Time Dependent Freezing of Water Under Multiple Shock Wave

TIME DEPENDENT FREEZING OF WATER UNDER MULTIPLE SHOCK WAVE

COMPRESSION

DANIEL H. DOLAN III

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

Doctor of Philosophy

WASHINGTON STATE UNIVERSITY Department of Physics

MAY 2003

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 assistance 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 phenomenological transition rate to demonstrate that water can freeze on nanosecond time scales.

Single pass optical transmission measurements (ns resolution) indicated that compressed 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 onset 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 heterogeneous nucleation at water-window interfaces.

Optical imaging measurements (0.01 mm spatial resolution, 20 ns exposures) revealed 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 References for Chapter 2 ...... 41

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 3.2.1 Overall configuration ...... 71 3.2.2 Mechanical components and assembly ...... 75 3.2.3 Instrumentation and optical components ...... 77 3.2.4 Experimental setup ...... 80 3.3 Wave profile experiments ...... 88 3.3.1 Overall configuration ...... 88 3.3.2 Mechanical components and assembly ...... 88 3.3.3 Instrumentation and optical components ...... 92 3.3.4 Experimental setup ...... 92 References for Chapter 3 ...... 95

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 4.2.2 Experimental results ...... 121 4.2.3 Summary ...... 129 4.3 Wave proﬁle measurements ...... 132 4.3.1 Particle velocity determination ...... 132 4.3.2 Experimental results ...... 135

vii 4.3.3 Summary ...... 142 References for Chapter 4 ...... 144

5. Time Dependent Continuum Model for Water ...... 145 5.1 EOS development ...... 145 5.1.1 Thermodynamic consistency ...... 146

5.1.2 The form of f (T,v) for constant cv ...... 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 of cv ...... 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 References for Chapter 5 ...... 182

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 References for Chapter 6 ...... 231

7. Summary and Conclusions ...... 235 7.1 Summary ...... 235 7.2 Conclusions ...... 238 7.3 Recommendations for future work ...... 239 References for Chapter 7 ...... 240

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 with gL − 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 4.15 Images obtained in experiment I2 ...... 127 4.16 Images obtained in experiment I3 ...... 128 4.17 Photodiode transmission record for experiment I4 ...... 130 4.18 Images obtained in experiment I4 ...... 131 4.19 Raw signals of the VISAR measurement ...... 134 4.20 Particle velocity history for experiment V1 ...... 137 4.21 Particle velocity history for experiment V2 ...... 138 4.22 VISAR measurement for experiment V3 ...... 140 4.23 VISAR measurements long after compression ...... 141 4.24 Particle velocity history for experiment V4 ...... 143

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 of cv 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 the P − 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 the T − P plane ...... 175 5.12 Multiple shock compression loading path in the P − 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

Introduction A material at thermodynamic equilibrium assumes the structure or phase that is most favorable under the applied conditions [1]. At a fixed temperature T and pressure P, the stable phase is one that has the minimum specific Gibb’s free energy g(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 on 10−10 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 typical 10−6 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 approach The 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 compression, 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 compression using plate impact. This process approximates isentropic compression and results in substantially lower temperatures than those in single shock compression. With this technique, 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 mechanical 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 dependent 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 propagation code [17] for modelling the reverberation experiments.

1.2 Organization of Subsequent Chapters Chapter 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 studies 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-induced phase 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. Oxford Press, New York, (1969).

[9] G.E. Duvall, K.M. Ogilvie, R. Wilson, P.M. Bellmany and P.S.P Wei. Optical spectroscopy in a shocked liquid. Nature 296, 846 (1982).

[10] J.M. Winey. Time-resolved optical spectroscopy to examine shock-induced decomposition 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 velocities of 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. Solids 35, 501 (1974).

5 [17] Y.M. Gupta. COPS wave propagation code (Stanford Research Institute, Menlo Park, CA, 1978), unpublished.

6 Chapter 2

Background This 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 understood...” –Felix Franks [1]

Although water is among the most familiar substances on Earth, the complete description 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 water. 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 0.957 A˚ 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 structure 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 macroscopic 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 properties [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 hydrogen 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 indicated with 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 heat cp [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∗ 18.182 MΩ · 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 18.182 MΩ · cm; for comparison, the resistivity of tap water is on the order of 10−3 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 from v(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 for cP, ( 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 reproduce v(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). µ ¶ µ ¶ ∂v v2 T ∂v 2 = − 2 − (2.1) ∂P c cP ∂T µ ¶T µ ¶ P ∂c ∂ 2v P = −T (2.2) ∂ ∂ 2 P T T P

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 energy f (T,v) with a set of adjustable parameters. The model parameters can then be adjusted to optimally match published thermodynamic measurements. 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

−αx α, where I = I0e . This coefﬁcient is related to the complex dielectric function ε = ε0 + iε00 [60].   Ã µ ¶ !1/2 8π2ε0 ε00 2 α2 =  1 + − 1 (2.3) λ 2 ε0

Light absorption occurs for nonzero values of ε00. Figure 2.3 shows the functions ε0(λ) and

ε00(λ) for liquid water. In the range of 0.2-1 µm, the value of ε00 is much small than ε0, so Equation 2.3 can be simpliﬁed.

2π ε00 α ≈ √ (2.4) λ ε0

In this region, α is on the order of 10−3 − 10−4 cm−1, which means that water samples √ less than about 100 cm are effectively transparent. The refractive index of water n ≈ ε0 is about 1.33-1.34 in the visible range (λ = 400 − 700 nm) [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 1-100 µm 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 1-10 µm region.

Molecular rotations contribute to absorption at 17 µm; at 62 µm, 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 10 µm. 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 under 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 refractive index of liquid water up to 1.1 GPa and found a linear relationship with the Eulerian

strain ε. · ¸ 1 ³v ´2/3 n(v) = n(v ) − Aε where ε ≡ 1 − 0 (2.5) 0 2 v

The value of A is approximately 1.01 for λ=546.1 nm.

2.2 Solid water Water 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.

17 800 500 750 450 700 critical 400 point 650 350 600 300 550

250 C) 500 vapor o T (K)

liquid T ( 200

450 18 150 400 100 350 VII ambient 50 V 300 conditions III VI 0 250 VIII Ih II −50 200 −7 −6 −5 −4 −3 −2 −1 0 1 10 10 10 10 10 10 10 10 10 P (GPa) Figure 2.4: Equilibrium phase diagram of water [29, 72–76] 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 mass M 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 area A and surface tension σ.

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

20 liquid (mass=M-m)

solid (mass=m)

Figure 2.5: Formation of a solid nucleus in a liquid

∆ G*

0 G(r) ∆

0 Solid radius r

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

21 If the nucleus is treated as a sphere of radius r, ∆G may be written in terms of the solid size. 4π ∆G(r) = − ρ (g − g )r3 + 4πσr2 (2.7) 3 S L S A plot of ∆G(r) is shown in Figure 2.6. Competition between the r2 and r3 terms creates a maximum value of ∆G∗ when r = r∗.

2σ r∗ = (2.8) ρS (gL − gS) 16π σ 3 ∆G∗ = (2.9) ρ2 2 3 S (gL − gS) Nuclei smaller than r∗ are not thermodynamically stable since small ﬂuctuations will tend to force the system back to the pure liquid state. When an embryo larger than r∗ 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 size r∗. The equilibrium number density of these nuclei N(r∗) is deﬁned by a Boltzmann distribution related to the magnitude of ∆G∗. µ ¶ ∆G∗ N(r∗) ∝ exp − (2.10) kT

Freezing from any speciﬁc nucleus is a random event, but an average nucleation rate per unit volume J can be deﬁned for a large ensemble of metastable water samples. This rate is proportional to N(r∗). µ ¶ −∆G∗ J = J exp 0 kT µ ¶ B 16πσ 3 = J exp − where B ≡ (2.11) 0 2 ρ2 (gL − gS) 3 S kT

22 The characteristic time for a single nucleation event tn is inversely proportional to the nucleation rate and volume V of each sample in the ensemble. µ ¶ 1 B tn = exp 2 (2.12) J0V (gL − gS)

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 of gL − gS. Along the equilibrium phase boundary, gL = gS, so the nucleation rate is zero and an inﬁnite time is required 1 for solid formation. As the liquid moves further into the solid domain, gL − gS increases , 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 of tn does not go to zero as gL −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

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

23 Nucleation time

Nucleation rate

g −g L S

Figure 2.7: Variation of nucleation time and rate with gL − 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 simple 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 freezing 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 effectiveness 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 water at atmospheric pressure [89,91–93,116]. In these measurements, liquid water is held at a temperature T < 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 average time dependence. Measured freezing times in such experiments range from 102 −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 conduction 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 of 107 − 1010 K/s can be

produced by spraying water droplets (1-10 µm 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 10-30 µs 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 water. 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 transient 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 on 10−1 − 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 water The term shock wave is used here to denote a rapid, planar compression that travels 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 experiments 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 temperature 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, where h 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 of 103 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 a T − 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), until 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 1200

1100

1000

900 Rice and Walsh

800

Liquid T (K) 700

Snay and 600 Rosenbaum

500 Ice VII

400

300 VI 0 2 4 6 8 10 12 14 P (GPa)

Figure 2.8: Calculated P,T states of shocked liquid water Dark 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 20 µs 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 linear Us − 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 reflections to diffuse scattering was observed 10−7 − 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 protective 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 microseconds. Given the complexity of the problem and the extremely limited experimental measurements, 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 approach The fundamental purpose of this thesis is to examine the lower time limits of freezing 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 than 10−7 s, and that there are certain 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 on 10−9 − 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 multiple 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 compression As 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 temperatures 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 thermodynamics 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 longitudinal stress Pi and particle velocity ui. The values of Pi and ui are defined by the intersection of the impactor and front window Hugoniots, which maintains continuity of stress and particle 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 to Pi 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 dependent 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 (a) sample (b) impact/peak state midpoint v

6 5

4 s

s e

r 3 t s

l a n i 2 IM loading/ d

u FW unloading t i

impactor g front liquid rear n

(IM) o

window sample window L 1 (FW) (LS) (RW) FW/RW loading

v/2 v Particle velocity

6 peak state 6 5 5 4 t n i 3 4 o p d i e m

m 3 i

e 2 l T p

2 a s

t a

1 s

1 s e

shock arrival r t

at sample S

i i 0 impact 0 Lagrangian position Time

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 diameter D, an experimental probe diameter of d, and a edge wave velocity c (which is the sound speed in the shocked state), the maximum duration of uniaxial strain is given by tmax. D − d tmax ≈ √ (2.13) 2 c 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 irreversible. 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]. µ ¶ 1 d2P ∆s = A(v − v)3 where A ≡ (2.14) single 0 12T dv2 0 v=v0 Multiple shock compression is simply a sequence of individual shock waves, so entropy is also generated in the process. For a series of N shocks of equal volume increment, the total entropy change is equal to a sum of terms similar to Equation 2.14

N−1 µ ¶3 N−1 3 v0 − v ∆sreverb = ∑ Ai(vi − vi+1) = ∑ Ai (2.15) i=0 N i=0

38 The sum in this relation scales with the number of shocks N, so the overall entropy change scales with 1/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). µ ¶ · µ ¶ ¸ ∂T ∂v dT = ∂ ds + ∂ dP (2.16) s P T P ∂ ∂ ∂ ∂ 3 ( T/ s)P and ( v/ T)P are positive , so temperature increases with both pressure and entropy. In multiple shock loading, ds ≈ 0, so only the right term contributes to the temperature. 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 isentropic 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.

3 ∂ ∂ It is possible for ( v/ T)P to be negative in liquid water, but not at temperatures relevant to this work.

39 800

750 single shock loading 700

650

600 Liquid

550

isentropic 500 loading Temperature (K)

450

400

Ice VII 350

300 VII 0 1 2 3 4 5 6 7 8 9 10 Pressure (GPa)

Figure 2.10: Temperature advantages of multiple shock compression Dark lines indicate the equilibrium phase boundaries of water [75,178]. The Hugoniot and isentrope curves are calculated from an equation of state presented in Chapter 5.

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52 Chapter 3

Experimental Methods This chapter describes the experimental techniques and instrumentation used in the present study. Optical transmission experiments are discussed in Section 3.1. Optical imaging 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 summarized 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 impact 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] reactions, 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 attached 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 streak CCD camera spectrometer

projectile transmission photodiode target chamber impactor liquidsample port

optical collection fiber

collimating lens 45°turning mirrors 45°turning mirror windows

target chamber port

flashlamp pulsedXeflashlamp monitor photodiode

parabolicturning mirror

Figure 3.1: Setup for optical transmission measurements Collimated light from a pulsed xenon ﬂashlamp was directed through the liquid sample with a series of mirrors. Light exiting the sample was collected with an optical ﬁber and recorded. The transmission measured with this technique was constant while the water sample remains a pure liquid. The formation of ice regions created scattering and reduced light 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 photodiode 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 experiments. 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 window. 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)

aperture mirrors impactor mirror mount

impactor mount 56

projectile body

Figure 3.2: Transmission experiment projectile (a) Schematic cross section. The impactor and turning mounts are ﬁxed to the projectile with screws (not shown). (b) Completed projectile 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 (a) lockring

brassstep frontwindow machinedtocontrol samplethickness second O-ring fill first tube O-ring fill channels

rearwindow (pressfit)

filltube

(b) frontwindow

brass O-ring cell fillchannel body cutinbrass step

rearwindow Teflonpad 0.03"thick lock ring

Figure 3.3: Liquid cell assembly (a) In design # 1, the rear window is held in place with a press ﬁt, while the front window fastened with a lock ring. (b) In design # 2, the front window is epoxied to the liquid cell, and the rear window held in 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) standoff liquid ring cell

aperture target plate

lens collimating 60 bracket lenstube

optic fiber

Figure 3.4: Transmission experiment target (a) Schematic cross section. Screws (not shown) are used to mount the liquid cell and lens bracket to target plate. The target plate is attached to the standoff ring with Belville washers (not shown) for alignment purposes. (b) Completed target (cell design 1) viewed from the impact side. A rear view of the target is shown in Figure 3.5. 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 5 µm 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 0.5 µF capacitor, charged to about 11 kV, generating a peak pulse energy of 30 J. This discharge generated light over approximately 10 µs; peak output occured 4-5 µs 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 mirrors (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, 400 µm 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 400 µm 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 15 µs, 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 (3-6 µs) 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 immediately 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 (1-3 µs) 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 covering 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 horizontally 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 setup The mirror mounted to the brass rod directs light from the xenon ﬂashlamp towards the projectile, which has been removed here. Transmitted light from the sample is collected by 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 400 µm 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 v

projectile impacting surfaces h h+D h trigger impact pin pin D1

pulse generator delay D3 generator

trig D4 D2

Diagnostic Digitizer

Xe flashlamp

trig

Data flashlamp Digitizer monitor photodiode spectrometer streak CCD camera transmission photodiode

Figure 3.6: Electronic setup for transmission measurements Adjustable delays D1-D4 are set to synchronize electronic equipment. h is the impactor height above the projectile face; D is the additional height of the standoff trigger pin; v is the 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 (1-2 µs) 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, 4-5 µs 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 distance h above the projectile face, so the impact shorting pin must be h 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 transmission until a external pulse was received. This pulse would alter the EOM to allow light

67 (a) simulated short D2 pulse generator

delay D3 generator

D1 CW laser Diagnostic Digitizer EOM

photodiode monitorat impactpoint spectrometer streak CCD camera

(b) 6800

6600 linear fit

6400

6200

6000 laser pulse delay (ns)

5800

5600 0 50 100 150 200 250 track on CCD

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 is adjusted 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 transmission 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 pulse at the target and the original trigger short (considering cable delays) was equal to the necessary standoff delay (∆t). Once ∆t was measured, the standoff trigger distance was found found from D = v∆t, where v 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 pixels (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 (a)

Hg 435.83 nm

Cd Cd Hg 479.99 nm 508.58 nm Hg 546.07 nm Cd 404.66 nm 467.81 nm Intensity (arbitrary units)

0 50 100 150 200 250 300 350 400 450 500 pixel on CCD

(b) 650

600 linear fit

550

500 (nm) λ

450

400

350 0 50 100 150 200 250 300 350 400 450 500 pixel on CCD

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 measurement. 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

3.2.1 Overall configuration Figure 3.9 shows the configuration for imaging a sample during an impact experiment. 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, imaging capabilities evolved considerably through changes in the optical relay and improvements 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 framing projectile camera

impactor liquidsample

optical relay

45°turning mirrors 45°turning mirror windows

flashlamp pulsedXeflashlamp monitor photodiode

parabolicturning mirror

Figure 3.9: General setup of optical imaging experiments Object illumination is identical to the conﬁguration used in optical transmission experiments (Figure 3.1).

