Hot spots in Ammonium Nitrate
Nicholas Taylor
Department of Physics
Churchill College, University of Cambridge
This dissertation is submitted for the degree of
Doctor of Philosophy
April 2011
ii
Acknowledgements
This research would not have been possible without the assistance of
others. Corporately, Orica Mining Services and the Atomic Weapons
Establishment envisioned and funded the project.
From Orica, Ian Kirby, Jim Chan, John Cooper and Richard Goodridge
all provided useful support, insight and questions. Caroline Handley
and Hugh James fulfilled a similar role on behalf of AWE, while Geoff
Wanstall, Jane Didham and Maurice Marshall did so for the Defence
Science and Technology Laboratory. Dr. Marshall is particularly to
be thanked for providing a sample of Nitram prills; high-purity am-
monium nitrate fertilizer prills are understandably difficult to obtain
from garden centres.
Without Ray Flaxman and Bob Marrah of the Cavendish workshops,
and Saevar Sigurdsson of the electronics workshop, it’s doubtful any of
the apparatus used in this research would have worked well enough, or
for long enough, to produce the results presented here. While David
Powell and Nigel Palfrey of the student workshop didn’t manage to
turn me into a competent machinist, thanks to their efforts I can now
see that state on a clear day. Perhaps more impressively, they also
managed to prevent my suffering any serious injury in the process.
Trevor Fairhead of the Cavendish operated the ESEM used in this
research. Peter Laity of Earth Sciences operated the X-ray tomogra-
phy machine. Stewart Palmer of the SMF group provided invaluable
assistance in the operation of the Instron.
David Williamson supervised the end of this research, having taken
over adroitly when my original supervisor, Bill Proud, left the Caven-
dish. John Field provided many useful discussions. I am indebted to
all three. David Chapman, David Bell, Adam Collins, Martin Green-
away, Tacye Phillipson, Chris Braithwaite, Stephen Walley and Mike
Morley, all of the SMF group, provided helpful comments and discus-
sion.
Last and by no means least, I wish to thank my long-suffering wife
Michelle. Without her support it’s unlikely this research would have
been completed.
Declaration
This dissertation is the result of my own work and includes nothing
which is the outcome of work done in collaboration except where
indicated in the text. No part of this dissertation has been submitted
for any other degree at any institution. The length of this dissertation
does not exceed the sixty thousand words permitted by the Physics
and Chemistry Degree Committee.
Nicholas Taylor
March 2011
Abstract
Ammonium nitrate (AN) is commonly used as an explosive and as a
fertilizer. In both roles it is provided as prills or pellets, approximately
spherical and a few millimetres in diameter. The microstructures of
several commercially-available AN compositions were investigated us-
ing environmental scanning electron microscopy (ESEM) and X-ray
microtomography. Those intended for explosive use were found to be
more porous than those intended for fertilizer use. The pores in explo-
sive prills were also found to form a connected network. The elemen-
tal composition of pellets of mixed AN and dolomite was investigated
using energy-dispersive X-ray spectroscopy (EDX); the dolomite ad-
ditive was found to take the form of grains roughly 50µm in size.
The compaction behaviour of confined cylindrical beds of these prills
and pellets was studied at strain rates between 4 × 10−4 s−1 and
200 s−1. Quasi-static experiments were performed using a screw-driven
instrumented press, while higher-rate experiments used a drop weight,
instrumented with a line laser and load cell. The resistance of a bed
to compaction was found to depend on the microstructure of its prills
in most cases. Denser prills offered greater resistance to compaction.
The exception to this rule was a pellet, rather than prill, formulation.
Beds were also found to offer more resistance to compaction at higher
strain rates. The Kawakita compaction model was found to agree well
with the experimental data.
A commercial fertilizer, not containing any AN, was assessed for use
as an inert mock for AN prills and pellets. Prills of a suitable size for
this purpose were found using EDX to consist of P2O5, with a coat-
ing of unknown composition. They were supplied mixed with smaller
K2CO3 and urea prills. The mixture was found to have comparable
compaction behaviour to AN compositions, indicating that it was use-
ful as a mock for those compositions. In a plate impact experiment
on a single layer of P2O5 prills, very little light was observed. This
indicated that these prills were sufficiently inert for these purposes.
The light produced by shocked granular ammonium nitrate beds and
single prill layers was investigated using high-speed framing photogra-
phy, photodiodes and gated visible-light spectroscopy. Framing pho-
tography of prill layers suggested that reaction in prill beds was dom-
inated by effects internal to prills. This was further supported by the
similarity between photodiode recordings of prill beds and beds of in-
ert prills containing a single reactive prills. Framing photography of
drop weight experiments searching for a mechanism for initiation of
reaction by interaction between prills found nothing.
Decay of the light output of the beds suggested that in both granu-
lar and prill beds this light output was due to small regions heated
to thousands of kelvin, which then cooled. Spectroscopic study con-
firmed this. These regions were found to reach a peak temperature
of 6660± 20K, well in excess of the approximately 2000 K predicted
by a simple chemical model. Investigation of spectral lines observed
during this study indicated that the exothermic reaction that led to
heating of these emitting regions involved NO.
vi
Contents
Contents vii
List of Figures xiii
List of Symbols xxvii
1 Introduction and literature review 1
1.1 Ammonium nitrate . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Phases of ammonium nitrate . . . . . . . . . . . . . . . . . 2
1.1.2 Thermal decomposition of ammonium nitrate . . . . . . . 4
1.1.2.1 Kinetics . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Hot spots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Crack growth . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Cavity collapse . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2.1 Plastic deformation . . . . . . . . . . . . . . . . . 7
1.2.2.2 Adiabatic compression . . . . . . . . . . . . . . . 8
1.2.2.3 Evaporation and stagnation . . . . . . . . . . . . 8
1.2.2.4 Jets . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.4 Shear bands . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.5 Vibrational up pumping . . . . . . . . . . . . . . . . . . . 9
1.2.6 Sensitization centres . . . . . . . . . . . . . . . . . . . . . 11
1.2.7 Hot spots in ammonium nitrate . . . . . . . . . . . . . . . 11
1.2.7.1 Framing photography . . . . . . . . . . . . . . . 11
1.2.7.2 Light output . . . . . . . . . . . . . . . . . . . . 13
vii
CONTENTS
1.2.7.3 Spectroscopy . . . . . . . . . . . . . . . . . . . . 16
1.3 Detonation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.1 Chapman-Jouguet detonation theory . . . . . . . . . . . . 18
1.3.1.1 Conservation relations . . . . . . . . . . . . . . . 19
1.3.1.2 The rear boundary conditions . . . . . . . . . . . 19
1.3.2 Zeldovich-von Neumann-Do¨ring detonation theory . . . . . 20
1.3.3 Non-ideal detonation . . . . . . . . . . . . . . . . . . . . . 21
1.3.4 Detonation of ammonium nitrate . . . . . . . . . . . . . . 22
1.4 Compaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.1 Experimental studies . . . . . . . . . . . . . . . . . . . . . 23
1.4.2 Compaction models . . . . . . . . . . . . . . . . . . . . . . 24
1.4.2.1 Heckel model . . . . . . . . . . . . . . . . . . . . 24
1.4.2.2 Kawakita equation . . . . . . . . . . . . . . . . . 25
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Experimental techniques 35
2.1 Environmental scanning electron microscopy . . . . . . . . . . . . 35
2.1.1 Secondary electron detection . . . . . . . . . . . . . . . . . 37
2.1.2 Beam damage . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2 Energy-dispersive X-ray spectroscopy . . . . . . . . . . . . . . . . 37
2.2.1 X-ray detection . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3 X-ray microtomography . . . . . . . . . . . . . . . . . . . . . . . 39
2.4 Screw-driven instrumented press . . . . . . . . . . . . . . . . . . . 40
2.4.1 Backlash . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5 Load cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5.1 Bending of the load cell . . . . . . . . . . . . . . . . . . . 46
2.6 Drop Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.6.1 Sensitivity of explosives . . . . . . . . . . . . . . . . . . . 47
2.6.2 The Cavendish drop weight . . . . . . . . . . . . . . . . . 48
2.6.3 Materials testing . . . . . . . . . . . . . . . . . . . . . . . 48
2.6.4 Calibrating the load cell . . . . . . . . . . . . . . . . . . . 50
2.6.5 Imaging through the sample . . . . . . . . . . . . . . . . . 51
2.7 Split Hopkinson Pressure Bar . . . . . . . . . . . . . . . . . . . . 51
viii
CONTENTS
2.7.1 The momentum trap . . . . . . . . . . . . . . . . . . . . . 53
2.7.2 Materials testing . . . . . . . . . . . . . . . . . . . . . . . 54
2.7.3 Strain measurement in the SHPB . . . . . . . . . . . . . . 56
2.7.4 Calibrating the SHPB . . . . . . . . . . . . . . . . . . . . 58
2.8 High speed photography . . . . . . . . . . . . . . . . . . . . . . . 59
2.8.1 Rotating-mirror . . . . . . . . . . . . . . . . . . . . . . . . 59
2.8.2 Image conversion . . . . . . . . . . . . . . . . . . . . . . . 61
2.9 Photodiodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.10 Plate impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.11 High-speed optical spectroscopy . . . . . . . . . . . . . . . . . . . 70
2.11.1 Dispersion stage . . . . . . . . . . . . . . . . . . . . . . . . 70
2.11.2 Image intensifier . . . . . . . . . . . . . . . . . . . . . . . 72
2.11.3 Detector array . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.12 Model fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.12.1 Justification for χ2 minimization . . . . . . . . . . . . . . 75
2.12.2 Inverse-Hessian method . . . . . . . . . . . . . . . . . . . 76
2.12.3 Steepest-descent method . . . . . . . . . . . . . . . . . . . 78
2.12.4 Levenberg-Marquardt algorithm . . . . . . . . . . . . . . . 78
2.12.5 Error estimation . . . . . . . . . . . . . . . . . . . . . . . 80
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3 Materials and microstructure 85
3.1 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.2 Orica prills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.3 Agricultural calcium ammonium nitrate pellets . . . . . . . . . . . 92
3.4 Nitram prills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.5 Westland pellets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4 Compaction 103
4.1 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . 103
4.1.1 Quasi-static compaction . . . . . . . . . . . . . . . . . . . 103
4.1.1.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . 105
ix
CONTENTS
4.1.2 Higher rate compaction: SHPB . . . . . . . . . . . . . . . 106
4.1.2.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . 107
4.1.3 Higher rate compaction: drop weight . . . . . . . . . . . . 107
4.1.3.1 The line laser . . . . . . . . . . . . . . . . . . . . 108
4.1.3.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . 111
4.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 111
4.2.1 Quasistatic results . . . . . . . . . . . . . . . . . . . . . . 111
4.2.1.1 Pressure oscillations in Instron traces . . . . . . . 114
4.2.2 SHPB results . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.2.3 Drop weight results . . . . . . . . . . . . . . . . . . . . . . 119
4.2.4 Effect of coating . . . . . . . . . . . . . . . . . . . . . . . . 120
4.2.5 Effect of strain rate . . . . . . . . . . . . . . . . . . . . . . 120
4.2.5.1 Momentum effects . . . . . . . . . . . . . . . . . 128
4.2.5.2 Fracture effects . . . . . . . . . . . . . . . . . . . 129
4.2.6 Effect of initial porosity . . . . . . . . . . . . . . . . . . . 130
4.2.7 Effect of microstructure . . . . . . . . . . . . . . . . . . . 131
4.2.8 Effect of composition . . . . . . . . . . . . . . . . . . . . . 132
4.2.9 Compaction modelling . . . . . . . . . . . . . . . . . . . . 134
4.2.9.1 Heckel model . . . . . . . . . . . . . . . . . . . . 135
4.2.9.2 Kawakita model . . . . . . . . . . . . . . . . . . 138
4.2.10 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.3 Conclusions and future work . . . . . . . . . . . . . . . . . . . . . 142
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5 Light output 145
5.1 Granular beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.1.1 Cell and flyer design . . . . . . . . . . . . . . . . . . . . . 146
5.1.1.1 Triggering . . . . . . . . . . . . . . . . . . . . . . 146
5.1.1.2 Pressure gauge . . . . . . . . . . . . . . . . . . . 147
5.1.1.3 Flyer . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.1.1.4 Bed construction . . . . . . . . . . . . . . . . . . 150
5.1.1.5 Projectile velocity . . . . . . . . . . . . . . . . . 150
5.1.1.6 Fibre and mirror mounts . . . . . . . . . . . . . . 151
x
CONTENTS
5.1.2 Framing photography . . . . . . . . . . . . . . . . . . . . . 153
5.1.3 Photodiode measurements . . . . . . . . . . . . . . . . . . 156
5.1.3.1 Induction time . . . . . . . . . . . . . . . . . . . 159
5.1.3.2 Decay time . . . . . . . . . . . . . . . . . . . . . 167
5.1.4 Particle size . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.2 Prill beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.2.1 Bed construction . . . . . . . . . . . . . . . . . . . . . . . 173
5.2.2 Westland prills . . . . . . . . . . . . . . . . . . . . . . . . 174
5.2.3 Orica prills . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.2.3.1 Reactive prill bed . . . . . . . . . . . . . . . . . . 174
5.2.3.2 Single reactive prill . . . . . . . . . . . . . . . . . 180
5.2.3.3 Direct impact . . . . . . . . . . . . . . . . . . . . 183
5.2.3.4 Drop weight experiments . . . . . . . . . . . . . . 184
5.2.4 Nitram prills . . . . . . . . . . . . . . . . . . . . . . . . . 187
5.2.4.1 Reactive prill bed . . . . . . . . . . . . . . . . . . 187
5.2.4.2 Single reactive prill . . . . . . . . . . . . . . . . . 190
5.2.5 Agricultural pellets . . . . . . . . . . . . . . . . . . . . . . 194
5.2.5.1 Reactive pellet bed . . . . . . . . . . . . . . . . . 194
5.2.5.2 Single reactive pellet . . . . . . . . . . . . . . . . 196
5.3 Conclusions and future work . . . . . . . . . . . . . . . . . . . . . 200
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6 Spectra 203
6.1 Apparatus and calibration . . . . . . . . . . . . . . . . . . . . . . 203
6.1.1 Dark current . . . . . . . . . . . . . . . . . . . . . . . . . 204
6.1.2 Wavelength calibration . . . . . . . . . . . . . . . . . . . . 206
6.1.3 Relative intensity calibration . . . . . . . . . . . . . . . . . 206
6.1.4 Fibre effects . . . . . . . . . . . . . . . . . . . . . . . . . . 210
6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
6.2.1 Reaction-like spectra . . . . . . . . . . . . . . . . . . . . . 214
6.2.1.1 Time-dependence of spectra . . . . . . . . . . . . 216
6.2.2 Cooling-like spectra . . . . . . . . . . . . . . . . . . . . . . 216
6.2.2.1 Variation of temperature with time . . . . . . . . 220
xi
CONTENTS
6.2.2.2 Uncertainty in temperature . . . . . . . . . . . . 222
6.2.2.3 Consequences of temperature measurement . . . 222
6.3 Conclusions and future work . . . . . . . . . . . . . . . . . . . . . 225
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
7 Conclusions 227
A Structures of ammonium nitrate 231
A.1 Phase I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
A.2 Phase II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
A.3 Phase III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
A.4 Phase IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
A.5 Phase V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
B Additional photography 249
xii
List of Figures
1.1 Ammonium nitrate pressure-temperature phase diagram . . . . . 4
1.2 Layout of cell for impact on granular ammonium nitrate bed. . . . 12
1.3 Framing photography on 2mm AN beds at a range of impact ve-
locities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Framing photography on a 0.4mm AN bed. . . . . . . . . . . . . 13
1.5 Light output of 2mm ammonium nitrate beds, under impact from
flyers of a range of velocities. . . . . . . . . . . . . . . . . . . . . . 14
1.6 Normalized light outputs from figure 1.5. . . . . . . . . . . . . . . 15
1.7 Light output of 2mm AN beds of varying density, under impact
from a flyer at 700m s−1. . . . . . . . . . . . . . . . . . . . . . . . 15
1.8 Light output of 2mm ammonium nitrate beds, under impact from
flyers of a range of velocities, to some of which chalk has been added. 16
1.9 Spectrum of 2mm AN bed under flyer impact, 60–560 ns after
impact. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.10 Spectrum of 2mm AN bed under flyer impact, 1–1.5µs after impact. 17
1.11 Spectrum of 2mm AN bed under flyer impact, 2–2.5µs after impact. 18
1.12 Intersection of Rayleigh lines with the reacted Hugoniot. . . . . . 20
2.1 Diagram of an ESEM. . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 Schematic showing X-ray tomography measurements. . . . . . . . 41
2.3 Penumbral broadening. . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 Schematic of a screw-driven instrumented press. . . . . . . . . . . 43
2.5 Arrangement of strain gauges on a Wheatstone bridge load cell. . 45
2.6 Circuit diagram of strain gauges in a Wheatstone bridge load cell 45
2.7 Usual arrangement of the Cavendish drop weight system. . . . . . 49
xiii
LIST OF FIGURES
2.8 Light path through the sample in drop weight. . . . . . . . . . . . 52
2.9 A compressive SHPB system. . . . . . . . . . . . . . . . . . . . . 52
2.10 Approximate gauge outputs for a compressive SHPB. . . . . . . . 53
2.11 Symbols used in SHPB analysis . . . . . . . . . . . . . . . . . . . 54
2.12 Potential divider circuit used in the SHPB. . . . . . . . . . . . . . 56
2.13 Schematic of a simple rotating-mirror camera. . . . . . . . . . . . 60
2.14 Optics of the C4 rotating-mirror camera. . . . . . . . . . . . . . . 62
2.15 System for measuring the rotation speed of the mirror in the C4
camera. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.16 Electron optics of the Ultranac 501 image conversion camera. . . . 64
2.17 Variation of potentials of shuttering system of the Ultranac 501
with time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.18 Electric potential energy for electrons in the vicinity of the shut-
tering system of the Ultranac 501. . . . . . . . . . . . . . . . . . . 66
2.19 Arrangement of output frames in the Ultranac 501 image conver-
sion camera, when configured to produce twelve frames. . . . . . . 67
2.20 PIN semiconductor junction under reverse bias. . . . . . . . . . . 68
2.21 Schematic of the firing chamber of the Cavendish small gas gun. . 69
2.22 Schematic of diffraction stage of Princeton Applied Research 1235
spectrograph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.23 Layout of the image intensifier in the Princeton Applied Research
1455B-700-HQ intensified detector. . . . . . . . . . . . . . . . . . 73
3.1 Photograph showing the various materials studied. . . . . . . . . . 86
3.2 ESEM image showing razor damage in uncoated Orica prill. . . . 87
3.3 ESEM image of outer surface of uncoated Orica prill. . . . . . . . 88
3.4 ESEM image of outer surface of coated Orica prill. . . . . . . . . 89
3.5 ESEM image of body of coated Orica prill, showing network of
voids which permeates the material. . . . . . . . . . . . . . . . . . 89
3.6 Tomographic slice approximately through the centre of an un-
coated Orica prill. . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.7 ESEM image of body of coated Orica prill, showing microstructure
around the central void. . . . . . . . . . . . . . . . . . . . . . . . 92
xiv
LIST OF FIGURES
3.8 ESEM image of body of coated Orica prill, showing microstructure
near the outer surface of the prill. . . . . . . . . . . . . . . . . . . 93
3.9 ESEM image of the interior of an agricultural calcium ammonium
nitrate pellet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.10 ESEM image of the interior of an agricultural calcium ammonium
nitrate pellet, covering the area shown in figure 3.11. . . . . . . . 95
3.11 Composite ESEM/EDX image of the interior of an agricultural
calcium ammonium nitrate pellet. . . . . . . . . . . . . . . . . . . 95
3.12 ESEM image of the microstructure of a Nitram prill. . . . . . . . 96
3.13 ESEM image of a void within a Nitram prill. . . . . . . . . . . . . 97
3.14 ESEM image of the microstructure of one of the larger Westland
pellets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.15 Spatially-unresolved EDX spectrum of one of the larger Westland
pellets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.16 Spatially-unresolved EDX spectrum of one of the small, pale West-
land pellets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.17 Spatially-unresolved EDX spectrum of one of the small, dark West-
land pellets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.18 ESEM image of the microstructure near the centre of a Westland
potassium carbonate pellet, showing central void. . . . . . . . . . 101
3.19 ESEM image of the microstructure near the centre of a Westland
urea pellet, showing central void. . . . . . . . . . . . . . . . . . . 101
4.1 Confinement cell used for compaction experiments. . . . . . . . . 104
4.2 Arrangement of sample and confinement for SHPB compaction
testing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.3 Arrangement of drop weight used for compaction experiments. . . 109
4.4 Arrangement of line laser used to measure position of drop weight. 110
4.5 Effect of adding an obstruction to the line laser system. . . . . . . 110
4.6 Pressure-porosity curves from two Instron experiments on uncoated
Orica prill beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.7 Pressure-porosity curves from two Instron experiments on coated
Orica prill beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
xv
LIST OF FIGURES
4.8 Pressure-porosity curves from two Instron experiments on agricul-
tural pellet beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.9 Pressure-porosity curves from two Instron experiments on Nitram
prill beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.10 Pressure-porosity curves from two Instron experiments on West-
land prill beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.11 Force oscillations in Instron experiments. . . . . . . . . . . . . . . 115
4.12 Optical microscope image of epoxy cast of interior of confinement
jacket, showing striations at various length scales. . . . . . . . . . 116
4.13 Pressure-porosity curves from repeated SHPB experiments on a
single uncoated Orica prill bed. Initial strain rate 450 s−1. . . . . 117
4.14 Sample geometry encountered in later SHPB compaction runs. . . 118
4.15 Pressure-porosity curve generated by ninth SHPB experiment on
uncoated Orica prill bed. . . . . . . . . . . . . . . . . . . . . . . . 119
4.16 Pressure-porosity curve from drop weight experiment on uncoated
Orica prill bed. Initial strain rate 170 s−1. . . . . . . . . . . . . . 120
4.17 Pressure-porosity curves from drop weight experiments on uncoated
Orica prill beds. Weight dropped from 17 cm. . . . . . . . . . . . 121
4.18 Pressure-porosity curves from drop weight experiments on uncoated
Orica prill beds. Weight dropped from 32 cm. . . . . . . . . . . . 121
4.19 Pressure-porosity curves from drop weight experiments on coated
Orica prill beds. Weight dropped from 17 cm. . . . . . . . . . . . 122
4.20 Pressure-porosity curves from drop weight experiments on coated
Orica prill beds. Weight dropped from 32 cm. . . . . . . . . . . . 122
4.21 Pressure-porosity curves from drop weight experiments on agricul-
tural pellet beds. Weight dropped from 17 cm. . . . . . . . . . . . 123
4.22 Pressure-porosity curves from drop weight experiments on agricul-
tural pellet beds. Weight dropped from 32 cm. . . . . . . . . . . . 123
4.23 Pressure-porosity curves from drop weight experiments on Nitram
prill beds. Weight dropped from 17 cm. . . . . . . . . . . . . . . . 124
4.24 Pressure-porosity curves from drop weight experiments on Nitram
prill beds. Weight dropped from 32 cm. . . . . . . . . . . . . . . . 124
xvi
LIST OF FIGURES
4.25 Pressure-porosity curves from drop weight experiments on West-
land prill beds. Weight dropped from 17 cm. . . . . . . . . . . . . 125
4.26 Pressure-porosity curves from drop weight experiments on West-
land prill beds. Weight dropped from 32 cm. . . . . . . . . . . . . 125
4.27 Pressure-porosity curves from Instron experiments on coated and
uncoated Orica prill beds. Initial strain rate 4× 10−4 s−1. . . . . . 126
4.28 Pressure-porosity curves from drop weight experiments on coated
and uncoated Orica prill beds. Initial strain rates 100 s−1 and
120 s−1 for coated and uncoated bed respectively. . . . . . . . . . 126
4.29 Pressure-porosity curves from both Instron and drop weight exper-
iments on uncoated Orica prill beds . . . . . . . . . . . . . . . . . 127
4.30 Pressure-porosity curves from drop weight experiments on uncoated
Orica prill beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.31 Pressure-porosity curves from drop weight experiments on Nitram
prill beds. All curves have been shifted by their initial fractional
TMD, greatly improving experiment repeatability. . . . . . . . . . 131
4.32 Pressure-porosity curves obtained by Instron experiments on beds
of Nitram and uncoated Orica prills, showing greater resistance to
compaction in Nitram prill bed. Initial strain rate 4×10−4 in both
cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.33 Pressure-porosity curves from drop weight experiments on West-
land prill beds. All curves have been shifted by their initial poros-
ity. This has not eliminated variation between experiments. . . . . 134
4.34 Heckel plot for quasistatic compaction of uncoated Orica prills . . 135
4.35 Pressure-porosity graph for quasistatic compaction of uncoated Or-
ica prills, showing limited agreement of Heckel model with experi-
ment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.36 Heckel plot for drop weight compaction of uncoated Orica prills . 137
4.37 Pressure-porosity graph for drop weight compaction of uncoated
Orica prills, showing discrepancy between Heckel model and ex-
periment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.38 Kawakita plot for quasistatic compaction of uncoated Orica prills 139
xvii
LIST OF FIGURES
4.39 Pressure-porosity graph for quasistatic compaction of uncoated Or-
ica prills, showing good agreement with Kawakita model. . . . . . 140
4.40 Kawakita plot for drop weight compaction of uncoated Orica prills 140
4.41 Pressure-porosity graph for drop weight compaction of uncoated
Orica prills, showing broad agreement between experiment and
Kawakita model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.1 The sample cell used for holding granular beds in shock-induced
reaction experiments. . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.2 Design of make trigger used in these experiments. . . . . . . . . . 147
5.3 Design of pressure gauge used in these experiments. . . . . . . . . 148
5.4 Circuit diagram of charge integrator for PVDF gauges. . . . . . . 148
5.5 Flyer design used in these experiments. . . . . . . . . . . . . . . . 149
5.6 Projectile velocity calibration graph for the Cavendish small gun. 151
5.7 Fibre mount design when framing photography is required. . . . . 152
5.8 Fibre mount design when framing photography is not required. . . 153
5.9 Framing photography of 150–212µm AN granular bed impacted
at 700± 40m s−1. 460 ns exposure time, 40 ns interframe time. . . 154
5.10 Photodiode output of experiment shown in figure 5.9. . . . . . . . 158
5.11 Figure 5.10, re-plotted with logarithmic y axis. . . . . . . . . . . . 159
5.12 Induction times for granular ammonium nitrate beds, impacted at
a range of velocities. . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.13 X-T diagram of the collision between the flyer plate and the front
shim of the experimental cell . . . . . . . . . . . . . . . . . . . . . 161
5.14 Induction times for granular ammonium nitrate beds, impacted
at a range of velocities. The calculated loading pulse duration for
each impact velocity has been subtracted from these induction times.164
5.15 Arrangement of modified AN cell used to measure the travel time
of pressure waves through the bed. . . . . . . . . . . . . . . . . . 165
5.16 Travel time of pressure wave through 3.5mm thick 150–212µm AN
granular bed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
xviii
LIST OF FIGURES
5.17 Induction times for granular ammonium nitrate beds, impacted
at a range of velocities. The calculated travel time of the input
pressure wave through the bed has been subtracted from these
induction times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.18 Comparison of light-emission model with average normalized pho-
todiode output, showing failure of the model. . . . . . . . . . . . . 170
5.19 Framing photography of 20–40µm AN granular bed impacted at
683± 3m s−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.20 Photodiode output of experiment shown in figure 5.19. . . . . . . 172
5.21 Static image of the bed impacted in figure 5.22. . . . . . . . . . . 174
5.22 Framing photography of a bed of 3.4–3.6mm Westland prills im-
pacted at 700± 40m s−1. . . . . . . . . . . . . . . . . . . . . . . . 175
5.23 Photodiode output of experiment shown in figures 5.21 and 5.22. . 175
5.24 Static image of the bed impacted in figure 5.25. . . . . . . . . . . 176
5.25 Framing photography of a bed of 1.9–2.1mm Orica prills impacted
at 700± 40m s−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.26 Illustration of a prill bed through some of which a shock has travelled.177
5.27 Photodiode output of experiment shown in figures 5.24 and 5.25. . 178
5.28 Photodiode output from bed of 1.9–2.1mm Orica prills, impacted
at 700± 40m s−1. Fibre end roughened with 30µm polishing paper.179
5.29 Static image of the bed impacted in figure 5.30. . . . . . . . . . . 181
5.30 Framing photography of a bed of 3.4–3.5mm Westland prills, with
a single 3.5mm Orica prill, impacted at 700± 40m s−1. . . . . . . 181
5.31 Photodiode output of experiment shown in figures 5.29 and 5.30. . 182
5.32 Sabot design for direct impact experiments. . . . . . . . . . . . . 183
5.33 Sample geometry for jet-based initiation. . . . . . . . . . . . . . . 185
5.34 Framing photography through sample in drop weight experiment
on 2mm ANFO prills. . . . . . . . . . . . . . . . . . . . . . . . . 186
5.35 Arrangement of prills for heat-sensitive film experiment. . . . . . 187
5.36 Layout of heat-sensitive film experiment. . . . . . . . . . . . . . . 187
5.37 Recovered heat-sensitive film after drop weight experiment on hexag-
onal arrangement of prills. . . . . . . . . . . . . . . . . . . . . . . 188
5.38 Static image of bed impacted in figure 5.39. . . . . . . . . . . . . 188
xix
LIST OF FIGURES
5.39 Framing photography of a bed of 3.4–3.6mm Nitram prills im-
pacted at 700± 40m s−1. . . . . . . . . . . . . . . . . . . . . . . . 189
5.40 Photodiode output of experiment shown in figures 5.38 and 5.39. . 190
5.41 Static image of bed impacted in figure 5.42. . . . . . . . . . . . . 191
5.42 Framing photography of a bed of 3.4–3.6mm Westland prills, with
a single 3.5mm Nitram prill, impacted at 700± 40m s−1. . . . . . 192
5.43 Photodiode output of experiment shown in figures 5.41 and 5.42. . 193
5.44 Static image of the bed impacted in figure 5.45. . . . . . . . . . . 194
5.45 Framing photography of a bed of 3.4–3.6mm agricultural pellets
impacted at 700± 40m s−1. . . . . . . . . . . . . . . . . . . . . . 195
5.46 Photodiode output of experiment shown in figure 5.44 and 5.45. . 196
5.47 Photodiode output of a bed of 3.4–3.6mm agricultural pellets im-
pacted at 700± 40m s−1. . . . . . . . . . . . . . . . . . . . . . . . 197
5.48 Static image of the bed impacted in figure 5.49. . . . . . . . . . . 197
5.49 Framing photography of a bed of 3.4–3.6mm Westland prills, with
a single 3.5mm agricultural pellet, impacted at 700± 40m s−1. . . 198
5.50 Photodiode output of experiment shown in figure 5.48 and 5.49. . 199
6.1 Example dark current spectrum. . . . . . . . . . . . . . . . . . . . 205
6.2 Illustration of repeatability of apparently-random noise in dark
current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
6.3 Recorded spectrum of fluorescent room lights. . . . . . . . . . . . 207
6.4 Arrangement used for relative intensity calibration of spectroscope. 208
6.5 Comparison of observed and calculated spectrum of a 2856K black
body source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
6.6 Sensitivity curve for apparatus. . . . . . . . . . . . . . . . . . . . 209
6.7 Attenuation coefficient α for RS 368-047 PMMA-cored fibre. . . . 212
6.8 Spectra illustrating effect of fibre variations on measured temper-
ature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
6.9 Spectrum of light emitted by AN granular bed approx. 500 ns be-
fore photodiode peak. . . . . . . . . . . . . . . . . . . . . . . . . . 215
6.10 Spectrum of light emitted by AN granular bed approx. 2µs after
photodiode peak. . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
xx
LIST OF FIGURES
6.11 Spectrum of light emitted by AN granular bed approx. 2µs before
photodiode peak. . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
6.12 Spectrum of light emitted by AN granular bed approx. 1µs after
photodiode peak. . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
6.13 Temperature of emitting regions in AN beds impacted at 700 ±
40m s−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
6.14 Probability distribution of fitted parameters for spectrum shown
in figure 6.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
6.15 Projection of figure 6.14 onto T axis . . . . . . . . . . . . . . . . 223
A.1 Centres of molecular groups in ammonium nitrate (I) unit cell,
looking down any of the three axes. . . . . . . . . . . . . . . . . . 232
A.2 Structure of ammonium nitrate (II) unit cell, looking down c axis,
with ions in first orientation. . . . . . . . . . . . . . . . . . . . . . 234
A.3 Structure of ammonium nitrate (II) unit cell, looking down a axis,
with ions in first orientation. . . . . . . . . . . . . . . . . . . . . . 235
A.4 Structure of ammonium nitrate (II) unit cell, looking down c axis,
with ions in second orientation. . . . . . . . . . . . . . . . . . . . 236
A.5 Structure of ammonium nitrate (II) unit cell, looking down a axis,
with ions in second orientation. . . . . . . . . . . . . . . . . . . . 237
A.6 Structure of ammonium nitrate (III) unit cell, looking down c axis,
with ammonium ions in first orientation. . . . . . . . . . . . . . . 238
A.7 Structure of ammonium nitrate (III) unit cell, looking down b axis,
with ammonium ions in first orientation. . . . . . . . . . . . . . . 239
A.8 Structure of ammonium nitrate (III) unit cell, looking down c axis,
with ammonium ions in second orientation. . . . . . . . . . . . . . 240
A.9 Structure of ammonium nitrate (III) unit cell, looking down b axis,
with ammonium ions in second orientation. . . . . . . . . . . . . . 241
A.10 Structure of ammonium nitrate (IV) unit cell, looking down a axis. 242
A.11 Structure of ammonium nitrate (IV) unit cell, looking down b axis. 243
A.12 Structure of ammonium nitrate (IV) unit cell, looking down c axis. 244
A.13 Structure of ammonium nitrate (V) unit cell, looking down c axis. 245
A.14 Structure of ammonium nitrate (V) unit cell, looking down a axis. 246
xxi
LIST OF FIGURES
B.1 Framing photography of 150–212µm ammonium nitrate granular
bed. 700 ± 40m s−1 impact velocity, 460 ns exposure time, 40 ns
interframe time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
B.2 Framing photography of 150–212µm ammonium nitrate granular
bed. 589 ± 2m s−1 impact velocity, 460 ns exposure time, 40 ns
interframe time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
B.3 Framing photography of 150–212µm ammonium nitrate granular
bed. 516 ± 18m s−1 impact velocity, 460 ns exposure time, 40 ns
interframe time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
B.4 Static image of the bed impacted in figure B.5. . . . . . . . . . . . 252
B.5 Framing photography of a bed of 1.9–2.1mm Orica prills impacted
at 700± 40m s−1. Variable exposure and interframe time. . . . . . 253
B.6 Static image of the bed impacted in figure B.7. . . . . . . . . . . . 253
B.7 Framing photography of a bed of 3.4–3.6mm agricultural pellets
impacted at 700± 40m s−1. . . . . . . . . . . . . . . . . . . . . . 254
B.8 Static image of the bed impacted in figure B.9. . . . . . . . . . . . 254
B.9 Framing photography of a bed of 3.4–3.6mm Nitram prills im-
pacted at 700± 40m s−1. . . . . . . . . . . . . . . . . . . . . . . . 255
B.10 Framing photography through sample in drop weight experiment
on hexagonal arrangement of 2mm ANFO prills with sapphire anvils.256
B.11 Framing photography through sample in drop weight experiment
on hexagonal arrangement of 2mm ANFO prills. . . . . . . . . . . 257
B.12 Framing photography through sample in drop weight experiment
on hexagonal arrangement of 2mm ANFO prills with annular con-
finement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
xxii
List of Symbols
Roman Symbols
A Area
A,B Some constants
A1, A2 Resistances of axial strain gauges in Wheatstone bridge load cell
a Parameters of a model
c The speed of light in a vacuum, approx. 2.998× 108ms−1
c0 Empirically determined parameter of Hugoniot curve
cb Wave speed along a bar
cl Longitudinal sound speed
D Hessian matrix of χ2 at point p, see equation (2.33)
d Value of ∇χ2 at point p, see equation (2.33)
D Detonation velocity
d Diameter
E Energy
e Engineering strain, ∆l
l
F Force
xxiii
LIST OF SYMBOLS
G Gauge factor of a resistive strain gauge
h Planck’s constant, approx. 6.626× 10−34 J s
i Index of a single data point
I(λ) Spectrum
J Impulse
K Bulk modulus, in chapter 4
k Position along linear detector
kB Boltzmann’s constant, approx. 1.381× 10−23 JK−1
k, l Indices of a single parameter
l Length
M Number of parameters in a model; equivalently, length of a
m Mass
N Some integer
P Pressure
Pθ(k) X-ray absorption
P1, P2 Resistances of circumferential strain gauges in Wheatstone bridge load
cell
P (a) Probability of data given parameters a, see equation (2.30)
p Point in parameter space about which χ2 is Taylor-expanded, see equa-
tion (2.33)
p Momentum
Q Heat energy
xxiv
LIST OF SYMBOLS
R Resistance
s Displacement
s Empirically determined parameter of Hugoniot curve, in chapter 5
S(λ) Wavelength-dependent sensitivity of a spectroscope
T Temperature
t Time
u Velocity
V Voltage
v Specific volume, 1
ρ
W˙ Power
xi Value of independent variable for i
th data point
Y Young’s modulus
yi Value of dependent variable for i
th data point
y(xi;a) Value of dependent variable of model for an independent variable xi
and parameters a
Z Atomic number
Greek Symbols
α Attenuation coefficient
α′ Modified Hessian matrix, see equation (2.44)
α Matrix defined for convenience in model fitting; −1
2
D
β Vector defined for convenience in model fitting; −1
2
d
γ Fracture surface energy, in chapter 4
xxv
LIST OF SYMBOLS
γ Value of χ2 at point p, see equation (2.33)
δa Iteration step in model-fitting techniques
ε Emissivity, in chapter 5
ε Strain
ε˙ Strain rate
ζ Parameter of heat conduction model: ζ =
2K
Cpρ
θ Angle
κ Factor by which ξ is multiplied or divided
λ Wavelength
ν Poisson’s ratio, in chapter 2
ν Vibrational energy quantum number, in chapter 6
ξ Switching factor in Levenberg-Marquardt algorithm
σ Stress
σB Stefan-Boltzmann constant, approx. 5.670× 10−8Wm−2K−4
σi Uncertainty in yi
φ Azimuthal angle, in chapter 5
χ2 Weighted sum of squared deviations between a model and data, see
equation (2.29)
ω Angular frequency
Acronyms
AN Ammonium nitrate
ANFO Ammonium nitrate/fuel oil explosive
xxvi
LIST OF SYMBOLS
boPET Biaxially orientated poly(ethylene terephthalate), a transparent plastic
film. Also known as Mylar.
CJ Chapman-Jouguet, a theory of detonation
EDX Energy-dispersive X-ray spectroscopy
ESEM Environmental scanning electron microscopy
MCP Microchannel plate
NPL National Physical Laboratory
P20 A particular scintillator material, used as the phosphor screen in some
image tubes.
PIN p-type, intrinsic, n-type semiconductor junction
PMMA Poly(methyl methacrylate), a transparent plastic. Also known as Per-
spex or Plexiglas.
PN p-type, n-type semiconductor junction
POM Poly(oxymethylene), a plastic. Also known as acetal or Delrin.
PVDF Poly(vinylidene difluoride), a piezoelectric polymer
S25 A particular material of low work function, used as the photocathode
in some image tubes.
SEM Scanning electron microscopy
SHPB Split Hopkinson pressure bar
TATB Tri-amino-tri-nitro-benzene, an explosive
TMD Theoretical maximum density
TTL Transistor-Transistor Logic; a TTL pulse is a +5V electrical signal.
ZND Zeldovich-von Neumann-Do¨ring, a theory of detonation
xxvii
xxviii
Chapter 1
Introduction and literature
review
Ammonium nitrate is a widely used agricultural fertilizer. As a component of
ammonium nitrate/fuel oil (ANFO) and emulsion explosives, it is a very popular
commercial explosive. Mechanisms by which it can be initiated are therefore of
interest both for safety of handling and for development of explosive formulations.
Hot spots are widely held to be the means by which a mechanical shock or
impact can cause detonation of a heterogeneous explosive. As such, they are
an interesting and useful field of study. Many potential hot spot mechanisms
have been proposed, but determining which are relevant in a particular situation
remains challenging.
1.1 Ammonium nitrate
Ammonium nitrate is a colourless, highly hygroscopic crystal. When crystallized
from aqueous solution, it tends to form needles. By careful choice of solvents
and the use of chelating agents, this can be modified (Doxsee and Francis, 2000).
