
700 1.8×10-5 4.3×10-1 69.2
800 4.7×10-5 7.7×10-1 31.2
Table 6.3 Variation of the diffusion coefficient, ionic and electronic conductivity with temperature
Assuming that the whole cross sectional area contributes to the conduction process,
the current density is equal to:
J = I / cross sectional area of sample = 3 A / 3×10-2 cm2 = 1.0 ×102 A/cm2.
Continuing the calculations, the values for the change of electric field with position in
the zone can be obtained using Ohm’s law. From the value obtained, the ion current
density and the current carried by the oxygen ions as they move along the sample can
be calculated:
Temperature
(°C)
Electric field
(V/cm)
Ionic current density
(A/cm2)
Ionic current
(A)
400 0.2 1.6×10-3 4.9×10-5
500 0.3 2.1×10-2 6.2×10-4
600 0.8 0.1 4.3×10-3
700 1.7 0.7 2.2×10-2
800 3.7 2.9 8.7×10-2
Table 6.4 Variation of the electric field, ionic current density and ionic current with temperature
The decrease in electronic conductivity with increasing ionic conductivity can be seen
very clearly in the graph below:
                                                
(1) Ionic conductivity (Eq. (6.16)) = DC(zF)2 /RT (Ω-1cm-1).
(2)
 The electronic conductivity has been calculated from Eq. (6.1)
(3)
 Temperatures from 400 °C to 800 °C are shown in the calculations, although the temperatures
reached in the zone are 600 °C to 800 °C (in bold). The other temperatures are presented as a reference
only in order to fully appreciate the variation with temperature of the diffusion coefficient, conductivity,
ionic current and number of mobile ions.
Chapter 6 Direct Current Zoning effect
191
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
300 400 500 600 700 800 900
Temperature (C)
Co
n
du
ct
iv
ity
 
(1/
o
hm
 
cm
) electronic conductivity
ionic conductivity
Fig. 6.24 Variation of total conductivity and ionic (O2- ) conductivity of YBa2Cu3O7-δ
Conductivity
Zone
σelectronic
σionic
Fig. 6.25 Schematic of variation in electrical and ionic conductivities
The values of the oxygen ion conductivity of YBCO in this temperature range are
comparable to those of other fast ionic conductors, as it can be seen in Fig. 6.26.
Fig. 6.26 Comparison between ionic conductivities of oxygen ion conductors (from Rickert (1982)).
Crosses show the values of the oxygen ion conductivity in YBCO calculated in this chapter (see Table
6.3)
Chapter 6 Direct Current Zoning effect
192
If the zone (at 800 °C) is taken to move at 1 mm/min (approximately 17 µm/s),
calculating the number of mobile ions in the whole volume of a slab 17 µm (1.7×10-3
cm) wide, the number of ions, Ni, that will have moved to the next 17 µm towards the
anode per unit time is:
Volume of a 17 µm slab = 1.7×10-3 × 3×10-2 = 5.1×10-5 cm3
Number of mobile ions per unit volume, ni = 2.4×1021 ions/cm3 (for 6.4)
Mobile ions in 17µm slab, Ni = 2.4×1021 × 5.1×10-5 = 1.2×1017 ions
~ 17 µm
anodecathode
Zone shift O2-
Fig. 6.27 Schematic of the slab at the zone moving along sample towards the cathode
Calculating the number of oxygen ions flowing due to the potential difference applied
across the YBCO bar:
Since 1A = 1/2e ions/s:
N = I /2e where: e = electronic charge (C)
I = ionic current (A)
N = number of mobile ions per second
Temperature (°C) Mobile ions/s
400 9.7×1013
500 1.3×1014
600 1.2×1015
700 7.6×1016
800 4.0×1017
Table 6.5 Variation in the number of mobile ions with temperature
It can be seen that calculations of mobile ions per unit time at different temperatures
and the value calculated above for the 17 µm slab agree very well.
It has been shown that the combination of chemical diffusion and drift of the oxygen
ions due to the presence of an electric field gives rise to the formation of a zone where
Chapter 6 Direct Current Zoning effect
193
the increase in temperature gives rise to an ionic contribution to the conduction
process that would not take place otherwise. The motion of oxygen ions due to the
potential difference and the concentration gradients present has been related to the
speed at which the zone moves. It has been proposed that the effects of ionic drift and
chemical diffusion on the distribution of oxygen in the sample can, under certain
circumstances, decrease the travelling speed of the zone, sometimes making it stop at
one point. This effect is going to be observed in the theoretical analyisis of DCZ
presented in the next section. This analysis shows the direct link between the fast ionic
conductor behaviour of YBCO and the DCZ effect.
Changes in the electric field along the sample due to alterations in the microstructure
should be monitored to assess the extent of the effect of these variations. Further
insight into the subject of conductivity and oxygen ion diffusion is required to carry
out a more accurate study of the DCZ phenomenon.
6.4.4 Computer model of the Direct Current Zoning
effect
To complete this initial study of the DCZ effect, a computer program was developed
to reproduce the experimental results theoretically. This model was written in
collaboration with Dr. Ilan Blech during his stay in the laboratory as a visiting
researcher, and it applies finite element analysis to calculate the I-V characteristic of
an YBCO bar across which a certain potential difference, V, is established. The value
of V can be kept constant or changed to suit the reproduction of the experimental
results in the same way as they were obtained.
The program calculates the variation in temperature with time resulting from drawing
a current through the sample. As it has been mentioned in the previous sections this
heating has a direct effect on the mobility of oxygen ions in the YBCO bar. The
increase in the value of the oxygen diffusion coefficient with temperature and the
presence of an electric field (electromigration) increase the mobility of mobile species
and their concentration. These two factors determine the electrical properties of the
material.
Chapter 6 Direct Current Zoning effect
194
The variation in the distribution of oxygen in the YBCO bar results in a change in the
value of the electronic and ionic conductivity (Fig. 6.21); this, in turn, alters the
temperature of the sample. At this point, the series of calculations starts again.
The computer model developed calculates the temperature variation with time of the
whole bar as well as the distribution of oxygen across it. This distribution determines
the total resistive behaviour of the sample, allowing one to follow the variation in the
amount of current passing through the sample at each moment in time for the voltage
applied.
Therefore, it is possible to reproduce theoretically the I-V characteristics obtained in
the experiments while being able to observe the changes in temperature and in oxygen
content experienced by the sample (discussed in the previous sections) that give rise to
the formation of a hot zone. Although the model is a highly simplified representation
of the real situation it will be seen that the main characteristic of zone formation, drift
and the tendency for zones to halt at high currents are successfully reproduced by it.
The computer algorithm was written in QBASIC. It calculates the variations in the
value of the temperature, oxygen ion concentration and resistivity of an YBCO long
cylindrical bar divided into a certain number of parallel segments of length seglen, as
shown in the figure:
seglen
length
Area
V
Fig. 6.28 Schematic of YBCO bar with segments
The geometrical factors characteristic of the bar: diameter, length, surface area, cross
sectional area, etc., the diffusion and the conductivity parameters are stored as
constants in the programme. Do and Ae are taken from the expression for diffusion in a
dense polycrystal presented in chapter 5 (section 5.2.5), and the conductivity
Chapter 6 Direct Current Zoning effect
195
parameters are taken from the average of the change in conductivity with temperature
calculated in section 6.2.1 (Eq. (6.1)).
The user then inputs the value of the initial oxygen concentration of the bar and the
initial value of the potential difference established across it. The value of V can be
altered once the program has started to simulate the experimental conditions of
increasing the voltage at a constant rate or reducing it to study the effects of the
variation in voltage on the concentration profile as well as the changes in temperature
associated to it.
The program starts calculating the resistance of the bar for the initial conditions of
temperature and oxygen concentration. It continues evaluating the heat generated in
each of the segments due to the Joule effect and calculating the total heating along the
bar by subtracting the total power from the heat dissipated by radiation, convection
and conduction. This heating results in an increase in temperature of the bar and the
temperature values are updated.
Heating the sample increases the value of the oxygen diffusion coefficient; this means
that some of the oxygen is going to diffuse out of the sample. The new equilibrium
concentration of oxygen in the bar due to the increase in temperature is then
calculated.
Oxygen ions also move in the bar due to the potential difference established across it
(electromigration), so the next step calculates the drift of oxygen ions towards the
positive electrode. A gradient in concentration is generated, and the resulting shift of
ions in this gradient is used to update once more the value of the concentration profile.
The change in the distribution of oxygen ions at the end of each time step alters the
conductivity of the segments, which in turn modifies the generation of heat with time
and current passing through each of them.
Each one of these blocks of calculations (power generated, temperature distribution,
variation in concentration, etc.) is carried out for each segment in the bar so that it is
possible to follow the evolution of the change in temperature and concentration along
the whole sample.
Chapter 6 Direct Current Zoning effect
196
Diffusion and conductivity parameters
Physical constants
Geometrical factors
InitConc, V
Initialising conditions
t = t + delta
Power, radiation,
conduction, heating
Temperature profile update
Cooling due to convection
Temperature profile update
Oxygen concentration update
due to temperature gradient
Drift left, drift right,delconc(i)
Oxygen concentration update
due to drift in electric field
t = mxdelta?
END
Yes
 (for t = to mxdelta)
V, I, seglen, t,
Tc(i), Conc(i),
No
Fig. 6.29 Flowchart of the algorithm developed to model the DCZ effect
The values of current and voltage, the variation in the concentration profile along the
bar and the temperature distribution are exported to a file that can be read in
spreadsheet form so that the theoretical results can be compared with the experimental
data. The listing of this program is included in Appendix 2C.
Chapter 6 Direct Current Zoning effect
197
The results obtained from the theoretical calculations using the model developed are a
good first approximation for the calculation of the I-V characteristic of the sample
during the DCZ effect. This model reproduces quite well the variations in temperature
occurring due to the movement of oxygen ions within the sample as a result of the
Joule effect. The changes in the oxygen content calculated confirm the presence of
oxygen-rich and oxygen-depleted areas at the zone that give rise to its movement
along the sample.
6.4.4.1 Comparison with experimental results
It has been observed that the initial slope of the I-V characteristic of the experimental
runs varies from one sample to another. This is thought to be due to microstructural
aspects such as the density of the polycrystalline bar, grain size, presence of
impurities, etc.
The resistance of the bars used depends on their density. In order to be able to
reproduce the experimental results it is therefore necessary to adjust the value of the
initial resistance to match the theoretical and experimental situations.
Fig. 6.30 shows one of the results obtained from modelling the increase in voltage to
4.5 V. It can be seen that the agreement between the curves is very good. The drop in
current is larger in the theoretical calculation. This could be due to the need for a more
accurate calculation of the resistivity values and the heat dissipation from the sample.
0
0.5
1
1.5
2
2.5
3
0 1 2 3 4 5
Voltage (V)
Cu
rr
en
t (A
)
Fig. 6.30 I-V characteristics of experimental run shown in Fig. 6.8 (in blue) and theoretical model of
the same experiment (in red)
Chapter 6 Direct Current Zoning effect
198
The variation in temperature with time during this theoretical experiment had the
form:
0
100
200
300
400
500
600
700
0 5 10 15 20 25 30 35 40 45 50
Length of bar (mm)
Te
m
pe
ra
tu
re
Zone becomes visible 
above this temperature
Zone movement
Fig. 6.31 Variation in temperature profile with time when a potential difference of 4.5 V is applied to a
polycrystalline YBCO bar. Time between temperature profiles: 30 s
It can be seen how the variation in temperature calculated (Fig. 6.31) is quite similar
to that measured experimentally (see Fig. 6.14). The decrease in the temperature at the
ends of the bar when the zone forms corroborates the calculations and suggestions
presented in section 6.4.2 concerning the variation in electric power supplied: the
formation of the zone is associated to a drop in the power dissipated by the sample.
This is accomodated by a reduction in the temperature of the regions away from the
zone.