72 Newtonian telescope

image plane

framing camera standard Pyrex window vacuum seal

Lexan window

targetchamber

collimated visible light plane turningmirror

optical fiber

Figure 3.10: Telescope base image relay system Note 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 produce 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 capabilities 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 combination used for imaging in this study.

3.2.2 Mechanical components and assembly 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 topview

sampleimage plane planeturningmirror SLR 8way 8MCP 8CCD lenses beam- 90/10 splitter beamsplitter

photodiode

polishedBK-7or fusedsilicawindow vacuum seal vertical tube

achromat lenses polishedBK-7or fusedsilicawindow

targetchamber

rear view collimated visible plane light turningmirror

Figure 3.11: Lens based image relay system With the obscuring mirror removed, collimated light can be centered on the image relay. A picture 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.

3.2.3 Instrumentation and optical components 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 of λ/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 splitter External optics (Figure 3.11) relay the target image to the input plane of the camera (right side). 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) mounted coaxially behind the beam splitter.

78 intensity. The optical images were directed onto a S25 photocathode for maximum sensitivity 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 2 µs.

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 3.2.4 Experimental setup

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 reflected 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 relay lens unit

turning mirrors

light from xenon flaslamp

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 sample 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 target chamber.)

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 mirror directs this light towards a pellicle beam splitter, which directs 90% of the light to the framing camera and 10% to a photodiode. A pair of SLR camera lenses are mounted to the framing 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 programmed 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.

Timing synchronization 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 output. 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 alignment 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. This chart 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 a 17.5 µm spatial period.

85 plot determined the scaling of pixel distance to actual sample distance. The limiting resolution 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 17.5 µm, so the line widths are about 9 µm. This distance is the minimum resolvable separation. Objects smaller than 9 µm 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 Measurements Ideal System 1.2

0.8

0.6

0.4

0.2 Average Framing Camera Extinction (base 10)

0 0 0.2 0.4 0.6 0.8 1 Filter Extinction (base 10)

Figure 3.16: Linearity of the optical imaging system Each 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

3.3.1 Overall configuration 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 experimental 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 reflected 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 system operation is given here; Appendix F contains a thorough discussion of the conversion between measured optical phase and the velocity of the target mirror.

3.3.2 Mechanical components and assembly

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 VISAR system

YAG laser mirror 532 nm EOM

1/8 wave plate mirror

laser monitor mirror photodiode BS with pinhole etalon

dichroic mirror BS

intensity monitor photodiode projectile Data photodiodes optical fiber

liquid sample plated Al mirror

imaging lens unit

impactor

windows

Figure 3.17: General setup of VISAR measurement

89 epoxy (a) rear buffer window

water VISAR sample system

(b) rear window

water VISAR sample system

Figure 3.18: Mirror configurations for VISAR experiments All 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” diameter × 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.

3.3.3 Instrumentation and optical components

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 10 µs 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 200 µm core with a 0.16 numerical aperture.

3.3.4 Experimental setup

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 time τ. 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.

Timing synchronization 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 1-1.5 µs 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 10 µs. Data from the

photodiodes were acquired for 5-12 µs.

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 v

projectile

h+D h h+D

trigger front trigger pin#1 window pin#2 Diagnostic pulse Digitizer generator

delay generator

CWlaser EOM

laser monitor photodiode

data1 photodiode

Data data2 Digitizer photodiode

BIM photodiode

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 spectroscopy in a shocked liquid. Nature 296, 846 (1982).

[2] C.F. Bohren and D.R. Huffman. Absorption and Scattering of Light by Small Parti- cles. John Wiley & Sons, New York, (1983).

[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-induced phase 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 beyond its elastic limit. M.S. thesis, Washington State University, (1990).

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

[7] 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).

[8] NIST. Physical reference data, (2002). www.physics.nist.gov.

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

[10] L.M. Barker. The development of the VISAR, and its use in shock compression science. In Shock 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 quartz shocked to 60 kbar. J. Appl. Phys. 88, 5671 (2000).

95 96 Chapter 4

Experimental Results This 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 measurements 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 necessary conditions for water to show signs of freezing. Finally, these results are summarized, making note of any experimental trends.

4.1 Optical transmission measurements The objective of the optical transmission experiments was to determine if liquid water remained transparent under multiple shock compression. Transparency was interpreted 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 Table 4.1: Summary of optical transmission experiments

Experiment Impact Impactor Front window** Sample* Rear Window** peak state 90% Exp. Changes number velocity Material h [D] Material h [D] h Material h [D] P T ρ level limit observed (km/s) (mm) (mm) (mm) (mm) (GPa) (K) (g/cc) (ns) (ns) T1 (00-056) 0.150 SLG 18.8 SLG 1.585 0.104 SLG 18.799 1.10 344 1.23 254 3100 no [44.5] [50.8] [63.5] T2 (00-038) 0.425 FS 12.7 FS 1.589 0.128 FS 12.705 2.57 392 1.37 164 2200 yes [31.8] [38.1] [31.8] T3 (00-030) 0.602 FS 12.7 FS 1.587 0.124 FS 12.705 3.55 417 1.43 96 2200 yes [31.8] [38.1] [31.8] T4 (01-041) 0.396 Q 12.7 Q 3.184 0.028 Q 12.720 3.47 422 1.43 34 1500 yes [25.4] [38.1] [31.8] T5 (00-051) 0.559 Q 12.7 Q 3.183 0.108 Q 12.720 5.00 463 1.51 111 2200 yes [31.8] [38.1] [31.8] T6 (01-013) 0.566 Q 12.7 Q 3.184 0.015 Q 12.718 5.06 464 1.51 16 2200 yes

[31.8] [38.1] [31.8] 98 T7 (01-MF02) 0.221 S 12.7 S 3.174 0.104 S 12.694 4.98 448 1.51 261 900 no [25.4] [31.8] [25.4] T8 (01-MF10) 0.223 S 12.7 S 3.175 0.015 S 12.687 5.03 448 1.52 37 900 no [25.4] [31.8] [25.4] T9 (01-033) 0.393 Q 12.7 Q 3.162 0.100 S 12.691 5.03 455 1.51 151 900 yes [25.4] [38.1] [25.4] T10 (01-010) 0.399 S 12.7 S 3.194 0.099 Q 12.723 5.11 458 1.52 147 900 yes [25.4] [31.8] [25.4] T11 (00-057) 0.445 S 12.7 S 3.170 0.108 S 12.691 10.15 528 1.71 153 900 ? [25.4] [31.8] [25.4] T12 (00-050) 0.448 S 12.7 S 3.170 0.110 S 12.691 10.22 529 1.71 154 900 ? [25.4] [31.8] [25.4] h= thickness D= diameter SLG= soda lime glass FS=fused silica Q=z-cut quartz S=a-cut sapphire * 0.005 mm uncertainty (0.010 for T1) in thickness ** 0.002 mm uncertainty in thickness Diameters are nominal values (Appendix B), and the water model discussed in Chapter 5. The peak pressure is determined 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 transmission 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 optical 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 dependent. This dependence was a combination of changes occurring in the sample and systematic 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 x

scattered light incident transmitted light light detector

reflected light

Figure 4.1: Transmission losses in a non-absorbing sample Transparency is reduced by scattering regions within the sample. Reﬂection also occurs at all optical index interfaces.

projectile reference rotation signal

shock Photodiode output wave impact arrives signal at water sample

Time

Figure 4.2: Measured photodiode outputs from experiment T6

100 samples. I − I T = impact bg (4.1) Ire f − Ibg

Iimpact is the measured light intensity during the impact experiment, Ire f is the measured light intensity at ambient conditions, and Ibg is the background intensity level of the detector. 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 that T = 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 intensity Imin and the minimum transmission value.

Imin Tmin = (4.2) Ire f − Ibg

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 impact experiments. The limitations of those measurements are described in the discussion of experiment T5. Smoothing in those measurements was performed by convolving the measured 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 collected 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, time t = 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 1

T1 0.9

0.8

0.7

0.6

0.5 Transmission 0.4 T2

0.3

0.2 T3

0.1

0 0 200 400 600 800 1000 1200 1400 1600 1800 Time (ns)

Figure 4.3: Photodiode transmission for experiments T1, T2, and T3 Experiment T1 used soda lime glass windows to a peak pressure of 1.1 GPa. T2 and T3 were 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 compressed liquid water enters the ice VII region somewhere above 2 GPa. Under such conditions, 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 isentropic 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 experiment 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 128 µm 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 124 µm, 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 water 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 performed 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 stability, about 26 K below the liquid-ice VII phase boundary. The initial sample thickness was 28 µm 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 directly 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.

Experiments T5 and T6 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 108 µm thick, and 90% of the peak pressure was reached in 111 ns. Initial sample thickness in T6 was 15

105 1

0.9

0.8

0.7

0.6

0.5 Transmisison 0.4

0.3

0.2

0.1

0 0 200 400 600 800 1000 1200 1400 1600 1800 Time (ns)

Figure 4.4: Photodiode transmission for experiments T4 Quartz 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 near T = 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 intensity Imin 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 = 1000 ns, 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 1

0.9

0.8

0.7

0.6

0.5 Transmission 0.4

0.3

0.2 T6

0.1 T5

0 0 200 400 600 800 1000 1200 1400 1600 1800 Time (ns)

Figure 4.5: Photodiode transmission records for experiments T5 and T6 Quartz windows were used to obtain a 5 GPa peak stress in both experiments. Initial sample thickness in T5 was 108 µm; in T6 it was 15 µm.

108 (a) 1 0.9 Each plot separated by 50 nm 0.8 0.7 0.6 0.5 0.4 600 nm Transmission 0.3 0.2 400 nm 0.1 0 0 200 400 600 800 1000 1200 1400 1600 1800 Time (ns) (b)

1 t=0 0.9 0.8 0.7 t=200 ns 0.6 0.5 0.4 t=400 ns 0.3 Transmission 0.2 t=1000 ns 0.1

0 T at 1000 ns min 400 425 450 475 500 525 550 575 600 Wavelength (nm)

Figure 4.6: Spectrally resolved transmission for experiment T5 Quartz windows were used to compress the 108 µm 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 Experiments T7 and T8 Since a signiﬁcant decrease in optical transmission was observed for water compressed 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 104 µm, and 90% peak

pressure was attained in 261 ns. Initial sample thickness in T7 was 15 µm, 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 1 T7 T8 0.9

0.8

0.7

0.6

0.5 Transmission 0.4

0.3

T5 0.2

0.1

0 0 100 200 300 400 500 600 700 800 Time (ns)

Figure 4.7: Photodiode transmission records for experiments T7 and T8 Sapphire windows were used in experiments T7 and T8 to compress water to a peak pressure of 5 GPa. The sample thicknesses for T7 and T8 were 104 and 15 µm, respectively. Experiment T5 (quartz windows, 108 µm sample) is shown for comparison.

111 Experiments T9 and T10 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 individual 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 window. 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 (≈100 µm) 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 5.5

4.5

4 hybrid cell (T10)

3.5

2.5 Pressure (GPa) 2 hybrid cell (T9) sapphire cell 1.5 (T7) quartz cell 1 (T5)

0.5

0 0 100 200 300 400 500 600 700 800 Time (ns) from impact

Figure 4.8: Loading history in quartz, sapphire, and hybrid liquid cells These curves show the pressure history at the midpoint of the water sample.

113 1

0.9

0.8 T9 0.7

0.6 T10

0.5 Transmission 0.4 T5

0.3

0.2

0.1

0 0 100 200 300 400 500 600 700 800 Time (ns)

Figure 4.9: Photodiode transmission records for experiments T9 and T10 Experiment T9 used a quartz front window and a sapphire rear window; in experiment T10, the order was reversed. Peak pressures in both experiments was 5 GPa. Initial water sample thicknesses were about 100 µm. Experiment T5 (quartz windows, 108 µm 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.

Experiments T11 and T12 Once window effects were established at 5 GPa, additional experiments were carried 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 using 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 (108-110 µm), 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 calculations 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 1.1

0.9 T12

0.8

0.7 T11

0.6

0.5 Transmission

0.4

0.3

0.2

0.1

0 0 100 200 300 400 500 600 700 800 Time (ns)

Figure 4.10: Photodiode transmission records for experiments T11 and T12 These 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 compression 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 dependent 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 measurements The purpose of the optical imaging experiments was to further examine the transparency 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 Table 4.2: Summary of optical imaging experiments

Experiment Impact Impactor Front window** Sample* Rear Window** peak state 90% Exp. Changes number velocity Material h [D] Material h [D] h Material h [D] P T ρ level limit observed (km/s) (mm) (mm) (mm) (mm) (GPa) (K) (g/cc) (ns) (ns) 12.7 3.202 12.722 I1 (02-016) 0.314 Q Q 0.137 Q 2.72 400 1.38 217 2200 yes [31.8] [38.1] [31.8] 12.7 3.203 12.727 I2 (02-017) 0.410 Q Q 0.132 Q 3.60 425 1.43 162 2200 yes [31.8] [38.1] [31.8] 118 12.7 3.200 12.727 I3 (02-009) 0.566 Q Q 0.109 Q 5.06 465 1.51 114 2200 yes [31.8] [31.8] [31.8] 12.7 3.188 12.709 I4 (02-MF02) 0.219 S S 0.108 S 4.94 448 1.51 273 900 no [25.4] [31.8] [25.4] h= thickness D= diameter Q=z-cut quartz S=a-cut sapphire * 0.005 mm uncertainty in thickness ** 0.002 mm uncertainty in thickness Diameters are nominal values 4.2.1 Determining sample transmission Spatial resolution and exposure time are two key differences between the transmission 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 profile T(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.

Iimpact − hIbgii Ti = (4.3) hIre f ii − hIbgii

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 that Tmin =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 Reference image

350

300

250

200

150

100

Impact image

350

300

250

200

150

100

Figure 4.11: Raw images from experiment I1 The top image was taken at ambient conditions, while the bottom image was acquired during 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 images Ti(x,y)

were nonlinearly mapped to gray scale values Gi(x,y) using the following function. · µ ¶¸ 1 T (x,y) − T G (x,y) = 1 + tanh i m (4.4) i 2 L

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, both Tm and L were equal to 0.5. The white level of the displayed images was then set at G = 1; the black level was set at G = 0.12, which corresponds to a completely opaque region (i.e. T = 0).

4.2.2 Experimental results 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 137 µm, and 90% peak pressure was reached in 217 ns.

Figure 4.12 shows the transmission history of the water sample measured with a

121 photodiode 1 Imacon 200

0.9

0.8

0.7

0.6

0.5 Transmission 0.4

0.3

0.2

0.1

0 0 200 400 600 800 1000 1200 1400 1600 1800 Time (ns)

Figure 4.12: Measured transmission for experiment I1 Quartz windows were used in this experiment to generate a peak pressure of 2.7 GPa.

