1. The Scaling of Nucleation Rates Barbara Hale Physics Department and Cloud and Aerosol Sciences Laboratory University of Missouri – Rolla Rolla, MO 65401 USA

2. Nucleation : formation of embryos of the new phase from the metastable parent phase K. Yasuoka and M. Matsumoto, J. Chem. Phys. 109, 8451 (1998 )

3. Molecular dynamics of homogeneous nucleation in the vapor phase: Lennard-Jones fluid, K. Yasuoka and M. Matsumoto, J. Chem. Phys. 109, 8451 (1998);  = 2.15ps; vol. = ( 60 x 60 x 60)  3 ; T = 80.3 K; S = 6.8

4. Estimating the nucleation rate, J, from the molecular dynamics simulation for (argon) LJ at T = 80.3K; S =6.8

5. Nucleation is generally treated theoretically as the decay of a metastable state – a non-equilibrium process. ●There is no “first principles” theory from which to determine the nucleation rate. ● Most models attempt to predict nucleation rates using properties of near-equilibrium metastable states. ● The classical nucleation theory (CNT) model was first developed in 1926 by Volmer and Weber, and by Becker and Döring in 1935 …. following a proposal by Gibbs. ● CNT treats nucleation as a fluctuation phenomenon in which small embryos of the new phase overcome free energy barriers and grow irreversibly to macroscopic size.

6. Classical Nucleation Theory  n (S, T) =  1 exp[- ∆G(n) /kT ]; S = P/P o (  includes effect of clusters near n*) ∆G(n) =  (n) – n  1 (free energy of formation) = G(n) surface + n  liq – n  1 = 4  r n 2  - nkTln(P/P o ) J classical = [  1 v 4  r n* 2 ] ·  n* (S, T)   = [Monomer flux] · [# Critical Clusters/Vol.] (vapor-to-liquid nucleation rate)

7. n* = critical sized cluster equal probability of growing or decaying at n = n*: d/dn[ 4  r n 2  - nkTln(P/P o )] = 0 d/dn [ An 2/3 - nlnS] = 0 ………………………………………….. A = [36  ] 1/3  /[  liq 2/3 kT ] ; S = P/P o  liq = n/[4  r n 3 /3]

8. Volume / Surface in ∆G(n*) d/dn [ An 2/3 - nlnS] n* = 0 (2/3)A n* -1/3 - lnS = 0 n* = [2A/ 3lnS] 3 ∆G(n*) /kT = (1/2) [2A/ 3lnS] 3 lnS ∆G(n*) /kT = [16  /3] [  /(  liq 2/3 kT) ] 3 / [lnS] 2

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

10. Homogeneous Nucleation rate data for water: classical nucleation rate model has wrong T dependence

11. Motivation for Scaling J at T << T c The CNT nucleation rate depends exponentially on  (T) 3 / [ln (P/P o (T))] 2. To obtain a physically realistic T dependence of J, a good starting point is to require functional forms for  (T) and P o (T) which reflect “universal” properties of surface tension and vapor pressure.

12. Scaling of the surface tension at T << T c Assume a scaled form for  :  =  o ’ [T c - T]  with  =1 for simplicity. Many substances fit this form and the critical exponent (corresponding to  ) is close to 1.  = excess surface entropy per molecule / k  2 for normal liquids  1.5 for substances with dipole moment (a law of corresponding states result; Eötvös 1869)

13. 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

14. Water nucleation rate data of Wölk and Strey plotted vs. lnS / [T c /T-1] 3/2 ; C o = [T c /240-1] 3/2 ; T c = 647.3 K J. Chem. Phys. 122, 204509 (2005)

15. 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

16. 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

17. Comparison of J scaled with water data from different experimental techniques: plot log[J/J 0,scaled ] vs. J 0,scaled  [2  mkT c /h 2 ] 3/2 s -1  10 26 cm -3 s -1 for most materials (corresponding states)

19. Missing terms in the classical nucleation rate energy of formation?

20. 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)

21. Nucleation rate via Monte Carlo Calculation of Nucleation rate from Monte Carlo free energy differences, -  f(n) : J n = [  1 v 1 4  r n 2 ]·  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.

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

24. 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?

25. Molecular Dynamics Simulations  Solve Newton’s equations, m i d 2 r i /dt 2 = F i = -  i  j≠i U(r j -r i ), iteratively for all i=1,2… n atoms;  Quench the system to temperature, T, and monitor cluster formation.  Measure J  rate at which clusters form