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
HUGONIOT CONSTRAINT MOLECULAR DYNAMICS STUDY OF
A TRANSFORMATION TO A METASTABLE PHASE IN SHOCKED
SILICON
Evan J. Reed1, J. D. Joannopoulos1, and Laurence E. Fried2
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
2
L-282, Chemistry and Materials Science Directorate, Lawrence Livermore National Laboratory,
Livermore, California 94550
Abstract. A molecular dynamics constraint based on the Hugoniot is formulated for the study of
shocked materials. The constraint allows the simulation of a small part of the system under shock
conditions for longer time periods than the use of a moving piston which creates a shock wave in a
large system. It also allows for the use of quantum mechanical methods for which the work scales
rapidly with system size. The constraint is used as a first approximation to study dynamical effects in
shocked silicon. Transformations through metastable states are found under some conditions for
shocks in the [111] direction using a tight-binding model. A new six-fold coordinated phase of silicon
is found to be metastable under hydrostatic conditions using density functional theory.
INTRODUCTION
Theoretical approaches to extreme atomic scale
phenomena in covalent materials like silicon
usually require a thorough treatment of the
quantum mechanical aspect of the problem.
Unfortunately, quantum approaches often require
computational work that increases rapidly with the
size of the computational system.
We focus here on crystalline silicon as an
exemplary material where the accurate treatment of
extreme atomic scale phenomena is difficult.
Silicon has a complex phase diagram with many
metastable structures, allowing for the possibility
that the system can be shocked into a metastable
state. Shock experiments that have been performed
on the diamond phase of silicon indicate an
orientational anisotropy in the shock wave
structure (1). Other experiments suggest that
shocked silicon exhibits a metallic conductivity at
pressures above the Hugoniot elastic limit
(pressure above which plastic deformation occurs)
and below the first equilibrium phase transition to a
metallic state (2).
Traditional approaches to atomistic simulations
of shock compression involve creating a shock in a
very large system and allowing it to develop until it
reaches the other side. This approach allows the
shock to evolve for a time that scales linearly with
the size of the system, i.e. the time required for the
shock to traverse the computational system.
Therefore the maximum simulation time is linked
to the scaling of the method used to evaluate the
forces on the atoms with system size. In the best
possible scenario, where the work required to
evaluate the atomic forces scales linearly with the
system size, the work required to simulate a shock
is quadratic in the evolution time. When quantum
mechanical methods with poorer scaling with
system size are employed, this approach to shock
simulations quickly becomes impossible.
velocity aim to simulate the conditions of the
material near an explosive charge or flyer plate.
The thermal constraint has the form,
HUGONIOT MOLECULAR DYNAMICS
CONSTRAINT
An alternative approach is to attempt to
simulate the effects of the shock wave passing over
a small piece of the material. As a first approach,
we employ the Hugoniot as an effective
thermodynamic constraint,
where the energy is added to the kinetic energy of
the system through velocity rescaling on a
timescale associated with YE- This a^s tne
irreversible thermal energy to the system. When
the piston velocity is used as the additional
constraint, the pressure of the system is changed
according to,
where volume and energy are per unit mass and p
corresponds to the component of the stress tensor
in the direction of shock propagation throughout
this paper. The thermodynamic state of the system
before the shock is described by pQ , VQ , and EQ .
This relation comes from the steady state solution
to the Euler equations of motion for compressible
flow, and holds at all points in a steady state shock
wave. Macroscopic definitions of the particle and
shock velocities respectively are,
_
V0-V
where
the associated time scale. We take
the pressure to be applied uniaxially, with fixed
computational cell dimensions transverse to the
shock direction. A damped equation of motion is
used for the cell dimension in the shock direction.
The time scales in both equations are chosen to be
100 fs to Ips.
Some calculations were performed with the
system confined in the direction of shock
compression between two "pistons" of silicon
atoms fixed in the diamond structure with periodic
boundary conditions applied in the direction
transverse to the shock. Other calculations were
performed with fully periodic boundary conditions.
Maillet et al. (3) have performed molecular
dynamics simulations with Lennard-Jones
potentials using a Hugoniot constraint for the
temperature similar to the one employed here.
They choose the system volume as the additional
constraint. They find good agreement between
shock temperatures calculated with the Hugoniot
constraint and actual shock molecular dynamics for
steady waves.
The initialization of the simulation can be
performed in a variety of ways. Since we hope to
accurately represent some of the dynamics of the
system, we wish to start the simulation in a
configuration that is not unphysical. This is
v.-v
For a given pressure and volume, the Hugoniot
relation provides a constraint on the temperature of
the system. We must choose an additional
thermodynamic variable to constrain. Constraint of
the shock velocity allows the simulation of a steady
state wave. However, phase transformations and
other phenomena can result in multiple shock
waves or unsteady waves. Since these are some of
the type of phenomena we would like to study, this
equilibrium approach is only an approximation to
an actual shock wave molecular dynamics
simulation.
