time dependent freezing of water under multiple

TIME DEPENDENT FREEZING OF WATER UNDER MULTIPLE SHOCK WAVE

COMPRESSION

By

DANIEL H. DOLAN III

A dissertation submitted in partial fulfillmentof the requirements for the degree of

Doctor of Philosophy

WASHINGTON STATE UNIVERSITYDepartment of Physics

MAY 2003

c©Copyright by DANIEL H. DOLAN III, 2003All Rights Reserved

c©Copyright by DANIEL H. DOLAN III, 2003All Rights Reserved

To the Faculty of Washington State University:

The members of the Committee appointed to examine the dissertation of

DANIEL H. DOLAN III find it satisfactory and recommend that it be accepted.

ii

ACKNOWLEDGEMENTS

I would like to thank my advisor, Dr. Yogendra Gupta, for suggesting, supporting,

and guiding this project. I am also grateful for the assistance and insight of Dr. James John-

son in developing the mixed phase water model. Many thanks go to Dr. Philip Marston for

providing the water purification system and serving on the thesis committee. Dr. Matthew

McCluskey and Dr. Jow-Lian Ding are also thanked for serving on the committee.

The wave code calculations in this work were aided by numerous discussions with

Dr. Michael Winey. I thank Dr. Oleg Fat’yanov and Dr. Scott Jones for their assis-

tance in the VISAR experiments. Kurt Zimmerman played a large role in the constructing

the optical imaging system and other instrumentation for this work. Dave Savage, Steve

Thompson, John Rutherford, and Gary Chantler assisted in building and performing the

experiments. Finally, I wish to acknowledge the support and understanding of my wife,

Elizabeth.

Funding for this research was provided by DOE Grant DE-FG03-97SF21388.

iii

TIME DEPENDENT FREEZING OF WATER UNDER MULTIPLE SHOCK WAVE

COMPRESSION

Abstract

by Daniel H. Dolan III, Ph.D.Washington State University

May 2003

Chair: Y.M. Gupta

Multiple shock wave compression experiments were performed to examine the time

dependence of freezing in compressed water. These experiments produced quasi-isentropic

compression, generating pressures and temperatures where liquid water is metastable with

respect to the ice VII phase. Time resolved optical and wave profile measurements were

used in conjunction with a thermodynamically consistent equation of state and a phe-

nomenological transition rate to demonstrate that water can freeze on nanosecond time

scales.

Single pass optical transmission measurements (ns resolution) indicated that com-

pressed water loses its transparency in a time dependent manner. This change is consistent

with the formation of ice regions that scatter light and reduce sample transparency. The on-

set of freezing and subsequent transition were accelerated as water was compressed further

past the ice VII phase boundary. Freezing was always observed when water was in contact

with a silica window; the transition did not occur if only sapphire windows were present.

iv

These observations suggest that freezing on nanosecond time scales begins through hetero-

geneous nucleation at water-window interfaces.

Optical imaging measurements (0.01 mm spatial resolution, 20 ns exposures) re-

vealed that freezing in compressed water is heterogeneous on 0.01-0.1 mm length scales

and begins at several independent sites. As freezing progressed over time, the water sample

became a complex network of opaque material separated by transparent regions of unfrozen

liquid. The transition growth morphology was consistent with freezing, which is limited

by the diffusion of latent heat.

Laser interferometry was used to measure particle velocity histories in compressed

water. The results were compared to calculated wave profiles to show that water remains

a pure liquid during the initial compression stages. As compression approached a steady

state, the measured particle velocity decreased when a silica window was present. This

decrease suggests that the frozen material is denser than the compressed liquid. Similar

particle velocity decreases were observed in the calculated wave profiles when water was

allowed to remain a metastable liquid for some time. These results are consistent with the

transparency loss described above, demonstrating that water can freeze on nanosecond time

scales.

v

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS iii

ABSTRACT iv

LIST OF TABLES xii

LIST OF FIGURES xvii

CHAPTER

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Objectives and approach . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Organization of Subsequent Chapters . . . . . . . . . . . . . . . . . . 3

References for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Liquid water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Microscopic structure . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 Thermodynamic and transport properties . . . . . . . . . . . . 10

2.1.3 Optical transmission properties . . . . . . . . . . . . . . . . . 15

2.2 Solid water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 The phase diagram of water . . . . . . . . . . . . . . . . . . . 19

2.2.2 Solid nucleation from the liquid phase . . . . . . . . . . . . . 20

2.2.3 Freezing time scales in water . . . . . . . . . . . . . . . . . . 26

2.3 Previous shock wave experiments on water . . . . . . . . . . . . . . . 28

2.3.1 Shock wave experiments in liquid water . . . . . . . . . . . . 29

2.3.2 Shock induced freezing . . . . . . . . . . . . . . . . . . . . . 30

2.4 Unresolved questions and approach . . . . . . . . . . . . . . . . . . . 34

2.5 Multiple shock wave compression . . . . . . . . . . . . . . . . . . . . 35

2.5.1 Multiple shock compression and plate impact . . . . . . . . . 36

2.5.2 Temperature advantages of multiple shock compression . . . . 38

vi


3. Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.1 Optical transmission experiments . . . . . . . . . . . . . . . . . . . . 53

3.1.1 Overall configuration . . . . . . . . . . . . . . . . . . . . . . 53

3.1.2 Mechanical components and assembly . . . . . . . . . . . . . 55

3.1.3 Instrumentation and optical components . . . . . . . . . . . . 61

3.1.4 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . 63

3.2 Optical imaging experiments . . . . . . . . . . . . . . . . . . . . . . 71





3.3 Wave profile experiments . . . . . . . . . . . . . . . . . . . . . . . . 88






4. Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.1 Optical transmission measurements . . . . . . . . . . . . . . . . . . . 97

4.1.1 Determining sample transmission . . . . . . . . . . . . . . . . 99

4.1.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 102

4.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.2 Optical imaging measurements . . . . . . . . . . . . . . . . . . . . . 117

4.2.1 Determining sample transmission . . . . . . . . . . . . . . . . 119


4.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.3 Wave profile measurements . . . . . . . . . . . . . . . . . . . . . . . 132

4.3.1 Particle velocity determination . . . . . . . . . . . . . . . . . 132


vii

4.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142


5. Time Dependent Continuum Model for Water . . . . . . . . . . . . . . . . 145

5.1 EOS development . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.1.1 Thermodynamic consistency . . . . . . . . . . . . . . . . . . 146

5.1.2 The form off (T,v) for constantcv . . . . . . . . . . . . . . . 146

5.2 Liquid water model . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.2.1 Choice of EOS . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.2.2 EOS formulation . . . . . . . . . . . . . . . . . . . . . . . . 148

5.2.3 Isentropic freezing and the value ofcv . . . . . . . . . . . . . 154

5.3 Solid water model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

5.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

5.3.2 EOS formulation . . . . . . . . . . . . . . . . . . . . . . . . 157

5.4 Mixed phase modelling . . . . . . . . . . . . . . . . . . . . . . . . . 160

5.4.1 Mixture rules . . . . . . . . . . . . . . . . . . . . . . . . . . 160

5.4.2 Time dependence of the freezing transition . . . . . . . . . . . 161

5.4.3 Limiting cases for isentropic compression . . . . . . . . . . . 162

5.5 Wave propagation calculations . . . . . . . . . . . . . . . . . . . . . 167

5.5.1 Calculation outline . . . . . . . . . . . . . . . . . . . . . . . 167

5.5.2 Enforcing the mixture rules . . . . . . . . . . . . . . . . . . . 168

5.5.3 Mixed phase calculations . . . . . . . . . . . . . . . . . . . . 171


6. Analysis and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

6.1 First order phase transition . . . . . . . . . . . . . . . . . . . . . . . 187

6.1.1 Latent heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

6.1.2 Volume change . . . . . . . . . . . . . . . . . . . . . . . . . 188

6.2 The importance of surface effects . . . . . . . . . . . . . . . . . . . . 190

6.2.1 Evidence for surface effects . . . . . . . . . . . . . . . . . . . 192

6.2.2 Surface initiated freezing . . . . . . . . . . . . . . . . . . . . 197

6.2.3 Ice nucleation at window surfaces . . . . . . . . . . . . . . . 206

viii

6.3 Freezing time scales . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

6.3.1 Apparent time scales . . . . . . . . . . . . . . . . . . . . . . 209

6.3.2 Incubation time analysis . . . . . . . . . . . . . . . . . . . . . 213

6.3.3 Transition time analysis . . . . . . . . . . . . . . . . . . . . . 218

6.4 Transition length scales . . . . . . . . . . . . . . . . . . . . . . . . . 220

6.4.1 Domains of the water sample . . . . . . . . . . . . . . . . . . 220

6.4.2 Lateral freezing variations . . . . . . . . . . . . . . . . . . . . 222

6.4.3 Composition of the transformed material . . . . . . . . . . . . 228


7. Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 235

7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

7.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

7.3 Recommendations for future work . . . . . . . . . . . . . . . . . . . 239


APPENDIX

A. Mechanical Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

References for Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . 268

B. Window Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

B.1 Soda lime glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

B.2 Fused silica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

B.3 z-cut quartz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

B.4 a-cut Sapphire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

References for Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . 273

C. Water Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

C.1 Contamination and treatment methods . . . . . . . . . . . . . . . . . 275

C.2 Sample purification . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

C.3 Filling the liquid cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

References for Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . 285

ix

D. Supplemental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

D.1 Soda lime glass experiments . . . . . . . . . . . . . . . . . . . . . . . 287

D.1.1 Water experiments . . . . . . . . . . . . . . . . . . . . . . . . 287

D.1.2 Transparency of shocked soda lime glass . . . . . . . . . . . . 289

D.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

D.2 Supplemental photodiode records . . . . . . . . . . . . . . . . . . . . 292

D.2.1 Imacon 200 demonstration experiments . . . . . . . . . . . . 292

D.2.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

References for Appendix D . . . . . . . . . . . . . . . . . . . . . . . . . . 297

E. Optical Extinction in a Scattering Medium . . . . . . . . . . . . . . . . . . 299

E.1 Single particle scattering . . . . . . . . . . . . . . . . . . . . . . . . . 299

E.2 Multiple scatterers and optical extinction . . . . . . . . . . . . . . . . 302

E.3 Limitations of optical transmission measurements . . . . . . . . . . . 305

E.4 Transmission interpretation . . . . . . . . . . . . . . . . . . . . . . . 306

References for Appendix E . . . . . . . . . . . . . . . . . . . . . . . . . . 309

F. Details of the VISAR Calculations . . . . . . . . . . . . . . . . . . . . . . 311

F.1 Apparent velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

F.2 Window corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

References for Appendix F . . . . . . . . . . . . . . . . . . . . . . . . . . 319

G. Details of the Mixed Phase Water Model . . . . . . . . . . . . . . . . . . . 321

G.1 Numerical stability of the mixed phase model . . . . . . . . . . . . . 321

G.2 Mixed phase model subroutines . . . . . . . . . . . . . . . . . . . . . 323

G.2.1 Alterations of the COPS source code . . . . . . . . . . . . . . 323

G.2.2 Source code for mixed phase water model . . . . . . . . . . . 324

G.2.3 Material list . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

References for Appendix G . . . . . . . . . . . . . . . . . . . . . . . . . . 333

H. Heat Dissipation in Solidification Processes . . . . . . . . . . . . . . . . . 335

H.1 Planar solidification of a supercooled liquid . . . . . . . . . . . . . . . 335

H.1.1 The Stefan problem . . . . . . . . . . . . . . . . . . . . . . . 335

x

H.1.2 Solid growth in reverberated water . . . . . . . . . . . . . . . 339

H.2 Growth stability in a supercooled liquid . . . . . . . . . . . . . . . . . 341

References for Appendix H . . . . . . . . . . . . . . . . . . . . . . . . . . 346

xi

LIST OF TABLES

Page

2.1 Physical properties of liquid water . . . . . . . . . . . . . . . . . . . . . . 11

4.1 Summary of optical transmission experiments . . . . . . . . . . . . . . . . 98

4.2 Summary of optical imaging experiments . . . . . . . . . . . . . . . . . . 118

4.3 Summary of wave profile experiments . . . . . . . . . . . . . . . . . . . . 133

5.1 b(v) fit parameters for liquid water . . . . . . . . . . . . . . . . . . . . . . 153

6.1 Measured incubation and transition times . . . . . . . . . . . . . . . . . . 212

C.1 Classes of water contaminants . . . . . . . . . . . . . . . . . . . . . . . . 276

C.2 Standard water purification techniques . . . . . . . . . . . . . . . . . . . . 276

C.3 Summary of water testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

D.1 Summary of soda lime glass experiments . . . . . . . . . . . . . . . . . . . 288

D.2 Supplemental photodiode measurements . . . . . . . . . . . . . . . . . . . 293

G.1 Summary of single phase EOS parameters . . . . . . . . . . . . . . . . . . 322

H.1 Summary of Stefan problem solutions . . . . . . . . . . . . . . . . . . . . 340

xii

LIST OF FIGURES

Page

2.1 Hydrogen bonding in water . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Crystal structure of ice Ih . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Dielectric functions of liquid water . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Equilibrium phase diagram of water . . . . . . . . . . . . . . . . . . . . . 18

2.5 Formation of a solid nucleus in a liquid . . . . . . . . . . . . . . . . . . . . 21

2.6 Stability of a solid nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.7 Variation of nucleation time and rate withgL−gS . . . . . . . . . . . . . . 24

2.8 Calculated P,T states of shocked liquid water . . . . . . . . . . . . . . . . . 31

2.9 Multiple shock compression . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.10 Temperature advantages of multiple shock compression . . . . . . . . . . . 40

3.1 Setup for optical transmission measurements . . . . . . . . . . . . . . . . . 54

3.2 Transmission experiment projectile . . . . . . . . . . . . . . . . . . . . . . 56

3.3 Liquid cell assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4 Transmission experiment target . . . . . . . . . . . . . . . . . . . . . . . . 60

3.5 Rear view of transmission setup . . . . . . . . . . . . . . . . . . . . . . . 64

3.6 Electronic setup for transmission measurements . . . . . . . . . . . . . . . 66

3.7 Timing calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.8 Wavelength calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.9 General setup of optical imaging experiments . . . . . . . . . . . . . . . . 72

3.10 Telescope based image relay system . . . . . . . . . . . . . . . . . . . . . 73

3.11 Lens based image relay system . . . . . . . . . . . . . . . . . . . . . . . . 76

3.12 Cross section of the DRS Hadland 8 way beam splitter . . . . . . . . . . . 78

3.13 Imaging experiment target chamber setup . . . . . . . . . . . . . . . . . . 81

3.14 Imaging experiment external optics setup . . . . . . . . . . . . . . . . . . . 82

3.15 Calibration with 1951 USAF resolution chart . . . . . . . . . . . . . . . . 85

3.16 Linearity of the optical imaging system . . . . . . . . . . . . . . . . . . . . 87

3.17 General setup of VISAR measurement . . . . . . . . . . . . . . . . . . . . 89

xiii

3.18 Mirror configurations for VISAR experiments . . . . . . . . . . . . . . . . 90

3.19 Electronic setup for VISAR measurements . . . . . . . . . . . . . . . . . . 94

4.1 Transmission losses in a non-absorbing sample . . . . . . . . . . . . . . . 100

4.2 Measured photodiode outputs . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.3 Photodiode transmission for experiments T1, T2, and T3 . . . . . . . . . . 103

4.4 Photodiode transmission for experiments T4 . . . . . . . . . . . . . . . . . 106

4.5 Photodiode transmission records for experiments T5 and T6 . . . . . . . . . 108

4.6 Spectrally resolved transmission for experiment T5 . . . . . . . . . . . . . 109

4.7 Photodiode transmission records for experiments T7 and T8 . . . . . . . . . 111

4.8 Loading history in quartz, sapphire, and hybrid liquid cells . . . . . . . . . 113

4.9 Photodiode transmission records for experiments T9 and T10 . . . . . . . . 114

4.10 Photodiode transmission records for experiments T11 and T12 . . . . . . . 116

4.11 Raw images from experiment I1 . . . . . . . . . . . . . . . . . . . . . . . 120

4.12 Measured transmission for experiment I1 . . . . . . . . . . . . . . . . . . . 122

4.13 Images obtained in experiment I1 . . . . . . . . . . . . . . . . . . . . . . . 123

4.14 Measured transmission for experiment I2 . . . . . . . . . . . . . . . . . . . 126



4.17 Photodiode transmission record for experiment I4 . . . . . . . . . . . . . . 130


4.19 Raw signals of the VISAR measurement . . . . . . . . . . . . . . . . . . . 134

4.20 Particle velocity history for experiment V1 . . . . . . . . . . . . . . . . . . 137


4.22 VISAR measurement for experiment V3 . . . . . . . . . . . . . . . . . . . 140

4.23 VISAR measurements long after compression . . . . . . . . . . . . . . . . 141


5.1 25 C isotherm for liquid water . . . . . . . . . . . . . . . . . . . . . . . . 150

5.2 P−v Hugoniot for liquid water . . . . . . . . . . . . . . . . . . . . . . . . 152

5.3 b(v) for liquid water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.4 The value ofcv and adiabatic freezing . . . . . . . . . . . . . . . . . . . . 155

xiv

5.5 300 K isotherm for ice VII . . . . . . . . . . . . . . . . . . . . . . . . . . 158

5.6 Liquid-ice VII coexistence curve . . . . . . . . . . . . . . . . . . . . . . . 159

5.7 Limits of isentropic compression . . . . . . . . . . . . . . . . . . . . . . . 164

5.8 Mixed phase compression in theP−v plane . . . . . . . . . . . . . . . . . 165

5.9 Multiple shock compression temperatures for various liquid cells . . . . . . 172

5.10 Thermodynamic histories under multiple shock compression . . . . . . . . 173

5.11 Multiple shock loading path in theT−P plane . . . . . . . . . . . . . . . . 175

5.12 Multiple shock compression loading path in theP−v plane . . . . . . . . . 177

5.13 Simulated particle velocity profiles . . . . . . . . . . . . . . . . . . . . . . 178

5.14 Incubation and transition time effects . . . . . . . . . . . . . . . . . . . . . 180

6.1 Latent heat and growth morphology . . . . . . . . . . . . . . . . . . . . . 189

6.2 Volume change and wave profiles . . . . . . . . . . . . . . . . . . . . . . . 191

6.3 Extinction histories for different window configurations . . . . . . . . . . . 194

6.4 Surface effects in wave profile measurements . . . . . . . . . . . . . . . . 196

6.5 Temperature of the water-window interface . . . . . . . . . . . . . . . . . 199

6.6 Optical transmission in surface initiated freezing . . . . . . . . . . . . . . . 203

6.7 Increasing extinction with sample thickness . . . . . . . . . . . . . . . . . 205

6.8 Time scales and peak pressure . . . . . . . . . . . . . . . . . . . . . . . . 210

6.9 Comparisons of the quartz cell photodiode experiments . . . . . . . . . . . 211

6.10 Fit of the T5 photodiode record . . . . . . . . . . . . . . . . . . . . . . . . 212

6.11 Incubation time and the metastable history . . . . . . . . . . . . . . . . . . 215

6.12 Incubation time and sample thickness . . . . . . . . . . . . . . . . . . . . 217

6.13 Transition time and sample thickness . . . . . . . . . . . . . . . . . . . . . 219

6.14 Longitudinal length scales of the freezing of water . . . . . . . . . . . . . . 221

6.15 Consistency of the optical transmission measurements . . . . . . . . . . . . 223

6.16 Initiation and growth in imaging measurements . . . . . . . . . . . . . . . 225

6.17 Nucleation rate and lateral freezing . . . . . . . . . . . . . . . . . . . . . . 227

6.18 Measured transmission profile and the Rayleigh scattering limit . . . . . . . 229

A.1 4” standard projectile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

A.2 4” monkey’s fist projectile . . . . . . . . . . . . . . . . . . . . . . . . . . 245

xv

A.3 2.5” projectile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

A.4 Projectile modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

A.5 Alignment projectile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

A.6 Soda lime glass impactor mount . . . . . . . . . . . . . . . . . . . . . . . 249

A.7 Universal impactor mount . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

A.8 Projectile turning mirror mounts (glass) . . . . . . . . . . . . . . . . . . . 251

A.9 Projectile turning mirror mounts (standard)) . . . . . . . . . . . . . . . . . 252

A.10 Impactor apertures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

A.11 Soda lime glass liquid cell . . . . . . . . . . . . . . . . . . . . . . . . . . 254

A.12 Silica window liquid cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

A.13 Sapphire window liquid cell . . . . . . . . . . . . . . . . . . . . . . . . . 256

A.14 Quartz liquid cell (VISAR without buffer) . . . . . . . . . . . . . . . . . . 257

A.15 Standoff target ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

A.16 Soda lime glass cell target plate . . . . . . . . . . . . . . . . . . . . . . . . 259

A.17 Quartz/fused silica cell target plate . . . . . . . . . . . . . . . . . . . . . . 260

A.18 Sapphire cell target plate . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

A.19 Unbuffered VISAR target plate . . . . . . . . . . . . . . . . . . . . . . . . 262

A.20 Lens bracket for glass target plate . . . . . . . . . . . . . . . . . . . . . . . 263

A.21 Lens bracket for quartz/fused silica target plate . . . . . . . . . . . . . . . 264

A.22 Lens bracket for sapphire target plate . . . . . . . . . . . . . . . . . . . . . 265

A.23 Transmission lens unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

A.24 VISAR lens unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

C.1 Water purification stages . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

C.2 High-Q 103S still . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

D.1 Photodiode transmission for experiments G1 and G2 . . . . . . . . . . . . 290

D.2 Transparency of shocked soda lime glass . . . . . . . . . . . . . . . . . . . 291

D.3 Photodiode transmission of experiment IS1 . . . . . . . . . . . . . . . . . 294

D.4 Photodiode transmission of experiment IS2 and IS3 . . . . . . . . . . . . . 296

E.1 Single scattering setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

xvi

E.2 Optical extinction from a collection of scattering particles . . . . . . . . . . 303

E.3 Extinction complications in a scattering system . . . . . . . . . . . . . . . 307

F.1 Simplified view of the VISAR measurement . . . . . . . . . . . . . . . . . 312

F.2 True versus apparent velocity in the VISAR system . . . . . . . . . . . . . 316

H.1 Layout of the Stefan problem . . . . . . . . . . . . . . . . . . . . . . . . . 336

H.2 Numerical solutions to Stefan problem . . . . . . . . . . . . . . . . . . . . 342

H.3 Uniform solid size limits in freezing water . . . . . . . . . . . . . . . . . . 343

H.4 Instabilities in supercooled solidification . . . . . . . . . . . . . . . . . . . 345

xvii

Chapter 1

IntroductionA material at thermodynamic equilibrium assumes the structure or phase that is

most favorable under the applied conditions [1]. At a fixed temperatureT and pressureP,

the stable phase is one that has the minimum specific Gibb’s free energyg(T,P); all other

phases present under such conditions will eventually convert to the stable phase. The time

required for this conversion varies greatly depending on the nature of the transition and the

conditions of interest. For example, polymorphic phase transitions in shocked cadmium

sulfide [2] occur on10−10 s times scales, while the transition from graphite to diamond

at ambient conditions takes much longer than hundreds of years (À 109 s) [3]. There is

a broad interest in understanding and controlling the time scales of phase transitions in

numerous system, such as atmospheric clouds [4] and biological tissues [5].

The study of phase transition dynamics requires that thermodynamic changes be

made on time scales less than or equal to the transformation time. Shock waves provide

a useful approach for generating rapid thermodynamic changes, and as such have been

used to study polymorphic and melting transitions [6]. In contrast, shock induced freezing

presents several fundamental difficulties. Shock compression in liquids leads to significant

temperature increases, often producing states that are too hot for freezing. Even when

freezing is thermodynamically possible, there is some time required for the transition to

take place. This time may be on the order of many seconds [7], which is much longer than

the typical10−6 s duration of a shock wave experiment. Whether freezing actually occurs

in shock wave loading is an issue that has not been resolved [6] and is the subject of the

present study.

1

1.1 Objectives and approachThe general objective of this work was to examine and understand shock wave

induced freezing in liquid water on nanosecond time scales. Liquid water was chosen

for a variety of reasons. Not only is water a material of general scientific interest [8], it

has an unusually large specific heat, which reduces the temperature rise caused by shock

compression. There has also been a long standing controversy regarding shock induced

freezing in liquid water [6].

The specific objectives of this work were:

1. To generate P,T states such that water may freeze using shock wave techniques.

2. To perform optical and wave profile measurements in shocked water to examine and

characterize changes due to freezing.

3. To construct a model that describes liquid and solid water under shock compression

as well as the mixed phase state.

4. To study the nucleation mechanism (i.e. homogeneous versus heterogeneous) gov-

erning shock induced freezing.

5. To assess the characteristic time and length scales for freezing.

High pressure states were generated in liquid water using multiple shock wave com-

pression, a method that has been applied to other liquids [9, 10]. In this technique, a thin

water sample confined between two optical windows is subjected to multiple shock com-

pression using plate impact. This process approximates isentropic compression and results

in substantially lower temperatures than those in single shock compression. With this tech-

nique, water can be compressed to a state where ice VII [11] is more stable than the liquid

phase.

2

Time resolved optical transmission measurements [9] were performed to detect the

presence of scattering caused by liquid-solid coexistence. Optical imaging experiments

were also performed to examine and characterize morphological features during freezing.

Wave profiles were measured using laser interferometry [12] to obtain time-resolved me-

chanical changes resulting from freezing.

A mixed phase water model was constructed to simulate reverberation loading and

to investigate time dependent changes caused by freezing. Complete equations of state

for the liquid and solid phases were constructed using published data and thermodynamic

consistency requirements [10]. Rules governing the mixed phase state and a time depen-

dent transformation law were incorporated in the water model to examine phase transition

dynamics [13–16]. The water model was incorporated into a one dimensional wave propa-

gation code [17] for modelling the reverberation experiments.

1.2 Organization of Subsequent ChaptersChapter 2 presents a general overview of the relevant properties of liquid and solid

water. A discussion of the freezing process is also given with a review of previous stud-

ies on rapid freezing in water. The specific scientific questions relevant to this work are

summarized with a discussion of the overall experimental approach.

Chapter 3 describes the experimental details of the optical transmission, optical

imaging, and wave profile measurements made in this work. Construction, setup, and

calibration procedures are given for each type of experiment. Chapter 4 presents the results

from these experiments.

Chapter 5 discusses a method for constructing the complete equation of state for a

single phase. This method is used to develop models for liquid and solid water. Rules for

treating a mixed phase system are postulated along with a time dependent transition rate to

model the freezing process. The use of this model in a one dimensional wave propagation

3

code is also discussed.

In Chapter 6, the experimental results are analyzed to show that freezing can occur

on nanosecond time scales if suitable nucleation sites are present. The relevant time and

length scales of the transition are also discussed. An overall summary of this work and the

resulting conclusions are given in Chapter 7.

Appendix A contains mechanical drawings for all components used in this work.

Appendix B presents specifications and mechanical models for all window materials used

in the experiments. Appendix C describes the preparation and characterization of water

samples. Appendix D contains supplemental data not presented in Chapter 4. A brief

review of optical scattering theory and its application to the transmission experiments is

presented in Appendix E. The calculation of particle velocity from laser interferometry

measurements is discussed in Appendix F. Details of the mixed phase water model are

contained in Appendix G. The effects of heat dissipation during the solidification of a

liquid are discussed in Appendix H.

4

References for Chapter 1[1] H.B. Callen. Thermodynamics and an introduction to statistical mechanics. Wiley,

New York, 2nd edition, (1985).

[2] M.D. Knudson.Picosecond electronic spectroscopy to understand the shock-inducedphase transition in cadmium sulfide. Ph.D. thesis, Washington State University,(1998).

[3] H.T. Hall. The synthesis of diamond.J. Chem. Ed.38, 484 (1961).

[4] B.J. Mason.Clouds, Rain, and Rainmaking. Cambridge University Press, London,(1962).

[5] R.E. Lee, G.J. Warren and L.V. Gusta, editor.Biological Ice Nucleation and Its Ap-plications. American Phytopathological Society Press, St. Paul, (1995).

[6] G.E. Duvall and R.A. Graham. Phase transitions under shock wave loading.Rev.Mod. Phys.49, 523 (1977).

[7] P.G. Debenedetti.Metastable Liquids. Princeton University Press, Princeton, (1996).

[8] D. Eisenberg and W. Kauzmann.The Structure and Properties of Water. OxfordPress, New York, (1969).

[9] G.E. Duvall, K.M. Ogilvie, R. Wilson, P.M. Bellmany and P.S.P Wei. Optical spec-troscopy in a shocked liquid.Nature296, 846 (1982).

[10] J.M. Winey.Time-resolved optical spectroscopy to examine shock-induced decompo-sition in liquid nitromethane. Ph.D. thesis, Washington State University, (1995).

[11] P.W. Bridgman. Phase diagram of water to 45,000 kg/cm2. J. Chem. Phys.5, 964(1937).

[12] L.M. Barker and R.E. Hollenbach. Laser interferometer for measuring high velocitiesof any reflecting surface.J. Appl. Phys.43, 4669 (1972).

[13] Y. Horie. The kinetics of phase change in solids by shock wave compression. Ph.D.thesis, Washington State University, (1966).

[14] D.J. Andrews.Equation of state of the alpha and epsilon phases of iron. Ph.D. thesis,Washington State University, (1970).

[15] D.B. Hayes.Experimental determination of phase transition rates in shocked potas-sium chloride. Ph.D. thesis, Washington State University, (1972).

[16] J.N. Johnson, D.B. Hayes and J.R. Asay. Equations of state and shock-induced trans-formations in solid I-solid II-liquid bismuth.J. Phys. Chem. Solids35, 501 (1974).

5

[17] Y.M. Gupta. COPS wave propagation code (Stanford Research Institute, Menlo Park,CA, 1978), unpublished.

6

Chapter 2

BackgroundThis chapter presents background information relevant to the present work. The

structure and properties of liquid water are discussed in Section 2.1. Solid phases of water

and the freezing transition are discussed in Section 2.2. Previous shock wave research on

liquid water and shock induced freezing are reviewed in Section 2.3. Unresolved questions

regarding the freezing of water on nanosecond time scales are summarized in Section 2.4.

The multiple shock wave compression technique used in this work is described in Section

2.5.

2.1 Liquid water“Of all known liquids, water is probably the most studied and least under-

stood...” –Felix Franks [1]

Although water is among the most familiar substances on Earth, the complete de-

scription of liquid water continues to be an outstanding problem in modern science. A

brief overview of the microscopic structure of liquid water is presented here to summarize

the challenges that make water so difficult to understand. The remainder of this section

discusses the thermodynamic, transport, and optical transmission properties of liquid wa-

ter. This discussion focuses on a small subset of past water research. Extensive reviews of

water may be found in the ongoing series by Franks [2]; more concise reviews are given in

References 3 and 4.

7

2.1.1 Microscopic structure

Ordinary water, H2O, has a molecular mass of 18.015 g/mol. Two hydrogens are

covalently bonded to an oxygen atom with a mean O-H distance of 0.957A and an H-O-H

bond angle of 104.52 [5,6]. Electrons are not distributed equally in these bonds, leaving a

residual negative charge on the oxygen atom and partial positive charges on the hydrogens.

Water molecules may form hydrogen bonds with one another, which link a hydrogen from

one molecule to the oxygen of another molecule with a mean O-O separation of 2.74-2.76

A [5, 6]. These bonds occur in a nearly tetrahedral fashion as shown Figure 2.1. In the

normal solid phase, these bonds lead to the open crystal structure shown in Figure 2.2.

When ice melts, water molecules infiltrate the open volume of the crystal, which leads to a

higher density in the liquid phase.

Liquid water has no permanent microscopic structure, although local molecular

positions are correlated at each instant through hydrogen bonding. The average local struc-

ture is well known from diffraction studies [9], but such information does not reveal the

temporal fluctuations of molecules in the liquid. Numerous models have been proposed

during the last century to describe the structural dynamics and understand water’s macro-

scopic properties. These models generally fall into one of two classes– mixture models

and the uniformist viewpoint [10]. Mixture models can be traced back to Rontgen [11],

who proposed that liquid water is composed of ice-like molecular clusters of various sizes.

Although this approach can be used to empirically model various thermodynamic proper-

ties [12], the presence of small ice clusters is inconsistent with the observed vibrational

spectra and x-ray diffraction data [5]. Other mixture models treat liquid water as a hydro-

gen bonded structure containing numerous unbonded molecules in the interstitial volume.

Interstitial models are generally consistent with vibrational spectra and x-ray diffraction

measurements [5], but there is some opposition to the concept of distinguishable bonded

8

Figure 2.1: Hydrogen bonding in water (from Ref. 7)Oxygen atoms are shown in red, hydrogen atoms in white. Hydrogen bonds are indicatedwith dashed lines; covalent O-H bonds are represented with solid rods.

Figure 2.2: Crystal structure of ice Ih (from page 26 of Ref. 8)Oxygen positions are shown as open circles; hydrogen locations have been omitted.

9

and unbonded molecules. The uniformist description, first suggested by Pople [13], views

liquid water as a perturbed form of ice Ih, where hydrogen bonds are severely distorted but

not broken. Recent mixture models have begun to incorporate aspects of the uniformist

viewpoint by treating liquid water as a combination of bonding structures from different

ice polymorphs [14]. Molecular dynamics and Monte Carlo simulations [15] have also

been performed to study the instantaneous structure of liquid water. Such studies typically

assume an empirical potential for water [16–19] to numerically simulate an ensemble of

molecules; first principles molecular dynamics have also been applied recently to liquid

water [20]. Although numerical simulations can generate reasonable radial distribution

functions [9], the results have been inconclusive since competing interpretations can been

applied [21].

To date, there is no single description of liquid water that can account for all of

its known properties. It is generally accepted that the instantaneous structure of liquid

water is similar to ice and has a lifetime of 1-5 ps [3]; beyond that are many competing

theories and unanswered questions. Since this work focused primarily on the macroscopic

properties of liquid water, an exact description of the microscopic structure was not needed.

A qualitative understanding of macroscopic properties of liquid water can be formulated in

terms of the presence or absence of hydrogen bonding. When these bonds are present,

liquid water shows a number of anomalous properties. As the bonds are destroyed and/or

distorted, water loses its anomalous behavior.

2.1.2 Thermodynamic and transport properties

Standard conditions

The distinction between liquid water and water vapor persists up to the critical point

at 22.064 MPa and 647.096 K [25]. At atmospheric pressure, liquid water is stable for

temperatures ranging from 0 to 100 C. Near the freezing point, liquid water is anomalous

10

Table 2.1: Physical properties of liquid water at 1 bar and 298.15 K

Specific heatcp [22] 4.18 J/g·K

Density [22] 0.997049 g/cc

Sound speed [23] 1.496687 km/s

Self diffusivity [24] 2.26×10−9m2/s

Thermal conductivity∗ 0.610 W/m·K

Viscosity∗ 1.0016×10−7 N·s

Surface tension∗ 0.07198 J/m2

Electrical resistivity∗ 18.182MΩ ·cm

∗ Technical reports from the International Association for the Properties of Water and Steam(IAPWS) given in Refs. 25–28.

11

in that the density increases with temperature, so(∂v/∂T)P< 0. This trend continues

to about 4 C, above which(∂v/∂T)P> 0. Table 2.1 summarizes the general properties

of water at ambient conditions. The specific heat of liquid water is considerably higher

than most materials and nearly twice that of ice or water vapor. This anomaly is tied to

the configurational energy of the fluctuating hydrogen bond network [4]. The density and

sound speed of water are comparable to other liquids, as are the self diffusivity, thermal

conductivity, and viscosity. The surface tension, however, is 3-5 times larger in water than

most liquids [29] due to extensive hydrogen bonding. Pure liquid water has a resistivity

of 18.182MΩ ·cm; for comparison, the resistivity of tap water is on the order of10−3

MΩ ·cm [30].

Static high pressure compression

There are several practical difficulties to studying liquid water at elevated pressures.

Water at room temperature solidifies at pressures above 1 GPa, limiting the range that

can be studied under static conditions. Hot, compressed water becomes quite corrosive,

and is known to attack the gaskets and ruby pressure standards used in diamond anvil cell

experiments above 200 C [31]. There is evidence that water can etch diamond at 5 GPa for

temperatures near 1300 C [32], completely dissolving diamond at 1500 C. It is generally

accepted that the application of high pressure disrupts the hydrogen bonding network in

water, eliminating many of the anomalies observed at atmospheric pressure [33].

An extensive compilation of thermodynamic measurements of liquid water prior to

1990 may be found in Ref 34. Much of that work focused on measuring specific volume

as a function of temperature and pressure. High pressure volumetric studies of liquid water

date back to the pioneering work of Bridgman [35], who measured the specific volume

of liquid water up to 3.5 GPa and 175 C [36]. Numerous other measurements of specific

volume have been reported, mostly restricted to pressures below 1 GPa. Aside fromv(T,P),

12

the thermodynamic properties of liquid water at high pressure are not well known beyond

1 GPa. Many thermodynamic response functions, such as specific heat, are difficult to

measure under high pressure and have not been extensively studied beyond 0.1-0.2 GPa.

One exception to this is the acoustic sound speed, which has been measured directly in

liquid water for pressures up to 1 GPa [37,38]. Optical measurements of sound speed have

been made in diamond anvil cell studies up to 3.5 GPa [31, 39]. The transport properties

of liquid water are largely unknown for pressures beyond 1 GPa. Measurements of the

transport properties below 1 GPa show a number of extrema that are not observed in other

liquids [33]. Overall, the viscosity and thermal conductivity of liquid water increase with

pressure, while the self diffusivity decreases with pressure. Recent measurements indicate

that the thermal conductivity of water increases with pressure up to 3.5 GPa. The thermal

conductivity of water at elevated temperatures is known to increase with pressure up to 3

GPa [40].

Numerical simulations have been used to estimate the thermodynamic properties

of liquid water at high pressures. Madura et al. [41] applied Monte Carlo techniques to

calculate the room temperature density to within 2% of the experimental measurements up

to 1 GPa; the correct qualitative pressure trends forcP, (∂v/∂T)P, and(∂v/∂P)T were also

obtained. Tse and Klein [42] used molecular dynamics to simulate liquid water over the

same range and were able to reproducev(T,P) within a few percent. Molecular dynamics

studies of liquid water above 1 GPa have been reported by Brodholt and Wood [43] and

Belonoshko and Saxena [44]. The former study covered the 300-2300 K and 0.05-4 GPa

range, while the latter dealt with the 700-4000 K and 0.5-100 GPa range. The validity of

these simulations is difficult to determine given the limited experimental data. Wiryana

et al. [39] used high pressure sound speed measurements to argue that the intermolecular

potential in numerical simulations must be refined to account for increased density of water

at high pressures. Since these potentials are typically optimized to match the properties of

13

liquid water at ambient conditions, it is not clear that numerical simulations can provide

accurate high pressure thermodynamic information.

Thermodynamic models of liquid water

Several thermodynamic tabulations are available for temperatures up to 1023 K

and pressures of 1 GPa [45, 46], although these are of limited use outside the tabulated

range. For temperatures much higher than 100 C, water has been modelled as a Redlich-

Kwong fluid [47, 48], an extension of the van der Waals fluid [49]. A variety of water

models have been constructed by assuming constant isobaric or constant isochoric specific

heats [50–53]. Thermodynamic properties of water may also be found by integrating the

thermodynamic consistency relationships involving the acoustic sound speed (c) and the

isobaric specific heat capacity (cP).

(∂v∂P

)

T= −v2

c2 −TcP

(∂v∂T

)2

P(2.1)

(∂cP

∂P

)

T= −T

(∂ 2v∂T2

)

P(2.2)

Vedam and Holton [54] first applied this method to liquid water at pressures below 1 GPa

and temperatures of 0-100 C. Similar constructions have been made for the 1 atm-0.1 GPa

pressure range and temperatures of 0-150 C [22, 55, 56]. Wiryana et al. [31, 39] extended

this calculation to pressures of 3.5 GPa and temperatures of 200 C.

Another approach to modelling liquid water is to construct a functional form for

the specific Helmholtz free energyf (T,v) with a set of adjustable parameters. The model

parameters can then be adjusted to optimally match published thermodynamic measure-

ments. This approach was used by Saul and Wagner [57] to create an equation of state for

water containing 58 adjustable parameters; an updated version of this model has since been

published [58, 59]. The model is reported to be valid for temperatures up to 1273 K and

pressures of 25 GPa. Within its intended range, this model is probably the most accurate

14

thermodynamic description of water. The accuracy and stability outside that range are not

immediately clear.

2.1.3 Optical transmission properties

Standard conditions

Light transmission through liquid water is determined by the absorption coefficient

α, whereI = I0e−αx. This coefficient is related to the complex dielectric functionε =

ε ′+ iε ′′ [60].

α2 =8π2ε ′

λ 2

(1+

(ε ′′

ε ′

)2)1/2

−1

(2.3)

Light absorption occurs for nonzero values ofε ′′. Figure 2.3 shows the functionsε ′(λ ) and

ε ′′(λ ) for liquid water. In the range of 0.2-1µm, the value ofε ′′ is much small thanε ′, so

Equation 2.3 can be simplified.

α ≈ 2πλ

ε ′′√ε ′

(2.4)

In this region,α is on the order of10−3− 10−4 cm−1, which means that water samples

less than about 100 cm are effectively transparent. The refractive index of watern≈ √ε ′

is about 1.33-1.34 in the visible range (λ = 400−700nm) [61,62].

Water strongly absorbs light on either side of the visible spectrum. Below 180 nm,

the optical response is dominated by electronic excitations [64]. The optical properties in

the 1-100µm range are determined by motions of the water molecule [63]. Vibrational

modes, which consist of a symmetric OH stretch, an anti-symmetric OH stretch, and a

bending mode and their overtones, contribute to optical absorption in the 1-10µm region.

Molecular rotations contribute to absorption at 17µm; at 62µm, translational vibrations

are active. At very long wavelengths (λ > 100 µm), dissipation occurs due to thermal

randomization of the permanent molecular dipoles. Ice thus absorbs far less light at long

15

Figure 2.3: Dielectric functions of liquid water (from pg. 275 of Ref. 63)Dotted lines mark the 400-700 nm visible range.

16

wavelengths due to the restricted dipole motion in the crystal lattice. The optical properties

of liquid and solid water are fairly similar at wavelengths shorter than about 10µm. In the

visible region, normal ice has a refractive index of about 1.30-1.32 [65], which is slightly

lower than that of liquid water. This difference results from lower density of ice, which

leads to a lower refractive index through the Clausius-Mossotti relation [60].

Static high pressure conditions

All available experimental evidence suggests that water retains its transparency un-

der static high pressure. Optical microscopy studies of water have verified transparency

for pressures of 0.1-3 GPa [66–68]. Careful studies of the refractive index of liquid water

under high pressure are more limited. Several empirical formulations have been developed

for the refractive index below 0.1 GPa [14,62]. Vedam et al. [69–71] measured the refrac-

tive index of liquid water up to 1.1 GPa and found a linear relationship with the Eulerian

strainε.

n(v) = n(v0)−Aε whereε ≡ 12

[1−

(v0

v

)2/3]

(2.5)

The value ofA is approximately 1.01 forλ=546.1 nm.

2.2 Solid waterWater exists in many different solid phases at different pressures and temperatures.

A brief overview of these phases is presented here; more general reviews are given in

Refs. 65 and 72. Solid nucleation is then discussed to demonstrate that freezing is an

activated process, which requires some time to initiate. Previous studies of the limiting

time scales of freezing in water are also summarized.

17

−50

0 50 100

150

200

250

300

350

400

450

500

T (oC)

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

200

250

300

350

400

450

500

550

600

650

700

750

800

P (G

Pa)

T (K)

vapo

rliq

uid

Ih

VII

VI

VIII

IIIII

V

critical p

oin

t

amb

ient

con

ditio

ns

Figure

2.4:E

quilibriumphase

diagramofw

ater[29,72–76]

18

2.2.1 The phase diagram of water

Figure 2.4 shows an equilibrium phase diagram for water [29, 72–76]. The term

equilibrium is used here to indicate phase stability. At equilibrium, water assumes the most

stable phase, which for fixed temperature and pressures is defined by the minimum in Gibbs

free energy [49]. The familiar form of ice is properly known as ice Ih due to its hexagonal

crystal structure. Ice Ih is less dense than liquid water and melts with the application of

pressure. With sufficient compression, water can transform to a number of different ice

phases that are denser than the liquid phase. Ices II, III and V, which exist at temperatures

below 0 C and pressures 0.2-0.5 GPa, were discovered in the early 20th century [35].