122 100 µm 123

Figure 4.13: Images obtained in experiment I1 Quartz windows were used to generate a peak pressure of 2.7 GPa. Initial sample thickness was 137 µm; 90% peak pressure was attained in 217 ns. The image sequence progress from left to right, starting from the upper left corner. The beginning of each image exposure was at -170, 230, 330, 530, 630, 730, 830, 930, 1030, 1130, 1330, and 1430 ns. Each exposure lasted for 25 ns. photodiode detector and the Imacon 200 framing camera. These two measurements are relatively 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 8.5 µm (11 camera pixels at ∼0.77 µm 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 > 330 ns) was largely perpendicular to the fast growth process, widening previously frozen regions. The widening process was much slower than the initial directional growth and continued throughout the experiment. On average, the darkened

regions are ≤ 100 µm in width and several hundred microns long. At the end of the image sequence, numerous liquid regions, 100 µm 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 132 µm thick, and 90% peak pressure was attained within 162 ns. The optical transmission records (photodiode and Imacon 200 results) for this experiment 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 8.3 µm (11 pixels at ∼0.75 µm sample distance/pixel). Freezing occurred 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 features; in this case, the overlap began to occur while new regions were still forming (as seen at times t = 225 and t = 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 ≤ 100 µm in width, with most features ≤ 50 µm, and only about 100 µm 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 109 µm, 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 8.8 µm (11 pixels at ∼0.80 µm sample distance/pixel).

Under these conditions, freezing appeared so rapidly that it is not possible to distinguish independent transition regions. In less than 200 ns, numerous darkened regions were formed,

125 photodiode 1 Imacon 200

0.9

0.8

0.7

0.6

0.5 Transmission 0.4

0.3

0.2

0.1

0 0 200 400 600 800 1000 1200 1400 1600 1800 Time (ns)

Figure 4.14: Measured transmission for experiment I2 Quartz windows were used in this experiment to generate a peak pressure of 2.7 GPa.

126 100 µm 127

Figure 4.15: Images obtained in experiment I2 Quartz windows were used to generate a peak pressure of 3.5 GPa. Initial sample thickness was 132 µm; 90% peak pressure was attained in 162 ns . The image sequence progress from left to right, starting from the upper left corner. The beginning of each image exposure was at 25, 75, 125, 175, 225, 275, 325, 375, 425, 475, 575, and 675 ns. Each exposure lasted for 25 ns. 100 µm 128

Figure 4.16: Images obtained in experiment I3 Quartz windows were used to a generate peak pressure of 5 GPa. Initial sample thickness was 109 µm; 90% peak pressure was reached in 114 ns. The image sequence progress from left to right, starting from the upper left corner. The beginning of each image exposure was at 55, 95, 135, 175, 285, 325, 365, 405, 445, 485, 525, and 565 ns. Each exposure lasted for 20 ns. 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 50-100 µm wide; transition regions that did not merge are conﬁned to widths less than 25 µm. The residual liquid areas were 50 µm 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 108 µm, and 90% of the peak state was reached in 273 ns. The optical transmission records (photodiode and Imacon 200 results) for this experiment 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 28 µm

(7 pixels at ∼3.94 µm sample distance/pixel). A few dark spots were observed in the images, 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 window 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 1

0.9 T7 0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1 photodiode Imacon 200 0 0 100 200 300 400 500 600 700 800

Figure 4.17: Photodiode transmission record for experiment I4 Sapphire 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 mm 131

Figure 4.18: Images obtained in experiment I4 Sapphire windows were used to generate a peak pressure of 5 GPa. Initial sample thickness was 108 µm; 90% peak pressure is attained in 273 ns. The image sequence progress from left to right, starting from the upper left corner. The beginning of each image exposure was at -85, 115, 415, 515, 565, 615, 665, 715, 765, 815, 865, and 915 ns. Each exposure lasted for 20 ns. 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 measurements The purpose of the wave profile measurements was to determine if optical transparency 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 Table 4.3: Summary of wave proﬁle experiments

Experiment Impact Impactor Front window** Sample* Buffer Rear Window** peak state Exp. Changes number velocity Material h [D] Material h [D] h [Epoxy] Material h [D] P T ρ limit observed (km/s) (mm) (mm) (mm) (mm) (mm) (GPa) (K) (g/cc) (ns) 12.7 3.202 0.531 12.723 V1 (01-538) 0.567 Q Q 0.103 Q 5.05 465 1.51 2200 yes [25.4] [38.1] [0.003] [31.8]

133 12.7 3.200 - 3.183 V2 (01-045) 0.563 Q Q 0.103 Q 5.03 464 1.51 950 yes [25.4] [31.8] - [38.1] 12.7 3.200 - 12.720 V3 (02-002) 0.568 Q Q 0.104 Q 5.08 465 1.51 2200 yes [31.8] [38.1] - [31.8] 12.7 3.188 - 12.705 V4 (02-MF02) 0.220 S S 0.100 S 4.96 448 1.51 900 no [25.4] [31.8] - [25.4] h= thickness D= diameter Q=z-cut quartz S=a-cut sapphire * 0.005 mm uncertainty in thickness ** 0.002 mm uncertainty in thickness Diameters are nominal values (a)

y polarization

x polarization

shock arrives at mirror Photodiode output

intensity monitor

Time (b) mirror at rest

optical phase constant mirror

y polarization motion

x 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].

4.3.2 Experimental results For consistency with the optical measurements, time in the VISAR signal was defined 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 100-104 µm. Since VISAR measurements using a liquid sample had not previously 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 531 µm thick piece of z-cut quartz. The epoxy bond between the buffer and the rear window was 2-3 µm thick. The VISAR system

135 was conﬁgured with a 8.0175” BK-7 etalon to obtain a delay of τ = 1.255 ns and a fringe constant F = 205 m/s (considering actual interferometer leg lengths). The particle velocity record for this experiment is shown in Figure 4.20. The measured 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 resulted 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 of τ = 1.255 ns and a fringe constant F = 205 m/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

136 325

300 liquid calculation 275

250 VISAR data

225 reflections 200 from epoxy layer 175

150

125 Particle velocity (m/s)

100

0 0 100 200 300 400 500 600 700 800 Time (ns)

Figure 4.20: Particle velocity history for experiment V1 (quartz windows)

137 325 4−5% lower than 300 liquid calculation calculation 275 VISAR 250 data free 225 surface releases 200 manual correction 175

150

125 Particle velocity (m/s)

100

0 0 100 200 300 400 500 600 700 800 Time (ns)

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 experimental 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 of τ = 1.251 ns and a fringe constant F = 205 m/s (considering actual interferometer leg lengths). Figure 4.22 shows the particle velocity history for this experiment. As in experiment V2, the measured and calculated particle velocities were consistent for several loading 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

139 325

300 liquid calculation

275

VISAR data 250

225 7−9% lower than 200 calculation

175 (m/s) p u 150

125

100

0 0 100 200 300 400 500 600 700 800 t (ns) from shock arrival

Figure 4.22: VISAR measurement for experiment V3 (quartz windows)

140 300 calculated steady state

250

V1 free surface 200 V3 releases

150 Particle velocity (m/s)

100

0 0 250 500 750 1000 1250 1500 1750 2000 Time (ns)

Figure 4.23: VISAR measurements long after compression

141 higher impedance of the sapphire windows and impactor. This led to lower particle velocities in the water sample. To compensate, the VISAR system used a 12.0199” BK-7 etalon,

yielding a time delay of τ=1.818 ns and a fringe constant F = 141 m/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 generated 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.

142 110 VISAR data 100

70 Liquid calculation 60 (m/s) p u 50

0 0 100 200 300 400 500 600 700 800 t (ns) from shock arrival

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, and sapphire. J. Appl. Phys. 41, 4208 (1970).

[3] R.L. Gustavsen. Time resolved reﬂection spectroscopy on shock compressed liquid carbon disulﬁde. Ph.D. thesis, Washington State Univesity, (1989).

[4] M.D. Knudson. Picosecond electronic spectroscopy to understand the shock-induced phase 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 under elastic, uniaxial strain compression along the a axis. J. Appl. Phys. 90, 4990 (2001).

144 Chapter 5

Time Dependent Continuum Model for Water This 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 thermodynamic 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 freezing 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 development The thermodynamic behavior of a pure fluid is completely specified by the specific Helmholtz free energy f (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

that cv is constant for the conditions of interest. The consequences of this assumption and

145 the development of an expression for f (T,v) are discussed in this section.

5.1.1 Thermodynamic consistency

cv is deﬁned as the isochoric temperature derivative of entropy: µ ¶ ∂s cv ≡ T ∂ (5.1) T v

The pressure-temperature coefﬁcient b is deﬁned as the isochoric temperature derivative of pressure: µ ¶ ∂P b ≡ ∂ (5.2) T v These two quantities are related through thermodynamic consistency: µ ¶ µ ¶ ∂b 1 ∂cv ∂ = ∂ (5.3) T v T v T ∂ ∂ For the case where cv is constant, ( b/ T)v = 0. Thus, b must be a function of volume only. Using this result, the pressure P(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 energy e(T,v) and entropy s(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. µ ¶ µ ¶ ∂e ∂e ∂ = cv ∂ = Tb(v) − P(T,v) T v v T Z v £ 0 0 ¤ 0 e(T,v) = e0 + T0 b(v ) − P(T0,v ) dv + cv(T − T0) (5.6) v0

146 s(T,v) can be determined using the same approach. µ ¶ µ ¶ ∂s cv ∂s ∂ = ∂ = b(v) T v T v T Z v 0 0 s(T,v) = s0 + b(v )dv + cv lnT/T0 (5.7) v0

Combining Equations 5.5, 5.6, and 5.7, f (T,v) can be written in the following form. · ¸ Z v Z v 0 0 0 0 T f (T,v) = e0 − Ts0 − (T − T0) b(v )dv − P(T0,v )dv + cv (T − T0) − T ln v0 v0 T0 (5.8)

In addition to cv, several other quantities are necessary to specify f (T,v). The values of e0

and s0 must be known for a reference state (T0,v0). An isotherm P(T0,v) passing through this state is also required. Finally, the function b(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 define cv(T,v), which was then used to constrain b(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 of cv is a good approximation to describe the behavior of liquid water. This approach, along with the assumption that b 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 that b can be treated as a constant [6]. Gurtman et al. [4] developed a constant cv EOS for liquid water while maintaining the volume dependence of b(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 of cv 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.

K h³v ´g i P(T ,v) = P + 0 − 1 (5.9) 0 0 g v

K is the ambient isothermal bulk modulus and g is the rate at which kT increases with pressure. 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 of K to be 2.210 GPa [12], leaving g as the only adjustable parameter. Figure 5.1 shows the best fit to the data using an optimal value of g = 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.

◦ ◦ ∗ c0(25 ) ∗ c0(25 ) up = up ◦ US = US ◦ (5.10) c0(20 ) c0(20 )

149 1 data fit 0.9

0.8

0.7

0.6

0.5

Pressure (GPa) 0.4

0.3

0.2

0.1

0 0.8 0.825 0.85 0.875 0.9 0.925 0.95 0.975 1 Specific volume (cc/g)

Figure 5.1: 25◦ C isotherm for liquid water Data shown here are from Grindley and Lind [13]. The ﬁt is a Murnaghan equation (Equa- tion 5.9) with K = 2.210 GPa and g = 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 nonlinear US − up relation [20]. To account for this behavior, the temperature corrected P − v data were ﬁtted using a polynomial in terms of the density

compression (µ ≡ ρ/ρ0 − 1).

2 3 P = P0 + a1µ + a2µ + a3µ (5.11)

To ensure that the shock speed matches the acoustic sound speed c0=1.496687 km/s [24] for 2 P → P0, a1 must be equal to c0/v0 = 2.233 GPa. The parameters a2 and a3 were optimized to match the corrected Hugoniot data. The best ﬁt to the data shown in Figure 5.2 was obtained for a2 =6.976 GPa and a3=21.15 GPa.

Determination of b(v)

b(v) can be determined from the Hugoniot PH(v) and the isotherm by inverting Equation 5.4 [6]. P (v) − P(T ,v) b(v) = H 0 (5.12) TH(v) − T0

This expression requires knowledge of temperature along the Hugoniot TH(v); hence, both

b and TH(v) must be determined simultaneously. An iterative calculation was performed

to determine b(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. µ ¶ N+1 v i−1 b(v) = ∑ Bi − 1 (5.13) i=1 v0

The best ﬁt parameters for this function are shown in Table 5.1.

151 16 data fit

8 Pressure (GPa) 6

0 0.6 0.7 0.8 0.9 1 Specific volume (cc/g)

Figure 5.2: P − v Hugoniot for liquid water Corrected data from Walsh and Rice [19] were used to ﬁt Equation 5.11.

152 −3 x 10 6

5.5

4.5

3.5

b(v) (GPa/K) 2.5

1.5

0.5 Iterative calculation Fit 0 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Specific volume (cc/g)

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) = −1.17590 B(2) = −1.95275 × 10−2 B(6) = −1.74669 B(3) = −1.54322 × 10−2 B(7) = −8.65802 × 10−1 B(4) = −2.35540 × 10−1

153 A maximum was observed in b(v) near v = 0.675 cc/g. A similar maximum was observed at v ≈ 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 in b(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 of cv In Section 2.5.2, it was proposed that freezing may be thermodynamically possible under isentropic loading. The P − 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 additional cv values were considered using the approach outlined in Section 5.2.2. The lowest possible temperatures were estimated by

setting cv to 4 J/g·K; the highest possible temperatures were determined by setting cv equal

to 2 J/g·K. Isentropic loading paths were calculated for all three cv 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 equation. µ ¶ Áµ ¶ dT ∂v ∂s = ∂ ∂ (5.14) dP T P T P

The partial derivatives in this expression were evaluated from the three different liquid water 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 (a) 700

600

2 J/g⋅K

500 Liquid 3 J/g⋅K T (K)

4 J/g⋅K

400 Ice VII

VI 300 0 1 2 3 4 5 6 7 8 9 10 (b) Pressure (GPa) 700 2 J/g⋅K 3 J/g⋅K 4 J/g⋅K

600

500 Liquid T (K)

400 Ice VII

VI 300 0 1 2 3 4 5 6 7 8 9 10 Pressure (GPa)

Figure 5.4: The value of cv and adiabatic freezing Dark lines indicate equilibrium phase boundaries [26]. (a) Isentropic loading states (b) Single shock loading states

155 above 2 GPa. Depending on the cv 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 under isentropic compression. The principle ambiguity created by the constant cv assumption is in determining the exact pressure needed to enter the ice VII phase. For single shock compression, temperature was determined by integrating the following equation [27]. Á dT dT dP = (5.15) dP dv dv · ¸ dT 1 dP bT where = P − P0 + (v0 − v) − dv 2cv dv cv b dP (∂P/∂v) + (P − P0) and = S 2cv dv 1 − b (v − v) 2cv 0 The results of these calculations are shown in Figure 5.4(b). Unlike isentropic loading, single 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 thermodynamically 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 of cv was chosen to be 2 J/g·K, which is similar to that of normal ice [10]. The reference state was chosen to be T = 300 K and P=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 were K=25.04 GPa, g=3.660, and v0=0.6795 cc/g.

To complete the EOS for solid water, the values of b, s0, and e0 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 20 Hemley data Fei data 18 fit

10 Pressure (GPa)

2 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.62 Specific volume (cc/g)

Figure 5.5: 300 K isotherm for ice VII Data were taken from Hemley et al. [30] and Fei et al. [31]. The ﬁt is a Murnaghan equation (Equation 5.9) with K=25.04 GPa, g=3.660, and v0=0.6795 cc/g.

158 700 Data Calculation

650

600

550

500 Temperature (K)

450

400

350 2 3 4 5 6 7 8 9 10 Pressure (GPa)

Figure 5.6: Liquid-ice VII coexistence curve Data were taken from Datchi et al. [41].