The increase in temperature is associated with faster chemical diffusion of the oxygen
ions and their electromigration due to the electric field. This modifies the initial
oxygen distribution in the sample as shown in Fig. 6.32. There is an accumulation of
oxygen ions in the region immediately to the right of the zone due to the combined
effects of the electromigration and Fick diffusion of ions. This region is at a lower
temperature, and chemical diffusion occurs more slowly. This creates an oxygen-rich
area with a low resistivity, whereas to the left of the zone the opposite effects of the
drift due to the potential difference and chemical diffusion to reduce the concentration
Chapter 6 Direct Current Zoning effect
199
gradient and away from the zone (see section 6.4.3 and Fig. 6.22), create an oxygen
depleted region where there is a less abrupt concentration gradient.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 5 10 15 20 25 30 35 40 45 50
Length of bar (mm)
Co
n
ce
n
tr
at
io
n
0
100
200
300
400
500
600
700
Te
m
pe
ra
tu
re
 (C
)
Higher resistivity
Zone movement
Fig. 6.32 Variation in the distribution of oxygen in the sample when a potential difference of 4.5 V is
applied to a polycrystalline YBCO bar. Time between lines: 30 s
This oxygen depleted region has a high resistivity and the slope of the concentration
gradient within it determines whether the zone remains mobile or immobile in the
sample. If the gradient is small, the electric field established across the sample is
going to be strong enough to drive oxygen ions across the gradient towards the anode.
In this case the zone moves. However, if the gradient is large (due to an increase in the
power supplied to the sample), the flux due to the electric field may not be strong
enough to overcome the difference in concentration and the ions will not move
towards the anode
It has been observed during the theoretical experiments that in certain cases the zone
did not move. This is probably due to the effect mentioned in section 6.4.3 where the
effects of chemical diffusion and ionic drift due to the potential difference give rise to
the formation of a sharp concentration gradient that the ionic drift cannot overcome.
The accumulation of ions at the cathode end of the zone in the example presented in
Fig. 6.32 is quite large; this gives rise to a steep concentration gradient. The zone was
observed to move very slowly in this case.
Chapter 6 Direct Current Zoning effect
200
The development of this model has for the first time given the opportunity to carry out
a calculation of the effects of the different processes that combine to produce the
Direct Current Zoning effect. There is still need for more improvement of the
modelling, evaluating the dependence of the conductivity on the density of the bar, as
well as the effects of other microstructural features on this value. For the purposes of a
simplified calculation it was assumed that the oxygen content of the sample segments
was always the equilibrium content for the temperature of the segment. It has been
mentioned in section 6.4.2 that the speed at which the ions move due to the change in
temperature is much slower than the speed at which the zone moves. This would
result in smaller changes in concentration than those calculated with this model.
Further development of the program to incorporate a more accurate calculation of the
chemical diffusion in the sample was beyond the scope of this thesis. For the
following studies of the DCZ effect it will be very interesting to introduce such
modification in the program in order to be able to calculate more accurately the
contribution of chemical diffusion in the process.
The study of diffusion in polycrystalline samples presented in chapter 5 has shown
that the microstructure has a very important effect on the chemical diffusion process.
In order to be able to model the DCZ effect more accurately it would be necessary to
know the dependence of the value of the oxygen diffusion coefficient on the density of
the samples used.
6.4.5 Summary
This section has presented a more complete analysis of the Direct Current Zoning
effect than those carried out so far. It has corroborated the suggestions made by other
groups concerning the motion of oxygen ions within the sample that result in the
formation of a zone when a potential difference is applied across a polycrystalline bar
of YBCO.
The formation of a zone in a sample is associated to a reduction in the power supplied
to it. This has been explained in terms of the variation in temperature of the sample
before and after the zone appeared.
Chapter 6 Direct Current Zoning effect
201
It has been shown that the chemical diffusion and drift of oxygen ions are directly
related to the speed at which the zone moves along the sample, and it has been
possible to suggest a reason for its remaining stationary considering the magnitude
and direction of these ionic fluxes.
The development of a computer model to reproduce this phenomenon has
corroborated the suggestions made concerning the changes in temperature within the
sample before and after the zone appears and it has also calculated the change in the
oxygen distribution within the sample once the zone is present and moving. This
evolution of the concentration gradients with time has corroborated the suggestion
made relating chemical diffusion and drift of the oxygen ions in the presence of an
electric field that cause the zone to stop at a certain point.
6.5 Conclusions
The oxygen non-stoichiometry of YBCO and its behaviour as a solid mixed conductor
at high temperatures are thought to be the reason behind what is known as the Direct
Current Zoning effect. This effect consists on the formation of a hot region in a bar of
polycrystalline YBCO when a DC voltage is applied across it.
This chapter has presented an initial study of the processes occurring during the
formation of this hot zone, combining the increase in temperature due to the Joule
effect with the variation in the conductivity of the sample and chemical diffusion of
oxygen within it because of the increase in temperature. This chapter has developed a
more thorough study than any of those carried out to date, and it presents an analysis
of the motion of oxygen ions due to electromigration and chemical diffusion that
supports the proposed mechanism for the motion of the zone.
A more exhaustive analysis of the conditions in which the zone forms will help to
characterise it fully (zone nucleation, speed, size and temperature), such as more
accurate measurements of the temperature of the bar (with an infrared pyrometer, for
example) and the effects of sample microstructure on ion conduction and chemical
diffusion.
Chapter 6 Direct Current Zoning effect
202
It is still unknown why the zone forms although it is thought that it occurs at some
point along the sample where there is a higher degree of porosity or to the presence of
impurities that increase the temperature of the sample locally, triggering the formation
of the zone but allowing it to move away from this point towards the negative
electrode.
To complete this preliminary study of the DCZ effect a computer model has been
developed taking into account the three processes that take part in the formation and
motion of the hot zone along the bar: temperature change (Joule effect), increase in
the oxygen conductivity and  variation in the value of oxygen chemical diffusion in
the sample. The theoretical I-V characteristics obtained match quite well the
experimental results. The model calculates the variation in temperature with time
during the theoretical reproduction of the experiments as well as the variation in the
distribution of oxygen due to the increase in the value of the chemical diffusion and
the electromigration of the oxygen ions due to the potential difference established
across the sample. This model has given the opportunity to study for the first time all
the processes that take part simultaneously in the Direct Current Zoning effect. The
results from the computer programme have corroborated the suggestions made during
the experimental stage of the study concerning the variation in temperature and
oxygen content in the sample. The program has also given an insight as to why the
zone sometimes stops at some point in the sample.
Chapter 7 Conclusions and further work
203
Chapter 7
Conclusions and further work
7.1 Conclusions
This thesis has presented a study of the phenomenon of oxygen diffusion in
YBa2Cu3O7-δ (YBCO). Analysis of the thermogravimetric results from experimental
isothermal oxygenations of YBCO powder has yielded an Arrhenius expression for
the variation of the oxygen diffusion coefficient with temperature in this material.
This study of the diffusion process includes the development of a computer model for
oxygen diffusion in YBCO particles.
YBCO was one of the first high temperature superconductors discovered, and its
superconducting properties are strongly dependent on its oxygen stoichiometry. A
large amount of work has been done on the variation of this stoichiometry and its
effect on the superconducting properties of the material. Since the critical temperature
(Tc) of YBCO increases with oxygen content (Fig. 1.4), processing of this material for
optimum superconducting properties concentrates on achieving the highest oxygen
content possible.
However, the manufacturing of YBCO samples for superconducting applications
requires very high temperatures, and in this region the equilibrium oxygen content is
low (Fig. 1.5). In order to increase the oxygen content of the material it is necessary to
include an oxygenation step at lower temperatures.
For a given sample size, the duration of this step is determined by the value of the
oxygen diffusion coefficient. This is crucial to optimise the superconducting
properties and it is one of the reasons why such a large amount of work has been
dedicated to the calculation of the value of the diffusion coefficient of oxygen in
YBCO.
Chapter 7 Conclusions and further work
204
However, in spite of all the work done, the results published in the literature are quite
scattered (see Fig. 7.1). The present study has tried to reconcile the data available,
taking into account the factors that may have affected these calculations of the oxygen
diffusion coefficient (chapter 3).
The first aspect to consider is the fact that depending on the technique used, the
diffusion coefficient measured is different. Chemical diffusion, self diffusion and
tracer diffusion are the three types of coefficients measured. The driving force for
diffusion is different in each case, and in order to compare them it is necessary to take
into account the factors that relate each of them with the rest. These factors have been
reviewed in chapter 2.
The second aspect concerning the analysis of diffusion results is the effect of the
sample microstructure on the results obtained with each measuring technique. Tracer
diffusion and internal friction methods are not affected by the type of sample used,
although there are some difficulties associated to tracer diffusion measurements in the
c direction due to the reduced size of the single crystals along this axis. However,
resistance measurements and thermogravimetry, for example, are highly sensitive to
the sample microstructure. The effect of microstructural features becomes more
important when using polycrystalline specimens. The microstructural features of these
samples introduce a series of factors that in most cases have unknown effects on the
diffusion process. The presence of grain boundaries introduces the possibility of grain
boundary diffusion occurring if the experiment is carried out at low temperature,
although the range of temperatures within which it is predominant over bulk diffusion
is not yet known.
Other microstructural features of polycrystalline samples, such as the presence of open
or closed porosity may also alter the diffusion results obtained. If the sample has a
high degree of porosity, oxygen will be able to reach the most inner regions of the
sample without any opposition (open porosity). This should be taken into account
when the diffusion distance is established in the analysis of diffusion data. In this case
the diffusion length will be one half of the distance between interconnected pores (if
the effect of grain boundaries is not taken into account). In the case of samples with
closed porosity, the diffusion length should be one half of the dimensions of the
Chapter 7 Conclusions and further work
205
sample (assuming again that grain boundary diffusion is not active in the temperature
range considered). The estimation of the diffusion distance is one of the main sources
of error in diffusion studies.
The effect of secondary phases and impurities introduced during the processing of
polycrystalline and single crystal specimens should also be taken into account when
analysing diffusion results.
After the analysis of the literature data it seemed clear that the use of a polycrystalline
sample to study the oxygen diffusion coefficient in YBCO would not help in the
clarification of the scattered results. The use of powder samples, however, would
eliminate the effects mentioned above from the analysis of the diffusion process, so it
was decided to carry out a series of oxygenations of powder samples to determine a
value of the diffusion coefficient that was not affected by microstructural defects.
The study presented in this thesis has been divided into two sections:
The first section consists in the analysis of isothermal oxygenations of powder
specimens using thermogravimetry. This technique allows the calculation of the value
of the chemical diffusion coefficient, D~ , because the weight changes of the powder
samples are directly related to the change in oxygen stoichiometry of the material. The
analysis of the data was made fitting the results of two different solutions of Fick’s
second law assuming that D~  was independent of the concentration of oxygen in the
sample. An Arrhenius expression relating the change in the value of the diffusion
coefficient with temperature was obtained with the calculated values of the chemical
diffusion coefficient at different temperatures.
It is possible to calculate the length of the oxygenation step needed to obtain an
YBCO sample with high oxygen content (above 6.9 for optimum superconducting
properties) knowing the value of the diffusion coefficient at each temperature. The
optimisation of the time it takes to oxygenate YBCO specimens has been a matter of
concern for large scale applications. The shorter diffusion distances of thin films mean
that full oxygenation can be completed during manufacturing without increasing
processing time considerably. However, due to the larger diffusion distances of bulk
samples, oxygenation times can increase to weeks, months or even years. The increase
Chapter 7 Conclusions and further work
206
in processing time due to the long oxygenation steps needed reduces the practical
applications of this type of samples.
The problem of increased oxygenation times has been solved in the second section of
this diffusion study:
In the course of the oxygenation experiments, it was observed that the weight gain
traces obtained during non-isothermal oxygenations (heating slowly to high
temperatures) had a very characteristic shape (Fig. 5.26) that could be divided in two
regions: an initial part where oxygen was absorbed by the specimen at low
temperatures (non-equilibrium regime) up to a certain temperature, Tmax, and a second
part at higher temperatures where the weight of the sample decreased following the
equilibrium oxygen content for each temperature (quasi-equilibrium regime). If the
sample was cooled at the same rate as it had been heated up, the weight change of the
sample followed this quasi-equilibrium trace until the temperature decreased to Tmax.
At this temperature D~  becomes too slow to oxygenate the whole sample in the time
set by the cooling rate. Provided the temperature of the specimen is always kept above
the Tmax of the rate at which it is cooled, its oxygen content will be distributed
uniformly in the sample and will always be at the equilibrium value for each
temperature.
