Pressure of fluids and solids composed of particles interacting
with a short-range repulsive potential
Siegfried Hess
1
and Martin Kro
¨
ger
1,2
1
Institut fu ¨r Theoretische Physik, Technische Universita ¨t Berlin, PN 7 – 1, Hardenbergstrasse 36, D– 10623 Berlin, Germany
2
Polymer Physics, Department of Materials, ETH Zu ¨rich, ML H 18, CH-8092 Zu ¨rich, Switzerland
Received 15 October 1999
A simple short range repulsive potential SR, with an even smoother cut off than the Weeks-Chandler-
Andersen WCA –Lennard–Jones potential, yields practically the same pressure, both in the fluid state and for
the fcc solid, when the potential parameters are chosen such that the forces are the same at the distance where
the the two potential curves are equal to k
B
T . The comparison of the pressure for the SR and the WCA systems
is based on molecular dynamics computer simulations. The fluid branch of the equation of state is rather well
described by a modified Carnahan-Starling expression.
PACS numbers: 47.10.+g, 62.10.+s, 64.10.+h, 66.20.+d
INTRODUCTION
A fluid composed of particles which interact via the 6-12
Lennard-Jones LJ potential cut off at its minimum and
shifted such that it vanishes at the cut off distance was con-
sidered by Weeks, Chandler, and Andersen WCA1 as a
‘‘hard-sphere-like’’ reference system. This interaction with a
purely repulsive force is frequently referred to as the WCA
potential. In Ref. 1, effects of the attractive force in a fluid
with the full LJ interaction potential were treated as a per-
turbation. Due to its short range of interaction and its smooth
cut off, the WCA potential, taken as a model as such, is quite
popular in equilibrium and nonequilibrium molecular dy-
namics MD and NEMD computer simulation studies. It is
the purpose of this note to point out that a computationally
still simpler short range repulsive potential SR, with an
even smoother cut off, yields practically the same pressure,
both in the fluid state and in the fcc solid, provided that the
forces are the same at the distance where the the two poten-
tial curves are equal to k
B
T . The comparison of the pressure
for the SR and the WCA systems is based on MD computer
simulations. The fluid branch of the equation of state is
rather well described by a modified Carnahan-Starling CS
expression 2,3.
I. THE POTENTIAL CURVES
The Lennard-Jones LJ potential, cut off in its minimum
at r
cut
and shifted such that it is zero at the cutoff distance
r
cut
, was used by Weeks, Chandler, and Anderson 1 as a
purely repulsive reference potential. The WCA potential, for
r r
cLJ
=2
1/6
r
0
1.122r
0
, is given by
WCA
r =4
0
r / r
0
-12
- r / r
0
-6
+
0
, 1
and
WCA
( r ) =0 for r r
cLJ
. The quantities
0
and r
0
set
the characteristic energy and length scales. A system com-
posed of WCA particles possesses solid and fluid phases but
no gas-liquid phase transition.
The SR potential, introduced in Ref. 4, is defined by
SR
r =
0
„9 -8 r / r
0
…
3
, r r
csh
=
9
8
r
0
=1.125r
0
,
2
and
SR
( r ) =0 for r r
csh
. The potential parameters in Eq.
2 have been chosen such that, at r =r
0
, the values of the
potential functions and of their first derivatives are equal,
viz., ( r
0
) =
0
and ' ( r
0
) =-24
0
/ r
0
, for both poten-
tials. In the following, results for the pressure are presented
for the temperature T =
0
/ k
B
.
In a quantitative comparison of analytic calculations with
results obtained by MD simulations for the structure of a
ferrofluid, an equivalent scaling was used to relate a r
-12
soft sphere potential to a screened Coulomb interaction 5.
In numerical calculations and in the graph displayed here, all
physical quantities are expressed in the standard LJ units of
Refs. 6–9, e.g., lengths and energies are given in units of
r
0
and
0
. When no danger of confusion exists, the dimen-
sionless variables will be denoted by the same symbols as
the original quantities. Then the WCA and SR potentials
read
WCA
( r ) =4( r
-12
-r
-6
) +1, r r
cLJ
=2
1/6
,
WCA
( r )
=0 for r r
cLJ
, and
SR
( r ) =(9 -8 r )
3
, r r
csh
=1.125,
SR
( r ) =0 for r r
csh
. Similarly, the number density n
=N / V , where N and V are the number of particles and the
volume of the system, and the temperature T are expressed in
units of n
ref
=r
0
-3
and T
ref
=
0
/ k
B
, respectively. The unit
for the pressure is p
ref
=
0
r
0
-3
.
II. PRESSURE VERSUS DENSITY
A. Remarks on MD simulations
Simulations at the constant temperatures T / T
ref
=1 and
constant number densities n =N / V ( NVT simulations in the
range n / n
ref
=0.1, . . . , 1.2 were performed for N =10
3
=1000 and 4 8
3
=2048 particles, where the initial posi-
tions were simple cubic and face centered cubic fcc lattice
sites. The equations of motion were integrated with the ve-
locity Verlet algorithm with the time step t / t
ref
=0.002. The
LJ reference time is t
ref
=r
0
( m /
0
)
(1/2)
, and m is the mass of
a particle. A cubic simulation box with volume V and peri-
odic boundary conditions were used. The temperature was
PHYSICAL REVIEW E APRIL 2000 VOLUME 61, NUMBER 4
PRE 61 1063-651X/2000/614/46293/$15.00 4629 © 2000 The American Physical Society
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