Cluster algorithm for nonadditive hard-core mixtures.

Abstract

In this paper, we present a cluster algorithm for the numerical simulations of nonadditive hard-core mixtures. This algorithm allows one to simulate and equilibrate systems with a number of particles two orders of magnitude larger than previous simulations. The phase separation for symmetric binary mixtures is studied for different nonadditivities as well as for the Widom-Rowlinson model [B. Widom and J. S. Rowlinson, J. Chem. Phys. 52, 1670 (1970)] in two and three dimensions. The critical densities are determined from finite size scaling. The critical exponents for all the nonadditivities are consistent with the Ising universality class.

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Cluster algorithm for nonadditive hard-core mixtures
Arnaud Buhot
UMR 5819 (UJF, CNRS, CEA) DRFMC/SI3M, CEA Grenoble, 17 rue des Martyrs,
38054 Grenoble cedex 9, France
?Received 16 September 2004; accepted 20 October 2004; published online 17 December 2004?
In this paper, we present a cluster algorithm for the numerical simulations of nonadditive hard-core
mixtures. This algorithm allows one to simulate and equilibrate systems with a number of particles
two orders of magnitude larger than previous simulations. The phase separation for symmetric
binary mixtures is studied for different nonadditivities as well as for the Widom–Rowlinson model
?B. Widom and J. S. Rowlinson, J. Chem. Phys. 52, 1670 ?1970?? in two and three dimensions. The
critical densities are determined from finite size scaling. The critical exponents for all the
nonadditivities are consistent with the Ising universality class.
Physics. ?DOI: 10.1063/1.1831274?
© 2005 American Institute of
I. INTRODUCTION
The Widom–Rowlinson ?WR? model1attracted a lot of
attention as a prototype for the liquid-vapor phase separation.
This simple model is composed of a two-component system
where likewise particles do not interact whereas unlike par-
ticles interact through a hard-core potential. It was shown to
present a phase separation at high density and a critical point
that belongs to the Ising universality class.2,3This mixture
exhibits a liquid-liquid critical point ?with large composition
fluctuations? in contrast to pure fluids that experience a
liquid-vapor critical point ?with large density fluctuations?.
However, the phase transitions are related4and believed to
depend on the same universality class.5
A straightforward generalization of the WR model is the
nonadditive hard-core ?NAHC? mixtures where the likewise
particles also experience a hard-core interaction. Two par-
ticles i and j present a minimal distance of approach ?XY
with X and Y representing, respectively, the A or B compo-
nent to which belong particles i and j. Nonadditive mixtures
are characterized by ?AB?(?AA??BB)(1??)/2 with ??0.
Additive mixtures correspond to ??0. For a negative non-
additivity ??0, particles tend to form heterocoordinations.6,7
For a positive nonadditivity ??0, an entropically driven
phase separation occurs between two phases at sufficiently
high density due to the extra repulsion between unlike
particles.8The two phases are chemically different, one is
rich in A particles whereas the other is rich in B particles.
This phase separation occurs even for the symmetric mix-
tures where ?AA??BB. The critical density as well as the
universality class have been determined from different nu-
merical simulations.5,9–12The special case of ?AA??BB?0
corresponding to the WR model has also been studied.2,3
Additive mixtures with a strong asymmetry ?AA??BBare
also of interest and present another kind of entropically
driven phase separation transition predicted for the first time
by Biben and Hansen13but only recently observed by nu-
merical simulations.14,15
Most recent simulations of the NAHC mixtures5,11,12
used a Monte Carlo algorithm within the semigrand canoni-
cal ensemble.16,17In addition to the simple moves of par-
ticles as simple Monte Carlo steps, some steps consist in
changing the nature ?or component? of a particle when the
hard-core interactions permit this modification. The resulting
ensemble corresponds to a fixed total number of particles but
a variable composition ?or fixed difference of chemical po-
tentials between particles of different components?. A de-
tailed description of the algorithm in the case of binary mix-
tures with squared-well interactions may be found in the
paper of de Miguel et al.17Working in the semigrand canoni-
cal ensemble gives access to the coexistence curve of the
model. However, the algorithm is limited to rather small sys-
tem sizes. The largest simulations concerned 16384
particles11but most of them were limited to a few
thousands.9,10,12Furthermore, only small nonadditivities
??1 have been simulated.11,12The main reason for this limi-
tation is the following: When a Monte Carlo step corre-
sponding to a change of components is tried, at high nonad-
ditivity there is a large probability of overlap with at least
one of the neighbors. This results in the rejection of the
change of components and a dramatic slowing down of the
equilibration of the numerical simulations.
In the present paper we consider a cluster algorithm pro-
posed by Dress and Krauth18for hard-core mixtures which
proved useful for the detection of the phase separation in
additive asymmetric mixtures14,19,20and for the analysis of
two-dimensional polydisperse hard-core mixtures.21The ad-
vantage of this algorithm is to allow one to equilibrate sys-
tems with up to 106particles ?two order of magnitude larger
than previous simulations?. Also, the slowing down of the
equilibration for large nonadditivity is completely avoided
with this cluster algorithm. This allows us to analyze the
limit of infinite nonadditivity ??→?? where the NAHC mix-
tures converge to the WR model. Furthermore, the coexist-
ence curve is accessible in contrary to the previous cluster
algorithm used for the WR model.3
The paper is organized as follows: In Sec. II, we present
the model of nonadditive hard-core mixtures and we describe
the cluster algorithm used for the numerical simulations. We
present the numerical results for the two- and three-
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Page 2

dimensional mixtures in Sec. III. From finite size scaling
analysis, we extract the critical densities and the critical ex-
ponents. Finally, we conclude by a discussion of the results
in Sec. IV.