159 Consistency with the liquid-ice VII phase boundary Along the phase boundary T(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 of b, s0, and e0. Values for b and s0 were chosen such that T(P) optimally ﬁtted the experimental data shown in Figure 5.6. The value of e0 was determined by requiring the liquid and solid Gibbs free energies to be equal at the liquid-ice VI-ice VII triple point, located at T ∗ = 354.8 K and P∗ = 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) v − v = L (5.16) dP s − sL

Here vL and sL refer to the liquid phase, while v and s refer to the solid phase. This curve was optimized using the measured phase boundary shown in Figure 5.6. The best ﬁt values

−3 for this curve were b = 6.829 × 10 GPa/K, s0 = −1.514 J/g·K, and e0 = −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 parameter w, which was deﬁned to be the local mass fraction of the liquid phase. With this deﬁnition, w = 1 for a pure liquid and w = 0 in a pure solid. The thermodynamic properties 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 denoted 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 pressure P.

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 of T, P, and w; 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 system g exceeds the equilibrium energy gmin for the current pressure and temperature conditions, driving the system towards its equilibrium configuration. The rate at which w changes should be related to the magnitude of g − gmin. The following phenomenological expression [42,43] can be used to meet these conditions.

dw M g − g = − min (5.21) dt RT τ

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, and τ 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.   dw 0 t ≤ tI = M g − g (5.22) dt  − min t > t RT τ I tI represents the incubation time for the transformation. As discussed in Chapter 6, tI is related to the P,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 of tI.

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 response. 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 minimize g (tI and τ1 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 curve T(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.

dT v − v = 1 2 (5.24) dP s1 − s2 dw w ds1 + (1 − w) ds2 = − dP dP (5.25) dP s1 − s2 µ ¶ µ ¶ dsi ∂si ∂si dT where = ∂ + ∂ dP P T T P dP

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 the T − 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, where P ≤ 10 GPa. To understand why complete freezing is so difﬁcult under isentropic compression,

163 (a) 800

Equilibrium 700 mixture Liquid 600 metastable liquid 500 Temperature (K) 400 Ice VII phase boundary 300 0 2 4 6 8 10 12 14 (b) Pressure (GPa)

1 metastable liquid 0.8

0.6 Equilibrium mixture 0.4

Liquid mass fraction 0.2

0 0 2 4 6 8 10 12 14 Pressure (GPa)

Figure 5.7: Limits of isentropic compression (a) T − P plane (b) w − P plane

164 10

isentropic 8 liquid

isentropic 7 mixture

Liquid 6

∆ ≈ Pressure (GPa) 5 v 2%

Ice VII Liquid−solid 4 mixture

3 isotherm

2 Ice VI and mixtures

0.55 0.6 0.65 0.7 0.75 specific volume (cc/g)

Figure 5.8: Mixed phase compression in the P − v plane

165 consider the equilibrium mixture P − 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 the T(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 the P − v plane. µ ¶ µ ¶ 0 ∂P ∂P > b2T = − (5.26) ∂ ∂ v s v T cv 2 b and cv in this expression represent properties of the mixed phase. The quantity b T/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 calculations The 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 step dt 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 computational water cells. During the time dt, a given water cell undergoes a volume change dv.

The new speciﬁc volume of that cell is then simply v = v0 + dv. During this compression, the liquid mass fraction of the cell changes at a rate dw/dt, deﬁned by the transition rate in

Equation 5.22. This rate is treated as a constant during the time step.

dw w = w + dt (5.27) 0 dt

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 pressure P and the viscous stress q.

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 of P is tied to the value of v, but also depends upon the cell temperature T and the mass fraction w.

5.5.2 Enforcing the mixture rules The values of P and T 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 of v, w, and e are known from the above discussion; Pi and ei are known in terms of T and vi from the pure phase models. Equations 5.29-5.31 can thus be solved for the remaining quantities: T, v1, and v2. The following vector notation is convenient for describing the mixed phase state.      v1   z1          x ≡  v2  z ≡  z2  (5.32)     T z3

For the proper choice of x, z = 0. The value of z may be described by a Taylor series

168 expansion about a point x0.

z = z0 + A dx (5.33)

 ¯ ³ ´ ³ ´ ¯ ∂P1 ∂P2 − b1 − b2 ¯  ∂v1 ∂v2 ¯  T T ¯  ¯ where A ≡  w 1 − w 0 ¯ (5.34)  ¯ ¯ w(Tb1 − P1(T,v1)) (1 − w)(Tb2 − P2(T,v2)) wcv1 + (1 − w)cv2 ¯ x=x0 Suppose for the moment that the mixture rules are satisﬁed at x. The necessary step dx to move from the original point x0 to x can be found from Equation 5.33.

−1 dx = −A z0 (5.35)

A−1 is the matrix inverse of A, 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 guess x0, an improved solution x = x0 + dx is determined. This solution is then used as a new guess and the procedure repeated until the value of x converges. The value of x deﬁnes v1, v2, and T as well as P, which is equal to P1(T,v1) and P2(T,v2).

All other quantities are deﬁned from lever rules (e.g. g = wg1(T,v1) + (1 − w)g2(T,v2)). The values of e and P 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 speed cF was used for the stability analysis [47, 48]. This speed is deﬁned using the usual isentropic derivative at constant mass fraction. µ ¶ µ ¶ ∂ ∂ 2 P 2 P cF ≡ ∂ρ = −v ∂ (5.36) s,w v s,w

169 The derivative was determined using a technique developed by Johnson et al. [55]. The deﬁnition of sound speed can be rewritten by considering P = P(e,v). " µ ¶ µ ¶ # ∂ ∂ 2 2 P P cF = v P ∂ − ∂ (5.37) e v,w v e,w

Since the two phases share common pressure, P can be deﬁned in terms of (T,v1). " µ ¶ µ ¶ µ ¶ # c2 ∂T ∂P ∂v F = P b + 1 1 2 1 ∂ ∂ ∂ v e v,w v1 T e v,w " µ ¶ µ ¶ µ ¶ # ∂T ∂P1 ∂v1 − b1 ∂ + ∂ ∂ (5.38) v e,w v1 T v e,w

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 iterations have been performed, the value of A−1 converges to a constant. The total volume derivative of dx can then be written in terms of the matrix elements of A−1 using Equation 5.35.   µ ¶ µ ¶  0  ∂ ∂   x −1 z0 −1   ∂ = −A ∂ = A  1  v e,w v e,w   0

    ∂ ∂ −1  ( v1/ v)e,w   A12         −1   (∂v2/∂v)  =  A  (5.39)  e,w   22  ∂ ∂ −1 ( T/ v)e,w A32

The total energy derivatives of dx can be found with the same method.     ∂ ∂ −1  ( v1/ e)v,w   A13         −1   (∂v2/∂e)  =  A  (5.40)  v,w   23  ∂ ∂ −1 ( T/ e)v,w A33

170 The frozen sound speed is therefore related to liquid phase variables and matrix elements of A−1. · µ µ ¶ ¶ µ µ ¶ ¶¸ ∂P ∂P 2 = 2 −1 + 1 −1 − −1 + 1 −1 cF v P b1A33 ∂ A13 b1A32 ∂ A12 (5.41) v1 T v1 T

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 compression states in the T − 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 100 µm 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 time tI was adjusted to postpone the transition until the end of compression; the

171 700

650 freezing not observed

600

550

single shock

500 isentrope

Temperature (K) 450

Liquid Ice VII

400

350 glass cell fused silica cell quartz cell Ice sapphire cell 300 VI 0 1 2 3 4 5 6 7 8 9 10 Pressure (GPa)

Figure 5.9: Multiple shock compression temperatures for liquid water in various liquid cells

172 1.5 5

1.4 4

1.3 3

1.2 2 Density (g/cc) Pressure (GPa) 1.1 1

1 0 0 100 200 300 400 500 0 100 200 300 400 500 Time (ns) Time (ns)

550 1 500 0.8 450 0.6 400 0.4 350 Temperature (K)

300 Liquid mass fraction 0.2

250 0 0 100 200 300 400 500 0 100 200 300 400 500 Time (ns) Time (ns)

Figure 5.10: Simulated thermodynamic histories of water under multiple shock compression Freezing is marked by the vertical dashed line.

173 transition time τ 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 the T −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 a few microns.

174 550

525

500 equilibrium isentrope Liquid 475

450

425 2nd shock metastable isentrope 400 Temperature (K) simulation

375 1st Ice VII shock

350 phase boundary 325

300 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Pressure (GPa)

Figure 5.11: Multiple shock loading path in the T − P plane Dots 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 the P − 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 the P−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 and τ was set to 50 ns. The equilibrium mixture profile was obtained by setting both time scales to zero. The

176 5.5

metastable isentrope 5

4.5 equilibrium Liquid isentrope

4 simulation

3.5 Pressure (GPa)

Ice 3 VII 2nd shock mixed 2.5 phase region

2 0.6 0.625 0.65 0.675 0.7 0.725 0.75 Specific volume (cc/g)

Figure 5.12: Multiple shock compression path in the P − v plane Dots show the steady shock states between shocks.

177 300 equilibrium mixture

delayed freezing 250

200

150 Particle velocity (m/s) 100

0 0 100 200 300 400 500 Time (ns)

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 the P −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 and τ. 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 of τ; the immediate transition profile was obtained by setting tI 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 transition 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 velocities between shock compressions. The particle velocity history of the immediate transition monotonically increased with time towards the peak state value. The delayed freezing profiles 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 parameter τ. Small values of τ led to faster particle velocity drops; the magnitude of these drops was largest for small values of τ. It was determined that the drops could only be observed if freezing was delayed until compression was complete. This feature did not

179 300 Immediate freezing Delayed freezing τ=50 ns τ=10 ns

275

Delayed freezing τ=50 ns 250

225 Particle velocity (m/s) 200

first particle velocity jump 175

150 50 100 150 200 250 300 350 400 Time (ns)

Figure 5.14: Incubation and transition time effects in mixed phase water simulations

180 occur in an immediate transition regardless of the value of τ. The particle velocity differences between an immediate and a delayed phase transition 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 sample 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.

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[30] R.J. Hemley, A.P. Jephcoat, H.K. Mao, C.S. Zha, L.W. Finger and D.E. Cox. Static compression of H2O-ice to 128 GPa (1.28 MPa). Nature 330, 737 (1987).

183 [31] Y. Fei, H. Mao and R.J. Hemley. Thermal expansivity, bulk modulus, and melting curve of H2O ice VII to 20 Gpa. J. Chem. Phys. 99, 5369 (1993). [32] B. Olinger and P.M. Halleck. Compression and bonding of ice VII and an empirical linear expression for the isothermal compression of solids. J. Chem. Phys. 62, 94 (1974). [33] L. Liu. Compression of ice VII to 500 kbar. Earth Planet. Sci. Lett. 61, 359 (1982). [34] R.G. Munro, S. Block, F.A. Mauer and G. Piermarini. Isothermal equations of state for H2O-VII and D2O-VII. J. Appl. Phys. 53, 6174 (1982). [35] G.E. Walrafen, M. Abebe, F.A. Mauer, S. Block, G.J. Piermarini and R. Munro. Ra- man and x-ray investigations of ice VII to 36.0 GPa. J. Chem. Phys. 77, 2166 (1982). [36] C.W. Pistorius, M.C. Pistorius, J.P. Blakey and L.J. Admiraal. Melting curve of ice VII to 200 kbar. J. Chem. Phys. 38, 600 (1963). [37] O. Mishima and S. Endo. Melting curve of ice VII. J. Chem. Phys. 68, 4417 (1978). [38] S.L. Wunder and P.E. Schoen. Pressure measurement at high temperatures in the diamond anvil cell. J. Appl. Phys. 52, 3772 (1981). [39] W. Wagner, A. Saul and A. Pruss. International equations for the presure along the melting and along the sublimation curve of ordinary water substance. J. Phys. Chem. Ref. Data 23, 515 (1994). [40] S.N. Tkachev, R.M. Nasimov and V.A. Kalinin. Phase diagram of water in the vicinity of the triple point. J. Chem. Phys. 105, 3722 (1996). [41] F. Datchi, P. Loubeyre and R LeToullec. Extended and accurate determination of the melting curves of argon, helium, ice (H2O) and hydrogen (H2). Phys. Rev. B 61, 6535 (2000). [42] Y. Horie. The kinetics of phase change in solids by shock wave compression. Ph.D. thesis, Washington State University, (1966). [43] D.J. Andrews. Equation of state of the alpha and epsilon phases of iron. Ph.D. thesis, Washington State University, (1970). [44] D.B. Hayes. Experimental determination of phase transition rates in shocked potas- sium chloride. Ph.D. thesis, Washington State University, (1972). [45] Y.M. Gupta. COPS wave propagation code (Stanford Research Institute, Menlo Park, CA, 1978), unpublished. [46] J. von Neumann and R.D. Richtmyer. A method for the numerical simulation of hydrodynamic shocks. J. Appl. Phys. (1950).

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185 186 Chapter 6

Analysis and Discussion The 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 transition. 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 dynamic 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 transition A ﬁrst order transition is characterized by a discontinuity in the ﬁrst derivatives of the Gibbs free energy [1]. µ ¶ µ ¶ ∂g ∂g s = − ∂ v = ∂ T P 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 images 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 continue 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 formation 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 times t =330, 530, 730, 930, 1130, and 1330 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 hvi to the particle velocities of the water-window interfaces:

dhvi u − u = R F (6.1) dt ρ0d0 where ρ0 is the initial sample density, d0 is the initial sample thickness, and uF and uR are the front and rear window velocities, respectively. When the sample is in the peak state, uR = uF , so the value of hvi is constant. The fact that uR subsequently drops below its steady state value is consistent with a density increase, i.e. dhvi/dt < 0.

6.2 The importance of surface effects When 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

190 300

VISAR 250 Mixed phase calc. data

200 freezing begins

150

Particle velocity (m/s) 100

0 0 100 200 300 400 500 600 t (ns) from shock arrival

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 velocity.

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 heterogeneous 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 surface, 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 decreases. 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 similar initial sample thicknesses (≈100 µm). 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 conduction 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 sapphire 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 experiments 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 2.5

2 quartz windows

1.5

sapphire front/ quartz rear

1 Extinction (base e) quartz front/ sapphire rear

0.5

sapphire windows 0 0 100 200 300 400 500 600 700 800 t (ns) from shock arrival

Figure 6.3: Extinction histories for different window conﬁgurations The peak state in each experiment was (5 GPa, 450-460 K). Initial sample thicknesses were similar (99-108 µm).

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 experiments, the magnitude and timing of the measured particle velocity steps matched the liquid state calculation during the initial compression. The sapphire liquid cell experiment continued 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

195 300 Liquid calc.

Quartz cell 250 VISAR data

200

150 (m/s) p u

Sapphire cell

100

0 0 100 200 300 400 500 600 700 800 t (ns) from shock arrival

Figure 6.4: Surface effects in wave proﬁle measurements Peak 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 into H+ and OH−, 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 temperatures 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

temperature TL by multiple shock compression, and at time t = 0 heat begins to ﬂow into

the window, which is initially at temperature TR. Window compression is negligible in

these experiments, so TR is approximately equal to room temperature. The water sample and the window are assumed to be semi-inﬁnite slabs in contact at x = 0. Far from this interface, each region is at the adiabatic compression temperature. The temperature proﬁle has the following form [2].

x T (x,t) = T + (T − T )er f 1 i i L 1/2 2(κ1t) x T2(x,t) = Ti + (TR − Ti)er f 2(κ t)1/2 2 s T + ηT K ρ c L R η 2 2 P2 Ti = η = ρ (6.3) 1 + K1 1 cP1 κ ρ The diffusivity is equal to the ratio of thermal conductivity and heat capacity ( i = Ki/ icPi ). ρ Parameters for the water sample (K1, 1, and cP1 ) are presented in Section H.1.2; the pa- ρ rameters (K2, 2, and cP2 ) 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 of K1. Since η > 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 generate TR ≈ 465, but the interface temperature would only be 352-365 K (79-92◦ C). The actual surface temperature 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 (a)

water sample window

T=T (x,t) T=T (x,t) 1 2 T → T at x → − ∞ T → T at x → ∞ L R

K , ρ , c K , ρ , c 1 1 P 2 2 P 1 2

(b) x=0

T L

t=0

T i t=2 µs Temperature (K) t=1 µs

T R

−10 −8 −6 −4 −2 0 2 4 6 8 10 x (µm)

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 time t = 0.