Non-isothermal oxygenations of powder samples at different rates showed that Tmax
decreased with decreasing heating rate. The possibility of cooling the samples
following the equilibrium oxygen content trace slowing the cooling rate with
decreasing temperature (thus decreasing the value of Tmax and broadening the range of
temperatures at which the sample is at equilibrium) offered an alternative way of
oxygenating YBCO samples. Cooling from a high temperature at a series of cooling
rates (faster at high temperatures and more slowly at low temperatures), as shown in
chapter 5, allows a reduction of oxygenation times of up to 12 hours for powder
particles ≈ 5 µm in diameter.
It has already been pointed out that the oxygenation of small powder particles or thin
films is not a problem because of their small dimensions. The situation becomes more
problematic when larger samples need to be oxygenated. To be able to apply the
quasi-equilibrium procedure developed to the oxygenation of bulk samples for large
Chapter 7 Conclusions and further work
207
scale superconducting applications, it was necessary to draw up the relationship
between cooling rates and Tmax for larger particle sizes (as the diffusion length
increases cooling rates become slower and loger times are required for full
oxygenation to be achieved). Carrying out experimentally the non-isothermal
oxygenations needed to obtain this data would have taken a very long time, so the
diffusion programme developed was used instead to calculate theoretically the results
from non-isothermal oxygenations of larger samples allowing one to obtain the
relationships between cooling rates and maximum temperatures for a series of sample
sizes. It has been shown in chapter 5 that these quasi-equilibrium cooling procedures
offer a fast alternative to isothermal oxygenation of bulk samples.
The diffusion programme developed has proved to be highly versatile. It reproduces
fractional weight changes under isothermal and non-isothermal conditions, and it
gives further insight to the oxygen diffusion process, calculating the evolution of the
concentration profile during these oxygenations. This is particularly interesting as it
could be of help when studying the superconducting behaviour of the samples. The
programme can also be used to predict oxygenation times of larger samples without
having to carry out trial-and-error experiments, reducing considerably research times.
This diffusion programme has also been applied to the study of the evolution of
oxygenations under non-isothermal conditions. It has been particularly useful in
calculating oxygenations of large samples to develop the quasi-equilibrium cooling
procedure proposed in this thesis. This novel oxygenation step can be introduced at
the end of the processing of YBCO bulk samples for large scale applications reducing
oxygenation times and producing samples with the required oxygen contents.
The following chapter (chapter 6) presents a preliminary analysis of the Direct Current
Zoning (DCZ) effect. The generation of a hot zone in a bar of polycrystalline
superconductor when a certain potential difference is applied across it is directly
linked to the motion of oxygen ions in the material due to the effects of
electromigration and diffusion in a temperature and concentration gradient. A small
number of groups have studied this effect. The work presented in this thesis includes a
more detailed analysis of the DCZ effect, studying the variation in electronic and ionic
conductivity and how these changes give rise to the presence of a hot mobile zone in
Chapter 7 Conclusions and further work
208
the bar. The work done has concentrated on the stability and motion of the zone along
the polycrystalline sample. It has been observed that in some samples the zone
remained immobile, and reasons for this behaviour have been suggested.
The study of this phenomenon includes the development of a computer model that
reproduces the DCZ effect. This is the first model that has been written to study the
effect theoretically, and it takes into account the combination between
electromigration and diffusion processes. The results obtained using this computer
programme have corroborated the proposed variation in temperature and oxygen
distribution along the polycrystalline sample. The model has also supported the
suggestions made concerning the stopping of the zone as it is moving along the bar.
7.2 Further work
With respect to the chemical diffusion study, the analysis of the data in the literature
has brought up a large number of questions that have not been looked into in detail
and that are very important if the diffusion of oxygen in YBCO is to be characterised
fully.
Most of these are related to the microstructure of the samples used. Oxygen diffusion
in polycrystalline materials has been studied undifferentiated from diffusion in
powders or single crystals, and this seems to be associating an highly complex
diffusion behaviour to the same group as single diffusion events.
The effects that grain boundary diffusion have on the diffusion process and the
temperature range over which this short-circuit diffusion occurs have not been studied,
and it would be very interesting to be able to study the possibility of combining the
effects of fast grain boundary and bulk diffusion cycling the temperature to design an
alternative oxygenation procedure.
The presence of open and closed porosity has also been mentioned. It would be very
interesting to carry out a thorough study of the effects of density on the value of the
diffusion coefficient. But great care will have to be taken in the correct
characterisation of the sample, taking into account the grain size and the degree of
open and closed porosity.
Chapter 7 Conclusions and further work
209
The possibility of blocking the diffusion process in certain regions of the sample due
to misaligned grains (c-axis oriented) combined with grain boundary diffusion seems
to be a matter that will only be possible to analyse theoretically. This brings up the
need to develop a computer model that considers the effects of grain boundary
diffusion and bulk diffusion in their respective temperature ranges as well as
introducing changes in grain size. This is a challenging project that could only be
carried out after experimental studies have been performed clarifying the points
mentioned above.
With respect to the Direct Current Zoning effect, once the matters of diffusion in a
polycrystalline material have been dealt with, it will be possible to carry out a
complete study of the phenomenon. It would also be necessary to clarify the electrical
behaviour of YBCO at high temperatures (> 300 K) in polycrystalline samples of
different densities. The continuation of the DCZ project would study the reasons
behind the generation of the zone, its motion down the polycrystalline bar and why, on
occasions, the zone remains immobile at one point in the sample.
Meanwhile, however, it would be very interesting to use the relationship between the
oxygen diffusion and oxygen ion conductivity (Nernst-Einstein equation) to control
the oxygen content of YBCO samples applying a potential difference across them
monitoring the temperature of the sample. This could be the topic of an interesting
study that would assess the potential use of electrochemistry as a faster means of
controlling the oxygen stoichiometry of superconducting samples.
Appendix 1
210
APPENDIX 1
1A Listing of the computer model developed to study oxygen
diffusion in a spherical powder particle
The following programme calculates the variation in weight change and the change in
the oxygen content distribution in a spherical particle. The results obtained with this
programme have been discussed in chapter 5 (see flowchart of the algorithm used in
section 4.2.4).
CLS  ’’ Initializing the screen and clearing
   DEFDBL A-H      ’’ Double precision numbers
   DEFDBL O-V      ’’ Double precision numbers
   DEFINT I-N      ’’ Integers
   DEFSNG W-Z  ’’ Single precision numbers
’’Variables:
   DIM C1(40, 40), V(40, 40), C2(40, 40), Se(40, 40), re(40, 40)
   DIM radio(40), alfa(40), Vr(40), suma(40), grade(40, 40),
gradi(40, 40)
   DIM WeightT(5000), Tst(10), Ten(10), tstart(10), tend(10)
’’ Diffusion parameters:
   B = 1.38066E-23                 ’’Boltzmann constant
   si = 8.617E-05 / 1.38066E-23     ’’Ratio eV/J
   d0 = .033                    ’’Diffusion pre-exponent cm2/s
   q = 1.24     ’’Activation energy for diffusion
in eV
   q = q / si                      ’’Activation energy in J
’’ Geometrical factors:
   Pi = 4 * ATN(1): rd = Pi / 180
   r = .00021                      ’’ Radius of particle in cm
   nr = 20: na = 20 ’’ Number of divisions of sphere:
’’ nr = number of slices in semisphere
’’ na = number of rings in each slice. 
   dh = r / nr    '' Divisi¢n of radius to
calculate the '' sections of
the semisphere
'' Calculation of the angles that limit each section of the sphere
considered: FOR kr = 1 TO nr
           cose = (kr - 1) * dh / r:  '' Sine of the angle
           GOSUB 1000:                '' To cos-1 subroutine:
           alfa(kr) = Pi / 2 - angu:
NEXT kr
alfa(nr + 1) = Pi / 2
Vu = 2 / 3 * Pi * r ^ 3: Su = 2 * Pi * r * dh
FOR kr = 1 TO nr
     F = SIN(alfa(kr + 1)) - SIN(alfa(kr))
     F = F - (SIN(alfa(kr + 1))) ^ 3 / 3 + (SIN(alfa(kr))) ^ 3 / 3
     Vr(kr) = Pi * r ^ 3 * F
     radio(kr) = (Vr(kr) / Pi / dh) ^ .5
NEXT kr
Appendix 1
211
’’ Data input:
        PRINT
        PRINT "Enter initial concentration, x, (YBCO(6+x))"; : INPUT
conc0
        PRINT
        PRINT "You have the choice of doing:"
        PRINT "a) a study of the change in concentration with
increasing T"
        PRINT "b) a study of the change in concentration at constant
T"
        PRINT "c) a combination of the above (ramp + isothermal)"
        PRINT
        PRINT "Data saved in file (enter filename - default:
matrix.dat)"; : INPUT file$
        PRINT "Concentration data saved in file (enter filename -
default: h:\conc.dat)"; : INPUT concfile$
        IF file$ = "" THEN file$ = "matrix.dat"
        IF concfile$ = "" THEN concfile$ = "h:\conc.dat"
        PRINT
        PRINT "Enter number of heat treatment cycles"; : INPUT Ncyc
        PRINT "Enter initial T"; : INPUT T0
        initialt = 0
        Ten(0) = T0
OPEN concfile$ FOR OUTPUT AS #3
        PRINT #3, "Initial Concentration=" + STR$(conc0)
        FOR m = 1 TO Ncyc
                PRINT "Step no. "; m
                PRINT "Enter delta time"; : INPUT dtime
                PRINT "Enter final T"; : INPUT T1
        PRINT #3, "Step no.=" + STR$(m) + CHR$(9) + "Delta
time=" + STR$(dtime) + CHR$(9) + "Final T=" + STR$(T1)
                Tst(m) = Ten(m - 1)
                Ten(m) = T1
                tstart(m) = initialt
                initialt = initialt + dtime
                tend(m) = initialt
        NEXT m
        PRINT #3, "ka"
        PRINT #3,
        FOR ka = 1 TO na
                PRINT #3, STR$(ka)
        NEXT ka
CLOSE #3
’’ Calculation of the total heat treatment time:
numheaders = Ncyc + 1
tiTot = tend(Ncyc) * 60
'' “Slice” section. Calculation of the change in concentration in
each slice
'' considered:
FOR kr = 1 TO nr
'' Geometric factors:
        dr = radio(kr) / na
'' Calculation of the external radius of each ring (re), volume of
each
'' ring (V), external surface of each ring (SE) and initial
concentration
'' (C1):
      FOR ka = 1 TO na
             V(kr, ka) = Vr(kr) * (2 * (na - ka) + 1) / (na ^ 2)
Appendix 1
212
             re(kr, ka) = radio(kr) - (ka - 1) * dr
             Se(kr, ka) = 2 * Pi * re(kr, ka) * dh
             C1(kr, ka) = conc0
      NEXT ka
NEXT kr
ti = 0                  ’’ Initial time in seconds
’’ Calculation of the total time of the heat treatment:
dti = .1                ’’ Time interval chosen (in s)
kescribe = 1   ’’ Each kescribe iterations C1 is displayed
on screen
’’ Main calculation:
FOR kr = 1 TO nr
      kr = 1
      dr = radio(kr) / na
      SCREEN 12                      ’’ Graphic screen 640x480
      CLS
      GOSUB 2000: GOSUB 3000         ’’ Draw grids
’’ Time count starts here:
FOR ti = 0 TO tiTot STEP dti
