A replica Ornstein-Zernike self-consistent theory for mixtures
in random pores
G. Pellicane, C. Caccamo
Dipartimento di Fisica, Università di Messina and Istituto Nazionale per la Fisica della
Materia (INFM)
D. S. Wilson and L. L. Lee
School of Chemical Engineering and Materials Science, University of Oklahoma,
Norman, Oklahoma 73019
We present a self-consistent integral equation theory for a binary liquid in equilibrium
with a porous medium, based on the formalism of the replica Ornstein-Zernike (ROZ)
equations. Specifically, we derive direct formulas for the chemical potentials and the
zero-separation theorems (which provide a connection between the chemical potentials
and the fluid cavity distribution functions). Next we solve a modified-Verlet closure to
the replica OZ equations, which has built-in parameters that can be adjusted to satisfy the
zero-separation theorems. The degree of thermodynamic consistency of the theory is kept
under control by monitoring all members of the Gibbs-Duhem relation. We model the
binary fluid in random pores as a symmetrical binary mixture of non-additive hard
spheres in a disordered hard-sphere matrix and consider two different values for the nonadditivity parameter and for the quenched matrix packing fraction, at different mixture
concentrations. We compare the theoretical structural properties with Percus-Yevick (PY)
and Martinov-Sarkisov (MS) integral equations and assess both structural and
thermodynamic properties by performing NVT, standard and biased VT Monte Carlo
simulations. Our theory appears superior to the other integral equation schemes here
1
examined and gives reliable estimates of the chemical potentials. This feature should be
useful in studying liquid-liquid demixing behavior in general.
2
I.
Introduction
Phase changes of liquids inside porous media are of both scientific and practical interests.
Porous materials, such as activated carbon, silica gels, zeolites, and pillared clay and
more recently, aerogels, aminosilicates, aluminophosphate, carbon nanotubes, Vycor
glass, microporous BN, and starburst dendrimers, have been extensively used in industry
for adsorption, dehumidification, catalysis, gas separation [1] and gas storage[2] [3].
Many of these materials have an amorphous structure, that is, they consist of micro-sized
pores that are irregularly distributed throughout the material. For example, aerogels have
a cobweb-like structure that is made up of cross interconnecting inorganic or organic
colloidal-like particles or polymeric chains with high porosity (75-99%)[4]. When fluids
invade the interior, the confinement in narrow dimensions affects the fluid phase
behavior, as has been well documented. Few studies have looked into binary fluid
mixtures in pores. The simplest model for such a mixture, exhibiting phase changes, is
the nonadditive hard sphere mixtures (NAHSM) in random pores. In this work we
employ both integral equations (IE) and molecular simulations to characterize the liquidliquid phase behavior of NAHSM under confinement. Such studies should shed light on
the size and nonadditivity effects on phase properties in general.
The integral equations of the Replica type will be solved for binary nonadditive hard
sphere mixtures in pores. For the systems chosen, these equations are afferent. We have
applied the replica Ornstein-Zernike (ROZ) [5] equations earlier for pure spheres in
pores [6] and obtained accurate results as compared to simulation data. A number of
3
other publications have also investigated the mixtures in pores [7][8]. In our approach,
we improve the performance of the OZ integral equations by requiring thermodynamic
consistency and conformity to the zero-separation theorems. This combined
methodology has been applied and tested in a number of previous studies: on pure hard
spheres in pores [9] and in bulk [10], additive [11] and nonadditive hard sphere mixtures
in bulk [12], Lennard-Jones molecules [13], and diatomic hard dumbbells [14] as well.
In all cases, close agreement was obtained between IE and simulation. The essence of
such an approach is the use of the zero separation theorems [15], in addition to the usual
thermodynamic consistencies [16], that the values of correlation functions at coincidence
(when two particles coincide and the separation distance r=0) obey certain exact
conditions that tie them to the thermodynamic properties of the system under study, such
as the chemical potentials and isothermal compressibility. This is, in a way, similar to the
contact value theorem (e.g., at g(d+), the contact value of the pair correlation g(r)) for
hard-core systems [17]. Since these relations are exact, there is all the reason that they
should be satisfied for consistency. The importance of ZST is in the “local” nature: that
specific values of the correlation functions are required at specific distances r (=0, or
=d), in contrast to the “global” (or integral or almost everywhere) nature of the
thermodynamic consistencies. Both global and local consistencies reinforce the
“accuracy” in the closure formulation.
In this work, we develop new zero-separation theorems for our particular system
(NAHSM gases in hard sphere pores). There will be five (5) monomer chemical
potentials, and seven (7) “dimer” chemical potentials. Consequently, there are eleven
4
(11) coincidence values for the cavity functions (twice for the y10(0) , y20(0) , and y12(0)
functions). These zero-separation values are used in constructing the closure relations in
the ROZ. In order to test the accuracy of the theoretical calculations, we have carried out
three types of Monte Carlo (MC) simulations: the grand canonical ensemble MC, the
NVT canonical ensemble MC, and the cavity-biased grand canonical ensemble MC.
These simulations are used to generate the correlation functions, chemical potentials, and
phase envelopes to check the theoretically derived behavior for the NAHSM in pores.
Fifteen conditions are investigated, representing different nonadditivities (Δ) of hard
sphere species, matrix porosities (ρ0), gas densities (ρ*), and compositions of mixtures
(x). The two hard sphere species, 1 and 2, are of equal size (diameters σ11 = σ22 = σ),
except for their nonadditive cross interaction (σ12 = ½(σ11 +σ22 )(1+ Δ)). Two values of
Δ = 0.2, and 0.4 are used (i.e., positive nonadditivity). The gas densities ρ* = ρσ3, range
from 0.18 (low porosity) to 0.3 (high porosity). The matrix number density ρ0 , reduced
by the fluid sphere diameter σ , varies from 0.1 to 0.3. The compositions (mole fractions)
are around 0.1, 0.3, and 0.5. This range of variables is needed to characterize the liquidliquid phase diagram. (Note that our conditions of simulation are similar to those of
Sierra and Duda [8]). However, we used different MC algorithms— other than the
semigrand canonical ensembles, and we used the ZSEP closures for the ROZ equations in
addition to the PY or MS closures.
In Section II, we present the theory, the Replica OZ as applied to our system, and the
ZSEP closure. We give detailed formulas for the zero-separation theorems on the cavity
functions, yij(0). The details of our simulation algorithms are also described. In Section
5
III, we show the results of both integral equations and MC simulation data. We also give
concluding remarks in the section.
II. The Model and the Methods
A. Model
We consider a binary mixture of non-additive hard-sphere particles:
Vij (r )  