When commercially produced for use in fertilizer and ANFO, it is usually in
the form of prills. These are pellets, roughly 5mm long, and generally with
some coating. The prills are usually produced by spraying molten ammonium
nitrate from a “shower head” at the top of a tower. An upwards draught of air
1
1 INTRODUCTION AND LITERATURE REVIEW
keeps these pellets aloft and cools them (Kjaergaard, 2000). This cooling tends
to leave prills with a void in the centre, as the outside solidifies first and the
inside contracts as it cools (Kwok et al., 2003). The coating is added afterwards,
to inhibit the ammonium nitrate’s absorption of moisture. For laboratory use,
ammonium nitrate is available as granules, produced by milling pure AN crystals
or prills.
1.1.1 Phases of ammonium nitrate
Ammonium nitrate has five stable phases at atmospheric pressure. They are
summarized in table 1.1. Details of their structures are given in appendix A. The
existence of a low-temperature phase VII has been suggested (Volfkovich et al.,
1954) but subsequently rejected (Ahtee et al., 1983b; Choi and Prask, 1983).
With the exception of those involving phase III, the phase transitions are
unremarkable. They all involve a loss of order as temperature rises, accompanied
by slight modifications of the unit cell (Brown and McLaren, 1962). The IV
 III
transition, however, merits special consideration. It involves a density change
(from 1.72 g cm−3 for phase IV to 1.66 g cm−3 for phase III). As a result, it tends
to break crystals and loosen polycrystalline mixtures. The resulting fragments
cause the material to cake, making handling more difficult (Oommen and Jain,
1999).
The IV
 III transition requires water, and appears to proceed by dissolution
of phase IV and recrystallization of phase III in a thin layer of water (Brown and
McLaren, 1962; Davey et al., 1991). As the phase transition temperature is
approached from below, phase IV becomes more soluble in water than phase III.
The phase change front moves at 100µms−1 at its fastest (Davey et al., 1991).
This is substantially slower than the other phase transitions, so phase III may be
bypassed altogether on sufficiently rapid heating or cooling, or in the absence of
water (Ingman et al., 1982). There has also been extensive work on suppressing
this phase transition. A variety of chemical additives have been found to stabilize
one or another phase of ammonium nitrate, preventing the awkward IV
 III
transition in a wide range around room temperature (Choi et al., 1980; Highsmith
et al., 1994; Mishra, 1985).
2
AMMONIUM NITRATE 1.1
Phase Symmetry, Lattice constants/A˚, Ordered? References
space group Z
I Cubic a = 4.3655(2) No a, b
Pm3m Z = 1
398K
II Tetragonal a = 5.7193(1) No c, d
P4¯21m c = 4.9326(1)
(at 355K) Z = 2
356K
III Orthorhombic a = 7.7184(3) No e, f
Pnma b = 5.8447(1)
(at 318K) c = 7.1624(2)
Z = 4
305K
IV Orthorhombic a = 5.7574(1) Yes g, d
Pmmn b = 5.4394(1)
(at 300K) c = 4.9298(1)
Z = 2
257K
V Orthorhombic a = 7.9804(1) Yes h
Pccn b = 8.0027(1)
(at 233K) c = 9.8099(1)
Z = 8
Table 1.1: The phases of ammonium nitrate. References: (a) Shinnaka (1959),
(b) Ahtee et al. (1979), (c) Shinnaka (1956), (d) Lucas et al. (1979), (e) Goodwin
and Whetstone (1947), (f) Lucas et al. (1980), (g) Choi et al. (1972), (h) Ahtee
et al. (1983a).
3
1 INTRODUCTION AND LITERATURE REVIEW
-50
0
50
100
150
200
250
300
350
0 0.5 1 1.5 2 2.5 3 3.5
T
em
p
er
at
u
re
/
◦ C
Pressure / GPa
Liquid
V
II
I
III
IV
VI
Figure 1.1: Ammonium nitrate pressure-temperature phase diagram, from
Rapoport and Pistorius (1966).
The pressure-temperature phase diagram of ammonium nitrate is shown in
figure 1.1. The structure of the high-pressure phase VI is poorly studied. Another
phase, labelled VIII, is accessible only by shear loading and is not shown on the
phase diagram (Adams and Sharma, 1976).
1.1.2 Thermal decomposition of ammonium nitrate
The mechanism and products of this decomposition change with heating rate and
pressure. When ammonium nitrate is heated slowly (∼ 50 ◦C/hour) to between
180 ◦C and 300 ◦C, it decomposes exothermically. The overall reaction (Friedman
and Bigeleisen, 1950; Saunders, 1922) is
NH4NO3 N2O+ 2H2O . (1.1)
This reaction requires trace amounts of water. Dry ammonium nitrate will not
decompose up to 300 ◦C. A small amount of water vapour allows it to decompose
at 180 ◦C. Using isotope tracing shows this water is not incorporated in the N2O
4
AMMONIUM NITRATE 1.1
Reaction Times/s ∆H/kcal
A 4[NH4NO3(`)→ HNO3(g) + NH3(g)
→ NH4NO3(solid aerosol)]
1–7 4(44)
B 3[5NH4NO3(`)→ 2HNO3 + 4N2 + 9H2O] 1–3 3(−35)
C 5[NH4NO3(`)→ N2O+ 2H2O] 2–7 5(−13)
D 4NH4NO3(`)→
2NH3 + 3NO2 +NO+N2 + 5H2O
2–7 81
Table 1.2: Proposed reaction scheme for decomposition of ammonium nitrate
under rapid heating at atmospheric pressure. ∆H for reaction A only includes
the dissociation step. From Brill et al. (1993).
produced (Friedman and Bigeleisen, 1950). It seems likely that the mechanism is
NH+4 +NO
−
3 H2N−NO2 +H2O , (1.2)
O
OH
H2N−NO2 HN−NO –2 +H+ HN−N
, (1.3)
N2O+H2O
or acid
catalysis
BasicO
OH
HN−N
. (1.4)
Other oxides of nitrogen are also produced, along with nitrogen gas.
At higher rates of heating, the reaction changes. On heating to ∼ 400 ◦C at
∼ 2000 ◦C/s in argon at atmospheric pressure, Fourier transform infrared spec-
troscopy suggests the system of reactions in table 1.2. These reactions are en-
dothermic overall. This is consistent with pure ammonium nitrate’s inability to
burn at atmospheric pressure. At higher pressures, the NH3 and NO2 produced
are confined, and can react exothermically:
2NH3 + 2NO2 N2O+N2 + 3H2O (∆H ≈ −148 kcal) . (1.5)
However, the overall reaction remains slightly endothermic up to at least 33
atmospheres (Brill et al., 1993).
5
1 INTRODUCTION AND LITERATURE REVIEW
1.1.2.1 Kinetics
Vyazovkin et al. (2001) studied the isothermal decomposition of AN using a
thermogravimetric technique. They determined that this decomposition was due
to the reaction
NH4NO3(s) NH3(g) + HNO3(g) , (1.6)
and that this had an activation energy of approximately 90 kJmol−1 for both
solid and liquid AN. They also reviewed existing work on the topic and found
a wide range of activation energies quoted for this process. This may be due to
flawed kinetic models, or to the complexity of the sequence of reactions involved
in AN decomposition.
1.2 Hot spots
Hot spots are regions of a material which become much hotter than the bulk
during shock or impact. They are generally thought to cause the onset of reaction
in shocked or impacted energetic materials. If a hot spot is large enough, and
remains hot for long enough, the heat produced by reaction of the material can
exceed that lost to conduction. Such a hot spot is called ‘critical’, and reaction
can grow from one throughout the material.
The minimum temperature of a critical hot spot will depend on the explosive.
It can be measured by adding grits to an explosive. If the melting point of the grit
exceeds the ignition temperature of the explosive, the grit’s addition sensitizes
the explosive to frictional loading. Otherwise, the grit has little effect. For most
secondary explosives, the minimum temperature for a critical hot spot is in the
region of 700 K (Bowden and Gurton, 1948, 1949).
The minimum size of a critical hot spot can be calculated using a heat-balance
argument. This depends on the thermal conductivity, density, heat capacity, and
heat of reaction of the explosive, and on the temperature of the hot spot. For most
secondary explosives, any hot spot smaller than about 1µm must be implausibly
hot to be critical (Bowden and Yoffe, 1952).
There are several proposed mechanisms by which the mechanical energy of
the shock is concentrated and converted to heat. They include the collapse of
6
HOT SPOTS 1.2
pores in the material, friction between grains or at surfaces produced by fracture,
the formation of shear bands, and the growth of cracks. These mechanisms do
not compete: their effects are additive.
1.2.1 Crack growth
Explosives are relatively weak, and therefore release relatively little energy in
fracture. A theoretical treatment shows that the energy released in the growth of
a single crack is insufficient to initiate a secondary explosive. It may, however, be
able to cause partial reaction (Holmes et al., 1999). Experimentally, even primary
explosives are not initiated by fracture (Chaudhri, 1989). Fracture can, however,
cause partial reaction (Fox and Soria-Ruiz, 1970).
A crack creates free surfaces, which may rub against each other. Together with
the products of partial reaction, this may sensitize the explosive to subsequent
stimuli. Also, while explosives are weak, the plastics used to bind polymer-bonded
explosives are stronger. A crack passing through such a material close to a grain
of explosive could conceivably ignite it.
1.2.2 Cavity collapse
There are several mechanisms by which the collapse of a cavity may generate heat.
In the collapse, the surrounding material must undergo plastic deformation, which
will raise its temperature. Any gas contained in the cavity will be adiabatically
compressed by the shock, reaching a high temperature. The shock may evaporate
explosive from the upstream wall of the cavity; this could cross the void, and
stagnate against the downstream wall, again reaching a high temperature. A
shock can also cause a jet to form from the upstream wall of the cavity, which
would strike the downstream wall with great force.
1.2.2.1 Plastic deformation
The plastic work around a symmetrically collapsing spherical void has been an-
alytically treated. The heat produced could cause a critical hot spot in a strong
shock (Khasainov et al., 1981). However, weaker shocks have been seen to cause
7
1 INTRODUCTION AND LITERATURE REVIEW
cavity collapse, leading to initiation.
1.2.2.2 Adiabatic compression
From study of bubbles in water near single crystals of primary explosives, it seems
adiabatic compression is the main mechanism by which cavity collapse causes
initiation in shocks of a few hundred megapascals (Chaudhri, 1989; Chaudhri and
Field, 1974). A collapsing bubble rebounds after the shock has passed, generating
an expansion shock. By holding the bubble close to the crystal, but not in thermal
contact with it, the expansion shock was eliminated as an initiation mechanism.
This also removed the possibility that radiation produced in the compression was
important. Using a two-dimensional bubble, the jet was visible. The explosive
crystal did not detonate when the jet struck it, but when the bubble had been
compressed significantly (Chaudhri and Field, 1974).
1.2.2.3 Evaporation and stagnation
Using streak photography, the evaporation and stagnation effect can be studied.
Two crystals of PETN were separated by vacuum, and the first detonated. The
detonation products were not visible while traversing the space between the crys-
tals. A bright flash of light was seen at the face of the second crystal, at a time
when the detonation products might reasonably have reached the second crystal
(Blackburn and Seely, 1965). Atomistic simulations show that a similar effect
can be seen when the incident shock is inert (Holian et al., 2002).
1.2.2.4 Jets
Though jets were shown not to be able to initiate primary explosives in weak
shocks (Chaudhri and Field, 1974), they are more important in strong shocks
of around a gigapascal (Bourne and Field, 1991; Chaudhri, 1989). Under these
circumstances, the jet reaches a speed of a few kilometres a second (Bourne and
Milne, 2003; Field, 1994). Fast framing photography shows that there is some
heating of gas trapped by the jet tip as it strikes the downstream wall of the
cavity. There is far more heating as the toroid cavity remaining around the jet
is compressed. When a cavity in ammonium nitrate solution was investigated,
8
HOT SPOTS 1.2
though, reaction broke out at the point of the jet impact, and not around the
torus. It is possible that the increase of the number and energy of inter-molecular
collisions by the jet, or hydrodynamic loading of the downstream wall by the jet,
is more important than the heating (Hatano, 2004).
1.2.3 Friction
The rubbing of grains converts mechanical energy to heat. Most secondary explo-
sives, including ammonium nitrate, melt below their ignition temperature, and
therefore cannot be initiated by friction between the grains. In primary explo-
sives, this is not the case.
If a flat-tipped needle is dropped into a granular bed of explosive, grains
adhere to it and rub against grains in the bed (Chaudhri, 1989). Under these
circumstances, initiation is predictable from the coefficient of friction and the ig-
nition temperature (Chaudhri, 1976). The coefficient of friction can be measured,
even when growing a large crystal is difficult (Amuzu et al., 1976).
1.2.4 Shear bands
When a material undergoes plastic deformation, heat is generated in the deform-
ing region. Many materials become softer as their temperature is raised. In
impact loading, this effect can be stronger than the work-hardening of the de-
forming region. This results in strain localization: the easiest region to deform is
one that has already deformed to some degree. This results in narrow bands of
material experiencing intense strain; these regions are referred to as shear bands.
The localized heating associated with this deformation is capable of initiating
some explosives (Winter and Field, 1975). It has been speculated that the high
density of dislocations may allow multi-phonon excitation in such a band (Coffey,
1991)
1.2.5 Vibrational up pumping
Heating excites the phonon modes of a material, but it is the excitation of the
internal vibrational modes which causes chemistry (Dlott and Fayer, 1990). The
9
1 INTRODUCTION AND LITERATURE REVIEW
rate at which energy can be transferred between these modes is therefore an
important factor in the response of a material to a hot spot.
For rapid transfer between phonons and internal vibrations, the modes cannot
be purely harmonic. Anharmonicity permits multiple phonons to couple to an
internal vibration. This is necessary, as the lowest-energy internal vibration is
usually more energetic than the highest-energy phonon.
One possible mechanism for energy transfer involves so-called ‘doorway modes’.
This model only applies to crystals in which the energy levels have certain proper-
ties. The lowest-frequency internal vibration must have less than twice the energy
of the highest frequency phonon. In addition, the energy difference between any
internal vibration and the next most energetic internal vibration must not exceed
the energy of the most energetic phonon mode. Most large organic molecular
crystals meet these criteria. Under these circumstances, only three-mode inter-
actions are needed to excite all the internal vibrational modes. The doorway
modes are those internal vibrations which can be excited by two phonons. The
rate at which they are excited can be calculated from relaxation times at low
temperatures.
The model is that both phonon and internal vibrational modes rapidly reach
thermal equilibrium among themselves, but energy transfer between these two
sets of modes is restricted by excitation of the doorway modes. Each set of modes
is assigned a quasi-temperature which describes its internal energy distribution,
and the problem of energy transfer to the internal modes is treated as one of
thermal conduction.
This mechanism provides an interesting new hot spot mechanism. Some de-
fects enhance the anharmonicity of vibrational modes in the region around the
defect. In such a region energy from the phonons would flow into the internal
vibrations more rapidly than it would in the bulk. The internal vibrational quasi-
temperature in that region would then overshoot the equilibrium temperature,
possibly allowing a reaction to start (Dlott and Fayer, 1990; Tokmakoff et al.,
1993).
An alternative model for vibrational up pumping involves direct excitation
of vibrational modes by many phonons. In this model, the phonon modes are
treated separately, rather than as a sea of uniform mode density. Resonance
10
HOT SPOTS 1.2
between overtones of the phonon modes and the internal vibrational modes be-
comes important, and is used in the model. An interaction between the N th
overtone of some phonon mode and an internal vibrational mode is equivalent
to an interaction between N phonons belonging to that mode, and the internal
vibrational mode. This model was used to find the energy transfer in the first
ten femtoseconds after impact. This quantity correlated well with the energy to
ignition of various energetic materials (McNesby and Coffey, 1997).
1.2.6 Sensitization centres
Sub-critical hot spots may allow partial reaction to take place. Many organic
secondary explosives begin to react by the molecules breaking into more reactive
fragments. In a critical hot spot, these would then go on to complete reaction. In
a sub-critical hot spot they remain unreacted. These reactive fragments provide
a point at which reaction can start more readily under subsequent stimulus than
in the bulk. Such sensitization centres have been seen in TATB (Sharma et al.,
1987).
1.2.7 Hot spots in ammonium nitrate
Most work on hot spots in ammonium nitrate has used flat granular beds under
projectile impact (8210X, 2003; Proud, 2002; Proud et al., 2003). The design of
the cell is shown in figure 1.2. The projectile was fired from a gas gun at speeds
between 400 and 700m s−1. The response of the bed was studied using framing
photography, a photodiode, and a spectrometer.
1.2.7.1 Framing photography
Figure 1.3 shows photographs of the impact process on 2mm beds at various flyer
speeds. The bright ring which develops is due to the copper front plate being
driven into the glass back plate. The edges of the projectile produce intense shear
in the front plate at this point. This heats it, producing light.
The light output clearly depends strongly on the flyer speed. However, evac-
uating the test chamber appears to have little effect on the light output. The
11
1 INTRODUCTION AND LITERATURE REVIEW
0.2 − 2 mm AN bed
(via optical fibre)
spectrometer
Photodiode and
O-ring
front-silvered mirror
25mm glass plate
Camera
(on nylon sabot)
2mm Cu flier plate
Steel support ring
2mm Cu front plate
Figure 1.2: Layout of cell for impact on granular ammonium nitrate bed used in
Proud (2002), 8210X (2003), and Proud et al. (2003).
Figure 1.3: Framing photography on 2mm AN beds from Proud et al. (2003).
Each column contains frames from a single experiment, and is labelled with the
impact velocity for that experiment in m s−1. The suffix “Vac” indicates that the
experiment took place under a rough vacuum of 1mbar. The row labels give the
times after impact over which the frames were exposed.
12
HOT SPOTS 1.2
Figure 1.4: Framing photography on a 0.4mm bed, from Proud et al. (2003).
Grain size 150–210µm. Exposure time of each frame 30 ns. Flyer velocity
700m s−1. Frames are in order left to right, top to bottom. Reaction region
viewed approx. 8mm diameter.
sequence labelled “600” was taken at atmospheric pressure, while that labelled
“600 Vac” was taken at 1mbar. The only difference is due to the magnification.
This suggests that gas trapped in the pores before the experiment is unimportant
to the process.
The 2mm bed was too thick to resolve individual hot spots. Any light relating
to a hot spot was scattered or absorbed. To avoid this, an 0.4mm bed was used
for framing photography. Figure 1.4 shows the sequence thus obtained. The
growth of critical hot spots and the quenching of sub-critical hot spots are visible
between the first two frames.
1.2.7.2 Light output
The photodiode gave a more quantitative measure of total light output than the
camera. The total light output in the experiment is a useful indicator of the
density of critical hot spots. Ammonium nitrate reacts quite slowly, so only a
certain volume of material surrounding each hot spot will have time to react
13
1 INTRODUCTION AND LITERATURE REVIEW
Figure 1.5: Light outputs of 2mm ammonium nitrate beds, under impact from
flyers of a range of velocities, from Proud et al. (2003). The legend gives the flyer
velocity that gave rise to each trace in m s−1. The suffix “Vac” indicates that the
experiment took place under a rough vacuum of 1mbar.
over the course of the experiment. Figure 1.5 shows the photodiode outputs
corresponding to the high-speed sequences in figure 1.3.
Figure 1.6 shows the normalized outputs. The output has been scaled so that
all the traces have the same maximum value. The time has been scaled to account
for the different shock travel times at different flyer velocities. The similarity of
the traces suggests that the kinetics of the situation are unchanged by the flyer
velocity.
Figure 1.7 shows the effect of the bed’s porosity on the light output. The
ammonium nitrate was pressed before putting it in the cell, to control its porosity.
Reducing the porosity will favour fracture as a hot spot mechanism. The lower
pore fraction naturally reduces the importance of cavity collapse. The decreased
scope for grain movement will also reduce the importance of friction. The marked
decrease in light output as the packing density increases suggests that fracture is
of little importance as a hot spot mechanism.
Figure 1.8 shows the effect of adding chalk to the bed. Chalk is commonly
added to ammonium nitrate to reduce caking and improve safety. It seems from
the traces that it acts as a heat sink, absorbing energy from critical hot spots
and thereby rendering them sub-critical. However, as the hot spot density rises,
14
HOT SPOTS 1.2
Figure 1.6: Normalized light outputs of 2mm ammonium nitrate beds, under im-
pact from flyers of a range of velocities, from Proud et al. (2003). The legend gives
the flyer velocity that gave rise to each trace in m s−1. The suffix “Vac” indicates
that the experiment took place under a rough vacuum of 1mbar. Corresponding
non-normalized light output shown in figure 1.5.
Figure 1.7: Light output of 2mm ammonium nitrate beds of varying density,
under impact from a flyer at 700m s−1, from Proud et al. (2003).
15
1 INTRODUCTION AND LITERATURE REVIEW
Figure 1.8: Light output of 2mm ammonium nitrate beds under flyer impact of
variable velocity, from Proud et al. (2003). Legend indicates the flyer velocity that
gave rise to each trace, in m s−1. The suffix “Vac” indicates that the experiment
was performed under a rough vacuum of 1mbar. The suffix “c” indicates that
the ammonium nitrate bed also contained 10% chalk by weight.
its finite heat capacity makes it less effective at preventing reaction.
1.2.7.3 Spectroscopy
Figures 1.9, 1.10 and 1.11 show the spectra recorded during an impact. Their
intensities have been normalized: the intensities of their peaks were originally
in the ratio 1:55:1000. Figure 1.9 shows peaks consistent with NO. This could
be some early partial reaction. The gas thus produced could fill the cavities
in the material, allowing adiabatic compression. In figure 1.10, the spectrum is
broadening, consistent with the sample warming up. In figure 1.11, the spectrum
has become that of a black body at ∼ 6000K.
1.3 Detonation
A detonation is a reactive shock. The shock causes chemical reaction in the mate-
rial through which it propagates. This reaction, in turn, sustains the shock wave.
Only one-dimensional, steady state detonation is analytically tractable. Deto-
nations normally deviate from this. The front often has some three-dimensional
16
DETONATION 1.3
Figure 1.9: Spectrum of 2mm ammonium nitrate bed under flyer impact, from
Proud et al. (2003). Impact velocity not reported. Spectrum was gathered be-
tween 60 ns and 560 ns after impact.
Figure 1.10: Spectrum of 2mm ammonium nitrate bed under flyer impact, from
Proud et al. (2003). Impact velocity not reported. Spectrum was gathered be-
tween 1µs and 1.5µs.
17
1 INTRODUCTION AND LITERATURE REVIEW
Figure 1.11: Spectrum of 2mm ammonium nitrate bed under flyer impact, from
Proud et al. (2003). Impact velocity not reported. Spectrum was gathered be-
tween 2µs and 2.5µs.
structure. The test charge is also of finite diameter. This introduces rarefaction
waves from the walls of the charge, which modify the process somewhat. Nonethe-
less, the models are still useful for a qualitative understanding of detonation.
1.3.1 Chapman-Jouguet detonation theory
Chapman-Jouguet (CJ) theory (Ficket and Davis, 1979) was the first analyt-
ical treatment of detonation, by two independent workers. It assumes a one-
dimensional steady detonation, an instantaneous reaction taking place at the
shock front, and that the material the shock front reaches is entirely unreacted.
It also assumes mass, momentum and energy are conserved across the shock.
Under these assumptions, the behaviour is entirely determined by the material’s
equation of state and the rear boundary conditions. The latter is treated as a
piston following the shock with constant velocity up.
18
DETONATION 1.3
1.3.1.1 Conservation relations
Conservation of mass and momentum give the Rayleigh line (Ficket and Davis,
1979). Any single shock is constrained by the relation
D2
v20
=
P − P0
v0 − v , (1.7)
where D is the speed of the detonation front, P is the pressure after the shock,
P0 is the pressure before the shock, v is the specific volume or reciprocal density
after the shock, and v0 is the specific volume before the shock.
The Rayleigh line is a straight line in P–v co-ordinates. Its gradient is
−
( v0
D2
)
, which must always be negative.
Conservation of energy gives the Hugoniot curve. To conserve energy the
shock must satisfy
E0 − E = 12(P0 + P )(v − v0) , (1.8)
where E is the internal energy in the shocked, reacted material and E0 is the
internal energy in the unshocked material.
In an inert shock, the Rayleigh line is a chord to the Hugoniot. It joins the
unshocked conditions to the shocked conditions, and its slope determines the
shock speed. In a detonation, the reaction moves the Hugoniot for the reacted
material. The Rayleigh line goes from P0 and v0 on the unreacted Hugoniot, and
touches the reacted Hugoniot at a tangent. The point of contact between the
Rayleigh line and the reacted Hugoniot is called the Chapman-Jouguet point.
The slope of the Rayleigh line is the Chapman-Jouguet detonation velocity, DCJ .
In the frame of reference of a detonation travelling at DCJ , the shocked material
retreats from it at the speed of sound in the material.
1.3.1.2 The rear boundary conditions
If up < DCJ , the detonation will travel at DCJ . It will be followed by a rarefaction
fan which takes the particle velocity down to up. If up > DCJ , the detonation
is over-driven. All the material behind the detonation will travel at up, and the
detonation velocity will be increased.
Over-driven detonation is not sustainable without the rear piston. In the
19
1 INTRODUCTION AND LITERATURE REVIEW
D′
v
P
(P0, v0)
C
S
W
DCJ
Figure 1.12: Intersection of Rayleigh lines with the reacted Hugoniot. C is the
Chapman-Jouguet point. S and W are the strong and weak points, respectively.
frame of an over-driven detonation, the shocked material is receding at less than
the sound speed. This means rarefaction waves can catch up with the detonation,
and reduce its velocity to DCJ .
As figure 1.12 shows, the Rayleigh line crosses the reacted Hugoniot twice
when the detonation is over-driven. The high- and low-pressure points are called
the strong and weak points, respectively. An over-driven detonation will assume
the state corresponding to the strong point. At the weak point, the shocked
material retreats from the detonation faster than the sound speed.
1.3.2 Zeldovich-von Neumann-Do¨ring detonation theory
This is an extension of CJ theory (Ficket and Davis, 1979). It no longer assumes
an instantaneous reaction. It does, however, assume that the reaction is a single,
irreversible reaction. In Zeldovich-von Neumann-Do¨ring (ZND) theory, the reac-
tion takes place over a reaction zone behind the detonation front. The reaction
zone moves with the detonation. The state in which the reaction is completed
is equivalent to the CJ point. This means that ZND theory and CJ theory both
20
DETONATION 1.3
predict the same detonation velocities.
The structure of the front is changed from that expected under CJ theory.
The products of the explosive reaction are usually more compressible than the
initial explosive. As the detonation is now compressing unreacted material, the
front now reaches a higher pressure than in CJ theory. This pressure drops over
the reaction zone, as more compressible products are formed. The high-pressure
region is referred to as a ‘von Neumann spike’.
Under ZND theory, the weak point becomes unattainable.
1.3.3 Non-ideal detonation
In the frame of the unshocked material, the shocked material is accelerated by
the detonation and moves in the same direction as it, but more slowly. As the
material gets further from the shock, it decelerates. In the detonation’s frame of
reference, the shocked material is moving backwards from it, and gets faster as
it becomes more distant. In the reaction zone, this motion is sub-sonic. Energy
released in the reaction zone can therefore reach the detonation and reinforce it.
At the CJ point, the material moves at the speed of sound; beyond that point, the
material’s retreat is supersonic. Material beyond the CJ point cannot, therefore,
contribute energy to the detonation.
In an ideal detonation, all the energy of the material is liberated before the
CJ point. A non-ideal detonation is one in which that is not the case (King and
Bauer, 1980; Luebcke, 1995). This could be caused by the slow reaction of the
explosive, by large explosive grains, or by rarefaction waves from the walls of the
charge.
Some energetic materials (including AN) give off relatively little energy on
reaction. This reduces the rate of reaction. Because reaction spreads only slowly
through such a material, the density of critical hot spots is very important to its
detonation characteristics.
Heat conduction from hot reaction products into grains of explosive is not
sufficiently fast that the entire grain will react simultaneously. Instead, the grain
burns in layers (Eyring et al., 1949). An explosive composed of larger grains will
present less surface area per unit volume for burning. It will, therefore, react
21
1 INTRODUCTION AND LITERATURE REVIEW
more slowly. This may cause the detonation to become non-ideal.
The rarefaction waves from the walls of a finite charge will, some time after
the detonation, reach the reaction zone. The reduction in pressure when they
arrive will reduce the reaction rate there. This effect will be more important
in smaller charges, as the rarefaction waves have less distance to travel. This
can change an ideal detonation to a non-ideal one. It can also prevent detonation
altogether. This leads to the concept of a critical diameter for an explosive, below
which detonation is impossible.
1.3.4 Detonation of ammonium nitrate
There has been little work on detonation of pure ammonium nitrate. King and
Bauer (1980) studied the detonation of prilled and molten ammonium nitrate.
They found the critical diameter for prilled ammonium nitrate was about one
metre. The detonation characteristics depend on the prill type. The porosity of
prills varies between manufacturers (Kwok et al., 2003). This will affect the hot
spot density. 11–68% of the prills failed to react in the reaction zone.
Molten ammonium nitrate forms bubbles from thermal decomposition. Above
220 ◦C these bubbles sensitize it enough that detonation is possible. The bubbles
also reduce the bulk energy density of the material, reducing the detonation
velocity.
Hollow glass spheres were used to introduce artificial bubbles to molten ammo-
nium nitrate at 190 ◦C. Enough spheres were added to reduce the density to that
of ammonium nitrate at 260 ◦C. This ammonium nitrate was found to have very
similar detonation characteristics to ammonium nitrate at 260 ◦C. This suggests
that gas cavities were the primary hot spot mechanism in molten ammonium
nitrate.
ANFO explosives based on agricultural ammonium nitrate fertilizer are pop-
ular terrorist weapons. Considerable effort has therefore been expended on mod-
ifying ammonium nitrate fertilizer to render it less effective as a component of
ANFO (EC2003; Porter, 1965). The most common approach involves the addi-
tion of an inert material, which absorbs energy from hot spots without reacting
exothermically. This should reduce the density of critical hot spots, and thereby
22
COMPACTION 1.4
render the explosive less sensitive. However, large-scale trials have found that the
additives have negligible effect on the performance of ANFO made with adulter-
ated ammonium nitrate (Reza and McCarthy, 1999). This suggests that these
additives merely increase the critical diameter, so that the small-scale trials used
in testing them fail to detonate.
1.4 Compaction
Compaction is the process by which a porous material loses porosity in response
to load. Since several hot spot mechanisms involve the collapse of pores in a
material, compaction may be relevant to the study of initiation.
Compaction is used as a manufacturing process for pharmaceuticals, and for
metal and ceramic parts. It is also relevant for understanding of soils and some
geological processes. It has therefore been widely studied and repeatedly re-
viewed (Cunningham et al., 2004; Denny, 2002; Kremer and Hancock, 2006; Sinka,
2007).
1.4.1 Experimental studies
The usual way in which compaction is studied is using a triaxial system, as de-
scribed in Sinka and Cocks (2007) and Pavier and Doremus (1999). This consists
of a uniaxial press whose working surfaces are mounted inside a hydrostatic cell.
The powder to be studied is placed into an elastic sleeve of negligible strength,
and the packaged powder is then placed between the working surfaces of the press.
The length and diameter of this package are then monitored as it is subjected to
superimposed hydrostatic and axial loads.
While this technique allows a wide range of stress states in the powder to
be explored, it is restricted to slow compaction of the powder. The uniaxial
press cannot exceed 1mms−1, while industrial compaction processes may reach
10m s−1. Many experiments therefore use a simple confined piston arrangement
(Adams et al., 1994; Doremus et al., 2010). Greenaway (2005) used a modified
split Hopkinson pressure bar apparatus to achieve even higher compaction rates.
Borg et al. (2005) used plate impact to probe compaction response under shock
23
1 INTRODUCTION AND LITERATURE REVIEW
loading.
1.4.2 Compaction models
A wide range of models are used in an attempt to understand compaction pro-
cesses. Given the wide range of materials and phenomena covered by the term
“powder compaction”, this is unsurprising. For compaction at low porosities, soil
models have proven popular (Drucker and Prager, 1952; Schofield and Wroth,
1968). Fleck (1995) modelled the compacting bed as a random array of identical
rigid-plastic spheres with some success. However, this model was unable to use
a single state variable for the compacting bed: it required the full stress history.
Cocks and Sinka (2007) observed that the stress history of the bed was of a simple
form, and that plastic work (rather than density, as is usual) could be used as a
single state variable for the model.
Two models, though, have proven especially popular. These are the Heckel
and Kawakita models, and are covered in more detail.
1.4.2.1 Heckel model
The Heckel model (Heckel, 1961a,b) is based on the differential equation
dφ
dP
= −αφ , (1.9)
where φ is the void fraction in the material, P is the applied pressure, and α is a
constant. Integrating gives
ln
1
φ
= ln
1
φ0
+ αP , (1.10)
where φ0 is the void fraction of the bed at zero applied pressure. A plot of ln
1
φ
against P is therefore expected to give a straight line of slope α and intercept
ln 1
φ0
. However, in most materials this plot shows some curvature at low pressure
(Denny, 2002). This is usually interpreted as the material deforming by some
means other than plastic deformation. The straight line region of such a graph
does not, therefore, agree with the initially observed φ0.
24
REFERENCES
Heckel (1961b) observed that the parameter α could be related to the yield
stress σ0 of the bed material by
α =
1
3σ0
. (1.11)
However, Sonnergaard (1999) finds this interpretation to be, at best, highly prone
to error.
1.4.2.2 Kawakita equation
Kawakita and Lu¨dde (1970) present the powder-compression relationship
e =
V0 − V
V0
=
ABP
1 +BP
, (1.12)
where V0 is the initial volume of the powder, V is its volume at applied pressure
P , and A and B are constants. For a uniaxial confined experiment, e is the
engineering strain ∆l
l
of the sample. Equation 1.12 may be rearranged to give
P
e
=
1
AB
+
P
A
. (1.13)
P
e
may then be plotted against P to give a straight line of slope A−1 and inter-
cept (AB)−1, in a fashion very similar to that of the Heckel equation. In confined
uniaxial compaction work on powder agglomerates, B−1 has been found to be
proportional to the failure stress of the agglomerated particles (Adams and McK-
eown, 1996; Adams et al., 1994). The constant of proportionality was found to
be very close to unity when friction within the testing apparatus was taken into
account.
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34
Chapter 2
Experimental techniques
This chapter provides a description of the established experimental techniques
used in this research. It also details the capabilities of the equipment used.
All this information is available from other sources. However, collecting this
information was not trivial, and therefore it is provided here for convenience.
2.1 Environmental scanning electron microscopy
Environmental scanning electron microscopy (ESEM) is a technique for examin-
ing the surface of a material at a resolution of up to a few nanometres, at up to
atmospheric pressure. Its chief advantages over conventional scanning electron
microscopy are that it can image uncoated, insulating specimens, and that it does
not require the specimen to be held in a vacuum (Danilatos, 1988). This research
used an FEG XL30 microscope, manufactured by FEI.
In ESEM, an electron beam is generated using an electron gun, typically of
either tungsten filament or field-emission design with an accelerating voltage in
the tens of kilovolts. This beam is then focused using magnetic lenses to generate
a small spot on the sample. The size of this spot will limit the resolution of the
microscope. This spot is deflected by electric fields to scan it across the sample
in a controlled fashion.
The high electric fields involved in these processes require that these stages be
performed in a high vacuum. To achieve this, while allowing the sample chamber
35
2 EXPERIMENTAL TECHNIQUES
Electron gun
PUMP
PUMP
PUMP ELECTRON
OPTICS
PLA 1
PLA 2
10 mbar
10−1 mbar
10−3 mbar
Sample
Electron beam
Figure 2.1: Diagram of an ESEM. “PLA” indicates a pressure-limiting aperture.
Pressures given are typical pressures in each section of the ESEM. Not to scale.
to be held at a much higher pressure, the beam passes through a series of pressure-
limiting apertures. These are essentially small holes, which restrict the rate of gas
flow through them but provide no obstacle to the electron beam. Each pressure-
reducing stage is connected to a separate vacuum pump, and consequently the
pressure is reduced dramatically over a few such stages. See figure 2.1 for a simple
arrangement involving two pressure-limiting apertures.
When an electron from the beam interacts with an atom in the sample, it
excites an electron, causing it to be ejected from the atom. These secondary
electrons form the basis for imaging in ESEM. Secondary electrons which are
captured by other atoms within the sample will not be detected, and the proba-
bility that they will be captured before escaping the sample depends strongly on
the distance they must travel. As the electron beam penetrates some finite dis-
tance into the sample before provoking secondary electron emission, this distance,
and hence the intensity of the secondary-electron signal recorded, will depend on
36
ENERGY-DISPERSIVE X-RAY SPECTROSCOPY 2.2
the angle between the sample surface and the beam. Surfaces normal to the beam
will produce the weakest signal, while those at a sharp angle to it will produce a
stronger signal.
Typically the secondary electron signal from a given point on the sample is
displayed as a single greyscale pixel, with higher signals assigned paler shades of
grey. An image is built up by scanning the electron beam across the sample, and
the eye usefully interprets this as an image of the sample surface.
2.1.1 Secondary electron detection
Secondary electrons are detected using avalanche multiplication (Danilatos, 1990).
An electric field is set up in the sample chamber, with the sample at the cath-
ode. Secondary electrons escaping from the sample are accelerated by the field
until they have sufficient energy that, on interaction with an atom in the gas,
they ionize it. This releases another electron which undergoes the same process,
amplifying the current due to the electrons emitted by the sample. This current
is recorded, and forms the secondary electron signal. Effectively the secondary
electrons from the sample cause a Townsend discharge between the sample and
the detector.
The ionized gas thus formed also neutralizes any build-up of charge on the
sample, eliminating the need to give a conductive coating to insulating samples
as in vacuum SEM.
2.1.2 Beam damage
The beam of high-energy electrons used in ESEM can induce chemical reactions
or decomposition of the material under the beam. This problem is particularly
pronounced when studying energetic materials, as they are of necessity chemically
unstable.
2.2 Energy-dispersive X-ray spectroscopy
Energy-dispersive X-ray spectroscopy (EDX) is a technique for determining the
distribution of elements on the surface of a sample. It has a spatial resolution
37
2 EXPERIMENTAL TECHNIQUES
of approximately 1µm, and used correctly can determine the constituents of an
unknown organic sample with an absolute error of 5–10%. The accuracy improves
to 1% for samples containing heavier elements. The FEG XL30 microscope used
for ESEM has an EDX system. This system was used in this research.
In EDX, an electron beam is scanned over the sample. The electrons in this
beam provoke secondary electron emission from the atoms in the sample. This
emission leaves the resulting ion in an excited state, with a vacancy in a low-
energy electron orbital. This can be resolved by an electron in a higher-energy
orbital undergoing a transition into this vacancy, emitting an X-ray photon in
the process. The photon can then be detected and its energy recorded. The
secondary electrons can also be detected, as in ESEM, and EDX and ESEM are
commonly combined to produce composite images.
The X-rays detected are typically from 2p→ 1s and 3p→ 1s transitions, often
referred to as Kα and Kβ respectively. The energies of these transitions are largely
independent of the chemical environment of the atom, but vary considerably
between atoms of different elements. The energies of the X-ray photons detected
when the electron beam is over a particular point in the sample therefore indicate
which elements are present at that point. These X-rays travel through the sample
more easily than the secondary electrons used for SEM and ESEM imaging. This
limits the resolution of EDX to the size of the volume in which electrons from
the probe beam interact with the sample. This volume is called the interaction
volume, and can be decreased by using a lower-energy electron beam or a thin
foil sample, if increased resolution is required.
2.2.1 X-ray detection
The X-rays are generally detected using a section of intrinsic semiconductor under
an electric field. The passage of the X-ray photon produces electron-hole pairs
in the semiconductor. The presence of these charge carriers allows a current to
flow through the semiconductor, and this current is detected. Each electron-hole
pair requires a well-known energy to produce, and the X-ray photon is assumed
to dissipate all its energy in the production of such pairs. The detected charge
pulse then indicates the energy of the photon.
38
X-RAY MICROTOMOGRAPHY 2.3
While this process is similar to that of ionization in a gas, the energy required
to create an electron-hole pair is roughly an order of magnitude smaller than
the energy required to create an electron-ion pair. This means that more charge
carriers are produced by a single photon in a semiconductor, and consequently the
energy resolution is improved. Modern semiconductor detectors typically have an
energy resolution of about 130 eV. This resolution is due in roughly equal parts
to statistical uncertainty in the energy of the charge-carrier pairs and to thermal
noise in the processing electronics.
The detector inevitably has a “dead region” some 100–200 nm thick in which
the electric field is zero, facing the sample. The lack of electric field means that X-
rays absorbed in this region will not be detected. The detector is also covered by a
10–20 nm thick gold electrode. This natural window absorbs all X-rays produced
by elements with atomic number Z < 5, so boron is the lightest element which
can be detected by EDX.
In addition to this natural window, the detector is usually covered by an
opaque window. This prevents light within the sample chamber saturating the
detector, and protects the detector from dirt and gases in the sample chamber.
