Phase equilibria of binary Lennard-Jones mixtures ...mlh/documentos/Miguel/ljmixt2.pdf · vapour-liquid equilibria, liquid-liquid equilibria, fluid-fluid equilibria, Lennard-Jones

Fluid Phase Equilibria 100 ( 1994) 153- 17t~

Phase equilibria of binary Lennard-Jones mixtures: simulation and van der Waals l-fluid theory

Aikaterini M. Georgoulaki a, Ioannis V. Ntouros a, Dimitrios P. Tassios a,*, Athanassios Z. Panagiotopoulos b

a Department of Chemical Engineering, National Technical University of Athens, Heroon polytechniou 9, Zographos, 15773 Athens, Greece

b School of Chemical Engineering, Cornell University, Ithaca, NY 14853-5201, USA

Received 23 September 1993; accepted in final form 14 April 1994

Abstract

The Gibbs ensemble simulation technique is used to investigate the ability of van der Waals l-fluid theory to predict phase equilibria for binary Lennard-Jones mixtures. Simulation data for highly asymmetric mixtures with size and energy parameter ratios equal to 1.00, 0.5, 0.4, 0.35 and to 0.5, 0.33, 0.25 respectively are compared to theoretical results for cases in which unlike-pair interactions follow the Lorentz-Berthelot combining rules. Additional comparisons are made for vapour-liquid and liquid-liquid equilibria of mixtures with energy parameter deviating from the geometric mean (Berthelot) rule, and for vapour-liquid and gas-gas equilibria of mixtures with size parameter deviating from the arithmetic mean (Lorentz) rule. Good agreement was found between theory and simulation when the energy parameter deviates from the Berthelot combining rule. The agreement is less satisfactory when the size parameter deviates from the Lorentz rule, especially for the case of gas-gas equilibria.

Keywords: Theory; Monte Carlo simulation; Gibbs ensemble; van der Waals l-fluid theory; equations of state, vapour-liquid equilibria, liquid-liquid equilibria, fluid-fluid equilibria, Lennard-Jones mixtures

1. Introduction

The phase behaviour of fluid mixtures has been the subject of many experimental and theoretical studies because of its importance for design and operation of industrial separation processes. Theories provide a simple and flexible way for obtaining phase coexistence envelopes

* Corresponding author.

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1.54 A.M. Georgoulaki et al. 1 Fluid Phase Equilibria 100 (1994) 153-I 70

of mixtures in a short time. However, their validity has to be tested in order to find the range of their applicability. When dealing with specific intermolecular potential models, such as the Lennard-Jones fluid, the only way to test theories is to compare them with simulation data, which can in principle represent exactly the behaviour of a system with given potential model.

The Gibbs ensemble Monte Carlo simulation method, proposed by Panagiotopoulos (1987), Panagiotopoulos et al. (1988) and Smit et al. ( 1989) has been widely applied for evaluating phase equilibria in mixtures of model substances, such as Lennard-Jones mixtures, hard sphere and soft disk mixtures, polydisperse fluids and hydrocarbon mixtures. A recent extensive review of the method and its applications is available (Panagiotopoulos, 1992). Harismiadis et al. ( 1991) used the Gibbs ensemble technique to test the applicability of the van der Waals l-fluid (vdWlf) theory in predicting phase equilibria of asymmetric Lennard-Jones mixtures obeying the Lorentz-Berthelot combining rules. In contrast to the earlier studies of Shing et al. (1988) and Shukla and Haile (1987, 1988), who suggested that for mixtures consisting of molecules with significant differences in size the van der Waals l-fluid theory fails in predicting Henry’s constants and excess properties, Harismiadis et al. (1991, 1994) found good agreement between theory and simulation for the phase envelopes of mixtures with size parameter ratio up to 2 and energy parameter ratio also up to 2 obeying the Lorentz-Berthelot rules.

Binary Lennard-Jones mixtures with unlike pair energy parameter deviating from the Berth- elot rule have been studied previously using molecular dynamics and Monte Carlo simulations (Singer and Singer, 1972; Torrie and Valleau, 1976; Schoen and Hoheisel, 1984; Panagiotopou- 10s et al., 1986; van Leeuwen et al., 1991). Torrie and Valleau (1976) studied the excess free energy of mixing of a mixture with the cross interaction parameter equal to 75% of the geometric mean and found good agreement between simulation and random mixing theory, which for the specific mixture was equivalent to the vdWlf theory (Torrie and Valleau, 1976).

The goal of the present study is to investigate further the applicability of conformal solutions theory in predicting the phase behaviour of binary Lennard-Jones mixtures using the Gibbs ensemble Monte Carlo technique. The study includes three parts:

(i) vapour-liquid equilibria of binary Lennard-Jones mixtures consisting of components with potential well depth ratio and size ratio with values up to 4 and up to 2.85 respectively, significantly extending the range of asymmetry studied previously by Harismiadis et al. (1991);

(ii) vapour-liquid and liquid-liquid equilibria of Lennard-Jones mixtures with unlike-pair potential parameter deviating from the Berthelot combining rule;

(iii) vapour-liquid and gas-gas equilibria of Lennard-Jones mixtures with unlike-pair potential parameter deviating from the Lorentz combining rule.

