Theoretical and computer simulation study of phase coexistence of nonadditive harddisk mixtures Giacomo Fiumara, Owen D. Pandaram, Giuseppe Pellicane, and Franz Saija Citation: The Journal of Chemical Physics 141, 214508 (2014); doi: 10.1063/1.4902440 View online: http://dx.doi.org/10.1063/1.4902440 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/21?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Isotropic-nematic phase equilibria of hard-sphere chain fluids—Pure components and binary mixtures J. Chem. Phys. 142, 064903 (2015); 10.1063/1.4907639 Fourth virial coefficients of asymmetric nonadditive hard-disk mixtures J. Chem. Phys. 136, 184505 (2012); 10.1063/1.4712035 Computational study of the melting-freezing transition in the quantum hard-sphere system for intermediate densities. I. Thermodynamic results J. Chem. Phys. 126, 164508 (2007); 10.1063/1.2718523 Phase coexistence in polydisperse multi-Yukawa hard-sphere fluid: High temperature approximation J. Chem. Phys. 125, 034501 (2006); 10.1063/1.2212419 Phase equilibria of model ternary mixtures: Theory and computer simulation J. Chem. Phys. 107, 6366 (1997); 10.1063/1.474297

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THE JOURNAL OF CHEMICAL PHYSICS 141, 214508 (2014)

Theoretical and computer simulation study of phase coexistence of nonadditive hard-disk mixtures Giacomo Fiumara,1 Owen D. Pandaram,2,3 Giuseppe Pellicane,2,3 and Franz Saija4,a) 1

Department of Mathematics and Computer Science, University of Messina, Viale F. Stagno D’Alcontres 31, I-98166 Messina, Italy 2 School of Chemistry and Physics, University of Kwazulu-Natal, Scottsville 3209, Pietermaritzburg, South Africa 3 National Institute for Theoretical Physics (NITheP), KZN Node, Pietermaritzburg, South Africa 4 CNR-IPCF, Viale F. Stagno d’Alcontres 37, 98158 Messina, Italy

(Received 8 October 2014; accepted 13 November 2014; published online 3 December 2014) We have studied the equation of state (EOS) and the equilibrium behavior of a two-component mixture of equal-sized, nonadditive hard disks with an interspecies collision diameter that is larger than that of each component. For this purpose, we have calculated the fifth virial coefficient by evaluating numerically the irreducible cluster integrals by a Monte Carlo method. This information is used to calculate both the virial equation of state and an equation of state based on a resummation of the virial expansion. Then, the fluid-fluid phase coexistence boundaries are determined by integrating the EOS so as to obtain the free energy of the system. Canonical and Gibbs ensemble Monte Carlo simulations over a wide range of nonadditivity are also performed in order to provide a benchmark to the theoretical predictions. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4902440] I. INTRODUCTION

The structural properties of real dense fluids depend mainly on short ranged repulsive intermolecular forces, which are adequately accounted for by hard-core models.1 In these systems exhibiting entropy-driven phase transitions,2 molecules do not interact at separations larger than a given distance and experience infinite repulsion if their separation is less than that distance.3 For a binary mixture confined on a substrate the simplest, but still nontrivial, model is represented by hard disks characterized by two different collision diameters (σ 11 , σ 22 ) and by an interspecies diameter σ 12 = (1/2)(σ 11 + σ 22 )(1 + ), where the dimensionless parameter  accounts for deviations from additive interactions.4 The sign of  is critical as to the type of thermodynamic behavior that is exhibited by the mixture. For  < 0 the model shows a tendency toward heterocoordination, while for  > 0 the mixture exhibits phase separation at high densities. This phenomenon is due to the extra repulsion between unlike spheres, which favours homocoordination in the system.4, 5 Many real phenomena can be modelled by a mixture of nonadditive hard-disks (NAHD). In a two-dimensional geometry, phase separation has been observed in mixtures of rare-gases physisorbed on graphite.6–8 In colloidal science, it has been shown that the interaction of two polymers dissolved in water can be mapped onto a nonadditive hard-disk potential.9 The NAHD mixtures have been also used to model the asphalten flocculation inhibition phenomenon.10 Recently, the model has been also tested to describe mixtures of ganglioside lipids and phospholipids arranged in monolayers or bilayers.11

a) Electronic mail: [email protected]

