Monte Carlo simulations of complete phase diagrams for binary Lennard–Jones mixtures

doi:10.1016/S0378-3812(01)00378-8

Fluid Phase Equilibria

Volume 182, Issues 1–2, 5 June 2001, Pages 37-46

https://doi.org/10.1016/S0378-3812(01)00378-8 Get rights and content

Abstract

Vapor–liquid, vapor–solid, liquid–liquid, and liquid–solid coexistence lines are calculated for binary mixtures of Lennard–Jones spheres with diameter ratio σ₁₁/σ₂₂=0.85, well-depth ratio ϵ₁₁/ϵ₂₂=0.45, and binary interaction parameters δ₁₂=1.0, 0.9, and 0.75, using Monte Carlo simulation and the Gibbs–Duhem integration technique. These calculations allow us to construct complete phase diagrams, i.e. showing equilibrium between vapor, liquid, and solid phases. For the mixture with δ₁₂=1, we find a completely miscible vapor–liquid coexistence region with a eutectic solid–liquid coexistence region. These two regions are separated by a completely miscible liquid phase. For the mixtures with δ₁₂<1, we find that the vapor–liquid and solid–liquid coexistence regions interfere. This interference results in a vapor–solid coexistence region bounded above and below by solid–liquid–vapor coexistence lines. We also find that the mixtures with δ₁₂<1 have a region of liquid–liquid immiscibility that is metastable with respect to the solid–fluid phase equilibria.

Introduction

Phase equilibria for binary mixtures of spherically symmetric molecules has been the subject of intensive investigation for decades. Van Konynenburg and Scott [1], [2] classified the phase diagrams predicted by the van der Waals equation of state for binary mixtures. Remarkably, the simple van der Waals equation of state was found to exhibit five of the six types of fluid phase behavior observed experimentally. This landmark study has been followed by similar analyses for other equations of state, such as the Redlich–Kwong [3], the Carnahan–Starling–Redlich–Kwong [4], the Guggenheim [5], and the Ree [6] equations of state.

Most of the research directed at understanding how intermolecular interactions affect phase behavior has focused exclusively on fluid phase equilibria. However, in real systems solid phases form and often interrupt the complex fluid phase behavior [7], [8]. Phenomenological descriptions of complete phase diagrams (i.e. showing equilibrium between vapor, liquid, and solid phases) have been given by Luks [9], Valyashko [10], [11], and Peters et al. [12]. Valyashko proposed a classification scheme for complete diagrams and introduced new types of complete phase behaviors that have not yet been observed in real systems. Garcia and Luks [13] recently examined the solid–liquid–vapor locus for a series of binary mixtures using the van der Waals equation of state and a simple solid state fugacity model. In their model calculations, they found examples of solid–fluid phase behavior in keeping with what has been observed in real systems, as well as solid–fluid phase behavior that has yet to be verified by experiment. The new possibilities for complete phase behavior proposed by Valyashko and by Garcia and Luks are intriguing and invite further investigation.

Molecular simulation has become a popular way to study the phase behavior of model systems [14]. Numerous simulations of binary mixture phase behavior for the Lennard–Jones intermolecular potential, the quintessential model of a spherically symmetric molecule, have been conducted for both fluid–fluid [15], [16], [17], [18], [19], [20], [21] and solid–liquid [22] phase equilibria. Vlot et al. [23] calculated complete phase diagrams for symmetric (equal diameters, σ₁₁=σ₂₂; equal attractions, ϵ₁₁=ϵ₂₂) Lennard–Jones mixtures by using Monte Carlo simulation for selected state points to determine the excess free energy as a function of composition. This free energy versus composition data was fitted with a two-parameter Redlich–Kister polynomial, and the convex envelope construction method was used to determine the phase diagram.

The Lennard–Jones intermolecular potential is given by $u_{ij} (r)=4ϵ_{ij} σ_{ij} r^{12} − σ_{ij} r^{6},$ where u_ij is the potential energy of interaction between particles i and j, r the distance between particles i and j, ϵ_ij the Lennard–Jones attractive well-depth, and σ_ij the Lennard–Jones diameter. The cross-species interaction parameters (σ₁₂,ϵ₁₂) are determined by the Lorentz–Berthelot combining rules [24] $σ_{12} = σ_{11} +σ_{22} 2,$ $ϵ_{12} =δ_{12} ϵ_{11} ϵ_{22},$ where δ₁₂, the binary interaction parameter, accounts for deviations of the unlike-pair attraction from the geometric mean of the like-pair attractions. According to the global phase diagrams associated with the various equations of state [2], [3], [4], [5], [6] for binary mixtures of equal-size, spherically symmetric molecules, liquid–liquid immiscibility occurs whenever the unlike-pair attractions are less than the arithmetic mean of the like-pair attractions. This means that liquid–liquid immiscibility is predicted to occur when $δ_{12} < ϵ_{11} +ϵ_{22} 2 ϵ_{11} ϵ_{22} .$ For binary mixtures of unequal-size, spherically symmetric molecules, liquid–liquid immiscibility is predicted to occur when $δ_{12} < ϵ_{11} +ϵ_{22} (σ_{11} /σ_{22})^{3} 2 ϵ_{11} ϵ_{22} (σ_{11} /σ_{22} +1)^{3} .$ Liquid–liquid immiscibility might not be observed however, because of intervening solid phases, which cannot be predicted by the equations of state. In fact, Rowlinson and Swinton [24] point out that unless the unlike-pair attractions are significantly weaker than the like-pair attractions, the liquid–liquid upper critical solution temperature resides at temperatures below the quadruple point (s₁s₂lv).