Bridgman [35,77] discovered two phases, ice VI and VII, which are stable above 0 C. At

ambient temperature, liquid water transforms to ice VI above 0.9 GPa. Ice VII is stable

above 2 GPa and for pressures up to at least 50 GPa. When cooled below 0 C, ice VII

transforms to ice VIII. Ice IX, which is not related to the fictional ice IX of Vonnegut [78],

is a form of ice III that exists below -100 C [79]. Ice phases X, XI, and XII have also

been reported, although there is some skepticism that all of these phases are distinct from

ice VII [80]. Computational studies have predicted new ice phases that have not been

experimentally observed [81]; such phases are not generally assigned a Roman numeral

until their crystal structure is established by diffraction and/or spectroscopy measurements

[72]. High pressure ices are optically transparent like ice Ih, but have larger refractive

indices than liquid water due to their higher density [82,83].

Water may also exist in a number of metastable phases that do not reflect the true

energy minimum. For example, liquid water can often be found in the solid domain of

P,T space [35], in which case the liquid is said to be in a supercooled state (i.e. below the

melting temperature at fixed pressure). Some high pressure ices can exist at atmospheric

pressure if cooled in liquid nitrogen prior to decompression [84]. There exist ice structures,

19

such as ice IV [35], which are not stable under any conditions. Ice Ic, a cubic form of ice

Ih, can be formed at atmospheric pressure by vapor deposition of water on a substrate

cooled to temperatures of∼133-173 K [65]; this phase is irreversibly converted to ice Ih

by heating. A variety of amorphous ice phases may also be produced with different cooling

and compression techniques [72, 85]. The existence of metastable phases implies that the

equilibrium phase diagram shown in Figure 2.4 is an incomplete description of water. The

microscopic structure of a water sample depends on its complete P,T history.

2.2.2 Solid nucleation from the liquid phase

Metastable phases are the result of an energetic barrier that prevents the stable phase

from forming. Classical nucleation theory relates this barrier to the Gibbs free energy

differences between the liquid and solid phases. A brief review of nucleation theory is

presented here; more extensive treatments may be found in References 86 and 87.

Classical nucleation theory

The following discussion follows the review by Walton [86]. Consider a massM

of liquid water that is metastable with respect to some solid phase at fixed temperature and

pressure. The specific Gibbs free energy of the liquid phase (gL) is higher than that of

the solid phase (gS), but the entire mass cannot freeze simultaneously because that would

require long range, coordinated molecular motion. Solid formation instead begins with

small nuclei that form within the liquid. Figure 2.5 shows a system where a nucleus of mass

m has formed. The stability of this system is related to the Gibbs free energy difference

∆G = GL+S−GL needed to create the nucleus. This energy is a combination of the bulk

energies of each phase and the energy required to maintain the solid-liquid interface, which

has an areaA and surface tensionσ .

∆G = [(mgS+(M−m)gL +σA]−MgL =−m(gL−gS)+σA (2.6)

20

r

solid(mass=m)

liquid(mass=M-m)

Figure 2.5: Formation of a solid nucleus in a liquid

0

0

Solid radius r

∆ G

(r)

r*

∆ G*

Figure 2.6: Stability of a solid nucleus as a function of size

21

If the nucleus is treated as a sphere of radiusr, ∆G may be written in terms of the solid

size.

∆G(r) =−4π3

ρS(gL−gS)r3 +4πσ r2 (2.7)

A plot of ∆G(r) is shown in Figure 2.6. Competition between ther2 andr3 terms creates a

maximum value of∆G∗ whenr = r∗.

r∗ =2σ

ρS (gL−gS)(2.8)

∆G∗ =16π

3σ3

ρ2S (gL−gS)2

(2.9)

Nuclei smaller thanr∗ are not thermodynamically stable since small fluctuations will tend

to force the system back to the pure liquid state. When an embryo larger thanr∗ is formed,

it will tend to grow larger and cause the entire system to solidify.

Freezing begins when a single water molecule adheres to a solid nucleus of sizer∗.

The equilibrium number density of these nucleiN(r∗) is defined by a Boltzmann distribu-

tion related to the magnitude of∆G∗.

N(r∗) ∝ exp

(−∆G∗

kT

)(2.10)

Freezing from any specific nucleus is a random event, but an average nucleation rate per

unit volumeJ can be defined for a large ensemble of metastable water samples. This rate

is proportional toN(r∗).

J = J0 exp

(−∆G∗

kT

)

= J0 exp

(− B

(gL−gS)2

)whereB≡ 16πσ3

3ρ2SkT

(2.11)

22

The characteristic time for a single nucleation eventtn is inversely proportional to the nu-

cleation rate and volumeV of each sample in the ensemble.

tn =1

J0Vexp

(B

(gL−gS)2

)(2.12)

J0 is related to microscopic details of the nucleation process and is assumed to be constant

here. For simplicity,B is also assumed to be constant. Figure 2.7 shows the variation

in nucleation rate and nucleation time as a function ofgL − gS. Along the equilibrium

phase boundary,gL = gS, so the nucleation rate is zero and an infinite time is required

for solid formation. As the liquid moves further into the solid domain,gL−gS increases1,

resulting in a higher nucleation rate and a shorter nucleation time. The dramatic decrease in

nucleation time creates an observational limit on supercooling possible in a liquid sample.

For example, water at atmospheric pressure freezes within a fraction of a second below

about -40 C, so this temperature is considered a lower limit for experiments with times

scales≥ 1 s. However, the value oftn does not go to zero asgL−gS→∞ because nucleation

always requires a finite time, as shown in Figure 2.7.

Thus far, it has been assumed that all portions of a metastable liquid are equivalent,

a situation known as homogeneous nucleation. Most phase transitions are not actually

initiated in this way, but instead rely upon heterogeneous nucleation that occurs in certain

regions of the sample [86]. The boundaries of the water sample may have a lower surface

tension with the solid phase than the liquid phase, thus reducing the value of∆G∗ (Equation

2.6) and making freezing more likely. The presence of dispersed foreign material may also

assist freezing in the same way. Experimental studies of metastable liquids are typically

limited by the most effective heterogeneous nucleator in the system. Only with extensive

care can all heterogeneous nucleations sites be removed to allow the system to transform

1To a first approximation, the value ofgL−gS increases linearly with temperature and pressure for isobaricand isothermal (respectively) supercooling [88].

23

gL−g

S

Nucleationtime

Nucleation rate

Figure 2.7: Variation of nucleation time and rate withgL−gS

24

via homogenous nucleation.

Heterogeneous nucleation in liquid water

A variety of substances assist freezing in water through heterogeneous nucleation.

Although freezing can occur at any temperature below 0 C (at 1 atm), foreign substances

typically induce freezing in supercooled water at a specific temperature [88–94]. This

temperature is defined by averaging the results from an ensemble of water samples or by

repeated freezing of a single sample. Silver iodide, first studied by Vonnegut [95], is the

best known freezing agent, inducing solidification at temperatures above -10 C (reports

vary from -8 to -4 C) at atmospheric pressure. Numerous inorganic crystals, particularly

iodides and sulfides, and various minerals can nucleate ice at temperatures above -20

C [96, 97]. For example, ice nucleation in natural snowflakes commonly occurs around

airborne silicate particles [98,99], although pure silica (SiO2) is considered to be a poor ice

nucleator unless it is exposed to certain chemical and/or heat treatments [100–102]. Water

soluble alcohols tend to inhibit freezing, but alcohol monolayers can be effective freezing

nucleators [90,103]. Amino acid crystals [104] as well as some bacteria, plants, and fungi

[105] have also been found to be suitable freezing agents. Various organic molecules are

able to nucleate ice phases II-VI at high pressures [106,107].

Understanding the actual mechanism for ice nucleation on a substrate is not a sim-

ple matter. Traditionally, it has been thought that water freezes effectively on hydrophillic

substrates that have crystal structures similar to ice. The nucleating effectiveness of silver

iodide and its similarity to ice Ih [95] partially motivates this viewpoint, although silver

iodide is largely hydrophobic [100]. Electric field effects are another consideration in

nucleator effectiveness. The strong dipole moment [4] of water molecules tend to align

with external electric fields, producing a more ordered structure that may assist the freez-

ing process. Externally applied fields have been shown to induce ice nucleation in cloud

25

chamber studies [108] as well as bulk water samples [65, 109]. The nucleating effective-

ness of silver iodide may also increase in the presence of an external field [110], although

some have argued that the highest effectiveness of silver iodide occurs when the surface

is uncharged [111, 112]. The latter argument is based on the fact that ice Ih is proton dis-

ordered, so the molecular dipoles have no common orientation, whereas an electric field

will tend to align these dipoles. This difficulty may be overcome, however, if the field is

nonuniform. Nucleation studies with ferroelectric barium titanate [113] and polar amino

acid crystals [104] indicate that alternating positive and negative surface charges that exist

in microscopic cracks of a solid substrate can promote freezing in supercooled water. For

many freezing substrates, there are reports of a “memory effect” [90, 114, 115], which is

observed when the nucleation effectiveness increases after previous freezing has occurred.

This observation suggests that a monolayer of solid ice resides on the substrate after the

water sample is melted. If the temperature is not raised too high above the melting point,

the substrate tends to nucleate ice more readily the next time it is exposed to supercooled

water.

2.2.3 Freezing time scales in water

Several freezing time measurements have been reported for supercooled liquid wa-

ter at atmospheric pressure [89,91–93,116]. In these measurements, liquid water is held at

a temperatureT < 0 C until freezing is observed. The time measured for a single freezing

event is random, so multiple samples or heating/cooling cycles are performed to find an av-

erage time dependence. Measured freezing times in such experiments range from102−104

seconds, even in the presence of a silver iodide nucleation seed [91]. The lower limit to the

freezing time is the time required to cool the liquid sample, which is on the order of several

seconds.

26

While isobaric cooling ultimately leads to freezing in liquid water, heat conduc-

tion limits the rate at which temperature can change, and thus governs the freezing time

scale. One way to circumvent this difficulty is by using small samples, which reduces the

time necessary for thermal diffusion. For example, cooling rates of107−1010 K/s can be

produced by spraying water droplets (1-10µm diameter) into a cryogenic medium [117].

Molecular water clusters also allow rapid cooling and can show signs of solid structure

(measured by electron diffraction) on 10-30µs time scales [118]. Mechanical compression

is another way to induce freezing in water. It is often observed that mechanical distur-

bances may initiate freezing in water clouds [119], although there is some debate on the

mechanism of this process. Vonnegut [119, 120] argued that the adiabatic expansions that

follow compression reduce temperatures to the homogeneous nucleation limit. Shock tube

measurements by Goyer et al. [121] suggested that rapid compression of supercooled water

causes freezing through cavitation in water droplets. Subsequent studies [122, 123] do not

support the cavitation conclusion, and suggest instead that supercooled water droplets in

shock tubes freeze in the presence of hydrophobic surfaces. Freezing has been reported

when bullets of sufficient velocity are fired into liquid water [124]. There are also claims

of freezing from shock waves created during the collapse of a gas bubble in liquid wa-

ter. Hickling [125] suggested that this collapse produced pressure states compatible with

high pressure ice phases on nanosecond time scales. In that work, it was argued that tran-

sient freezing during bubble collapse could explain the enhanced sonoluminescence and

the reduced substrate erosion for liquid water cooled near 0 C. Ohsaka and Trinh [126]

obtained microscope images of bubble collapse with a high speed video camera. These

images indicate a solidification process on10−1−10−2 s time scales.

The shortcoming of these dynamic cooling and compression experiments is that the

thermodynamic history of the water sample is poorly characterized. For example, the peak

pressure generated by bubble collapse is thought to be anywhere from 1-10 GPa, depending

27

on how the collapse is modelled [126]. In the process, water is exposed to a complex series

of compression and tension states. Similar uncertainties exist in fast cooling experiments,

where direct measurements of temperature have not been made. Real time diagnostics

are not employed in most dynamic cooling or compression experiments, so the true time

dependence of freezing is not known.

Molecular dynamics simulations represent another approach to studying the lower

time limits of freezing in liquid water. Svishchev and Kusalik [127,128] demonstrated that

water molecules simulated with the TIP4P [16] potential can freeze within 200 ps in the

presence of an external electric field; similar results were obtained by Borzsak and Cum-

mings [129]. Simulations of water confined within a small pore can also solidify on 1-10 ns

time scales [130]. Xia and Berkowitz [131] found that freezing occurred in less than 1 ns

for water confined between platinum substrates if the substrates carried a sufficient electric

charge. Matsumoto et al. [132] reported that water can form stable ice clusters about 200

ns after being cooled to 230 K without an external field or surface perturbations. These

clusters grow and cover the entire sample within about 100 ns after their first appearance.

Freezing has recently been observed in simulated water on 3-30 ns time scales [133] using

the TIP5P potential [19], although there have been some suggestions that this model does

not represent water as well as the TIP4P potential [134].

2.3 Previous shock wave experiments on waterThe term shock wave is used here to denote a rapid, planar compression that trav-

els with a constant velocity. Such a compression is described by the velocity of the shock

front and the particle velocity, density, longitudinal stress, and internal energy behind the

front. These variables are linked to the conditions ahead of the front through the Rankine-

Hugoniot jump conditions [135]. States obtained by shock compression lie along the Hugo-

niot curve [136], which may be derived from a series of shock wave experiments or from

28

a complete equation of state [137]. The rapid mechanical changes produced by shock

compression are useful for studying the dynamics of phase transitions [138]. Two types

of shock wave experiments are reviewed in this section. First, general shock wave ex-

periments on liquid water are summarized. A review of previous shock induced freezing

studies is then presented.

2.3.1 Shock wave experiments in liquid water

Numerous shock wave experiments have been performed on liquid water. The most

notable of these is the work of Walsh and Rice [139], who reported the first Hugoniot

measurements for liquid water. Those measurements showed a smooth Hugoniot in the

3-42 GPa range. Al’tshuler et al. [140] disputed this conclusion and claimed that there

was break in the Hugoniot near 11 GPa. Subsequent Hugoniot measurements [141–146]

generally agree with the Walsh and Rice data and do not suggest any such discontinuity.

Hugoniot measurements provide a mechanical description of the shocked state,

but do not provide information about the temperature in that state. Very few tempera-

ture measurements have been reported for shocked water. Optical pyrometry was used by

Kormer [147] to infer shock temperatures in water in the 30-40 GPa range. This method

was also used by Lyzenga et al. [148] for water shocked to pressures of 50-80 GPa. Flu-

orescence thermometry [149] results have been reported for water shocked below 1 GPa.

No measurements have been reported for intermediate pressures (1-30 GPa), where shock

temperatures must be calculated from a complete equation of state. These calculations

can be constrained using the results of double shock compression experiments. Walsh and

Rice [139] used this method to determine the quantity(∂h/∂v)p, whereh is the specific

enthalpy, from double shock compressions and calculated shock temperatures for pressures

in the 0-45 GPa range [50]. A similar method was used by Mitchell and Nellis [145] to

specify(∂P/∂e)v in the 30-230 GPa range.

29

Shock temperatures of water compressed beyond 10 GPa are on the order of103

K, so it is possible for water molecules to decompose into ionic species. The presence of

these ions can be detected by their contribution to the electrical conductivity of the water

sample. Conductivity measurements [145, 150–152] demonstrate that the conductivity of

liquid water increases dramatically for shock pressures in the 10-20 GPa range. Raman

spectroscopy of shocked water [153, 154] indicates that this conductivity arises from a

decomposition of water into ionic hydrogen H+ and hydroxol OH−. That work also found

significant hydrogen bonding in water compressed to 12 GPa, whereas these bonds are

largely destroyed at 26 GPa.

2.3.2 Shock induced freezing

There has been a long standing debate as to whether it is possible for water to

freeze under shock compression. Snay and Rosenbaum [155] calculated aT−P Hugoniot

curve that suggests that shock compression crosses the ice VII phase boundary at roughly

2.7 GPa. Figure 2.8 shows this curve along with the calculations of Rice and Walsh [50],

which showed that the Hugoniot might access the ice VII phase near 3.5-4.2 GPa. The

differences between these curves lead to some uncertainty about the possibility of freezing

under shock wave compression. Even if the proper P,T conditions could be attained, it does

not immediately follow that freezing will be observed in a shock experiment. Freezing is

preceded by a metastable period, which may be on the order of seconds (Section 2.2.3), un-

til stable ice nuclei form. Since time durations in shock experiments are limited to the order

of microseconds, it is quite possible that only the metastable liquid state can be observed

in these experiments.

The first experimental study of shock induced freezing in water was made by Walsh

and Rice [139]. In their experiments, light was passed through the shocked liquid and

reflected from the metal driver plate, which was covered with a rectangular grid. The

30

0 2 4 6 8 10 12 14300

400

500

600

700

800

900

1000

1100

1200

P (GPa)

T (

K)

Liquid

Ice VII

VI

Snay andRosenbaum

Rice and Walsh

Figure 2.8: Calculated P,T states of shocked liquid waterDark lines indicate the equilibrium phase boundaries of water [74,75].

31

image of this grid was used to determine the state of the water sample. Clear images

of the grid were expected while the sample remained a pure liquid, while image quality

degradation would suggest freezing in the sample. No loss in image clarity was detected

for water shocked to 3-10 GPa over 20µs time scales, suggesting that either the Hugoniot

does not access the ice VII region or that the metastable lifetime is much longer than 20

µs. These results were challenged by Al’tshuler et al. [140], who reported a loss in optical

transparency and a discontinuity in the Hugoniot slope near 11 GPa. These observations

have been used to argue that liquid water coexists with ice VII at pressures above 10 GPa,

which is consistent with reports of unusually large viscosity for shock states near 15 GPa

[156]. Revised Hugoniot calculations [157] predict that shock loading crosses the ice VII

phase boundary somewhere between 4 and 8 GPa, creating a mixed phase state to 13 GPa.

Rybakov [158] has suggested that the Hugoniot of liquid water is composed of three distinct

regions: a liquid region below 3 GPa, a mixed liquid-ice VII region between 3 and 10 GPa,

and another pure liquid region above 10 GPa.

The difficulty with the shock freezing discussion above is that it assumes that single

shock loading crosses the ice VII boundary near 3 GPa. However, such a crossing has

never been observed in optical studies. Furthermore, the arguments for freezing have relied

on a piecewise linearUs− up Hugoniot, but it is well established that most liquids have

nonlinear Hugoniots [159]. Hamann and Linton [151] have made an alternate suggestion

that the observed changes are the result of increasing ionization in water rather than a phase

transition. Furthermore, shock wave studies of water saturated porous rocks do not show

signs of freezing, although freezing in such systems is readily observed under hydrostatic

compression [160].

The argument for freezing under shock compression is further weakened by the

fact that the optical changes reported by Al’tshuler et al. [140] have never been repro-

duced [161]. Zel’dovich et al. [162] found no change in optical transparency for water

32

shocked to 4-14 GPa. A similar conclusion was reached by Kormer et al. [147, 163], who

found that optical changes could only be observed when water was exposed to two shock

compressions. In measurements by Kormer et al. [147,163], a change from specular reflec-

tions to diffuse scattering was observed10−7−10−6 s after double shock compression to

the ice VII region. Experiments by Yakushev et al. [164] also showed that light reflected

from the metal drive plate was attenuated in doubly shocked water, but only when a bare

lithium fluoride window was exposed to the water. If this window was covered with a pro-

tective lacquer, the reflection change was not observed. Those results were interpreted as

an indication of heterogeneous ice nucleation or dissolution at the lithium fluoride-water

interface2. Both Kormer et al. [147, 163] and Yakushev et al. [164] reported that optical

changes observed in liquid water were not seen in other doubly shocked materials such as

alcohol, glycerine, nitrobenzene, and Plexiglas.

Overall, there is little evidence to suggest that water or any other liquid freezes

under shock compression [138]. In the case of liquid water, it is not clear that single shock

loading from ambient conditions is even thermodynamically possible; the water model

developed in Chapter 5 suggests that it is not. The strongest support to date for freezing is

from the double shock experiments of Kormer et al. [147,163], although the optical results

are crude and somewhat ambiguous. These results are also unclear on the nature of surface

effects, which Yakushev et al. [164] suggest are necessary for optical changes to occur.

Even if one accepts that freezing can happen in double shock loading, no indication of a

first order phase transition has been reported. Sheffield [166] measured particle velocity in

a doubly shocked water sample, but did not find any evidence of a volume change.

There is some evidence that shock wave loading of ice Ih can cause melting and

refreezing to a high pressure ice phase. Using embedded electromagnetic particle velocity

gauges, Larson [167] found that plate impact experiments with single and polycrystalline

2LiF dissolves in water at ambient conditions [165].

33

ice samples produced time dependent wave profiles. Using a Lagrangian analysis [168] of

the data, he argued that these changes were caused by the melting of ice Ih and subsequent

refreezing to ices VI and VII. Tchijov [169] performed wave propagation calculations with

a water model containing the liquid phase as well as ices Ih, II, III, V, and VI; this model

has since been extended to include ice VII [170]. Those calculations predict that shock

loading in ice Ih produces a dynamic mixture of all of these phases over several microsec-

onds. Given the complexity of the problem and the extremely limited experimental mea-

surements, it is difficult to support or deny these claims. A key difference between shocked

ice research and the issue of shock induced freezing in liquid water is that solid ice phases

are always present in the former case, which can provide nucleation sites for the refreezing

process.

Based on an independent examination of all relevant data, it may be concluded that

freezing in shock compressed water is an open question. This is similar to the conclusions

reached by Duvall and Graham [138], who stated that “The case for freezing in shock is

not a strong one...”.

2.4 Unresolved questions and approachThe fundamental purpose of this thesis is to examine the lower time limits of freez-

ing in liquid water. The discussion in Sections 2.2.3 and 2.3.2 indicate that there is no

conclusive evidence for freezing on time scales less than10−7 s, and that there are cer-

tain obstacles in producing the proper conditions for freezing on such time scales. The

following questions were posed in this work.

1. Using shock wave techniques, is it possible to create a state where liquid water can

freeze?

2. Do consistent signs of freezing occur on10−9− 10−6 s time scales? Are these

34

changes consistent with a first order phase transition?

3. Does freezing occur through heterogeneous or homogeneous nucleation on these

times scales?

Question #1 is addressed with quasi-isentropic compression obtained through mul-

tiple shock compression, a technique described in Section 2.5. With the development of

a complete equation of state for water, it is demonstrated that multiple shock compression

can produce states of possible freezing while single shock loading cannot.

Question #2 is addressed with the use of optical transmission, optical imaging, and

wave profile measurements during multiple shock compression experiments. The optical

diagnostics show that water changes from a clear liquid to a complex liquid-solid mixture.

Wave profile measurements indicate that water initially remains a pure liquid, but then

transforms into a material that is denser than the compressed liquid.

Question #3 is addressed with multiple shock compression experiments for water

confined in different types of windows. It is shown that heterogeneous nucleation occurs at

the water-window surfaces, but that the transformation is not confined to these surfaces.

2.5 Multiple shock wave compressionAs discussed in Section 2.3.2, it is unclear whether shock compression can produce

temperatures where freezing is thermodynamically possible. The difficulty arises from the

fact that shock compression is an irreversible, adiabatic process [137], which leads to sig-

nificant temperature increases. Multiple shock wave compression can be used to minimize

the entropy produced during rapid compression, leading to considerably lower tempera-

tures than single or double shock compression. This technique has been applied in other

high pressure liquid studies [171–175] and was used in this work to generate thermody-

namics states compatible with freezing. The following discussion explains how multiple

35

shock wave compression loading is obtained in a plate impact experiment. The temperature

differences between multiple and single shock compression are also discussed.

2.5.1 Multiple shock compression and plate impact

Figure 2.9(a) shows a schematic configuration for a multiple shock experiment,

where a liquid sample is confined by two solid windows. A light gas gun [176] accelerates

an impactor that strikes the front window of the liquid cell, generating an impact longitudi-

nal stressPi and particle velocityui . The values ofPi andui are defined by the intersection

of the impactor and front window Hugoniots, which maintains continuity of stress and par-

ticle velocity at the impact interface [137]. This construction is shown in Figure 2.9(b) for

the case where the impactor, front window, and rear window are all composed of the same

material. After impact, a right going shock wave propagates through the front window and

reaches the liquid sample as shown in Figure 2.9(c). This creates a new state(u1,P1) at the

intersection of the front window unloading curve and the liquid Hugoniot. The new shock

wave propagates through the water sample and reaches the rear window, generating a re-

flected shock(u2,P2). This shock wave travels back to the front window, creating another

reflected shock(u3,P3). The process continues as shown in Figures 2.9(b) and 2.9(c) until a

peak state is reached, which is identical to the original impact state(ui ,Pi). Pressure in the

liquid sample increases in stages toPi as shown in Figure 2.9(d). The number and timing

of loading stages in this history depends upon the thickness and properties of the liquid

sample, but the peak pressure depends only upon the impact velocity and window/impactor

materials. The steady state particle velocity is also independent of the liquid sample and

equals half of the original impact velocity. Should the sample undergo some time depen-

dent transition that leads it away from(ui ,Pi), subsequent wave reflections from the front

and rear windows will lead the system back to the original steady state.

With multiple shock wave compression, it is possible to generate peak pressures

36

impactor

impact

Tim

e

(c)

shock arrivalat sample

Lagrangian position

0ii

RW

1

2

3

4

LSFW

5

6

impactor(IM)

v

(a)

liquidsample(LS)

frontwindow

(FW)

rearwindow(RW)

samplemidpoint

peak state

Stre

ss a

t sam

ple

mid

poin

t

0

Time

(d)

3

2

1

6

54

FW/RW loading

Long

itudi

nal s

tress

2

Particle velocity

v/2

65

4

3

(b) impact/peak state

v

1

IM loading/FW unloading

Figure 2.9: Multiple shock compression(a) Plate impact experiment setup(b) Loading path for a symmetric impact(c) Shock waves generated in multiple shock compression(d) Stress history at the liquid sample midpoint.

37

from 1 GPa to more than 10 GPa with a single stage gas gun (v = 0.2−1 km/s). The use of

plate impact generates a state of uniaxial strain (εi j = 0 for i, j 6= 1) throughout the system

for a short time due to inertial confinement [137]. This state is preserved until the arrival of

release waves from the free edges of the impactor. Estimates of the edge wave arrival can

be made by assuming that these waves travel at an angle of 45 from the impact direction.

For an impactor diameterD, an experimental probe diameter ofd, and a edge wave velocity

c (which is the sound speed in the shocked state), the maximum duration of uniaxial strain

is given bytmax.

tmax≈ D−d√2 c

(2.13)

More precise estimates of the uniaxial strain duration can be made using two dimensional

wave calculations [175,177].

2.5.2 Temperature advantages of multiple shock compression

At this point, it is important to distinguish an adiabatic process, which occurs in

the absence of heat flow, from an isentropic process, which occurs at fixed entropy. All

reversible adiabatic processes are isentropic, but there are adiabatic process which are irre-

versible. Shock wave compression is an example of such a process, where the magnitude

of the entropy increase is related to the change in specific volume change [137].

∆ssingle= A(v0−v)3 whereA≡ 112T0

(d2Pdv2

)

v=v0

(2.14)

Multiple shock compression is simply a sequence of individual shock waves, so entropy is

also generated in the process. For a series ofN shocks of equal volume increment, the total

entropy change is equal to a sum of terms similar to Equation 2.14

∆sreverb=N−1

∑i=0

Ai(vi−vi+1)3 =(

v0−vN

)3 N−1

∑i=0

Ai (2.15)

38

The sum in this relation scales with the number of shocksN, so the overall entropy change

scales with1/N2. In the limit N → ∞, the total entropy change goes to zero. In reality,

some entropy is produced because the first few compressions have a nonzero magnitude,

but the total entropy production is substantially lower than in single shock compression.

Multiple shock compression is therefore a quasi-isentropic process.

The difference in entropy production for multiple and single shock leads to very

different temperatures for the same peak pressure. To demonstrate this difference, consider

T = T(s,P).

dT =(

∂T∂s

)

P

[ds+

(∂v∂T

)

PdP

](2.16)

(∂T/∂s)P and (∂v/∂T)P are positive3, so temperature increases with both pressure and

entropy. In multiple shock loading,ds≈ 0, so only the right term contributes to the temper-

ature. However,ds> 0 in shock compression, so both terms contribute to the temperature

increases. Thus isentropic and quasi-isentropic compression lead to lower temperatures

than single shock compression to the same pressure. Figure 2.10 shows calculated Hugo-

niot and isentrope states for liquid water based on a model presented in Chapter 5. Single

shock loading does not enter the ice VII domain at any pressure in this range, while isen-

tropic compression produces states of possible freezing for pressures above 2 GPa. There

is no indication that isentropic loading will reenter the liquid domain at high pressures, so

pressure can be increased as necessary to reduce the metastable lifetime of the liquid phase.

Since isentropic loading produces the lowest possible temperatures for an adiabatic

compression, it provides a limit on the possibility of freezing on short time scales. In other

words, if freezing does not occur under isentropic loading, one can conclude that it will not

occur in any other kind of adiabatic compression.

3It is possible for(∂v/∂T)P to be negative in liquid water, but not at temperatures relevant to this work.

39

0 1 2 3 4 5 6 7 8 9 10300

350

400

450

500

550

600

650

700

750

800

Pressure (GPa)

Tem

per

atu

re (

K)

Liquid

Ice VII

VII

isentropicloading

single shock loading

Figure 2.10: Temperature advantages of multiple shock compressionDark lines indicate the equilibrium phase boundaries of water [75,178]. The Hugoniot andisentrope curves are calculated from an equation of state presented in Chapter 5.

40

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[177] J.M. Winey, (2002). Private communication regarding edge wave arrival estimates.

[178] Carl W.F.T. Pistorius and W.E. Sharp. Properties of water. Part VI. Entropy andGibbs free energy of water in the range 10-1000 C and 1-250,000 bars.AmericanJournal of Science258, 757 (1960).

52

Chapter 3

Experimental MethodsThis chapter describes the experimental techniques and instrumentation used in the

present study. Optical transmission experiments are discussed in Section 3.1. Optical imag-

ing experiments are covered in Section 3.2. Wave profile experiments are described in Sec-

tion 3.3. Technical drawings for all parts used in this study are presented in Appendix A.

Specifications and mechanical models for the window materials used in this work are sum-

marized in Appendix B. Water treatment, testing, and handling is described in Appendix

C.

3.1 Optical transmission experiments

3.1.1 Overall configuration

Optical transmission studies have been used to probe material response in plate im-

pact experiments for some time [1]. Optical transmission may be reduced due to either

absorption or scattering [2]; such changes can indicate shock induced chemical [3] reac-

tions, phase transitions [4], or inelastic deformation [5]. Since there is no optical absorption

in water for the pressures of interest (Section 2.1.3), a decrease in optical transparency in

shocked liquid water can be attributed to light scattering from liquid-solid coexistence.

Figure 3.1 shows the experimental configuration [6] used for optical transmission

measurements in the present work. Reverberation loading (Section 2.5) was obtained by

impacting a transparent window onto a liquid cell containing water. The impactor was at-

tached to a projectile, which was accelerated to the desired impact velocity using a 4” gas

gun [7]. Visible light, generated by a xenon flashlamp, was collimated with a parabolic

53

45° turning mirror

45° turning mirrors

pulsed Xe flashlamp

projectile

impactor

CCD

collimating lens

flashlampmonitorphotodiode

parabolic turning mirror

windows

targetchamberport

liquid sample

transmissionphotodiode

spectrometerstreakcamera

opticalcollection

fiber

targetchamberport

Figure 3.1: Setup for optical transmission measurementsCollimated light from a pulsed xenon flashlamp was directed through the liquid samplewith a series of mirrors. Light exiting the sample was collected with an optical fiber andrecorded. The transmission measured with this technique was constant while the watersample remains a pure liquid. The formation of ice regions created scattering and reducedlight transmission.

54

turning mirror and directed through the impactor with a series of planar turning mirrors.

This light passed through the liquid cell and was focused into an optic fiber. The optical

signal in this fiber was split into two fibers, each connected to a detection system. A photo-

diode detector provided a wavelength integrated measurement of transmitted light. Wave-

length dispersed light from the spectrometer was coupled to a streak camera and recorded

on a CCD to provide a time resolved measurement of the transmitted light spectrum. Op-

tical transmission was calculated by comparing light levels during the impact experiment

with levels measured at ambient conditions.

3.1.2 Mechanical components and assembly

Projectile construction

Two different projectile designs were used in this study. Impact velocities above

0.3 km/s were obtained using a standard 4” projectile (Figure A.1); below 0.3 km/s, a

modified projectile was used (Figure A.2). Mounting holes (Figure A.4) were drilled into

the projectile face, which was lapped perpindicular (≤ 0.1 mrad) to the projectile axis.

Figure 3.2(a) shows a cross section view of a projectile used in transmission exper-

iments. Two 45 mirrors were positioned on the projectile with aluminum mounts (Figures

A.6-A.9). The central mount also had a brass aperture (Figure A.10) and the impactor win-

dow. This aperture was spray painted black to minimize stray light reflections. The mirrors,

aperture, and impactor were bonded to the mounts with Shell 815 epoxy. After the epoxy

had set, the aluminum base of the impactor mount was machined parallel to the impactor

surface and attached to the projectile with screws. Impactor surface tilt (with respect to

the projectile axis) was checked using an autocollimator. If significant tilt was measured,

the impactor mount was removed and preferentially sanded to correct the problem. The

impactor was reattached to the projectile and checked again. Corrections to the impactor

mount were made until the measured tilt was about 0.1 mrad. Next, the turning mirror

55

(a)(b)

impactor

impactor

mount

aperture

projectilebody

mirrors

mirror

mount

Figure

3.2:T

ransmission

experimentprojectile

(a)S

chematic

crosssection.

The

impactor

andturning

mounts

arefixed

tothe

projectilew

ithscrew

s(notshow

n).(b)

Com

pletedprojectile

56

mount was fixed to the projectile. Epoxy was applied around all screws to completely seal

the projectile face. An assembled projectile is shown in Figure 3.2(b).

Liquid cells

Water samples were confined in liquid cells to provide a well defined thickness and

to seal the sample from the evacuated target chamber. Two different cell designs were used

in this study. The first configuration, cell design #1, used a press fit to hold the rear window

in the cell. Cell design #2 was used in experiments where the rear window could not be

press fitted.

Figure 3.3(a) shows a cross-sectional view of cell design #1. Cell assembly was

started by soldering 1/16” stainless steel fill tubes to the brass cell. To prevent strain on

these joints, each tube was soldered to two points on the brass body. Oxidized brass formed

during soldering was abrasively cleaned from the cell. The inside diameter of the cell was

then machined for press fitting of the rear window. For sapphire windows, the undersizing

was 0.0002-0.0003” from the actual window diameter. Undersizing for fused silica and

quartz windows was 0.0001-0.0002” to prevent damage during press fitting. Liquid fill

channels (∼0.050” wide× 0.025” deep) were cut in the cell interior from the fill tubes to

the top of the brass step. After machining, the cell was cleaned with alcohol and fitted with

a Viton O-ring. The rear window was then slowly pressed into the cell from the rear. Final

sample sizing was done by machining the brass step of the cell with respect to interior rear

window surface. The front O-ring groove was also machined with respect to brass step

(0.061” deep). 1/16” stainless steel Swagelocks connectors were added to the fill tubes.

The cell was flushed with alcohol and cleaned in an ultrasonic alcohol bath. To ensure a

good seal on the rear window, a bead of epoxy was run around the window/cell edge. The

front O-ring (Viton) and window were cleaned with alcohol and held in the cell using a

brass lock ring.

57

lock ring

rear window (press fit)

front window

fill channels

front window

rear window

brasscell body

Teflon pad0.03" thick

(b)

fill tube

O-ring

filltube

second O-ring

first O-ring

(a)

fill channelcut in brass

step

lock ring

brass stepmachined to control

sample thickness

Figure 3.3: Liquid cell assembly(a) In design # 1, the rear window is held in place with a press fit, while the front windowfastened with a lock ring.(b) In design # 2, the front window is epoxied to the liquid cell, and the rear window heldin place with a lock ring.

58

Figure 3.3(b) shows the cross-sectional view construction for cell design #2. In this

configuration, 1/16” stainless steel tubing was soldered to the front of the cell rather than

the side. Fill channels were cut perpendicular to the fill tubes. After the cell was abrasively

roughened and cleaned, the front window was set in place using Shell 815 or Epotech 301

epoxy. Care was taken to ensure that epoxy did not flow up the sides of the window and

into the sample region. Once the window was set in place, a bead of epoxy was applied

to the front side for additional strength and better sealing. Sample sizing was done by

machining the interior brass step with respect to the interior surface of the front window.

1/16” stainless steel Swagelocks were added to the fill tubes, and the cell was cleaned with

alcohol. A single nitrile O-ring was held in place with the rear window and a lock ring. A

Teflon pad was placed between the lock ring and the glass window to avoid chipping and

cracking.

All liquid cells were leak tested by applying 1 atm of overpressure for at least 12

hours. If the cell successfully maintained overpressure, the assembled cell thickness was

measured with a supermicrometer1. Sample thickness was determined from the difference

in assembled cell thickness and individual window thicknesses (measured when cell is

disassembled). Water was injected into the cell following the procedures given in Appendix

C. Once filled with water, the cell was placed in a vacuum chamber for 5-10 minutes. If no

visible changes occurred in the cell, final sample thickness measurements were made. The

liquid cell was then mounted to the target as described below.

Target construction

Figure 3.4(a) shows a schematic view of a target used in transmission experiments.

The standoff ring (Figure A.15) was used to mount the entire assembly to the muzzle of

the 4” gas gun. Both surfaces of this ring were rough lapped; the front surface was also

1Soda lime glass cells were measured when filled with liquid and sealed to minimize bowing of the 1/16”front window.

59

(a)(b)

targetplate

lensbracket

standoffring

collimating

lens tube

opticfiber

aperture

liquidcell

Figure

3.4:T

ransmission

experimenttarget

(a)S

chematic

crosssection.

Screw

s(not

shown)

areused

tom

ountthe

liquidcelland

lensbracket

totarget

plate.T

hetarget

plateis

attachedto

thestandoffring

with

Belville

washers

(notshown)

foralignm

entpurposes.(b)

Com

pletedtarget(celldesign

1)view

edfrom

theim

pactside.A

rearview

ofthetargetis

shown

inF

igure3.5.

60

fine lapped. A rough lapped target plate (Figures A.16 - A.18) was separated from the

ring by flexible Belville washers (120 apart). The liquid cell and the lens bracket (Fig-

ures A.20-A.21) were held on opposite sides of the target plate with screws. The Belville

washers were used to align the front window surface of the liquid cell to within 0.1 mrad

of the standoff ring surface. A collimating lens unit (Figure A.23) was held within the lens

bracket. This unit located the tip of an optical fiber at the focal plane of a converging lens,

so that collimated light incident on the unit was focused into the fiber. A completed target

is shown in Figure 3.4(b).

Just before the experiment was performed, two shorting pins were mounted within

the target plate for triggering and diagnostic purposes. These pins were constructed from

RG 174A/U coaxial cable mounted within a 3/32” brass tubing. The outer conductor was

soldered to the tube; the inner conductor was separated from the tube with epoxy. The

3/32” tube was soldered into a larger brass tube with a threaded exterior, which was used to

control the pin height with respect to impact surface of the liquid cell. The striking surface

of the pins were lapped with respect the axis of the threaded section. Surface variations of

the pin tips were on the order of 5µm or less.

3.1.3 Instrumentation and optical components

Flashlamp

A pulsed xenon flashlamp (Xenon Corporation model 457) provided illumination

for transmission measurements in this work. The xenon filled flashtube (Xenon Corp #S-

1156D) was powered by a 0.5µF capacitor, charged to about 11 kV, generating a peak

pulse energy of 30 J. This discharge generated light over approximately 10µs; peak output

occured 4-5µs after the lamp was triggered. The arc within the flashtube was about 3”

long and 3 mm wide. The lamp produced broad spectral output covering the ultraviolet and

visible ranges.

61

Mirrors, lenses and fibers

A 35×35 mm plane turning mirror (Edmund Scientific #45519, 4-6 waves flatness)

was used to direct collimated light toward the projectile. Two 12.5×12.5 mm turning mir-

rors (Edmund Scientific #43790, 4-6 waves flatness) were mounted on the projectile. These

mirrors were made from aluminum plated glass and optimized for visible light reflection.

The collection lens was plano-convex, 12 mm in diameter, and had a 24 mm focal length

(Edmund Scientific #32011). The lens was made from uncoated BK-7 glass. The collection

fiber (3M FT-1.0-UMN) had a 1 mm core and numerical aperture of 0.39. Smaller fibers

(Mitsubishi #STU400E-SY, 400µm core, 0.2 NA) were used to carry light to the detectors.

Photodiode detector

THOR photodiodes (Models DET200 and DET210) were used to record provide

a wavelength integrated transmission profile. The detector was sensitive to light in the

185-1100 nm range with peak sensitivity around 950 nm. Sensitivity at 600 nm was ap-

proximately twice that at 400 nm. Light from the 400µm optical fiber was either directly

incident on the detector or coupled with a 1:1 lens imaging system. This light was typically

attenuated by a factor of 10 using neutral density filters to maintain detector linearity.

Digitizers

Tektronix TDS 654C or 684C models were used to digitize electronic data and

diagnostic signals. Both digitizers collected 15000 sample points. The diagnostic digitizer

sweep was 15µs, much longer than the useful experiment duration, to acquire the entire

sequence of events (trigger pulses, flashlamp pulse, etc). The data digitizer operated for a

shorter duration (3-6µs) near the time of impact.

62

Spectrometer

Initially, a SPEX 1681C spectrometer was used to disperse light in a horizontal

plane. This was later changed to an imaging spectrometer (Spectra Pro 150, Acton Re-

search Corp). Vertical spot size was controlled by the input fiber (∼400 µm); horizontal

spot size was set by an adjustable slit (∼50 µm). The 300 lines/mm diffraction grating

(600 nm blaze) was oriented to center the 350-650 nm range on the CCD.

Streak camera and CCD

Horizontally dispersed spectrometer output was directed onto the photocathode (S-

20 with Corning 9823 UV transmitting glass) of a Cordin 160-5B streak camera. To protect

the photocathode, a mechanical shutter was placed between the spectrometer and streak

camera. This shutter was manually opened just before each acquisition and closed immedi-

ately afterwards. The photocathode output was swept vertically and converted to an optical

image output. The streak rate of the camera was adjusted to cover the desired experimental

duration (1-3µs) on the CCD.

The two dimensional optical output from the streak camera was imaged onto an air

cooled Princeton Instruments TE/CCD-512-TKBM detector. A mechanical shutter cover-

ing the CCD was opened just prior to each acquisition and is closed approximately 1 ms

after the CCD was triggered. The detector was oriented so that wavelength varies horizon-

tally and time vertically. The CCD had a 16 bit dynamic range and was binned to produce

a 256×512 (vertical×horizontal) image.

3.1.4 Experimental setup

Optical alignment

Setup of the optical transmission experiment (Figure 3.1) required that the projectile

be located near its impact position. The target was centered with respect to the projectile

63

Figure 3.5: Rear view of transmission setupThe mirror mounted to the brass rod directs light from the xenon flashlamp towards theprojectile, which has been removed here. Transmitted light from the sample is collectedby the brass lens unit and carried to the detectors through optical fiber. Two shorting pins(right side) are used for triggering and impact diagnostics.

64

and secured to the muzzle of the gas gun. Continuous laser light was coupled into an

optic fiber and sent into the collimating lens unit. This light travelled opposite to the light

path shown in Figure 3.1, passing through the sample, striking the projectile mirrors, and

emerging off-center from the gun barrel axis. The remaining planar and parabolic mirrors

were adjusted to focus this light upon the filament of the xenon flashlamp.

When the flashlamp was fired, collimated light emerged from the parabolic mirror,

passed through the sample, and was collected by the lens unit. Sample illumination was

restricted to the central 1/4” diameter by the aperture mounted behind the impactor. A

second aperture on the lens unit restricted light collection to the same region. Transmitted

light collected by the lens unit was taken out of the target chamber by 1 mm core fiber and

split into two 400µm fibers coupled to the optical detection systems. Figure 3.5 shows the

target prior to impact.

Although the impact tilt magnitude is quite small (∼0.5 mrad), the projectile can

rotate in the 4” gun barrel. If the rotation is large enough (>45), light will not strike the

projectile turning mirror and no measurement is obtained. Typically, rotation angles are

on the order of 10-15. In each experiment, the projectile was rotated through this range

to determine if adequate light intensity was obtained. If these minor rotations resulted in

insufficient light intensity, the optical system was realigned.