The main body of this paper is organized as follows. We begin with a short discussion of van der Waals l-fluid theory and its application to a Benedict- Webb-Rubin type and a Peng- Robinson type equation of state. We then describe the molecular simulation method used in our calculations. The performance of van der Waals l-fluid theory for asymmetric mixtures, mixtures deviating from the Berthelot rule and mixtures deviating from the Lorentz rule is then compared with simulation results and discussed in detail. We close with a summary of our findings, which are that in most cases the predictions of theory are in satisfactory agreement with simulation data, except for systems with large differences in energy parameter values and systems deviating from the Lorentz rule.

A.M. Georgoulaki et al. 1 Fluid Phase Equilrhria 100 (1994) 153-l 70 155

2. Van der Waals l-fluid theory

The van der Waals l-fluid theory is a special case of conformal solutions theories, which are applicable to mixtures with conformal pair-wise additive intermolecular interactions (Rowlinson and Swinton, 1982). Such interactions occur in a binary Lennard-Jones mixture for which the potential model has the form

(1) where U,, is the potential energy between molecules i and j, E, and c,] are the energy and size parameters respectively for the same pair and r,, is the distance between the two molecules.

The van der Waals l-fluid (vdWlf) theory is a simple and very successful version of conformal solution theories in which a real mixture is considered as a single hypothetical pure fluid (Leland et al., 1968). The mixing rules used to obtain the size and energy parameter of this hypothetical fluid are (Rowlinson and Swinton, 1982)

(2)

where xk is the mole fraction of component k. The cross coefficient parameter E, is usually given by the geometric mean of the parameters E,,

and eJJ . Since one of our aims is to study mixtures that deviate from the Berthelot combining rule, we introduce an interaction parameter &, and the conventional Berthelot combining rule takes the form

CJ = &fi (4)

Similarly, for the mixtures with the cr,, parameter deviating from the arithmetic mean, an interaction parameter qlJ is introduced in the Lorentz combing rule:

3. Equations of state used

In order to apply the van der Waals l-fluid theory, the thermodynamic properties of the pure fluid are required as input. The description of the pure fluid properties is usually accomplished via an equation of state. Two empirical equations of state for the pure Lennard-Jones fluid were considered in this study, a Peng-Robinson type cubic equation proposed recently by Harismi- adis et al. (1994) and a Benedict-Webb-Rubin type equation proposed by Nicolas et al. (1979). The latter equation is considered one of the most successful in describing the properties of the Lennard-Jones fluid (Johnson, 1992). We used parameters recently proposed by Johnson et al. (1993) using a much greater range of simulation data for regression and constraining the equation to reproduce critical temperature and density very close to Smit ( 1990) estimates based

156 A.M. Georgoulaki et al. 1 Fluid Phase Equilibria 100 (1994) 153-l 70

on Gibbs ensemble simulation data and to the estimates of Lotfi et al. (1992) who used the NpT + test particle method. These new parameters improve the prediction of vapour-liquid equilibrium both for pure fluid and binary mixtures in comparison with the Nicolas et al. equation and with recent simulation data (Johnson et al., 1993). The resulting equation is abbreviated as MBWR-J in the present paper.

The mixing rules for the van der Waals l-fluid approximation, as they are usually applied to cubic equations of state have the following form (Rowlinson, 1990):

bm = C~x,xjb, * J

where a and b are the attractive and covolume parameters respectively used in two-parameter cubic equations of state. The cross coefficients a, are usually given by a geometric-mean combining rule, often involving an adjustable parameter k,, while b, are given by an arithmetic-mean combining rule. However, in order to use properly a cubic equation of state for the study of the behaviour of Lennard-Jones mixtures, the mixing rules used should be equivalent to the ones given by Eqs. (2) (3), (4) and (5). I n a recent study, Harismiadis et al. (1994) demonstrated that the combining rule for energy parameter equivalent to the Berthelot rule as written in Eq. (4) is

a, = &,b,Ja,@,lb,b, (8) while the combining rule equivalent to the Lorentz rule as written in Eq. (5) is

(b,) 1’3 = q,[(bj’3 + b;“>/2] (9)

In the same study, Harismiadis et al. proposed a modified Peng-Robinson equation of state for the Lennard-Jones fluid through a volume translation (abbreviated hence to t-PR-LJ) in order to give correct saturated liquid densities of the pure fluid. We used the t-PR-LJ equation in addition to the MBWR-J equation for comparisons with simulation data. The following mixing rule was used for the volume translation parameter t (Peneloux et al., 1982; Magoulas and Tassios, 1990).

trn = C&t*

where t, is the translation parameter of component i at the temperature of the mixture and x, is its mole fraction.

4. Molecular simulation methodology

The Gibbs ensemble method proposed by Panagiotopoulos (1987) was used in this study. The method allows the direct determination of two-phase equilibria of mixtures. The simulated system includes two regions, which are in thermodynamic equilibrium both internally and with each other. Three types of step are performed:

(i) displacements within each individual region for achieving internal equilibrium; (ii) volume changes in order to attain equality of pressures (mechanical equilibrium);

A.M. Georgoulaki et al. 1 Fluid Phase Equilibria 100 (1994) 153-170 157

(iii) particle transfers between the two regions which lead to equality of chemical potential of each component in the two phases (chemical equilibrium).