where
 r ij
0 r ij
 12  (1   )
(1) ,
( 11   22 )
  11 ,  22
2
and  is the non-
additive parameter. The fluid mixture is confined in a matrix of random, non-overlapping
hard-sphere particles. Within this model for the fluid mixture, a positive non-additivity in
the cross interaction between the two species favours homocoordination in comparison to
hetero-coordination. As a consequence of the extra-repulsion from unlike interactions,
non-additive hard sphere mixtures are expected to separate into two phases at sufficiently
high density.
We considered this model because it is the simplest mixture possible to exhibit a
thermodynamically stable liquid-liquid transition. In all the calculations we assume that
the size of the two species is equal to the diameter of the matrix sphere, i.e. 11 = 22=0,
briefly stated we considered symmetrical mixtures. Then, we study the effect of the
6
increasing tendency to homocoordynation at 20 % and 40 % compositions (
on the structure and the thermodynamics of the system.
B. Theory
The formalism of the ROZ equations was derived for the first time for pure fluids by
Madden and Gland and Given and Stell [5], and recently extended to binary liquids by
Paschinger and Kahl. [7] These equations describe the structure of a binary liquid inside
a porous matrix and are formally derived after a partial quenching of one of the
components and in the limiting case for s=0 of a fully equilibrated (2s + 1)-component
mixture, where s is the number of identical copies of the liquid mixture (the replicas);
thus, within this formalism, the mixture confined into the rigid matrix is depicted as a
semi-quenched system (partly quenched and partly annealed). The label 0 indicates the
matrix species; 1 and 2 the liquid mixture species; 3 and 4 any replica of 1 and 2 (odd
numbers for replica of 1, and even numbers for replicas of 2), respectively. In the
following, we adopt the matrix notation of [8] for the correlation functions hij(r) and
cij(r), and we suppress the argument r in order to improve readability; the Replica OZ
equations are so written as (“T” denotes the transpose of a vector and  is a convolution
):
h00  c00   0 c00  h00
h 01  c 01  h 01    c00  1h 11  c 01  1h 12c 01
h 11  c11  h 01    c T0 1  1h 11  c11  1h 12c12
h 12  c12  h 01    c T0 1  1h 12  c11  1h 12c12
 21h 12  c12
where
7
(2)
h 
c 
ρ 0 
, h 01   01 , c 01   01  (3)
1   1
 0 ρ2 
 h 02 
 c 02 
h
h 01   11
 h 12
h
h 12   13
 h 14
h 12 
c
, c11   11
h 22 
 c12
h 14 
c
, c12   13
h 24 
 c14
c12 
,
c 22 
(4)
c14 
.
c 24 
The total number of the coupled ROZ equations is 8, while the first one is the bulk OZ
equation for the matrix particles. The ROZ equations are solved under different closures,
for example, the modified-Verlet closure [18]:
Bij (r )  
ij 
2
 (5) ,
 *ij 1  ij 
* 