              Minconc = 0
              Maxconc = .987#
              FOR m = 1 TO Ncyc
    IF ti < tend(m) * 60 AND ti >= tstart(m) * 60 THEN
stepno = m
              NEXT m
        deltat = ti - tstart(stepno) * 60
        tinterval = tend(stepno) * 60 - tstart(stepno) * 60
        tempint = Ten(stepno) - Tst(stepno)
        T = Tst(stepno) + tempint / tinterval * deltat
’’ Calculation of diffusion coefficient:
              D = d0 * EXP(-q / B / (T + 273))
              SELECT CASE T
                        CASE IS < 396
                                Equconc = -.0000705 * T + 1
CASE 396 TO 535
                                Equconc = -.0006 * T + 1.21
                        CASE 535 TO 783
                                Equconc = -.0015 * T + 1.692
                        CASE IS > 783
                                Equconc = -.001 * T + 1.3
              END SELECT
’’ Concentration of outer ring(ka=1):
              C2(kr, 1) = Equconc
              IF C2(kr, 1) > Maxconc THEN C2(kr, 1) = Maxconc
              IF C2(kr, 1) < Minconc THEN C2(kr, 1) = Minconc
’’ Concentration of the rest of the rings:
              FOR ka = 2 TO na - 1
     grade(kr, ka) = (C1(kr, ka - 1) - C1(kr, ka)) / dr
     gradi(kr, ka) = (C1(kr, ka) - C1(kr, ka + 1)) / dr
     C2(kr, ka) = C1(kr, ka) + Se(kr, ka) * D * dti *
grade(kr, ka) / V(kr, ka)
     C2(kr, ka) = C2(kr, ka) - Se(kr, ka + 1) * D * dti
* gradi(kr, ka) / V(kr, ka)
                    IF C2(kr, ka) > Maxconc THEN C2(kr, ka) = Maxconc
                    IF C2(kr, ka) < Minconc THEN C2(kr, ka) = Minconc
              NEXT ka
Appendix 1
213
’’ Concentration of innermost ring (ka=na):
              FOR ka = na TO na
grade(kr, ka) = (C1(kr, ka - 1) - C1(kr, ka)) / dr
                  gradi(kr, ka) = 0
C2(kr, ka) = C1(kr, ka) + Se(kr, ka) * D * dti *
grade(kr, ka) / V(kr, ka)
C2(kr, ka) = C2(kr, ka) - Se(kr, ka + 1) * D * dti
* gradi(kr, ka) / V(kr, ka)
                  IF C2(kr, ka) > Maxconc THEN C2(kr, ka) = Maxconc
                  IF C2(kr, ka) < Minconc THEN C2(kr, ka) = Minconc
              NEXT ka
’’ Change C1 to C2 so that this value can be carried into
the next ’’ iteration:
              FOR ka = 1 TO na
                      C1(kr, ka) = C2(kr, ka)
              NEXT ka
              k$ = INKEY$             ’’waiting for trapping
keystrokes
              IF LEN(k$) = 1 THEN
                      IF LEN(k$) = 1 AND ASC(k$) = 27 THEN
                              END                      ’’ Exit
                      END IF
                      IF LEN(k$) = 1 AND k$ = "c" OR k$ = "C" THEN
                            GOSUB 4000: GOSUB 3000     ’’ Clear
screen
                      END IF
                      IF LEN(k$) = 1 AND ASC(k$) = 13 THEN
                            INPUT c        ’’ Stop until key is
pressed
                      END IF
              END IF
’’ Print label:
      LOCATE 1, 12
        
PRINT USING "Temperature= ####.#"; T;
      PRINT USING "oC   time=##"; FIX(ti / 3600);
      PRINT " hours ";
      PRINT USING "##.#"; (ti / 60) - 60 * FIX(ti / 3600);
      PRINT " min";
      LOCATE 2, 15: PRINT "Exit press ESC";
      LOCATE 2, 34: PRINT "Clear the top screen press C";
’’ (Draw line every kescribe iteration)
IF INT(ti/dti)/INT(kescribe/(dti)) =
INT(INT(ti/dti)/INT(kescribe/(dti))) THEN    
GOSUB 5000              ’’ Open a data file
            COLOR 2
            LOCATE 23, 1
            IF T > 1000 THEN
                    COLOR 31
  PRINT "The temperature is too close to the
melting point"
                    END
                  END IF
’’ Plot concentration profile:
FOR ka = 1 TO na - 1
LINE (offsetx + ka * sx, offsety+900 * C1(kr,
ka) * sy)-(offsetx + (ka + 1) * sx, offsety+900 *
C1(kr, ka + 1) * sy)
Appendix 1
214
      NEXT ka
COLOR 15
’’ Calculation of the weight change of the sphere with
oxygenation:
      n = INT(ti / kescribe)
      eightA = dr * C1(kr, 1)
      FOR ka = 2 TO na
      WeightA = WeightA + dr * (C1(kr, ka - 1) +
C1(kr, ka))/2
      NEXT ka
      WeightT(n) = WeightA
      WeightT(0) = 0
’’ Plot Mass-t curve
           COLOR 2
            IF n = 0 THEN
LINE (x01 + 0 * dx1, y02 - WeightT(0) / dy1)-
(x01 + 0 * dx1, y02 - WeightT(0) / dy1)
            END IF
            IF n = 1 OR n > 1 THEN
LINE (x01 + (n - 1) * dx1, y02 - WeightT(n -
1) / dy1)-(x01 + (n) * dx1, y02 - WeightT(n) / dy1)
            END IF
            GOSUB 6000       ’’ Save data in file
            GOSUB 7000       ’’ Close the data file
            COLOR 15
 ’’ Export data:
            OPEN concfile$ FOR INPUT AS #2
          OPEN "C:\temp\conctmp.dat" FOR OUTPUT AS #3
’’ Read headers and write
out again:
            FOR ix = 1 TO numheaders
                  LINE INPUT #2, inline$
                  PRINT #3, inline$
            NEXT ix
            LINE INPUT #2, inline$
            PRINT #3, inline$ + CHR$(9) + STR$(ti)
            LINE INPUT #2, inline$
            PRINT #3, inline$ + CHR$(9) + STR$(T)
            FOR ka = 1 TO na
                 LINE INPUT #2, inline$
                  PRINT #3, inline$ + CHR$(9) + STR$(C1(kr,
ka))
            NEXT
            CLOSE #2
CLOSE #3
            KILL concfile$
            SHELL "copy c:\temp\conctmp.dat " + concfile$
      END IF
NEXT ti
NEXT kr
END
1000 ’’ cos-1
 subroutine
     IF ABS(cose) > 10 ^ (-9) THEN 1010
     angu = Pi / 2: RETURN
1010 IF 1 - ABS(cose) > 10 ^ (-6) THEN 1030
     angu = Pi / 2 * (1 - SGN(cose)): RETURN
1030 angu = ATN((1 - cose * cose) ^ .5 / cose)
     IF angu < 0 THEN angu = Pi + angu: RETURN
2000 ’’ Sub to draw axes for Mass-t, labels etc.
CLS
sy = -.27               ’’ Scale factor y direction
sx = 500 / na           ’’ Scale factor x direction
Appendix 1
215
offsety = 280
offsetx = 35
COLOR 15
dy1 = 6.5 * 10 ^ -7#
dx1 = 3
x01 = 150: x02 = 450    ’’ Scale factor x direction
y01 = 300: y02 = 475    ’’ Scale factor y direction
LINE (x01, y01)-(x01, y02)
LINE (x01, y02)-(x02, y02)
RETURN
3000 ’’ Draw grid
COLOR 15
FOR tt = 0 TO 900 STEP 90
       LINE (offsetx + sx, offsety + tt * sy)-(offsetx + (na + 1) *
sx, offsety + tt * sy)
NEXT tt
FOR j = offsetx + sx TO offsetx + (na + 1) * sx + 1 STEP na / 10 * sx
       LINE (j, offsety)-(j, offsety + 900 * sy)
NEXT j
SCREEN 12
COLOR 15
LOCATE 10, 1: PRINT "   x "
LOCATE 3, 3: PRINT " 7.0 "
LOCATE 6, 3: PRINT " 6.8 "
LOCATE 9, 3: PRINT " 6.6 "
LOCATE 12, 3: PRINT " 6.4 "
LOCATE 15, 3: PRINT " 6.2 "
LOCATE 18, 3: PRINT " 6.0 "
LOCATE 20, 37: PRINT "radius (cm)";
LOCATE 19, 7: PRINT USING "#.####"; (re(kr, 1));
LOCATE 19, 38: PRINT USING "#.####"; (re(kr, 1) / 2);
LOCATE 19, 70: PRINT (0);
RETURN
4000 ’’ Clear the top of the screen
FOR u = 1 TO 30
        LOCATE u, 1: PRINT TAB(78);
NEXT u
RETURN
5000
OPEN "h:\" + file$ FOR APPEND AS #1
RETURN
6000
WRITE #1, ti, WeightT(n) * 10 ^ 4, T
RETURN
7000
CLOSE #1
RETURN
Appendix 2
216
APPENDIX 2
2A Preparation of mixture for polymer processing of YBCO
tapes
The mixture of YBCO powder, plasticiser and binder used had the following
proportions;
110 g of YBa2Cu3O7-δ
5.6 g of PBV
6.7 g of cyclohexanone
2 b of dibutylphthalate
The components were mixed together and worked through a rolling mill until the
tapes had a uniform appearance and strong consistency.
2B Calculation of heat dissipation in the YBCO bar used in the
computer model developed
During the process of formation and movement of the zone along the bar, three types
of heat transfer take place that determine the final temperature of the sample for a
given power VI supplied to the sample: radiation, convection and conduction. The
program developed to model the motion of the zone in the sample (see section 2C
below) takes them into account in order to calculate the variation in temperature along
the specimen. They are described by the following equations:
Radiation: QR = Aεσ(T4 - T4ambient) 
where:  ε = emissivity of the material (0.915)
 σ = Stefan-Bolzmann constant = 5.67×10-8 W/m2K4
 A = area through which the heat is dissipated
 T = temperature of the surface (in K)
 Tambient = ambient (room) temperature (in K)
Conduction: QC = Ack( T - T initial)/x 
           where: k = thermal conductivity of the material (W/m2)
Ac = cross sectional area through which the heat is dissipated
x = distance over which the temperature difference is
measured.
Appendix 2
217
Convection: Q = Ash( Tsample - T atmosphere) 
           where: h = convective heat transfer coefficient (W/m2 °C)
As = surface area through which the heat is dissipated
Tambient = ambient (room) temperature (in K)
The latter form of heat dissipation in this case is natural convection of air. In this
situation the transfer of heat occurs at a low rate since the natural convection currents
move rather slowly. The flow of air around the YBCO bar is assumed to be non-
turbulent near the surface. The thermal conductivity of a horizontal pipe for streamline
flow of air takes the form (Coulson and Richardson (1977)):
h = 1.18 (∆T/d)1/4.
It has been assumed in the calculations below that the thermal conductivity varies in
0.0001 every time the temperature of the sample increases 1 °C.
The calculation of the increase in temperature in the program is carried out subtracting
the heat lost by radiation, convection and conduction from the power supplied, VI.
This allows to estimate the difference in temperature after each time step. This new
temperature profile is used in the next set of calculations: oxygen diffusion
coefficient, resistivity and once more the new temperature profile of the sample.
2C Listing of the computer model developed to study the Direct
Current Zoning effect
The following programme calculates the variation in temperature and the change in
the oxygen content distribution in a polycrystalline rod during the Direct Current
Zoning effect. The results obtained with this programme have been discussed in
chapter 6 (see flowchart of the algorithm used in section 6.4.4).
SCREEN 0: CLS ’’Initializing the screen and clearing
’’ Geometrical factors:
diameter = .0023                          ’’ Diameter in m
area = 3.14159 * diameter ^ 2 / 4         ’’ Cross sectional area
length = .05                              ’’ Total length in m
surface = 3.14159 * diameter * length     ’’ Wire surface area in m^2
’’ Physical constants:
emissivity = .915
Density = 5.8                             ’’ Density in gr/cm3
Density = Density * .001 * 10 ^ 6         ’’ Density in Kg/m3
ThermalCond = 2.67                        ’’ Thermal conductivity in
W/(m K)
SpecificHeat = .431                       ’’ Specific heat in J/(g K)
Appendix 2
218
SpecificHeat = SpecificHeat * 1000        ’’ Specific heat J/(kg K)
ThermalTempCof = .0001              '' 1/ºC assumed temp coefficient
'' of the thermal conductivity
RT = 25                             '' Room temperature in ºC
concoefficient = 1.18               '' Convection coefficient in
W/m2K
'' Convection is assumed to follow:
'' Convection loss=coefficient*(temp-
ambient)^0.25/diameter^0.25
convection = concoefficient / diameter ^ .25
'' Convection is now used in the following equation -
'' Convection loss = convection*(temp-ambient)^0.25
endeffect = 1   '' This allows a larger loss at the ends to be
effected
    '' endeffect=1 means no increased loss at the end
    '' endeffect>1 means increased loss at the end
'' Physical constants:
stefan = 5.67E-08                         '' Stefan-Boltzmann
constant W/(m2K4)
si = 8.617E-05 / 1.38066E-23              '' Ratio eV/J
k = 1.38066E-23                           ''Boltzmann constant J/K
'' The ion concentration in x-6 units the oxygen ion concentration is
'' assumed to obey the relation:
'' IF T<400 THEN
''      ionconc = InitConc
''      ELSEIF ionconc < -0.015*Tc(i) +1.55057 THEN
''              ionconc = InitConc
''      ELSE ionconc = -0.015*Tc + 1.55057
ze = 2 * 1.6E-19          '' Charge transported by the oxygen ions
conc = 3.408 * 10 ^ 27    '' Mobile ions/m3 (for x=6.6, or x-6=0.6)
Maxconc = .987#           '' Maximum oxygen concentration
firstread = 0           '' A flag
'' The first reading occurs when the temperature
tempdiff = 5      '' of the zone is within tempdiff of the last
reading
'' after which the 'firstread' flag is set to 1.