0021-9606/2014/141(21)/214508/9/$30.00

At variance with the three-dimensional case,12 there are very few theoretical and computational studies on NAHD mixtures. Dickinson13, 14 reported molecular dynamics simulations of NAHD mixtures in which the compressibility factor and the radial distribution functions for a few size ratios and some nonadditivities were computed. Tenne and Bergmann developed a scaled-particle theory (SPT)15 which was subsequently modified by Bearman and Mazo16 to study fluidfluid phase equilibria for positive values of the nonadditivity parameter.17, 18 The compressibility factors obtained from the resulting equations of state were compared with molecular dynamics (MD) and Monte Carlo (MC) computer simulations of two-dimensional mixtures of hard-soft spheres.19 A new equation of state was proposed both for symmetric and asymmetric mixtures and compared with MD simulations performed over a range of size ratios from 1 to 4.20, 21 However, the comparison between the numerical simulation results and the theoretical approach is rather satisfactory only in the additive case ( = 0) and for slightly nonadditive mixtures ( < 0.1). A first-order perturbation theory, the so-called MIX1 approximation, already successfully applied to the case of a symmetric nonadditive hard-sphere mixture,22 was used to trace out the fluid-fluid coexistence line for some values of .23 Santos et al. developed a general and unified theoretical approach for the equation of state for multicomponent mixtures of nonadditive hard-spheres in d dimensions.24 Recently, one of us25 computed the fourth virial coefficient of symmetric NAHD mixtures assessing the fluid-fluid coexistence curve, derived from two equations of state based on it, against some already published simulation results. In a more recent article26 also the fourth virial coefficient in the asymmetric case (size ratio q = σ 2 /σ 1 different from unity)

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was used and the performance of the resulting equation of state (EOS) assessed. Computer simulation results of the fluid-fluid phase separation of symmetrical hard-core mixtures can be performed via the Gibbs ensemble Monte Carlo (GEMC) method that has been widely adopted for 3D mixtures.27, 28 For symmetric NAHD mixtures, GEMC simulations have been recently reported,29, 30 also using cluster algorithms which allow to equilibrate systems with a very large number of particles.31 However, both these studies are carried out for NAHD mixtures with moderate or big values of the nonadditivity parameter ( ≥ 0.5). More recently, Munoz-Salazar and Odriozola performed semigrand canonical ensemble Monte Carlo of a NAHD mixtures only for a value of  = 0.2, tracing out the fluid-fluid coexistence line.32 In this paper, we study the fluid-fluid phase separation of a symmetric NAHD mixture over a wide range of values of the nonadditivity parameter  by means of GEMC simulations. At the same time we compute the fifth virial coefficients evaluating numerically—by using Monte Carlo methods—the independent irreducible cluster integrals. The computation of virial coefficients for hard-disk mixtures has been limited to the additive case up to the sixth order.33–35 From the knowledge of these coefficients it is possible to build up an equation of state based on a resummation of the virial expansion that is separately assessed against novel MC simulations in the canonical ensemble. The resulting expression of the Helmholtz free energy is then used to calculate the coexistence line and the line of critical points as a function of the nonadditivity parameter. Then, these theoretical predictions are compared with GEMC computer simulation results. The paper is organized as follows: In Sec. II, we introduce the model, the GEMC simulation, and the numerical method to calculate the fifth virial coefficient. In Sec. III, we show the results for the fluid-fluid coexistence lines over a wide range of  and the comparison between theoretical and numerical simulation results. Finally, we provide some concluding remarks in Sec. IV.

