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
ATOMISTIC MODELING OF ORIENTATION
DEPENDENCE OF SHOCK WAVE PROPERTIES
IN DIAMOND
S. V. Zybina, M. L. Elert6, J. A. Harrison6, and C. T. White0
a
Department of Chemistry, The George Washington University, Washington D. C. 20052 USA
b
Chemistry Department, U. S. Naval Academy, Annapolis, MD 21402-5026 USA
c
Code 6189, Naval Research Laboratory, Washington D.C. 20375-5000 USA
A series of molecular dynamics (MD) simulations using a reactive empirical bond order (REBO)
potential for carbon were performed to study the orientation dependence of elastic-plastic shock
waves structure in a single diamond crystal. The calculated Hugoniot relation between shock
and piston velocities shows reasonable agreement with experimental results. The mechanism
of shock-induced plasticity for shock waves propagating along the {110} and (111) directions
is quite different. In the {110} case non-equality of transverse components of the stress tensor
stimulates formation of layered carbon structures in the shock layer, while the (111) shock
reveals bulk martensitic-like deformation with formation of diamond micrograins separated by
amorphous carbon inclusions.
INTRODUCTION
Previous MD simulations have shown significant orientation dependence of the shock layer
structure even in a simple Lennard-Jones solid
[1,2]. On the other hand, ab initio studies of uniaxial compression have demonstrated evidence of
orientation effects in the mechanism of diamond
plasticity [3-8], providing insight into the stressstrain relation beyond previous finite-elasticity
estimates obtained in Ref. [9]. These results
motivate our MD study of orientation-dependent
phenomena in shock waves in a diamond crystal.
The remarkable stability of diamond subjected
to extremely high temperatures and pressures is
important for technological applications and has
received considerable theoretical attention. The
strong covalent bonding in diamond makes the
study of its properties and structural transformar
tions under extreme conditions challenging. The
microstructure, failure mechanism, and phase
transitions of diamond under high pressure are
still not well understood. Shock-induced compression of diamond is of special interest because
shock waves in solids have been shown to exhibit
a considerably richer variety of phenomena than
fluid shocks. Atomistic modeling techniques are
appropriate methods for the microscopic investigation of such phenomena as structural transformation, fracture, and elastic-plastic transition
under shock-induced compression.
THE POTENTIAL
In the MD simulation of highly nonequilibrium phenomena such as shock waves in solids,
the number of atoms must be rather large to accurately capture the formation of shock-induced
defects and dislocations. At the present stage,
this suggests the use of classical MD rather than
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ab initio methods which are limited to relatively
small system sizes. In this work, a reactive anar
lytic bond-order REBO potential, originally developed to study diamond CVD [10,11] and recently modified for better reproduction of diar
mond elastic properties [12] was used to model
the interatomic forces.
A REBO potential is based upon an extended
two-body interaction formalism, similar to the
embedded-atom method (EAM), with regard to
the local environment of atom pair. This formalism, which can be motivated by the second moment approximation to the local electronic density of states, models the interatomic energy as
a sum of an inner repulsive t4walP VR and an attractive term VA at slightly larger distances representing bond formation. However, the strength
of the attractive term is modulated by an analytic bond order function Bij according to the
number of nearest neighbors of the atom pair in
question, bond angles, radical character of the
bond, and an approximation to conjugation effects arising from adjacent unsaturated atoms:
rest in the computational box for an arbitrarily
long period of time. The simulations were performed within a Lx x Ly x Lz box with periodic
boundary conditions applied in the transverse a:,
y directions. The z axis of shock propagation
is oriented along {110} and (111) directions in a
sample with approximately 90,000 atoms and a
cross-section of about 5 x 5 nm at the density
Po = 3.52 g/cm3.
Calculated shock us and piston up velocities
are in reasonable agreement with experimental
measurements [17-19] on diamond but provide
relatively larger shock velocities (Fig. 1). An
obvious explanation for this difference is that no
high-pressure data was included in fitting the
REBO potential, resulting in too strong repulsion upon uniaxial {110} and (111) compression.