'' Diffusion parameters:
d0 = .23                '' Diffusion pre-exponent cm2/s (from chapter
5)
d0 = d0 / 10000         '' m2/s
q = .8    ''Activation energy for diffusion in eV (from chapter
5)
q = q / si     '' Activation energy in J
'' Total = total conductivity data
Total4 = 34100!               '' Conductivity at 400ºC (from chapter
6)
Total8 = 2093!                '' Conductivity at 850ºC (from chapter
6)
'' Since the experimental conductivity is 8 times lower we
'' divide the conductivity by 8
Total4 = Total4 / 8             '' Conductivity at 400ºC
Total8 = Total8 / 8             '' Conductivity at 850ºC
Appendix 2
219
b = LOG(Total4 / Total8) / 450 ’’ Find exponential to fit the
following
 ’’ equation: conductivity=c*exp(-b*t)
c = Total4 / EXP(-b * 450)  ’’ Find the pre-exponent
’’ Linear simulation of a wire divided by n segments:
N = 50                     ’’ Number of segments, maximum 100
   '' Tc = Remperature in ºC
   '' Tk = Temperature in K
   '' heating = heat production
   '' cooling = heat loss
   '' conc = oxygen concentration in x-6
   '' delconc = change in concentration
DIM Tc(101), heating(101), cooling(101), conc(101), delconc(101)
PRINT "Enter initial oxygen concentration (in x-6 units):";
INPUT InitConc
'' Uniform distribution of Tc, the temperature at room temp
'' and concentration at InitConc
FOR i = 1 TO N
Tc(i) = RT
conc(i) = InitConc
NEXT i
PRINT "Do you want to enter a depleted oxygen region? (y/n)";
INPUT YesNo$
IF YesNo$ = "y" THEN GOSUB 5000 '' Create an area with a different
concentration
'' Initialising:
oldcurrent = 0
oldvoltage = 0
'' Calculate temperature rise with time time sequence starts at t=0
and
'' continues with m steps of delta until time totaltime (in sec) is
reached
totaltime = 5000                 '' Total length of calculation in
seconds
delta = .1                       '' Interval in seconds
m = totaltime / delta            '' m*delta = total time
seglen = length / N              '' Segment length
PRINT "Enter voltage in V:"; : INPUT v0
mm = -1                          '' mm is a counter
SCREEN 12                        '' Graphic screen 640x480
GOSUB 1000: GOSUB 2000           '' Draw grids
ss = 1 / SpecificHeat / Density / area / seglen * delta '' Temp,
increase ºC/W
OPEN "h:\qbprogs\Tprodcz.dat" FOR OUTPUT AS #2
PRINT #2, "initial concentration= " + STR$(InitConc)
PRINT #2, "i"
FOR i = 1 TO N
PRINT #2, STR$(i)
NEXT i
CLOSE #2
Appendix 2
220
OPEN "h:\qbprogs\VIdcz.dat" FOR OUTPUT AS #3
PRINT #3, "initial voltage =" + STR$(v0)
PRINT #3,
PRINT #3, "T, V, current"
CLOSE #3
OPEN "h:\qbprogs\condcz.dat" FOR OUTPUT AS #8
PRINT #8, "initial concentration= " + STR$(InitConc)
PRINT #8, "i"
FOR i = 1 TO N
PRINT #8, STR$(i)
NEXT i
CLOSE #8
FOR T = 0 TO m * delta STEP delta        ’’ Start main loop
Resistance0 = 0
’’ Find the resistance: Total=total conductivity
FOR i = 1 TO N
Tk = 273 + Tc(i)             ’’ Temperature in K
D = d0 * EXP(-q / k / Tk)    ’’ Diffusion coefficient
’’ ionconc is the ion concentration (max=maxconc-6)
SELECT CASE Tk
CASE IS <= 435 + 273
     ionconc = -.00009 * Tc(i) + 1
CASE (435 + 273) TO (705 + 273)
     ionconc = -.001653 * Tc(i) + 1.68
CASE IS >= (705 + 273)
     ionconc = -.001163 * Tc(i) + 1.334
END SELECT
IF ionconc > 1 THEN ionconc = Maxconc
IF Tk < 673 THEN
     electr = Total4
ELSE electr = c * EXP(-b * Tc(i))
END IF
’’Ionic is now the conductivity due to the conc(i)
ionic = (D / k / Tk) * ze ^ 2 * conc(i) * conc / .6
’’ Total is now the real total conductivity
Total = electr + ionic
resistivity = 1 / Total ’’ Resistivity in 1/Ωm
Resistance0 = Resistance0 + resistivity * seglen / area
’’ Sum all the individual segments to give the value of
the
’’ total resistance resistance0 of the wire.
NEXT i
current = v0 / Resistance0
V = 0
FOR i = 1 TO N
’’ Heat generation in the segments and radiation + convection loss:
Tk = 273 + Tc(i)
D = d0 * EXP(-q / k / Tk)
SELECT CASE Tk
CASE IS <= 435 + 273
     ionconc = -.00009 * Tc(i) + 1
CASE (435 + 273) TO (705 + 273)
     ionconc = -.001653 * Tc(i) + 1.68
CASE IS >= (705 + 273)
     ionconc = -.001163 * Tc(i) + 1.334
END SELECT
IF Tk < 673 THEN
Appendix 2
221
     electr = Total4
ELSE electr = c * EXP(-b * Tc(i))
END IF
ionic = (D / k / Tk) * ze ^ 2 * conc(i) * conc / .6
Total = electr + ionic
resistivity = 1 / Total
resistance = resistivity * seglen / area
power = current ^ 2 * resistance         ’’ Joule heating
V = V + current * resistance             ’’ Voltage drop
radiation = emissivity * stefan * surface / N * ((Tc(i) +
273) ^ 4 - (RT + 273) ^ 4)
’’ Radiation loss
’’ Add convection term:
’’ if Tc(i)-RT<0 then Tc(i)=RT
radiation = radiation + convection * surface / N * (Tc(i)
- RT)^.25
’’ add convection term
’’ special treatment for ends
IF i = 1 OR i = N THEN
radiation = radiation + emissivity * stefan * area
* ((Tc(i) + 273) ^ 4 - (RT + 273) ^ 4)
’’add convection term
radiation = endeffect * radiation + convection *
area * (Tc(i) - RT) ^ .25
END IF
’’ ss = tempearture increase in oC/W after delta seconds
heating(i) = (power - radiation) * ss’’’oC
Tc(i) = Tc(i) + heating(i)  ’’ Update temperature profile
    ’’ if Tc(i)<RT then Tc(i)=RT
NEXT i
’’ Cooling due to thermal conduction along the rod
FOR i = 2 TO N - 1
cof = 1 - ThermalTempCof * Tc(i)
’’ Thermal conductivity at Tc(i) in W/(Km)
cooling = -ThermalCond * cof * area / seglen * (2 * Tc(i)
- Tc(i - 1) - Tc(i + 1))         ’’W
cooling(i) = cooling * ss           '' cooling in ºC
NEXT i
'' Special treatment for the first and last segments:
'' Conduction into the rod only:
cof = 1 - ThermalTempCof * Tc(i)
'' Thermal conductivity at Tc(1) W/(mK)
cooling = -ThermalCond * cof * area / seglen * (Tc(1) - Tc(2))
  '' W
cooling(1) = cooling * ss         '' ºC
cof = 1 - ThermalTempCof * Tc(i)
'' Thermal conductivity at Tc(n) W/(mK)
cooling = -ThermalCond * cof * area / seglen * (Tc(N) - Tc(N -
1))  '' W
cooling(N) = cooling * ss         '' ºC
'' Update the temperature profile:
FOR i = 1 TO N
Tc(i) = Tc(i) + cooling(i)   '' if Tc(i)<RT then Tc(i)=RT
NEXT i
Appendix 2
222
’’ Calculate concentration changes:
FOR i = 1 TO N
Tk = Tc(i) + 273
D = d0 * EXP(-q / k / Tk)
SELECT CASE Tk
CASE IS <= 435 + 273
 ionconc = -.00009 * Tc(i) + 1
CASE (435 + 273) TO (705 + 273)
     ionconc = -.001653 * Tc(i) + 1.68
CASE IS >= (705 + 273)
     ionconc = -.001163 * Tc(i) + 1.334
END SELECT
’’ ionconc is the equilibrium oxygen concentration in x-6 units:
tau = (diameter / 2) ^ 2 / D ’’Diffusivity time constant
conc(i) = conc(i) + (ionconc - conc(i)) * (delta / tau)
IF conc(i) > ionconc THEN conc(i) = ionconc
NEXT i
’’ Ion drift calculation:
FOR i = 1 TO N
Tk = Tc(i) + 273
D = d0 * EXP(-q / k / Tk)
Tkleft = Tc(i - 1) + 273
IF i = 1 THEN
Dleft = 0
ELSE Dleft = d0 * EXP(-q / k / Tkleft)
END IF
SELECT CASE Tkleft
CASE IS <= 435 + 273
 ionconc = -.00009 * Tc(i - 1) + 1
CASE (435 + 273) TO (705 + 273)
     ionconc = -.001653 * Tc(i - 1) + 1.68
CASE IS >= (705 + 273)
     ionconc = -.001163 * Tc(i - 1) + 1.334
END SELECT
IF Tkleft < 673 THEN
     electr = Total4
ELSE electr = c * EXP(-b * Tc(i))
END IF
ionic = (Dleft / k / Tkleft) * ze ^ 2 * conc(i - 1) *
conc / .6
Total = electr + ionic
resistivityl = 1 / Total
Vleft = current / area * resistivityl
driftleft = (Vleft) * Dleft / k / Tkleft * ze
Tkright = Tc(i + 1) + 273
IF i = N THEN
Dright = 0
ELSE Dright = d0 * EXP(-q / k / Tkright)
END IF
SELECT CASE Tkright
CASE IS <= 435 + 273
 ionconc = -.00009 * Tc(i + 1) + 1
CASE (435 + 273) TO (705 + 273)
     ionconc = -.001653 * Tc(i + 1) + 1.68
CASE IS >= (705 + 273)
     ionconc = -.001163 * Tc(i + 1) + 1.334
END SELECT
Appendix 2
223
IF Tkright < 673 THEN
     electr = Total4
ELSE electr = c * EXP(-b * Tc(i))
END IF
ionic = (Dright / k / Tkright) * ze ^ 2 * conc(i + 1) *
conc / .6
Total = electr + ionic
resistivityr = 1 / Total
Vright = current / area * resistivityr
driftright = (Vright) * Dright / k / Tkright * ze
delconc(i) = (conc(i - 1) * driftleft - conc(i + 1) *
driftright) * delta / seglen
’’ Chemical gradients:
fick = (-(conc(i) - conc(i - 1)) * Dleft + (conc(i + 1) -
conc(i)) * Dright) / seglen
delconc(i) = delconc(i) + fick * delta / seglen
NEXT i
’’ Update concentration due to drift:
FOR i = 1 TO N
conc(i) = conc(i) + delconc(i)
IF conc(i) > Maxconc THEN
conc(i) = Maxconc
END IF
IF conc(i) < .2 THEN
conc(i) = .2
END IF
NEXT i
k$ = INKEY$
IF LEN(k$) = 1 THEN
IF LEN(k$) = 1 AND ASC(k$) = 27 THEN
END     ’’exit
END IF
IF LEN(k$) = 1 AND ASC(k$) = 13 THEN
INPUT c     ’’ Hold calculation until next
keystroke
END IF
IF LEN(k$) = 1 AND k$ = "c" OR k$ = "C" THEN