We consider instead the constraint of the
pressure and the piston velocity separately as the
additional constraint on the system with the hopes
that some of the dynamical processes that occur in
the shock wave will be reflected in the constrained
simulation. The constraints of pressure and piston
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which is a view down the [110] axis similar to that
in Fig. 2.
accomplished by dynamically compressing the
system from its starting value on a time scale of
100 fs to 1 ps. This also allows for some nonequilibrium effects associated with dynamical
compression like excessively high temperatures in
anharmonic regions like defects, which might be
important for phase transformations.
The
constraints above are not applied until the end of
the dynamical compression.
TRANSFORMATION TO A
METASTABLE PHASE
The tight binding model of Sawada (4) was
employed with the Hugoniot constraint. A
computational cell with 108 atoms was used to
simulate shocks in the [111] direction. The volume
as a function of time for the 1.75 km/sec fixed
piston velocity run is shown in Fig. 1. This
roughly corresponds to a pressure of 35 GPa.
Upon compression, the system initially undergoes a
martensitic phase transformation into a 6-fold
coordinated structure, shown in Fig. 2 looking
FIGURE 2: A view down the [110] axis of the six-fold
coordinated structure which exists from 0 to 20 ps in the
simulation of Fig. 1. Bonds have been created between atoms
along the [110] direction.
109
106
20
40
60
80
Time (ps)
FIGURE 1: The volume of the system as a function of
time for a tight-binding Hugoniot constraint simulation of a
shock in the [111] direction with piston velocity 1.75 km/sec.
down the [110] axis. This phase has the same
bonding structure as the diamond phase but with
additional bonds connecting all atoms down the
[110] axis. At 20 ps, another martensitic
transformation breaks some of those bonds. A
similar martensitic transformation occurs at 60 ps
which breaks more of the bonds in the [110]
direction. This phase is depicted in Figure 3,
FIGURE 3: A view down the [110] axis of the structure
which exists after 60 ps in the simulation of Fig. 1. Some of the
bonds down the [110] axis in Fig. 2 have broken.
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lifetime on the order of 10-100 picoseconds, which
might make them difficult to detect in
experimentally.
These transformations were most clearly
observed when using the diamond structure
"piston" boundary conditions described above.
This suggests that the nearby presence of diamond
structure silicon may aid in the dynamical
evolution of the system into the six-fold
coordinated state. A similar transformation to the
six-fold coordinated state was observed in
Hugoniot constraint simulations with piston
velocities between 1.5 and 1.75 km/sec. These
piston velocities roughly correspond to 22-35 GPa.
Some of these simulations decayed to a disordered
phase after going through the 6-fold coordinated
phase, indicating that it is merely metastable. The
formation of these phases was not observed for
Hugoniot constraint simulations for shocks in the
[001] or [110] directions.
The metastability of these phases for short
times in the simulations in non-hydrostatic stress
conditions does not necessarily imply they are also
metastable under hydrostatic stress. Density
functional theory calculations on the six-fold
coordinated phase indicate that it is metastable
under hydrostatic stress. It appears to be a
previously unidentified metastable high-pressure
phase of silicon. The other phases with some of
the bonds in the [110] direction missing are
currently being investigated for metastability with
density functional theory.
The empirical nature of most tight binding
models may lead to incorrect predictions of phase
transformations, but the transformation observed
here suggests that there may be a dynamical
pathway that favors it. The question of the
formation of this phase will not be resolved until
density functional theory molecular dynamics are
performed with the Hugoniot constraint. Some
preliminary density functional theory calculations
are currently being performed.
Shock experiments on perfect crystals of silicon
have been performed in various orientations (1).
However, the crystal phases responsible for the
various shock waves observed are not known. The
metastable phases observed here seem to have a
CONCLUSION
We have developed a Hugoniot constraint
approach which offers some insight into the
dynamical process of shock compression of silicon.
It is an approximation which allows the simulation
of the shocked system for orders of magnitude
longer than is possible with direct shock wave
simulations. We have observed events on the 10100 ps timescale with tight binding potentials
which would not be possible to observe by doing
shock wave molecular dynamics. A dynamical
pathway to a martensitic phase transformation of
silicon to a new six-fold coordinated phase may
exist for shocks in the [111] direction. Density
functional theory simulations using the Hugoniot
constraint may help resolve this question, and are
currently underway.
ACKNOWLEDGEMENTS
We wish to thank Jerry Forbes for helpful
discussions. Evan Reed acknowledges support
from the Department of Defense NDSEG
fellowship and Lawrence Livermore National
Laboratory.
REFERENCES
1. W. H. Gust and E. B. Royce, J. App. Phys. 42, 18971905 (1971).
2. N. L. Coleburn, J. W. Forbes, and H. D. Jones, J.
Appl. Phys., 43, 5007-5011 (1972).
3. J. B. Maillet, M. Mareschal, L. Soulard, R. Ravelo,
P. S. Lomdahl, T. C. Germann, and B. L. Holian,
Phys. Rev. E 63, 016121 (2001).
4. S. Sawada, Vacuum 42, 612-614.
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