199 the temperature profiles near the water-window interface at various times for the heat conduction 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 temperatures 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 the 10−9 − 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, reducing 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 pressure 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 seconds 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 of 10−6 seconds, so there is insufficient time for water to infiltrate the silica windows. Even if such infiltration 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 experiments 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 transformed material in the water samples. The optical extinction of a material is often modeled

by Beer’s law [14]:

X = αd (6.4) where d is the sample thickness and α is an extinction coefficient characteristic of the material. In general, both absorption and scattering losses contribute to α, 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.

Z d X = α(x,t)dx (6.5) 0

201 For certain special cases, α(x,t) can be determined from some assumptions about the size,

shape, and distribution of scatterers. Appendix E develops α(x,t) for the case where scattering 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 scattering 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 straightforward. 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 measurement 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 transition reaches a steady state, no new scattering objects are formed, so the transformed layer thickness H 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 estimate 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 hxi. This distance varies as the square root of time and the self diffusivity of water D ≈ 2.26 × 10−9

2 m /s [14]. r Dt hxi = 2 π (6.6)

During a 1 µs experiment, the mean diffusion distance is about 53 nm, so H ≤ 50 nm

202 d

detector

input transmitted light light

silica H window transformed liquid material water

Figure 6.6: Optical transmission in surface initiated freezing

203 because D 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 eventually 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 extinction should be independent of sample thickness for d ≥ 50 nm. However, if the transformation 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 thickness d, which is related to d0 by mass conservation:

ρ0 d = d0 ρ (6.7)

The steady state sample density, neglecting the volume change due to freezing, was approximately ρ =1.51 g/cc; the ambient value ρ0 was 0.99705 g/cc. The compressed sample thickness were therefore 10, 21, and 71 µm (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 21 µm), there is clear break between the rapid extinction changes and a steady state behavior. This break suggests that the transition spanned the complete thickness 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 21 µm 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 3 steady state T5 (71 µm) 2.75

2.5

2.25 IS3 (21 µm) 2

1.75 T6 (10 µm)

1.5

1.25 Extinction (base e) 1

0.75

0.5

0.25

0 0 250 500 750 1000 1250 1500 t (ns) from shock arrival

Figure 6.7: Increasing extinction with sample thickness These experiments were performed with quartz windows to a 5 GPa peak pressure using different sample thicknesses. The compressed thickness of each measurement is shown in the legend.

205 Because the optical changes in water are not conﬁned only to silica window surfaces, it is necessary to distinguish the nucleation of ice from the subsequent growth process. 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 surface initiated 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 effectiveness of silica and sapphire windows, but do not explain why 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 following 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 the work 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 spectroscopy 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 isoelectric 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 function 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 intermediate 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 surfaces 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 molecular 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 nucleating 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, homogenous 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 scales The time scales of freezing should be linked to the thermodynamic conditions created 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 1

0.9 T1 (1.1 GPa, 104 µm)

0.8

0.7

0.6 I1 (2.7 GPa, 137 µm)

0.5 Transmission 0.4 I2 (3.6 GPa, 132 µm)

0.3

0.2

T5 (5.0 GPa, 108 µm) 0.1

0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 t (ns) from shock arrival

Figure 6.8: Time scales and peak pressure Quartz windows were used for all experiments except T1, which used soda lime glass windows. The dashed lines show the transformation time for experiment I2.

210 (a) 1

0.8 I2 (132 µm)

0.6 T4 (28 µm) IS1 (19 µm) 0.4 Transmission

0.2

0 0 100 200 300 400 500 600 700 800 900 t (ns) from shock arrival (b) 1

0.8

T5 (108 µm) 0.6 IS3 IS2 (107 µm) (32 µm) 0.4

Transmission T6 (15 µm) 0.2

0 0 100 200 300 400 500 600 700 800 900 t (ns) from shock arrival

Figure 6.9: Comparisons of the quartz cell photodiode experiments (a) 3.5 GPa peak state (b) 5.0 GPa peak state

211 1.1

1 97 ns 0.9 apparent incubation time 0.8

0.7

Transmission 0.6

0.5

0.4 data fit 0 50 100 150 200 250 t (ns) from shock arrival

Figure 6.10: Fit of the T5 photodiode record Equation 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 < t  0" # I T(t) = µ ¶2 (6.8)  t −tI  Ts + (T0 − Ts)exp − t ≥ tI tL

T0 is equal to the initial transmission (≈ 1); Ts is the steady state transmission for t → ∞. T0,

Ts, tI, and tL 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 of τ is linked to tL by the following relation. ¯ ¯ ¯dT(t )¯−1 1/2 τ ¯ in f lect ¯ tLe = ¯ ¯ = √ (6.9) dt 2(T0 − TS)

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 time t = 0, and freezing becomes thermodynamically possible at the front window interface when t = tP.

Freezing is observed at a later time tI. 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 that tI 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 contribute 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 coordinate q that tracks the metastable sample history. This coordinate is fixed

for t ≤ tP because freezing is not possible; for t ≥ tP, the value of q changes until it reaches

some critical value at t = tI. The value of q is coupled to the liquid mass fraction w (Section

5.4) so that the freezing occurs at t = tI. This description treats freezing as a two stage process– nucleation (changing q) and growth (changing w). Dynamic loading during the incubation period can be accounted for by deﬁning q as the area under the curve of some

thermodynamic variable η. The value of tI is then given by a the time required for a ﬁxed amount of area A to be swept out.

Z t 0 0 q(t) ≡ η(t )dt q(tI) = A (6.10) tp

The function η represents the extent to which the liquid phase is driven into the metastable

region. Deeply supercooled states would therefore have large values of η and require short

incubation times, while states near the coexistence curve, where η = 0, require long incu-

bation times. A reasonable deﬁnition for η is the normalized Gibbs free energy difference between solid and liquid phases.

M (g (t) − g (t)) η =? − L S (6.11) RT

A general η(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 transition begins t transition I possible η t s p (t) η

0 t (ns) from shock arrival

Figure 6.11: Incubation time and the metastable history

215 sum.

A = ηS(tI −tp) − ∑(ηS − ηi)∆ti (6.12) i

ηi and ∆ti represent the magnitude and duration of the i-th shock after freezing becomes

possible; ηS is the steady state value of η(t) that the compression progresses towards. ∆ti is determined by the sample thickness during the i-th shock, which must scale with the

original sample thickness d0 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.

∆t = c d t = bd i Ã i 0 p ! 0 A ηi η = tI − d0 b + ∑(1 − η )ci = tI − d0B (6.13) S i S

The values of b, ηi, and ci (and thus B) 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.

A min tI = + d0B = tI + Bd0 (6.14) ηS

Thus a plot of measured incubation times versus initial sample thicknesses should lie along a straight line. Regardless of the actual deﬁnition of η(t), the intercept of this line indicates

min the shortest possible incubation time for that peak state. tI 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

min used to determine tI . 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, where tn is the characteristic time required to form an ice nucleus. A relationship

216 170 160 150 140 130 120 110 3.5 GPa: t =41 ± 2 ns 100 min 90 80 70 60 Apparent incubation time (ns) 50 40 5 GPa: 30 t =7 ± 2 ns min 20 10 0 0 20 40 60 80 100 120 140

Initial sample thickness (µm)

Figure 6.12: Incubation time versus sample thickness for different peak states

217 between tn 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.

min B lntn = lntn + 2 (6.15) (gL − gS)

The minimum nucleation time can thus be estimated from the linear intercept of a ln(tn) 2 versus 1/(gL − gS) plot. Since tn was only measured for two peak pressures (3.5 and 5 min GPa), tn was determined by solving Equation 6.15 directly. The energy difference gL −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 (≈ 298 K). 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 affect τ in generally the same

way as tI. Thus, it would seem that τ should increase with pressure and decrease with sample thickness.

Figure 6.13 contains plots of τ 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

218 650

600

550

500

450

3.5 GPa 400

350

300 Overall transition time (ns) 250

200 5.0 GPa

150

100 0 20 40 60 80 100 120 140

Initial sample thickness (µm)

Figure 6.13: Transition time versus sample thickness for different peak states Straight 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 of τ is dominated by the fastest growth 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 scales This section discusses several length scales associated with the freezing transition. First, a discussion of the different domains in the water sample is presented to summarize 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 temperature Ts 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 temperature Tm. 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 above Ts. 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 vicinal region

window solid liquid supercooled cooled material water liquid water (near T ) (T>T ) (T=T ) material m S S

transformed material

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 1

0.9

0.8

0.7

0.6

0.5

Transmission 0.4

0.3 T5

0.2 IS2

0.1

0 0 200 400 600 800 1000 1200 1400 t (ns) from shock arrival

Figure 6.15: Consistency of the optical transmission measurements Experiments T5 and IS2 were performed in quartz windows to a peak stress of 5 GPa. Initial sample thicknesses were 107-108 µm.

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 200 µm at 2.5 GPa to less than 50 µm at 5.0 GPa. The number of nucleation events n∗ is related to the nucleation rate J deﬁned in

Equation 2.11. If nucleation occurs in a vicinal layer of thickness d∗, then the average number of nucleation events observed in an area A is given by n∗.

µ ∗ ¶ ∗ ∗ ∗ B n = J0 Ad exp − 2 (6.16) (gL − gS)

Given the consistency of the optical transmission records, it is reasonable to conclude that

∗ ∗ ∗ the vicinal parameters J0 , B and d are relatively constant for the z-cut quartz windows −2 used in this work. The value of n∗ is thus dominated by the value of e−(gL−gS) for 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 2.7 GPa (I1) t=330 ns 3.5 GPa (I2) t= 225 ns 5.0 GPa (I3) t= 135 ns

t=1430 ns t=675 ns t= 565 ns 225

Figure 6.16: Initiation and growth in imaging measurements Initial stages of the transition are shown in the upper three images. The lower three images show the sample as it approaches the steady state. The size of each image is approximately 920 × 750 µm (horizontal × vertical). The spatial resolution limit for these images is about 10 µm. 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 30-100 µm wide and 100-200 µm long. These features were reduced to 15-70 µm widths and ≤150 µm lengths at 3.5 GPa. By 5.0

GPa, transition regions are 10-70 µm wide and less than 100 µm 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 is growth dominated, 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 is nucleation 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 Nucleation Growth (a)

liquid water

frozen silica material surface

(b)

(c)

Figure 6.17: Nucleation rate and lateral freezing (a) When nucleation is rare, ice growth occurs both along and away from the silica window without encountering other ice formations. (b) As more nucleation events occur, there is a period of independent growth before overlap occurs. (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.

λ λ 4 λ ( 0/ ) T( ) = T0 (6.17)

As shown in Figure 6.18, this proﬁle is inconsistent with the spectrally resolved transmission measurements made during experiment T5. There are a several possible reasons for

228 1

0.9

0.8

0.7 Measurement @ t=200 ns

0.6

0.5

0.4 Transmission

0.3

0.2 Rayleigh profile

0.1

400 425 450 475 500 525 550 575 600 Wavelength (nm)

Figure 6.18: Comparison of measured transmission proﬁle with Rayleigh scattering limit The 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 by X = 2λ κt, where λ 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 0.3-1.7 µm (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 interfaces (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 formation is commonly found in the solidiﬁcation of supercooled liquids [3, 4] and may have a fractal nature [34].

230 References for Chapter 6 [1] H.B. Callen. Thermodynamics and an introduction to statistical mechanics. Wiley, New York, 2nd edition, (1985).

[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 freezing processes. Hemisphere Publishing, Washington, (1993).

[6] Y.M. Gupta. Lecture notes for “Fundamentals of large amplitude plane stress wave propagation in condensed materials”, Physics 592 (Washington State University), (2001).

[7] W.A. Bassett, A.H. Shen and M. Bucknum. A new diamond anvil cell for hydrothermal 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 anvil cell 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. The α −β quartz transition at high temperatures and pressures in a diamond-anvil cell by laser interferometry. Am. Mineralogist 78, 694 (1993).

[10] G.W. Morey, R.O. Fournier and J.J. Rowe. The solubility of quartz in water in the temperature interval from 25-300◦ C. Geochim. Cosmochim. Acta 26, 1029 (1962).

[11] R.O. Fournier and J.J. Rowe. The solubility of amorphous silica in water at high temperatures and high pressures. Am. Mineralogist 62, 1052 (1977).

[12] C.E. Manning. The solubility of quartz in H2O in the lower crust and upper mantle. Geochim. Cosmochim. Acta 58, 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).

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

[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 heterogeneous 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 ice crystals grown from vapor. J. Crys. Grow. 160, 162 (1996). [20] T. Gonda and S. Nakahara. Dendritic ice crystals with faceted tip growing from the vapor phase. J. Crys. Grow. 173, 189 (1997). [21] G. Peschel and K.H. Adlfinger. Viscosity anomalies in liquid surface zones. J. Colloid Interface Sci. 34, 505 (1970). [22] P.A. Sermon. Interaction of water with some silicas. J.C.S. Faraday I 76, 885 (1980). [23] A. Fouzri, R. Dorbez-Sridi and M. Oumezzine. Water confined in silica gel and in vycor 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 Surface Properties, 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 at quartz/water interfaces. Phys. Rev. Lett. 72, 238 (1994). [27] M.S. Yeganeh, S.M. Dougal and H.S. Pink. Vibrational spectroscopy of water at liquid/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-electrolyte interface. Nature 368, 444 (1994). [29] S.-B. Zhu and G.W. Robinson. Structure and dynamics of liquid water between plates. J. Chem. Phys. 94, 1403 (1991). [30] H.R. Pruppacher and J.C. Pflaum. Some characteristics of ice-nucleation active sites derived from experiments with a ferroelectric substrate. J. Colloid Interface Sci. 52, 543 (1975). [31] M. Gavish, J.L. Wang, M. Eisenstein, M. Lahav and L. Leiserowitz. The role of crystal polarity in alpha amino acid crystals for induced nucleation of ice. Science 256, 815 (1992).

232 [32] H.C. van de Hulst. Light Scattering by Small Particles. Dover, New York, (1957).

[33] C.F. Bohren and D.R. Huffman. Absorption and Scattering of Light by Small Parti- cles. John Wiley & Sons, New York, (1983).

[34] T. Vicsek. Fractal Growth Phenomena. World Scientiﬁc, Singapore, (1992).

233 234 Chapter 7

Summary and Conclusions The 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 demonstrate that liquid water undergoes time dependent freezing under quasi-isentropic compression. 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 previous studies [1], these results show unambiguous evidence for freezing under shock wave compression.

7.1 Summary Multiple shock wave compression was used to quasi-isentropically compresses liquid 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 transparency, indicating that water no longer exists in a pure liquid state. Real time optical images revealed that these changes were not uniform across the water sample and that freezing occurred heterogeneously. Measured wave proﬁles also differed from the expected behavior 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 10 µm 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 (incubation 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 temperature conditions of the water sample, but the temporal nature of multiple shock compression 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 limiting 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 10 µm 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 Conclusions The 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 transformation 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 below 10−6 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 work Although 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 initiating 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 at quartz/water interfaces. Phys. Rev. Lett. 72, 238 (1994).

[3] M.S. Yeganeh, S.M. Dougal and H.S. Pink. Vibrational spectroscopy of water at liquid/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-electrolyte interface. Nature 368, 444 (1994).

