Critical point calculation for binary mixtures of symmetric non-additive hard disks
We have calculated the values of critical packing fractions for the mixtures of symmetric non-additive hard disks. An interesting feature of the model is the fact that the internal energy is zero and the phase transitions are entropically driven. A cluster algorithm for Monte Carlo simulations in a...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Critical point calculation for binary mixtures of symmetric non-additive hard disks / W.T. Gózdz, A. Ciach // Condensed Matter Physics. — 2016. — Т. 19, № 1. — С. 13002: 1–8. — Бібліогр.: 41 назв. — англ. |
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Gózdz, W.T. Ciach, A. 2019-06-17T12:21:56Z 2019-06-17T12:21:56Z 2016 Critical point calculation for binary mixtures of symmetric non-additive hard disks / W.T. Gózdz, A. Ciach // Condensed Matter Physics. — 2016. — Т. 19, № 1. — С. 13002: 1–8. — Бібліогр.: 41 назв. — англ. 1607-324X DOI:10.5488/CMP.19.13002 arXiv:1512.08284 PACS: 05.10.Ln, 05.20.Jj, 05.70.Jk https://nasplib.isofts.kiev.ua/handle/123456789/155779 We have calculated the values of critical packing fractions for the mixtures of symmetric non-additive hard disks. An interesting feature of the model is the fact that the internal energy is zero and the phase transitions are entropically driven. A cluster algorithm for Monte Carlo simulations in a semigrand ensemble was used. The finite size scaling analysis was employed to compute the critical packing fractions for infinite systems with high accuracy for a range of non-additivity parameters wider than in the previous studies. Обчислено значення критичних фракцiй заповнення для сумiшей симетричних неадитивних твердих дискiв. Цiкавою рисою даної моделi є факт, що її внутрiшня енергiя є нульовою i фазовi переходи мають ентропiйну природу. Використано кластерний алгоритм для моделювання Монте Карло у напiв-великому ансамблi. Застосовано аналiз скiнченно-вимiрного скейлiнгу для прецизiйного обчислення критичних фракцiй заповнення нескiнчених систем для ширшої областi параметра неадитивностi порiвняно iз попереднiми дослiдженнями. The support from NCN grant No 2012/05/B/ST 3/03302 is acknowledged. We would like to thank Noe Almarza and Paweł Rogowski for helpful discussions. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Critical point calculation for binary mixtures of symmetric non-additive hard disks Обчислення критичної точки для бiнарних сумiшей симетричних неадитивних твердих дискiв Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Critical point calculation for binary mixtures of symmetric non-additive hard disks |
| spellingShingle |
Critical point calculation for binary mixtures of symmetric non-additive hard disks Gózdz, W.T. Ciach, A. |
| title_short |
Critical point calculation for binary mixtures of symmetric non-additive hard disks |
| title_full |
Critical point calculation for binary mixtures of symmetric non-additive hard disks |
| title_fullStr |
Critical point calculation for binary mixtures of symmetric non-additive hard disks |
| title_full_unstemmed |
Critical point calculation for binary mixtures of symmetric non-additive hard disks |
| title_sort |
critical point calculation for binary mixtures of symmetric non-additive hard disks |
| author |
Gózdz, W.T. Ciach, A. |
| author_facet |
Gózdz, W.T. Ciach, A. |
| publishDate |
2016 |
| language |
English |
| container_title |
Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Обчислення критичної точки для бiнарних сумiшей симетричних неадитивних твердих дискiв |
| description |
We have calculated the values of critical packing fractions for the mixtures of symmetric non-additive hard disks. An interesting feature of the model is the fact that the internal energy is zero and the phase transitions are entropically driven. A cluster algorithm for Monte Carlo simulations in a semigrand ensemble was used. The finite size scaling analysis was employed to compute the critical packing fractions for infinite systems with high accuracy for a range of non-additivity parameters wider than in the previous studies.
