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/εk = 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 Nn clusters
• full LJ interaction potential
• separable classical Hamiltonian
Cluster Free Energy
Differences
n = number of atoms in cluster
- δfn = ln Qn – ln(Qn-1Qn)
n→ ∞
- δfn → ln [ ρliquid /ρ1,vapor]
Canonical Configuration Integral
Qn = ∫∫∫…∫ exp[-Σi >j VLJ(|ri-rj|)/kT]dnri
Monte Carlo Bennett method
is used to calculate ratios:
Qn/ [Qn-1 Q1]
Schematic of Monte Carlo Simulations
Ensemble B:
n cluster with normal
probe interactions
-δfn =
lnQn
Ensemble A:
(n -1) cluster with monomer
probe interactions turned off
-
ln(Qn-1Q1)
LJ n-cluster Free Energy Differences
T* = 0.335 (40K)
T* = 0.415 (50K)
T* = 0.503 (60K)
T* = 0.700 (83.6K)
16
- δfn
12
8
4
0
0
0.2
0.4
0.6
-1/3
n
0.8
1
LJ n-cluster Free Energy Differences
y = -22.062x + 17.6
2
R = 0.9974
16
y = -15.715x + 12.93
2
R = 0.999
-δfn
12
y = -11.447x + 9.9
2
R = 0.9988
8
y = -6.4432x + 6.1
2
R = 0.9955
4
0
0
0.2
0.4
n-1/3
0.6
0.8
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
[ρliq ([T/Tc])/ρc]LJ ≈ [ρliq(T/Tc)/ρc]Argon
Using this approximation, an estimate of the full LJ potential vapor density can
be obtained from the small cluster free energy difference intercepts, lnIo:
ln ρvapor,LJ ≈ ln ρliq,LJ – lnIo
LJ System Vapor Pressure: T*c =1.313
20
ln(Po/Pc) vapor pressure
formula
-ln ( Po / Pc )
16
experimental data
B. Chen et al.
12
Present work
8
4
0
0
1
2
Tc / T - 1
3
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/εk = 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/εk = 0.7.