GOSUB 3000: GOSUB 2000                     ’’ Clear
END IF
IF LEN(k$) = 1 AND k$ = "a" OR k$ = "A" THEN
GOSUB 1000: GOSUB 2000                  ’’ clear
all
END IF
IF LEN(k$) = 1 AND k$ = "h" OR k$ = "H" THEN  ’’ Hot spot
GOSUB 5000
END IF
 END IF
 IF LEN(k$) = 1 AND k$ = "p" OR k$ = "P" THEN
GOSUB 4000                ’’ Draw concentration profile
 END IF
IF LEN(k$) = 2 THEN
IF ASC(MID$(k$, 2, 1)) = 80 THEN
v0 = v0 - .1        ’’ Down arrow, decrease voltage
END IF
IF ASC(MID$(k$, 2, 1)) = 72 THEN
v0 = v0 + .1        ’’ Up arrow, increase voltage
END IF
END IF
Appendix 2
224
mm = mm + 1                     ’’ mm = number of iterations
IF mm / 300 - INT(mm / 300) = 0 THEN
’’ Draw line every 50th
calculation
mmprint = 6             ’’ Calculate every mmprint cycles
’’ Print label
LOCATE 1, 1: PRINT TAB(78);
LOCATE 1, 1
PRINT "V=";
PRINT USING "####.##"; V;
PRINT "V    I=";
PRINT USING "####.##"; current;
PRINT "A    R=";
PRINT USING "####.##"; Resistance0;
PRINT "ˆ  ";
PRINT "T=";
PRINT USING "####.#"; Tc(N / 2);
PRINT "oC   time=";
PRINT USING "####.#"; T;
GOSUB 6000
IF mm / mmprint - INT(mm / mmprint) = 0 THEN
       '' Find location of max
temp
oldmaxtemp = maxtemp     '' Old value of
temperature
maxtemp = -1E+37
FOR i = 1 TO N
IF Tc(i) > maxtemp THEN
maxtemp = Tc(i): maxtemploc = i
END IF
NEXT i
IF mm > mmprint THEN
'' First time to start calculations:
 '' First time (zero point) to start measuring average drift
 '' when the zone temp is stable within tempdiff remember firsttime
 '' and firsloc=segment when max temp occured at time=firsttime
 '' then set firstread=1
IF ABS(oldmaxtemp - maxtemp) < tempdiff AND
firstread = 0 THEN
firstread = 1: firsttime = T: firstloc
= maxtemploc
END IF
END IF
IF mm > mmprint THEN
speed = (maxtemploc - oldmaxtemploc) * seglen
/ (delta * mmprint) * 1000 * 60            '' In
mm/min
END IF
LOCATE 18, 40: PRINT "Hot zone velocity=";
PRINT USING "####.##"; speed;
PRINT " mm/min  ";
IF firstread > 0 AND T > firsttime THEN
LOCATE 19, 40: PRINT "Av. zone velocity=";
avspeed = (maxtemploc - firstloc) * seglen /
(T - firsttime) * 1000 * 60          '' In mm/min
PRINT USING "####.##"; avspeed;
PRINT " mm/min   ";
END IF
Appendix 2
225
LOCATE 20, 40: PRINT "At";
PRINT USING "####"; INT(Tc(maxtemploc));
PRINT "oC, Ionic drift=";
Tk = Tc(maxtemploc) + 273
D = d0 * EXP(-q / k / Tk)
SELECT CASE Tk
    CASE IS <= 435 + 273
  ionconc = -.00009 * Tc(i) + 1
    CASE (435 + 273) TO (705 + 273)
   ionconc = -.001653 * Tc(i) + 1.68
    CASE IS >= (705 + 273)
 ionconc = -.001163 * Tc(i) + 1.334
END SELECT
IF Tk < 673 THEN
electr = Total4
ELSE electr = c * EXP(-b * Tc(i))
END IF
ionic = (D / k / Tk) * ze ^ 2 * conc(maxtemploc) *
conc / .6
Total = electr + ionic
resistivity = 1 / Total
Vseg = current / area * resistivity
PRINT USING "####.##"; -(Vseg) * D / k / Tk * ze *
1000 * 60;
PRINT " mm/min";
IF mm > 0 THEN
oldmaxtemploc = maxtemploc
’’old value of segment# where max temp
occurred
END IF
END IF
’’ Plot temperature profile:
COLOR 14
FOR i = 1 TO N - 1
LINE (offsetx + i * sx, offsety + Tc(i) * sy)-
(offsetx + (i + 1) * sx, offsety + Tc(i + 1) * sy)
NEXT i
OPEN "h:\qbprogs\Tprodcz.dat" FOR INPUT AS #1
OPEN "c:\temp\dcz1.dat" FOR OUTPUT AS #2
’’ Read headers and write out again
LINE INPUT #1, inline$
PRINT #2, inline$
LINE INPUT #1, inline$
PRINT #2, inline$ + CHR$(9) + STR$(T)
FOR i = 1 TO N
LINE INPUT #1, inline$
PRINT #2, inline$ + CHR$(9) + STR$(Tc(i))
NEXT i
CLOSE #1
CLOSE #2
KILL "h:\qbprogs\Tprodcz.dat"
SHELL "copy c:\temp\dcz1.dat " + "h:\qbprogs\Tprodcz.dat > nul"
OPEN "h:\qbprogs\condcz.dat" FOR INPUT AS #8
OPEN "c:\temp\dczcon.dat" FOR OUTPUT AS #7
’read headers and write out again
LINE INPUT #8, inline$
PRINT #7, inline$
LINE INPUT #8, inline$
PRINT #7, inline$ + CHR$(9) + STR$(T)
Appendix 2
226
FOR i = 1 TO N
LINE INPUT #8, inline$
PRINT #7, inline$ + CHR$(9) + STR$(conc(i))
NEXT i
CLOSE #8
CLOSE #7
KILL "h:\qbprogs\condcz.dat"
SHELL "copy c:\temp\dczcon.dat " + "h:\qbprogs\condcz.dat >
nul"
’’ Plot V-I curve:
COLOR 10
IF oldcurrent < 8 AND current < 8 THEN
LINE (x01 + oldvoltage * dx1, y02 - oldcurrent *
dy1)-(x01 + V * dx1, y02 - current * dy1)
END IF
oldvoltage = V: oldcurrent = current
GOSUB 7000 ’’ To save V-I values
GOSUB 8000 ’’ Close V-I file values
’’ Draw strip every 50 calculations
FOR i = 1 TO N
COLOR 0
IF Tc(i) > 600 THEN
COLOR 4 ’’brownish
END IF
IF Tc(i) > 700 THEN
COLOR 12 ’’reddish
END IF
IF Tc(i) > 800 THEN
COLOR 14 ’’yellow
END IF
IF Tc(i) > 950 THEN
COLOR 15 ’’white
END IF
LINE (offsetx + i * 2, offsety + 20)-(offsetx + i *
2, offsety + 30)
LINE (offsetx + i * 2 + 1, offsety+20)-(offsetx + i
* 2 + 1, offsety + 30)
NEXT i
END IF
COLOR 15
NEXT T
END
1000              ’’ sub to draw axes for V-I, labels, etc.
CLS
sy = -.215        ’’ Scale factor y direction
sx = 500 / N      ’’ Scale factor x direction
offsety = 215
offsetx = 10
COLOR 15
LINE (offsetx, offsety)-(offsetx + N * sx, offsety)
’’ Outline of the rod:
leftx = 1
LINE (offsetx + leftx, offsety + 19)-(offsetx + leftx, offsety + 31)
LINE (offsetx + leftx + 1 + 2 * N, offsety + 19)-(offsetx + leftx + 1
+ 2 * N, offsety + 31)
LINE (offsetx + leftx, offsety + 19)-(offsetx + leftx + 1 + 2 * N,
offsety + 19)
Appendix 2
227
LINE (offsetx + leftx, offsety + 31)-(offsetx + leftx + 1 + 2 * N,
offsety + 31)
’’ V-I axes drawing:
maxa = 4: maxv = 5         ’’ max I, max V
dy1 = 40: y01 = 280: y02 = y01 + maxa * dy1
dx1 = 50: x01 = 10: x02 = x01 + maxv * dx1
FOR i = 0 TO maxv
LINE (x01 + i * dx1, y01)-(x01 + i * dx1, y02)  ’’ 1 V steps
NEXT i
FOR j = 0 TO maxa STEP 1                              ’’ 1 A steps
LINE (x01, y02 - j * dy1)-(x02, y02 - j * dy1)
NEXT j
COLOR 15
LOCATE 17, 1: PRINT "Current  (A)"
LOCATE 18, 1: PRINT "4"
LOCATE 23, 1: PRINT "2"
LOCATE 29, 30: PRINT "Voltage (V)";
LOCATE 29, 1: PRINT "0";
LOCATE 29, 8: PRINT "1";
LOCATE 29, 15: PRINT "2";
LOCATE 29, 21: PRINT "3";
LOCATE 29, 27: PRINT "4";
’’ Instructions:
xr = 41                                ’’ Left column
yr = 22                                ’’ Top row
LOCATE yr, xr: PRINT "Exit press ESC";
LOCATE yr + 1, xr: PRINT "Increase voltage press "; CHR$(24);
LOCATE yr + 2, xr: PRINT "Decrease voltage press "; CHR$(25);
LOCATE yr + 3, xr: PRINT "Clear the top screen press C";
LOCATE yr + 4, xr: PRINT "Create a hot spot press H";
LOCATE yr + 5, xr: PRINT "Concentration profile press P";
x1 = 640 * xr / 80 - 20                 ’’ 640x480 pixels are 80x30
col*rows
y1 = 480 * yr / 30 - 20
x2 = 640 * (xr + 29) / 80
y2 = 480 * (yr + 5) / 30 + 5
LINE (x1, y1)-(x2, y1)
LINE (x2, y1)-(x2, y2)
LINE (x2, y2)-(x1, y2)
LINE (x1, y2)-(x1, y1)
’’ Drift information border
yr = 18
xr = 41
x1 = 640 * xr / 80 - 20                 ’’ 640x480 pixels are 80x30
col*rows
y1 = 480 * yr / 30 - 20
x2 = 640 * (xr + 35) / 80
y2 = 480 * (yr + 2) / 30 + 5
LINE (x1, y1)-(x2, y1)
LINE (x2, y1)-(x2, y2)
LINE (x2, y2)-(x1, y2)
LINE (x1, y2)-(x1, y1)
RETURN
2000               ’’ Draw grid
COLOR 15
FOR tt = 0 TO 900 STEP 100
Appendix 2
228
LINE (offsetx, offsety + tt * sy)-(offsetx + N * sx, offsety +
tt * sy)
NEXT tt
FOR j = offsetx TO offsetx + N * sx + 1 STEP N / 10 * sx
LINE (j, offsety)-(j, offsety + 900 * sy)
NEXT j
COLOR 15
LOCATE 2, 65: PRINT "900oC"
LOCATE 6, 65: PRINT "600oC"
LOCATE 10, 65: PRINT "300oC"
LOCATE 14, 65: PRINT "0oC"
LOCATE 15, 47: PRINT "Distance  mm";
LOCATE 15, 31: PRINT INT(length * 500 + .5);
T$ = "TEMPERATURE"
FOR j = 1 TO LEN(T$)
LOCATE j + 2, 72
PRINT MID$(T$, j, 1)
NEXT j
RETURN
3000
FOR u = 1 TO 14          ’’ Clear the top of the screen
LOCATE u, 1: PRINT TAB(78);
NEXT u
RETURN
4000
GOSUB 3000: GOSUB 2000
LOCATE 2, 65: PRINT "6.9   "
LOCATE 6, 65: PRINT "6.6   "
LOCATE 10, 65: PRINT "6.3   "
LOCATE 14, 65: PRINT "6.0   "
T$ = "    x      "
FOR j = 1 TO LEN(T$)
LOCATE j + 2, 72
PRINT MID$(T$, j, 1)
NEXT j
COLOR 14
FOR i = 1 TO N - 1
LINE (offsetx + i * sx, offsety + 1000 * conc(i) * sy)-(offsetx
+ (i + 1) * sx, offsety + 1000 * conc(i + 1) * sy)
NEXT i
111 k$ = INKEY$: IF k$ = "" THEN 111
 GOSUB 3000
 GOSUB 2000
RETURN
5000             ’’ Create an area with a different concentration
LOCATE 1, 1: PRINT TAB(78);
LOCATE 1, 1: PRINT "Enter start,end of spot in % of length:"; : INPUT
stpct, endpct
LOCATE 1, 1: PRINT TAB(78);
LOCATE 1, 1: PRINT "Enter oxygen concentration (in x-6):"; : INPUT
spot
FOR sp = INT(stpct / 100 * N) TO INT(endpct / 100 * N)
conc(sp) = spot
NEXT sp
RETURN
6000
OPEN "h:\qbprogs\VIdcz.dat" FOR APPEND AS #3
RETURN
Appendix 2
229
7000
WRITE #3, T, V, current
RETURN
8000
CLOSE #3
RETURN
References
230
REFERENCES
A.A. Abrikosov, Sov. Phys. JETP, Vol. 5 (1957) pp. 1174-1182.