Обчислено значення критичних фракцiй заповнення для сумiшей симетричних неадитивних твердих дискiв. Цiкавою рисою даної моделi є факт, що її внутрiшня енергiя є нульовою i фазовi переходи мають ентропiйну природу. Використано кластерний алгоритм для моделювання Монте Карло у напiв-великому
ансамблi. Застосовано аналiз скiнченно-вимiрного скейлiнгу для прецизiйного обчислення критичних
фракцiй заповнення нескiнчених систем для ширшої областi параметра неадитивностi порiвняно iз попереднiми дослiдженнями.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/155779 |
| citation_txt |
Critical point calculation for binary mixtures of symmetric non-additive hard disks / W.T. Gózdz, A. Ciach // Condensed Matter Physics. — 2016. — Т. 19, № 1. — С. 13002: 1–8. — Бібліогр.: 41 назв. — англ. |
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2025-11-25T20:31:28Z |
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2025-11-25T20:31:28Z |
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| fulltext |
Condensed Matter Physics, 2016, Vol. 19, No 1, 13002: 1–8
DOI: 10.5488/CMP.19.13002
http://www.icmp.lviv.ua/journal
Critical point calculation for binary mixtures
of symmetric non-additive hard disks
W.T. Góźdź, A. Ciach
Institute of Physical Chemistry Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland
Received November 6, 2015, in final form November 29, 2015
We have calculated the values of critical packing fractions for the mixtures of symmetric non-additive hard
disks. An interesting feature of the model is the fact that the internal energy is zero and the phase transitions
are entropically driven. A cluster algorithm for Monte Carlo simulations in a semigrand ensemble was used. The
finite size scaling analysis was employed to compute the critical packing fractions for infinite systems with high
accuracy for a range of non-additivity parameters wider than in the previous studies.
Key words: phase coexistence, critical point, finite size scaling, Monte Carlo simulations
PACS: 05.10.Ln, 05.20.Jj, 05.70.Jk
1. Introduction
Two-dimensional fluid mixtures are quite common in soft-matter and in biological systems. Impor-
tant examples are particles adsorbed at interfaces, on surfaces of pores in porousmaterials and biological
membranes. When the adsorbed fluid forms a monolayer, it may be modelled as a two-dimensional sys-
tem. The phenomenon of adsorption has been intensively studied, and one of the main contributors to
the field is Stefan Sokolowski [1–6]. An interesting question that is not fully solved yet is the phase sepa-
ration in a binary mixture on surfaces with different curvatures. This question is important not only for
the adsorption on curved surfaces present in porous materials, but also for the properties of biological
membranes surrounding organelle. In living organisms, the membranes are close to the critical point of
the demixing transition [7, 8]. Therefore, the phase behavior of multicomponent two dimensional flu-
ids and, particularly, the critical behavior may be of biological importance. During the phase transition,
a small change of thermodynamic parameters causes a large change of the composition. It may result
in significant shape transformations of the biological membranes since their shape is linked with their
composition [9]. Especially interesting are the membranes which form triply periodic bilayers [10, 11], as
well as porous materials whose internal surfaces are similar to minimal surfaces (the average curvature
vanishes at every point, and the Gaussian curvature is negative).
The phase separation in binary fluidmixtures belongs to the Ising universality class, and the universal
properties of the two-dimensional Ising model are well known from exact results [12]. The nonuniversal
properties, however, should be determined for each particular system, and the only feasible method for a
surface with arbitrary curvature is by computer simulations. Thus, it is important to develop a simulation
procedure that is fast, efficient and accurate. Moreover, it is important to very accurately determine the
critical parameters of a generic model on a flat surface, so that the results of approximate theories or
the results of simulations on curved surfaces could be compared with reliable data. The lattice model is
not appropriate for investigations of the properties of curved surfaces. Therefore, in this work we have
chosen a generic continuous model of non-additive hard core mixtures. Some real phenomena, which
can be modelled by a mixture of non-additive hard disks, are discussed, for example, in reference [13]
and in references cited herein. The behavior of the hard core mixtures has been studied in bulk [14–21]
and in restricted geometry [3, 22–24] in three dimensional systems. Much less attention was paid to the
two dimensional systems [13, 20, 24–27].