II. NONADDITIVE HARD-CORE MIXTURES
A. Description of the model
We consider NAHC mixtures in two and three dimen-
sions. The system is made up of two components A and B
with, respectively, NAand NBparticles. The particles expe-
rience hard-core interactions. No overlap is possible if the
distance d between the center of the particles i and j is lower
than ?XY, where X and Y represent the components of par-
ticles i and j, respectively. For convenience, the particles are
placed in two identical boxes of equal volume V. Periodic
boundary conditions are assumed on each of the boxes. From
the following description of the cluster algorithm, the reason
for considering two boxes will become obvious. Notice that
the consideration of two boxes is of common use for the
determination of the coexistence curve in semigrand canoni-
cal ensemble simulations.
In the following, we restrict ourselves to symmetric mix-
tures with ?AA??BB??, however, we allow for a general
positive nonadditivity ???AB/??1. Due to the hard-core
interactions, the temperature plays a trivial role and the
phase diagram is determined only by the number density ?
??A??B?NA/2V?NB/2V and the composition xA?1?xB
??A/(?A??B) of the system. In the particular case of the
symmetric NAHC mixtures, the critical point (?c,xc) is de-
termined in composition (xc?1/2) due to the symmetry.
Thus, in the following, we will consider NA?NB. Above the
critical density ?c, the system separates in two phases I and
II. One phase is rich in A particles and the other is rich in B
particles. Furthermore, due to the symmetry, those phases are
symmetric in composition (xA
The determination of the coexistence curve xA
possible from the use of the two equivalent boxes. The over-
all composition xA?NA/(NA?NB) is fixed during the
simulations whereas the particular compositions xA
(NA
free to fluctuate. NX
of component X?A or B inside the box Y?I or II. Notice
that the number of particles in each box NI?NA
NII?NA
thus intermediate between the grand canonical ensemble and
the semigrand canonical ensemble. We will discuss in the
following the consequences of the slight density fluctuations
inside each box.
In complement to the number density ?, it is interesting
to introduce the scaled packing fraction ??vAB? with vAB
the volume of unlike particles. vAB???AB
sions and ??AB
packing fraction which takes into account the nonadditivity
allows us to compare the critical packing fraction of the WR
model with those of the NAHC mixtures. Notice that from
this definition, the packing fraction may exceed 1.
I?xB
IIand xA
II?xB
I).
Iand xA
IIis
I?NA
I/
I?NB
I) and xA
II?NA
Ycorresponds to the number of particles
II/(NA
II?NB
II) of the boxes I and II are
I?NB
Iand
II?NB
IIalso fluctuates. The ensemble considered is
2/4 in two dimen-
3/6 in three dimensions. This definition of the
B. Description of the cluster algorithm
A cluster algorithm has been introduced recently by
Dress and Krauth18for the simulation of hard-core mixtures.
Inspired by the lattice cluster algorithms of Swendsen–
Wang22and Wolff,23it allows one to equilibrate large off-
lattice systems ?up to 106particles? thanks to a nonlocal
move of a large number of particles at each Monte Carlo
step. The general idea of the algorithm is to take advantage
of the hard-core interactions between particles to construct a
cluster of particles that will be moved at each Monte Carlo
step satisfying the detailed balance while keeping ergodicity.
The Monte Carlo step is constructed as follows:
?i? We select randomly one of the two boxes.
?ii? A second box is chosen randomly ?it can be the
same as the previous one?.
?iii? An inversion symmetry around a randomly chosen
pivot point is performed on all the particles of the second
box.
?iv? The two boxes with their particles are then super-
imposed on top of each other resulting in a set of clusters of
overlapping particles ?in the sense of the hard-core interac-
tions?.
?v? A particle is randomly selected and the cluster from
which it belongs is flipped. Each particle belonging to this
cluster is moved from its initial box to the other box in the
position corresponding to the superimposed configuration.
For a clear graphical representation of a Monte Carlo
step see Fig. 1 in Dress and Krauth paper.18
Let us first prove that the new configuration obtained
after the Monte Carlo step satisfies all the hard-core interac-
tions between particles:
?a? For two particles outside the flipped cluster there is
no overlap since those particles did not move and there was
no overlap before the Monte Carlo step.
?b? For two particles inside the cluster, they were flipped
keeping there relative positions in such a way that there is
still no overlap.
?c? Finally, by construction of the cluster, there is no
overlap between a particle belonging to the cluster and one
outside the cluster when the two boxes are superimposed and
a fortiori after the Monte Carlo step.
Those arguments justify the fact that the new configura-
tion obtained satisfies all the hard-core interactions between
particles in each box.