It also allows the detector to be removed without breaking the sample chamber’s
vacuum. These were historically made of 7.6µm thick beryllium, which absorbed
all X-rays produced by elements lighter than fluorine. Modern windows consist
of a nickel or silicon grid, the gaps in which are filled by a thin layer of some
carefully chosen material. They have sufficient strength to withstand a 1 atm
pressure difference, are opaque to light, and do not prevent detection of any
elements not already blocked by the natural window of the detector. Knoll (1989)
and Goldstein et al. (2003) contain more information.
2.3 X-ray microtomography
X-ray microtomography is a non-destructive technique for studying planes in the
interior of a sample. It has a minimum spatial resolution of approximately 1µm.
A Skyscan 1072 microtomograph was used for this research.
Multiple views of the X-ray absorption of the sample at different angles θ
are taken, as shown in figure 2.2. These are then combined to produce a two-
39
2 EXPERIMENTAL TECHNIQUES
dimensional image of the sample in the plane perpendicular to the axis of rotation,
using a filtered back-projection algorithm (Kak and Slaney, 1988). This compu-
tationally expensive process is referred to as reconstruction.
In practice, most X-ray sources are small points, from which X-rays radi-
ate. The back-projection algorithm can be modified to allow reconstruction with
diverging X-ray beams. The technique can be further modified to allow recon-
struction of the full 3D structure of a sample, in which the X-ray absorption of
the sample at each angle θ is measured by a 2D sensor array. This technique is
referred to as cone-beam reconstruction (Feldkamp et al., 1984; Kak and Slaney,
1988), and is used in the Skyscan 1072.
Three factors limit the resolution of X-ray microtomography. The first is the
resolution of the detector. This limits the density with which Pθ(k) is sampled,
and hence the resolution of the reconstruction. The second, closely related, is the
size of the sample. Because X-ray absorption across the entire sample must be
measured, the resolution with which a tomographic slice of a sample of maximum
diameter d can be reconstructed using a detector with Nd elements cannot exceed
d
Nd
. The third factor affecting resolution of the technique is the finite size of the
X-ray source. Penumbral broadening, as illustrated in figure 2.3, will blur Pθ(k),
and hence the reconstruction.
2.4 Screw-driven instrumented press
A screw-driven instrumented press is a tool for applying a uniaxial force to a
sample, at a known strain rate. The strain rate can vary from 10−7 s−1 to 1 s−1.
Screw driven instrumented presses applying a maximum force of around 106N
are commercially available. This research used an Instron 4466 press, capable of
applying a maximum force of 104N. Figure 2.4 shows such a press schematically.
The cross-head rests on the thread of the screws. Motors in the base of the
apparatus turn the screws and thereby move the cross-head up or down. Since
the pitch of the screws is known, the rotation of the motors gives the position
of the cross-head. A load cell is mounted on the cross-head, and the sample is
attached to that.
In practice, the forces applied cause some elastic deformation of the cross-
40
SCREW-DRIVEN INSTRUMENTED PRESS 2.4
θ0
y
x
absorption
X-ray
Detector
Sample
a)
Incident X-rays
k
Pθ0(k)
y
x
θ1
X-ray
Detector
Sample
Incident
X-rays
absorption
k
Pθ1(k)
b)
Figure 2.2: Schematic showing X-ray tomography measurements. Pθ(k) is mea-
sured for Nm angles θ between 0
◦ and 180◦. In practice Nm is usually in the
hundreds and θ is evenly distributed, so Pθ(k) is measured every few tenths of a
degree.
41
2 EXPERIMENTAL TECHNIQUES
b)
Detector
Sample
Source
P (k)
Umbra Penumbra
k
k
Umbra
Source
Sample
Detector
P (k)
a)
Figure 2.3: Penumbral broadening. (a) With a point source of X-rays, the edges
of the sample recorded at the detector are perfectly sharp. (b) With a source of
finite size, a penumbral region exists in which the sample only partially obscures
the source.
42
SCREW-DRIVEN INSTRUMENTED PRESS 2.4
Load cell
Screws
Cross head
Sample
MOTOR BASE MOTOR
Figure 2.4: Schematic of a screw-driven instrumented press, such as the Instron
4466 used in this research. The motors turn the screws to move the cross head.
This in turn deforms the sample. The load cell records the force applied to the
sample during this process. Not to scale.
head, drive screws and load cell. It is therefore necessary to measure the reported
motion, as a function of measured force, in the absence of any sample. This
motion is called the compliance of the system. The compliance can then be
subtracted from the reported motion in the presence of a sample to recover the
actual sample deformation.
2.4.1 Backlash
Another potential issue with this class of apparatus is that of backlash. There
must be some empty space between the threads of screws to allow motion. If
the direction of motion is reversed, this free space is rearranged. In the case of
this apparatus, this results in uncertainty as to the position of the cross-head.
As several useful experiments require the direction of motion of the cross-head
to be reversed (for example, any cyclical loading experiment), most instrumented
presses take steps to eliminate backlash.
Backlash is usually eliminated by preloading. The cross-head contacts each
43
2 EXPERIMENTAL TECHNIQUES
drive screw in two locations. These positions are held apart by a spring, applying
a force greater than any the drive screws are expected to exert on the cross head.
With this arrangement there is sufficient free space for the screws to function, but
this space will not be rearranged when the direction of motion of the cross-head
is reversed. The drawback to this approach is that the contact forces between
the cross-head and the drive screw are considerably increased. This increases
the friction between the drive screws and the cross head, and hence reduces the
efficiency of the motors in moving the cross-head. To ameliorate this problem, a
highly efficient screw design, such as the ball screw, is used for the drive screws.
2.5 Load cell
A load cell, also known as a force transducer, is a device for measuring force along
an axis. The most common design is a Wheatstone bridge strain gauge load cell.
This design is incorporated into the drop weight and Instron 4466 used in this
research. The SHPB uses a much simpler load cell design discussed in section 2.7.
The principle underlying the strain gauge load cell is that a stressed material
will deform. A cylindrical block of some material, chosen to remain elastic at the
forces the load cell will measure, is instrumented with uniaxial resistive strain
gauges as shown in figure 2.5. These gauges act as resistors, with a resistance
R = R0(1 + Gε), where ε is the strain along the gauge axis and R0 and G are
properties of the gauge. They are arranged in a Wheatstone bridge (Horowitz
and Hill, 1989), as shown in figure 2.6.
The variable resistor Rv is set so that the output voltage Vo is zero when the
applied force is zero. This is referred to as balancing the bridge. The strain gauges
are typically manufactured to identical specifications with very tight tolerances,
so that Rv is very small. It is henceforth ignored, and the four strain gauges
assumed to have identical values of R0 and G.
When the load cell is subjected to a force, it will deform. The axial strain
gauges A1 and A2 will record a different strain to the circumferential gauges P1
and P2. We assume the load cell does not bend. Then, if the strain recorded by
the axial gauges is ε, that recorded by the circumferential gauges will be −νε. ν
is Poisson’s ratio for the material of which the load cell is made.
44
LOAD CELL 2.5
P1
P1 P2
P2
A1
A2 A2
A1
P2
A2
Figure 2.5: Arrangement of strain gauges on a Wheatstone bridge load cell. Not
to scale. The gauges are only sensitive to strain along the indicated axes. Labels
are as used in figure 2.6 and equation 2.1.
P1
A2
Rv
P2
A1
VoVB
Figure 2.6: Circuit diagram of strain gauges in a Wheatstone bridge load cell.
Labels are as used in figure 2.5 and equation 2.1.
45
2 EXPERIMENTAL TECHNIQUES
This difference will unbalance the bridge, resulting in a non-zero output volt-
age which is amplified and recorded. The output of an unbalanced Wheatstone
bridge is
Vo = VB
(
P2
P2 + A1
− A2
A2 + P1
)
. (2.1)
Substituting in the resistance of the gauges, this gives
Vo = −VB
(
Gε(1 + ν)
2R0 +Gε(1− ν)
)
. (2.2)
Using binomial expansion to convert this to a linear form gives
Vo ≈ −VBG(1 + ν)
2
ε± G
2(1− ν2)
4
ε2. (2.3)
The load cell material and strain gauges are usually chosen so that the error
due to assuming Vo ∝ ε is very small. For example, in the drop weight load cell the
fractional error from this source is approximately 10−5. As the load cell material
is elastic, the force applied to the cell is proportional to the strain. Hence, to
good approximation, the output voltage is proportional to the force on the cell.
A load cell is typically calibrated by applying a known force and recording the
output, rather than by calculation from the material and gauge properties.
2.5.1 Bending of the load cell
The gauges are mounted in pairs to reduce the extent to which flexing of the load
cell will affect the output voltage. If the load cell bends such that one gauge is
in compression, the opposite gauge will be in tension. Assuming that the load
cell is initially balanced, and it flexes so that gauge A1 experiences a small strain
ε, gauge A2 will experience the strain −ε. From equation 2.1, the output of the
bridge will be
Vo = VB
(
R0
R0 +R0(1 +Gε)
− R0(1−Gε)
R0(1−Gε) +R0
)
. (2.4)
46
DROP WEIGHT 2.6
This simplifies to
Vo =
VBG
2
4−G2ε2 ε
2. (2.5)
As we have already assumed that ε is small, this gives Vo ∝ ε2, which will be
very small.
2.6 Drop Weight
The drop weight is a tool for applying uniaxial stress to a sample, at a strain
rate of order 103 s−1. The stress that can be applied is limited by the materials of
which the sample-facing surfaces are constructed. Typically the maximum stress
is a few gigapascals. The advantages of a drop weight system over an SHPB are
the high maximum strain to which the sample can be taken, the robustness of
the system, and the relative ease of imaging through the sample.
As its name suggests, the central feature of a drop weight is a weight that
falls on the sample, crushing it. The weight typically weighs a few kilograms
and runs on rails to prevent off-axis motion of the weight and ensure a planar
impact. The sample rests on top of an anvil, which remains elastic throughout
the experiment. The impact velocity of the weight is commonly measured using
a light gate system. The anvil often rests on top of a load cell, to measure the
force applied to the sample. If the sample is not expected to be approximately in
mechanical equilibrium during the experiment, an accelerometer may be added
to the weight.
2.6.1 Sensitivity of explosives
Drop weights are routinely used to measure the sensitivity of explosives. The
standard approach is to test multiple samples of the explosive of interest to find
the drop height which has a 50% probability of causing explosion (AMCP 706-
180). However, not all of the potential energy of the weight will be transferred
to the explosive sample: some will be stored as elastic deformation of the weight.
The fraction of the weight’s energy transferred to the explosive at the moment
of ignition will depend on several factors, including the construction of the drop
47
2 EXPERIMENTAL TECHNIQUES
weight. The drop weight used for sensitivity testing must therefore be standard-
ized. Refinements have been suggested (Coffey and DeVost, 1995) to eliminate
some of the factors affecting energy deposition in the explosive sample.
2.6.2 The Cavendish drop weight
Figure 2.7 shows the usual arrangement of the Cavendish drop weight. In this
configuration, two maraging steel anvils and a load cell are restricted to vertical
movement by a cylindrical anvil guide. The sample is placed between the anvils,
and the weight falls on the upper anvil. The attraction of this approach is that the
sample is guaranteed to experience uniaxial stress. In order to allow the weight
to move freely, the carriage guides are not pressed tightly against the guide rails.
The weight can therefore tip slightly during impact. Were the impact plate to
strike the sample directly, this tipping would result in off-axis forces being applied
to the sample. The upper anvil and anvil guide practically eliminate these forces.
2.6.3 Materials testing
In this configuration, the drop weight is conventionally used to find the stress-
strain response of a material at a particular strain rate. The sample is usually a
cylinder with height and diameter both of a few millimetres, and is assumed to
have its top and bottom faces in mechanical equilibrium. Given this assumption,
and the combined mass m of the weight and top anvil, the acceleration of the
weight can be recovered from the force recorded by the load cell:
a =
F
m
. (2.6)
If the velocity of the weight at impact is measured, its position, and hence
the length of the sample, can be recovered by numerically integrating the force
48
DROP WEIGHT 2.6
Guide rail
Sample
Upper anvil
Lower anvil
Anvil guide
Viewing slit
Load cell
Impact face
(removable)
Weight
Carriage guide
Figure 2.7: Usual arrangement of the Cavendish drop weight system. Not to
scale.
49
2 EXPERIMENTAL TECHNIQUES
twice:
l = l0 −
∫
v dt,
l = l0 −
∫ (
vi −
∫
F
m
dt
)
dt,
l = l0 − vit+ 1
m
∫∫
F d2t. (2.7)
The strain ε(t) = ln l(t)
l0
is then easy to find. To find the true stress, the
sample’s cross-sectional area at a given time is required; the strains involved are
too great to assume that the cross-sectional area is constant. To find the cross-
sectional area, we assume first that the sample deforms uniformly along its length,
and second that the sample’s volume is conserved. To achieve the first condition,
the faces of the upper and lower anvils are usually lubricated. In the absence
of this lubrication, the top and bottom faces of the sample stick to the anvil,
and the sample preferentially expands at its mid-point. Lubrication eliminates
this “barrelling”. Under these assumptions, the cross-sectional area A is given in
terms of the initial, measured cross-sectional area A0 by
Al = A0l0,
A =
A0l0
l
. (2.8)
As l is already known from equation 2.7, it is now easy to find the stress
σ = F
A
.
2.6.4 Calibrating the load cell
The load cell in the drop weight is calibrated by dropping the weight on the anvils
in the absence of a sample. The impact and rebound velocities ui and ur of the
weight are measured. Given the mass m of the weight, the impulse J applied to
the weight in this impact can then be calculated:
J = ∆p = m(ui − ur). (2.9)
50
SPLIT HOPKINSON PRESSURE BAR 2.7
As established in section 2.5, the load cell’s output Vo(t) may be assumed to
be proportional to the force applied to the cell. Given this, we have
J =
∫
F (t) dt = A
∫
Vo(t) dt, (2.10)
where A is a constant. From equations 2.9 and 2.10 we get the calibration factor
A =
m(ui − ur)∫
Vo(t) dt
. (2.11)
2.6.5 Imaging through the sample
The drop weight can also be configured with a light path through the sample
area, allowing images to be taken along the loading axis as the sample deforms
(Field et al., 1982; Heavens and Field, 1974). Figure 2.8 shows the light path in
this case. This configuration is often useful in studying the mechanical processes
which lead to ignition in energetic materials. In this research, the camera was a
C4 rotating mirror camera, described in section 2.8.1.
2.7 Split Hopkinson Pressure Bar
The split Hopkinson Pressure Bar (SHPB), also known as a Kolsky bar, is a
tool for applying uniaxial stress to a sample, at a strain rate of order 103 s−1.
In this respect it is much like the drop weight. Also like the drop weight, the
maximum stress that can be applied is limited by the yield stress of the sample-
facing surfaces, and is typically a few gigapascals. The chief advantage of an
SHPB over a drop weight is the SHPB’s greater accuracy in measuring stress
and strain. While elastic deformation of a drop weight introduces uncertainty to
measurements made with that apparatus, this deformation is a central feature of
an SHPB (Gray, 2000).
A compressive SHPB is illustrated schematically in figure 2.9. The striker bar
is accelerated by some means to a few metres per second, and strikes the input
bar. All the bars are made of the same material and are of equal diameter, and the
51
2 EXPERIMENTAL TECHNIQUES
Weight
Mirror
Sample
Camera
Base
Prism
anvils
Glass
Flash
Figure 2.8: Light path through the sample in drop weight. Not to scale.
Strain gauges
Striker bar Input bar Sample Output bar Momentum trap
Figure 2.9: A compressive split Hopkinson pressure bar system (SHPB). Not to
scale. In practice the ratio of bar length to bar diameter is of order 100:1.
striker bar velocity is chosen to ensure that they behave elastically throughout
the experiment. The impact will therefore produce a well-defined, very nearly
rectangular pressure pulse in the input bar, and bring the striker bar to rest.
This pressure pulse will be recorded by the strain gauge on the input bar. This
is the incident pulse.
When the incident pulse reaches the sample, some fraction is reflected and
some is transmitted. The transmitted pulse is recorded by the strain gauge on
the output bar. The reflected pulse is recorded by the strain gauge on the input
bar. This requires that the incident pulse is shorter than the round trip between
the input gauge and the sample, so that the incident and reflected pulses do
52
SPLIT HOPKINSON PRESSURE BAR 2.7
Figure 2.10: Approximate gauge outputs for a compressive SHPB. In this case,
as is usual for SHPB experiments, the sample impedance is lower than the bar
impedance.
not overlap. Figure 2.10 shows an example of the gauge outputs for this situa-
tion. Note that the tensile reflected pulse indicates that the sample was of lower
impedance than the bars; this is the default situation in an SHPB experiment.
2.7.1 The momentum trap
The transmitted pulse enters the momentum trap. There is no impedance mis-
match between the momentum trap and the output bar. There will, therefore, be
no reflected wave from the interface between the output bar and the momentum
trap. The reflected wave from the far end of the momentum trap will, however,
be tensile. As the momentum trap and the output bar are not secured to each
other, this tensile wave will be unable to pass the interface between the two. The
momentum trap will therefore be ejected with a large fraction of the momentum
of the striker bar.
The momentum trap serves two purposes. First, it prevents the reflection
of the transmitted pulse separating the interface between the sample and the
output bar, which could interfere with recovery of the sample for subsequent
tests. Second, by isolating most of the bulk motion of the apparatus into the
53
2 EXPERIMENTAL TECHNIQUES
εr
εi
εt
lInput bar Output barSample
s1 s2
Figure 2.11: Symbols used in SHPB analysis. εi, εr and εt are the strains associ-
ated with the incident, reflected and transmitted waves, respectively. s1 and s2
are the displacements of the sample faces from their original positions. l is the
thickness of the sample.
inexpensive and robust momentum trap, the momentum trap extends the lifetime
of the apparatus. In the original pressure bar experiments of Hopkinson (1914),
which predated high-speed data capture, the velocity of the momentum trap was
the output of the experiment.
2.7.2 Materials testing
The SHPB, like the drop weight, is normally used to find the stress-strain response
of a material. Also like the drop weight, the analysis to extract this from the
strain gauge readings is not entirely trivial. Throughout this analysis, the bars
are assumed to be elastic and non-dispersive. While it is possible to correct for
dispersion, the error due to ignoring dispersion is small (Siviour, 2005). This
research therefore ignores dispersion. Figure 2.11 shows the symbols used in this
analysis.
First, consider the one-dimensional wave equation
∂2s
∂x2
=
1
c2b
∂2s
∂t2
, (2.12)
where s is the displacement of a point in the bar, x is the position along the bar,
and cb is the wave speed along the bar. Solutions of this equation are
s = f(x− cbt) + g(x+ cbt), (2.13)
54
SPLIT HOPKINSON PRESSURE BAR 2.7
where f corresponds to a wave travelling in the positive direction, and g to one
travelling in the negative direction. In the input bar, these will be si and sr, the
incident and reflected waves, respectively. In the output bar, the wave travelling
in the negative direction will be removed by the momentum trap and only the
transmitted wave st = h(x− cbt) will be present.
By definition, the strain
ε =
∂s
∂x
. (2.14)
From this, the strain in the bars will be
ε = f ′ + g′ = εi + εr , Input bar (2.15)
ε = h′ = εt . Output bar (2.16)
Using the chain rule, differentiate s with respect to time:
s˙ = cb(−f ′ + g′) = cb(εr − εi), Input bar (2.17)
s˙ = −cbh′ = −cbεt . Output bar (2.18)
Equations 2.15, 2.16, 2.17 and 2.18 are true throughout their respective bars.
This allows the strain rate in the sample to be calculated:
ε˙ =
s˙1 − s˙2
l
=
cb(εr − εi + εt)
l
. (2.19)
Given the initial length of the sample l0, and the measurements of εi(t), εr(t)
and εt(t) from the strain gauges, this can be numerically integrated to find the
strain ε in the sample.
To find the stress in the sample, observe that the forces in the bars are
F = AY ε = AY (εi + εr), Input bar (2.20)
= AY εt , Output bar (2.21)
where Y is the Young’s modulus of the bar material and A is the cross-sectional
area of the bar.
Averaging these two forces will give the force on the sample. To find the true
55
2 EXPERIMENTAL TECHNIQUES
Vtot
Rg
Rg
Rc Vout
Figure 2.12: Potential divider circuit used in the SHPB. The resistors Rg are
strain gauges; the resistor Rc is constant.
stress requires the cross-sectional area of the sample. As with the drop weight,
this is obtained by assuming that the sample’s volume is conserved, and that
it deforms uniformly. Also as with the drop weight, lubrication of the interfaces
between the sample and the bars is required to satisfy the second criterion. Then,
using equations 2.8, 2.20 and 2.21, the stress σ in the sample is given by
σ =
AY l(εi + εr + εt)
2A0l0
, (2.22)
where A0 and l0 are the measured initial sample area and length, respectively.
The instantaneous sample length l is known from numerical integration of equa-
tion 2.19.
2.7.3 Strain measurement in the SHPB
The strain gauges in the SHPB are not arranged as a conventional Wheatstone
bridge load cell. Instead a single pair of strain gauges is mounted parallel to the
bar axis at each measurement point. The gauges are placed on opposite sides of
the bar to minimize the effect of bar flexing on the reported strain. They are
then placed in a simple potential divider circuit as shown in figure 2.12.
These strain gauges, like those used in a conventional load cell, have a resis-
56
SPLIT HOPKINSON PRESSURE BAR 2.7
tance Rg = R0(1 +Gε), so this circuit’s output will be given by
Vout = Vtot
Rc
Rc + 2Rg
,
Vout = Vtot
Rc
Rc + 2R0(1 +Gε)
. (2.23)
The oscilloscope recording Vout is AC coupled, so it records only the deviation
V from the output in the absence of strain:
V0 ≡ Vout(ε = 0),
V0 = Vtot
Rc
Rc + 2R0
. (2.24)
This gives the recorded output on the oscilloscope as a function of strain:
V = VtotRc
(
1
Rc + 2R0(1 +Gε)
− 1
Rc + 2R0
)
. (2.25)
By simple but tedious algebra, this can be rearranged to give the strain as a
function of the oscilloscope output:
ε = −Rc + 2R0
2R0G
· V
V +
VtotRc
Rc + 2R0
. (2.26)
This design of strain gauge is much less sensitive to strain than a Wheatstone
bridge arrangement. It was chosen because fast-acting amplifiers, of the kind
used to amplify a Wheatstone bridge’s output, were unavailable when the system
was designed. To compensate for its reduced sensitivity, semiconductor strain
gauges are used. The gauge factor G of such gauges is typically two orders of
magnitude greater than that of the foil gauges used in most Wheatstone bridge
load cells. However, semiconductor gauges are also considerably more sensitive
to temperature variations than foil gauges. The potential divider system used
here ameliorates this temperature sensitivity.
The choice of semiconductor strain gauges has other benefits. Semiconductor
gauges are physically smaller than foil gauges, and therefore sample the strain
57
2 EXPERIMENTAL TECHNIQUES
over a smaller length of the bar. As the temporal structure of the wave is spread
along the bar, this smaller sampling range translates directly to improved tem-
poral resolution. This smaller size also makes them more suitable for use in a
miniaturized SHPB system (Gorham et al., 1992).
2.7.4 Calibrating the SHPB
Rather than substituting measured values into equation 2.26 to calculate the
strain that corresponds to a given oscilloscope output, a dynamic calibration
technique similar to that used for the drop weight is employed. First, observe
that, if the bars are elastic, equation 2.26 is of the form
F = A · V
V +B
, (2.27)
where A and B are constants to be determined. To do so, the bar to be calibrated
is set in the place of the input bar, with no sample or output bar. The striker
bar is then fired at it at a velocity measured using a light gate system.
By an impedance-matching argument, the pressure pulse produced in the bar
must correspond to an impulse exactly sufficient to bring the striker bar to rest.
As the striker bar’s mass ms and velocity us are known, this impulse is known.
If the strain gauge output is approximated as a square wave of duration t and
height V , the force applied by the striker bar is given by
F =
msus
t
. (2.28)
The pressure pulse in the bar then gives a single (F, V ) data point. Assuming
total reflection at the end of the bar, the tensile pulse reflected from that end will
give another such point. Several of these points are gathered. A and B can then
be determined by model fitting, as described in section 2.12.
Another aspect of the bars that must be calibrated is the wave speed in the
bar, cb. This can in principle be calculated from the properties of the bar material
and the geometry of the bars. However, it can be highly sensitive to a variety of
factors which are poorly known, such as the thermal and mechanical histories of
the bars.
58
HIGH SPEED PHOTOGRAPHY 2.8
Calibration of cb is achieved simply by placing the input and output bars
in contact and firing the striker bar at the input bar. The pressure pulse will
propagate down the bars at cb. If the distance between the strain gauges is
known, the time taken for the pressure pulse to pass from the input strain gauge
to the output strain gauge gives cb.
It is possible to measure cb in each bar independently by measuring the time
taken for the reflection of the incident pulse to return to the bar’s strain gauge.
In practice, as the bars are manufactured, stored and used as a pair, there is
rarely a measurable difference between cb in the two bars.
2.8 High speed photography
The term “high speed photography” refers to any photographic technique with
a frame rate higher than 128 frames per second. There are several high speed
photographic techniques (Ray, 1997). This research used high speed cameras of
the rotating-mirror and image conversion designs.
2.8.1 Rotating-mirror
A rotating-mirror camera typically records a few tens of frames, with an inter-
frame time of a few microseconds. Unlike a conventional film camera, the film in
a rotating-mirror camera remains stationary. The rotating mirror for which the
camera design is named moves the image across the film, as shown in figure 2.13.
The mirror rotates at a constant speed, typically a few hundred r. p. s., and is
the only moving part in the system. This rapid rotation subjects the mirror to
considerable centrifugal force. The temporal resolution of a rotating-mirror cam-
era will be limited by the mirror’s mechanical strength: above a certain rotation
speed deformation of the mirror will prevent image formation.
This design of rotating-mirror camera is unable to record continuously: for
some time during the mirror’s rotation the light from the experiment will fall
on the mirror’s edge, and not be reflected. The experiment must therefore be
carefully synchronized with the mirror’s rotation, or the camera will fail to record
for some unpredictable part of the experiment. The C4 rotating mirror camera
59
2 EXPERIMENTAL TECHNIQUES
Object
Lens
Light path
Mirror
Shutter strip
ω
Film
Image
Past images
Figure 2.13: Schematic of a simple rotating-mirror camera. The mirror rotates
with angular velocity ω. Not to scale.
60
HIGH SPEED PHOTOGRAPHY 2.8
(Coleman, 1959) uses more complicated optics, shown in figure 2.14, to avoid this
problem. Two film tracks are used, and a beam-splitter divides the light from
the experiment into two paths. These are arranged so that, when one light path
reaches the end of its film track, the other reaches the start of the other film
track. While light from the experiment will still strike the edge of the mirror and
be lost, this will never occur while that light would otherwise be recorded. Note
that this optical arrangement records a full set of images in half a rotation of the
central mirror. The design of the C4 camera allows it to record 140 frames.
In this research, the exposure of the film was controlled by use of a short-
duration xenon flash. The mirror was rotated at approximately 500 r. p. s. and
its precise speed measured using the rotating mirror and photodiode arrange-
ment shown in figure 2.15. The shutter was then opened and the experiment
activated. The experiment triggered the flash, which provided illumination for
approximately one millisecond. During this time each frame would be exposed
once. Some frames received a double exposure, overprinting the initial and final
states of the experiment. This can be used to compare these two states directly,
and is a useful feature of the C4 camera.
One disadvantage of the C4 camera is that it weighs over a tonne and is
effectively immobile. The experiment, therefore, must be brought to the camera,
rather than the reverse. This contrasts with an image converter camera, which
typically weighs about 20 kg and can be moved to the event being studied.
2.8.2 Image conversion
An image converter camera records roughly ten frames, with an interframe time
as low as a few tens of nanoseconds. The camera converts an optical image to
an electron beam. This beam is then manipulated electrostatically by altering
the charge on deflecting plates. The elimination of moving parts dramatically
reduces the minimum interframe time. The temporal resolution is limited by the
time taken to charge the deflecting plates (Zavoisky and Fanchenko, 1965).
Figure 2.16 shows the image tube of the Ultranac 501 image conversion camera
used in this research (Garfield and Riches, 1991). Light from the experiment is
focused using optical lenses (not shown) onto the photocathode. The photocath-
61
2 EXPERIMENTAL TECHNIQUES
I′2
Experiment
A G H
B
Pv
Pw
Lw
Lv
Fw
Fv
Tw
Tv
I1
I2
M
I3
Figure 2.14: Optics of the C4 rotating-mirror camera. Not to scale. The beam
splitter B divides the light into two fractions, v and w. These are reflected from
their respective fixed mirrors P and focused by lenses L to form an image I2 on
the rotating mirror M. This image is then reflected and refocused by a lens in the
static array of lenses F to form an image I3 on the film track T. Each lens array
contains 70 lenses, allowing the system to record a total of 140 images. P, L, F
and T are displaced out of the plane of the paper, so that Fw does not obscure Tv
and Fv does not obscure Tw. The shutter H controls exposure of the film track.
This design allows continuous access: provided the shutter is open, the camera is
guaranteed to be able to record an image.
62
HIGH SPEED PHOTOGRAPHY 2.8
M′
Laser
Photodiode
Figure 2.15: System for measuring the rotation speed of the mirror in the C4
camera. Not to scale. The mirror M′ is affixed to the same axle that drives
the rotating mirror M in the camera (see figure 2.14). With each rotation, the
photodiode will produce a pulse; the frequency of these pulses is measured with
a counter-timer.
63
2 EXPERIMENTAL TECHNIQUES
Deflection
plates
Phosphor
screen
Focus
cone
Shutter
electrodes
Photocathode
Drift region
plate
Aperture
31 2
Figure 2.16: Electron optics of the Ultranac 501 image conversion camera. Not
to scale. The photocathode and shutter electrodes are held near to 0V (see
figures 2.17 and 2.18 for details). The focus cone is held at +450V, and the
aperture plate and phosphor screen at +15 kV.
ode is constructed from a mixture of compounds of alkali metals and antimony,
referred to as S25. This material has a very low work function, and will emit
photoelectrons when exposed to visible light (Holtom et al., 1979). Ignoring, for
now, the shutter electrodes, these photoelectrons will be accelerated and focused
by the focus cone and aperture plate. These two components act as an electro-
static lens, transferring the electron image at the photocathode to the phosphor
screen. Within the drift region, there is no axial electric field. The deflection
plates move the electron beam across the phosphor screen to produce distinct
images or a streak record, as required. The phosphor screen is a thin layer of a
mixture of zinc sulphide, cadmium sulphide and silver, referred to as P20. When
struck by the accelerated electrons from the photocathode, it emits light. It is
coupled by optical fibres to a flat back plate, against which photographic film is
pressed to record the images.
Figures 2.17 and 2.18 show the operation of the shutter electrodes. Before
64
HIGH SPEED PHOTOGRAPHY 2.8
t1 + e1 + tp t2 + e2 t2 + e2 + tp
t1 t1 + tp t2 t2 + tp
a) 50 V
20 V
0 V
b) 40 V
20 V
0 V
t1 + e1
−10 V
Figure 2.17: Potential of (a) the photocathode and (b) shutter electrode 2 in the
Ultranac 501 image conversion camera, as a function of time. Also shown is the
constant 20V potential of shutter electrode 1. Shutter electrode 3 is held at a
constant 120V potential (not shown). Frame n starts at time tn, and is exposed
for a time en. tp is chosen so that tn + en + tp < tn+1, and may vary over the
course of an experiment.
the exposure, there is a potential energy minimum at electrode 1. A 50V pulse
of width tp is generated at the start of an exposure, and fed both to electrode 2
and to a delay generator. The change to the potential at electrode 2 eliminates
the potential energy minimum at electrode 1, and admits electrons from the pho-
tocathode to the accelerating region of the image tube. The delay generator
delays the pulse for the exposure time en, then applies it to the photocathode.
This produces a potential energy barrier at electrode 1 and consequently pre-
vents electrons passing from the photocathode into the image tube, ending the
exposure.
When the camera is used in imaging mode, the distance moved by the electron
beam between frames is minimized. This shortens the minimum inter-frame time
that can be achieved without image distortion or misalignment. However, it leads
65
2 EXPERIMENTAL TECHNIQUES
Electrode 2
a) 10 eV
0 eV
−20 eV
b) 0 eV
−20 eV
−40 eV
c) 0 eV
−20 eV
−40 eV
−50 eV
Photocathode Electrode 1
Figure 2.18: Electric potential energy for electrons in the vicinity of the shutter
grids in the Ultranac 501 image conversion camera. Potential energy is shown
(a) immediately before an exposure, (b) during an exposure, and (c) immediately
after an exposure. Horizontal axis not to scale.
66
PHOTODIODES 2.9
12
TITLE
1
2
3 4
5
6 7
8
9 10
11
Figure 2.19: Arrangement of output frames in the Ultranac 501 image conversion
camera, when configured to produce twelve frames.
to a somewhat counter-intuitive layout of frames on the output film. Figure 2.19
shows the layout of frames produced when the camera is used to produce twelve
frames. This mode was used throughout this research.
2.9 Photodiodes
A photodiode is a device for measuring the intensity of light. They typically have
a temporal resolution less than a nanosecond, and an intensity resolution of a few
milliwatts per square metre, if they are attached to suitably sensitive measure-
ment electronics. An Electro-Optics Technology, Inc. ET-2030 photodiode was
used in this research.
The ET-2030 photodiode is a silicon p-type, intrinsic, n-type (PIN) photo-
diode, operated with a 9V reverse bias. The photodiode is based on a PIN
semiconductor junction, as shown in figure 2.20. The intrinsic semiconductor
serves to extend the depletion region that forms at a p-type, n-type (PN) semi-
67
2 EXPERIMENTAL TECHNIQUES
−
p-type n-type
intrinsic
+
Figure 2.20: PIN semiconductor junction under reverse bias, as used in ET-2030
photodiode. Not to scale.
conductor junction. A photon incident on this extended depletion region excites
an electron-hole pair, which allows a certain charge to flow. The current flowing
through the photodiode will therefore indicate the photon flux at the photodiode.
In practice the average response of the photodiode to a given photon is a function
of that photon’s energy. The system must therefore be calibrated with light of
varying wavelengths. The variation of photon energy with wavelength is typically
included in this calibration, giving a wavelength-dependent relationship between
output current and incident light intensity.
2.10 Plate impact
Plate impact is a technique for introducing a shock wave to a material. The
amplitude of the shock wave may be a few tens of gigapascals, and its duration
may be several microseconds. The technique works by accelerating a flat plate
of some material (the “flyer plate”) to several hundred meters per second and
colliding it with a suitably designed sample cell. This research used the small gas
gun at the Cavendish to accelerate the flyer plate.
The small gas gun consists of an eight litre gas reservoir, connected to one of
a set of interchangeable barrels. The barrel used throughout this research had
a 19.6mm bore. Figure 2.21 shows the gas reservoir. To prepare the system for
firing, chamber A is filled to 5 bar above the intended firing pressure using the
68
PLATE IMPACT 2.10
Polycarbonate rod
A
BB
Barrel mount
Priming valve Main valve
Firing valve Release valve
Figure 2.21: Schematic of the firing chamber of the Cavendish small gas gun.
Not to scale. The main valve and priming valve are connected to a high-pressure
helium cylinder; the release valve and firing valve are connected to atmosphere.
The priming chamber, A, is filled through the priming valve and emptied through
the firing valve. The firing chamber, B, is in fact a single continuous volume; the
two apparently separate regions are connected outside the plane of the diagram.
The firing chamber is filled through the main valve; in normal operation it is
emptied into the barrel when the gun is fired. The release valve is used to empty
the firing chamber without firing the gun.
priming valve. This presses the polycarbonate rod against the inside of chamber
B near the barrel mount, sealing chamber B. Chamber B is then filled to the
intended firing pressure using the main valve. To fire the gun, the firing valve
is rapidly opened. This allows the polycarbonate rod to retreat into chamber A.
This in turn opens chamber B to the barrel, firing any projectile loaded therein.
The release valve is used to interrupt this process and make the system safe
without firing the projectile.
69
2 EXPERIMENTAL TECHNIQUES
2.11 High-speed optical spectroscopy
Optical spectroscopy is a technique for measuring the energy distribution, as
a function of frequency, of electromagnetic radiation. The temporal resolution
of the technique is determined by the time for which radiation is gathered. A
UV/visible gated spectroscope system produced by Princeton Applied Research
Corporation was used in this research. This system was able to record a single
spectrum with an exposure time no shorter than 50 ns. It was sensitive to light
with wavelengths in the range 200–800 nm.
The system accepts light through a bundle of quartz optical fibres. This light
is then dispersed into its spectral components. The spectrum thus formed falls
on an image intensifier. An array of photodiodes is coupled to the output of the
image intensifier with optical fibres. The output of these photodiodes is recorded.
These stages are discussed in more detail below.
2.11.1 Dispersion stage
In this spectroscope system, spectral dispersion is performed by a Princeton Ap-
plied Research model 1235 spectrograph. The 1235 is a slight modification of a
standard Czerny-Turner monochromator (Czerny and Turner, 1930; Fastie, 1952).
The modifications consist of the removal of the output slit, and the addition of
plane mirrors R so that the input and output are on opposite sides of the device.
Figure 2.22 shows the optics of the spectrograph. The fibre bundle illuminates
the input slit. This light is collimated into a parallel beam by the collimating
mirror C. This strikes the diffraction grating on one face of the grating turret T
and is diffracted. The grating is angled so that the first-order diffracted spectrum
strikes the re-focusing mirror F. The dispersed spectrum is then focused on the
input plane of the image intensifier.
The grating turret may be rotated. This serves two purposes. First, it allows
the grating to be changed, by presenting a different face of the turret to the
incident light. Second, it allows the spectral region of interest to be centred on
the re-focusing mirror. The re-focusing mirror itself may be moved along its
axis to adjust the focus of the spectrum. The gratings themselves are “blazed”:
their grooves have been cut at an angle, to maximize the energy reflected at that
70
HIGH-SPEED OPTICAL SPECTROSCOPY 2.11
Light out
Light in
R
R
T
C
F
Figure 2.22: Schematic of diffraction stage of Princeton Applied Research model
1235 spectrograph used in this research. The plane mirrors R allow the input
and the output to be on opposite sides of the device. The collimating mirror C is
arranged so that a parallel beam of light strikes the grating. The grating turret
T has a grating on each face, and can be rotated either to change gratings, or to
move the diffracted light over the detector. The re-focusing mirror F focuses the
diffracted light on the detector plane (not shown). Not to scale.
71
2 EXPERIMENTAL TECHNIQUES
angle. This angle is chosen to maximize the brightness of the first-order diffracted
spectrum; this minimizes the light wasted in spectra which do not strike the re-
focusing mirror and are not recorded. The wavelength for which this process
is optimized is the blaze wavelength. The spectrograph used in this research
was supplied with two gratings, etched with 150 and 600 grooves per millimetre
respectively. Both were etched with a 500 nm blaze wavelength. Only the first
grating was used here.
2.11.2 Image intensifier
The image intensifier and detector array were supplied as a single unit. The
combined unit was a Princeton Applied Research model 1455B-700-HQ intensified
detector. The intensifier is essentially a generation II image intensifier. This has
several similarities with the image conversion camera described in section 2.8.2.
Both systems convert the incident optical image to an electron image, manipulate
this electron image, then use a phosphor screen to convert it back to an optical
image.
Figure 2.23 shows the layout of the intensifier. The interior of the intensifier
is evacuated at the time of manufacture, then sealed. The spectrum produced
by the spectrograph is focused on the input face. The incident light causes the
photocathode to emit photoelectrons. These are accelerated by the 200V poten-
tial difference between the photocathode and the input face of the microchannel
plate.
The microchannel plate acts as an array of electron multipliers (Wiza, 1979).
The plate consists of an array of glass tubes, aligned at an angle to the plate’s
normal. This angle is typically a few degrees, and the tubes typically have internal
diameters of a few microns. An electron striking the wall of one of these tubes
provokes secondary electron emission. These secondary electrons are accelerated
by the 1 kV potential difference across the plate. When they strike the tube wall,
they also provoke secondary electron emission. This electron cascade results in
a gain between 103 and 105, depending on the design of the microchannel plate
and the potential difference applied across it. In this intensifier, the potential
difference across the microchannel plate may be varied. This allows the gain of
72
HIGH-SPEED OPTICAL SPECTROSCOPY 2.11
+1kV
Photocathode Phosphor
−200V +5 kV
OutputInput
MCP
Figure 2.23: Layout of the image intensifier in the Princeton Applied Research
model 1455B-700-HQ intensified detector. “MCP” denotes the micro-channel
plate. Not to scale. Horizontal axis disproportionately expanded.
the system to be adjusted. Electrons typically exit the plate with energies of a
few tens of electron-volts.
The multiplied electrons leaving the microchannel plate are accelerated by the
4 kV potential difference between the output face of the microchannel plate and
the phosphor screen. They strike the screen, causing it to emit light. This light
is transferred by optical fibre to the detector array. Each stage of the system is
proximity focused: the short distance the electrons travel removes the need for
any focusing electron optics.
