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
CONTINUUM PROPERTIES FROM MOLECULAR SIMULATIONS
Robert J. Hardy1, Seth Root1, and David R. Swanson2
Department of Physics and Astronomy, University of Nebraska, Lincoln, Nebraska 68588-0111
2
Department of Computer Science, University of Nebraska, Lincoln, Nebraska 68588-0115
Abstract. Space averaged values of the local density, velocity, temperature, and pressure in the vicinity of a
shock front are obtained from a two-dimensional molecular dynamics simulation. Different types of space
averages are considered. It is found that a cosine localization function evaluated on a fine grid gives smooth
results with a spatial resolution of a few Angstroms.
INTRODUCTION
Discussions of shock waves based on the
Rankine-Hugoniot equations and similar relationships utilize the continuum concepts of local density,
velocity, temperature, and pressure, while moleculardynamics simulations consider atoms and molecules,
which are discrete. This note discusses the
relationship between the continuum description and
the discrete description of shock phenomena. We
demonstrate that smooth values for the local
properties with a resolution of a few Angstroms can
be obtained from atomic level simulations with a
cosine localization function that goes smoothly to
zero. The resulting densities and fluxes are fully
consistent with the continuum expressions of the
conservation of mass, momentum, and energy.
In continuum descriptions the conservation laws
are expressed in terms of continuity equations,
(1)
where D is a density and / is the associated flux.
The equations of hydrodynamics are obtained by
transforming these total fluxes, which are rates of
transfer across a fixed surface, to local fluxes, which
are the fluxes in the frame of reference moving with
the local velocity of the medium. Simplifying
assumptions are often made. For example, the
Rankine-Hugoniot relations are obtained after
assuming a steady state and vanishing heat flux. As
indicated in the fundamental paper by Irving and
Kirkwood in 1950,1 the conservation laws of
continuum theory are exact analytic consequences of
the atomic level equations of motion. Time
averaging of the atomic motion is not required, but
space averaging is. The exact relationship between
the continuum and discrete descriptions does not
restrict the size of the averaging region used, but the
type of results obtained does. Using too small a
region causes large fluctuations that can obscure the
behavior being investigated.
THEORY
The formulas relating the densities and fluxes to
the atomic behavior of Irving and Kirkwood are not
in a form convenient for simulations. Hardy has
reformulated them for use in molecular dynamics.2
These results are restricted by the assumption that
the potential energy is a sum of two-body potentials
<K/jy-) j where jjy is the distance between particles i
and j. This restriction is not essential. However, it is
essential that the resulting forces are consistent with
Newton's third law,
(2)
The first subscript on F^ indicates the particle acted
on by the force and the second identifies the particle
exerting it. Any potential energy function that
depends on the (vector) positions
(scalar) interatomic distances r
only through the
r-
, (6)
yields
the localization function becomes a function of one
variable A j ( x ) , and the function fp(x) is
forces that are consistent with the third law. Unlike
the situation with two-body potentials, potentials can
depend on the interatomic separation of any number
of atomic pairs. The many-body reactive bond order
(REBO) potentials of White, et al.3>4 are functions of
this type.
The local mass, momentum, and kinetic-energy
densities are sums of the corresponding particle
properties that are associated with specific points in
space by a localization function A(r) that is peaked
at r = 0. It is normalized so the integral of A(r)
over all space equals one and its dimensions are
(volume)"1. The momentum density at point R is
(7)
The kinetic part of the pressure tensor is
PK(X) = X"WA(*i - JO-p(Jr)[>(A-)]2 (8)
where pVxVx is the contribution to the momentum
flux from the motion of the medium.
Two choices for the one-dimensional location
function are considered. The simplest is the
rectangular function of width W that equals (AW}~1
i
where mi is the mass of particle i and rt = (x,y,z)
and rt are its position and velocity. The mass density
p(R) is given by a similar expression with the
velocity omitted, and the local velocity is
= p(R)/p(R).
(4)
The formulas for the fluxes are obtained by rewriting
the time derivatives of the densities as minus the
gradients of quantities, which are the fluxes. This is
done with the aid of Newton's laws of motion and a
simple identity that involves the gradient of an
integral.2 The procedure yields exact relationships
between the discrete and continuum descriptions.
The pressure tensor is the negative of the momentum
flux and is a sum of kinetic and potential parts. The
joe-element of the potential part of the pressure tensor
is
(5)
where xtj = xt-Xj. The function B(rt,rj,R) arises
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from rewriting the time derivative.2
For systems whose properties vary in only one
direction (the ^f-direction) the potential part of the
pressure atJfis
80
Angstroms
Figure 1. Views at 0.0,0.3,0.6 picoseconds of a shock wave in a
reactive binary system. At O.Ops the atoms left of the dotted line
at 20A have a velocity of 20km/s.
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for |jc|<FT/2 and vanishes elsewhere. The crosssectional area A is needed for normalization. The
other choice, which goes smoothly to zero, is the
cosine localization function
A (x) = J_{l[1+cos(^/^)]>if \X\^W\ (9\
1
AW\
0,
otherwise]'
RESULTS
To illustrate the effect of different choices of
localization function on predicted properties, a
simulation of a two-dimensional system of 1518
atoms with periodic boundaries y = 0 and y = 99A
was performed. The simulation used a REBO
potential for an AB model with an equilibrium
spacing between bond atoms of l.OA and the weak
Van der Waals interaction between molecules turned
off.3'4 Both type A and type B atoms had masses of
14 amu. The initial structure, shown in the top
section of Fig. 1, was a close-packed lattice of
randomly oriented binary AB molecules. The system
initially extended in the ^-direction from zero to
81 A. The atoms were at rest, except for the 378
atoms to the left of X = 20A. These were given a
velocity of 8.0 km/s in the +x direction and initiated
a shock wave that traveled at approximately 8 km/s.