II. THEORY AND NUMERICAL SIMULATION METHODS A. Virial expansion

The virial expansion arrested to the fifth order can be written as βP = ρ + Bρ 2 + Cρ 3 + Dρ 4 + Eρ 5 + . . . ,

(1)

where P is the pressure, β = 1/kB T is the inverse of the temperature in units of the Boltzmann constant kB , and N +N ρ = ρ1 + ρ2 = 1 V 2 , with V being the volume and the ρ i are, respectively, the total and ith species number densities. In the remaining part of the paper, we shall refer to the virial expansion arrested to the third (D = E = 0 in Eq. (1)), fourth (E = 0 in Eq. (1)), and fifth order (see Eq. (1)) as, respectively, Zvir3, Zvir4, and Zvir5. In a mixture, at variance with the one-component case, the virial coefficients B, C, D, E, . . . do also depend on the relative concentration of the two species. In particular, the

fifth-order coefficient reads E = E11111 x15 + 5E11112 x14 x2 + 10E11122 x13 x22 +10E11222 x12 x23 + 5E12222 x1 x24 + E22222 x25 ,

(2)

N

i where xi = N +N are the mole fractions of species labeled 1 2 as i = 1, 2. The terms E11111 and E22222 can be calculated through the expressions of the fifth virial coefficient for a monodisperse fluid of particles with diameters equal to, respectively, σ 11 or σ 22 . As for the 3D case, ten classes of irreducible cluster integrals contribute to this coefficient whose 8 = 2.03071192(25).36 The current best estimate is E11111 /σ11 partial coefficients E11112 , E11122 are represented by 24 and 45 classes of distinct cluster integrals, respectively. For the sake of completeness they are thoroughly reported in the supplementary material.37 The remaining two coefficients E11222 , E12222 are obtained from the graphical expressions of E11112 , E11122 , respectively, after interchanging the larger particles with the smaller ones and vice versa. In the symmetric case of equal size components (σ 11 = σ 22 ) E11112 = E12222 and E11222 = E11122 . By the knowledge of the virial coefficients it is possible to build up some refined equations of state that in principle should improve over the accuracy of the virial expansion. Hence, we will consider the rescaled virial expansion (RVE) proposed by Baus and Colot38 to obtain an (approximate) equation of state for a symmetric NAHD mixture. The RVE equation of state truncated to the fifth order has the following form:38, 39

Z=

c + c1 η + c2 η2 + c3 η3 + c4 η4 βP = 0 , ρ (1 − η)2

(3)

where Z = βP/ρ is the compressibility factor. Similarly as for the virial expansions related to Eq. (1), in the remaining part of the paper we shall refer to the rescaled virial expansions arrested to the third (c3 = c4 = 0 in Eq. (3)), fourth (c4 = 0 in Eq. (3))), and fifth order (see Eq. (3)) as, respectively, Zsc3, Zsc4, and Zsc5. The coefficients c0 , c1 , c2 , c3 , and c4 depend only on the mole fraction and reproduce the first n virial coefficients exactly. In general we have39 cn = bn − 2bn−1 + bn−2 ,

(4)

where bn are the reduced virial coefficients so defined: b0 = 1, b1 = Bξ , b2 = ξC2 , b3 = ξD3 , and b4 = ξE4 where, for a symmetric mixture ξ = π4 and the packing fraction is η = 2 2 + (1 − x1 )σ22 )] = (π/4)ρσ 2 (σ 11 = σ 22 = σ ). (π/4)ρ[x1 σ11 In order to trace the phase-coexistence curve, we calculated numerically the Helmholtz free energy according to the formula  ρ βP   βA = dρ N ρ 2 0 = [ln(ρ  3 ) − 1] + 

ρ

+ 0



2 

xk ln(xk )

k=1

 βP /ρ  − 1 dρ  , ρ

(5)