When the constant volume (NVT) Gibbs ensemble is considered, in which the total volume of the simulated system remains constant, the second step leads to a volume change of equal value but opposite sign for the two regions. In the constant pressure (NPT) Gibbs ensemble the pressure is specified in advance while now the volume change in each region is carried out independently. A detailed presentation of the method is given by Panagiotopoulos (1987, 1989a, 1992) along with a review of its applications. For mixtures with size ratio far from unity the modification of the Gibbs ensemble technique for highly asymmetric mixtures was used (Panagiotopoulos 1989a). In this modification the particle transfer step is performed only for the component with smaller size, while equilibration of the chemical potential of the “larger” component is achieved by species identity exchange. At low pressures the NPT Gibbs ensemble was used, whereas at higher pressures great uncertainties did not allow us to approach the critical point of the mixtures. This was accomplished by using the NVT Gibbs ensemble, which proved to be more stable, as van Leeuwen et al. (1991) had also observed. For a better determination of the densities and mole fractions of the two phases close to the critical point, probability plots were made as described by Smit et al. (1989).

The species identity exchange technique for asymmetric mixtures was also used for determin- ing liquid-liquid equilibrium for systems with 5, different from unity. These mixtures have identical pure components, with unfavourable interaction energy parameter between unlike component particles. The symmetry of these mixtures made possible the equilibration of chemical potentials only for one of the components since the two phases have complementary compositions so the chemical potential of component 1 in one liquid phase would be the same with the chemical potential of component 2 in the other phase, provided the densities of the two phases are identical. A similar method was used by Mountain and Harvey (1991) in a study of a symmetric mixture of non-additive soft disks. In addition, no volume change moves are required since the symmetry of the mixture impose the equality of densities of the two coexisting phases. This technique was also applied by Amar (1989) in a study of a mixture of symmetric non-additive hard spheres. In the case of gas-gas equilibria of mixture with unlike size parameter deviating from the arithmetic mean, the technique of species identity exchange was also used. However, the particle transfer step for the second component was now performed, since it was observed that smaller uncertainties in pressure and mole fraction were achieved.

5. Results and discussion

5.1. Asymmetric systems obeying the Lorentz - Berthelot combining rules

In order to extend the asymmetry of the two components relative to the earlier study of Harismiadis et al. (1991), we examined mixtures with ratio of well depth parameters up to 4 and ratio of size parameters up to 3. Well depth ratios of up to 4 can be found in mixtures of real fluids (Reid et al., 1987). The potential parameters of the “larger” component ell and cl, were taken equal to unity. In both the theoretical and simulation method the reduced variables P* = Po3/eI,, T* = ~T/E~~ and p* = po3 are defined in terms of the pair potential parameters

158 A.M. Georgoulaki et al. 1 Fluid Phase Equilibria 100 (1994) 153- 170

0.05 0.05 0'151

\ \

8"";b

\

0.10

0.00 0.00 0.0 0.2 0.4 0.6 0.6 1.0 0.0 0.2 0.4 0.6 0.6 1.0

X1*Y1 P*

Fig. 1. Reduced pressure vs. mole fraction and vs. reduced density, for system 1 (g’2/c,, = 1.00. G~/E,, = 0.50, T* = 1.15): 0, Monte Carlo results; -, vdWlf theory with MBWR-J equation; ~ - -, vdWlf theory with t-PR-LJ equation.

2.0

0.0 0.0 0.2 0.4 0.6 0.6 1.0

1.5

'h 1.0

0.5

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3

Xl *Y1 P*

Fig. 2. Reduced pressure vs. mole fraction vs. reduced density, for system 2 (gzz/g,, = 0.50, G~/E,, = 0.33, T* = 0.75); 0, Monte Carlo results; -, vdWlf theory with MBWR-J equation; - - -, vdWlf theory with t-PR-LJ equation,

E, , and Do,. Systems of 500 particles were used in each simulation run. Simulation details were similar to the previous study of Harismiadis et al. (1991). The probability of attempted particle transfers for each component was adjusted to lead approximately to the same number of successful transfers for each component in the range 2500-3500. 3 x lo’-lo6 equilibration configurations and 106-4 x lo6 production configurations were performed for each run.

Results for a number of highly asymmetric systems are given in Figs. l-5 as reduced pressure versus mole fraction of the larger component and reduced pressure versus reduced density. Numerical values of the results are presented in Table 1. The quantities in parentheses in the tables are the statistical uncertainties of the observed values during a simulation, in units of the last decimal point of the corresponding quantity shown. For example, 0.148 (5) means that the measured quantity is 0.148 + 0.005. Quantities that are specified as input parameters in a run,

A.M. Georgoulaki et al. 1 Fluid Phase Equilibria 100 (1994) 153-I 70 159

2.5

2.0

b 1.5

1.0

0.5

0.0 0 1 2 3 4 5

P*

Fig. 3. Reduced pressure vs. mole fraction and vs. reduced density, for system 3 (elz/~,, = 0.35, .+/E,, = 0.66, T* = 1.15): 0, Monte Carlo results; --, vdWlf theory with MBWR-J equation; - - -, vdWlf theory with t-PR-LJ equation.