After the liquid cell was filled, mounted, and aligned with the target ring, it was

placed in the target chamber for final testing. All movable joints in the optical system are

epoxied and reference signals recorded. The projectile was then removed from the barrel

and loaded in its firing position.

Timing synchronization

The electronic setup shown schematically in Figure 3.6 permitted the following

sequence of events during the optical transmission experiments.

65

delaygenerator


transmissionphotodiode

DataDigitizer

trig

h

impactpin

impactingsurfaces h

trig

Diagnostic Digitizer

pulsegenerator

D4

trigger pin

projectile

v

h+D

streakcamera

spectrometer

Xeflashlamp

CCD

D2

D1

D3

Figure 3.6: Electronic setup for transmission measurementsAdjustable delays D1-D4 are set to synchronize electronic equipment.h is the impactorheight above the projectile face;D is the additional height of the standoff trigger pin;v isthe projectile velocity.

66

1. Several microseconds prior to impact, the projectile face contacted the first shorting

pin. This event generates a triangular pulse which is used to activate the diagnostic

digitizer and a delay generator.

2. Once the delay generator becomes active, delays D1 and D2 trigger the flashlamp

and CCD controller (within∼20-30 ns considering cable delays). Shortly after the

flashlamp was triggered, discharge occurred and light emission started. A shutter

covering the CCD was left open just before the experiment; triggering the controller

started a countdown for closing this shutter (∼1 ms later).

3. Roughly 2 microseconds later, delay D3 triggered the streak camera. The sweep

region of the streak camera was larger than the CCD detector, so some additional

time (1-2µs) passed before any data were recorded.

4. Delay D4 activated the data digitizer, which covered the time duration near impact.

5. The flashlamp light output reached its maximum, 4-5µs after being triggered.

6. Shortly after the flashlamp output peaked, the impactor struck the front window. This

occurred about 20% through the streak camera record. Impact appeared in the optical

record as an intensity jump resulting from from the closing of reflecting interfaces.

A second shorting pin was struck at the moment of impact to produce an electrical

fiducial.

The impactor surface was a distanceh above the projectile face, so the impact short-

ing pin must beh above the front window. The trigger shorting pin must then be a distance

h+D above the front window.D was determined using the configuration shown in Figure

3.7(a). The electro-optic modulator (EOM) was used to generate a brief light pulse from

the continuous laser emission. The EOM controller was adjusted to minimize light trans-

mission until a external pulse was received. This pulse would alter the EOM to allow light

67

delaygenerator

photodiodemonitor at impact point

(a) simulated short


pulsegenerator

streakcamera

spectrometer CCD

D1 CWlaser

EOM

D2

D3

0 50 100 150 200 2505600

5800

6000

6200

6400

6600

6800

track on CCD

lase

r p

uls

e d

elay

(n

s)

(b)

linear fit

Figure 3.7: Timing synchronization/calibration for transmission shots(a) Electronic setup. Delays D2 and D3 are fixed as described in Section 3.1.4. D1 isadjusted to place a laser pulse at various times of the streak camera record.(b) Verification of streak camera sweep linearity

68

transmission for 100-1000 ns. The rising edge of this light pulse was used as an optical

fiducial that simulates impact. The pulse was coupled into the collection fiber of the trans-

mission system and directed onto the streak camera2. Delay D1 was adjusted so that the

pulse edge appeared in the desired portion of the streak camera record (typically the first

15-25%). The time between the arrival of the laser pulseat the targetand the original trig-

ger short (considering cable delays) was equal to the necessary standoff delay (∆t). Once

∆t was measured, the standoff trigger distance was found found fromD = v∆t, wherev is

the projectile velocity.

Timing calibration

Streak camera timing calibration was performed with the setup shown in 3.7(a).

A 100-1000 ns laser pulse, created by modulating continuous laser light, was adjusted so

that it appeared on the streak camera record. Delay D1 was adjusted to place the laser

pulse at different locations of the streak camera record. The laser delay and pulse location

(track number) were recorded for each setting and plotted as shown in Figure 3.7(b). The

streak camera sweep rate was equal to the slope of laser delay versus track number, which

was quite constant across the CCD. This calibration was performed for every transmission

experiment.

Wavelength calibration

Wavelength calibration was performed with emission lines from a Hg-Cd lamp [8].

The streak camera was operated in focus mode, resulting in a fixed horizontal image on the

CCD. A cross section of this image is shown in Figure 3.8(a). A typical linear calibration is

shown in Figure 3.8(b). This calibration was performed for every transmission experiment.

Spots on the CCD while the camera was in focus mode extended over several pix-

els (both vertically and horizontally). A variety of factors contributed to the spot width,

2Minimal laser intensity was used to avoid damaging the streak camera cathode.

69

0 50 100 150 200 250 300 350 400 450 500pixel on CCD

Inte

nsi

ty (

arb

itra

ry u

nit

s)

0 50 100 150 200 250 300 350 400 450 500350

400

450

500

550

600

650

pixel on CCD

λ (n

m)

(a)

Hg404.66 nm

Hg435.83 nm

Cd467.81 nm

Cd479.99 nmCd

508.58 nmHg546.07 nm

(b)

linear fit

Figure 3.8: Wavelength calibration(a) Hg-Cd spectrum (cross section of the CCD image)(b) Pixel-wavelength calibration

70

including optical fiber diameter, spectrometer slit width, and geometric dispersion in the

camera. Variations of the CCD image on scales smaller than these widths were not physi-

cally meaningful, so smoothing was performed using convolution kernels based upon these

CCD spots to eliminate high frequency noise.

Diagnostics

Several diagnostic measurements were made with every experiment. A photodiode

placed near the xenon flashlamp established that light was generated during the measure-

ment. The monitor pulse generated by the streak camera was also recorded to verify that

this system was activate. The impact pin fiducial was compared with the transmission jump

of the photodiode record to ensure impact occurred near the expected time. The photodiode

transmission jump rise time was also used to verify planar impact.

3.2 Optical imaging experiments


Figure 3.9 shows the configuration for imaging a sample during an impact exper-

iment. This setup is quite similar to the optical transmission experiments described in

Section 3.1, where an image relay replaces the collimating lens/optical fiber and a framing

camera is used in place of the spectrometer/streak camera system. During this study, imag-

ing capabilities evolved considerably through changes in the optical relay and improve-

ments in the framing camera flexibility and resolution. The following discussion highlights

these developments.

Telescope relay system

The original imaging system used a Newtonian telescope to relay the target image

as shown in Figure 3.10. The imaging region was a 1×6 mm horizontal slit, defined by

71

45° turning mirror

45° turning mirrors

pulsed Xe flashlamp

projectile

impactor


parabolic turning mirror

windows

liquid sample

optical relay

framing camera

Figure 3.9: General setup of optical imaging experimentsObject illumination is identical to the configuration used in optical transmission experi-ments (Figure 3.1).

72

Newtoniantelescope

vacuum seal

standard Pyrexwindow

collimated visible light

target chamber

optical fiber

planeturning mirror

Lexanwindow

imageplane

framingcamera

Figure 3.10: Telescope base image relay systemNote that central light rays from the target are obscured by the telescope turning mirror.

73

an aperture mounted on the rear of the sample. An optical fiber was mounted directly to

the aperture to provide an optical impact fiducial. Light passing through the aperture was

reflected into a vertical tube using a 45 turning mirror. To prevent impact debris from

entering the tube, a 1/4” thick piece of Lexan was taped to the top of the target chamber. A

second 1/4” window, made from Pyrex, was mounted at the top of the tube for additional

debris protection and to seal the tube and target chamber. The telescope, located above the

second window, images the target onto the photocathode of a DRS Hadland Imacon 790

camera. The optical image was converted into an electronic image, which was directed

at a portion of a single phosphor screen for a 10 ns exposure. 40 ns later, another 10 ns

exposure was produced on a different portion of the phosphor. With this method, images

were rastered into a N×2 array with the same exposure and interval time. Optical output

from the phosphor screen was coupled to a single image intensifier and a CCD chip for

acquisition. 1×6 mm target images were necessary to prevent image overlap on the CCD.

Though this system acquired images, the limitations of the framing camera were

quite severe. While it was possible to start imaging during any time of an experiment

with an adjustable trigger pulse, interframe and exposure times were completely fixed.

Since all images were placed on the same CCD, the spatial resolution of each image was

quite limited. Internal variations of the framing camera placed images in slightly different

positions during each acquisition, making it difficult to compare images taken during that

experiment with those acquired at ambient conditions.

During July 1999 and December 1999-January 2000, an improved framing camera

(DRS Hadland Imacon 468) was available to us. This and all later models of Hadland Ima-

con cameras used independent optical channels to record the output of an eight way beam

splitter (Figure 3.12). The Imacon 468 model had seven imaging channels and one streak

channel. Imaging channels used a gated intensifier to control CCD exposure, producing a

single image frame. The streak channel swept a slit of the image across a CCD. Since each

74

image was on a separate CCD, there is no possibility of overlap and larger images could be

obtained.

With the framing camera improvements, shortcomings of the telescope relay system

became apparent. Since the telescope was placed outside the target chamber, there was

a large object distance that led to a demagnification of the target image, limiting spatial

resolution. Collimated light coming through the sample was confined to a small diameter

(∼1/4”), so it was necessary to direct the light at an angle to avoid the telescope turning

mirror (Figure 3.10), producing off axis image distortion. Further image degradation was

found to occur due to the 45 turning mirror and protective windows.

Lens relay systems

Replacing the telescope system with a lens based optical relay solved the difficulties

described above. A schematic diagram of the optical system is shown in Figure 3.11. No

obscuring turning mirrors are present, and lenses are placed closer to the the target to pro-

duce higher magnification. Extremely flat turning mirrors were used in this system, and the

protective windows were replaced with highly polished Pyrex or fused silica. Continuous

transmission measurements were also made during the experiment with a photodiode.

An upgraded framing camera (DRS Hadland Imacon 200) was tested with the lens

relay system in April 2001. This model improved upon the timing controls and CCD ca-

pabilities of the Imacon 468. The demonstration camera generated eight image frames. A

similar model, acquired in August 2001, produced sixteen images from eight channels by

double exposure. This combination of lens relay and framing camera was the final combi-

nation used for imaging in this study.


The projectiles and targets for imaging experiments were nearly identical those used

in transmission experiments (Section 3.1). The only difference was that the collimating lens

75

90/10 beamsplitter

polished BK-7 orfused silica window

polished BK-7 orfused silica window

sample imageplane

achromatlenses

target chamber

collimated visible light

planeturning mirror

vacuum seal

plane turning mirror

photodiode

verticaltube

top view

rearview

SLRlenses

8 waybeam-splitter

8 MCP 8 CCD

Figure 3.11: Lens based image relay systemWith the obscuring mirror removed, collimated light can be centered on the image relay. Apicture of the rear view is shown in Figure 3.13; the top view is shown in Figure 3.14.

76

unit and bracket were omitted.


With the exception of the image relay and framing camera, all instrumentation and

optical components in an imaging experiment were identical to those used in a transmission

experiment (Section 3.1.3). This included the flashlamp, photodiodes, digitizers, and all

turning mirrors between the flashlamp and target (3 planar, 1 parabolic).

Image relay

Initially, the target image was directed outside of the target chamber with the same

type of mirror used to direct light toward the projectile (4-6 waves flatness). This mirror

was upgraded to a flatness ofλ /10 (Newport #10D20BD.1, 1” O.D.) to minimize wavefront

distortion. Flatness and clarity were also issues for the protective windows of the image

relay tube. Lexan was no longer used– both windows were made from highly polished

fused silica (VLOC) or BK-7 glass (Melles Griot). The diameter of the lower window was

2”, while the upper window was 3”. Both windows were 1/4” thick.

Sample imaging was done with two achromatic lenses. The lower lens (Melles

Griot # 01LAO355) had a 700 mm focal length and 50 mm diameter. The upper lens

(Melles Griot # 01LA0369) had a 1185 mm focal length and 80 mm diameter. Additional

Nikon SLR camera lenses (f=28, 50, or 85 mm) were mounted directly to the framing

camera in pairs for additional magnification control.

Framing camera

The DRS Hadland Imacon 200 framing camera used an eight way beam splitter

(UK patent 9324459.8) to separate the input image into eight channels (Figure 3.12). Con-

sidering losses in the 8-way split, each channel received about 10% of the original light

77

Figure 3.12: Cross section of the DRS Hadland 8 way beam splitterExternal optics (Figure 3.11) relay the target image to the input plane of the camera (rightside). Internal optics relay this image to an eight way beam splitter designed by DRS Had-land (UK patent 9324459.8). Only two facets of the beam splitter are shown in this figure.Eight images of the original input are sent to independent channels (left side) mountedcoaxially behind the beam splitter.

78

intensity. The optical images were directed onto a S25 photocathode for maximum sen-

sitivity in the visible spectrum. Individual exposures were controlled with a gated image

intensifier coupled to a 10 bit CCD detector (1280×1024 pixels, each> 6.7 µm square).

The system clock ran at 200 MHz, giving 5 ns adjustment in exposure and delay times of

each channel. All eight channels of the framing camera were exposed twice, yielding total

of sixteen image frames. This was achieved by transferring the contents of each CCD to

a temporary buffer before taking a second exposure. Some limitations resulted from this

doubling procedure.

1. To allow buffer transfer and CCD cleaning between exposures, no two image frames

could be obtained on the same channel less than 500 ns apart.

2. The minimum internal delay of the camera was roughly 50 ns (time from trigger

pulse to camera activation). However, to ensure proper behavior of the second eight

frames, a longer time was needed between camera activation and the first frame.

Typically, this time was set to be about 2µs.

3. To minimize artifacts of the first image on the second exposure, all CCD’s were

exposed to no more than half the full dynamic range (512 counts out of a possible

1024 maximum).

The Hadland Imacon 200 camera produced programmable electronic monitor pulses

for diagnostic purposes. Although in principle the pulses were adjustable to within a clock

cycle of the camera, the pulses became quite erratic when separated by less than∼50 ns.

In cases where image frames were widely spaced in time (>100 ns), monitor pulses were

aligned with each frame. For shorter interframe times, only a few pulses were generated at

key moments of the experiment, such as camera activation, anticipated impact, and first/last

image frame times.

79


Optical alignment

Setup of the optical system shown in Figure 3.11 began with rough placement of

the turning mirrors, lenses, and framing camera. This placement was performed with an

alignment projectile (Figure A.5) in the barrel of the 4” gun, which produced a collimated

beam of laser light centered and aligned with the gun axis. The first turning mirror (located

inside the target chamber) was positioned and oriented to direct this laser beam up the

center of the vertical tube. The achromat lens pair was mounted so that the first lens was

approximately one focal length away from the target. Protective windows were added

below the lenses and at the top of the vertical tube. A second turning mirror was positioned

to reflect the laser beam to the input plane of the framing camera. The front window of the

liquid cell was replaced with a resolution chart (chart facing cell interior), and the entire

target was mounted to the gun muzzle, keeping the alignment projectile in place. Operating

the framing camera in its focus mode, the camera was positioned to obtain a focused image

of the resolution chart3. Timing tests (discussed below) were performed at this stage.

The alignment projectile was then removed, and the projectile used the experiment

was placed in the gun barrel. The illumination optics were oriented to gather light from

the xenon flashlamp as described in Section 3.1.4. Figure 3.13 shows a rear view of the

complete target chamber setup. Light entered the target chamber (left side of the picture)

and was directed through the sample with three turning mirrors (left mirror in picture plus

two projectile mirrors which are not visible). The right turning mirror in Figure 3.13 re-

flected transmitted light towards the lens unit at the the top of the target chamber. This

lens pair relayed an image of the target up a to a point outside the target chamber. Figure

3.14 shows a top view of the optics outside of the target chamber. Light from the lens relay

3Laser speckle limits the precision of this focus.

80

relaylens unit

light from xenon flaslamp

turning mirrors

Figure 3.13: Imaging experiment target chamber setup (rear view Figure 3.11)The right turning mirror (mounted to adjustable turning mount) directs light from the sam-ple towards a lens unit mounted at the top of the target chamber. A pair of achromatic lens,protected by a 1/4” window, relays an image of the sample to a point outside the targetchamber.)

81

Figure 3.14: Imaging experiment external optics setup (top view of Figure 3.11)Light from the target chamber (Figure 3.13) exits a port on the floor. A turning mirrordirects this light towards a pellicle beam splitter, which directs 90% of the light to theframing camera and 10% to a photodiode. A pair of SLR camera lenses are mounted to theframing camera for magnification adjustment.

82

was directed towards a the framing camera with a 45 mirror. A 90%/10% beam splitter

directed a portion of this light towards a photodiode detector while the majority was passed

to the camera. The sample image was formed at a point to the left of the framing camera; a

pair of SLR camera lenses were used to transfer this image to the input plane of the camera

and adjust the total image magnification.

A flashlight was briefly inserted in the illumination path to provide a steady, inco-

herent source for additional adjustment of the framing camera focus and alignment. Final

focusing was performed with the flashlamp. The image frames of the camera were pro-

grammed with the appropriate delays, exposures, and gain settings for the experiment.

Spatial calibration (discussed below) were performed at this stage. The resolution chart

was then removed from the liquid cell, which was again mounted to the gun muzzle for

additional testing. If sufficiently bright images were obtained with minor rotations of the

projectile (Section 3.1.4), all optical joints were epoxied in place. The liquid cell was

filled with water, and reference images and signals were acquired. The projectile was then

extracted and loaded into the gun breech.


The timing sequence for an imaging experiment was nearly the same as that for a

transmission experiment (Section 3.1.4). Since the framing camera had an internal clock,

the image frames could be adjusted individually. The delay generator (Figure 3.7(a)) sent

an initial trigger pulse (D3) to activate the framing camera early in the experiment. Individ-

ual image timing was determined within the camera control software. As with transmission

experiments, a standoff pin was set so that impact occurred just after peak flashlamp out-

put. At least one image was timed to occur prior to impact to establish proper flashlamp

behavior and sample clarity. Subsequent frame timing was based upon the nature of the

experiment.

83

Coupling the laser light pulse into the framing camera was somewhat more difficult

in the imaging setup than the transmission experiment. This was achieved using the align-

ment projectile, so that the laser pulse traversed the optical relay and was incident on the

framing camera. A single SLR lens was used to image the laser fiber tip. The pulse delay

was then adjusted so that the a brief (∼10 ns) image of the fiber appeared at the desired

moment of impact on the framing camera record. The arrival of this pulse in the target

chamber was then used to determine the standoff trigger pin height (Section 3.1.4).

Timing calibration

The internal clock of the Imacon 200 camera operates at 200 MHz. Timing intervals

on the camera should be accurate to within one clock cycle (∼5 ns). When the camera was

received, the system timing was tested with a configuration similar to Figure 3.7(a). In this

test, a light pulse was located on an image frame with a 5 ns exposure. The light pulse was

then shifted in time by a known amount, and the image timing adjusted to locate the laser

pulse. It was determined that time durations on the camera were indeed accurate to within

5 ns. Electronic monitor pulses produced by the camera were tested in a similar fashion

and appeared to be aligned within thd 5 ns tolerance, provided monitor spacing was greater

than 50 ns (otherwise the pulses are erratic as noted above).

Spatial calibration

1951 USAF resolution charts were used for focusing the imaging system as well

as for calibration. The charts were made from round pieces of soda lime glass (1”-1.25”

diameter× 1/8” thick). The chromium patterns were deposited by Sine Patterns LLC. Fig-

ure 3.15 shows a scale image of one chart and a focused image of the chart taken with

the framing camera. Spatial calibration was performed by comparing the known line pair

spacings to the measured values taken from average cross sections of the image. A linear

84

Figure 3.15:(a) Scale image of a glass slides plated with a positive 1951 USAF resolution chart. Thischart was used in place of the liquid cell front window for focusing and spatial calibration.The red box shows the line pairs displayed in (b).(b) Optimally focused image of the positive resolution chart on the Imacon 200 camera.The smallest resolvable line pair, highlighted in blue, is element 6 of group 5, which has a17.5µm spatial period.

85

plot determined the scaling of pixel distance to actual sample distance. The limiting reso-

lution was found by locating the smallest line pair that could be resolved in the image. In

the image shown in Figure 3.15, this would occur at element 6 of group 5. The line pairs

for these lines are spaced by 17.5µm, so the line widths are about 9µm. This distance is

the minimum resolvableseparation. Objects smaller than 9µm may be detectable if the

surrounding area is clear, while objects separated by less than∼9 µm will not be resolved

clearly as separate entities.

Framing camera linearity and consistency

System linearity was tested by inserting neutral density filters behind the liquid cell

window, which attenuated light by a known amount. If the system behaves linearly, then a

plot of measured extinction to the known filter extinction should be linear with a slope of

unity. Figure 3.16 shows frame averaged extinctions against the known filter extinctions for

the final imaging system. For extinctions (base 10) below 0.5, the match is relatively good.

This corresponds to transmission values≥30%. For large extinctions (X ≈ 1→ T ≈10%),

the system response is not quite so linear as the measured intensity levels approach the

background noise level.

Individual pixel intensity readings were found to scale within 10% of the frame

average in any given image. These fluctuations were caused within the image intensifier and

were random between exposures. The second eight frames, obtained by double exposure,

often showed remnant features of the first exposure. This was minimized by preventing the

first exposure from exceeding 50% of its full dynamic range, but some artifacts remained.

Diagnostics

All diagnostic measurements for the imaging experiments were similar to those for

transmission experiments. The programmed monitor pulses from the framing camera were

recorded for timing analysis. The photodiode measurement that accompanies each image

86

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

Filter Extinction (base 10)

Ave

rag

e F

ram

ing

Cam

era

Ext

inct

ion

(b

ase

10)

MeasurementsIdeal System

Figure 3.16: Linearity of the optical imaging systemEach measurement is averaged over all 16 frames of the Imacon 200 camera.

87

set was used for timing and as a general check of image intensity.

3.3 Wave profile experiments


VISAR (Velocity Interferometer System for Any Reflector) [9] was used in this

work to measure particle velocity profiles in a shock reverberated liquid. This diagnostic

is commonly used in shock wave experiments [10]. Figure 3.17 shows the overall exper-

imental configuration. Mechanical aspects of the impact were identical to the description

in Section 3.1. To measure particle velocity histories of the liquid sample, light was re-

flected from a surface near the back of the sample. The optical phase of the reflected light

was changed by the motion of the mirror. These changes were recorded with the VISAR

system and used to calculate the velocity of the target mirror. A brief description of sys-

tem operation is given here; Appendix F contains a thorough discussion of the conversion

between measured optical phase and the velocity of the target mirror.


Projectile construction

Projectile construction for VISAR experiments was similar to that for the optical

experiments described earlier. The turning mirror mount was omitted from the side of the

projectile, and no mirror or aperture was epoxied to the impactor mount.

Liquid cells

In all VISAR experiments, a reflecting surface was created by vapor plating a thin

(< 1 µm) aluminum layer onto the rear window of the liquid cell. Since multiple waves

were generated by reverberation, this layer was placed close to the water sample to prevent

subsequent loading stages from overtaking the initial compressions. Figure 3.18 illustrates

88

liquid sample

Data photodiodes

YAG laser 532 nm

impactor

projectile

1/8 wave plate

mirror

etalonBS

dichroic BS

plated Al mirror

windows

imaginglens unit

mirror with pinhole

intensitymonitor

photodiode

mirror

VISAR system

EOM

lasermonitorphotodiode

opticalfiber

mirror

Figure 3.17: General setup of VISAR measurement

89

epoxy

watersample

(b)

watersample

(a)

buffer

VISARsystem

VISARsystem

rear window

rear window

Figure 3.18: Mirror configurations for VISAR experimentsAll clear pieces represent z-cut quartz(a) Buffered configuration(b) Unbuffered configuration

90

two basic approaches that were tried in this work. Initially, a thin quartz buffer was used to

protect the mirror. Later on, this buffer was removed, placing the mirror in direct contact

with the water sample. The specific configurations are described below.

1. Buffered mirror configuration

In this configuration, a thin (0.5 mm× 1.00” O.D.) z-cut quartz buffer was epoxied

onto the plated rear window of liquid cell design #1 (Section 3.1.2). Sizing of the

brass step (Figure 3.3(a)) was done with respect to the top of the buffer.

2. Unbuffered mirror configuration #1

This configuration used a custom liquid cell (Figure A.14) based upon design #2

(Section 3.1.2). The aluminum mirror was plated onto the rear window (1.5” diame-

ter× 1/8” thick), which was submerged in a closed bath of distilled water for at least

12 hours to completely oxidize the aluminum surface.

3. Unbuffered mirror configuration #2

This configuration used liquid cell design #1 (Section 3.1.2). The intentional oxidiza-

tion of the aluminum mirror needed to be done prior to press fitting the rear window

in the liquid cell. If the operations were reversed, considerable electrochemical cor-

rosion occurred as the entire brass cell and aluminum mirror were soaked in distilled

water for≥12 hours. The exposed mirror was protected from metal debris during the

press fit and machining using a layer of Parafilm. Non-contact cleaning with ethanol,

isopropanol, and distilled water was also necessary.

Target construction

Aside from the liquid cell alterations indicated above, the only target modification

to the optical transmission configuration was in the lens unit. In VISAR experiments, a

different lens unit (Figure A.24) was used with a pair of lenses to image the fiber tip onto

91

the sample mirror. The optical fiber was held and polished at a slight angle to minimize the

collection of reflected light at the fiber tip.


VISAR system

Continuous laser light was produced by a Coherent Verdi 5 W YAG laser operating

at 532 nm. Laser output power was set at 0.75-1.5 W. A Conoptics model 100 Pockels cell

was used as a high speed shutter, opening for about 10µs during the impact experiment

to minimize sample heating. Light signals in the VISAR system were fiber coupled to

Thorlabs SV2-FC silicon photodiodes (2 GHz bandwidth, 120 ps rise time). The detectors

were connected directly to a TDS-694 digitizer (120,000 samples at 1010 samples/s) to

minimize signal distortion. Electronic diagnostic signals were recorded with a Tektronix

TDS 654C or 684C, which acquired 15,000 data points at 109 samples/s.

Relay optics

Uncoated, plano-convex lenses, 12 mm in diameter with a 24 mm focal length

(Edmund Scientific #32011), were used in the lens unit (Figure A.24). The optical fiber

carrying light to and from the target had a 200µm core with a 0.16 numerical aperture.


Optical alignment

VISAR measurements (Figure 3.17) were made with a single fiber, which coupled

light to and from the target [11]. Light from a continuous YAG laser was passed through

an electro-optic modulator (EOM) to provide controlled illumination. A pair of lenses

imaged the optical fiber tip onto the mirror placed within the target. The reflected image

was relayed back into the fiber and passed into a conventional VISAR [9]. Light exiting

the fiber was nominally unpolarized. Some of this light was directed to a photodiode to

92

monitor the intensity profile; the remainder was sent to the interferometer system. One leg

of the interferometer contained an etalon and a 1/8 wave plate; the other leg was empty.

Light from both legs was collected and then separated using a polarizing beam splitter. The

intensity of each linear light polarization was acquired by a separate photodiode detectors

and digitized. These detectors observed a superposition of light from each leg, where one

leg was delayed by a timeτ. The magnitude of this delay was defined by optical path length

of the etalon. By adjusting the orientation of the 1/8 wave plate, one of optical polarizations

also experienced a 90 phase shift. The signals from all three photodiode detectors were

used to determine the optical phase shifts of light reflected from the target mirror. The

velocity of the target mirror was then determined from the analysis described in Appendix

F.


Electrical connections for a VISAR measurement are shown in Figure 3.19. All

equipment was triggered from a pulse generated by a standoff shorting pin. The height

of the trigger pin was adjusted so that impact occurred roughly 1-1.5µs after the pin was

struck. A second pin was set at the same height in case the first pin failed. Both digitizers

and an adjustable delay generator were activated by the resulting trigger pulse. The delay

generator activated the EOM, allowing light illumination for roughly 10µs. Data from the

photodiodes were acquired for 5-12µs.

Diagnostics

A monitor pulse from the EOM was recorded to ensure proper opening during the

experiment. Stray reflected light prior to fiber coupling was also collected with a photo-

diode (laser monitor in Figure 3.17) to verify that the EOM opens during the experiment.

The BIM channel was examined for signs of light loss in the sample rear window or failure

of the reflecting mirror.

93

data 1photodiode

data 2photodiode

BIM photodiode

DataDigitizer

CW laser

triggerpin #1

h+D

frontwindow

pulsegenerator

trigger pin #2

projectile

h h+D

v

laser monitorphotodiode

EOM

delaygenerator


Figure 3.19: Electronic setup for VISAR measurements

94

References for Chapter 3[1] G.E. Duvall, K.M. Ogilvie, R. Wilson, P.M. Bellmany and P.S.P Wei. Optical spec-

troscopy in a shocked liquid.Nature296, 846 (1982).


[3] K.M. Ogilvie. Time resolved spectoscopy of shock compressed liquid carbon disulfide.Ph.D. thesis, Washington State University, (1982).

[4] M.D. Knudson.Picosecond electronic spectroscopy to understand the shock-inducedphase transition in cadmium sulfide. Ph.D. thesis, Washington State University,(1998).

[5] R.L. Webb.Transmission of 300-500 nm light through z-cut sapphire shocked beyondits elastic limit. M.S. thesis, Washington State University, (1990).


[7] G.R. Fowles, G.E. Duvall, J. Asay, P. Bellamy, F. Feistmann, D. Grady, T. Michaelsand R. Mitchell. Gas gun for impact studies.Rev. Sci. Instrum.41, 984 (1970).

[8] NIST. Physical reference data, (2002). www.physics.nist.gov.

[9] L.M. Barker and R.E. Hollenbach. Laser interferometer for measuring high velocitiesof any reflecting surface.J. Appl. Phys.43, 4669 (1972).

[10] L.M. Barker. The development of the VISAR, and its use in shock compression sci-ence. InShock Compression of Condensed Matter,M.D. Furnish, editor, 11 (Ameri-can Institute of Physics, Snowbird, UT, 1999).

[11] S.C. Jones and Y.M. Gupta. Refractive index and elastic properties of z-cut quartzshocked to 60 kbar.J. Appl. Phys.88, 5671 (2000).

95

Chapter 4

Experimental ResultsThis chapter presents the results of optical and wave profile measurements obtained

during multiple shock compression of water. These experiments used consistent sample

preparation (Appendix C) and detection systems; other experiments relevant to this study

are presented in Appendix D. The results are organized by the type of measurement made.

Optical transmission measurements are presented in Section 4.1. Optical imaging measure-

ments are described in Section 4.2. Wave profile measurements are in Section 4.3. Each

section begins with a brief overview of the measurements and a discussion of the relevant

calculations. Next, results from specific experiments are presented to establish the neces-

sary conditions for water to show signs of freezing. Finally, these results are summarized,

making note of any experimental trends.

4.1 Optical transmission measurementsThe objective of the optical transmission experiments was to determine if liquid

water remained transparent under multiple shock compression. Transparency was inter-

preted as a sign that water samples remained in the liquid state, whereas transparency loss

was viewed as evidence for freezing. Table 4.1 summarizes the experiments performed to

study freezing in water. Columns two through four list the velocity and construction of the

impactor. Columns five through nine describe the front window, water sample, and rear

window of the impact target. The peak compression state in the water sample is shown in

columns ten through twelve. This state was determined from wave propagation simulations

using the COPS code [1], previously established material models for the windows/impactor

97

Table4.1:

Sum

mary

ofopticaltransmission

experiments

Experim

entIm

pact Im

pactor F

ront window

**Sam

ple* R

ear Window

**peak state

90%E

xp.C

hangesnum

bervelocity

Material

h [D]

Material

h [D]

hM

aterialh [D

]P

T

ρρρ ρlevel

limit

observed(km

/s)(m

m)

(mm

)(m

m)

(mm

)(G

Pa)

(K)

(g/cc)(ns)

(ns)T

1 (00-056)0.150

SLG

18.8

SLG

1.5850.104

SLG

18.7991.10

3441.23

2543100

no[44.5]

[50.8][63.5]

T2 (00-038)

0.425F

S 12.7

FS

1.5890.128

FS

12.7052.57

3921.37

1642200

yes[31.8]

[38.1][31.8]

T3 (00-030)

0.602F

S 12.7

FS

1.5870.124

FS

12.7053.55

4171.43

962200

yes[31.8]

[38.1][31.8]

T4 (01-041)

0.396Q

12.7

Q

3.1840.028

Q

12.7203.47

4221.43

341500

yes[25.4]

[38.1][31.8]

T5 (00-051)

0.559Q

12.7

Q

3.1830.108

Q

12.7205.00

4631.51

1112200

yes[31.8]

[38.1][31.8]

T6 (01-013)

0.566Q

12.7

Q

3.1840.015

Q

12.7185.06

4641.51

162200

yes[31.8]

[38.1][31.8]

T7 (01-M

F02)

0.221S

12.7S

3.1740.104

S 12.694

4.98448

1.51261

900no

[25.4][31.8]

[25.4]T

8 (01-MF

10)0.223

S 12.7

S 3.175

0.015S

12.6875.03

4481.52

37900

no[25.4]

[31.8][25.4]

T9 (01-033)

0.393Q

12.7

Q

3.1620.100

S12.691

5.03455

1.51151

900yes

[25.4][38.1]

[25.4]T

10 (01-010)0.399

S12.7

S3.194

0.099Q

12.723

5.11458

1.52147

900yes

[25.4][31.8]

[25.4]T

11 (00-057)0.445

S 12.7

S 3.170

0.108S

12.69110.15

5281.71

153900

?[25.4]

[31.8][25.4]

T12 (00-050)

0.448S

12.7S

3.1700.110

S12.691

10.22529

1.71154

900?

[25.4][31.8]

[25.4]

h= thickness

D=

diameter

SLG

= soda lim

e glassFS=

fused silicaQ

=z-cut quartz

S=a-cut sapphire

* 0.005 mm

uncertainty (0.010 for T1) in thickness

** 0.002 mm

uncertainty in thicknessD

iameters are nom

inal values

98

(Appendix B), and the water model discussed in Chapter 5. The peak pressure is deter-

mined solely by the impact velocity and the mechanical properties of the impactor and rear

window. Temperature and density are related to the details of the water model. For these

calculations, the water is assumed to remain a liquid during compression. A characteristic

compression time was estimated from the simulations by determining the time required for

pressure at the sample midpoint to be 90% of its peak value. That time is shown in column

thirteen. Column fourteen shows the maximum experiment duration, defined by the preser-

vation of uniaxial strain in the water sample and the transparency of the impactor/windows.

The final column indicates whether optical transmission changes were observed during the

experiment.

4.1.1 Determining sample transmission

Figure 4.1 shows a schematic view of an optical transmission experiment. Visible

light is directed through a sample, and the transmitted light is recorded by a detector. The

transmission of a sample is defined by the ratio of the transmitted intensity to the incident

light intensity; thus, a perfectly transparent sample has a transmission of unity. The trans-

mission of real samples is less that unity due to absorption, reflection, and scattering of

incident light. None of the materials used in this work absorb visible light, so that type

of loss can be neglected. Optical reflections are present in the system, but account for

only a few percent of the measured transmission after impact. Hence, the measured opti-

cal changes can be attributed to scattering that occurs within the water sample due to the

coexistence of liquid and solid phases (Appendix E) when the sample begins to freeze.

The transmitted light intensity measured during the impact experiment was time de-

pendent. This dependence was a combination of changes occurring in the sample and sys-

tematic variations in the optical system. Since water is transparent at ambient conditions,

the following transmission definition was used to extract information about the compressed

99

y

scatteredlight

reflected light

incident light

d

detector

transmitted light

z

x

Figure 4.1: Transmission losses in a non-absorbing sampleTransparency is reduced by scattering regions within the sample. Reflection also occurs atall optical index interfaces.

Time

Ph

oto

dio

de

ou

tpu

t

reference signal

impact signal

shock wave arrives at watersample

projectile rotation

Figure 4.2: Measured photodiode outputs from experiment T6

100

samples.

T =Iimpact− Ibg

Ire f − Ibg(4.1)

Iimpact is the measured light intensity during the impact experiment,Ire f is the measured

light intensity at ambient conditions, andIbg is the background intensity level of the de-

tector. Figure 4.2 shows one example of the systematic intensity variations that make this

definition necessary. At ambient conditions, the measured light intensity had a downward

slope due to the time dependent light output of the xenon flash lamp. A similar downward

slope also appeared in the impact measurements prior to the arrival of the shock wave at

the water sample. The transmission defined by Equation 4.1 corrects for this variation in

the impact measurement. In most experiments, the overall amplitudes of the reference and

impact intensities were not the same due to projectile rotation effects. To account for the

rotation, transmission signals were normalized so thatT = 1 just prior to the arrival of the

shock wave at the water sample.

A photodiode detector was used in all optical experiments to provide a wavelength

integrated measurement of sample transmission. The noise in this measurement limited the

minimum measurable intensityImin and the minimum transmission value.

Tmin =Imin

Ire f − Ibg(4.2)

Tmin was less that 1% for the photodiode measurements discussed here. Prior to calculat-

ing T, the photodiode records were smoothed with a local averaging filter. For reference

signals, a 10 ns duration was used for smoothing; for impact signals this was reduced to 2

ns. Spectrally resolved transmission measurements were also made during a few of the im-

pact experiments. The limitations of those measurements are described in the discussion of

experiment T5. Smoothing in those measurements was performed by convolving the mea-

sured signals with a low pass kernel based upon the results of the wavelength calibration

101

procedure (Section 3.1.4). Both the photodiode and spectrally resolved measurements col-

lected transmitted light from the entire illuminated region (≈1/4” diameter), so the present

results represent an average of the optical changes in this area.

4.1.2 Experimental results

The following results are organized by the peak pressure state and the liquid cell

construction. Similar experiments are grouped together so that only one variable (peak

pressure, window type, or sample thickness) is significantly different. Unless otherwise

noted, timet = 0 is defined to be the time when the shock wave first enters the water

sample.

Experiment T1

A basic assumption of this study was that compressed liquid water remains optically

transparent. To test that assumption, a water sample was compressed to a state where the

liquid phase was stable. This was achieved by compressing water to 1.10 GPa in a liquid

cell constructed from soda lime glass windows. The calculated peak temperature was 344

K, which is approximately 40 K above the liquid-ice VI phase boundary (at fixed pressure).

As such, there was no reason for water to change from the liquid state, and the sample was

expected to remain completely transparent.

The photodiode transmission measurement for this experiment is shown in Figure

4.3. As expected, no change in the optical transmission was detected. This measurement

confirms the expectation that compressed liquid water remains optically transparent.

102

0 200 400 600 800 1000 1200 1400 1600 18000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (ns)

Tra

nsm

issi

on

T1

T2

T3

Figure 4.3: Photodiode transmission for experiments T1, T2, and T3Experiment T1 used soda lime glass windows to a peak pressure of 1.1 GPa. T2 and T3were both performed using fused silica windows to a peak pressure of 2.6 and 3.6 GPa,respectively.

103

Experiments T2 and T3

Temperature calculations discussed in Section 5.2.3 indicate that isentropically com-

pressed liquid water enters the ice VII region somewhere above 2 GPa. Under such con-

ditions, a water sample will eventually solidify and thus lose optical transparency. Experi-

ments T2 and T3 were performed to verify this transparency loss. The liquid cells in these

experiments were constructed with fused silica windows, which creates slightly different

loading conditions than the other experiments. After impact, the fused silica front window

does not propagate a shock wave, but produces a continuous ramp wave compression [2].

Compression of the water sample is also continuous and closely approximates true isen-

tropic compression. Thus, it was expected that water compressed in fused silica windows

should freeze at pressures above 2 GPa. A 2.57 GPa peak stress was achieved in experi-

ment T2; the calculated peak temperature state was 392 K, about 5 K above the liquid ice

VII phase boundary. The water sample in that experiment was initially 128µm thick, and

it took approximately 164 ns to reach the 90% level. In experiment T3, the peak pressure

was 3.55 GPa and the calculated peak temperature 417 K (about 36 K below the phase

boundary). The initial sample thickness was 124µm, and compression was 90% complete

in about 96 ns.

Photodiode transmission records for experiments T2 and T3 are also shown in Fig-

ure 4.3. Both measurements showed an incubation period of about 200 ns during which

the optical transmission was equal to unity. After that time, the optical transmission was

observed to decrease gradually over hundreds of nanoseconds, indicating that liquid wa-

ter was not stable under these conditions. The incubation period in experiment T3 was

shorter than in T2; the overall transmission loss was faster in T3 than in T2. These results

demonstrate that freezing is enhanced as liquid water is compressed further into the solid

domain.

104

Experiment T4

Once it was established that water could lose transparency in fused silica windows,

similar measurements were made with z-cut quartz windows. Experiment T4 was per-

formed to a peak pressure of 3.47 GPa and a calculated peak temperature of 422 K. This

temperature is higher than that in experiment T3 because the front quartz window sustains

a single shock compression, creating multiple shock compressions rather than continuous

compression. The steady state in experiment T4 was still outside the domain of liquid sta-

bility, about 26 K below the liquid-ice VII phase boundary. The initial sample thickness

was 28µm thick, 90% of the peak pressure was reached in 34 ns.

Figure 4.4 shows the photodiode transmission record for this experiment. As in

experiment T3, optical transmission was lost, indicating freezing. The loss occurred more

quickly in this experiment than in T3, but it is difficult to compare these experiments di-

rectly since the loading paths were different. In addition to the loading differences between

fused silica and quartz, the water sample was much thinner in this experiment, so the peak

state was achieved more quickly. The fact that experiments T3 and T4 show extinction is

significant because it eliminates the crystalline or amorphous nature of the windows as a

variable. In other words, freezing was not sensitive to the presence or absence of long range

molecular order in the windows.


Further transmission measurements for water compressed in z-cut quartz windows

were made in experiments T5 and T6. Similar impact velocities were used to generate peak

pressures of about 5 GPa. The calculated peak temperature was 463 K, which is about 64

K below the liquid-ice VII phase boundary. Different initial thicknesses were used to vary

the loading time in each experiment. In T5, the water sample was initially 108µm thick,

and 90% of the peak pressure was reached in 111 ns. Initial sample thickness in T6 was 15

105

0 200 400 600 800 1000 1200 1400 1600 18000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (ns)

Tra

nsm

isis

on

Figure 4.4: Photodiode transmission for experiments T4Quartz windows were used to compress liquid water to a pressure of 3.47 GPa.

106

µm, and only 15 ns was required to reach the 90% level.

Figure 4.5 shows photodiode transmission records for experiments T5 and T6. Op-

tical transmission loss was observed sooner in the thin sample (T6) than in the thick sample

(T5). This indicates that the rate of transmission loss depends upon the compression time.

Both measurements showed a continuous loss of transmission for some time, but the thin

sample transmission approached a constant at a value nearT = 0.15, whereas the thick

sample transmission continued to decrease. Thus, the extent of freezing was related to

sample thickness.

Spectrally resolved optical transmissions for experiment T5 are shown in Figure

4.6; similar wavelength profiles were observed in T6 (not shown). The time dependence of

transmission at each wavelength was similar to the photodiode measurement. The initial

transmission was uniform across the visible spectrum. At later times, a slight wavelength

dependence did appear, but this variation was an artifact of the limited dynamic range of

the streak camera system. This limitation arose from the light induced background caused

by incomplete photon conversion at the streak camera photocathode [3]. The unconverted

photons were reflected from surfaces in the streak tube and struck the photocathode again,

producing electrons in regions not directly illuminated by the spectrometer. This limited the

minimum intensityImin to about 500-1000 counts [4]. Since the reference intensity was a

function of both wavelength and time,Tmin (Equation 4.2) also varied with wavelength and

time. Early in the experiment, this value is less than 5% across the entire visible spectrum,

but it increases over time, particularly at long wavelengths. Figure 4.6(b) shows an estimate

of Tmin at t = 1000ns, indicating that the apparent spectral profile in the measurements is

largely a result of dynamic range limitations. Therefore, the loss of light in the compressed

water samples was essentially independent of wavelength.

107

0 200 400 600 800 1000 1200 1400 1600 18000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (ns)

Tra

nsm

issi

on

T6

T5

Figure 4.5: Photodiode transmission records for experiments T5 and T6Quartz windows were used to obtain a 5 GPa peak stress in both experiments. Initial samplethickness in T5 was 108µm; in T6 it was 15µm.

