CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Hone
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
UNIAXIAL HUGONIOSTAT: METHOD AND
APPLICATIONS
Jean-Bernard Maillet and Stephane Bernard
CEA, Departement de Physique Theorique et Appliqu.ee, BP12, 91680 Bruyeres-le-Chatel, France
Abstract. The uniaxial Hugoniostat method is an equilibrium Molecular Dynamics technique
used to study the properties of shocked materials. Its application to Lennard-Joiies systems has
been successful from both structural and thermodynamic point of view. For metals, the classical
description falls down at high pressure and an ab initio approach is necessary. The shock-hugoniot
curve of Tin has been computed, and a good agreement is found with experimental data.
INTRODUCTION
UNIAXIAL HUGONIOSTAT-THEORY
An equilibrium MD technique has been proposed wrhich simulates both thermodynamic and
structural properties of shocked crystalline solids
and fluids. This method is based on the validity
of the Hugoniot relations when a steady planar
shock wave passes through a material. The properties of the initial and final states are linked by
the so-called Hugoniot equations of mass, momentum and energy across the shock front:
Computer simulation of shock waves at the
atomic scale is a field of growing interest [I].
This is due to the increasing computer power and
efficiency of the codes. Simulations of sample
containing up to 10s particles are now feasible.
Polycristalline material can now be included, the
interaction of which with a shock wave may differ
from those of a monocristal. The Lennard-Jones
material is particularly appreciated because of its
simplicity of implementation. Other materials,
like metals, are more difficult to simulate, because the classical potentials available in the literature are generally fitted to experimental properties at ambient conditions. There is no guarantee that such potentials reproduce correctly the
properties of the material under pressure. In this
context, ab initio calculations appear as a solution. We have implemented the uniaxial Hugoniostat method in an ab initio code and calculated the shock hugoniot ab initio curve of tin.
We finally use the optimal potential method to
compute the hugoniot curve of tin using locally
optimal MEAM potentials fitted on ab initio simulations.
mass:
pQus — p(us — up)
=>e = i-vyvb = t* p /«<i
momentum: Pzz — P0 + PQUSUP
Us =
energy:
E = EQ +
- PQ
U
=
(Pzz + PQ)(VQ - V}.
where E is the internal energy per unit mass, Pzz
is the normal component of the pressure tensor
in the direction of the shockwave and V = l/p is
the volume per unit mass of the shocked material; subscript "0" refers to these quantities in the
initial unshocked state. us is the shock velocity
in the material produced by a piston moving into
it at velovity up. 6 is the compressive volumetric
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strain (compression).
The implementation of the uniaxial Hugoniostat
method in ab initio calculation is straightforward: it follows the same formalism as for a thermostat [2] i.e. the new term in the new equations
of motions is applied to the ions, the calculation
concerning the electrons remains unchanged.
These relations are valid when a shockwave trav-
els through a material. Given the thermodynamic parameters of the initial unshocked state,
all the other parameters are determined provided
one of them is known (piston velocity or compression for example). In the Uniaxial Hugoniostat
method, initial uniaxial compression c = V/Vo is
fixed and applied to the system. The dynamical evolution of the system is then computed using new equations of motion, which constrain the
system to satisfy the Hugoniot relations. These
APPLICATIONS
This method has been applied to a LennardJones crystal shocked in the [100] direction, and
a comparison was made between the hugoniostat and a direct simulation of a shock wave
(non-equilibrium MD). Results have shown that
not only the shock-hugoniot curve has been reproduced up to the melting transition but also
the defective structure of the material above the
plasticity threshold [3]. When the material is
shocked in other directions, the analysis of the
results is more complex: it exists a range of pressure (just above the plastic threshold) in which
the shock wave is split into two waves: the elastic precursor and the plastic wave. The former
propagates faster than the second, leading to the
existence of a two stages process. In this domain,
the hugoniostat method does not produce similar result as the NEMD method [4]. For stronger
shock wave, a good agreement between the two
methods is found.
Our concern is to apply the Uniaxial Hugoniostat
method in order to compute the shock hugoniot
curve of tin. A classical description of the interaction between atoms have been chosen, using
MEAM potentials fitted at different thermodynamic conditions (from ambient up to 400 Gpa)
on ab initio calculations by a force matching procedure [5]. Results are shown in Fig. 1. From
Fig. 1, it can be seen that none of the MEAM potentials is capable of describing the whole range
of pressure. An alternative was to calculate the
hugoniot curve fully ab initio. It appeared to be
feasible as the pseudopotential for tin requires
a small energy cutoff (12 Ryd) and thus allows
the study of large systems (up to 64 atoms), so
as to minimize size effects in the ab initio simulations. The initial structure of the material is
/?—tin and the simulation cell contains 64 atoms.
Simulations are performed for different compression factors up to V/VQ = 0.55. Results are dis-
equations are:
dt
EL
m
Pi = Fi ~ XPi
(1)
(2)
where r?; and pi are the particle coordinates and
momenta respectively. F,- is the total force applied on the particle i. The parameter \ is chosen such that the derivative of the constrain is
zero; it is a Lagrange multiplier which minimizes
the differences between the newtonian and constrained trajectories. Its value is given by:
P(V* ~ V
(3)
where P is the time derivative of pressure and Ec
is the instantaneous kinetic energy of the system.
Simulations are run at constant volume (the vol-
ume of the final state is fixed at the begining
of the simulation by applying an instantaneous
uniaxial compression to the system). The system is given an initial kinetic energy such that
the hugoniot relation is satisfied. The dynamical
evolution of the system is then computed using
the new equations of motions, which preserve the
validity of the hugoniot relation at each timestep.
