Jave Kane

Raymond Smith

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UCRL-PROC-215696
Modeling non-equilibrium phase
transitions in isentropically
compressed Bi
Jave Kane, Raymond Smith
September 27, 2005
14th APS Topical Conference on Shock Compression of
Condensed Matter
Baltimore,, MD, United States
August 1, 2005 through August 5, 2005
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Modeling non-equilibrium phase transitions in isentropically compressed Bi
J. O. Kane1, R. F. Smith2
1Lawrence Livermore National Laboratory, L-473, 7000 East Avenue, Livermore CA 94551
2 Lawrence Livermore National Laboratory, L-286, 7000 East Avenue, Livermore CA 94551
Abstract. We report here on modeling of non-equilibrium phase transitions in Bi samples
isentropically compressed to 120 GPa by a ramped drive, which is produced using the Janus laser. In
the experiments, the Bi samples are attached to windows of LiF or sapphire, and the velocity history of
the sample-window interface is recorded with line VISAR. The 1D response of the targets is modeled
using a multiphase Bi EOS, the Andrews-Hayes method for non-equilibrium transitions, and a
Boettger-Wallace kinetics model. The pressure drive is deduced by back integration of VISAR data
from shots performed with Al samples.
Keywords: Phase transitions, rate dependence, Bi, Andrews-Hayes, laser ICE, isentropic compression.
PACS: 64.70.Kb, 64.70.Dv.
INTRO
Isentropic compression experiments (ICE)
have been performed with a variety of drivers,
including pulsed power machines, gas guns, and
lasers [1, 2, 3, 4]. At the strain rates produced in a
compressed sample in any of these experiments,
first-order phase transitions are easily overdriven.
Furthermore, the strain rates produced in an
isentropically compressed sample can vary by
orders of magnitude across these different
experimental facilities.
As a result, the pressure and temperature
trajectories of a compressed sample can deviate
significantly from the equilibrium isentrope, and
the particle velocity of the sample material can
exhibit rate-dependence effects. Therefore,
modeling of wave propagation in the compressed
sample must account for non-equilibrium phase
transitions and involve rate-dependent kinetics.
Recently, Bi ICE experiments were performed
at the Sandia Z machine [1]. In the present
discussion we present modeling of Bi ICE
experiments performed at the Janus laser at
Lawrence Livermore National Laboratory. One
goal of the experiments modeled here is to allow
cross-platform study of rate-dependence in non-
equilibrium phase transitions of Bi and other
materials.
Figure 1. Experiment. The reservoir is 75 µm polyimide,
the gap width is 425 µm, and the Bi samples are from
10–30 µm thick. The on-target intensity was 3.8 x 1012
W/cm2 in a 2ns square pulse of 527 nm light.
EXPERIMENTAL PROCEDURE
The Janus experiment is depicted
schematically in Fig. 1. The sample is loaded with
a ramped laser ICE drive. The drive is produced
laser
VISAR
reservoir
window
gap
Bi sample
when shock loaded reservoir material is released
across a gap and then stagnates on the gap-facing
side of the sample. In the present experiments, the
peak pressure of the drive is 12 Gpa (120 kbar).
The velocity history of the sample-window
interface is recorded using line VISAR [5, 6]. A
cartridge heating system is employed to select the
initial temperature (between room temperature and
melting) of the sample. The experiment is
described in detail in [7].
Figure 2. Bi EOS used for present modeling. Shaded
regions: stability fields of indicated phases. White lines:
equilibrium isentropes. Solid black line: equilbrium
isentrope through atmospheric pressure and 200 °C.
Dashed black line: non-equilibrium model trajectory of
Bi zone next to window, with Bi initially at atmospheric
pressure and 200 °C.
MODELING
To model 1D wave propagation in the Bi
samples, we use the Andrews-Hayes method [8, 9,
10] to advance material zones in (P, T, x) space,
where P  is pressure, T  is temperature, and the
components of the vector x are the mass fractions
of the individual phases. This method assumes that
all phases within a zone are individually in
thermodynamic equilibrium, and all phases are at a
common P, T. Within a zone, extensive quantities
such as energy and entropy are the mass fraction-
weighted sums of the quantities corresponding to
the individual phases. We do not model strength, or
effects of grain size and orientation. Steep waves
are stabilized by Von Neumann quadratic and
linear artificial viscosity.
In the modeling, we apply a pressure drive to
the gap-facing side of the target. The drive is
deduced by back integration [11] of VISAR data
from shots performed with Al samples in place of
the Bi.
The Bi EOS (Fig. 2) contains five solid phases
and a liquid phase. Phases I, II and the liquid are
the Johnson Bi EOS [12]. Phases III, IV and V
have the same form as phase II, with Johnson
parameters chosen to reproduce roughly phase
boundaries, Clapeyron slopes, volume changes and
cold curves prescribed in [13, 14, 15].
In the present discussion, we use a Boettger-
Wallace [16] kinetics model, which balances the
Gibbs driving force with a reverse stress due to the
