1. Computer Simulations, Scaling and the Prediction of Nucleation Rates
Barbara Hale
Physics Department and
Cloud and Aerosol Sciences Laboratory University of Missouri – Rolla Rolla, MO USA

2. Nucleation : formation of embryos of the new phase from the metastable (supersaturated) 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);

4. Estimating the nucleation rate, J, from the molecular dynamics simulation at T = 80.3K. Supersaturation ratio = P/Po = 6.8
vol. = ( 60 x 60 x 60) 3;

5. Nucleation is a non-equilibrium process!
●There is no “first principles” theory from which to determine the nucleation rate.
● Most models attempt to treat nucleation as the decay of a near-equilibrium metastable (supersaturated) state.
● The classical nucleation theory (CNT) model was first developed in by Volmer and Weber, and by Becker and Döring in …. 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
(vapor-to-liquid)
Jclassical = [N1 v 4rn*2/3 ] · Nn*
= [Monomer flux] · [# Critical Clusters/Vol.]

7. Estimating Nn Nn / N1 = exp [–Work of formation / kT]
Work of formation of cluster from vapor:
W(n) = 4 rn2  n kT ln S
S = P/Po

8. n* = critical sized cluster has equal probability of growing or decaying

9. n* = critical sized cluster
at n = n*:
dW(n) / dn = 0
Let
W(n) = An2/3 -nlnS
where A = [36]1/3 liq-2/3  /kT
liq= liquid number density

10. Volume / Surface in W(n*)
d/dn [ An2/3 - nlnS]n* = 0
(2/3)A n*-1/3 = lnS
n* = [2A/ 3lnS]3
W(n*) /kT = ½ n* lnS
= [16/3] [/(liq2/3 kT) ]3 / [lnS]2
liq= liquid number density

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

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

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

14. Scaling of the surface tension at T << Tc
Assume a scaled form for :
 = o’ [Tc- 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
 for substances with dipole moment
(a law of corresponding states result; Eötvös 1869)

15. Scaled Nucleation Rate at T << Tc B. N. Hale, Phys
Scaled Nucleation Rate at T << Tc B. N. Hale, Phys. Rev A 33, 4156 (1986); J. Chem. Phys. 122, (2005)
J0,scaled  [thermal (Tc)] -3 s-1
“scaled supersaturation”  lnS/[Tc/T-1]3/2

16. Water nucleation rate data of Wölk and Strey plotted vs
Water nucleation rate data of Wölk and Strey plotted vs. lnS / [Tc/T-1]3/2 ; Co = [Tc/240-1]3/2 ; Tc = K J. Chem. Phys. 122, (2005)

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

18. Nonane (C9H20) nucleation data of Adams et al. plotted vs
Nonane (C9H20) nucleation data of Adams et al. plotted vs. scaled supersaturation ; Co = [Tc/240-1]3/2 ; Tc = 594.6K

19. for most materials (corresponding states)
Comparison of Jscaled with water data from different experimental techniques: plot log[J/J0,scaled] vs.
J0,scaled  cm-3 s-1
for most materials (corresponding states)

21. Missing terms in the classical work of formation?

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

23. 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)

24. Nucleation rate via Monte Carlo
Calculation of Nucleation rate from Monte Carlo -f(n) :
Jn = flux · Nn* Monte Carlo
= [N1v1 4rn2 ] · N1 exp 2,n(-f(n´) – ln[liq/1o]+lnS)
J = [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.

25. Monte Carlo TIP4P nucleation rate results for experimental water data points (Si,Ti)

27. 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?

28. Molecular Dynamics Simulations
Solve Newton’s equations,
mi d2ri/dt2 = Fi = -i j≠i U(rj-ri),
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