Gating is accomplished by altering the potential at which the photocathode
is held. When the intensifier is to be operated in gated mode, the photocathode
is held at the same potential as the input face of the microchannel plate. Pho-
toelectrons from the photocathode will therefore not travel to the microchannel
plate. To activate the intensifier, a −200V pulse is applied to the photocathode
for the desired duration. This allows electrons to travel from the photocathode
to the input face of the microchannel plate, and the intensifier then acts in the
fashion described above. The intensifier was capable of a minimum exposure
time of 50 ns. The gating pulse was provided by a Princeton Applied Research
model 1304 pulse amplifier. This could be triggered by a 5V TTL pulse from the
73
2 EXPERIMENTAL TECHNIQUES
experiment. It also provided a 5V monitoring output. This was high while the
intensifier was active. This eliminated uncertainty due to delays in the internal
electronics of the pulse amplifier.
2.11.3 Detector array
The detector array is the other component of the Princeton Applied Research
model 1455B-700-HQ intensified detector. It is an array of 1024 photodiodes.
Seven hundred of these are coupled by optical fibres to a line across the centre of
the intensifier’s output plate. The remainder are not illuminated. Photodiodes
are described in section 2.9. The photodiodes in this array are reverse biased to
5V, then disconnected from their power source. In this state they are effectively
a charged capacitor. Each photon that is absorbed in the depletion region will
generate a charge-carrier pair. These charge carriers will act to discharge the
capacitor. After the array has been exposed, the potential difference across each
photodiode is read in sequence. This will give a quantity proportional to the
number of photons absorbed, and hence to the intensity with which the intensi-
fier’s output plate was lit. After a photodiode has been read, it is re-biased to
5V to prepare it for the next exposure. This arrangement requires only a single
accurate amplifier to read the entire photodiode array.
The detector array is equipped with a Peltier device for cooling. Cooling the
detector reduces the rate at which thermally-generated charge carriers discharge
the photodiodes. This “dark current” causes a constant offset in the recorded
spectrum. Its magnitude depends on the detector temperature and the time for
which the array was exposed between measurements. It cannot be eliminated
altogether at non-zero temperature, and must be accounted for.
2.12 Model fitting
Model fitting describes a family of techniques for determining the parameters
of a model from some data. It is also known as curve fitting. The technique
used in this research was the Levenberg-Marquardt algorithm (Marquardt, 1963;
Press et al., 1992). This is an iterative algorithm which attempts to minimize the
74
MODEL FITTING 2.12
χ2 deviation between the model and the data to be fitted. It does not assume
that the model depends linearly on its parameters. This research used the
implementations of this algorithm in the SciPy and NumPy suite (Ascher et al.,
2001; Jones et al., 2001–), and in the gnuplot software (Williams et al., 2010).
2.12.1 Justification for χ2 minimization
If the model depends on M parameters, these parameters may be written as the
M -element vector a. If the independent variable is x, the value of the model
is y(x;a). The observed data, to which the model is to be fitted, consists of N
data points of the form (xi, yi); each yi has some error σi which is assumed to be
normally distributed. The deviation which is to be minimized is then
χ2 =
N∑
i=1
[
yi − y(xi;a)
σi
]2
. (2.29)
Finding the parameters a for which χ2 is minimized is an attractive approach
to model-fitting. First, it reduces the problem to a special case of function min-
imization. Second, it is a form of maximum-likelihood method. The probability
P (a) that a particular set of parameters will produce the set of data (xi, yi) is
P (a) =
N∏
i=1
1√
2piσ2i
exp
[
−(yi − y(xi;a))
2
2σ2i
]
. (2.30)
This is just the product of the Gaussian probabilities of each data point, assuming
that they are centred on y(xi,a) and have standard deviations σi. This will be
the case if the measurement errors have been correctly assessed, and the model
is, in fact, the model that gave rise to these observed data. Equation 2.30 may
be rearranged to give
P (a) =
1√
2pi
N
exp
[
N∑
i=1
−
[
yi − y(xi;a)
2σi
]2
− lnσi
]
. (2.31)
This, the probability of the data given some set of parameters a, may be
75
2 EXPERIMENTAL TECHNIQUES
thought of as the likelihood that this set of data was generated by a model with
those parameters. The set of parameters which maximizes this likelihood will,
assuming the form of the model is correct and subject to some uncertainty due to
the measurement errors σi, be that responsible for the observed data. The pre-
factor and lnσi terms in equation 2.31 do not depend on a, and may therefore
be neglected. Further, maximizing a function is equivalent to maximizing its
logarithm. Taking these into account, the function to be maximized is
− 1
4
N∑
i=1
[
yi − y(xi;a)
σi
]2
= −1
4
χ2 . (2.32)
Therefore maximizing the likelihood that a model gave rise to some data is equiv-
alent to minimizing the χ2 deviation between the model and the data.
2.12.2 Inverse-Hessian method
The Levenberg-Marquardt algorithm interpolates smoothly between two func-
tion-minimization techniques. The first of those, the inverse-Hessian technique,
is based on a Taylor expansion to quadratic order of the function to be minimized.
Taking a point p as the origin, and expanding about that,
χ2(a) ≈ γ − d · a+ 1
2
a ·D · a , (2.33)
where
γ = χ2(p) , dk =
∂χ2
∂ak
∣∣∣∣
p
, Dkl = − ∂
2χ2
∂ak∂al
∣∣∣∣
p
.
Differentiating this gives
∇χ2(a) =D · a− d . (2.34)
If the approximation were exact, this equation would be zero at the value of a
that minimized χ2(a). The approximation becomes increasingly accurate as the
minimum is approached. It can therefore be used as the basis for an iteration,
an+1 = an + δa , (2.35)
76
MODEL FITTING 2.12
where δa is found using equation 2.34:
δa = −D−1 · d . (2.36)
D is referred to as the Hessian matrix of χ2 at the point p. It must be
recalculated at each step of the iteration, as a changes. This calculation is not
entirely trivial. The form of χ2 is given in equation 2.29. Differentiating this with
respect to the component of a indexed by k gives
∂χ2
∂ak
= −2
N∑
i=1
yi − y(xi;a)
σ2i
∂y(xi;a)
∂ak
. (2.37)
Differentiating this, in turn, by the component of a indexed by l gives
∂2χ2
∂ak∂al
= 2
N∑
i=1
1
σ2i
[
∂y(xi;a)
∂ak
∂y(xi;a)
∂al
− (yi − y(xi;a))∂
2y(xi;a)
∂ak∂al
]
. (2.38)
The last term of equation 2.38 is multiplied by yi − y(xi;a). This is simply
the residual between the model and the observed data. If the model fits well, this
will be zero when summed over i. The second derivatives of y(xi,a) can therefore
be neglected, giving
∂2χ2
∂ak∂al
= 2
N∑
i=1
1
σ2i
∂y(xi;a)
∂ak
∂y(xi;a)
∂al
. (2.39)
Conventionally, a matrix α and a vector β are defined as
βk ≡ −1
2
∂χ2
∂ak
, αkl ≡ 1
2
∂2χ2
∂ak∂al
.
This eliminates the factor of 2 in equations 2.37 and 2.39. Equation 2.36 is also
re-written as a set of linear equations,
M∑
l=1
αklδal = βk , (2.40)
77
2 EXPERIMENTAL TECHNIQUES
which are solved for δal.
2.12.3 Steepest-descent method
The steepest-descent method is a much simpler approach than the inverse-Hessian
method. At each iteration step, the gradient of the function is computed. The
parameter vector a is then adjusted to take a step in the direction of the gradient.
In the notation used above,
δal = Aβl , (2.41)
where A is a constant chosen to avoid overshooting the downward slope.
This method is very simple, and does not assume the form of χ2 may be
approximated by a quadratic. It is therefore superior to the inverse-Hessian
method far from the minimum. However, close to the minimum it tends to
converge unacceptably slowly, and the inverse-Hessian method is preferable.
2.12.4 Levenberg-Marquardt algorithm
The Levenberg-Marquardt algorithm combines the inverse-Hessian and steepest-
descent methods, switching between them to exploit their strengths and avoid
their weaknesses.
The first stage is to consider the dimensionality of the constant A in equa-
tion 2.41. χ2(a) is dimensionless, and therefore βk will have dimensions of a
−1
k .
A must therefore have dimensions of a2k. There is exactly one component of the
Hessian matrix α which has dimensions purely in terms of a2k; this is αkk. This
has dimensions of a−2k . α
−1
kk is therefore a suitable value for A. To err on the side
of caution, though, it is divided by a factor ξ, giving
δal =
βl
ξαll
. (2.42)
This may be rearranged to give
ξαllδal = βl , (2.43)
78
MODEL FITTING 2.12
which closely resembles equation 2.40. The inverse-Hessian and steepest-descent
methods may be combined by defining a new matrix α′ such that
α′jj = (1 + ξ)αjj ,
α′jk = αjk (j 6= k) .
(2.44)
Equation 2.40 is used to compute δa, using α′ instead of α. When ξ is large,
the off-diagonal terms of α′ become less significant and the system tends towards
the steepest-descent method. When ξ is small, α′ becomes close to α and the
system tends towards the inverse-Hessian method.
There remains the problem of setting ξ so that it is small when close to the
value of a which will minimize χ2(a), but large when far from that value. The
technique used is as follows.
1. Choose an initial value for ξ. 0.001 is usually sensible.
2. Choose a multiplier κ > 1. 10 is usually sensible.
3. Compute χ2(a).
4. Solve for δa and compute χ2(a+ δa).
5. If χ2(a + δa) ≥ χ2(a) (i.e. this step has probably been away from the
minimum), increase ξ by a factor κ and go back to step 4.
6. If χ2(a + δa) < χ2(a) (i.e. this step has probably been towards the min-
imum), decrease ξ by a factor κ. Update a to use the new value a + δa.
Go back to step 4 with the new χ2(a).
Thus, if the iteration is converging, ξ will decrease and the inverse-Hessian
method, which performs well near the minimum, will come to dominate. If the
iteration is failing to converge, ξ will increase until the steepest-descent method,
which reliably moves the iteration towards a minimum, comes to dominate.
This method will, by default, never terminate. It is necessary to set a halting
condition. If subsequent iterations are changing χ2 very little, then they are
providing no statistically significant improvement in the fit. The algorithm is
79
2 EXPERIMENTAL TECHNIQUES
therefore halted if iterations are changing χ2 by only some small amount. This
is normally a fractional rather than an absolute change. Were it an absolute
change, rounding errors in χ2 could prevent the algorithm from ever terminating.
Fractional changes in χ2 of order 10−5 are commonly used as halting conditions.
2.12.5 Error estimation
Without some measure of uncertainty, the parameters found by model fitting
are of limited use. A wide range of techniques for estimating this uncertainty are
available. The bootstrap method was used in this research (Efron and Tibshirani,
1986; Press et al., 1992). This is a Monte Carlo technique which probes the
probability distribution of the model parameters. It is attractive because it does
not require information on the error associated with each data point, nor does
it make any assumption about the relationship between the parameters and the
model. It is also straightforward to implement. It requires that the data used are
independent and identically distributed.
The bootstrap method operates on a data set y = (x1,x2, . . . ,xn), N ele-
ments long. When subjected to the model-fitting technique of choice (Levenberg-
Marquardt here, but the technique is irrelevant), this produces model parame-
ters a. The bootstrap method consists of generating some large number B of
simulated N -element data sets ysi by randomly drawing, with replacement, N
members from y. Each of these simulated data sets is then subjected to the
same model-fitting technique as the original data set, producing B sets of model
parameters asi . The probability distribution of these simulated model parameters
may be shown to approximate the true probability distribution of the real model
parameters a.
For a model with a single parameter, the standard deviation of the simulated
parameter set approximates the standard error in that parameter. For models
withM parameters, theM -dimensional probability distribution may be projected
onto each of the axes in turn, reducing the problem to that of a single-parameter
model. This is the approach normally used in this research. While simple to
implement, it neglects interactions between the various model parameters. The
simulated probability distribution is discussed qualitatively, where appropriate.
80
REFERENCES
No pre-packaged code for the bootstrap method was available. It was, however,
simple to implement.
References
D. Ascher, P. F. Dubois, K. Hinsen, J. Hugunin, and T. Oliphant. Numerical
python. Technical Report UCRL-MA-128569, Lawrence Livermore National
Laboratory, 2001. http://numpy.scipy.org.
C. S. Coffey and V. F. DeVost. Impact testing of explosives and propellants.
Propellants, Explosives, Pyrotechnics, 20(3):105, 1995.
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83
84
Chapter 3
Materials and microstructure
The following sample materials were used in this research. Figure 3.1 shows these
materials.
• Low density pure ammonium nitrate prills, produced by the prilling pro-
cess described in chapter 1, section 1.1. These are made for use in ANFO
explosives. They were supplied by Orica as approximately 2 mm roughly
spherical pellets, both with and without an organic coating (Lilamin AC-
59P, a blend of anionic and cationic surface-active agents) to reduce caking.
They are referred to as coated or uncoated Orica prills.
• Mixed ammonium nitrate and dolomite pellets, produced using fluid-bed
granulation. These are intended for use as fertilizer, and meet the definition
of “Calcium Ammonium Nitrate” in EEC directive 76/116/EEC. They were
purchased from a garden centre, and take the form of approximately 4 mm,
irregular, rounded pellets. They are referred to as agricultural pellets.
• High density pure ammonium nitrate prills. These are intended for use as
fertilizer. Due to their popularity in improvised explosive devices, they are
difficult to acquire in the UK. They were supplied by [dstl], as approximately
4 mm, irregular, rounded pellets. They are referred to by their brand name,
Nitram.
• Fertilizer pellets containing a variety of fertilizers, not including ammonium
nitrate. These were purchased from a garden centre, and used as an inert
85
3 MATERIALS AND MICROSTRUCTURE
Figure 3.1: Photograph showing the various materials studied. a) Orica explosive
prill. b) Agricultural Calcium Ammonium Nitrate fertilizer pellet. c) Nitram fer-
tilizer prill. d) Westland fertilizer pellet. e) Analytical reagent grade ammonium
nitrate crystals.
mock for the other pellets. They take the form of approximately 4 mm
roughly spherical pellets. They are referred to by their brand name, West-
land Feed-All Plant Food, or simply as Westland.
• Analytical reagent grade ammonium nitrate. This was purchased from
Aldrich. It was supplied as approximately 1 mm crystals, which were typi-
cally ground and sieved before use. This is referred to as ground ammonium
nitrate.
If cavity collapse is an important hot-spot mechanism in these materials, the
microstructure of the materials studied will be important to understanding their
behaviour. The primary tool used to study this was environmental scanning
electron microscopy (ESEM). Some initial studies with X-ray microtomography
were also performed. The distribution of the different components of the West-
land and agricultural materials was studied using energy-dispersive X-ray spec-
86
SAMPLE PREPARATION 3.1
Figure 3.2: ESEM image showing razor damage in uncoated Orica prill. The
damage is visible as the region of parallel stripes to the top left.
troscopy (EDX).
3.1 Sample preparation
To image the interior of prills with ESEM and EDX, they were bisected with a
razor blade. They were then glued, cut side up, to a stud which was held by
the sample stage of the microscope. The glue was a conductive paste known as
silver dag. The razor blade caused considerable damage to the microstructure of
the prill, as shown in figure 3.2. This damage was, however, localized, and easily
identified by the pattern of parallel lines left by imperfections in the razor’s edge.
To image the surface of prills, the initial sectioning was omitted and the intact
prill was glued to the sample stud.
No preparation was required for microtomography.
87
3 MATERIALS AND MICROSTRUCTURE
Figure 3.3: ESEM image of outer surface of uncoated Orica prill.
3.2 Orica prills
Figures 3.3 and 3.4 show ESEM images of the outer surface of uncoated and
coated Orica prills, respectively. The organic coating is visible as a change in the
texture of the surface.
The internal microstructures of coated and uncoated Orica prills are very
similar. This is unsurprising, as the coating is applied after the prilling process,
and the latter is identical for both kinds of prill. Figure 3.5 shows a typical section
of the internal structure of an Orica prill. It has a very high density of voids with
sizes of the order of tens of microns. These voids are large enough to form critical
hot spots when they collapse, as discussed in chapter 1, section 1.2.2. A high
density of such hot spots will spread reaction through the material more quickly
than a low density. This is relevant to the performance of a non-ideal explosive,
such as ammonium nitrate, as the duration of the reaction zone is comparable
to the time taken for reaction to spread. A higher density of critical hot spots is
therefore likely to increase the detonation velocity of the explosive.
The voids in the Orica prills appear to form an interconnected network. This
88
ORICA PRILLS 3.2
Figure 3.4: ESEM image of outer surface of coated Orica prill.
Figure 3.5: ESEM image of body of coated Orica prill, showing network of voids
which permeates the material.
89
3 MATERIALS AND MICROSTRUCTURE
may allow fuel oil to penetrate the prill easily, reducing the average distance
between the oxidizing ammonium nitrate and the fuel. This will in turn increase
the heat of reaction within the reaction zone, again increasing the detonation
velocity.
Figure 3.6 shows a tomographic slice approximately through the centre of an
Orica uncoated prill. It shows the interconnected network of voids, as well as the
expected central void. Figure 3.7 uses ESEM to show the microstructure near that
void in another Orica prill. The plates of ammonium nitrate that make up the
prill appear to have aligned with the wall of the central void. The microstructure
also appears to have become considerably more dense. Figure 3.8 shows a similar
effect at the outer edge of the prill.
Although X-ray microtomography is intended to show the interior structure
of a sample, figure 3.6 shows a few indentations in the exterior. Marked with a
J on figure 3.6, these indentations could, if shocked from the correct direction,
produce a jet (see chapter 1, section 1.2.2.4) directed away from the prill. While
this effect could not cause a hot spot in the originating prill, many prills are often
combined to form a bed. In such a prill bed, the jet from an indentation on one
prill could strike an adjacent prill, creating a hot spot.
Figure 3.7 also shows some pitting of the ammonium nitrate surface. This
could be due to beam damage. To determine the effect of beam damage, the
beam current was increased and the beam was restricted to a small area of the
sample. This would be expected to increase the frequency of any artefacts of
beam damage in that area. No such increase in the density of pits was observed.
This leaves two possible explanations for the pits. First, they could be small
pores present within the prill structure. Second, they could be due to corrosion
of the ammonium nitrate surface by the low-pressure water vapour in the ESEM.
Unfortunately the X-ray microtomograph has insufficient resolution to detect
these pores and discriminate between these two possibilities.
90
ORICA PRILLS 3.2
Figure 3.6: Tomographic slice approximately through the centre of an uncoated
Orica prill, showing the central void (arrowed) and exterior indentations (marked
with J).
91
3 MATERIALS AND MICROSTRUCTURE
Figure 3.7: ESEM image of body of coated Orica prill, showing microstructure
around the central void. The void itself is to the bottom left of the image.
3.3 Agricultural calcium ammonium nitrate pel-
lets
Figure 3.9 shows a typical region of the interior of an agricultural pellet. Here
there are fewer, larger voids, and the voids do not appear to form any kind of
network. It is possible that these “voids” are in fact caused by the material failing
along a line of weakness during sample preparation. The pitting observed in the
Orica prills is also present here. There is also no sign of the plate-like structures
present in the Orica prills, with blocky crystals dominating instead. There was
no sign of a large central void. As the pellets were not produced by prilling, this
is unsurprising.
These pellets contain a mixture of ammonium nitrate and dolomite grains,
to reduce explosive sensitivity. Dolomite is a calcium magnesium carbonate. It
can therefore be distinguished from ammonium nitrate using element mapping:
dolomite contains no nitrogen, and ammonium nitrate contains no calcium or
92
AGRICULTURAL CALCIUM AMMONIUM NITRATE PELLETS
3.3
Figure 3.8: ESEM image of body of coated Orica prill, showing microstructure
near the outer surface of the prill. The main body of the prill is left of the field
of view.
93
3 MATERIALS AND MICROSTRUCTURE
Figure 3.9: ESEM image of the interior of an agricultural calcium ammonium
nitrate pellet.
magnesium. The sizes of the dolomite and ammonium nitrate crystals may be
relevant to the energetic properties of these pellets. Figure 3.10 shows an ESEM
image of a region within an agricultural pellet. Figure 3.11 shows a composite
image of the same region generated using EDX. It shows a pair of dolomite grains
roughly 50µm in size in the centre, surrounded by larger ammonium nitrate
grains. There are a few smaller dolomite grains scattered around the field of
view.
3.4 Nitram prills
Figure 3.12 shows a typical patch of the microstructure of a Nitram prill. The
microstructure appears to be very dense, with very few passages down which fuel
oil or hot gases could pass. The few passages that exist are also considerably
narrower than those in the Orica prills. Although this material was produced by
a prilling process, there was no evidence of a central void. There were, however,
94
NITRAM PRILLS 3.4
Figure 3.10: ESEM image of the interior of an agricultural calcium ammonium
nitrate pellet. This image covers the area shown in the composite EDX image,
figure 3.11.
Figure 3.11: Composite image of the interior of an agricultural calcium ammo-
nium nitrate pellet. The background image was generated using ESEM. The green
pixels indicate that magnesium was detected in that pixel; the blue, calcium; and
the red, nitrogen. Green and blue pixels therefore correspond to dolomite, and
red pixels to ammonium nitrate.
95
3 MATERIALS AND MICROSTRUCTURE
Figure 3.12: ESEM image of the microstructure of a Nitram prill.
several smaller voids of the kind shown in figure 3.13. As with the calcium
ammonium nitrate pellets, these voids could be an artefact of sample preparation.
3.5 Westland pellets
The advertised composition of these pellets is shown in table 3.1. This brand was
chosen because its low nitrate content implied a low ammonium nitrate content,
which in turn suggested it might provide a usable inert mock for ammonium
nitrate prills and pellets. The product as purchased consisted of orange-brown
roughly spherical pellets, roughly 4 mm in diameter, mixed with black irregular
pellets a few millimetres long. These black pellets were ignored, as their shape
rendered them obviously unsuitable mocks for the roughly spherical AN pellets
and prills. Their size and distinctive colour made them easy to remove from any
experiments using this material.
Figure 3.14 shows a patch of the microstructure of one of the larger orange
pellets. It appears to contain very few voids, and there is, again, no evidence of a
96
WESTLAND PELLETS 3.5
Figure 3.13: ESEM image of a void within a Nitram prill. The void is to the
lower left of the image.
connected network of voids. The orange colour of the pellet was due entirely to a
coating applied to the outside; the interior of the pellet was white. This coating
was probably intended to slow the release of nutrients from the fertilizer pellet.
Figure 3.15 shows the spatially-unresolved EDX spectrum for this pellet. This
suggests that the pellet is almost pure P2O5. Close inspection of the Westland
pellets suggested that there were at least three visually distinct types of orange-
brown roughly spherical pellet: the large orange ones studied here, smaller, paler
ones, and smaller, darker ones. Figures 3.16 and 3.17 show the EDX spectra
for the paler and darker small pellets, respectively. These suggest that the paler
pellets consist of potassium carbonate1, and that the darker consist of urea.
Figure 3.18 shows a region near the centre of one of the potassium carbonate
pellets. The large void suggests that the pellet was produced by prilling. There
1Note that, while the advertised composition includes potassium oxide, this is simply an
assaying convention; typically other potassium compounds are substituted (potassium carbon-
ate in this case), and the mass of potassium oxide which would contain an equal amount of
potassium is quoted.
97
3 MATERIALS AND MICROSTRUCTURE
Component Percentage
Nitric nitrogen 1.3%
Ammoniacal nitrogen 8.3%
Ureic nitrogen 5.4%
Phosphorous pentoxide 6.0%
Potassium oxide 30.0%
Water soluble boron 0.025%
Total copper 0.012%
Total iron 0.150%
Total manganese 0.075%
Total zinc 0.150%
Total chlorine < 0.600%
Table 3.1: Advertised composition of Westland Feed-All Plant Food.
Figure 3.14: ESEM image of the microstructure of one of the larger Westland
pellets.
98
WESTLAND PELLETS 3.5
Figure 3.15: Spatially-unresolved EDX spectrum of one of the larger Westland
pellets. Horizontal scale is X-ray energy in keV. Labels indicate the element to
which each peak corresponds.
Figure 3.16: Spatially-unresolved EDX spectrum of one of the small, pale West-
land pellets. Horizontal scale is X-ray energy in keV. Labels indicate the element
to which each peak corresponds.
99
3 MATERIALS AND MICROSTRUCTURE
Figure 3.17: Spatially-unresolved EDX spectrum of one of the small, dark West-
land pellets. Horizontal scale is X-ray energy in keV. Labels indicate the element
to which each peak corresponds.
appears to be very little porosity apart from this void. Figure 3.19 shows a similar
region on one of the urea pellets, again suggesting that the pellet is a prill and
indicating very low porosity.
3.6 Conclusions
• Orica’s explosive prills have a much more open microstructure than agri-
cultural grades of ammonium nitrate, with a network of connected voids.
• This open microstructure allows good mixing between fuel and ammonium
nitrate in ANFO formulations.
• This open microstructure also increases the density of hot spots when Orica
prills are shocked.
• Features of the exterior of Orica prills could lead to inter-prill jetting and
hot spot formation when beds of these prills are shocked.
• The dense microstructures of agricultural ammonium nitrate formulations
increase their mechanical strength and packing density, both desirable fea-
tures in a commercial material.
• EDX is a useful technique for studying the physical distribution of chemicals
in these samples.
100
CONCLUSIONS 3.6
Figure 3.18: ESEM image of the microstructure near the centre of a Westland
potassium carbonate pellet, showing central void.
Figure 3.19: ESEM image of the microstructure near the centre of a Westland
urea pellet, showing central void.
101
3 MATERIALS AND MICROSTRUCTURE
• It is difficult to study internal voids using ESEM, as the sectioning technique
used may introduce false voids.
• The resolution of X-ray microtomography is sufficient to study most, but
not all, internal voids in the prilled samples studied here.
• Westland plant food is likely to be a suitable mock for agricultural grades
of ammonium nitrate. It may be less suitable for explosive grades.
102
Chapter 4
Compaction
Compaction describes the various processes by which a porous material becomes
less porous in response to load. Several of these processes (for example, cavity
collapse) are also hot spot mechanisms. Study of the compaction behaviour of
ammonium nitrate is therefore relevant to understanding hot spots in ammonium
nitrate.
Compaction is generally considered in terms of the relationship between ap-
plied pressure and porosity. Porosity can be expressed either as void fraction (the
fraction of the bulk material that is empty) or as fractional theoretical maximum
density (the density of the bulk material as a fraction of the density the material
would have, were all the voids filled).
Several techniques developed for compressive testing of materials can be mod-
ified to study compaction. This research used a screw-driven instrumented press
to study compaction at a strain rate of roughly 10−4 s−1 and a drop weight to
study compaction at a strain rate of roughly 100 s−1. This research also used a
split Hopkinson pressure bar system (SHPB), with limited success.
4.1 Experimental methods
4.1.1 Quasi-static compaction
A confinement cell of the form shown in figure 4.1 was constructed. Initially,
the end cap and outer jacket were made from PMMA, and the plunger was made
103
4 COMPACTION
55mm
55mm
30mm
10mm
10mm
15mm
13mm
Sample
Outer jacket
End cap
Ball bearing
Plunger
Figure 4.1: Confinement cell used for compaction experiments. All measurements
are approximate. Load is applied vertically to the ball bearing.
from mild steel. Early experiments, though, found that the plunger would plough
into the soft outer jacket, damaging it. This also invalidated the results from that
experiment: the quantity probed was the resistance of PMMA to cutting by mild
steel, rather than the resistance of the sample to compaction. The confinement
cell was therefore remade using stainless steel for all components, which was found
to eliminate the problem.
The ball bearing was intended to ensure there was a single point of contact
between the plunger and the driving force. This would ensure that the loading was
entirely axial. However, it was found that the ball bearing indented the sample-
facing surface used. It was therefore removed for most of these experiments.
For some of the later experiments, the sample-facing surface was replaced
with a synthetic diamond anvil and the ball bearing replaced. This presented the
reverse problem, that the ball bearing deformed. If the ball bearing was rotated
slightly between tests so that the flat surface thus produced did not contact the
diamond anvil, this deformation was found to be repeatable. The deformation
was therefore simply included in the compliance measurement for the system.
Whether data were gathered with or without a ball bearing is indicated in the
results.
104
EXPERIMENTAL METHODS 4.1
The load was provided by an Instron 4466 as described in chapter 2, sec-
tion 2.4. All experiments were performed at a constant loading rate of 0.333mmmin−1,
and a maximum force of 10 kN. The compliance was obtained by placing the con-
finement cell in the Instron without any sample. This ensured that any elastic
deformation of the plunger would be accounted for in the compliance.
A known mass ms of the sample material was poured into the confinement
cell, and the plunger added gently. The plunger was then held in place as the
system was shaken several times, then allowed to fall onto the sample bed. The
protrusion of the plunger from the confinement cell was measured; combined with
the known dimensions of the various components of the confinement cell, this gave
the initial sample length l0.
The filled confinement cell was then placed in the Instron and loaded. The
moment when the recorded force first exceeded the load cell’s noise level of 0.27N
was taken to be the moment of contact between the loading surface and the
plunger. Given that the weight of the plunger was approximately 6N, this seemed
both reasonable and unlikely to lead to undetected deformation of the sample bed.
It was possible to recover the compacted pellet after an experiment. The
length of the recovered pellet was measured with calipers to assess the reliability
of the length measured during the experiment.
4.1.1.1 Analysis
The goal is to relate the pressure applied to the sample to the fractional theoretical
maximum density (TMD) of the sample. This is quite straightforward in the
Instron, which gives the applied force F and the change in length ∆l directly.
Having measured the initial length l0 of the sample, the mass ms of the sample,
and the cross-sectional area A of the confinement, the bulk density ρ of the sample
is given by
ρ =
ms
A(l0 −∆l) . (4.1)
The fractional TMD can be recovered easily if the TMD is known. For Nitram
and Orica prills, this TMD is that of ammonium nitrate, which is well known. For
Westland and agricultural pellets, the TMD was assumed to be the bulk density
after pressing to 50 kN.
105
4 COMPACTION
bar
Output
bar
Sample
jacket
OuterInput
15mm
55mm
13mm 30mm
Figure 4.2: Arrangement of sample and confinement for SHPB compaction test-
ing. All measurements are approximate.
Recovering the applied pressure is simple: the applied force F and the area
A over which that force is applied are known, so the pressure P is given by
P =
F
A
. (4.2)
4.1.2 Higher rate compaction: SHPB
An SHPB system, as described in section 2.7, was used in an attempt to probe
compaction at higher strain rates than the Instron. To achieve this, a tubu-
lar PMMA confinement was constructed and arranged as shown in figure 4.2.
Aluminium bars were used throughout this experiment.
The confinement was filled with a known mass ms of sample material, and the
bars of the SHPB inserted to seal it. The bars were then gently pressed together
as the confinement was rotated, until the sample bed was firmly held in place.
The length of the sample l0 was then measured by aligning calipers with the bar
ends visually. The lensing effect of the PMMA ameliorated the associated parallax
errors, but they still reduced the accuracy of the measurement considerably. This
somewhat unsatisfactory measurement technique was chosen because it did not
permit rearrangement of the sample bed.
106
EXPERIMENTAL METHODS 4.1
The confined sample bed was then subjected to an SHPB test. The degree of
compaction achieved during this test was small, so the experiment was repeated
on the compacted specimen. After a few such tests, the sample was sufficiently
compacted that it held firmly to the walls of the confinement. Its length could
then be measured using the more precise approach of plunger protrusion employed
in the quasi-static tests. Typically, ten tests on a given sample were required to
reach 85–90% TMD.
4.1.2.1 Analysis
Much of the analysis from chapter 2, section 2.7.2 is applicable to the problem
of recovering a pressure-porosity relation from these experiments. Equation 2.19
gives the strain rate in the bed:
ε˙ =
s˙1 − s˙2
l
=
cb(εr − εi + εt)
l
. (2.19)
Just as in conventional materials testing, this can be numerically integrated to
find the sample length l as a function of time, given the measured initial sample
length l0. The bulk sample density ρ, and hence the fractional TMD, can then
be recovered, almost exactly as in section 4.1.1.1.
The force applied to the sample can be found from equations 2.20 and 2.21:
F = 1
2
AY (εi + εr + εt). (4.3)
This analysis differs from the conventional SHPB analysis, though, in that it
assumes the cross-sectional area As of the sample is constant and equal to the bar
diameter, and that the volume Asl of the sample is not conserved. The applied
pressure P is then given by
P = 1
2
Y (εi + εr + εt). (4.4)
4.1.3 Higher rate compaction: drop weight
The short duration of the loading pulse in the SHPB severely limited the degree
of compaction reached in a single experiment. Repeating the SHPB experiments
107
4 COMPACTION
to achieve a higher degree of compaction produced results which were difficult to
interpret and suffered from a low signal-to-noise ratio. The drop weight, with its
much longer loading pulse, seemed an obvious way to avoid this problem.
The confinement cell shown in figure 4.1, without a ball bearing, was used
for these experiments. The sample was weighed, loaded and measured as in
section 4.1.1. The loaded confinement was placed in the drop weight as shown in
figure 4.3. The length of the confinement cell required the removal of the upper
anvil. The confinement cell was held securely by the anvil guide; combined with
the length of the outer jacket and plunger, this reduced the off-axis forces applied
to the system.
Initially, the analysis presented in chapter 2, section 2.6.3 was used to find the
sample length, and the fractional TMD was calculated from this. This analysis
showed that the sample was being compacted past its TMD by quite modest
forces. While this is not impossible for samples of a sufficiently small bulk mod-
ulus K, it was thought far more likely that accumulating errors in the analysis
resulted in the length of the sample being underestimated. A separate method
for measuring the sample length was therefore sought.
4.1.3.1 The line laser
A line laser projects a fan of laser light. With appropriate optics, this can be
used to measure the position of an opaque obstruction, such as the falling mass
in a drop weight. The line laser used in this research was a Power Technology,
Inc. PM01(635-5)L71. This produces a fan whose intensity is uniform along its
length, apart from lobes at the ends which are easily eliminated with a mask. It
was arranged using optical benches as shown in figure 4.4.
When the weight passes through this fan, it will reduce the intensity of the
light falling on the photodiode as shown in figure 4.5. The photodiode will then
produce an output dependent on the position of the weight. Assuming the optics
are ideally aligned, as depicted here, the output will be proportional to the height
of the weight above the bottom of the fan. In practice such alignment was difficult
to achieve.
The quantity to be measured by the line laser is the total length of the sample.
108
EXPERIMENTAL METHODS 4.1
Confinement
Plane of
line laser
Sample
Guide rail
Lower anvil
Anvil guideLoad cell
Impact face
(removable)
Weight
Carriage guide
Figure 4.3: Arrangement of drop weight used for compaction experiments. Not
to scale.
109
4 COMPACTION
Laser Plane of weight Photodiode
Focusing lensCollimating lens
Figure 4.4: Arrangement of line laser used to measure position of drop weight.
Weight passes along the plane indicated, partially blocking the laser beam. Not
to scale.
Laser Weight Photodiode
Collimating lens Focusing lens
Figure 4.5: Effect of adding an obstruction to the line laser system. Obstruction
partially blocks laser beam, reducing light incident on photodiode. Not to scale.
110
RESULTS AND DISCUSSION 4.2
It was therefore aligned to measure in the plane indicated in figure 4.3. To
calibrate the output of the photodiode, the confinement was placed, with no
sample, into the drop weight. The weight was then lowered to rest on it. Various
spacers of known thickness were added between the confinement and the weight,
and the photodiode output corresponding to that thickness recorded.
As with the quasi-static experiments, it was desirable to measure the length of
the pellet after each experiment and compare this against that recorded during
the experiment. However, recovery of the pellet proved more difficult in these
experiments. The plunger protrusion was therefore measured with calipers; as
the plunger did not undergo plastic deformation, this was equivalent to measuring
the pellet thickness.
4.1.3.2 Analysis
The output of the calibrated photodiode directly measures the length l of the
sample, and the load cell measures the force F applied. A pressure-porosity
relationship can therefore be recovered almost exactly as in section 4.1.1.1.
4.2 Results and discussion
4.2.1 Quasistatic results
Figures 4.6 to 4.10 show the pressure-porosity curves obtained by Instron exper-
iments on beds of various prills. There are a few noteworthy features at this
stage.
Most obviously, in several of the experiments, the sample bed is pressed past
its TMD. Measurements of the bed length after the load is removed suggest that
this is due to elastic compression of the bed. Attempts to assess this compression
using these measurements, however, had an uncertainty in excess of 80%. No
attempt was therefore made to correct for it.
After the experiments on agricultural pellet beds, a clear, colourless, oily
liquid was found on the surface of the compacted bed. This is believed to be oil,
added to the pellets to reduce caking.
111
4 COMPACTION
0
10
20
30
40
50
60
70
80
30 40 50 60 70 80 90 100 110
P
re
ss
u
re
/
M
P
a
% TMD
Experiment 1
Experiment 2
Figure 4.6: Pressure-porosity curves from two Instron experiments on uncoated
Orica prill beds. Initial strain rate 4 × 10−4 s−1. Vertical dashed lines indicate
fractional TMD of recovered pellets.
0
10
20
30
40
50
60
70
80
40 50 60 70 80 90 100 110
P
re
ss
u
re
/
M
P
a
% TMD
Experiment 1
Experiment 2
Figure 4.7: Pressure-porosity curves from two Instron experiments on coated
Orica prill beds. Initial strain rate 4 × 10−4 s−1. Vertical dashed lines indicate
fractional TMD of recovered pellets.
112
RESULTS AND DISCUSSION 4.2
0
10
20
30
40
50
60
70
80
40 50 60 70 80 90 100 110
P
re
ss
u
re
/
M
P
a
% TMD
Experiment 1
Experiment 2
Figure 4.8: Pressure-porosity curves from two Instron experiments on agricul-
tural pellet beds. Initial strain rate 4 × 10−4 s−1. Vertical dashed lines indicate
fractional TMD of recovered pellets.
0
10
20
30
40
50
60
70
80
50 55 60 65 70 75 80 85 90 95 100
P
re
ss
u
re
/
M
P
a
% TMD
Experiment 1
Experiment 2
Figure 4.9: Pressure-porosity curves from two Instron experiments on Nitram
prill beds. Initial strain rate 4×10−4 s−1. Vertical dashed lines indicate fractional
TMD of recovered pellets.
113
4 COMPACTION
0
10
20
30
40
50
60
70
80
45 50 55 60 65 70 75 80 85 90 95
P
re
ss
u
re
/
M
P
a
% TMD
Experiment 1
Experiment 2
Figure 4.10: Pressure-porosity curves from two Instron experiments on Westland
prill beds. Initial strain rate 4×10−4 s−1. Vertical dashed lines indicate fractional
TMD of recovered pellets. Ball bearing used for uniaxial loading. Note the poor
repeatability; this is discussed in section 4.2.8.
4.2.1.1 Pressure oscillations in Instron traces
Figure 4.11 shows small-amplitude pressure oscillations in the results of Instron
experiments. Fourier analysis using the NumPy software (Ascher et al., 2001)
shows these oscillations to have a frequency of 0.5143± 0.0010Hz, whether a ball
bearing was used or not. To assess the amplitude of these oscillations, a high-pass
filter with a cut-off frequency of 0.4Hz was applied to the data. The oscillations
were found to follow a linearly growing sinusoid, of the form (At+B) sin(ωt+ θ).
The rate A at which the amplitude of the oscillations increased was found to
be 25.8 ± 1.1Pa s−1. The initial amplitude B was found to be less repeatable:
2.0± 1.0 kPa.
In these experiments, the plunger was driven into the confinement cell at a
constant rate of 5.55µms−1. The contact area between the jacket of the confine-
ment cell and the plunger will therefore increase linearly with time. That the
amplitude of these oscillations increases linearly with time suggests that some
feature of the interaction between the jacket and the plunger may be responsi-
114
RESULTS AND DISCUSSION 4.2
3.66
3.68
3.7
3.72
3.74
3.76
3.78
3.8
3.82
3.84
3.86
2 2.02 2.04 2.06 2.08 2.1
P
re
ss
u
re
/
M
P
a
∆l / mm
Agricultural (no BB, shifted)
Westland (BB)
Figure 4.11: Force oscillations in Instron experiments, showing structure and re-
peatability of oscillations. A ball bearing was used to achieve uniaxial loading for
the experiment on a Westland prill bed, while none was used for the experiment
on an agricultural pellet bed. The latter has been vertically shifted by 2.65MPa
to aid comparison.
115
4 COMPACTION
Figure 4.12: Optical microscope image of epoxy cast of interior of confinement
jacket, showing striations at various length scales. The axis of the confinement
cell is horizontal.
ble for them. From the known frequency of the oscillations, and the constant
speed of the plunger, the features responsible are expected to be separated by
10.79± 0.02µm. Such features would be detectable by optical microscopy.
