Liquid-vapor coexistence properties obtained by grand-canonical transition-matrix Monte Carlo and histogram re-weighting over the reduced temperature range 0.70 to 1.05 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 and histogram re-weighting [1, 8-12] |
V/σ^{3} | 512 |
TRUNCATION | Linear Force Shifted at 3.0σ |
Prob. of Disp. Move | 0.4 |
Prob. of Ins/Del Move | 0.6 |
Biasing Function Update Frequency | 1.0E6 trial moves |
Simulation Length | 4.0E10 trial moves |
T* |
ρ_{vap}* |
+/- |
ρ_{liq}* |
+/- |
p_{sat}* |
+/- |
U_{vap}* |
+/- |
U_{liq}* |
+/- |
lnz_{sat}* |
+/- |
0.70 | 7.501E-03 | 8.399E-07 | 7.852E-01 | 8.085E-05 | 4.931E-03 | 5.616E-07 |
-8.026E-02 |
1.200E-05 | -4.880E+00 | 1.079E-03 | -5.014E+00 | 3.579E-05 |
0.75 | 1.260E-02 | 9.754E-07 | 7.582E-01 | 5.995E-05 | 8.603E-03 | 9.141E-07 | -1.272E-01 | 2.613E-05 | -4.675E+00 | 5.116E-04 | -4.553E+00 | 1.019E-04 |
0.80 | 2.000E-02 | 3.176E-06 | 7.294E-01 | 4.892E-05 | 1.398E-02 | 1.457E-06 | -1.918E-01 | 6.606E-05 | -4.462E+00 | 6.670E-04 | -4.164E+00 | 7.423E-05 |
0.85 | 3.043E-02 | 2.874E-06 | 6.981E-01 | 6.105E-05 | 2.146E-02 | 2.037E-06 | -2.789E-01 | 3.193E-05 | -4.239E+00 | 6.362E-04 | -3.833E+00 | 3.196E-05 |
0.90 | 4.502E-02 | 1.127E-05 | 6.632E-01 | 4.728E-05 | 3.147E-02 | 3.614E-06 | -3.959E-01 | 1.371E-04 | -4.001E+00 | 4.514E-04 | -3.549E+00 | 7.106E-05 |
0.95 | 6.578E-02 | 9.238E-06 | 6.229E-01 | 8.581E-05 | 4.447E-02 | 4.035E-06 | -5.562E-01 | 8.421E-05 | -3.738E+00 | 6.128E-04 | -3.304E+00 | 4.753E-05 |
1.00 | 9.728E-02 | 3.577E-05 | 5.727E-01 | 9.086E-05 | 6.099E-02 | 8.989E-06 | -7.930E-01 | 3.541E-04 | -3.432E+00 | 5.604E-04 | -3.091E+00 | 6.316E-05 |
1.05 | 1.567E-01 | 1.848E-04 | 4.959E-01 | 1.935E-04 | 8.184E-02 | 9.093E-06 | -1.228E+00 | 1.810E-03 | -3.015E+00 | 8.225E-04 | -2.905E+00 | 3.964E-05 |
Remarks:
Uncertainties were obtained from five independent simulations and represent 95% 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, linear force shifted at 3.0σ, the critical properties were estimated to be T_{c}*=1.079, ρ_{c}*=0.321, and p_{c}*=0.096. 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.)