2
1   ij  ij 

 ij
where Bij(r) and ij*(r) are the bridge and the renormalized indirect correlation functions,
respectively; the renormalization of the indirect correlation function was achieved by
adding a soft-Weeks Chandler Andersen (WCA) potential [19] to the ij function
1
 * (r )   (r )  f (r ) , where f(r) is the Mayer factor of the repulsive WCA 6:3 potential.
2
The introduction of this term circumvents the discontinuities in the bridge functions when
combined with the closure; we slightly modified the indirect correlation function by
setting the pseudo-temperature in the Mayer factor to 9999. The 24 adjustable parameters
ij, ij and ij (3 parameters for each of the 8 ROZ equations) appear to be excessive and
we drastically reduce the number via a consistent procedure by letting them assume
“reasonable” values according to the following criteria: while we always set ij=1, we use
8
a two different sets of parameters for the equimolar and asymmetric concentrations,
respectively. For the equimolar concentrations we set 11SYM=22 SYM, 12 SYM
=14 SYM =12 SYM =14 SYM, 10 SYM =20 SYM ,33 SYM =44 SYM, 11 SYM =22
SYM
, 10 SYM =20 SYM ,33 SYM =44 SYM; for the asymmetric concentrations we fix
the values of the ij to the symmetrical concentration ones, ij=ijSYM and we allow all the
ij to vary under the constraint 12=14. In both cases, we end up with 7 parameters to be
determined by enforcing self-consistency theorems. We used zero separation theorems,
which relate the cavity functions values at zero distance to the proper expression for the
chemical potentials. In a previous paper, one of us reported the formulas of the chemical
potentials for common mixtures [12][20]; here we derive these formulas for the system
representing the matrix and the mutually noninteracting s copies (replicas) of the fluid
molecules. We introduce the following short-hand notation:
1


 k  ij    k  dr log y ij  hij  hij  ij  hij Bij  S ij  (6) ,
2


where ij(r) and Sij(r) are respectively the indirect correlation functions and the Star
functions introduced earlier [20], and k is the number density of specie k. Thus, the
monomer chemical potentials are easily written as (all chemical potentials are
configurational quantities, in excess of the ideal part):
0   0  00  s 1  01  s 2  02
1   0  10   1  11  (s 1 - 1) 3  13   2  12  (s 2 - 1) 4  14
 2   0  20   2  22  (s 2 - 1) 4  24   1  21  (s 1 - 1) 3  23
 