108

0 200 400 600 800 1000 1200 1400 1600 18000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (ns)

Tra

nsm

issi

on

400 425 450 475 500 525 550 575 600

00.10.20.30.40.50.60.70.80.9

1

Wavelength (nm)

Tra

nsm

issi

on

(a)

(b)

t=0

t=200 ns

t=400 ns

t=1000 ns

T

min at 1000 ns

600 nm

400 nm

Each plot separated by 50 nm

Figure 4.6: Spectrally resolved transmission for experiment T5Quartz windows were used to compress the 108µm water sample to 5 GPa peak stress.90% peak stress was obtained in 111 ns.(a) Transmission records for several different wavelengths(b) Spectral profile snapshots

109


Since a significant decrease in optical transmission was observed for water com-

pressed beyond 2 GPa in both fused silica and z-cut quartz windows, the window materials

did not appear to be relevant to the observed changes. To further test this idea, experiments

T7 and T8 were performed using a-cut sapphire windows. The impact velocities in these

experiments were adjusted to produce similar peak pressure states to experiments T5 and

T6 (≈ 5 GPa). The calculated peak temperature was 488 K, which is about 79 K below

the liquid-ice VII phase boundary. The reduced temperatures in these experiments, com-

pared to T5 and T6, arises from the different loading path for water contained in sapphire

windows. The impedance of sapphire is higher than quartz, producing more loading steps

in the water sample that result in a better approximation to isentropic compression. As in

the quartz cell experiments, two initial sample thicknesses were used to produce different

loading times. The initial sample thickness in experiment T7 was 104µm, and 90% peak

pressure was attained in 261 ns. Initial sample thickness in T7 was 15µm, and 90% peak

pressure was reached in 37 ns.

Photodiode measurements of transmission for experiments T7 and T8 are shown in

Figure 4.7. Although water was compressed into the ice VII domain, no significant optical

changes took place. The minor variations of transmission about unity are comparable to

shot to shot variations in flashlamp intensity; these variations are negligible compared to the

losses observed in experiment T5. The initial sample thickness did not affect transmission

in any obvious way. Since transmission loss was observed for water at similar conditions

in quartz windows, it is concluded that freezing must somehow be related to the nature of

the window materials.

110

0 100 200 300 400 500 600 700 8000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (ns)

Tra

nsm

issi

on

T5

T7

T8

Figure 4.7: Photodiode transmission records for experiments T7 and T8Sapphire windows were used in experiments T7 and T8 to compress water to a peak pres-sure of 5 GPa. The sample thicknesses for T7 and T8 were 104 and 15µm, respectively.Experiment T5 (quartz windows, 108µm sample) is shown for comparison.

111


One explanation for the differences between quartz and sapphire liquid cells is that

transparency loss may require shock waves of a certain magnitude. This would occur if

changes were initiated by conditions within the shock front, rather than the thermodynamic

state behind the front. Such a hypothesis is consistent with the observations thus far since

sapphire windows lead to smaller compression increments than quartz windows. Transmis-

sion loss might also be linked to surface interactions at the water-window interface which

occur for fused silica and quartz but not for sapphire. To determine which of these expla-

nations is correct, a series of hybrid liquid cell experiments were performed. These cells

were constructed with one quartz window and one sapphire window. Calculated pressure

histories for quartz, sapphire, and hybrid liquid cells are shown in Figure 4.8. For a given

peak pressure state, the magnitude of loading steps in a hybrid liquid cell is closer to the

quartz cell than the sapphire cell. If optical transparency loss is tied the magnitude of indi-

vidual loading steps, then the hybrid cell experiments should be very similar to the quartz

cell results.

Experiment T9 was constructed with a quartz front window and sapphire rear win-

dow. This order was reversed in experiment T10 (sapphire front window, quartz rear win-

dow). The impactor material was identical to the front window material for both config-

urations to prevent wave reflections at the impact surface. The calculated peak state in

these experiments was approximately (5 GPa, 457 K), roughly 70 K below the liquid-ice

VII phase boundary. Initial sample thicknesses (≈100µm) and 90% level times (≈150 ns)

were comparable for both configurations.

Photodiode transmission records for experiments T9 and T10 are shown in Figure

4.9. Transmission loss was observed in both hybrid configurations, although the magnitude

of this loss is not nearly as large as in the quartz cell experiments of similar thickness.

112

0 100 200 300 400 500 600 700 8000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Time (ns) from impact

Pre

ssu

re (

GP

a)

hybrid cell (T10)

hybrid cell (T9)

sapphire cell(T7)

quartz cell (T5)

Figure 4.8: Loading history in quartz, sapphire, and hybrid liquid cellsThese curves show the pressure history at the midpoint of the water sample.

113

0 100 200 300 400 500 600 700 8000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (ns)

Tra

nsm

issi

on

T9

T10

T5

Figure 4.9: Photodiode transmission records for experiments T9 and T10Experiment T9 used a quartz front window and a sapphire rear window; in experimentT10, the order was reversed. Peak pressures in both experiments was 5 GPa. Initial watersample thicknesses were about 100µm. Experiment T5 (quartz windows, 108µm sample)is shown for comparison.

114

This difference eliminates the magnitude of individual shocks as an explanation for the

difference between quartz and sapphire experiments. The observed optical changes for

water compressed in quartz windows must be therefore be linked to interactions at the

water-window interface.


Once window effects were established at 5 GPa, additional experiments were car-

ried to determine if transmission loss would occur in sapphire windows at significantly

higher pressure. Peak pressures of 10.2 GPa were achieved in experiments T11 and T12 us-

ing a-cut sapphire windows. The calculated liquid temperature at that pressure was 530 K,

which is about 161 K below the liquid-ice VII phase boundary. Initial sample thicknesses

in both experiments were similar (108-110µm), and 90% peak pressure was reached in

about 150 ns.

Photodiode records for experiments T11 and T12 are shown in Figure 4.10. Both

measurements showed a brief loss in transmission, followed by a recovery back to the

transparent state. Loss began 122-128 ns after the shock arrival, reached a minimum at

132-140 ns, and then returned to almost unity at 146-154 ns. Wave propagation calcula-

tions indicated that this process was synchronous with the third shock reflection at the rear

window, which occurred at 123 ns. The state behind this shock was approximately (8 GPa,

505 K). The return shock (reflected from the front window) reached the rear window at

148 ns, nearly the same time when transmission was fully restored. The state behind the

reflected shock was (8.6 GPa, 513 K). Although the transmission loss and recovery was

repeatable in these two experiments, the phenomena was not observed in any other type of

measurement. If the transmission loss is interpreted as onset of freezing, it is unclear if the

recovery signifies reversal or completion of the phase transition.

115

0 100 200 300 400 500 600 700 8000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Time (ns)

Tra

nsm

issi

on

T11

T12

Figure 4.10: Photodiode transmission records for experiments T11 and T12These experiments used sapphire windows to a peak pressure of 10 GPa.

116

4.1.3 Summary

Optical transmission measurements through liquid water under multiple shock com-

pression generated the following results.

1. Water remained transparent when compressed to a state where the liquid phase was

stable.

2. When liquid water was compressed to a metastable state (P > 2GPa), a time depen-

dent loss in optical transparency occurred when a quartz or fused silica window was

present. This transparency loss is consistent with the onset of freezing. Only one

silica window was necessary for freezing to occur.

3. The onset of freezing depended upon both the peak pressure state and the initial

sample thickness. The extent of freezing, as measured by total transmission loss,

increased with water sample thickness.

4. At pressures of 8-10 GPa, water in sapphire windows briefly lost and then regained

transparency. The interpretation of this transmission recovery is not clear.

4.2 Optical imaging measurementsThe purpose of the optical imaging experiments was to further examine the trans-

parency loss in water under multiple shock compression. This technique allowed direct

observation of the time dependent changes that occurred in the water samples. Table 4.2

summarizes the imaging experiments; the information is organized in the same manner as

for the optical transmission experiments (Table 4.1).

117

Table4.2:

Sum

mary

ofopticalimaging

experiments

Experim

entIm

pact Im

pactor F

ront window

**Sam

ple* R

ear Window

**peak state

90%E

xp.C

hangesnum

bervelocity

Material

h [D]

Material

h [D]

hM

aterialh [D

]P

T

ρρρ ρlevel

limit

observed(km

/s)(m

m)

(mm

)(m

m)

(mm

)(G

Pa)

(K)

(g/cc)(ns)

(ns)12.7

3.20212.722

[31.8][38.1]

[31.8]12.7

3.20312.727

[31.8][38.1]

[31.8]12.7

3.20012.727

[31.8][31.8]

[31.8]12.7

3.18812.709

[25.4][31.8]

[25.4]

h= thickness

D=

diameter

Q=

z-cut quartzS=

a-cut sapphire* 0.005 m

m uncertainty in thickness

** 0.002 mm

uncertainty in thicknessD

iameters are nom

inal values

I1 (02-016)0.314

Q

Q

0.137Q

2.72

4001.38

2172200

yes

I2 (02-017)0.410

Q

Q

0.132Q

3.60

4251.43

1622200

yes

I3 (02-009)0.566

Q

Q

0.109Q

5.06

4651.51

1142200

yes

I4 (02-MF

02)0.219

S S

0.108S

4.94448

1.51273

900no

118

4.2.1 Determining sample transmission

Spatial resolution and exposure time are two key differences between the transmis-

sion and imaging experiments. In the former case, a single detector continuously receives

light from all parts of the illuminated water sample; in the latter case, light from different

parts of the sample is relayed onto a spatially resolved detector for a fixed exposure time.

Otherwise, the interpretation of the two measurements is quite similar.

Two images taken from experiment I4 are shown in Figure 4.11. One image was

acquired at ambient conditions, the other was taken during the impact experiment. It was

initially thought that Equation 4.1 could be applied to these images at each pixel location

to determine a transmission profileT(x,y), but this process did not remove the random

fluctuations from the image intensifier of the framing camera. These fluctuations generated

point to point intensity variations of 10-20%. The use of Equation 4.1 only amplified the

problem by dividing two images containing these variations. To avoid this problem, the

transmission of a single image (Ti) was defined in terms of the the average reference and

background values of that frame.

Ti =Iimpact−〈Ibg〉i〈Ire f〉i−〈Ibg〉i

(4.3)

In each experiment, the image sequence was normalized so that the average value of the first

frame taken prior to shock arrival was equal to unity. The dynamic range of the Imacon

200 framing camera was limited by the image intensifier noise (Imin ≈30-50 counts) so

thatTmin =10-20%. The average transmission of any particular image frame was typically

within 5-10% of the value measured with a photodiode detector.

To enhance the visibility of changes in the water sample, the optical data were

processed as follows. All images were smoothed using a median filter [5] to remove high

frequency fluctuations. This filter replaced each image pixel with the median value of pixels

119

0

50

100

150

200

250

300

350

Reference image

0

50

100

150

200

250

300

350

Impact image

Figure 4.11: Raw images from experiment I1The top image was taken at ambient conditions, while the bottom image was acquiredduring the impact experiment.

120

in a N×N neighborhood, which preserves image feature edges better than convolution

filters. The size of the neighborhood was determined by the limiting resolution found

during spatial calibration (Section 3.2.4). The contrast of these images was also enhanced

to compensate for the limited intensity range of the human eye [5]. To maximize the gray

levels available to regions of changing intensity, the calculated transmission imagesTi(x,y)

were nonlinearly mapped to gray scale valuesGi(x,y) using the following function.

Gi(x,y) =12

[1+ tanh

(Ti(x,y)−Tm

L

)](4.4)

The logic behind this mapping is as follows: at regions of high or low transmission, the

gray level of the image approaches a constant value (1 or zero, respectively). Regions of

intermediate transmission (T ≈ Tm) are given the greatest range of gray levels and have

the most contrast. For this work, bothTm andL were equal to 0.5. The white level of

the displayed images was then set atG = 1; the black level was set atG = 0.12, which

corresponds to a completely opaque region (i.e.T = 0).


The following discussion is organized by peak pressure. As in the previous section,

time is defined to be zero when the first shock wave enters the water sample.

Experiment I1

Based on the observation of transparency loss in experiment T2, this experiment

was performed to examine the spatial variations when liquid water is compressed to a state

near the liquid-ice VII phase boundary. A peak pressure of 2.7 GPa was generated in a

water sample using a z-cut quartz liquid cell. The calculated sample temperature was 400

K, which is about 5 K above the phase boundary. The initial sample thickness was 137µm,

and 90% peak pressure was reached in 217 ns.

Figure 4.12 shows the transmission history of the water sample measured with a

121

0 200 400 600 800 1000 1200 1400 1600 18000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (ns)

Tra

nsm

issi

on

photodiodeImacon 200

Figure 4.12: Measured transmission for experiment I1Quartz windows were used in this experiment to generate a peak pressure of 2.7 GPa.

122

100

µm

Fig

ure

4.13

:Im

ages

obta

ined

inex

perim

entI

1Q

uart

zw

indo

ws

wer

eus

edto

gene

rate

ape

akpr

essu

reof

2.7

GP

a.In

itial

sam

ple

thic

knes

sw

as13

7µm

;90%

peak

pres

sure

was

atta

ined

in21

7ns

.T

heim

age

sequ

ence

prog

ress

from

left

torig

ht,

star

ting

from

the

uppe

rle

ftco

rner

.T

hebe

ginn

ing

ofea

chim

age

expo

sure

was

at-1

70,2

30,3

30,5

30,6

30,7

30,8

30,9

30,1

030,

1130

,133

0,an

d14

30ns

.E

ach

expo

sure

last

edfo

r25

ns.

123

photodiode detector and the Imacon 200 framing camera. These two measurements are rel-

atively consistent with one another, and match the general transmission profiles obtained

in Section 4.1.2. Figure 4.13 shows the images obtained in this experiment, which had a

limiting resolution of about 8.5µm (11 camera pixels at∼0.77µm sample distance/pixel).

These images provided important details about the changes in the water sample. Obviously,

the sample opacity was not uniform. Instead, some regions became extremely dark while

others remained completely transparent. Since the sample was originally transparent, the

clear regions represent areas where water remains in the liquid phase. The darkened re-

gions represent areas where freezing has occurred. The transition can be traced back to a

few independent sites that formed between 230 and 330 ns. As time progressed, freezing

propagated in a highly irregular fashion. This growth occurred in two stages. The initial

growth stage (t ≈ 330− 630 ns) was confined to narrow channels that extended rapidly

across the sample. These channels ceased growing once they overlapped with one another.

The second growth stage (t > 330ns) was largely perpendicular to the fast growth process,

widening previously frozen regions. The widening process was much slower than the ini-

tial directional growth and continued throughout the experiment. On average, the darkened

regions are≤ 100µm in width and several hundred microns long. At the end of the image

sequence, numerous liquid regions, 100µm size and larger, persisted in the sample.

Experiment I2

Based on the results from transmission experiments T2 and T3, it was expected that

water compressed to 3.5 GPa would lose transparency faster and more completely than for

pressures at the liquid-ice VII phase boundary. This experiment was performed to examine

how such a pressure change affects the features observed in experiment I1. A z-cut quartz

cell was used to compress liquid water to a peak pressure of 3.60 GPa. The calculated

temperature of this state was 425 K, which is approximately 31 K below the liquid-ice VII

124

phase boundary. Initially, the water sample was 132µm thick, and 90% peak pressure was

attained within 162 ns.

The optical transmission records (photodiode and Imacon 200 results) for this ex-

periment are shown in Figure 4.14. As before, these measurements are consistent with one

another. Figure 4.15 shows the images obtained in this experiment, which had a limiting

resolution of about 8.3µm (11 pixels at∼0.75 µm sample distance/pixel). Freezing oc-

curred earlier and originated from more independent sites than in the previous experiment.

There was still a distinction between rapid initial growth and the slower widening of fea-

tures; in this case, the overlap began to occur while new regions were still forming (as

seen at timest = 225 andt = 275 ns). After 375 ns, the only significant changes in the

images were the result of the features widening. More liquid regions were observed in this

experiment than in I1, but they were considerably smaller (≤ 100 µm). On average, the

frozen areas are≤ 100µm in width, with most features≤ 50 µm, and only about 100µm

in length.

Experiment I3

This experiment was performed to further examine the pressure trends observed in

experiment I1 and I2. A z-cut quartz cell was used to compress liquid water to a pressure

of 5.06 GPa. The calculated temperature of the water was 465 K, which is approximately

64 K below the liquid-ice VII phase boundary. The initial sample thickness was 109µm,

and 90% peak pressure was attained in 114 ns.

No photodiode record was obtained in this experiment due to a detector failure. Im-

ages were successfully recorded and are shown in Figure 4.16. The limiting resolution of

these images was approximately 8.8µm (11 pixels at∼0.80 µm sample distance/pixel).

Under these conditions, freezing appeared so rapidly that it is not possible to distinguish in-

dependent transition regions. In less than 200 ns, numerous darkened regions were formed,

125

0 200 400 600 800 1000 1200 1400 1600 18000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (ns)

Tra

nsm

issi

on


Figure 4.14: Measured transmission for experiment I2Quartz windows were used in this experiment to generate a peak pressure of 2.7 GPa.

126

100

µm

Fig

ure

4.15

:Im

ages

obta

ined

inex

perim

entI

2Q

uart

zw

indo

ws

wer

eus

edto

gene

rate

ape

akpr

essu

reof

3.5

GP

a.In

itial

sam

ple

thic

knes

sw

as13

2µm

;90%

peak

pres

sure

was

atta

ined

in16

2ns

.T

heim

age

sequ

ence

prog

ress

from

left

torig

ht,

star

ting

from

the

uppe

rle

ftco

rner

.T

hebe

ginn

ing

ofea

chim

age

expo

sure

was

at25

,75,

125,

175,

225,

275,

325,

375,

425,

475,

575,

and

675

ns.

Eac

hex

posu

rela

sted

for

25ns

.

127

100 µm

Figure

4.16:Im

agesobtained

inexperim

entI3Q

uartzw

indows

were

usedto

agenerate

peakpressure

of5G

Pa.

Initialsample

thicknessw

as109

µm

;90%peak

pressurew

asreached

in114

ns.T

heim

agesequence

progressfrom

leftto

right,starting

fromthe

upperleft

corner.T

hebeginning

ofeach

image

exposurew

asat55,95,135,175,285,325,365,405,445,485,525,and

565ns.

Each

exposurelasted

for20

ns.

128

and these regions are extensively linked with one another. After 200 ns, new features were

formed as separate transition regions merged with one another. In areas where merging

occured, the frozen feature widths were 50-100µm wide; transition regions that did not

merge are confined to widths less than 25µm. The residual liquid areas were 50µm or

less in size.

Experiment I4

The optical transmission measurements demonstrated that water would only lose

transparency in the presence of a quartz or fused silica window. That observation was

tested in this experiment, had a-cut sapphire windows. The peak pressure was 4.94 GPa,

similar to experiment I3. The calculated peak water temperature 448 K, about 76 K below

the liquid-ice VII phase boundary. The initial sample thickness was 108µm, and 90% of

the peak state was reached in 273 ns.

The optical transmission records (photodiode and Imacon 200 results) for this ex-

periment are shown in Figure 4.17. These measurements are consistent with each other

and with previous optical transmission measurements. Images from this experiment I4 are

shown in Figure 4.18. The limiting resolution of these images was approximately 28µm

(7 pixels at∼3.94µm sample distance/pixel). A few dark spots were observed in the im-

ages, but these appeared prior to impact and remained constant throughout the experiment.

These features were probably the result of some debris striking the impactor or front win-

dow surfaces prior to impact. As expected, no optical changes in the water sample were

detected.

4.2.3 Summary

Optical imaging measurements of liquid water under multiple shock compression

generated the following results.

129

0 100 200 300 400 500 600 700 8000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


T7

Figure 4.17: Photodiode transmission record for experiment I4Sapphire windows were used to compress liquid water to a pressure of 4.94 GPa. Experi-ment T7, which used a similar impact configuration, is shown for comparison.

130

1 m

m

Fig

ure

4.18

:Im

ages

obta

ined

inex

perim

entI

4S

apph

irew

indo

ws

wer

eus

edto

gene

rate

ape

akpr

essu

reof

5G

Pa.

Initi

alsa

mpl

eth

ickn

ess

was

108

µm;9

0%pe

akpr

essu

reis

atta

ined

in27

3ns

.T

heim

age

sequ

ence

prog

ress

from

left

torig

ht,

star

ting

from

the

uppe

rle

ftco

rner

.T

hebe

ginn

ing

ofea

chim

age

expo

sure

was

at-8

5,11

5,41

5,51

5,56

5,61

5,66

5,71

5,76

5,81

5,86

5,an

d91

5ns

.E

ach

expo

sure

last

edfo

r20

ns.

131

1. Freezing did not occur uniformly in the water sample, but was confined to regions

that originated from a number of initial sites. Growth away from the initiation sites

was highly directional and irregular.

2. The peak pressure affected the morphology of the freezing process. At pressures near

the liquid-ice VII phase boundary, only a few initial sites were formed and much of

the water sample remained clear. At higher pressures, the number of initiation sites

increased and the amount of clear sample decreased.

3. Freezing was observed only when water was compressed in quartz windows. Use of

sapphire windows did not result in freezing.

4.3 Wave profile measurementsThe purpose of the wave profile measurements was to determine if optical trans-

parency loss in water was accompanied by mechanical changes. Table 4.3 summarizes the

wave profile experiments in this work. The organization of this table is similar to those

in previous sections except for two changes. First, an extra column is added to the target

configuration to describe the protective buffer layer used in one type of liquid cell. Second,

the 90% level time is omitted because the loading stages are obvious in the wave profile

measurements.

4.3.1 Particle velocity determination

Particle velocity histories were determined using the VISAR system. Raw data

from experiment V3 are shown in Figure 4.19(a). One of these signals monitored the

light intensity profile entering the VISAR; the other two signals measured the intensity of

each linear light polarization passing through the interferometer. The latter signals were

smoothed with a five point neighborhood averaging filter, normalized with the intensity

132

Tabl

e4.

3:S

umm

ary

ofw

ave

profi

leex

perim

ents

Exp

erim

ent

Impa

ct

Impa

ctor

Fro

nt w

indo

w**

Sam

ple*

Buf

fer

Rea

r W

indo

w**

peak

sta

teE

xp.

Cha

nges

num

ber

velo

city

Mat

eria

lh

[D]

Mat

eria

lh

[D]

h[E

poxy

]M

ater

ial

h [D

]P

T

ρ ρρρlim

itob

serv

ed(k

m/s

)(m

m)

(mm

)(m

m)

(mm

)(m

m)

(GP

a)(K

)(g

/cc)

(ns)

12.7

3.20

20.

531

12.7

23[2

5.4]

[38.

1][0

.003

][3

1.8]

12.7

3.20

0-

3.18

3[2

5.4]

[31.

8]-

[38.

1]12

.73.

200

-12

.720

[31.

8][3

8.1]

-[3

1.8]

12.7

3.18

8-

12.7

05[2

5.4]

[31.

8]-

[25.

4]

h= th

ickn

ess

D=

dia

met

erQ

=z-

cut q

uart

zS=

a-cu

t sap

phir

e*

0.00

5 m

m u

ncer

tain

ty in

thic

knes

s**

0.0

02 m

m u

ncer

tain

ty in

thic

knes

sD

iam

eter

s ar

e no

min

al v

alue

s

1.51

900

no0.

100

S4.

9644

8V

4 (0

2-M

F02

)0.

220

SS

1.51

2200

yes

0.10

4Q

5.

0846

5V

3 (0

2-00

2)0.

568

Q

Q

1.51

950

yes

0.10

3Q

5.

0346

4V

2 (0

1-04

5)0.

563

Q

Q

1.51

2200

yes

0.10

3Q

5.

0546

5V

1 (0

1-53

8)0.

567

Q

Q

133

Time

Ph

oto

dio

de

ou

tpu

t

x polarization

y p

ola

riza

tio

n

(a)

(b)

shock arrivesat mirror

mirror at rest

optical phase

constantmirror motion

intensity monitor

x polarization

y polarization

Figure 4.19:(a) Raw VISAR signals from experiment V3(b) Optical phase ellipse

134

monitor profile, and plotted as shown in Figure 4.19(b). The resulting ellipse defined the

optical phase of the reflected light, which can be related to particle velocity through a

calculation discussed in Appendix F. This calculation is based on the fact that motion of the

target mirror changes the optical phase of the reflected light. When the mirror is at constant

velocity, the optical phase is constant; when the mirror accelerates, both polarization signals

in the interferometer change, moving the optical phase angle around the ellipse. Figure

4.19(b) indicates the first acceleration of the target mirror as an example. The conversion

from the optical phase to particle velocity was performed using a program developed at

Washington State University [6].


For consistency with the optical measurements, time in the VISAR signal was de-

fined to be zero when the shock wave first arrived at the water sample. The measured

particle velocity history was manually aligned with the first particle velocity jump of the

wave propagation calculation performed for each experiment. The peak stress in all VISAR

experiments was about 5 GPa, a state where optical changes were readily observed in trans-

mission and imaging experiments. The sample thicknesses in these experiments were in the

range of 100-104µm. Since VISAR measurements using a liquid sample had not previ-

ously been made at Washington State University, a large portion of this work was spent in

developing the experimental techniques. The experiments are presented here in chronolog-

ical order to show the evolution of the VISAR measurements.

Experiment V1

This experiment used the buffered configuration (Section 3.3.2) in an attempt to

protect the VISAR mirror from damage. The mirror was plated on the rear window of a

standard quartz liquid cell and covered with a 531µm thick piece of z-cut quartz. The

epoxy bond between the buffer and the rear window was 2-3µm thick. The VISAR system

135

was configured with a 8.0175” BK-7 etalon to obtain a delay ofτ = 1.255ns and a fringe

constantF = 205m/s (considering actual interferometer leg lengths).

The particle velocity record for this experiment is shown in Figure 4.20. The mea-

sured and calculated magnitudes of the first particle jump were similar, but in subsequent

jumps the measured value decayed below the calculated signal. This behavior indicated

some transition from pure liquid behavior in the water sample. After the measured signal

approached the steady state, a series of spikes appeared in the record. These features re-

sulted from multiple wave reflections between the epoxy bond and the water sample. The

calculated particle velocity contained similar features when an epoxy layer [7] was added to

the simulation. The steady state particle velocity should be unaffected by these reflections,

but their presence was an undesirable complication. As a result, the buffered configuration

was abandoned in later experiments.

Experiment V2

A second experiment was constructed using an unbuffered configuration (Section

3.3.2) to eliminate the epoxy layer and the extraneous wave reflections. The mirror was

plated on the 3.183 mm thick z-cut quartz rear window. Cell design #2 was used to keep

this window separate from the cell body to minimize mirror damage during construction.

The VISAR system again used with a 8.0175” BK-7 etalon to obtain a delay ofτ = 1.255

ns and a fringe constantF = 205m/s (considering actual interferometer leg lengths).

The particle velocity history for this experiment is shown in Figure 4.21. Measured

and calculated particle velocities were consistent through the first two jumps. During and

after the third compression, the measured particle velocity decreased from the calculated

value. Rapid decelerations were observed in the measurement due to changes in the VISAR

window correction (Appendix F); those artifacts are manually removed in Figure 4.21.

Even after the correction, the measured signal was 4-5% lower than the calculated steady

136

0 100 200 300 400 500 600 700 8000

25

50

75

100

125

150

175

200

225

250

275

300

325

Time (ns)

Par

ticl

e ve

loci

ty (

m/s

)

reflectionsfrom epoxy

layer

liquid calculation

VISAR data

Figure 4.20: Particle velocity history for experiment V1 (quartz windows)

137

0 100 200 300 400 500 600 700 8000

25

50

75

100

125

150

175

200

225

250

275

300

325

Time (ns)

Par

ticle

vel

ocity

(m

/s)

manual correction

free surface releases

4−5% lower than

calculationliquid calculation

VISARdata

Figure 4.21: Particle velocity history for experiment V2 (quartz windows)

138

state value. At about 950 ns, waves from the rear window free surface reached the water

sample, releasing the sample pressure. This release made it impossible to determine if the

measured signal would have returned to the calculated steady state value at a later time.

Experiment V3

Although the epoxy reflections were removed in experiment V2, the short experi-

mental duration was not satisfactory. To extend this duration, a standard quartz liquid cell

was used in an unbuffered configuration. The rear window was now 12.720 mm thick,

preventing free surface release waves for 1500 ns longer than in experiment V2. Since

the exposed mirror was contained with the liquid cell body, extreme care was necessary to

avoid mirror damage during construction (Section 3.3.2). As in experiment V1 and V2, a

8.0175” BK-7 etalon was used in the VISAR system to obtain a delay ofτ = 1.251ns and

a fringe constantF = 205m/s (considering actual interferometer leg lengths).

Figure 4.22 shows the particle velocity history for this experiment. As in experi-

ment V2, the measured and calculated particle velocities were consistent for several load-

ing steps. After about the fourth velocity jump, the measured particle velocity once again

dropped below the calculated steady state. The measured value was as much as 7-9% lower

than the calculated value. The complete records for this experiment and V1 are shown in

Figure 4.23 to compare the measured particle velocities long after the initial compression

period. Both measurements continued to fluctuate throughout the experiment and never

returned to the calculated steady state value.

Experiment V4

A final experiment was performed using a sapphire liquid cell to determine whether

the deviations noted above required the presence of a quartz window. The liquid cell was

constructed in the unbuffered configuration, as in experiment V3. The projectile velocity

in this experiment was considerably slower than in the quartz experiments because of the

139

0 100 200 300 400 500 600 700 8000

25

50

75

100

125

150

175

200

225

250

275

300

325

t (ns) from shock arrival

up (

m/s

)

7−9% lower than

calculation

liquid calculation

VISAR data

Figure 4.22: VISAR measurement for experiment V3 (quartz windows)

140

0 250 500 750 1000 1250 1500 1750 20000

50

100

150

200

250

300

Time (ns)

Par

ticl

e ve

loci

ty (

m/s

)

free surface releases

calculated steady state

V1

V3

Figure 4.23: VISAR measurements long after compression

141

higher impedance of the sapphire windows and impactor. This led to lower particle veloci-

ties in the water sample. To compensate, the VISAR system used a 12.0199” BK-7 etalon,

yielding a time delay ofτ=1.818 ns and a fringe constantF = 141m/s.

The result for this experiment is shown in Figure 4.24. Unlike experiments V1-V3,

the measured particle velocity in this experiment always matched the liquid calculation. No

decreases from the calculated steady state were observed. A slight upward or downward

slope was observed during some of the loading stages of compression. These features were

artifacts of birefringence in the a-cut sapphire rear window, which resulted from a partial

polarization of the laser light used in the VISAR system [8]. This experiment suggested

that water remained in the pure liquid state, which is consistent with the optical experiments

using sapphire windows.

4.3.3 Summary

Wave profile measurements of liquid water under multiple shock compression gen-

erated the following results.

1. There was a period during the initial stages of compression where particle velocity

followed the behavior of a pure liquid.

2. The rear side of a water sample compressed in quartz windows showed a particle

velocity decrease near the end of compression. After that time, the measured particle

velocity was unsteady and did not return the calculated value within the time duration

of this work.

3. Measured particle velocity history of water compressed in a sapphire cell closely

matched the calculated history for a pure liquid sample.

142

0 100 200 300 400 500 600 700 8000

10

20

30

40

50

60

70

80

90

100

110


up (

m/s

)

Liquid calculation

VISAR data

Figure 4.24: Particle velocity history for experiment V4 (sapphire windows)

143

References for Chapter 4[1] Y.M. Gupta. COPS wave propagation code (Stanford Research Institute, Menlo Park,

CA, 1978), unpublished.

[2] L.M. Barker and R.E. Hollenbach. Shock-wave studies of PMMA, fused silica, andsapphire.J. Appl. Phys.41, 4208 (1970).


[4] M.D. Knudson. Picosecond electronic spectroscopy to understand the shock-inducedphase transition in cadmium sulfide. Ph.D. thesis, Washington State University, (1998).

[5] J.C. Russ.The Image Processing Handbook. CRC Press, Boca Raton, 2nd edition,(1995).

[6] S.C. Jones. VISAR 2000 code, Washington State University (unpublished), (2000).

[7] J.M. Winey, (2001). Epoxy model for the COPS code, Washington State University(unpublished).

[8] S.C. Jones, B.A.M. Vaughan and Y.M. Gupta. Refractive indices of sapphire underelastic, uniaxial strain compression along the a axis.J. Appl. Phys.90, 4990 (2001).

144

Chapter 5

Time Dependent Continuum Model for WaterThis chapter describes the development of a continuum model for simulating the

response of water in shock wave experiments. The model contains equation of state (EOS)

information for pure liquid and solid phases, and describes the behavior of a mixture of

these phases. Numerical simulations were performed using this model to calculate ther-

modynamic properties (e.g. density, temperature) for reverberated water samples and to

estimate the duration of the loading steps. By postulating a rate of transition for the freez-

ing process, time dependent changes in the thermodynamic states were also calculated.

Both liquid and solid phases were described in terms of the Helmholtz free energy,

derived using thermodynamic consistency requirements. Section 5.1 discusses the overall

approach for the EOS development. Sections 5.2 and 5.3 present the EOS details for pure

liquid and solid water, respectively. Mixed phase considerations are discussed in Section

5.4. Numerical simulations of the water response under multiple shock compression are

presented in Section 5.5.

5.1 EOS developmentThe thermodynamic behavior of a pure fluid is completely specified by the spe-

cific Helmholtz free energyf (T,v) [1]. f (T,v) can be determined through integration of

thermodynamic response functions such as(∂s/∂T)v and(∂P/∂T)v. These quantities are

related to one another through thermodynamic consistency, placing certain constraints on

the EOS. This approach has been used previously to develop models for liquid water [2–4]

and other materials [5–7] under shock wave loading. In the present work, it was assumed

thatcv is constant for the conditions of interest. The consequences of this assumption and

145

the development of an expression forf (T,v) are discussed in this section.

5.1.1 Thermodynamic consistency

cv is defined as the isochoric temperature derivative of entropy:

cv≡ T

(∂s∂T

)

v(5.1)

The pressure-temperature coefficientb is defined as the isochoric temperature derivative of

pressure:

b≡(

∂P∂T

)

v(5.2)

These two quantities are related through thermodynamic consistency:

(∂b∂T

)

v=

1T

(∂cv

∂v

)

T(5.3)

For the case wherecv is constant,(∂b/∂T)v = 0. Thus,b must be a function of volume

only. Using this result, the pressureP(T,v) can be written by integration of Equation 5.2.

P(T,v) = P(T0,v)+b(v)(T−T0) (5.4)

5.1.2 The form of f (T,v) for constant cv

f (T,v) is related to the specific energye(T,v) and entropys(T,v) by a Legendre

transformation [1].

f (T,v) = e(T,v)−Ts(T,v) (5.5)

e(T,v) can be defined by integration of its temperature and volume derivatives.

(∂e∂T

)

v= cv

(∂e∂v

)

T= Tb(v)−P(T,v)

e(T,v) = e0 +∫ v

v0

[T0 b(v′)−P(T0,v

′)]dv′+cv(T−T0) (5.6)

146

s(T,v) can be determined using the same approach.

(∂s∂T

)

v=

cv

T

(∂s∂v

)

T= b(v)

s(T,v) = s0 +∫ v

v0

b(v′)dv′+cv lnT/T0 (5.7)

Combining Equations 5.5, 5.6, and 5.7,f (T,v) can be written in the following form.

f (T,v) = e0−Ts0− (T−T0)∫ v

v0

b(v′)dv′−∫ v

v0

P(T0,v′)dv′+cv

[(T−T0)−T ln

TT0

]

(5.8)

In addition tocv, several other quantities are necessary to specifyf (T,v). The values ofe0

ands0 must be known for a reference state(T0,v0). An isothermP(T0,v) passing through

this state is also required. Finally, the functionb(v) must be determined for the entire

volume range of interest.

5.2 Liquid water model

5.2.1 Choice of EOS

As indicated in Section 2.1.2, a number of models have been constructed for liquid

water. The most extensive continuum liquid water model constructed to date is the one by

Saul and Wagner [8]; an updated version has been released recently [9]. In this approach,

a form of f (T,v) was chosen with a set of 58 adjustable parameters. These parameters

were optimized to match an array of published thermodynamic data, such as volume and

heat capacity. This model was not used here because of the large number of adjustable

parameters, and because its validity and stability for the states of interest in the present

work was unclear. Instead, the EOS described in Section 5.1 was used to describe liquid

water.

147

Determination of the isochoric specific heat is an important first step in specify-

ing the f (T,v) EOS. For example, Winey [6] treated liquid nitromethane as a collection

of Einstein oscillators to definecv(T,v), which was then used to constrainb(T,v) through

thermodynamic consistency (Equation 5.3). In liquid water, this approach is not as useful

because only about half of the heat capacity comes from molecular vibrations, while the

other half arises from the configurational energy of hydrogen bonding [10]. In the present

work, it was assumed that a constant value ofcv is a good approximation to describe the

behavior of liquid water. This approach, along with the assumption thatb is also constant,

has been used previously [2, 3] to model liquid water. Since liquids may undergo large

volume changes (>10%) under shock wave loading, it is not clear thatb can be treated as a

constant [6]. Gurtman et al. [4] developed a constantcv EOS for liquid water while main-

taining the volume dependence ofb(v), which was determined from two isothermal curves.

Following the approach of Winey [6],b(v) was determined in this work by combining a

single isotherm with the Hugoniot curve.

5.2.2 EOS formulation

At ambient conditions, the heat capacity of liquid water is about twice that of ice

due to thermal contributions from hydrogen bonds [10]. When liquid water is subjected to

high pressure, these bonds become distorted or broken, so the heat capacity decreases [11].

Thus, the specific heat of liquid water at high compression approach to that of ice. For this

work, the value ofcv was chosen to be 3 J/g·K as a compromise between the specific heat

at ambient conditions (4.186 J/g·K [10]) and that of ice (2 J/g·K [10]).

Ambient conditions (T0 = 25 C, P0 = 1 bar) were used as the reference state for

liquid water, where the specific volume is 1.00296 cc/g [12]. The reference entropy and

energy were defined to be zero. Isothermal data at 25 C were taken from the work of

Grindley and Lind [13], who measured specific volume for pressures from 0.05 to 0.8 GPa;

148

isothermal data beyond 0.9 GPa are not available as liquid water transforms to ice VI [14].

The P− v data [13] were fit using the Murnaghan equation [15] to permit extrapolation

beyond the equilibrium freezing pressure value.

P(T0,v) = P0 +Kg

[(v0

v

)g−1

](5.9)

K is the ambient isothermal bulk modulus andg is the rate at whichkT increases with pres-

sure. To ensure the correct low pressure behavior, the fit in Equation 5.9 was constrained

to match the isothermal bulk modulus at ambient conditions. This forced the value ofK to

be 2.210 GPa [12], leavingg as the only adjustable parameter. Figure 5.1 shows the best fit

to the data using an optimal value ofg = 6.029.

For the purposes of this work, Hugoniot data of interest were limited to the 0-20

GPa pressure range. Several Hugoniots for liquid water have been reported, but there are

some difficulties with much of the published data. Low pressure (<1 GPa) shock speed

measurements reported by Nagayama et al. [16] do not extrapolate correctly to the ambient

pressure value(i.e. Us(P→ P0) 6= c0). Lysne [17] performed shock wave measurements

in the 0.3-2.2 GPa range with water samples cooled to 0 C prior to shock compression;

the scatter in these measurement was extremely high. Large data scatter is also present in

the 0-16 GPa Hugoniot data of Cook et al. [18]. Shock speed measurements from Walsh

and Rice [19] in the 3.2-20 GPa pressure range were used here because they cover most of

the pressure range of interest with acceptable precision. The initial temperature reported in

their work was 20 C, whereas the ambient state here was 25 C. Based on the universal

liquid Hugoniot proposed by Woolfolk et al. [20], temperature corrections were made to

the data in Ref. [19] by scaling the measured shock (US) and particle (up) velocities with

the acoustic sound speed ratio.

u∗p = upc0(25)c0(20)

U∗S = US

c0(25)c0(20)

(5.10)

149

0.8 0.825 0.85 0.875 0.9 0.925 0.95 0.975 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Specific volume (cc/g)

Pre

ssu

re (

GP

a)

datafit

Figure 5.1: 25 C isotherm for liquid waterData shown here are from Grindley and Lind [13]. The fit is a Murnaghan equation (Equa-tion 5.9) withK = 2.210GPa andg = 6.029.

150

The starred quantities were combined with the Rankine-Hugoniot jump conditions [21] to

generate shock states in the pressure-volume plane. These states are shown in Figure 5.2.

Despite some previous reports [22,23], the Hugoniot of liquid water does not have

to be linear (or piece-wise linear) in any coordinate system. In fact, it has been shown

that most liquids have a nonlinearUS−up relation [20]. To account for this behavior, the

temperature correctedP− v data were fitted using a polynomial in terms of the density

compression (µ ≡ ρ/ρ0−1).

P = P0 +a1µ +a2µ2 +a3µ3 (5.11)

To ensure that the shock speed matches the acoustic sound speedc0=1.496687 km/s [24] for

P→ P0, a1 must be equal toc20/v0 = 2.233GPa. The parametersa2 anda3 were optimized

to match the corrected Hugoniot data. The best fit to the data shown in Figure 5.2 was

obtained fora2 =6.976 GPa anda3=21.15 GPa.

Determination of b(v)

b(v) can be determined from the HugoniotPH(v) and the isotherm by inverting

Equation 5.4 [6].

b(v) =PH(v)−P(T0,v)

TH(v)−T0(5.12)

This expression requires knowledge of temperature along the HugoniotTH(v); hence, both

b andTH(v) must be determined simultaneously. An iterative calculation was performed

to determineb(v) for v≤ v0 using the isotherm and Hugoniot fits described above. The

results of the iterative calculation are shown in Figure 5.3. These values were fitted with a

polynomial function to allow smooth integration and differentiation as needed.

b(v) =N+1

∑i=1

Bi

(vv0−1

)i−1

(5.13)

The best fit parameters for this function are shown in Table 5.1.

151

0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

16


Pre

ssu

re (

GP

a)

datafit

Figure 5.2: P−v Hugoniot for liquid waterCorrected data from Walsh and Rice [19] were used to fit Equation 5.11.

152

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6x 10

−3


b(v

) (G

Pa/

K)

Iterative calculationFit

Figure 5.3: b(v) for liquid water

Table 5.1: b(v) fit parameters for liquid water (units of GPa/K)

B(1) = +4.83640×10−4 B(5) =−1.17590B(2) =−1.95275×10−2 B(6) =−1.74669B(3) =−1.54322×10−2 B(7) =−8.65802×10−1

B(4) =−2.35540×10−1

153

A maximum was observed inb(v) nearv = 0.675 cc/g. A similar maximum was

observed atv≈ 0.6 cc/g in the EOS formulation of Gurtman et al. [4]. This behavior has

also been observed in ab initio analysis of compressed water [25]. The decrease inb(v) at

high compressions is probably caused by a collapse of the hydrogen bonding structure of

liquid water, which allows water molecules to pack more densely.

5.2.3 Isentropic freezing and the value ofcv

In Section 2.5.2, it was proposed that freezing may be thermodynamically possi-

ble under isentropic loading. TheP−T states for these compressions may be calculated

from the EOS for liquid water, but there is a potential problem with the assumption of

cv = 3 J/g·K. Because this value is only an approximation to the true behavior of liquid

water, temperatures calculated using this EOS may not be accurate. Thus, the existence

of possible freezing states is somewhat ambiguous. To estimate the effect of variations

in cv on the calculated temperatures, two additionalcv values were considered using the

approach outlined in Section 5.2.2. The lowest possible temperatures were estimated by

settingcv to 4 J/g·K; the highest possible temperatures were determined by settingcv equal

to 2 J/g·K. Isentropic loading paths were calculated for all threecv values to determine the

pressure range where freezing is possible. Hugoniot states were also calculated to assess

the possibility of freezing under single shock compression.

Temperature along the isentrope was calculated by integrating the following equa-

tion.

dTdP

=(

∂v∂T

)

P

/(∂s∂T

)

P(5.14)

The partial derivatives in this expression were evaluated from the three different liquid wa-

ter models (cv=2, 3, and 4 J/g·K). The results of these calculations are shown in Figure

5.4(a). In all three cases, isentropic compression of liquid water enters the ice VII phase

154

0 1 2 3 4 5 6 7 8 9 10300

400

500

600

700

Pressure (GPa)

T (

K)

0 1 2 3 4 5 6 7 8 9 10300

400

500

600

700

Pressure (GPa)

T (

K)

(a)

(b)

Liquid

Ice VII

VI

Liquid

Ice VII

VI

4 J/g⋅K

3 J/g⋅K

2 J/g⋅K

2 J/g⋅K 3 J/g⋅K 4 J/g⋅K

Figure 5.4: The value ofcv and adiabatic freezingDark lines indicate equilibrium phase boundaries [26].(a) Isentropic loading states(b) Single shock loading states

155

above 2 GPa. Depending on thecv value chosen, the exact crossing point of these curves

with the phase boundary varies from 2.5 to 3.4 GPa. After crossing the phase boundary, the

isentropes continue into the ice VII region and do not return to the stable liquid domain.