© W.T. Góźdź, A. Ciach, 2016 13002-1
http://dx.doi.org/10.5488/CMP.19.13002
http://www.icmp.lviv.ua/journal
W.T. Góźdź, A. Ciach
We study the mixture of symmetric non-additive hard disks with the interaction potential defined by:
Uαγ(r ) =
{
∞ if r <σαγ,
0 if r >σαγ,
(1.1)
where r is the distance between the centers of two disks, indexes α ∈ {A,B} and γ ∈ {A,B} describe the
species. The length scale is set by the A-component hard-disks diameter, σA A = 1. For symmetric non-
additive mixtures,
σBB =σA A (1.2)
and
σAB =
1
2
(σA A +σBB )(1+∆), (1.3)
where ∆ is the non-additivity parameter. We study the mixtures of positive non-additivity with different
values of the parameter ∆. This potential is an idealization of interactions in amixture of identical colloid
particles with surfaces covered with polymeric brushes of two types, A and B . The polymeric brushes of
different type effectively repel each other, but the polymers of the same type can interpenetrate. For this
reason, the separation between the like particles can be shorter than between different particles.
Quite evidently, this is an athermal model, because all the allowed configurations are of the same
energy. Nevertheless, such mixtures are capable of separating into two phases, i.e., one phase rich in
component A and the other one rich in component B . The phase separation is not induced by competition
between the internal energy and the entropy as in standard systems, but rather by competition between
the entropy of mixing and the entropy associated with the area available for the particles. An increase of
the packing fraction defined as
η= ηA +ηB =
π
4
NAσA A
S
+
π
4
NBσBB
S
, (1.4)
where S is the surface area of the system and NA and NB are the numbers of the A and B particles, leads
to a larger decrease of the available area in a homogeneous mixture than in a phase-separated system.
This effect starts to dominate over the decrease of the entropy of mixing in a phase-separated system
for η > ηc. Thus, ηc plays the role analogous to the critical temperature Tc, and (ηc−η)/ηc plays a role
analogous to (T −Tc)/Tc in standard mixtures with interacting particles. Simulation results [19] show
that this model system belongs to the Ising universality class. The universal properties of the model are
known from the exact solution of the Ising model in two dimensions, but the nonuniversal properties,
such as ηc, should be obtained by simulations.
At the absence of an external field, the symmetry of interactions implies, for the two coexisting phases
I and II, the following relations:
xI
A = xII
B , xI
B = xII
A , (1.5)
and
µI
A =µII
A =µI
B =µII
B , (1.6)
where µI
α and xI
α are the chemical potential and the composition of the component α in the i -th phase,
respectively [19, 28]. Here, the difference of the chemical potentials h = µA −µB plays the role of an
external field.
We are interested in the critical properties of a mixture when the external field is zero. Along the
symmetry line h = 0, the composition of the coexisting phases is symmetric, and therefore the critical
point is at the concentration xc = NA/(NA +NB ) = 0.5.
2. The method
Wemodel an open system in contact with a reservoir by using the Semigrand [19, 29] Monte Carlo [30,
31] simulation method. Using this method, the system is simulated under a constant total number of
particles N , total volume V , temperature T , and the difference of the chemical potential of one species
with respect to an arbitrarily chosen species∆µ. Thus, the number ofmolecules of each species fluctuates,
13002-2
Critical point calculation for binary mixtures of symmetric non-additive hard disks
while the total density remains constant. The semigrand ensemble is superior to the grand ensemble
in simulating dense fluids because the particle insertion moves are inefficient for dense fluids. For a
symmetric binary hard disks mixture, the internal energy and the chemical potential difference are both
zero.
