1. Scaled Nucleation in Lennard-Jones System Barbara Hale and Tom Mahler Physics Department Missouri University of Science & Technology Jerry Kiefer Physics Department St. Bonaventure University

2. Motivation To understand how scaling of the nucleation rate is related to the microscopic energies 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. Kinetic Nucleation Rate Formalism 1/J =  n = 1,M 1/J n ; M large J n =  n (N 1 S) 2  j=2,n S[ N 1  j-1 /  j ] growth/decay rate constants  S = N exp 1 /N 1  P/P o

6. Growth/Decay Rate Constants Detailed balance:  n-1 N n-1 N 1 =  n N n from Monte Carlo: ln[Q n /(Q n-1 Q 1 n)]= ln[N n /(N n-1 N 1 )] = ln(  n-1 /  n ) = -  f n

7. 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

10. Scaling of free energy differences for small Lennard-Jones clusters

13. Comments & Conclusions Experimental data  J ( lnS/[T c /T-1] 3/2 ). Source of scaling? Monte Carlo LJ small cluster simulations  scaled energies of formation. Scaling appears to emerge from [T c /T-1] dependence of the  f n.

16. 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

17. 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, J, from kinetic nucleation rate formalism;  compare logJ vs lnS and logJ vs lnS/[T c /T-1] 3/2

18. 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

19. 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

20. 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.

27. Classical Nucleation Rate  (T)  a – bT is the bulk liquid surface tension ;

28. Scaled Nucleation Rate at T << T c B. N. Hale, Phys. Rev A 33, 4156 (1986); J. Chem. Phys. 122, 204509 (2005) J 0,scaled  [ thermal (T c )] -3 s -1 “scaled supersaturation”  lnS/[T c /T-1] 3/2

29. Toluene (C 7 H 8 ) nucleation data of Schmitt et al plotted vs. scaled supersaturation, C o = [T c /240-1] 3/2 ; T c = 591.8K

30. Nonane (C 9 H 20 ) nucleation data of Adams et al. plotted vs. scaled supersaturation ; C o = [T c /240-1] 3/2 ; T c = 594.6K

32. Missing terms in the classical work of formation?

33. Monte Carlo Helmholtz free energy differences for small water clusters:  f(n) =[F(n)-F(n-1)]/kT B.N. Hale and D. J. DiMattio, J. Phys. Chem. B 108, 19780 (2004)

34. Nucleation rate via Monte Carlo Calculation of Nucleation rate from Monte Carlo -  f(n) : J n = flux · N n* Monte Carlo = [N 1 v 1 4  r n 2 ] · N 1 exp  2,n (-  f(n´) – ln[  liq /  1o ]+lnS) J -1 = [  n J n ] -1 The steady-state nucleation rate summation procedure requires no determination of n* as long as one sums over a sufficiently large number of n values.

35. Monte Carlo TIP4P nucleation rate results for experimental water data points (S i,T i )

37. Comments & Conclusions Experimental data indicate that J exp is a function of lnS/[T c /T-1] 3/2 A “first principles” derivation of this scaling effect is not known; Monte Carlo simulations of  f(n) for TIP4P water clusters show evidence of scaling; Temperature dependence in pre-factor of classical model can be partially cancelled when energy of formation is calculated from a discrete sum of  f(n) over small cluster sizes. Can this be cast into more general formalism?