These results demonstrate that liquid water becomes metastable with respect to ice VII un-

der isentropic compression. The principle ambiguity created by the constantcv assumption

is in determining the exact pressure needed to enter the ice VII phase.

For single shock compression, temperature was determined by integrating the fol-

lowing equation [27].

dTdP

=dTdv

/dPdv

(5.15)

wheredTdv

=1

2cv

[P−P0 +

dPdv

(v0−v)]− bT

cv

anddPdv

=(∂P/∂v)S+ b

2cv(P−P0)

1− b2cv

(v0−v)

The results of these calculations are shown in Figure 5.4(b). Unlike isentropic loading, sin-

gle shock compression states do not generally enter the ice VII region. In the case where

cv = 4 J/g·K, single shock compression intersects the phase boundary near 4.5 GPa, but at

higher pressures the Hugoniot returns to the liquid region. These results demonstrate that

freezing is highly unlikely in singly shocked water since the liquid phase is always ther-

modynamically favorable. Using a slightly different EOS, Rice and Walsh [28] estimated

that the liquid water Hugoniot enters the ice VII region in the 3.5-4.0 GPa pressure range.

There is little experimental evidence to support freezing in single shock compression [29].

Thus, the optimal approach to examine freezing is to subject liquid water to isentropic or

quasi-isentropic compression.

156

5.3 Solid water model

5.3.1 Assumptions

Because no direct information was available about the solid phase formed under

multiple shock compression, several assumptions were made to develop a solid water

model. Solid water was assumed to have negligible shear modulus and strength. The

EOS developed in Section 5.1 was used to model the solid phase with the assumption that

b is constant. Finally, it was assumed that the solid phase is similar to ice VII, which is

consistent with the P,T conditions generated by multiple shock compression.

5.3.2 EOS formulation

The value ofcv was chosen to be 2 J/g·K, which is similar to that of normal ice

[10]. The reference state was chosen to beT = 300K andP=0. Since solid water is not

stable under these conditions, the volume and bulk modulus in the reference state were

determined by fitting the isothermal data on ice using a Murnaghan equation (Equation

5.9). A variety of isotherms have been reported for ice VII [30–35]; the most consistent

are those of Hemley et al. [30] and Fei et al. [31]. These data and the fitted curve are

shown in Figure 5.5. The best fit parameters for this curve wereK=25.04 GPa,g=3.660,

andv0=0.6795 cc/g.

To complete the EOS for solid water, the values ofb, s0, ande0 are needed. These

values were determined by requiring the solid and liquid models to be consistent with the

equilibrium liquid-ice VII coexistence curve for water. Several measurements of this curve

are available in the literature [14, 26, 36–41]. The most accurate data to date appear to be

from the work of Datchi et al. [41], which are shown in Figure 5.6.

157

0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.622

4

6

8

10

12

14

16

18

20


Pre

ssu

re (

GP

a)

Hemley dataFei datafit

Figure 5.5: 300 K isotherm for ice VIIData were taken from Hemley et al. [30] and Fei et al. [31]. The fit is a Murnaghan equation(Equation 5.9) withK=25.04 GPa,g=3.660, andv0=0.6795 cc/g.

158

2 3 4 5 6 7 8 9 10350

400

450

500

550

600

650

700

Pressure (GPa)

Tem

per

atu

re (

K)

DataCalculation

Figure 5.6: Liquid-ice VII coexistence curveData were taken from Datchi et al. [41].

159

Consistency with the liquid-ice VII phase boundary

Along the phase boundaryT(P), the Gibbs free energy of the solid and liquid phases

are equal [1].T(P) can be calculated using the liquid water EOS (Section 5.2) and the solid

water EOS for some choice ofb, s0, ande0. Values forb ands0 were chosen such thatT(P)

optimally fitted the experimental data shown in Figure 5.6. The value ofe0 was determined

by requiring the liquid and solid Gibbs free energies to be equal at the liquid-ice VI-ice VII

triple point, located atT∗ = 354.8 K andP∗ = 2.17 GPa [41]. The coexistence curve was

found by integrating the Claperyon equation [1] from(P∗,T∗) over the pressure range 0-13

GPa.dT(P)

dP=

v−vL

s−sL(5.16)

HerevL andsL refer to the liquid phase, whilev ands refer to the solid phase. This curve

was optimized using the measured phase boundary shown in Figure 5.6. The best fit values

for this curve wereb = 6.829×10−3 GPa/K,s0 =−1.514J/g·K, ande0 =−160.2 J/g.

5.4 Mixed phase modelling

5.4.1 Mixture rules

The coexistence of liquid and solid phases was described in terms of a single pa-

rameterw, which was defined to be the local mass fraction of the liquid phase. With this

definition,w = 1 for a pure liquid andw = 0 in a pure solid. The thermodynamic proper-

ties in the mixed phase region were formulated by combining the pure phase models from

Sections 5.2 and 5.3 with several mixture rules. For clarity, liquid phase variables are de-

noted with a subscript “1” and solid phase variables are denoted with a subscript “2”. The

overall approach and assumptions used here are similar to mixture model developments in

previous studies of phase transitions under shock loading [5,42–44].

Within a small mass element, it is assumed that the liquid and solid water have a

160

common particle velocity so that they do not mechanically separate. The phases are also

assumed to be in intimate contact with one another, and thus share a common temperature

T and pressureP.

T = T1 = T2 (5.17)

P = P1(T,v1) = P2(T,v2) (5.18)

The extensive properties of a mass element (e.g. volume, internal energy) are defined in

terms of lever rules [1].

v = wv1 +(1−w)v2 (5.19)

e = we1(T,v1)+(1−w)e2(T,v2) (5.20)

The mixed phase state can be described completely in terms ofT, P, andw; all other

variables can be determined from Equation 5.18 and the lever rules.

5.4.2 Time dependence of the freezing transition

The optical and wave profile measurements presented in Chapter 4 clearly indicate

that the transition from liquid to solid water is a time dependent process. As such, the

continuum model describing water should incorporate a time dependent phase transition

that satisfies the following conditions. Changes in the liquid mass fraction should occur

when the Gibbs free energy of the systemg exceeds the equilibrium energygmin for the

current pressure and temperature conditions, driving the system towards its equilibrium

configuration. The rate at whichw changes should be related to the magnitude ofg−gmin.

The following phenomenological expression [42,43] can be used to meet these conditions.

dwdt

=− MRT

g−gmin

τ(5.21)

161

M is the molar mass of water (18.015 g/mole),R is the gas constant (8.314510 J/mole·K),

T is the current temperature, andτ is some characteristic time scale associated with the

transition.

The transition rate defined in Equation 5.21 describes a continuous transformation

from one phase to another, and does not allow the liquid to remain in a metastable state

during the incubation period observed in the optical transmission experiments. The sim-

plest way to account for the incubation period is to force the transition rate to be zero until

some predefined time.

dwdt

=

0 t ≤ tI

− MRT

g−gmin

τt > tI

(5.22)

tI represents the incubation time for the transformation. As discussed in Chapter 6,tI is

related to theP,T loading history and the presence of nucleation sites in the liquid sample.

For the purposes of modeling, the metastable period of a water sample was described with

a single value oftI .

5.4.3 Limiting cases for isentropic compression

In Section 5.2.3, it was shown that isentropic compression of liquid water accesses

the ice VII phase region. Once this occurs, there are two limiting cases for the water re-

sponse. In the metastable limit, water remains in the liquid phase indefinitely (tI → ∞ in

Equation 5.22). In the equilibrium limit, the water maintains the most favorable configu-

ration, so the solid phase appears in such a way to minimizeg (tI andτ1 are both zero).

All time dependent transitions under isentropic loading lie somewhere between these two

limits.

The isentropic loading path for metastable water is identical to the pure liquid phase

calculation discussed in Section 5.2.3. For an equilibrium mixture, both phases must have

the same Gibbs free energy (g1 = g2); if this were not the case, freezing or melting would

162

immediately occur to make this condition valid. The temperature of the loading path is thus

defined by the equilibrium coexistence curveT(P). The equilibrium liquid mass fraction

is dictated by the condition that the total system entropy remains fixed.

s = ws1 +(1−w)s2

ds = wds1 +(1−w)ds2 +(s1−s2)dw= 0 (5.23)

This relationship can be used to determine the pressure derivative of the mass fraction. Isen-

tropic mixed phase compression is thus defined by integration of two coupled differential

equations.

dTdP

=v1−v2

s1−s2(5.24)

dwdP

=−wds1dP +(1−w)ds2

dP

s1−s2(5.25)

wheredsi

dP=

(∂si

∂P

)

T+

(∂si

∂T

)

P

dTdP

The calculated loading paths for an equilibrium mixture are shown in Figure 5.7 along

with the metastable liquid loading path. Isentropic compression of an equilibrium mixture

follows the liquid-ice VII phase boundary in theT −P plane. As the pressure increases,

the liquid mass fraction decreases and more of the water sample becomes frozen. However,

the liquid mass fraction does not reach zero in the 3-14 GPa pressure range, which means

that isentropically compressed water does not become a pure solid under these conditions.

If the integration is extended beyond 80 GPa, the liquid mass fraction eventually reaches

zero, although such pressures are far beyond the intended domain of the liquid and solid

water models. Regardless of the exact pressure value, it is clear that complete freezing is

not possible in the experiments performed in this work, whereP≤ 10GPa.

To understand why complete freezing is so difficult under isentropic compression,

163

0 2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

1

Pressure (GPa)

Liq

uid

mas

s fr

acti

on

0 2 4 6 8 10 12 14300

400

500

600

700

800

Pressure (GPa)

Tem

per

atu

re (

K)

(a)

(b)

Liquid

Ice VII phase boundary

metastableliquid

Equilibriummixture

metastableliquid

Equilibriummixture

Figure 5.7: Limits of isentropic compression(a)T−P plane(b) w−P plane

164

0.55 0.6 0.65 0.7 0.75

2

3

4

5

6

7

8

9

10

specific volume (cc/g)

Pre

ssur

e (G

Pa)

Liquid

Ice VII Liquid−solid mixture

∆ v≈2%

Ice VI and mixtures

isotherm

isentropicliquid

isentropic mixture

Figure 5.8: Mixed phase compression in theP−v plane

165

consider the equilibrium mixtureP−v loading path shown in Figure 5.8. Once isentropic

compression enters the mixed phase region, the metastable liquid and equilibrium mixture

paths separate due to the volume contraction caused by freezing. As pressure increases,

the equilibrium mixture path slowly progresses towards the pure solid region. However,

the equilibrium mixture isentrope steepens with pressure, so the intersection of this curve

with the pure solid boundary occurs at pressures much higher than the pressure needed

to initiate freezing. This behavior is very different from isothermal freezing, which is also

shown in Figure 5.8. Since isotherms cross theT(P) coexistence curve at a single point, the

pressure of an isothermal mixture must be fixed, so(∂P/∂v)T = 0. Complete freezing at

fixed temperature is always possible given sufficient compression. That is not necessarily

true at fixed entropy because mixed phase compression has a nonzero slope in theP− v

plane.(

∂P∂v

)

s=

½½

½½½>

0(∂P∂v

)

T− b2T

cv(5.26)

b andcv in this expression represent properties of the mixed phase. The quantityb2T/cv

is always positive, so the isentrope is steeper than the isotherm. Hence, complete freezing

under isentropic loading is more difficult than complete freezing under isothermal loading.

The equilibrium mixture loading path shown in Figure 5.8 provides an upper limit to

the volume change a water sample may experience during freezing. The volume difference

between an isentropically compressed liquid and an equilibrium mixture at 5 GPa is about

2%. This is considerably smaller than the total volume change between liquid and solid

phases, which is about 8-10% at the same pressure. These volume changes are relevant

for the wave profile measurements discussed in Chapter 4 because volume contraction is

related to particle velocity histories in the water sample. This relationship is discussed

further in Sections 5.5 and 6.1.

166

5.5 Wave propagation calculationsThe COPS wave propagation code [45] was used to simulate shock compression

of liquid water samples. These simulations demonstrate that multiple shock compression

in liquid water is quasi-isentropic. A brief outline of the simulations is presented here to

describe the mixed phase calculation. Details and source code for the mixed phase model

are contained in Appendix G. Thermo-mechanical changes associated with freezing are

also discussed.

5.5.1 Calculation outline

The COPS code simulates one dimensional impact experiments as an array of cells

that define the local thermo-mechanical variables. During a time stepdt of the simulation,

mass and momentum conservation laws [27] are applied to each cell, which relates the new

cell density to the previous density as well as the previous particle velocity and longitudinal

stress profiles. Artificial viscosity [46] is used to prevent discontinuities from forming in the

simulation. Material model routines calculate the longitudinal stress for the current density.

The next time step for the calculation is determined from stability conditions [47,48]. The

above calculations are then repeated as required.

Material models for the window and impactor materials are described in Refs. 49

and 50. The remainder of this discussion focuses on the calculations that occur in compu-

tational water cells. During the timedt, a given water cell undergoes a volume changedv.

The new specific volume of that cell is then simplyv = v0 +dv. During this compression,

the liquid mass fraction of the cell changes at a ratedw/dt, defined by the transition rate in

Equation 5.22. This rate is treated as a constant during the time step.

w = w0 +dwdt

dt (5.27)

167

Since the compression is adiabatic, the local internal energy is increased by the work done

on the cell. This work is determined by the sum of the thermodynamic pressureP and the

viscous stressq.

e= e0− ((P+P0)/2+q)dv (5.28)

The average pressure during the time step(P+ P0)/2 was used to improve the accuracy

of the calculation. The value ofP is tied to the value ofv, but also depends upon the cell

temperatureT and the mass fractionw.

5.5.2 Enforcing the mixture rules

The values ofP andT are constrained by the mixtures rules specified in Section

5.4.1. The mixture rules can be rewritten into the following form [51].

z1 ≡ P1(T,v1)−P2(T,v2) = 0 (5.29)

z2 ≡ wv1 +(1−w)v2−v = 0 (5.30)

z3 ≡ we1(T,v1)+(1−w)e2(T,v2)−e= 0 (5.31)

The values ofv, w, ande are known from the above discussion;Pi andei are known in

terms ofT andvi from the pure phase models. Equations 5.29-5.31 can thus be solved for

the remaining quantities:T, v1, andv2.

The following vector notation is convenient for describing the mixed phase state.

x≡

v1

v2

T

z≡

z1

z2

z3

(5.32)

For the proper choice ofx, z = 0. The value ofz may be described by a Taylor series

168

expansion about a pointx0.

z = z0 +A dx (5.33)

whereA ≡

(∂P1∂v1

)T

−(

∂P2∂v2

)T

b1−b2

w 1−w 0

w(Tb1−P1(T,v1)) (1−w)(Tb2−P2(T,v2)) wcv1 +(1−w)cv2

∣∣∣∣∣∣∣∣∣∣x=x0

(5.34)

Suppose for the moment that the mixture rules are satisfied atx. The necessary step dx to

move from the original pointx0 to x can be found from Equation 5.33.

dx =−A−1z0 (5.35)

A−1 is the matrix inverse ofA, which was determined here by the cofactor method [52].

Equation 5.35 is simply a multi-dimensional version of Newton’s root finding method [53].

Given an initial guessx0, an improved solutionx = x0 +dx is determined. This solution is

then used as a new guess and the procedure repeated until the value ofx converges. The

value ofx definesv1, v2, andT as well asP, which is equal toP1(T,v1) andP2(T,v2).

All other quantities are defined from lever rules(e.g.g = wg1(T,v1)+ (1−w)g2(T,v2)).

The values ofe andP are coupled through energy conservation (Equation 5.28), so the

calculation was performed iteratively to ensure that the two variables were consistent.

Local sound speed

Following the discussion of Fickett and Davis [54], the frozen sound speedcF was

used for the stability analysis [47, 48]. This speed is defined using the usual isentropic

derivative at constant mass fraction.

c2F ≡

(∂P∂ρ

)

s,w=−v2

(∂P∂v

)

s,w(5.36)

169

The derivative was determined using a technique developed by Johnson et al. [55]. The

definition of sound speed can be rewritten by consideringP = P(e,v).

c2F = v2

[P

(∂P∂e

)

v,w−

(∂P∂v

)

e,w

](5.37)

Since the two phases share common pressure,P can be defined in terms of(T,v1).

c2F

v2 = P

[b1

(∂T∂e

)

v,w+

(∂P1

∂v1

)

T

(∂v1

∂e

)

v,w

]

−[

b1

(∂T∂v

)

e,w+

(∂P1

∂v1

)

T

(∂v1

∂v

)

e,w

](5.38)

The total volume and energy derivatives in Equation 5.38 can be determined from

the Newton step dx used in enforcing the mixture rules (Equation 5.35). After many iter-

ations have been performed, the value ofA−1 converges to a constant. The total volume

derivative of dx can then be written in terms of the matrix elements ofA−1 using Equation

5.35.

(∂x∂v

)

e,w=−A−1

(∂z0

∂v

)

e,w= A−1

0

1

0

(∂v1/∂v)e,w

(∂v2/∂v)e,w

(∂T/∂v)e,w

=

A−112

A−122

A−132

(5.39)

The total energy derivatives of dx can be found with the same method.

(∂v1/∂e)v,w

(∂v2/∂e)v,w

(∂T/∂e)v,w

=

A−113

A−123

A−133

(5.40)

170

The frozen sound speed is therefore related to liquid phase variables and matrix elements

of A−1.

c2F = v2

[P

(b1A−1

33 +(

∂P1

∂v1

)

TA−1

13

)−

(b1A−1

32 +(

∂P1

∂v1

)

TA−1

12

)](5.41)

5.5.3 Mixed phase calculations

Multiple shock compression temperatures

The mixed phase model was used to calculate metastable liquid temperatures for

the impact experiments discussed in Chapter 4. Figure 5.9 shows multiple shock com-

pression states in theT −P plane for various window materials used in the liquid cells.

Single shock and isentropic loading states are also shown in the figure for comparison. As

expected, multiple shock compression in liquid water is similar to the isentrope, leading

to considerably lower temperatures than single shock compression to the same pressure.

Multiple shock compression in water is therefore a quasi-isentropic process. Under such

conditions, liquid water becomes metastable with respect to ice VII somewhere above 2

GPa. This is consistent with the optical transmission experiments presented in Section 4.1,

which showed that water remained transparent for pressures below 2 GPa.

Mixed phase loading path

The formation of ice within a water sample changes the thermodynamic state in a

time dependent manner. To study these changes, numerical simulations were performed for

an experimental configuration where freezing was known to occur. These simulations were

performed for a 100µm water sample, confined in z-cut quartz windows, and impacted

with a z-cut quartz impactor travelling at 0.56 km/s. The resulting peak pressures were

5 GPa, which are well within the ice VII region (Figure 5.9). Thermodynamic histories

taken from the sample midpoint of a typical simulation are shown in Figure 5.10. The

incubation timetI was adjusted to postpone the transition until the end of compression; the

171

0 1 2 3 4 5 6 7 8 9 10300

350

400

450

500

550

600

650

700

Pressure (GPa)

Tem

per

atu

re (

K)

glass cellfused silica cellquartz cellsapphire cell

sing

le s

hock

isentrope

Liquid Ice VII

IceVI

freezingnot

observed

Figure 5.9: Multiple shock compression temperatures for liquid water in various liquidcells

172

0 100 200 300 400 5000

0.2

0.4

0.6

0.8

1

Time (ns)

Liq

uid

mas

s fr

acti

on

0 100 200 300 400 500250

300

350

400

450

500

550

Time (ns)

Tem

per

atu

re (

K)

0 100 200 300 400 5000

1

2

3

4

5

Time (ns)

Pre

ssu

re (

GP

a)0 100 200 300 400 500

1

1.1

1.2

1.3

1.4

1.5

Time (ns)

Den

sity

(g

/cc)

Figure 5.10: Simulated thermodynamic histories of water under multiple shock compres-sionFreezing is marked by the vertical dashed line.

173

transition timeτ was set at 50 ns. For consistency with the experimental results in Chapter

4, time was defined to be zero when the shock wave first enters the water sample. During

compression, the density, particle velocity, and temperature of the sample increased in a

stepwise fashion; the liquid mass fraction was fixed at unity during the incubation period.

Once freezing began, the sample density increased slightly, and at the same time there was

a brief pressure decrease. Pressure later returned to its original steady state value, which is

defined by the intersection of the window Hugoniots rather than the properties of the water

sample (Section 2.5). The temperature of the water sample increased with freezing due to

the release of latent heat; this increase was more dramatic than the density and pressure

changes. Freezing continued for more than 100 ns, reducing the liquid mass fraction to

about 0.7. This value indicates that the water sample was mostly liquid, with only 30% of

the sample frozen. The lack of complete freezing under these conditions is a result of the

adiabatic1 nature of the multiple shock compression calculation. If heat could be extracted

from the water on these time scales, complete freezing might be possible.

Figure 5.11 shows the results of the above simulation in theT−P plane along with

the isentropic case considered in Section 5.4.3. The first two compressions were somewhat

hotter than isentropic compression because of entropy production (Section 2.5.2). Subse-

quent temperature increases were comparable to isentropic compression. It is clear that

freezing became thermodynamically possible after at least two shock compressions. Dur-

ing the incubation period, multiple shock compression continued parallel to the isentrope

into the ice VII region. Once freezing began, the temperature increased until the water

sample reached the liquid-ice VII phase boundary. This loading path demonstrates that

time dependent freezing in a reverberated water sample begins near the metastable liquid

limit and moves toward the equilibrium mixture limit. The incubation time determines

1In an actual experiment, multiple shock compression is not truly adiabatic due to heat flow at the water-window interfaces. The discussion in Chapter 6 demonstrates that these effects are limited to the range of afew microns.

174

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5300

325

350

375

400

425

450

475

500

525

550

Pressure (GPa)

Tem

per

atu

re (

K)

Liquid

Ice VII 1st shock

2nd shock

phase boundary

metastableisentrope

equilibriumisentrope

simulation

Figure 5.11: Multiple shock loading path in theT−P planeDots show the steady shock states between shocks.

175

how quickly multiple shock compression deviates from the liquid isentrope, whereas the

transition rate dictates how quickly the system approaches the equilibrium limit.

Figure 5.12 shows the multiple shock compression path in theP− v plane. Here

the volume contraction and the corresponding pressure drop caused by freezing are clearly

visible. The volume change of a multiply shocked sample is about 1.5%, which is smaller

than the volume difference between an isentropic liquid and equilibrium mixture at the

same pressure. This reduction in volume change is due to entropy production, which leads

to a higher liquid mass fraction than a true isentropic compression. Once again it is clear

that the multiple shock peak state is bounded by the metastable and equilibrium isentropes.

This figure also reinforces the difficulty of complete freezing under isentropic compression

due to the adiabatic nature of the process. The specific volume of the mixed phase isentrope

is considerably higher than the pure solid volume because of the nonzero loading slope in

theP−v plane. Without the removal of heat, it is not possible to achieve complete freezing

under these conditions.

Wave profiles in freezing water

The time dependent mechanical changes noted in the previous discussion are linked

to the particle velocity profiles in a reverberating water sample. To determine how freezing

affects these profiles, numerical simulations were performed with the same configuration

(impact velocity, sample thickness, etc.) as described above. The particle velocity histories

in these simulations were recorded at the rear window interface rather than the sample

midpoint for consistency with the wave profile measurements discussed in Chapter 4.

Simulated wave profiles for two different types of transitions are shown in Figure

5.13. The delayed freezing profile is similar to the thermodynamic calculations discussed

above, where freezing was prevented until the end of compression andτ was set to 50

ns. The equilibrium mixture profile was obtained by setting both time scales to zero. The

176

0.6 0.625 0.65 0.675 0.7 0.725 0.752

2.5

3

3.5

4

4.5

5

5.5


Pre

ssu

re (

GP

a)

Liquid

Ice VII

mixed phase region

2nd shock

simulation

equilibriumisentrope

metastableisentrope

Figure 5.12: Multiple shock compression path in theP−v planeDots show the steady shock states between shocks.

177

0 100 200 300 400 5000

50

100

150

200

250

300

Time (ns)

Par

ticl

e ve

loci

ty (

m/s

)

delayedfreezing

equilibriummixture

Figure 5.13: Simulated particle velocity profiles

178

difference between these two cases during compression is somewhat subtle. The particle

velocity steps for an equilibrium mixture are slightly smaller than for the delayed transition;

the duration of each step is also somewhat longer in the equilibrium transition. The most

significant difference between these profiles occurs in the peak state, when the delayed

transition profile first begins to freeze. The density increase and transient pressure decrease

seen in theP−V plane (Figure 5.12) correspond to this drop in particle velocity. Like

pressure, particle velocity returned to its original steady state value at later times.

To determine the effect of freezing times on the calculated wave profiles, several

other simulations were performed for the same experimental layout using different values

of tI andτ. Figure 5.14 shows a few of the calculated profiles. In the delayed transition

profiles, freezing was prevented until the end of compression for different values ofτ; the

immediate transition profile was obtained by settingtI equal to zero but using a finite value

of τ. All wave profiles were identical through the first particle velocity jump, which is

actually the second compression of the water sample since particle velocity histories were

recorded at the rear window interface. Subsequently, the immediate and delayed phase

transition profiles became different. Like the equilibrium mixture, the immediate transi-

tion calculation showed a lower particle velocity history than found in the delayed freezing

profiles. Later compression stages of the immediate transition model showed continuous

increases in particle velocity while the delayed freezing profiles maintained steady veloci-

ties between shock compressions. The particle velocity history of the immediate transition

monotonically increased with time towards the peak state value. The delayed freezing pro-

files also increased towards the peak state, but then showed a decrease in particle velocity

when the transition began. The magnitude and duration of these decreases were tied to the

rate parameterτ. Small values ofτ led to faster particle velocity drops; the magnitude of

these drops was largest for small values ofτ. It was determined that the drops could only

be observed if freezing was delayed until compression was complete. This feature did not

179

50 100 150 200 250 300 350 400150

175

200

225

250

275

300

Time (ns)

Par

ticl

e ve

loci

ty (

m/s

)

first particle velocity jump

Immediate freezingτ=50 ns

Delayed freezingτ=50 ns

Delayed freezingτ=10 ns

Figure 5.14: Incubation and transition time effects in mixed phase water simulations

180

occur in an immediate transition regardless of the value ofτ.

The particle velocity differences between an immediate and a delayed phase transi-

tion are important for understanding the wave profile measurements presented in Chapter

4. In those measurements, the freezing appeared as a drop in particle velocity near the end

of sample compression. Based on the discussion above, it is not possible for the measured

velocity decreases to be caused by an immediate phase transition. Instead, the water sam-

ple must remain a metastable liquid for several compression steps before freezing takes

place. This emphasizes the importance of an incubation time in the freezing process, since

a single transition time cannot reproduce the observed wave profiles.

181



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[36] C.W. Pistorius, M.C. Pistorius, J.P. Blakey and L.J. Admiraal. Melting curve of iceVII to 200 kbar.J. Chem. Phys.38, 600 (1963).

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185

Chapter 6

Analysis and DiscussionThe experimental results reported in the previous two chapters are analyzed and

discussed in this Chapter. The optical results and wave profile measurements are analyzed

in Section 6.1 to show that the changes observed are consistent with a first order phase

transition. The remainder of the chapter focuses on different aspects of the freezing transi-

tion. Section 6.2 describes the importance of surface effects for initiating freezing. Section

6.3 discusses the observed time dependence of freezing and considers the effects of the dy-

namic P,T conditions created by multiple shock compression. Section 6.4 reviews various

length scales associated with the freezing transition.

6.1 First order phase transitionA first order transition is characterized by a discontinuity in the first derivatives of

the Gibbs free energy [1].

s=−(

∂g∂T

)

Pv =

(∂g∂P

)

T

The discontinuity in entropy is linked to the release of latent heat, which is caused by

the formation of permanent hydrogen bonds when water freezes. Latent heat emission

related to the growth morphology observed in the optical imaging measurements. The

discontinuity in volume is inferred through the particle velocity decrease observed in wave

profile measurements.

187

6.1.1 Latent heat

The release of latent heat during freezing has several consequences during solid

formation. Appendix H discusses a few aspects of this problem to demonstrate that latent

heat affects both the solidification rate as well as the solid morphology. The growth rate

of a uniform solid interface is dictated by the conduction of heat from that interface, a

situation commonly referred to as the Stefan problem [2]. The morphological stability of

the solid-liquid interface becomes an issue when the liquid phase is supercooled [3–5]. The

experimental results in this work show both phenomena, indicating that latent heat release

accompanied the changes reported here.

Figure 6.1 shows images from experiment I1 to demonstrate the effects of latent

heat emission; similar effects were also observed in other imaging experiments. These im-

ages reveal that solid growth is a two stage process. Initially, the transition proceeds along

narrow channels that rapidly overlap one another. Afterwards, the transformed regions con-

tinue to widen, generally preserving the original shape, at a much slower rate. Both stages

are consistent with two consequences of latent heat production. Unstable ice growth occurs

in the first stage because the liquid sample is highly supercooled, which favors the forma-

tion of sharp solid interfaces to maximize heat dissipation into the liquid, similar to the

formation of natural snowflakes [3–5]. As these features coalesce, unstable growth ceases

because the temperature ahead of the growth tips is close to the melting point. Stable ice

growth then becomes dominant, which is more planar (in a local sense) and much slower

than the initial growth stage.

6.1.2 Volume change

Although lateral variations are clearly visible at the mesoscale in Figure 6.1, these

variations are not included in the continuum model used to describe freezing (Chapter

188

100 µm

Figure 6.1: Latent heat and growth morphology (images from experiment I1)These images were taken from experiment I1 at timest =330, 530, 730, 930, 1130, and1330 ns (starting from the top left corner).

189

5). For the present discussion, the details of solid formation and growth are neglected,

treating the water sample is a one dimensional system under uniaxial strain [6]. The effects

of freezing on the particle velocity can then be calculated using the mixed phase model

developed in Chapter 5. These calculations are compared with the measured wave profiles

to demonstrate that water transforms to a denser state than the compressed liquid.

Figure 6.2 shows the results of experiment V3 as an example of the changes caused

by freezing. Also shown in the figure is a simulated particle velocity history obtained using

the mixed phase model, where freezing was postponed until the end of compression. Ini-

tially, the measured velocity history is similar to that expected for pure liquid compression

(Section 2.5), indicating that the sample did not freeze during this time. Once the peak state

was reached, there was a subsequent drop in particle velocity that is similar to the results of

the mixed phase calculation. In the mixed phase calculation, this drop occurs when liquid

water transforms to ice VII, which is denser than the liquid phase. It is therefore expected

that a drop in rear window particle velocity corresponds to an increase in sample density

and a decrease in specific volume. Mass conservation relates the average specific volume

〈v〉 to the particle velocities of the water-window interfaces:

d〈v〉dt

=uR−uF

ρ0d0(6.1)

whereρ0 is the initial sample density,d0 is the initial sample thickness, anduF anduR are

the front and rear window velocities, respectively. When the sample is in the peak state,

uR = uF , so the value of〈v〉 is constant. The fact thatuR subsequently drops below its

steady state value is consistent with a density increase, i.e.d〈v〉/dt < 0.

6.2 The importance of surface effectsWhen liquid water is compressed into the ice VII region, there is a characteristic

time required for the formation of a stable ice nucleus. This time is related to an energetic

190

0 100 200 300 400 500 6000

50

100

150

200

250

300


Par

ticl

e ve

loci

ty (

m/s

) freezingbegins

VISAR data

Mixed phasecalc.

Figure 6.2: Volume change and wave profiles (data from experiment V3)The dashed line shows the predicted steady state value, which equals half the impact ve-locity.

191

barrier that prevents the formation of small solid regions (Section 2.2.2). In the presence of

heterogeneous nucleation sites, the energetic barrier is reduced and freezing occurs more

readily. Several water interfaces were present in this work, creating potential sites for het-

erogeneous nucleation. Residual impurities in the water sample are one possible source

of nucleation sites; residual gas bubbles (≤ 10 µm diameter) present within most water

samples are another. The most important nucleation sites, however, were at the liquid cell

windows. With the proper choice of window material, freezing was consistently observed;

without these windows, the transition was not detected. This finding suggests that impu-

rities and residual gas bubbles did not play any significant role in the present work. This

section describes the experimental evidence for surface effects, demonstrating that these

effects are necessary for nucleation. It is shown that once ice nucleates at a window sur-

face, it then grows into the sample interior. A brief discussion of the nucleation mechanism

is also presented.

6.2.1 Evidence for surface effects

Optical measurements

For this discussion, the optical transmission results are shown in terms of extinction,

X, defined as the negative logarithm of transmission (Equation 4.1).

X ≡− lnT (6.2)

X = 0 for a completely transparent sample and increases as the sample transmission de-

creases. This definition is useful because extinction increases with the amount of scattering

in the water sample, and thus provides a measure of the extent of the phase transition.

Optical extinction was observed in all multiple shock compression experiments where the

peak pressure was greater than 2 GPa and a silica (SiO2) window was present. The exact

structure of the silica was unimportant– experiments using soda lime glass, fused silica,

192

and z-cut alpha quartz all showed optical extinction. When only a-cut sapphire (Al2O3)

windows were present, no significant optical changes were observed. Since windows in

this work remain transparent for the stress range of their use here (Appendix B), the loss of

transparency must be associated with the water samples.

A number of quartz and sapphire liquid cells experiments were performed at a peak

pressure of 5 GPa. Figure 6.3 shows results from a few of those experiments that used sim-

ilar initial sample thicknesses (≈100µm). Although optical extinction became significant

for water compressed in quartz windows, only negligible changes occurred in the presence

of sapphire windows. Quartz and sapphire windows result in slightly different loading

paths in water, so the temperature states in the two experiments were different. However,

this difference should only enhance freezing in the case of sapphire, which more closely

approximates the isentrope than quartz (Figure 5.9). The thermal conductivity of sapphire

is also about three times higher than quartz (Appendix B), resulting in greater heat con-

duction at the water-window interface and, therefore, lower temperatures. If temperature

effects were the sole factor for solid nucleation, freezing should be more readily observed

in sapphire liquid cells than quartz liquid cells. This is exactly opposite to the experimental

observations, so other factors must be considered for solid nucleation.

Hybrid window experiments were also performed at 5 GPa to further examine the

role of surface effects in freezing. These results are compared with the pure quartz and sap-

phire experiments in Figure 6.3. The hybrid experiments showed an intermediate extinction

history, which at later times appears to be about half that of a pure quartz liquid cell. This

indicates that the amount of transformed material in a hybrid liquid cell is roughly half that

of a quartz liquid cell. The most reasonable explanation for such behavior is that the quartz

window is necessary for starting the transition. The onset of extinction in the hybrid exper-

iments supports this explanation. The first shock compression does not produce conditions

compatible with freezing– at least two shocks are required (Section 5.5.3). Freezing thus

193

0 100 200 300 400 500 600 700 8000

0.5

1

1.5

2

2.5


Ext

inct

ion

(b

ase

e)

quartz windows

sapphire windows

quartz front/sapphire rear

sapphire front/quartz rear

Figure 6.3: Extinction histories for different window configurationsThe peak state in each experiment was (5 GPa, 450-460 K). Initial sample thicknesses weresimilar (99-108µm).

194

becomes thermodynamically possible only after the initial shock wave traverses the sample

and reflects from the rear window. When this window is composed of quartz, freezing can

start soon afterwards. In the opposite case (quartz front window), the reflected shock must

return to the front window to reach the nucleating silica surface, adding an extra delay to

the onset of freezing. This delay can be seen in Figure 6.3.

Wave profile measurements

Figure 6.4 shows results from experiments V3 and V4, which were performed to a

5 GPa peak pressure using a quartz and a sapphire liquid cell, respectively. In both experi-

ments, the magnitude and timing of the measured particle velocity steps matched the liquid

state calculation during the initial compression. The sapphire liquid cell experiment contin-

ued to match the liquid calculation while the quartz cell experiment showed a decrease in

particle velocity. Since this decrease can be associated with the volume change caused by

freezing, it is clear that no phase change occurred when water was compressed in sapphire

windows. The sapphire cell data also make a good case for the validity of the liquid EOS

because the magnitude and timing of compressed steps are well matched in the calculation

and experiment.

Although the wave profile and optical results are consistent in terms of window

effects, there are several differences between the measurements. Optical measurements

probed the entire sample thickness, whereas the wave profiles were measured at a single

point. There were also differences in the interfaces present in VISAR experiments V2-V4,

where water was directly exposed to a vapor plated aluminum mirror. These mirrors were

intentionally oxidized in an attempt to make the surfaces chemically similar to sapphire (i.e.

aluminum oxide). The lack of particle velocity deviations in experiment V4 (1 sapphire

+ 1 oxidized aluminum surface) was consistent with this expectation. The wave profile

experiments in quartz liquid cells (1 quartz + 1 oxidized aluminum surface) were thus

195

0 100 200 300 400 500 600 700 8000

50

100

150

200

250

300


up (

m/s

)

Quartz cell

Sapphire cell

VISARdata

Liquid calc.

Figure 6.4: Surface effects in wave profile measurementsPeak pressure in both experiments was about 5 GPa.

196

chemically similar to the hybrid liquid cells but underwent the same loading history as the

standard quartz liquid cells. These differences do not change the fact that freezing requires

a silica surface, but direct comparisons between the optical and wave profile measurements

are difficult.

6.2.2 Surface initiated freezing

Since the optical and wave profile changes in water occur only in the presence of

a silica window, it could be argued that other phenomena besides freezing are responsible

for the experimental observations. For example, optical transparency might be reduced

by chemical reactions or physical damage at the silica surface. However, these alternative

processes do not explain the consistency of the optical measurements with the equilibrium

phase diagram of water. The following discussion presents several other arguments to

eliminate these alternatives, leaving freezing as the only viable process that can cause the

changes observed in the present work.

Chemical reactions

There are few chemical reactions that could occur in the water samples that might

account for the observed optical changes. One possibility is the dissociation of water

molecules intoH+ andOH−, which would increase the electrical conductivity and might

allow for optical absorption and/or reflection in the visible spectrum. However, previous

shock wave research on liquid water (Section 2.3.1) indicates that at least 10 GPa single

shock compression is needed to generate significant ionization. The pressures and tem-

peratures obtained in this study are not sufficient to cause significant dissociation, so this

reaction cannot account for the observed transparency loss. Another possible reaction is

dissolution of the silica window into the water sample. If material is removed from the

window unevenly, the window-water interface will become roughened, creating optical

scattering and reducing light transmission. Amorphous and crystalline silica can dissolve

197

in water, a process that depends on the temperature of the water-silica interface.

To estimate the temperature of the water-silica interface, it is necessary to consider

the effects of heat conduction at the water-window interface. A diagram of this interface

is shown in Figure 6.5(a). To simplify the problem, heat conduction is assumed to be

decoupled from the mechanical loading. The water sample is compressed to a uniform

temperatureTL by multiple shock compression, and at timet = 0 heat begins to flow into

the window, which is initially at temperatureTR. Window compression is negligible in

these experiments, soTR is approximately equal to room temperature. The water sample

and the window are assumed to be semi-infinite slabs in contact atx = 0. Far from this

interface, each region is at the adiabatic compression temperature. The temperature profile

has the following form [2].

T1(x, t) = Ti +(Ti−TL)er fx

2(κ1t)1/2

T2(x, t) = Ti +(TR−Ti)er fx

2(κ2t)1/2

Ti =TL +ηTR

1+ηη =

√K2

K1

ρ2

ρ1cP2

cP1

(6.3)

The diffusivity is equal to the ratio of thermal conductivity and heat capacity (κi = Ki/ρicPi ).

Parameters for the water sample(K1, ρ1, andcP1) are presented in Section H.1.2; the pa-

rameters(K2, ρ2, andcP2) for a quartz window are summarized in Section B.3. The value

of η ranges between 1.5 and 2.1, depending on the exact value ofK1. Sinceη > 1, the

interface temperature is closer to the initial window temperature than the adiabatic water

temperature. For example, an 5 GPa multiple shock compression would generateTR≈ 465,

but the interface temperature would only be 352-365 K (79-92 C). The actual surface tem-

perature is still lower since the water model developed in this study tends to overestimate

the temperature during the initial compression stages (Section 5.2.3). Figure 6.5(b) shows

198

x=0

water sample

T=T1(x,t)

T → TL at x → − ∞

K1, ρ

1, c

P1

window

T=T2(x,t)

T → TR at x → ∞

K2, ρ

2, c

P2

−10 −8 −6 −4 −2 0 2 4 6 8 10x (µm)

Tem

per

atu

re (

K)

(a)

(b)

t=0

t=1 µs t=2 µs

Ti

TL

TR

Figure 6.5: Temperature of the water-window interface(a) Layout of the heat conduction problem at the water-window interface(b) Temperature profile at the water-window interface, where heat flow begins at timet = 0.

199

the temperature profiles near the water-window interface at various times for the heat con-

duction problem. The cooling effects of the window are relevant within a few microns of

the water sample, while the remainder of the sample stays at the adiabatic compression

temperature.

Previous static high pressure studies [7–9] have shown that the solubility of silica in

water becomes significant at temperatures above 400 C. Since the window interface tem-

peratures in this work are less than≤100 C, there is little chance that significant window

dissolution can occur in these experiments. Equilibrium solubility of silica in water at these

conditions is on the order of 100 ppm, but this concentration is attained only when water is

exposed to silica at high pressures and temperatures for several days [10–12]. It is unlikely

that any dissolution could occur during the10−9−10−6 s time scales of this work.

Window surface damage

Physical damage to the silica windows during compression and release is another

possible explanation for the experimental observations in this work. This effect occurs

when pressurized liquid water infiltrates microscopic cracks in the silica window, causing

the material to fracture. Optical scattering sites are formed as the window shatters, re-

ducing the sample transparency. Infiltration damage in silica was studied by Poulter and

Wilson [13], who reported that glass and quartz pieces immersed in liquid water sometimes

shattered during compression and release experiments. In those experiments, samples were

immersed in water and compressed to a pressure of 3 GPa within 30 seconds. If the pres-

sure was released immediately after reaching the peak state, returning to ambient pressure

in about 5 seconds, the glass and quartz pieces were recovered intact. However, when the

pressure was maintained for 5-20 minutes, then released within 5 seconds, the recovered

pieces were shattered. Unbroken pieces could be obtained after 20 minutes of high pressure

exposure if the release was performed over the course of 7 days. These observations lead

200

Poulter and Wilson to conclude that high pressure water infiltration requires several sec-

onds to occur. Once this infiltration takes place, the window does not immediately shatter,

but only breaks during rapid pressure release.

The experimental time scales of this work were on the order of10−6 seconds, so

there is insufficient time for water to infiltrate the silica windows. Even if such infiltra-

tion were to occur, there were no decompressions at water-window interfaces during the

experiments, so surface fracture is unlikely. Thus, the explanation that extinction arises

from physical damage caused by water infiltration is not consistent with the experimental

results.

Surface versus bulk optical extinction

Physical and chemical modifications to the silica surface by liquid water cannot

account for the observations of the present work. However, the differences between exper-

iments using quartz and sapphire windows indicate that surface effects must play a role.

To determine the role of surface effects, it is necessary to establish the extent of the trans-

formed material in the water samples. The optical extinction of a material is often modeled

by Beer’s law [14]:

X = αd (6.4)

whered is the sample thickness andα is an extinction coefficient characteristic of the

material. In general, both absorption and scattering losses contribute toα, but only the

latter process is significant for this work. Under dynamic conditions,α may be a function

of space and time, so extinction is defined by an integral.

X =∫ d

0α(x, t)dx (6.5)

201

For certain special cases,α(x, t) can be determined from some assumptions about the size,

shape, and distribution of scatterers. Appendix E developsα(x, t) for the case where scat-

tering occurs from a distribution of independent spherical bodies. However, it is not clear

that such a formulation is valid for this work (Section 6.4.3).

A qualitative interpretation of optical extinction can be developed by assuming that

freezing creates some average collection of scattering objects, where the density and cross

sections of these objects is some function of the thermodynamic conditions in the sample.

