SAT-TMMC: Liquid-vapor coexistence properties - Long-Range Corrections at 3σ

Grand-canonical Transition-Matrix Monte Carlo (GC-TMMC) simulations [1, 8-12] were performed at k_BT/ϵ = 0.70, 0.85, 1.10, and 1.20 with a cubic simulation box of V=512 σ³ and N_max=510. The particle number range was divided into 32 windows and expanded ensemble trial moves were also performed to ensure adequate sampling at high densities. Each particle number window was swept a minimum of 100 times. Within a window, half of a sweep was defined as the condition where each bin in a visited-states histogram had a minimum of 100 hits. The main result of a GC-TMMC simulation is the particle number probability distribution, which is constructed by stitching together the particle number distributions from each window. Physically, this distribution corresponds to the probability of finding N particles in a simulation box of volume V, temperature T, and chemical potential μ (or activity ln⁡ z). At each value of N, average total potential energies U and average square total potential energies U² were collected. Knowledge of the particle number probability distribution can be used to determine liquid-vapor coexistence properties. Other key simulation details are given below:

Interested readers can download raw data and contact us if there are questions. We note that the analysis used to extract thermophysical properties from the macrostate distributions used N_max=485 due to the system’s tendency to crystallize at high densities. This value of N_max ensured that the temperature and macrostate range was reflective of an amorphous fluid phase.

The particle number probability distribution (or the macrostate distribution) at temperatures not simulated directly were determined using statistical mechanical extrapolation methods [13, 14] along with interpolating polynomials [15]. A second order temperature extrapolation was used here. This provides a data of liquid-vapor coexistence data from k_BT/ϵ = 0.7 – 1.20 at increments of 0.001.

Data Availability:

Particle number probability distributions and canonical energy averages used to generate the table and plots above are available in a Git repository: https://github.com/dwsideriusNIST/NIST_SRSW_Data/tree/master/LJ_PURE/LJ_3sigma_LRC_TMMC

Remarks:

Uncertainties were obtained from four 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_BT) (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.)

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

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).
8. V. K. Shen and D. W. Siderius, J. Chem. Phys., 140, 244106, 2014.
9. V. K. Shen and J. R. Errington, JPC B 108, 19595, 2004.
10. V. K. Shen and J. R. Errington, JCP 122, 064508, 2005.
11. V. K. Shen, R. D. Mountain, and J. R. Errington, JPC B 111, 6198, 2007.
12. D. W. Siderius and V. K. Shen, JPC C 117, 5681, 2013.
13. N. A. Mahynski, M. A. Blanco, J. R. Errington, and V. K. Shen, JCP 146, 074101 (2017).
14. N. A. Mahynski, J. R. Errington, and V. K. Shen, JCP 147, 054105 (2017).
15. J. I. Monroe, H. W. Hatch, N. A. Mahynski, M. S. Shell, and V. K. Shen, JCP 153, 144101 (2020).