1. A Study of Scaled Nucleation in a Model Lennard-Jones System Barbara Hale and Tom Mahler Physics Department University of Missouri – Rolla Jerry Kiefer Physics Department St. Bonaventure University

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

3. Scaling: Wölk and Strey Water Data C o = [T c /240-1] 3/2 ; T c = 647.3 K B. Hale, J. Chem. Phys. 122, 204509 (2005)

4. Schmitt et al Toluene (C 7 H 8 ) data C o = [T c /240-1] 3/2 ; T c = 591.8K

5. Study of Scaling in LJ System  calculate rate constants for growth and decay of model Lennard-Jones clusters at three temperatures;  determine model nucleation rates from kinetic nucleation rate formalism;  compare logJ vs lnS and logJ vs lnS/[T c /T-1] 3/2

6. Model Lennard-Jones System  Law of mass action  dilute vapor system with volume, V;  non-interacting mixture of ideal gases;  each n-cluster size is ideal gas of N n particles ;  full atom-atom LJ interaction potential;  separable classical Hamiltonian

7. Law of Mass Action N n /[N n-1 N 1 ] = Q(n)/[Q(n-1)Q(1)n] Q(n) = n-cluster canonical configurational partition function

8. Relation to Growth & Decay  (n-1)N n-1 N 1 =  (n)N n we calculate: Q(n)/[Q(n-1)Q(1)n]= N n /[N n-1 N 1 ] =  (n-1)/  (n)

9. Kinetic Nucleation Rate Formalism 1/J =  n = 1,M 1/J n ; M large J n =  (n) (N 1 S) 2  j=2,n [ N 1 S  (j-1) /  (j)] S = N 1 exp /N 1

10. Free Energy Differences -  f(n) = ln [Q(n)/(Q(n-1)Q(1))] calculated = ln [ (ρ o liq /ρ o vap )  (j-1)/  (j) ] Use Monte Carlo Bennett technique.

11. 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

14. The nucleation rate can be calculated for a range of supersaturation ratios, S. 1/J =  n = 1,M 1/J n ; M large J n =  (n) (N 1 S) 2  j=2,n [ N 1 S  (j-1) /  (j)] S = N 1 exp /N 1

17. Comments & Conclusions Experimental data indicate that J exp is a function of lnS/[T c /T-1] 3/2. No “first principles” derivation of scaling exists. Monte Carlo LJ cluster simulations show evidence of scaling. Scaling appears to emerge from [T c /T-1] dependence of the  f(n).