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where is the de Broglie thermal wavelength. The first two terms on the right-hand side of Eq. (5) represent the ideal-gas (kinetic and mixing) contributions. In symmetric mixtures the mole fraction of the two coexisting phases (I, II) should satisfy the relations x1I = x2I I and x2I = x1I I . As a result, the condition of material stability (equality of the chemical potentials of two distinct phases at equilibrium) leads to the following identity:   ∂A =0 (6) ∂xi T ,ρ and the critical point falls at x1c = x2c = 1/2. After fixing the value of  one can find a solution xi (with i = 1, 2) of Eq. (6) and the coexistence lines can be easily traced using, for instance, the double tangent method (see Refs. 40–42 for details). B. Monte Carlo simulation

The equations of state as obtained by means of the different theoretical approaches were assessed against NVT MC43 computer simulations, in which the virial pressure was calculated according to the formula βP 2π  =1+ ρ xi xj σij3 gij (σij ), (7) ρ 3 ij where gij (σ ij ) are the contact values of the radial distribution functions of the two species labeled as i and j. Initial N V T configurations were generated by performing a short simulation in the grand canonical Monte Carlo (GCMC43 ) ensemble that was stopped upon reaching the desired density in the two boxes. Then, the system was equilibrated along 106 MC cycles, where each cycle consisted in a number of attempts to displace a particle equal to the total number of particles N = 1000. The acceptance rate for particle displacements was fixed at 40% and cubic periodic boundary conditions were applied to the simulation box.43 Phase coexistence properties were calculated by means of GEMC43–45 simulations. The total number of disks in the two boxes was fixed to 2000, and the attainment of conditions of phase coexistence was checked by verifying the equality of the pressures in the two boxes according to Eq. (7), and the equality of the chemical potentials of the two species, as calculated according to the formula  V 1 , (8) μi = −kB T ln 3 i Ni + 1 V

where V and Ni are, respectively, the area of one of the two boxes and the number of particles of species i, i is the De Broglie thermal wavelength of species i, and . . .  is a statistical average over the fluctuating V and Ni . From now on, we will always omit to report the term ln 3i , which is identical across the two coexisting phases. The system was equilibrated by using 5 × 105 –106 GEMC cycles. Subsequently, statistical averages were collected following 20 cumulation runs of 105 cycles. Each GEMC cycle consists of a number of attempts to displace particles in the two boxes equal to the total number of particles, an attempt to change the total area of the two

boxes, and a number of particle exchanges between the two boxes between 1% and 5% of the total number of particles. The critical density ρ cr of the systems considered was estimated by using the critical power-law x1 − x1cr ∝ |ρ − ρ cr |β ,

(9)

where β is the critical exponent related to the order parameter, whose exact value for the two-dimensional case is β = 1/8,31 and x1cr = 0.5 for symmetrical mixtures. All the numerical simulation data (both NVT-MC and GEMC) have been included in tabular form in the supplementary material.37 III. RESULTS AND DISCUSSION

The MC results for the composition-independent coefficients E11112 and E11122 are given in Table I. In the limit of  → 0 the partial coefficients tend to the value of the fifth virial coefficient of a one-component hard-disk fluid 8 ). The behavior of these quanti2.03071192...(in units of σ11 ties is qualitatively very similar to the nonadditive hard-sphere mixture case.42 We will deal with the positive value of nonadditivity only. The behavior of E(x1 ) when  > 0 is shown in Fig. 1. The cumulative fifth virial coefficient E(x1 ) is symmetric around x1 = 1/2. For values of  < 0.4, the curves of E(x1 ) present one central maximum only that is replaced by a minimum, bounded by two symmetric side maxima for larger values of the nonadditivity parameter. Furthermore, for  > 0.5, E(x1 ) becomes negative within a range of mole fractions which widens for increasing values of the  parameter. In Figs. 2–6, we plot the compressibility factor as a function of the packing fraction for different values of mole TABLE I. The partial coefficients E11112 and E11122 for a symmetric NAHD mixture tabulated as a function of the nonadditivity parameter  > 0. The 8 . The error on the last significant numerical values are given in units of σ11 figure is enclosed in parentheses.  0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