Now let us justify the detailed balance of the cluster
algorithm. Each Monte Carlo step has its symmetric step in
the sense that if we start from the final configuration there is
a pivot point that gives back the initial configuration. The
pivot points being randomly selected with a uniform distri-
bution, there are equality between the probabilities to go
from the initial to the final configuration and vice versa. The
detailed balance is thus satisfied noting that due to the hard-
core interactions both configurations have the same equilib-
rium probability. In the case of a general potential of inter-
action between particles, the cluster algorithm may be
generalized with the extent of some rejections of the Monte
Carlo steps.24
024105-2Arnaud Buhot J. Chem. Phys. 122, 024105 (2005)
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Page 3

On the point ?ii? of a Monte Carlo step, either the same
box or the other one may be selected. This leads to intrabox
or interbox Monte Carlo steps. The reason for this choice is
to decrease the equilibration time and as we will see to sat-
isfy the ergodicity of the cluster algorithm. With intrabox
Monte Carlo steps ?same box selected twice?, it has been
shown already14,18that the algorithm satisfies internal ergod-
icity like the usual Monte Carlo algorithm. By internal er-
godicity, we consider the fact that for the given composition
of the box, all possible configurations of the particles are
attainable by the cluster algorithm, In fact, if the pivot point
is chosen sufficiently close to a particle position and this
particle is considered as the starting point for the cluster
construction, the Monte Carlo step corresponds to a slight
move of this particle without affecting the other particles.
This move corresponds to a usual Monte Carlo move in a
general algorithm. This argument justifies the internal ergod-
icity. However, from those moves, the composition of the
box is kept constant. In order to change the composition or
the relative number of particles of components A or B in each
box, interbox Monte Carlo steps are necessary. Those inter-
box Monte Carlo steps are performed to exchange the com-
ponents of particles. If the pivot point is selected such that
two particles from different components and boxes superim-
posed exactly and if one of those particles is selected as the
cluster starting point, the Monte Carlo step reduces to the
exchange of those particles. The Monte Carlo move then
corresponds to the usual change of components considered
by the Monte Carlo algorithm in the semigrand canonical
ensemble. In summary, both usual Monte Carlo steps ?the
move of a particle and the change of components? are pos-
sible moves in this cluster algorithm justifying the ergodicity
if this ergodicity is assumed for the usual Monte Carlo algo-
rithm in the semigrand canonical ensemble.
As can be seen from the discussion on the ergodicity, the
use of the two boxes is useful for the nonadditive hard-core
mixtures. It has another strong advantage since it allows one
to determine the coexistence curve or the relative composi-
tion of the two separated phases above the critical density.
Due to the symmetry of the problem, the critical point cor-
responds to an equal partition in particles A and B (xc
?1/2) and above the critical density the coexistence curve is
symmetric around xc?1/2. The use of identical boxes is thus
justified since the densities ?Iand ?IIof the two phases are
equal. As previously discussed, the total number of particles
in each box is slightly fluctuating with this cluster algorithm
in contrary to the simulations in the semigrand canonical
ensemble. However, those fluctuations of the density are not
related to the composition fluctuations inside each box. More
importantly, around the phase separation transition, the com-
position fluctuations diverge whereas the density fluctuations
stay insensitive to the transition. As a consequence the slight
density fluctuations do not affect the coexistence curve de-
termined from the cluster algorithm.
The last question concerning the cluster algorithm con-
cerns the equilibration time or the number of Monte Carlo
steps necessary to equilibrate the system. The Swendson-
Wang cluster algorithm was introduced to simulate systems
of Ising spins with ferromagnetic interactions between neigh-
bor spins on a lattice.22In contrary to the simple Monte
Carlo algorithm, this cluster algorithm does not suffer from a
critical slowing down at the phase transition due to the fact
that the flipped clusters are then directly related to the spin
clusters observed around the phase transition. In the case of
the present cluster algorithm, such direct relation of the
flipped clusters with the configurations of particles is not
demonstrated. However, the number of Monte Carlo steps
necessary for the equilibration of the system does not seem
to increase significantly when approaching the phase separa-
tion transition. It is also interesting to note that this number
is roughly independent of the system size as observed for
cluster algorithms on lattices and in contrary to usual Monte
Carlo algorithms where this number usually increases
strongly with the system size. Another important point is that
there is no critical slowing down for the equilibration when
the nonadditivity is increased. Thus, this cluster algorithm is
well adapted for large nonadditivities ? and for the WR
model in comparison to the Monte Carlo simulations in the
semigrand canonical ensemble. In fact, the critical slowing
down is observed for small nonadditivities when the phase
separation transition occurs at high densities close to a fluid-
solid transition.
III. RESULTS OF THE NUMERICAL SIMULATIONS
A. Critical exponents and critical packing fractions
As all second-order phase transitions, the critical point in
NAHC mixtures is characterized by different critical expo-
nents. From finite size scaling of equilibrium thermodynamic
quantities close to the critical point (?c,xc), it is possible to
determine those exponents. For the NAHC mixtures and for
the WR model, the critical exponents are supposed to belong
to the Ising universality class ?see numerical values of ?, ?,
and ? in Table I?. In the following, we define different ways
to determine or test the value of the critical exponents. We
also describe four different ways to define a finite size criti-
cal packing fraction ?c(L) from numerical simulations.