The realization of the semigrand ensemble Monte Carlo simulation for symmetric non-additive hard
disks requires two kinds of moves: translation and identity change. The identity change moves can be
implemented in the following way. A molecule can be chosen randomly from all the molecules, and the
identity change move is always accepted if there is no overlap between the particles after the identity
change. Such a procedure works quite well for a small number of molecules, up to a few hundred. The
simulations near the critical point might be, however, time consuming. Therefore, we use a cluster al-
gorithm to perform the simulations for larger system sizes [16, 24]. The idea of the cluster moves is as
follows. The system is divided into a set of clusters. The molecules belong to a cluster if the distance from
any molecule to any other molecule is smaller than σA A +∆. When the clusters are formed, the identity
of all molecules in each cluster is randomly changed with the probability p = 0.5. We identify the clus-
ters using the SLINK algorithm [32–34], which is fast and does not require a huge amount of computer
memory. We have performed the calculations using either local MC moves or cluster moves. The results
obtained by both methods were consistent, but the time of calculations was much shorter for the cluster
algorithm.
For the translation moves, the maximum displacement is chosen to obtain 50% acceptance ratio. The
identity-change and the translation moves are chosen randomly with the ratio of N translations per one
cluster identity change move. We have performed calculations with a square or with a rectangular simu-
lation box. The aspect ratio of the rectangular box is taken as
p
3 : 2 to allow for the arrangement of the
molecules into a triangular lattice. Periodic boundary conditions are employed. We have not observed
any dependence of the results of calculations on the shape of the simulation box when the simulations
were performed for fluid mixtures. The averages are taken over 106 Monte Carlo cycles, where a cycle
consists of N translation and one cluster identity change move.
In molecular simulations, the results of calculations depend on a system size. The dependence ismore
pronounced for the calculations near the critical point since the correlation length becomes larger and
larger when the system is closer and closer to the critical point. When the correlation length is larger
than the size of the computational box, the results of the calculations become biased. That is why it may
be very difficult to perform exact calculations of the critical point parameters.
When the system is in the two phase region, we do not always have only one phase in the simulation
box during the simulation in the semigrand Monte Carlo method. It is possible that in the simulation box
we will have either the first or the second phase. With some frequency, the first phase disappears and the
second phase appears and vice versa. The higher is the frequency of this change, the closer the system is
to the critical point. That is why it is hard to calculate the concentration at the coexistence as an ensemble
average. One may try to overcome this problem by taking the most probable value of the concentration
instead of the ensemble average, but near the critical point, this proceduremay be problematic especially
for two dimensional fluids. The shape of the coexistence curve for the two dimensional fluid is much
flatter than for the three dimensional fluids. The critical exponent for the two dimensional fluids is β =
1/8 while for the three dimensional fluids, it is β ≈ 0.3258. The distribution of the concentration is not
sharply peaked and it is difficult to obtain themost probable value of the concentration with a sufficiently
high accuracy.
Fortunately, it is possible to use the calculations performed for the systems of finite size to get the
data on the infinite systems by using the concept of finite-size scaling [19, 35–37]. For each value of the
non-additivity parameter ∆, the critical packing fraction of the infinite system, ηc(∞), can be calculated
from the set of apparent critical packing fractions in systems with N particles, η∗c (N ), according to the
relation [19, 36, 37]:
η∗c (N )−ηc(∞) ∝ N−1/(dν)
, (2.1)
where the critical exponent ν is equal to 1 for the 2D Ising universality class, and d = 2 for the two dimen-
sional systems. The apparent critical point in the finite-size system can be determined from the distribu-
tion of the order parameter, PN (m) calculated in the simulations as a histogram. The order parameter in
the case of the non-additive hard disks system is the concentration, m = x − xc, where x = NA/(NA +NB )
13002-3
W.T. Góźdź, A. Ciach
and xc is the critical concentrationwhich is exactly 1/2 for symmetric mixtures. The distribution PN (m) is
rescaled in such away as to have a unit norm and a unit variance, and the rescaled distribution is denoted
by P∗
N (y), where the rescaled order parameter is y = a−1
m Nβ/(dν)m with am denoting a proportionality
constant. For the apparent critical packing fraction η∗c (N ), P∗
N (y) has an universal shape P∗(y) [19, 37].