0.4

0.3

b 0.2

0.1

0.0 I 0 0.2 0.4 0.6 0.6 1.0

X1*Y1

0.4

0.3

b 0.2

0.1

0.0 3 0.2 0.4 0.6 0.8 1

P*

Fig. 4. Reduced pressure vs. mole fraction and vs. reduced density, for system 4 (g2&cr,, = 1.00, G*/E,, = 0.33, T* = 1.15): 0, Monte Carlo results; -, vdWlf theory with MBWR-J equation; - - -, vdWlf theory with t-PR-LJ equation.

such as pressure in a constant-pressure Gibbs simulation or the densities in a simulation of liquid-liquid equilibria of symmetric systems at a fixed volume do not have associated statistical uncertainties.

Consideration of the performance of the vdWlf theory in predicting the phase behaviour of the asymmetric systems, including those studied earlier by Harismiadis et al. (1991) suggests that in many cases results of the theory are satisfactory as illustrated in Figs. l-3. The component size difference does not seem to have a significant influence on the performance of theory as suggested by the results for system 3, for which the size ratio is equal to 0.35, which corresponds to a volume ratio of the large to the small component equal to 23.3.

160 A.M. Georgoulaki et al. 1 Fluid Phase Equilibria 100 (1994) 153- 170

Xl *Y1

l a 1.0

Fig. 5. Reduced pressure vs. mole fraction and vs. reduced density, for system 5 ((T?~/(T,, = 0.50, G~/E,, = 0.25, T* = 1 .OO) : 0, Monte Carlo results; -, vdWlf theory with MBWR-J equation; - - -, vdWlf theory with t-PR-LJ equation.

Table 1 Simulation results for systems following the Lorentz-Berthelot rules

P* AP* -XI AX, )II AY,

System 1: ~/a ,, = 1.00, E&,, =0.50. T* = 1.15 0.100 0.919 (7) 0.?39 (14) 0.070 0.980 (2) 0.904 (8) 0.148 (5) 0.820 (20) 0.600 (20) 0.157 (12) 0.751 (21) 0.622 (35)

PT AP: P& AP%

0.580 (13) 0.138 (11) 0.959 ( 12) 0.089 (4) 0.53 (4) 0.253 (9) 0.48 (4) 0.30 (3)

System 2: aZ2/o,, = 0.50, E&,, =0.33, T* =0.75 0.0110 (2) 0.960 (6) 0.146 (11) 0.833 (12) 0.0148 (2) 0.0226 (2) 0.920 (9) 0.078 (5) 0.872 (15) 0.0305 (2) 0.0385 (3) 0.860 (14) 0.055 (5) 0.907 (16) 0.052 (3) 0.0597 (12) 0.773 (19) 0.025 (8) 0.964 (24) 0.0805 (13) 0.0685 (5) 0.776 (18) 0.034 (11) 1.00 (2) 0.0932 (9) 0.286 0.462 (17) 0.013 (5) 1.43 (4) 0.401 (5) 0.476 0.339 (20) 0.015 (8) 1.72 (6) 0.694 ( 16) 0.606 0.264 ( 14) 0.012 (4) 1.92 (4) 0.914 (23) 0.763 0.233 (12) 0.028 ( 17) 2.08 (2) 1.26 (11) 0.945 0.182 (16) 0.019 (4) 2.25 (5) 1.48 (4) 1.000 0.153 (8) 0.020 (5) 2.29 (5) 1.58 (5) 1.050 0.146 (8) 0.027 (8) 2.36 (6) 1.74 (11) 1.100 0.144 (18) 0.024 (12) 2.35 (4) 1.75 (9) 1.21 (4) 0.087 (31) 0.051 (29) 2.35 (19) 2.13 (18)

System 3: a,,/a,, = 0.35, E’&,, = 0.66, T* = 1.15 0.12 0.810 (10) 0.47 (4) 0.73 (2) 0.125 (4) 0.20 0.641 (9) 0.268 (19) 0.92 (3) 0.202 (4) 0.50 0.367 (18) 0.085 (18) 1.52 (6) 0.495 (15) 1.00 0.208 (6) 0.037 (8) 2.50 (8) 1 .oo (3)


Table 1 (continued)

P* AP* Xl Ax, Yl AYI Pt AP:

1.50 0.146 (5) 0.034 2.00 0.102 (6) 0.017 2.56 (9) 0.062 (6) 0.008 2.72 (7) 0.057 (4) 0.009 3.04 (4) 0.026 (8) 0.014

System 4: IJ~JG,, = 1 .OO, l JE,, = 0.33, T* = 1.15 0.07 0.991 (2) 0.910 0.193 (5) 0.867 (17) 0.613 0.14 0.925 (13) 0.661 0.22 0.827 (22) 0.625 0.24 0.73 (5) 0.71 0.25 0.867 (17) 0.613 0.25 0.867 (17) 0.613