From finite size scaling, it is possible to extract the ?in-
finite size? critical packing fraction ?c:12,25
?c?L???c?A/L1/?.
?1?
The finite size of the system L is defined as NA?Ld, where
d is the dimension of the system. The critical exponent ? is
independent of the definition of the finite size critical pack-
ing fraction considered in contrary to the coefficient A. Due
to the corrections to scaling for small system sizes, it is usu-
ally difficult to extract the critical exponent ?. However, a
linear behavior of the finite size critical packing fraction with
TABLE I. Critical exponents for the Ising universality class in two and three
dimensions.
Exponent
???
2D ?exact?
3D ?num.?
1
0.627
7/41/8
1.2390.326
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Page 4

the rescaled system size 1/L1/?is a good test of the univer-
sality class of the model considered and allows one to extract
the critical packing fraction ?c.
The phase separation transition is characterized by the
order parameter m?2(xA?xc) with ?1?m?1. Its prob-
ability distribution P(m;?,L) for a packing fraction ? and a
system size L changes form around the critical packing frac-
tion ?c. In the thermodynamic limit (L??), the distribution
of the order parameter presents ? picks. It has a single pick at
m?0 below the critical packing fraction (???c). However,
above the critical packing fraction (???c), the distribution
is double picked at values ?mmaxwith mmax?(???c)?for
???c. For finite size systems, the picks broaden but the
change of the distribution remains and a maximum
mmax(?,L)?0 may be defined for each packing fraction ?
and size L above the size dependent critical packing fraction
?c
ing fraction at a fixed system size is:25
max(L). The dependence of the maximum mmaxon the pack-
mmax?????c
max?L???
?2?
for ???c
to extract the finite size critical packing fraction ?c
from a linear fit of mmax
fraction ?, mmaxsaturates to 1. This limits the range of pack-
ing fractions for which the linear fit is valid. Due to this
limited range of the power law behavior, it is difficult to
extract the critical exponent ?. However, a linear dependence
obtained for the expected value of the critical exponent ? is
still a strong confirmation of the universality class.
Due to the symmetry of the model, the average of the
order parameter is zero. However, the mean absolute value of
the order parameter is another possibility to define a finite
size order parameter:
??m????,L???
?1
max(L). Knowing the exponent ?, it is thus possible
max(L)
1/?as function of ?. For high packing
1
?m?P?m;?,L?dm.
?3?
This definition suffers from an additional drawback com-
pared to mmax. The average absolute value of the order pa-
rameter does not vanish at the finite size critical packing
fraction ?c
av(L). Thus, the power law behavior,25
??m???????c
av?L???,
?4?
presents corrections to scaling not only for large packing
fraction but also around ?c
packing fraction ?c
fit of ??m??1/?on a limited range of packing fractions. The
small value of the critical exponent ? in two dimensions
leads to a sufficiently large range, however, in three dimen-
sions, the larger value of ? renders the range of power law
behavior ?4? too small to be able to extract ?c
The maximum of the modified susceptibility ?(?,L)
??m2????m??2is a third possibility to define a finite size
critical packing fraction. This modified susceptibility pre-
sents a single maximum in contrary to the real susceptibility
?m2???m?2. The general form of the modified susceptibility
is25
av(L). This finite size critical
av(L) may still be extracted from a linear
av(L).
???,L??L?/??d? ¯?L1/?????c??
?5?
with ? ˜(x) a function with a single maximum ? ˜max. The
packing fraction at the maximum of the modified suscepti-
bility defines the finite size critical packing fraction ?c
The modified susceptibility may also be used to extract the
ratio of critical exponents ?/?. Its maximum depends alge-
braically on the system size with an exponent ?/??d:25
?(L).
?max??c
??L?,L??L?/??d? ˜max.
?6?
The ratio of critical exponents ?/? is thus simply determined
by the slope of a linear fit in a log-log scale. The results for
the two- and three-dimensional systems are presented in
Table II in comparison with the Ising universality prediction.
Those results are discussed later.
Another possibility to determine the critical packing
fraction concerns the Binder parameter:25
U??,L??1??m4?
3?m2?
.
?7?
The Binder parameter saturates to 2/3 for large packing frac-
tions and vanishes for small ones. It also presents the inter-
esting property to intersect at the critical packing fraction at
leastfor sufficiently large
U(?c,L)?U* at this intersection is expected to be univer-
sal. The monotonous behavior of the Binder parameter may
be used to define a finite size critical packing fraction ?c
as the value for which U??c
value 1/2 is arbitrary and could be modified as soon as it is
sufficiently different from the boundary values 0 and 2/3 and
from the intersection value U*. The particular choice of U*
would render the finite size packing fraction independent of
the system sizes. This point will be discussed further for the
three-dimensional systems.