[6] M.F. Reedijk, J. Arsic, F.F.A. Hollander, S.A. de Vries and E. Vlieg. Liquid order at the interface of KDP crystals with water: evidence for icelike layers. Phys. Rev. Lett. 90, 66103 (2003).

[7] H.R. Pruppacher and J.C. Pﬂaum. Some characteristics of ice-nucleation active sites derived from experiments with a ferroelectric substrate. J. Colloid Interface Sci. 52, 543 (1975).

[8] M. Gavish, J.L. Wang, M. Eisenstein, M. Lahav and L. Leiserowitz. The role of crystal polarity in alpha amino acid crystals for induced nucleation of ice. Science 256, 815 (1992).

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

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

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

[12] 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. Solids 35, 501 (1974).

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

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

240 [15] T. Mashimo, A. Nakamura and M. Kodama. Yielding and phase transition under shock 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. Science 290, 783 (2000).

241 242 Appendix A

Mechanical Drawings This 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 4.026+0.000 -0.0005"

0.50" 0.5±0.01" 1.0"

3.25"

45°taper (interior)

1/8"radius

PARKER#2-342 O-rings

8.0± 0.062"

3.650+0.000 -0.050"

6061-T651Alalloy 1/8"radius O.D.taper<0.001" .xx=±0.02" .xx=±0.002"

0.87" 0.37" 3.25"

Figure A.1: 4” standard projectile This 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 breech of the 4” gas gun [1].

244 4.026+0.000 -0.0005

0.38"

1/4"radius

0.100+0.010 2-240O-rings -0.000"

6.00"

45°cut

6061-T651or 0.38" 7075-T651 Alalloy O.D.taper<0.001" 3.23" .xx=±0.02" .xx=±0.002"

Figure A.2: 4” monkey’s fist projectile: This design was necessary for 5 GPa experiments using sapphire windows. Modifications shown in Figure A.4 were made to the projectile face. Velocities of 0.180-0.250 km/s are obtained using the “monkey’s fist” configuration of the 4” gas gun [2].

245 2.45" 8-32bodydrill on0.750"diameer projectile 0.25" 2places centerline 0.75" 0.25" 0.121±0.001" 1.75" 2.062-16thread

0.100" 0.50" 1.0" 0.937"

2-228O-RINGGROOVE

4.50" 0.125+0.010 -0.000"

10-24drill/tap 1/2"deep 1/8"radius

0.25" 7075-T6ALUMINUM .XX=±0.02" .XXX=±0.002" 1/8"radius 0.38" 1/16"radius 0.5" 2.464+0.0000 -0.0005"

Figure A.3: 2.5” projectile and cap This projectile design was used for only one VISAR experiment (V1).

246 8-32drill/tap 3/8"deep

(c)

(b)

1.700" 1.345" 8-32bodydrill throughprojectile face

(a)

0.375" 0.375"

Figure A.4: Projectile modiﬁcations for mounted pieces All projectiles had impactor mount holes at (a) Optical experiments using quartz, fused silica, and sapphire impactors had turning mirror mounted at (b) Optical experiments with glass impactors had turning mirror mounted at (c)

247 0.75"diameter

5/8-27thread

0.25"

5-40drill/tap 1.250"diameter 3places

bodydrill5-40 1.250"diameter 3places

1.20"

1.50"

Aluminumring 1/4-36drill/tap 1.20"fromcenter 1/2"diameter x0.3"deep countersink

Figure A.5: Modiﬁcations made to the standard 4” projectile used for alignment during setup of imaging experiments. The aluminum ring is mounted to the projectile face with Belville washers and holds a collimating lens unit (Figure A.23). Optical ﬁber carries light into the side hole and connects with the lens unit from the back side of the projectile (not shown). Collimated light is emitted from the lens unit. Screws holding the plate to the projectile are adjusted to align light output with the projectile.

248 0.44"

0.520" 45.00°

1.00"

0.50"

8-32drill/tap 0.375" 0.134" 2places 0.088"

Material:Aluminum Tolerances: .xx=0.005" .xxx=0.001"

Figure A.6: Soda lime glass impactor mount

249 1.50"

0.75"diameter

0.44"

0.520" 45.00°

1.00"

0.50"

8-32drill/tap 0.375" 0.134" 2places 0.088"

Material:Aluminum Tolerances: .xx=0.005" .xxx=0.001"

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 45° 0.22" 0.28"

0.43"

0.440" 8-32bodydrill 0.520" 0.375"fromcenterline 2places

0.088"at45° 0.43"

1.125"

Figure A.8: Projectile turning mirror mounts (glass impactor) Outer corners should be ﬁled down as this mount rests near the projectile edge.

251 1.125"

0.75"diameter

0.44"

0.520" 45.00°

0.50"

8-32drill/tap 0.375" 0.134" 2places 0.088"

Material:Aluminum Tolerances: .xx=0.005" .xxx=0.001"

Figure A.9: Projectile turning mirror mounts for non-glass experiments

252 Material:Brass Tolerances: .xx=0.005" 1.00" .xxx=0.001" (a) 0.050"

0.25" ~3/8"

Reliefcut0.01-0.02"deep topreventepoxyflowinto centralregion 1.25" (b)

0.25" ~3/8"

Figure A.10: Impactor apertures (a) Sapphire impactor (b) Quartz/fused silica impactor (c) Glass impactor

253 3.50" Tolerances: .xx=0.005" .xxx=0.001" 1/x=1/64" 6-32drill/tap 2.750"diameter 6places 0.25"deep

10-32drill/tap 9/64"drillthrough 1.81" 3.250"diameter through 3places 3.250"diameter 2places 90°apart

#52drillthrough 2.125"diameter 2places 0.05"groove 2.5"ODO-ringgroove 0.136"wideX0.074"deep 1/8"

0.08" 1/8" 1/8" 0.90" brass 2.01" cellbody 2.51" Teflonpad 0.03"thick 2.31" 1/8" brasslockring 3.00"

Figure A.11: Liquid cell for soda lime glass windows

254 Twofillchannels #52Drill(0.0635") 1.75"

11/2"OD1/16" O-ringgroove (Parker#2-028) centeredoncell

Lockring 1.375" 1.550"-40thread 1/8"

0.25" 0.38" 0.455" 0.71" 3-48drill/tap 3placesonø1.530" 0.150"deep 1.23" 1"ID1/16" Material:Brass O-ringgroove Tolerances: (Parker#2-026) .xx=0.005" centeredoncell .xxx=0.001" Innerdiameterforpressfit O-ringgroovesinitially 0.052"deep0.085"wide

Figure A.12: Liquid cell used for silica windows This 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 smaller O-ring (see Figure A.13).

255 Twofillchannels #52Drill(0.0635") 1.50"

11/4"OD1/16"O-ringgroove (2-024centeredoncell)

Lockring 1.08" 1.290"-40thread 1/8"

0.25" 0.38" 0.455" 0.71" 5-40drill/tap 3placesonø1.300" 5/16"deep 0.990" 1ID1/16"O-ring groove(2-022)centered Material:Brass oncell Tolerances: .xx=0.005" innerdiameterforpressfit .xxx=0.001" O-ringgroovesinitially 0.052"deep0.085"wide

Figure A.13: Liquid cell for sapphire windows This 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 5-40drill/tap #70drillthrough on1.800"diameter on1.32"diameter 3/8"deep 2places 1.11" 3places

1.5"OD1/16"O-ringgroove (2-024centeredoncell)

2.00"

#52drill 0.1"deep 0.15" 2places 0.50" 1.26" 0.25" 1/8" 1 .550"-40thread 1.375" 3/16"deeponcellbody lock ring

Material:Brass Tolerances: .xx=0.005" .xxx=0.001" O-ringgrooveinitially 0.052"deep0.085"wide

Figure A.14: Quartz cell used for VISAR experiment V2. A Viton O-ring was used to seal the cell.

257 diameter=5.96" #23drillthrough on5.50"diameter diameter=5.00" 3places

diameter=4.50"

sideview 0.25" 1.00"

0.38"wideX1.5"slot

Material:AluminumHolobar 6"ODX3/4"wall Insidediameterleftrough Scale=1/2 Tolerances: .xx=0.005"

Figure A.15: Standoff target ring All target plates (Figures A.16-A.19) were attached to this type of target ring, which was then mounted to the gun.

258 5.00"

radius=2.24"

radius=1/4"

100°

diameter=3.01"

8-32drill/tap throughon 3.250"diameter 2places

drilledthrough1/4" on3.250"diameter 2places 9/64drillthrough 90°apart on4.750"diameter 3places

Material:1/4"aluminumplate Tolerances: .xx=0.005" 5-40drill/tapthrough .xxx=0.001" on3.250"diameter 3places

Figure A.16: Soda lime glass cell target plate

259 radius=2.24"

1/4"radius 100°

3-48bodydrill onø1.530" 3Places

radius=0.68" 9/64"drill onø4.750" 8-32drill/tap 3places onø2.250" 2places

10-32drill/tap onø3.25" 2places(90°)

Material:1/4"thickaluminumplate Tolerances: .xx=0.005" .xxx=0.001"

Figure A.17: Quartz/fused silica cell target plate

260 9/64"drill on4.75"ø 3places radius=2.240"

120° radius=1/4"

radius=0.77"

5-40drill/tap on1.300"diameter 8-32drill/tap 3places 2.75"apart (centered) 2places

diameter=1.04" 9/64"drill ON1.300"diameter 3places

20°

10-32drill/tap 3.12"diameter 2place(90°apart) Material:1/4"thickaluminumplate Tolerances: .xx=0.005" .xxx=0.001"

Figure A.18: Sapphire cell target plate

261 radius=2.25"

100° radius=1/4"

5-40bodydrill ø1.800" 3places

5.00" radius=0.785" 8-32drill/tap ø2.750 2places

9/643holes onø4.75

drill/tap10-32 2places(90°)ONø3.25"

Material:1/4"aluminumplate Tolerances: .xx=0.005" .xxx=0.001"

Figure A.19: Target plate used with the unbuffered VISAR cell of Figure A.14

262 Material:Aluminum Tolerances: .xx=0.005"

5/8-27drill/tapthrough

1.50"

8-32bodydrill 1.625fromcenter 2places

3.25"

3.50"

0.25"

1.00"

3.00"

Figure A.20: Lens bracket for glass target plate

263 0.75" 0.38"

2.25"

2.50"

1.00"

11/64"drillthrough 2placesoncenterline 5/8-27drill/tapthrough oncenterline

Material:Aluminum Tolerances: .xx=0.005"

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

264 3.00"

0.75" 0.38"

2.25"

2.75"

1.00"

11/64"drillthrough 2placesoncenterline 5/8-27drill/tapthrough oncenterline

Material:Aluminum Tolerances: .xx=0.005"

Figure A.22: Lens bracket for sapphire target plate

265 5/8-27thread 1.35" 1.19" 1/16"drill

7/16"drill 1/16"DRILL 0.475"

0.332" DRILLPOINTANGLE

1/4-36drill/tap Lenstube(brass)

5/8-27tap

3/4" ID=7/16" hex 0.470" 0.70" 0.25" INTERNAL Locknut(brass) CHAMFER 0.20"

Spacerring(brassoraluminum)

Aperture(brass) Thickness=0.05" OD=0.472" Tolerances: .xx=0.005" .xxx=0.001"

ID=0.25"

Figure A.23: Lens unit used in transmission experiments to collect collimated light. The aperture is placed above a single plano convex lens using the spacer ring.

266 0.125" 0.31" 7/16-27thread 5/8"hex 0.62" 0.36"

0.25" 0.60" 0.08"

spotface3/8" 0.75" drill/tap1/4-36 LOCKNUT 3.6°offaxis

FIBEROPTICHOLDER

7/16-27thread 5/8-27thread 5/8-27thread 0.17" #67drill 3/4"hex

0.330" 0.70" 0.25" 0.80" 0.475" 5/8LOCKNUT LENSHOLDERBODY Tolerances: .xx=0.005" .xxx=0.001"

Figure A.24: Lens unit used to image laser ﬁber tip onto VISAR mirror. 1:1 imaging is performed with a pair of identical plano-convex lenses separated by a spacer ring (see Figure 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 Materials This 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 variables of interest. In the following discussion, the mean stress in denoted by P and the shear

modulus by G. Sample compression is deﬁned as µ = ρ/ρ0 − 1.

B.1 Soda lime glass Soda 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 average 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 silica Dynasil 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:

2 3 4 σx = 77.60ε − 415.9ε + 3034.0ε − 6926.0ε where stress has units of GPa. This relation must be modiﬁed to match the measurements

270 of Rigg [6]:

2 3 4 σx = 77.60ε − 400.0ε + 2934.0ε − 6926.0ε

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 quartz Electrical grade quartz was obtained from Boston Piezo-Optics with a crystal orientation 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 = 2.6485 g/cc

P = 43.19µ + 156.2µ2 + 48.60µ3

G = 46.92 + 1.873P + 0.3459P2 where both P and G are in GPa. Quartz remains optically transparent for shock compressions 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 Sapphire Sapphire 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 = 3.985 g/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 shock compression. M.S. thesis, Washington State University, (2001).

[3] R. Feng and Y.M.Gupta. Material models for sapphire, α-quartz, lithium ﬂuoride, and fused silica for use in shock wave experiments and wave code calculations. Technical Report 96-XX, Shock Dynamics Center (unpublished), (1996).

[4] M.P. Conner. Shear wave measurements to determine the nonlinear elastic response of 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, and sapphire. J. Appl. Phys. 41, 4208 (1970).

[6] P.A. Rigg. Real-time x-ray diffraction to examine lattice deformation in shocked lithium 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).

[8] R.L. Gustavsen. Time resolved reﬂection spectroscopy on shock compressed liquid carbon disulﬁde. Ph.D. thesis, Washington State Univesity, (1989).

[9] Norman Beniot, (2002). Boston Piezo-Optics, private communication regarding quartz polishing.

[10] J.M. Winey, R. Feng and Y.M. Gupta. Isotropic material models for the elastic response of sapphire and quartz single crystals under shock wave compression. Tech- nical report, Insitute for Shock Physics, (2001).

[11] S.C. Jones and Y.M. Gupta. Refractive index and elastic properties of z-cut quartz shocked to 60 kbar. J. Appl. Phys. 88, 5671 (2000).

[12] R.L. Webb. Transmission of 300-500 nm light through z-cut sapphire shocked beyond its elastic limit. M.S. thesis, Washington State University, (1990).

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

273 274 Appendix C

Water Sample Preparation This 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 contamination and treatment of ultrapure water is presented in Section C.1. The speciﬁc treatments 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 methods Virtually 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 impurities Miscible organic Alcohol Miscible inorganic Hydrogen peroxide Immiscible organic Oils Immiscible inorganic Mercury

Solid impurities Molecular organic Sugars Molecular inorganic Salts Colloidal organic Viruses Colloidal inorganic Silica Suspended organic Biological matter Suspended inorganic Metal oxides

Gaseous impurities Volatile 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 purification Preliminary 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, releasing H+ ions; anions are exchanged for OH− ions at different sites in the resin. This process removes virtually all ionic contamination from water. However, uncharged impurities, including organics and microorganisms, are largely unaffected. 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 (a)

Deionization Distillation Degassing

input feed water

collection +testing

(b)

Deionization Filtration Distillation

input feed water

sealed storage

collection +testing

Figure C.1: Water puriﬁcation stages (a) Initial ultrapure system (b) Final ultrapure system

278 insect entered here

insect died here

growth found 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 the rising vapor, stripping away residual volatile impurites. The boiler automatically drains periodically 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 its resistance 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 loading 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 samples. 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 measurements 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 density 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 measurements. 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 measurements Performed on site with Thornton 200CR Conductivity/Resistivity Meter Detection limits: 0.01 MΩ · cm Range of measured values: 1.9-2.0 MΩ · cm

Dissolved oxygen (DO) measurements Performed on site with CHEMetrics visual DO kit Detection limit: 1 ppm Range of measured values: 4-5 ppm

Total organic carbon measurements Performed 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 silica Performed 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 measurement 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 25◦C. The readings varied from 1.8-2.0 MΩ · cm for all samples taken from the puriﬁcation system. In the absence of charged impurities, the resistivity of water at 25◦C 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 cell Filling liquid cells with water proved to be more difficult than other liquids. The initial 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. In Ultrapurity: 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 Chromatagraphy 6, 2565 (1983).

