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3A. SAT-TMMC: Liquid-Vapor coexistence properties - Long-Range Corrections applied at r=3.0σ
Liquid-vapor coexistence properties obtained by grand-canonical transition-matrix Monte Carlo and histogram re-weighting over the reduced temperature range 0.70 to 1.20 at increments of 0.05. Mean values of the saturation pressure, density, potential energy per molecule, and activity (chemical potential- see below) for each phase are reported.
Uncertainties were obtained from five independent simulations and represent 67% confidence limits based on a standard t statistic. Liquid-vapor coexistence was determined by adjusting the activity such that the pressures of the liquid and vapor phases were equal. Here, the pressure is not the conventional virial pressure [2,3] but is the actual thermodynamic pressure, based on the fact that the absolute free energies can be obtained from the distributions determined from simulation . Alternative methods, for example Gibbs-ensemble Monte Carlo and combination grand-canonical Monte Carlo and histogram re-weighting, can be used to determine liquid-vapor coexistence. A review of standard methods of phase equilibria simulations can be found in Ref. 5.
z = Λ-3 exp(βμ)
where Λ is the de Broglie wavelength, β = 1/(kBT) (where kB is Boltzmann's constant), and μ is the chemical potential. It is sometimes more convenient to work with ln z in the simulations and in post-processing. (NOTE: The reported activity is dimensionless, having been scaled by the LJ length cubed.)
Phase-coexistence energies were obtained by determining the mean potential energy at a given value of N for an additional 40 billion MC trials. Combining this information with the particle number probability distribution, the mean potential energy of the coexisting phases can be calculated .
For the Lennard-Jones fluid, cut at 3.0σ with analytic long-range corrections, the critical properties were estimated to be Tc*=1.291, ρc*=0.317, and pc*=0.117. Estimates were found via rectilinear diameter analysis of TMMC data computed with V*=512 close to the critical point . (Finite-size scaling analysis has not been completed, so these critical properties should be taken simply as estimates.)
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3. D. Frenkel and B. Smit, Understanding Molecular Simulation, 2nd ed. (Academic, San Diego, 2002)., pp.37-38.
4. J. R. Errington and A. Z. Panagiotopoulos, J. Chem. Phys., 109, 1093 (1998).
5. A. Z. Panagiotopoulos, J. Phys.: Condens. Matter, 12, R25-R52, (2000).
6. J. R. Errington and V. K. Shen, J. Chem. Phys., 123, 164103 (2005).
7. B. Smit and C. P. Williams, J. Phys.: Condens. Matter, 2, 4281-4288 (1990).