We defined different critical packing fractions for finite
size systems from the maximum of the modified susceptibil-
ity ?c
coexistence curve either from the maximum of the distribu-
tion of the order parameter ?c
solute value of the order parameter ?c
results obtained from the cluster algorithm are analyzed in
the following sections.
system sizes.The value
B(L)
B(L),L??1/2. The choice of the
?(L), from the Binder parameter ?c
B(L) and from the
max(L) or from the average ab-
av(L). The numerical
B. Two-dimensional NAHC mixtures
In the two-dimensional model, we considered four dif-
ferent nonadditivities ??0.5, 1.0, 2.0, and 4.0 as well as the
WR model ?????. The system sizes ranged from L?NA
?10–400.The largestsystems
?320000 particles. Numerical simulations were divided in
five consecutive runs of 105–106Monte Carlo steps depend-
1/2
contained
NA?NB
TABLE II. Numerical results for the ratio of critical exponents ?/? for
different nonadditivities and for the WR model ????? in two and three
dimensions. Numbers in parentheses correspond to the error on the last
digits. The data in column Ising are the predictions of the Ising universality
class.
?
0.5 1.02.0
?
Ising
2D
3D
1.749?8?
2.03?8?
1.749?8?
2.05?8?
1.746?7?
2.05?8?
1.742?8?
2.05?8?
7/4
1.98
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Page 5

ing on the system sizes with longest runs for the largest
systems. The first run is kept for equilibration of the initial
configuration and the last four runs for the data collection
and error estimation.
On Fig. 1, we plot equilibrium configurations for two
different packing fractions. The configurations correspond to
systems with NA?NB?3200 particles and a nonadditivity
??2. The likewise hard-core diameter ?AAand ?BBare rep-
resented, respectively, by white and black disks whereas the
unlike diameter ?ABis represented by light and dark gray
disks, respectively, for the A and B particles. No overlaps
between unlike particles are present but overlaps of likewise
particles are observed on the gray scale. However, no overlap
is present between black and white particles justifying that
both configurations satisfy all the hard-core interactions. For
the top configuration above the phase separation transtion,
??1.03??c(L)?0.95, we observe a large difference be-
tween the number of A and B particles. The large percolating
cluster of B particles ?the dark gray particles? is a clear evi-
dence of the phase separation. On the contrary, on the bottom
configuration, ??0.72??c(L), the A and B particles are
roughly in equal number and perfectly mixed. In this sys-
tems, the phase separation did not occur, however, due to the
stronger unlike particles hard-core interactions, a local clus-
tering of likewise particles is present.
The modified susceptibility ? is determined as function
of the packing fraction ? for different system sizes. It pre-
sents a single maximum ?max??c
cal packing fraction ?c
maximum depends on the system size with a power law be-
havior. From a linear fit in a log-log scale, we deduce the
ratio of critical exponents ?/? ?see Table II?. Due to the cor-
rections to scaling observed for small system sizes, the three
smaller sizes (L?10, 12, and 15? where removed for the
linear fit to extract ?/?. This ratio of critical exponents com-
pare nicely with the Ising universality class prediction 7/4 for
all the nonadditivities considered and for the WR model. The
estimation for the errors presented on Table II comprises the
error on the maximum of the modified susceptibility and the
error coming from the linear fit. The relative error on the
ratio of critical exponents ?/? is smaller than 1%.
The confirmation of the critical exponent ? is obtained
from the two different definitions of the finite size order pa-
rameter: mmaxand ??m??. First, we plot the rescaled maxi-
mum of the distribution of the order parameter mmax
tion of ? for different system sizes and for a nonadditivity
??2 on Fig. 3?a?. Second, we plot the rescaled average order
parameter ??m??1/?for the same system sizes but for a non-
additivity ??1 on Fig. 3?b?. The linear behavior observed on
both figures for a critical exponent ??1/8 confirms the Ising
universality class prediction. Similar results are obtained for
all the nonadditivities and for the WR model. The range of
the linear regime is still rather small and limited for large
packing fractions to mmax
?(L),L? at the finite size criti-
?(L). As can be seen on Fig. 2, this
1/?as func-
1/??0.7 and ??m??1/??0.6. Further-
FIG. 1. Equilibrium configurations for a system above ?top? and below
?bottom? the phase separation transition. The systems correspond to NA
?NB?3200 particles with a nonadditivity ??2. White and black disks
correspond to likewise diameter ?AAand ?BBwhereas the light and dark
gray disks correspond to the unlike diameter ?AB. With this representation,
the overlaps are only allowed by the hard-core interactions to the likewise
particles on the gray scale disks.
FIG. 2. Maximum of the modified susceptibility ?maxas function of the
system size L in a log-log scale for three different nonadditivities ? and for
the WR model ?Widom?. The ratio of critical exponents ?/? is determined
from linear fits. The three smallest system sizes are removed from the linear
fit due to the corrections to scaling effects for those small sizes. The data
have been shifted for clarity.
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Page 6

more, due to the strictly positive average of the absolute
order parameter, the power law behavior is also limited from
below to ??m??1/??0.1. The restriction on the range of the
power law behavior prevents from a direct determination of
the critical exponent ?.
The critical packing fraction ?cis then determined from
the plot of the finite size critical packing fraction ?c(L) as
function of the rescaled system size 1/L1/?with the expected
exponent ??1. On Fig. 4, the numerical results for ?c
?c
for ?c
The error bars are smaller than the symbols and thus not
represented on the figure. As can be seen on Fig. 4, ?c
?c
This confirms the value of the critical exponent ? predicted
by the Ising universality class. However, ?c
rections to scaling and a quadratic fit is necessary to extract
the critical packing fraction ?c
similar results are obtained for all the nonadditivities as well
as the WR model.
