CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
A KINETIC MODEL OF MULTIPLE PHASE TRANSITIONS IN ICE
Gloria Cruz Leon, Suemi Rodriguez Romo, and Vladimir Tchijov
Centra de Investigations Teoricas, Universidad National Autonoma de Mexico, Campus Cuautitldn,
Av. Iro de Mayo, s/n, Cuautitldn Izcalli, Edo. Mexico, C.P. 54700 MEXICO
Abstract. Kinetic model for multiple non-equilibrium phase transitions in ice is used to study shockwave compression of ice. We use a one-dimensional wave-propagation code to describe the evolution
of shock waves in a specimen of ice and calculate the profiles of thermodynamic quantities.
model was used to study dynamic phase changes in
ice during the loading stage of the compression
cycle in the "0-dimensional" problem, i.e., the
problem of adiabatic compression of a Lagrangian
particle of ice. In [5], the model was developed by
including the equations of state of ices II and VII, in
order to cover a complete range of pressures
(0...3.6 GPa) obtained in the experiments of
Larson. Both loading and unloading stages of the
compression cycle, as well as the hysteresis on the
pressure-volume diagram,
were studied for
different initial conditions and rising times of the
pressure.
In the present paper, we first briefly describe our
kinetic model and then use it in one-dimensional
wave-propagation code. We present new results on
the flow-field profiles of pressure, particle velocity
and density in the compressed ice sample, and
discuss further verification of the model by the
method of nanoshocks and coherent anti-Stokes
Raman spectroscopy.
INTRODUCTION
Experimental studies of the response of water ice
to shock-wave loading indicate multiple nonequilibrium phase transitions in ice during the
compression cycle [1,2]. According to Larson's [1]
interpretation of his experimental data based on the
densities of shock states, (a) the onset of melting of
ice Ih occurs at pressures 0.15 to 0.2 GPa; (b) for
pressures between 0.2 and 0.5 GPa, the data suggest
the existence of a mixed phase of ice Ih and water1;
(c) for loading stresses between 0.6 and 1.7 GPa,
the final state is predominantly ice VI; (d) the
results at 2.4 and 3.6 GPa suggest a mixed phase of
ice VI and ice VII; (e) the unloading paths indicate
that ice VI remains in a metastable form during
most of the unloading stage; the latter gives rise to a
considerable hysteresis in the compression cycle.
A kinetic model for multiple phase changes in ice
subjected to shock compression was proposed by
Tchijov [3] and further developed by Tchijov et al
[4,5]. The model is based on the set of the
equations of state and thermodynamic functions of
liquid water and solid ice polymorphs (namely, ices
Ih, II, III, V, VI, and VII), analytical description of
the pressure-temperature diagram of water
substance, and kinetic equations for the rates of
transitions between pairs of ice phases. In [3,4], the
KINETIC MODEL
Let P, V, and T be pressure, specific volume, and
temperature, respectively. The region Q = {(P, 7):
0 < P < 4.0 GPa, 230 K < T < 500 K} on the P-T
diagram of water substance is composed of the
It should be mentioned that there exists another interpretation
of Hugoniot data for water ice at low pressures (see [2]).
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domains of stability of liquid water and seven solid
ice polymorphs, ices Ih, II, III, V, VI, VII, and VIII2
[6]. On the other hand, the region Q is of our
special interest since it covers the intervals of
pressure and temperature of the shock-wave
experiments of Larson [1]. Let *F be a set of indices
1,2,3,5,6,7, and w which refer to ices Ih, II, III, V,
VI, VII, and liquid water, respectively. The region
of thermodynamic stability of the phase /e*F is
denoted hereafter as Q/.
A general approach to investigation of the
propagation of shock waves in a condensed matter
with N transforming phases has been proposed by
Hayes [7] and developed by Tchijov et al. [3-5] as
applied to ice (see [4] for the details of the
assumptions of the model). Let jc/ (0 < #, < 1) be a
mass fraction of the phase / e *F in a nonequilibrium mixture of transforming phases. The
specific internal energy E and the specific volume V
of the mixture are calculated by the formulae
£ = £*,£,. (P,r),
?eT
K = ]£*,F,(^n
belongs to Q;. Otherwise, yy- = 0. In other words, all
ice phases that exist in a non-equilibrium mixture at
a given moment of time transform into an ice phase
to whose region of thermodynamic stability belong
the current values of P and T. In (3), Ay and By are
constants, and fy(T) are known functions of
temperature. The values of Ay and By are chosen
such that the results of computer simulations of the
"0-dimensional" problem match the P-V data of
Larson (see [3,4] for details).