[4] P.M. Pichal and R.B. Dooley. Electrolytical conductivity (speciﬁc conductance) of liquid and dense supercritical water from 0◦ C to 800◦ C and pressures up to 1000 MPa. 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 atmospheric carbon dioxide. J. Am. Chem. Soc. 38, 1480 (1916).

285 286 Appendix D

Supplemental Data This 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 experiments A 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 Experiment Impact Impactor Front window Sample Rear Window peak Changes number velocity Material h [D] Material h [D] h Material h [D] P T observed (km/s) (mm) (mm) (mm) (mm) (GPa) (K) 18.8 1.6 18.8 G1 (99-066) 0.487 SLG SLG 0.13 SLG 3.61 429 yes [44.5] [50.8] [63.5] 18.8 1.6 18.8 G2 (00-020) 0.485 SLG SLG 0.13 SLG 3.58 428 yes [44.5] [50.8] [63.5] 18.8 - 18.8 G3 (99-069) 0.491 SLG - - SLG 3.62 - no

[44.5] - [63.5] 288 18.8 1.6 18.8 G4 (99-065) 0.489 SLG SLG 0.13 SLG 3.61 - no [44.5] [50.8] [63.5] h= thickness D= diameter SLG=soda lime glass Diameters and thicknesses are nominal values Table D.1: Summary of soda lime glass experiments 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 vacuum. 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 somewhat 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 1 µs [2–4], so no

289 1

0.9

0.8 G1 0.7

0.6

G2 0.5 Transmission

0.4

0.3

0.2

0.1

0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Time (ns) from impact

Figure D.1: Photodiode transmission for experiments G1 and G2 In both experiments, water was compressed in soda lime glass windows to a peak pressure of 3.6 GPa. Experiment G1 used a deionized water sample, while experiment G2 used deionized, distilled, and degassed water.

290 G4

G3 0.9

0.8

0.7

0.6

0.5 Transmission

0.4

0.3

0.2

0.1 G3 G4 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Time (ns) from impact

Figure D.2: Transparency of shocked soda lime glass

291 optical changes were expected here. No signiﬁcant changes were observed in the photodiode 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 loading 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, time t = 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 19 µm. This is similar to experiment T4 (page 105), which used a 28 µm sample. The photodiode record is shown in Figure D.3. This experiment was generally consistent with transmission experiment T4.

292 Table D.2: Supplemental photodiode measurements

293 [31.8] [38.1] [31.8] 12.7 3.184 12.720 IS2 (10-019) 0.566 Q Q 0.107 Q 5.06 465 1.51 112 900 yes [31.8] [38.1] [31.8] 12.7 3.164 12.719 IS3 (01-021) 0.565 Q Q 0.032 Q 5.05 464 1.51 33 2200 yes [31.8] [38.1] [31.8] h= thickness D= diameter Q=z-cut quartz * 0.005 mm uncertainty in thickness ** 0.002 mm uncertainty in thickness Diameters are nominal values 1

0.9

0.8

0.7

0.6

0.5 Transmisison 0.4 T4

0.3 IS1

0.2

0.1

0 0 200 400 600 800 1000 1200 1400 1600 1800 Time (ns)

Figure D.3: Photodiode transmission of experiment IS1 z-cut quartz windows were used to achieve a 3.5 GPa peak pressure. The initial sample thickness was 19 µm. Experiment T4 (3.5 GPa, 28 µm 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 107 µm, and 112 ns were required to reach the 90% pressure level. In experiment

IS3, the water sample was initially 32 µm 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 IS2 1 IS3

0.9

0.8

0.7

0.6

0.5 Transmission 0.4

0.3

0.2 IS2 0.1 IS3 0 0 200 400 600 800 1000 1200 1400 1600 1800 Time (ns)

Figure D.4: Photodiode transmission of experiment IS2 and IS3 z-cut quartz windows were used to achieve a 5 GPa peak pressure. The initial sample thickness in experiment IS2 was 107 µm; in IS3 it was 32 µm.

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 dilution on the ultraviolet and visible absorptivity of CS2 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 on the 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 decomposition in liquid nitromethane. Ph.D. thesis, Washington State University, (1995).

297 298 Appendix E

Optical Extinction in a Scattering Medium This 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 scattering A 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 radius a embedded 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. −iωt ikz Polarized plane waves of light (field amplitude Eliquid = E0e e , wave number k = 2ω/c), incident from the left, produce a polarization field Psolid in the sphere. Electric

(Esolid) and displacement (Dsolid) ﬁelds within the sphere are related by the permittivity.

2 Dsolid = εsolidEsolid = m Esolid (E.1)

299 E0 scattered light

k γ

P solid a

liquid water

collimated incident light

Figure E.1: Single scattering setup The sphere represents a solid region that has formed from the liquid. Plane waves of light, incident from the left, induce a polarization P within the sphere, creating a scattering ﬁeld.

300 The polarization ﬁeld (Psolid) is given by:

1 m2 − 1 P = (D − E ) = E (E.2) solid 4π solid solid 4π solid

Integration of Psolid over the entire sphere gives the total dipole moment p. Z m2 − 1 p = E dV (E.3) 4π solid

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.

2 2 m − 1 ω m − 1 ω p ≈ E e−i tV = E a3e−i t (E.4) 4π 0 3 0

This time dependent dipole moment leads to a scattering ﬁeld of magnitude Es:

k2 p sinγ E = e−ikr (E.5) s r

where r is the observation distance from the dipole. The angle between the observation

angle and the induced dipole moment is γ. The average radiated intensity from this dipole

2 is given by Is = c|Es| /8π; the total scattered power Ps is deﬁned by integrating Is over a sphere of radius r.

µ ¶2 Z c m2 − 1 (m2 − 1)2 P = E2a6 dΩsin2 γ = cE2 k4a6 (E.6) s 8π 3 0 0 27

The scattering cross section, σs, is deﬁned as the ratio of the scattered power and the inci- 2 dent intensity I0 = c|E0| /8π.

8π 32π σ = (m2 − 1)2k4a6 ≈ (m − 1)2k4a6 (ka << 1) (E.7) s 27 27

The ﬁnal approximation reﬂects the assumption that m 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 evaluate σs 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 refractive 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 transmitted light may be ambiguous.

E.2 Multiple scatterers and optical extinction In 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 material dz (Figure E.2) normal to an incident beam of collimated light. Within that layer, all scattering particles in area A are illuminated with the same light intensity I. The scattered light power Ps (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 within dz, each with different cross sections, so the total loss

302 scattered A=crosssection powerloss area

I(z) I(z+dz)

Figure E.2: Optical extinction from a collection of scattering particles An area A of the sample is illuminated with collimated light. Within a thin layer dz, all scattering particles are subjected to a local ﬁeld intensity I. Light entering the sample I(z) is attenuated by scattering losses, resulting in a weaker intensity I(z + dz).

303 must be integrated over this distribution: Z Ps = IA dz da n(a)σs(a) (E.8)

where n(a) is the number density of scatters of size a. The optical power in the layer IA is

attenuated by PS. Z dI d(IA) = −IAdz da n(a)σs(a) → = −Iα (E.9) Z dz where α ≡ da n(a)σs(a) (E.10)

The extinction coefﬁcient α thus increases with the number density of scatters in the system. In general, α is a function of wavelength, position, and time. For a one dimensional material, the transmitted light intensity becomes: µ Z ¶ Iout = Iin exp − dz α (E.11)

If α is constant throughout the sample, this reduces to Beer’s law (page 458 of Ref. [3]):

−αL Iout = Iine (E.12) where L is the sample thickness. The general results for a one dimensional scattering system are related to the transmission and extinction deﬁnitions used in Chapters 4 and 6. · Z ¸ T(t,λ) = exp − dz α(z,t,λ) (E.13) Z X(t,λ) = dz α(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. µ Z Z ¶ A T(t,λ) = exp − dz da n(a,z,t) a6 (E.15) λ 4 Z Z A X(t,λ) = dz da n(a,z,t) a6 (E.16) λ 4

For such a system, the total extinction has a 1/λ 4 wavelength dependence regardless of the details of n(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 measurements The 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 if σ(a) is known, the function n(a,z,t) is also needed. This expression is complicated since ice nucleation requires a silica surface (Section 6.2), so n(a,z,t) is not simply a function 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 spherical 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 scattered into 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 interpretation Despite 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 (c) (b) (a) (d)

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 scattering Not shown in the diagram are complications of single scatterer cross sections (size and shape) 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, increases 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-induced phase transition in cadmium sulﬁde. Ph.D. thesis, Washington State University, (1998).

309 310 Appendix F

Details of the VISAR Calculations

F.1 Apparent velocity The 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 velocity u(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 frequency f0 is incident on a mirror within the target; reﬂected light is Doppler shifted to a frequency f (t): µ ¶ c + u(t) u(t) f (t) = f ≈ f 1 + 2 (F.1) 0 c − u(t) 0 c

The approximation made here is that particle velocity u 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 of u/c is relevant. Light entering the VISAR interferometer is nominally unpolarized and has a fre-

quency f (t). Consider first a single polarization with field amplitude EP. The electric field

at the entrance plane of the system is given by Ein.

−i2π f (t)t Ein(t) = EP e (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 time τ longer than the other path, so the phase of one beam lags the other by τ. The optical phase Φ(t)

311 target mirror

u(t)

VISARinput f(t)

path1 VISARoutput I(t) polarization sensitive detector

stationary mirror

etalon path2 (delayed)

1/8wave plate

stationary mirror

Figure F.1: Simpliﬁed view of the VISAR measurement The detector receives superimposed light from both paths, one of which is delayed by a time τ.

312 of each beam depends on the time dependent frequency f (t).

Z t Φ(t) ≡ 2π f (t0)dt0 (F.3) 0

The total electric ﬁeld, Eout, is a linear sum of the waves from each path. ³ ´ −i2π f (t)t iΦ(t) iΦ(t−τ) Eout = EP e ae + be (F.4) a and b 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.

2 ³ ´ E ∆ ∆ I ∝ P a∗a + b∗b + ab∗ ei + a∗b e−i (F.5) out 2 ∆ ≡ 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. µ ¶ Z t Z t 0 0 0 u(t ) 0 ∆ = 2π f (t )dt = 2π f0 1 + 2 dt τ τ c t−µ ¶ t− 2 4πτ ≈ 2π f0τ 1 + u(t) ∆ = ∆0 + u c λ0 ∆ − ∆ → u = F 0 (F.7) 2π

The approximation made by reducing the integral is discussed in Goosman [2]. ∆0 is a constant phase shift that can be determined from knowledge of u(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 many cycles.

313 Two detectors are used in the actual VISAR system, each sensitive to one polariza-

tion. For one polarization, a = a1 and b = b1, where a1 and b1 are real.

E2 ¡ ¢ I ∝ P a2 + b2 + 2a b cos∆ (F.8) 1 2 1 1 1 1

Within the delay path is a 1/8 wave plate that adds an additional e−iπ/2 of phase (in two

passes) to one polarization. Thus a = a2 and b = −ib2, where a2 and b2 are both real.

E2 ¡ ¢ I ∝ P a2 + b2 − 2a b sin∆ (F.9) 2 2 2 2 2 2

Redeﬁning the detector outputs as X ≡ I1 (unaffected by 1/8 wave plate) and Y ≡ I2 (affected by 1/8 wave plate) gives the following result.

X = A1 + B1 cos∆ (F.10)

Y = A2 + B2 sin(∆ + φR) (F.11)

2 φR is a constant parameter that may be necessary when the round trip phase shift through 1/8 plate is not exactly π/4. The following steps are taken to calculate u(t) from VISAR data signals X and Y.

1. Both X and Y 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 data X and Y 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 parameters A1, A2, B1, B2, and φR. The values of these parameters are chosen to optimally ﬁt the data ellipse.

2 During experimental setup, the VISAR system is optimized so that φR is nearly zero, producing an ellipse oriented 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 of A1, A2, B1, B2, and φR .

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 of 2π 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 corrections The 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, located 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 path Pop that is longer than the physical distance. This is a result of the fact that light travels more slowly in the window than

in vacuum, leading to an apparent position x = −Pop. The apparent mirror velocity uv is therefore the time derivative of the apparent position. Z dP d u = − op = − n(x0,t)dx0 (F.12) v dt dt

315 free truemirror surface velocity velocity

VISARmeasures apparentvelocity sample mirror Q

L(t) rearwindow

Figure F.2: True versus apparent velocity in the VISAR system To an observer at point Q, the mirror appears to be at position x = −Pop, where Pop is the optical path length. The refractive index of the window is larger than unity, so the apparent distance is larger than the actual distance. The VISAR system measures apparent velocity uv. Window corrections must be made to determine the true velocity uT .

316 The integral is carried out from the true mirror position xT (t) to the point Q. Let the rear

window free surface position be xFS(t) and the distance from this free surface to point Q be L(t). The total optical path is then the sum of the window and free space propagation.

Z x FS 0 0 Pop = n(x ,t) dx + L(t) xT Z x FS 0 0 = n(x ,t) dx + L0 − xFS(t) + xFS(0) (F.13) xT

L0 is the distance between the rear window free surface and point Q at time t = 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.

Z x Z x FS 0 0 FS 0 Pop = a ρ(x ,t)dx + b dx + L0 − xFS(t) + xFS(0) xT xT

= 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 area A, the total mass within the window is m. 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) (u (t) + (b − 1)u (t)) → u (t) = v FS (F.16) T b

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 of b 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.

[6] S.C. Jones and Y.M. Gupta. Refractive index and elastic properties of z-cut quartz shocked to 60 kbar. J. Appl. Phys. 88, 5671 (2000).

[7] S.C. Jones, B.A.M. Vaughan and Y.M. Gupta. Refractive indices of sapphire under elastic, uniaxial strain compression along the a axis. J. Appl. Phys. 90, 4990 (2001).

