Introduction
One of Nature’s most valued components is the life-sustaining chemical water. This familiar yet unusual substance not only plays an important part in all biological processes but is important to geological processes as well. Water’s geological role is most dramatically demonstrated in the Earth’s cold regions or permafrost country. Although water and its large number of high-pressure phases (Reference BridgmanBridgman, 1911, Reference Bridgman1912, Reference Bridgman1914, Reference Bridgman1937) are intriguing from a pure science point of view, our main motivation for studying the behavior of ice under shock-wave loading stems from seismic monitoring application. Our interests are in understanding the mechanisms of the processes occurring in ice and frozen soil that control the coupling of explosion energy and therefore the seismic signal amplitude that propagates through the Earth from frozen regions.
The objective of this paper is to report results from an experimental study of plane shock-wave loading of single and polycrystalline samples of ice which were initially at ambient pressure and a temperature of −10 ± 2° C. The complexities associated with water are illustrated by the phase diagram shown in Figure 1 (Reference BridgmanBridgman, 1942). Ice I is the familiar hexagonal form and ice VII is believed to be the ultimate close-packed high-pressure phase (Reference Kamb and DavisKamb and Davis, 1964). In an earlier paper (Reference Larson, Larson,, Bearson, and TaylorLarson and others, 1973), we presented data that suggested that ice I at −10° C under shock loading is transformed to the liquid phase at stresses above approximately 0.2 GPa. Evidence was also given for the possible existence of a high-pressure solid phase. The research reported in this paper, with its more extensive study of ice I at −10° C under dynamic loading, suggests the composition at most shock states is complex with the onset of melting and the formation of a mixed phase probable above 0.15 – 0.2 GPa. Moreover, these data suggest that at stress levels between 0.6 and 2.2 GPa, there is a transformation to ice VI which is nearly complete and is held in a metastable state during most of the unloading process.
Fig. 1. Phase diagram of water.
Experimental Procedures and Results
Shock-wave experiments were conducted using a light-gas gun to obtain loading of the ice samples under conditions of uniaxial strain. Flat-faced projectiles were accelerated with compressed gas before they impacted the target assembly shown in Figure 2. The shock wave produced upon impact loads the sample; several microseconds later, a rarefaction from the free surface causes unloading, whereby the sample is relaxed to ambient stress conditions. The diagnostic velocity gages used were thin (2.5 × 10−2 mm) brass foils that were placed on the surface and also sandwiched between the plates of the target material. Because these gages were thin, they equilibrated to the surrounding material velocity within nanoseconds, thus following the motion of the sample material as both loading and unloading occurred. The foils were also constrained to move perpendicular to the field produced by a large external electromagnet. Under these conditions, the history of particle velocity with time,U p(τ) in kilometers per second, is proportional to the recorded electromotive force E (τ) in volts :
where B is the magnetic field strength in kilo-gauss and L is the gage-element length in centimeters. Both B and L are measured and have a combined uncertainty of less than 1% for these experiments. During the fabrication of the experimental assemblies, the gages were placed on carefully machined surfaces (flat and parallel to approximately 0.008 cm) of plates of ice. A thin film of water was used as a bonding agent to hold the plates together and fill any voids between plates.
Fig. 2. Schematic of gas-gun experiment showing projectile, target assembly, and poles of external electomagnet.
The projectile insert and target buffer plate were made of polymethylmethacrylate (PMMA). The temperature was maintained at −10 ± 2° C by the controlled circulation of cold nitrogen vapors through a cooling coil mounted in an aluminum housing. The PMMA buffer plate and the aluminum vacuum plate were used to insulate the ice surfaces. A thermocouple was mounted in the ice to record sample temperature. In the enclosed assembly, a void between the rear plate of the ice and the rear of the target assembly provided the free-surface conditions necessary to unload the sample in uniaxial strain. The experimental assemblies were mounted mechanically between the magnet poles near the end of the barrel and in the vacuum chamber.
The data from each of the experiments were recorded as voltage-time histories of oscilloscope traces on Polaroid film. These records were digitized and then converted from voltage-time histories to particle-velocity-time histories using Equation (1). The conversion of these particle-velocity-time histories to equation-of-state data requires expressions relating particle-velocity history to the material properties of interest (i.e. stress and specific volume). These expressions can be derived by using the conservation equations for uniaxial flow. The equation for conservation of mass and linear momentum can be expressed as (Reference Cowperthwaite and WilliamsCowperthwaite and Williams, 1971).
where ρ is the density (the subscript 0 refers to initial conditions), V is specific volume, σ is the axial stress, h is the Lagrangian space coordinate, t is the time, and
In the analysis, it was assumed that the non-steady flow observed in ice samples was that of simple waves. In the case of non-steady simple waves, the phase velocities at constant stress and constant particle velocity, Cσ
and 
respectively, are equal but are functions of particle velocities rather than constants. In this case, the digitized particle-velocity-time data from the experiments can be used to compare Cσ
and 
and the numerical integration of Equations (2) and (3) can be used to obtain the relationship between stress and specific volume through the non-steady wave.
