The extrapolation of phase equilibrium curves of mixtures in the isobaric-isothermal Gibbs ensemble

A powerful extrapolation scheme is proposed to determine the vapour-liquid and liquid-liquid equilibrium curves of mixtures by performing a single isothermal-isobaric Gibbs ensemble Monte Carlo (GEMC) simulation. The coexistence curves for the mole fraction and the density are extrapolated as functions of the temperature and the pressure by second-order Taylor series. The coefficients of the Taylor series, which are the temperature and pressure derivatives of these quantities along the coexistence curves, can be calculated from the data produced by a single GEMC simulation on the basis of fluctuation formulas. We show that the application of a Pade approximant considerably widens the temperature and pressure range where the extrapolation is accurate. Using Lennard-Jones mixtures as test systems, we show that the technique is able to produce quite accurate equilibrium curves at fixed temperature in the function of the pressure and vice versa. The procedure yields good results not only for vapour-liquid but also for liquid-liquid coexistence curves. The calculation of the vapour pressure curves at a fixed composition of the liquid side is straightforward with the method.