Design and modelling
of ultra-high strength steels:
nanoprecipitation and plasticity
Bij-Na Kim
Lucy Cavendish College
University of Cambridge
A dissertation submitted for the degree of
Doctor of Philosophy at the University of Cambridge
September 2013
Preface
This dissertation is submitted for the degree of Doctor of Philosophy at the Univer-
sity of Cambridge. The research reported herein was conducted under the super-
vision of Dr Pedro Rivera in the Department of Materials Science and Metallurgy,
University of Cambridge, between October 2010 and September 2013.
This work is to the best of my knowledge original, except where acknowledgment
and references are made to previous work. This dissertation is the result of my own
work and includes nothing which is the outcome of work done in collaboration except
where specifically indicated in the text. Neither this, nor any substantially similar
dissertation has been, or is being, submitted for any other degree, diploma, or other
qualification at any other university. This dissertation does not exceed 60,000 words
in length.
Some of the work described herein has been presented in the following publications:
• B. Kim, D. San Mart´ın, J. Chao and P. E. J. Rivera-Dı´az-del-Castillo, The
effect of silicon on the ε → θ transformation in ultra-strong spring steels, in
Proceedings of the international conference Materials Science and Technology,
7-11 October 2012, Pittsburgh PA: 1086-1093
• B. Kim, C. Celada, D. San Mart´ın, J. Chao, J. Vara and P. E. J. Rivera-
Dı´az-del-Castillo, Interrupted ageing in steels: Hardness improvement and
microstructural stabilization, Scripta Materialia, 68:945-948, 2013
• B. Kim, C. Celada, D. San Mart´ın, T. Sourmail and P. E. J. Rivera-Dı´az-
del-Castillo, The effect of silicon on the nanoprecipitation of cementite, Acta
Materialia, 61:6983-6992, 2013
i
• B. Kim, E. Boucard, T. Sourmail, D. San Mart´ın, N. Gey and P. E. J. Rivera-
Dı´az-del-Castillo, Understanding the microstructure-properties relationship in
tempered martensite: 0.5–0.6 wt.% C steels, to be submitted in Acta Materi-
alia.
Bij-Na Kim
September 2013
ii
Acknowledgement
I am indebted to Professor A. L. Greer for the provision of laboratory facilities in the
Department of Materials Science and Metallurgy at the University of Cambridge.
I would like to express my sincere thanks to my supervisor, Dr Pedro Rivera,
for his scientific guidance, enthusiasm, encouraging support and mentoring during
the project, as well as Dr David San Mart´ın and colleagues at CENIM, particularly
Carola, Javi and Nacho, for the tremendous amount of work carried out in collab-
oration, and for all that I have learned during my visit to the Materalia group in
April 2012.
I am indebted to ASCOmetal for their financial support in the research, espe-
cially to Dr Thomas Sourmail, Dr Fabien Perrard and Dr Simon Frappart, for all
the stimulating discussion meetings and feedback.
In addition I wish to thank numerous individuals in the department for all the
help I have received from the beginning, in particular those who have given me im-
mense support on a daily basis: JeeHyun Kang, Kunio Hayashi (‘uncle Martensite’),
Isaac Toda (‘uncle Matlab’) and Enrique Galindo. I would like to extend this grati-
tude to all of my friends I have met over these past 3 years: PhD life in Cambridge
wouldn’t have been the same without you.
I would also like to thank my first year CPGS examiners, Dr Howard Stone and
Professor Harry Bhadeshia, as their comments and feedback resulted in exploring
new directions in research, such as the synchrotron work. I am indebted to the I11
(high resolution powder diffraction) beamline team led by Professor Chiu Tang, for
their assistance before and during beamtime.
Finally, I would like to express my deepest gratitude to my dearest family: my
father, my mother and my sister Hanna- thank you for everything!
iii
Abstract
Understanding the changes occurring in the mechanical properties during marten-
site tempering is essential in the development of new industrial grades. The aim
of this research was to develop new ultra-high strength steels via nanoprecipitation
control, which requires an understanding of the processing-microstructure-property
relationship in medium carbon (0.5–0.6 wt.%) steels throughout tempering. Much
of the work has been centred in understanding the role of silicon at the precipitation
level and in the recovery of martensite.
By using an existing spring steel grade, the effect of interrupted ageing (IA) in
tempered martensite has been studied. In IA, an intermediate step between quench-
ing and tempering is introduced, where quenched martensite is left to rest at room
temperature for a defined period of time. By allowing carbon segregation into dis-
location cores, the incorporation of IA resulted in a more stable microstructure and
hardness improvement.
The effect of silicon in the epsilon to cementite carbide transition has also been
studied. The classical nucleation theory was applied in order to model cementite
formation under paraequilibrium conditions, thus incorporating silicon during nu-
cleation. Characterisation using high energy X-rays showed the inhibiting effect of
silicon in the overall cementite precipitation.
The second effect of silicon was observed in the martensite recovery. A series of
experiments were carried out in order to capture the various microstructural changes
taking place during tempering: precipitation, grain size and dislocation density evo-
lution. It was observed that the addition of silicon reduces the rate of martensite
recovery, owing to the reduced cross-slip in the ferrite lattice.
A plasticity model based on irreversible thermodynamics and EBSD character-
isation was applied to identify the effective grain size. The results from these two
techniques require further research. Nevertheless, based on the post-failure analysis
iv
by TEM, it appears that at relatively early tempering stages, even low angle lath
boundaries can contribute to strengthening, where piled-up dislocations have been
observed at lath boundaries.
v
Contents
1 Introduction 1
2 Literature review:
carbide precipitation in martensite 6
2.1 Existing spring steels grades . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Tempered martensite . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Austenitisation . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Quenching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.3 Tempering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Carbides in martensite . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.1 Characterising carbides . . . . . . . . . . . . . . . . . . . . . . 23
3 Interrupted ageing in steels 26
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.1 Ageing in steels . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.2 Thermoelectric power . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.1 Heat treatment . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.2 Determining the prior austenite grain size . . . . . . . . . . . 32
3.2.3 Mechanical testing . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.4 Microstructural characterisation . . . . . . . . . . . . . . . . . 34
3.2.5 Thermoelectric power . . . . . . . . . . . . . . . . . . . . . . . 35
vi
3.3 Characterisation by dilatometry . . . . . . . . . . . . . . . . . . . . . 35
3.3.1 Summary of critical temperatures . . . . . . . . . . . . . . . . 35
3.3.2 PAG size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Hardness and microstructural analysis . . . . . . . . . . . . . . . . . 37
3.4.1 Hardness testing . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4.2 TEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5 Modelling carbon segregation during interrupted ageing . . . . . . . . 45
3.5.1 The Kalish-Cohen model . . . . . . . . . . . . . . . . . . . . . 45
3.5.2 Thermoelectric power . . . . . . . . . . . . . . . . . . . . . . . 47
3.5.3 Semi quantitative analysis . . . . . . . . . . . . . . . . . . . . 50
3.6 Tensile and compression testing . . . . . . . . . . . . . . . . . . . . . 53
3.6.1 Tensile testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.6.2 Compression testing . . . . . . . . . . . . . . . . . . . . . . . 55
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 Modelling θ nucleation: the Fe-C-Si model alloy system 58
4.1 The Fe-C-Si model alloy system . . . . . . . . . . . . . . . . . . . . . 58
4.1.1 Mechanical properties . . . . . . . . . . . . . . . . . . . . . . 62
4.1.2 Microstructural analysis . . . . . . . . . . . . . . . . . . . . . 63
4.1.3 X-ray diffraction . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Modelling cementite nucleation using the classical nucleation theory . 67
4.2.1 Thermodynamics- ∆GV . . . . . . . . . . . . . . . . . . . . . 70
4.2.2 Misfit strain- ∆GS . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2.3 Computing the effective driving force (∆GV –∆GS) . . . . . . 76
4.3 Characterising carbides using high energy synchrotron radiation . . . 78
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5 The effect of silicon on tempered martensite 83
5.1 Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
vii
5.2 Microstructure and properties . . . . . . . . . . . . . . . . . . . . . . 85
5.2.1 Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2.2 Optical and scanning electron microscopy . . . . . . . . . . . 87
5.2.3 Transmission electron microscopy . . . . . . . . . . . . . . . . 88
5.3 The effect of silicon on cementite growth kinetics: in situ characteri-
sation using high energy X-rays . . . . . . . . . . . . . . . . . . . . . 91
5.3.1 Preliminary characterisation . . . . . . . . . . . . . . . . . . . 92
5.3.2 Experimental procedure . . . . . . . . . . . . . . . . . . . . . 94
5.3.3 Temperature calibration . . . . . . . . . . . . . . . . . . . . . 95
5.3.4 Reference samples . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3.5 In situ tempering . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.3.6 Cementite growth kinetics . . . . . . . . . . . . . . . . . . . . 100
5.4 The effect of silicon on martensite recovery . . . . . . . . . . . . . . . 103
5.4.1 Martensite recovery . . . . . . . . . . . . . . . . . . . . . . . . 103
5.4.2 Dislocation density throughout tempering . . . . . . . . . . . 104
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6 Plasticity 113
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Tensile testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.3 Fitting σy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3.1 Precipitation hardening, σp . . . . . . . . . . . . . . . . . . . 118
6.3.2 Solid solution strengthening, σs . . . . . . . . . . . . . . . . . 121
6.3.3 Forest dislocation hardening, σρ . . . . . . . . . . . . . . . . . 122
6.3.4 Grain size effect . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.4 Brief introduction to plasticity theory . . . . . . . . . . . . . . . . . . 129
6.4.1 Kinematic hardening . . . . . . . . . . . . . . . . . . . . . . . 130
6.4.2 Isotropic hardening . . . . . . . . . . . . . . . . . . . . . . . . 131
6.4.3 Example 1: HS-250 condition . . . . . . . . . . . . . . . . . . 132
viii
6.4.4 Example 2: HS-450 condition . . . . . . . . . . . . . . . . . . 134
6.4.5 Example 3: LS alloy . . . . . . . . . . . . . . . . . . . . . . . 136
6.5 Grain size study by EBSD . . . . . . . . . . . . . . . . . . . . . . . . 137
6.5.1 Experimental procedure . . . . . . . . . . . . . . . . . . . . . 138
6.5.2 Grain size determination based on misorientation angle . . . . 140
6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7 General conclusions and scope for future work 149
7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.2 Scope for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
ix
Chapter 1
Introduction
The progress in materials science has led to the development of advanced engineered
materials. Taking the example of the automotive steel industry, one of the main driv-
ing forces has been the quest for stronger and lighter materials. The development
of such advanced structural materials provide car manufacturers greater flexibility
in the design. The main focus of this research has been centered around spring
steels technology. Suspension springs act together with shock absorbers, with their
main role consisting in damping vibrations that may be transferred from the road
to the passengers. For instance, in the case of spring steels, the development of spe-
cial high-strength grades allows a reduction in the number of coils used, as shown in
Figure 1.1. For example, by reducing the number of coils by one, a significant spring
weight reduction of over 20% can be achieved [1]. From the consumers perspective,
lower vehicle weight results in reduced fuel consumption. This also contributes to
the ever-increasing demand for lower carbon dioxide emissions.
1
Figure 1.1: Weight reduction achieved by the use of improved grades in spring steels
[1].
In compensating for the reduction in weight, the component must meet higher
mechanical properties demands. One important family of engineered materials used
in structural applications is tempered martensite. Martensite tempering is a very
complex topic on its own, being accompanied by a series of microstructural changes
which is reflected in the mechanical properties. During the early stages of temper-
ing, martensite exhibits ultra-high strength, but the alloy is extremely brittle for
commercial use. On the other hand, as tempering progresses, the initial ultra-high
strength is lost while ductility is recovered. In order to understand the trade-off
between these two properties, it is necessary to understand the changes occurring
at the microstructural level.
One particular phenomenon that has attracted immense attention is precipitation
hardening. When steel is quenched, carbon atoms initially occupy the interstitial
sites within the Fe-lattice. Once tempering occurs, carbon participates in the pre-
cipitation of carbides.
The effect that the presence of dispersed phases have on the strength increment,
2
∆τ , can be described by the Ashby-Orowan relationship:
∆τ = 0.84
(
1.2µb
2piL
)
ln
(
x
2b
)
, (1.1)
where µ is the shear modulus of the matrix, b is the magnitude of the Burgers vector,
L is the spacing between particles, and x is the particle diameter.
From Equation (1.1), it is possible to establish that the principal microstructural
parameter that dictates the degree of precipitation hardening is the spacing between
the particles, followed by the size of particles.
Described by the three-link chain model, Figure 1.2, microstructure tailoring is
achieved by controlling the processing stage, which in turn determines the mechani-
cal properties and final performance. In the processing stage, the two main variables
are heat treatment and alloying. All the possible combinations of heat treatments
and alloying systems give rise to a very large number of possible microstructural
outcomes.
Figure 1.2: Three-link chain model of the central paradigm of materials science and
engineering [2].
In spring steels technology, usually medium carbon steels containing 0.5 to 0.6 wt.%
carbon, are used. The typical industrial process takes place in the 350–450 ◦C tem-
3
perature range. This corresponds to the relatively early stages of martensite tem-
pering, involving the ε- and cementite θ-carbides, where the former is the preceding
metastable phase that forms before the latter. Early literature has attributed the
exceptional ultra-high strength in tempered martensite to the presence of ε-carbide
within martensitic laths [3]. Furthermore, silicon has been reported to delay the ε
→ θ carbide transition. In order to relate to the mechanical properties of the steel,
the effect of silicon on the aforementioned transition is of great technological impor-
tance. Chapter 2 provides a literature review centred around carbide precipitation
during martensite tempering, and the effect of silicon in carbide precipitation.
In understanding the effect of heat treatment in carbide precipitation, Chapter
3 explores the effect of interrupted ageing in martensitic steels. During interrupted
ageing, the quenched martensite is left to age at room temperature for a specified
period of time, prior to tempering. Using a single grade, a series of heat treatments
have been designed to understand the effect of interrupted ageing time, tempering
time and temperature on carbide precipitation. By allowing carbon segregation dur-
ing interrupted ageing, the number of effective nucleation sites is increased, which
aids in improving and stabilising the microstructure and hardness properties (∼10%
increase). Such carbon segregation behaviour was modelled using the Kalish-Cohen
model [4] and thermoelectric power measurements.
Aiming at understanding the sequence of precipitation in tempered martensite,
Chapter 4 looks into the early precipitation stages of ε and θ. In order to under-
stand the effect of silicon on carbide precipitation, a model alloy of composition
Fe–0.55C–2Si wt.% was studied. The nucleation of cementite was modelled under
paraequilibrium conditions, in order to account for the silicon entrapment in cemen-
tite during low-temperature tempering. Carbides were characterised using electron
and X-ray diffraction methods, where trial runs were carried out using high energy
X-rays from a synchrotron source. The knowledge built from the ternary model
alloy was then applied to more complex alloy compositions in Chapter 5. In situ
4
tempering was characterised using synchrotron radiation, and showed that silicon
inhibits cementite precipitation by enveloping the growing precipitate. Gathering
data from literature, a new mechanism by which silicon retards cementite growth is
suggested.
Although precipitation hardening accounts for a significant strengthening contri-
bution in tempered martensite, there are other microstructural properties evolving
during martensite tempering that determine the overall mechanical properties. In
order to model stress-strain behaviour of martensitic steels in Chapter 6, an irre-
versible thermodynamics-based approach to plasticity has been taken in order to
bring multiscale plasticity phenomena together. Experimentally determined mi-
crostructural features in Chapter 5 such as lath size, precipitate size and volume
fraction, dislocation density, and carbon content in solid solution, are incorporated,
thus integrating the role of silicon on the mechanical properties of martensitic steels.
Finally Chapter 7 gives the general conclusions, highlighting the contributions
of this PhD thesis in further understanding the properties of tempered martensitic
steels. Furthermore, a series of suggestions for further work are given, raising further
topics of interest that are beyond the scope of this work.
5
Chapter 2
Literature review:
carbide precipitation in martensite
2.1 Existing spring steels grades
This section aims at giving a brief overview of the physical metallurgy of spring
steels, giving examples of high performance grades and details on typical chemical
compositions and heat treatments. Two common grades used for making spring
steels within the European market are 54SiCr6 and 54SiCrV6 [5], where the chemical
composition specified by the European norm is detailed in Table 2.1.
Table 2.1: 54SiCr6 and 54SiCrV6 compositions specified by the EN10089 [5], (in
wt.%, Fe to balance).
C Si Mn Cr V
54SiCr6 0.51–0.59 1.2–1.6 0.5–0.8 0.5–0.8 -
54SiCrV6 0.51–0.59 1.2–1.6 0.5–0.8 0.5–0.8 0.1–0.2
6
Table 2.2: Summary of heat treatment and minimum mechanical properties specified
by the European norm [5].
‡ Tγ Quenching Ttemp σ0.2 σUTS A Z I
( ◦C) media ( ◦C) (MPa) (MPa) (%) (%) (J)
54SiCr6 860 oil 450 1300 1450-1750 6 25 8
54SiCrV6 860 oil 400 1600 1650-1950 5 35 8
‡ Tγ and Ttemp are the austenitising and tempering temperatures respectively, and σ0.2, σUTS,
A, Z and I refer to 0.2% proof yield stress, ultimate tensile stress, elongation, reduction of area
and impact energy respectively.
In the 54SiCrV6 grade, vanadium is added to the base 54SiCr6 grade. Usually
vanadium is present as a microalloying addition, where the formation of precipitates
would account for an improvement in mechanical properties. The aimed mechanical
properties for both alloys are summarised in Table 2.2.
One alloying element that has drawn significant industrial interest is silicon.
In studying the effect of silicon for spring steel applications, Choi [6] compared
two steel grades: SAE9254 (Fe–0.55C–1.5Si–0.7Mn–0.7Cr wt.%), and POSHIS120D
(Fe–0.48C–2.1Si–0.65Mn–0.7Cr wt.%). Improved mechanical properties were ob-
tained in terms of both tensile strength and area reduction in the higher silicon
grade, POSHIS120D. The microstructural feature responsible for such increased
strengthening was the precipitation of fine Fe2.4C carbides, and the suppression of
the transition from this carbide to spheroidised cementite. Therefore in optimis-
ing the mechanical properties of spring steels, understanding carbide precipitation
throughout processing becomes essential.
7
2.2 Tempered martensite
The typical heat treatment of tempered martensitic steels is represented in Figure
2.1. The main stages encountered are: austenitisation, quenching and tempering.
Each of these stages are very important, and are carefully controlled in industry.
Time 
Te
m
pe
ra
tu
re
 
Austenitisation 
Quenching 
Tempering 
Figure 2.1: Typical heat treatment for tempered martensitic steels.
2.2.1 Austenitisation
The austenitisation stage has two principal objectives: (i) complete austenitisation
so there is no ferrite left in the microstructure prior to quenching, and (ii) carbon lev-
elling, in order to ensure homogeneity in composition. The latter point is especially
important, as uneven distributions of carbon would cause martensite transformation
in carbon-rich areas, whereas lower bainite formation is promoted in carbon-poor
regions [7]. Such inhomogeneity in the microstructure has a detrimental impact on
the mechanical properties of the steel. Although sufficient time is required for such
even distribution of carbon in the austenite, another factor that needs to be consid-
ered is the growth of austenite grains. In literature, austenite grain size has been
reported to affect the martensitic start temperature [8, 9], as well as the mechanical
properties. After quenching, smaller prior austenite grains (PAGs), would lead to a
higher density of ‘barriers’ to dislocation motion [7]. This last point will be further
explored in Chapter 6.
8
2.2.2 Quenching
When steel is quenched from its austenite state to room temperature, it undergoes
a martensitic transformation. The stable form of iron at low temperatures is ferrite
(α), a body centred cubic (bcc) structure; the presence of carbon atoms occupying
the interstitial sites gives rise to tetragonality. Hence, as-quenched martensite (α′)
adopts a body centred tetragonal (bct) structure. Carbon being insoluble in α-Fe,
the matrix is initially in a super saturated solid solution (s.s.s.s.) form.
The quenching process is very important, as the cooling rate needs to be suf-
ficiently high in order to avoid pearlite formation and to minimise the degree of
autotempering. As-quenched martensite has a distinct microstructure, consisting
of needle- or lath-like structures that are reported to have formed following the
Kurdjumov-Sachs orientation relationship with respect to the previous austenite
phase [10]. The martensitic microstructure is dependent on the carbon content of
the alloy, Figure 2.2 [10]. On closer inspection, martensite consists of a series of
sub-structures, as illustrated in Figure 2.3. Within a prior austenite grain, packets
are observed, which are composed of blocks that contain individual laths with small
angle misorientations. The effect of each constituent substructure will be further
explored in Chapter 6.
9
Figure 2.2: Optical micrographs of lath martensite for Fe–C alloys containing (a)
0.0026, (b) 0.18, (c) 0.38, (d) 0.61 wt.% carbon [10].
Figure 2.3: Schematic illustration of lath martensite structure for (a) low carbon
(0–0.4 wt.% C) and (b) high carbon (0.6 wt.% C) alloys [10].
10
2.2.3 Tempering
Table 2.3: Sequence of the changes occurring in the microstructure during the tem-
pering of martensite [11].
Stage Temperature
range ( ◦C)
Changes occurring throughout stage
T1 100–250 Decomposition of supersaturated martensite to low-
carbon martensite and transition carbides (ε or η)
T2 200–300 Decomposition of retained austenite to bainite
T3 250–350 Transformation of the reaction products of T1 and T2
into ferrite and/or cementite
T4 >350 Growth and spheroidisation of cementite
T5 >500 Intermetallic precipitation and further carbide forma-
tions in alloy steels
Table 2.3 summarises the typical sequence of events occurring during the tempering
of martensite. Throughout martensite tempering, carbon leaves the interstitial sites
and takes part in carbide precipitation. Consequently, a reduction in tetragonality
is observed, until the matrix reverts back to its original bcc form. Furthermore,
although the main microstructural feature is martensite, there will always be some
retained austenite (γR) left on the as-quenched condition. It is thought that its
presence even at room temperatures, is due to the enrichment of the interstitials,
e.g. carbon, during isothermal holding [12]. The amount of retained austenite can
be estimated from the Koistinen-Marburger equation [13]:
1− Vα′ = exp
(
β(Ms − Tq)
)
, (2.1)
11
where V α′ is the volume fraction of martensite, β=−0.011 K−1, M s is the martensite
start temperature and T q is the temperature to which the sample is cooled. From
(2.1), typically in medium carbon martensitic steels, γR is under 10%. Owing to
the enrichment of carbon in the austenite phase, the decomposition of the retained
phase is thought to lead to further carbide precipitation [12, 11].
At this stage, it is important to note that there is an intermediate stage between
quenching and tempering, called ageing [11]. As most alloy have an M s temper-
ature above room temperature, inevitable martensite ageing takes place at room
temperature. During ageing, only interstitial elements, such as carbon atoms, are
sufficiently mobile. The structural rearrangements of carbon atoms taking place
during martensite ageing have been extensively studied by Ge´nin [14], and Taylor
and Cohen [15]. Investigating the effect of ageing prior to tempering on carbide
precipitation, constitutes the base of Chapter 3.
2.3 Carbides in martensite
From a precipitation hardening perspective it is of great interest to characterise
and understand the carbide phases present within the matrix. The principal iron
carbides that have been reported in literature are the ε and the θ carbides, where
the former is the preceding metastable phase that forms before cementite θ-carbide.
The basic properties of these two carbides are summarised in Table 2.4.
Typically in martensitic steels, during early stages of tempering, the material
exhibits ultra-high strength. However, its brittleness limits its industrial usabil-
ity. Throughout continuous tempering, ductility of the material is recovered at the
expense of strength. Therefore, throughout the heat treatment, there is always a
compromise between strength and ductility. This softening transition, which typi-
cally occurs around 300–350 ◦C, also corresponds to the temperature regime where
the ε → θ carbide transition takes place.
12
Table 2.4: Summary of properties of cementite and ε-carbides.