All the critical packing fractions from the four different
definitions of their finite size analog are presented on Table
III for all the nonadditivities considered and for the WR
model. It is interesting to notice that the four different defi-
nitions are not identical since the numerical values obtained
for finite sizes differ ?see Fig. 4?. However, the thermody-
namic limit results for the critical packing fractions are con-
sistent for all the four definitions. The estimation of the rela-
tive error which is smaller than 1% for all the critical
packing fractions combines the error on the finite size esti-
mates and the error coming from the finite size scaling to
extract the thermodynamic limit results.
Let us now compare our results to previous simulations.
Our prediction for the critical density ?c?AB
the WR model is in good agreement with the value 1.566?3?
obtained by Johnson et al.3For the NAHC mixtures in two
dimensions, the determination of the critical packing frac-
tions have been recently obtained from numerical simula-
tions by Saija and Giaquinta.10Only small nonaddtivities
??1 have been considered and the system was limited to
800 particles. The critical packing fraction was determined
without finite size scaling and may thus be considered only
as a lower bound. Their critical packing fraction obtained for
??1 is ?c?0.118(1??)2?0.472. A scaled particle theory
predicts ?c?0.096(1??)2?0.385 ?Ref. 26? whereas a
virial expansion 0.324. The different definitions of the pack-
ing fraction in both cases explain the ? dependence intro-
duced to compare to our prediction ?c?0.897(2). The
strong difference should be contrasted by the fact that the
prediction of a first-order perturbation theory usually referred
to as MIX1 gives higher values for the critical packing frac-
tions especially for large nonadditivities ???0.4?.10It could
be interesting to pursue the comparison of this theory for
higher nonadditivities ???1? with our numerical predictions.
max,
av, and ?c
?close to those for ?c
Bare plotted for a nonadditivity ??1. The results
Bhave been removed for clarity.
maxand
avpresent a linear behavior with respect to 1/L1/?for ??1.
B(L) presents cor-
B. The same is true for ?c
?and
2?1.560(10) of
FIG. 4. Finite size critical packing fractions ?c
function of the rescaled system size 1/L1/?. The nonadditivity considered is
??1. Linear fits of ?c
Ising universality class and allow us to extract the critical packing fractions
?c
critical packing fraction ?c
finite size effects.
av(L), ?c
max(L), and ?c
B(L) as
avand ?c
maxconfirm the critical exponent ??1 of the
avand ?c
max. A quadratic fit is necessary for ?c
Bdue to the corrections to scaling and stronger
B(L) in order to extract the
TABLE III. Numerical results for the critical packing fraction for the two-
dimensional NAHC mixtures for different nonadditivities and for the WR
model ?????. The numbers in parentheses correspond to the error on the
last digits.
?
?c
av
?c
max
?c
B
?c
?
0.5
1.0
2.0
4.0
?
0.8141?5?
0.8988?8?
1.0215?18?
1.147?4?
1.228?3?
0.8138?11?
0.8976?20?
1.0179?45?
1.141?5?
1.222?6?
0.811?2?
0.896?4?
1.017?9?
1.141?5?
1.224?8?
0.811?4?
0.895?5?
1.017?6?
1.140?5?
1.225?5?
FIG. 3. ?a? Rescaled maximum of the distribution of the order parameter
mmax
??m??1/?for a nonadditivity ??1 as function of the packing fraction ? for
different system sizes. Linear fits confirm the critical exponent ??1/8 pre-
dicted by the Ising universality class and allow us to extract the finite size
critical packing fractions ?c
1/?for a nonadditivity ??2 and ?b? rescaled average order parameter
max(L) and ?c
av(L), respectively.
024105-6 Arnaud Buhot J. Chem. Phys. 122, 024105 (2005)
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Page 7

C. Three-dimensional NAHC mixtures
In three dimensions, we considered a large number of
different nonadditivities ??0.5, 0.6, 0.8, 1.0, 2.0, 3.0, 4.0,
and 9.0 as well as the WR model ?????. The system sizes
ranged from L?NA
NA?NB?250000 particles. Systems with up to 2?106?or
L?100) were simulated for some nonadditivities. The equi-
librium time was then close to the day on simple personal
computers. Since a large number of densities need to be
simulated to extract the critical density, we are for L?100 at
the limit of the numerical capabilities. As for the two-
dimensional systems, five consecutive runs were simulated.
The first one was used to equilibrate the initial configuration
and the four remaining ones served for the collection of data
and the estimation of errors. 105–106Monte Carlo steps for
each run were sufficient for the equilibration and in order to
obtain a small error.
The modified susceptibility ? is determined as function
of the packing fraction ? for different system sizes. It pre-
sents a single maximum ?max??c
cal packing fraction ?c
maximum depends on the system size with a power law be-
havior. From a linear fit of the maximum of the modified
susceptibility as function of the system size in a log-log
scale, we deduce the ratio of critical exponents ?/? ?see Table
II?. Those ratios for all nonadditivities are systematically
larger than the Ising universality class prediction but still
inside the error bars. The relative errors were estimated as
the sum of the average relative error on the maximum ?max
and the error due to the linear fit.