 44     43  (s
(7 )
 
 43 
3   0  30   3  33  (s 1 - 1) 3'  33'   4  34  (s 2 - 1) 4'  34 '
 4   0  40   4  44  (s 2 - 1) 4'
'
3
9
1
- 1) 3'
'
where 3’ and 4’ are replicas of 1 and 2 other than 3 and 4, respectively, and =1/kBT is
the inverse temperature in Boltzmann constant kB units.
Now, the reversible works of insertion of dimers made up from pairs of species 1+1, 2+2
, 1+2, 1+0,2+0,1+3,2+4 merged up to zero bond length (at infinite dilution in the
mixture) are the chemical potentials for dimers of the corresponding pairs at close
contact:
11   0  1  1 0   1  1  11  (s 1 - 1) 3  1  1 3   2  1  1 2  (s 2 - 1) 4  1  1 4
 2 2   0  2  2 0   2  2  2 2  (s 2 - 1) 4  2  2 4   1  2  2 1  (s 1 - 1) 3  2  2 3
1 0   0  1  0 0   1  1  0 1  (s 1 - 1) 3  1  0 3   2  1  0 2  (s 2 - 1) 4  1  0 4
 2 0   0  2  0 0   2  2  0 2  (s 2 - 1) 4  2  0 4   1  2  0 1  (s 1 - 1) 3  2  0 3
1 2   0  1  2 0   1  1  2 1  (s 1 - 1) 3  1  2 3   2  1  2 2  (s 2 - 1) 4  1  2 4
(8)


13   0  1  3 0   1  1  3 1   3  1  3 3   2  1  3 2   4  1  3 4  (s 1 - 2) 3'  1  3 3' 

(s 2 - 2) 4'  1  4 4 '



 2 4   0  2  4 0   2  2  4 2   4  2  4 4   1  2  4 1   3  2  4 3  (s 2 - 2) 4'  2  4 4 ' 

(s 1 - 2) 3'  2  4 3 '

These formulas are general and are applicable to any type of pair potential of interaction.
Now we shall specialize the above relations to the case of non-additive hard-sphere
mixture in the hard-sphere matrix and we shall also formulate the zero-separation
theorems. For the 11 cavity functions:
log y11 0  21  11  1 (9) .
Thus, we have (upon taking the limit s→0):
log y11 0   0  10  1  11 -  3  13   2  12   4  14 (10) .
10
In a similar way, we can build up the zero-separation theorems for the other cavity
functions:
log y 22 0    0  20   2  22 -  4  24   1  21 -  3  23
log y13 0    0  10
(11) .
log y 24 0    0  20
We note that the coincidence value of the mixture-replica cavity functions depends only
on the matrix-fluid correlations, similar to the pure case as reported in literature. The case
of log y10(0), log y20(0) and log y12(0) is a bit more complicated and we have to
distinguish between different cases. For the unlike cavity functions we have:
log y12 0   0  20   1  11   2  22   2  24  1  23
log y12 0   0  20   1  11   2  22   1  13   2  14
1   2
1   2
(12) ,
while for the particle-matrix cavity functions:
log y10 0    0  00   1  13   2  14   1  01   2  02
log y10 0    0  01   1  11  1  13   2  12   2  14
log y 20 0    0  00   2  24  1  23   2  02  1  01
log y 20 0    0  02   2  22   2  24   1  12  1  23
1   0
1   0
2  0
(13) .
2  0
Thus, the closure equation () to the ROZ equations is solved numerically under the
constraint determined by the 7 eq.s (11)-(14). We also checked the ability of this closure
to guarantee internal thermodynamic consistency. Unfortunately, the virial equation of
state for the quenched-annealed system is not straightforwardly related to structural
functions as for bulk mixtures [21]; in fact for hard-core interactions:
11
 P 

