Computer Simulations, Scaling and the Prediction of Nucleation Rates Barbara Hale Physics Department and Cloud and Aerosol Sciences Laboratory University of Missouri – Rolla Rolla, MO USA

Nucleation : formation of embryos of the new phase from the metastable (supersaturated) parent phase K. Yasuoka and M. Matsumoto, J. Chem. Phys. 109, 8451 (1998 )

“Molecular dynamics of homogeneous nucleation in the vapor phase: Lennard-Jones fluid”, K. Yasuoka and M. Matsumoto, J. Chem. Phys. 109, 8451 (1998);

Nucleation rate, J, in the argon - LJ molecular dynamics simulation ( T = 80.3K, S = P/P o = 6.8)

Nucleation is a non-equilibrium process ● Most models attempt to treat nucleation as the decay of a near-equilibrium metastable (supersaturated) state. ● The classical nucleation theory (CNT) model was first developed in 1926 by Volmer and Weber, and by Becker and Döring in 1935 …. following a proposal by Gibbs. ● CNT treats nucleation as a fluctuation phenomenon in which small embryos of the new phase overcome free energy barriers and grow irreversibly to macroscopic size.

Classical Nucleation Theory ( vapor-to-liquid ) J classical = [N 1 v 4  r n* 2/3 ] · N 1 e - W(n*)/kT = [Monomer flux] · [# Critical Clusters/Vol.]

Homogeneous Nucleation rate data for water: classical nucleation rate model has wrong T dependence

Scaling of the Nucleation Rate at T << T c J(S, T) = J( lnS/[T c /T-1] 3/2 ) S = P/P 0 T c = critical temperature “scaled supersaturation”  lnS/[T c /T-1] 3/2 B. N. Hale, Phys. Rev A 33, 4156 (1986); B. Hale, J. Chem. Phys. 122, (2005)

Scaling: Wölk and Strey Water Data ( T c = K ) B. Hale, J. Chem. Phys. 122, (2005)

Schmitt et al Toluene (C 7 H 8 ) data (T c = 591.8K)

Motivation:  To understand how temperature scaling of the nucleation rate is related to the microscopic energy of formation, W(n), of small clusters ;  To examine how scaling can be used to make nucleation rate predictions in more complex systems (atmosphere, biological systems).

Research Projects:  Monte Carlo simulation calculations of small cluster free energies of formation from the vapor: H 2 O clusters Argon LJ clusters Binary H 2 O/H 2 SO 4 clusters Methanol clusters  Scaling model analysis of nucleation rate data  Monte Carlo simulation studies of cluster impurity effects on T dependence.

Monte Carlo Simulations Ensemble A : (n -1) cluster plus monomer probe interactions turned off Ensemble B: n cluster with normal probe interactions Calculate:  f(n) = [F(n)-F(n-1)]/ kT ≈ - ln(ρ liq /ρ vap ) + 2/3An -1/3

Water Cluster Simulations  60 water molecules;  TIP4P intermolecular potential;  60M Monte Carlo steps  280K

Binary H 2 O- H 2 SO 4 Clusters  H 2 O 16 [H 2 SO 4 ] 16  Kathmann-Hale interatomic potentials  8M Bennett Monte Carlo steps  273K

Argon Lennard-Jones Clusters  60 atoms;  Lennard-Jones potential;  60M Monte Carlo steps;  40K

Monte Carlo Helmholtz free energy differences for small water clusters: B.N. Hale and D. J. DiMattio, J. Phys. Chem. B 108, (2004)

Experimental Water Nucleation Rate Data & Simulation Results

Scaling of Ar-LJ free energy differences

Monte Carlo results: argon-LJ nucleation rates

Data + Fladerer □ Iland, Strey X Wu et al. Calculations: ● ten Wolde ■ Yasuoka and Matsumoto ▼ Senger, Reiss, et al. ▲ Oh and Zeng  Chen et al. ■ Hale et al. ■ Classical model ■ Lauri et al.

Comments Experimental data indicate J is a function of lnS/[T c /T-1] 3/2. Monte Carlo LJ cluster simulations show evidence of scaling. Scaling appears to emerge from [T c /T-1] dependence of small cluster free energy differences. Problems remain with Argon nucleation rate magnitude predictions: impurity effects?

Motivation To understand how scaling of the nucleation rate is related to the microscopic energy of formation of small clusters.