Since this model belongs to the Ising universality class, the universal function P∗(y) is known [38, 39].
3. Results
The order-parameter distribution PN (m) is calculated in the simulations as a histogram, with the
number of bins equal to the number of particles N . The function PN (m) obtained in simulations for fixed
N and various η is then rescaled in such a way as to have a unit norm and a unit variance. The rescaled
function P∗
N (y) is compared with P∗(y) for several values of η, and the best fit gives us η∗c (N ). Figure 1
shows the order-parameter distribution function P∗
N (y) calculated in the simulations for the system with
∆ = 0.2 and N = 324 disks, compared with the distribution function for the 2D Ising model [39]. P∗
N
(y)
-2 -1 0 1 2
y
0
0.5
1
1.5
P
* (y
)
Figure 1. (Color online) The normalized distribution of the order parameter P∗
N
(y) (open circles) for ∆=
0.2 and
p
N = 18, expressed as the function of the scaling variable y = a−1
m Nβ/(dν)m, at the packing
fraction η = 0.5384. The solid line is the universal function P∗(y) for the Ising model [39]. The critical
exponent ν is equal to 1, and am is a proportionality constant.
agrees verywell with the universal distribution P∗(y) for the value of η that we identify with the apparent
critical volume fraction η∗c (N ). In the same way, we obtain apparent critical packing fractions for a set of
systems with a different size. In figure 2, we present the plot of apparent critical densities for different
system sizes calculated for the non-additivity parameter ∆= 0.2. The apparent critical packing fractions
are estimated for a set of finite systems with
p
N = {18,20,22,24,26, 28,30}. The critical packing fraction
as a function of N−1/2 for an infinite system was obtained by fitting the set of apparent critical packing
fractions to a straight line and extrapolating the value of the critical packing fraction at infinity. The same
procedure was used to determine the critical packing fractions for all the values of the non-additivity
parameter ∆.
In practice, the shape of the rescaled distribution function P∗
N (y) is compared with the universal dis-
tribution of the order parameter, P∗(y), for the first estimation of the apparent critical packing fraction.
In order to determine the precise value of the apparent critical packing fraction, the fourth order cumu-
lants,
UN = 1−
〈m4〉
3〈m2〉2
. (3.1)
are calculated for the values of the packing fraction η for which the distribution functions are the most
similar to the universal distribution P∗(y). To calculate the matching point, the values of the cumulant
UN (η) have been interpolated near the universal value U∗
N . We use the fourth order cumulant UN [40],
because its value at a fixed point has been already calculated for the 2D Ising universality class.
13002-4
Critical point calculation for binary mixtures of symmetric non-additive hard disks
0 0.01 0.02 0.03 0.04 0.05 0.06
N
-1/2
0.538
0.539
0.54
0.541
0.542
η
Figure 2. Scaling of the apparent critical packing fraction with the system size
p
N =
{18,20,22,24,26,28, 30} for ∆ = 0.2. The open circles represent the apparent critical packing frac-
tions for different system sizes obtained by matching P∗
N
(y) with the universal function as in figure 1.
The solid circle is the critical packing fraction for the infinite system [see equation (2.1)]. The dashed line
is the least square fit of the apparent critical packing fractions to a straight line.