Although there are potential complexities in the optical transmission of a scattering medium

(Appendix E), the total amount of light scattered should scale with the total number of scat-

tering objects present in the sample. The measured optical extinction thus increases with

the amount of frozen material in the sample, even if the exact relationship is not straight-

forward. From the discussion in Section 6.2.1, freezing is initially confined to regions in

contact with a silica window. Figure 6.6 shows a schematic drawing of an optical measure-

ment for water confined in two silica windows. The sample is composed of transformed

layers in contact with the windows and untransformed liquid in the interior. When the tran-

sition reaches a steady state, no new scattering objects are formed, so the transformed layer

thicknessH must be constant. There are two limiting cases to the transformation thickness–

surface confined and sample confined. In the surface confined limit, the phase transition

would only involve water molecules that directly contact a silica window. An upper esti-

mate of the range for this type of transition can be made by considering the range of mass

diffusion, which carries water molecules to the surface from a mean distance〈x〉. This

distance varies as the square root of time and the self diffusivity of waterD≈ 2.26×10−9

m2/s [14].

〈x〉= 2

√Dtπ

(6.6)

During a 1µs experiment, the mean diffusion distance is about 53 nm, soH ≤ 50 nm

202

liquidwater

transformedmaterial

silicawindow

inputlight

d

detector

H

transmittedlight

Figure 6.6: Optical transmission in surface initiated freezing

203

becauseD decreases with pressure [14]. In the sample confined limit, the transformation

begins at the silica windows but can propagate beyond the range of surface effects and even-

tually cover the entire sample thickness. For a liquid cell containing two silica windows,

H → d/2 in the steady state;H → d for a sample with only one silica window.

If the freezing in water is confined to the silica surface, then the steady state extinc-

tion should be independent of sample thickness ford ≥ 50 nm. However, if the transfor-

mation can propagate from the surface, the measured extinction must increase with sample

thickness. Figure 6.7 shows extinction records from three different transmission experi-

ments (T6, IS3, and T5), all performed in quartz windows to 5 GPa peak pressure. The

initial sample thicknesses were 15, 32, and 108 (±5) µm, respectively. After compression,

the samples were compressed to a thicknessd, which is related tod0 by mass conservation:

d = d0ρ0

ρ(6.7)

The steady state sample density, neglecting the volume change due to freezing, was ap-

proximatelyρ =1.51 g/cc; the ambient valueρ0 was 0.99705 g/cc. The compressed sam-

ple thickness were therefore 10, 21, and 71µm (same order as above). The plots in Figure

6.7 indicates that the optical extinction increased with sample thickness. For the thinner

samples (10 and 21µm), there is clear break between the rapid extinction changes and a

steady state behavior. This break suggests that the transition spanned the complete thick-

ness of the water sample. Such a sharp break is not observed in the thickest sample, so it is

not clear whether the transition reached completion during that experiment. These results

make a strong case that the freezing transition can span at least 21µm of the sample during

the limits of the multiple shock compression experiment. This distance is three orders of

magnitude larger than the diffusion limited layer; hence, freezing must extend well beyond

the range of surface effects.

204

0 250 500 750 1000 1250 15000

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

2.75

3


Ext

inct

ion

(b

ase

e)

steady state

T6 (10 µm)

IS3 (21 µm)

T5 (71 µm)

Figure 6.7: Increasing extinction with sample thicknessThese experiments were performed with quartz windows to a 5 GPa peak pressure usingdifferent sample thicknesses. The compressed thickness of each measurement is shown inthe legend.

205

Because the optical changes in water are not confined only to silica window sur-

faces, it is necessary to distinguish the nucleation of ice from the subsequent growth pro-

cess. Since the transition only starts in the presence of silica, the nucleation process is

heterogeneous (on the time scales of this work) and is caused by some interaction at the

water-silica surface. Once ice nuclei are formed at the interface, they may then grow into

the bulk sample, allowing material far from the silica interface to be transformed. The

freezing process is thus a surfaceinitiated phase transition. The need for these surface

effects suggests that reverberated liquid water is in a metastable state. In other words,

multiple shock compression does not cross a spinodal limit [15] for pressures up to 5 GPa.

6.2.3 Ice nucleation at window surfaces

The present data demonstrates a significant difference in the ice nucleating effec-

tiveness of silica and sapphire windows, but do not explainwhy water behaves so dif-

ferently for each substrate. Both substances are considered to be poor ice nucleators at

ambient pressure– silica induces freezing in water supercooled to -20 to -25 C [16–18],

and sapphire1 does not induce freezing for temperatures well below -15 C. However, the

nucleation of high pressure ice phases may be very different than that of ice Ih. The follow-

ing discussion attempts to link the results of this work to known surface effects that occur

at the water-substrate interface.

Water at solid surfaces

When liquid water is in contact with a solid surface, there is a vicinal layer formed

near the interface that is very different from the bulk liquid. Numerous experimental studies

of the water-silica interface indicate a variety of differences between vicinal and bulk liquid

1The exact ice nucleation temperature for sapphire is not clear, but must be less than -15 C based on thework of Gonda and Nakahara [19,20].

206

water. For instance, viscosity in the vicinal layer is higher than in the bulk liquid [21]. Pro-

ton magnetic resonance studies [22] indicate that water molecules are ordered within the

vicinal layer; this ordering is also observed in x-ray diffraction studies [23]. The thickness

of the vicinal layer is roughly a few molecular layers (< 1 nm) [24]. Vibrational spec-

troscopy indicates that water molecules adsorbed onto silica are oriented with the oxygen

towards the surface [25]. The situation is somewhat more complicated when a bulk water

sample is in contact with silica because the surface may become charged. The extent to

which the surface is charged depends upon the pH of the liquid. There exists a specific pH,

known as the isoelectric point [24], where the surface is uncharged. At a pH above the iso-

electric point, the surface is negatively charged as protons are drawn into the liquid. Below

the isoelectric point, a positive surface charge is formed. Optical sum frequency generation

(SFG) studies of the quartz-water interface have revealed surface structure trends as a func-

tion of pH [26]. At the isoelectric point (pH≈2), water is oriented so that the oxygen atom

faces the surface, forming hydrogen bonds with the silica substrate. As the pH increases,

electrostatic repulsion tends to rotate the water molecules to an oxygen up configuration;

for pH >10, nearly all surface water molecules are in the oxygen up orientation. At inter-

mediate pH, there is a competition between the two configurations. There is also a strong

similarity between cases of extreme pH (strong ordering) and SFG measurements of the

quartz-ice interface [26].

Like silica, sapphire also attains a surface charge in the presence of liquid water.

However, the isoelectric point of sapphire is much higher (pH≈8), so that under normal

conditions, the surface charge is relatively small. Otherwise, SFG studies of the molecular

orientation of water at sapphire surfaces [27] show similar features to that of the silica [26].

At normal conditions (pH≈7), water molecules sit oxygen down on sapphire substrates.

207

Ice nucleation in this work

Due to the presence of atmospheric carbon dioxide, the water samples had a pH

of about 6 (Appendix C), which is closer to the isoelectric point of sapphire (pH≈8) than

of silica (pH≈2). Given that pH is a logarithmic scale, the surface charge on silica sur-

faces should be significantly higher than on sapphire surfaces. This extra charge produces

a stronger electric field and thus more ordering of water molecules. In addition to molecu-

lar ordering, there is experimental [28] and computational [29] evidence that water near an

electrified surface has a higher density than the bulk liquid. A possible explanation for the

surface effects observed in this work is that water near silica surfaces has certain similar-

ities to a high pressure ice phase. Under normal conditions, this phase is only stable near

the charged silica surface, but this phase may nucleate bulk freezing when the sample is

compressed. Since the induced surface charge on sapphire is lower than silica, these effects

are much weaker and may be insufficient to induce freezing. The importance of charged

surfaces in freezing water has been reported in other studies [30, 31], so it is possible that

such phenomena may play a role in the present work.

Although it has been demonstrated that silica acts as a suitable nucleator, the nu-

cleating capacity of sapphire is not clearly understood. Freezing might occur at sapphire

interfaces on time scales beyond the experimental limits of this work; pressures higher than

5 GPa may also be required for the transition to occur. At very high pressures, homoge-

nous nucleation should become possible, eliminating the need for window surface effects

to start the transition. Ultimately, it may be possible to exceed a thermodynamic stability

limit [15], at which point the liquid transforms without an energetic barrier. Either of these

phenomena might may contribute to the results of experiments T11 and T12 (page 115),

where a repeatable loss and recovery of transmission occurred as the pressure reached 8

GPa. The recovery of optical transmission is consistent with the formation of a uniform

208

solid phase, which would not scatter light. The interpretation of optical measurements then

becomes more complicated as optical transmission is no longer a unique function of the

transition extent. Further study of water at pressures above 5 GPa is needed to understand

these results.

6.3 Freezing time scalesThe time scales of freezing should be linked to the thermodynamic conditions cre-

ated by multiple shock compression. Based on the mixed phase calculations in Chapter 5,

it is expected that freezing will occur more rapidly as pressure is increased beyond 2 GPa.

This trend can be seen in the optical transmission results shown in Figure 6.8. The loss

of optical transparency occurs more quickly with increasing peak pressure for samples of

similar initial thickness, which is expected since the water is compressed further into the

ice VII region. However, the time scales of optical transmission loss are also a function

of sample thickness. This trend is shown in Figure 6.9 for experiments of similar peak

pressures. The purpose of this section is to separate these effects, allowing the fundamental

freezing time scales to be determined.

6.3.1 Apparent time scales

The optical transmission results were analyzed in terms of two apparent time scales–

incubation and transformation. Incubation time (tI ) was defined by the period where optical

transmission remains near unity. Transformation time (τ) was defined by the inverse slope

of the optical transmission curve at the inflection point as shown in Figure 6.8. To measure

these times in a systematic fashion, optical transmission curves were fitted with a piecewise

209

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Tra

nsm

issi

on

T1 (1.1 GPa, 104 µm)

I1 (2.7 GPa, 137 µm)

I2 (3.6 GPa, 132 µm)

T5 (5.0 GPa, 108 µm)

Figure 6.8: Time scales and peak pressureQuartz windows were used for all experiments except T1, which used soda lime glasswindows. The dashed lines show the transformation time for experiment I2.

210

0 100 200 300 400 500 600 700 800 9000

0.2

0.4

0.6

0.8

1


Tra

nsm

issi

on

0 100 200 300 400 500 600 700 800 9000

0.2

0.4

0.6

0.8

1


Tra

nsm

issi

on

(a)

(b)

IS1 (19 µm) T4 (28 µm)

I2 (132 µm)

T6(15 µm)

IS3 (32 µm)

IS2 (107 µm)

T5 (108 µm)

Figure 6.9: Comparisons of the quartz cell photodiode experiments(a) 3.5 GPa peak state(b) 5.0 GPa peak state

211

0 50 100 150 200 250

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1


Tra

nsm

issi

on

datafit

97 ns apparent incubation

time

Figure 6.10: Fit of the T5 photodiode recordEquation 6.8 was used to determine the optimal incubation (tI ) and transition (τ) times.

Table 6.1: Measured incubation and transition times

Exp. ID Pmax (GPa) d0 (µm) tI (ns) τ (ns)

I1 2.7 137 192.3 1236

IS1 3.5 19 55.4 234

T4 3.5 28 70.3 248

I2 3.6 132 163.8 583

T6 5.0 15 19.9 182

IS3 5.0 32 32.6 184

IS2 5.0 107 92.7 149

T5 5.0 108 97.4 163

212

continuous function.

T(t) =

T0 t < tI

Ts+(T0−Ts)exp

[−

(t− tI

tL

)2]

t ≥ tI(6.8)

T0 is equal to the initial transmission (≈ 1); Ts is the steady state transmission fort →∞. T0,

Ts, tI , andtL were used as adjustable parameters to match the measured optical transmission

from t = 0 until a short time after the inflection point. An example of this fit for experiment

T5 is shown in Figure 6.10. The value ofτ is linked totL by the following relation.

τ =∣∣∣∣dT(tin f lect)

dt

∣∣∣∣−1

=tLe1/2

√2(T0−TS)

(6.9)

Apparent incubation and transition times for all quartz cell experiments are shown in Table

6.1. As expected, both time scales decrease with increasing peak pressure.

6.3.2 Incubation time analysis

Freezing in compressed liquid water is nucleated by surface effects (Section 6.2),

so the incubation times listed in Table 6.1 are dominated by the thermodynamic conditions

at the water-quartz interfaces. For the purpose of clarity, it is assumed here that freezing

begins at the front window of the water sample, although the following analysis can readily

be applied to freezing at the rear window. Sample compression begins at timet = 0, and

freezing becomes thermodynamically possible at the front window interface whent = tP.

Freezing is observed at a later timetI . It is assumed that compression is essentially complete

prior to tI .

The difficulty in analyzing the incubation times lies in the time duration of multiple

shock compression. Since several shocks are required to cross the phase boundary (Sec-

tion 5.5.3), the measured incubation time includes a duration where freezing cannot occur,

which means thattI is affected by factors unrelated to the thermodynamics of freezing (e.g.

213

sample thickness). Once freezing becomes thermodynamically possible, there are several

additional compressions before a steady state is reached. Several intermediate states are

compatible with freezing, so it is difficult to associate the measured incubation time with

any specific set of P,T conditions. One would expect that these intermediate states con-

tribute to the overall nucleation process, so the fundamental problem in determining the

incubation time is how the material “remembers” its metastable history.

To deal with the time dependent conditions of multiple shock compression, consider

a transition coordinateq that tracks the metastable sample history. This coordinate is fixed

for t ≤ tP because freezing is not possible; fort ≥ tP, the value ofq changes until it reaches

some critical value att = tI . The value ofq is coupled to the liquid mass fractionw (Section

5.4) so that the freezing occurs att = tI . This description treats freezing as a two stage

process– nucleation (changingq) and growth (changingw). Dynamic loading during the

incubation period can be accounted for by definingq as the area under the curve of some

thermodynamic variableη . The value oftI is then given by a the time required for a fixed

amount of areaA to be swept out.

q(t)≡∫ t

tpη(t ′)dt′ q(tI ) = A (6.10)

The functionη represents the extent to which the liquid phase is driven into the metastable

region. Deeply supercooled states would therefore have large values ofη and require short

incubation times, while states near the coexistence curve, whereη = 0, require long incu-

bation times. A reasonable definition forη is the normalized Gibbs free energy difference

between solid and liquid phases.

η ?=−M (gL(t)−gS(t))RT

(6.11)

A generalη(t) history for a reverberating system is shown in Figure 6.11. The intermediate

steady states in this loading allow the integral in Equation 6.10 to be reduced to a discrete

214

0t (ns) from shock arrival

η (t

)

transitionbegins

tI

transitionpossible

tp

ηs

Figure 6.11: Incubation time and the metastable history

215

sum.

A = ηS(tI − tp)−∑i(ηS−ηi)∆ti (6.12)

ηi and∆ti represent the magnitude and duration of thei-th shock after freezing becomes

possible;ηS is the steady state value ofη(t) that the compression progresses towards.∆ti

is determined by the sample thickness during thei-th shock, which must scale with the

original sample thicknessd0 through mass conservation.tP is defined by the traversal time

of one or more shocks, each of which must also scale linearly with the original sample

thickness. Equation 6.12 then reduces to the following form.

∆ti = cid0 tp = bd0

AηS

= tI −d0

(b+∑

i(1− ηi

ηS)ci

)= tI −d0B (6.13)

The values ofb, ηi , andci (and thusB) are related to the compression path. For a given

impact velocity and window configuration, these parameters are constant. Equation 6.13

may then be rewritten so that the incubation time is a function of sample thickness.

tI =AηS

+d0B = tminI +Bd0 (6.14)

Thus a plot of measured incubation times versus initial sample thicknesses should lie along

a straight line. Regardless of the actual definition ofη(t), the intercept of this line indicates

the shortest possible incubation time for that peak state.tminI is equivalent to the apparent

incubation time in the limit of zero sample thickness, where compression is instantaneous.

Plots of apparent incubation times versus sample thickness for 3.5 and 5.0 GPa peak

states are shown in Figure 6.12. Along with the data for each pressure state is a linear fit

used to determinetminI . At 3.5 GPa, this time was 41±2 ns; at 5 GPa, it was 7±2 ns. If

it is assumed that the incubation time is dominated by the formation of stable nuclei, then

tI ≈ tn, wheretn is the characteristic time required to form an ice nucleus. A relationship

216

0 20 40 60 80 100 120 1400

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

Initial sample thickness (µm)

Ap

par

ent

incu

bat

ion

tim

e (n

s)

3.5 GPa: tmin

=41 ± 2 ns

5 GPa: tmin

=7 ± 2 ns

Figure 6.12: Incubation time versus sample thickness for different peak states

217

betweentn and the liquid-solid Gibbs free energy difference was derived in Section 2.2.2

(Equation 2.12). That expression can be written in the following form.

ln tn = ln tminn +

B(gL−gS)2 (6.15)

The minimum nucleation time can thus be estimated from the linear intercept of aln(tn)

versus1/(gL−gS)2 plot. Sincetn was only measured for two peak pressures (3.5 and 5

GPa),tminn was determined by solving Equation 6.15 directly. The energy differencegL−gS

was calculated from the liquid and solid water models developed in Chapter 5. However,

there is an ambiguity regarding the temperature of the nucleating window surface. Heat

conduction was ignored in multiple shock compression temperature calculations, but is

significant for samples thinner than a few microns. To determine the limiting values of

tn, gL − gS was calculated using both the adiabatic temperature as well as the window

temperature (≈ 298K). The result of this calculation indicates that ice nucleation in liquid

water requires a minimum of 2-5 ns.

6.3.3 Transition time analysis

Unlike the incubation time, there is no simple interpretation of the transition time.

The growth of ice in the water sample is a complex process that depends on many different

phenomena, preventing a quantitative description of the measured transition time. One

might expect that peak pressure and sample thickness should affectτ in generally the same

way astI . Thus, it would seem thatτ should increase with pressure and decrease with

sample thickness.

Figure 6.13 contains plots ofτ for fixed peak pressures as a function of initial

sample thickness. At 3.5 GPa, transition time increased with sample thickness, which

seems reasonable for a surface initiated transformation, where time is required for freezing

to infiltrate the bulk sample. Transition times were also generally smaller at 5 GPa than 3.5

218

0 20 40 60 80 100 120 140100

150

200

250

300

350

400

450

500

550

600

650

Initial sample thickness (µm)

Ove

rall

tran

siti

on

tim

e (n

s)

3.5 GPa

5.0 GPa

Figure 6.13: Transition time versus sample thickness for different peak statesStraight lines are shown as a visual guide.

219

for any sample. However, transition times were nearly constant at 5 GPa, which seems to

contradict the surface initiation theory developed in Section 6.2.2. However, this analysis

has ignored the possibility of multiple transition time scales in the problem. By defining

transmission time at the inflection point of the optical transmission record, the value ofτ is

dominated by thefastestgrowth processes. The possibility of multiple transition time scales

is discussed further in the next section using the results of optical imaging experiments.

6.4 Transition length scalesThis section discusses several length scales associated with the freezing transition.

First, a discussion of the different domains in the water sample is presented to summa-

rize the processes that affect freezing. Next, the variations in the initiation and growth of

ice perpendicular to the impact direction are described. Finally, the distribution of solid

material in the transformed region is discussed.

6.4.1 Domains of the water sample

Figure 6.14 shows various domains of a compressed water sample that had a tem-

peratureTs prior to the onset of freezing. Within about 1 nm of the silica window, there is a

vicinal region where freezing is initiated. Once the transition begins, the window provides

a heat sink that absorbs the latent heat, so the frozen material is cooler than the melting tem-

peratureTm. The cooling effect spans a few microns of the sample; beyond that range, heat

conduction through the window becomes negligible, so the transformed material is nearly

at Tm. Heat conduction persists in the liquid material just ahead of the transition, which

raises the temperature aboveTs. Very far ahead of the transition is the liquid unaffected by

latent heat release.

The mixed phase model presented in Chapter 5 treats all portions of a water sample

in the same fashion, but this is clearly inconsistent with the various domains discussed

220

window cooled

material

solidmaterial(near T )

transformedmaterial

m

vicinalregion

liquidwater

(T>T )

supercooledliquid water

(T=T )S S

Figure 6.14: Longitudinal length scales of the freezing of water

221

above. For example, the transition rate proposed in Equation 5.22 describes a process that

can occur at any location if the local P,T conditions allow freezing. In reality, freezing

should only be allowed if a computational cell is near a silica window or another cell that

has already transformed. The lack of heat conduction in the wave code calculations has

several additional shortcomings. For certain pressures near the phase boundary, it may be

possible for freezing to occur in the window cooled region of the water sample while the

rest of the sample is too hot to freeze. Since heat is removed from this region, complete

freezing may be possible (Section 5.4.3), allowing a larger volume change than in the

sample interior. Future mixed phase models need to consider the domains shown in Figure

6.14 to describe the freezing process in water. In essence, the model must consider each

cell’s location in addition to its thermodynamic history.

6.4.2 Lateral freezing variations

The optical imaging measurements shown in Chapter 4.2 demonstrate that freezing

in water is not uniform across the sample. However, these variations did not affect the

optical transmission results because the probe size in those experiments was large enough

(≈6 mm) to average over the lateral heterogeneities in the sample, producing consistent

transmission results. Figure 6.15 shows an example of this consistency for two similar

optical transmission experiments. The similarity of these two curves implies that many

independent nucleation events are measured, which leads to a well defined nucleation rate.

Thus the optical transmission records provide a consistent measure of the freezing time

scales.

For length scales shorter than 1 mm, there is considerable structure in the phase

transition, revealing stark differences in the transformed and untransformed material. The

complex structure of these regions prohibits systematic analysis as applied to the optical

transmission results, but there is qualitative information about the initiation and growth of

222

0 200 400 600 800 1000 1200 14000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Tra

nsm

issi

on

T5

IS2

Figure 6.15: Consistency of the optical transmission measurementsExperiments T5 and IS2 were performed in quartz windows to a peak stress of 5 GPa.Initial sample thicknesses were 107-108µm.

223

ice within the images. These stages are shown for pressures of 2.7, 3.5, and 5.0 GPa in

Figure 6.16. The remainder of this discussion shows how these stages are consistent with

the time scale analysis of Section 6.3.

Initiation

The unconnected darkened regions in the top set of images in Figure 6.16 indicates

that freezing begins at several independent locations. Each independent feature should

be the result of at least one nucleation event, so the number of these features gives an

overall measure of the number of nucleation events in the sample. At 2.7 GPa, about 14

independent nucleation events were observed in a 0.69 mm2 area. This number increased

at 3.5 GPa, but its actual value is somewhat ambiguous because some features began to

overlap while new nucleation events occurred. By 5.0 GPa, it was nearly impossible to

identify specific nucleation sites because initiation happened so quickly throughout the

sample. The separation of nucleation events ranged from about 200µm at 2.5 GPa to less

than 50µm at 5.0 GPa.

The number of nucleation eventsn∗ is related to the nucleation rateJ defined in

Equation 2.11. If nucleation occurs in a vicinal layer of thicknessd∗, then the average

number of nucleation events observed in an areaA is given byn∗.

n∗ = J∗0 Ad∗ exp

(− B∗

(gL−gS)2

)(6.16)

Given the consistency of the optical transmission records, it is reasonable to conclude that

the vicinal parametersJ∗0, B∗ andd∗ are relatively constant for the z-cut quartz windows

used in this work. The value ofn∗ is thus dominated by the value ofe−(gL−gS)−2for a fixed

observation area. The increase in independent freezing features should thus increase with

pressure, which is consistent with the discussion above. This increase in nucleation rate

leads to a decrease in the incubation time, which is consistent with the analysis in Section

224

t=33

0 n

s

t=14

30 n

s

t= 2

25 n

s

t=67

5 n

s

t= 1

35 n

s

t= 5

65 n

s

2.7

GP

a (I

1)

3.5

GP

a (I

2)

5.0

GP

a (I

3)

Fig

ure

6.16

:In

itiat

ion

and

grow

thin

imag

ing

mea

sure

men

tsIn

itial

stag

esof

the

tran

sitio

nar

esh

own

inth

eup

per

thre

eim

ages

.T

helo

wer

thre

eim

ages

show

the

sam

ple

asit

appr

oach

esth

est

eady

stat

e.T

hesi

zeof

each

imag

eis

appr

oxim

atel

y92

0×

750

µm(h

oriz

onta

l×ve

rtic

al).

The

spat

ialr

esol

utio

nlim

itfo

rth

ese

imag

esis

abou

t10µm

.

225

6.3.2.

Transformation growth

The lower set of images in Figure 6.16 were taken when optical changes in the

water sample approached a steady state. The samples were not completely frozen in this

state due to growth instabilities resulting from latent heat release. The overall size of the

individual transformation regions decreased with the peak pressure state. At 2.5 GPa, most

of the independent transition features were 30-100µm wide and 100-200µm long. These

features were reduced to 15-70µm widths and≤150 µm lengths at 3.5 GPa. By 5.0

GPa, transition regions are 10-70µm wide and less than 100µm long. This decrease

in feature size is a direct consequence of the higher nucleation rate, which reduces the

space available for independent feature growth. The smaller, denser features produced at

high pressure more thoroughly cover the sample, which is consistent with the observation

that increases in peak pressure reduce the measured optical transmission (Figure 6.8). The

dominant mechanism in freezing is thus related to the the nucleation rate, which is defined

by the pressure in the water sample. For low pressure states, freezing isgrowthdominated,

since most of the transmission loss is due to ice propagation from a limited number of

initiation initiation sites. As pressure increases, optical transmission is reduced more by the

formation of freezing sites rather than their subsequent growth, so freezing isnucleation

dominated.

The extent to which a particular transformation region can grow is determined by

the separation of independent nucleation sites. Separate transformation regions can only

grow until they overlap one another, at which point freezing ceases in that direction. This

is important here because ice only nucleates on the window surfaces, which may result

in different overlap times for growth along and away from the windows. A schematic

diagram of this problem is shown in Figure 6.17. When only a few transition regions

226

(c)

silicasurface

(b)

(a)

frozenmaterial

liquid water

Nucleation Growth

Figure 6.17: Nucleation rate and lateral freezing(a) When nucleation is rare, ice growth occurs both along and away from the silica windowwithout encountering other ice formations.(b) As more nucleation events occur, there is a period of independent growth before overlapoccurs.(c) When nucleation is very common, lateral coverage is rapid.

227

are formed, some time passes before separate regions interact, so considerable growth can

occur both along and away from the window. The situation changes at high nucleation

rates, where independent regions quickly interact with one another. During that time, very

little longitudinal growth occurs, so the initial stages of freezing are largely independent of

the total sample thickness. The complete description of freezing would therefore require

both a lateral and a longitudinal time transition scale. For low nucleation rates, these time

scales may be comparable, but at high nucleation rates, the lateral time scale dominates the

observed transition time scale. This behavior is consistent with the transition time scale

analysis in Section 6.3.3.

6.4.3 Composition of the transformed material

Although the images in Figure 6.16 reveal both frozen and unfrozen material, the

actual composition of the transformed material is not known. At length scales below 10

µm, the limiting resolution of the optical images, the opaque regions may be a complex

mixture of liquid and solid. The following discussion presents several conceptual models

to assess the distribution of solid material in the transformed material.

A natural starting point for visualizing the transformed material is to assume that

ice exists as a collection of independent spheres located throughout the water sample. Each

sphere scatters some of the light passing through the sample, so the total transmission is

governed by the solid mass fraction (Appendix E). In the Rayleigh limit [32,33], where all

the spheres are much smaller than the wavelength of visible light, the optical transmission

profile becomes a simple function of wavelength.

T(λ ) = T(λ0/λ )4

0 (6.17)

As shown in Figure 6.18, this profile is inconsistent with the spectrally resolved transmis-

sion measurements made during experiment T5. There are a several possible reasons for

228

400 425 450 475 500 525 550 575 600

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Wavelength (nm)

Tra

nsm

issi

on

Rayleighprofile

Measurement@ t=200 ns

Figure 6.18: Comparison of measured transmission profile with Rayleigh scattering limitThe measured transmission was taken 200 ns after shock arrival in experiment T5.

229

this deviation (Appendix E), so it is difficult to draw conclusions about the solid size based

on optical transmission alone. Results of the Stefan problem (Appendix H) can also be

used to estimate the characteristic sizes of ice particles. The thickness of frozen material is

given byX = 2λ√

κt, whereλ is determined by the boundary conditions at the origin of

freezing. For a 5 GPa multiple shock compression, the characteristic solid size for 1000 ns

of growth is 0.3-1.7µm (Figure H.3)

Since many features in the optical imaging experiments were much larger than 10

µm, uniform ice growth cannot adequately describe the freezing process. Instead, much of

the transformation must occur through unstable growth that favors sharp liquid-solid inter-

faces (Section H.2). Additionally, ice cannot exist as a collection of independent spheres

because that would require nucleation sites to be scattered throughout the water sample,

which is inconsistent with the surface effects described in Section 6.2. Ice growth probably

resembles a tree-like entity that is anchored to a silica window. Such complex pattern for-

mation is commonly found in the solidification of supercooled liquids [3,4] and may have

a fractal nature [34].

230



[2] H.S. Carslaw and J.C. Jaeger.Conduction of heat in solids. Oxford University Press,Oxford, 2nd edition, (1959).

[3] B. Chalmers.Principles of Solidification. John Wiley & Sons, New York, (1964).

[4] J.S. Langer. Instabilities and pattern formation in crystal growth.Rev. Mod. Phys.52,1 (1980).

[5] V. Alexiades and A.D. Solomon.Mathematical modelling of melting and freezingprocesses. Hemisphere Publishing, Washington, (1993).


[7] W.A. Bassett, A.H. Shen and M. Bucknum. A new diamond anvil cell for hydrother-mal studies to 2.5 GPa and from -190 to 1200 C. Rev. Sci. Instrum.64, 2340 (1993).

[8] W.A. Bassett, A.H. Shen and I. Chou. Hydrothermal studies in a new diamond anvilcell up to 10 GPa and from -190 C to 1200 C. Pure Appl. Geophys.141, 487 (1993).

[9] A.H. Shen, W.A. Bassett and I. Chou. Theα−β quartz transition at high temperaturesand pressures in a diamond-anvil cell by laser interferometry.Am. Mineralogist78,694 (1993).

[10] G.W. Morey, R.O. Fournier and J.J. Rowe. The solubility of quartz in water in thetemperature interval from 25-300 C. Geochim. Cosmochim. Acta26, 1029 (1962).

[11] R.O. Fournier and J.J. Rowe. The solubility of amorphous silica in water at hightemperatures and high pressures.Am. Mineralogist62, 1052 (1977).

[12] C.E. Manning. The solubility of quartz inH2O in the lower crust and upper mantle.Geochim. Cosmochim. Acta58, 4831 (1994).

[13] T.C. Poulter. Apparatus for optical studies at high pressure.Phys. Rev.40, 860 (1932).

[14] P. Atkins.Physical Chemistry. W.H. Freeman, New York, 6th edition, (1998).


[16] A.C. Zettlemoyer, J.J. Chessick and N. Tcheurekdjian. Nucleating process, (1966).U.S. Patent 3,272,434.

231

[17] L.H. Seeley, G.T. Seidler and J.G. Dash. Apparatus for statistical studies of heteroge-neous nucleation.Rev. Sci. Instrum.70, 3664 (1999).

[18] L.H. Seeley.Heterogeneous nucleation of ice from supercooled water. Ph.D. thesis,Unversity of Washington, (2001).

[19] T. Gonda and H. Nakahara. Formation mechanism of side branches of dendritic icecrystals grown from vapor.J. Crys. Grow.160, 162 (1996).

[20] T. Gonda and S. Nakahara. Dendritic ice crystals with faceted tip growing from thevapor phase.J. Crys. Grow.173, 189 (1997).

[21] G. Peschel and K.H. Adlfinger. Viscosity anomalies in liquid surface zones.J. ColloidInterface Sci.34, 505 (1970).

[22] P.A. Sermon. Interaction of water with some silicas.J.C.S. Faraday I76, 885 (1980).

[23] A. Fouzri, R. Dorbez-Sridi and M. Oumezzine. Water confined in silica gel and invycor glass at low and room temperature, x-ray diffraction study.J. Chem. Phys.116,791 (2002).

[24] R.K. Iler. The Chemistry of Silica. Solubility, Polymerization, Colloid and SurfaceProperties, and Biochemistry. John Wilery and Sons, New York, (1979).

[25] K. Klier, J.H. Shen and A.C. Zettlemoyer. Water on silica and silicate surfaces. I.Partially hydrophobic surfaces.J. Phys. Chem.77, 1458 (1973).

[26] Q. Du, E. Freysz and Y.R. Shen. Vibrational spectra of water molecules atquartz/water interfaces.Phys. Rev. Lett.72, 238 (1994).

[27] M.S. Yeganeh, S.M. Dougal and H.S. Pink. Vibrational spectroscopy of water atliquid/solid interfaces: crossing the isoelectric point of a solid surface.Phys. Rev.Lett.83, 1179 (1999).

[28] M.F. Toney, J.N. Howard, J. Richer, G.L. Borges, J.G. Gordon, O.R. Melroy and D.G.Wiesler. Voltage-dependent ordering of water molecules at an electrode-electrolyteinterface.Nature368, 444 (1994).

[29] S.-B. Zhu and G.W. Robinson. Structure and dynamics of liquid water between plates.J. Chem. Phys.94, 1403 (1991).



232

[32] H.C. van de Hulst.Light Scattering by Small Particles. Dover, New York, (1957).


[34] T. Vicsek.Fractal Growth Phenomena. World Scientific, Singapore, (1992).

233

Chapter 7

Summary and ConclusionsThe overall objective of this study was to observe and characterize freezing in water

on nanosecond time scales. Optical and wave profile measurements were used to demon-

strate that liquid water undergoes time dependent freezing under quasi-isentropic compres-

sion. Although the crystal structure of the solid phase was not determined in this work, the

freezing transition observed here is consistent with the formation of ice VII. Unlike previ-

ous studies [1], these results show unambiguous evidence for freezing under shock wave

compression.

7.1 SummaryMultiple shock wave compression was used to quasi-isentropically compresses liq-

uid water. This method reduced the temperature increase during compression, creating

thermodynamic conditions that favor ice VII over the liquid phase for pressures greater

than 2 GPa. Samples compressed above that pressure showed a clear loss of optical trans-

parency, indicating that water no longer exists in a pure liquid state. Real time optical im-

ages revealed that these changes were not uniform across the water sample and that freezing

occurred heterogeneously. Measured wave profiles also differed from the expected behav-

ior of a pure liquid. Both optical and mechanical changes were observed after an incubation

period, where water remained in a metastable liquid state. The growth morphology of the

subsequent transformation was consistent with thermal diffusion limited growth, which

suggests a release of latent heat. There was also evidence that the transformed state was

denser than the compressed liquid water, which is consistent with the formation of a high

pressure ice phase.

235

Surface effects were found to play an important role in the freezing process. The

optical and mechanical changes described above only occurred in the presence of a silica

(crystalline or amorphous) window; water compressed in sapphire windows did not show

signs of freezing. When the freezing transformation was observed, optical losses spanned

more than 10µm of the sample, so these changes cannot be confined to short range surface

effects. Silica must therefore provide heterogeneous nucleation sites for freezing. Once

freezing is initiated at a silica surface, it propagates into the bulk water sample. The need for

surface nucleation sites implies that liquid water is metastable when compressed to 5 GPa.

The present results do not provide direct information about the microscopic nucleation

mechanism, but it is well known that substantial electric fields are present at the water-

silica interface [2, 3]. These fields alter the local water structure, producing a vicinal layer

that is similar to ice [4–6]. Solid nucleation would be more likely in the vicinal region than

in the bulk liquid, making freezing possible on nanosecond time scales. This hypothesis

is consistent with ambient pressure studies that show enhanced ice nucleation on polar

substrates [7,8].

Thermodynamic states produced by multiple shock compression were calculated

using a mixed phase water model. Pure liquid and solid models were coupled with several

mixture rules to describe the mixed phase state [9–12]. This model showed that isentropic

and quasi-isentropic compression can lead to freezing in water for pressures above 2 GPa.

Complete solidification, however, may be difficult without the removal of heat. A time

dependent transition rate [9, 10] was proposed to describe the dynamics of freezing, in-

corporating both incubation and transformation time scales. The mixed phase model was

implemented in a wave propagation code [13] to study the effects of freezing on particle

velocity histories. Simulated particle velocities were generally consistent with the initial

loading stages measured in wave profile experiments. When freezing was delayed (incuba-

tion time) until the end of compression, the simulations showed a brief decrease in particle

236

velocity, a feature that was observed in the experimental wave profiles. This decrease did

not appear if the simulation allowed the sample to freeze immediately after entering the

solid phase region (no incubation time). The presence of an incubation time was observed

in all three types of measurements reported here.

The incubation and transformation time scales are linked to the pressure and tem-

perature conditions of the water sample, but the temporal nature of multiple shock com-

pression complicates this relationship. The dynamic aspects of the problem were resolved

by comparing experiments of different sample thickness but similar peak pressures. Mea-

sured incubation times were extrapolated to a zero thickness sample to remove the temporal

features associated with multiple shock compression. This approach was used for several

peak pressures and was coupled with classical nucleation theory [14] to estimate the lim-

iting nucleation time, which was on the order of 2-5 ns for water confined in z-cut quartz

windows. After the incubation period, freezing occurred over time scales of 150-200 ns.

There was evidence that multiple transition time scales are necessary to fully describe the

freezing process.

Near the water-window interface, short range effects in the vicinal region (< 1

nm) and window cooled region (≈1 µm) enhance ice nucleation. The transition spanned

more than 10µm of the sample thickness with lateral variations of tens to hundreds of

microns. The density of independent freezing events was consistent with expectations from

classical nucleation theory, i.e. more freezing events were observed as liquid water was

compressed deeper into the ice VII region. Subsequent ice growth was morphologically

unstable due to the limited dissipation of latent heat. A quantitative measurement of the

solid size was not made from the optical measurements due to complexities of the mixed

phase state. However, it was determined that the transformed sample cannot be composed

of a collection of small spheres. Instead, there must be an irregular network of solid regions

that are connected in some way to a nucleating silica surface.

237

7.2 ConclusionsThe present study on the freezing of water under rapid compression has resulted in

the following conclusions:

1. Quasi-isentropic compression of water leads to pressure and temperature conditions

where the liquid phase is metastable with respect to ice VII.

2. Using quasi-isentropic compression, it is possible to solidify water on nanosecond

time scales if a heterogeneous nucleation surface (SiO2) is present. This transfor-

mation produces measurable changes in the optical transmission, the optical images,

and the mechanical wave profiles.

3. Freezing is not uniform in compressed water, but occurs heterogeneously within the

sample. The variations are averaged out for lateral dimensions greater than 1 mm,

leading to reproducible optical transparency losses.

4. Crystalline and amorphous silica are effective ice nucleators, while sapphire does not

promote ice growth on time scales below10−6 s.

5. To a first approximation, freezing is described by an incubation and a transformation

time scale. Measured incubation times can be extrapolated to zero sample thickness

to determine the limiting incubation time, which is 2-5 ns for z-cut quartz windows.

Multiple transition time scales may be needed to describe ice growth along and away

from the nucleating window surface.

6. The growth of ice in compressed water is highly irregular due to heat dissipation

effects. There is also strong evidence that the density of the solid phase is higher

than the compressed liquid.

238

7. Measured wave profiles in reverberated water match calculated histories from the

mixed phase water model during the initial stages of compression. The calculated

and measured wave profiles agree during and after compression in the absence of

freezing. When freezing does occur, the mixed phase model predicts a transient

decrease in particle velocity, a feature observed in the wave profile measurements.

7.3 Recommendations for future workAlthough the present work has demonstrated shock wave induced freezing and

many aspects of the freezing transition, many issues need to be addressed in future work.

Foremost among these issues is understanding the importance of silica windows in initiat-

ing freezing. Since this work only used silica and sapphire windows, it is unclear which

material’s behavior is unusual. Cubic zirconia [15] and moissanite [16] are examples of

window materials that may be suitable for further multiple shock compression experiments

to investigate surface effects in freezing water. The effect of physical and chemical mod-

ifications on the nucleating properties of a window surface is another area of potential

research. More extensive studies of water at pressures above 5 GPa are also needed to

understand the loss and recovery of optical transmission observed at 10 GPa (experiments

T11 and T12).

239

References for Chapter 7[1] G.E. Duvall and R.A. Graham. Phase transitions under shock wave loading.Rev.

Mod. Phys.49, 523 (1977).

[2] Q. Du, E. Freysz and Y.R. Shen. Vibrational spectra of water molecules atquartz/water interfaces.Phys. Rev. Lett.72, 238 (1994).

[3] M.S. Yeganeh, S.M. Dougal and H.S. Pink. Vibrational spectroscopy of water atliquid/solid interfaces: crossing the isoelectric point of a solid surface.Phys. Rev.Lett.83, 1179 (1999).

[4] S.-B. Zhu and G.W. Robinson. Structure and dynamics of liquid water between plates.J. Chem. Phys.94, 1403 (1991).

[5] M.F. Toney, J.N. Howard, J. Richer, G.L. Borges, J.G. Gordon, O.R. Melroy and D.G.Wiesler. Voltage-dependent ordering of water molecules at an electrode-electrolyteinterface.Nature368, 444 (1994).

[6] M.F. Reedijk, J. Arsic, F.F.A. Hollander, S.A. de Vries and E. Vlieg. Liquid order atthe interface of KDP crystals with water: evidence for icelike layers.Phys. Rev. Lett.90, 66103 (2003).









240

[15] T. Mashimo, A. Nakamura and M. Kodama. Yielding and phase transition undershock compression of yttria-doped cubic zirconia single crystal and polycrystal.J.Appl. Phys.77, 5060 (1995).

[16] J. Xu and H. Mao. Moissanite: a window for high-pressure experiments.Science290,783 (2000).

241

Appendix A

Mechanical DrawingsThis appendix contains mechanical drawings for all parts used in this research.

Projectile design and modifications are shown in Figures A.1-A.5. Pieces mounted to the

projectile are shown in Figures A.6-A.9. Figures A.11-A.14 show the various liquid cell

designs. All other target pieces are shown in Figures A.15-A.24. Additional machining,

lapping, and assembly of these pieces is described in Chapter 3.

243

8.0±0.062"

4.026+0.000 -0.0005"

0.50" 0.5±0.01"

1/8" radius

3.650+0.000 -0.050"

3.25"

1.0"

3.25"

0.37"

6061-T651 Al alloyO.D. taper <0.001".xx=±0.02".xx=±0.002"

PARKER #2-342O-rings

45° taper (interior)

0.87"

1/8" radius

Figure A.1: 4” standard projectileThis projectile was used for most experiments with the modifications shown in Figure A.4.Velocities of 0.290-0.700 km/s are obtained with this design using the wrap around breechof the 4” gas gun [1].

244

4.026+0.000 -0.0005

3.23"

6061-T651 or7075-T651Al alloy O.D. taper <0.001".xx=±0.02".xx=±0.002"

45° cut

0.100+0.010 -0.000"

2-240 O-rings

1/4" radius

6.00"

0.38"

0.38"

Figure A.2: 4” monkey’s fist projectile:This design was necessary for 5 GPa experiments using sapphire windows. Modificationsshown in Figure A.4 were made to the projectile face. Velocities of 0.180-0.250 km/s areobtained using the “monkey’s fist” configuration of the 4” gas gun [2].

245

0.25"

1/8" radius

0.38"

0.5"

projectile centerline

0.25"

2.464+0.0000 -0.0005"

1.75"

2.45"

8-32 body drillon 0.750" diameer2 places

7075-T6 ALUMINUM.XX=±0.02".XXX=±0.002"

10-24 drill/tap1/2" deep

1/16" radius

1/8" radius

0.50"

0.121±0.001"

2-228 O-RING GROOVE

0.937"

0.125+0.010 -0.000"

0.100"

0.75"

1.0"

0.25"

4.50"

2.062-16 thread

Figure A.3: 2.5” projectile and capThis projectile design was used for only one VISAR experiment (V1).

246

8-32 body drillthrough projectileface

8-32 drill/tap3/8" deep

1.700"

1.345"

0.375"

(a)

0.375"

(c)

(b)

Figure A.4: Projectile modifications for mounted piecesAll projectiles had impactor mount holes at (a)Optical experiments using quartz, fused silica, and sapphire impactors had turning mirrormounted at (b)Optical experiments with glass impactors had turning mirror mounted at (c)

247

0.75" diameter

5-40 drill/tap1.250" diameter3 places

1/4-36 drill/tap1.20" from center

1/2" diameterx 0.3" deepcountersink

1.20"

0.25"

Aluminum ring

1.50"

body drill 5-401.250" diameter3 places

5/8-27 thread

Figure A.5: Modifications made to the standard 4” projectile used for alignment duringsetup of imaging experiments. The aluminum ring is mounted to the projectile face withBelville washers and holds a collimating lens unit (Figure A.23). Optical fiber carries lightinto the side hole and connects with the lens unit from the back side of the projectile (notshown). Collimated light is emitted from the lens unit. Screws holding the plate to theprojectile are adjusted to align light output with the projectile.