The original uniaxial Hugoniostat method makes
use of the extended system technique to calculate
the parameter \- at each timestep. This method
is convenient for classical MD simulations but
not for ab initio calculations because of the time
needed to equilibrate the two systems (the mate-
rial and the hugoniostat). The advantage of the
Gauss technique is to reduce this time, making
the method of practical use in ab initio calculations.
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0.7
V/V0
v/vo
0.8
FIG. 1. Hugoniot curve plotted in a (P,V r ) diagram using the MD hugoniostat technique. The
potentials used are fitted at P = 0 (dashed line),
P = 100 (dot-dashed) and P = 400 (long dashed).
All these curves have been computed with 108 atoms
in the simulation cell. The plus symbols correspond
to experimental data and the continuous line to the
EOS 2.
payed in Fig. 2. There is a good agreement between sesame-like EOS, experimental and ab initio results for the shock hugoniot curve plotted in
a (P, V) diagram. Ab initio calculations predict
much higher temperatures on the hugoniot than
assumed in the sesame-like EOS (but almost the
same as Mabire's EOS [6]). However, temperature is not a direct measurement of shock experiments, so the reliability of EOS temperatures is
questionable. Also questionable is the transferability of our pseudopotential to pressure of the
order of 2 Mbar and temperatures of 15000 K.
We have also tested a pseudopotential which includes the 4d electron in the valence (and so has
more flexibility) but the temperatures are not
changed.
At low pressure, some discrepencies between ab
initio and EOS results are also observed. They
come from the fact that the system is supposed
to be solid and to exhibit a phase transition from
the initial /?—tin structure to the bet structure.
Ab initio simulations do not reproduce this solidsolid phase transition, and instead exhibit the
melting of the material at lower pressure than experiments. This may be due to the fact that the
800
1000
1200
Pression (kbar)
1400
1600
1800
2000
FIG. 2. Hugoniot curve plotted in a (P, V) and
in a (T, P} diagram using the ab initio hugoniostat
technique. The full and dashed curves represent the
EOS 1 and 2, empty and filled triangles represent
experimental data and ab initio results respectively.
The dotted curve in the (T, P) diagram represents
the model of Mabire (see text).
simulation box was not allowed to change shape.
It could be argued also that the simulation cell
in our simulations is not large enough to allow
accurate calculations of thermodynamic properties of the system. Classical simulations were
then performed using up to 512 atoms, with a
potential fitted on a set of configurations coming from ab initio calculations at a particular
point of the hugoniot curve. A potential fitted in
this wra.y is non transferable to other conditions,
but produces accurate values of the thermody369
16000
CONCLUSION
The uniaxial Hugoniostat method has been
used in ab initio calculations in order to compute
the shock hugoniot curve of tin. A good agreement with different EOS and experimental data
has been found for the (P, V) curve. However,
our ab initio simulations predict much higher
temperatures than a majority of EOS. Classical
potentials for tin have been fitted to the ab initio
simulations at high pressure. These potentials,
valid on a limited range of pressure and temperature, produce the same results as the ab initio
calculations. Work is in progress to determine
wether or not the same procedure can be applied
to molecular systems.
15000
14000
13000
1500
1700
1600
1800
P/kbar
FIG. 3. full curve: ab initio trajectory calculated
with the hugoniostat method at V/V® = 0.56. The
triangles represent the time average of a classical MD
trajectory computed with several potentials fitted on
the ab initio trajectory.
REFERENCES
namic properties at the considered point. In fact
it exists an infinity of set of parameters for the
MEAM potential that satisfy the required criteria, namely a small error on the difference between the stress tensor and on the force of each
particle calculated using ab initio simulation and
the classical potential. This method of fit for a
potential has been successfully employed to computed the melting curve of iron [7] and tin [8].
[1] B.L. Holian and P.S. Lomdahl, Science 80, 2085
(1998); T.C. Germann, B.L. Holian, P.S. Lomdahl and R. Ravelo, Phys. Rev. Let. 84, 5351
(2000)
[2] S. Nose, J. Chem. Phys. 81, 511 (1984);W. G.
Hoover, Phys. Rev. A 31, 1695 (1985).
[3] J.-B. Maillet, L. Soulard, M. Mareschal, R, Ravelo, P. S. Lomdahl, T. Germann and B. L. Holian,
Phys. Rev. E, 63,016121 (2001).
[4] T.C. Germann, B.L. Holian, private communications.
[5] F. Ercolessi, J.-B. Adams, Europhys. Lett. 26,
583 (1994)
[6] C. Mabire, PhD thesis, Poitiers university (1999)
[7] A. Laio, S. Bernard, G. Chiarotti, S. Scaiidolo
and E. Tosatti, Science 287, 1027 (2000)
[8] J.-B. Maillet et al, to be published.
A set of thirteen configurations was picked up
from a 1500 timesteps ab initio trajectory at a
compression of 0.56. We assumed these configurations to be uncorrelated. Thirty different potentials were then fitted on this set of configurations and classical MD trajectories were run
starting from the same initial configuration (corresponding to the final configuration of the ab
initio trajectory). Only the time averaged values
of the pressure and the temperature have been
shown. The results are displayed in Fig. 3. It
can be seen that there is a very good agreement
between the classical MD results and the ab initio trajectory. The fluctuations of the classical
simulations results due to the use of different potentials seems to be lower than the natural fluctuations of the ab initio trajectory. This method
has been used for other compression factors, and
the same adequacy between classical MD and ab
initio results have been found.
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