Phase equilibria of asymmetric hard sphere mixtures
N. G. Almarza and E. Enciso
Departamento de Quı ´mica Fı ´sica I, Facultad de Ciencias Quı ´micas, Universidad Complutense, E-28040 Madrid, Spain
Received 4 September 1998; revised manuscript received 3 December 1998
The phase diagram of mixtures of hard spheres with additive diameters is studied. The case of very different
sizes is treated by means of mapping the two component system on a one component problem. In this
monocomponent system large particles are explicitly considered, whereas the effects of the small component
are included through an additional effective interaction potential between large particles. The effective poten-
tial is used to analyze the phase diagram of the mixture by means of computer simulation techniques. Results
for the behavior at low density of small spheres seem to indicate that no fluid-fluid equlibria occur. On the
other hand, the results show how this kind of mixture can exhibit equilibria between isostructural crystalline
phases. S1063-651X9908204-5
PACS numbers: 05.20.-y, 64.75.+g, 82.70.Dd
I. INTRODUCTION
The phase equilibria of asymmetric hard sphere mixtures
AHSM has become a problem which has received a lot of
attention in recent years 1. The existence of fluid-fluid
equilibria in AHSM has been predicted from theoretical ap-
proaches; however, the results of different approximate theo-
ries are very discrepant 1. Standard computer simulation
methods are not very effective when applied to these sys-
tems: the interesting results are supposed to appear at high
packing fractions of both components when the particle sizes
are quite different, this leads us to consider systems with a
high number of particles of small species in order to have a
sensible number of large particles, furthermore, the systems
suffer from a serious additional problem: the diffusion of
large particles is quite hindered by the presence of many
small particles which makes equilibration an impossible task.
One of the most popular methods to simulate fluid-fluid equi-
libria, the so called Gibbs ensemble Monte Carlo method 2,
is not useful in this context. In this case the problem is due to
the difficulties in designing effective methods to insert large
particles in the system.
It is known that in a system composed of a solute of large
particles and a solvent of small particles the difference in
size can induce some attraction between large particles 3,4,
due to excluded volume effects which produce the so called
depletion forces. These effects are supposed to be short
ranged in the large particle size scale. Some effort has been
devoted to parametrizing the form of these induced interac-
tions between solute particles 5–10. The influence of such
effects on the phase equilibria of the system is the main point
in this work. On the other hand, in recent years some atten-
tion has been devoted to evaluating the effect of the potential
range in the phase diagram of simple fluids. One of the most
interesting findings is that, for systems interacting through a
hard core potential plus a very short ranged attractive inter-
action, the liquid does not appear as stable phase 11,12. In
addition, for very short range potentials, equilibria between
two crystalline phases with the same symmetry have been
found 13. Such equilibria end for high temperatures at a
critical point. One question that arises after watching these
facts is whether the phase diagram of very asymmetric hard
sphere mixtures could show this kind of phenomenology.
Preliminary results 14 seem to indicate such a behavior. In
this work we explore a combination of different methods of
statistical mechanics to determine the phase diagram of these
systems. The procedure lies in the possibility of mapping the
two component system in a one component system with an
effective pair potential which depends on the activity of the
solvent. This mapping procedure is based on the theoretical
framework of the statistical mechanics of simple fluids in
external fields. The use of an effective potential of a mono-
component system allows us to use a number of simulation
techniques to explore the phase diagram without needing to
use a very large number of particles or very long runs.
The paper is sketched as follows. In Sec. II we show an
accurate route to map the two component system in a one
component problem. Section III is devoted to the procedures
used to determine the phase diagram. In Sec. IV we mention
the main simulation details. In Sec. V the main results are
shown, including the phase diagram and some tests to check
the ability of the effective potential formalism to reproduce
the results of the binary system. Finally, in Sec. VI the main
conclusions are collected.
II. STATISTICAL MECHANICS
We will deal with binary mixtures of hard spheres. The
hard sphere HS diameters will be
s
and
l
(
l
s
). The
size ratio is defined as R =
s
/
l
. Components s and l will
be referred to as the solvent and solute, respectively. The aim
of this work is to solve the statistical problem of the mixtures
by considering a fluid of solute particles interacting through
some effective potential which will depend on the chemical
potential or other related property of the solvent.
The partition function Q in an ensemble defined by
=1/( k
B
T ), p , z
s
, and N
l
, where k
B
is the Boltzmann con-
stant, T is the absolute temperature, p is the pressure, and
z
s
e
s
is the activity of the solvent with
s
being the
chemical potential of the solvent reads
Q =
1
l
3 N
l
N
l
!
dV V
N
l
e
- pV
d q
l
e
-U
LL
q
l
, V
e
- z
s
, V, q
l
,
2.1
PHYSICAL REVIEW E APRIL 1999 VOLUME 59, NUMBER 4
PRE 59 1063-651X/99/594/44268/$15.00 4426 ©1999 The American Physical Society