Examination of the plunger revealed no suitable features. Directly imaging
the interior of the confinement jacket was difficult. Instead, a cast was made
using fast cure epoxy (National Adhesives Bondmaster Double Bubble Mix &
Fix). This was left for twelve hours to cure, then broken from the jacket. The
resulting cast was suitable for microscopy. Figure 4.12 shows an image of this
cast. Features normal to the axis of the confinement cell, formed when the jacket
was produced, are visible. The broad dark lines, spaced irregularly a few hundred
microns apart, were caused by the manual removal of the cutting tool after the
jacket’s fabrication. They are too widely-spaced and irregular to be responsible
for the oscillations observed in this experiment. The finer lines between them were
caused by the cutting tool as it hollowed out the confinement jacket. During this
process it was moved automatically, explaining the regular spacing of these lines.
This spacing was assessed by averaging the rows of the image to find its average
brightness as a function of horizontal position, then taking the Fourier transform
116
RESULTS AND DISCUSSION 4.2
-10
-5
0
5
10
15
20
25
30
35
40
40 45 50 55 60 65 70 75 80 85
P
re
ss
u
re
/
M
P
a
% TMD
Run 1
Run 2
Run 3
Run 4
Run 5
Run 6
Run 7
Run 8
Run 9
Figure 4.13: Pressure-porosity curves from repeated SHPB experiments on a
single uncoated Orica prill bed. Initial strain rate 450 s−1.
of that function. This clearly shows a feature with period 20.3± 1.7µm, double
that of the force oscillations when error is considered. The most likely explanation
for the doubled frequency of the oscillations involves the helical nature of these
features. The plunger could be pushed across the bore of the confinement jacket,
interacting with the peaks of these grooves on diametrically opposed sides of the
bore. This would result in the frequency-doubling seen here.
4.2.2 SHPB results
Figure 4.13 shows the results from a series of nine experiments on an uncoated
Orica prill bed. Each experiment compacts the bed for the subsequent experi-
ment. In principle, these traces should give the complete pressure-porosity curve
for the bed. Examination, however, suggests quite strongly that this is not the
case. Most notably, the end of one trace does not match the beginning of the
subsequent trace.
Several of these mismatches, for example the gap between runs 1 and 2, can
be explained by reloading. The compressive incident pulse will reflect from the
sample as a tensile wave. This tensile wave will reflect from the striker end of the
117
4 COMPACTION
Sample
Dust
jacket
OuterInput
bar
Output
bar
Figure 4.14: Sample geometry encountered in later SHPB compaction runs. Dust
falls from the sample after each experiment. This produces a buffer which pre-
vents the bars from probing the true response of the sample. Not to scale.
bar as a compressive wave, and apply load to the sample after data recording has
ceased. This will result in unrecorded compaction of the sample.
Several other traces, however, overlap the previous trace. The most extreme
example is that of runs 4 and 5, which overlie each other completely. While some
reversible elastic compression will occur, it seems unlikely that this can account
for such a great degree of overlap.
A more likely explanation for this overlap involves dust. At the high strain
rates used in this experiment, some of the prills in the sample bed will fracture,
producing dust. At the end of each run, the compressive force will be removed and
dust at the end of the compacted bed will fall down. Rather than compacting
a cylindrical bed, the next experiment encounters a bed of the form shown in
figure 4.14. The thin layer of loosely-packed dust will extend the initial sample
length, leading to overlap of the kind seen between several of the later runs.
The presence of dust also explains the low applied pressure at the start of the
later runs. Figure 4.15 shows run 9 from figure 4.13. Before about 81% TMD it
demonstrates much less resistance to compaction than would be expected. This
would be consistent with rearrangement of a thin layer of dust.
Figures 4.13 and 4.15 also demonstrate the very low signal-to-noise ratio of
this technique. The low impedance of the beds relative to the bars results in
118
RESULTS AND DISCUSSION 4.2
-10
-5
0
5
10
15
20
25
30
35
40
78 78.5 79 79.5 80 80.5 81 81.5 82 82.5
P
re
ss
u
re
/
M
P
a
% TMD
Figure 4.15: Pressure-porosity curve generated by ninth SHPB experiment on
uncoated Orica prill bed.
a very high fraction of the incident pulse being reflected. This leads naturally
to a poor signal-to-noise ratio after analysis. The conventional solution to this
problem is to use lower-impedance bars. However, these data were taken using
the lowest-impedance solid metal bars available. While it is possible to achieve
lower impedances, by using tubular or polymer bars, dispersion in such bars is
significant.
While none of the problems with this technique are insurmountable, it was
deemed more efficient to employ a technique which circumvented them than to
attempt to overcome them.
4.2.3 Drop weight results
Figure 4.16 shows a pressure-porosity curve generated using the drop weight, with
sample length measured by the line laser. The low spatial resolution of the line
laser is apparent from the large error in the fractional TMD. The output of the
photodiode, even when the laser fan was unobstructed, was roughly 80mV. This
led to considerable digitization error when the output was recorded. However,
the temporal resolution of the technique was far higher than necessary: 10µs in
119
4 COMPACTION
-10
0
10
20
30
40
50
60
70
80
90
100
30 40 50 60 70 80 90 100
P
re
ss
u
re
/
M
P
a
% TMD
Figure 4.16: Pressure-porosity curve from drop weight experiment on uncoated
Orica prill bed. Initial strain rate 170 s−1.
an experiment lasting a few milliseconds. A moving average, with a window size
of five samples, was therefore applied to the data, sacrificing temporal for spatial
resolution.
Figures 4.17 to 4.26 show the pressure-porosity curves generated using this
technique. To vary the strain rate, the height from which the weight was dropped
was varied. All the results presented here were produced by dropping the weight
from either 32 cm or 17 cm above the top of the plunger.
4.2.4 Effect of coating
Figure 4.27 compares pressure-porosity curves for coated and uncoated Orica prill
beds. There is no significant difference between these curves. Figure 4.28 shows
that this is also the case at higher strain rates.
4.2.5 Effect of strain rate
Figure 4.29 shows the effect of a large change in strain rate on the compaction
response of a bed of uncoated Orica prills. The other bed materials demonstrated
120
RESULTS AND DISCUSSION 4.2
-10
0
10
20
30
40
50
60
30 40 50 60 70 80 90
P
re
ss
u
re
/
M
P
a
% TMD
ε˙0 = 130 s
−1
ε˙0 = 120 s
−1
ε˙0 = 105 s
−1
Figure 4.17: Pressure-porosity curves from drop weight experiments on uncoated
Orica prill beds. Weight dropped from 17 cm. Vertical lines indicate fractional
TMD measured after experiment.
-10
0
10
20
30
40
50
60
70
80
90
100
30 40 50 60 70 80 90 100
P
re
ss
u
re
/
M
P
a
% TMD
ε˙0 = 170 s
−1
ε˙0 = 170 s
−1
ε˙0 = 140 s
−1
Figure 4.18: Pressure-porosity curves from drop weight experiments on uncoated
Orica prill beds. Weight dropped from 32 cm. Vertical lines indicate fractional
TMD measured after experiment.
121
4 COMPACTION
-10
0
10
20
30
40
50
60
30 40 50 60 70 80 90
P
re
ss
u
re
/
M
P
a
% TMD
ε˙0 = 120 s
−1
ε˙0 = 130 s
−1
ε˙0 = 100 s
−1
Figure 4.19: Pressure-porosity curves from drop weight experiments on coated
Orica prill beds. Weight dropped from 17 cm. Vertical lines indicate fractional
TMD measured after experiment.
-20
0
20
40
60
80
100
30 40 50 60 70 80 90 100
P
re
ss
u
re
/
M
P
a
% TMD
ε˙0 = 160 s
−1
ε˙0 = 160 s
−1
ε˙0 = 150 s
−1
Figure 4.20: Pressure-porosity curves from drop weight experiments on coated
Orica prill beds. Weight dropped from 32 cm. Vertical lines indicate fractional
TMD measured after experiment.
122
RESULTS AND DISCUSSION 4.2
-10
0
10
20
30
40
50
60
70
30 40 50 60 70 80 90 100
P
re
ss
u
re
/
M
P
a
% TMD
ε˙0 = 130 s
−1
ε˙0 = 120 s
−1
ε˙0 = 140 s
−1
Figure 4.21: Pressure-porosity curves from drop weight experiments on agricul-
tural pellet beds. Weight dropped from 17 cm. Vertical lines indicate fractional
TMD measured after experiment.
-20
0
20
40
60
80
100
120
140
30 40 50 60 70 80 90 100 110
P
re
ss
u
re
/
M
P
a
% TMD
ε˙0 = 150 s
−1
ε˙0 = 170 s
−1
ε˙0 = 170 s
−1
Figure 4.22: Pressure-porosity curves from drop weight experiments on agricul-
tural pellet beds. Weight dropped from 32 cm. Vertical lines indicate fractional
TMD measured after experiment.
123
4 COMPACTION
-10
0
10
20
30
40
50
60
70
80
30 40 50 60 70 80 90 100
P
re
ss
u
re
/
M
P
a
% TMD
ε˙0 = 110 s
−1
ε˙0 = 130 s
−1
ε˙0 = 130 s
−1
Figure 4.23: Pressure-porosity curves from drop weight experiments on Nitram
prill beds. Weight dropped from 17 cm. Vertical lines indicate fractional TMD
measured after experiment.
-20
0
20
40
60
80
100
120
140
160
40 50 60 70 80 90 100 110
P
re
ss
u
re
/
M
P
a
% TMD
ε˙0 = 190 s
−1
ε˙0 = 170 s
−1
ε˙0 = 200 s
−1
Figure 4.24: Pressure-porosity curves from drop weight experiments on Nitram
prill beds. Weight dropped from 32 cm. Vertical lines indicate fractional TMD
measured after experiment.
124
RESULTS AND DISCUSSION 4.2
-10
0
10
20
30
40
50
60
70
80
90
40 50 60 70 80 90 100
P
re
ss
u
re
/
M
P
a
% TMD
ε˙0 = 130 s
−1
ε˙0 = 140 s
−1
ε˙0 = 160 s
−1
Figure 4.25: Pressure-porosity curves from drop weight experiments on Westland
prill beds. Weight dropped from 17 cm. Vertical lines indicate fractional TMD
measured after experiment.
-20
0
20
40
60
80
100
120
140
160
180
30 40 50 60 70 80 90 100 110 120
P
re
ss
u
re
/
M
P
a
% TMD
ε˙0 = 170 s
−1
ε˙0 = 160 s
−1
ε˙0 = 190 s
−1
Figure 4.26: Pressure-porosity curves from drop weight experiments on Westland
prill beds. Weight dropped from 32 cm. Vertical lines indicate fractional TMD
measured after experiment.
125
4 COMPACTION
0
10
20
30
40
50
60
70
80
40 50 60 70 80 90 100 110
P
re
ss
u
re
/
M
P
a
% TMD
Coated
Uncoated
Figure 4.27: Pressure-porosity curves from Instron experiments on coated and
uncoated Orica prill beds. Initial strain rate 4× 10−4 s−1.
-5
0
5
10
15
20
25
30
35
40
45
50
30 35 40 45 50 55 60 65 70 75 80 85
P
re
ss
u
re
/
M
P
a
% TMD
Coated
Uncoated
Figure 4.28: Pressure-porosity curves from drop weight experiments on coated
and uncoated Orica prill beds. Initial strain rates 100 s−1 and 120 s−1 for coated
and uncoated bed respectively.
126
RESULTS AND DISCUSSION 4.2
-10
0
10
20
30
40
50
60
70
80
90
100
30 40 50 60 70 80 90 100
P
re
ss
u
re
/
M
P
a
% TMD
ε˙0 = 170 s
−1
ε˙0 = 4× 10−4 s−1
Figure 4.29: Pressure-porosity curves from both Instron and drop weight experi-
ments on uncoated Orica prill beds, showing enhanced resistance to compaction
at higher strain rate.
a similar response. At a higher rate, the bed demonstrates markedly increased
resistance to compaction.
Compaction necessarily involves rearrangement of material within the bed.
Material must be accelerated to move it from one location to a vacancy, so that
the bed is more efficiently packed. At the quasistatic strain rates achieved in
Instron experiments, the acceleration involved is negligibly small. At the higher
strain rates of the drop weight experiments, though, the force required to provide
this acceleration could conceivably increase the pressure required to achieve a
given degree of compaction. This mechanism is considered in section 4.2.5.1,
below.
Second, the higher-rate experiments caused considerably more fracture in the
bed materials than the quasi-static experiments, as observed in section 4.2.2.
This energy-absorption mechanism is considered in section 4.2.5.2, below.
A smaller relative change of strain rate, as shown in figure 4.30, has very little
effect on the compaction behaviour of a bed. Again, all the sample materials ex-
hibited this behaviour. This lack of change may indicate that the processes which
increase resistance to compaction at higher strain rate are sensitive to relative,
127
4 COMPACTION
-10
0
10
20
30
40
50
60
70
80
90
100
30 40 50 60 70 80 90
P
re
ss
u
re
/
M
P
a
% TMD
ε˙0 = 170 s
−1
ε˙0 = 120 s
−1
Figure 4.30: Pressure-porosity curves from drop weight experiments on uncoated
Orica prill beds, showing no significant variation in bed response with strain rate.
The weight was dropped from a different height for each experiment to vary the
strain rate.
rather than absolute, changes in strain rate. Alternatively, it may indicate that
there exists some threshold strain rate, between those probed with Instron and
with drop weight experiments, above which an additional compaction-resisting
mechanism becomes active. Distinguishing between the two would require addi-
tional experiments at intermediate strain rates.
4.2.5.1 Momentum effects
To within an order of magnitude, the effect of momentum on the resistance of
the bed to compaction can be found by considering the applied pressure required
to accelerate the bed from rest to the plunger velocity over the duration of the
experiment. This gives a stress
σ =
mv
1
4
pid2t
, (4.5)
where m is the bed mass, v is the plunger velocity, d is the plunger diameter,
and t is the duration of the experiment. For a typical drop weight experiment,
128
RESULTS AND DISCUSSION 4.2
m = 1.5 g, v = 3ms−1, d = 13mm, and t = 3ms. This gives σ = 1.1MPa.
This is approximately the difference between high-rate and low-rate com-
paction curves in the early stages of compaction; however, it is also approximately
the size of the uncertainty in the high-rate compaction curves at this stage. As
the compaction process continues, the difference between high-rate and low-rate
compaction curves increases until it is much greater than can be accounted for
by momentum considerations.
4.2.5.2 Fracture effects
The higher-rate experiments caused considerably more fracture in the bed mate-
rials than the quasi-static experiments, as observed in section 4.2.2. While the
drop weight experiments do not suffer from the presence of dust as the SHPB
experiments did, this fracture is an energy-absorption mechanism. The surface
energy for the new surfaces opened by fracture must be supplied by the kinetic
energy of the falling weight. A simple calculation can assess the likely effect of
this fracture on the compaction behaviour of the bed.
First, note that during a compaction experiment nearly all the kinetic energy
of the weight is dissipated in compaction of the bed. This kinetic energy
Ek =
1
2
mu2. (4.6)
In these experiments, the mass of the weight m ≈ 6 kg, and the impact velocity
u ≈ 2m s−1. This gives Ek ≈ 12 J.
To assess the energy dissipated in fracture, assume the sample is composed of
cubic crystallites, exactly filling the sample volume, each of which is bisected by
fracture. The surface energy required to bisect a single such crystallite will be
E = γa2, (4.7)
where γ is the fracture surface energy of the sample material and a is the length of
one side of a crystallite. The total number of crystallites, and hence the number
129
4 COMPACTION
of these fracture events, will be
N =
Volume
a3
=
pild2
4a3
, (4.8)
where l and d are the length and diameter of the confinement cell. This gives the
energy dissipated by fracture in this simple model as
Efrac = NE =
γpild2
4a
. (4.9)
Based on the electron micrographs in chapter 3, 100µm is a reasonable value
for a. For all these experiments, l ≈ 15mm and d ≈ 13mm. Due to the dif-
ficulty of obtaining large, high-quality ammonium nitrate crystals, the fracture
surface energy of ammonium nitrate is unknown. However, several other explo-
sive materials have been investigated (Palmer and Field, 1982) and found to have
fracture surface energies of order 0.1 Jm−2. Taking this as a value for γ gives
Efrac = 2.0mJ. This is a negligibly small fraction, 1.7 × 10−4, of the kinetic
energy Ek of the weight.
4.2.6 Effect of initial porosity
The results from drop weight experiments on beds of Nitram prills, shown in fig-
ures 4.23 and 4.24, appear to be highly variable. This does not seem to be an effect
of the strain rate at which the experiment was carried out. As figure 4.31 shows,
this variability is almost eliminated by simply shifting the pressure-porosity curve
by the initial porosity of the bed.
This indicates that the pressure required to compact these beds depends pri-
marily on the amount of compaction already performed. How near they are to
their theoretical maximum density appears to be of secondary importance. This
is discussed further in section 4.2.10.
This effect is most noticeable in Nitram prill beds, probably because the large,
irregular Nitram prills are capable of a wider range of initial porosities than the
other bed materials. However, it seems to be present to some degree in beds of
all materials.
130
RESULTS AND DISCUSSION 4.2
-20
0
20
40
60
80
100
120
140
160
-20 -10 0 10 20 30 40 50 60
P
re
ss
u
re
/
M
P
a
Increase in % TMD
ε˙0 = 190 s
−1
ε˙0 = 110 s
−1
ε˙0 = 170 s
−1
ε˙0 = 130 s
−1
ε˙0 = 200 s
−1
ε˙0 = 130 s
−1
Figure 4.31: Pressure-porosity curves from drop weight experiments on Nitram
prill beds. All curves have been shifted by their initial fractional TMD, greatly
improving experiment repeatability.
4.2.7 Effect of microstructure
As discussed in chapter 3, Nitram prills and Orica prills are very similar chem-
ically, being composed of nearly pure ammonium nitrate. However, their mi-
crostructure differs, with Nitram prills being considerably denser than Orica prills.
This increases the initial fractional TMD of Nitram beds, which in turn makes
direct comparison of the compaction data shown in figures 4.6 and 4.9 difficult.
Following the reasoning in section 4.2.6, though, figure 4.32 compares the results
of Instron experiments on Nitram and Orica prill beds, shifted by their initial
porosity.
The bed of Nitram prills is clearly more resistant to compaction than the
bed of Orica prills, despite their similar chemical composition. This suggests
that the denser microstructure of Nitram prills must enhance their resistance
to compaction. Denser prills are likely to be mechanically stronger and more
resistant to disintegration; if the primary mechanism by which prill beds undergo
compaction is one of prill disintegration, this alone would explain the Nitram
beds’ increased resistance to compaction.
131
4 COMPACTION
0
10
20
30
40
50
60
70
80
40 50 60 70 80 90 100 110
P
re
ss
u
re
/
M
P
a
% TMD
Nitram
Orica uncoated
Figure 4.32: Pressure-porosity curves obtained by Instron experiments on beds
of Nitram and uncoated Orica prills, showing greater resistance to compaction in
Nitram prill bed. Initial strain rate 4× 10−4 in both cases.
Other results in this section, however, suggest that compaction is dominated
by plastic flow of the bed material. The enhanced resistance of beds of denser
prills to compaction can also be explained in this case. In order for a bed to
compact, material must flow into voids in the bed’s structure. The dense mi-
crostructure of Nitram prills increases the mean distance between material and
void. In order to achieve a given rate of compaction, the rate at which material
enters voids must be constant. Therefore, at a given compaction rate, a bed of
denser prills must contain more material in motion, or that material must be
moving faster, than a bed of less dense prills. This will increase the amount of
energy dissipated by plastic work in the flowing material, and hence the bed’s
resistance to compaction.
4.2.8 Effect of composition
Comparing the compaction properties of agricultural pellets and Westland prills
with those of the pure AN prills is challenging. Both the chemical content and
microstructure of these materials differ from those of the pure AN prills.
132
RESULTS AND DISCUSSION 4.2
Agricultural pellet beds offer very similar resistance to compaction to coated
Orica prill beds (see figures 4.7, 4.8, 4.19, 4.20, 4.21, and 4.22). This is surprising
for a few reasons. First, the ESEM images presented in chapter 3 indicate that
the microstructure of the agricultural pellets is roughly as dense as that of the
Nitram prills, and considerably denser than that of the Orica prills. Following the
reasoning in section 4.2.7, this should give beds of agricultural pellets considerably
greater resistance to compaction than beds of Orica prills. Second, the inert
additive in these agricultural pellets is dolomite, a mechanically stronger material
than ammonium nitrate (Robertson, 1955; Ross et al., 1945). It would be unusual
for the addition of a stronger material to make the resulting composite weaker,
all else being equal.
One possible explanation for the anomalously low resistance a bed of agricul-
tural pellets offers to compaction involves the fluid-bed granulation method by
which the pellets are manufactured. This method is essentially one of controlled
caking, in which dissolution and recrystallization are used to bond individual
crystals together into a pellet. The resulting pellet will be markedly less resistant
to disintegration than the interlocking network of crystals produced by prilling.
This may, in turn, lower a pellet bed’s resistance to compaction. It is also possible
that oil in the pellet beds, noted in section 4.2.1, is relevant. By lubricating the
motion of crystallites over each other, this oil could reduce the bed’s resistance
to compaction.
Beds of Westland prills offer more resistance to compaction than most other
beds studied, and very similar resistance to beds of Nitram prills (see figures 4.10,
4.25 and 4.26). This is not unexpected given the dense microstructure of Westland
prills (see chapter 3). The pressure-porosity traces from experiments on Westland
prill beds also appear to be more variable than those from experiments on other
beds. As figure 4.33 shows, this variability is not eliminated by the shifting
technique used in section 4.2.6. This is probably because of the multiple different
kinds of prill present in a sample of Westland prills, as determined by EDX in
chapter 3. While these prills are easily distinguishable by sectioning and EDX,
they are harder to distinguish by eye. A sample contains roughly a hundred prills,
and therefore the composition of each sample could vary significantly. This would
naturally affect the sample’s compaction behaviour.
133
4 COMPACTION
-20
0
20
40
60
80
100
120
140
160
180
-10 0 10 20 30 40 50 60 70
P
re
ss
u
re
/
M
P
a
Increase in % TMD
ε˙0 = 170 s
−1
ε˙0 = 130 s
−1
ε˙0 = 140 s
−1
ε˙0 = 160 s
−1
ε˙0 = 190 s
−1
ε˙0 = 160 s
−1
Figure 4.33: Pressure-porosity curves from drop weight experiments on Westland
prill beds. All curves have been shifted by their initial porosity. This has not
eliminated variation between experiments.
The shape of the pressure-porosity curves for quasistatic compaction of West-
land prill beds shown in figure 4.10 is interesting. While the curves for other
materials increase smoothly at an ever-increasing rate, bumps and plateaux are
visible on the Westland curve. The most obvious such feature is between 55%
and 60% TMD on the red curve in figure 4.10. This behaviour is likely due to
the heterogeneous nature of the prill bed. Some prills will yield at considerably
lower stresses than others; once the bed is sufficiently compacted that such a prill
is loaded, that prill’s deformation will briefly dominate the compaction behaviour
of the bed until stronger prills support the load.
4.2.9 Compaction modelling
As discussed in chapter 1, section 1.4.2, several models of compaction exist. It is
useful to compare two of the most popular to these data.
134
RESULTS AND DISCUSSION 4.2
0
1
2
3
4
5
6
0 10 20 30 40 50 60 70 80
−
ln
φ
Pressure / MPa
Experiment
Heckel fit
Figure 4.34: Heckel plot for quasistatic compaction of uncoated Orica prills,
showing linear region between 10 and 40MPa. Initial strain rate 4 × 10−4 s−1.
Heckel parameters φ0 = 49.232± 0.014× 10−2, α = 62.059± 0.013GPa−1. Note
severe deviation from linearity near 60MPa; this corresponds to the bed reaching
its ambient-pressure TMD, causing the Heckel model to fail.
4.2.9.1 Heckel model
The Heckel model is described in chapter 1, section 1.4.2.1. It gives the pressure-
porosity relationship
ln
1
φ
= ln
1
φ0
+ αP , (1.10)
where φ is the void fraction of the material at applied pressure P , φ0 is the void
fraction at zero applied pressure, and α is a constant. Plots of ln 1
φ
against P ,
referred to as Heckel plots, typically show a curved region at low applied pressure,
then become straight. This curved region means that the intercept of the straight
region is not, in fact, the observed φ0.
Figure 4.34 shows a typical Heckel plot for quasistatic compaction of a prill
bed. There is clearly a region in which the Heckel model applies. However, it is
just as clear that this is only a limited region of the compaction curve. Figure 4.35
reproduces this plot in the more usual pressure-porosity space.
At higher strain rates, the disagreement between the Heckel model and the
135
4 COMPACTION
-10
0
10
20
30
40
50
60
70
80
90
100
40 50 60 70 80 90 100 110
P
re
ss
u
re
/
M
P
a
% TMD
Experiment
Heckel fit
Figure 4.35: Pressure-porosity graph for quasistatic compaction of uncoated Orica
prills, showing limited agreement of Heckel model with experiment. Initial strain
rate 4× 10−4 s−1. Heckel parameters φ0 = 49.232± 0.014× 10−2, α = 62.059±
0.013GPa−1.
data becomes more pronounced. Figure 4.36 shows a typical Heckel plot for drop
weight compaction of a prill bed. As in the quasi-static case, there exists a region
over which the linear relationship predicted by the Heckel model appears to be
obeyed. However, as figure 4.37 shows, outside this region the predictions of the
Heckel model match the data very poorly.
The Heckel model also predicts that the slope α of the linear region is related
to the yield stress σ0 of the material by
σ0 =
1
3α
. (4.10)
Apart from the Westland prills, all the prills studied here are made mostly or en-
tirely of ammonium nitrate. They should all, therefore, have the same value of σ0
as given by equation 4.10. Table 4.1 collects these measurements. The measured
yield stress clearly varies with prill type and strain rate. This provides further
indication that the Heckel model does not adequately predict the behaviour of
these prill beds.
136
RESULTS AND DISCUSSION 4.2
0
0.5
1
1.5
2
2.5
3
-10 0 10 20 30 40 50 60 70 80 90 100
−
ln
φ
Pressure / MPa
Experiment
Heckel fit
Figure 4.36: Heckel plot for drop weight compaction of uncoated Orica prills,
showing linear region between 40 and 80MPa. Initial strain rate 170 s−1. Heckel
parameters φ0 = 37.4± 1.2× 10−2, α = 10.8± 0.6MPa−1.
0
20
40
60
80
100
30 40 50 60 70 80 90 100
P
re
ss
u
re
/
M
P
a
% TMD
Experiment
Heckel fit
Figure 4.37: Pressure-porosity graph for drop weight compaction of uncoated
Orica prills, showing discrepancy between Heckel model and experiment. Initial
strain rate 170 s−1. Heckel parameters φ0 = 37.4 ± 1.2 × 10−2, α = 10.8 ±
0.6GPa−1.
137
4 COMPACTION
Material Yield stress / MPa
Quasistatic Drop weight
17 cm 32 cm
Orica uncoated 5.1± 0.3 20.7± 1.6 29.4± 0.5
Orica coated 4.61± 0.19 19.2± 0.8 24± 3
Agricultural 4.78± 0.08 19± 2 22± 3
Nitram 9.5± 0.5 24± 3 28± 7
Westland 17± 4 21± 2 21± 7
Table 4.1: Table indicating the yield stress measured by Heckel’s model from
compaction experiments on a range of prills. Apart from the Westland prills,
all prills studied are composed primarily or wholly of ammonium nitrate. This
suggests that they should all have the same yield stress. That they do not is
interesting.
The Heckel model was developed for metal powders, and assumes that the
deformation of the bed is dominated by particles undergoing plastic yield at their
contact points. Its failure to model AN prill beds suggests that the behaviour of
these beds is not so dominated.
4.2.9.2 Kawakita model
The Kawakita model is described in chapter 1, section 1.4.2.2. It gives the
pressure-volume relation
P
e
=
1
AB
+
P
A
, (1.13)
where e is the engineering strain ∆l
l
of the confined uniaxial bed, P is the applied
pressure, and A and B are constants.
Figure 4.38 shows a typical Kawakita plot for quasistatic compaction of a prill
bed. Most Kawakita plots show deviation from linearity at low pressures, and this
bed is no exception. However, usually P
e
is lower than expected at low pressures;
here, it is higher. It is likely that this unusual behaviour is due to the tendency
of ammonium nitrate to “cake”, forming a rigid network of bonds at contact
points between particles. Until the applied force becomes large enough to break
this network, an ammonium nitrate bed will show anomalously high resistance
to compaction. As figure 4.39 shows, this discrepancy does not seriously affect
the agreement between the Kawakita model and experiment in pressure-porosity
138
RESULTS AND DISCUSSION 4.2
0
20
40
60
80
100
120
0 10 20 30 40 50 60 70 80
P e
/
M
P
a
Pressure / MPa
Experiment
Kawakita fit
Figure 4.38: Kawakita plot for quasistatic compaction of uncoated Orica prills,
showing linear relationship above 10MPa. Initial strain rate 4 × 10−4 s−1.
Kawakita parameters A = 63.800± 0.003× 10−2, B = 241.97± 0.09GPa−1.
space.
At higher strain rates, the model still performs quite well. Figure 4.40 shows a
typical Kawakita plot for drop weight compaction of a prill bed. It is interesting
to note that the low-pressure behaviour of the bed is the reverse of that for
quasistatic compaction: the bed offers less resistance to compaction than the
Kawakita model predicts. It is possible that the caking network of the ammonium
nitrate breaks as the bed is achieving mechanical equilibrium, and therefore does
not have any effect on the recorded results. Figure 4.41 shows this plot in pressure-
porosity space. Unsurprisingly, the Kawakita model agrees very well with the
data over the linear region of figure 4.40. Its agreement is less good outside
that range, but it captures the broad shape of the data up to 85% TMD. Above
that the model fails completely to follow the data. This relatively flat region of
the drop weight compaction curve is interesting, and will be discussed further in
section 4.2.10. However, it has little bearing on the model.
The Kawakita model appears to provide good agreement with experimental
data overall. Adams and McKeown (1996) found that the Kawakita parameter B
for a bed of agglomerates was related to the failure stress of a single agglomerate.
139
4 COMPACTION
0
10
20
30
40
50
60
70
80
40 50 60 70 80 90 100 110
P
re
ss
u
re
/
M
P
a
% TMD
Experiment
Kawakita fit
Figure 4.39: Pressure-porosity graph for quasistatic compaction of uncoated
Orica prills, showing good agreement with Kawakita model. Initial strain
rate 4 × 10−4 s−1. Kawakita parameters A = 63.800 ± 0.003 × 10−2, B =
241.97± 0.09GPa−1.
0
20
40
60
80
100
120
140
160
0 10 20 30 40 50 60 70 80 90 100
P e
/
M
P
a
Pressure / MPa
Experiment
Kawakita fit
Figure 4.40: Kawakita plot for drop weight compaction of uncoated Orica prills,
showing linear region from 20 to 80MPa, and small deviations from linearity
outside that range. Initial strain rate 170 s−1. Kawakita parameters A = 61.30±
0.18× 10−2, B = 130± 2GPa−1.
140
RESULTS AND DISCUSSION 4.2
0
20
40
60
80
100
30 40 50 60 70 80 90 100
P
re
ss
u
re
/
M
P
a
% TMD
Experiment
Kawakita fit
Figure 4.41: Pressure-porosity graph for drop weight compaction of uncoated
Orica prills, showing broad agreement between experiment and Kawakita model.
Model reliably over-predicts required pressure from 45% to 60% TMD, and fails
to reproduce tail at 85% to 95% TMD. Initial strain rate 170 s−1. Kawakita
parameters A = 61.30± 0.18× 10−2, B = 130± 2GPa−1.
141
4 COMPACTION
As this closely resembles the situation here, it would be interesting to study
the failure stress of individual prills and compare this with that found from bed
compaction.
4.2.10 Discussion
These compaction data suggest that the compaction behaviour of prill beds is
dominated by viscoplastic flow of material over distances comparable to the prill
diameter.
The enhanced resistance of the bed to compaction at higher strain rates (sec-
tion 4.2.5) may be explained in this light. At higher strain rates, the material
must flow more rapidly to compact, and will consequently dissipate more energy
as plastic work. A similar argument applies to the enhanced resistance to com-
paction of beds of denser prills (section 4.2.5): denser prills require material to
flow further to accommodate compaction.
Finally, and most tellingly, the bed undergoes compaction during the unload-
ing cycle of a drop weight experiment. This is most clearly shown in figure 4.41
as a flat region at high pressure and fractional TMD, which the Kawakita model
does not follow. Here the bed density is increasing at constant applied pressure,
in a clear indication of plastic flow.
4.3 Conclusions and future work
• The short duration of the loading in an SHPB experiment renders the tech-
nique unsuitable for investigation of compaction.
• The addition of a line laser system to a drop weight produces a suitable
technique for investigation of compaction at a strain rate of around 100 s−1.
• A screw-driven instrumented press is a suitable technique for investigation
of compaction at a strain rate of around 10−4 s−1.
• Increasing the rate at which compaction occurs increases the resistance of
the bed to compaction. This is probably due to the increased speed at which
142
CONCLUSIONS AND FUTURE WORK 4.3
the material flows, and thus the increased energy dissipation in plastic work
of the material.
• The increased force required to accelerate the bed material, and the in-
creased degree of fracture seen at higher compaction rates, are unlikely to
have any effect on the bed’s resistance to compaction.
• Further compaction experiments at rates intermediate between those chosen
here would help to quantify this effect.
• The porosity of the bed at the start of a compaction experiment has min-
imal effect on the shape of the pressure-porosity curve produced by that
experiment.
• Beds of ammonium nitrate prills become more resistant to compaction if the
microstructure of the prills is denser. This is probably due to the greater
distance material must move to achieve compaction, and is likely to be true
regardless of the material from which the prills are made.
• Beds of agricultural ammonium nitrate pellets show anomalously little re-
sistance to compaction, compared to beds of ammonium nitrate prills. This
is likely to be due partly to the pellets’ relatively low resistance to disinte-
gration, and partly to the lubricant effect of oil added to the pellets by the
manufacturer as an anti-caking agent.
• Heckel’s compaction model does not agree with the compaction data gath-
ered here.
• Kawakita’s compaction model agrees quite well with the compaction data
gathered here.
• Experiments on the failure stress of individual prills would help assess the
physical relevance of Kawakita’s model to these data.
143
4 COMPACTION
References
M. J. Adams and R. McKeown. Micromechanical analyses of the pressure-
volume relationships for powders under confined uniaxial compression. Powder
Technology, 88:155–163, 1996.
D. Ascher, P. F. Dubois, K. Hinsen, J. Hugunin, and T. Oliphant. Numerical
python. Technical Report UCRL-MA-128569, Lawrence Livermore National
Laboratory, 2001. http://numpy.scipy.org.
S. J. P. Palmer and J. E. Field. The deformation and fracture of β-HMX.
Proceedings of the Royal Society of London, Series A, 383:399–407, 1982.
E. C. Robertson. Experimental study of the strength of rocks. Bulletin of the
Geological Society of America, 66:1275–1314, October 1955.
W. H. Ross, J. Y. Yee, and S. B. Hendricks. Properties of granular and monocrys-
talline ammonium nitrate. Industrial and Engineering Chemistry, 37(11):1079–
1083, 1945.
144
Chapter 5
Light output
As an explosive, ammonium nitrate reacts rapidly and exothermically. Reacting
regions will therefore become very hot, and consequently emit light. The initi-
ation, spread and rate of reaction can then be assessed by following this light
output. In this research, plate impact was used to initiate reaction in beds of
ammonium nitrate a few millimetres thick. High speed photography and photo-
diodes were then used to study the light output of, and hence reaction propagation
within, these beds.
5.1 Granular beds
The light emitted during shock-induced reaction of granular beds of ammonium
nitrate was studied. To facilitate this, a copper-fronted, glass-backed cell was
constructed to hold a bed a few millimetres thick. A shock was introduced to
the bed by plate impact of a copper-fronted flyer on the copper front of the cell.
The glass window permitted light to leave the back face of the bed; this light was
then captured and analysed. To contain harmful debris, the experiments took
place in a sealed chamber. This chamber was evacuated to 100 mbar to prevent
a dangerous overpressure developing.
145
5 LIGHT OUTPUT
4.3mm
11.9mm
Make trigger
Toughened glass plate
Pressure gauge
3.5mm
0.92mm
Steel holding ring
Nitrile O-ring
Copper front plate
41
.7
m
m
49
.6
m
m
51
.0
m
m
53
.1
m
m
62
.0
m
m
18.3mm
AN bed
Figure 5.1: The sample cell used for holding granular beds in shock-induced
reaction experiments. The projectile approaches the cell from the left.
5.1.1 Cell and flyer design
Figure 5.1 shows the design of the cell used in these experiments. The O-ring
was a nominally 3.5mm thick, 13
4
′′
(44.45mm) internal diameter nitrile O-ring,
supplied by Polar Bearings. The toughened glass plate was a nominally 19mm
thick, 50mm diameter toughened float glass anvil, supplied by W. H. Constable
& co. The steel holding ring was machined from 21
2
′′
(63.5mm) bright mild steel
bar. The copper front plate was made from copper sheet using a 2′′ (50.8mm)
circular punch.
5.1.1.1 Triggering
The make trigger serves to provide a triggering pulse to the various instruments
directed at the experiment. It consists of layered one thou (25.4µm) copper foil
and double-sided Sellotape. The arrangement is shown in figure 5.2. The make
146
GRANULAR BEDS 5.1
Sellotape
+5V
0V
Copper
Surface
Figure 5.2: Design of make trigger used in these experiments. Not to scale.
trigger is connected to the input of a delay generator, set to a 5V trigger level and
a 500Ω input impedance. The delay generator is configured to be triggered by a
negative-going edge. When the projectile strikes the cell, it will cut through the
make trigger. Its copper front plate will form an electrical connection between the
two copper layers of the make trigger, causing the potential difference between
them to fall. This in turn will trigger the delay generator, the outputs of which
may be connected to instruments. It was found that arranging the make trigger
with the positive foil away from the sample cell, as shown in figure 5.2, reduced
the triggering system’s sensitivity to electrical noise.
5.1.1.2 Pressure gauge
While the make trigger is suitable for triggering instrumentation, it is valuable
to know the time at which a shock wave is introduced to the AN bed. The
delay between the projectile striking the make trigger and the trigger activating is
subject to considerable uncertainty. To eliminate this uncertainty, a thin pressure
gauge was placed between the AN bed and the copper front plate. The design of
the pressure gauge was based on that used in Obara et al. (1995), and is shown
in figure 5.3.
The sensing element of this pressure gauge is a small disc of polyvinylidene
difluoride (PVDF). PVDF is a piezoelectric polymer. It was biaxially stretched
and poled in a strong electric field to form a thin film with a well-defined piezo-
electric direction normal to that film. This film was then coated with an alloy
of gold and platinum, to provide a good electrical connection to the PVDF film.
The resulting sheet was 25µm thick and provided by Piezotech S. A. S. Discs
were cut from this sheet by pressing a 1.6mm diameter circular cutter through
the sheet and into a layer of paper. The same cutter was used to produce a
147
5 LIGHT OUTPUT
Conductive
tape
Front Side
Copper
PVDF
boPET
Figure 5.3: Design of pressure gauge used in these experiments. The PVDF is
the sensing element. Not to scale. The gauge was assembled on double-sided
Sellotape (not shown).
Vout
50Ω
100 nFVin
Figure 5.4: Circuit diagram of charge integrator for PVDF gauges.
hole in a 24µm sheet of biaxially orientated polyethylene terephthalate (boPET;
Mylar). This sheet served to insulate the two copper legs of the completed gauge
from each other. 3M conductive double-sided tape was applied to one side of this
pierced sheet. The PVDF disc was aligned with the hole by feel and eye, with
its positive face arranged so that it would face the shock in the completed gauge.
The conductive tape was then used to affix this sheet to the legs, cut from one
thou (25.4µm) copper foil.
As a piezoelectric sensor, the PVDF gauge will produce a charge dependent on
the pressure applied to it. To convert this to a voltage, the gauge was connected
to a charge integrator (figure 5.4). The output of this integrator was recorded
with an oscilloscope.
148
GRANULAR BEDS 5.1
12
.7
m
m
19
.6
m
m
16
m
m
16mm
20mm
2.0mm
Flyer plate Sabot
Figure 5.5: Flyer design used in these experiments. The flyer plate strikes the
front plate of the sample cell shown in figure 5.1. The flyer plate is made of
copper, and the sabot is made of POM.
5.1.1.3 Flyer
The design of the flyer is shown in figure 5.5. The purpose of the sabot is to
carry the flyer plate along the gun barrel, and ensure that the plate strikes the
sample cell at normal incidence. The flyer plate was made from 14 gauge copper
sheet using a circular punch. This punching left a burr on the flyer plate, which
interfered with attaching the plate to the sabot. The burr was therefore filed off.
The sabot was made from 20mm diameter polyoxymethylene (POM; acetal) rod,
using a lathe. It was hollowed out to reduce the flyer’s mass, and hence increase
the impact velocity achievable with the small gun. The production technique left
a nipple on the front face of the sabot, which was removed using a knife. The
flyer plate was attached to the sabot using double-sided Sellotape. To align the
plate with the sabot axis, the assembled flyer was placed, plate down, on a flat
surface. Firm pressure was then applied to the rear of the sabot by hand. This
was found to align the plate and sabot to within 3 milliradians. Each flyer had a
mass of 6.3± 0.3 g.
