Liquid-vapor coexistence properties obtained by grand-canonical transition-matrix Monte Carlo and histogram re-weighting over the reduced temperature range 0.80 to 1.15 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, 7-11] |
V/σ^{3} | 729 |
TRUNCATION | Not applicable |
Prob. of Disp. Move | 0.3 |
Prob. of Ins/Del Move | 0.7 |
Biasing Function Update Frequency | 1.0E6 trial moves |
Simulation Length | ≥1.0E9 trial moves |
T* |
ρ_{vap}* |
+/- |
ρ_{liq}* |
+/- |
p_{sat}* |
+/- |
U_{vap}* |
+/- |
U_{liq}* |
+/- |
lnz_{sat}* |
+/- |
0.80 | 5.805E-03 | 6.184E-06 | 7.314E-01 | 2.794E-04 | 4.362E-03 | 4.376E-06 | -1.044E-01 | 1.310E-04 | -5.827E+00 | 2.669E-03 | -5.270E+00 | 8.043E-04 |
0.85 | 9.352E-03 | 2.089E-06 | 7.128E-01 | 1.951E-04 | 7.266E-03 | 1.920E-06 | -1.572E-01 | 4.228E-04 | -5.664E+00 | 2.749E-03 | -4.843E+00 | 2.429E-04 |
0.90 | 1.443E-02 | 8.262E-06 | 6.930E-01 | 7.121E-04 | 1.146E-02 | 4.768E-06 | -2.284E-01 | 1.658E-04 | -5.491E+00 | 6.341E-03 | -4.473E+00 | 3.590E-04 |
0.95 | 2.153E-02 | 5.073E-06 | 6.712E-01 | 1.748E-04 | 1.727E-02 | 5.016E-06 | -3.228E-01 | 1.087E-04 | -5.303E+00 | 1.521E-03 | -4.149E+00 | 1.856E-04 |
1.00 | 3.140E-02 | 1.591E-05 | 6.467E-01 | 1.561E-04 | 2.507E-02 | 6.127E-06 | -4.478E-01 | 2.779E-04 | -5.097E+00 | 1.443E-03 | -3.865E+00 | 1.516E-04 |
1.05 | 4.523E-02 | 1.653E-05 | 6.187E-01 | 1.431E-04 | 3.527E-02 | 9.471E-06 | -6.144E-01 | 3.262E-04 | -4.867E+00 | 5.220E-03 | -3.614E+00 | 1.614E-04 |
1.10 | 6.528E-02 | 9.247E-06 | 5.845E-01 | 1.932E-04 | 4.837E-02 | 7.734E-06 | -8.439E-01 | 2.079E-04 | -4.599E+00 | 1.460E-03 | -3.391E+00 | 1.305E-04 |
1.15 | 9.723E-02 | 1.027E-04 | 5.384E-01 | 1.655E-04 | 6.500E-02 | 1.839E-05 | -1.192E+00 | 1.363E-03 | -4.266E+00 | 1.308E-03 | -3.194E+00 | 1.234E-04 |
Remarks:
Uncertainties were obtained from four 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. Combining this information with the particle number probability distribution, the mean potential energy of the coexisting phases can be calculated [6].