The ice used in these experiments was from two sources. The polycrystalline ice samples were made from blocks of commercial ice, while the single crystal ice was obtained from Dr K. Itagaki of the U.S. Army Cold Regions Research and Engineering Laboratory, Hanover, New Hampshire.
A typical series of particle-velocity-time histories for a single experiment is shown in Figure 3. The first gage, located at the interface between the buffer plate and ice, shows the particle velocity imparted to the ice. The subsequent gages show the development of the wave profile, including the rarefaction from the free surface. The precursor wave which develops as the wave propagates in ice is attributed to the onset of melting. This wave propagates at a velocity of 3.63 km/s which is about 10% greater than the longitudinal sound speed. The dispersive-like feature of the wave following the precursor is associated with compaction of the material. This feature is followed by a steeply rising steady wave which takes the material to its final loading state. The last feature of the wave profile is the rarefaction wave with its ramp-like characteristics which result from unloading the ice beginning at the free surface.
Fig. 3. Particle velocity time histories recorded at ice-buffer-plate interface and at depths of 3.139 mm and 6.268 mm in polycrystalline ice.
Figure 4 shows particle-velocity-time histories for two gages embedded in single-crystal ice and loaded perpendicular to the c-axis along the a-axis or <1010>. In these records, the precursor moving at a velocity of 4.00 km/s is clearly defined. The precursor is followed by a ramp-like feature. However, the final loading stress is not large enough nor the compaction great enough to establish conditions where a second shock is obtained. The final feature on these records is the rarefaction or unloading wave.
Fig. 4. Particle-velocity history of two embedded gages in single-crystal ice loaded uniaxially perpendicular to c-axis (arrows indicate arrival of rarefaction wave at gages).
In Table I and Figures 5 and 6, the data from the various experiments are presented based upon analysis using Equations (2) and (3) and the assumption of simple non-steady waves. The data from single-crystal samples show little or no effect of crystal orientation on either the stress amplitude of the precursor or the specific volume at which it initiates. The implication is that differences in strength due to crystal orientation play little or no part in the development of the precursor at 0.16 to 0.19 GP a.
Fig. 5. Shock-wave compression data for single-crystal ice shocked (a) along 
and (b) along <0001> axis, starting at −10° C.
Fig. 6. Stress versus specific volume data for ice (dashed lines show uniaxial loading and unloading paths followed in each experiment; solid line is average precursor path for all experiments).
In Figure 6, the data from all experiments both single crystal and polycrystalline are shown. The figure shows not only the development of the cusp around 0.2 GPa but also the tremendous von that occurs at loading stresses above about 0.6 GPa. This compaction is followed by an abrupt stiffening of the material for stresses from about 0.6 to 3.6 GPa. The unloading paths measured at these larger stresses, i.e. greater than 0.6 GPa, indicate that the high-stress component produced upon loading is maintained in a metastable state through most of the unloading process.
Discussion of Results
Some insight into the significance of the data shown in Figure 6 can be obtained through comparison with the −10° C isotherm for water. Figure 7 shows this comparison and indicates various high-pressure phases that have been identified from static compression studies.
Fig. 7. Comparison of shock-wave data and −10° isotherm for ice.
