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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.


METHOD Grand-canonical transition-matrix Monte Carlo [1] and histogram re-weighting
V/σ3 512
TRUNCATION 3.0σ + standard long-range corrections
Prob. of Disp. Move
Prob. of Ins/Del Move
Biasing Function Update Frequency
1.0E6 trial moves
Simulation Length
8.0E9 trial moves
















0.70 1.996E-03 1.422E-05 8.437E-01 2.49E-04 1.370E-03 9.507E-07 -2.500E-02 3.323E-05 -6.106E+00 1.832E-03 -6.257E+00 4.639E-04
0.75 3.630E-03 1.049E-06 8.219E-01 2.678E-04 2.635E-03 6.607E-07 -4.250E-03 2.120E-05 -5.908E+00 1.731E-03 -5.683E+00 4.312E-04
0.80 6.100E-03 2.698E-06 7.999E-01 2.700E-04 4.645E-03 1.294E-06 -6.758E-02 6.177E-05 -5.716E+00 1.868E-03 -5.196E+00 3.535E-04
0.85 9.640E-03 2.726E-06 7.769E-01 2.622E-04 7.636E-03 1.769E-06 -1.109E-01 5.454E-05 -5.518E+00 1.985E-03 -4.779E+00 6.406E-05
0.90 1.451E-02 3.913E-06 7.528E-01 1.185E-04 1.185E-02 2.172E-06 -1.474E-01 3.697E-05 -5.316E+00 9.197-04 -4.419E+00 3.020E-05
0.95 2.102E-02 5.631E-06 7.277E-01 2.032E-04 1.754E-02 3.707E-06 -2.060E-01 8.411E-05 -5.110E+00 1.569E-03 -4.107E+00 1.547E-04
1.00 2.957E-02 7.326E-06 7.010E-01 9.555E-05 2.496E-02 4.827E-06 -2.805E-01 1.133E-04 -4.896E+00 4.103E-04 -3.834E+00 1.114E-04
1.05 4.065E-02 9.121E-06 6.722E-01 1.136E-04 3.436E-02 6.927E-06 -3.744E-01 7.623E-05 -4.670E+00 6.910E-04 -3.595E+00 9.919E-05
1.10 5.508E-02 8.878E-06 6.408E-01 1.185E-04 4.602E-02 6.581E-06 -4.934E-01 1.399E-04 -4.431E+00 9.164E-04 -3.384E+00 7.073E-05
1.15 7.412E-02 8.935E-06 6.052E-01 9.148E-05 6.021E-02 7.403E-06 -6.468E-01 1.452E-04 -4.168E+00 6.971E-04 -3.197E+00 5.917E-05
1.20 1.003E-01 2.998E-05 5.632E-01 5.223e-05 7.723E-02 1.416E-05 -8.552E-01 2.949E-04 -3.871E+00 4.561E-04 -3.031E+00 7.606E-05

Temperature-density projection of the Lennard-Jones phase diagramPressure-density projection of the Lennard-Jones phase diagram


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 [4]. 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.

As introduced in Refs. 2 and 3, the activity, z, is defined as

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 [6].

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 [7]. (Finite-size scaling analysis has not been completed, so these critical properties should be taken simply as estimates.)


1. J. R. Errington, J. Chem. Phys. 118, 9915 (2003).
2. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, New York, 1989).
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).