The n-th moment 〈mn〉 can be easily calculated from the distribution of the order parameter PN (m):
〈mn〉 =
∑
m mnPN (m)
∑
m PN (m)
. (3.2)
The apparent critical packing fractions η∗c (N ) satisfy the equation UN (η∗c ) = U∗, and have been read
off from the interpolated line for the value of U∗ = 0.61069 [41]. In figure 3, we present the cumulants,
UN (m), as functions of the packing fraction for different system sizes N . The curves cross at the values of
the cumulant higher than the universal value U∗. Similar behavior was observed in references [20, 24].
In figure 4we present the results of our calculation of the critical packing fractions for a set of the non-
additivity parameter ∆, compared with the results already reported in the literature. The calculations
reported in reference [26] significantly overestimate and in reference [27] significantly underestimate
the results of other works [13, 20, 24, 25]. This is most probably the result of inappropriate simulation
methods employed in those calculations. It can be easily noticed that the results of our calculations are
in very good agreement with the results of the calculations reported in reference [20] and reference [24].
In all these works, the authors use cluster algorithms in the simulations and the critical packing frac-
tions are calculated for infinite systems where different kinds of finite size scaling analysis were used.
0.48 0.5 0.52 0.54
η
0
0.1
0.2
0.3
0.4
0.5
0.6
U
N
Figure 3. (Color online) Fourth order cumulant, UN (η), as a function of the packing fraction η for the
non-additivity parameter ∆ = 0.2 and
p
N = {18,20,22,24,26, 28,30}. The solid circles are the results of
the simulations. The dashed line is just to guide the eye. The dash-dotted line is plotted at the universal
value of the cumulant U∗ = 0.61069.
13002-5
W.T. Góźdź, A. Ciach
0 2 4 6 8
∆
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
η
0 1
∆
0.2
0.3
0.4
0.5
0.6
η
Figure 4. The critical packing fraction as a function of the non-additivity parameter ∆. The open circles
show the calculations from this work, the triangles up — reference [26], the stars — [27], the triangles
down — [13], the squares — [24], the diamonds — [20], the triangle right-hand — [25]. The dashed line is
just to guide the eye.
In reference [25], the critical point packing fraction was obtained using the method of crossing the re-
duced second moment. This method is not as precise as the method of finite size scaling analysis. In
reference [13], the critical packing fraction was estimated from the simulation of finite systems of the
order of N = 2000 molecules. Thus, the values of the critical packing fractions should be lower than the
values of the critical packing fractions obtained for infinite systems.
In our work, we employ the same cluster algorithm as in reference [24]. We use, however, a differ-
ent numerical algorithm to identify the clusters, and different scaling procedure to obtain the critical
packing fraction. In reference [20], a different cluster algorithm is used, where the clusters are built
by superimposing two configurations and identifying the clusters by selecting the groups of overlap-
ping molecules in the sense of hard core interactions. This method works very well for sufficiently large
values of the non-additivity parameter ∆ but is not as good for small ∆. In reference [20], calculations
were performed for ∆ ∈ {0.5,1.0,2.0,4.0}. The cluster algorithm used in reference [24] and in our calcula-
tions works very well for any value of the non-additivity parameters. In reference [24], calculations were
performed for ∆ ∈ {0.1,0.2,0.5,1.0}. In this work, we have performed calculations for all the values of
the non-additivity parameter ∆ for which the simulation results already existed, and we have extended
the calculation to additional large and small non-additivities. We have performed the calculation for
∆ ∈ {0.1,0.15,0.2,0.3,0.4,0.5, 0.6,0.7, 0.8, 0.9,1.0, 1.5, 2.0,3.0, 4.0, 6.0,8.0}. The critical packing fractions for
this set of parameters are presented in table 1. In figure 4, the results of our calculations are indistin-
guishable from the results of calculations in references [20] and [24]. An increase of the non-additivity
parameter ∆ results in a decrease of the critical packing fraction as shown in figure 4. In the mixture
of non-additive hard disks, we have a competition between the entropy of mixing and the excluded vol-
ume effects. When the non-additivity parameter is large, the excluded volume effects are strong and the
Table 1. The critical packing fractions ηc for infinite systems for different values of the non-additivity
parameter ∆.