ε-carbide θ-cementite
Chemical composition Fe2.4C Fe3C
Crystal structure hexagonal close packed orthorhombic
Space group P6322 Pnma
OR‡ Jack [16] Bagaryatski [17]
(0001)ε ‖ (011)α′ [100]θ ‖ [01¯1]α′
(101¯0)ε ‖ (21¯1)α′ [010]θ ‖ [11¯1¯]α′
(101¯1)ε ‖ (101)α′ [001]θ ‖ [211]α′
‡ OR refers to the most commonly reported orientation relationship existing between the carbide
and the matrix.
In early literature, the ultra-high strength observed in tempered martensite has
been attributed to the presence of microstrains between ε and the matrix [3]. Early
literature has reported the effect that silicon has in delaying the aforementioned
transition [18, 19]. Numerous researches have reported strength increments at higher
temperatures, as a consequence of increasing silicon content within the alloy [18, 19,
20, 21, 22, 23, 24, 25].
Gordine and Codd [22] proposed a mechanism by which silicon retards the ε→ θ
carbide transition: the third stage of tempering (T3 on Table 2.3) originally taking
place in two steps:
• Stage IIIA- the remaining carbon atoms in the matrix are taken up for cemen-
tite precipitation while ε-carbide persists in the matrix. At this stage, both
carbides co-exist in the material
• Stage IIIB- once the matrix becomes depleted, the ε-carbide decomposes in
order to assist in the further precipitation of cementite.
The rate-controlling factors were seen to be the diffusion of carbon through the
13
low-carbon martensite for IIIA, and the mobility of substitutional elements or the
self-diffusion of Fe for step IIIB. On the other hand, it was proposed that in the
presence of silicon, steps IIIA and IIIB would merge, and shift the reactions to
higher temperatures, consequently stabilising ε at higher temperatures [22].
In understanding the carbide transition and the influence of silicon, numerous
studies have been carried out. Miyamoto et al. [26] used electron diffraction tech-
niques in order to characterise the carbides present within martensite throughout
tempering under the influence of manganese and silicon as alloying elements, sum-
marised in Table 2.5.
Table 2.5: Carbide transition seen in the tempering of martensite in Fe-0.6C-
2(Si,Mn), from Miyamoto et al. [26].
Alloy 250 ◦C 450 ◦C
system 30 s 1200 s 30 s 120 s 300 s 1200 s
Fe-0.6C ε θ θ - - θ
Fe-0.6C-2Mn ε θ θ - - θ
Fe-0.6C-2Si ε ε ε ε/θ θ θ
Comparing Fe–0.6C with Fe–0.6C–2Mn, the addition of manganese does not appear
to have a significant effect on the carbide transition. However a marked effect is
seen when silicon is added, as ε is seen throughout longer tempering conditions,
both time- and temperature-wise. Taking the Fe–0.6C–2Si alloy tempered at 450
◦C, according to Table 2.5, 30 s tempering results in ε being formed, whereas after
300 s, θ is seen. Figure 2.4 shows the SAD patterns for the ε and θ conditions. The
measured interplanar spacing (shown in Figure 2.4) were 1 4.36 and 2 2.40 A˚,
for ε, and 1 4.10 and 2 2.29 A˚, for cementite.
14
Figure 2.4: SAD patterns for the Fe-0.6C-2Si alloy tempered at 450 ◦C for (a) 30 s,
and (b) 300 s, where the present carbide phases are ε and θ, respectively [26].
Furthermore in the work by Miyamoto et al. [26], by means of atom probe to-
mography (APT), there was a slight indication of silicon being built up at the
cementite/matrix interface after the material was tempered at 450 ◦C for 1200 s.
This result is in agreement with the earlier work by Chang and Smith [20], where a
Si-rich region of 1–2 nm was observed at the cementite/matrix interface using APT,
Figure 2.5. Such partitioning behaviour of silicon from the carbide to the matrix
has been frequently reported in the APT literature, e.g. [26, 27, 28, 29].
Such findings led to the conclusion that silicon inhibits cementite growth. The
suggested mechanism presumes that the Si-rich region at the interface acts as a
barrier, where carbon diffusion is slowed down, reducing the overall carbide growth
kinetics [20]. This further adds to the inhibited cementite growth observed due to
15
the relatively slow diffusion of silicon during partitioning.
Figure 2.5: Composition profiles across the cementite/matrix interface after 1 hour
tempering at 400 ◦C for the EN45 commercial grade containing 0.6 carbon and 2
silicon by weight % [20].
For silicon partitioning to take place during the growth stage, there should have
been silicon present during the initial cementite nucleation process. This concept
of cementite nucleation under paraequilibrium conditions has been well-received in
literature. During nucleation under paraequilibrium conditions, cementite traps the
inherent chemical composition of the matrix phase, including silicon. However, as
tempering progresses, the silicon content within the carbide decreases, as observed
by Reisdorf [3]. Given that silicon is not present under equilibrium conditions within
cementite, it is inferred that silicon partitioning takes place as the system tends to
equilibrium.
Based on the growth kinetics experimentally observed, it is possible to infer the
conditions at which cementite nucleated in the first place. Extensive effort has been
16
put into understanding θ-nucleation, and different conditions under which cementite
was seen to nucleate have been reported, out of which, three are listed as examples:
Figure 2.6: TEM diffraction work by Nam et al. on a Si–Cr steel, where (a) ε carbide
is seen at 250 ◦C, (b) the corresponding diffraction pattern of (a) on the [001]α zone,
(c) schematic representation of the diffraction pattern of (b), (d) tempered at 350
◦C, (e) dark field image of (d) using (11¯01)ε reflection, (f) cementite seen at 400 ◦C,
(g) diffraction pattern of (f), and (h) schematic representation of diffraction pattern
of (g) zone axes [100]α and [011]θ [25].
17
• Nam et al. [25] have shown that ε carbide forms when a spring steel is tem-
pered at 250 ◦C, and as temperature increases to 350 ◦C, further ε-carbide
precipitates. Simultaneously, θ nucleation takes place, either at dislocations
or at the ε/matrix interface. As cementite nucleates, dissolution of the ε phase
takes place. See Figure 2.6.
• Perez and Deschamps [30] have based their work on the condition that co-
precipitation of ε and θ carbides occurs during tempering, and that solute
competition defines the carbide being precipitated. See Figure 2.7.
R
ad
iu
s 
(n
m
) 
100 
10 
1 
0.1 
Time (min) 
10-4 10-2 1 102 104 
R(ε) 
R(θ) 
R*(ε) 
R*(θ) 
R(ε) 
R(θ) 
R*(ε) 
R*(θ) 
Figure 2.7: Simulation of carbide growth for the metastable ε-phase and the stable
θ precipitates at 200 ◦C. Initially, both phases grow simultaneously in a similar
manner. Once the critical radius for dissolution of the metastable phase, R*(ε),
exceeds the value of the mean radius, R(ε), it destabilises the metastable phase, and
dissolution occurs in order to allow the growth of the stable phase θ [30].
• Andrews [31] stated that cementite can precipitate in steels at temperatures
below 200 ◦C. The proposed mechanism consisted of a ‘zig-zag’ movement
by which a slight displacement of iron atoms breaks the cubic symmetry of
the ferrite phase and builds the orthorhombic unit cell. Furthermore, in high
18
carbon steels, it has been observed that defects play a significant role in offering
powerful nucleation sites where, for instance, cementite has been reported to
nucleate along twin boundaries [11].
Due to its metastable nature, ε carbide phase is absent from commercial thermo-
chemical databases. Instead, most of the computational thermodynamics work has
been on the formation of cementite from ferrite, under both ortho- and paraequi-
librium conditions, such as the work by Ghosh and Olson [32], and Kozeschnik and
Bhadeshia [33].
Paraequilibrium (PE) is a kinetically constrained form of growth, where only
interstitial atoms are able to diffuse and the diffusivity of substitutional species is
considered negligible. Thus, a growing phase under PE conditions would inherit
the substitutional alloy content from the parent phase. Since the substitutional el-
ements cannot partition, instead of considering the individual chemical potential of
each element, instead they are referred to collectively by one hypothetical element,
Z.
Figure 2.8 shows the method of determining the driving force of cementite under
paraequilibrium conditions [32]. The thermodynamic driving force, ∆GN is given
by:
∆GN = (µN,PE−θZ − µN,PE−αZ )xN,PE−θZ + (µN,PE−θC − µN,PE−αC )xN,PE−θC , (2.2)
where xN,φi is the mole fraction of element i in the critical nucleus, and µ
N,φ
i is the
corresponding chemical potential for phase φ, where φ = PE–α, PE–θ.
The chemical potential of Z, µφZ, is given by the sum of each individual substitu-
tional alloying element:
µφZ ≡
∑
yjµ
φ
j , (2.3)
where y is the site fraction for the substitutional alloying element j.
19
Z C
PE‒α 
PE‒θ 
Z3C
µC
N, PE−α
µC
N, PE−θ
µC
PE−α,µCPE−θ
µZ
PE−α,µZPE−θ
µZ
N, PE−α
µZ
N, PE−θ
Mole fraction C 
XCPE−α XC0,α
G
ΔG
N
 
Figure 2.8: Schematic plot of Gibbs energy vs. composition for determining the
driving force for cementite nucleation under paraequilibrium conditions based on
the parallel tangent construction, where ∆GN is the driving force for nucleation, µ
is the chemical potential for paraequilibrium ferrite (PE–α) and cementite (PE–θ).
XPE−αC is the initial carbon content in the alloy and X
0,α
C is the carbon content in
PE–α after complete precipitation of PE–θ. After Ghosh and Olson [32].
Kozeschnik and Bhadeshia [33] modelled cementite precipitation under parae-
quilibrium conditions for two 0.4 wt.% C alloys containing 1.6 wt.% (high) and 0.3
wt.% (low) silicon. Inconsistent with the widely accepted effect of silicon in retarding
the formation of cementite during martensite tempering, there was no meaningful
difference between the high and low silicon alloys. The only appreciable retarding
effect of silicon was seen at low carbon contents only, Figure 2.9 (a). An example of
a set of calculations under paraequilibrium conditions are shown in Figure 2.9 (b),
for the high silicon alloy containing 1.6 wt.% silicon.
20
(a) (b) 
Figure 2.9: (a) Kinetics of paraequilibrium cementite precipitation at 315 ◦C for the
low silicon alloy (solid line) and high silicon alloy (dotted) as a function of carbon
content, and (b) computed ∆G for α → α + θ at 315 ◦C for the 1.6 wt.% Si alloy.
After Kozeschnik and Bhadeshia [33].
They went further in explaining the counterintuitive results of the effect of silicon
in the tempering kinetics. Following the work by Kalish and Cohen [4], where it
was shown that carbon preferentially segregates to dislocations rather than forming
carbides, Kozeschnik and Bhadeshia have postulated that the apparent effect of sili-
con, in reality is due to the time taken for the dissolution of defects in order to make
the carbon available for precipitation. Therefore, as less carbon will be available for
precipitation (assuming that the majority has been segregated to dislocations), then
the silicon will greatly affect tempering kinetics, as shown by the 0.01 wt.% case in
Figure 2.9 (a).
In further understanding the effect of substitutional elements such as silicon, alu-
minium and manganese, Jang et al. [34] have performed Density Functional Theory
(DFT) calculations to obtain the formation energy for ε and θ carbides. In this
modelling approach, the formation energy is given by the energy difference between
21
the products and the reactants:
∆U = E(FelCmXn)− lE(Fe)−mE(C)− nE(X), (2.4)
where E(FelCmXn) is the total energy at the corresponding equilibrium lattice pa-
rameters, E(Fe) is the total energy of the bcc phase, and E(C) is the total energy
of the carbon (graphite) phase. X is the substitutional element, such as Si, Al or
Mn. E(X) is the total energy of the substitutional element at the corresponding
equilibrium lattice constants.
The substitution of silicon into Fe3C-θ was simulated with an orthorhombic struc-
ture of composition (Fe11Si)C4, which corresponds to 4.07 wt.% silicon content, i.e.
one out of the twelve iron atoms constituting the unit cell is substituted by silicon
[35]. The structure is shown in Figure 2.10. In the case of ε-carbide, one substitu-
tional atom was introduced into a unit cell containing 6 Fe atoms [34]. The values
are summarised in Table 2.6.
Figure 2.10: (a) Unit cell of cementite (orthorhombic, Pnma) containing 12 Fe
atoms and 4 C atoms [35], and (b) ε-carbide (hexagonal, P6322), with the basal
plane shown on the left and the unit cell on the right [34].
22
Table 2.6: Formation energies (kJ mol−1 of unit cells containing 12 Fe atoms. The
values in brackets are the differences between the pure carbide and when one Fe
atom per unit cell is substituted by an alloying element [34].
System ε-carbide Cementite
Pure carbide 106.0 86.1
Si-substituted 154.4 (+48.4) 123.2(+37.1)
Al-substituted 84.7 (-21.3) 72.5 (-13.6)
Mn-substituted 74.8 (-31.2) 81.1 (-5.0)
From the DFT results, it appears that it is less favourable to substitute silicon
into ε than in cementite. Jang et al. have interpreted these results in light of the
enhancement of ε precipitation when silicon is incorporated into the lattice not being
a thermodynamic effect. Instead, they have postulated that since silicon substitution
in the iron lattice leads to a contraction in lattice parameters, the presence of silicon
would reduce the c-axis, improving the coherency between the ε-carbide and the
matrix during nucleation.
2.3.1 Characterising carbides
Two recent techniques that have been developed over the past decade for charac-
terising carbides have been 3D atom probe tomography (3D ATP), and high energy
X-ray diffraction methods using synchrotron radiation sources. The former has been
considered useful especially in studying the alloying elements distribution across the
matrix/carbide interface, as exemplified by the work carried out by Miyamoto et al.
[26], Zhu et al. [28, 29] and Caballero et al. [27]. Nevertheless, care should be taken
when identifying carbides using this method, as this particular technique does not
provide direct evidence of the crystallographic structure of the phase. In the case
of the APT technique, the detected carbon level is used to distinguish the carbides
23
present.
Characterising carbides becomes a particular challenge, as often in literature,
different chemical compositions have been reported for the ε-carbide phase. If the
chemical composition of the iron-carbide is to be expressed in the form of FexC, in
most cases, the orthorhombic cementite case is reported to have a value of x= 3.
However, in the case of ε, many researches have found that the value of x can range
between 2 and 2.4, as reviewed by Ge´nin [14]. Furthermore, Ge´nin proposed a Fe9C4
structure, as this would ensure the highest occupancy of carbon sites within a hcp
ordered structure.
Additional transitional carbides that have been reported in the literature are η-
and χ-carbides. The former, an orthorhombic structure of composition Fe2C, has
been reported be an intermediate phase between ε and θ. The latter carbide, χ, also
known as the Ha¨ag carbide, was hypothesised to have a monoclinic crystal structure,
of composition (Fe,Mn)5C2 [36, 37], which was thought to occur in Mn-containing
alloys, also as an intermediate carbide between ε and θ.
Table 2.7 summarises the carbides that have been reported, with their corre-
sponding carbon content in weight percent. Note that in the case of ε-carbide, the
most widely accepted form of Fe2.4C has been considered. Given the similarity in
carbon content, it thus becomes a challenge to distinguish the carbides from each
other solely based on the carbon content.
Table 2.7: List of possible carbides occurring in tempered martensite.
Phase ε η χ θ
Carbon content in wt.% 8.2 8.7 7.9 6.7
The other experimental technique that has gained strength is high-energy X-rays
from a synchrotron source. As it is based on crystal diffraction methods, it over-
comes the challenge of distinguishing the carbides phases. Given its high energy
24
X-rays, it allows further penetration into the sample, offering a more representative
analysis of the bulk material. Although for precise details such as size of the car-
bides, the TEM would be the technique of choice, X-ray diffraction (XRD) methods
allow a better estimate of the volume fraction of the carbide phase present. Fur-
thermore, there are other aspects that are better resolved using XRD methods than
TEM, such as dislocation density and tetragonality of the matrix phase. Given the
abundant information that can be extracted from XRD methods, in the current
work X-ray diffraction plays a significant role in understanding the effect of silicon
in carbide precipitation and throughout martensite tempering.
The ultimate aim of this research is to elucidate the microstructural parame-
ters that determine strengthening in martensitic steels. The research carried out by
Choi [6] showed that by optimising the microstructure via controlled precipitation,
outstanding mechanical properties of 2350 MPa tensile strength and 25% reduction
area have been achieved. In his work, a combination of heat treatment and silicon
additions to retard the ε → θ carbide transition was explored.
25
Chapter 3
Interrupted ageing in steels
This chapter looks at the effect of interrupted ageing in tempered martensite. In-
terrupted ageing is an intermediate step between quenching and tempering, where
quenched martensite is left to rest at room temperature for a defined period of time,
during which carbon segregates into dislocation cores. As exemplified by aluminium
metallurgy, the incorporation of interrupted ageing (IA) allows microstructural tai-
loring, where Lumley et al. [38, 39] reported simultaneous improvements of both
strength and ductility through interrupted ageing in aluminium alloys.
This chapter presents the effect of interrupted ageing in martensitic steels, and
models carbon segregation to dislocations with the aim of understanding carbon be-
haviour throughout the heat treatment. The study has been conducted in a commer-
cial spring steel grade, where for a single alloy, by manipulating the heat treatment
a wide range of microstructural and hardness properties have been obtained.
3.1 Introduction
3.1.1 Ageing in steels
As-quenched martensite has a very high density of crystal defects, such as dislo-
cations. Dislocations act as carbon-sinks, where the strain fields surrounding the
26
dislocation cores attract carbon atoms [40], forming Cottrell atmospheres that can-
cel the dislocation strain. Initially, carbon would be present in interstitial sites,
forming a super-saturated solid solution; however, the metastability of these struc-
tures causes redistribution of carbon atoms during ageing and tempering [41]. Thus,
freshly quenched martensite is known to undergo ageing even at room temperature
[11]. During room temperature ageing, the diffusion of solutes is restricted due to
slow kinetics, and only interstitials such as nitrogen and carbon diffuse appreciably.
Understanding such redistribution of interstitials is of significant interest as the dis-
tribution of carbon atoms immediately after quenching has a marked effect on the
subsequent heat treatment [4].
Figure 3.1: XRD patterns obtained using synchrotron radiation, (λ= 0.189821 nm)
for Fe–1.13C wt.% aged at room temperature for 3.5 years, where prolonged room
temperature ageing (solid line) shows peaks corresponding to transitional carbides
[41].
Van Genderen et al. [41] reported room temperature formation of transitional
carbides (either ε or η) in a binary alloy of composition Fe–1.13C wt.% that has
27
been aged for 3.5 years, Figure 3.1. It is evident that appreciable microstructural
changes occur even at room temperature. By introducing an intermediate stage
between quenching and tempering called interrupted ageing (IA), the aim of this
chapter is to explore the effect IA can have on carbide precipitation, in particular
microstructural properties such as particle spacing and size.
3.1.2 Thermoelectric power
A technique that has been employed in order to study carbon-controlled reactions,
such as precipitation, is thermoelectric power (TEP) [42, 43, 44, 45, 46, 47]. The
main benefit of TEP is attributed to its sensitivity in detecting variations in the
concentration of atoms in solid solution. This feature can be exploited for studying
the kinetics of carbon diffusion during IA. TEP operation consists in clamping the
two extremes of the material in the form of a rectangular block on two metallic
reference blocks (copper was used in this work). One block is kept at 15 ◦C, while
the other is at 25 ◦C, and given the temperature gradient, ∆T , of 10 ◦C, the response
in voltage, ∆V is recorded. The set-up is illustrated in Figure 3.2.
Figure 3.2: Experimental set-up of the TEP system [43].
The TEP, S∗, of the tested material is given by:
S = S∗ − S∗Cu =
∆V
∆T
, (3.1)
28
where S is the observed TEP, S∗Cu is the TEP of the reference blocks (pure copper,
99.99 wt.%), S∗Cu= 1.91 µV K
−1 at 20 ◦C.
In steels, the observed TEP is the result of various additive terms [48]:
S∗ = S∗0 + ∆Sss + ∆Sd + ∆Spcpt, (3.2)
where S∗0 is the TEP of the base metal, ∆Sss is due to the elements in solid solution,
∆Sd is due to dislocations, and ∆Spcpt is due to the presence of precipitates. Such
an observation has been related to the interaction between free carbon atoms in
solid solution and dislocations in pre-strained ultra-low carbon steels [47].
Figure 3.3: (a) Room temperature TEP signal after tempering 25 minutes at dif-
ferent temperatures for an Fe–1.23C–0.64Cr–0.34Mn–0.22Si–0.07Ni–0.12Mo–0.10Cu
wt.% alloy [49], and (b) Room temperature TEP evolution after various strain age-
ing treatments on an Fe–4C–7N–225Mn–22Al–11P–9S (×10−3 wt.%) alloy, where
∆Sa is the TEP value after the strain-ageing treatment [47].
Tkalcec et al. [49] recorded the TEP value after 25 minutes tempering at different
temperatures, for a Fe–1.23C–0.64Cr–0.34Mn–0.22Si–0.07Ni–0.12Mo–0.10Cu wt.%
alloy (Figure 3.3 (a)). It was observed that the TEP signal varied inversely with
the amount of carbon in solid solution. Furthermore, Lavaire et al. [47] recorded the
room temperature TEP evolution after strain ageing experiments. For an ultra low
carbon (ULC) steel of composition Fe–4C–7N–225Mn–22Al–11P–9S (×10−3 wt.%)
29
that had been cold-rolled to a 50% reduction, the TEP evolution differed between
tempering conditions, Figure 3.3 (b).
Lavaire et al. [47] attributed the observed TEP differences to the ability of the
strained steel to trap carbon atoms on dislocations, and concluded that the observed
TEP evolution was due to the diffusion of carbon in ferrite. All conditions satu-
rate after some time, which appears to be shorter as the tempering temperature
increases. This saturation level would infer complete carbon segregation. Although
the dislocation density was not stated in their work, and some carbon segregation
would have inevitably occurred immediately after cold-rolling, it is clear that TEP
evolution can be used for studying carbon segregation to dislocations during marten-
site ageing.
TEP measurements, combined with the model originally devised by Kalish and
Cohen [4], are used for gaining further understanding of the interrelationship be-
tween processing, microstructure and the mechanical properties of mid-carbon steels
undergoing IA.
3.2 Experimental procedure
The 54SCV6 automotive grade used in spring steels was selected for this current
study. Table 3.1 summarises the measured chemical composition.
Table 3.1: Measured chemical composition of the 54SCV6 grade used, with Fe to
balance.
C Si Mn Cr Cu Ni Mo V
wt. % 0.56 1.59 0.71 0.54 0.12 0.06 0.02 0.13
The as-received material was in the form of rods, with an initial hardness of 317
HV1.
30
3.2.1 Heat treatment
Cylindrical rod-like specimens of 3 mm in diameter and 12 mm in length were
machined. The heat treatment of these specimens was carried out in an Adamel
Lhomargy dilatometer (model DT1000) as shown in Figure 3.4. The conventional
heat treatment consists in tempering immediately after quenching. In the heat
treatment involving interrupted ageing, a dwell time is introduced between quench-
ing and tempering, where the sample is removed from the dilatometer and kept at
room temperature for a prescribed time.
T 
t 
IA 
IA: 
Interrupted 
Ageing 
Conventional 
HT 
T 
t 
(a) (b) 
Figure 3.4: (a) Conventional heat treatment for tempered martensite, and (b) heat
treatment used for interrupted ageing.
A series of heat treatments were designed in order to study the effect of:
• The duration of the interruption.
• The duration and temperature of the tempering stage.
• Austenitisation temperature.
Table 3.2 shows a summary of all the 11 conditions studied, where T is the temper-
ature, t is time and subscripts 1 and 2 refer to the austenitisation and the tempering
stages respectively. tIA is the duration of the interruption time at room temperature.
The choice of the austenitisation conditions reflects the typical industial process
for spring steels, where austenitisation is carried out at 880 ◦C for 180 s. Tempering
31
is usually carried out in the range of 350–450 ◦C for 1800–3600 s. However lower
tempering temperatures were selected in this study in order to optimise hardness
properties. Furthermore, a relatively low tempering temperature would ensure in-
complete reactions, where carbon would still remain in solid solution. This gives
greater flexibility in studying the effects of interrupted ageing, as will be shown
further on.
Table 3.2: Summary of heat treatments in order to study the effect of interrupted
ageing.