The existence of corrections to scaling may explain the
over estimation of ?/?. Removing the data from the small
size systems for the linear fits reduces the discrepancy with
the Ising universality class prediction. A possible source for
the corrections to scaling observed is the existence of fluc-
tuations on the density inside each box during the simula-
tions. Even though those fluctuations are small they may
affect slightly the scaling behavior especially for small sys-
tems where the fluctuations are stronger. Some constrained
simulations have been done where the density in each box
1/3?5–50. The largest systems contained
?(L),L? at the finite size criti-
?(L). As can be seen on Fig. 5, this
was kept constant. The clusters were moved only if they did
not change the total number of particles inside each box.
This modification of the algorithm leads to similar results
with less pronounced corrections to scaling. However, errors
were also stronger since due to the rejections the equilibra-
tion time was sensibly increased.
The confirmation of the critical exponent ? is obtained
from the plot of the rescaled maximum of the distribution of
the order parameter mmax
? for different system sizes. A nonadditivity ??0.8 has been
considered on Fig. 6 but similar results are obtained for all
nonadditivities and for the WR model. The linear behavior
observed on the figure for a critical exponent ??0.326 con-
firms the Ising universality class prediction. The range of the
linear regime is still rather small and limited for large pack-
ing fractions to mmax
??m??1/?is even smaller and cannot allow us the confirmation
of the critical exponent ? neither the determination of the
finite size critical packing fraction ?c
The critical packing fraction ?cis determined from the
plot of the finite size critical packing fraction ?c(L) as func-
tion of the rescaled system size 1/L1/?with the expected
exponent ??0.627. The numerical results for ?c
7?a?? and for ?c
additivities and for the WR model. The error bars are smaller
than the symbols and thus not represented. As can be seen,
?c
for ??0.627. This confirms the value of the critical exponent
? predicted by the Ising universality class. The results are
similar for ?c
The corrections to scaling observed for ?c
dimensions are not present in the three-dimensional systems.
It is possible to notice a change in the slope of the linear fit
around ??1. Already evident on Fig. 7, this is confirmed by
the results for the nonadditivities ??0.6 and 0.8 ?data not
shown on the figure?. For ??1 the slope is independent of
the nonadditivity whereas for ??1 the slope is increasing
with the nonadditivity. This change may be due to the onset
of the fluid-solid transition that the model experience for
1/?as function of the packing fraction
1/??0.5. The range of the linear regime for
av(L).
max(L) ?Fig.
?(L) ?Fig. 7?b?? are plotted for different non-
maxand ?c
?present a linear behavior with respect to 1/L1/?
Band for the other nonadditivities considered.
?and ?c
Bin two
FIG. 5. Maximum of the modified susceptibility ?maxas function of the
system size L in a log-log scale for different nonadditivities ? and for the
WR model ?Widom?. The ratio of critical exponents ?/? is extracted from
linear fits. Data have been shifted for clarity.
FIG. 6. Rescaled maximum of the distribution of the order parameter mmax
as function of the packing fraction ?. The nonadditivity considered is
??0.8. The critical exponent ??0.326 predicted for the Ising universality
class has been used and the critical packing fractions ?c
system sizes L are extracted from linear fits.
1/?
max(L) for different
024105-7Cluster algorithm for mixturesJ. Chem. Phys. 122, 024105 (2005)
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Page 8

large packing fractions. At sufficiently low nonadditivity, the
phase separation transition is expected to be preempted by
the fluid-solid transition.26
All the critical packing fractions for the different nonad-
ditivities and for the WR model are presented on Table IV.
The relative errors are less than 0.5% and the three defini-
tions of the critical packing fraction ?c
to identical estimations in the thermodynamic limit.
Let us now compare our results with others. Our predic-
tion for the critical density of the WR model is ?c?AB
?0.7470(8) which is in strong agreement with the value
max, ?c
?, and ?c
Blead
3
0.748?2? obtained by Johnson et al.3with another cluster al-
gorithm. Our result improves by a factor 2 the error on this
critical density. The critical packing fraction has been deter-
mined from different numerical simulations for a large set of
nonadditivities. For a nonadditivity ??1, our prediction
?c/(1??)3?0.04515(8) is slightly higher than the value
0.04484?20? proposed by Go ´z ´dz ´.12However, it strongly con-
firms the previous underestimation of 0.0288 by Saija et al.9
The nonadditivity dependence introduced is due to a differ-
ent definition for the packing fraction. Go ´z ´dz ´12also pre-
dicted the critical packing fraction 0.0866?2? and 0.0611?2?
for the nonadditivity ??0.6 and 0.8, respectively. His pre-
dictions compare nicely with our results ?c/(1??)3
?0.0871(1) and 0.0614?1?. It is important to notice that de-
termining the critical packing fraction has been possible
thanks to the absence of a critical slowing down of the simu-
lations for large nonadditivities with the cluster algorithm in
comparison to the simulations in the semigrand canonical
ensemble.
From the determination of the critical packing fraction
for large nonadditivities, it is possible to consider the con-
vergence of the NAHC mixtures to the WR model which
corresponds to an infinite nonadditivity. It is evident from
Fig. 8 that the critical packing fraction ?c(?)??c(?)