System 5: +JcT,, = 0.50, EJE,, = 0.25, T* = 1.00 0.162 0.825 (15) 0.201 0.302 0.705 (12) 0.177 0.440 0.596 (10) 0.126 0.601 0.487 (8) 0.107 0.802 0.40 (3) 0.14 0.840 0.39 (2) 0.12 0.960 0.33 (4) 0.15 1.040 0.20 (5) 0.26

(6) (4) (6) (4) (8)

(4) (17) (14) (10) (4) (17) (17)

(17) (9) (18) (13) (3) (2) (3) (7)

3.30 (9) 3.96 (14) 2.9 (3) 4.7 (4) 4.3 (5)

0.59 (2) 0.56 (2) 0.58 (2) 0.55 (3) 0.47 (5) 0.565 (21) 0.565 (21)

0.801 (10) 0.909 ( 18) 1.002 (12) 1.11 (3) 1.22 (3) 1.24 (3) 1.28 (7) 1.24 (9)

P:; AP?,

1.60 (5) 2.21 (4) 4.6 (3) 3.15 (14) 3.8 (4)

0.085 (4) 0.274 (8) 0.191 (15) 0.33 (2) 0.45 (3) 0.274 (8) 0.274 (8)

0.172 (3) 0.333 (7) 0.479 (11) 0.64 (2) 0.89 (6) 0.90 (4) 1.08 (4) 1.32 (8)

On the other hand, increased energy parameter asymmetry does affect the performance of theory as suggested by the results shown in Figs. 4 and 5. In these cases the theory overestimates the critical point. This overestimation of the critical point was also observed by Harismiadis et al. (1991) where it was suggested that this trend for the original MBWR equation (Nicolas et al., 1979) might arise from the fact that the equation does not reproduce the true critical point of the pure fluid. Nevertheless, this is not the case for the MBWR-J equation (Johnson et al., 1993) since it was constrained to give critical properties (Tz = 1.313, pr = 0.310) very close to the Smit et al. (1989) and Lotfi et al. (1992) estimates. Thus this overestimation might not be due to the specific equation used but due to intrinsic problems of the vdWlf theory.

The pressure-mole fraction phase envelopes are generally predicted better than the pressure- density phase envelope. The cubic equation of state (t-PR-LJ) performs surprisingly well considering its simplicity when compared to the MBWR-J equation. Actually there is no significant difference in the performance of the two equations, with a slight advantage - as expected - for the MBWR-J one.

5.2. Systems deviating from the Berthelot combining rule

In this part of the study simple mixtures are studied in order to investigate the effect of deviations from the Berthelot combining rule. Since we are interested in studying phase

162 A.M. Georgoulaki et al. / Fluid Phase Equilibria 100 (1994) 153- 170

separation, values of t i2 less than unity are used, which lead to a decrease of the attractive forces between unlike molecules. One may well wonder whether the large deviations from the Berthelot rule for the systems studied here can be observed in physical systems. For highly asymmetric noble gas systems such as Xe/Ne, in a previous simulation study (Panagiotopoulos, 1989a) we had to use large deviation from the Berthelot rule (l,? = 0.688) as well as small deviations from the Lorentz rule to obtain agreement between simulation and experimental results. Large deviations from the Berthelot rule also occur for strongly interacting systems, such as water with supercritical fluids (Panagiotopoulos, 1989b), although in such cases the apparent large deviations are party due to the use of a Lennard-Jones type model to describe polar or hydrogen bonding components. It is clear, however, that for strongly immiscible liquid-liquid or fluid- fluid systems large deviations from the Berthelot rule do occur if one considers effective potentials.

The mixture chosen consisted of molecules of the same size and energy. The potential parameters of the first component were set equal to unity. The above system with a value of cl2 equal to 0.75 had been studied by Torrie and Valleau (1976) in a wide range of temperatures. Panagiotopoulos et al. (1986) applied a Monte Carlo simulation method, making use of the Widom test-particle method and confirmed their suggestion of liquid-liquid immiscibility at reduced temperature T* = 1.15. Since we wanted to study liquid-liquid equilibria, we main- tained this temperature for all values of k,, examined. The system with (I? equal to 0.7 was also studied at a temperature of T* = 0.95 in order to observe the effects of temperature on the agreement between theory and simulation. The values of the interaction parameter <iZ used are 0.7, 0.6 and 0.5.

Vapour-liquid and liquid-liquid equilibria data for these systems are presented in Figs. 6-9 and listed in Table 2. The coexisting liquid phases have the same densities, as expected from the symmetry of the mixture. This symmetry was also imposed during simulation by keeping the total volume of the system constant while the particle transfer step was performed only for the first component as explained in Section 3. Simulations were done with a total of 500 particles, whereas the number of attempted particle interchanges were between 8 and 300 per cycle of 500 Monte Carlo steps.