149
5 LIGHT OUTPUT
5.1.1.4 Bed construction
To obtain consistent particle sizes, the raw bed material was ground and passed
through a pair of sieves of different mesh size. The fraction passing through the
coarser sieve and retained by the finer was used to manufacture the bed. The
sizes of these two sieves were used to identify the distribution of particle sizes
in a given bed. For most of these experiments, 150–212µm particles were used.
These were ground using a pestle and mortar. Some experiments used 20–40µm
particles. These were ground by rotating a cylinder containing the bed material
and steel ball bearings at 60 r. p. m. for eight hours.
The sample cell was constructed, face down, with the make trigger, glass
plate, and rear holding ring left off. The ground and sieved bed material was
poured into the sample cell, and the glass plate and rear holding ring attached.
This process will reduce the porosity of the bed. The confinement cell shown
in chapter 4, figure 4.1 was used to assess this effect. The free-pour density of
150–212µm AN was found to be 54± 4% TMD. Inserting the plunger with force
similar to that used to apply the back plate of the sample cell increased the
density to 58.2± 1.3% TMD.
5.1.1.5 Projectile velocity
The muzzle of the gun was equipped with a pair of light gates, 14.955±0.009mm
apart. This allowed the velocity of the projectile to be measured. The velocity
is expected to depend on the mass of the projectile and the firing pressure of the
gun. While the barrel diameter and length will also affect the projectile velocity,
these are constant throughout this research.
Neglecting friction between the barrel and projectile, the pressure in the target
chamber, and the volume of the barrel, the muzzle velocity of the projectile will
be
u =
√
2PAl
m
, (5.1)
where P is the firing pressure, A is the cross-sectional area of the barrel, m is the
projectile mass, and l is the barrel length. As noted above, A and l are constant
150
GRANULAR BEDS 5.1
400
450
500
550
600
650
700
750
16000 18000 20000 22000 24000 26000 28000 30000 32000
V
el
o
ci
ty
/
m
s−
1
√
P
m
/ Pa
1
2 kg−
1
2
Experiment
m = 0.0233± 0.0003
c = 0
m = 0.0175± 0.0005
c = 142± 12
Figure 5.6: Projectile velocity calibration graph for the Cavendish small gun,
using helium propellant and the 19.6mm barrel. Two best-fit lines, one of the
form y = mx and one of the form y = mx+ c, are shown.
between shots, giving
u ∝
√
P
m
. (5.2)
Figure 5.6 shows the projectile velocity as a function of this mass-adjusted
firing pressure
√
P
m
. The large y-intercept of the best-fit line suggests that this
model’s assumptions are not wholly justified. However, it serves as a useful aid
when selecting firing pressures.
5.1.1.6 Fibre and mirror mounts
Capturing and analysing the light leaving the rear of the sample cell required
that mirrors and fibres be mounted on the cell. Two designs of mount were used,
depending on whether framing photography was required. The first is shown
in figure 5.7. This design was used with framing photography. A large batch
of these mounts was cut from a single 9.8mm PMMA sheet using a bandsaw
and appropriately angled jigs. The hole was then drilled using a pillar drill and
another suitable jig. In use, the surface labelled ∗ was attached to the rear of
151
5 LIGHT OUTPUT
9.8mm 2.2mm
45◦
45◦
∼ 20mm
†
∗
9.
8
m
m
∼ 7.5mm
‡
Figure 5.7: Fibre mount design when framing photography is required. Measure-
ments prefixed with ∼ are approximate, and vary between individual mounts.
The surface labelled † is attached to the rear of a front-silvered mirror using
double-sided tape. The surface labelled ∗ is similarly attached to the rear steel
holding ring of the sample cell. An optical fibre is inserted through the hole
labelled ‡, and held in place by friction.
a front-silvered mirror using double-sided tape. The surface labelled † was then
similarly attached to the rear steel holding ring shown in figure 5.1, so that the
mirror would reflect light from the AN bed. This allowed the camera to be placed
out of the line of fire of the gun. An optical fibre was then fed through the drilled
hole labelled ‡, and held in place by friction with the hole walls. This arrangement
pointed the end of the fibre at the centre of the AN bed, to improve light capture.
The second design of mount was used to affix fibres to the rear of the toughened
glass plate at normal incidence. When the mount was placed at the centre of the
plate, this improved light capture. However, this arrangement was incompatible
with framing photography. Figure 5.8 shows this design. The mounts were, again,
made in a batch. The starting material in this case was 1
2
′′
(12.7mm) square-
section aluminium-alloy bar. A groove was cut in this using a rotary side-and-face
cutter, and individual mounts were cut from the resulting grooved bar using a
bandsaw. Holes to accept the fibres were then drilled. The surfaces labelled ∗
were affixed to the back plate using double-sided tape, and fibres fed into the
holes. As with the first mount design, friction held the fibres in place.
152
GRANULAR BEDS 5.1
2.2mm3
8
′′
1
2
′′
1 4
′′
∼
1.
3
m
m∼
1.
3
m
m
1 2
′′
∼ 6.5mm
∗
∗
Figure 5.8: Fibre mount design when framing photography is not required. Mea-
surements prefixed with ∼ are approximate, and vary between individual mounts.
The surfaces labelled ∗ are attached to the rear of the glass back plate of the sam-
ple cell using double-sided tape.
5.1.2 Framing photography
Framing photography using an Ultranac 501 camera provided a useful sanity
check for other diagnostics and models. Figure 5.9 shows framing photography
typical of this experiment. It clearly shows an induction time of 2.5 ± 0.5µs
between the arrival of the shock in the bed and the peak light output of the bed.
This is followed by a decay of similar duration, leaving very little light output
6µs after impact. Saturation of the detector, and the low temporal resolution of
the technique, leads to large uncertainty in the induction and decay times.
During the decay, structure can be seen within the emitted light. The most
obvious is a ring around the bright region. This is most likely to be due to the
construction of the projectile (figure 5.5). The hollow centre of the projectile will
cause the centre of the flyer plate to be released more quickly, and to a lower
pressure, than the edges of the flyer plate. This in turn will lead to regions of
the bed near the edges of the flyer plate experiencing high pressure for longer
than regions near the centre. This could easily cause those regions to emit more
brightly than the centre in the late stages of the experiment. In frame 8, when
this structure is first visible, the ring is 11.0± 0.5mm in diameter. Subsequently
it grows, reaching 15.3 ± 0.8mm across in frame 11, the last frame in which
153
5 LIGHT OUTPUT
Figure 5.9: Framing photography of 150–212µm ammonium nitrate granular bed.
Camera was triggered 110±30 ns before shock entered bed. 700±40m s−1 impact
velocity, 460 ns exposure time, 40 ns interframe time. Taken using Ultranac 501
camera; frame order as shown in figure 2.19.
154
GRANULAR BEDS 5.1
it is sufficiently visible to measure. This suggests that the flyer is spreading
laterally under the applied load. Recovered flyers suggest that this interpretation
is reasonable: they are broader and thinner than pristine flyers, and dished with
the concave face towards the direction of travel.
It is useful to consider the very similar experiments of Proud et al. (2003).
By performing the experiment on a bed of 150–212µm sugar, they were able to
eliminate triboluminescence and compression of trapped gas as significant sources
of light in this experiment. Despite increasing the exposure time to 10µs and
employing image intensification, very little light was seen from the sugar bed.
Sugar was chosen for its strong triboluminescence and similar density to AN.
The density of sugar is 1587 kgm−3, versus 1720 kgm−3 for AN.
The speckled pattern visible from frame 8 onward is interesting. A similar
pattern was seen in Proud et al., but could not be distinguished from noise due
to the image intensifier. Here, comparison with frame 2 suggests that this speckle
is not an artefact of the photographic film on which the image is recorded. It
has a scale of the order of a few hundred microns. This is comparable to that of
the ammonium nitrate grains in the bed; it seems reasonable to assume that the
grains are the source of this structure. It is also apparent that features of this
structure are persistent between frames. The loop highlighted with a red circle in
frame 9 is the most obvious of these. It is visible from frame 7 to frame 11. The
presence of such apparent structures is unremarkable: a random speckle pattern is
likely to contain apparent structures simply because human perception is adapted
for detecting such structures. Their persistence, however, is interesting for two
reasons. First, it provides a method for assessing lateral expansion of the bed.
The distance between two features which remain visible from frame 9 onward can
be measured, and changes in this distance can be converted to a strain. This
indicates a strain of 0.11 ± 0.07 between frames 9 and 10, with no measurable
lateral motion from frame 10 onward. Second, it indicates that no new light
sources are appearing; each emitting region is simply dimming.
The behaviour of this speckle pattern suggests that it is caused by the re-
gion of reacting material surrounding a hot spot. Each hot spot will cause the
surrounding material to react, emitting light. The reaction then ceases, and the
emitting region dims over time. A success of this technique is that the resolution
155
5 LIGHT OUTPUT
is good enough to see the individual grains of AN, and observe the late evolution
of these emitting regions. However, the early evolution of the emitting region is
obscured. In the earlier frames, unreacted AN between the emitting regions and
the camera prevents observation. Once the AN at the face of the sample towards
the camera starts to react, the light output from previously ignited AN saturates
the camera, preventing observation. Proud et al. avoided this problem by using a
0.4mm thick bed. By using very short exposure times and image intensification,
they were able to observe the formation and growth of emitting regions. Prema-
ture decay of some emitting regions, believed to be associated with a sub-critical
hot spot, was also observed.
The 501 camera’s variable exposure time could be used to reduce the un-
certainty in measurements of induction and decay time. However, the temporal
resolution would be limited by the time-dependent light output of the reacting
bed. Optimizing temporal resolution, and avoiding saturation of the detector,
would require careful synchronization between the bed’s light output and the
camera’s exposure. In light of the variable delay between the make trigger and
the arrival of the shock in the bed, this synchronization would depend heavily on
luck. The higher dynamic range, finer temporal resolution, and longer recording
time of a photodiode would avoid these problems.
5.1.3 Photodiode measurements
Optical fibre was used to connect an ET-2030 photodiode to the rear of the sample
cell. This allowed the photodiode to be placed outside the capture chamber in
which the experiment took place, thereby reducing the chance that part of the
sample cell would damage the photodiode. The fibre used was cut with a razor,
then polished using diamond polishing paper. The fibre was polished first on
30µm paper, then on 3µm, 1µm and finally 0.3µm paper. The output of the
photodiode was connected to a digital oscilloscope. Initially, the oscilloscope
input was set to 1MΩ impedance. The resulting photodiode traces indicated
that bright light was emitted for tens of microseconds following the impact. This
is inconsistent with the framing photography described above, which shows the
light fading several microseconds after impact.
156
GRANULAR BEDS 5.1
Further investigation found that this result was due to the photodiode’s in-
ternal capacitance of 146.4 ± 0.7 pF discharging across the oscilloscope’s input
impedance. The resulting RC circuit had a time constant of 146.4 ± 0.7µs, and
was therefore unsuitable for following an experiment whose duration is of the
order of microseconds. The oscilloscope’s input impedance was therefore set to
50Ω. This reduced the time constant of the system to 7.32 ± 0.08 ns, which
was suitable for this experiment. The sensitivity of the photodiode-oscilloscope
system was, however, reduced. This is because the photodiode produces a cur-
rent dependent on the intensity of the light incident on it, while the oscilloscope
records the potential difference produced when that current flows through its in-
put impedance. The limiting factor on the system’s sensitivity will be electrical
noise in the oscilloscope, which is not affected by the input impedance. Therefore,
reducing the oscilloscope’s input impedance by a factor of 20,000 will reduce the
system’s sensitivity by a like factor. This reduced sensitivity was, however, still
adequate for this experiment.
Figure 5.10 shows the photodiode output corresponding to the framing pho-
tography shown in figure 5.9. This shows the same growth and fall noted in
section 5.1.2. The induction time can be seen from the photodiode trace to be
2.49 ± 0.05µs. This agrees with that found using framing photography, but is,
as expected, more precise. The photodiode output clearly shows the decay of
the light output. It also shows a bump in that decay from 3.5 to 4µs, in which
the light output rises again. Comparison with framing photography indicates
that this coincides with the formation of a bright ring structure, also discussed
in section 5.1.2. Interestingly, this bump was not observed in every experiment –
of the 52 granular-bed experiments for which photodiode data were gathered, 11
showed this bump.
The decay of the bed’s light output is best illustrated by figure 5.11. This
shows the same data as figure 5.10, but with a logarithmic y axis. That the
decay follows a straight line when so plotted strongly suggests that the decay is
exponential. Such a decay may be written
V (t) = V0e
− t
τ , (5.3)
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5 LIGHT OUTPUT
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 1 2 3 4 5 6
O
u
tp
u
t
/
V
Time / µs
Figure 5.10: Photodiode output of 150–212µm ammonium nitrate granular bed
under 700± 40m s−1 flyer impact. Superimposed images show framing photogra-
phy of the experiment. Horizontal extent of image indicates duration for which
shutter was open. Experiment also shown in figure 5.9.
where V0 is the output at t = 0 and τ is the time constant of the decay. V0 is of
little interest: it depends upon the precise time at which the trigger is activated,
and the efficiency with which light from the experiment is gathered. τ , however,
is purely a property of the light output of the experiment. Taking logarithms,
equation 5.3 may be re-written as
lnV = lnV0 − 1
τ
t . (5.4)
Using equation 5.4, V0 and τ were found by a linear regression fit to the
photodiode output data. Errors were assessed by using the bootstrap method.1
For this experiment, V0 = 125± 5V and τ = 547± 4 ns.
The induction time, and the time constant of the decay, were investigated for
150–212µm granular beds of ammonium nitrate.
1See chapter 2, section 2.12.5
158
GRANULAR BEDS 5.1
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0 1 2 3 4 5 6
ln
V V
p
Time / µs
Figure 5.11: Photodiode output of 150–212µm ammonium nitrate granular bed
under 700 ± 40m s−1 flyer impact, with logarithmic y axis. Vp is the maximum
output recorded in the experiment. In this case Vp = 1.18V. Experiment also
shown in figures 5.9 and 5.10.
5.1.3.1 Induction time
The induction time is the delay between the arrival of the shock wave in the bed,
and the peak of the bed’s light output. Given the observations in section 5.1.2,
this peak marks the time at which ignition of fresh AN grains ceases. Several shots
on cells of the form shown in figure 5.1, filled with 150–212µm ground ammonium
nitrate, were performed. The induction time was taken as the time between the
first rise of the front PVDF gauge output and the peak of the photodiode output.
Uncertainty in the induction time was due to the duration of these features.
Figure 5.12 shows the results of these experiments. At impact velocities be-
low 450m s−1 no light was seen; this could be because the granular bed did not
react. Alternatively, the slow-moving projectile may not have activated the make
trigger while the bed was reacting. Recovered debris from these low-velocity ex-
periments indicated that the projectile did not breach the copper front shim. As
the make trigger must be cut through to activate, this lends some support to
the latter suggestion. Rather than re-design the trigger mechanism, the exper-
159
5 LIGHT OUTPUT
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
450 500 550 600 650 700 750 800
In
d
u
ct
io
n
ti
m
e
/
µ
s
Impact velocity / m s−1
Figure 5.12: Induction times for granular ammonium nitrate beds, impacted at
a range of velocities.
iment was restricted to velocities in excess of 450m s−1. The results show that
the induction time falls as the projectile velocity increases. This is intuitively
attractive: supplying more shock energy to the explosive causes it to react more
quickly. However, there are competing explanations to consider.
The variable induction time may be due to variation in the duration of the
pressure pulse in the bed. This pulse is caused by the collision between the flyer
plate and the front shim of the sample cell. When the pulse ends, it is possible
that ignition of AN ceases. After this time, the light output of the bed will decay
as reacting regions cool down and are not replaced. This will produce the peak
and decay seen in photodiode records of the bed’s light output.
If lateral release waves from the edge of the flyer plate are neglected, this
problem is analytically tractable. Figure 5.13 shows the X-T diagram for the
experiment. The flyer approaches the front shim from the left at a velocity ui.
On impact, shock waves are propagated forward into the shim and backward
into the flyer. After some time ts the forward-going shock reaches the interface
between the shim and the AN bed. This is the start of the pressure pulse in the
bed. The release wave reflected from this interface is neglected in this treatment,
160
GRANULAR BEDS 5.1
t = ts
t = ts + tp
t = ti
t = 0
lsli
void
Figure 5.13: X-T diagram of the collision between the flyer plate and the front
shim of the experimental cell, in the lab’s frame of reference. li is the initial
thickness of the flyer plate. ls is the initial thickness of the front shim. ts is the
time taken for the shock to travel through the front shim, while ti is the time
taken for the shock to travel through the flyer plate. tp is the duration for which
the back face of the front shim is held at its shock pressure, and is to be found.
because its properties depend strongly on the unknown properties of the reacting
AN bed. Meanwhile, the backward-going shock from the impact is travelling
towards the rear face of the flyer. It reaches this face at a time ti, and is reflected
as a release wave. This travels through the flyer and front shim, which are now
in a uniform shocked state. The release wave reaches the bed-shim interface at a
time ts + tp, and ends the pressure pulse in the AN bed.
Copper is widely used in plate impact experiments, and therefore its shock
properties are well known. For this problem, the relationship between shock
velocity and the particle velocity behind the shock is required. This is simply
us = c0 + sup (5.5)
where us and up are the shock and particle velocities respectively, and c0 and s
161
5 LIGHT OUTPUT
are experimentally-determined constants. The speed with which a release wave
propagates through the material is also required. During this experiment, the
copper experiences pressures of approximately 10GPa. In this regime, the release
wave travels at approximately the longitudinal sound speed cl of the material.
For copper, c0 = 3.940 km s
−1, s = 1.489 (Asay and Shahinpoor, 1993) and cl =
4.76 km s−1 (Marsh, 1980). These references did not state any errors; for purposes
of error analysis the error has been assumed to be 5 in the least significant figure.
The velocity of the interface between the flyer and the shim is also required. Since
the two are of the same material, a simple symmetry argument shows this to be
uj =
1
2
ui (5.6)
where ui is the impact velocity.
Calculating the travel times of the various shock and release waves in this
system is simplified by careful choice of reference frames. The simplest reference
frame in which to consider a shock is the rest frame of the unshocked material.
The simplest reference frame in which to consider a release wave is the rest frame
of the unreleased material. Since the velocities of these frames relative to the lab
frame are not relativistic, travel times calculated in these frames remain valid in
the lab frame without further transformation.
Bearing this in mind, the time ts for the forward-going shock to reach the
bed-shim interface is easily calculated. The rest frame of the unshocked material
is also the lab frame, so the particle velocity behind the shock will simply be the
interface velocity uj. Equation 5.5 then gives the velocity of this shock
uf = c0 +
1
2
sui . (5.7)
This shock must cross the thickness ls of the front shim, giving
ts =
ls
c0 +
1
2
sui
. (5.8)
The time ti for the backward-going shock to reach the rear of the flyer is only
slightly more challenging. In this case the simplest frame is the rest frame of the
162
GRANULAR BEDS 5.1
incoming flyer, which is moving at ui in the positive x direction. In this frame,
the particle velocity behind the shock will be
up = ui − uj
= 1
2
ui (5.9)
in the negative x direction. By equation 5.5, the shock velocity in this frame will
then be
ub = c0 +
1
2
sui (5.10)
in the negative x direction. In this frame of reference, the rear of the flyer is
stationary. The shock must therefore cover a distance li to reach the back face,
giving
ti =
li
c0 +
1
2
sui
. (5.11)
The time tr for the release wave to travel from the rear of the flyer to the
bed-shim interface requires a little more thought. All the material through which
the release travels will have been shocked to a uniform state. This material will
be travelling at 1
2
ui in the positive x direction, providing an obvious frame of
reference in which the release will simply travel at cl through stationary material
to reach the stationary bed-shim interface. However, the material will also have
been compressed by the shock waves. The distance the release must travel will
therefore not be simply ls+ li. The new thickness of the material can most easily
be found by considering the shock wave itself. Both the shocks considered in this
problem have travelled at a velocity us into stationary material of some density
ρ0, accelerating it to a velocity
1
2
ui and compressing it to a density ρs. Conserving
mass across the shock gives
ρ0us = ρs
(
us − 12ui
)
ρ0
(
c0 +
1
2
sui
)
= ρs
(
c0 +
1
2
(s− 1)ui
)
from equations 5.5 and 5.6
ρ0
ρs
=
c0 +
1
2
(s− 1)ui
c0 +
1
2
sui
. (5.12)
This compression ratio immediately gives the distance the release wave must
163
5 LIGHT OUTPUT
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
450 500 550 600 650 700 750 800
In
d
u
ct
io
n
ti
m
e
/
µ
s
Impact velocity / m s−1
Figure 5.14: Induction times for granular ammonium nitrate beds, impacted at a
range of velocities. The calculated loading pulse duration for each impact velocity
has been subtracted from these induction times.
travel, and hence the time it takes doing so. This time
tr =
li + ls
cl
· c0 +
1
2
(s− 1)ui
c0 +
1
2
sui
. (5.13)
The total pulse duration tp can thus be found:
tp = ti + tr − ts
=
li
c0 +
1
2
sui
+
li + ls
cl
· c0 +
1
2
(s− 1)ui
c0 +
1
2
sui
− ls
c0 +
1
2
sui
=
1
c0 +
1
2
sui
(
li + ls
cl
[
c0 +
1
2
(s− 1)ui
]
+ li − ls
)
. (5.14)
Figure 5.14 shows the effect of applying this correction to the experimental
results. Induction time still clearly decreases with increasing impact velocity.
This is inconsistent with the hypothesis that the variation in induction time is due
to reaction ceasing when the loading pulse ends. Were that the case, subtracting
the duration of the loading pulse from the induction time should give a constant.
164
GRANULAR BEDS 5.1
Make trigger
Pressure gauges
3.5mm
0.92mm
Steel holding ring
Nitrile O-rings
AN bed
41
.7
m
m
11.9mm
4.3mm
Copper plates
51
.0
m
m
53
.1
m
m
62
.0
m
m
Figure 5.15: Arrangement of modified AN cell used to measure the travel time of
pressure waves through the bed.
Another possible explanation for this variability in induction time is based
on the opacity of the ammonium nitrate bed. The bed contains many AN-air
interfaces which will scatter light. This will attenuate light emitted far from the
glass back plate of the bed. If this attenuation is significant on the scale of the
bed’s thickness, only the region of the bed close to the back plate will contribute
to the recorded light output. The pressure wave introduced by the impact will
take some time to reach this observable volume. This delay between impact
and observation may be sufficient to explain the variation in induction time with
impact velocity.
The relationship between impact velocity and travel time in a reactive granular
bed is difficult to model. However, the sample cell used in these experiments may
be modified to measure this relationship directly. Figure 5.15 shows the modified
sample cell. This design was chosen to avoid the need to produce new parts,
while also eliminating the glass back-plate.
Figure 5.16 shows the results of this experiment. The vertical error bars are
165
5 LIGHT OUTPUT
1.6
1.8
2
2.2
2.4
2.6
2.8
450 500 550 600 650 700
T
ra
ve
l
ti
m
e
/
µ
s
Impact velocity / m s−1
Experiment
m = −0.0027± 0.0012µs (m s−1)−1
c = 3.7± 0.7µs
Figure 5.16: Travel time of pressure wave through 3.5mm thick 150–212µm AN
granular bed. A best-fit line of the form y = mx+ c is plotted. Note uncertainty
on fit parameters.
based on uncertainty in the individual measurement. The broad spread of the
data suggests that there is some source of shot-to-shot variation for which this
does not account. The best-fit line shown in figure 5.16 was used to calculate
the travel time for each experiment in figure 5.12. This travel time was then
subtracted from the induction time.
Figure 5.17 shows the result of this subtraction. The large error bars are
due to uncertainty in the best-fit line calculated in figure 5.16. Subject to these
errors, however, the induction time is entirely eliminated by accounting for this
travel time. This suggests that the opacity of the granular AN bed is significant
on the 3.5mm length-scale of this experiment, and that this opacity is sufficient
to explain the observed dependence of induction time on impact velocity.
The large uncertainty in the travel time of pressure waves through the bed
weakens this conclusion considerably. One simple way to reduce this uncertainty
would be to take additional measurements of the travel time. A more elegant
approach, though, would be to take travel time and induction time data simul-
taneously. This would eliminate the shot-to-shot variation seen in figure 5.16.
However, the sample cell would need to be re-designed. Both light output and
166
GRANULAR BEDS 5.1
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
450 500 550 600 650 700 750 800
In
d
u
ct
io
n
ti
m
e
/
µ
s
Impact velocity / m s−1
Figure 5.17: Induction times for granular ammonium nitrate beds, impacted at a
range of velocities. The calculated travel time of the input pressure wave through
the bed has been subtracted from these induction times.
back-face pressure should be measured close to the loading axis. The PVDF
gauges used in this experiment are opaque. The gauges would therefore need
to be slightly re-designed to place the sensing element as close as possible to
the edge of the copper. By careful placement of this modified PVDF gauge and
the photodiode fibre, off-axis effects on both the light output and the rear-face
pressure reading would be minimized.
5.1.3.2 Decay time
After the light output has reached a maximum, it decays. As is observed in
chapter 6, the bed is emitting as a black body during this time. This decay can
be interpreted as the hot regions cooling, and therefore emitting less light.
To probe this, all emitting regions were taken to be identical. This allowed
a single emitting region to be considered: the output of the bed as a whole will
be a constant multiplier of the output of this one region. Further, this emitting
region was taken to be defined by an instantaneous release of heat at a single
point in the bed: a delta function in space and time. This was assumed to cool
167
5 LIGHT OUTPUT
by conduction into the surrounding bed. It is immediately apparent that such a
delta function does not, in fact, exist. The infinite spike in temperature would
be quite obvious. However, the heat profile of any small heat source cooling by
conduction will quickly converge to that of a delta function. The time taken for
this convergence can be accounted for by shifting the initial delta function in
time.
The heat equation for an isotropic material (Carslaw and Jaeger, 1947) is
∂T
∂t
=
K
ρCp
∇2T , (5.15)
where K is the conductivity of the material, ρ is its density, and Cp is its spe-
cific heat capacity at constant pressure. Given the problem, spherical polar co-
ordinates are appropriate. In this system,
∇2T = 1
r2
∂
∂r
(
r2
∂T
∂r
)
+
1
r2 sin θ
∂
∂θ
(
sin θ
∂T
∂θ
)
+
1
r2 sin2 θ
∂2T
∂φ2
. (5.16)
Since the problem is spherically symmetrical, only the term in
∂T
∂r
is non-zero.
This gives
∂T
∂t
=
K
ρCp
· 1
r2
∂
∂r
(
r2
∂T
∂r
)
. (5.17)
The appropriate solution to this (Carslaw and Jaeger, 1947) is
T = Q
(
1
2piζt
) 3
2
exp
(
− r
2
2ζt
)
, (5.18)
where Q is the heat energy in the emitting region and ζ =
2K
Cpρ
. The peak
temperature in the emitting region, T (r = 0), will therefore be
T (r = 0) =
(
1
2piζ
) 3
2 1
t
3
2
∝ t− 32 . (5.19)
Assuming the light is black-body radiation, the power radiated will follow the
Stefan-Boltzmann law
W˙ = AεσBT
4 , (5.20)
168
GRANULAR BEDS 5.1
where A is the emitting area, T is the temperature of the black body, ε is its
emissivity, and σB is the Stefan-Boltzmann constant
2pi5k4B
15c2h3
. The expected form
of T (t) is given by equation 5.19. The variation of area with time must also be
considered. The Gaussian component of equation 5.18 provides a means to do
so. Its characteristic length
〈r2〉 = 2ζt ∝ t, (5.21)
and thus we expect A ∝ t, dimensionally. The power radiated will therefore be
W˙ ∝ t ·
(
1
t
3
2
)4
= t−5 . (5.22)
Assuming the photodiode captures some constant fraction of the light output,
the photodiode output should follow this t−5 dependence. It is worth noting that
this is not the exponential decay expected based on figure 5.10. Figure 5.18 shows
that there is a significant difference between these two models. The agreement is
superficially convincing, but the model is clearly failing to capture some of the
physics of the situation.
There are three possible explanations for this. The first is that the tempera-
ture distribution in an emitting region takes non-negligible time to relax to the
gaussian of equation 5.18. Probing this would require a more sophisticated heat-
conduction model, or direct observations of the evolution of an emitting region.
The second possibility is that the emitting regions of the bed are initiated at a
range of times, with a range of thermal energies. In this case, the photodiode
output could be used to test a statistical model for the distribution of initiation
times and energies. Finally, the release of heat in each emitting region may not
be instantaneous: as the energy comes at least partially from chemical reaction
of AN, the kinetics of that reaction may distort the light output.
5.1.4 Particle size
Some work was performed on 20–40µm ground AN, produced as described in
section 5.1.1.4. This production technique was somewhat unsatisfactory. The
only grinding balls available were steel, and the resulting rust contamination was
sufficient to impart a visible pink colour to the ground AN. Figure 5.19 shows
169
5 LIGHT OUTPUT
0.01
0.1
1
0 0.2 0.4 0.6 0.8 1
N
or
m
al
iz
ed
p
h
ot
o
d
io
d
e
ou
tp
u
t
Time / µs
Photodiode
A(t− t0)−5
Figure 5.18: Log-linear plot comparing light-emission model with average nor-
malized photodiode output, and showing failure of the model. Note systematic
under-prediction between 0.08 and 0.4µs and after 0.9µs, and over-prediction be-
fore 0.08µs and between 0.4 and 0.9µs. A = 0.136±0.003, t0 = −0.632±0.003µs.
170
GRANULAR BEDS 5.1
Figure 5.19: Framing photography of 20–40µm ammonium nitrate granular bed.
Camera was triggered 274±17 ns before shock entered bed. 683±3m s−1 impact
velocity, 460 ns exposure time, 40 ns interframe time. Taken using Ultranac 501
camera; frame order as shown in figure 2.19.
framing photography of a 700± 40m s−1 impact on a 3.5mm bed of this ground
AN, and figure 5.20 shows the corresponding photodiode output.
These differ from those of a 150–212µm AN bed (figures 5.9 and 5.10) in a
few interesting respects. The light output of the 20–40µm bed is much dimmer
than that of the 150–212µm bed. It also rises and falls more quickly. The ring
structure seen in the late stages of the light output of both beds is sharper in the
finer bed.
All these features are consistent with increased bed opacity in the finer bed.
A more opaque bed will permit light from a narrower region near the glass back
plate to escape and be detected. This will result in the dimmer light output
observed. As the reaction front travels through the bed at a finite speed, sampling
a narrower region of the bed will naturally lead to the faster rise and fall in light
171
5 LIGHT OUTPUT
-10
0
10
20
30
40
50
60
70
80
0 1 2 3 4 5 6
O
u
tp
u
t
/
V
Time / µs
Figure 5.20: Photodiode output of 20–40µm ammonium nitrate granular bed
under 683±3m s−1 flyer impact. Superimposed images show framing photography
of the experiment. Horizontal extent of image indicates duration for which shutter
was open. Experiment also shown in figure 5.19.
output observed. The light emitted by the ring structure will not be scattered as
far by a more opaque bed, explaining its sharper definition.
There is also a faint speckle pattern with a length scale of approximately
200µm visible in frame 5 (2–2.5µs). Comparison with early frames of figure 5.9
suggests that this may be due to the grain of the film used here.
The increased opacity of the finer bed is readily explained. Ammonium nitrate
is a transparent material. The opacity of the bed will therefore be dominated by
scattering from AN-air interfaces. This effect could in principle be quantified and
compared to experiment. However, both the time and voltage resolution of the
photodiode data obtained in this experiment are too coarse for this to be useful.
Given the shortcomings of the milling technique used, further experiments were
not performed. This could prove an interesting avenue of study in future.
172
PRILL BEDS 5.2
5.2 Prill beds
While granular beds are a useful system to experiment on, ammonium nitrate
is normally used in prill form. Some experiments were therefore performed on
prill beds. Prills are typically a few millimetres in diameter. The small-scale
experiments performed here only contain tens of prills. The conclusions drawn
from them are therefore mostly qualitative.
Beds of each prilled formulation were investigated using framing photography
and photodiode output. Beds containing a single reactive prill were investigated
using the same techniques. The bulk of these beds consisted of Westland prills.
These served as an inert mechanical mock for the reactive prills. The various prill
types used here are described in chapter 3.
5.2.1 Bed construction
The cells and flyers used in these experiments were nearly identical to those
used for granular beds. See section 5.1.1 for details. The bed itself was a sin-
gle randomly-packed layer of prills. Prills with diameters within 0.1mm of the
thickness of the bed were chosen by sieving through circular holes. As with the
granular bed, the sample cell was constructed, face down, with the make trigger,
glass plate, and rear holding ring left off. The sieved prills were poured in and
spread to form a uniform layer. The remainder of the cell was then assembled.
The nitrile O-rings used as spacers for the granular bed experiments were
3.5mm thick. Most of the prilled formulations here had a significant population of
prills with diameters within 0.1mm of this thickness. For these formulations, the
cell design used for granular bed experiments was not modified. The Orica prills,
however, were smaller. A ring was punched from 2mm copper sheet. This was
used instead of the nitrile O-ring for experiments on Orica prills. This fabrication
technique was chosen simply to avoid the delay associated with ordering different
O-rings.
173
5 LIGHT OUTPUT
Figure 5.21: Static image of the bed of 3.4–3.6mm Westland prills impacted in
figure 5.22. To scale with, and taken with the same apparatus as, figure 5.22.
5.2.2 Westland prills
Westland Feed-All Plant Food was used as an inert mechanical mock for the reac-
tive prills studied here. It was necessary to check that these prills were in fact inert
in this experiment. A bed consisting entirely of Westland prills was constructed.
Figure 5.21 shows the bed. This bed was then impacted at 700 ± 40m s−1, and
framing photography and photodiode readings recorded. Figure 5.22 shows the
framing photography, and figure 5.23 shows the photodiode output.
A very small amount of light is emitted by the bed. The light is dim enough
that it could be due to compression of trapped gas or triboluminescence of frac-
turing prills, rather than reaction of the bed material. Most relevantly, it is much
dimmer than the light emitted by the reactive prills studied later in this chapter.
Therefore Westland prills are suitable for use as an inert mock of reactive prills
in these experiments.
5.2.3 Orica prills
There was no discernible difference between coated and uncoated Orica prills.
They are therefore discussed together.
5.2.3.1 Reactive prill bed
The bed shown in figure 5.24 was shocked; figure 5.25 shows the resulting framing
photography. Due to the small size of the Orica prills, the beds used here were
2mm thick. Many of the features observed when impacting granular beds are
repeated here; see section 5.1.2 for discussion. Note that the work of Proud et al.
(2003) discussed there is not directly comparable to these results: that work was
174
PRILL BEDS 5.2
Figure 5.22: Framing photography of a bed of 3.4–3.6mm Westland prills im-
pacted at 700 ± 40m s−1. 460 ns exposure, 40 ns interframe time. Taken using
Ultranac 501 camera; frame order as shown in figure 2.19. Static image of bed,
to the same scale, shown in figure 5.21.
-6
-4
-2
0
2
4
6
8
10
12
-1 0 1 2 3 4 5 6 7 8 9
P
h
ot
o
d
io
d
e
ou
tp
u
t
/
m
V
Time / µs
Figure 5.23: Photodiode output from a bed of 3.4–3.6mm Westland prills im-
pacted at 700± 40m s−1. Experiment also shown in figure 5.22.
175
5 LIGHT OUTPUT
Figure 5.24: Static image of the bed of 1.9–2.1mm Orica prills impacted in
figure 5.25. To scale with, and taken with the same apparatus as, figure 5.25.
concerned solely with granular beds. The behaviour observed here deviates in a
few interesting ways from that of a granular bed.
First, frames 1 and 2 (0–1µs) show the undisturbed prill bed, with light be-
tween the prills. There are two possible explanations for this. The first is that
reaction starts in the inter-prill region; the light from this initial reaction would
therefore be visible only in the voids between prills. The second involves the finite
speed with which the shock wave travels through the prill bed. Figure 5.26 shows
the situation when the shock has recently entered the prill bed. Reaction will
only start in those regions into which the shock wave has passed. The unshocked
material will neither move nor react, and will block light from the reacting ma-
terial. The camera will therefore observe these unshocked prills, with light from
the reaction visible only between them.
The second interesting feature is the structure of the light output in frames
7 onward (3µs onward). Heterogeneity of a scale comparable to that of the
prills persists. This suggests that at least substantial fragments of the prills, and
possibly entire prills, survive the reaction. This in turn indicates that the reaction
of the bed was arrested before consuming all available AN. This is unsurprising:
the scale of the experiment is much smaller than the critical diameter for pure
AN, and the loading pulse provided by the flyer is of very short duration. It is
also interesting to note that the prills are not entirely reduced to dust over the
timescale studied here. Inspection of the debris after the experiment, however,
found no prill fragments.
Finally, frame 12 (5.5–6µs) appears to show limited re-ignition of the central
region of the bed. This is rather surprising. While the unreacted prill fragments
provide reagent, there is no clear energy input to initiate reaction at this time.
176
PRILL BEDS 5.2
Figure 5.25: Framing photography of a bed of 1.9–2.1mm Orica prills impacted
at 700± 40m s−1. 460 ns exposure, 40 ns interframe time. Taken using Ultranac
501 camera; frame order as shown in figure 2.19. Static image of this bed, to the
same scale, shown in figure 5.24.
Copper front plate
Shock front
Reacting Unreacted
Glass back plate
prills prills
Figure 5.26: Illustration of a prill bed through some of which a shock has travelled.
177
5 LIGHT OUTPUT
0
100
200
300
400
500
0 1 2 3 4 5 6 7
P
h
ot
o
d
io
d
e
ou
tp
u
t
/
m
V
Time / µs
Figure 5.27: Photodiode output of a bed of 1.9–2.1 mm Orica prills impacted at
700 ± 40m s−1. Superimposed images show framing photography of the experi-
ment. Horizontal extent of image indicates duration for which shutter was open.
Experiment also shown in figure 5.25.
The most likely explanation is that these unreacted fragments have “cooked off”:
the uniformly high temperature in the sample cell has caused them to decom-
pose exothermically. Cook-off events are associated with a random delay, which
would explain why this re-ignition is not directly associated with any event in the
experiment. It would also make a detailed study exceedingly difficult.
Figure 5.27 shows the output of the photodiode attached to this experiment.
The reignition seen in frame 12 of the framing photography is visible here. It
causes a slight rise in the photodiode output between 5.5µs and 7µs.
The overall form of the photodiode output is similar to that produced by a
granular bed. It rises to a peak, then decays. The bump in the decay at 3µs is
similar to that seen in some of the granular bed experiments. However, here it is
much more pronounced.
The photodiode output also shows a plateau in light output between 1.2 and
1.7µs. The photodiode viewed the experiment through an optical fibre. It was
thought that the narrow acceptance angle of this fibre might be responsible for
both this plateau and the bump observed at 3µs. Therefore a polished fibre was
178
PRILL BEDS 5.2
-20
0
20
40
60
80
100
120
0 1 2 3 4 5 6 7
P
h
ot
o
d
io
d
e
ou
tp
u
t
/
m
V
Time / µs
Figure 5.28: Photodiode output from bed of 1.9–2.1mm Orica prills, impacted
at 700± 40m s−1. Fibre end roughened with 30µm polishing paper.
deliberately roughened using 30µm polishing paper, and the experiment repeated.
Figure 5.28 shows the result of this experiment. While the duration of the plateau
is reduced, it is still clearly present. Its brightness relative to the bed’s peak light
output is increased. The bump during the light’s decay also remains, and has
also become relatively more prominent.
One possible interpretation of the plateau region is that the light output
indicates the rate at which AN is reacting. Under this interpretation, the plateau
indicates that the reaction temporarily reaches a steady state. This may be due
to some feature of the kinetics of AN. However, in that case it would also be
observed in granular beds of AN. Figure 5.10 shows that this is not the case. A
more likely explanation involves the structure of the prills themselves. As the
framing photography shows, significant fragments of the prills survive until late
in the experiment. Their structure is therefore likely to be relevant in the early
stages of the experiment. In chapter 3 it was seen that the outside of Orica prills
is denser and less permeable than the interior. It is likely, therefore, that reaction
will spread through the interior of the prills before the outside begins to react.
During this time, the opaque outer shell of the prill will block the light produced
179
5 LIGHT OUTPUT
by the reacting interior. The light output will therefore reach a plateau. This
will end when the shells of the prills either react or break to such an extent that
they no longer block light.
5.2.3.2 Single reactive prill
A bed of inert Westland prills was constructed, and a single Orica prill placed
near the centre. The size distributions of the two prill types had very little
overlap. Therefore the bed was constructed of 3.4–3.6mm prills: it was easier to
find a single oversized Orica prill than to find enough undersized Westland prills
to fill an entire bed. It was possible that the prill thus selected would have an
unusual microstructure. Unfortunately tomographic imaging was not available at
the time, and bisecting the prill to investigate would have rendered it unsuitable
for this experiment.