In the next section, we use kinetic model (2), (3)
of multiple phase transitions in ice together with
one-dimensional hydrocode to study pressure,
particle velocity, and density profiles of a shock
wave propagating through a specimen of ice.
ONE-DIMENSIONAL SHOCK WAVE
Let time t and spatial coordinate z (0 < z < LQ) be
Lagrangian variables. The mass and momentum
balance equations, and the energy balance equation
for a mixture of ice phases in the adiabatic
approximation can be written as
(1)
/e^
where EfJPJ) and VfJPJ) are the specific internal
energy and the specific volume of the /-th phase,
respectively. All the necessary P-V-T equations of
state and thermodynamic quantities of ices Ih, II,
III, V, VI, VII, and liquid water are now available
and can be found in [8-12].
Denote by y// the rate of transition of the phase /
into the phase j (/, j E ^P). Assuming y/, > 0, we can
write a kinetic equation for jc/:
^
dt
— ——— — —— ^
— —— — — ——— ^
——— — — f ——— ^
vv
where u stands for the particle velocity. The
specific volume V and the internal energy E of a
mixture are given by the relationships (1).
Equations (4), (2), and (1) form a system of
differential and algebraic equations for unknown
functions w, F, £, P, r, x\y ..., xj, xw and are used to
calculate the profiles of pressure, density, particle
velocity, and the phase composition of an ice
sample compressed by shock waves.
To describe the process of propagation of a shock
wave through a specimen of ice, we consider the
following initial boundary value problem. For t > 0,
0 < z < LQ, find the functions w, F, E, P, T, xt (i e T)
that satisfy the system of equations (4), (2), and (1)
together with the boundary condition w(0,0 = UQ (t
> 0) and initial conditions u(zJS) = 0, F(z,0) = F0,
£(z,0) = £o, />(*,<)) = PO, T(z,0) = TO, (0 < z < L0)9
(2)
where t stands for time. The functions y// are defined
by the relationships:
(3)
where the first line in (3) holds if #,- > 0 and at the
same time the point (P,7) on the P-T diagram
T (7 0^ := T
f Z ^ M"'^
The partial differential equations (4) are
integrated by the method of finite differences [13].
In the present paper, ice VIII is not considered.
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After the new values of u, F, and E are deduced at
time t + Ar from the state of the body at time t, the
semi-explicit index one differential-algebraic
system (1), (2) is solved [14] to find the new values
of P and T and the new phase composition jt7- of the
mixture at time t + At. The values of the time step
A/ are chosen according to stability conditions.
and the density p = \IV (Fig. 2) are given as
functions of the Lagrangian variable z, at different
moments of time.
One can see a complex
multiwave structure of a shock wave propagating
through a specimen of ice. The first wave that
carries ice to its melting point is followed by a
centered compression fan
which is due to
anomalous compressibility of the ice I - water
mixture [8]. The final values of pressure and
density obtained in this computational experiment
are P = 1.20 GPa, p = 1.338xl03 kg/m3. The
corresponding values reported by Larson [1] are P
= 1.27 GPa andp= 1.355xl03 kg/m3.
Figure 3 displays the calculated values of the
particle velocity u(z, f) at depths z\ = 3.139 mm and
z2 = 6.268 mm. The values of z\ and z2 are chosen
equal to the coordinates of the gages reported in [1].
The profiles of u(z\J) and u(z2,i) are quite similar to
those of Larson (see Fig. 3 in [1]). Note that the
digitized particle-velocity-time data that would
permit a more detailed comparison between the
calculated and the measured values of u(z, t) are not
presented in [1].
Calculations performed for other values of UQ give
qualitatively similar profiles of pressure, particle
RESULTS AND DISCUSSION
In accordance with the flow-field data of Larson
[1], we assume the initial thickness of an ice
specimen LQ = 10~2 m, initial pressure P 0 = 0.1 MPa,
initial temperature T0 = 263 K, and initial fractions
of ice phases jc10 = 1, # 2 o = 0, ... , xw0 = 0. The values
of FO and EQ are calculated using the relationships
(1). The values of the particle velocity UQ at the
boundary z - 0 are taken from the experimental data
[1]. The calculations are performed until the shock
front reaches the rear surface z - L0.