A strip of the system at 0.3 ps is shown in the middle
section of the figure, while the bottom section shows
the system at 0.6 ps. The properties were all
calculated for the configuration at 0.6 ps.
Figure 2 was obtained with the cosine localization
function in Eq. (9) with a width of W = 2.0A and
shows the local density, velocity, temperature, and
pressure. Both the kinetic part and the sum of the
two parts are given. Figure 3 shows the pressure for
the same set of atomic positions and velocities as
obtained with three different localization schemes of
the same width, W= 2.5A. The localization functions
used are included at the left of figures. Their plotted
width is consistent with the scale on the horizontal
axis.
A simple way to calculate the potential part of the
pressure P<j?(X) is to find the total force exerted on
all particles to the right of the plane at X by all
particles to the left of it and divide the result by the
Pressure/kinetic part
Pressure/kinetic part
A
20
20
40
60
80
40
Angstroms
60
80
Figure 3. Local properties at 0.6 ps obtained with a rectangular
location function (shown at left) of width W= 2.5A. Potential part
of pressure is calculated across a plane [top] and with Eq. (6)
[middle and bottom]. Grid is 2.5 A at top and middle and is 0.5A at
bottom.
Angstroms
Figure 2. Local properties of 0.6 ps obtained with a cosine
localization function (shown at lower left) of width W = 2.0A
calculated on a 0.5A grid.
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The fluctuations were greatly reduced when the
width of the localization was increased to W= 2.5A,
as shown in Fig. 3. Much of the improvement,
however, was the result of drawing straight lines
between widely spaced data points. The resolution
was significantly improved by reducing the spacing
between data points from 2.5A to 0.5A, while
keeping W the same, as was done in the lower
section of Fig. 3 and in all of Fig. 2. Although not
particularly clear in the small figures included,
cosine localization (Fig. 2) gave noticeably better
results that rectangular localization (Fig. 3), and the
improvement was achieved with a localization
function with an 80% smaller width at halfmaximum.
It is expected that the advantages of cosine
localization on a fine grid will be applicable to threedimensional simulations when similar numbers of
atoms are within the range of localization and when
the material is at a similar temperature. Before the
arrival of the shock there were 19 atoms per
Angstrom of length in the simulated system. Thus, at
that density there are about 40 atoms in the 2.0A
width of the cosine localization function of Fig 2. As
indicated in the figure, the arrival of the shock
causes a doubling of the density followed by a
continuous drops to zero. Behind the shock the
temperature fluctuate about 3000K.
cross-sectional area A. This is equivalent to replacing
the localization function At(jc) by a Dirac 8function. The consistent treatment of the kinetic part
of the pressure would then require that P™ (X) be
zero except when a particle is passing through the
plane at X, at which time P™ would be infinite. This
would cause fluctuations of infinite magnitude. To
overcome this difficulty, such planar calculations of
the potential part of the pressure are sometimes
combined with a kinetic part determined by a
procedure that is equivalent to using a rectangular
localization function of finite width. The results of
the procedure are shown in the top section of Fig. 3.
In the middle section of Fig. 3 the potential part of
the pressure is calculated consistently with the
kinetic part by using Eq. (6). The properties in the
top two sections were calculated on a grid of X
values separated by 2.5A, which is the same as the
width of the localization function. The bottom
section of the figure illustrates the effect of
evaluating properties on a finer grid. The same width
of rectangular location function was used, but the
pressure was evaluated at X values separated by
0.5A.
The effect of using a cosine localization function,
instead of a rectangular one, can be seen by
comparing the bottom sections of Figs. 2 and 3, both
of which were calculated with the same 0.5 A grid of
X values. If the resolving power of the localization
function is measured by its full width at halfmaximum, the cosine localization of 2.0A in Fig. 2
has better resolution than the rectangular localization
of 2.5A in Fig. 3. If the localization functions are
treated as probability densities and standard
deviation is used as its measure, the resolving power
of the localizations functions in Figs. 2 and 3 are the
same.
ACKNOWLEDGEMENTS
We are grateful to Carter White for the programs
used, and to Chad Petersen for valuable assistance.
REFERENCES
1. Irving, J. H. and Kirkwood, J. G., J. Chem. Phys. 18,
817-829(1950).
2. Hardy, R. J., J. Chem. Phys. 76, 622-628 (1982).
3. White, C. T., Robertson, D. H., Elert, M. L., and
Brenner, D. W. in Macroscopic Simulation of Complex
Hydrodynamic Phenomena, edited by M. Mareschal and
B. L. Holian, Plenum Pess, New York, 1992, p.l 11.
4. Brenner, D. W., Robertson, D. H., Elert, M. L., and
White, C. T., Phys. Rev Lett. 70, 2174-2177 (1993), ibid
76,2202(1996).
DISCUSSION
We found that averaging with a localization width
of W = l.OA gave fluctuations that obscured all but
the major trends in the development of the shock.
This was true independent of the averaging scheme
used, although the problem was greatest when the
potential part of the pressure was calculated by
simply summing the forces exerted across a plane.
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