319 320 Appendix G

Details of the Mixed Phase Water Model This 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 model Although convergence of Newton’s method tends to be rapid, there are issues regarding its numerical stability. If the initial conditions are not close enough the actual solution, 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 between time steps. Using the previous value of x as the guess value for the current time step, only minor changes in x 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)

where λ 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 K 6 2 2 P01 = 10 dyne/cm P02 = 0 dyne/cm

Reference state v01 =1.00296 cc/g v02 =0.679497 cc/g 7 e01 = 0 e02 = −1.601827 × 10 erg/g 7 s01 = 0 s02 = −1.513880 × 10 erg/g·K

10 2 10 2 Isotherm K1 = 2.21008 × 10 dyne/cm K2 = 25.03581 × 10 dyne/cm parameters g1=6.02913 g2= 3.66007

6 B1 = 4.836403 × 10 8 B2 = −1.952754 × 10 8 B3 = −1.543223 × 10 9 7 2 Pressure-temperature B4 = −2.355402 × 10 b2 = 6.828800 × 10 dyne/cm 10 derivative B5 = −1.175901 × 10 10 B6 = −1.746693 × 10 B = −8.658027 × 109 7 µ ¶ 7 v i−1 b1(v) = ∑ Bi − 1 i=1 v0 (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 of x was limited so that T, v1, and v2 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 the setup subroutine to allow the user to access this material model (not shown). A modification of the sweep subroutine was also made to prevent large changes in the water layers during single time step (not shown). The subroutine dtcheck21 (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 fraction w 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 none include ’cops_rev.common’ integer mat,k1,k2 double precision vold,eold,wold,pold double precision vnew,enew,wnew,pnew double precision v1old,v2old,told double precision v1new,v2new,tnew double precision vmid,tmid,pmid,cv,dpdt,kt double precision eosparams(3),xeos(3),eosout(8),gibbs1,gibbs2 double precision meanrho,dexx,dvfull double precision bulk,shear,pcut,q,tau1,tau2,dwdt integer ii,jj double precision wpureliquid parameter (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 strain meanrho=(varold(ii,5)+var(ii,5))/2.0d0 dexx=var(ii,15)-varold(ii,15) dvfull=dexx/meanrho vnew=vold+dvfull xeos=[v1old,v2old,told] call water_rate(wold,xeos,tau1,dwdt) if (time.lt.tau2) dwdt=0.0d0 wnew=wold+dwdt*dt

324 enew=eold tnew=told vmid=(vold+vnew)/2.0d0 pmid=pold tmid=told

do jj=1,10 eosparams=[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)/2.0d0 enew=eold-(pmid+q)*dvfull end do

* update cops variables var(ii,5)=1/vnew var(ii,23)=tnew var(ii,50)=wnew var(ii,48)=v2new var(ii,49)=v1new var(ii,7)=pnew var(ii,11)=-pnew-q var(ii,12)=var(ii,11) var(ii,13)=var(ii,11) var(ii,14)=0.0d0 var(ii,9)=enew

* gibbs free energy difference call 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 cops shear=0.0d0 call timestep(ii,bulk,shear) end do return end

* time step check subroutine dtcheck21(mat,k1,k2,dteos) implicit none include ’cops_rev.common’ integer mat,k1,k2 double precision dteos,meanrho,dexx,dvfull

325 double precision xeos(3),w,tau,dwdt,dvdt double precision dwmax,dvmax parameter (dwmax=0.001,dvmax=0.0001) integer ii

tau=prop(mat,4) dteos=1.0d-6 do ii=k1,k2 if(var(ii,6) .lt. 1.e-10) cycle xeos(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/2 end do meanrho=(varold(ii,5)+var(ii,5))/2.0d0 dexx=var(ii,15)-varold(ii,15) dvfull=dexx/meanrho dvdt=dvfull/dteos do while (dabs(dvdt*dteos) .gt. dvmax) dteos=dteos/2 end do end do return end

* transformation rate routine subroutine water_rate(w,x,tau,dwdt) implicit none double precision w,x(3),tau,dwdt double precision v1,v2,t double precision eosout(8),g1,g2,gtot,dg double precision R,MW parameter (R=8.314510D7,MW=18.02D0)

v1=x(1) v2=x(2) t=x(3) * current gibbs free energy call watereos1(1,[t,v1],eosout) g1=eosout(8) call watereos1(2,[t,v2],eosout) g2=eosout(8) gtot=w*g1+(1-w)*g2

dg=0.0d0 if (gtot .gt. g1) dg= -(g1-gtot) if (gtot .gt. g2) dg= +(g2-gtot) dwdt=dg*MW/(R*t)/tau

326 return end

* 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 none double precision fixedparams(3),x(3),bulk double precision x0(3),dx(3),f0x(3),axinv(3,3) double precision v,e,w,v1,v2,t double precision eosout(8),b1,kt1,p double 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,L2 integer maxiter,ii,jj logical done

* some error checking v=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 method do ii=1,3 x0(ii)=x(ii) end do maxiter=1000 L1=0.1 L2=20 done=.false. jj=1 do while (.not. done) if (jj.gt.maxiter) then write(*,*) ’max newton steps exceeded’ write(*,*) fixedparams write(*,*) x0 write(*,*) f0x stop ’water/ice model not converging’ end if call 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)*lambda if (x(ii).lt.xmin(ii)) then x(ii)=(x0(ii)+xmin(ii))/2.0d0 write(*,*) ’Lower limits of water model exceeded!’ write(*,*) ’ Newton iteration reset’ end if if (x(ii).gt.xmax(ii)) then x(ii)=(x0(ii)+xmax(ii))/2.0d0 write(*,*) ’Upper limits of water model exceeded!’ write(*,*) ’ Newton iteration reset’ end if x0(ii)=x(ii) eqnerr=dmax1(eqnerr,dabs(f0x(ii))/eqntol(ii)) end do if (eqnerr .lt. 1.0d0) then done=.true. end if jj=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*v return end

********************************************************************* * Water matrices subroutine watermatrices(fixedparams,x,fx,dx,axinv) implicit none double precision fixedparams(3),x(3),fx(3),dx(3) double precision ax(3,3),axinv(3,3),det double precision v,e,w,v1,v2,t,eosout(8) double precision cv1,cv2,b1,b2,kt1,kt2 double precision p1,p2,e1,e2 integer 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-p2 fx(2)=w*v1+(1.0d0-w)*v2-v fx(3)=w*e1+(1.0d0-w)*e2-e

ax(1,1)=-kt1/v1 ax(1,2)=kt2/v2 ax(1,3)=b1-b2 ax(2,1)=w ax(2,2)=1.0d0-w ax(2,3)=0.0d0 ax(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))/det axinv(1,2)=(ax(1,3)*ax(3,2)-ax(1,2)*ax(3,3))/det axinv(1,3)=(ax(1,2)*ax(2,3)-ax(1,3)*ax(2,2))/det axinv(2,1)=(ax(2,3)*ax(3,1)-ax(2,1)*ax(3,3))/det axinv(2,2)=(ax(1,1)*ax(3,3)-ax(1,3)*ax(3,1))/det axinv(2,3)=(ax(1,3)*ax(2,1)-ax(1,1)*ax(2,3))/det axinv(3,1)=(ax(2,1)*ax(3,2)-ax(2,2)*ax(3,1))/det axinv(3,2)=(ax(1,2)*ax(3,1)-ax(1,1)*ax(3,2))/det axinv(3,3)=(ax(1,1)*ax(2,2)-ax(1,2)*ax(2,1))/det do ii=1,3 dx(ii)=-axinv(ii,1)*fx(1)-axinv(ii,2)*fx(2)-axinv(ii,3)*fx(3) end do

return end

*********************************************************************

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 none integer phase double precision eosin(2),eosout(8) double precision t,v,cv,b,kt,p,e,s,f,g double precision x,ib,it0,pt0,dbdv,kt0,t0,e0,s0 integer nb1,ii parameter (nb1=7) * liquid phase params double precision v01,t01,p01,cv1,k01,g1,b1(nb1),e01,s01 parameter( + 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 params double precision v02,t02,p02,cv2,k02,g2,b2,e02,s02 parameter( + 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=0.0d0 ib=0.0d0 dbdv=0.0d0 select case (phase) * liquid phase case (1)

330 t0=t01 cv=cv1 x=v/v01-1.0d0 do ii=1,nb1 b=b+b1(ii)*x**(ii-1) ib=ib+b1(ii)*v01/dfloat(ii)*x**ii if (ii.gt.1) then dbdv=dbdv+b1(ii)/v01*dfloat(ii-1)*x**(ii-2) end if end do it0=(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)**g1 e0=e01 s0=s01 * ice VII phase case (2) t0=t02 cv=cv2 b=b2; dbdv=0.0d0 ib=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)**g2 e0=e02 s0=s02 case default stop ’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)+ib f=e-t*s g=f+p*v

eosout=[cv,b,kt,p,e,s,f,g] return end

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 = 3 ieos = 1 icut = 1 G0 = 46.92e+10 G1 = 1.873e+00 G2 =0.3459e-10 Yo = 1.000e+30 M = 0.000e+00 K0 = 43.19e+10 K1 = 156.2e+10 K2 = 48.60e+10 K3 = 0.000e+00 gamma = 0.675 pcut =-1.250e+09 cv = 0.670e+07 Fused silica density = 2.201 nmdef = 8 ieos = -1 icut = 1 g01 = 3.062e+11 g11 =-1.199e+12 g12 = 6.684e+11 gcut = 7.600e-02 K0 = 7.760e+11 K1 =-4.000e+12 K2 = 2.934e+13 K3 =-6.926e+13 pcut =-2.900e+08 Soda lime glass density = 2.493 nmdef = 3 ieos = 0 icut = 1 G0 = 2.813e+11 G1 = 0.000e+00 G2 = 0.000e+00 Yo = 1.000e+30 M = 0.000e+00 bulk = 4.793e+11 a-cut sapphire density = 3.985 nmdef = 3 ieos = 1 icut = 1 G0 = 178.9e+10 G1 = 1.512e+00 G2 =-7.260e-12 Yo = 1.000e+30 M = 0.000e+00 K0 = 259.1e+10 K1 =+5.203e+12 K2 =-2.061e+13 K3 = 0.000e+00 gamma = 1.280 pcut =-1.250e+09 cv = 0.670e+07 Metastable water density = 0.99705 nmdef = 21 ieos = -1 icut = 1 T0 =2.9815e+02 tau1 =1.0000e+00 tau2 =1.0000e+00 pcut =-1.000e+09 Equil water/ice density = 0.99705 nmdef = 21 ieos = -1 icut = 1 T0 =2.9815e+02 tau1 =1.0000e-09 tau2 =1.0000e-09 pcut =-1.000e+09 Normal ice/water density = 0.99705 nmdef = 21 ieos = -1 icut = 1 T0 =2.9815e+02 tau1 = 50.00e-09 tau2 =650.00e-09 pcut =-1.000e+09 PMMA density = 1.186 nmdef = 2 ieos = 1 icut = 1 K0 = 8.820e+10 K1 = 8.015+10 K2 = 3.838e+11 K3 = 0.000e+00 gamma = 1.130 pcut =-1.250e+09 Epoxy density = 1.186 nmdef = 2 ieos = 1 icut = 1 K0 = 8.820e+10 K1 = 0.00e+00 K2 = 0.000e+00 K3 = 0.000e+00 gamma = 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, 2nd edition, (1992).

333 334 Appendix H

Heat Dissipation in Solidification Processes Since 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 liquid This 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 heat cp 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 region x > 0 (Figure H.1). At time t = 0, all

335 T

Solid Liquid T T 1 2

X (t) T f 0

x x=0

Figure H.1: Layout of the Stefan problem Solidiﬁcation begins at the x = 0 interface at time t = 0. The x = 0 boundary is in contact with a thermal reservoir (T = TR). The phase interface (Xf (t)) moves the right, liberating latent heat as the solid grows.

336 portions of the liquid are supercooled to a temperature T0 < Tm, where Tm is the equilibrium melting temperature at the current pressure. Let solidiﬁcation begin on the x = 0 interface

at t = 0 and proceed to the right. The position of this interface is Xf (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:

∂ 2T ∂T κ i − i = 0 (H.1) i ∂x2 ∂t

κ ρ where i = Ki/ icpi is the thermal diffusivity of region i. 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.

dX ∂T ∂T ρL f = K 1 − K 2 (x = X ) (H.3) dt 1 ∂x 2 ∂x f

Here ρ is mass density and L 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.

x Ti(x,t) = Ai + Bi er f √ (H.4) 2 κit

Far ahead of the transition front (x → ∞), the liquid region remains at the original super-

cooled temperature T0, which gives the following form for T2.

x T2 = T0 − B2 er f c √ (H.5) 2 κ2t

337 Applying the ﬁrst interface boundary condition (Equation H.2) gives the following result. · ¸ Xf Tm = T0 − B2 er f c √ (H.6) 2 κ2t

For this condition to be valid at all times, Xf must be restricted to the form

√ Xf = 2λ κ2t (H.7) where λ is a constant. The growth velocity of the phase interface is thus nonlinear. r dX κ v = f = λ 2 (H.8) dt t

Rewriting the solution T2 in terms of λ:

T0 − Tm λx T2 = T0 − er f c (H.9) er f cλ Xf

In order to construct T1, the boundary condition at x = 0 must be speciﬁed. Two different cases are considered below.

Adiabatic case

If no heat may ﬂow out of the x = 0 boundary, the temperature at that point is set

by the transition front at t = 0. Since the front is at Tm for t > 0, the value of T2 must be a constant to satisfy Equation H.1.

T2 = Tm (H.10)

A relation for λ can be obtained using the second interface boundary condition (Equation

H.3). √ λ 2 c (T − T ) π λ er f cλe = P2 m 0 (H.11) L

338 Isothermal case In this case, it is assumed that the the x = 0 boundary is ﬁxed at a “reservoir” temperature TR. A form of T1 could be constructed using Equation H.4 as was done for T2 above. This will alter the equation that deﬁnes λ and change the rate at which the phase interface moves. To allow for simpler interpretation, consider instead the following solid temperature function. x T = T + (T − T ) (H.12) 1 R m R X 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. · ¸ √ 2 1 K c (T − T ) c (T − T ) π λ λ λ 1 P2 m R P2 m 0 er f c e 1 − 2 = (H.13) 2λ K2 L L

Table H.1 summarizes the results of the isothermal and adiabatic cases. In reverberation experiments, the x = 0 boundary would be a silica window that is initially at temperature TR ≈ 300 K. The two limits describe bounding cases because heat ﬂow from the sample would raise the interface temperature to a point between T0 and Tm. The true value of λ 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 of T0 at this pressure is 446 K, while the value of Tm 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 that cp2 ≈ 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 condition Result Adiabatic Isothermal Liquid temperature T = T − T0−Tm er f c λx 2 0 er f cλ Xf

x Solid temperature T1 = Tm T1 ≈ TR + (Tm − TR) X √ Interface position Xf = 2λ κ2t q κ λ 2 Growth velocity v f = t

λ solution Equation H.11 Equation H.13

340 The value of λ 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 of λ 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 of λ 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 of λ 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. When T0 = 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 1 µs 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 liquid In 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

1 This is certainly true while cP2 (Tm −T0) ≤ L. For very low values of T0, the system becomes hypercooled (i.e. the release of latent heat does not bring the material to its melting temperature), changing the nature of the solidiﬁcation problem [2].

341 F ad F (K =1 W/m⋅K) 1 iso 2 F (K =2 W/m⋅K) iso 2 0.9

0.8

0.7

0.6

F 0.5 iso F (K =2 W/m⋅K) iso 2 (K =1 W/m⋅K) 0.4 2

0.3

F 0.2 ad

0.1

0 0.001 0.01 0.1 1 10 λ

Figure H.2: Numerical solutions to Stefan problem The black line shows the right hand side of Equations H.11 and H.13 for an isentropic compression of liquid water to 5 GPa. Increases in the Fad shows the left hand side of Equation H.11 (adiabatic limit); Fiso shows the left hand side of Equation H.13 (isothermal limit).

342 1600

1400

1200

1000

Isothermal 800 X (nm) limit

600

Adiabatic 400 limit

200

0 0 200 400 600 800 1000 t (ns)

Figure H.3: Uniform solid size limits in freezing water The 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 tension γ required to form the solid-liquid interface. A characteristic length scale created by the these competing effects can be deﬁned as: γ l = (H.14) ρL 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 of l is on the order of 10−11 m. Thermal diffusion will thus play a considerable role even at the earliest stages of growth. Theoretical calculations suggest that growth instabilities become an issue 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 of 10−9 m, then it would be expected that stable growth will not be observed in the 10−5 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) T >T 0 m

heat flow

x=0

(b) T

heat flow

x=0

Figure H.4: Instabilities in supercooled solidiﬁcation Lines show snapshots of solid-liquid interface position. (a) Planar freezing is stable if the liquid ahead of the solid interface is warmer than the melting temperature. (b) Instabilities form when the liquid ahead of the interface is cooler than the melting temperature.

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 freezing processes. 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 pressures. J. Chem. Phys. 30, 643 (1959).

[5] G.S. Kell. Thermodynamic and transport properties of ﬂuid water. In Water: 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 high pressures 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 diffusion or heat ﬂow. J. Appl. Phys. 34, 323 (1963).

346