∆ 0.1 0.15 0.2 0.3 0.4 0.5
ηc 0.6607(30) 0.5928(20) 0.5416(30) 0.4644(30) 0.4069(30) 0.3615(40)
∆ 0.6 0.7 0.8 0.9 1.0 1.5
ηc 0.3246(40) 0.2936(30) 0.2674(40) 0.2447(40) 0.2250(30) 0.1554(30)
∆ 2.0 3.0 4.0 6.0 8.0
ηc 0.1140(40) 0.0686(30) 0.0455(30) 0.0241(20) 0.0148(20)
13002-6
Critical point calculation for binary mixtures of symmetric non-additive hard disks
demixing process takes place at a lower packing fraction. When the non-additivity parameter is small,
the entropy of mixing dominates and the demixing takes place at a higher packing fraction.
4. Summary and conclusions
We have calculated the critical packing fractions with high accuracy for the non-additive hard disks
mixtures for a wide range of the non-additivity parameter. We have used Monte Carlo simulations with
cluster algorithm which resulted in rejection-free moves. We have used finite size scaling analysis based
on the universal distribution of the order parameter to determine the critical packing fraction for infi-
nite systems. The results of our calculations agree very well with the results of the previous calculations
where the finite size scaling analysis was used. The proposed simulation method allows for accurate and
unambiguous determination of the critical packing fraction for any value of the non-additivity param-
eter ∆. We hope that the results of our calculations will be the reference point for testing the results of
approximate theories for hard disks systems. They should also allow one to compare the critical proper-
ties of the particles adsorbed at flat or at curved surfaces, and to determine in this way the effect of the
curvature of the underlying surface on the critical properties of the adsorbed fluid mixture.
Acknowledgement
The support from NCN grant No 2012/05/B/ST 3/03302 is acknowledged. We would like to thank Noe
Almarza and Paweł Rogowski for helpful discussions.
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Обчислення критичної точки для бiнарних сумiшей
симетричних неадитивних твердих дискiв
В.Т. Гуздзь, А. Цях
Iнститут фiзичної хiмiї Польської академiї наук, Варшава, Польща
Обчислено значення критичних фракцiй заповнення для сумiшей симетричних неадитивних твердих ди-
скiв. Цiкавою рисою даної моделi є факт, що її внутрiшня енергiя є нульовою i фазовi переходи мають ен-
тропiйну природу. Використано кластерний алгоритм для моделювання Монте Карло у напiв-великому
ансамблi. Застосовано аналiз скiнченно-вимiрного скейлiнгу для прецизiйного обчислення критичних
фракцiй заповнення нескiнчених систем для ширшої областi параметра неадитивностi порiвняно iз по-
переднiми дослiдженнями.
Ключовi слова: фазове спiвiснування, критична точка, скiнченно-вимiрний скейлiнг, моделювання
Монте Карло
13002-8
http://dx.doi.org/10.1007/BF01448218
http://dx.doi.org/10.1063/1.1501126
http://dx.doi.org/10.1080/00268979500100581
http://dx.doi.org/10.1080/00268978800100743
http://dx.doi.org/10.1080/01621459.1949.10483310
http://dx.doi.org/10.1063/1.1699114
http://dx.doi.org/10.1093/comjnl/16.1.30
http://dx.doi.org/10.1103/PhysRevLett.28.1516
http://dx.doi.org/10.1103/PhysRevE.52.602
http://dx.doi.org/10.1088/0305-4470/21/1/028
http://dx.doi.org/10.1063/1.3377089
http://dx.doi.org/10.1103/PhysRevLett.47.693
http://dx.doi.org/10.1088/0305-4470/26/2/009
Introduction
The method
Results
Summary and conclusions
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