T1 (
◦C) t1 (s) tIA T2 ( ◦C) t2 (s)
L1 880 180 7.2 hours 250 1800
L2 880 180 3 days 250 1800
L3 880 180 30 days 250 1800
L4 880 180 30 days 250 300
L5 880 180 - - -
L6 880 180 - 250 1800
L7 880 180 - 250 5400
L8 880 180 - 300 1800
L9 880 180 30 days 300 1800
L10 1000 180 - 250 1800
L11 1000 180 - 300 1800
3.2.2 Determining the prior austenite grain size
As there are two austenitisation temperatures being used, 880 and 1000 ◦C, this will
result in a microstructural difference. The conditions austenitised at 1000 ◦C are
bound to have a larger prior austenite grain (PAG) size, which is likely to affect the
32
subsequent martensite formation [7].
Typically, in industry, austenitisation is carried out at temperatures near 880 ◦C.
1000 ◦C was chosen to account for the dissolution of ‘reluctant carbides’ [50] such as
vanadium carbide, V4C3. Given the list of alloying elements of the 54SCV6 grade,
it is likely that vanadium carbide is present within the as-received state. Thermo-
Calc predicts the dissolution of vanadium carbide at 983 ◦C for the given alloying
system. Although grain coarsening will only take place after the dissolution of the
carbide phases [51], it is still worth comparing the PAG sizes for the austenitisation
conditions.
The PAG boundaries were revealed by the thermal etching method [52], where
samples of 5 mm diameter and 12 mm length were polished down to a 1 µm dia-
mond paste and etched with 2% nital solution. Austenitisation was carried out in an
805 Ba¨hr dilatometer at the respective temperatures, and quenched. The specimens
were later examined under an optical microscope, at which the grain boundaries
were visible. The average grain size was estimated using the mean linear intercept
method [53].
3.2.3 Mechanical testing
Hardness
Dilatometry samples were polished to a 1 µm diamond paste finish. Two sets of
hardness measurements were carried out. The first set shortly after heat treatment
in the dilatometer, and the second set after a six months period from the first set of
measurements. Vickers hardness tests were carried out using a Mitutoyo MVK-H2
microhardness indenter with a 1 kg load. 10 measurements were taken from each
condition.
33
Tensile testing
Tensile tests were performed on dilatometry samples of 3 mm diameter and 15 mm
length. The heat treatments described on Table 3.4 were followed and the tensile
tests were carried out immediately after removal from a LK02 dilatometer. 3 runs
were made per condition, using a Microtest tensile machine.
Compression testing
In addition to the tensile tests, the conditions L3, L5 and L8 were selected for
compression testings. Dimensions of 5 mm diameter and 8 mm length were used.
3.2.4 Microstructural characterisation
Microstructural characterisation was carried out using transmission electron mi-
croscopy (TEM) on a Philips CM30 model. 3 mm discs were cut from the dilatome-
try samples, and were electropolished using a solution composed of 15 ml percholoric
and 85 ml ethanol. A twinjet electropolisher Struers Tenupol-5 was used, operated
under a voltage of 20.5 V, at 16 ◦C, and a flow rate set at 12. Over 500 intralath
precipitates were analysed per condition, taken from three to four different TEM
frames, all under the same magnification, ×52 k. Characteristics such as particle
length, width and centre-to-centre spacing were manually measured using ImageJ.
Using bright field imaging techniques, features down to ∼5 nm could be resolved
by TEM. Stereological corrections were applied [54]. Furthermore, for comparison
purposes, given the length, l, and width, w, an expression for the equivalent radius,
requiv was derived:
requiv =
(
3
16
(waverage)
2laverage
) 1
3
, (3.3)
where the subscript ‘average’ stands for average values.
34
3.2.5 Thermoelectric power
Thermoelectric power (TEP) measurements were carried out on quenched specimens
of dimensions 30×2×1 mm3, using a TechLab-GEM-PPM INSA Lyon model. The
TEP evolution in the as-quenched state was recorded over a period of 3 months.
For the austenitisation at 880 ◦C condition, 3 runs were made for reproducibility.
3.3 Characterisation by dilatometry
3.3.1 Summary of critical temperatures
The dilatometry curves obtained during the heat treatment were analysed in or-
der to extract the critical transformation temperatures, such as the austenitisation
start and finish temperatures (Ac1 and Ac3, respectively) and the martensitic start
temperature (Ms). These are summarised in Table 3.3.
Table 3.3: Summary of critical temperatures obtained from dilatometry.
Ac1 Ac3 Ms
798 ± 3 ◦C 832 ± 3 ◦C 238 ± 9 ◦C
Given that Ac3 is below the austenitisation temperature, complete austenitisa-
tion is guaranteed during the heat treatment. A typical dilatometry curve is shown
in Figure 3.5 for the austenitisation and quenching part. The transformation tem-
peratures were manually measured, where the point at which a change in slope
occured was estimated by eye, as shown in the magnified view in Figure 3.5, The
values in Table 3.3 represent an average of 11 readings.
35
Figure 3.5: Typical dilatometry curve showing the transformations for the austeni-
tisation stage.
3.3.2 PAG size
Figure 3.6 shows typical optical microgaphs after thermal etching for 3 minutes at
(a) 880 and (b) 1000 ◦C. The PAG boundaries are hardly visible for the 880 ◦C
austenitisation temperature due to the small grain size. The average PAG size was
measured to be 5.1 ± 0.4 µm for 880 ◦C and 23.3 ± 0.6 µm for 1000 ◦C. Increasing
the austenitisation temperatures results in a five-fold increase in the PAG size.
36
Figure 3.6: Optical micrograph revealing prior austenite grain size by the method
of thermal etching for 3 minutes austenitisation at (a) 880 and (b) 1000 ◦C, and (c)
880 ◦C taken at a magnification of ×1000.
3.4 Hardness and microstructural analysis
3.4.1 Hardness testing
The hardness for the various heat treatment conditions listed in Table 3.2 are shown
in Figure 3.7. Three immediate trends can be spotted. The first is that the initial
hardness (dark grey) is higher for the conditions undergoing interrupted ageing (L1–
L4, with the exception of L9). The second is that hardness properties become stable
via interrupted ageing, whereas for those undergoing direct tempering an increase
in hardness is observed. Finally, the third is that after the 6 months following the
initial readings, the conditions undergoing direct tempering (L6–L8) as well as L9,
37
reach hardness values approaching the interrupted ageing ones (∼ 700HV1).
600 
650 
700 
750 
800 
850 
900 
L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 
M
ic
ro
ha
rd
ne
s 
(H
V1
) 
Condition 
Initial 
After 6 months 
-1.3 % -1.8 % 0.2 % -1.8 % 0.1 % 9.5 % 8.2 % 5.9 % 5.9 % 6.3 % 5.1 % 
Austenitisation T ( °C) 
Duration of IA (days) 
Tempering t (s) 
Tempering T ( °C) 
880 
1/3 
250 
1800 
880 
3 
250 
1800 
880 
30 
250 
1800 
880 
30 
250 
300 
880 
30 + 
- 
- 
880 
- 
250 
1800 
880 
- 
250 
5400 
880 
- 
300 
1800 
880 
30 
300 
1800 
1000 
- 
250 
1800 
1000 
- 
300 
1800 
KEY 
Figure 3.7: Hardness measurements after heat treatment (dark grey) and 6 months
after (light grey). The percentage increase in hardness is shown at the bottom of
the bars, and the error bars indicate one standard deviation.
Upon quenching, the formation of martensite is accompanied by a lattice expan-
sion. During tempering, decomposition of martensite takes place, as the initially
trapped carbon diffuses out. The tetragonality is reduced and the process is accom-
panied by the following: lattice contraction due to the decomposition of martensite,
and a corresponding expansion due to the decomposition of retained austenite [55].
On the series of work by Averbach et al. where the dimensional stability of compo-
nents has been studied, it was shown that even after the final manufacturing stage,
further dimensional changes were observed due to transformations taking place at
room temperature: “In all cases, the martensite contractions are quite additive in
that if the transformation at a higher temperature is stopped, it proceeds at room
temperature at the same rate it would have if all the shrinkage had occurred at room
temperature” [55].
This seems to confirm the postulate that at low temperatures such as 250 ◦C, the
precipitation process does not come to a completion, but instead continues at room
38
temperatures, although most probably at a reduced kinetics. For example, studying
the effect of tempering in the dimensional stability of martensite, Averbach et al.
showed that tempering 1 hour at 120 ◦C causes the same contraction as tempering
1050 days (∼35 months) at 20 ◦C.
By incorporating the IA stage before the conventional tempering, a higher de-
gree of reaction completion is achieved, as during the IA stage, ‘ordering’ of carbon
atoms takes place. In this sense, having an interruption stage prior to tempering
provides a more efficient solute take-up. This could also be the reason why in indus-
try, tempering is conventionally carried out at higher temperatures between 400 and
450 ◦C, as from the dimensional stability point of view, this would ensure reaction
completion.
For condition L9, despite the interrupted ageing, the hardness is significantly
lower than conditions L1–L4, and it is not stabilised even after the 30 days of in-
terruption time. A possible explanation for this discrepancy is presented in the
forthcoming microstructural analysis.
A second set of experiments for conditions L1 and L6 were performed using
a lower quenching rate of 15 ◦C/s. In this case, the measured hardness for L115
and L615 were 661.8 ± 3.1 and 659.9 ± 5.8 HV1, respectively. Comparing these
values with the ones obtained using the maximum quenching rate achieved by the
dilatometer, 75 ◦C/s (Figure 3.8), this shows that the effect of the interrupted ageing
is limited to high quenching rates.
So far, only the effects of precipitates have been considered for understand-
ing hardness properties. Nevertheless, there are other microstructural aspects that
must be taken into account: carbon content in solid solution, dislocation density and
structure, and whether dislocations are mobile or locked due to segregated carbon
atoms. Understanding the different strengthening contributions will be explored in
detail in Chapter 6 on Plasticity.
39
0 
200 
400 
600 
800 
1000 
265 270 275 280 285 
Te
m
pe
ra
tu
re
 ( 
°C
) 
Time elapsed during heat treatment (s) 
A
us
te
ni
tis
at
io
n 
Quenching 
Figure 3.8: Representative dilatometry curve showing temperature-time during sam-
ple heat treatment.
3.4.2 TEM
Selected bright field images are shown in Figures 3.9 and 3.10. The differences in
the intralath carbides are not very obvious at first, but quantitative metallography
shows obvious trends. The values obtained for length, l, and width, w, of intralath
carbides, and the corresponding aspect ratio for each condition are summarised in
Table 3.4. Furthermore the extracted equivalent radius, requiv, in Equation (3.3)
and the centre-to-centre spacing between carbides are also given. Conditions L10
and L11 have been omitted due to the different austenitisation conditions and L5
as no intralath carbides were observed for the as-quenched condition.
40
Figure 3.9: Representative bright field images showing intralath carbides for con-
ditions L3, L6, L8 and L9, where the effect of interrupted ageing and tempering
temperature are compared.
Figure 3.10: Bright field images for conditions L10 and L11 (austenitisation at
1000 ◦C). Notice the inhomogeneity of the microstructure.
41
Table 3.4: Summary of the microstructural analysis.
l (nm) w (nm) Aspect ratio requiv (nm) Spacing (nm)
L1 51.5 8.8 5.9 9.1 36.4
L2 51.7 9.1 5.7 9.3 33.2
L3 48.0 7.0 6.9 7.6 29.0
L4 40.6 7.8 5.2 7.7 28.4
L6 56.1 8.7 6.5 9.2 32.2
L7 48.2 8.4 5.8 8.6 39.9
L8 46.9 7.1 6.6 7.6 33.1
L9 42.9 5.4 7.9 6.2 29.1
Histograms for the length, width and spacing measurements for condition L3 are
shown in Figure 3.11.
Figure 3.11: Histograms for (a) length, (b) width, and (c) spacing measurements
for condition L3.
Kolmogorov–Smirnov two sample test
A statistical method is needed in order to systematically compare the size and
spacing distributions of carbides. The Kolmogorov–Smirnov (K-S) statistical test
[56] is carried out between two conditions in order to compare the distribution of
two sample data sets, x1 and x2. In the K-S test, the null hypothesis states that
x1 and x2 are from the same continuous distribution. The output of the test is as
42
follows:
• 1: reject null hypothesis, i.e. x1 and x2 are not from the same continuous
distribution;
• 0: valid hypothesis.
The null hypothesis is rejected when the p-value is less than the significance level
(5%), i.e. p ≤ 0.05. The K-S test can be applied to distinguish and compare the
microstructure of two given conditions. The results from the test are summarised
in Table 3.5.
The observed trend is that IA results in a finer microstructure in terms of smaller
particle size and spacing (L3 vs. L6 and L9 vs. L8), and that the duration of temper-
ing after 30 days of IA does not influence the microstructure (L3 vs. L4). Tempering
at a higher temperature after IA leads to smaller carbides, without altering their
spacing (L3 vs. L9). In general, the incorporation of the IA stage leads to finer
microstructures.
The last two columns from Table 3.4 are represented in Figure 3.12 on a requiv
vs. spacing plot.
Figure 3.12: Summary for intralath carbide equivalent radius and spacing for con-
ditions L1 to L9, except L5.
43
Table 3.5: Summary of the outcome from the K-S test, where p-values are given in
brackets (when p-values are less than 0.001 they are indicated as ∼0).
Conditions Particle size Particle spacing Interpretation
L3 vs. L6 1 ( ∼0) 1 ( ∼0) 30 days of IA gives a finer
microstructure (both precipitate
size and spacing)
L1 vs. L2 0 (0.541) 1 (0.0013) Size does not change but the spac-
ing decreases between 1/3 and 3
days of IA
L3 vs. L4 0 (0.121) 0 (0.5535) After 30 days of IA, the temper-
ing time does not influence pre-
cipitation
L2 vs. L6 0 (0.633) 0 (0.2827) 3 days of IA does not alter the
microstructure significantly
L6 vs. L7 1 (0.003) 1 ( ∼0) In direct tempering, longer tem-
pering decreases carbide size and
increases spacing
L6 vs. L8 1 ( ∼0) 0 (0.1736) In direct tempering, increasing
tempering temperature decreases
carbide size
L3 vs. L9 1 ( ∼0) 0 (0.6750) After 30 days of IA, increasing
tempering temperature decreases
carbide size
A clear trend is observed in Figure 3.12: as the duration of the interrupted ageing
increases, the spacing between carbides decreases; whereas as tempering tempera-
ture increases, the size of the carbides decreases. Hardness measurements showed
44
that the hardness value was significantly lower than the conditions undergoing inter-
rupted ageing. Condition L9 presents the finest microstructure in terms of particle
size and spacing. The low hardness could be due to a lesser degree of precipitation
hardening if particle shearing mechanism is assumed [57]. The increase in hardness
in L9 over the 6 months period is not understood yet. However, an initial postulate
is that due to the small spacing between carbides, dissolution of carbides or carbon
clusters may have taken place, increasing the carbon content in matrix.
Overall, the reason for the finer microstructures obtained via IA is based on the
postulate that IA provides a more efficient take-up of carbon atoms from the solid
solution. The following sections aims at understanding and modelling carbon seg-
regation to dislocations by means of thermoelectric power measurements, adopting
the Kalish-Cohen model [4] for carbon segregation.
3.5 Modelling carbon segregation during inter-
rupted ageing
3.5.1 The Kalish-Cohen model
In understanding the microstructural changes occurring during strain-ageing of marten-
site, Kalish and Cohen [4] presented a model to describe the process by which carbon
segregates into dislocation fields. Consider a carbon atom in solid solution in freshly
quenched martensite. Due to the strain fields spanning around dislocation lines, the
carbon atom will be attracted to the core of the dislocation. At this stage it is
necessary to state the assumption made that the only carbon-sink acting within the
microstructure are dislocations.
If the spacing between two adjacent dislocations is given by d⊥ then, as illus-
trated in Figure 3.13 (a), the maximum distance that a carbon atom would need to
diffuse before reaching the nearest trapping site is given by xmax.
45
During dilatometry experiments, the Ms was determined to be 238 ± 9 ◦C. This
would correspond to a dislocation density, ρ, of around 8×1015 m−2 [58]. Assum-
ing a homogeneous distribution of dislocations throughout the material, the average
spacing between two adjacent dislocations, Figure 3.13 (a), is given by ρ−
1
2 ; hence
d⊥ ≈ 11.2 nm, yielding a value of 5.6 nm for xmax. For simplicity, the ρ value for
the as-quenched condition is used. Nevertheless, this value will vary throughout
different heat treatment conditions. For instance, in the case of direct tempering,
the dislocation density will reduce after tempering and this will slightly change the
process of carbide precipitation.
The interaction energy between the carbon atom and the dislocation strain fields
at the dislocation core has been stated to be 0.46 eV at a distance of one Burgers
vector (1b) [4], where b is taken to be 0.25 nm. Furthermore, it is known that at the
limit of interaction radius, ri, Figure 3.13 (b), the interaction energy experienced by
the carbon atom is equivalent to the thermal energy kT = 0.026 eV, where k= 8.62
× 10−5 eV K−1 and T = 298 K, equivalent to a distance of 17.8b. [4].
Figure 3.13: (a) Illustration of the distance that a carbon atom needs to travel before
being segregated into the core of the dislocation, and (b) schematic diagram of the
dislocation core, where rs and ri are the segregation and the interaction radius,
respectively.
46
The process suggested by Kalish and Cohen for carbon segregation into disloca-
tion strain fields is summarised in Figure 3.14.
Figure 3.14: Illustration of the carbon segregation process by Kalish and Cohen [4].
It has been suggested that initially carbon within the interaction region, ri,
segregates into the segregation zone, rs. This leaves a temporary carbon-depleted
region within the ri − rs zone, while the dislocation core remains carbon rich. This
establishes a diffusion gradient between the ri < r < rs region and the region beyond
ri, which will draw the carbon into the depleted zone in order to even out the carbon
content within the matrix. The following section attempts at explaining the process
occurring during interrupted ageing by applying the Kalish-Cohen model interpreted
in light of TEP measurements.
3.5.2 Thermoelectric power
The TEP evolution recorded over a 3 month period is shown in Figure 3.15.
47
Figure 3.15: TEP evolution for as-quenched martensite over a 3 month period for
austenitisation at 880 and 1000 ◦C.
The TEP signal for the 1000 ◦C condition is higher than for 880 ◦C. TEP is very
sensitive to the state of the microstructure [49]. Differences in the microstructure
between the two samples such as prior austenite grain size, and carbon concentration
in solid solution after the dissolution of secondary carbides, could contribute to
a difference in TEP signal. Recalling Equation (3.2), the total TEP signal is a
contribution from solid solution (∆Sss), dislocations (∆Sd) and precipitate effects
(∆Spcpt).
During room temperature ageing of quenched martensite, carbon segregation
into dislocation cores would cancel out the dislocation strain fields, hence the ∆Sd
term would change. Similarly, the signal due to solid solution would also evolve,
as less free carbon in solid solution remains. The ∆Spcpt term becomes significant
only in the case of small, coherent precipitates [45]. For the sake of simplicity, the
following assumptions are made in interpreting TEP evolution:
• Quenching is sufficiently fast to avoid the formation of small coherent carbide
precipitates during cooling.
• The carbon/dislocation interaction energy remains constant until the disloca-
tion core becomes saturated; i.e., ∆Sd remains constant during early segrega-
48
tion stages.
• Following the work by Lavaire et al. [47], in the first instances after quenching,
the decrease in TEP magnitude is due to C segregation alone.
Studying the TEP response for the individual effects of each microstructural con-
tribution goes beyond the scope of the current research. Instead, focus has been on
the 880 ◦C condition with the aim to study the evolution of the TEP throughout
room temperature ageing. Three runs have been made for reproducibility and the
curves are shown in Figure 3.16.
Figure 3.16: TEP evolution for as-quenched martensite from austenitisation at 880
◦C, where the inset shows a detailed view of the three runs (a), (b) and (c).
From the central frame in Figure 3.16, it can be seen that the gradient of the curve
changes twice, where the changing points are marked with an arrow. It appears that
first there is an increase (slightly linear) in TEP signal, although some scattering
of the data occurs at first quenching instances. At (1), the TEP curve flattens out,
as it approximately reaches a plateau for some time, followed by further increase in
49
TEP from (2). The aim of the following sections is to extract from the TEP curves
semi-quantitative information regarding carbon segregation.
The first step is to mark the moment at quenching, time= 0 seconds, as the
beginning of carbon segregation. Therefore, the time elapsed since quenching is an
indication of the carbon segregation in martensite. Then, given the time lapse, and
with a suitable diffusivity value, the distance diffused by a carbon atom could be
determined.
3.5.3 Semi quantitative analysis
The time elapsed at (1) and (2) (Figure 3.16) were found to be 1.8 ± 0.2 and 5.5 ±
0.6 hours, respectively. Given the times at points (1) and (2), the diffusing distance
of the carbon atom can be determined. Using reported value from literature for the
diffusivity of carbon in ferrite at room temperature, DC,α = 2.8 × 10−22 m2 s−1 [59],
the diffusing distance, <x>, can be determined:
< x >=
√
2DC,αt, (3.4)
Moreover, by finding the equivalent distance in terms of b units, and assuming
that the interaction energy between the core of the dislocation and at the ri limit
varies linearly, it is possible to estimate the interaction energy between the diffusing
carbon atom at the dislocation. Knowing that at a distance of 1b the interaction
energy is 0.46 eV, and that at a distance of 17.8b the interaction energy decreases
to 0.026 eV [4], a linear decrease is assumed between these two ranges. The values
are summarised in Table 3.6.
50
Table 3.6: Estimating diffusing distances and carbon-dislocation interaction energies
from TEP results.
Time (h) <x> (nm) <x> (b) IE (eV)
(1) 1.8 ± 0.2 1.9 7.7b 0.060
(2) 5.5 ± 0.6 3.3 13.3b 0.034
At time (2), the estimated interaction energy is 0.034 eV. Previously it was de-
termined that at the limit of ri the interaction energy is 0.026 eV, equivalent to a
distance of 17.8b. Thus it can be inferred that at even after 5.5 hours of quenching,
the carbon within the rs < r < ri region is still reaching the dislocation core.
This is consistent with the microstructural observations made by TEM. Results
from the Kolmogorov-Smirnov statistical test on Table 3.5 show that when com-
paring conditions L1 vs. L2, increasing the interruption time from 7.2 hours to 3
days the average spacing between intralath carbides becomes smaller, also shown in
Figure 3.12.
Initially, the plateau region of the TEP curve (marked by (1) and (2)) was
thought to correspond to the stage in the Kalish-Cohen model where the rs < r
< ri region temporarily becomes depleted, and further carbon from the matrix is
drawn due to the established diffusion gradient. Thus, it would relate to a period
of time where minimum carbon diffusion takes place due to carbon competition be-
tween neighbouring dislocations. However, between (1) and (2) there is a time lapse
of ∼ 3.7 hours, which appears to be rather high, especially given that one would
expect such flux of carbon should be continuous instead of maintaining the depleted
region for 3.7 hours.
Nonetheless, the quantification is sensitive to the value of DC,α used for estimat-
ing carbon diffusion. For instance, Veiga et al. [60] have computed DC,α values using
different computational methods (atomistic simulations, elasticity calculations and
51
simple random walk) for both screw and edge dislocations, in order to determine
the kinetics of Cottrell atmosphere formation at 300 K. The values are summarised
in Table 3.7.
Table 3.7: Diffusion coefficient values from literature [60].
Dislocation Method DC,α (m
2 s−1)
Edge Atomistic simulations 4.2 × 10−21
Edge Elasticity calculations 2.5 × 10−21
Edge Simple random walk 2.6 × 10−21
Screw Atomistic simulations 2.3 × 10−21
Screw Elasticity calculations 1.7 × 10−21
Screw Simple random walk 2.6 × 10−21
The values reported by Veiga et al. [60] are approximately one order of magni-
tude lower than the value used [59]. The use of smaller DC,α values would slightly
overestimate the diffusing distances. For instance, using the value from elasticity
calculations for an edge dislocation, the diffusing distances at points (1) and (2) will
be 22.8 and 39.8b respectively. The diffusing distance at (2) almost corresponds to
the actual distance between adjacent dislocations, d⊥= 11.2 nm, equivalent to 44.8b.