E11112

E11122

2.368(2) 2.730(2) 3.120(2) 3.535(2) 3.977(2) 4.444(2) 4.938(2) 5.458(2) 6.001(3) 6.569(3) 7.168(3) 7.789(3) 8.438(3) 9.113(3) 9.813(3) 10.538(3) 11.288(3) 12.066(3) 12.869(3) 13.698(3)

2.529(2) 3.030(2) 3.494(3) 3.863(3) 4.066(4) 4.007(5) 3.571(7) 2.60(1) 0.93(4) − 1.67(2) − 5.45(1) − 10.778(6) − 17.962(5) − 27.464(4) − 39.800(3) − 55.561(2) − 75.429(2) − 100.200(2) − 130.777(2) − 168.195(2)

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FIG. 1. The fifth virial coefficient plotted as a function of the mole fraction for positive values of the nonadditivity parameter .

fraction and . As a general trend, we note that the rescaled EOS (RVE) provide the higher values for every considered value of , and in general, they look very similar to each other for the low values of the nonadditivity parameter  ≤ 0.2. There is a moderate tendency of RVE to become slightly different by increasing the mole fraction, and also a non-monotonic behavior observed for the higher considered nonadditivity parameters, i.e., their predictions do not follow the series Zsc3 < Zsc4 < Zsc5 (adding more virial coefficients to Eq. (1) does not necessarily bring to higher values of the compressibility factor Z); see for instance bottom panels of Figs. 4–6 (we did not report the values of Zsc3 in the figures in order to avoid making them too crowded with data). It is interesting to note that for  ≥ 0.3 the predictions of Zsc5 are in better agreement with the exact MC result, which is a clear indication of a well-behaved convergence of the rescaled EOS when more virial coefficients are added to it. While this feature is somewhat expected and clearly shows how beneficial the rescaling procedure can be, this is still a non-trivial result due to the limitation of the RVE, which predicts an unphysical divergence of the compressibility factor when the total packing fraction becomes equal to 1. We argue that the reason of a clear emergence of well-behaved convergence of the RVE at the higher considered values of the nonadditivity parameter ( ≥ 0.3) is due to the fact that the binary mixture is found to be in a homogeneous state for lower values of the packing fraction as compared to the cases where the nonadditivity parameter  is small (i.e., we are evaluating the RVE at quite smaller values of the packing fraction when  is increased).

J. Chem. Phys. 141, 214508 (2014)

In fact, we note as a general rule that the critical packing fraction of the binary mixture tends to become smaller as the nonadditivity parameter is increased. On the contrary, the virial EOS of Eq. (1) appear wellseparated from each other also at the lower mole fractions, and they strictly follow the series Zvir3 < Zvir4 < Zvir5 , i.e., adding more virial coefficients to Eq. (1) always increases the value of the compressibility factor Z. Now, it is interesting to assess the performance of the two schemes as compared to NVT MC computer simulation. At the lower values of the nonadditivity parameters  = 0.1, 0.2 the virial EOS underestimates the MC EOS irrespective of the retained number of virial coefficients (see Figs. 2 and 3), while approaching the MC result monotonously, which means that Zvir5 is the most accurate one. Remarkably, for  = 0.1, 0.2 all of the three rescaled EOS reproduce the MC EOS quantitatively. For  = 0.2 the three rescaled EOS are still very accurate up to x = 0.1. Then, for the compositions closer to the equimolar one, they start to slightly overestimate the EOS at the higher densities (the higher the mole fraction, the larger the discrepancy). It is worth to note that even if all of the virial EOS underestimate the MC EOS, their predictions appear to improve when increasing the nonadditivity parameter and the mole fraction, so that Zvir5 reproduces the MC EOS better at  = 0.2 than at  = 0.1. This feature is a consequence of another effect: the apparent shifting-up of the virial EOS predictions on increasing  and x. This effect can also be read as a progressive lowering of the MC EOS when increasing  and x that makes the RVE predictions slightly overestimating the MC result. The trend noted at  = 0.1–0.2 persists when the nonadditivity parameter is increased to  = 0.3, making Zvir5 quantitative at x = 0.2, while at x = 0.4 the Zvir4 EOS begins to approach the MC EOS (see Fig. 4). As a consequence, the MC EOS is located between Zvir4 and Zvir5 , and Zvir5 slightly overestimates the MC EOS at high packing fraction, whereas Zvir4 is slightly underestimating it. The RVE predictions are still quantitative at the lower considered mole numbers x = 0.05, 0.1. Then, at  = 0.4, Zvir5 becomes quantitative at x = 0.1, Zvir4 becomes quantitative at x = 0.2, and the MC EOS is sitting between Zvir4 and Zvir3 for x = 0.4 at the high packing fraction (see Fig. 5). In this case, the RVE predictions are quantitative for the lowest mole number x = 0.05 only. Finally, at  = 0.5 Zvir5 is predictive at the lowest composition x = 0.05, Zvir4 becomes predictive at x = 0.1, Zvir3 becomes predictive at x = 0.4, while at x = 0.2 the MC EOS sits in the middle between Zvir3 and Zvir4 (see Fig. 6). At this value of , the RVE are clearly less predictive than the virial EOS, since even at low mole numbers they do not match the MC EOS profile. The observed difficulty in finding a general trend arises because the convergence of the virial expansion is not well understood for binary mixtures. While for a pure system as, e.g. the hard-sphere model, the virial coefficients are always positive and adding more of them to the virial expansion is expected to improve its convergence to the exact result, this in general does not hold true. In fact, as we showed in Fig. 1 for a binary mixture the virial coefficients do not