The cusp or “break” in the shock-wave compression curve at 0.15 to 0.2 GPa has been interpreted (Reference Larson, Larson,, Bearson, and TaylorLarson and others, 1973; Reference LarsonLarson, 1978) as arising from the onset of melting of ice I. An alternative interpretation would be the onset of dynamic yielding. However, as mentioned in the previous papers, the precursor which gives rise to the cusp has a wave velocity which is significantly (10% or more) greater than longitudinal sound speed. This is inconsistent with other observations of dynamic yielding where the precursor velocity is equal to longitudinal sound speed. Moreover, the data on single crystals show that the precursor amplitude is unaffected by orientation. If the precursor were caused by dynamic yielding, such in-sensitivity would not be expected to occur because during static loading the strengths of single crystals differ by a factor of two or more depending upon orientation (Reference ChambeotainChamberlain, 1967). Furthermore, static values for strength at −10° C are of the order of 10 MPa (Reference Shōji and HigashiShōji and Higashi, 1978) and are not likely to increase to the order of 0.2 GPa even at the strain-rates of these experiments. However, recent data by Reference Durham, Durham,, Heard, and KirbyDurham and others (in press) indicate that ice near −10° C shows more than a doubling of strength as strain-rate increases from 3.5 × 10−6 s−1 to 3.5 × 10−4 s−1. Whether this strain-rate effect can be extrapolated to strain-rates of 10+5 s−1 is conjecture; thus dynamic yielding remains as an alternative explanation for the precursor wave. However, even with these new data, our interpretation remains the same, i.e. the precursor observed when shock loading ice I is attributed to the onset of melting of ice I. The onset of melting creates a mixed phase and absorbs energy from the system as latent heat of melting. This “loss” of energy results in a lowering of the average temperature of the system which may explain the higher threshold stress for transformation observed in these dynamic experiments.
TABLE I. Shock-Wave Data for Ice
The final loading states between 0.6 and 1.6 GPa seem to suggest near complete transformation to ice VI. In a previous paper (Reference Larson, Larson,, Bearson, and TaylorLarson and others, 1973) we suggested that ice V may be the final state for the point near 0.7 GPa but re-analysis of those data and these new results suggest that ice VI not ice V is the final state. The small but consistent offset between the isotherm for ice VI and the final states deduced from the shock data is consistent with the expected higher temperatures that would occur along the Hugoniot. However, an alternative but less likely explanation would be incomplete transformation to ice VI. The unloading paths determined from these data show a strong hysteresis and imply that the ice VI state is maintained as a “frozen” or metastable state during most of the unloading process.
The comparison made in Figure 7 suggests that stresses above 2.2 GPa should produce the onset of a transformation to ice VII. The shock-wave data at 2.4 and 3.6 GPa are consistent with that expectation since they are displaced to the left of the lower-stress ice VI data but also suggest the possibility of a partial transformation or a mixed-phase region.
Figure 8 shows a possible path on the phase diagram of water for shock loading of ice I from −10° C. Also shown is a Hugoniot for liquid water calculated by Anderson (Reference Larson and AndersonLarson and Anderson, 1979). Transformation of liquid water into the high-pressure phase is unlikely unless the water is cooled to near 0° C and then only at stresses of the order of 1.5 to 2.0 GPa would one expect to observe the formation of ice VI or ice VII. Reference Kormer, Kormer,, Yushko, and KrishkevichKormer and others (1968) report the formation of ice VII but they could obtain this phase only by double shock-wave compression of liquid water.
Fig. 8. Phase diagram for water with Hugoniot calculated for water at 20° C and possible path for shock-loading of ice I from −10° C.
Summary and Conclusions
All of the experiments described in this study show the onset of melting at 0.15 to 0.2 GPa. Experiments with maximum stress states between 0.2 and 0.5 GPa yield results which suggest that a mixed phase of ice I and liquid water exists at these confining stresses. For maximum loading stresses between 0.6 and 1.7 GPa the experimental results suggest that the final state is predominantly ice VI. In these experiments the specific volume upon compression is changed from 1.09 m3/Mg to approximately 0.76 m3/Mg, which represents compaction of approximately 30%. The unloading paths determined from these experiments indicate that ice VI remains in a “frozen” or metastable state during most of the unloading process. This hysteresis in the compression cycle gives rise to a large “loss” of shock-wave energy to the transformation process. At stress levels above 2.2 GPa, ice VII should be the stable form for water according to static compression measurements. Experimental data at 2.4 and 3.6 GPa suggest that ice VII may be formed but these results indicate a mixed phase of ice VI and ice VII rather than complete transformation to ice VII.
Dynamic or shock-wave loading of ice form −10° C produces high-pressure phases which in general remove a large fraction of the energy from the propagating shock wave. These processes begin at a stress of about 0.15 GPa with the onset of melting and continue with transformations to ice VI and ice VII for stresses in excess of 0.6 GPa. Liquid water, on the other hand, is unlikely to transform along the Hugoniot unless cooled to near 0° C and then only at stresses of the order of 1.5 to 2.0 GPa where transformation to ice VI or ice VII may occur.
Acknowledgements
I would like to acknowledge the invaluable support of J. Taylor, G. Hickman, G. Bearson, and E. Britton for their outstanding work in fabrication and assembly of the various experiments. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. I also appreciate the financial support of the Defense Advanced Research Projects Agency.