Figure 3.17: Schematic representation of the region around dislocations.
52
Figure 3.17 schematically represents the distances between neighbouring dislo-
cations. At high dislocation densities there is solute competition between adjacent
dislocations [61]. A carbon atom sitting in the the region beyond ri will experience
a ‘pull’ from two dislocation fields simultaneously, which would reduce the effective
segregation.
The accuracy of the numerical results presented here strongly depends on the
validity of the assumptions made for the analysis. For the sake of simplicity, it has
been assumed that the dislocation interaction energy varies linearly with distance,
although it has been reported that the variation is parabolic [4]. Furthermore, the
change in TEP signal has been attributed solely to the diffusion of carbon. However,
as carbon segregates into dislocations, the TEP will gradually change as dislocation
strain fields become relaxed due to carbon migration to sites where energy is min-
imised.
Nonetheless, the conclusion that can be reached remains the same: the duration
of the interruption time is related to the process of carbon segregation to dislocation
sites. Hence longer interruption time leads to increased carbon segregation, which
increases the number of effective nucleation sites. By ‘concentrating’ the carbon
atoms enables a more homogeneous dispersion of carbides, manifested by smaller
size and spacing between carbides.
3.6 Tensile and compression testing
3.6.1 Tensile testing
The results for the tensile tests are shown in Figure 3.18 and detailed in Table 3.8.
All conditions showed very little ductility and, on average, one out of three samples
per condition failed before reaching the elastic limit. In the case of conditions L5
and L10, all three samples failed prematurely. Part of the premature failure might
be due to poor surface finish of the tensile specimens. Nevertheless, a comparative
53
study can still be made between the conditions.
The general trend observed is that samples undergoing interrupted ageing show
very little ductility. This is due to the fact that there will always be some carbon
remaining within the dislocation cores that does not precipitate [62, 63]. This point
will be further explored in Chapter 6 (Plasticity).
Figure 3.18: Summary plot of tensile properties.
54
Table 3.8: Summary of tensile properties. The heat treatment for each condition is
given within brackets, in the form of (‘austenitisation temperature (◦C)’ - ‘duration
of IA (hours)’ - ‘tempering temperature (◦C)’ - ‘tempering time (s)’).
Condition σy (MPa) σUTS (MPa) Elongation (%)
L1 (880 - 1
3
- 250 - 1800) 2134 ± 50 2244 ± 14 0.54 ± 0.2
L2 (880 - 3 - 250 - 1800) 2121 ± 30 2251 ± 20 0.45 ± 0.2
L3 (880 - 30 - 250 - 1800) 2140 ± 12 2263 ± 100 0.73 ± 0.6
L4 (880 - 30 - 250 - 300) 2079 ± 2 2098 ± 190 0.50 ± 0.4
L5 (880 - 30+ - x - x) - - -
L6 (880 - x - 250 - 1800) 2087 ± 11 2073 ± 80 0.57 ± 0.3
L7 (880 - x - 250 - 5400) 2073 ± 110 2291 ± 70 0.99 ± 0.6
L8 (880 - x - 300 - 1800) 2151 ± 40 2308 ± 30 2.07 ± 0.2
L9 (880 - 30 - 300 - 1800) 2234 ± 20 2265 ± 11 0.27 ± 0.03
L10 (1000 - x - 250 - 1800) - - -
L11 (1000 - x - 300 - 1800) 2070 ± 42 2120 ± 195 0.54 ± 0.45
3.6.2 Compression testing
Further to the tensile tests, conditions L3, L5 and L8 were tested under compression.
The yield stress obtained has been compared with the tensile properties, Table 3.9.
In comparing the σy values for conditions L3 and L8, the results are inconclusive,
as they are within experimental error. Although the hardness properties show a
marked difference between the two conditions (Figure 3.7), there is no appreciable
difference in the yield stress.
55
Table 3.9: Comparison of compression and tensile results.
Condition Compression σy (MPa) Tensile σy (MPa)
L3 2467 ± 80 2140 ± 16
L5 3047 ± 12 -
L8 2390 ± 17 2151 ± 60
The linear relationship between Vickers hardness and yield stress initially pro-
posed by Tabor [64], and then confirmed by Hutchinson et al. [65] with a propor-
tionality constant of 3, is not evident in this case (Figure 3.19).
L1 
L2 L3 
L4 
L6 
L7 
L8 
L9 
L11 
1800 
1900 
2000 
2100 
2200 
2300 
620 640 660 680 700 720 
σ y
 fr
om
 te
ns
ile
 te
st
 (M
Pa
) 
Hardness (HV1) 
σy= 3
HV 
Figure 3.19: Plot of σy vs. Vickers hardness.
What becomes evident is that in tempered martensitic steels different terms con-
tribute to the yield stress, such as precipitate hardening and carbon in solid solution,
as well as the initial dislocation density in the material. A quantification of the in-
fluence of the different contributions is presented in Chapter 6.
56
3.7 Conclusion
The effect of interrupted ageing in a mid-carbon commercial steel was studied. It
was found that interruption increases hardness by ∼10%, and stabilises properties.
Microstructural characterisation by TEM shows that a finer precipitation distribu-
tion is obtained through interrupted ageing.
From the systematic microstructural analysis using the Kolmogorov-Smirnov
analysis, some important conclusions can be drawn:
• 30 days of IA leads to a finer microstructure, both in precipitate size and
spacing.
• After 30 days of IA, the duration of the subsequent tempering stage does not
influence precipitation; while increasing the tempering temperature, decreases
carbide size.
• In direct tempering, longer tempering time decreases carbide size and increases
spacing; whereas increasing the tempering temperature, decreases carbide size.
With the aid of TEP measurements, and invoking the Kalish-Cohen model, it is un-
derstood that for freshly quenched martensite the interruption at room temperature
allows carbon migration towards dislocation cores. Such uniform redistribution is
consistent with the formation of nuclei, where the spacing between precipitates is
reduced with interruption time. Therefore by introducing interrupted ageing into
the heat treatment, effective and efficient carbon distribution takes place, reflected
on the microstructure and properties. Although there is a limit on the scaling-up
due to component size and quenching rates required, this may be exploited in appli-
cations where dimensional stability and constant hardness are required throughout
component life.
57
Chapter 4
Modelling θ nucleation: the
Fe-C-Si model alloy system
This chapter looks at the effect of silicon in cementite nucleation during marten-
site tempering. The literature review presented in Chapter 2 showed that there is
still some controversy around the nucleation and growth stages in cementite pre-
cipitation, especially under the influence of silicon. Aiming to further understand
the cementite nucleation process, a model based on the classical nucleation theory
(CNT) is presented. To account for the effect of silicon, paraequilibrium condi-
tions have been set for cementite nucleation. The driving force for nucleation was
evaluated from the chemical and misfit energy terms.
4.1 The Fe-C-Si model alloy system
Preliminary studies were carried out on a Fe-C-Si ternary alloy system in order to
observe carbide transition under the influence of silicon. The measured chemical
composition of the steel is shown in Table 4.1.
58
Table 4.1: Measured chemical composition of model alloy (Fe to balance).
C Si Mn Cr Mo V Ti S P Ni
wt.% 0.57 1.97 0.002 0.005 0.005 0.000 0.002 0.009 0.006 0.000
The model alloy was cast using an Arc Melter AM (Edmund Buhler GmbH).
A laboratory cast of 65 g was produced, which was then homogenised at 1200 ◦C
for 48 hours in vacuum and then furnace cooled to room temperature. Figure 4.1
shows an image taken by optical microscopy of the as-received state, etched using
2% nital. Notable features include the pearlite microstructure, some ferrite shown
in a lighter shade near prior austenite grain boundaries, and circular dark regions
most probably owing to impurities or porosity. Prior austenite grain boundaries
were also visible, and the PAG size was ∼300 µm of the initial microstructure.
Figure 4.1: Optical micrograph of the initial microstructure of the as-cast material
following homogenisation treatment, where the image on the right is a magnified
view of the marked pearlite microstructure on the right.
59
Table 4.2: Heat treatment for the model alloy.
Condition Tγ (
◦C) tγ (s) Ttemp ( ◦C) ttemp (s)
250 ◦C 880 180 250 1800
300 ◦C 880 180 300 1800
350 ◦C 880 180 350 1800
400 ◦C 880 180 400 1800
The conditioning of the specimens was carried out using an Adamel Lhomargy
DT1000 dilatometer. The list of heat treatments is summarised in Table 4.2. These
were in the shape of cylindrical rods of 3 mm diameter and 12 mm length. Quenching
was done using a helium flux, potentially achieving cooling rates of about 300-500
◦C per second, although a close inspection of the dilatometry curve in the previous
chapter showed an actual quenching rate of ∼75 ◦C/s.
During the heat treatment, the critical transformation temperatures Ac1, Ac3
and Ms were recorded, and are summarised in Table 4.3.
Table 4.3: Summary of critical temperatures.
Condition Ac1 (
◦C) Ac3 ( ◦C) Ms ( ◦C)
250 ◦C 810 853 280
300 ◦C 814 855 275
350 ◦C 815 851 276
400 ◦C 814 850 269
Average 813 852 275
Standard deviation 2 2 4
Given that Ac3 is below the austenitisation temperature, the austenitisation condi-
tion chosen for this steel (880 ◦C) ensures complete austenitisation. Furthermore,
from the dilatometry curves, some unexpected slope changes were detected, Figure
4.2.
60
Figure 4.2: Segment of the dilatometry curve, where the martensitic transformation
has been highlighted.
The strongest signal detected at 275 ◦C has been taken as the martensitic start
temperature. However, in some conditions an early slope change was detected. In
Figure 4.2, this was seen near 348 ◦C. This early detection of the Ms has been re-
ported in the literature for steels containing Cr. The observed splitting phenomenon
was associated with an incomplete dissolution of secondary phases [66]. This is un-
likely to be the case in the model alloy as the only alloying elements are silicon and
carbon.
Another possible cause is the inhomogeneous distribution of carbon. Given that
the Ms temperature strongly depends on the carbon content, the local concentra-
tion of carbon may cause fluctuations in the Ms. Andrews [67] derived an empirical
expression for determining the Ms (in
◦C) based on the chemical composition of the
alloys (in wt.%):
Ms = 539 – 423C – 30.4Mn – 17.7Ni – 12.1Cr – 7.5Mo (4.1)
The change in signal observed near 117 ◦C is likely to correspond to the end of the
martensitic transformation, Mf.
61
4.1.1 Mechanical properties
Vickers hardness tests were carried out on dilatometry specimens, as they provide
fast and relatively accurate results for the mechanical properties of the samples. The
measurements were carried out using an LH Vickers Hardness Tester with a 5 kg
load, and an indentation dwelling time of 10 s. The reported value for each condition
is an average obtained from 5 readings, as shown in Figure 4.3 (a). Furthermore,
compression tests were also carried out using a MicroTest model at room tempera-
ture, where the strain rate used was 7×10−4 s −1, Figure 4.3 (b). The compressive
yield stress, σ(y), was recorded, as well as the maximum engineering stress before
rupture, σ(e,max), and the corresponding strain values, ε(e) at σ(e,max), were also
recorded. Strictly speaking, the last two values do not represent a true property of
the material. However, they have been reported for comparative purposes between
the four conditions.
Figure 4.3: (a) Hardness values and (b) summary from compression testing for the
model alloy.
In both tests, the yield strength was seen to decrease with increasing tempering
temperature. In the case of hardness, a dramatic decrease is seen between 350 and
400 ◦C, whereas during compression testing, the softening transition was seen to
occur near 350 ◦C. The properties are further elucidated in light of microstructural
analysis.
62
4.1.2 Microstructural analysis
Representative micrographs for all conditions are shown in Figure 4.4. The same
microstructural analysis of the intralath carbides has been followed as in Chapter 3,
where over 250 particles have been analysed, coming from 3-4 TEM frames, under
the same magnification (×52k). The principal features are summarised in Table 4.4.
Figure 4.4: Representative bright field TEM micrographs for each condition, where
intralath carbides are shown.
Table 4.4: Summary of the microstructural analysis.
Condition Aspect ratio requiv (nm) Spacing (nm)
250 ◦C 7.9 10.4 37.3
300 ◦C 7.4 10.7 37.3
350 ◦C 7.0 11.1 40.8
400 ◦C 7.2 11.7 50.3
Notice that through tempering, the size of intralath carbides does not change
significantly. This will be considered in Chapter 5, where silicon is shown to have a
pronounced effect on the growth kinetics of cementite. One thing to notice is that
the spacing between intralath carbides increases, while the size remains relatively
constant with increasing temperature. This implies a decrease in the volume fraction
63
of intralath carbides. During microstructural analysis, only the intralath carbides
were characterised. It is hypothesised that at increasing tempering temperatures,
there are other favourable nucleation sites in competition, such as lath and other
subgrain boundaries. The decreasing volume fraction of intralath carbides is consis-
tent with the drop in hardness and strength observed in Figure 4.3.
Figure 4.5: Representative bright field images for the 250 ◦C condition, showing (a)
intralath carbides within the lath, where the inset shows a higher resolution image
of the corresponding carbides, and (b) considerably high dislocation density within
the lath unit.
Other noticeable features observed on the TEM are presented in Figure 4.5,
where a high dislocation density can be seen within a single martensitic lath. Figure
4.6 shows the diffraction patterns taken from representative areas. One observation
is the confirmation of the presence of ε-carbide after tempering at 300 ◦C, where the
reported Jack orientation relationship (OR) between the carbide and the matrix has
been observed (Table 2.4). Furthermore, cementite was also observed near twinned
regions for the 250 ◦C condition, where the Bagaryatski OR exists. Therefore it
becomes evident that although the ε → θ transition occurs throughout tempering,
there are circumstances under which cementite nucleates independent of ε.
64
Figure 4.6: The presence of ε obeying Jack OR at 300 ◦C was confirmed using
diffraction pattern analysis, and cementite satisfying Bagaryatski OR was seen to
be present after tempering 1800 s at 250 ◦C.
Retained austenite was not detected during TEM work. In order to obtain a
more representative analysis of the bulk material, X-ray diffraction was carried out.
4.1.3 X-ray diffraction
A Phillips X’Pert PW3020 model equipped with a copper radiation source (λ ≈
1.540598 A˚) was used. Scans were taken from 35 to 105◦ 2ϑ, using a step size of
0.035◦ and a dwell time of 15 s. Note that in order to distinguish the cementite
carbide from the Bragg angle, θ is used for the former, while ϑ is used for the latter.
65
Diffraction analysis and Rietveld refinement were carried out using HighScore Plus
software.
Dilatometry samples were used for X-ray analysis. Cylinders of 3 mm diameter
were sliced into several discs. The surface of the samples was polished to a colloidal
silica finish (5 minutes), in order to relieve the residual stresses from the surface
during sample preparation. The 3 mm discs were then stuck to the surface of a
silicon sample holder, which has no background noise, so that the reflection from the
sample holder does not interfere with the diffraction pattern from the specimens.
The scans for conditions 250 and 400 ◦C have been plotted together, for easier
comparison, Figure 4.7.
2ϑ 
Figure 4.7: Diffraction patterns for conditions 250 and 400 ◦C.
A clear difference between the two conditions is that at 400 ◦C the retained
austenite peaks no longer become visible. Furthermore, slight sharpening of the
ferrite peaks also occurs at 400 ◦C, which is attributed to the decrease in the dislo-
cation density during martensite recovery at higher temperatures. The method of
quantifying dislocation density by XRD methods will be discussed in Chapter 5.
66
Rietveld refinement was carried out with two phases present: γ-austenite (Fm3m)
and α-ferrite (Im3m). For the 250 ◦C condition, some tetragonality was observed,
refinement was carried out with phase α′-martensite (I4/mmm) instead of Im3m,
to account for the tetragonality of the matrix phase. The lattice parameters for
condition 250 ◦C were determined to be a= 0.2862 and c= 0.2876 nm, giving a
tetragonality of 1.005. The measured volume fraction of retained austenite is sum-
marised in Table 4.5.
Table 4.5: Retained austenite volume fraction during tempering.
Condition Volume fraction of γR (%) ‡
250 ◦C 6.8
300 ◦C 5.5
350 ◦C 4.3
400 ◦C -
‡ The values obtained are after refining to a goodness of fit
(
GOF= Rwp
Rexp
)
< 2%.
4.2 Modelling cementite nucleation using the clas-
sical nucleation theory
In order to model cementite nucleation, the classical nucleation theory is consid-
ered. Typically in the case of homogeneous nucleation, the overall Gibbs free energy
change per nucleus, ∆G, is defined as:
∆G = −V (∆GV −∆GS) + Aγ, (4.2)
where GV , GS and γ are the chemical, strain and interfacial energy terms for the
newly forming nucleus of volume V and area A.
Differentiating Equation (4.2), an expression for the activation energy barrier for
67
the nucleation process (∆G∗) can be obtained.
∆G∗ =
16piγ3
3(∆GV −∆GS)2 , (4.3)
From Equation (4.3), it appears that maximising the denominator would minimise
the activation energy barrier. The denominator evaluates the quantity by which the
chemical driving force, ∆GV , is reduced by the counteracting strain energy, ∆GS,
term. The effective driving force for nucleation is therefore given by (∆GV –∆GS)
[68]. In other words, nucleation may theoretically occur when ∆GV > ∆GS, but
will most likely require ∆GV – ∆GS >> 0.
The process of nucleation is very sensitive to the value of γ. Perez and Deschamps
[30] reported the interfacial energy of ε and θ for the nucleation process based on
the Becker-Do¨ring theory: γ(α′/ε) = 0.147 J m−2 and γ(α′/θ)=0.174 J m−2. Never-
theless if one were to consider that the ε→ θ transition is a two-step process, where
the intralath cementite nucleates at the α′/ε boundary [25], in order to successfully
model the interfacial energy there are three interfaces to be considered: α′/ε, ε/θ
and α′/θ, Figure 4.8.
Figure 4.8: Representation of the three interfaces involved during the ε → θ transi-
tion.
68
Figure 4.9: Four different types of interfaces presented depending on the degree of
coherency: fully coherent, coherent with strain, semicoherent and incoherent. After
Porter, Easterling and Sherif [68].
The interface between two phases, α and β, can be of four types, depending on
the degree of coherency, Figure 4.9. During early stages of nucleation, the coherency
between the matrix and the precipitate phase is full, yielding a very low interfacial
energy, γ between 1–200 mJ m−2 [68]. However as a precipitate grows, coherency
is lost. Due to discontinuity in the lattice parameter across the interface, strain-
ing occurs. During continuous growth, there comes a point where it becomes more
favourable to replace interfacial strain by the introduction of a dislocation. For
semicoherent interfaces, the interfacial energy is in the range of 200–500 mJ m−2
[68], whereas for incoherent interfaces, the value of γ can go up to 1000 mJ m−2 [68].
By the time the carbide transition takes place, ε reaches a considerable size, as
seen by TEM. Supposing that a semicoherent interface exists between the matrix
69
and ε prior to θ nucleation, then γ(α′/ε)semicoherent > γ(α′/θ)coherent, as the nucle-
ating cementite will be coherent with the matrix during such early stages. Thus
during the ε to θ transition, the formation of the new interfaces given by the sum
of γ(ε/θ)coherent plus γ(α
′/θ)coherent might counterbalance the annihilation of the ex-
isting interface γ(α′/ε)semicoherent.
Due to the absence of the necessary interfacial energy values, only the (∆GV –
∆GS) contribution for ∆G
∗ will be evaluated. The values for GV and GS are eval-
uated in sections 4.2.1 and 4.2.2 respectively.
4.2.1 Thermodynamics- ∆GV
At the early stages of carbide precipitation during martensite tempering (generally
temperatures below 350 ◦C), the diffusivity of most elements is substantially low.
The only elements that will be able to diffuse significantly are interstitials such as
carbon. Therefore paraequilibrium conditions need to be considered.
Consider the process of nucleation of cementite (θ) occurring in ferrite (α) under
paraequilibrium conditions, in a ternary system Fe–C–Si. At the carbide/matrix
interface, due to the relatively high mobility of carbon, the chemical potential of
carbon, µC, will have equilibriated in a short time, i.e. µ
α
C = µ
θ
C. On the other
hand, iron and silicon would have different chemical potentials [69], i.e. µαFe 6= µθFe
and µαSi 6= µθSi. At this stage, given the restricted diffusivities of Fe and Si, X is used
to collectively refer to Fe and Si. This state of ‘constrained’ equilibrium is defined
as paraequilibrium.
The driving force for nucleation, ∆GN of paraequilibrium cementite is given by
[32]:
∆GN = (µθX − µαX)xθX + (µθC − µαC)xθC (4.4)
where xφi is the mole fraction of element i= C, X, within phase φ, where φ= α, θ.
Furthermore, since paraequilibrium cementite inherits the matrix composition, the
70
concentration of the elements will be the same across the interface, thus the u-
fraction is considered, which is defined as [69]:
ui =
xi∑
xX
, (4.5)
where xi is the mole fraction of element i, and
∑
xX is the sum taken for Fe and the
other substitutional elements. In order to achieve paraequilibrium conditions at the
interface, the product of chemical potential µX and uX must be the same in both
phases.
Having defined the alloy system and conditions, the u-fraction was determined
in order to calculate the paraequilibrium driving force. The TCFE6 (v 6.2) database
[70] was used in ThermoCalc to obtain the ∆GN term. Once obtained, the volume
Gibbs free energy can be determined by:
∆GV =
∆GN
Vm,θ
, (4.6)
where ∆GN is the chemical Gibbs free energy obtained from ThermoCalc expressed
in J mol−1, and Vm,θ is the molar volume of cementite, which will be determined
in section 4.2.2. The values obtained for ∆GN are shown in Figure 4.10. Both
paraequilibrium and full equilibrium conditions have been computed.
71
Figure 4.10: Driving force for nucleation obtained using ThermoCalc for cementite
nucleation under both para and full equilibrium conditions for a Fe–0.55C–(0–2.2)Si
wt.% ternary system.
Table 4.6: Computed equilibrium composition (in mass fraction) of cementite for
Fe–0.55C–2.0Si wt.% system, where silicon is seen to have negligible solubility in
cementite.
Temperature Fe C Si
250 ◦C 9.33 × 10−1 6.69 × 10−2 ∼ 0
Table 4.6 summarises the chemical composition for cementite under equilibrium con-
ditions for the Fe–0.55C–2.0Si wt.% alloy. It can be seen that under equilibrium
conditions in ThermoCalc, silicon has negligible solubility in cementite.
72
The driving force for cementite nucleation under paraequilibrium conditions is
lower than for full equilibrium. These values are comparable with those reported by
Kozeschnik and Bhadhesia [33] and Ghosh and Olson [32, 71].
4.2.2 Misfit strain- ∆GS
The second aspect to consider in determining the effective driving force is the ∆GS
term. The elastic strain energy stored in the matrix surrounding an inclusion is
approximated by [72]:
∆GS = 6µδ
2 · E
(
H
D
)
, (4.7)
where µ is the matrix shear modulus, δ is the volumetric misfit and the term within
brackets indicate functional dependence of E with H
D
, accounts for the shape factor
of the inclusion, where D is the equatorial diameter and H is the polar diameter of
the inclusion. The volumetric misfit is a function of the lattice parameters:
δ =
1
3
(
V1 − V2
V2
)
, (4.8)
where V i refers to the atomic volumes for phases i= 1 and 2. The atomic volume
of a phase is given by:
Vatomic =
Vunit cell
n
, (4.9)
where V unit cell is given by the product of lattice parameters a×b×c, and n is the
number of atoms sitting within the defined unit cell.
The presence of silicon leads to contraction in the Fe-lattice, owing to its small size.
In order to incorporate the effect of silicon during nucleation, one must consider the
variation in the matrix properties, as this will have an impact on the δ misfit value.
One method of estimating the lattice parameters of ferrite based on chemical com-
position is by empirical formulae, such as the one shown below, where the elements
73
within the brackets refer to the alloying element content in atomic percent [73]:
aα,0 = 0.28664 + 0.00006(Mn) – 0.00003(Si) – 0.00005(Cr), (4.10)
The lattice parameter for Fe–0.55C–2.0Si wt.% was determined to be 0.2826 nm us-
ing XRD methods. This is used as a benchmark. From Equation (4.10), the presence
of silicon leads to a contraction of -0.00003 per 1 at.% silicon, therefore the lattice
parameter, aα,0, as a function of silicon content can be determined. Furthermore,
since carbide precipitation occurs at temperatures above ambient, temperature de-
pendency of the shear modulus and the lattice parameters should be considered.