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FIG. 2. Compressibility factors as a function of the total packing fraction at different values of the  parameter and mole fraction x. Legend: NVT-MC represents the Monte Carlo computer simulation (canonical ensemble); Zvir3 represents the virial expansion incorporating the first three virial coefficients; Zvir4 represents the virial expansion incorporating the first four virial coefficients; Zvir5 represents the virial expansion incorporating the first five virial coefficients; Zsc4 represents the scaled EOS incorporating the first four virial coefficients; Zsc5 represents the scaled EOS incorporating the first five virial coefficients.

necessarily get positive ones only, and this feature originates a non-monotonic convergence of the virial expansion to the exact result. In fact, it is no coincidence that the larger discrepancy for the virial expansions retaining more virial coefficients (i.e., Zvir4 –Zvir5 ) is observed for the higher values of the nonadditivity parameter , where the range of mole fractions for the negative values of E(x1 ) widens. Now we have a look at the theoretical predictions for the fluid-fluid coexistence lines, which for a symmetric mixture are traced out just by examining the Helmholtz free energy. In fact, once the Helmholtz free energy is calculated by integrating the theoretical EOS according to Eq. (5), the binodal line for many  values can be calculated by using Eq. (6). In Fig. 7, the curves resulting from the virial expansion and the RVE approaches are shown. We also report the results obtained from a first-order perturbation theory, the so-called MIX1 approximation implemented with the SPT equation of state18 of the pure fluid of hard-disks as a reference system.22, 23 Actually, the SPT equation is not very accurate. There are available more accurate equations of state for the HD fluid,46, 47 but as a matter of fact their use does not seem to improve remarkably the theoretical estimates with respect to the simulation data. In order to avoid overburdening the figure with irrelevant information, we reported the virial expansion and RVE using the fifth virial coefficient only. A

complex scenario shows up: at  = 0.1 (see top panel of Fig. 7), none of the theoretical approaches seem to be predictive, but the virial coexistence line on one side, and the RVE coexistence line/MIX1 on the other side, bracket the computer simulation GEMC data with the virial results overestimating, and the RVE/MIX1 underestimating them. At  = 0.2 (see second panel from the top), theories generally tend to follow the GEMC data at mole fractions

Theoretical and computer simulation study of phase coexistence of nonadditive hard-disk mixtures.

We have studied the equation of state (EOS) and the equilibrium behavior of a two-component mixture of equal-sized, nonadditive hard disks with an int...
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