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

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

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

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

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

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

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

Scaling of free energy differences for small Lennard-Jones clusters

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.

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

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

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

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

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.

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

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

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

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

Missing terms in the classical work of formation?

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, (2004)

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.

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

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?