Those will be explored next.
Shear modulus
The temperature dependency of the shear modulus has been extracted from a similar
alloy in literature (Fe–0.4C–0.14Si–0.86Mn wt.%) [74]. The values are listed in
Table 4.7.
Table 4.7: Matrix shear modulus variation as a function of temperature [74].
T ( ◦C) 25 250 300 350 400
µ (GPa) 81 75 73 71 68.5
These values are to be substituted into Equation (4.7) in order to obtain more
accurately the strain energy during actual tempering conditions.
Lattice parameters
The aα,0 values need further readjustments in order to incorporate the effect of tem-
perature. The effect of temperature on the lattice parameter should be incorporated
74
in the following manner [75]:
aα = aα,0[1 + β(T − 800 K)], (4.11)
where β is the linear expansion coefficient and T is the absolute temperature.
The value of β was reported to be 17.5×10−6 K−1 for ferrite containing up to 0.6
wt.% carbon [75] (NB. although these reported values were determined for a tem-
perature range between 800 and 1200 K (527 and 927 ◦C), where 800 K was used as
the reference temperature, they will still be used given the similar carbon contents
in the matrix).
Furthermore, at the early stages of tempering, the tetragonality of the matrix
phase also needs to be considered. The lattice parameters obtained by XRD showed
that at 250 ◦C the tetragonality of the matrix was 1.005, while at higher tem-
peratures, no sign of tetragonality was observed. The tetragonality has also been
considered when computing the atomic volume of the matrix phase.
The tetragonality in the as-quenched state can be estimated by [76]:
c/a = 1 + 0.045xw, (4.12)
where xw is the mass % of carbon. Thus, the expected tetragonality in the as-
quenched state is calculated to be 1.025. There might be some confusion as to
which tetragonality value for the matrix should be used for estimating the lattice
parameters of the matrix phase. However, it is assumed that cementite forms after
ε-carbide, by which time, considerable reduction of matrix tetragonality would have
already occurred due to the precipitation of ε [77].
Carbides
The lattice parameters for the carbide phases were obtained from literature: a=
0.2752 and c= 0.4353 nm for ε-carbide and a= 0.4516, b= 0.5077 and c= 0.6727
75
nm for θ-carbide [78], giving an atomic volume of 0.0101 and 0.0096 nm3 respectively.
Using these parameters, the molar volume of cementite term from Equation (4.6),
V m,θ, was determined to be 5.86 × 10−6 m3 mol−1. Although strictly speaking the
lattice parameters of the carbide phases would also change due to thermal expansion,
a constant value has been used for their atomic volume.
Table 4.8: The misfit between the different phases in Fe–0.55C–2.0 wt.% at 250 ◦C.
Interface α′/ε α′/θ ε/θ
δ 4.52 ×10−2 6.32 ×10−2 2.2 ×10−2
The estimated misfit δ values at 250 ◦C for the different interfaces are summarised in
Table 4.8. Closer inspection of these values shows that δ(α′/θ) ≈ δ(α′/ε) + δ(ε/θ),
as represented in Figure 4.11. This could be a way of understanding the role of
a metastable phase, as the transitional phase offers an intermediate misfit energy
path. This is consistent with Nam’s postulate, that the nucleation of cementite has
been suggested to take place at the matrix-epsilon interface [25].
Figure 4.11: The presence of the metastable ε phase offers an intermediate misfit
energy path during cementite precipitation.
4.2.3 Computing the effective driving force (∆GV –∆GS)
Now that ∆GV and ∆GS have been determined, the effective driving force can
be computed, as shown in Figure 4.12 for θ-nucleation. It should be noted that
76
for nucleation to take place, the chemical driving force should be higher than the
opposing strain term, i.e. for nucleation to be favourable, the effective driving force
(∆GV – ∆GS) > 0. The x-axis, shown in red in Figure 4.12, corresponds to (∆GV –
∆GS) = 0. Therefore the horizontal axis can be used as a criterion to decide whether
nucleation is favourable or not. With the incorporation of ∆GS, the total driving
force for nucleation is reduced. The contribution from each individual energy term
is shown in Figure 4.13.
|Δ
G
V 
| -
 |Δ
G
S 
| (
J 
m
-3
) 
Figure 4.12: Driving force for cementite nucleation, ∆GN under paraequilibrium
conditions.
|Δ
G
V 
| -
 |Δ
G
S 
| (
J 
m
-3
) 
-5.0E+08 
0.0E+00 
5.0E+08 
1.0E+09 
1.5E+09 
2.0E+09 
1.0 1.2 1.4 1.6 1.8 2.0 2.2 
ΔG
 (J
 m
-3
) 
Silicon content (wt.%) 
Chemical contribution (-ΔGV) 
Strain contribution (ΔGS) 
Overall (|ΔGV| - |ΔGS|) 
(a) (b) 
Figure 4.13: (a) Driving force for cementite nucleation as a function of Si content
at 300 ◦C, and (b) contribution from each of the energy terms.
For instance, at 300 ◦C (Figure 4.13 (a)), it appears that when the silicon content is
above ∼1.95 wt.%, cementite nucleation is not favoured. This is consistent with the
77
diffraction pattern work in Figure 4.6, where ε-carbide was seen present at 300 ◦C.
One aspect to notice is that, in the presence of crystal defects, an extra energy
term needs to be considered. Thus, Equation (4.2) becomes:
∆G = −V (∆GV −∆GS) + Aγ −∆Gd, (4.13)
where the ∆Gd term accounts for the reduction in the nucleation energy barrier
owing to the presence of crystal defects in the material.
The incorporation of the ∆Gd term will further reduce the overall ∆G
∗ value in
Equation (4.3), facilitating the nucleation event by reducing the activation barrier.
This will favour the nucleation of θ in higher silicon contents. This is observed in
Figure 4.6, where cementite was seen present near twin regions at 250 ◦C. Further-
more, this agrees with results presented by Andrews [31], where it was demonstrated
the formation of cementite at 250 ◦C is possible, as cementite nucleation is strongly
related to twin density [79].
In order to refine the model, precise lattice parameters of the carbide phases
need to be determined.
4.3 Characterising carbides using high energy syn-
chrotron radiation
Given the limited resolution of laboratory X-ray diffractometers, trial runs were
made using a high energy source on the I11 beamline (high resolution powder diffrac-
tion) [80] of Diamond Light Source (Didcot, UK). The sample, needle in shape,
was spun and measured under transmission mode. A 15 keV beam was used (λ≈
0.817157 A˚) for the trial tests. The conditions that were analysed ex situ were 250
and 400 ◦C after tempering for 1800 s. The dilatometry samples were manually
round until they were 0.8 mm diameter and 6 mm length.
78
Trial runs were made using both high resolution MAC detectors and the time-
resolved position sensitive detector (PSD) [81], where room temperature measure-
ments were taken. Out of the two detectors, the latter gave a better signal-to-noise
ratio. The diffraction peaks obtained using the PSD detector are shown in Figure
4.14.
Figure 4.14: Diffraction peaks for conditions 250 and 400 ◦C using an exposure time
of 120 s with the PSD detectors.
In order to account for the straining on the sample surface during sample prepara-
tion, the peak positions were analysed. The ferrite peaks can be used as a benchmark
in order to estimate the peak shift that occurred due to sample straining. Bragg’s
equation (4.14) can be used in order to determine the dhkl-spacing of a peak of index
(hkl), taking n=1 and given that λ and ϑ are known:
nλ = 2dhklsinϑ, (4.14)
For cubic structures, the dhkl-spacing and the lattice parameter a have the following
79
relationship:
dhkl =
a√
h2 + k2 + l2
, (4.15)
For instance, taking the (110) reflection of the ferrite peak, λ= 0.817157 A˚, and
2ϑ= 23.588 ◦, which by Equation (4.14), this corresponds to a dhkl-spacing of 1.999
A˚. This value is slightly smaller than the one obtained from laboratory XRD mea-
surements, where for λ= 1.540598 A˚, and 2ϑ= 44.880 ◦, dhkl = 2.018 A˚, which would
be equivalent to an angle of 2ϑ= 23.360 ◦ on the synchrotron results. The sample
preparation for the latter case was done carefully to remove the residual stress built
on the surface due to mechanical grinding. Therefore due to the straining of the
material, in the 2ϑ ∼ 23 ◦ region, the peaks from the synchrotron measurements
will be shifted to slightly higher angles by ∼0.2 ◦ to the right.
Whilst bearing this in mind, the region 27 ◦ < 2ϑ < 32 ◦ is considered, as in this
interval no ferrite and austenite peaks are present and so the peaks will occur only
due to carbides. Three carbides have been considered from the Inorganic Crystal
Structure Database (ICSD) [82]: ε-carbide (collection code 44354, source Nagakura
[78]), η-carbide (collection code 87128, source Hirotsu and Nagakura [83]) and θ-
carbide (collection code 99017, source Wood et al. [84]). The relevant carbide peaks
in the range 27 ◦ < 2ϑ < 32 ◦ are summarised in Table 4.9, and their positions
labelled in Figure 4.15.
The peaks were further matched against possible forms iron oxides from the
ICSD database (e.g. collection codes 26410, 15840, 35000). Within the 27 ◦ < 2ϑ
< 32 ◦ region, relatively strong oxide peaks should be observed at 2ϑ∼ 31.9 ◦, with
additional peaks at 2ϑ∼ 27.6 ◦ and 29.3 ◦. The absence of these oxide peaks shows
that the peaks in Figure 4.15 are better indexed as carbide peaks.
80
Table 4.9: Carbide reflections in the range of 27 ◦ < 2ϑ < 32 ◦, where only those
peaks that are strong have been included (> 10% of the most intense peak, which
are the (111)ε, (211)η and (031)θ peaks).
Carbide Structure Space group hkl reflection 2ϑ dhkl I
ε h.c.p. P6322 (112) 29.45
◦ 1.6074 16 %
η orthorhombic Pnnm (211) 29.36 ◦ 1.6125 100 %
η orthorhombic Pnnm (220) 29.77 ◦ 1.5905 47.5 %
θ orthorhombic Pnma (230) 28.07 ◦ 1.6848 11 %
θ orthorhombic Pnma (301) 29.86 ◦ 1.5857 17 %
θ orthorhombic Pnma (222) 31.41 ◦ 1.5095 11 %
Figure 4.15: Carbides within the 27 ◦ < 2ϑ < 32 ◦.
81
From Figure 4.15, most peaks can be indexed as either ε, η and θ carbides, al-
though due to the inappropriate sample preparation, precise indexing is not possible.
Nevertheless, this shows that the resolution of the synchrotron is potentially good
enough for detecting carbides. This led to the planning of experiments involving in
situ tempering of martensite on the I11 in order to observe carbide precipitation in
alloys containing a range of silicon contents. This will be explored in Chapter 5.
4.4 Conclusion
Preliminary work was carried out on an Fe-0.55C-2.0Si wt.% model alloy, in order to
become familiar with cementite precipitation in martensitic steels. Cementite nucle-
ation was modelled within the classical nucleation theory framework, where it was
observed that as silicon content increased, the driving force for cementite decreased.
According to the theoretical result at 300 ◦C, a silicon content of 2.0 wt.% would
not favour cementite nucleation, which was consistent with the electron diffraction
work from TEM, where ε was seen to be stable at 300 ◦C. This could explain the
peak in mechanical properties observed for the 300 ◦C condition during compression
testing. On the other hand, it was also seen that cementite nucleation is favoured
under the presence of defects, such as twins, at temperatures as low as 250 ◦C.
Furthermore, simulation of the misfit at the carbide/matrix interface, showed
that the precipitation of ε before cementite lowered the misfit at the interface.
This is a way of understanding the carbide precipitation sequence, as the preceding
metastable phase offers an alternative path that reduces the misfit energy during
nucleation.
82
Chapter 5
The effect of silicon on tempered
martensite
Following the ternary model alloy system, this chapter further elaborates on the
understanding of the effect of silicon during martensite tempering on more complex
alloying systems. The studied compositions have been inspired from commercial al-
loys. In situ tempering was carried on the synchrotron, in order to study cementite
growth under the influence of silicon. Furthermore, based on the broadening of the
ferrite peaks, the dislocation density throughout tempering was measured. Com-
bined with microstructural analysis, this experimental chapter provides a clearer
picture of the microstructural transitions occurring during martensite tempering.
5.1 Alloys
Three alloys containing different silicon contents have been considered, the measured
chemical compositions are listed in Table 5.1, as determined by supplier ASCOmetal.
The notation used throughout is HS, MS and LS, for the high (2.3 wt.%), medium
(1.7 wt.%) and low (1.4 wt.%) silicon contents, respectively.
83
Table 5.1: Measured chemical composition of alloys, in weight %, Fe to balance.
C Si Mn Cr Mo V Ti Ni Cu S P
HS 0.56 2.30 0.69 0.89 0.02 0.10 0.03 0.20 0.15 0.02 0.01
MS 0.55 1.71 0.71 0.89 0.02 0.10 0.03 0.20 0.16 0.03 0.01
LS 0.55 1.43 0.72 0.91 0.02 0.10 0.03 0.21 0.16 0.02 0.01
These were casts made specifically for this study, where laboratory heats were man-
ufactured as 20 kg ingots using a vacuum induction furnace. The ingots, of section
100 × 100 mm, were left to cool to room temperature. Approximately 2/10 and
1/10 of the total material was cut from the top and bottom of the ingot respectively,
keeping the more homogeneous and sound material from the remaining 7/10. This
was then homogenised at 1200 ◦C, and forged into a bar of 40 mm diameter. To
reduce hardness and to ease cutting, the bars thus forged were slowly cooled in a
furnace rather than in air.
Heat treatment and dilatometry
The heat treatment consisted in austenitising at 880 ◦C for 180 s, quenching
and tempering. Two sets of heat treatments were carried out. The first set was heat
treated in a DT1000 dilatometer, where quenching was done using helium flux. The
samples were characterised using microscopy (both optical and electron), hardness
and high energy X-rays using synchrotron radiation. Tempering was carried out at
250, 300, 350, 400 and 450 ◦C for 1800 s. The critical transformation temperatures
determined by dilatometry are summarised in Table 5.2.
84
Table 5.2: Summary of critical temperatures obtained by dilatometry.
Ac1
◦C Ac3 ◦C Ms ◦C
HS 822 ± 2 857 ± 3 222 ± 10
MS 809 ± 1.5 843 ± 1.7 233 ± 6
LS 802 ± 1.8 838 ± 1.4 231 ± 8
Given that for all three alloys, the Ac3 is below the austenitisation temperature,
complete austenitisation is ensured prior quenching.
The second set of heat treatments were performed using laboratory furnaces.
The samples were wrapped within a stainless steel foil, and quenching was carried
out using ice brine. The samples for synchrotron use were left in the as-quenched
form, whereas those used for lab-based X-ray diffraction were tempered as follows:
• 1800 s at 300 ◦C
• 3600 s at 300 ◦C
• 1800 s at 350 ◦C
• 3600 s at 350 ◦C
• 1800 s at 400 ◦C
5.2 Microstructure and properties
5.2.1 Hardness
Vickers hardness measurements were carried out using a microhardness machine
(Wilson Wolpert model 401 MVA), with a 1 kg load, where average values from 5
measurements have been determined per condition. The as-received microstructure
consisted of pearlite, with an initial hardness of 328 ± 3, 327 ± 2 and 311 ± 4
85
HV1 for alloys HS, MS and LS, respectively. The hardness evolution throughout
tempering is summarised in Figure 5.1.
500 
550 
600 
650 
700 
750 
250 300 350 400 450 
H
ar
dn
es
s 
(H
V1
) 
Tempering temperature ( °C) 
HS MS LS 
Figure 5.1: Average hardness properties (error bars within marker size).
It is observed that the hardness decreases with tempering temperature. However,
there is a slight difference in the trend between the HS and the other two alloys. In
the case of HS, the hardness remains relatively constant on tempering up to 350 ◦C.
At 400 and 450 ◦C, there is a dramatic drop in HS hardness. On the other hand,
in both MS and LS, the decrease in hardness occurs gradually from 250 ◦C. This
observation is consistent with the reported cases in literature [23], where this delay
in the softening transition has been attributed to the role of silicon in delaying the
ε → θ transition.
86
5.2.2 Optical and scanning electron microscopy
Figure 5.2: (a) Optical micrograph, (b) SEM micrograph, and (c) SEM micrograph
with a magnified view of the inclusion, for HS tempered at 250 ◦C.
Given the complexity of the alloys, these were first characterised by optical mi-
croscopy and by SEM, using a MEB Hitachi S-4800 field emission gun operated at
15 keV. Under the optical microscope, several globular structures ranging from 1
to 5 µm in size were observed, as indicated by an arrow in Figure 5.2 (a). Further
examination was performed under SEM. EDS scans were carried out on regions 1
and 2 marked on Figure 5.2 (b), where region 2 covers the area within the globular
structure. The chemical composition is listed in Table 5.3. Values for carbon have
been omitted, as precise carbon contents cannot be resolved using EDS. From the
obtained information, the inclusions marked in Figure 5.2 correspond to manganese
sulphide, MnS.
Table 5.3: Chemical composition of regions 1 and 2 (in wt.%, Fe to balance).
Si Cr Mn S
Region 1 2.5 1.1 0.7 trace
Region 2 trace trace 44 24
The formation of MnS inclusions is unavoidable in industrial processing of steel. The
presence of such non-metallic inclusions, are thought to act as the source of crack
nucleation in fatigued components [85, 86]. Notice that the chemical composition
87
measured by EDS gives similar results as the actual chemical composition of the
alloy.
5.2.3 Transmission electron microscopy
Figure 5.3 shows low magnification TEM images of the microstructures, highlighting
the lath structures. In trying to determine the hypothetical role that individual laths
play during work hardening in lath martensite, a similar microstructural analysis
as the one carried out for intralath carbides was performed to measure lath width
(note that although referred to as lath martensite, from TEM it was observed that
the microstructure consisted of mixed lath and plate martensite). 25 measurements
per condition were taken, from 2 TEM frames taken at ×21k magnification. The
averages are represented in Figure 5.4.
Given that measurements were taken from random sections, stereological correc-
tions have been applied. For a bainitic microstructure, Chang and Bhadeshia [54]
derived the following:
L¯T =
pi
2
t, (5.1)
where L¯T is the measured linear intercept in the plane of observation across the lath
thickness, and t is the thickness of the plate. Therefore the lath width data have
been corrected by a factor of pi
2
[54].
88
Figure 5.3: Low magnification images showing the lath structures (tempered at
250 ◦C).
Figure 5.4: Lath width as a function of temperature and silicon content.
Table 5.4 reports the results numerically. Based on the TEM characterisation, it
appears that the lath width increases the least for the highest silicon content, HS.
In the case of the LS alloy, the increase is twofold.
In tempered martensite, the laths are heavily dislocated. Therefore, given that
the lath size and the rate of increase is dependent on silicon content, it suggests
that the recovery of martensite is a function of silicon. This point will be revisited
when the synchrotron results are presented, where the dislocation density has been
measured.
89
Table 5.4: Lath width (in nm) as a function of temperature and silicon content.
250 ◦C 350 ◦C 450 ◦C
HS 91 ± 25 96 ± 28 122 ± 41
MS 94 ± 23 89 ± 28 154 ± 46
LS 74 ± 16 105 ± 29 154 ± 63
Microstructural analysis of the intralath carbides was carried out, as shown in Figure
5.5. The extracted equivalent radius and spacing between carbides is summarised
in Figure 5.6. No results are available for the 450 ◦C condition, as the intralath
carbides were hardly visible, so this condition was left out.
Figure 5.5: Intralath carbides shown for conditions 250, 350 and 450 ◦C.
Based on the carbide statistics, no obvious trends are seen, since the measure-
ment error overlap between conditions. On the other hand, recalling the results
90
obtained for Model Alloy (Chapter 4), it was seen that although size did not vary
much, the spacing between precipitates increased significantly: from 37.3 nm at
250 ◦C to 50.3 nm at 400 ◦C. This is opposite to what is observed in HS, MS and
LS. Therefore, it appears that the presence of Mn, Cr, and other alloying elements,
stabilises intralath carbides. This has also been observed by Barrow and Rivera-
Dı´az-del-Castillo [87].
Figure 5.6: Microstructural analysis of intralath carbides.
5.3 The effect of silicon on cementite growth ki-
netics: in situ characterisation using high en-
ergy X-rays
Sample preparation
The samples for synchrotron studies were in the form of thin round needles of ∼0.5
mm diameter and 6 mm length. Initially, the sample consisted of ∼1.6 mm diame-
ter and 12 mm length. While fixing one end of the sample, sand paper was applied
to the other end of the material. Material was successively removed by grinding,
until the diameter of the ground section was reduced to 0.5 mm, represented by the
91
shaded section in Figure 5.7 (a).
Once this form of specimen was obtained, thermocouples were attached to the
thicker section, and conditioned in the dilatometer. Once heat treated, the sample
was then cut, keeping the thinner cross-section for beamtime.
Figure 5.7: (a) Diagram illustrating sample preparation (image courtesy of David
San Mart´ın, CENIM), and (b) representation of the two heat treatments followed.
Figure 5.7 (b) shows the two types of heat treatments followed: as-quenched
martensite for in situ tempering, and reference dilatometry samples. For the in
situ tempering samples, the needles were austenitised using laboratory furnaces and
quenched in ice brine, four days prior beamtime. The reference samples were heat
treated in a DT1000 dilatometer.
5.3.1 Preliminary characterisation
In order to ensure that a martensitic microstructure has been obtained during
quenching, the as-quenched condition was investigated by means of XRD, hard-
ness and optical microscopy, for alloys HS and LS.
92
Flat-plate specimens were investigated by conventional XRD methods, which
had a surface area of 20×20 mm2, and a thickness of ∼ 1 mm. A Phillips X’Pert
PW3020 model equipped with a copper radiation source (λ ≈ 1.540598 A˚) was used
for the preliminary characterisation. Scans were taken from 40 to 105◦ 2ϑ, using
a step size of 0.035◦ and a dwell time of 15 seconds. Diffraction analyses and Ri-
etveld refinement were carried out using HighScore Plus software. The obtained
XRD spectra are shown in Figure 5.8 (a).
Figure 5.8: (a) As-quenched XRD pattern for flat-plate specimens, and (b) optical
microscopy revealing martensitic microstructure for alloy LS needle specimen.
Rietveld refinement was carried out with two phases present: γ-austenite (Fm3m)
and α′-martensite (I4/mmm) (Figure 5.8 (a)), and was refined until a good fit was
obtained, where Rwp < 0.15. Rwp is the weighted profile factor, and is thought to
be the most meaningful value in following the progress of the refinement. It ex-
presses the agreement between the observed (obs) and the calculated (cal) intensity
distributions, which is determined by the following [88]:
Rwp =
(∑
wi(yi(obs)− yi(cal))2∑
wi(yi(obs))2
) 1
2
, (5.2)
where wi is the observation weight, yi(obs) and yi(cal), are the observed and calcu-
lated intensity distributions, respectively.
93
The phase fraction of retained austenite, γR, found was 4.9 and 2.5 % for alloys
HS and LS, respectively. The lattice parameters were: aγ= 3.5925 nm, aα= 2.8550
nm and cα= 2.8978 nm for alloy HS, and aγ= 3.5870 nm, aα= 2.8666 nm and cα= 2.9003
nm for alloy LS. The tetragonality of the matrix was 1.015 and 1.012 for alloys HS
and LS, respectively.
The needle-shaped specimens were also examined. Etching using 2% nital re-
vealed a martensitic microstructure (Figure 5.8 (b)). Hardness averages for alloys
HS and LS were 812 ± 4 and 786 ± 6 HV2, respectively. Therefore it appears that
the austenitisation and quenching processes were carried out adequately.