A series of calculations has been performed for
different values of the particle velocity UQ. As an
example, Fig. 1-3 display the results obtained for u0
= 650 m/s. The profiles of the pressure P (Fig. 1)
1.2-1
1.4-1
1.30.8-
S.
o
I
0.4-
1-
I
0.2
1
I
0.4
'
I
T
0.6
0.8
0.9-
I
0.2
I
0.4
0.6
0.8
Sample size, cm
Sample size, cm
FIGURE 1. Calculated pressure profiles of shock wave traveling
through a specimen of ice. The numbers 1 through 7 correspond
to the state of compressed ice at different moments of time: 1 4.29xl(T7 s, 2 - 8.53xl(T7 s, 3 - 1.27xlO'6 s, 4 - \.69xW6 s, 5 , 6-2.53xlO' 6 s, and 7- 2.95xl(T 6 s.
FIGURE 2. Calculated density profiles of shock wave traveling
through a specimen of ice. The numbering of the curves is the
same as in Fig.l.
243
by the grant No. INI01598. VT wishes to thank Dr.
Edward S. Gaffiiey for fruitful discussions.
0.8-1
0.6-
REFERENCES
E
1. Larson, D. B., J. Glaciology, 30, 235-240 (1984).
2. Gaffney E. S., In: Ices in the Solar System, J, Klinger
et al (eds.), D. Reidel Publishing Company, 1985, pp.
119-148.
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Phys. 36, 933-938 (1995) (translated from Russian
Prikl. Mekh, i Tekhn. Fiz, 36, 158-164 (1995)).
4. Tchijov, V., Keller, J., Rodriguez Romo, S., and
Nagornov, O., J. Phys. Chem. B, 101, 6215-6218
(1997).
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Tchijov, V., Entropie (in press).
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Oxford University Press, Oxford, 1999, pp. 252-286.
0.4-
CO
Q.
0.2-
1
2
'
I
3
Time, |is
FIGURE 3. Calculated particle velocity time histories at
distances z\ = 3.139 mm (curve 1) and z2 = 6.268 mm (curve 2)
from the front surface of an ice sample.
7. Hayes, D. B., J. Appl Phys,, 46, 3438-3443
(1975).
8. Nagornov, O. V., and Chizhov (Tchijov)V. E., J.
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Phys. 34, 253-263 (1993) (translated from Russian
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J. Phys. Chem. Solids (in press).
velocity, and density (not presented here because of
the restricted volume of this paper). The final P-V
states of compressed ice are in reasonable
agreement with the experimental data of Larson [1].
Concluding the paper, we note that a kinetic
model of multiple phase transitions in ice provides a
far more detailed information on the processes
behind the shock front than that obtained from the
conventional flow-field measurements. It predicts,
in particular, the dynamics of phase composition
Xi(z,f) (i € *F) in a specimen of ice compressed by a
shock wave. The inference of phase composition
directly from the raw flow-field measurements is
risky for shocks in ice because of the complexity of
the phase diagram of water substance. A new
experimental technique of laser-driven nanoshocks
and coherent anti-Stokes Raman spectroscopy [15]
permits to determine the shock front rise time,
shock temperature, pressure, velocity, and phase
composition of a compressed material. The
application of this technique to ice will permit a
definitive verification of our kinetic model and its
parameters.
11. Fei, Yingwei, Mao, Ho-kwang, Hemley, R. J., J.
Chem. Phys. 99, 5369-5373 (1993).
12. Saul, A., and Wagner, W., J. Phys. Chem. Ref. Data,
18, 1537-1564(1989).
13. Ames, W. F., Numerical Methods for Partial
Differential Equations, 3rd Edition, Academic Press,
San Diego, 1992, pp. 315-317.
14. Brenan, K. E., Campbell, S. L., and Petzold L. R.,
Numerical Solution of Initial-Value Problems in
Differential-Algebraic Equations, SIAM, Philadelphia,
1996, pp. 115-148.
15. Dlott, D. D,, Hambir, S., and Franken, J., J. Phys.
Chem. B, 102, 2121-2130 (1998).
ACKNOWLEDGMENTS
The authors are grateful to PAPIIT DGAPA
UNAM, Mexico for financial support of this work
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