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

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Presentation transcript:

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

Motivation To understand how scaling of the nucleation rate is related to the microscopic energy 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

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

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

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

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)

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

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.

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

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

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).