M.A. Alario-Franco and C. Chaillout, Solid State Ionics, Vol. 32/33 (1989) pp. 1056-
1063.
M. Asta, D. de Fontaine, G. Ceder, E. Salomons and M. Kraitchman, J. Less-Common
Metals, Vol. 168 (1991) pp. 39-51.
M. Ausloos and A. P NDOVNLPhys. Rev. B, Vol. 52 (1995) pp. 4577-4582.
H. Bakker, J.P.A. Westerveld and D.O.Welch, Physica C, Vol. 153-155 (1988) pp.
848-849.
H. Bakker, J.P.A. Westerveld, D.M.R. Lo Caschio and D.O.Welch, Physica C, Vol.
157 (1989) pp. 25-36.
J.G. Bednorz and K.A. Müller, Z. Phys. B, 64 (1986), pp. 189-193.
A. Belzner, T.M. Gür and R.A. Huggins, Solid State Ionics Vol. 40-41 (1990) pp. 535-
538.
V.S. Bobrov, E.A. Brener, S.E Esipov, A.N. Izotov, Yu A Koval', V.V. Korshunov,
R.K. Nikolaev, L.A. Novomlimskii, Yu A Ossipyan, N.S. Sidorov and V Sh
Shekthtman. B.C.H.Steele. R (1990)/01 (private communication).
A. Bourdillon and N.X. Tan Broudillon, “High temperature superconductors:
processing and science”, Academic Press (1994).
V.A.M. Brabers, W.J.M. de Jonge, L.A. Bosch, C. van der Steen, A.M.W. de Groote,
A.A. Verheyen and C.W.H.M. Vennix, Mat. Res. Bull., Vol. 23 (1988) pp. 197-207.
S.I. Bredikhin, G.A. Emel’chenco, V.Sh. Shechtman, A.A. Zhokhov, S. Carter, R.J.
Chater, J.A. Kilner and B.C.H. Steele, Physica C, Vol. 179 (1991) pp.286-290.
J.J. Capponi, C. Chaillout, A.W. Hewat, P. Lejay, M. Marezio, N. Nguyon, B.
Raveau, J.L. Soubeyroux, J.L. Tholence and R. Tournier, Europhys. Lett., Vol. 3
(1987) pp. 1301-1307.
W. Carrillo-Cabrera, H-D. Wiemhöfer and W. Göpel, Solid State Ionics,Vol. 32-33
(1989) pp. 1172-1178.
H.S. Carslaw and J.C. Jaeger, “Conduction of heat in solids”, Oxford Science
Publications, 2nd Ed. (1959).
References
231
R.J. Cava, B. Batlogg, C.H.Chen, E.A. Tietman, S.M. Zahurak and D. Wereder,
Nature, Vol. 329 (1987) pp. 423-425.
R.J. Cava, A.W. Hewat, E.A. Hewat, B. Batlogg, M. Marezio, K.M. Rabe, J.J.
Krajewski, W.F. Peck Jr. and L.W. Rupp Jr., Physica C, Vol. 165 (1990) pp. 419-433.
G. Ceder, M. Asta, W.C. Carter, M. Kraitchman, D. de Fontaine, M.E. Mann and M.
Sluiter, Phys. Rev. B, Vol. 41 (1990) pp. 8698-8701.
E.K. Chang, D.J.L. Hong, A. Mehta and D.M.Smyth, Mat. Lett., Vol. 6 (1988), pp.
251-253.
J-S. Choi, M. Sarikaya, I.A. Aksay and R. Kikuchi, Phys. Rev. B, Vol. 42 (1990) pp.
4244-4254.
K. Conder, E. Kaldis, M. Maciejewski, K.A. Müller and E.F. Steigmeier, Physica C,
Vol. 210 (1993) pp. 282-288.
K. Conder, Ch. Krüger, E. Kaldis, D. Zech and H. Keller, Physica C, Vol. 225 (1994)
pp. 13-20.
J.R. Cost and J.R. Stanley, J. Mater. Res., Vol 6 (1991) pp. 232-243.
J.M. Coulson and J.F. Richardson, “Chemical engineering. Volume 1. Fluid flow,
heat transfer and mass transfer”, 3rd Ed., Pergamon Press, (1977).
J. Crank, “The mathematics of diffusion”, 2nd Ed., Oxford Science Publications,
(1975).
M. Cyrot and D. Pavuna, “Introduction to superconductivity and high Tc materials”,
World Scientific, (1992).
T. Datta, “Concise encyclopedia of magnetic and superconducting materials”, Ed.
J.E. Evetts, Pergamon Press, (1992).
A. Davidson, P. Santhanam, A. Palevski and M.J. Brady, Phys. Rev. B, Vol. 38 (1988)
pp. 2828-2831.
S. Elschner, W. Becker, H. Bestgen and M. Brand, Physica C, Vol. 202 (1992) pp.
401-407.
A. Erb, B. Greb and G. Müller-Vogt, Physica C, Vol. 259 (1996) pp. 83-91.
F. Faupel and T. Hehenkamp, Z. Metallkd., Vol. 84 (1993) pp. 529-533.
A.T. Fiory, M. Gurvitch, R.J. Cava and G.P. Espinosa, Phys. Rev. B, Vol. 36 (1987)
pp. 7262-7265.
J.C. Fisher, J. Appl. Phys., Vol. 22 (1951) pp. 74-77.
References
232
B. Fisher, E. Polturak, G. Koren, A. Kessel, R. Fischer and L. Harel, Solid State
Commun., Vol 64 (1987) pp. 87-88.
D. de Fontaine, M.E. Mann and G. Ceder, Phys. Rev. Lett., Vol. 63 (1989) pp. 1300-
1303.
D. de Fontaine, G. Ceder and M. Asta, Nature, Vol. 343 (1990-a) pp. 544-546.
D. de Fontaine, G. Ceder and M. Asta, J. Less-Common Metals Vol. 164-165 (1990-b)
pp. 108-122.
P.P. Freitas and T.S. Plaskett, Phys. Rev. B, Vol. 36 (1987) pp. 5723-5726.
P.P. Freitas and T.S. Plaskett, Phys. Rev. B, Vol. 37 (1988) pp. 3657-3659.
T. Furukawa, T. Shigematsu and N. Nakanishi, Physica C, Vol 204 (1992) pp. 103-
108.
P.K. Gallagher, Adv. Cer. Mat., Vol. 2 (1987) pp. 632-639.
%$*ÆRZDFNL5-+LJKPRUH.) 3HWHUV$/*UHHU DQG -( (YHWWV. Supercond.
Sci. Technol., Vol 1 (1988) pp. 7-11.
G.S. Grader, P.K. Gallagher and E.M. Gyorgy, Appl. Phys. Lett., Vol. 51 (1987) pp.
1115-1117.
M. Gurvitch and A.T. Fiory, Phys. Rev. Lett., Vol. 59 (1987) pp. 1337-1340.
I. Haller, M.W. Shafer, R. Figat and D.B. Goland, “Chemistry of oxide
superconductors”, Ed. C.N.R. Rao, Blackwell Scientific Publications IUPAC (1988)
pp. 93-100.
M.S. Islam, Supercond. Sci. Technol., Vol. 3 (1990) pp. 531-536.
Y. Ikuma and S. Akiyoshi, J. Appl. Phys., Vol. 64 (1988) pp. 3915-3917.
J.A. Jacobson, J.M. Newsam, D.C. Johnston, J.P. Stokes, S. Bhattacharya, J.T.
Lewandowski, D.P. Goshorn, J.M. Higgins and M.S. Alvarez, “Chemistry of oxide
superconductors”, Ed. C.N.R. Rao, Blackwell Scientific Publications IUPAC (1988)
pp. 43-61.
J.D. Jorgensen, M.A. Beno, D.G. Hinks, L.Solderholm, K.J.Volin, R.L. Hitterman,
J.D. Grace, I. K.Schuller, C.U. Segre, K. Zhang and M.S. Kleefisch, Phys. Rev. B,
Vol. 36 (1987-a) pp. 3608-3616.
J.D. Jorgensen, B.W. Veal, W.K.Kwok, G.W. Crabtree, A. Umezawa, L.J. Nowicki
and A.P.Paulikas, Phys. Rev. B, Vol. 36 (1987-b) pp. 5731-5734.
J.D. Jorgensen, B.W. Veal, A.P. Paulikas, L.J. Nowicki, G.W. Crabtree, H. Claus and
W.K. Kwok, Phys. Rev. B, Vol. 41 (1990) pp. 1863-1877.
References
233
K. Kishio, J. Shimoyama, T. Hasegawa, K. Kitazawa and K. Fueki, Jpn. J. App. Phys.,
Vol. 26, (1987), pp. L1228-L1230.
K. Kishio, K. Kitazawa, K. Suzuki, T. Hasegawa and K. Fueki, Physica C, Vol. 162-
164 (1989-a) pp. 55-56.
K. Kishio, K. Suzuki, T. Hasegawa, T. Yamamoto and K. Kitazawa, J. Solid State
Chemistry, Vol. 82 (1989-b) pp. 192-202.
C. Krauns and H-U. Krebs, Z. Phys. B, Vol. 92 (1993) pp. 43-46.
Ch. Krüger, K. Conder and E. Kaldis, Physica C, Vol. 213 (1993) pp. 219-223.
T. Kwok, P. S. Ho and S. Yip, Phys. Rev. B, Vol. 29 (1984-a) pp. 5354-5362.
T. Kwok, P. S. Ho and S. Yip, Phys. Rev. B, Vol. 29 (1984-b) pp. 5363-5371.
J.R. LaGraff, P.D. Han and D.A. Payne, Physica C, Vol. 169 (1990) pp. 355-360.
J.R. LaGraff, P.D. Han and D.A. Payne, Phys. Rev. B, Vol. 43 (1991) pp. 441-447.
J.R. LaGraff and D.A. Payne, "Physics and Materials Science of High Temperature
Superconductors, II", R. Kossowski et al. (eds.), Kuwler Academic Publishers (1992),
pp. 225-246.
J.R. LaGraff and D.A. Payne, Phys. Rev. B, Vol. 47 (1993-a) pp. 3380-3390.
J.R. LaGraff and D.A. Payne, Physica C, Vol. 212 (1993-b) pp. 470-477.
J.R. LaGraff and D.A. Payne, Physica C, Vol. 212 (1993-c) pp. 478-486.
S-F. Lau, A.B. Rosenthal, N.P. Pyrros, J.A. Graham and H.N. Cheng, J. Mater. Res.,
Vol. 6 (1991) pp. 227-231.
A.D. Le Claire, "Treatise in Solid State Chemistry, Vol. 4, Reactivity of Solids” N.B.
Hannay (Ed.), Plenum (1975), p.1.