5.3.2 Experimental procedure
In situ synchrotron experiments were carried out at the I11 beamline at the Diamond
Light Source, Didcot, UK, under transmission. In situ tempering was carried out by
heating continuously using the Cyberstar hot-air blower provided by the beamline.
The sample was mounted at the tip of a brass holder and fixed with phenol aldehyde
epoxy resin. Once the resin was cured, the specimen was then placed horizontally
on the spinner, and aligned with the incoming beam using the sample alignment
cameras. A 15 keV beam energy (λ≈ 0.827134 A˚) was used for this study. The
set-up is shown in Figure 5.9.
Figure 5.9: (a) Set-up of the beamline, and (b) needle-shaped samples mounted on
brass holders.
94
Test runs were carried out at the start of the beamtime. Due to time constraints,
and also because temperature control of the hot-air blower becomes less precise below
250 ◦C, the heat treatment was carried out starting from 300 ◦C. Moreover it was
also found that a significant temperature gradient existed between the nozzle and
the sample. Although it is not possible to state the carbide transition temperature
with absolute certainty, the relative transition temperatures can be estimated.
While the furnace was set running at 300 ◦C, and the sample had been aligned,
the hot-air blower was brought to the heating position. To ensure temperature
homogeneity, the sample was submitted to isothermal holding for 10 minutes before
ramping up the temperature from 300 to 900 ◦C at a rate of 0.1 ◦C/s. Due to the
nature of this work, the time-resolved PSD detector was used in order to take rapid
scans continuously throughout tempering, giving an exposure time of 60 seconds in
total for each scan. It is worth noting that between the start and end of one scan,
the temperature of the nozzle would have increased by approximately 6 ◦C.
5.3.3 Temperature calibration
During the preliminary run, it was noticed that a considerable gradient existed
between the temperature of the furnace and that of the sample. Given its relatively
simple chemical composition, the ternary Model alloy system was first assessed in
order to become acquainted with the in situ technical aspects.
Figure 5.10 shows the diffraction peaks obtained for Model alloy, where cementite
peaks have been indexed.
95
Figure 5.10: Shown 20 ◦ < 2ϑ < 30 ◦ for the in situ tempering of the Model alloy
(NB. Temperature indicated refers to that of the nozzle).
By means of dilatometry, the austenitisation starting temperature of the alloy, Ac1,
was determined to be 808.3 ± 2.0 ◦C. Although the temperature of the nozzle,
T(nozzle), indicated 900 ◦C, there was no evidence of the material undergoing ferrite-
austenite phase transformation. Thus confirming a significant thermal gradient be-
tween the sample and the nozzle exit.
Figure 5.11: (a) Peak shift due to thermal expansion shown for Model alloy where
the inset shows a magnified view of the (200) ferrite reflection (NB. Temperature
indicated refers to that of the nozzle), and (b) (200) peak position at different
temperatures.
96
One way of estimating the thermal gradient is by assuming that the peak shift
is caused by thermal expansion of the lattice, as shown in Figure 5.11 (a), and
quantified in (b). Hence, given its peak shift, the “true” temperature difference,
∆T true, can be estimated from:
∆L
L
= αL∆Ttrue, (5.3)
where ∆L/L is the strain caused by lattice expansion, and αL is 11.8×10−6 ◦C−1
[89]. However, in this case, a slight shift of the ferrite peaks could also be due to a
reduction in tetragonality within the matrix phase.
Using Equation (5.3) and the data from Figure 5.11, it is observed that while
the nozzle temperature increases from 300 to 900 ◦C, i.e. temperature increment
of 600 ◦C, the actual temperature increment experienced by the sample is of ∼447
◦C. Hence, the actual tempering would have been carried out until ∼750 ◦C instead
of 900 ◦C. Although this is still an approximation, it provides a more realistic view
of the actual tempering conditions. Therefore, from here onwards, this correction
factor will be considered.
5.3.4 Reference samples
The diffraction spectra for reference samples prepared by dilatometry are shown
in Figure 5.12. The retained austenite seems to be stable even after 1800 s at
temperatures up to 400 ◦C in the lower silicon alloy, and 450 ◦C in the higher silicon
contents. This observation is surprising as generally the decomposition of retained
austenite (Stage II of tempering) is believed to take place between 200 and 300 ◦C
[11].
In the case of tempering at 500 ◦C, carbide peaks in LS alloy are stronger than
in the higher silicon alloy (dotted green circles in Figure 5.12).
97
2ϑ 
20 22 24 26 28 30 
2ϑ 
20 22 24 26 28 30 
In
te
ns
ity
 (a
.u
.) 
In
te
ns
ity
 (a
.u
.) 
HS alloy LS alloy 
Figure 5.12: XRD spectra shown for the reference samples for alloys HS and LS.
Closer inspection of the 22 ◦ < 2θ < 25 ◦ region indicates that the (111)γ peak
shows higher stability at high silicon content. This observation is consistent with
the trend reported in 0.2 wt.%C steels undergoing quenching and partitioning heat
treatments, reported by Santofimia et al. [90], where by adding 1.5 wt.% silicon, a
higher volume fraction of retained austenite was observed when tempered at 350 ◦C
for various time lengths.
HS (2.3 wt.% silicon) LS (1.4 wt.% silicon) 
In
te
ns
ity
 (a
.u
.) 
In
te
ns
ity
 (a
.u
.) 
22 22.5 23 23.5 24 24.5 25 
2ϑ 
22 22.5 23 23.5 24 24.5 25 
2ϑ 
300 °C 
400 °C 
500 °C 
300 °C 
400 °C 
500 °C 
Figure 5.13: Earlier martensite recovery in lower silicon content, as well as higher
stability of the retained austenite phase are observed.
Furthermore, it is worth noting that considerable sharpening of the ferrite peaks
takes place. Given that peak broadening is attributed to the presence of defects
98
in the lattice, e.g. dislocations, it is also possible to calculate the evolution in
dislocation density during martensite tempering. Moreover, peak sharpening occurs
more readily in the lower silicon case, implying a dependency in silicon content for
martensite recovery. This will be discussed in §5.4.2.
Absence of ε peaks
Close analysis of the XRD spectra did not show any sign of ε-carbide peaks. One
possible explanation is that, owing to its small crystallite size and its low volume
fraction, considerable broadening takes place, beyond detection limit. Another pos-
sible explanation is that the formation of ε-carbide is supressed, and instead cemen-
tite is directly precipitated. Kalish and Cohen [4] stated that in highly dislocated
martensite, carbon preferentially segregates into dislocation cores. As tempering
progresses, dislocations loose the ability to retain the carbon and cementite is pre-
cipitated, skipping the intermediate ε-phase. Following the work on interrupted
ageing (Chapter 3), 4 days of room temperature ageing would have been sufficient
for significant microstrucrucal change in the as-quenched state. One further possi-
ble explanation is that the sample undergoes ε → θ during the initial isothermal
holding at 300 ◦C. Although, as seen in Chapter 4, the Model alloy showed that
after tempering at 300 ◦C for 1800 s, ε-carbide was obtained.
5.3.5 In situ tempering
Although it has not been possible to follow the early nucleation stage of cementite
during synchrotron analysis, the growth stage has been studied. In both HS and LS
alloys, similar cementite peaks are already present by the time the first measurement
had been taken at 300 ◦C. Nevertheless, during continuous heating cementite peaks
appear much stronger in the lower silicon alloy. The relative intensities for the
cementite peaks are much stronger in LS than in HS, implying faster precipitation
rates for the lower silicon case during continuous tempering. Since the first obtained
99
spectra are for T= 300 ◦C, the inhibiting effect of silicon on the ε → θ carbide
transition remains unproven in situ at this moment.
HS alloy LS alloy 
In
te
ns
ity
 (a
.u
.) 
In
te
ns
ity
 (a
.u
.) 
2ϑ 
20 22 24 26 28 30 
2ϑ 
20 22 24 26 28 30 
Figure 5.14: In situ runs for alloys HS and LS.
5.3.6 Cementite growth kinetics
In the previous chapter, cementite nucleation was modelled under paraequilibrium
conditions. Reisdorf [3] showed that, throughout tempering, the silicon content
within cementite decreased. This observation was supported by Chang et al. [20],
where after prolonged tempering, a rich silicon layer was seen to form surround-
ing the carbide. Based on this, there is a tendency to ascertain that as cementite
forms under constrained paraequilibrium conditions; as the conditions become re-
laxed throughout tempering and orthoequilibrium can be considered, as silicon is
eventually rejected from θ.
3D-APT (atom probe tomography) results presented by Caballero et al. [27] are
charted in Figure 5.15, where it has been observed that the measured silicon content
within cementite decreased with increasing tempering temperature.
100
Figure 5.15: Average silicon content in ferrite and cementite for different tempering
conditions, from tabulated data in [27].
When considering cementite formation under paraequilibrium conditions at rela-
tively low temperatures, the diffusion of the substitutional elements is negligible.
One can assume that, at this stage, only carbon content will vary across the ma-
trix/carbide interface, whereas the content of the rest of the elements remains un-
changed. However, under prolonged tempering, the trapped silicon within cementite
will begin to diffuse out of the carbide, as the carbide tends to equilibrium. Zhu et
al. [28] observed that the rejected silicon builds up a Si-rich layer at the interface,
as schematically illustrated in Figure 5.16 (a).
101
Figure 5.16: (a) Possible growth mechanism for cementite, and (b) silicon and carbon
content variation throughout tempering, where JC and JSi refer to the carbon and
silicon flux across the interface.
For continuous carbide growth, further carbon is needed. However, the carbon
flux into the carbide across the interface will be reduced due to the Si-rich layer, as
illustrated in Figure 5.16 (b). Therefore the diffusion rate of silicon within cementite
is likely to be the rate-controlling factor in the growth of carbides. This mechanism
has also been hinted by Barrow and Rivera-Dı´az-del-Castillo [87]. This view agrees
with the carbide growth kinetics seen during in situ synchrotron analysis (Figure
5.14).
The slow growth of cementite might also be due to the slow diffusion of carbon
through the ferritic matrix. However, it is known that the presence of silicon in α-
102
ferrite raises the chemical potential of carbon [68]. This causes the carbon to diffuse
to more favourable sites within the material, e.g. carbides. This can be interpreted as
an additional driving force for the carbon to participate during carbide precipitation.
Despite the former, the fact that slower growth kinetics was observed in the higher
silicon alloy, hints that the rate is governed by the 1-2 nm Si-rich layer reported by
Chang et al. [20]. An additional growth rate controlling factor is the partitioning of
silicon from the carbide.
5.4 The effect of silicon on martensite recovery
5.4.1 Martensite recovery
Throughout martensite tempering, several changes may occur, which include car-
bon segregation (Chapter 3), carbide precipitation (Chapter 4), retained austenite
decomposition and recovery and recrystallisation of the ferrite matrix [91].
Caron and Krauss [92] observed that during the early stages of tempering there
is a significant drop in hardness, as well as in the total grain boundary surface per
unit volume. Their method for determining the number of grain boundaries con-
sisted of obtaining micrographs, all of equal dimensions, from different tempering
conditions. Concentric circles were drawn on each photograph, where the number
of intersections between boundaries and the drawn circles were counted, which gave
an indication of the grain boundary content. They observed a reduction in the grain
boundary content during prolonged tempering times, which was related to the elim-
ination of low angle boundaries and boundary readjustment.
Recovery is described as the process by which the stored energy in the material
is decreased by a change in the dislocation structure. A lower energy configuration
is achieved by dislocation motion such as glide, climb and cross-slip. Recovery is
known to consist of three components [93]:
• Dislocation annihilation.
103
• Dislocation rearrangement.
• Subgrain growth.
One characteristic of as-quenched martensite is its very high dislocation density,
which can be up to 1016 m−2 for medium carbon steels. Upon tempering, due to the
recovery process, there will be some changes in the microstructures. At this stage,
the focus will be on dislocation annihilation and subgrain growth.
Speich and Leslie [94] stated that in Fe-C alloys, recovery becomes an important
process at temperatures above 400 ◦C, whereas recrystallisation generally occurred
between 600-700 ◦C. During recovery, it is believed that the as-quenched lath struc-
ture, initially consisting of low angle dislocation cells and high angle lath boundaries,
undergoes annihilation of the boundaries and dislocations [94].
In Table 5.4 there already was a hint of recovery manifested in lath thickening
as tempering progressed. The thickness increase was lowest for HS, and highest for
LS. An initial hypothesis is established: if the apparent change in lath width is to be
described as part of the subgrain growth process during recovery, then high silicon
content seems to inhibit recovery in martensite.
The effect of silicon in retarding martensite recovery is also hinted in Figure 5.13,
where peak sharpening occurs more readily when the silicon content is low. There-
fore in order to measure the extent of dislocation annihilation throughout tempering,
the dislocation density is measured in the forthcoming section.
5.4.2 Dislocation density throughout tempering
Extensive crystallographic information can be obtained from analysing XRD data.
104
Figure 5.17: Origin of strain in crystals, adapted from Cullity [95].
Figure 5.17 (a) represents a perfect crystal lattice, with no strain present in
the material. However, there are various defects that distort the perfect lattice,
and influence the peak position and broadening. For instance, (b) shows the case
where the presence of alloying elements causes peak shifting due to the dilation in
lattice parameters. For example, in ferrite, the presence of silicon would lead to a
contraction in the α-Fe lattice, which would shift the peak to the right. On the
other hand, the presence of carbon atoms at the interstitial sites, would lead to an
expansion in the lattice, shifting the peak to the left. Another cause for uniform
strain is the thermal expansion effect experienced at higher temperatures due to a
volume expansion of the lattice.
The next phenomenon to consider is peak broadening, Figure 5.17 (c), which
is caused by the presence of crystal imperfections such as a small crystallite size,
strains and faulting [96, 95]. Quantifying the different effects has been researched
extensively, as exemplified by the seminal work by Hall, Williamson and Smallman
[97, 98, 99], Warren and Averbach [100, 101] and, later on, further developed by
Unga´r et al. [102, 103]. Many researchers have applied these methods to determine
105
the dislocation density in ferritic martensite, [104, 105, 106, 107]. Recently, Christien
et al. have measured in situ the dislocation density evolution on a Fe–0.03C–0.3Si–
0.8Mn–4.8Ni–15.6Cr–3.1Cu wt.% steel, in both austenite and martensite phases
during martensitic transformation by neutron diffraction [108]. It was seen that in
both phases the dislocation density increased progressively throughout the austenite
to martensite phase transformation, Figure 5.18.
Figure 5.18: Measured dislocation density evolution on a Fe–0.03C–0.3Si–0.8Mn–
4.8Ni–15.6Cr–3.1Cu wt.% steel during martensitic transformation (a) during cool-
ing, and (b) near the M s, adapted from Christien et al. [108].
Often in literature, the two methods used for measuring dislocation densities are
TEM and XRD. The general agreement is that TEM is suitable for low dislocation
densities (∼1012 m−2), whereas for high dislocation densities (1014–1015 m−2), XRD
methods are preferred. In this work, it was not possible to observe individual dislo-
cations by TEM, perhaps owing to the very high dislocation density in the sample.
Therefore XRD methods have been used.
The microstrain resulting from defects within the sample can be obtained by
differentiating Bragg’s equation at constant λ (NB. n=1, where n is the integer
106
indicating the order of diffraction):
λ = 2dsinϑ (5.4a)
1
d
=
2 sinϑ
λ
(5.4b)
−∆d
d2
= 2∆ϑ
cosϑ
λ
(5.4c)
∆d
d
= 2d∆ϑ cosϑ
−1
2d sinϑ
(5.4d)
∆d
d
=
−∆ϑ
tanϑ
(5.4e)
∆d
d
term represents the average strain within the material, εmicro. The broadening
due to strain (βstrain) is given by the ∆ϑ term, which is obtained by the peak’s
full-width at half maximum (FWHM).
One other factor that contributes to broadening of the diffracted peaks is crys-
tallite size, which can be described by the Scherrer formula [95]:
t =
Kλ
β cosϑ
, (5.5)
where t is the crystallite size, K is shape factor, which often is taken to be 1, λ is
the X-ray wavelength, β is the line broadening at FWHM, and ϑ is the Bragg angle.
From Equations (5.4e) and (5.5), it can be seen that strain varies with tan ϑ,
whereas crystallite size varies as a function of 1/cos ϑ. The Williamson-Hall method
allows the separation of these two properties [109], where the total broadening of
the peaks βtotal is given by the sum of strain and size contributions, βstrain and
βsize, respectively. Instrumental broadening also contributes to βtotal. Nevertheless,
the instrumental broadening at the synchrotron facility on beamline I11, where this
work was carried out, is assumed to be negligible. Therefore peak broadening is
107
given by:
βtotal = βstrain + βsize (5.6a)
βtotal = εmicro tanϑ+
kλ
t cosϑ
(5.6b)
βtotal cosϑ = εmicro sinϑ+
kλ
t
(5.6c)
Plotting (βtotal cos ϑ) vs. sinϑ would give a linear plot, where the gradient would
give the strain and the intercept would provide information on the crystallite size.
Figure 5.19: Separation of the size and strain effects on a Williamson-Hall plot.
The above remains true only for Lorentzian peaks. For Gaussian peaks, Equation
(5.6a) becomes:
β2total = β
2
strain + β
2
size, (5.7)
Nevertheless, it is often the case that the curves are best fitted by a Voigt function,
which consists of a convolution of Lorentzian and Gaussian. The “double-Voigt”
method after Langford [110] models both size- and strain-broadened profile, and
has been proven to be more accurate [111].
Based on the strain broadening, Christien et al. derived an expression for the
dislocation density [108]:
ρ =
3E
µb2(1 + 2ν2)
ε2micro, (5.8)
where E is the Young’s modulus, µ is the shear modulus, b is the Burgers vector
108
and ν is the Poisson’s ratio.
In this work, Rietveld refinement was performed on the XRD spectra using
MAUD software [112], which uses a pseudo-Voigt (PV) function for curve fitting.
The microstrain values, εmicro, were extracted from the refinement and were substi-
tuted into Equation 5.8. The values used were as follows: E= 211 GPa, µ= 80.3
GPa, ν= 0.3. To determine the value of b, consider the 1
2
a〈111〉 Burgers vector in
ferrite. This corresponds to a magnitude given by
√
3
2
a. Based on the lattice pa-
rameters previously extracted from XRD work in §5.3.1, Table 5.5 summarises the
Burgers vector for the two alloy systems.
Table 5.5: Burgers vectors for the different alloys.
aα by XRD (A˚) b (A˚)
HS 2.8550 2.473
LS 2.8666 2.483
The microstrain, εmicro, values obtained from MAUD and the calculated dislocation
density values are summarised in Table 5.6. The dislocation density values are sum-
marised in Figure 5.20. It is seen that for all tempering conditions, the dislocation
density in HS is higher than in LS, as hinted by Figure 5.13. It is worth noting
that instrumental broadening is assumed to be negligible in the current analysis.
However, any broadening due to the instrument will overestimate the dislocation
density reported in Figure 5.20. Nonetheless the order of magnitude ∼1015 m−2
corresponds to the typically reported value for martensitic steels [108, 105].
109
Table 5.6: Summary of the microstrain and dislocation density values obtained.
Ttempering HS alloy LS alloy
( ◦C) εmicro (×103) ρ (×1015 m−2) εmicro (×103) ρ (×1015 m−2)
250 7.8 6.6 8.0 7.1
300 7.2 5.6 7.4 6.0
350 5.8 3.7 6.4 4.4
400 4.4 2.1 6.0 3.9
450 3.9 1.6 5.9 3.8
500 2.6 7.4 4.1 1.9
0 
1 
2 
3 
4 
5 
6 
7 
8 
250 300 350 400 450 500 
D
is
lo
ca
tio
n 
de
ns
ity
, ρ
 ×
 1
01
5  m
-2
 
Temperature ( °C) 
Corr'd 
HS 
LS 
HS 
Figure 5.20: Dislocation density evolution with tempering temperature.
Additional XRD was carried out using lab-based samples. These samples, as
mentioned in §5.1, were heat treated using lab furnaces. For surface preparation,
the samples were polished down to 1 micron using diamond paste, with an OPS
colloidal silica finish. The same procedure was used as in §5.3.1, except for the
dwell time was decreased to 5 seconds. Results are shown in Figure 5.21.
110
Figure 5.21: Diffraction spectra for the (100) and (211) ferrite peaks. The legend
summarises the heat treatment combinations investigated, e.g. 300-1800 means
tempered at 300 ◦C for 1800 seconds.
The first observation that can be made is that, throughout tempering, peak
sharpening occurs. This sharpening occurs between 300 and 350 ◦C for both HS
and LS alloys. Nevertheless, based on peak broadening effects, it is observed that
in the case of LS, 3600 s tempering at 300 ◦C has already the same effect as 1800 s
tempering at 350 ◦C for HS. Thus, based on these findings, the kinetics of martensite
recovery appears to be influenced by the silicon content.
At this stage, it is not possible to know whether this is a specific property of
silicon, or if it is merely due to a solid solution effect in general. Nevertheless,
Wittig and Frommeyer [113] reported that the presence of silicon in the bcc-iron
lattice caused a reduction in dislocation cross-slip with increasing silicon content.
When alloyed with silicon, additional to the Peierls stress from the ferrite lattice,
there is a stress increment in moving a dislocation, owing to an increase in the
free energy due to the silicon-silicon interaction. Hence, dislocation cross-slip will
111
be lower in HS than in LS, causing lower recovery kinetics with increasing silicon
contents.
5.5 Conclusion
The effect of silicon in martensite tempering has been studied in this chapter. Two
main effects have been discussed: its inhibiting effect in cementite growth and its
retarding effect in the rate of martensite recovery.
The first mentioned effect has been based on in situ synchrotron observations,
where initially cementite would nucleate under paraequilibrium conditions, trapping
the silicon within. However, during growth, as the system tends to orthoequilibrium,
the silicon is rejected from within the carbide. The rejected silicon, as shown by
literature, is said to form a Si-rich layer, which inhibits further carbon intake, nec-
essary for continuous carbide growth.
The second effect is related to martensite recovery during the tempering stage.
Considerable microstructural changes related to the recovery process were observed.
By TEM and XRD methods it was observed how, when silicon content is increased,
martensite recovery kinetics is inhibited, as revealed by lath width increase and
peak broadening during tempering. This has been related to a reduced cross-slip
rate with increasing silicon.
The microstructural features quantified in this chapter will serve as a foundation
to the forthcoming chapter, particularly in understanding the plasticity behaviour
in medium carbon tempered martensite.
112
Chapter 6
Plasticity
The characterisation work presented in Chapter 5 identified the several microstruc-
tural changes taking place throughout martensite tempering, where the features
that most noticeably changed during heat treatments were intralath carbide pre-
cipitation, dislocation density and lath width. Literature on plasticity modelling of
0.5–0.6 wt.% carbon martensitic steels is scarce, and it is even a bigger challenge
due to the complex microstructures obtained in medium-carbon martensite. A few
modifications are introduced into existing plasticity models in order to customise
these for the HS, MS and LS alloys.
6.1 Introduction
Strengthening in tempered martensite results from the addition of several contribut-
ing effects such as precipitation hardening, solid solution strengthening, forest dislo-
cation hardening and grain size refinement. The aim of the first part of this chapter
is to quantitatively determine, with the aid of experimentally determined values,
each individual strengthening term, in order to observe how these evolve as a func-
tion of both temperature and chemical composition.
The second part deals with reproducing the stress-strain curves that were ob-
tained experimentally. Existing models based on irreversible thermodynamics were
113
applied, where a few modifications have been incorporated in order to adapt them
to medium carbon martensitic steels. Lastly, the third part presents the EBSD work
carried out in order to determine the “effective” grain size based on the misorienta-
tion between grains.
6.2 Tensile testing
A total of four conditions were tested for all three alloys HS, MS and LS: as-
quenched, and tempered 1800 s at 250, 350 and 450 ◦C. Test specimens were cylindri-
cal, and were heat treated after machining. The tests were performed in accordance
with ISO 6892 standard [114]. The machined test samples are shown in Figure 6.1,
where L0 is the original gauge length, Lt is the total length of test piece, S0 is the
original cross-sectional area of the parallel length, Lu is the final gauge length after
fracture, and Su is the minimum cross-sectional area after fracture.