volume change, creating ‘metastable surfaces’ that
essentially replace the equilibrium (P, T) phase
lines.
The experimental trajectories potentially
involve numerous stable and metastable phase
transformations. Since there is little guidance in
seeking a detailed multi-parameter Boettger-
Wallace kinetics model, for simplicity we set all
rate constants τ to a single value, all activation
energies AF to zero, and all energy scale parameters
BF to a single value.
We model the LiF using a Mie-Gruneisen
EOS.
Figure 3. Modeling (heavy line) vs. VISAR data for Bi
next to window, with Bi initially at atmospheric pressure
and 200 °C.
RESULTS
In Fig. 3 we compare a typical piece of
VISAR-derived particle velocity data to our
modeling. The Bi sample was initially at
atmospheric pressure and a temperature of 200°C.
Time (ns)
Pa
rt
ic
le
 v
el
oc
ity
 (
km
/s
)
T
em
pe
ra
tu
re
 (
K
)
Pressure (kbar)
(kbar)
The window was LiF. The particle velocity history
shows a familiar two-wave structure, with a
‘plateau’ and subsequent rise. In Fig. 2 we show
the predicted (P, T) trajectory of the Bi
immediately adjacent to the window, as well as the
equilibrium isentrope derived from the EOS.
In Fig. 2, the model trajectory is initially
isentropic. However, once the phase I stability field
is exited, a non-equilibrium mixture of phases is
present, and in (P, T) space the material follows a
non-equilibrium isentrope instead of the
equilibrium isentrope. The particular non-
equilibrium isentrope followed depends upon the
compression history of the material, the equation of
state, and the kinetics model. The steep rise in
particle velocity produces entropy, after which the
material cannot return to the equilibrium isentrope.
As Fig. 3 shows, the model reproduces
features of the observed VISAR particle velocity
record, including the sloped plateau and the
subsequent rise. Equilibrium modeling — with
infinitely fast transition to the phase having lowest
Gibbs free energy (not shown) predicts a flat
plateau; see [1] for another perspective on the slope
of the plateau. Generally for these experiments, the
rise in the data has a more abrupt onset and a
steeper slope (shock-like) than the modeling
predicts. Finding the source of this disagreement
may be useful in evaluating candidate EOS’s, the
applicability of the Andrews-Hayes method (see
discussions in [9, 10]), and possible effects related
to mechanisms of elastic or plastic deformation.
It should also be noted that while a two-wave
structure occurs naturally when there is a single
phase transition, for these experiments, the
modeling predicts that the material trajectory will
cross more than one equilibrium phase boundary,
as well as various metastable boundaries. Thus, in
general a multi-wave structure is potentially
possible, and its genesis may involve both stable
and metastable transitions. For examples and
discussion relevant to these points, see [9].
Therefore, in the present modeling, the
apparent two-wave structure is a complex product
of the EOS, the kinetics model, and the
compression history. Furthermore, reflected waves
at the sample-window interface somewhat modify
the trajectory from the in-situ (no window)
trajectory.
CONCLUSIONS
We have reported on modeling of laser ICE
experiments in which Bi was isentropically
compressed to 12 Gpa starting from atmospheric
pressure and initial temperatures between room
temperature and melting. The very simple non-
equilibrium modeling and multi-phase Bi EOS
discussed here appear to qualitatively reproduce
features of the VISAR data.
ACKNOWLEDGEMENTS
This work was performed under the auspices of the
U.S. Department of Energy by the University of
California, Lawrence Livermore National
Laboratory under Contract No. W-7405-Eng-48.
REFERENCES
1. M. Bastea, S. Bastea, J.A. Emig, P.T. Springer,
D.B. Reisman, “Kinetics of propagating phase
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2. L.C. Chhabildas and L.M. Barker, Sandia Report
SAND86-1888 (1986)
3. J.R. Asay, Int. J. Impact Engng. 20, 27 (1997)
4. J. Edwards et al., Phys. Rev. Lett., 92 (7), 075002
(2004).
5. Barker, L. M., and Hollenback, R. E., “Laser
Interferometer for Measuring High Velocities of
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4669, 1972.
6. Bloomquist, D. D. and Sheffield, S. A., “Optically
recording interferometer for velocity-measurements
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7. Smith, R. F. et al., in preparation for submittal to
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8. D.J. Andrews, J. Comp. Phys. 7, 310–326 (1971).
9. D.B. Hayes, J Appl. Phys. 46, 3438–3443 (1975).
10. Duvall, G. E. Duvall and R. A. Graham, Rev. Mod.
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11. D.B. Hayes, Backward Integration of Equations of
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12. J. N. Johnson et al., J. Phys. Chem. Solids 35, 501
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13. Klement et al., PR 131 (2), 632 (1963).
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Modeling Non-Equilibrium Phase Transitions

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Modeling Non-Equilibrium Phase Transitions

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