Estimates of vapor pressure for the LJ system below

the triple point

Barbara N. HalePhysics Department

Missouri University of Science & TechnologyRolla, 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 Nn clusters

full LJ interaction potential

separable classical Hamiltonian

Cluster Free Energy Differences

n = number of atoms in cluster

- fn = ln Qn – ln(Qn-1Qn)

n - fn ln [ ρliquid /ρ1,vapor]

Canonical Configuration Integral

Qn = … exp[-i >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 A:

(n -1) cluster with monomer probe interactions turned off

Ensemble B:

n cluster with normalprobe interactions

-fn = lnQn - ln(Qn-1Q1)

LJ n-cluster Free Energy Differences

0

4

8

12

16

0 0.2 0.4 0.6 0.8 1

n-1/3

- f

nT* = 0.335 (40K)

T* = 0.415 (50K)

T* = 0.503 (60K)

T* = 0.700 (83.6K)

LJ n-cluster Free Energy Differences

y = -15.715x + 12.93

R2 = 0.999

y = -6.4432x + 6.1

R2 = 0.9955

y = -22.062x + 17.6

R2 = 0.9974

y = -11.447x + 9.9

R2 = 0.9988

0

4

8

12

16

0 0.2 0.4 0.6 0.8 1

n-1/3

-f n

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

0

4

8

12

16

20

0 1 2 3Tc / T - 1

-ln

( P

o /

Pc

)ln(Po/Pc) vapor pressureformula

experimental data

B. Chen et al.

Present work

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.