248

1.00"

8-32 drill/tap2 places

0.375"

0.44"

0.520"

Material: Aluminum Tolerances: .xx=0.005" .xxx=0.001"

0.134"

0.50"

0.088"

45.00°

Figure A.6: Soda lime glass impactor mount

249

0.75" diameter

0.50"


0.375"

0.44"

0.520"

1.50"

1.00"


0.088"

0.134"

45.00°

Figure A.7: Universal impactor mount for quartz, fused silica, and sapphire experiments.Additional boring is performed to match the size of the impactor piece.

250

0.43"

0.43"

8-32 body drill0.375" from center line2 places

45°

1.125"

0.520"

0.440"

0.22" 0.28"

0.088" at 45°

Figure A.8: Projectile turning mirror mounts (glass impactor)Outer corners should be filed down as this mount rests near the projectile edge.

251

0.75" diameter

0.50"


0.375"

0.44"

0.520"

1.125"


0.088"

0.134"

45.00°

Figure A.9: Projectile turning mirror mounts for non-glass experiments

252

Relief cut 0.01-0.02" deepto prevent epoxy flow intocentral region

1.25"

1.75"(c)

1.00"

(b)

(a)

0.25"

0.25"

~3/8"

~3/8"

Material: BrassTolerances: .xx=0.005" .xxx=0.001"

0.050"

0.25" ~3/8"

Figure A.10: Impactor apertures(a) Sapphire impactor(b) Quartz/fused silica impactor(c) Glass impactor

253

2.01"

1.81"

3.50"

2.31"

3.00"

2.51"

#52 drill through2.125" diameter2 places0.05" groove

Teflon pad0.03" thick

brass lock ring

brass cell body

1/8"

1/8"

9/64" drill through3.250" diameter3 places

6-32 drill/tap2.750" diameter6 places0.25" deep

1/8"

1/8"

2.5" OD O-ring groove0.136" wide X 0.074" deep

0.90"

0.08"

10-32 drill/tap through3.250" diameter2 places 90° apart

Tolerances: .xx=0.005" .xxx=0.001" 1/x=1/64"

Figure A.11: Liquid cell for soda lime glass windows

254

1.75"

1.23"

1.375"

Inner diameter for press fit

0.38"0.25"

Material: BrassTolerances: .xx=0.005" .xxx=0.001"O-ring grooves initially0.052" deep 0.085" wide

3-48 drill/tap3 places on ø1.530"0.150" deep

Two fill channels#52 Drill (0.0635")

1 1/2" OD 1/16" O-ring groove(Parker #2-028) centered on cell

1.550"-40 thread

0.71"0.455"

1" ID 1/16" O-ring groove (Parker #2-026)centered on cell

1/8"

Lock ring

Figure A.12: Liquid cell used for silica windowsThis design was slightly modified for the quartz front/sapphire rear window experiment(T13) by machining the inner diameter to 0.990” rather than 1.23” and using a smallerO-ring (see Figure A.13).

255

1.50"

0.990"

1.08"

0.25"

0.455"

inner diameter for press fit

0.38"

Material: BrassTolerances: .xx=0.005" .xxx=0.001"O-ring grooves initially0.052" deep 0.085" wide

5-40 drill/tap3 places on ø1.300"5/16" deep

1 1/4" OD 1/16" O-ring groove(2-024 centered on cell)

Two fill channels#52 Drill (0.0635")

1.290"-40 thread

0.71"

1 ID 1/16" O-ring groove (2-022) centeredon cell

1/8"

Lock ring

Figure A.13: Liquid cell for sapphire windowsThis design was also used in the sapphire front/quartz rear experiment (T12) with a 1.00”O.D. X 0.50” piece of quartz.

256

1.375"

1.26"

2.00"

1.11"

Material: BrassTolerances: .xx=0.005" .xxx=0.001"O-ring groove initially0.052" deep 0.085" wide

lockring

0.50"

0.25"

0.15"

#70 drill throughon 1.32" diameter2 places

1.550"-40 thread3/16" deep on cell body

#52 drill 0.1" deep2 places

1/8"

5-40 drill/tapon 1.800" diameter3/8" deep3 places

1.5" OD 1/16" O-ring groove(2-024 centered on cell)

Figure A.14: Quartz cell used for VISAR experiment V2. A Viton O-ring was used toseal the cell.

257

#23 drill throughon 5.50" diameter3 places

1.00"

0.38" wide X 1.5" slot

Material: Aluminum Holobar6" OD X 3/4" wallInside diameter left roughScale=1/2Tolerances: .xx=0.005"

diameter=5.00"

diameter=4.50"

diameter=5.96"

side view

0.25"

Figure A.15: Standoff target ringAll target plates (Figures A.16-A.19) were attached to this type of target ring, which wasthen mounted to the gun.

258

100°

5.00"

diameter=3.01"

8-32 drill/tapthrough on3.250" diameter2 places

5-40 drill/tap throughon 3.250" diameter3 places

9/64 drill throughon 4.750" diameter3 places

drilled through 1/4"on 3.250" diameter2 places 90° apart

Material: 1/4" aluminum plateTolerances: .xx=0.005" .xxx=0.001"

radius=2.24"

radius=1/4"

Figure A.16: Soda lime glass cell target plate

259

100°

radius=2.24"

radius=0.68"9/64" drill on ø4.750"3 places

Material: 1/4" thick aluminum plateTolerances: .xx=0.005" .xxx=0.001"

10-32 drill/tapon ø3.25"2 places (90°)

8-32 drill/tapon ø2.250"2 places

1/4" radius

3-48 body drillon ø1.530"3 Places

Figure A.17: Quartz/fused silica cell target plate

260

radius=2.240"

20°

diameter=1.04"

120°

Material: 1/4" thick aluminum plateTolerances: .xx=0.005" .xxx=0.001"

5-40 drill/tapon 1.300" diameter3 places

8-32 drill/tap2.75" apart(centered)2 places

10-32 drill/tap3.12" diameter2 place (90 ° apart)

9/64" drillON 1.300" diameter3 places

radius=1/4"

9/64" drill on 4.75"ø3 places

radius=0.77"

Figure A.18: Sapphire cell target plate

261

100°

drill/tap 10-322 places (90°) ON ø3.25"

radius=0.785"

radius=2.25"

9/64 3 holeson ø4.75

Material: 1/4" aluminum plateTolerances: .xx=0.005" .xxx=0.001"

radius=1/4"

5-40 body drill ø1.800"3 places

5.00"

8-32 drill/tapø2.7502 places

Figure A.19: Target plate used with the unbuffered VISAR cell of Figure A.14

262

3.00"

3.25"

3.50"

8-32 body drill1.625 from center2 places

1.00"

5/8-27 drill/tap through

1.50"

0.25"

Material: AluminumTolerances: .xx=0.005"

Figure A.20: Lens bracket for glass target plate

263

2.50"

2.25"


11/64" drill through2 places on center line

0.38"

5/8-27 drill/tap throughon center line

1.00"

0.75"

Figure A.21: Lens bracket for quartz/fused silica target plate

264

3.00"

2.75"

2.25"

11/64" drill through2 places on center line


0.38"

1.00"

5/8-27 drill/tap throughon center line

0.75"

Figure A.22: Lens bracket for sapphire target plate

265

5/8-27 thread

Thickness=0.05"

DRILL POINT ANGLE

0.25"

0.332"

1/16" DRILL 0.475"

5/8-27 tap

3/4"hex

OD=0.472"

ID=0.25"

0.70"

Aperture (brass)

Lock nut (brass)

1/4-36 drill/tap

1/16" drill

1.19"

7/16" drill

1.35"

Spacer ring (brass or aluminum)

0.20"

INTERNALCHAMFER

Tolerances: .xx=0.005" .xxx=0.001"

ID=7/16"0.470"

Lens tube (brass)

Figure A.23: Lens unit used in transmission experiments to collect collimated light. Theaperture is placed above a single plano convex lens using the spacer ring.

266

7/16-27 thread

#67 drill

0.475"

LENS HOLDER BODY

7/16-27 thread

0.80"

5/8-27 thread

0.17"

0.330"

FIBER OPTIC HOLDER

spot face 3/8"drill/tap 1/4-363.6° off axis

0.62"

0.75"

0.125"

0.31"

0.36"

0.25"

0.25"

3/4" hex

5/8 LOCKNUT

Tolerances: .xx=0.005" .xxx=0.001"

5/8-27 thread

0.70"

LOCKNUT

5/8" hex

0.08"

0.60"

Figure A.24: Lens unit used to image laser fiber tip onto VISAR mirror. 1:1 imagingis performed with a pair of identical plano-convex lenses separated by a spacer ring (seeFigure A.23).

267

References for Appendix A[1] G.R. Fowles, G.E. Duvall, J. Asay, P. Bellamy, F. Feistmann, D. Grady, T. Michaels

and R. Mitchell. Gas gun for impact studies.Rev. Sci. Instrum.41, 984 (1970).

[2] D. Erlick. SRI International, private communication.

268

Appendix B

Window MaterialsThis section describes the solid materials used in water reverberation experiments.

Each of the materials listed below was ordered in large batches to ensure consistency.

Sound speed and density were measured for a few pieces in each batch for comparison

with published values. X-ray Laue patterns were also taken for the single crystal windows

to ensure the correct orientation was obtained.

All windows were cleaned in an ultrasonic bath of isopropanol, liberally flushed

with both ethanol and isopropanol, and blown dry with inert gas. Surfaces that contacted the

water sample were given a final flush with ultrapure water (Appendix C) and again blown

dry. Except for the aluminum plating in VISAR experiments (Section 3.3), no surface

modifications were made to the windows.

To eliminate the possibility of window contributions to optical measurements, win-

dows were never stressed beyond their established elastic limits. For modeling purposes,

all of the windows in this work were treated as isotropic, elastic materials. These models

were implemented in COPS [1] to calculate longitudinal stress and other continuum vari-

ables of interest. In the following discussion, the mean stress in denoted byP and the shear

modulus byG. Sample compression is defined asµ = ρ/ρ0−1.

B.1 Soda lime glassSoda lime glass was obtained from HP Scientific Glass Services. It was unpolished

and used as received. X-ray fluorescence measurements indicated that the the sample com-

position was roughly 73%Si02, 14%Na20, 9% CaO, and 4% MgO with variety of trace

269

contaminants. Where possible, glass pieces were obtained from the same sheet. The aver-

age sample density was 2.493 g/cc. Based on the work of Hegewald [2], soda lime glass is

elastic below 4 GPa with the following mechanical response:

P = (47.93 GPa)µ G = 28.13 GPa

Experiments described in Appendix D indicate that soda lime glass is optically transparent

for stresses below 4 GPa.

B.2 Fused silicaDynasil 1000 grade fused silica was obtained from United Lens Company. Di-

ameter tolerances were±0.005” and thickness tolerances were±0.001”. For press fitted

windows, the roundest pieces were chosen. If roundness varied by more than±0.0005”,

cell sizing was done to the largest diameter. Both sides were polished to 60/40 optical finish

with 1/2 wave flatness over 80% of the window with a cerium oxide grit (by the vendor).

Soap and water were used for initial cleansing; final cleaning was done ultrasonically in

water.

The density of fused silica is 2.201 g/cc. The shear modulus of fused silica was

based upon the report by Feng and Gupta [3] and the data by Conner [4].

G = 30.62−119.9µ µ ≤ 0.076

G = 21.5076+66.84(µ−0.0776) 0.076< µ < 0.012

where G is in GPa. Barker and Hollenbach [5] gave a stress-strain relation for fused silica:

σx = 77.60ε−415.9ε2 +3034.0ε3−6926.0ε4

where stress has units of GPa. This relation must be modified to match the measurements

270

of Rigg [6]:

σx = 77.60ε−400.0ε2 +2934.0ε3−6926.0ε4

Optical clarity is maintained in fused silica to stresses at least 4 GPa [5]. The thermal

conductivity of fused silica is approximately 1.4 W/m·K [7, 8]. The specific heat capacity

of fused silica is about 9/2 R≈ 0.62 J/g K.

B.3 z-cut quartzElectrical grade quartz was obtained from Boston Piezo-Optics with a crystal orien-

tation within 30 arc minutes of the z-axis. Thickness tolerances were±0.001”. The initial

batch of quartz had a diameter tolerance of±0.005”; subsequent batches were sized with

±0.0005” diameter tolerances. Both sides were polished to 20/10 optical quality with≤5

arc seconds parallelism (by the vendor). Rough polishing was performed with an abrasive

grit. The final polish stage was done using a colloidal silica suspension that removes an

amorphous layer left by the heat and pressure of normal polishing [9]. Acetone and alcohol

were used for cleaning.

The material response of z-cut quartz is based on the summary by Winey [10]:

ρ0 = 2.6485g/cc

P = 43.19µ +156.2µ2 +48.60µ3

G = 46.92+1.873P+0.3459P2

where bothP and G are in GPa. Quartz remains optically transparent for shock com-

pressions up to 6 GPa [11]. The thermal conductivity of z-cut quartz is approximately 12

W/m·K [7]; the specific heat is about 9/2 R≈ 0.62 J/g K.

271

B.4 a-cut SapphireSapphire boules were obtained from Union Carbide or Crystal Systems. They were

free of visible pink tint (a result of chromium contamination) and internal scattering defects.

The boules were cut and polished by Meller Optics with a crystal orientation within 3

arc minutes of the a-axis. Thickness tolerances were±0.001”. Diameter tolerances were

typically±0.001” except on press fit pieces, where the diameter tolerance was±0.0005”.

Both faces of the window were polished (by the vendor) to 20/10 optical quality with a

flatness of 5 waves over 85% of the central aperture. Parallelism of the windows was within

3 arc minutes. The pieces were polished with diamond paste and cleaned with alcohol.

The a-axis of sapphire remains elastic to 17-19 GPa [12]. The material response is

based on the summary by Winey [10].

ρ0 = 3.985g/cc

P = 259.1µ +520.3µ2−2061µ3

G = 178.9+1.512P−0.0726P2

This orientation of sapphire is commonly used as a spectroscopy window at Washington

State University [13]. Optical clarity is preserved in the orientation for stresses greater than

10 GPa [12]. Thermal conductivity in sapphire is about 38 W/m·K [8]; the specific heat is

approximately 15/2 R≈ 0.61 J/g·K.

272

References for Appendix B[1] Y.M. Gupta. COPS wave propagation code (Stanford Research Institute, Menlo Park,

CA, 1978), unpublished.

[2] H.D. Hegewald.Time-dependent longitudinal response of soda lime glass under shockcompression. M.S. thesis, Washington State University, (2001).

[3] R. Feng and Y.M. Gupta. Material models for sapphire,α-quartz, lithium fluoride, andfused silica for use in shock wave experiments and wave code calculations. TechnicalReport 96-XX, Shock Dynamics Center (unpublished), (1996).

[4] M.P. Conner.Shear wave measurements to determine the nonlinear elastic responseof fused silica under shock loading. M.S. thesis, Washington State University, (1988).

[5] L.M. Barker and R.E. Hollenbach. Shock-wave studies of PMMA, fused silica, andsapphire.J. Appl. Phys.41, 4208 (1970).

[6] P.A. Rigg. Real-time x-ray diffraction to examine lattice deformation in shockedlithium fluoride windows. Ph.D. thesis, Washington State University, (1999).

[7] D.E. Gray, editor.American Institute of Physics Handbook. McGraw-Hill, New York,3rd edition, (1972).


[9] Norman Beniot, (2002). Boston Piezo-Optics, private communication regardingquartz polishing.

[10] J.M. Winey, R. Feng and Y.M. Gupta. Isotropic material models for the elastic re-sponse of sapphire and quartz single crystals under shock wave compression. Tech-nical report, Insitute for Shock Physics, (2001).


[12] R.L. Webb.Transmission of 300-500 nm light through z-cut sapphire shocked beyondits elastic limit. M.S. thesis, Washington State University, (1990).


273

Appendix C

Water Sample PreparationThis section describes the preparation and characterization of the water samples

used in the experiments reported in Chapter 4 and Appendix D. A general review of con-

tamination and treatment of ultrapure water is presented in Section C.1. The specific treat-

ments and testing used in this work are summarized in Section C.2. The procedures for

filling the liquid cells with ultrapure water is discussed in Section C.3.

C.1 Contamination and treatment methodsVirtually all materials are soluble in water to some degree, and the solvating power

of water increases with its purity [1]. Solid, liquid, and gaseous contamination affects

all liquids, but water treatment systems must also deal with the possibility of microbial

growth and the resulting organic byproducts. An overview of different classes of water

contamination is given in Table C.1. The relevance of each contamination class varies

greatly depending on needs of a specific application. For this work, it is unclear which

types of contamination are of greatest concern, so the goal was to achieve an combination

of overall purity and sample consistency.

There are a number of different approaches for producing ultrapure water [1, 2],

each having certain advantages limitations. The most rigorous approach to obtain ultrapure

water is to directly combine pure hydrogen and oxygen by combustion or within a fuel

cell [1], though this is usually not feasible for laboratory scale systems. Most practical

approaches involve a series of purification stages, each aimed at reducing different classes

of contaminants. Common purification techniques are listed in Table C.2. Most of these

methods are extremely adept at removing a single class of contaminants but ineffective at

275

Table C.1: Classes of water contaminants (Adapted from Ref. 1)

Classification Example

Liquid impuritiesMiscible organic Alcohol

Miscible inorganic Hydrogen peroxideImmiscible organic Oils

Immiscible inorganic Mercury

Solid impuritiesMolecular organic Sugars

Molecular inorganic SaltsColloidal organic Viruses

Colloidal inorganic SilicaSuspended organic Biological matter

Suspended inorganic Metal oxides

Gaseous impuritiesVolatile organic Methane

Volatile inorganic Oxygen

Table C.2: Standard water purification techniques

Method Typical Application

Chemical Chlorination (kills microorganisms)

Electrochemical Deionization (removes charged contaminants)

Filtration Particulate removal

Selective membranes Reverse osmosis

Phase transition Distillation/crystallization

Irradiation Ultraviolet illumination (kills microorganisms)

Vacuum degassing Dissolved gas extraction

276

removing others. Some approaches add potential contaminants (such as chlorine) or only

break down contaminants without removing the byproducts. Other approaches, such as

distillation, require water that has already been treated to some degree.

C.2 Sample purificationPreliminary freezing experiments were performed using water from a Culligan

mixed bed deionizer, where water is passed over beds of ion-exchange resins. Cations

are absorbed by the resin, releasingH+ ions; anions are exchanged forOH− ions at dif-

ferent sites in the resin. This process removes virtually all ionic contamination from water.

However, uncharged impurities, including organics and microorganisms, are largely unaf-

fected. Bacteria and other microorganisms often thrive in resin beds, and the resin itself

tends to add organic material to the output. Therefore, additional treatment is necessary for

deionized water.

Distillation removes a large spectrum of contaminants, charged or uncharged, so it

was used to refine water that had been deionized. The schematic in Figure C.1(a) shows

the initial setup using distillation. In this system, the distillate was collected and placed

in an oil-free vacuum system to remove dissolved gas. Typically, a piece of Teflon was

necessary during degassing to provide nucleation sites for the gas bubbles. This system

was used for roughly six months (December 1999-April 2000), at which point the still

ceased functioning. A new still (High-Q 103S) was purchased to replace the old unit for all

experiments after April 2000. Figure C.1(b) shows the complete purification system; the

operation of the High-Q still is shown in Figure C.2. The still included a 350 nm prefilter

(High-Q, Inc. Model 200PT Media Filter), placed between the deionized water input and

the boiler of the still, to remove particulate matter. The degas stage of the original system

was omitted because the usable volume was too small to provide copious amounts of water

for flushing the liquid cell (Section C.3). When initially set up, the still was cleaned with

277

inputfeedwater

(b)

Deionization

inputfeedwater

(a)

Deionization

Distillation

collection+ testing

Filtration

sealedstorage

Degassing

collection+ testing

Distillation

Figure C.1: Water purification stages(a) Initial ultrapure system(b) Final ultrapure system

278

growthfound here

insectenteredhere

insectdied here

Figure C.2: High-Q 103S still (taken from http://www.high-q.com)The mist trap above the boiler prevents liquid from splashing into the distillate collection.A hydrophobic barrier prevents liquid from seeping out of the boiler along the glass walls.Steam condenses into liquid at the top of the still and rolls back down against the flow of therising vapor, stripping away residual volatile impurites. The boiler automatically drains pe-riodically to prevent excessive buildup of contaminants. The entire still and collection tank(not shown) are constructed from borosilicate glass that has been acid etched to increase itsresistance to water attack.

279

deionized water and run continuously for several weeks. Water from the still was stored in a

sealed glass container (High-Q 103C-G). This container was constructed from borosilicate

glass and acid etched to reduce the effects of ultrapure water attack. Special valves and seals

were used to prevent external contamination from entering the container. Even with these

precautions, water quality tends to degrade over the course of hours [3] due to leaching

from the container and microbial growth. It is likely that the water used in experiments and

testing was of lower quality than the direct output of the purification system.

Several contamination events occurred during the course of this project (Figure

C.2). In one case, an insect was found dead in the condenser after a long idle period. To

remove all contamination, the still was disassembled, cleaned with chromic acid, flushed

with deionized water, and run for a week before further experiments were performed. From

that point on, all exhaust ports were covered when the still was not in use. During another

long idle period, a barely visible growth was observed in a section near the condenser.

Afterwards, this section was flushed with ethanol periodically. Before any water from the

still was tested or used in an experiment, the still was run continuously for at least three

days to ensure all surfaces were flushed many times with clean water. Experiments were

synchronized with the purification system so that water placed in the cell was as freshly

distilled as possible.

Although three different configurations (deionized water and two different stills)

were used to provide purified water, freezing was always initiated under reverberation load-

ing if the pressure was above 2 GPa and a silica window was present. After the system in

Figure C.1(b), periodic testing was performed to check the consistency of the water sam-

ples. Rather than measuring every conceivable contaminant in the water, three basic tests

were used to determine the purity of the water.

• Electric resistivity

280

• Total organic carbon (TOC)

• Dissolved oxygen

Resistivity and dissolved oxygen measurements were performed on site; TOC measure-

ments were done by an outside agency (Chemtrace). This combination is certainly not

exhaustive, but does give a general sense of purity. For example, highly charged impurities

(metals or polar compounds) will radically change the resistivity of pure water, so it is not

necessary to directly measure the levels of every possible impurity of this class. Organic

molecules tend to be only weakly charged or nonpolar and have large molecular weights,

so their contribution to the resistivity is quite low. TOC measurements give the total den-

sity of organic carbon atoms present in the water, detecting much of the contamination that

does not contribute to conductivity. This value is a combined measurement of the number

of impurities and their relative size. Dissolved oxygen measurements give a measure of the

residual gas within the water, which is not detected by either resistivity or TOC measure-

ments. Purity testing procedures followed the guidelines set by Chemtrace. Exterior and

interior surfaces of the water tank port were flushed with isopropanol; excess alcohol was

allowed to drain off. The port was opened and adjusted so that no bubbles were present

in the interior of the port. The port was then closed for a few minutes to allow surface

alcohol to diffuse into the water. The port was reopened and water flowed continuously for

at least ten minutes to remove all traces of isopropanol. Resistivity and dissolved oxygen

measurements were made during the next five to ten minutes, with the water flowing for

most of that period. After completing these tests, water was collected for TOC testing or

an experiment. Table C.3 summarizes all purity measurements made on the final water

treatment system.

Resistivity measurements were made with a Thornton 200CR Conductivity/Resis-

tivity meter. The resistance cell was cleaned with distilled water and submerged completely

281

Table C.3: Summary of water testing

Resistivity measurementsPerformed on site with Thornton 200CR Conductivity/Resistivity MeterDetection limits: 0.01MΩ ·cmRange of measured values: 1.9-2.0MΩ ·cm

Dissolved oxygen (DO) measurementsPerformed on site with CHEMetrics visual DO kitDetection limit: 1 ppmRange of measured values: 4-5 ppm

Total organic carbon measurementsPerformed by ChemTrace NORTHWEST (Portland, OR)Detection limits: 3.0 ppb

Date Measurement (ppb) Comment

6/19/2000 120 original setup

10/19/2000 26 chromic acid clean

9/14/2001 78 alcohol clean

average 75

Total silicaPerformed by ChemTrace NORTHWEST (Portland, OR)Detection limits=1.0 ppb

Date Measurement (ppb) Comment

6/19/2000 14 original setup

282

in a stream of distilled water. All visible bubbles were shaken out of the cell. The mea-

surement was recorded after the temperature reading of the meter reached a steady state.

The meter was temperature compensated to shift the reading to a standard temperature of

25C. The readings varied from 1.8-2.0MΩ ·cmfor all samples taken from the purification

system. In the absence of charged impurities, the resistivity of water at 25C is about 18.2

MΩ ·cm [4]. Carbon dioxide in the atmosphere dissolves in water [5, 6] ; a small portion

of this ionizes to form carbonic acid, which lowers resistivity.

CO2 +H2O→ HCO3−+H+ (C.1)

This reaction reaches equilibrium over the span of minutes and is unavoidable for water

distilled in air. Distillation in other atmospheres was deemed too inconvenient to pursue at

this time.

Dissolved oxygen measurements were performed using a visual chemical test from

CHEMetrics. Readings indicated that dissolved oxygen levels were between 4-5 ppm for

water extracted directly from the system.

TOC measurements were performed on samples shipped to CHEMtrace. In these

measurements,CO2 gas was vacuum purged from the sample. A series of treatments (UV

irradiation, chemical attack, etc) were then applied to break all molecules containing carbon

into CO2, which was then collected and measured. On average, carbon levels in the water

were about 75 ppb.

After the initial setup and cleaning of the still, water samples were shipped to

CHEMtrace for dissolved silica measurements. It was determined that the water output

had roughly 14 ppb of dissolved silica.

283

C.3 Filling the liquid cellFilling liquid cells with water proved to be more difficult than other liquids. The ini-

tial approach was to close the cell down to the final desired thickness and inject water with

a syringe. Water has a rather large surface tension, and as a result it was often difficult to

fill the cell uniformly. In some areas the liquid would flow rapidly, leaving unwetted areas

behind. Once a channel of liquid spanned the cell, it was impossible to fill the remaining

voids completely. This effect became worse as thinner samples were attempted.

The following procedure was developed to avoid this problem and to maximize the

final cleanliness of the cell. The entire cell (windows, brass surfaces, and O-ring) was

first cleaned with ethanol and isopropanol, then dried with inert gas. Each surface was

then flushed liberally with pure water and blown dry. The liquid cell was assembled as

described in Section 3.1.2, but tightened only slightly so that the removable window made

contact with the O-ring, but did not compress completely. The gap between the windows

was now on the order of 1/2 mm. Freshly distilled water was injected into the cell with

a syringe, easily wetting all surfaces. A valve on the syringe was closed to prevent back

flow before tightening the cell to the desired thickness. Once tightening was complete,

the syringe valve was reopened and the remaining pure water injected into the cell. This

displaced all the previous volume, along with impurities from the initial fill, and ensured

that the sample was as clean as possible. All efforts were made to ensure that the liquid

spends as little time in the cell as possible to minimize any interaction of the pure water

with the cell. This time never exceeded 3 hours.

284

References for Appendix C[1] V.C. Smith. Preparation of Ultrapure Water. InUltrapurity: Methods and Techniques,

M. Zief and R. Speights, editors. Marcel Dekker, New York (1972).

[2] R.C. Hughes, P.C. Murau and G. Gundersen. Ultra-pure water: preparation and quality.Anal. Chem.43, 691 (1971).

[3] R. Gabler, R. Hegde and D. Hughes. Degradation of high purity water on storage.Journal of Liquid Chromatagraphy6, 2565 (1983).

[4] P.M. Pichal and R.B. Dooley. Electrolytical conductivity (specific conductance) ofliquid and dense supercritical water from 0 C to 800 C and pressures up to 1000MPa. Technical report, IAPWS, (1990).

[5] J. Kendall. Review: the preparation of conductivity water.J. Am. Chem. Soc.38, 2460(1916).

[6] J. Kendall. The specific conductivity of pure water in equilibrium with atmosphericcarbon dioxide.J. Am. Chem. Soc.38, 1480 (1916).

285

Appendix D

Supplemental DataThis appendix contains experimental data not presented in from Chapter 4. Section

D.1 discusses some preliminary experiments involving water and soda lime glass. Section

D.2 contains additional photodiode transmission records used in the analysis of Chapter 6.

D.1 Soda lime glass experimentsA number of preliminary water experiments were performed using soda lime glass

windows. A few of those experiments are presented here to demonstrate that water loses

optical transmission when compressed in soda lime glass. In the process, it was shown

that shock compressed glass remains transparent in the elastic domain below 4 GPa [1].

Table D.1 summarizes the experiments presented in this section. Columns two through four

describe the impactor; columns five through nine characterize the target. Peak pressure

is given in column ten. For experiments that used a water sample, peak temperature is

shown in column eleven. The final column indicates whether optical transmission loss was

detected during the experiment.

D.1.1 Water experiments

Experiment G1

This experiment was among the first optical transmission measurements of water

under reverberation loading. A peak pressure of 3.6 GPa was used to produce a state of

possible freezing while maintaining optical clarity in the soda lime glass windows. The

water sample in this experiment was treated by deionization only (no distillation or filtra-

tion). The photodiode transmission measurement for this experiment is shown in Figure

287

Experim

entIm

pact Im

pactor F

ront window

Sample

Rear W

indow peak

Changes

number

velocityM

aterialh [D

]M

aterialh [D

]h

Material

h [D]

P

Tobserved

(km/s)

(mm

)(m

m)

(mm

)(m

m)

(GP

a)(K

)18.8

1.618.8

[44.5][50.8]

[63.5]18.8

1.618.8

[44.5][50.8]

[63.5]18.8

-18.8

[44.5]-

[63.5]18.8

1.618.8

[44.5][50.8]

[63.5]

h= thickness

D=

diameter

SLG

=soda lim

e glassD

iameters and thicknesses are nom

inal values

yes0.13

SLG

3.61429

G1 (99-066)

0.487SL

GSL

G

yes0.13

SLG

3.58428

G2 (00-020)

0.485SL

GSL

G

no0.13

SLG

3.61-

G4 (99-065)

0.489SL

GSL

G

no-

SLG

3.62-

G3 (99-069)

0.491SL

G-

TableD

.1:S

umm

aryofsoda

lime

glassexperim

ents

288

D.1. Time dependent transmission loss was observed, suggesting that water could freeze

under reverberation loading.

Experiment G2

This experiment was similar to G1 in all ways except the preparation of the water

sample. In addition to deionization, the water was distilled and then degassed under vac-

uum. The photodiode transmission measurement for this experiment is shown in Figure

D.1. Transmission loss occurred earlier in this experiment than in G1, which was some-

what surprising since it was expected that additional purification would remove potential

nucleation sites from the water sample. No characterization of the water purity was made

in either experiment, so it is not clear what the actual impurity levels were. The results of

these experiments were sufficient to motivate further study of the problem and led to the

experiments discussed in Chapter 4.

D.1.2 Transparency of shocked soda lime glass

Experiment G3

This experiment was performed to verify the transparency of soda lime glass under

single shock compression. The experiment was similar to experiments G2 and G3 with

the omission of the front window and water sample. The photodiode transmission for this

experiments is shown in Figure D.2. No optical changes were detected, so it was concluded

that glass remains transparent under this level of single shock compression.

Experiment G4

This experiment was performed to verify optical transparency in soda lime glass

exposed to multiple shock compression. The impactor and liquid cell were similar to that

in experiments G1 and G2, but a hexane sample was used in place of water. Previous

work established transparency in shocked hexane to 10 GPa for at least 1µs [2–4], so no

289

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Tra

nsm

issi

on

G1

G2

Figure D.1: Photodiode transmission for experiments G1 and G2In both experiments, water was compressed in soda lime glass windows to a peak pressureof 3.6 GPa. Experiment G1 used a deionized water sample, while experiment G2 useddeionized, distilled, and degassed water.

290

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Tra

nsm

issi

on

G3G4

G3

G4

Figure D.2: Transparency of shocked soda lime glass

291

optical changes were expected here. No significant changes were observed in the photodi-

ode measurement shown in Figure D.2, which confirmed that the glass windows were not

responsible for the results of experiments G1 and G2.

D.1.3 Summary

The soda lime glass experiments generated the following results.

1. Water compressed to 3.6 GPa using reverberation loading lost optical transparency.

This loss was not tied to the exact preparation of the water sample.

2. Soda lime glass windows remained transparent under single and multiple shock load-

ing to 3.6 GPa.

D.2 Supplemental photodiode records

D.2.1 Imacon 200 demonstration experiments

The following experiments were performed using a demonstration model of the

Imacon 200 framing camera. The image intensities from that detector were inconsistent

with the photodiode measurements and are omitted here. Table D.2 summarizes these

experiments in the same format discussed in Section 4.1. Each of the experiments used z-

cut quartz windows. For consistency with Chapter 4, timet = 0 is defined by shock arrival

at the water sample.

Experiment IS1

The peak pressure in this experiment was 3.50 GPa; the initial sample thickness

was 19µm. This is similar to experiment T4 (page 105), which used a 28µm sample. The

photodiode record is shown in Figure D.3. This experiment was generally consistent with

transmission experiment T4.

292

Tabl

eD

.2:

Sup

plem

enta

lpho

todi

ode

mea

sure

men

ts

Exp

erim

ent

Impa

ct

Impa

ctor

Fro

nt w

indo

w**

Sam

ple*

Rea

r W

indo

w**

peak

sta

te90

%E

xp.

Cha

nges

num

ber

velo

city

Mat

eria

lh

[D]

Mat

eria

lh

[D]

hM

ater

ial

h [D

]P

T

ρ ρρρle

vel

limit

obse

rved

(km

/s)

(mm

)(m

m)

(mm

)(m

m)

(GP

a)(K

)(g

/cc)

(ns)

(ns)

12.7

3.18

412

.718

[31.

8][3

8.1]

[31.

8]12

.73.

184

12.7

20[3

1.8]

[38.

1][3

1.8]

12.7

3.16

412

.719

[31.

8][3

8.1]

[31.

8]

h= th

ickn

ess

D=

dia

met

erQ

=z-

cut q

uart

z*

0.00

5 m

m u

ncer

tain

ty in

thic

knes

s**

0.0

02 m

m u

ncer

tain

ty in

thic

knes

sD

iam

eter

s ar

e no

min

al v

alue

s

IS2

(10-

019)

0.56

6Q

Q

0.

107

Q

5.06

465

1.51

112

900

yes

IS1

(01-

022)

0.40

0Q

Q

0.

019

Q

3.50

423

1.43

2322

00ye

s

IS3

(01-

021)

0.56

5Q

Q

0.

032

Q

5.05

464

1.51

3322

00ye

s

293

0 200 400 600 800 1000 1200 1400 1600 18000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (ns)

Tra

nsm

isis

on

IS1

T4

Figure D.3: Photodiode transmission of experiment IS1z-cut quartz windows were used to achieve a 3.5 GPa peak pressure. The initial samplethickness was 19µm. Experiment T4 (3.5 GPa, 28µm sample) is shown for comparison.

294

Experiment IS2 and IS3

The peak pressure in these experiments was 5 GPa. Different sample thicknesses

were used to obtain different loading rates. The original water sample in experiment IS2

was 107µm, and 112 ns were required to reach the 90% pressure level. In experiment

IS3, the water sample was initially 32µm thick, and 90% peak pressure was reached in 33

ns. The photodiode transmission measurement for these experiments in shown in Figure

D.4. The effect of thickness was the same as noted in experiments T5 and T6 (page 105).

Transmission was lost first for the thin sample, but thick sample ultimately reached a lower

transmission value.

D.2.2 Summary

The Imacon 200 demonstration experiments generated the following results.

1. Transmission measurements made with the imaging relay system were consistent

with fiber coupled measurements.

2. Initial thickness had two effects on sample transmission. Thin samples lost light

more quickly, but thick samples eventually have lower transmissions.

295

0 200 400 600 800 1000 1200 1400 1600 18000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (ns)

Tra

nsm

issi

on

IS2IS3

IS3

IS2

Figure D.4: Photodiode transmission of experiment IS2 and IS3z-cut quartz windows were used to achieve a 5 GPa peak pressure. The initial samplethickness in experiment IS2 was 107µm; in IS3 it was 32µm.

296

References for Appendix D[1] H.D. Hegewald.Time-dependent longitudinal response of soda lime glass under shock

compression. M.S. thesis, Washington State University, (2001).

[2] C.S. Yoo, J.J. Furrer, G.E. Duvall, S.F. Agnew and B.I. Swanson. Effects of dilutionon the ultraviolet and visible absorptivity ofCS2 under state and shock compression.J. Phys. Chem.91, 6577 (1987).

[3] C.S. Yoo, G.E. Duvall, J. Furrer and R. Granholm. Effects of pressure and dilution onthe visible and ultraviolet spectrum of carbon disulfide under shock compression.J.Phys. Chem.93, 3012 (1989).

[4] J.M. Winey. Time-resolved optical spectroscopy to examine shock-induced decompo-sition in liquid nitromethane. Ph.D. thesis, Washington State University, (1995).

297

Appendix E

Optical Extinction in a Scattering MediumThis appendix outlines optical scattering theory and its relation to the transmission

measurement. Section E.1 covers some basic aspects of scattering from a non-absorbing

body embedded in a dielectric medium. An ideal multiple particle system is treated in Sec-

tion E.2 to build some intuitive relationships between optical transmission and the extent of

a phase transition that causes scattering. Limitations of optical transmission measurements

are given in Section E.3.

E.1 Single particle scatteringA brief overview of optical scattering theory is given here; more extensive dis-

cussion may be found in Refs. [1, 2]. This treatment closely follows that of Ref. [1] for

a non-absorbing system, which is reasonable for the 400-700 nm spectral range in water

(Section 2.1.3). The problem considered here is a dielectric sphere of radiusa embed-

ded in a uniform dielectric medium (Figure E.1). The sphere represents the a solid region

within the liquid phase. Both phases are assumed to be optically linear and isotropic as

well as non-magnetic (µ → 1). m is defined as the refractive index ratio of each phase, i.e.

nsolid/nliquid.

Polarized plane waves of light (field amplitudeEliquid = E0e−iωteikz, wave number

k = 2ω/c), incident from the left, produce a polarization fieldPsolid in the sphere. Electric

(Esolid) and displacement (Dsolid) fields within the sphere are related by the permittivity.

Dsolid = εsolidEsolid = m2Esolid (E.1)

299

a

γ

collimatedincidentlight

liquidwater

E

k

0

Psolid

scatteredlight

Figure E.1: Single scattering setupThe sphere represents a solid region that has formed from the liquid. Plane waves of light,incident from the left, induce a polarizationP within the sphere, creating a scattering field.

300

The polarization field (Psolid) is given by:

Psolid =1

4π(Dsolid−Esolid) =

m2−14π

Esolid (E.2)

Integration ofPsolid over the entire sphere gives the total dipole momentp.

p =∫

m2−14π

EsoliddV (E.3)

If it is assumed that the sphere is very small (ka << 1) and that the optical difference

between the phases is also small (m≈ 1), then the electric field within the sphere is nearly

constant and approximately equal to the liquid electric field. The integration for the total

dipole moment reduces to the following form.

p≈ m2−14π

E0e−iωtV =m2−1

3E0a3e−iωt (E.4)

This time dependent dipole moment leads to a scattering field of magnitudeEs:

Es =k2p sinγ

re−ikr (E.5)

wherer is the observation distance from the dipole. The angle between the observation

angle and the induced dipole moment isγ. The average radiated intensity from this dipole

is given byIs = c|Es|2/8π; the total scattered powerPs is defined by integratingIs over a

sphere of radiusr.

Ps =c

8π

(m2−1

3

)2

E20a6

∫dΩsin2γ = cE2

0(m2−1)2

27k4a6 (E.6)

The scattering cross section,σs, is defined as the ratio of the scattered power and the inci-

dent intensityI0 = c|E0|2/8π.

σs =8π27

(m2−1)2k4a6≈ 32π27

(m−1)2k4a6 (ka<< 1) (E.7)

The final approximation reflects the assumption thatm is close to unity.

301

The result in Equation E.7 is typically referred to as Rayleigh scattering and is only

valid in the limit when the solid region is very small (ka<< 1). Some improvements can

be made by using Rayleigh-Gans scattering theory (Ch. 7 of Ref. [1]), which treats larger

bodies as collections of Rayleigh scatterers. It is possible to evaluateσs for any size sphere

using Mie scattering theory (Ch. 4 of Ref. [2]), although this approach is quite numerically

intensive. Certain non-spherical geometries (Ch. 8 of Ref. [2]) can also be treated.

There is a key distinction between a scattering and an absorption system. The

amount of light removed by an absorbing sphere will increase with the sphere volume;

once the entire volume is transformed, the absorbed power will be constant. This is not

necessarily true for a scattering system because a completely solid region contains no re-

fractive index variations. Thus, a completely solid sample should be transparent aside for

local defects in the solid. As such, the connection between solidification extent and trans-

mitted light may be ambiguous.

E.2 Multiple scatterers and optical extinctionIn real samples, optical extinction typically occurs from multiple scattering sites.

Light exiting the sample along the input direction will therefore have a lower intensity

than the incident beam. With a collection of particles, the scattered light power increases

with the number of scatterers, but there are potential complications if light scattered from

one region interacts with others areas of the system. This complication will be suspended

momentarily, so light is assumed to exit the system once scattered.

Consider a thin layer of materialdz (Figure E.2) normal to an incident beam of

collimated light. Within that layer, all scattering particles in areaA are illuminated with

the same light intensityI . The scattered light powerPs (for a specific wavelength of light)

from the layer is the sum of scattering from each individual particle. There may exist a

distribution of scattering sizes withindz, each with different cross sections, so the total loss

302

dz

scatteredpower loss

I(z)

A= cross section area

I(z+dz)

Figure E.2: Optical extinction from a collection of scattering particlesAn areaA of the sample is illuminated with collimated light. Within a thin layerdz, allscattering particles are subjected to a local field intensityI . Light entering the sampleI(z)is attenuated by scattering losses, resulting in a weaker intensityI(z+dz).

303

must be integrated over this distribution:

Ps = IA dz∫

da n(a)σs(a) (E.8)

wheren(a) is the number density of scatters of sizea. The optical power in the layerIA is

attenuated byPS.

d(IA) =−IAdz∫

da n(a)σs(a)→ dIdz

=−Iα (E.9)

whereα ≡∫

da n(a)σs(a) (E.10)

The extinction coefficientα thus increases with the number density of scatters in the sys-

tem. In general,α is a function of wavelength, position, and time. For a one dimensional

material, the transmitted light intensity becomes:

Iout = Iin exp

(−

∫dzα

)(E.11)

If α is constant throughout the sample, this reduces to Beer’s law (page 458 of Ref. [3]):

Iout = Iine−αL (E.12)

whereL is the sample thickness.

The general results for a one dimensional scattering system are related to the trans-

mission and extinction definitions used in Chapters 4 and 6.

T(t,λ ) = exp

[−

∫dzα(z, t,λ )

](E.13)

X(t,λ ) =∫

dzα(z, t,λ ) (E.14)

If α can be determined as a function of wavelength, position, and time, then transmission

and extinction can be calculated for comparison to experimental results. For example, a

sample containing a collection of Rayleigh scatterers has the following transmission and

304

extinction.

T(t,λ ) = exp

(− A

λ 4

∫dz

∫da n(a,z, t) a6

)(E.15)

X(t,λ ) =Aλ 4

∫dz

∫da n(a,z, t) a6 (E.16)

For such a system, the total extinction has a1/λ 4 wavelength dependence regardless of the

details ofn(a,z). In other words, the total optical loss is a multiple of the single particle

cross section. The multiplication factor is a function of the size distribution, so increasing

the number of scatters or their size results in higher system extinction. If wavelength-size

separation is not possible, the extinction calculation is more difficult, but may be performed

numerically.

E.3 Limitations of optical transmission measurementsThe discussion above suggests a link between the microscopic distribution of solid

material(n(a,z, t)) to measurable optical quantities(X(λ , t)). This is similar to the method

used by Knudson [4] to study a polymorphic phase transition in shocked cadmium sulfide.