For <i2 equal to 0.75 (Fig. 6) the simulation data obtained by Panagiotopoulos et al. ( 1986) making use of the Widom test-particle method, were reproduced using now the Gibbs ensemble. The Gibbs ensemble technique leads to almost identical results for the vapour-liquid equilibria, while there is a slight discrepancy between the two Monte Carlo methods for the liquid-liquid equilibria at higher pressure. The phase behaviour of this system can be classified as a symmetric II-A according to the classification of van Konynenburg and Scott (1980) at this specific temperature. The same classification arises from the relation of the potential parameters E,,, cZ2 and cl2 according to Boshkov and Mazur ( 1986). Good agreement is observed between theory and molecular simulation results for the vapour-liquid phase envelope while the agreement becomes worse for the liquid-liquid equilibrium. The MBWR-J equation underestimates the liquid-liquid lower critical solution pressure. It was impossible to obtain liquid-liquid predictions with the t-PR-LJ. This failure must be due to the inadequacy of t-PR-LJ for PVT predictions at high densities (Harismiadis et al., 1994).

A similar mixture with a slightly lower value of <i2 = 0.70 was studied at a lower temperature of T* = 0.95 (Fig. 7 and Table 2) in order to investigate the effect of temperature on the

A.M. Georgoulaki et al. / Fluid Phase Equilibria 100 (1994) 153-170 163

2.c

1.5

1.0

'6

0.06

0.06

0.’

0.:

0.2

0.1

b

0.04c

0.035

0.030

0.025

0.020

0.015 I 0.2 0.4 0.6 0.6 1.

X1*Y1

I 0.2 0.4 0.6 0.6 1

%tY1

Fig. 6. Reduced pressure vs. mole fraction for system 6 ((rz2 = g,, , cz2 = E,, , <,2 = 0.75, T* = 1.15): 0, Monte Carlo

results; -, vdWlf theory with MBWR-J equation; - - -, vdWlf theory with t-PR-LJ equation.

Fig. 7. Reduced pressure vs. mole fraction for system 7 (ez2 = cr, , . cz2 = cl,, (,? = 0.70, T* = 0.95): 0, Monte Carlo


agreement between theory and simulation. The MBWR-J equation does well with vapour-liquid and liquid-liquid predictions while the t-PR-LJ equation deviates significantly from the molecular simulation data, especially in the liquid-liquid case.

Decreasing the parameter 5 ,2 to the value of 0.60 (Fig. 8 and Table 2) the unlike-pair interactions become even less favourable. This leads to a much wider vapour-liquid phase envelope and a lower liquid-liquid critical immiscibility pressure. According to the classification of van Konynenburg and Scott (1980) the mixture phase behaviour changes from symmetric type II-A with 4,2 = 0.75 to symmetric type III-HA, or equivalently to type III-H according to Boshkov and Masur (1986). The MBWR-J equation follows excellently this transition presenting the three-phase LlL2G line and predicts the vapour-liquid equilibrium very well while for the liquid-liquid equilibrium it gives more distant Ll and L2 branches. On the other hand, the cubic equation (t-PR-LJ) fails completely in predicting this change of classification of the mixture, continuing to predict a type II-A symmetric mixture. It appears to be that t-PR-LJ underestimates the reduction of unlike-pair attraction forces.

164 A.M. Georgoulaki et al. /Fluid Phase Equilibria 100 (1994) 153-l 70

i \

.O 0.2 0.4 0.6 0.6 1.

Fig. 8. Reduced pressure vs. mole fraction for system 8 (oz2 = e,, , c2* = E,, , 1,2 = 0.60, T* = 1.15): 0 Monte Carlo


Fig. 9. Reduced pressure vs. mole fraction for system 9 (ez2 = o, , , eJZ = E, , , (,2 = 0.50, T* = 1.15): 0. Monte Carlo


Table 2 Simulation results for systems deviating from the Lorentz-Berthelot rules

P* AP* XI Ax, Yl AYI P: AP: P:, AP;,

System 0.0670 0.0780 0.0860 0.0990 1.86 1.63 1.53

6: g12 = e,, = , cz2 E, ,

0.032 0.068 0.099 0.233

(7) 0.708 (6) 0.701 (6) 0.692

<,2 = 0.75, T* = 1.15

(5) 0.122 (5) 0.195

(11) 0.268 (20) 0.392 (28) 0.292 (16) 0.308

(16) 0.324

System 7: Q = 0, ,, Ed? = E, , = , <,2 = 0.70, T* 0.95

0.0220 0.987 (3) 0.802 0.0250 0.983 (3) 0.731 0.0300 0.962 (14) 0.581 0.0350 0.942 (10) 0.551 0.13 (5) 0.938 (2) 0.062 0.21 (3) 0.941 (3) 0.059 0.30 (5) 0.944 (3) 0.056

(4) 0.595 (16) 0.084 (5)

(6) 0.581 (17) 0.104 (6)

(19) 0.572 (17) 0.119 (9)

(13) 0.521 (25) 0.151 (17)

(28) 0.787 0.787

(16) 0.773 0.773

(16) 0.756 0.756

(5) 0.723 (6) 0.0266 (3)

(13) 0.730 (10) 0.0310 (4)

(24) 0.720 (9) 0.0379 (9)

(20) 0.712 (10) 0.0462 (10)

(2) 0.731 0.731

(3) 0.738 0.738

(3) 0.750 0.750


Table 2 (continued)