?B/(1??)2for large nonadditivities. A linear fit for the six
largest nonadditivities leads
?0.3907(5) for the WR model in good agreement with the
direct simulations. The coefficient B was estimated to be
1.21. This interesting dependence of ?c(?) on ? may be a
good test for the different theories developed to determine
the critical packing fraction of the NAHC mixtures.
The Binder parameter offers another test of the univer-
sality class from its intersection value U*. In Fig. 9, we have
plotted the Binder parameter as function of the packing frac-
tion for different system sizes and for the nonadditivity ??2.
It can be seen that the intersections between the different
curves occur at smaller and smaller values as the system size
increases but still at higher values than the expected univer-
sal one U*?0.47. Similar results are observed for the other
nonadditivities and for the WR model. This result was al-
ready observed in Go ´z ´dz ´ simulations.12Corrections to scal-
to anestimate of
?c
FIG. 7. Finite size critical packing fractions ?c
function of the rescaled system size 1/L1/?with ??0.627 as predicted by the
Ising universality class. Different nonadditivities are considered as well as
the WR model ?same legends for both figures?. Linear fits of ?c
?c
packing fractions ?c
max(L) ?a? and ?c
?(L) ?b? as
maxas well as
?confirm the critical exponent ??0.627 and allow us to extract the critical
maxand ?c
?.
TABLE IV. Numerical results for the critical packing fractions in three
dimensions for the different nonadditivities considered and for the WR
model ?????. The numbers in parentheses correspond to the error on the
last digits.
?
?c
max
?c
B
?c
?
0.5
0.6
0.8
1.0
1.5
2.0
3.0
4.0
9.0
?
0.3577?3?
0.3572?4?
0.3583?5?
0.3610?5?
0.3706?5?
0.3758?5?
0.3827?10?
0.3862?10?
0.3896?9?
0.3910?4?
0.3574?3?
0.3564?4?
0.3581?4?
0.3614?3?
0.3703?5?
0.3766?5?
0.3839?6?
0.3871?8?
0.3908?6?
0.3912?4?
0.3574?3?
0.3568?4?
0.3580?5?
0.3614?5?
0.3705?8?
0.3766?8?
0.3843?9?
0.3872?11?
0.3914?9?
0.3912?4?
FIG. 8. Critical packing fraction for the different nonadditivities ?c(?) as
function of 1/(1??)2?disks? and for the WR model ?square?. The line
corresponds to a linear fit for the six largest nonadditivities and leads to an
estimation of ?c?0.3907(5) for the WR model.
024105-8Arnaud BuhotJ. Chem. Phys. 122, 024105 (2005)
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Page 9

ing may explain such behavior. For our definition of the criti-
cal packing fraction ?c
arbitrary choice was U??c
was expected to decrease with increasing system size. The
opposite behavior is found. The absence of corrections to
scaling in the dependence of ?c
thus an argument against this explanation for the high inter-
section in Fig. 9. However, no other explanations have been
found.
Bbased on the Binder parameter, the
B(L),L??1/2?U*. Thus, ?c
B(L)
B(L) with the system size is
IV. DISCUSSION
In this paper we present a cluster algorithm for the nu-
merical simulations of the NAHC mixtures as well as for the
WR model. This cluster algorithm allows one to simulate
large systems ?up to 106particles?. Each Monte Carlo step
corresponds to the nonlocal move of a large number of par-
ticles reducing strongly the equilibration time. The absence
of critical slowing down for increasing nonadditivities ? al-
lows us to study large nonadditivities and the convergence of
the NAHC mixtures to the WR model when ?→?.
Two- and three-dimensional systems have been consid-
ered. In both cases, the models were found to belong to the
Ising universality class for all the nonadditivities considered
as well as for the WR model. The critical packing fractions
?chave been determined from four different finite size defi-
nitions. All the definitions lead to identical predictions in the
thermodynamical limit with a high precision.
Different theories have been used to determine the criti-
cal packing fractions ?c(?) of NAHC mixtures,27from the
scaled particle theory26to the virial expansion28,29or the den-
sity functional theory.30A comparison with our precise nu-
merical predictions of the critical packing fractions for a
large set of nonadditivities ? is possible. Even if a quantita-
tive comparison between theory and simulations is difficult,
the ? behavior of the critical packing fraction ??c??c(?)
?B/(1??)2in three dimensions? for large nonadditivities
? is a good test for a theory.
Recently, Jagannathan and Yethiraj5have studied the dy-
namical behavior of the WR model close to the phase sepa-
ration transition. The cluster algorithm could be used to con-
sider larger systems in order to be closer to the critical
density. The cluster algorithm could also be used to study the
stability and interfacial properties in confined geometries.31
ACKNOWLEDGMENT
W. Krauth is thanked for useful discussions and a careful
reading of the manuscript.
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FIG. 9. Binder parameter for different system sizes as function of the pack-
ing fraction ? for the nonadditivity ??2.0. The horizontal line represents
the universal intersection value U*?0.47 expected for the Ising universality
class.
024105-9 Cluster algorithm for mixtures J. Chem. Phys. 122, 024105 (2005)
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