## Introduction

Understanding the vapor-, liquid-, and solid-phase behavior of mixtures is a prerequisite for designing extraction and crystallization processes. Most of the research aimed at understanding how intermolecular interactions affect phase behavior has focused on fluid-phase equilibria, although in real systems solid phases form and often interrupt the complex phase behavior exhibited by fluids (Schneider, 1978; Scott, 1987). Phenomenological descriptions of complete phase behavior (that is, showing equilibrium between vapor, liquid, and solid phases) were given by Luks (1980), Peters et al. (1986), and Valyashko (1986, 1990). Luks and Peters et al. described four types of complete phase diagrams observed for binary mixtures of solvent (methane, ethane, carbon dioxide) and a homologous series of solutes (*n*-alkanes). Valyashko proposed a classification scheme for 12 types of complete diagrams. Eight of these types result from the analysis of experimental data for water–inorganic salt mixtures. The remaining four types were deduced by the method of continuous topological transformation, which involves making educated guesses about the transitions in topography between the eight known types. This is based on the likely-correct assumption that there are continuous transitions between all types of phase behavior (Schneider, 1970).

A quantitative description of complete phase behavior was given by Luks and coworkers (Garcia and Luks, 1999; Labadie et al., 2000). Garcia and Luks (1999) calculated the solid–liquid–vapor locus for binary mixtures of solvent and a homologous series of solutes using the van der Waals equation of state for the fluid phase and a simple fugacity model for the solid phase. This work was extended by Labadie et al. (2000) who calculated the fluid-phase critical loci for these mixtures, offering a picture of how the multiphase topography progresses with changes in the solute properties. 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. As with any analytical equation of state, there exists the possibility that the new topographies are mathematical artifacts stemming from the approximations made in the development of the equation of state. Nonetheless, the new possibilities for complete phase behavior calculated by Luks and coworkers are intriguing and invite further investigation.

In a previous article (Lamm and Hall, 2001) we used the Gibbs-Duhem integration method combined with semigrand canonical Monte Carlo simulations to calculate complete phase diagrams for binary Lennard-Jones mixtures over a range of diameter ratios σ_{11}/σ_{22} = 0.85 − 1.0 and well-depth ratios ϵ_{11}/ϵ_{22} = 0.625 − 1.6 at a single reduced pressure. The cross-species interaction parameters were calculated using the Lorentz-Berthelot (Rowlinson and Swinton, 1982) combining rules. We restricted our study to diameter ratios ranging from 0.85 to 1.0, because calculations on binary hard-sphere mixtures have shown that the stable solid phase in this region is a substitutionally disordered fcc crystal (Barrat et al., 1986, 1987; Kranendonk and Frenkel, 1991; Cottin and Monson, 1995). At diameter ratios of less than 0.85, the phase-equilibrium calculation becomes more complex, because several ordered crystalline phases are possible, necessitating the calculation of each phase's free energy to determine the most stable crystalline structure. We found that for well-depth ratios of unity (equal attractions among species) there is no interference between the vapor–liquid and solid–liquid coexistence regions. As the well-depth ratio increases or decreases from unity, the vapor–liquid and solid–liquid phase envelopes become wider and interfere with each other, leading to a solid–vapor coexistence region. For all well-depth ratios and a diameter ratio of 0.95, the solid–liquid lines have a shape characteristic of a solid solution (with or without a minimum melting temperature); as the diameter ratio decreases, the solid–liquid lines fall to lower temperatures until they eventually drop below the solid–solid coexistence region, resulting in either a eutectic or peritectic three-phase (solid–solid–liquid) line.

In this article we explore the effect of pressure on the complete phase behavior of a mixture. The complete phase diagrams are calculated using Gibbs-Duhem integration combined with semigrand canonical Monte Carlo simulation; this procedure has been described in detail elsewhere (Lamm and Hall, 2001). We present phase diagrams for six binary Lennard-Jones mixtures with σ_{11}/σ_{22} = 0.85, 0.9, and 0.95, and ϵ_{11}/ϵ_{22} = 0.45 and 1.6. We begin with a detailed description of the *T* − *x* phase diagrams for the binary Lennard-Jones mixture with diameter ratio σ_{11}/σ_{22} = 0.85 and well-depth ratio ϵ_{11}/ϵ_{22} = 1.6 at reduced pressures *P** = 0.002, 0.01, 0.025, 0.05, and 0.1. These results are then summarized on a *P*–*T* projection that identifies the three-phase coexistence features of the mixture (solid–liquid–vapor and solid–solid–liquid) in addition to the pure component vapor–liquid, solid–liquid, and vapor–solid coexistence curves. We then present similar *P*–*T* projections for the remaining mixtures and discuss how the three-phase lines (solid–liquid–vapor and solid–solid–liquid) change with variations in diameter ratio σ_{11}/σ_{22} and well-depth ratio ϵ_{11}/ϵ_{22}.

Highlights of our simulation results are the following. For the mixture with σ_{11}/σ_{22} = 0.85 and ϵ_{11}/ϵ_{22} = 1.6, the vapor–liquid and solid–liquid coexistence regions interfere on the *T*–*x* diagram at low pressure (*P** = 0.002). As pressure increases, the vapor–liquid coexistence region first shifts to higher temperatures and then begins to disappear as the pressure approaches critical conditions. These results are summarized on a pressure–temperature phase diagram that identifies the three-phase coexistence features of the mixture (solid–liquid–vapor, solid–solid–liquid) in addition to the pure-component vapor–liquid, solid–liquid, and vapor–solid coexistence curves. Pressure–temperature diagrams are presented for five additional mixtures. For one mixture, σ_{11}/σ_{22} = 0.9 and ϵ_{11}/ϵ_{22} = 1.6, we locate a quadruple (solid(1)–solid(2)–liquid–gas) coexistence point. Upon comparing the *P*–*T* projections for each mixture, we find that as the diameter ratio decreases, the maximum in the locus of solid–liquid–vapor coexistence pressures decreases and the locus of solid(1)–solid(2)–liquid temperatures shifts from temperatures above the solid–liquid temperature of pure component 1 to temperatures below the solid–liquid coexistence temperature of pure component 1. We find that as well-depth ratio decreases, the coexistence curves for pure component 2 shift from temperatures and pressures below those of pure component 1 to temperatures and pressures above those of pure component 1 and that the maximum in the locus of solid–liquid–vapor coexistence pressures increases.

This article is organized as follows. The Gibbs-Duhem integration method is outlined for two- and three-phase coexistence lines. The results of our calculations of complete T-X phase diagrams are presented for the binary mixture, σ_{11}/σ_{22} = 0.85, ϵ_{11}/ϵ_{22} = 1.6, at five reduced pressures, *P** = 0.002, 0.01, 0.025, 0.05, and 0.1, in order to observe how the shape of the curves on the *T*–*x* diagram change with variation in pressure. *P*–*T* projections are also presented for binary mixtures with σ_{11}/σ_{22} = 0.85, 0.9, and 0.95, and ϵ_{11}/ϵ_{22} = 0.45 and 1.6 to examine how these diagrams change with variations in Lennard-Jones diameter ratio and well-depth ratio. We conclude with a brief summary and further discussion.