Systematic studies of the effect of δ₁₂ on fluid phase behavior have been conducted for two Lennard–Jones binary mixtures [15], [17] using Gibbs ensemble simulation. In both cases, δ₁₂ was varied over the range 1.0–0.7 to observe the effect of the unlike-pair attractions on the constant temperature P–x phase diagram. In the first mixture (σ₁₁/σ₂₂=0.82, ϵ₁₁/ϵ₂₂=0.52), Panagiotopoulos et al. [15] found that liquid–liquid immiscibility occurred for δ₁₂≤0.8 at kT/ϵ₁₁=1.348. In the second mixture (σ₁₁/σ₂₂=0.94, ϵ₁₁/ϵ₂₂=0.73), van Leeuwen et al. [17] found that liquid–liquid immiscibility occurred for δ₁₂≤0.75 at kT/ϵ₁₁=1.218. A possible explanation for why liquid–liquid immiscibility was not observed for δ₁₂=1.0 in these two studies is that the simulations were conducted at fixed temperatures that were higher than the upper critical solution temperature.

In this paper, we calculate complete phase diagrams for binary Lennard–Jones mixtures using Gibbs–Duhem integration combined with semigrand canonical Monte Carlo simulation. Our objective is to explore the effect of δ₁₂ on the phase behavior of a mixture when solid phase formation is included in the calculation. We present complete phase diagrams for binary Lennard–Jones mixtures with diameter ratio σ₁₁/σ₂₂=0.85, well-depth ratio ϵ₁₁/ϵ₂₂=0.45, and binary interaction parameters δ₁₂=1.0, 0.9, and 0.75 at reduced pressure, $P^{∗} =Pσ_{11}^{3} /ϵ_{11} =0.05$ .

The remainder of this paper is organized as follows. We first outline the Gibbs–Duhem integration method and describe how we applied the procedure to the calculation of complete phase behavior. We then present the resulting complete phase diagrams.

Section snippets

Integration

The coexistence lines were calculated using Gibbs–Duhem integration [25], [26], [27]. In this method, phase coexistence is determined by numerically integrating the Clapeyron differential equation appropriate to the system of interest. Clapeyron equations describe how field variables (variables that must be equal among coexisting phases) change along the phase equilibrium line. The Clapeyron equation for equilibrium between two phases (α and γ) of a binary mixture containing components 1 and 2

Results

In this section, we present the results of our Gibbs–Duhem integration calculations of complete phase behavior for binary Lennard–Jones mixtures.

We calculated three complete phase diagrams at $P^{∗} =Pσ_{11}^{3} /ϵ_{11} =0.05$ for diameter ratio σ₁₁/σ₂₂=0.85, well-depth ratio ϵ₁₁/ϵ₂₂=0.45, and binary interaction parameters δ₁₂=1.0, 0.9, and 0.75. According to Eq. (3) (based on equation of state predictions), liquid–liquid immiscibility is predicted to occur for this mixture whenever δ₁₂<1.002. Fig. 2 shows a

Summary

The Gibbs–Duhem integration technique was combined with semigrand canonical Monte Carlo simulations to calculate complete T–x phase diagrams for binary Lennard–Jones mixtures. We studied the effect of unlike-pair attractions on the mixture phase diagram by varying the binary interaction parameter over the range, δ₁₂=1–0.75, while holding the diameter ratio and well-depth ratio constant.

For the mixture with δ₁₂=1, we found a completely miscible vapor–liquid coexistence region and a eutectic

Acknowledgments

We gratefully acknowledge helpful discussions with Professor John M. Prausnitz. This work was supported by the GAANN Computational Sciences Fellowship of the U.S. Department of Education and the Office of Energy Research, Basic Sciences, Chemical Science Division of the U.S. Department of Energy under contract no. DE-FG05-91ER14181. Acknowledgement is made to the Donors of the Petroleum Research Fund administered by the American Chemical Society for partial support of this work.

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Monte Carlo simulations of complete phase diagrams for binary Lennard–Jones mixtures

Abstract

Introduction

Section snippets

Integration

Results

Summary

Acknowledgments

Fluid Phase Equilib.

Fluid Phase Equilib.

Fluid Phase Equilib.

Fluid Phase Equilib.

Fluid Phase Equilib.

Fluid Phase Equilib.

Fluid Phase Equilib.

Chem. Eng. Sci.

Discuss. Faraday Soc.

Phil. Trans. R. Soc. London A

J. Chem. Phys.

J. Chem. Phys.

Mol. Phys.

Phys. Lett.

Acc. Chem. Res.