P* AP* Xl Ax, YI

System 8: Q=Q,,. Q~=E,,, 5,2=0.60, T*=l.l5 0.070 0.988 (2) 0.895 0.110 0.931 (9) 0.667 0.135 (6) 0.909 (21) 0.58 0.20 (3) 0.889 (8) 0.111 0.32 (4) 0.912 (6) 0.088 0.48 (3) 0.931 (2) 0.069

System 9: oz2 = CJ,, , cz2 = E,, , 5,2 = 0.50, T* = 1.15 0.065 0.9964 (8) 0.934 0.084 0.981 (3) 0.772 0.123 0.968 (14) 0.650 0.145 0.957 (12) 0.565 0.21 (3) 0.9627 (10) 0.0373 0.36 (4) 0.9708 ( 13) 0.0292 0.49 (3) 0.9751 (5) 0.0249

System 10: g2* = D,, , cZ2 =E,,, r/,1 = 1.30, T* = 1.15 0.039 0.380 (4) 0.258 0.042 0.290 (6) 0.162 0.046 0.202 (10) 0.069 0.050 0.167 (5) 0.052 0.056 0.090 (7) 0.018

System 11: cz2 = o,, , cz2 = E,, , q,2 = 1.30, T* = 0.95 0.0115 0.374 (14) 0.190 0.0120 0.311 (14) 0.107 0.0125 0.263 (15) 0.061 0.0130 0.243 (10) 0.043 0.0140 0.171 (14) 0.018 0.0150 0.124 (8) 0.008 0.0160 0.093 (7) 0.006

System 12: uz2 = 0,, , l z2 = E,,, tj,* = 1.20, T* = 3.00 10.5 (2) 0.54 (6) 0.46 10.8 (2) 0.72 (2) 0.28 11.2 (5) 0.89 (3) 0.11

(8) 0.599 (15) 0.087 (3)

(16) 0.580 (16) 0.163 (13)

(4) 0.58 (3) 0.24 (2) (8) 0.606 0.606

(6) 0.642 0.642

(2) 0.675 0.675

(4) 0.615 (16) 0.078 (4)

(18) 0.593 (15) 0.101 (4)

(28) 0.61 (4) 0.177 (13)

(19) 0.601 (24) 0.232 (22)

(10) 0.631 0.631

(13) 0.667 0.667

(5) 0.692 0.692

(23) 0.516 (11) 0.038 (5)

(16) 0.530 (10) 0.049 (3)

(12) 0.534 (6) 0.055 (3)

(5) 0.543 (9) 0.059 (3)

(3) 0.554 (17) 0.068 (4)

(27) 0.611 (7) 0.0137 (3)

(16) 0.614 (8) 0.0142 (4)

(15) 0.618 (4) 0.0145 (2)

(8) 0.612 (6) 0.0152 (2)

(6) 0.628 (6) 0.0165 (1)

(3) 0.647 (8) 0.0177 (2)

(2) 0.656 (4) 0.0191 (3)

(6) (2) (4)

Pt APt

0.770 0.770 0.780 0.780 0.807 0.807

AP :;

Only for the value of li2 equal to 0.5 (Fig. 9 and Table 2), the cubic equation predicts a symmetric III-A phase diagram. However, the agreement is still not good especially for the liquid-liquid equilibrium branch. On the other hand, the MBWR-J equation continues to describe this mixture well, presenting smaller deviations from molecular simulation data in describing the liquid-liquid equilibrium in comparison with the mixtures with smaller deviations from the Berthelot rules.

166 A.M. Georgoulaki et al. 1 Fluid Phase Equilibria 100 (1994) 15% 170

5.3. Systems deviating from the Lorentz combining rule

In order to investigate the effect of the deviation of the unlike size parameter from the arithmetic mean, a parameter q i2 was introduced into the Lorentz combining rule. To study

phase separation, the interaction parameter II i2 was assigned values greater than unity (qlr = 1.20 and 1.30) which meant an increase in the repulsive forces between unlike molecules. The mixture studied was again symmetric consisting of molecules of the same size and energy. Many real mixtures of small molecules show small deviations from the diameter additivity rule (e.g. He/H*, Schouten et al. 1991). Larger deviations from diameter additivity occur in colloidal systems in the presence of polymers (e.g. Gast et al., 1983).

Vapour-liquid equilibria were studied at reduced temperatures T* equal to 1.15 and 0.95. Liquid-liquid equilibria could not be studied at those temperatures because the systems solidified before phase separation in the liquid phase. At a higher temperature (T* = 3) at which both components are supercritical, gas-gas or (fluid-fluid) equilibria were observed. Such a phenomenon for mixtures with ylZ greater than unity was also investigated by Schoen and Hoheisel (1984) and Schaink and Hoheisel (1992) using molecular dynamics. The phase diagrams for vapour-liquid and gas-gas equilibrium are presented in Figs. lo- 13. For the gas-gas equilibrium we observe that densities are equal within statistical uncertainties, as expected from the symmetry of the mixture.