I.A. Leonidov, Ya.N. Blinovskov, E.E. Flyatau, P.Ya. Novak and V.L. Kozhevnikov,
Physica C, Vol. 158 (1989) pp. 287-292.
T.B. Lindemer, F.A. Washburn, C.S. MacDougall, R. Feenstra and O.B. Cavin,
Physica C, Vol. 178 (1991) pp. 93-104
F. London and H. London, Proc. Roy. Soc. (London), Vol. A155 (1935) pp. 71.
J.R. Manning, “Diffusion kientics for atoms in crystals”, Van Nostrand (1968) p. 95.
J.L. McManus, D.J. Fray and J. Evetts, Supercond. Sci. Technol. Vol. 1 (1989) pp.
291-295.
References
234
J.L. McManus, D.J. Fray and J. Evetts, Physica C, Vol. 169 (1990) pp. 193-200.
J.L. McManus, D.J. Fray and J. Evetts, Physica C, Vol. 190 (1992) pp. 511-521.
W. Meissner and R. Ochsenfeld, Naturwissenschaften, Vol. 21 (1933) pp. 787.
R. Mogilevsky, R. Levi-Stetti, B. Pashmakov, L. Liu, K. Zhang, H.M. Jaeger, D.B.
Buchholz, R.P.H. Chang and B.W. Veal, Phys. Rev. B, Vol. 49 (1994) pp. 6420-6423.
A.P. Mozahev and S.V. Chernyaev, J. Mater. Chem., Vol 4 (1994) pp. 1107-1110.
A. P. Mozhaev, S.V. Chernyaev, T.I. Udal’tsova and N.M. Kotov, Russ. J. Inorg.
Chem., Vol. 37 (1992) pp. 1111-1113.
F. Munakata, K. Shinohara, H. Kanesaka, N. Hirosaki, A. Okada and M. Yamanaka,
J. Jnl. Appl. Phys., Vol. 26 (1987) pp. L1292-L1293.
V.I. Nefedov, A.N. Solokov. A.R. Vinogradov. T.A. Fomina, N.G. Shuranova, K.G.
Oksenoid, G.I. Ramendik and E.A.Volkov, Medeleev Commun., (1992) pp. 21-22.
J. Nowotny and M. Rekas, J. Am. Ceram. Soc., Vol. 73 (1990) pp. 1054-1061.
H.M. O’Bryan and P.K. Gallagher, Adv. Cer. Mat., Vol. 2 (1987) pp. 640-648.
H.M. O’Bryan and P.K. Gallagher, Solid State Ionics, Vol. 32-33 (1989) pp. 1143-
1148.
T. Okamoto, B. Huybrechts and M. Takata, Jpn. J. Appl. Phys., Vol. 33 (1994-a) pp.
L1212-L1214.
T. Okamoto, B. Huybrechts and M. Takata, MRS Fall Meeting ’94 (1994-b).
H.K. Onnes, Leiden Comm., Vol. 120b, 122b, 124c (1911).
Yu.A. Osip’yan, R.K. Nikolaev, N.S. Sidorov, V.S. Bobrov and V.S. Tsoi, JETP Lett.,
Vol. 47 (1988) pp. 310-314.
E.J.M. O’Sullivan and B.P. Chang, Appl. Phys. Lett., Vol. 52 (1988) pp. 1441-1443.
G. Ottaviani, C. Nobili, F. Nava, M. Affronte, T. Manfredini, F.C. Matacotta and E.
Galli, Phys. Rev. B, Vol. 39 (1989) pp. 9069-9073.
T. Ozawa, J. Thermal Analysis, Vol. 2 (1970) p. 301.
M.V. Patrakeev, I.A. Leonidov, V.L. Kozhevnikov, V.I. Tsidilkovskii and A.K.
Demin, Physica C, Vol. 210 (1993), pp. 213-220.
J. S. Park and H.G. Kim, Jpn. J. Appl. Phys., Vol. 27 (1988) pp. L191-L194.
J-H. Park, P. Kostic and J.P.Singh, Mat. Lett., Vol. 6 (1988) pp. 393-397.
References
235
N.L. Peterson, “Diffusion in solids, Recent developments”, A.S. Nowick and J.J.
Burton (Eds.), Academic Press (1975), pp. 115-170.
H.F. Poulsen, N.H. Andersen, J.V. Andersen, H. Bohr and O.G. Mouritsen, Nature,
Vol. 349 (1991) pp. 594-596.
H. Rickert, “Electrochemistry of solids. An introduction”, Springer Verlag (1982).
G.P. Rodríguez-Donoso, J. Ruiz, B.J. Fernández and A.J. Vázquez-Vaamonde, Mat.
and Design, Vol. 16 (1995) pp. 163-166.
M. Ronay, Phys. Rev. B, Vol. 36 (1987) pp. 8860-8862.
M. Ronay and P. Nordlander, Physica C, Vol. 153-155 (1988) pp. 834-835.
A.C. Rose-Innes, E.H. Rhoderick, “Introduction to Superconductivity”, Pergamon,
(1988).
S.J. Rothman, J.L. Roubort and J.E. Baker, Phys. Rev. B, Vol. 40 (1989) pp. 8852-
8860.
S.J. Rothman, J.L. Roubort, U. Welp and J.E. Baker, Phys. Rev. B, Vol. 44 (1991) pp.
2326-2333.
J.L. Roubort, N. Chen. K.C. Goretta and S.J. Rothman, ICMC '90 Conference "High-
Temperature Superconductors, Materials Aspects", (1990).
J.L. Roubort and S.J. Rothman, J. Appl. Phys., Vol. 76 (1994) pp. 5615-5628.
J. Ruiz, J.B. Fernández and J.M Belló, 2nd Seminar of Surface Engineering with High
Energy Beams Sci. and Techn. (1989) pp. 161-172.
E. Saiz and J.S. Moya, Mater. Lett., Vol. 6 (1988) pp. 369-373.
J. Sabras, G. Peraudeau, R. Berjoan and C. Monty, J. Less Common Metals, Vol. 164-
165 (1990) pp. 239-246.
K. Salama and D.F. Lee, Supercond. Sci. Technol., Vol. 7 (1994) pp. 177-193.
E. Salomons and D. de Fontaine, Phys. Rev. B, Vol. 41 (1990) pp. 11159-11167.
I.K. Schuller, D.G. Hinks, M.A. Beno, D.W. Capone II, L. Soderholm, J.-P. Locquet,
Y. Bruynseraede, C.U. Segre and K. Zhang, Solid State Comm., Vol. 63 (1987) pp.
385-388.
Y. Scolnik, E.Sabatani and D. Cahen, Physica C, Vol. 174 (1991) pp. 273-279.
S. Semenovskaya and A.G. Khachaturyan, Phys. Rev. B, Vol. 46 (1992) pp. 6511-
6534.
References
236
H. Shaked, J.D. Jorgensen, J. Faber Jr., D.G. Hinks and B. Dabrowski, Phys. Rev. B,
Vol. 39 (1989) pp. 7363-7366.
D. Shi, Phys. Rev. B, Vol. 39 (1989) pp. 4299-4305.
D. Shi “High temperature superconducting materials science and engineering: new
concepts and technology”, Pergamon, (1995).
E.D. Specht, C.J. Sparks, A.G. Dhere, J. Brynestead, O.B. Cavin, D.M. Kroeger and
H.A. Oye, Phys. Rev. B, Vol 37 (1988) pp. 7426-7434.
H. Strauven, J.P. Locquet, O.B. Verbeke and Y. Bruynseraede, Solid State Comm.,
Vol. 65 (1988) pp. 293-296.
P. Strobel, J.J. Capponi, M. Marezio and P. Monod. Solid State Comm., Vol. 64
(1987) pp. 513-515.
M.-Y. Su, S.E.Dorris and T.O.Mason, J. of Solid State Chem., Vol. 75 (1988) pp. 381-
389.
T. Suzuoka, J. Phys. Soc. Jap., Vol. 19 (1964) pp. 839-851.
J.L. Tallon and M.P. Staines, J. Appl. Phys., Vol. 68 (1990) pp. 3998-4001.
T.B. Tang and W. Lo, Physica C, Vol. 174 (1991) pp. 463-466.
S. Tsukui, T. Yamamoto, M. Adachi, K. Kawabata, N. Fukuoka, A. Yanase and Y.
Yoshioka, Physica C, Vol. 185-189 (1991-a) pp.929-930.
S. Tsukui, T. Yamamoto, M. Adachi, Y Shono, K. Kawabata, N. Fukuoka. S.
Nakanishi, A. Yanase and Y. Yoshioka, Jpn. J. App. Phys., Vol. 30 (1991-b) pp.
L973-L976.
K.N. Tu, C.C. Tsuei, S.I. Park and A. Levi, Phys. Rev. B, Vol. 38 (1988-a) pp. 772-
775.
K.N. Tu, N.C. Yeh, S.I. Park and C.C. Tsuei, Phys. Rev. B, Vol. 38 (1988-b) pp.
5118-5121.
K.N. Tu, N.C. Yeh, S.I. Park and C.C. Tsuei, Phys. Rev. B, Vol. 39 (1989) pp. 304-
314.
X. Turrillas, J.A. Kilner, I. Kontoulis and B.C.H. Steele, J. Less Common Metals, Vol.
151 (1989) pp. 229-236.
T. Umemura, K. Egawa, M. Wakata and K Yoshizaki, Jpn. J. of App. Phys., Vol. 28
(1989) pp. L1945-L1947.
B.W. Veal, H. You, A.P. Paulikas, H. Shi, Y. Fang and J.W. Downey, Phys. Rev. B,
Vol. 42 (1990-a) pp. 4770-4773.
References
237
B.W. Veal, H. You, A.P. Paulikas, H. Shi, Y. Fang and J.W. Downey, Phys. Rev. B,
Vol. 42 (1990-b) pp. 6305-6316.
D.J. Vischjager, A.A. van Zomeren, J. Schoonman, I. Kontoulis and B.C.H. Steele,
Solid State Ionics, Vol. 40-41 (1990) pp. 810-814.
R.T.P. Whipple, Phil. Magazine, Vol. 45 (1954) pp. 1225-1236.
W.K. Wu, J.R. Ashburn, C.J. Torng, P.H. Hor, R.L. Meng, L. Gao, Z.J. Huang, Y.Q.
Wang and C.W. Chu, Phys. Rev. Lett., Vol. 58 (1987), pp. 908.
X.M. Xie, T.G. Chen and Z.L.Wu, Phys. Rev. B, Vol. 40 (1989) pp. 4549-4556.
S. Yamaguchi, K. Terabe, A. Imai and Y. Iguchi, Jpn. J. of App. Phys., Vol. 27 (1988)
pp. L220-L222.
E. Yanmaz, I.R. Harris and J.S. Abell, J. of Alloys and Compounds, Vol. 185 (1992)
pp. 311-320.
E. Yanmaz, K. Donnelly, I.R. Harris and J.S. Abell, J. of Alloys and Compounds, Vol.
195 (1993) pp. 39-42.
H-I. Yoo, J. Mater. Res., Vol 4 (1989) pp. 23-27.
H-I. Yoo and S-M. Lee, J. Am. Ceram. Soc., Vol. 77 (1994) pp. 3131-3136.
J. Zaanen, A.T. Paxton, O. Jepsen and O.K. Andersen, Phys. Rev. Lett. Vol. 60 (1988)
pp. 2685-2688.
J.X. Zhang, G.M. Lin, W.G. Zeng, K.F. Liang, Z.C. Lin, G.G. Siu, M.J. Stokes and
P.C.W. Fung, Supercond. Sci. Technol., Vol. 3 (1990-a) pp. 113-117.
J.X. Zhang, G.M. Lin, W.G. Zeng, K.F. Liang, Z.C. Lin, G.G. Siu, M.J. Stokes and
P.C.W. Fung, Supercond. Sci. Technol., Vol. 3 (1990-b) pp. 163-172.
Z. Zhao, L. Chen, Q. Yang, Y. Huang, G. Chen, R. Tang, G. Liu, C. Cui, L. Chen, L.
Wang, S. Guo, S. Li and J. Bi, Kexue Tongbao, Vol. 32 (1987) pp. 1098.