P   0 

  0  V T 1  2

 (14)
dg 00  00 ; s 
2  2 3
   i   0  00 lim
   i2  3ii g ii ( ii )  2   0  i  30i g 0i ( 0i )  21 2 123 g 12 (12 )
s

0
3
ds
i
i 1, 2
i 1, 2


,
where the first term is the infinitesimal change of the virial pressure with respect to the
matrix density at constant values of volume, temperature and chemical potentials of the
dg 00  00 ; s 
is the infinitesimal change of the matrix-matrix radial
s 0
ds
two species, and lim
distribution function in the replicated system with respect to the number of replicas in the
limit of the quenched-annealed system. Both the two quantities should be evaluated
numerically, and in particular the first one implies the calculation of the product of two
numerical derivatives as a standard mathematical manipulation of thermodynamic
derivatives shows:
 P 
 P 



 
  0 V T 1  2   0 V
T N1 N 2
 P 

  
i 1, 2   i V

  i 




0

V
T N1 N 2

 (15)
T N1 N 2 

Thus, we resorted to a different strategy in order to monitor the thermodynamic
consistency of the theory, by using the Gibbs-Duhem relation [13]:
 P 



  V T N 0

 i




j  i 
i 1, 2   j
j 1, 2




V

~c
  k BT T 1  1    i  j  c ij (0) (16) ,

i , j 1, 2
T N k  j 0 
where the chemical potentials of the two species are differentiated numerically in order to
~c
obtain the inverse of the isothermal compressibility and where the c ij (0) are the Fourier
Transforms at zero wave vector of the connected part of the direct correlation functions,
within the ROZ formalism.
12
The ROZ equations were also solved under the Percus-Yevick (PY) [22] closure:

cij (r )  exp
 Vij ( r )

 1 1   ij (r )

(17) ,
and the Martinov-Sarkisov (MS) [23] closure:
cij (r )  exp
 Vij ( r )
1 10.5  ij ( r )
(18) .
We solved numerically the different closures to the ROZ equations with a Picard method
and adopted a standard mixing procedure for the direct correlation function in order to
ensure convergence; normally, 1024 grid points with grid interval 0.010 were used but
we also checked the stability of the solution with higher grids of 2048, 4096 points for
some thermodynamic state points. Zero separation theorems and the thermodynamic
constraint (16) were enforced in the modified-Verlet closure (ZSEP) with a tolerance of 1
%.
C. Simulation
We have performed Monte Carlo simulations with three algorithms [24]: the grand
canonical (T, V, 1, 2) ensemble (GCMC) [25], the standard NVT canonical ensemble,
and the cavity-biases grand canonical (CB-GCMC) ensemble [26]. The GCMC is used in
order to sample different matrix realizations and fluid mixture particle configurations.
Three types of particles moves (displacement, creation or destruction of a particle) were
performed randomly with equal probability inside a cubic box with cubic periodic
boundaries; typically, averages were taken over 30 matrix realizations and for each
realization not less than 100000 trial moves per particle were generated.
13
We also increased the efficiency of the importance sampling by performing simulations
in the cavity biased grand canonical (T, V, 1, 2) ensemble (CB-GCMC), where the
acceptance probability of a molecular configuration is biased so to attempt insertions of
new particles into existing cavities in the system instead of at randomly selected points.
In particular, insertion and deletion attempts were accepted with the probabilities:



exp    V (r N )  V (r N 1 )
Pi  min 1,VPcN (r N )
3 ( N  1)




exp     V (r N )  V (r N 1 )
Pd  min 1, N3
V PcN 1 (r N -1 )





