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/σ^{3} | 512 |
TRUNCATION | 3.0σ + standard long-range corrections |
Prob. of Disp. Move | 0.75 |
Prob. of Ins/Del Move | 0.25 |
Biasing Function Update Frequency | 1.0E6 trial moves |
Simulation Length | 8.0E9 trial moves |
T* |
ρ_{vap}* |
+/- |
ρ_{liq}* |
+/- |
p_{sat}* |
+/- |
U_{vap}* |
+/- |
U_{liq}* |
+/- |
lnz_{sat}* |
+/- |
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 |
Remarks:
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
$$ z = \dfrac{\exp \left( \beta \mu \right)}{\lambda^3}$$
where Λ is the de Broglie wavelength, β = 1/(k_{B}T) (where k_{B} 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 T_{c}*=1.291, ρ_{c}*=0.317, and p_{c}*=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.)
References
- J. R. Errington, J. Chem. Phys. 118, 9915 (2003).
- M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, New York, 1989).
- D. Frenkel and B. Smit, Understanding Molecular Simulation, 2nd ed. (Academic, San Diego, 2002)., pp.37-38.
- J. R. Errington and A. Z. Panagiotopoulos, J. Chem. Phys., 109, 1093 (1998).
- A. Z. Panagiotopoulos, J. Phys.: Condens. Matter, 12, R25-R52, (2000).
- J. R. Errington and V. K. Shen, J. Chem. Phys., 123, 164103 (2005).
- B. Smit and C. P. Williams, J. Phys.: Condens. Matter, 2, 4281-4288 (1990).