Estimates of vapor pressure for the LJ system below the triple point Barbara N. Hale Physics Department Missouri University of Science & Technology Rolla,
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Estimates of vapor pressure for the LJ system below the triple point Barbara N. Hale Physics Department Missouri University of Science & Technology Rolla, MO 65401
Motivation One of the challenges in predicting nucleation rates from potential models is obtaining a reliable equilibrium vapor density at low temperatures, where the experimental rate data exist. Equally troublesome for experimentalists is extrapolating measured vapor pressure data far below the freezing point. In this work we present “small cluster Monte Carlo simulation based” estimates of the LJ system vapor pressure at reduced temperatures, T/ k = 0.33, 0.42, 0.50 and 0.70. The results are presented in a “Dunikov” corresponding states analysis together with an extrapolated vapor pressure formula, vapor pressure data (at high temperatures) and results from other MC simulations.* * B. Hale and Mark Thomason, Scaled Nucleation in a LJ System, submitted.
Model Lennard-Jones System dilute LJ vapor system with volume, V non-interacting mixture of ideal gases each n-cluster size is ideal gas of N n clusters full LJ interaction potential separable classical Hamiltonian
Cluster Free Energy Differences n = number of atoms in cluster - f n = ln Q n – ln(Q n-1 Q n ) n - f n ln [ liquid /ρ 1,vapor ]
Canonical Configuration Integral Q n = … exp[- i >j V LJ (|r i -r j |)/ kT ]d n r i Monte Carlo Bennett method is used to calculate ratios: Q n / [Q n-1 Q 1 ]
Schematic of Monte Carlo Simulations Ensemble A : (n -1) cluster with monomer probe interactions turned off Ensemble B: n cluster with normal probe interactions - f n = ln Q n - ln( Q n-1 Q 1 )
Dunikov Corresponding States Approach D. O. Dunikov, S. P. Malyshenko and V. V. Zhakhovskii, J. Chem. Phys. 115, 6623 (2001) demonstrated that LJ potential model systems (full potential and cutoff models) and experimental argon data display corresponding states properties. That is, for the LJ liquid number density [ρ liq ([T/T c ])/ ρ c ] LJ [ ρ liq (T/T c )/ ρ c ] Argon Using this approximation, an estimate of the full LJ potential vapor density can be obtained from the small cluster free energy difference intercepts, lnI o : ln ρ vapor,LJ ln ρ liq,LJ – lnI o
References Argon vapor pressure formula: A. Fladerer and R. Strey, J. Chem. Phys. 124, 164710 (2006); K. Iland, J. Wolk and R. Strey, J. Chem. Phys. 127, 54506 (2007). Monte Carlo simulations for LJ vapor number density at T * = 0.7 : B. Chen, J. I. Siepmann, K. J. Oh, and M. L. Klein, J. Chem. Phys. 115, 10903 (2001) Argon experimental vapor pressure data: R. Gilgen, R. Kleinram and W. Wagner, J. Chem. Therm. 22, 399 (1994) Monte Carlo simulations of small LJ clusters: B. N. Hale and M. Thomason, “Scaled Nucleation in a Lennard-Jones System”, submitted for publication.
Summary & Comments Estimates of vapor pressures for the full LJ potential system at reduced temperatures, T/ k = 0.33, 0.42, 0.50 and 0.70, are obtained from small cluster free energy differences. Using a corresponding states approach, the results are compared with extrapolations of an argon vapor pressure formula, experimental data at high temperature, and MC simulation results at T/ k = 0.7.