(19) ,


where PcN (r N ) is the configuration-dependent probability of finding a cavity of diameter
c or larger, provided the system consists of N particles and is the DeBroglie thermal
wavelength. We estimated PcN (r N ) by implementing a cavity search using a finite grid.
We always used a box of side length equal to 13.5, and a uniform grid of 1003 points;
the grid size was chosen so as to be computationally manageable while generating a
negligible error in the limiting distribution of the Markov chain, since the deletion can
occur at any point in the box but the insertions are restricted to the grid points.
Finally, we performed standard constant number of particles, constant volume and
constant temperature (NVT) simulations from previously equilibrated GCMC simulations
in order to evaluate the radial distribution functions.
14
III. Results and discussion
We started to assess the structure of integral equation closures against standard NVT
simulations. In figures 1-2, from left to right panels, we report radial distribution
functions for the particle one-particle one, particle one-particle two and particle onematrix short-range correlations at a equimolar composition.
It appears that for a moderate non-additivity and two different matrix porosities, ZSEP
gives rise to the best performances, quantitatively describing the simulation profiles for
all the correlations. These very good performances by ZSEP maintain unaltered also at a
as much asymmetric concentration as 10% (see fig. 3) , with a slight shift up at longer
distances for the more dilute specie like-correlations and close to contact for the crossed
correlation; however, ZSEP seems to capture the correct phase and oscillations, in
agreement with the other integral equation schemes. Moreover, as shown in fig. 4, the
particle 2-particle 2 and particle 2-matrix correlations are nicely described by ZSEP.
We also monitored the effect of increasing non-additivity in fig. 5, that means increasing
the tendency of particles to homocoordination. While ZSEP shows a slight tendency to
overestimate and underestimate the like-like and the crossed correlations at longer
distances, respectively, it remains definitely the most reliable integral equation closure
between the ones here considered. These performances by ZSEP have been checked also
at a lower degree of matrix porosity, as reported in fig. 6.
After considering structure, we turned our attention to thermodynamics and, in particular,
to the ability of the theory to give reliable estimates of the chemical potential, that is a
primary ingredient for the determination of liquid-liquid phase separation envelopes.
15
As it appears in fig. 7, where the calculations were performed at fixed equimolar
concentration, ZSEP is able to reproduce accurately simulation results up to the mixture
densities close to the region of immiscibility in the phase diagram, with a discrepancy not
exceeding 2-3 %.
This result was confirmed by adopting a different strategy for comparison in figures 8-9:
we used as a starting guess for simulations, the theoretical chemical potentials and tried
to look at the simulation mean total mixture densities and the mean particle
concentrations in comparison with the theoretical values. Again, at the worst case
reported in the upper panel of fig. 8, i.e. for the higher matrix density and for the lower
mixture non-additivity considered in this work, the percentage discrepancy with
simulation was inferior to 3%.
In an attempt to reduce this error from theory in
comparison with simulation results, we also monitored the effect of imposing the
thermodynamic constraint of eq. (16) just for the case =0.2, 0=0.3. To this aim, we
used as an additional variational parameter the quantity and it emerged that decreasing
the degree of thermodynamic inconsistency by ZSEP from 10% to less than 1% brought
a slight change in the calculated chemical potentials (see upper panel of fig. 8). For this
reason, we have not tried to enforce systematically the internal thermodynamic
consistency of eq. (16) for all the cases considered in this paper. However, as
documented in tables I-II, ZSEP was not found to be thermodynamically inconsistent in a
significant way, with a percentage discrepancy (%TU) which on the average was of the
order of 3-4% and in the worst cases was never exceeding 10-15%.
Finally, we calculated the liquid-liquid coexistence lines for the worst case reported in the
upper panel of fig. 8 and assessed ZSEP predictions with Cavity-Biased Monte Carlo
16
simulations. Theoretical coexistence lines reported in fig. 10 were calculated by equating
the chemical potentials of the two species (second and third equation of (7), respectively),
at different concentrations. After performing such a match, we were able to establish the
lower consolute critical point, with an error of ~ 3%, as compared to the simulation
results. This is in conformity with the earlier observation on the accuracy of the
calculated chemical potentials.
17

