Scaled Nucleation in Lennard-Jones System

Scaled Nucleationin Lennard-Jones System Barbara Hale and Tom Mahler Physics Department Missouri University of Science & TechnologyJerry 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 DataCo = [Tc/240-1]3/2 ; Tc = 647.3 K B. Hale, J. Chem. Phys. 122, 204509 (2005)

Schmitt et al Toluene (C7H8) data Co = [Tc /240-1]3/2 ; Tc = 591.8K

Kinetic Nucleation Rate Formalism 1/J = n=1,M 1/Jn ; M large Jn = n (N1S)2 j=2,n S[N1j-1/j] growth/decay rate constants  S = Nexp1 /N1  P/Po

Growth/Decay Rate Constants Detailed balance: n-1 Nn-1N1= n Nn from Monte Carlo: ln[Qn/(Qn-1 Q1 n)]= ln[Nn/(Nn-1N1)] = ln(n-1 /n)= - fn

Monte Carlo Simulations Ensemble A: (n -1) cluster plus monomer probe interactions turned off Ensemble B: n cluster with normal probe interactions Calculate fn = [Fn – Fn-1 ]/kT

Scaling of free energy differences for small Lennard-Jones clusters

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

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 Nn 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/[Tc/T-1]3/2

Law of Mass Action Nn/[Nn-1N1] = 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/Jn ; M large Jn = (n)(N1S)2j=2,n [N1S(j-1)/(j)] S = N1exp/N1

Free Energy Differences - f(n) = ln [Q(n)/(Q(n-1)Q(1))]calculated = ln [ (ρoliq/ρovap)(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 << TcB. N. Hale, Phys. Rev A 33, 4156 (1986); J. Chem. Phys. 122, 204509 (2005) J0,scaled [thermal (Tc)] -3 s-1 “scaled supersaturation” lnS/[Tc/T-1]3/2

Toluene (C7H8) nucleation data of Schmitt et al plotted vs. scaled supersaturation, Co = [Tc /240-1]3/2 ; Tc = 591.8K

Nonane (C9H20) nucleation data of Adams et al. plotted vs. scaled supersaturation; Co = [Tc/240-1]3/2;Tc= 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, 19780 (2004)

Nucleation rate via Monte Carlo Calculation of Nucleation rate from Monte Carlo -f(n) : Jn = flux · Nn* Monte Carlo = [N1v1 4rn2 ] · N1exp 2,n(-f(n´) – ln[liq/1o]+lnS) J -1 = [n Jn ]-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 resultsfor experimental water data points (Si,Ti)

Comments & Conclusions • Experimental data indicate that Jexp is a function of lnS/[Tc/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?

Scaled Nucleation in Lennard-Jones System