Liquid-vapor coexistence properties obtained by grand-canonical transition-matrix Monte Carlo and histogram re-weighting over the reduced temperature range 0.60 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 and histogram re-weighting [1, 8-12] |
V/σ^{3} | 729 |
TRUNCATION | Linear Force Shifted at 4.0σ |
Prob. of Disp. Move | 0.4 |
Prob. of Ins/Del Move | 0.6 |
Biasing Function Update Frequency | 1.0E6 trial moves |
Simulation Length | 2.0E10 trial moves |
T* |
ρ_{vap}* |
+/- |
ρ_{liq}* |
+/- |
p_{sat}* |
+/- |
U_{vap}* |
+/- |
U_{liq}* |
+/- |
lnz_{sat}* |
+/- |
0.60 | 8.450E-04 | 2.761E-07 | 8.643E-01 | 1.437E-04 | 5.018E-04 | 1.784E-07 | -1.149E-02 | 9.239E-06 | -5.962E+00 | 2.057E-03 | -7.097E+00 | 2.043E-04 |
0.65 | 1.828E-03 | 2.863E-07 | 8.426E-01 | 1.105E-04 | 1.165E-03 | 1.145E-07 | -2.287E-02 | 9.803E-06 | -5.771E+00 | 2.118E-03 | -6.343E+00 | 1.534E-04 |
0.70 | 3.508E-03 | 1.070E-06 | 8.203E-01 | 1.110E-04 | 2.377E-03 | 5.199E-07 | -4.095E-02 | 3.185E-05 | -5.579E+00 | 2.208E-03 | -5.717E+00 | 2.383E-04 |
0.73 | 4.958E-03 | 8.281E-07 | 8.064E-01 | 2.177E-04 | 3.466E-03 | 6.408E-07 | -5.580E-02 | 3.102E-05 | -5.464E+00 | 3.478E-03 | -5.391E+00 | 1.980E-04 |
0.75 | 6.146E-03 | 1.447E-06 | 7.970E-01 | 1.683E-04 | 4.379E-03 | 1.035E-06 | -6.767E-02 | 3.534E-05 | -5.385E+00 | 1.785E-03 | -5.192E+00 | 1.235E-04 |
0.80 | 1.004E-02 | 2.556E-06 | 7.728E-01 | 1.043E-04 | 7.446E-03 | 1.906E-06 | -1.050E-01 | 4.604E-05 | -5.186E+00 | 1.815E-03 | -4.747E+00 | 1.436E-04 |
0.85 | 1.553E-02 | 4.011E-06 | 7.474E-01 | 6.274E-05 | 1.187E-02 | 2.526E-06 | -1.555E-01 | 5.087E-05 | -4.982E+00 | 2.032E-03 | -4.367E+00 | 1.176E-04 |
0.90 | 2.304E-02 | 6.700E-06 | 7.203E-01 | 5.083E-05 | 1.795E-02 | 4.003E-06 | -2.218E-01 | 9.956E-05 | -4.771E+00 | 7.566E-04 | -4.039E+00 | 1.218E-04 |
0.95 | 3.314E-02 | 1.205E-05 | 6.192E-01 | 1.313E-04 | 2.600E-02 | 5.282E-06 | -3.080E-01 | 1.331E-04 | -4.550E+00 | 1.180E-03 | -3.756E+00 | 9.539E-05 |
1.00 | 4.664E-02 | 8.095E-06 | 6.592E-01 | 4.684E-05 | 3.633E-02 | 5.230E-06 | -4.193E-01 | 1.072E-04 | -4.316E+00 | 7.062E-04 | -3.510E+00 | 8.772E-05 |
1.05 | 6.489E-02 | 1.549E-05 | 6.233E-01 | 5.901E-05 | 4.930E-02 | 7.871E-06 | -5.653E-01 | 1.396E-04 | -4.061E+00 | 7.023E-04 | -3.293E+00 | 4.871E-05 |
1.10 | 9.047E-02 | 1.573E-05 | 5.807E-01 | 1.314E-04 | 6.528E-02 | 3.457E-06 | -7.644E-01 | 1.618E-04 | -3.774E+00 | 1.036E-03 | -3.103E+00 | 2.168E-05 |
1.15 | 1.310E-01 | 6.111E-05 | 5.238E-01 | 1.179E-04 | 8.474E-02 | 6.193E-06 | -1.078E+00 | 6.530E-04 | -3.417E+00 | 5.594E-04 | -2.934E+00 | 7.240E-05 |
1.20 | 2.047E-01 | 5.207E-04 | 4.367E-01 | 6.416E-04 | 1.084E-01 | 1.759E-05 | -1.589E+00 | 4.357E-03 | -2.914E+00 | 3.161E-03 | -2.785E+00 | 4.626E-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 4.0σ, the critical properties were estimated to be T_{c}*=1.207, ρ_{c}*=0.319, and p_{c}*=0.112. Estimates were found via rectilinear diameter analysis of TMMC data computed with V*=729 close to the critical point [7]. (Finite-size scaling analysis has not been completed, so these critical properties should be taken simply as estimates.)