Su 
Lu 
S0 
L0 
Lt 
(a) Before testing (b) After testing 
0 
Figure 6.1: Diagrams illustrating tensile specimens (a) before and (b) after failure,
adapted from ISO 6892 [114].
Figure 6.2 shows typical stress-strain curves for alloy HS. All as-quenched (AQ)
conditions failed before reaching the elastic limit. The other three conditions showed
appreciable necking in the region where failure occurred. Two runs per alloy were
tested, and the properties are summarised in Table 6.1.
114
  800 
  600 
  400 
  200 
      0 
1200 
1000 
En
gi
ne
er
in
g 
st
re
ss
 (M
Pa
) 
  500 
      0 
2500 
En
gi
ne
er
in
g 
st
re
ss
 (M
Pa
) 
2000 
1500 
1000 
  500 
      0 
2500 
En
gi
ne
er
in
g 
st
re
ss
 (M
Pa
) 
2000 
1500 
1000 
  500 
2000 
En
gi
ne
er
in
g 
st
re
ss
 (M
Pa
) 
1500 
1000 
0 0.02 0.04 0.06 0.08 0.1 
Engineering strain 
0 0.02 0.04 0.06 0.08 0.1 
Engineering strain 
0 1 2 3 4 5 
Engineering strain 
6 7 
× 10-3 
0 0.02 0.04 0.06 0.08 0.10 
Engineering strain 
0.12 
HS 
As-quenched 
HS 
1800 s at 250 °C 
HS 
1800 s at 350 °C 
HS 
1800 s at 450 °C 
      0 
Figure 6.2: HS alloy tensile curves, for the as-quenched condition, and 250, 350, and
450 ◦C tempering. The two runs per condition are shown in red and blue.
The properties which are usually reported in industrial spring steels specifications
are σy, σUTS, A and Z (Chapter 2), where A and Z are the percentage elongation
after fracture (Equation (6.1)), and the percentage area reduction (Equation (6.2)),
respectively.
A =
Lu − L0
L0
, (6.1)
Z =
S0 − Su
S0
, (6.2)
Although industrially, A and Z are part of the specification, the current project is
limited to stress modelling only.
115
Table 6.1: Summary of tensile properties.
Condition σy,0.2 (MPa) σUTS (MPa) A (%) Z (%)
HS - 250 2038 ± 2 2382 ± 11 8.8 33
HS - 350 2039 ± 1 2324 ± 10 6.5 22
HS - 450 1602 ± 8 1866 ± 8 9.2 39.8
MS - 250 1991 ± 0.2 2311 ± 7 2.3 16.1
MS - 350 1972 ± 3 2233 ± 6 2 38.4
MS - 450 1550 ± 4 1727 ± 5 3.7 34.5
LS - 250 1936 ± 52 2275 ± 17 2.7 38.0
LS - 350 1916 ± 3 2187 ± 0.2 1.6 19.6
LS - 450 1511 ± 7 1672 ± 10 3.6 35.8
6.3 Fitting σy
Morito et al. [115] showed that the general equation for estimating the yield stress
of lath martensite in 0.2 wt.% C takes the following form:
σy = σ0 + σp + σs + σρ + kHPd
− 1
2 , (6.3)
where σ0 is the friction stress for pure Fe, an intrinsic property of the material, σp
is the precipitation hardening contribution, σs is the solid solution strengthening
factor, σρ is the forest hardening of dislocations effect within the laths and in the
low angle boundaries, and kHPd
− 1
2 term is the grain boundary strengthening term,
which accounts for the Hall-Petch grain size effect [115].
The biggest challenge is in determining the last term, which accounts for the Hall-
Petch effect. In tempered lath martensite, owing to its very complex microstructure,
116
it becomes difficult to establish the ‘effective’ grain size. Revisiting Figure 2.3, the
microstructural features in martensite were: PAG, packet, block, lath; each of them
is respectively larger than the following.
While for 0.2 wt.% C, Morito et al. [115] have concluded that the block size acts
as an ‘effective’ grain size, the literature on medium-C steels is scarce. Figure 5.4
gives some indication on the changes occurring at the microstructural level. During
continuous tempering, from the moment the steel is quenched until when it becomes
polygonal ferrite, there is a gradual change in the microstructure, where the initially
formed martensite laths undergo rearrangement and recombination during recovery
and recrystallisation. It is therefore, the starting point of this section to compare
existing models with the tensile experiments in §6.2, in order to examine the ‘effec-
tive’ grain size term for tempering conditions 250, 350 and 450 ◦C in HS, MS and
LS alloys.
Equation 6.3 is slightly modified, where the kHPd
− 1
2 term is replaced by a sub-
grain strengthening term, σsg:
σy = σ0 + σp + σs + σρ + σsg (6.4)
In literature, by experimentally determining σy and estimating the σ0, σp, σs and
σρ terms, Hutchinson et al. [65] calculated the ‘unaccounted strength’, which they
found was a function of interstitial carbon content. However, two points should
be brought forward related to their method: first, in their calculation of the σρ
term, the Taylor factor, M was omitted, and second, their σy value was obtained by
simply multiplying the Vickers hardness by 3. In Chapter 3, it was shown that in
many cases, the actual σy value was less than the predicted value from the Tabor
relationship. Therefore, the ‘unaccounted strength’ determined by Hutchinson et
al. clearly is overestimated. Nevertheless, their method of subtraction still remains
valid, where if all terms except for one are known; then, by simple subtraction, the
117
unknown can be determined.
Using the experimentally determined values in Chapter 5, we can estimate σ0,
σp, σs and σρ. These values are then to be subtracted from the σy values listed in
Table 6.1, in order to estimate the σsg term.
6.3.1 Precipitation hardening, σp
Failed tensile specimens were analysed under TEM. The condition HS-250 and 450
were characterised, where two sections were analysed: the region near necking, would
have undergone the most plastic deformation, and the region near the gripping ends
is where least deformation would have occurred. In Figure 6.3, these are labelled A
and B, respectively.
In Figure 6.3 (HS-250), two interesting observations are made from comparing
the microstructure from these two regions. The first is that the intralath carbides
appear to remain unchanged after failure, implying that precipitation hardening is
more probable to follow a by-pass mechanism, rather than particle shearing. The
second is that in the microstructure of region A, the laths appear to be both wider
in size, and aligned in a particular orientation, whereas region B shows the more
customary random orientation of the laths. If the microstructure in B resembles
that of packets of martensite, the post-failure microstructure in A appears to have
a more defined ‘blocky’ structure.
118
Figure 6.3: TEM micrographs for the HS-250 sample after failure.
Figure 6.4: TEM micrographs for the HS-450 sample after failure.
The same analysis was carried out for HS-450, shown in Figure 6.4. Quite differ-
119
ent to the post-failure microstructure pattern observed in HS-250, when tempered
at 450 ◦C, the ferrite laths at the localised necking region tend to form a random
structure, with less-defined laths, almost adopting a cellular structure.
It is obvious to see that, despite the same chemical composition, under dif-
ferent tempering conditions a different failure mechanism operates, hinting on the
progressive change in the effective grain size throughout tempering. Furthermore,
the intralath carbides appear to remain intact during tensile loading; thus, particle
by-passing is assumed. The original Orowan equation can be used to describe the
degree of precipitation hardening:
∆τp = 0.6Gb
√
Vf
r
, (6.5)
where ∆τp is the shear strength increment due to precipitation hardening, r is the
radius of the particles, and V f is the volume fraction of the precipitate phase. How-
ever, the Orowan equation relies on the assumption that the particles are arranged
on a square grid in the slip plane [57]. In order to allow a random dispersion of the
particles, the Ashby-Orowan equation (6.6) has been chosen for modelling precipi-
tation hardening, as the original Orowan equation leads to an overestimate of the
hardening term, especially for small particle size [57]. This is illustrated in Figure
6.5.
∆σp =
(
0.538Gb
√
Vf
X
)
ln
(
X
2b
)
, (6.6)
where X is the diameter of the particle, taken to be the equivalent spherical diameter
of the rod-shaped intralath carbides.
Based on the information presented in §5.2.3, and using Equation (6.6), Table 6.2
summarises values for X and V f , and the corresponding σp values for all conditions.
Given the absence of data for the 450 ◦C condition, those for 400 ◦C are used, and
instead a lower volume fraction is input.
120
Figure 6.5: Comparison of the Orowan vs. Ashby-Orowan equations for precipitation
hardening.
The values used for b were 2.483, 2.480 and 2.473 nm for alloys LS, MS and HS,
respectively (Table 5.5). In the case of the shear modulus, Ghosh and Olson reported
that in the case of lath microstructure, the high dislocation density decreases the
shear modulus by ∼5% compared to the cubic ferrite counterpart, thus µferrite= 80
GPa corresponds to µlath= 76 GPa [116].
6.3.2 Solid solution strengthening, σs
Given the chemical composition of an alloy, the solid solution strengthening term
can be obtained by empirical means [117]. The values obtained for HS, MS and LS
are summarised in Table 6.3, where only the elements for which data are available
are considered. Notice that the interstitial carbon content has a drastic effect on σs.
For the sake of simplicity, the tetragonality is assumed to be 1, as most carbon would
have precipitated during the 1800 s tempering. Attention is drawn to chromium.
In literature, its effect in solid solution is still not clear, where some have reported
contradictory effects [118, 119]. Kelley and Stoloff [119] have concluded that between
147 and 300 K, up to 10 wt.% Cr in high-purity iron has little effect on the yield
strength, given the small atomic misfit between Cr and Fe, and their same bcc
crystal structure. Therefore the effect of Cr has been omitted.
121
Table 6.2: Estimating precipitation hardening.
Condition X (2×requiv) (mm) V f ∆σp (MPa)
HS - 250 1.58 × 10−5 0.08 628
HS - 350 1.40 × 10−5 0.08 683
HS - 450 1.43 × 10−5 0.05 540
MS - 250 1.38 × 10−5 0.08 693
MS - 350 1.21 × 10−5 0.08 757
MS - 450 1.35 × 10−5 0.05 555
LS - 250 1.43 × 10−5 0.08 676
LS - 350 1.31 × 10−5 0.08 717
LS - 450 1.25 × 10−5 0.05 586
Table 6.3: Estimating solid solution strengthening contribution [117].
Element i ∆σ per 1 wt.% of i (MPa) i wt. % in alloy ∆σs,i (MPa)
C 5544 - -
Si 83 2.3/1.7/1.4 190.9/141.1/116.2
Mn 32 0.7 22.4
Cr -31 0.9 –27.9
Cu 39 0.15 5.85
Ni 0 0.2 0
6.3.3 Forest dislocation hardening, σρ
The hardening contribution from dislocations can be estimated by the following:
∆σρ = MαGb
√
ρ, (6.7)
122
where M is the Taylor factor, taken to be 3, α= 0.25, and ρ is the initial dislocation
density following heat treatment. During the synchrotron measurements, only alloys
HS and LS were characterised (Figure 5.20). The dislocation density for the MS is
interpolated from the previous two values. Table 6.4 provides a summary of ρ and
∆σρ.
Table 6.4: Estimating forest dislocation hardening.
Condition ρ ×1015 m−2 ∆σρ (MPa)
HS - 250 7.1 1184
HS - 350 4.4 939
HS - 450 3.8 866
MS - 250 6.8 1161
MS - 350 3.9 885
MS - 450 2.2 682
LS - 250 6.6 1150
LS - 350 3.7 856
LS - 450 1.6 568
6.3.4 Grain size effect
Substituting the numerical values into (6.4), the grain size effect contribution can
be worked out by simple subtraction, as summarised in Table 6.5.
123
Table 6.5: Unaccounted stress: grain size effect.
Condition σy (MPa) σ0 (MPa) σp (MPa) σs (MPa) σρ (MPa) σsg (MPa)
HS - 250 2038 ± 2 54 628 219 1184 (46)
HS - 350 2039 ± 1 54 683 219 939 144
HS - 450 1602 ± 8 54 540 219 866 (77)
MS - 250 1991 ± 0.2 54 693 169 1161 (87)
MS - 350 1972 ± 3 54 757 169 885 106
MS - 450 1550 ± 4 54 555 169 682 89
LS - 250 1936 ± 52 54 676 144 1150 (88)
LS - 350 1916 ± 3 54 717 144 856 145
LS - 450 1511 ± 7 54 568 144 568 158
On Table 6.5, the values within brackets on the σsg represent negative values.
Given that it has no physical meaning to have a negative stress contribution, an
alternative σy expression is employed. Often, in plasticity modelling, the flow stress
is given by [120]:
τ = τ0 + τs + τb +
√
τ 2in + τ
2
p , (6.8)
where τ is the shear stress, and the subscripts ‘b’ and ‘in’ refer to the kinematic and
isotropic hardening, respectively. The former represents the long range back stress
due to the pile-up of dislocations a the grain boundary, shown in Equation (6.9),
where n is the number of pile-up dislocations and D is the ‘effective’ grain size;
whereas, the latter is the contribution due to the dislocations in the grain interior,
equivalent to the previously defined σρ term, shown in Equation (6.10) [121, 122].
τb =
Gb
D
n, (6.9)
τin = αGb
√
ρinitial, (6.10)
124
Therefore, an alternative expression for σy is:
σy = σ0 + σs + σsg +
√
σ2ρ + σ
2
p, (6.11)
This form of expression, which relies in a quadratic mixture law, is also consistent
with the observations made by Queyreau et al. [123], where they have evaluated
both the individual and the superposed contribution of these two mechanisms. It is
thought that these dislocations impinge on each other, while the rest of the terms
remain additive. Applying (6.11), a new set of σsg values are obtained, summarised
in Table 6.6 and illustrated in Figure 6.6.
300 
400 
500 
600 
700 
250 350 450 
~σ
sg
 c
on
tr
ib
ut
io
n 
(M
Pa
) 
Tempering temperature ( °C) 
HS MS LS 
Figure 6.6: Estimated contribution of the σsg term throughout tempering.
125
Table 6.6: Unaccounted stress: grain size effect.
Condition σy (MPa) σ0 + σs +
√
σ2ρ + σ
2
p (MPa) σsg (MPa)
HS - 250 2038 1585 425
HS - 350 2039 1406 605
HS - 450 1602 1265 309
MS - 250 1991 1548 415
MS - 350 1972 1360 584
MS - 450 1550 868 447
LS - 250 1936 1504 404
LS - 350 1916 1287 601
LS - 450 1511 987 496
The following section looks into the grain size effect, in order to understand the
trends observed in Figure 6.6.
Carbon in solid solution
Given the strong effect that carbon has on solid solution strengthening (Table 6.3), it
deserves further attention. During the XRD peak analysis of the model alloy which
contained the same carbon content (Fe-0.55C-2.0Si wt.%), it showed that after 1800
s tempering at 250 ◦C, the ferritic phase remained tetragonal (c/a = 1.005), while
at 300 ◦C and higher tempering conditions, the tetragonality reduced to 1.
From Equation 4.12, it can be estimated that a tetragonality of 1.005 corresponds
to 0.11 wt.% C in solid solution. Substituting this value into Table 6.3, the strength
increment corresponding to the carbon alone would be of 610 MPa. This value ap-
pears to be rather high, as it would account for almost 30 % of the experimental σy.
Furthermore, taking HS-250 as an example, incorporating the effect of the carbon
in solid solution in Equations (6.4) and (6.11) would give σsg = –656 and –185 MPa,
respectively.
126
It is also possible that the reported value of 5544 MPa per 1 wt.% C is an
overestimate. Capturing the effects of carbon in ferrite is challenging, as finding
the isolated solid solution effect without interference from neighbouring carbides,
grain boundaries and dislocation will be difficult. Therefore, although modelling
the strengthening effect of carbon in solid solution is crucial, currently, challenges
remain in determining the exact carbon concentration in pure solid solution, as well
as determining an accurate value of ∆σ per 1 wt.% C.
Expressions for grain size effect
The widely-accepted Hall-Petch effect shows that the increase in strength is inversely
proportional to the square root of the grain size, d:
∆σy ∝ 1√
d
, (6.12)
However, in lath martensitic steels containing 9–12% Cr, Ghassemi-Armaki et al.
[124] have made use of the following expression for subgrain boundary hardening:
σsg =
10Gb
λlath
, (6.13)
where λlath is the width of the elongated subgrains, referring to individual laths,
typically reported to be 0.3–0.5 µm in this family of steels. Applying (6.13), a value
for σsg in the range of 530–320 MPa was obtained. Based on this, they had concluded
that the contribution from the lath boundaries was predominant in martensite, even
over the precipitation hardening term.
Daigne et al. [125] based their work in a much simplified version of Equation
(6.4), where the σρ term was omitted. Basing their work on a Fe–0.4C–0.2Si–
0.9Mn–0.1Ni–0.2Cr wt.% alloy, their proposed expression for the σy (MPa) was:
σy = 290 +
116
wlath
+
61
L
, (6.14)
127
where wlath is the lath width (µm) and L is the average distance between a precipitate
and its near neighbours, which according to Kocks’s [126] formulation, is determined
by:
L = 1.18r
(
2pi
3Vf
) 1
2
, (6.15)
where r and Vf are the radius and volume fraction, respectively, of precipitates.
What becomes obvious is that grain size hardening is inversely proportional to
grain size. In Chapter 5, it was observed that the average lath width increased with
increasing tempering temperature. This was explained in terms of the martensite
recovery process. Observing the changes from 250 to 350 ◦C in Figure 6.6, despite the
apparent increase in the lath width observed in TEM with increasing temperature,
rather counterintuitively, the grain size effect contribution also increases, albeit the
decrease in the overall σy. This is worth further consideration.
For better visualisation, the percentage strengthening contributions from each
of the terms are summarised in Figure 6.7. As tempering temperature increases,
there appears to be a net decrease in the combined effect of forest dislocation and
precipitation hardening, and consequently, the grain size effect gains importance.
0% 
20% 
40% 
60% 
80% 
100% 
250 350 450 %
 s
tr
en
gt
he
ni
ng
 c
on
tr
ib
ut
io
n 
Tempering temperature (°C) 
0% 
20% 
40% 
60% 
80% 
100% 
250 350 450 
Tempering temperature (°C) 
σ0 σss √(σρ+σp) σsg 
HS MS LS 
0% 
20% 
40% 
60% 
80% 
100% 
250 350 450 
Tempering temperature (°C) 
Figure 6.7: Percent strengthening contribution as a function of tempering temper-
ature and silicon content.
If one were to suppose that the width of individual martensitic laths acts as
the ‘effective’ grain size, then it will fail to explain the increasing grain size effect
128
with tempering temperature between 250 and 350 ◦C. Given that usually the grain
size strengthening effect is inversely proportional to the grain size, the increase in
lath width observed experimentally will not explain the simultaneous increase in the
grain size effect.
When increasing the tempering temperature from 350 to 450 ◦C, there is a re-
duction in the σsg term. The decrease in strength can be explained by the increasing
lath size, which is the case as observed by TEM. However, the increase in σsg con-
tribution from 250 to 350 ◦C cannot be explained by the present TEM results.
In understanding the σsg contribution, one approach is by characterising the
nature of the grain boundaries within the material. EBSD would be a suitable tech-
nique, as the characterisation and location of high and low angle boundaries, based
on the misorientation angle between adjacent grains, could provide some clues. Gen-
erally, it is thought that low angle boundaries are considered to be transparent to
dislocation motion, whereas high angle boundaries tend to hinder dislocation mo-
tion. By using a specific misorientation angle as the criterion for the ‘effective’ grain
boundary, it would be possible to determine the effective grain size.
Now that most of the strengthening contributions have been determined, the
second part of this chapter aims at reproducing the stress-strain curves that were
obtained experimentally.
6.4 Brief introduction to plasticity theory
Considerable amount of work on plasticity modelling can be found in literature. For
instance, Fribourg et al. developed a model for precipitation hardening in aluminium
alloys, where they have modelled Orowan loop storage in precipitates [127].
For ultrafine grains, Huang et al. [122] have developed a model based on irre-
versible thermodynamics. For the detailed specifics, their original work is referred.
Previously, it was defined that the yield strength of the alloy can be expressed in
129
terms of additive terms. In determining the flow stress, Huang et al. modelled the
evolution of the terms n and ρ in Equations (6.9) and (6.10), respectively, during
plastic deformation.
6.4.1 Kinematic hardening
During plastic deformation, as dislocations pile-up at the boundary, the value of n
in the kinematic hardening expression in Equation (6.9) is bound to change. The
evolution of n with respect to the applied plastic shear strain, γp, is given by:
dn
dγp
=
λslip
b
(
1− n
n∗
)
, (6.16)
where λslip is the mean spacing between slip planes at the grain boundaries, and n
∗
is the maximum number of dislocations that can pile up at the grain boundary on
a given slip plane. Beyond this number, the accumulated stress is large enough for
climb or cross-slip to occur. N.B. in order to distinguish the austenite phase, γ, used
in earlier chapters with the shear strain, the latter is accompanied by a subscript
‘p’.
By combining Equations (6.9) and (6.16) and integrating whilst assuming n=0 when
γp=0, the kinematic hardening term becomes:
τb =
Gb
D
n∗
[
1− exp
(
− λslip
bn∗
γp
)]
, (6.17)
The correct use of n∗ value is crucial. In interstitial free steels, this value was between
3 and 4 [122], although in precipitation-hardened aluminium steels, this value was
found to be 9 and 60 for the 7xxx and 6xxx alloys, respectively [127, 128].
In order to obtain an estimate value for D, Equation (6.13) can be applied using
σsg values. Most values lie within the 309–605 MPa range, which would correspond
to an average value of D≈ 0.4 µm. Although this is a crude approximation, it serves
as a starting point in obtaining the right order of magnitude.
130
6.4.2 Isotropic hardening
Much work has been carried out in defining the evolution of the dislocation density
with shear strain, dρ
dγp
, mostly originally based on the Kocks-Mecking approach:
dρ
dγp
=
k1
b
√
ρ− k2ρ, (6.18)
where k1 is the storage coefficient and f is the dynamic recovery term.
During plastic deformation, the following cases must be considered regarding dis-
locations: generation, glide along slip systems and annihilation. The change in
entropy due to these processes is given by:
diS =
dWge
T
+
dWgl
T
+
dWan
T
, (6.19)
where dWge, dWgl and dWan are the energies dissipated due to the processes of dis-
location generation, glide and annihilation, respectively.
Based on the theory of irreversible thermodynamics, Huang et al. [122] derived
an expression for the change in entropy during plastic deformation in terms of dis-
location density:
dS =
1
T
[
µb2 +
τb
(ρin)1/2
]
dρin +
1
T
[
µb2 +
τb
(ρin)1/2
]
ν0
γ˙p
exp
(
− ∆G
kBT
)
dρindγ − 1
T
(τdγ),
(6.20)
where ν0 is the atomic vibration frequency, ∆G is the effective actiation energy for
dislocation annihilation.
Rivera-Dı´az-del-Castillo and Huang [129] showed that the entropy is related to
the shear stress by a proportionality constant C:
dS =
(
C
T
)(
b
l
)
dτin, (6.21)
131
where l is the average distance between dislocations. Thus Equation (6.21) becomes:
dS =
Cb
T
ρ
1/2
in dτin, (6.22)
From the Taylor relation (Equation (6.10)), the evolution of τ in with shear strain can
be written with constant α as dτin
dγ
= ( αµb
2ρ
1/2
in
) (dρin
dγ
), an expression for the evolution of
ρ is obtained, where the effects of precipitation hardening has also been incorporated
[120]:
dρ
dγp
=
1
1 + ( τ
Gb
√
ρ
)− 1
2
Cαc
[√τ 2in + τ 2p
Gb2
−
(
1 +
τb
Gb
√
ρ
)
ν0
γ˙p
exp
(
− ∆G
kBT
)]
ρ, (6.23)
where C and αc are material constants, which were taken from the literature to be
−100 and 0.25, respectively, and ∆G is the effective activation energy for dislocation
annihilation accounting for the average of the activation energies for dislocation
climb or cross-slip, which was taken to be 0.766 eV [122]. k and T are the Boltzmann
constant and the absolute temperature, respectively, and γ˙ is the shear strain rate,
taken to be 0.002 s−1 for the following set of calculations.