This is also responsible for the calculated elastic constant 044 = 7.2 Mbar being considerably
higher than the experimental value of 5.79 Mbar.
I
REBO potential
28
~
p = 3.52 g/cm3
-*
^ 24
, Pavlovskii
As a result, it allows the potential energy function to model realistic bond-breaking and bondforming reactions. Recently, the REBO potential
has been extended to yield the AIREBO (adaptive intermolecular REBO) potential which includes non-bonded interactions [13].
MD simulations employing chemically realistic REBO and AIREBO potentials are capable of
providing information about atomic-scale details
of shock-induced chemistry such as "7r-bonding
chemistry," which is important for carbon phase
transitions. REBO potentials have a long history of use in shock simulations [14], and recently have been successfully applied in simular
tions of the fracture of polycrystalline diamond
[15] as well as the diamond melting line and
liquid-liquid carbon phase transition [16].
p = 3.51 g/cm3
McQueen, Marsh
I
- p = 3.191
...
g/cm3
V
<110>fit
<111>
ll I l l l III III I l l l III I
6
PajtticXe velocity
8
9
10
U (km./s)
FIGURE 1. Simulated (dashed linear fit) and experimental [17,18] (solid lines) Hugoniot relationship
between plastic shock velocity us and particle velocity Up. Low Up simulation data corresponding to an
elastic shock were excluded from a linear fit.
The covalent bonding of diamond atoms significantly affects the mechanism of shock-induced
elastic-plastic transformation. Figure 2 shows
a typical snapshot of atom planes within
the shock layer for {110} steady shock wave
at shock velocity us = 25 km/s, piston velocity up = 6 km/s, and compression ratio
PQ/PI = 0-77. The shock layer consists of three
regions shown in Fig. 2(a): an elastic region
RESULTS
To study the shock wave structure in diamond
we have employed a stationary-state approach
[2], in which the shock front is maintained at
356
——
.M '>:*3:t:€*w«*^^
elastic compression region
ffi-bonding chemistry | shear in X-Z planes
formation
of layered
structures
FIGURE 2. Structure of shock wave in the (110) direction of diamond crystal, where (a) y-z planes and
(b) x-z planes are depicted; (c) shows formation of layered carbon structures in the rectangle inset from (a).
compression. For this reason, the shear deformar
tion in the {110} shock wave develops only for the
lateral direction with larger shear stress.
However, the situation is completely different for {111} uniaxial compression, where the
transverse stress components are equal. As a result, the mechanism of shock-induced plasticity
in {111} shock waves is changed. Namely, instead of the formation of layered structures, the
lattice exhibits martensitic-like deformation with
formation of diamond-like grains separated by
amorphous carbon inclusions as shown in Fig. 3.
where the lattice is uniaxially compressed
through oscillatory relaxation due to the elastic
bouncing of atomic planes off one another
in the z direction; a shear region where the
shear deformation is developed; and, a region
of n-bonding chemistry with tetrahedral bond
breaking and formation of new bonds (including
ap2).
The remarkable feature of the {110} shock
wave is that the shear deformation develops
only in the x-z planes but not in the y-z planes
as shown in Fig. 2(a,b). This provokes lattice
instability followed by bond breaking and formation of graphite-like layered structures as in Fig.
2(c). These layers are clearly shown in Fig. 2(b)
by depicting only a couple of corresponding x-z
planes. As it turns out, the stress tensor has nonequal transverse components (and thus different
shear stresses), ayy > axx , under uniaxial {110}
ACKNOWLEDGMENTS
This work was supported by the Office of
Naval Research. We thank V.V.Zhakhovskii for
several useful discussions.
357
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R.J. Colton Thin Solid Films 206, 220-223
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333-337 (1989); D.H. Robertson, D.W. Brenner, and C.T. White, Phys. Rev. Lett. 67,
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FIGURE 3. Structure of shock wave in diamond
along (111) axis, where (a) y-z planes and (b) x-z
planes are depicted,
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