However, the optical changes in that system were largely the result of optical absorption,

whereas the changes in this work are caused by optical scattering. This section describes

the difficulties in scattering systems that prevent such analysis.

As pointed out in Section E.1, the scattering cross section for an individual particle

may be complicated if the scatterer size is comparable or larger than the probe wavelength.

Even ifσ(a) is known, the functionn(a,z, t) is also needed. This expression is complicated

since ice nucleation requires a silica surface (Section 6.2), son(a,z, t) is not simply a func-

tion of the local thermodynamic history. The growth rate of spherical ice bodies cannot be

constant due to heat dissipation effects (Section H.1); these effects may also make spher-

ical growth impossible (Section H.2). Thus the conceptual model proposed in the above

305

discussion is not valid for freezing water.

Practical issues add to the complexity of extinction measurements in a scattering

system. Figure E.3 illustrates several of these difficulties. In obtaining Equation E.10, it

was assumed that all light entering the sample is collimated. This is particularly important

for scattering measurements because uncollimated light may be scatteredinto the collection

system, an effect that cannot occur in a purely absorbing material. A related difficulty

occurs on the other side of the sample, where light scattered close to the forward direction

is collected by the detector. The transmission measurement described in Chapter 3 attempts

to minimize these effects with a series of apertures, although it is generally difficult to

completely reject the forward scattered light [2]. There are also problems in the single

scattering assumption used in Section E.2. In order for the total scattered power to equal

the sum of the individual scattering cross sections, the scatters must be sparsely located

throughout the material. Correlated scattering occurs when particles are close enough for

coherence to be an issue (separations of 3 diameters or less [1]), so that the scattering

fields add rather than the scattering intensities. Multiple scattering effects may also direct

scattered light into the collection angle of the detector. Single scattering is only dominant

of extinctions below 0.1; for much larger extinctions, the problem becomes the far more

complicated problem of radiative transfer [1]. These effects cast some doubt on the general

form of Beer’s law (Equation E.12) for a highly scattering material. Even when such a

material is uniform, it is unclear if a simple relation between the sample thickness and

measured extinction exists.

E.4 Transmission interpretationDespite the complications of scattering noted above, several firm assertions can

be made about the optical transmission measurements in this work. These assertions are

sufficient for the analysis presented in Chapter 6.

306

(a)

(d)

(c)

(b)

Figure E.3: Illustration of some the complications in scattering systems:(a) Improper collimation(b) Collection of forward scattered light(c) Correlated scattering(d) Multiple scatteringNot shown in the diagram are complications of single scatterer cross sections (size andshape) and size distributions.

307

1. A uniform liquid has unity transmission

In the absence of refractive index variation, scattering cannot occur, so all light is

transmitted through the sample.

2. The time scales of the freezing transition are related to time scales observed in optical

transmission record

The onset of freezing is readily determined by transmission values lower than unity.

As the sample freezes, the optical transmission decreases. When freezing reaches

a steady state configuration, no new scatters are formed, so the optical transmission

must be constant. Thus, there is correlation between the time dependence of the

transmission and the phase transition.

3. For a steady state density of scatters, the optical extinction should increase with the

amount of frozen material

Although effects such as multiple scattering can reduce the measured extinction, the

optical attenuation at any point in the water sample is a positive quantity. Thus, in-

creases in the transformed sample thickness must add to the total value of extinction.

308

References for Appendix E[1] H.C. van de Hulst.Light Scattering by Small Particles. Dover, New York, (1957).

[2] C.F. Bohren and D.R. Huffman.Absorption and Scattering of Light by Small Particles.John Wiley & Sons, New York, (1983).

[3] P. Atkins. Physical Chemistry. W.H. Freeman, New York, 6th edition, (1998).

[4] M.D. Knudson. Picosecond electronic spectroscopy to understand the shock-inducedphase transition in cadmium sulfide. Ph.D. thesis, Washington State University, (1998).

309

Appendix F

Details of the VISAR Calculations

F.1 Apparent velocityThe following discussion is a brief overview of the VISAR measurement; a more

thorough discussion is given in Refs. [1, 2]. The VISAR system (Figure 3.17) measures

particle velocityu(t) from the change in optical phase of light reflected from a mirror

embedded in the target. A simplified drawing of the measurement is shown in Figure

F.1. Laser light of frequencyf0 is incident on a mirror within the target; reflected light is

Doppler shifted to a frequencyf (t):

f (t) = f0c+u(t)c−u(t)

≈ f0

(1+2

u(t)c

)(F.1)

The approximation made here is that particle velocityu is always much less than the speed

of light c. Even for particle velocities of 1 km/s,u/c≈ 10−6, so only the first order ofu/c

is relevant.

Light entering the VISAR interferometer is nominally unpolarized and has a fre-

quencyf (t). Consider first a single polarization with field amplitudeEP. The electric field

at the entrance plane of the system is given byEin.

Ein(t) = EP e−i2π f (t)t (F.2)

The input beam is split into two portions, each portion traversing a different path before

being recombined at the detector. One path is adjusted to make the traversal timeτ longer

than the other path, so the phase of one beam lags the other byτ. The optical phaseΦ(t)

311

targetmirror

stationarymirror

path 1

etalon

1/8 wave plate

polarizationsensitivedetector

stationarymirror

VISAR outputI(t)

path 2(delayed)

u(t)

VISAR inputf(t)

Figure F.1: Simplified view of the VISAR measurementThe detector receives superimposed light from both paths, one of which is delayed by atime τ.

312

of each beam depends on the time dependent frequencyf (t).

Φ(t)≡ 2π∫ t

0f (t ′)dt′ (F.3)

The total electric field,Eout, is a linear sum of the waves from each path.

Eout = EP e−i2π f (t)t(

aeiΦ(t) +beiΦ(t−τ))

(F.4)

a andb are (possibly complex) scalar factors that include changes in amplitude or phase

that occur in either leg. The output intensity read by the detector is proportional to the

square field amplitude1.

Iout ∝E2

P

2

(a∗a+b∗b+ab∗ ei∆ +a∗b e−i∆

)(F.5)

∆≡Φ(t)−Φ(t− τ) (F.6)

Using Equation F.3 and F.1,∆ can be written in terms of the velocity of the target mirror.

∆ = 2π∫ t

t−τf (t ′)dt′ = 2π f0

∫ t

t−τ

(1+2

u(t ′)c

)dt′

≈ 2π f0τ(

1+2c

u(t))

∆ = ∆0 +4πτλ0

u

→ u = F∆−∆0

2π(F.7)

The approximation made by reducing the integral is discussed in Goosman [2].∆0 is a con-

stant phase shift that can be determined from knowledge ofu(t = 0). The fringe constant

F ≡ λ0/2τ relates the magnitude of phase shift with changing velocity.

1This is a time averaged intensity. Although the mirror velocity may change on picosecond time scales,the period of visible light is at least three orders of magnitude shorter, so the detector averages over manycycles.

313

Two detectors are used in the actual VISAR system, each sensitive to one polariza-

tion. For one polarization,a = a1 andb = b1, wherea1 andb1 are real.

I1 ∝E2

P

2

(a2

1 +b21 +2a1b1cos∆

)(F.8)

Within the delay path is a 1/8 wave plate that adds an additionale−iπ/2 of phase (in two

passes) to one polarization. Thusa = a2 andb =−ib2, wherea2 andb2 are both real.

I2 ∝E2

P

2

(a2

2 +b22−2a2b2sin∆

)(F.9)

Redefining the detector outputs asX ≡ I1 (unaffected by 1/8 wave plate) andY ≡ I2 (af-

fected by 1/8 wave plate) gives the following result.

X = A1 +B1cos∆ (F.10)

Y = A2 +B2sin(∆+φR) (F.11)

φR is a constant parameter2 that may be necessary when the round trip phase shift through

1/8 plate is not exactlyπ/4. The following steps are taken to calculateu(t) from VISAR

data signalsX andY.

1. BothX andY are normalized by the unpolarized beam intensity monitor (BIM). This

eliminates slow intensity changes from the mirror and light collection system at the

target. Five point nearest neighbor averaging is performed on each signal

2. The normalized dataX andY are plotted as a Lissajous figure, forming an ellipse.∆

is eliminated from Equations F.10 and F.11 to give a function Y(X) that depends on

the parametersA1, A2, B1, B2, andφR. The values of these parameters are chosen to

optimally fit the data ellipse.

2During experimental setup, the VISAR system is optimized so thatφR is nearly zero, producing an ellipseoriented along the x or y axis.

314

3. ∆ is determined by solving Equation F.10 or Equation F.11 for every(X,Y) data

point, using the optimized values ofA1, A2, B1, B2, andφR .

4. Particle velocity is then found by using the approximation from Equation F.7. Some

additional logic is necessary to properly interpret the result because∆ is periodic. In

this work, the VISAR traces maintained sufficient contrast that it was not necessary

to manually add fringes, i.e. shift∆ by integer multiples of2π during fast changes in

u(t).

These procedures are incorporated into a VISAR analysis code [3] that was used in all

experiments.

F.2 Window correctionsThe VISAR system measures an apparent velocity, which is identical to the actual

velocity in the case of a free mirror (Figure F.1). However, if the VISAR beam passes

through dynamically compressed dielectrics, these two velocities are not the same. Hayes

[4] reviews one approach for making the connection between true and apparent velocity; a

simpler method is developed below [5].

Consider the system shown in Figure F.2. Here the VISAR system does not directly

view the mirror as in Figure F.1, but instead looks through a transparent window. In shock

wave measurements, this window is necessary to preserve compression in the sample, lo-

cated to the left of the mirror. Light travelling from the mirror to a fixed point Q (located

to the right of the window at all times) experiences a pathPop that is longer than the phys-

ical distance. This is a result of the fact that light travels more slowly in the window than

in vacuum, leading to an apparent positionx = −Pop. The apparent mirror velocityuv is

therefore the time derivative of the apparent position.

uv =−dPop

dt=− d

dt

∫n(x′, t)dx′ (F.12)

315

samplemirror

rear window

true mirrorvelocity

free surface velocity

L(t)

Q

VISAR measuresapparent velocity

Figure F.2: True versus apparent velocity in the VISAR systemTo an observer at point Q, the mirror appears to be at positionx = −Pop, wherePop is theoptical path length. The refractive index of the window is larger than unity, so the apparentdistance is larger than the actual distance. The VISAR system measures apparent velocityuv. Window corrections must be made to determine the true velocityuT .

316

The integral is carried out from the true mirror positionxT(t) to the point Q. Let the rear

window free surface position bexFS(t) and the distance from this free surface to point Q

beL(t). The total optical path is then the sum of the window and free space propagation.

Pop =∫ xFS

xT

n(x′, t) dx′+L(t)

=∫ xFS

xT

n(x′, t) dx′+L0−xFS(t)+xFS(0) (F.13)

L0 is the distance between the rear window free surface and point Q at timet = 0; xFS is

the free surface position. The integral can be solved if the refractive index profile in the

window is known, although in practice this may quite complicated.

For certain window materials, such as z-cut quartz [6] or a-cut sapphire [7], the

relationship between refractive index and density is linear.

n = aρ +b (F.14)

Using this relationship simplifies the general form of the optical phase considerably.

Pop = a∫ xFS

xTρ(x′, t)dx′+b

∫ xFS

xT

dx′+L0−xFS(t)+xFS(0)

= a m/A+b(xFS(t)−xT(t))+L0−xFS(t)+xFS(0) (F.15)

The first term is simply a statement of mass conservation in a one dimensional system.

Given an arbitrary cross sectional areaA, the total mass within the window ism. The

quantity m/A is therefore constant in time. Apparent velocity can be determined using

Equation F.12.

uv(t) = −b(uFS(t)−uT(t))+uFS(t)

→ uT(t) =(uv(t)+(b−1)uFS(t))

b(F.16)

317

For this class of window materials, the VISAR correction becomes straightforward. When

the rear surface is not moving (uFS= 0), the correction is trivial. For z-cut quartz, the value

of b is 1.08253 [6]. In a-cut sapphire, the value ofb is 1.8693 [7].

318

References for Appendix F[1] L.M. Barker and R.E. Hollenbach. Laser interferometer for measuring high velocities

of any reflecting surface.J. Appl. Phys.43, 4669 (1972).

[2] D.R. Goosman. Analysis of the laser velocity interferometer.J. Appl. Phys.46, 2516(1975).

[3] S.C. Jones. VISAR 2000 code, Washington State University (unpublished), (2000).

[4] D. Hayes. Unsteady compression waves in interferometer windows.J. Appl. Phys.89,6484 (2001).

[5] S.C. Jones, (2001). Private communication regarding VISAR window corrections.


[7] S.C. Jones, B.A.M. Vaughan and Y.M. Gupta. Refractive indices of sapphire underelastic, uniaxial strain compression along the a axis.J. Appl. Phys.90, 4990 (2001).

319

Appendix G

Details of the Mixed Phase Water ModelThis appendix contains details of the two phase water model described in Chapter 5.

A summary of the parameters for the pure liquid and solid phase models is given in Table

G.1. Section G.1 discusses numerical stability issues of the mixed phase water model.

Section G.2 describes changes made to the COPS source code and presents the FORTRAN

source code for the mixed phase model.

G.1 Numerical stability of the mixed phase modelAlthough convergence of Newton’s method tends to be rapid, there are issues re-

garding its numerical stability. If the initial conditions are not close enough the actual so-

lution, the algorithm may never converge properly. Undesirable results, such as unbounded

progression, multiple point cycling and fractal behavior, are known to occur with Newton’s

method [1]. Several tactics were used in to avoid these problems.

1. The COPS code was modified to prevent large changes in any material element be-

tween time steps. Using the previous value ofx as the guess value for the current

time step, only minor changes inx are necessary, reducing the chances of failure.

2. Equation 5.35 describes a full Newton step, which may take the system in the right

direction, but can overshoot the solution and make the iteration unstable. An altered

form of this equation was used to prevent such overshoot:

x = x0 +λdx (G.1)

whereλ is a parameter that varies between 0 and 1. During the first few iterations,

321

Property Liquid phase (n=1) Solid phase (n=2)

Specific heat cv1=3 J/g·K cv2=2 J/g·K

T01=298 K T02=300 KP01 = 106 dyne/cm2 P02 = 0 dyne/cm2

Reference state v01=1.00296 cc/g v02=0.679497 cc/ge01 = 0 e02 =−1.601827×107 erg/gs01 = 0 s02 =−1.513880×107 erg/g·K

Isotherm K1 = 2.21008×1010 dyne/cm2 K2 = 25.03581×1010 dyne/cm2

parameters g1=6.02913 g2= 3.66007

B1 = 4.836403×106

B2 =−1.952754×108

B3 =−1.543223×108

Pressure-temperature B4 =−2.355402×109 b2 = 6.828800×107 dyne/cm2

derivative B5 =−1.175901×1010

B6 =−1.746693×1010

B7 =−8.658027×109

b1(v) =7

∑i=1

Bi

(vv0−1

)i−1

(units of dyne/cm2)

Table G.1: Summary of single phase EOS parameters

322

λ is set to be very small to move towards to the solution without dramatically over-

shooting it. After several iterations,λ was increased towards unity to regain the rapid

convergence of Newton’s method.

3. The range ofx was limited so thatT, v1, andv2 could not exceed a preset range of

validity. Realistic minimum and maximum values were specified for each variable

(e.g. T > 0). If these bounds were exceeded for any variable, this variable was

replaced with the mean of the previous value and the limiting value. A maximum of

1000 iterations were allowed for any single optimization.

For all reverberation calculations, the optimization procedure appeared to be stable. In

cases of large (> 8 GPa) single shock waves, the allowed temperature range was extended

beyond 1500 K to allow the mixture rules to converge properly.

G.2 Mixed phase model subroutines

G.2.1 Alterations of the COPS source code

Changes were made in thesetup subroutine to allow the user to access this material

model (not shown). A modification of thesweep subroutine was also made to prevent large

changes in the water layers during single time step (not shown). The subroutinedtcheck21

(included below) calculates the rate of volume and liquid mass fraction changes at the

present time step, then calculates the time step required to keep the absolute change within

some tolerance (0.001 for mass fraction, 0.0001 for specific volume).

This model utilizes several variables that are unused or undefined in most other

material models of COPS. Temperature is stored in variable number 23. Variable 47 stores

the specific Gibb’s free energy difference between the phases (g2−g1). Variables 48 and

49 keep track of the specific volume for the solid and liquid phases, respectively. Liquid

mass fractionw is stored in variable 50.

323

G.2.2 Source code for mixed phase water model********************************************************************** Two phase (liquid-ice VII) model for water* n=1 -> liquid phase* n=2 -> solid phase*********************************************************************

********************************************************************** Interface to COPS sweep routine*********************************************************************

subroutine mat21(mat,k1,k2)implicit noneinclude ’cops_rev.common’integer mat,k1,k2double precision vold,eold,wold,polddouble precision vnew,enew,wnew,pnewdouble precision v1old,v2old,tolddouble precision v1new,v2new,tnewdouble precision vmid,tmid,pmid,cv,dpdt,ktdouble precision eosparams(3),xeos(3),eosout(8),gibbs1,gibbs2double precision meanrho,dexx,dvfulldouble precision bulk,shear,pcut,q,tau1,tau2,dwdtinteger ii,jjdouble precision wpureliquidparameter (wpureliquid=0.9999)

tau1=prop(mat,4)tau2=prop(mat,5)pcut=prop(mat,6)do ii=k1,k2

if(var(ii,6) .lt. 1.e-10) cycle* values from previous time step

vold=1/varold(ii,5)eold=varold(ii,9)wold=varold(ii,50)v1old=varold(ii,49)v2old=varold(ii,48)told=varold(ii,23)pold=varold(ii,7)q=var(ii,8)

* determine volume increment from uniaxial strainmeanrho=(varold(ii,5)+var(ii,5))/2.0d0dexx=var(ii,15)-varold(ii,15)dvfull=dexx/meanrhovnew=vold+dvfullxeos=[v1old,v2old,told]call water_rate(wold,xeos,tau1,dwdt)if (time.lt.tau2) dwdt=0.0d0wnew=wold+dwdt*dt

324

enew=eoldtnew=toldvmid=(vold+vnew)/2.0d0pmid=poldtmid=told

do jj=1,10eosparams=[vnew,enew,wnew]call watermix(eosparams,xeos,bulk)v1new=xeos(1)v2new=xeos(2)tnew=xeos(3)call watereos1(1,[tnew,v1new],eosout)pnew=eosout(4)pmid=(pold+pnew)/2.0d0enew=eold-(pmid+q)*dvfull

end do

* update cops variablesvar(ii,5)=1/vnewvar(ii,23)=tnewvar(ii,50)=wnewvar(ii,48)=v2newvar(ii,49)=v1newvar(ii,7)=pnewvar(ii,11)=-pnew-qvar(ii,12)=var(ii,11)var(ii,13)=var(ii,11)var(ii,14)=0.0d0var(ii,9)=enew

* gibbs free energy differencecall watereos1(1,[tnew,v1new],eosout)gibbs1=eosout(8)call watereos1(2,[tnew,v2new],eosout)gibbs2=eosout(8)var(ii,47)=gibbs2-gibbs1

* update sound speed for copsshear=0.0d0call timestep(ii,bulk,shear)

end doreturnend

* time step checksubroutine dtcheck21(mat,k1,k2,dteos)implicit noneinclude ’cops_rev.common’integer mat,k1,k2double precision dteos,meanrho,dexx,dvfull

325

double precision xeos(3),w,tau,dwdt,dvdtdouble precision dwmax,dvmaxparameter (dwmax=0.001,dvmax=0.0001)integer ii

tau=prop(mat,4)dteos=1.0d-6do ii=k1,k2

if(var(ii,6) .lt. 1.e-10) cyclexeos(1)=varold(ii,49)xeos(2)=varold(ii,48)xeos(3)=varold(ii,23)w=varold(ii,50)call water_rate(w,xeos,tau,dwdt)do while (dabs(dwdt*dteos) .gt. dwmax)

dteos=dteos/2end domeanrho=(varold(ii,5)+var(ii,5))/2.0d0dexx=var(ii,15)-varold(ii,15)dvfull=dexx/meanrhodvdt=dvfull/dteosdo while (dabs(dvdt*dteos) .gt. dvmax)

dteos=dteos/2end do

end doreturnend

* transformation rate routinesubroutine water_rate(w,x,tau,dwdt)implicit nonedouble precision w,x(3),tau,dwdtdouble precision v1,v2,tdouble precision eosout(8),g1,g2,gtot,dgdouble precision R,MWparameter (R=8.314510D7,MW=18.02D0)

v1=x(1)v2=x(2)t=x(3)

* current gibbs free energycall watereos1(1,[t,v1],eosout)g1=eosout(8)call watereos1(2,[t,v2],eosout)g2=eosout(8)gtot=w*g1+(1-w)*g2

dg=0.0d0if (gtot .gt. g1) dg= -(g1-gtot)if (gtot .gt. g2) dg= +(g2-gtot)dwdt=dg*MW/(R*t)/tau

326

returnend

* mixed phase routine* fixedparams(1)=v* fixedparams(2)=e* fixedparams(3)=w* x(1)=v1* x(2)=v2* x(3)=v3* bulksound=bulk sound speed

subroutine watermix(fixedparams,x,bulk)implicit nonedouble precision fixedparams(3),x(3),bulkdouble precision x0(3),dx(3),f0x(3),axinv(3,3)double precision v,e,w,v1,v2,tdouble precision eosout(8),b1,kt1,pdouble precision eqntol(3),xmin(3),xmax(3)parameter (eqntol=[1.0d4,1.0d-6,1.0d4],+ xmax=[1.1d0,1.0d0,1500.0d0],+ xmin=[0.55d0,0.5d0,273.0d0])double precision eqnerr,lambda,L1,L2integer maxiter,ii,jjlogical done

* some error checkingv=fixedparams(1)w=fixedparams(3)if (v .le. 0.0d0) stop ’water has negative volume!’if (w .lt. 0.0d0) stop ’water has overfrozen!’if (w .gt. 1.0d0) stop ’water has overmelted!’

* newton’s methoddo ii=1,3

x0(ii)=x(ii)end domaxiter=1000L1=0.1L2=20done=.false.jj=1do while (.not. done)

if (jj.gt.maxiter) thenwrite(*,*) ’max newton steps exceeded’write(*,*) fixedparamswrite(*,*) x0write(*,*) f0xstop ’water/ice model not converging’

end ifcall watermatrices(fixedparams,x0,f0x,dx,axinv)eqnerr=0.0d0

327

lambda=L1+(1-L1)*(1-exp(-(jj-1)/L2))do ii=1,3

x(ii)=x0(ii)+dx(ii)*lambdaif (x(ii).lt.xmin(ii)) then

x(ii)=(x0(ii)+xmin(ii))/2.0d0write(*,*) ’Lower limits of water model exceeded!’

write(*,*) ’ Newton iteration reset’end ifif (x(ii).gt.xmax(ii)) then

x(ii)=(x0(ii)+xmax(ii))/2.0d0write(*,*) ’Upper limits of water model exceeded!’

write(*,*) ’ Newton iteration reset’end ifx0(ii)=x(ii)eqnerr=dmax1(eqnerr,dabs(f0x(ii))/eqntol(ii))

end doif (eqnerr .lt. 1.0d0) then

done=.true.end ifjj=jj+1

end do* 20 format(’cycles = ’,i5)* determine sound speed

v=fixedparams(1)v1=x(1)t=x(3)call watereos1(1,[t,v1],eosout)b1=eosout(2)kt1=eosout(3)p=eosout(4)bulk=p*(b1*axinv(3,3)-kt1/v1*axinv(1,3))+ +kt1/v1*axinv(1,2)-b1*axinv(3,2)bulk=bulk*vreturnend

********************************************************************** Water matrices

subroutine watermatrices(fixedparams,x,fx,dx,axinv)implicit nonedouble precision fixedparams(3),x(3),fx(3),dx(3)double precision ax(3,3),axinv(3,3),detdouble precision v,e,w,v1,v2,t,eosout(8)double precision cv1,cv2,b1,b2,kt1,kt2double precision p1,p2,e1,e2integer ii

v=fixedparams(1)e=fixedparams(2)w=fixedparams(3)v1=x(1)

328

v2=x(2)t=x(3)

call watereos1(1,[t,v1],eosout)cv1=eosout(1)b1=eosout(2)kt1=eosout(3)p1=eosout(4)e1=eosout(5)

call watereos1(2,[t,v2],eosout)cv2=eosout(1)b2=eosout(2)kt2=eosout(3)p2=eosout(4)e2=eosout(5)

fx(1)=p1-p2fx(2)=w*v1+(1.0d0-w)*v2-vfx(3)=w*e1+(1.0d0-w)*e2-e

ax(1,1)=-kt1/v1ax(1,2)=kt2/v2ax(1,3)=b1-b2ax(2,1)=wax(2,2)=1.0d0-wax(2,3)=0.0d0ax(3,1)=w*(t*b1-p1)ax(3,2)=(1.0d0-w)*(t*b2-p2)ax(3,3)=w*cv1+(1.0d0-w)*cv2

det=+ax(1,1)*(ax(2,2)*ax(3,3)-ax(2,3)*ax(3,2))+ -ax(1,2)*(ax(2,1)*ax(3,3)-ax(3,1)*ax(2,3))+ +ax(1,3)*(ax(2,1)*ax(3,2)-ax(3,1)*ax(2,2))axinv(1,1)=(ax(2,2)*ax(3,3)-ax(2,3)*ax(3,2))/detaxinv(1,2)=(ax(1,3)*ax(3,2)-ax(1,2)*ax(3,3))/detaxinv(1,3)=(ax(1,2)*ax(2,3)-ax(1,3)*ax(2,2))/detaxinv(2,1)=(ax(2,3)*ax(3,1)-ax(2,1)*ax(3,3))/detaxinv(2,2)=(ax(1,1)*ax(3,3)-ax(1,3)*ax(3,1))/detaxinv(2,3)=(ax(1,3)*ax(2,1)-ax(1,1)*ax(2,3))/detaxinv(3,1)=(ax(2,1)*ax(3,2)-ax(2,2)*ax(3,1))/detaxinv(3,2)=(ax(1,2)*ax(3,1)-ax(1,1)*ax(3,2))/detaxinv(3,3)=(ax(1,1)*ax(2,2)-ax(1,2)*ax(2,1))/detdo ii=1,3

dx(ii)=-axinv(ii,1)*fx(1)-axinv(ii,2)*fx(2)-axinv(ii,3)*fx(3)end do

returnend

*********************************************************************

329

* Single phase EOS routine (cgs units)* Final form 2/2/2002 by Dan Dolan* watereos1(phase,eosin,eosout)* phase (integer) :* phase=1 -> liquid* phase=2 -> solid ice VII* eosin (double precision) :* eosin(1)=temperature* eosin(2)=specific volume* eosout (double precision) :* eosout(1)=cv (specific heat)* eosout(2)=b (dp/dt)v* eosout(3)=kt (isothermal bulk modulus)* eosout(4)=p (pressure)* eosout(5)=e (specific energy)* eosout(6)=s (specific entropy)* eosout(7)=f (specific helmholtz free energy)* eosout(8)=g (specific gibbs free energy)*********************************************************************

subroutine watereos1(phase,eosin,eosout)implicit noneinteger phasedouble precision eosin(2),eosout(8)double precision t,v,cv,b,kt,p,e,s,f,gdouble precision x,ib,it0,pt0,dbdv,kt0,t0,e0,s0integer nb1,iiparameter (nb1=7)

* liquid phase paramsdouble precision v01,t01,p01,cv1,k01,g1,b1(nb1),e01,s01parameter(+ v01=1.00296d0, t01=298.15d0, p01=1.01325d6, cv1=3.0d7,+ k01=2.21008d10, g1=6.02913d0,+ b1=[4.836403d6, -1.952754d8, -1.543223d8, -2.355402d9,+ -1.175901d10, -1.746693d10, -8.658027d9],+ e01=0.0d0, s01=0.0d0)

* ice VII paramsdouble precision v02,t02,p02,cv2,k02,g2,b2,e02,s02parameter(+ v02=0.679497d0, t02=300.0d0, p02=0.0d0, cv2=2.0d7,+ k02=25.03581d10, g2=3.66007d0, b2=6.8288d7,+ e02=-1.6018266d9, s02=-1.5138799d7)

t=eosin(1)v=eosin(2)b=0.0d0ib=0.0d0dbdv=0.0d0select case (phase)

* liquid phasecase (1)

330

t0=t01cv=cv1x=v/v01-1.0d0do ii=1,nb1

b=b+b1(ii)*x**(ii-1)ib=ib+b1(ii)*v01/dfloat(ii)*x**iiif (ii.gt.1) then

dbdv=dbdv+b1(ii)/v01*dfloat(ii-1)*x**(ii-2)end if

end doit0=(k01/g1-p01)*(v01-v)

+ -k01*v01/g1/(g1-1.0d0)*((v01/v)**(g1-1.0d0)-1.0d0)pt0=p01+k01/g1*((v01/v)**g1-1.0d0)kt0=k01*(v01/v)**g1e0=e01s0=s01

* ice VII phasecase (2)

t0=t02cv=cv2b=b2;dbdv=0.0d0ib=b*(v-v02)it0=(k02/g2-p02)*(v02-v)

+ -k02*v02/g2/(g2-1.0d0)*((v02/v)**(g2-1.0d0)-1.0d0)pt0=p02+k02/g2*((v02/v)**g2-1.0d0)kt0=k02*(v02/v)**g2e0=e02s0=s02

case defaultstop ’unknown phase of water’

end select

kt=kt0-dbdv*v*(t-t0)p=pt0+b*(t-t0)e=e0-it0+t0*ib+cv*(t-t0)s=s0+cv*log(t/t0)+ibf=e-t*sg=f+p*v

eosout=[cv,b,kt,p,e,s,f,g]returnend

331

G.2.3 Material list

Below is a complete listing of the material parameters passed to COPS. Detailed

descriptions the window materials are given in Appendix B.

z-cut quartz density = 2.6485 nmdef = 3ieos = 1 icut = 1G0 = 46.92e+10 G1 = 1.873e+00 G2 =0.3459e-10Yo = 1.000e+30 M = 0.000e+00K0 = 43.19e+10 K1 = 156.2e+10 K2 = 48.60e+10 K3 = 0.000e+00gamma = 0.675 pcut =-1.250e+09 cv = 0.670e+07Fused silica density = 2.201 nmdef = 8ieos = -1 icut = 1g01 = 3.062e+11 g11 =-1.199e+12 g12 = 6.684e+11 gcut = 7.600e-02K0 = 7.760e+11 K1 =-4.000e+12 K2 = 2.934e+13 K3 =-6.926e+13pcut =-2.900e+08Soda lime glass density = 2.493 nmdef = 3ieos = 0 icut = 1G0 = 2.813e+11 G1 = 0.000e+00 G2 = 0.000e+00Yo = 1.000e+30 M = 0.000e+00bulk = 4.793e+11a-cut sapphire density = 3.985 nmdef = 3ieos = 1 icut = 1G0 = 178.9e+10 G1 = 1.512e+00 G2 =-7.260e-12Yo = 1.000e+30 M = 0.000e+00K0 = 259.1e+10 K1 =+5.203e+12 K2 =-2.061e+13 K3 = 0.000e+00gamma = 1.280 pcut =-1.250e+09 cv = 0.670e+07Metastable water density = 0.99705 nmdef = 21ieos = -1 icut = 1T0 =2.9815e+02 tau1 =1.0000e+00 tau2 =1.0000e+00 pcut =-1.000e+09Equil water/ice density = 0.99705 nmdef = 21ieos = -1 icut = 1T0 =2.9815e+02 tau1 =1.0000e-09 tau2 =1.0000e-09 pcut =-1.000e+09Normal ice/water density = 0.99705 nmdef = 21ieos = -1 icut = 1T0 =2.9815e+02 tau1 = 50.00e-09 tau2 =650.00e-09 pcut =-1.000e+09PMMA density = 1.186 nmdef = 2ieos = 1 icut = 1K0 = 8.820e+10 K1 = 8.015+10 K2 = 3.838e+11 K3 = 0.000e+00gamma = 1.130 pcut =-1.250e+09Epoxy density = 1.186 nmdef = 2ieos = 1 icut = 1K0 = 8.820e+10 K1 = 0.00e+00 K2 = 0.000e+00 K3 = 0.000e+00gamma = 1.130 pcut =-1.250e+09

332

References for Appendix G[1] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery.Numerical Recipies

in C: The Art of Scientific Computing. Cambridge University Press, New York, 2ndedition, (1992).

333

Appendix H

Heat Dissipation in Solidification ProcessesSince freezing is a first order phase transition, there is a latent heat emitted due to the

formation of bonds that stabilize the liquid into its solid form. In a closed system, this heat

increases the local temperature, which hinders further freezing. An important limitation in

freezing is thus the conduction of latent heat into the liquid ahead of the solid interfaces.

This limitation has several effects during solidification. Section H.1 describes the case of

1D planar solidification in a supercooled liquid, often referred to as the Stefan problem.

Section H.2 discusses the morphological stability of the solid interface, demonstrating that

heat dissipation creates instabilities that dominate the growth process.

H.1 Planar solidification of a supercooled liquidThis section describes one dimensional solidification in a supercooled liquid. For

simplicity, it is assumed that density and pressure of the system are held fixed (see Refs.

[1, 2] for more a more general treatment). The isobaric specific heatcp is also assumed to

be constant for both solid and liquid phases.

H.1.1 The Stefan problem

In a heat conduction problem with a first order phase transition, there is a moving

heat source located at the transition interface. This situation, commonly referred to as the

Stefan problem [1, 2], has several cases that can be solved analytically. Two related cases

will be treated here; the only distinction between these cases is the nature of the boundary

condition at the initiation point of freezing.

Consider a semi-infinite liquid in the regionx > 0 (Figure H.1). At timet = 0, all

335

x=0

SolidT

1

LiquidT

2

Xf(t)

x

T

T0

Figure H.1: Layout of the Stefan problemSolidification begins at thex = 0 interface at timet = 0. Thex = 0 boundary is in contactwith a thermal reservoir (T = TR). The phase interface (Xf (t)) moves the right, liberatinglatent heat as the solid grows.

336

portions of the liquid are supercooled to a temperatureT0 < Tm, whereTm is the equilibrium

melting temperature at the current pressure. Let solidification begin on thex = 0 interface

att = 0 and proceed to the right. The position of this interface isXf (t). Behind the interface

(region 1), the system is pure solid; ahead of the interface (region 2), the system is pure

liquid. On either side of the interface, there are no heat sources, so temperature must follow

thermal diffusion:

κi∂ 2Ti

∂x2 −∂Ti

∂ t= 0 (H.1)

whereκi = Ki/ρicpi is the thermal diffusivity of regioni. At the the interface between the

phases, two boundary conditions must be satisfied. Both solid at liquid coexist at that point,

so the temperature must be equal to the equilibrium melting temperature.

T1 = T2 = Tm (x = Xf ) (H.2)

As the front moves, latent heat is liberated. If the front is to remain at fixed temperature,

this heat must be conducted into the liquid ahead and the solid behind the front.

ρLdXf

dt= K1

∂T1

∂x−K2

∂T2

∂x(x = Xf ) (H.3)

Hereρ is mass density andL is the specific latent heat.

Carslaw and Jaeger [1] demonstrate that the following solution is compatible with

the diffusion equation in a semi-infinite domain.

Ti(x, t) = Ai +Bi er fx

2√

κit(H.4)

Far ahead of the transition front (x→ ∞), the liquid region remains at the original super-

cooled temperatureT0, which gives the following form forT2.

T2 = T0−B2er f cx

2√

κ2t(H.5)

337

Applying the first interface boundary condition (Equation H.2) gives the following result.

Tm = T0−B2

[er f c

Xf

2√

κ2t

](H.6)

For this condition to be valid at all times,Xf must be restricted to the form

Xf = 2λ√

κ2t (H.7)

whereλ is a constant. The growth velocity of the phase interface is thus nonlinear.

v =dXf

dt= λ

√κ2

t(H.8)

Rewriting the solutionT2 in terms ofλ :

T2 = T0− T0−Tm

er f cλer f c

λxXf

(H.9)

In order to constructT1, the boundary condition atx = 0 must be specified. Two different

cases are considered below.

Adiabatic case

If no heat may flow out of thex = 0 boundary, the temperature at that point is set

by the transition front att = 0. Since the front is atTm for t > 0, the value ofT2 must be a

constant to satisfy Equation H.1.

T2 = Tm (H.10)

A relation forλ can be obtained using the second interface boundary condition (Equation

H.3).√

π λ er f cλeλ 2=

cP2(Tm−T0)L

(H.11)

338

Isothermal case

In this case, it is assumed that the thex = 0 boundary is fixed at a “reservoir”

temperatureTR. A form of T1 could be constructed using Equation H.4 as was done forT2

above. This will alter the equation that definesλ and change the rate at which the phase

interface moves. To allow for simpler interpretation, consider instead the following solid

temperature function.

T1 = TR+(Tm−TR)xX

(H.12)

This form maintains the proper boundary conditions and is a reasonable approximation

when the solid layer is relatively thin. Using the second interface condition, the solution

for λ becomes slightly more complicated than Equation H.11.

√π λ er f cλeλ 2

[1− 1

2λ 2

K1

K2

cP2(Tm−TR)L

]=

cP2(Tm−T0)L

(H.13)

Table H.1 summarizes the results of the isothermal and adiabatic cases. In rever-

beration experiments, thex = 0 boundary would be a silica window that is initially at

temperatureTR≈ 300K. The two limits describe bounding cases because heat flow from

the sample would raise the interface temperature to a point betweenT0 andTm. The true

value ofλ must therefore lie between solutions of the adiabatic and isothermal cases.

H.1.2 Solid growth in reverberated water

Consider an isentropic compression of liquid water to a peak pressure of 5 GPa.

The value ofT0 at this pressure is 446 K, while the value ofTm is 527 K (Chapter 5). The

latent heat of water at 5 GPa is 760 J/g; the density is approximately 1.5 g/cc. It assumed

here thatcp2 ≈ cv2 = 3 J/g·K. The thermal conductivity of liquid water at high pressures is

in the range of 1-2 W/m·K [3–6]. Solid is assumed to be similar to ordinary ice (Ih), which

has a thermal conductivity of 2.25 W/m·K [7].

339

Table H.1: Summary of Stefan problem solutions

x = 0 conditionResult Adiabatic Isothermal

Liquid temperature T2 = T0− T0−Tmer f cλ er f cλx

Xf

Solid temperature T1 = Tm T1≈ TR+(Tm−TR) xX

Interface position Xf = 2λ√

κ2t

Growth velocity vf = λ√

κ2t

λ solution Equation H.11 Equation H.13

340

The value ofλ can be determined using the parameters above and the graphical

construction shown in Figure H.2. In the adiabatic limit, Equation H.11 is satisfied for

λ=0.330. The value ofλ in the isothermal limit ranges from 0.941-1.28, depending on

the thermal conductivity of the liquid phase. From this construction, it is clear that large

supercooling results in higher values ofλ 1, and thus a faster transformation front. In the

adiabatic case,λ approaches zero as the supercooled temperature approaches the melting

temperature, while there is a finite value ofλ at T0 = Tm in the isothermal case. This

is merely a reflection of the requirement that freezing only advances due a difference in

Gibb’s free energy. WhenT0 = Tm, no driving force exists unless heat is drawn out of the

window.

Figure H.3 shows limiting sizes as a function of time for both cases for a 5 GPa

reverberation. These results indicate that after 1µs of growth, uniform solid ice features

should at least 340 nm, but no larger than a 1-2 micron. The real situation is somewhere in

between, so the limiting size is probably around 400-800 nm. This assumes, of course, that

planer growth is maintained throughout the transformation.

H.2 Growth stability in a supercooled liquidIn Section H.1, it was assumed that a supercooled liquid solidifies through planer

ice growth, but this type of growth is typically unstable. A detailed stability analysis of such

problems may be found in Refs. [2,8]. The following discussion is a qualitative description

of the formation of growth instabilities in a supercooled liquid.

First consider freezing in a “normal” situation, where water is not supercooled. To

induce freezing, heat must be extracted from liquid water through an exterior interface,

forming some solid. If heat extraction is uniform across a large plane, the situation should

1This is certainly true whilecP2(Tm−T0)≤ L. For very low values ofT0, the system becomes hypercooled(i.e. the release of latent heat does not bring the material to its melting temperature), changing the nature ofthe solidification problem [2].

341

0.001 0.01 0.1 1 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

λ

Fad

Fiso

(K2=1 W/m⋅K)

Fiso

(K2=2 W/m⋅K)

Fad

Fiso

(K

2=1 W/m⋅K)

Fiso

(K

2=2 W/m⋅K)

Figure H.2: Numerical solutions to Stefan problemThe black line shows the right hand side of Equations H.11 and H.13 for an isentropiccompression of liquid water to 5 GPa. Increases in theFad shows the left hand side ofEquation H.11 (adiabatic limit);Fiso shows the left hand side of Equation H.13 (isothermallimit).

342

0 200 400 600 800 10000

200

400

600

800

1000

1200

1400

1600

Isothermallimit

Adiabaticlimit

t (ns)

X (

nm

)

Figure H.3: Uniform solid size limits in freezing waterThe actual limiting size is somewhere between these extreme cases.

343

look something like Figure H.4(a). Water ahead of the front is too warm to freeze, so

growth only occurs at the ice-liquid interface as heat is drawn out. In this circumstance,

planar growth is completely stable. If water is allowed to supercool, the situation is quite

different. From the discussion in Section H.1.1, it is clear that solidification rate is limited

by the dissipation of latent heat. Small perturbations in the planar interface are exposed to

more supercooled liquid, enhancing heat dissipation and increasing the growth rate at that

point. Sharp solid projections shown in Figure H.4(b) are thus more likely than the uniform

growth.

It might seem that freezing ought to create smaller and smaller features to maximize

heat dissipation. This never occurs because there is surface tensionγ required to form the

solid-liquid interface. A characteristic length scale created by the these competing effects

can be defined as:

l =γ

ρL(H.14)

Solid bodies beneath this length scale are dominated by the surface energy, which stabi-

lizes uniform growth. Above this length scale, heat dissipation instabilities are expected.

Estimating the liquid-solid surface tension to be 30×10−3 J/m2 [7], the value ofl is on the

order of10−11 m. Thermal diffusion will thus play a considerable role even at the earliest

stages of growth. Theoretical calculations suggest that growth instabilities become an is-

sue when a solid sphere is approximately 7 times the critical nucleus radius [8, 9]. If the

critical nucleus size for freezing is on the order of10−9 m, then it would be expected that

stable growth will not be observed in the10−5 m resolution limit of the imaging measure-

ments (Section 4.2). In such unstable growth, the solid interface grows quickly in certain

directions but slowly in others, leaving large regions of unfrozen liquid behind.

344

(a)

(b)

x=0

x=0

T0>T

m

T0<T

m

heatflow

heatflow

Figure H.4: Instabilities in supercooled solidificationLines show snapshots of solid-liquid interface position.(a) Planar freezing is stable if the liquid ahead of the solid interface is warmer than themelting temperature.(b) Instabilities form when the liquid ahead of the interface is cooler than the meltingtemperature.

345

References for Appendix H[1] H.S. Carslaw and J.C. Jaeger.Conduction of heat in solids. Oxford University Press,

Oxford, 2nd edition, (1959).

[2] V. Alexiades and A.D. Solomon.Mathematical modelling of melting and freezingprocesses. Hemisphere Publishing, Washington, (1993).

[3] P.W. Bridgman. The thermal conductivity of liquids under pressure.Proc. Am. Acad.Arts. Sci.59, 141 (1923).

[4] A.W. Lawson, R. Lowell and A.L. Jain. Thermal conductivity of water at high pres-sures.J. Chem. Phys.30, 643 (1959).

[5] G.S. Kell. Thermodynamic and transport properties of fluid water. InWater: A Com-prehensive Treatise v. 1,F. Franks, editor. Plenum, New York (1972).

[6] E.H. Abramson, J.M. Brown and L.J. Slutsky. The thermal diffusivity of water at highpressures and temperatures.J. Chem. Phys.115, 10461 (2001).

[7] P.V. Hobbs.Ice Physics. Claredon Press, Oxford, (1974).

[8] J.S. Langer. Instabilities and pattern formation in crystal growth.Rev. Mod. Phys.52,1 (1980).

[9] W.W. Mullins and R.F. Sekerka. Morphological stability of a particle growing by dif-fusion or heat flow.J. Appl. Phys.34, 323 (1963).

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time dependent freezing of water under multiple

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