6.4.3 Example 1: HS-250 condition
The first step, was to convert the engineering stress-strain values obtained from the
tensile tests into true stress–strain values. As this model is only concerned with the
plastic behaviour of steels, it was necessary to modify the kinematic strengthening
term in Equation (6.17). The reason being that at the beginning of the plastic
deformation, i.e. when γp= 0, this would give τ b= 0. However, up to the yielding
point, there is already some contribution from the ‘effective’ grain size, which was
determined by σsg in §6.3.4 earlier on. Therefore in order to carry this term forward,
132
the following is proposed:
τb = τb,0 +
Gb
D
n∗
[
1− exp
(
− λ
bn∗
γs
)]
, (6.24)
where τ b,0 would be the equivalent shear stress for the σsg obtained previously, and
has been treated as a constant.
Conside`re’s criterion: necking
It is a widely established concept that the onset for necking begins when the following
condition is achieved, which is known as the Conside`re’s criterion:
σ =
dσ
d
, (6.25)
n*= 4, D= 0.4 µm n*= 10, D= 1 µm 
Figure 6.8: Modelling plasticity for the HS-250 condition, where different parameters
have been used.
All the parameters that were obtained experimentally have been used as input to
the model. Figure 6.8 shows preliminary fittings made for the HS-250 condition.
On the left panel, the initial values of n∗= 4 and D= 0.4 µm were used. However,
the shape of the tensile curves do not match. Furthermore, Conside`re’s criterion
predicts very early necking. These two parameters were changed, as shown on the
right panel, to n∗= 10 and D= 1 µm, and a better fitting was obtained. D appears
to be rather high, thus EBSD would be a very useful technique in evaluating the
‘effective’ grain size. Nevertheless, a better fit with respect to the experimental data
133
is obtained.
The failed region (that underwent necking) was observed for this particular con-
dition, using an F20 Tecnai FEG-TEM (a higher resolution TEM), operated at
200 kV, under STEM mode. Zoomed regions are shown in Figure 6.9. The light
region represents an individual lath, where piled-up dislocations are clearly visible
at the lath boundary.
Figure 6.9: HRTEM images showing piled-up dislocations at lath boundaries on a
sample failed by necking.
6.4.4 Example 2: HS-450 condition
The same parameters as in Figure 6.8 were applied to the HS-450 condition, as
shown in Figure 6.10. The first set of results were obtained using n∗= 10 and D=
1 µm. However, the model overestimated the strengthening, therefore a larger grain
size was used, D= 1.5 µm. Although a better fit was obtained, when Conside`re’s
criterion was applied, early necking was predicted. By increasing n∗ to 12, the fit
was improved.
134
n*= 12, D= 1.5 µm n*= 10, D= 1.5 µm 
n*= 10, D= 1.0 µm 
Figure 6.10: Modelling plasticity for the HS - 450 condition.
0 
Pile-up Source 
x 
Barrier 
τ 
τ 
L 
Figure 6.11: Illustration of dislocations pile-up against barriers, after Hull and Ba-
con, [130].
During plastic deformation, as dislocations are generated from sources within
grains (Figure 6.11), the number of dislocation pile-ups is given by [130]:
n =
Lτ
A
, (6.26)
where L is the distance between the source and the boundary, and the x in the
diagram defines the pile-up spreading region, 0 ≤ x ≤ L. A is Gb
pi
or Gb
pi(1−ν) for screw
and edge dislocations, respectively.
135
If the assumption is made that the dislocation generation source is located in
the middle of the grain, then it might be possible to state that for a bigger grain
size, i.e. a larger L value, the higher the n is. In other words, larger grains might
be able to retain more dislocations, increasing the value of n∗. Therefore, it appears
that n∗ increases with increasing D.
6.4.5 Example 3: LS alloy
The model was also applied to the lower silicon alloy. The values for n∗ and D were
adjusted until a reasonable fit was obtained.
n*= 9, D= 1 µm n*= 12, D= 2 µm 
Figure 6.12: Model applied to LS alloys.
For the LS-250 condition, n∗= 9 and D= 1 µm, while these values increased to 12
and 2 µm, respectively, for the LS-450 condition.
If a comparative study is made between HS and LS for the different tempering
conditions, it appears that the increase in the ‘effective’ grain size, D, is bigger in
the LS than in the HS alloy case. Relating this observation back to the findings in
Chapter 5, this could be explained in terms of the recovery process being slowed
down in the presence of silicon.
Although this model still remains at a preliminary stage, where there are still
other refinable values that need to be adjusted, such as λslip and ∆G as a function
of chemical composition, for instance, the plasticity model reinforces the effect of
silicon on the different strengthening contributions in tempered martensite.
136
6.5 Grain size study by EBSD
Hutchinson et al. carried out a grain size study for as-quenched martensite using
EBSD techniques [65]. Figure 6.13 (a) shows EBSD maps for four different steels,
where the black continuous lines represent the PAG boundary (defined by a misori-
entation angle of 45◦). The misorientation distributions of grain boundaries for all
steels are summarised in Figure 6.13 (b).
Figure 6.13: Misorientation angles in as-quenched martensite for four different chem-
ical composition steels, after Hutchinson et al. [65].
On their analysis, Hutchinson et al. showed that ‘block’ structure formation be-
comes more prominent with increasing carbon content, as shown in Figure 6.13 (a).
Based on their description, a packet structure has been highlighted by the black
dotted line in Figure 6.13 for steel D, which consists of several blocks of alternating
colours.
The increasing complexity of the lath martensite microstructure with increasing
carbon content poses a challenge in determining the grain size needed for predicting
137
mechanical behaviour. One way of estimating the grain size is by considering the
misorientation angle between adjacent grains, as generally, the high angle bound-
aries are likely to inhibit dislocation motion, while low angle boundaries remain
transparent to dislocations.
Figure 6.13 (c) shows the variation in the grain size as a function of the mis-
orientation angle, which has been used as a criterion for defining grain boundaries.
According to this figure, for an as-quenched medium carbon steel the grain size is
likely to be between 0.4 and 1 µm. However, as tempering progresses, and recovery
of the matrix phase takes place, these values will change.
EBSD has proven to be a powerful analytical technique, as it allows mapping of
the misorientation angles between grains. The grain boundary angle can then be
used to estimate the effective grain size. Therefore, the aim of the EBSD work was
to measure the grain size evolution throughout martensite tempering under the in-
fluence of silicon. Previously, in Chapter 5, the individual laths were found to be in
the range of ∼0.1 µm. However, given the nature of martensite microstructure, it is
unlikely that the individual laths contribute significantly to grain boundary strength-
ening. Thus, using the misorientation angle as a criterion, ‘larger’ microstructural
units will be considered.
6.5.1 Experimental procedure1
The samples that were previously used for lab-based XRD in Chapter 5, were cut
into four square sections of 10×10×1 mm3. The samples analysed were: HS-250,
HS-450, LS-250 and LS-450. Samples HS-250, HS-450 and LS-250 were prepared to
an OPS colloidal silica finish, whereas another sample of LS-250 and LS-450 were
electrolytically etched (40 V for 12 s).
1This work has been carried out in collaboration with Elodie Boucard and Nathalie Gey (Uni-
versite´ de Lorraine), and Thomas Sourmail (ASCOmetal).
138
Figure 6.14: IQ and IPF images (after some cleaning) for all samples.
The EBSD patterns were acquired using a Zeiss SUPRA40 SEM. All measure-
ments were done with the Channel 5 system (Oxford Instrument), operated at an
accelerating voltage of 20 kV. The step size used varied between 50 and 75 nm.
The image quality (IQ), or band contrast (BC), maps are shown in Figure 6.14,
where the initial index rate at the moment of acquisition is shown for each sample
(NB. LS-250 prepared by electropolishing gave a poor index rate of 47%, therefore
the sample prepared using OPS is considered from here onwards). The inverse pole
139
figure (IPF) maps are also shown after some cleaning. Cleaning is a noise reduc-
tion process, where non-indexed regions are assigned a crystallographic orientation
based on that of their neighbouring grains, hence artificially increasing the index
rate. Some non-indexed regions still remain, shown as black regions on the IPF
maps.
6.5.2 Grain size determination based on misorientation an-
gle
Post-processing of the data has been carried out in Tango (HKL Channel 5). Us-
ing the software, a histogram plot of the misorientation angle, ϕ, can be obtained.
The histograms for all samples are shown in Figure 6.15 (this histogram has been
obtained after a first correction has been aplied). The results follow the same trend
as those reported by Hutchinson et al. where most angles are in the range of 5–15◦,
and in the range of ϕ > 45◦. The peaks in the region of 54–60◦ is characteris-
tic of certain orientation relationships between austenite and martensite, such as
Nishiyama-Wassermann and Kurdjumov-Sachs [10, 65].
Figure 6.15: Histogram plotting frequency of misorientation angle
It is worth noting that HS-250 and HS-450 overlap at all angles. When comparing
140
HS-250 and LS-250, the main difference is observed in the range of 20 < ϕ < 30 ◦.
In the case of LS-450, most angles are either 5 < ϕ < 10 ◦, and at ϕ> 50◦. Most
angles are found at ϕ< 10◦ and ϕ> 50◦, with almost no misorientation angles in the
middle. This gives further supporting evidence on the retarding effect silicon plays
during martensite recovery.
There are two methods of measuring grain size: linear intercept and grain area.
Before taking size measurements, the first step was to define the criterion for a grain
boundary based on the misorientation between two adjacent grains. Morito et al.
showed that lath boundaries have a misorientation angle, ϕ of 2.8–2.9◦, whereas
sub-blocks boundaries had a characteristic value of 6 ◦ [10]. Given that the EBSD
resolution limit is of 5◦, this technique is not ideal for characterising individual laths.
However, it is a useful technique for determining other structures in martensite.
Having defined the criterion for a grain boundary based on the misorientation,
the second step was to correct for noise reduction, artificially increasing the index
rate. This was done iteratively, until the index rate approached ∼100 %. This was
a necessary step, since the non-indexed regions would be interpreted as an intercept
and count as a separate grain. Therefore in order to overcome with this problem,
the index rate was artificially increased to ∼100 %, as shown in Figure 6.16.
Figure 6.16: Sequence of noise reduction.
141
Linear intercept method
This particular method has its drawback, as it is sensitive to how the measurements
are taken. For instance, the grain size results are dependent on the number of
intercept lines drawn and their orientation. Table 6.7 summarises the differences
observed in the average grain size for HS-250.
Table 6.7: Disparity in mean grain size (ϕ= 5◦, index rate 67 %).
No. of lines= 6 No. of lines= 10
Horizontal lines 0.72 µm 0.69 µm
Vertical lines 0.79 µm 0.77 µm
Values can fluctuate ∼0.1 µm when using the linear intercept method, perhaps due
to the highly irregular shapes. Nevertheless, in order to obtain approximate values,
the average grain size is summarised in Figure 6.17, where 10 horizontal lines were
drawn, on IPF maps with index rate ∼100 %.
Figure 6.17: Average grain size based on ϕ using the linear intercept method.
Notice that the trend observed in Figure 6.17 contradict the trends observed by
TEM and plasticity modelling. Throughout tempering, as martensite recovery takes
place, a larger grain size is expected at higher tempering conditions. Furthermore,
this increase is expected to be less with increasing silicon content. However, the
142
results in Figure 6.17 showed smaller grain sizes for LS than for HS.
The values of Dmean between ϕ= 20 and 50
◦ are smaller than expected, especially
when considering that ϕ= 45◦ corresponds to prior austenite grain boundaries [65],
which are likely to be one order of magnitude greater than the value predicted by
the linear intercept method.
Grain area method
An alternative method of measuring grain size in Tango is by the ‘detect grain’
function. The equivalent grain diameter is extracted from the measured area. Figure
6.16 showed that, as image correction is applied, grain size overestimation occurs.
Bearing this in mind, two sets of measurements have been taken: before correction
and after correction (index rate ∼100%). The former, is taken to be the lower limit,
whereas the latter would represent the upper limit; the actual value being within
these limits.
Figure 6.18: Grain size measurement by ‘detect grain’ method in Tango, where the
filled markers represent the upper boundary and the open markers show the lower
boundary for grain size measurements.
From Figure 6.18, it appears that for HS, the grain size actually decreases with
increasing tempering temperature, and that for LS, grain size does not vary much
throughout tempering. These are in contradiction with the trend observed from
143
plasticity modelling. The two methods for measuring grain size are compared in
Table 6.8 for ϕ> 15◦. It is reported in the literature that the transition between low
to high angle boundary is in the region of 10–20◦, with 15◦ being generally accepted
as the threshold [131, 132].
Table 6.8: Differences observed in grain size (ϕ> 15 ◦).
Grain area method Linear intercept method
HS-250 0.37 – 0.58 µm 0.71 µm
HS-450 0.38 – 0.56 µm 0.59 µm
LS-250 0.44 – 0.66 µm 0.41 µm
LS-450 0.42 – 0.66 µm 0.49 µm
Except for HS-250, the values obtained from the linear intercept method fall close to
the range obtained by the grain area method. Also, notice that in all cases, the aver-
age grain size is below the 1 µm value predicted by the plasticity model. A possible
reason for the significantly smaller grain size measured by EBSD is the irregularity
of the shapes observed in 2D. The results presented here are average values, and
the presence of numerous smaller grains will substantially decrease the average value.
D and n∗
Having determined D from the EBSD measurements, the values were plotted back
into the plasticity model, where only the value of n∗ was adjusted. The values
obtained by the linear internet method were used from Table 6.8. The plots are
summarised in Figure 6.19. Compared to the previous Figures 6.8, 6.10 and 6.12,
the fitting between the model and the experimental curve is not improved by using
the D values obtained from EBSD.
The slight disagreement observed between the EBSD values and the model will be
commented. Although questions remain on the validity of the parameters used in the
144
model (§6.4.5), the EBSD results are counterintuitive. In the HS alloys, as tempering
temperature increases, one would expect an increase in grain size. Nevertheless, D
was seen to drop from 0.71 to 0.59 µm when the temperature increased from 250 to
450 ◦C. Furthermore, as explored in Chapter 5, silicon is thought to decrease the
rate of recovery, thus a bigger grain size is expected in the LS alloy.
Figure 6.19: PAG boundaries revealed by thermal etching.
Prior austenite grain size
Using the same procedure as in Chapter 3, the prior austenite grain (PAG) size
was revealed by the method of thermal etching. Figure 6.20 shows representative
micrographs obtained for all three alloys.
145
Figure 6.20: PAG boundaries revealed by thermal etching.
Notice that the PAG boundaries could not be revealed for the HS alloy. The mean
PAG size was found to be 11.4 ± 1.1 and 12.3 ± 1.1 µm for MS and LS, respectively.
These values are comparable with those that can be obtained by EBSD. Hutchinson
et al. showed that prior austenite grains can be revealed by high angle boundaries
∼45◦. In Figure 6.21, a prior austenite grain is highlighted, where the thick dotted
lines have been drawn in order to highlight the boundaries.
Figure 6.21: PAG visible for condition LS-450.
From Figure 6.21, the PAG appears to be ∼15 µm, which agrees with the value
found by thermal etching.
146
6.6 Discussion
It becomes clear that martensite possesses a very complex microstructure, which
makes it challenging for mechanical properties modelling. Based on the analysis
carried out, Figure 6.22 gives an idea of the dimensions of the substructures involved
in martensite.
Figure 6.22: Representation of the dimensions of the hierarchical structures within
martensite.
Determining the ‘effective’ grain size still remains a challenge. Conventionally, it
is thought that only high angle boundaries contribute to the strengthening effect.
Nonetheless, Figure 6.9 shows a network of ‘tangled’ dislocations built-up at the
lath boundary, for a HS-250 sample that failed by necking. It might be the case
that at the early tempering stage, there has not been sufficient time for dislocations
to rearrange, and so even individual lath units that are heavily dislocated, might
add to the strengthening effect.
6.7 Conclusions
Using the experimental results obtained in Chapter 5, the key microstructural pa-
rameters contributing to strengthening in 0.55 wt.% steels have been unveiled. The
high dislocation density in these steels contribute significantly to the ultra-high
strength. An attempt has been made in revealing the ‘effective’ grain size using
different methods.
The method of determining the grain size based on plasticity modelling sup-
ported the idea presented in Chapter 5, where silicon inhibits the recovery process
of martensite. Since grain growth is impeded under the influence of silicon, this
147
resulted in finer sub-units as the silicon content increased within the alloy. The
predicted grain size for LS was bigger than HS.
The misorientation angle histogram plots obtained from EBSD also confirm the
effect of silicon in martensite recovery. Nevertheless, obtaining a suitable grain size
using EBSD remains a challenge.
The strengthening contribution due to sub-grain boundary hardening, σsg, was
also computed. From the post-fracture TEM analysis of sample HS-250, it appears
that at low tempering temperatures, given the very high dislocation density, even the
low angle lath boundaries can contribute to strengthening. In order to successfully
model the grain size effect, a more exhaustive study of the kinetics of martensite
recovery is needed.
148
Chapter 7
General conclusions and scope for
future work
7.1 Conclusion
In developing new grades of martensitic steels, the microstructure-property rela-
tionship must be understood. This is a challenging study, owing to the complex
microstructure of martensite, especially with increasing carbon content, and the nu-
merous processes occurring. At the temperature range where industrial processes
usually occur, the changes taking place are carbon segregation, carbide precipitation
and recovery of the ferrite matrix. Each of these phenomena was studied in detail.
Interrupted ageing
Grades used for spring steels contain a significant amount of carbon. Due to its
interstitial nature, it undergoes immediate reactions as it tends to precipitate as
carbides. Controlling carbon-related reactions from the start of a heat treatment is
crucial, as this will have a marked effect on the final properties.
This has been the starting point for the work in understanding the effect of
interrupted ageing (IA). By incorporating an intermediate stage between quench-
149
ing and tempering, where quenched martensite is left to age at room temperature,
changes in the precipitation process were observed, causing a response in the hard-
ness properties. Generally an increase of ∼10% in Vickers hardness was seen via
interrupted ageing. Thermoelectric power was used to model carbon segregation to
dislocations, where it was postulated that room temperature ageing increased the
number of effective nucleation sites for the subsequent tempering stage, reflected in
a more stable microstructure and the formation of finer precipitates.
Therefore, the incorporation of interrupted ageing (IA) during the heat treat-
ment is a process that can potentially be implemented in industry. In applications
where high hardness is the priority, the process of IA offers microstructural and
dimensional stability and an increase of ∼10% HV over conventional quenching and
heat treatment. Nonetheless, a restriction on the component size applies, given the
relatively high quenching rates required in order to control carbon segregation.
Furthermore, building upon existing models and techniques, an understanding of
the time-temperature aspects of the segregation/nucleation has been gained. Placing
the conventional heat treatment as a reference, the effects on carbide precipitation
of the duration of the IA time, and the time and temperature of the subsequent
tempering stage, was quantified. The incorporation of IA resulted in a finer mi-
crostructure, where the duration of the IA influenced the spacing between intralath
carbides.
The effect of silicon during martensite tempering
The effect of silicon has been a major point of interest shared by industrial and
academic researchers. The presence of silicon as an alloying element has led to im-
provements in mechanical properties. Despite decades of research, there have been
unanswered questions that require further exploration. In this work, the effect of
silicon has been studied from the microstructural (carbide precipitation) and me-
chanical property points of view.
150
The effect of silicon in carbide precipitation was examined. By observing the
evolution of carbide peaks in situ using high-energy X-rays, it was seen how the
presence of silicon inhibited the growth of cementite. Due to technical difficulties
it was not possible to characterise the early nucleation stage. Nevertheless, based
on the classical nucleation theory, where cementite formation was modelled under
paraequilibrium conditions, higher silicon content was predicted to retard nucle-
ation.
Although precipitation hardening plays a major role in the properties of marten-
sitic steels, there are other contributing factors that need to be considered, such
as solid solution strengthening, forest dislocation hardening and grain boundary
strengthening. Based on the microstructural analysis by TEM and line broaden-
ing by XRD methods, the strengthening contributions of several features were de-
termined. Owing to the high dislocation density in martensitic steels, the forest
dislocation hardening term was seen to be the primary contributor, followed by pre-
cipitation hardening.
As a result of dislocation density measurements, there have also been indications
on the possible role of silicon in retarding martensite recovery during tempering.
This point is also reinforced by TEM analysis. It was observed that the width of the
individual lath units increased during tempering. However, this increase was least
for high contents of silicon.
Modelling strengthening effects
In order to successfully model plasticity, it is important to determine the grain size
effect. This is a particular challenge in tempered martensite, due to the complex
microstructure. The microstructural subunits within martensite have been studied
extensively, especially for carbon contents up to 0.2 wt.%. However for higher con-
tents, such as 0.5–0.6 wt.% C, the literature on these alloys is scarce. During tensile
loading, the matrix phase participates actively in the kinematic hardening. There-
151
fore, it becomes necessary to characterise the boundaries present within martensite.
Generally, it is believed that high angle boundaries arrest dislocation motion, while
low angle boundaries remain transparent to them. In order to study the grain bound-
aries, a few selected conditions were studied using EBSD techniques. Assuming that
the transition from low to high angle boundary is defined by a critical misorientation
angle of 15◦, this boundary condition was set in defining the grain size. Grain size
measurements gave an equivalent diameter value in the range of 0.4–0.6 µm.
By the method of thermal etching, the prior austenite grain (PAG) size was mea-
sured to be in the order of 10 µm, while by low magnification TEM, the individual
lath sizes were seen to be ∼0.1 µm. Given that the building units in martensite
consist of PAG > packet > block > lath, considering the order of magnitude, it is
likely that the grain boundary contributing to strengthening correspond to that of
block and/or packet structures.
Using a plasticity model based on irreversible thermodynamics, the tensile curves
obtained for different tempering conditions were reproduced for a corresponding
grain size. The value was in the order of 1–2 µm, which is slightly higher than the
ones found by EBSD.
152
Figure 7.1: Main ideas and contributions represented on the chain model, and ap-
plied in this thesis.
Figure 7.1 summarises the findings in this work, which can be interlinked at the
processing, microstructural and properties levels. As a consequence of the various
studies presented in this thesis, a better understanding of medium (0.5 wt.%) car-
bon tempered martensite has been gained. This will serve as a strong foundation
and potentially become a powerful tool in assisting the design of new martensitic
alloys for the industry, where depending on the desired property that needs to be
optimised, a specific set of microstructural features can be targeted and tailored
through processing.
7.2 Scope for future work
Based on the present work, there are several interesting points that deserve further
investigation:
• During mechanical modelling, the main focus has been on optimising the ten-
sile properties. However, grades used for spring steel applications also have
153
a minimum impact energy requirement. During the interrupted ageing work,
conditions undergoing interrupted ageing displayed relatively poor ductility,
hinting the following: (a) a possible relationship between carbon content in
‘true’ solid solution and brittleness, or (b) unfavourable carbon atom config-
uration as a consequence of the atmosphere formations around dislocations
during IA, that may dramatically reduce dislocation motion. However, very
little improvements were observed on σy and σUTS during tensile loading. This
is counterintuitive, as one would assume an increase in σ when the dislocation
motion is hindered, and given the significant hardness improvement.
• A further in-depth EBSD study of the steels would be extremely valuable, in
order to develop a firm understanding of the microstructure and the various
sub-units involved. Carrying out a study of how each building unit evolves
during tempering, understanding the nature of the boundaries involved, and
relating this to martensite recovery is key to an accurate plasticity model.
• The significance of the retained austenite phase during martensite tempering
is worth exploring. Given that the volume fraction of retained austenite in
this family of steels is under 10%, its presence has been neglected so far.
Nevertheless, owing to its relative carbon enrichment, the decomposition of the
metastable phase could lead to further carbide precipitation, or solid solution
hardening.
• One aspect of the carbide precipitation that needs further clarification is the
role of silicon in the ε → θ carbide transition. Given the technical difficulties
experienced during synchrotron heat treatments in situ, it would be advisable
to perform ex situ a series of different time-temperature combinations and
carry them out as room temperature measurements.
• The process of martensite recovery is worth further exploration. In this work,
it was suggested that silicon plays a role in the recovery of martensite. In order
154
to confirm that the retarding effect is specific to silicon rather than a general
observation due to the presence of alloying elements in the ferrite lattice, silicon
could be substituted by another alloying element. A comparative study, where
by measuring the same properties, such as dislocation density and sub-units
size, would clarify this issue.
155
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