This bed is shown in figure 5.29. The red circle highlights the Orica prill.
The dark mark on its surface was added with a pen to identify it. The bed was
then impacted at 700± 40m s−1. Figure 5.30 shows framing photography of this
impact. The light output seen here gives some support to the conclusions of
section 5.2.3.1. Apart from a little light in frame 1 (0–0.5µs), the bed is dark
until frame 4 (1.5–2µs). By frame 5 (2–2.5µs) the light output is clear, and there
are visible fragments where the Orica prill has disintegrated. The light output in
frame 4 may be due to break-up of the side of the prill away from the camera. In
frames 4–8 (1.5–4µs), light can clearly be seen away from the reacting prill. This
may be due to hot, glowing gas from the prill spreading through the inert prill
bed. Alternatively it may be because the light from the prill illuminates distant
parts of the inert prill bed.
Figure 5.31 shows the output of the photodiode attached to this experiment.
The initial plateau is again reduced in size but increased in prominence, relative to
figure 5.28. The plateau appears to coincide with frame 4. This further supports
the explanation that the plateau is due to the outer layers of the prill blocking
light from reaction within the prill. The bump during decay of light output is
also present.
180
PRILL BEDS 5.2
Figure 5.29: Two copies of a static image of the bed of 3.4–3.6mm Westland
prills, with a single 3.5mm Orica prill, impacted in figure 5.30. To scale with,
and taken with the same apparatus as, figure 5.30. Position of Orica prill is
highlighted with a red circle in the right-hand image.
Figure 5.30: Framing photography of a bed of 3.4–3.6mm Westland prills, with
a single 3.5mm Orica prill, impacted at 700 ± 40m s−1. 460 ns exposure, 40 ns
interframe time. Taken using Ultranac 501 camera; frame order as shown in
figure 2.19. Static image of bed, to the same scale, shown in figure 5.29.
181
5 LIGHT OUTPUT
-10
0
10
20
30
40
50
60
70
80
0 1 2 3 4 5 6
P
h
ot
o
d
io
d
e
ou
tp
u
t
/
m
V
Time / µs
Figure 5.31: Photodiode output of a bed of 3.4–3.6 mm Westland prills, with a
single 3.5 mm Orica prill, impacted at 700±40m s−1. Superimposed images show
framing photography of the experiment. Horizontal extent of image indicates
duration for which shutter was open. Experiment also shown in figures 5.29 and
5.30.
182
PRILL BEDS 5.2
16mm
Sabot
19
.6
m
m
16
m
m
∼
2
m
m
Prill
20mm
∼ 1mm
Figure 5.32: Sabot design for direct impact experiments. Measurements prefixed
with ∼ are approximate, and vary between individual sabots. Sabot is made from
POM.
5.2.3.3 Direct impact
Several features of these experiments involved activity on the side of the prill
facing the shock wave. The design of the experiment prevented observation of
this part of the prill. Therefore a direct impact experiment was designed, with
the intention of circumventing this limitation.
In this experiment, the flyer was re-designed. The copper front plate was
removed, and the acetal sabot re-designed to carry a single prill. A small inden-
tation was made in the centre of the front face using the tip of a drill bit. The
prill was then held in this indentation with a small amount of silicone vacuum
grease. This was chosen to avoid potentially introducing an organic fuel to the
prill. Figure 5.32 shows the re-designed sabot, with prill attached. The sample
cell was also re-designed: the bed and front face were removed, so that the prill
would strike the glass rear plate directly. As in the prill and granular bed exper-
iments, framing photography and a photodiode were directed at the rear of the
glass plate.
Triggering the diagnostics presented a problem. The addition of an opaque
make trigger would prevent direct observation of the prill’s impact, and thus
defeat the point of the experiment. Instead, a time-of-flight computer was used.
This took input from the light gate used to measure the flyer’s velocity. It then
183
5 LIGHT OUTPUT
waited some pre-defined multiple of the delay between the light gates’ activation
before triggering the apparatus. The distance between the light gates is known,
and the projectile is assumed to travel at constant velocity for the duration of
the experiment. The position of the projectile when the apparatus is triggered
can then be controlled by modifying the multiplier.
This experiment produced extremely bright light, saturating the photodiode
and camera. Further investigation showed that this light coincided with the
arrival of the acetal sabot at the glass target plate. The light was found to be
independent of the presence or otherwise of the prill, or the silicone grease used
to attach the prill. It was therefore interpreted as being due to compression of air
trapped between the flyer and the glass target plate. There was no visible light
coinciding with the arrival of the prill at the target plate. This implies that the
prill disintegrated or was dislodged before impact. In the absence of any way to
reduce the loading experienced by the prill, the technique was abandoned.
5.2.3.4 Drop weight experiments
The early stages of the reactive prill bed experiment showed light between the
prills (see figure 5.25). One possible explanation for this light was inter-prill
reaction. It was therefore interesting to search for possible mechanisms for this
reaction. Any such mechanism will involve material being ejected from the prill’s
surface. A drop weight experiment, with imaging through the sample, was used to
search for this ejected material. The general technique is described in chapter 2,
section 2.6.5.
Experiments were performed on 1.9–2.1mm uncoated Orica prills, mixed with
5.8% diesel by weight. The fuel was added to sensitize the prills. The loading
in the drop weight experiments was slower than that in the plate impact exper-
iments, and reached a lower maximum pressure. It was hoped that sensitization
would allow the prills to react under these circumstances.
Figure 5.33 shows the arrangement of prills initially used. In similar experi-
ments (Field et al., 1982; Me-Bar and Walley, 1994) on liquid propellants, a jet
formed from the interface of drops 1 and 2. This jet then struck drop 3, causing
reaction. This geometry was chosen because jets were thought to be the most
184
PRILL BEDS 5.2
1
2
3
Figure 5.33: Sample geometry for jet-based initiation. The jet forms between
objects 1 and 2, and follows the path indicated by the arrow.
likely cause of inter-prill initiation. If a jet forms, this geometry should make
it visible. Figure 5.34 shows the result of this experiment. No jet is visible:
the prills simply disintegrate. There is no evidence of reaction or melting in the
resulting mass of prill fragments. The open microstructure of the prills will dif-
fuse the shock wave resulting from contact between the prills. The geometry of
the experiment also differs from that of a liquid-drop experiment: the prills are
spherical at the time of contact, while the liquid drops are cylindrical. Both these
factors will inhibit jet formation in this experiment.
The fashion in which the prills disintegrate is interesting. In the early stages
(42–105µs) cracks appear to propagate from the edges towards the centre of
the prills. These separate the prills into a small number of cohesive fragments,
which separate (140–245µs) and are crushed (350µs onward). This behaviour is
due to the inhomogeneity of the prills. As figure 3.6 in chapter 3 shows, Orica
prills consist of a porous centre surrounded by a dense outer shell. In uniaxial
compression, this shell will experience tensile loading around its circumference (as
seen by the camera). Referring again to chapter 3, figure 3.6, we see numerous
indentations in the circumference. These could initiate crack growth in the outer
shell, resulting in the form of disintegration seen here.
It is possible that limited reaction took place, but was obscured by unreacted
AN. This was tested in a second experiment. Maraging steel anvils were used,
as described in chapter 2, section 2.6. The arrangement of prills shown in fig-
ure 5.35 was placed between the anvils. A layer of heat-sensitive film was placed
in contact with the prill layer, as shown in figure 5.36. Heat-sensitive film is a
transparent plastic that undergoes a permanent colour change when heated above
some temperature (Swallowe et al., 1986). This temperature depends on the time
185
5 LIGHT OUTPUT
−7µs 0µs 21µs 42µs
63µs 105µs 140µs 175µs
245µs 350µs 490µs 518µs
546µs 595µs 735µs 847µs
Figure 5.34: Photography through sample in drop weight experiment on 2mm
ANFO prills taken using C4 camera. Time is relative to first detectable deforma-
tion of prills. No jetting or reaction is visible.
186
PRILL BEDS 5.2
Figure 5.35: Arrangement of prills for heat-sensitive film experiment.
Upper anvil
Prills
Lower anvil
Heat-sensitive
film
Figure 5.36: Layout of heat-sensitive film experiment. The anvils are those of the
drop weight. Not to scale.
for which the film is heated, but is always at least 180 ◦C. As seen in chapter 1,
section 1.1.2, AN will thermally decompose at this temperature. Figure 5.37
shows the film recovered from this experiment. The darkening of the film clearly
shows that it has been heated. There must therefore have been limited reaction
in the mass of disintegrated AN. However, there is no evidence that this reaction
began in the inter-prill spaces. The darker spots at the right-hand edge of the
film are similar in size and position to the prills. This suggests that there is
significant heating within the prills themselves, before they fragment.
5.2.4 Nitram prills
5.2.4.1 Reactive prill bed
The bed shown in figure 5.38 was shocked; figure 5.39 shows the resulting fram-
ing photography. The bed behaves very similarly to the Orica bed studied in
section 5.2.3.1. The structure of the light output is coarser, but this is due to
the larger prill size used here. There is also no evidence of re-ignition of the
bed, as was seen in section 5.2.3.1. Given the random nature of cook-off, this is
unremarkable.
Figure 5.40 shows the output of the photodiode attached to this experiment.
187
5 LIGHT OUTPUT
Figure 5.37: Recovered heat-sensitive film after drop weight experiment on hexag-
onal arrangement of prills.
Figure 5.38: Static image of the bed of 3.4–3.6mm Nitram prills impacted in
figure 5.39. To scale with, and taken with the same apparatus as, figure 5.39.
188
PRILL BEDS 5.2
Figure 5.39: Framing photography of a bed of 3.4–3.6mm Nitram prills impacted
at 700± 40m s−1. 460 ns exposure, 40 ns interframe time. Taken using Ultranac
501 camera; frame order as shown in figure 2.19. Figure 5.38 shows a static image
of this bed, to the same scale.
189
5 LIGHT OUTPUT
0
100
200
300
400
500
0 1 2 3 4 5 6
P
h
ot
o
d
io
d
e
ou
tp
u
t
/
m
V
Time / µs
Figure 5.40: Photodiode output of a bed of 3.4–3.6 mm Nitram prills impacted
at 700± 40m s−1. Superimposed images show framing photography of the exper-
iment. Horizontal extent of image indicates duration for which shutter was open.
Experiment also shown in figures 5.38 and 5.39.
This differs quite markedly from figure 5.27, the equivalent for a bed of Orica
prills. No plateau region is visible. After rising to a peak at 1.75µs, the light
output falls in three stages, arrested at 2.4µs and 3.9µs. The second of these
closely resembles the bump in light output seen in both granular beds and Orica
prill beds.
Nitram prills have a uniformly dense microstructure. Unlike Orica prills, they
do not become more dense near the surface of the prill. The absence of a plateau
in figure 5.40 therefore provides some support for the second possible explanation
of the plateau in section 5.2.3.1. There is no low-density interior to the Nitram
prills, through which reaction may spread without contributing to the observed
light output of the bed.
5.2.4.2 Single reactive prill
A bed of Westland prills was constructed, and a single Nitram prill placed near
the centre. This bed is shown in figure 5.41. The dark mark on the Nitram prill
190
PRILL BEDS 5.2
Figure 5.41: Two copies of a static image of the bed of 3.4–3.6mm Westland
prills, with a single 3.5mm Nitram prill, impacted in figure 5.42. To scale with,
and taken with the same apparatus as, figure 5.42. Position of Nitram prill is
highlighted with a red circle in the right-hand image.
was added with a pen to identify it. The bed was then impacted at 700±40m s−1.
Figure 5.42 shows framing photography of this impact. This reaction is noticeably
brighter than that induced in an Orica prill (figure 5.30). Because of this enhanced
brightness, the spread of light through the bed is visible as highlights on the inert
prills in frames 5 to 9 (2–4.5µs). Some disintegration of prills near the reactive
prill is visible in frame 5 (2− 2.5µs).
Figure 5.43 shows the photodiode output of this experiment. It is broadly
similar to that seen in the prill-bed experiment (figure 5.40). The first fall after
the peak (1.75–2.5µs) is steeper than that seen in the prill-bed experiment.
The enhanced light output of a Nitram prill relative to an Orica prill is in-
teresting. Nitram prills have a denser microstructure than Orica prills. A single
Nitram prill will therefore contain more energetic material; assuming that it reacts
completely, it will therefore produce more light. However, prill-bed experiments
indicate that this is not the case: unreacted prill fragments are visible throughout
the experiments (figures 5.25 and 5.38). The open microstructure of the Orica
prills should introduce additional hot spots, resulting in more complete reaction
of the Orica prills. This in turn would result in greater energy release and brighter
light output from Orica prills.
One possible explanation of this discrepancy involves the mechanical strengths
of the prills involved. It was found in chapter 4 that Orica prills were less resistant
to compaction than Westland prills. If this is the case at the higher strain rate of
these experiments, the inert Westland prills will support the Orica prill. Nitram
prills, meanwhile, show very similar compaction resistance to Westland prills.
191
5 LIGHT OUTPUT
Figure 5.42: Framing photography of a bed of 3.4–3.6mm Westland prills, with
a single 3.5mm Nitram prill, impacted at 700± 40m s−1. 460 ns exposure, 40 ns
interframe time. Taking using Ultranac 501 camera; frame order as shown in
figure 2.19. Figure 5.41 shows a static image of the bed, to the same scale.
192
PRILL BEDS 5.2
0
50
100
150
200
0 1 2 3 4 5 6
P
h
ot
o
d
io
d
e
ou
tp
u
t
/
m
V
Time / µs
Figure 5.43: Photodiode output of a bed of 3.4–3.6 mm Westland prills, with a
single 3.5 mm Nitram prill, impacted at 700 ± 40m s−1. Superimposed images
show framing photography of the experiment. Horizontal extent of image indi-
cates duration for which shutter was open. Experiment also shown in figures 5.41
and 5.42.
193
5 LIGHT OUTPUT
Figure 5.44: Static image of the bed of 3.4–3.6mm agricultural pellets impacted
in figure 5.45. To scale with, and taken with the same apparatus as, figure 5.45.
More shock energy would therefore be deposited in the Nitram prill than in the
Orica prill, even if the beds were loaded identically. This could account for the
brighter light output of the Nitram prill in this experiment.
5.2.5 Agricultural pellets
5.2.5.1 Reactive pellet bed
The bed shown in figure 5.44 was shocked; figure 5.45 shows the resulting framing
photography. The structure of the light output is similar to that from a Nitram
prill bed (figure 5.39). However, the light fades more rapidly. Agricultural AN
pellets contain additives intended to absorb heat and hence inhibit reaction. This
rapid fading indicates that the additives are behaving as intended. It also provides
additional confirmation that the light seen here is due to reaction of AN, and that
this reaction is thermally activated. There is no evidence of the ring structure
seen in Nitram and Orica prill beds, probably because of the rapid decay of the
light output.
Figure 5.46 shows the photodiode output of this experiment. As in other AN
experiments, the light output rises to a peak, in this case at 1.7µs, then falls.
Unlike other experiments, this fall is linear, with gradient −295± 5Vms−1. The
gradient of the fall appears to reduce after 3µs, but digitization of the photodiode
output makes this difficult to assess.
The photodiode output of a repeat experiment is shown in figure 5.47. In this
experiment, the region of linear fall runs from the peak at 1.55µs to 2.15µs. The
linear region has gradient −329 ± 9Vms−1. It is succeeded by a non-linear fall
with a bump at 3.75µs, as expected in prill-bed experiments.
194
PRILL BEDS 5.2
Figure 5.45: Framing photography of a bed of 3.4–3.6mm agricultural pellets
impacted at 700± 40m s−1. 460 ns exposure, 40 ns interframe time. Taken using
Ultranac 501 camera; frame order as shown in figure 2.19. Figure 5.44 shows a
static image of the bed, to the same scale.
195
5 LIGHT OUTPUT
0
100
200
300
400
500
600
700
0 1 2 3 4 5 6
P
h
ot
o
d
io
d
e
ou
tp
u
t
/
m
V
Time / µs
Figure 5.46: Photodiode output of a bed of 3.4–3.6 mm agricultural pellets im-
pacted at 700±40m s−1. Superimposed images show framing photography of the
experiment. Horizontal extent of image indicates duration for which shutter was
open. Experiment also shown in figures 5.44 and 5.45.
The linear fall seen in figures 5.46 and 5.47 may be due to the dolomite added
to these pellets. It is added to absorb heat without reacting. This absorption,
and later re-radiation, may account for the linear region. The greater extent and
prominence of the linear region in figure 5.46 compared with figure 5.47 may be
due to excess dolomite in the pellets used in the former experiment.
5.2.5.2 Single reactive pellet
A bed of Westland prills was constructed, and a single agricultural pellet placed
near the centre. This bed is shown in figure 5.48. The dark mark on the agricul-
tural pellet was added with a pen to identify it. The bed was then impacted at
700± 40m s−1. Figure 5.49 shows framing photography of this impact. The light
output appears similar to that from a single Nitram prill (figure 5.42). However,
in frames 6 onward (2.5µs onward), the reactive prill appears to have remained
mostly intact. There is also a small region near the reactive prill which remains
brightly lit for significantly longer than in the Nitram experiment. Both are likely
196
PRILL BEDS 5.2
0
100
200
300
400
500
600
700
0 1 2 3 4 5 6
P
h
ot
o
d
io
d
e
ou
tp
u
t
/
m
V
Time / µs
Figure 5.47: Photodiode output of a bed of 3.4–3.6mm agricultural pellets im-
pacted at 700± 40m s−1.
due to the inert additives in the agricultural pellet. These additives will reduce
the rate and extent of reaction in the pellet. This will prevent the pellet’s disin-
tegration, and preserve reactive material later in the experiment. This will result
in the small, long-lasting bright region seen here.
Figure 5.50 shows the output of the photodiode attached to this experiment.
Digitization and noise in the output make it difficult to determine whether the
linear fall observed in agricultural pellet bed experiments is repeated here.
Figure 5.48: Two copies of a static image of the bed of 3.4–3.6mm Westland
prills, with a single 3.5mm agricultural pellet, impacted in figure 5.49. To scale
with, and taken with the same apparatus as, figure 5.49. Position of agricultural
pellet is highlighted with a red circle in the right-hand image.
197
5 LIGHT OUTPUT
Figure 5.49: Framing photography of a bed of 3.4–3.6mm Westland prills, with
a single 3.5mm agricultural pellet, impacted at 700± 40m s−1. 460 ns exposure,
40 ns interframe time. Taken using Ultranac 501 camera; frame order as shown
in figure 2.19. Figure 5.48 shows a static image of the bed, to the same scale.
198
PRILL BEDS 5.2
0
20
40
60
80
100
0 1 2 3 4 5 6
P
h
ot
o
d
io
d
e
ou
tp
u
t
/
m
V
Time / µs
Figure 5.50: Photodiode output of a bed of 3.4–3.6 mm Westland prills, with
a single 3.5 mm agricultural pellet, impacted at 700 ± 40m s−1. Superimposed
images show framing photography of the experiment. Horizontal extent of im-
age indicates duration for which shutter was open. Experiment also shown in
figures 5.48 and 5.49.
199
5 LIGHT OUTPUT
5.3 Conclusions and future work
• Using the projectile and bed design detailed in section 5.1.1, a flyer velocity
of 450m s−1 is sufficient to induce reaction in a bed of 150–212µm granular
ammonium nitrate.
• The induction time between introduction of a shock to a granular ammo-
nium nitrate bed, and the outbreak of light in that bed, may be explained
by the travel time of the shock through the bed.
• A slightly re-designed sample cell would allow the above conclusion to be
probed more rigorously, by allowing simultaneous observation of light out-
put and shock travel time.
• The decaying light output of a shocked granular ammonium nitrate bed
is not fully explained by conductive cooling of a large number of identical
instantaneous delta-function heat sources.
• Therefore, either the heat sources are not approximately delta functions,
they are not identical, or they are not instantaneous.
• Direct measurement of the evolution of one of these heat sources would
determine how closely they approximate a delta function.
• The decaying light output could be used to test a model of the population
distribution of these heat sources, or the chemical kinetics involved in their
generation.
• In prill and pellet bed experiments, and in some granular bed experiments,
the decay of light output was interrupted by a rise. This “bump” remains
unexplained, and bears further study.
• Westland prills are suitable for us as an inert mock for agricultural pellets
and Nitram prills.
• Differences in size distribution and mechanical strength make Westland
prills less suitable as an inert mock for Orica prills.
200
REFERENCES
• Beds of Orica and Nitram prills, and agricultural pellets, undergo incom-
plete reaction when impacted with a flyer velocity of 700± 40m s−1.
• The variation of light output with time for such beds shows features charac-
teristic of the type of prill or pellet from which the bed is constructed. For
Orica and Nitram prills, these features are visible when a single prill is used
in an inert bed. This indicates that the features originate in the structure
of the prills, rather than in inter-prill interactions. For agricultural pellets,
the data neither support nor rule out this conclusion.
• More sensitive light-output experiments on inert beds would determine
whether the above conclusion also applied to agricultural pellets.
• The light output from both beds of Orica prills and individual Orica prills
reaches a plateau before growing to a peak. This plateau is due to spread
of reaction through the porous interior of the prills.
• The peak of the light output of beds of Nitram prills and agricultural pellets
shows structure which remains unexplained and bears further study.
• Direct prill impact experiments would provide useful information on the
early stages of reaction in individual prills. Such experiments, however,
require impractically high vacuum.
• Drop weight experiments find no indication that inter-prill jetting is respon-
sible for reaction of Orica prills.
References
J. R. Asay and M. Shahinpoor, editors. High-Pressure Shock Compression of
Solids. Springer-Verlag, 1993.
H. S. Carslaw and J. C. Jaeger. Conduction of Heat in Solids. Oxford University
Press, 1947.
201
5 LIGHT OUTPUT
J. E. Field, G. M. Swallowe, and S. N. Heavens. Ignition mechanisms of ex-
plosives during mechanical deformation. Proceedings of the Royal Society of
London, Series A, 382:231, 1982.
S. P. Marsh. LASL Shock Hugoniot Data. University of California Press, Berke-
ley, 1980.
Y. Me-Bar and S. M. Walley. Impact ignition of liquid explosives. Technical Re-
port R&D 6541-AN-09, European Research Office, United States Army, London,
W1, England, 1994.
T. Obara, N. K. Bourne, and Y. Mebar. The construction and calibration of an
inexpensive PVDF stress gauge for fast pressure measurements. Measurement
Science and Technology, 6:345–348, 1995.
W. G. Proud, E. J. W. Crossland, and J. E. Field. High-speed photography and
spectroscopy in determining the nature, number and evolution of hot-spots in
energetic materials. In C. Cavailler, G. P. Haddleton, and M. Hugenschmidt,
editors, 25 th International Congress on High-Speed Photography and Photonics,
volume 4948 of SPIE, pages 510–518, Physics and Chemistry of Solids Group,
Cavendish Laboratory, University of Cambridge, 2003. SPIE.
G. M. Swallowe, J. E. Field, and L. A. Horn. Measurements of transient high
temperatures during the deformation of polymers. Journal of Materials Science,
21:4089–4096, 1986.
202
Chapter 6
Spectra
It was seen in chapter 5 that reacting regions in ammonium nitrate beds reach a
high temperature. To measure this temperature, the spectrum of the light emitted
by these reacting regions was studied using a high-speed optical spectroscope. If
they are black- or grey-body emitters, this spectrum will have a characteristic
shape determined by their temperature (Planck, 1901). Curve-fitting tools were
used to extract the emitter temperature from the recorded spectra.
6.1 Apparatus and calibration
The cell and flyer used were described in chapter 5, section 5.1.1. The fibre mount
shown in chapter 5, figure 5.8 was used; no framing photography was taken during
these experiments. Two optical fibres were taken from the rear of the sample cell.
One of these led to a photodiode, as discussed extensively in chapter 5. The other
led to the light input of the spectroscope.
The spectroscope is described in chapter 2, section 2.11. It was configured
with a 0.21 s exposure time. Since reading the photodiode array takes 30ms,
this gave a 1
7
probability that the photodiode array would be read during an
experiment. In this case, one photodiode would report an erroneous value. This
was deemed acceptable. This exposure time is the time between array readings,
and should not be confused with the duration for which the image intensifier is
active. The detector was cooled to −15 ◦C; this required water cooling of the
203
6 SPECTRA
Peltier device.
The image intensifier was triggered by a TTL pulse, remaining active for as
long as the pulse was high. This pulse also signalled that the controlling computer
should record the output of the photodiode array when next it was read. In this
experiment, the pulse was produced by a delay generator, which in turn was
triggered by the make trigger on the sample cell. The delay and duration of
the pulse could be adjusted. This allowed the spectrum of light emitted by the
experiment at a particular time after impact to be recorded.
6.1.1 Dark current
The dark current in the photodiode array adds an approximately constant inten-
sity offset to the recorded spectrum. This offset depends on the array’s exposure
time, the temperature of the array, and the construction of the photodiodes in the
array. There is also a random factor due to the probabilistic nature of thermal
charge-carrier generation. The simplest way to account for this offset is to model
it as exactly constant. The array is configured as for an experiment, and a sin-
gle spectrum captured with the image intensifier gated off. Such a dark-current
spectrum is shown in figure 6.1. The average intensity of this spectrum would
then be taken as the dark current offset for these array parameters.
However, it was found that there was some repeatable structure to the spectra
produced for this purpose. This is likely to be due to variations between the
photodiodes in the array. Figure 6.2 illustrates this repeatability. Id(λ) is a dark
current spectrum, while Ie(λ) is some experimental spectrum. For illustrative
purposes, another dark current spectrum has been chosen for Ie(λ). Subtracting
each element of Id from its corresponding element in Ie produces a markedly
smaller spread of points1 than subtracting the mean dark current I¯d from each
element of Ie(λ). The former approach was therefore adopted for this research.
The outlier at approximately 630 nm, −60 counts in figure 6.2 bears some
mention. Occasionally, with probability of order 10−3, a photodiode will record
an anomalously low or high dark current. The element-wise subtraction approach
used here does not suppress this anomaly, and it will propagate to later stages of
1The standard deviation σ(Ie(λ)− I¯d) = 20.3 counts, while σ(Ie(λ)− Id(λ)) = 2.43 counts.
204
APPARATUS AND CALIBRATION 6.1
0
50
100
150
200
250
0 100 200 300 400 500 600 700 800
In
te
n
si
ty
/
co
u
n
ts
Wavelength / nm
Figure 6.1: Example dark current spectrum. Exposure time 0.21 s, detector tem-
perature −15 ◦C.
-60
-40
-20
0
20
40
60
0 100 200 300 400 500 600 700 800
In
te
n
si
ty
d
iff
er
en
ce
/
co
u
n
ts
Wavelength / nm
Ie(λ)− I¯d
Ie(λ)− Ie(λ)
Figure 6.2: Illustration of repeatability of apparently-random noise in dark cur-
rent for two dark current spectra, Id(λ) and Ie(λ). That the green points are much
more tightly grouped than the red points indicates that the structure of the dark
current spectrum, despite appearing random, is in fact highly repeatable.
205
6 SPECTRA
analysis. As those later stages are robust to a single point suffering an error of a
few tens of counts, these anomalies are ignored.
6.1.2 Wavelength calibration
For the output of the detector array to be useful, the wavelength of light corre-
sponding to each detector must be determined. The design of the array is such
that the wavelength change between two adjacent detectors is constant. Therefore
any source with multiple sharp spectral features at known wavelengths is suitable
for this calibration. A mercury vapour lamp is recommended in the spectroscope
documentation. A calibration using such a source was carried out long before this
research began. It was then found that the fluorescent room lights used in the
laboratory also produced light suitable for this calibration1. The wavelengths of
several sharp features of the spectrum from these lights were recorded with the
spectroscope. These recorded wavelengths were then used as the basis of subse-
quent wavelength calibrations of the spectroscope, including that performed in
this research.
Figure 6.3 shows the spectrum of the room lights used for this calibration.
In all cases the observed spectral features differed by less than 0.6 nm from the
recorded values for calibration. As the resolution of the spectroscope with this
grating is 0.6 nm, no further adjustment was deemed necessary.
6.1.3 Relative intensity calibration
Several components of the experiment reduce the intensity of light passing through
them in a wavelength-dependent fashion. As this experiment is based on fitting
a model to the relationship between intensity and wavelength of light emitted by
the experiment, this wavelength-dependence must be accounted for. To do so,
the light path of the experiment between the granular bed and the spectroscope
was constructed. In place of the reacting granular bed, though, a light source of
continuous, known spectrum was used.
1Unsurprisingly, given the similarity between a standard fluorescent lamp and a mercury
vapour lamp.
206
APPARATUS AND CALIBRATION 6.1
0
200
400
600
800
1000
1200
1400
1600
100 200 300 400 500 600 700 800
In
te
n
si
ty
/
co
u
n
ts
Wavelength / nm
Figure 6.3: Recorded spectrum of fluorescent room lights. Dark current correction
not applied. Superimposed lines are the nominal wavelengths of the features used
for calibration, at 404.7, 535.8, 546.1 and 578.2 nm.
Figure 6.4 shows the arrangement used for this calibration. The light source
was an incandescent bulb, determined by the National Physical Laboratory (NPL)
to produce a 2856K black body spectrum when a current of 3.549A was passed
through it. This current was supplied by a stabilized power supply and monitored
with a multi-meter. The potential difference across the bulb was monitored with a
second multi-meter, to ensure it remained at the 7.29V observed by the NPL. This
bulb would not fit into the confinement used for the granular bed. It was therefore
possible for light to bypass the glass back-plate and distort the calibration. The
shroud shown in figure 6.4 was used to eliminate this possibility. A windowless
room was used for this calibration, and the room lights were switched off; no
other steps were taken to eliminate stray light.
To record the spectrum of the light source, the image intensifier was gated on.
The detector array was then exposed and read 50 times, and the resulting outputs
automatically added. This reduced the random error in the recorded spectrum.
This composite spectrum was then corrected for dark current and compared to
the calculated black-body spectrum emitted by the light source. Figure 6.5 shows
such a comparison. At each wavelength λ for which the spectroscope recorded an
207
6 SPECTRA
To spectroscope
Glass plate
Fibre holder
Fibre
Shroud
Light source
Figure 6.4: Arrangement used for relative intensity calibration of spectroscope.
Fibre holder, glass plate and fibre are identical to those used in plate impact
experiments (see chapter 5, figures 5.1 and 5.8). Shroud is black paper, folded
and taped to exclude light from rear of glass plate.
intensity, this intensity was divided by the intensity predicted for that wavelength
for a black body at 2856 K. The set of ratios thus produced was referred to
as the sensitivity curve S(λ). This process was repeated three times to assess
repeatability. Figure 6.6 shows the sensitivity curve calculated by this method.
The sensitivity of the apparatus was very low below 380 nm and above 615 nm.
Therefore, only the spectral region between these two values was considered.
Note that the vertical axis is now in arbitrary units. The fraction of the radiation
emitted by the calibration source that is captured by the spectroscope is unknown,
as is the emitting area of the calibration source. Therefore it is not possible to
find the absolute sensitivity of the spectroscope without considerably refining the
system used for calibration. Such refinement was unnecessary, as the analysis
techniques used assumed a grey-body emitter. Calibration of the variation in
sensitivity with wavelength was therefore sufficient.
It was found that the sensitivity S(λ) varied as the gain of the intensifier
was varied. It was therefore decided that only integer numbers of turns of the
ten-turn gain-control potentiometer would be used in experiments, and a separate
calibration would be performed for each such gain. This presented a new problem.
While continuous illumination of the detector was possible at low gain, at high
gain this would saturate the detector. This prevented acquisition of a useful
spectrum and risked damage to the detector. To avoid this, the detector was
208
APPARATUS AND CALIBRATION 6.1
0
1
2
3
4
5
6
7
8
250 300 350 400 450 500 550 600 650 700 750
0
1
2
3
4
5
6
7
8
In
te
n
si
ty
/
10
5
co
u
n
ts
S
p
ec
tr
al
en
er
gy
d
en
si
ty
/
(J
m
−3
)
p
m
−1
Wavelength / nm
Observed
Calculated
Figure 6.5: Comparison of the observed spectrum of a 2856K black body source
(left axis) with the calculated spectral energy density of such a source (right axis).
At each wavelength of interest, the value of the red curve is divided by the value of
the green curve. This gives the sensitivity S of the apparatus at that wavelength.
Observed spectrum gathered with 50 exposures each of 0.21 s duration, detector
temperature −15 ◦C, intensifier gain set to “0”.
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
350 400 450 500 550 600 650
S
/
co
u
n
ts
A
U
−1
Wavelength / nm
Figure 6.6: Sensitivity curve for apparatus with 0.21 s exposure time, detector
temperature −15 ◦C, intensifier gain set to “0”.
209
6 SPECTRA
used in gated mode. A pulse generator was used to produce a TTL pulse of a
certain duration in response to user input. This was connected to the input of
the pulse amplifier, so that the intensifier would be active as long as the output of
the pulse generator was high. The duration of the TTL pulse was chosen so that,
at the intensifier gain under investigation, the detector would produce a clear
spectrum without saturation. The detector was then exposed and read multiple
times, just as for continuous illumination. During this time, the pulse generator
was manually triggered ten times, taking care to wait at least one exposure time
(0.21 s) between triggers.
Given an experimental spectrum I(λ) and the corresponding sensitivity curve
S(λ), the sensitivity-corrected spectrum
Icorr(λ) =
I(λ)
S(λ)
, (6.1)
i.e. for each wavelength of interest, the intensity reported by the spectroscope
is divided by the previously-obtained wavelength-dependent sensitivity for that
wavelength, S(λ). Assuming that no properties of the spectroscope or the optical
path leading to it have changed since S(λ) was measured, this will eliminate all
wavelength dependence of the observing apparatus from the corrected spectrum
Icorr. As previously discussed, relating this to the actual energy output of the
source requires multiplication by some unknown value. The calibration performed
here, though, ensures that this multiplier is independent of wavelength.
6.1.4 Fibre effects
In these experiments, inexpensive optical fibre was used to carry light from the
experiment to the spectroscope. This protected the spectroscope from being
physically damaged by the experiment. The fibre used was supplied by RS Com-
ponents, with stock number 368-047, and had a 1mm PMMA core. The degree
to which this fibre attenuates light passing through it depends on the wavelength
of that light. Altering the length of the fibre will therefore alter the shape of the
spectrum as received by the spectroscope. This in turn can modify the tempera-
ture the system records. The effect of the fibre must therefore be quantified.
210
APPARATUS AND CALIBRATION 6.1
If one end of a fibre of attenuation coefficient α(λ) is exposed to a spectrum
of intensity I0(λ), the spectrum emitted from the other end will be
I(λ) = I0(λ)e
−α(λ)l. (6.2)
The dependence of the attenuation coefficient α on wavelength can be deter-
mined by a simple experiment. A fibre of length l0 is used to carry light from
some suitable source (in this case, the 2856K lamp used for calibration) to the
spectroscope, and the transmitted spectrum recorded. This is then repeated using
a fibre of length l1. By rearranging equation 6.2,
α(λ) =
1
l1 − l0 lnA
I(λ, l0)
I(λ, l1)
,
=
lnA
l1 − l0 +
1
l1 − l0 ln
I(λ, l0)
I(λ, l1)
, (6.3)
where A is a constant to account for the varying alignment between the fibre and
the light source.
Figure 6.7 shows α(λ) for l0 = 1.79 ± 0.01m, l1 = 4.43 ± 0.01m. Note the
large error in α, particularly at longer wavelengths. This is due to the multimodal
nature and relatively short length of the fibre. There are multiple modes by which
light may propagate down the fibre, and each will impose a different wavelength-
dependent attenuation on that light. The differences will arise for two reasons.
First, light propagating in different modes will pass through different amounts
of the core material. Second, the different modes will couple differently with the
cladding and sheath of the fibre. The extent to which the various modes of the
fibre are excited will, in short fibres such as those used in this research, depend
strongly on the precise orientation of the fibre itself, and on the alignment between
the fibre and the light source (Wilson and Hawkes, 1989). Rolling the fibre into a
coil of ∼ 0.2m diameter eliminated this variability in attenuation measurements,
as would be expected if it was due to variable mode populations. However, as
the arrangement of the experiment itself required a length of un-coiled fibre, it
was more appropriate to treat this variability as a random error.
In practice, the length of fibre used varied by ∼ 0.5m from the length for
211
6 SPECTRA
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
350 400 450 500 550 600 650
α
/
m
−1
Wavelength / nm
Figure 6.7: Attenuation coefficient α as a function of wavelength for RS 368-047
PMMA-cored fibre, measured using 1.79m and 4.43m fibres. All traces had lnA
l1−l0
set so that α = 0 at the lowest point of the curve.
which the spectroscope was calibrated. This allows the error in temperature in-
troduced by the variable fibre length and mode excitation to be assessed. Using
equation 6.2, the recorded spectrum from an experiment can be adjusted to sim-
ulate the addition or removal of 0.5m of fibre. Black-body curves can then be
fitted to the two modified spectra thus produced, providing two temperatures.
These temperatures then set bounds on the error introduced to temperature mea-
surements by fibre variations in that experiment. Figure 6.8 demonstrates the
technique. This shows that the temperature error introduced by fibre variations
is of similar magnitude to that introduced by the fitting program.
6.2 Results
The spectra observed during these experiments fell into one of two categories,
approximately divided by the peak light output of the reacting bed. “Reaction-
like” spectra were captured before this peak, while “cooling-like” spectra were
captured after this peak. Reaction-like spectra were dominated by spectral lines
212
RESULTS 6.2
0.7
0.75
0.8
0.85
0.9
0.95
1
400 450 500 550 600
N
or
m
al
iz
ed
in
te
n
si
ty
Wavelength / nm
0.5m removed, T = 5677± 16K
As recorded, T = 5689± 16K
0.5mm added, T = 5703± 16K
Figure 6.8: Spectrum captured 0.5 ± 0.3µs after photodiode peak during AN
impact (700±40m s−1 impact on 3.5mm thick cell of 150–212µm AN) adjusted to
simulate addition and removal of 0.5m of fibre and with black-body curves fitted
between 400 nm and 530 nm, illustrating the effect of fibre variations on measured
temperature. Quoted errors found using bootstrap method. Accounting for both
errors, the measured temperature is 5690± 20K.
213
6 SPECTRA
and were unsuitable for temperature measurements. Cooling-like spectra followed
a black-body shape and were useful for temperature measurements.
6.2.1 Reaction-like spectra
Figure 6.9 shows a typical reaction-like spectrum. It shows clear peaks at 385,
424, 557 and 591 nm. Apart from these peaks, the light output increases with
wavelength. This underlying continuum could be due to the chemiluminescent
reaction
O + NO NO2 , (6.4)
(Young and Sharpless, 1963) or to radiation from a black body at ∼ 103K.
Disambiguating the two would require spectroscopy in the range 600–1000 nm;
this may also allow temperature measurements in the latter case.
The peaks at 557 and 591 nm are consistent with the N2 first positive bands
(Becker et al., 1972; Guerra et al., 2004; Kurzweg and Broida, 1959; Young and
Sharpless, 1963), caused by the radiative transition N2(B
3Πg → A 3Σ+u ). The
upper state of this transition is populated by the combination of atomic nitrogen
to form an excited N2 molecule. This transition also involves a change in the
vibrational energy quantum number of the N2 molecule. The broad peak at
557 nm is due to transitions in which this number falls by 5, while the sharper
peak at 591 nm is due to transitions in which this number falls by 4. The sharpness
of the 591 nm peak is interesting: it implies that the ν ′ = 9 vibrational energy
level of the excited N2 state is much more populous than the other vibrational
energy levels. Apart from this anomaly, these two peaks simply confirm that N2
is being formed in this reaction.
The peaks at 385 and 424 nm may be due to NO β bands (Young and Sharp-
less, 1963). These are caused by the radiative transition NO(B 2Π→ X 2Π).
Similarly to the N2 first positive bands, the upper state of this transition is
populated by the combination of atomic nitrogen and oxygen to form an ex-
cited NO molecule. Again, the transition involves a change in the vibrational
energy of the NO molecule. The peak at 424 nm could be due to either the
NO(B 2Π, ν ′ = 3→ X 2Π, ν ′′ = 15) or the NO(B 2Π, ν ′ = 0→ X 2Π, ν ′′ = 13) tran-
sition, the peaks for which overlap. The peak at 385 nm could be due to the
214
RESULTS 6.2
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
350 400 450 500 550 600 650
N
or
m
al
iz
ed
in
te
n
si
ty
Wavelength / nm
a)
-50
0
50
100
150
200
250
300
350
400
450
0 1 2 3 4 5 6
P
h
ot
o
d
io
d
e
ou
tp
u
t
/
m
V
Time / µs
b)
Figure 6.9: (a) Spectrum of light emitted by 3.5mm bed of 150–212µm milled
AN impacted at 700 ± 40m s−1. (b) Photodiode output from this bed; shaded
region indicates duration for which spectroscope recorded.