0 *

TU 






 0.3   1.8
0.6868 1.0111
0.77995 0.8325
0.9553 0.9853 1.1792
 0.3   0.5
0.6868 1.0111
0.77995 0.8325
0.9553 0.9848 1.1722
0.2 0.1 0.35
0.5 2.5
0.6726 1.1113
0.8565
1.74695 0.9407 1.0591 2.3981
0.4 0.3 0.18
0.5 2.8
0.5637 1.0687
0.7406
0.7361
0.4 0.1 0.18
0.5 0.7
0.6032 1.6598
1.1328
-0.1468 0.9394 1.5889 0.02915
0.9198 1.0276 1.2120
Table I. Consistency parameters and percentage thermodynamic unconsistency
( for different mixtures at the equimolar concentration. In the top of the table
V)V]=N1/(
1
18

0 *

 0.3 0.3078 
TU




8.6
0.9566
1.0077
0.9530
0.9237
1.0092
13
0.9560
0.3
 0.3 0.3090  



0.9931 0.9763
1.0905
1.2428
0.9530
0.9941 0.9758
1.08446 1.2443
0.9988
0.9500
1.010
0.9679
0.9922
1.2789
0.7045
1.005
0.9495
1.0087 0.9684
1.0051
1.2776

 0.3 0.309- 0.10.31
0.11
 0.3 0.3095 0.105
(0.1961)
 0.1 0.35
0.3
6.1
0.9427
1.10045 0.9315
1.1004 1.0352
1.9334
2.7364
0.2 0.1 0.35
0.1
4.7
0.9291
1.07895 0.8929
1.1664 0.9702
1.4872
2.7578
0.4 0.3 0.179
0.335
2.9
0.9223
1.0592
0.9098
1.0584 0.9969
1.1145
1.3092
0.4 0.3 0.178
0.1105 4.8
0.9322
1.0360
0.8957
1.1124 0.9616
0.9833
1.3565
0.4 0.1 0.1795 0.310
2.9
0.95505 1.5906
0.92335 1.7837 1.42965 3.9580
-3.1866
0.4 0.1 0.179
1.5
0.9710
0.8856
-3.7338
0.104
1.4072
2.0898 1.2515
8.5566
Table II. Consistency parameters and percentage thermodynamic unconsistency
(%TU) for different mixtures. See Table I for the meaning of the symbols.
19
Figure 1. Radial distribution functions for the mixture at =0.2, 0=0.3, =0.5. Full
line: ZSEP; long-dashed line: MS; dotted line: PY; open circles: NVT Monte Carlo
simulation.
20
Figure 2. Radial distribution functions for the mixture at =0.2, 0=0.1, =0.5. See
Fig. 1 for the meaning of the symbols.
21
Figure 3. Radial distribution functions for the mixture at =0.2, 0=0.3, =0.1. See
Fig. 1 for the meaning of the symbols.
22
Figure 4. Radial distribution functions for the mixture at =0.2, 0=0.3, =0.1. See
Fig. 1 for the meaning of the symbols.
23
Figure 5. Radial distribution functions for the mixture at =0.4, 0=0.3, =0.5. See
Fig. 1 for the meaning of the symbols.
24
Figure 6. Radial distribution functions for the mixture at =0.4, 0=0.1, =0.5. See
Fig. 1 for the meaning of the symbols.
25
Figure 7. Chemical potential versus total mixture density at =0.2, 0=0.2, =0.5.
Triangles: Grand canonical Monte Carlo simulation; circles: Cavity-biased Grand
canonical Monte Carlo simulation; full line: ZSEP.
26
Figure 8. Total mixture density versus particle concentration at fixed chemical
potentials. =0.2. Triangles: Grand Canonical Monte Carlo simulation; open
circles: Cavity-biased Grand Canonical Monte Carlo simulation; diamonds: ZSEP;
crosses: ZSEP with thermodynamic consistency.
27
Figure 9. Total mixture density versus particle concentration at fixed chemical
potentials. =0.4. See Fig. 7 for the meaning of the symbols.
28
Figure 10. Liquid-liquid phase coexistence boundary in the total mixture densityparticle concentration plane. Error bars: Grand Canonical Monte Carlo simulation;
open circles: ZSEP; full line: curve fitting ZSEP points.
29
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31