For the mixture with qlZ = 1.30 at T* = 1.15 (Fig. 10 and Table 2) the performance of vdW 1 f theory is not satisfactory. Both the MBWR-J and the t-PR-LJ equations predict the negative azeotrope at a higher pressure than the one resulting from molecular simulation. Results for the same mixture at T* = 0.95 (Fig. 11) suggest that the deviations between theory and simulation increase significantly at the lower temperature. It should be mentioned here that values of ~1,~ greater than unity mean an unlike-pair diameter greater than the like-pair diameter, which in the

Xl IYl

0.01 a

0.016

0.014

b 0.012

0.010

0.008 0 0.2 0.4 0.6 0.6 1

X1*Y1

Fig. 10. Reduced pressure vs. mole fraction for system 10 (cr22 = g,, , +=E~,, ~,~=1.30, T*=1.15): 0, Monte Carlo results; -, vdWlf theory with MBWR-J equation; - - -, vdWlf theory with t-PR-LJ equation.

Fig. 11. Reduced pressure vs. mole fraction for system 11 (cT~? = CT,, , E,?= E,,, q,* = 1.30, T* =0.95): 0, Monte Carlo results: -, vdWlf theory with MBWR-J equation: - - -, vdWlf theory with t-PR-LJ equation.


0.0 0.2 0.4 0.6 0.6

%*Y1

Fig. 12. Reduced pressure vs. mole fraction for system 12 (cT~? = G,,, cz2 = E],, q,? = 1.20, T* = 3.00): 0, Monte Carlo results; -, vdWlf theory with MBWR-J equation; - - -, vdWlf theory with t-PR-LJ equation.

absence of attractive interactions would lead to greater repulsive forces between unlike molecules. One might expect therefore the presence of a positive azeotrope. However, this is not the case. It appears that the increased surface area for the unlike pair interactions overcomes the repulsion at the temperatures studied.

At higher temperatures we expect that the repulsive part of the potential becomes dominant (Hirschfelder et al., 1954), leading to stronger repulsive forces between the unlike than the like molecules. As a result, phase separation occurs which is known as gas-gas or fluid-fluid equilibria. In Fig. 12 we present such results for a mixture with q12 = 1.20 at T* = 3.00. The performance of the vdWlf theory is completely unsatisfactory as shown in the figure for the MBWR-J equation. This failure may be due to the fact that the MBWR-J equation does not predict thermodynamic properties as well in the region of high temperatures and high densities (Johnson, 1992). The same applies, to a greater extent, to the t-PR-LJ equation. At lower temperatures (T* equal to 1.50 or 2.00), and because of the high value of densities involved, simulation resulted in a homogeneous solid phase.

6. Conclusions

In this study we investigated the applicability of van der Waals l-fluid theory in predicting phase equilibrium of binary Lennard-Jones mixtures, using the MBWR-J equation and a cubic equation of state (t-PR-LJ) along with simulation data. For asymmetric systems with energy parameter ratios down to 0.25 and size parameter ratios down to 0.35 (volume ratios of component 1 to component 2 up to 24) both equations gave good overall results, with a slight advantage of the MBWR-J equation. The ratio of energy parameters influences the performance of the vdWlf theory more than the ratio of size parameters. This suggests that the frequent comment that cubic equations fail as the component asymmetry in a mixture increases, reflects the increased difference in energy parameters rather than in the size ones.

168 A.M. Georgoulaki et al. 1 Fluid Phase Equilibria 100 (1994) 153-l 70

For mixtures with the cross energy parameter deviating from the Berthelot combing rule, the agreement between the MBWR-J equation and simulation is satisfactory for vapour-liquid equilibrium, but substantial deviations are observed for liquid-liquid equilibrium. Theory seems to predict a lower critical liquid-liquid immiscibility pressure and more distant liquid Ll and L2 branches than simulation indicates. For the t-PR-LJ equation, the performance is similar to that of the MBWR-J equation in predicting vapour-liquid equilibrium, but poorer in the case of liquid-liquid equilibrium.

The performance of both equations is not satisfactory in the case of vapour-liquid of mixtures with the cross size parameter deviating from the Lorentz combining rule. The vdWlf theory predicts a narrower phase envelope than simulation indicates. However, theory fails completely in predicting gas-gas equilibria of such a mixture at higher temperatures, possibly because of an inadequacy of the equation to describe the region of high temperatures and densities of the pure fluid.

Acknowledgements

The authors would like to acknowledge travel support for this work provided by NATO (grant CRG-910580). A.M.G. and I.V.N. would like to thank Professor Panagiotopoulos for his hospitality during their two-month stay at Cornell.

List of Symbols

a b P P”

t T*

u XI, Yz

energy parameter in cubic equation of state volume parameter in cubic equation of state pressure reduced pressure, P* = Po:,/E,, volume translation parameter of the t-PR-LJ equation of state reduced temperature, TX = ktlel 1 intermoleculear potential mole fractions of component i in two coexisting phases (not necessarily gas and liquid)

Greek symbols

A denotes statistical uncertainty of a quantity, in units of the last decimal point of the quantity it references (Tables 1 and 2) Lennard-Jones energy potential parameter for interaction between components i and j. parameter describing deviations from the Berthelot combining rule (Eq. (4)) parameter describing deviations from the Lorentz combining rule (Eq. (5)) Lennard-Jones size potential parameter for interaction between components i and j.

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