215
6 SPECTRA
NO(B 2Π, ν ′ = 1→ X 2Π, ν ′′ = 12) transition. It is interesting that the other
peaks in the ∆ν = 12 and ν ′ = 1 systems are not observed.
6.2.1.1 Time-dependence of spectra
Only the spectral line at 591 nm was visible in all reaction-like spectra. This line
was so persistent that it was visible in some cooling-like spectra, as discussed
in section 6.2.2. It was also visible well after the bed had cooled, as shown in
figure 6.10. This indicates that the recombination of atomic nitrogen was taking
place throughout the experiment. This in turn indicates that this recombination
was at least partly independent of whichever reaction led to heating of the bed.
The broad 557 nm peak of nitrogen recombination is absent in figure 6.10. This
may simply be due to its low intensity, compared with the 591 nm peak.
The spectral lines at 385 and 424 nm varied more with time. Figure 6.11
shows the earliest spectrum obtained, which lacks these lines. This indicates that
NO was not formed until later in the experiment. The first occurrence of the NO
spectral lines was in spectra taken between 1.1 and 0.6µs before the peak light
output of the bed. This suggests that NO production is associated with a rapid
rise in the light output of the bed. The photodiode peak is believed, based on
section 6.2.2 and chapter 5, section 5.1.3.2 to be thermal in origin. This suggests
that whichever exothermic reaction is heating the bed involves NO.
6.2.2 Cooling-like spectra
Figure 6.12 shows a typical cooling-like spectrum. The uncertainty in the mea-
sured spectrum is due to the fibre length and orientation effects discussed in
section 6.1.4. The 591 nm peak of N2 formation is visible in this spectrum. How-
ever, the continuum is suitable for temperature measurement.
To measure the temperature of the emitting regions of the bed, they were
assumed to emit as black bodies. The spectrum of the light emitted by the bed
would then be given by
I(λ) = A · 2hc
2
λ5
[
exp
(
hc
λkBT
)
− 1
] , (6.5)
216
RESULTS 6.2
0
0.2
0.4
0.6
0.8
1
350 400 450 500 550 600 650
N
or
m
al
iz
ed
in
te
n
si
ty
Wavelength / nm
a)
-50
0
50
100
150
200
250
300
350
0 1 2 3 4 5 6
P
h
ot
o
d
io
d
e
ou
tp
u
t
/
m
V
Time / µs
b)
Figure 6.10: (a) Spectrum of light emitted by 3.5mm bed of 150–212µm milled
AN impacted at 700 ± 40m s−1. (b) Photodiode output from this bed; shaded
region indicates duration for which spectroscope recorded.
217
6 SPECTRA
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
350 400 450 500 550 600 650
N
or
m
al
iz
ed
in
te
n
si
ty
Wavelength / nm
a)
-40
-20
0
20
40
60
80
100
120
0 1 2 3 4 5 6
P
h
ot
o
d
io
d
e
ou
tp
u
t
/
m
V
Time / µs
b)
Figure 6.11: (a) Spectrum of light emitted by 3.5mm bed of 150–212µm milled
AN impacted at 700 ± 40m s−1. (b) Photodiode output from this bed; shaded
region indicates duration for which spectroscope recorded. Digitization indicates
inept fibre polishing and oscilloscope configuration, rather than low bed light
output.
218
RESULTS 6.2
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
350 400 450 500 550 600 650
N
or
m
al
iz
ed
in
te
n
si
ty
Wavelength / nm
a)
Experiment
T = 4783± 13K
0
100
200
300
400
500
0 1 2 3 4 5 6
P
h
ot
o
d
io
d
e
ou
tp
u
t
/
m
V
Time / µs
b)
Figure 6.12: (a) Spectrum of light emitted by 3.5mm bed of 150–212µm milled
AN impacted at 700 ± 40m s−1. Black body fit to shaded region superimposed.
(b) Photodiode output from this bed; shaded region indicates duration for which
spectroscope recorded.
219
6 SPECTRA
where h, kB and c are Planck’s and Boltzmann’s constants and the speed of light,
respectively, T is the temperature of the emitting regions, and A is the constant
multiplier discussed in section 6.1.3. A and T can then be found using a two-
parameter Levenberg-Marquardt fit of equation 6.5 to the data, as described in
chapter 2, section 2.12. To avoid distortion due to the N2 recombination spectral
lines, the fit considered only wavelengths between 400 and 530 nm. Uncertainty
in the fitted parameters was assessed using the bootstrap method, as described in
chapter 2, section 2.12.5. The uncertainty in T due to fibre effects was assessed
as described in section 6.1.4. These two uncertainties were added in quadrature
to give the quoted uncertainty in T .
6.2.2.1 Variation of temperature with time
The spectroscope could only record a single spectrum. Assessing the variation
of bed temperature with time therefore required multiple experiments. Between
experiments there was some variation in projectile velocity, and thus induction
time between the shock entering the bed and the peak of the bed’s light output.
To reduce the effect of this on these results, the peak of the photodiode output
in each experiment was taken as the origin of the time axis. Figure 6.13 shows
the results of this series of experiments.
These results are broadly as expected. The temperature falls steadily from
the photodiode peak, consistent with the theory that the decaying light output
of the bed is thermal in origin. The earliest and highest temperature recorded
was 6661± 20K, recorded 0.2± 0.3µs after the photodiode peak. However, the
temperature fell more slowly than would be expected based on the photodiode
data, subject to the assumptions in chapter 5, section 5.1.3.2. It was assumed
there that the photodiode captured a constant fraction of the bed’s total radiation
output over the course of an experiment. Therefore, by rearranging the Stefan-
Boltzmann law (chapter 5, equation 5.20) and taking the photodiode output to
be proportional to the total power radiated by the bed, we find
T ∝ 4
√
V , (6.6)
where T is the temperature of the emitting regions and V is the photodiode
220
RESULTS 6.2
0
1000
2000
3000
4000
5000
6000
7000
-0.5 0 0.5 1 1.5 2 2.5
T
em
p
er
at
u
re
/
K
Time / µs
Photodiode
Temperature
Exponential
Figure 6.13: Data points show temperature of emitting regions in ammonium
nitrate beds impacted at 700 ± 40m s−1. Origin of time axis taken as peak of
photodiode output in each experiment. Horizontal error bars indicate duration
over which spectroscope integrated. Blue line is a decaying exponential through
these data points to aid comparison. Red line is the fourth root of the normalized
average photodiode output for these experiments, multiplied by 7000 to aid com-
parison. If the photodiode captures a constant fraction of the bed light output,
the red line and the blue line should lie near each other. This is clearly not the
case.
221
6 SPECTRA
output. As figure 6.13 shows, 4
√
V falls noticeably faster than T .
The most likely explanation for this discrepancy is that it is not safe to neglect
the wavelength-dependent sensitivity of the photodiode. As the temperature of
the bed changes, both the total radiated energy and the wavelength distribution
of that energy change. The output of the photodiode will depend on both of
these, rather than only the former as was assumed in chapter 5.
6.2.2.2 Uncertainty in temperature
As figure 6.13 shows, the uncertainty in the fitted temperature of a spectrum is
significantly smaller than the vertical scatter of the points at similar times. The
most likely explanation is that there is significant shot-to-shot variation in the
temperature reached by the reacting regions of the bed. However, it is possible
that the probability distribution of T differs markedly from a normal distribution.
The bootstrap method, described in section 2.12.5, generates a probability
distribution of the fitted parameters, given the data. Figure 6.14 shows the
probability distribution of A and T corresponding to the black body fit shown
in figure 6.12, while figure 6.15 shows its projection onto the T axis. A strong
negative correlation between T and A is apparent. This is to be expected: if
the range of wavelengths used for fitting does not contain the peak of the black-
body spectrum, the primary effect of varying T will be to vary the total intensity
of the spectrum. This will require an opposing change in A if the fitted and
experimental spectra are to agree. The error in T should therefore be reduced
when the black body peak is within the fitting region, and this is observed in
figure 6.13. Figure 6.15, meanwhile, shows no evidence that the distribution of T
differs significantly from a gaussian. It is therefore most likely that the vertical
spread of points seen in figure 6.13 is due to shot-to-shot variation. Further
experiments would be required to better characterize this effect.
6.2.2.3 Consequences of temperature measurement
These temperature measurements indicate that, in the system studied here, some
fraction of the bed reaches a temperature in excess of 6000 K. The chemical
energy released by reaction of ammonium nitrate can only heat its products to
222
RESULTS 6.2
4755
4760
4765
4770
4775
4780
4785
4790
4795
4800
4805
8.8 8.9 9 9.1 9.2 9.3 9.4 9.5
T
/
K
A× 1014
Figure 6.14: Probability distribution of fitted parameters for spectrum of light
gathered approximately 1µs after peak light output from a 3.5 mm bed of 150–
212µmmilled AN impacted at 700±40m s−1. Fit in question shown in figure 6.12.
0
10
20
30
40
50
60
4755 4760 4765 4770 4775 4780 4785 4790 4795 4800 4805
P
op
u
la
ti
on
d
en
si
ty
T / K
Figure 6.15: Projection of probability distribution of fitted parameters onto T
axis for spectrum of light gathered approximately 1µs after peak light output
from a 3.5 mm bed of 150–212µm milled AN impacted at 700 ± 40m s−1. Fit
in question shown in figure 6.12. Un-projected probability distribution shown in
figure 6.14. Gaussian fit to distribution superimposed.
223
6 SPECTRA
between 1500 K and 2500 K, depending on the model used for the chemistry
and for the heat capacity of the products. The discrepancy between the two is
interesting, as well as presenting problems for models of AN detonation based
entirely on chemical considerations.
The experimental set-up used here simulates detonation in many respects. In
detonation, a shock propagates through the material, initiating reaction which
then drives the shock. Here, a shock is introduced to the material and initi-
ates reaction; the thin bed avoids the problems of lateral release waves usually
encountered in small-scale tests of non-ideal explosives.
One possible explanation for the high temperatures seen in this experiment
is that they are purely physical in origin. The various hot spot mechanisms, by
definition, convert mechanical energy into local heating. However, subjecting an
inert bed of sugar to this experiment resulted in no detectable light emission. This
indicates that purely physical heating cannot account for the 4000 K difference
between the observed temperature and the temperature due to chemistry. Were
hot-spot mechanisms heating the sugar to 4000 K, some light would surely have
been visible.
Another possible explanation for these high temperatures is that the heated
material’s heat capacity diverges drastically from that expected. Were the heat
capacity in the emitting material to fall, the same amount of chemical energy from
AN decomposition would raise it to a much higher temperature. There are a wide
range of effects at work which could conceivably eliminate degrees of freedom from
the system, reducing its heat capacity. Probing these quantitatively, however, is
challenging.
It is possible that the emissivity of the emitting regions has a strong wave-
length dependence. This could cause the colour temperature of these regions to
over-estimate their actual temperature. However, the shape of the observed spec-
tra agrees well with that of a black-body spectrum at a range of temperatures.
This could not be produced by a wavelength-dependent emissivity: it would also
require a very specific (and very unlikely) temperature dependence.
A final possibility is that physical and chemical heating effects couple non-
additively. For example, consider the situation where chemistry pre-heats a cavity
to 2000K, and this cavity then undergoes adiabatic compression. Under a very
224
CONCLUSIONS AND FUTURE WORK 6.3
crude treatment, this cavity reaches approximately seven times the temperature of
an otherwise identical cavity which was at room temperature before compression.
A more rigorous treatment would almost certainly reduce this effect, but it could
still be sufficient to explain the apparent discrepancy between chemical energy
and the temperature observed in this experiment.
6.3 Conclusions and future work
• The spectroscopic technique described in this chapter is suitable for mea-
suring the temperature of emitting regions in a granular AN bed impacted
at 700± 40m s−1 after the peak light output of that bed.
• N2 formation in such an AN bed begins within 800 ns of impact and con-
tinues until at least 3µs after impact. The vibrational energy distribution
of these N2 molecules does not appear to be consistent with that observed
at room temperature and low pressure.
• Study of the N2 spectral lines at higher wavelength resolution may allow
the vibrational energy distribution of the N2 molecules to be measured.
• NO formation in such an AN bed begins between 1.1 and 0.6µs before the
peak light output of the bed, and is associated with an increase in the light
output of the bed. It is therefore believed to be involved in the exothermic
reactions that heat the bed.
• Study of the emission spectrum of such a bed in the 600–1000 nm range is
necessary to determine whether NO2 is formed during the reaction.
• The fraction of the bed’s light output detected by the photodiodes used in
chapter 5 is a function of temperature.
• The uncertainty in the temperature found using the spectroscopic tech-
nique described in this chapter is significantly smaller than the shot-to-shot
variation in this temperature.
225
6 SPECTRA
• This variation could be ameliorated by reducing the shot-to-shot variation
of the impact velocity.
• During this experiment some regions of the bed reach a temperature of
6660± 20 K. Chemical decomposition of AN can only reach a temperature
of approx. 2000 K. This presents a problem for models of AN detonation
that depend solely on chemistry.
References
K. H. Becker, E. H. Fink, W. Groth, W. Jud, and D. Kley. N2 formation in
the Lewis-Rayleigh afterglow. Faraday Discussions of the Chemical Society, 53:
35–51, 1972.
V. Guerra, P. A. Sa´, and J. Loureiro. Kinetic modeling of low-pressure nitrogen
discharges and post-discharges. European Physical Journal: Applied Physics,
28:125–152, 2004.
U. H. Kurzweg and H. P. Broida. Vibrational intensity distributions in the
nitrogen afterglow. Journal of Molecular Spectroscopy, 3:388–404, 1959.
M. Planck. Ueber das Gesetz der Energieverteilung im Normalspectrum. An-
nalen der Physik und Chemie, 4(4):553–563, 1901.
J. Wilson and J. F. B. Hawkes. Optoelectronics: An Introduction, chapter 8.
Prentice Hall, second edition, 1989.
R. A. Young and R. L. Sharpless. Chemiluminescent reactions involving atomic
oxygen and nitrogen. Journal of Chemical Physics, 39(4):1071–1102, 1963.
226
Chapter 7
Conclusions
Both granular ammonium nitrate (AN) and a variety of prill and pellet formula-
tions of AN were studied in this research. Their microstructure and compaction
response were investigated, as was the light they produced during shock-induced
reactions.
The main technique used for investigating microstructure was environmental
scanning electron microscopy (ESEM). X-ray microtomography was also found
to be a suitable technique, but was not used extensively in this research. ESEM
showed that formulations intended for explosive use contained a network of con-
nected voids with sizes of order tens of microns, while those intended for agricul-
tural use contained very few voids. The void network in explosive formulations
would serve two purposes. First, it would allow intimate mixing of fuel and AN.
Second, collapse of these voids under shock loading would produce a high den-
sity of hot spots. Conversely, the lack of such voids in agricultural formulations
would render them less useful as explosives. It would also improve the handling
and storage properties of the formulation.
One such low-void-fraction agricultural formulation of AN contained a mixture
of AN and dolomite. The dolomite was added to render the formulation harder to
detonate. The distribution of the two components was investigated using spatially
resolved energy-dispersive X-ray spectroscopy (EDX) in conjunction with ESEM.
This indicated that dolomite was present as blocks approximately 50µm in size.
A prilled fertilizer formulation containing no AN was found which could be
used as an inert mechanical mock for the various prill and pellet formulations of
227
7 CONCLUSIONS
AN studied here. This enabled study of light emission from individual shocked
AN prills in later experiments.
The compaction behaviour of beds of various AN formulations was studied
at strain rates of 10−4 s−1 and 100 s−1. The former was achieved using an exist-
ing screw-driven instrumented press. The latter required the addition of a line
laser system to a drop weight. Several results were found. First, the beds were
more resistant to compaction at the higher strain rate. Second, dense AN prills
intended for agricultural use were more resistant to compaction than low-density
AN prills intended for explosive use. This is probably due to the greater distance
material must move in the dense prills to fill a void. Mixed AN and dolomite
pellets did not follow this trend: despite their dense microstructure, they offered
approximately the same resistance to compaction as explosive AN prills. This
was probably caused by the pellets’ relatively low resistance to disintegration,
and lubrication from oil added to the pellets by the manufacturer. Third, the
initial porosity, or void fraction, of a bed had little effect on the shape of the
pressure-porosity curve found during a compaction experiment. All these phe-
nomena suggest that compaction of ammonium nitrate prill beds is dominated
by plastic flow of the prills over length-scales similar to those of prills.
Two simple and popular models, the Heckel and the Kawakita model, were
compared with the compaction data. The Heckel model was found to agree very
poorly with the data, while the Kawakita model agreed very well.
A copper-fronted flyer was used to introduce a shock to beds of AN, ground
and sieved to a particle size of 150–212µm. This shock induced reaction in
the beds, and the light emitted by this reaction was studied using high-speed
photography, a photodiode, and gated spectroscopy. It was found that a flyer
velocity of 450m s−1 was sufficient to induce reaction, while lower velocities were
not. Most investigation took place with a flyer velocity of 700± 40m s−1. There
was a delay between the introduction of the shock to the bed and the maximum
light output of the bed. This was found to be due to the travel time of the shock
through the bed.
The spectrum of light emitted by the shocked bed was studied. A black-body
fit to this spectrum found that some regions of the bed reached a peak tempera-
ture of 6660±20 K. The energy released by reaction of ammonium nitrate can only
228
raise its products to a temperature of approximately 2000 K. This discrepancy
may prove problematic for purely chemical models of ammonium nitrate deto-
nation. Some possible causes for the difference were suggested, but no definite
statements can be made.
This spectroscopic study also found spectral features. These indicate that N2
formation begins within 800 ns of impact and continues until at least 3µs after
impact. NO formation, meanwhile, begins between 1.1µs and 0.6µs before the
peak light output of the bed. It therefore seems likely that NO formation is
involved in the exothermic reactions which heat the bed. The vibrational energy
distribution of the N2 molecules differs from that observed in low pressure gas-
discharge experiments near room temperature, and may merit further study.
The light output of prill and pellet beds during shock-induced reaction was
also studied. For low-density prills intended for explosive use, there was evidence
that reaction propagated through their porous interior before their dense outer
shell failed. For higher-density agricultural prills, there was no such evidence.
Photodiode output from these beds showed unexplained features. These features
were consistent between reactive prill beds and beds of inert prills with a single
reactive prill added. This indicates that effects internal to the prills are domi-
nant in shock-induced reaction. This was supported by drop weight experiments
which found no evidence that inter-prill jetting was responsible for reaction of
low-density explosive prills. These drop weight experiments also gave some infor-
mation on the mechanical failure of such prills under uniaxial compression. The
fragmentation of the prills suggested that their strength was due to the outer
layer of the prill. This layer appeared to fail in a brittle fashion.
There are many possible mechanisms by which hot spots may form in AN
prills and granular beds. As these effects are additive rather than competitive,
no one effect can be identified as the sole mechanism for initiation in AN: even
cosmic rays will have some non-zero (though negligible in practice) contribution.
However, the relative importance of these mechanisms can be assessed.
Given that all the prills and beds studied here included cavities, the family
of mechanisms related to cavity collapse will clearly be significant. Determining
the degree to which each of those mechanisms contributes to reaction is, however,
challenging. A wide range of void geometries and sizes is encountered within a
229
7 CONCLUSIONS
prill. Therefore none of the cavity-collapse mechanisms (jetting, adiabatic com-
pression of gas, viscoplastic work, or evaporation and stagnation) can be ruled
out on grounds of geometry or scale.
In common with other secondary explosives, the energy release of fracture in
AN is unlikely to be sufficient to cause reaction; crack growth is therefore not a
significant hot spot mechanism. Unlike many other secondary explosives, how-
ever, AN undergoes some thermal decomposition at its melting point. Frictional
heating may therefore contribute to reaction, though it is unlikely to be sufficient
to cause detonation alone.
The unusual distribution of vibrational energy levels in N2 formed during
reaction of AN granular beds suggests that some form of vibrational pumping
mechanism may be relevant in the initiation of reaction. The simplicity of AN
molecules, meanwhile, makes the existence of partially-reacted sensitization cen-
tres from previous damage an unlikely contributor to reaction.
230
Appendix A
Structures of ammonium nitrate
The structures and lattice parameters of the various phases of ammonium nitrate
have been known for some time. They are collected here.
A.1 Phase I
This phase has a cubic structure, with space group Pm3m and lattice param-
eter a = 4.3655(2) A˚. Diffuse X-ray scattering shows that the nitrate group is
in slightly restricted rotation, with either eight or twelve possible orientations
(Shinnaka, 1959). In the model with eight orientations, the O−N−O bond angle
is exactly 120◦. This is not the case in the model with twelve orientations. There
is some correlation between the orientations of nearby nitrate groups.
Neutron powder diffraction on deuterated ammonium nitrate was used to find
atomic positions, but was unable to determine which model was correct (Ahtee
et al., 1979). Multipole analysis of the nuclear smearing functions gave a slight
preference for the model with twelve possible orientations. The ammonium group
assumes one of two possible orientations with equal probability.
Figure A.1 shows the positions of the nitrogens, which are very similar in each
model. Tables A.1 and A.2 show the positions of the atoms in the unit cell. They
use, respectively, the models with eight and twelve possible nitrate orientations.
231
A STRUCTURES OF AMMONIUM NITRATE
Nitrate
Ammonium
N(2)
N(1) ∼ 1
2
Figure A.1: Centres of molecular groups in ammonium nitrate (I) unit cell, look-
ing down any of the three axes.
x y z
N(1) 0.5179 0.5179 0.5179
O 0.4570(36) 0.4570(36) 0.7936(17)
N(2) 0 0 0
H 0.1265(9) 0.1265(9) 0.1265(9)
Table A.1: Positions of atoms in the ammonium nitrate (I) unit cell, under a
model in which there are eight possible nitrate orientations.
x y z
N(1) 1
2
1
2
0.55
O(1) 1
2
1
2
0.829
O(2) 1
2
0.7725(20) 0.3925
N(2) 0 0 0
H 0.1248(6) 0.1248(6) 0.1248(6)
Table A.2: Positions of atoms in the ammonium nitrate (I) unit cell, under a
model in which there are twelve possible nitrate orientations.
232
PHASE II A.2
x y z
N(1) 0 0 1
2
H(1) 0.1404(4) −0.0171(16) 0.3850(4)
N(2) 0 1
2
0.0230(26)
O(1) 0 1
2
0.2742(9)
O(2) 0.6287(4) 1
2
+ x 0.1029(10)
Table A.3: Atomic positions in the ammonium nitrate (II) unit cell.
A.2 Phase II
This structure was found by neutron diffraction on powdered, deuterated ammo-
nium nitrate. It is tetragonal, with space group P4¯21m. Both ammonium and
nitrate ions are disordered, each assuming one of two possible orientations. The
two nitrate orientations leave the plane of the ion unchanged (Lucas et al., 1979).
a = 5.7193(1) A˚
c = 4.9326(1) A˚
Z = 2
Figures A.2 and A.3 show the structure with both ammonium and nitrate
ions in their first orientation. Figures A.4 and A.5 show the structure with both
ions in their alternative orientation. Table A.3 gives the atomic positions.
A.3 Phase III
This structure was found by neutron diffraction on powdered, deuterated ammo-
nium nitrate. It is orthorhombic, with space group Pnma. The ammonium ions
are disordered, assuming one of two orientations with equal probability (Lucas
et al., 1980). This disorder is long-range; the ammonium ions assume a fixed
orientation over small regions (Kearley and Kettle, 1982).
233
A STRUCTURES OF AMMONIUM NITRATE
O(2)
O(1)
N(2)
H
b
a
N(1)
O
N behind O
N
H(1)
Figure A.2: Structure of ammonium nitrate (II) unit cell, looking down c axis,
with ions in first orientation.
234
PHASE III A.3
H(1)
N(1)
O(1)
b
N
N(2)
O(2) H
c
O
Figure A.3: Structure of ammonium nitrate (II) unit cell, looking down a axis,
with ions in first orientation.
235
A STRUCTURES OF AMMONIUM NITRATE
H′(1) H
N′(2)
N behind O
O
N
O′(1)
b
a
O′(2)
N′(1)
Figure A.4: Structure of ammonium nitrate (II) unit cell, looking down c axis,
with ions in second orientation.
236
PHASE III A.3
b
N′(2)
O′(1)
N′(1)
H′(1)
c
H
N
O
O′(2)
Figure A.5: Structure of ammonium nitrate (II) unit cell, looking down a axis,
with ions in second orientation.
237
A STRUCTURES OF AMMONIUM NITRATE
H(3)
H(1), H(2)
N(1)
H
O
N(2) O(2)
O(1)
b
a
N
Figure A.6: Structure of ammonium nitrate (III) unit cell, looking down c axis,
with ammonium ions in first orientation.
a = 7.7184(3) A˚
b = 5.8447(1) A˚
c = 7.1624(2) A˚
Z = 4
The structure is shown in figures A.6, A.7, A.8 and A.9. The atomic positions
are given in table A.4.
A.4 Phase IV
This structure was found by neutron diffraction on a single crystal. It is or-
thorhombic, space group Pmmn. Hydrogen bonding is significant in this struc-
ture. There are hydrogen-bonded sheets parallel to the (001) plane. These sheets
238
PHASE IV A.4
O(1)
N(2)
H(3)
N(1) H(2)H(1)
a N
c
H
O
O(2)
Figure A.7: Structure of ammonium nitrate (III) unit cell, looking down b axis,
with ammonium ions in first orientation.
x y z
N(1) 0.5105(3) 1
4
0.8177(3)
H(1) 0.5806(4) 1
4
0.7282(6)
H(2) 0.5806(4) 1
4
0.9118(6)
H(3) 0.4374(4) 0.03624(7) 0.8177(3)
N(2) 0.6557(4) 1
4
0.3729(4)
O(1) 0.5662(6) 1
4
0.2367(7)
O(2) 0.7004(4) 0.4351(5) 0.4442(5)
R =
∑ |sF 20 − F 2c |∑
F 20
= 0.095
Table A.4: Atomic positions in ammonium nitrate (III), from neutron diffraction
on powder.
239
A STRUCTURES OF AMMONIUM NITRATE
b
H
O(2)
N(2)
N
O(1)
a
O
H′(3)
H′(1), H′(2)
N(1)
Figure A.8: Structure of ammonium nitrate (III) unit cell, looking down c axis,
with ammonium ions in second orientation.
240
PHASE IV A.4
O
N
N(2)
O(2)
O(1)
H′(2)
H′(3)
c
N(1)
H′(1)
a
H
Figure A.9: Structure of ammonium nitrate (III) unit cell, looking down b axis,
with ammonium ions in second orientation.
241
A STRUCTURES OF AMMONIUM NITRATE
β
αα
α
α α
β
H(2)
H(1)
N(1)
H(2)
O(2)
N(2)
O(1)
ββ
β 2.161 A˚ hydrogen bond
β
c
β
N
Strike through
H
b
O
α 2.050 A˚ hydrogen bond
Figure A.10: Structure of ammonium nitrate (IV) unit cell, looking down a axis.
The struck through hydrogen bonds must be broken to transform the structure
to ammonium nitrate (III).
are held together by van der Waals forces. There is no rotational disorder. In
order to transform this structure into phase III, it is necessary to break some of
the hydrogen bonds (Choi et al., 1972).
a = 5.745 A˚
b = 5.438 A˚
c = 5.942 A˚
Z = 2
The structure is shown in figures A.10, A.11 and A.12. The atomic positions
are given in table A.5.
242
PHASE IV A.4
α 2.050 A˚ hydrogen bond
α β
α
β
β
α
H(1)
N(1)
O
N(2) N
β
H
O(2)
α
α
α
a
β
O(2)
O(1)
β α
β
2.161 A˚ hydrogen bond
α
H(2)
H(1)
Strike through
c
Figure A.11: Structure of ammonium nitrate (IV) unit cell, looking down b axis.
The struck through hydrogen bonds must be broken to transform the structure
to ammonium nitrate (III).
x y z
N(1) 3
4
1
4
−0.0836(4)
H(1) 0.6045(12) 1
4
−0.1898(17)
H(2) 3
4
0.1011(16) 0.0324(16)
N(2) 1
4
1
4
0.5067(3)
O(1) 1
4
1
4
0.7629(6)
O(2) 0.4342(5) 1
4
0.3832(5)
Table A.5: Atomic positions in ammonium nitrate (IV), from 1.232 A˚ neutron
diffraction on a single crystal.
243
A STRUCTURES OF AMMONIUM NITRATE
β
α
a
N
H
O
Strike through
H(2) β
2.161 A˚
hydrogen bond
N(1)H(1)H(1)
b
α
2.050 A˚
hydrogen bond
N behind O
β
β
α
H(2)
α
α
α
β
O(2) N(2) O(2)
O(1)
β
α
β
Figure A.12: Structure of ammonium nitrate (IV) unit cell, looking down c axis.
The struck through hydrogen bonds must be broken to transform the structure
to ammonium nitrate (III).
244
PHASE V A.5
N
H
O
O from next layer
N behind O
b
NO3(1)
OB
OA
a
OBOA
NO3(2)
Figure A.13: Structure of ammonium nitrate (V) unit cell, looking down c axis.
A.5 Phase V
This structure was found by neutron diffraction on powdered, deuterated ammo-
nium nitrate. It is orthorhombic, with space group Pccn (Ahtee et al., 1983).
a = 7.9804(1) A˚
b = 8.0027(1) A˚
c = 9.8099(1) A˚
Z = 8
The structure is shown in figures A.13 and A.14. The atomic positions are
given in table A.6.
245
A STRUCTURES OF AMMONIUM NITRATE
. . . below parent N
. . . above parent N
O from next layer
N
H
O
NO3(1)
c
b
NO3(2)
Figure A.14: Structure of ammonium nitrate (V) unit cell, looking down a axis.
x y z B/A˚2
N(1) 1
4
1
4
−0.0186(5) 0.5(1)
OA(1) 1
4
1
4
−0.1440(9) 1.1(2)
OB(1) 0.2701 0.3825 0.042(6) 1.9(1)
N(2) 1
4
3
4
0.0173(6) 1.1(1)
OA(2) 1
4
3
4
0.1399(10) 2.0(2)
OB(2) 0.3846(6) 0.7774(8) −0.0405(7) 2.5(1)
N(3) −0.0177(3) 0.0023(9) 1
4
1.6(1)
H(1) 0.0634(10) 0.0535(11) 0.1844(9) 3.4(2)
H(2) 0.0443(9) −0.0603(11) 0.3246(10) 4.9(2)
H(3) −0.0917(9) −0.0772(9) 0.2020(7) 2.2(1)
H(4) −0.0841(11) 0.1004(12) 0.2890(7) 5.1(2)
Table A.6: Atomic positions in deuterated ammonium nitrate (V), from 1.909 A˚
neutron diffraction on powder.
246
REFERENCES
References
M. Ahtee, K. Kurki-Suonio, B. W. Lucas, and A. W. Hewat. Determination of
molecular orientations in cubic ND4NO3. Acta Crystallographica A, 35:591–597,
1979.
M. Ahtee, K. J. Smolander, B. W. Lucas, and A. W. Hewat. The structure of
the low-temperature phase V of ammonium nitrate, ND4NO3. Acta Crystallo-
graphica C, 39(6):651–655, 1983.
C. S. Choi, J. E. Mapes, and E. Prince. The structure of ammonium nitrate
(IV). Acta Crystallographica B, 28:1357–1361, 1972.
G. J. Kearley and S. F. A. Kettle. Recent advances in the Raman study of
phases II, III and IV of ammonium nitrate. Journal of Molecular Structure, 79:
319–321, 1982.
B. W. Lucas, M. Ahtee, and A. W. Hewat. The crystal structure of phase II
ammonium nitrate. Acta Crystallographica B, 35:1038–1041, 1979.
B. W. Lucas, M. Ahtee, and A. W. Hewat. The structure of phase III ammonium
nitrate. Acta Crystallographica B, 36:2005–2008, 1980.
Y. Shinnaka. X-ray study on the molecular rotation in cubic ammonium nitrate.
Journal of the Physical Society of Japan, 14(8):1073–1083, 1959.
247
248
Appendix B
Additional photography
The majority of the photography taken in the course of this research was not
included in the main body of this document. This was primarily because only a
single image was necessary to illustrate most discussions. The remaining images
are presented here for completeness. The majority of these images were taken
using an Ultranac 501 image conversion camera; see chapter 2, figure 2.19 for the
order of frames in those images.
249
B ADDITIONAL PHOTOGRAPHY
Figure B.1: Framing photography of 150–212µm ammonium nitrate granular
bed. 700± 40m s−1 impact velocity, 460 ns exposure time, 40 ns interframe time.
Taken using Ultranac 501 camera; frame order as shown in figure 2.19.
250
Figure B.2: Framing photography of 150–212µm ammonium nitrate granular
bed. 589 ± 2m s−1 impact velocity, 460 ns exposure time, 40 ns interframe time.
Taken using Ultranac 501 camera; frame order as shown in figure 2.19.
251
B ADDITIONAL PHOTOGRAPHY
Figure B.3: Framing photography of 150–212µm ammonium nitrate granular
bed. 516± 18m s−1 impact velocity, 460 ns exposure time, 40 ns interframe time.
Taken using Ultranac 501 camera; frame order as shown in figure 2.19. Illumi-
nated frames are numbers 5 and 6.
Figure B.4: Static image of the bed of 1.9–2.1mm Orica prills impacted in fig-
ure B.5. To scale with, and taken with the same apparatus as, figure 5.25.
252
79–579 ns 619–1079 ns 1119–1579 ns 1619–1704 ns
1744–1829 ns 1869–1954 ns 1994–2079 ns 2119–2329 ns
2369–2579 ns 2619–2829 ns 2869–3329 ns 3369–3829 ns
Figure B.5: Framing photography of a bed of 1.9–2.1mm Orica prills impacted
at 700± 40m s−1. Taken using Ultranac 501 camera with variable exposure and
interframe time. Times are relative to shock entering bed; all times are subject
to the 13 ns uncertainty in shock arrival time.
Figure B.6: Static image of the bed of 3.4–3.6mm agricultural pellets impacted
in figure B.7. To scale with, and taken with the same apparatus as, figure B.7.
253
B ADDITIONAL PHOTOGRAPHY
Figure B.7: Framing photography of a bed of 3.4–3.6mm agricultural pellets
impacted at 700± 40m s−1. 460 ns exposure, 40 ns interframe time. Taken using
Ultranac 501 camera; frame order as shown in figure 2.19. White region is caused
by damage to the photographic film.
Figure B.8: Static image of the bed of 3.4–3.6mm Nitram prills impacted in
figure B.9. To scale with, and taken with the same apparatus as, figure B.9.
254
Figure B.9: Framing photography of a bed of 3.4–3.6mm Nitram prills impacted
at 700± 40m s−1. 460 ns exposure, 40 ns interframe time. Taken using Ultranac
501 camera; frame order as shown in figure 2.19.
255
B ADDITIONAL PHOTOGRAPHY
−7µs 0µs 35µs 70µs
154µs 245µs 301µs 399µs
427µs 448µs 490µs 581µs
588µs 595µs 602µs 609µs
616µs 644µs 721µs 728µs
Figure B.10: Photography through sample in drop weight experiment on a hexag-
onal arrangement of 2mm ANFO prills taken using C4 camera. Anvils were
sapphire rather than the usual float glass. Time is relative to first detectable
deformation of prills.
256
−7µs 0µs 14µs 56µs
105µs 126µs 245µs 308µs
350µs 420µs 560µs
Figure B.11: Photography through sample in drop weight experiment on a hexag-
onal arrangement of 2mm ANFO prills taken using C4 camera. Time is relative
to first detectable deformation of prills.
257
B ADDITIONAL PHOTOGRAPHY
−6µs 0µs 25µs 82µs
126µs 152µs 190µs 221µs
284µs 379µs 506µs
Figure B.12: Photography through sample in drop weight experiment on a hexag-
onal arrangement of 2mm ANFO prills taken using C4 camera. Prills have been
surrounded with nylon ring to provide annular confinement. Time is relative to
first detectable deformation of prills.
258
List of corrections
Fig. 1.1, p. 4: AN phase diagram.
p. 2: Discussion of high-pressure AN phases added.
Section 1.1.2.1, p. 6: Short section on AN reaction kinetics added.
Table 1.2, p. 5; fig. 1.2, p. 12; fig. 1.7, p. 15: Citation added to captions.
Fig. 1.3, p. 12; fig. 1.4, p. 13; fig. 1.5, p. 14; fig. 1.6, p. 15; fig. 1.8, p. 16; fig. 1.9,
p. 17; fig. 1.10, p. 17; fig. 1.11; p. 18: Captions expanded, citations added.
Fig. 2.1, p. 36; fig. 2.4, p. 43; fig. 2.5, p. 45; fig. 2.6, p. 45; fig. 2.11, p. 54; fig. 2.21,
p. 69; fig. 2.22, p. 71; fig. 3.2, p. 87; fig. 3.8, p. 93; fig. 3.10, p. 95; fig. 3.13, p. 97;
fig. 3.15, p. 99; fig. 3.16, p. 99; fig. 3.17, p. 100; fig. 3.18, p. 101; fig. 3.19, p. 101;
fig. 4.27, p. 126; fig. 4.28, p. 126; fig. 4.4, p. 110; fig. 4.5, p. 110; fig. 4.11, p. 115;
fig. 4.14, p. 118; fig. 4.29, p. 127; fig. 4.30, p. 128; fig. 4.31, p. 131; fig. 4.33,
p. 134; fig. 5.13, p. 161; fig. 5.3, p. 148; fig. 5.5, p. 149; fig. 5.7, p. 152; fig. 5.8,
p. 153; fig. 5.10, p. 158; fig. 5.11, p. 159; fig. 5.16, p. 166; fig. 5.20, p. 172; fig. 5.22,
p. 175; fig. 5.23, p. 175; fig. 5.25, p. 177; fig. 5.27, p. 178; fig. 5.30, p. 181; fig. 5.31,
p. 182; fig. 5.34, p. 186; fig. 5.36, p. 187; fig. 5.37, p. 188; fig. 5.39, p. 189; fig. 5.40,
p. 190; fig. 5.42, p. 192; fig. 5.43, p. 193; fig. 5.45, p. 195; fig. 5.46, p. 196; fig. 5.49,
p. 198; fig. 5.50, p. 199; fig. 6.2, p. 205; fig. 6.7, p. 212; fig. 6.14, p. 223; fig. 6.15,
p. 223: Captions expanded.
p. 19: Added references for conservation relations.
Section 1.4, pp. 23–25: Short review of compaction work added.
Fig. 2.8, p. 52: Label moved.
p. 75: Added references for Levenberg-Marquardt implementations.
p. 81: Note that bootstrap method needed to be implemented.
p. 105 and 111: Mention length measurements after compaction experiments.
p. 106: Mentioned that bars used were aluminium.
259
B ADDITIONAL PHOTOGRAPHY
Fig. 4.6, p. 112; fig. 4.7, p. 112; fig. 4.8, p. 113; fig. 4.9, p. 113; fig. 4.17, p. 121;
fig. 4.18, p. 121; fig. 4.19, p. 122; fig. 4.20, p. 122; fig. 4.21, p. 123; fig. 4.22,
p. 123; fig. 4.23, p. 124; fig. 4.24, p. 124; fig. 4.25, p. 125; fig. 4.26, p. 125: Added
length measurements of recovered samples.
Section 4.2.4, p. 120: Removed assertion that coating affects compaction, and
associated discussion.
Section 4.2.5.1, p. 128: Evaluated effect of momentum on strain rate dependence
of compaction.
Section 4.2.6, p. 130: Moved speculation on implications of initial porosity effects
to section 4.2.10, p. 142.
Fig. 4.32, p. 132: Raw, rather than shifted, traces plotted. Caption expanded.
Section 4.2.9, pp. 134–142: Other compaction models considered. φ2 model re-
moved.
Section 4.2.10, p. 142: Added some speculation about mechanisms underlying
compaction.
Section 4.3, p. 142: Re-written to agree with corrections.
p. 153: Clarified meaning of “induction time”.
p. 153: Clarified explanation of bright ring.
Section 5.1.3.2, pp. 167–169; section 5.3, p. 200; sections 6.2.2.1–6.3, pp. 220–226:
Rewritten to incorporate revised heat-conduction model.
Fig. 5.18, p. 170: Original figure replaced with illustration of revised heat-
conduction model.
pp. 178, 180, 190, 191, 194, and 197: Removed references to previous heat-
conduction model.
pp. 207 and 210, and fig. 6.5 on p. 209: Clarified calibration procedure for spec-
troscope.
Fig. 6.13, p. 221: Figure re-drawn to account for revised heat-conduction model.
Caption expanded.
Section 6.2.2.3, pp. 222–225: Highlighted implications of temperature measure-
ment, and speculated on possible causes.
Fig. 6.14, originally on p. 217; fig. 4.28, originally on p. 118; fig. 4.29, originally
on p. 119; fig. 4.34, originally on p. 124: removed.
Chapter 7, p. 227: Re-written to mention revised compaction model, not mention
260
removed thermal-conduction model, and mention discrepancy between observed
temperature and chemistry.
Appendix B, “Monte Carlo simulations”, removed. Appendix C, “Additional
photography”, re-labelled Appendix B.
261