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The Scaling of Nucleation Rates

The Scaling of Nucleation Rates. Barbara Hale Physics Department and Cloud and Aerosol Sciences Laboratory University of Missouri – Rolla Rolla, MO 65401 USA. Nucleation : formation of embryos of the new phase from the metastable parent phase. K. Yasuoka and M. Matsumoto,

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The Scaling of Nucleation Rates

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  1. The Scaling of Nucleation Rates Barbara Hale Physics Department and Cloud and Aerosol Sciences LaboratoryUniversity of Missouri – RollaRolla, 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. ClassicalNucleation Theory n(S, T) = 1 exp[- ∆G(n) /kT ]; S = P/Po ( includes effect of clusters near n*) ∆G(n) = (n) – n1 (free energy of formation) = G(n)surface + n liq – n1 = 4rn2 - nkTln(P/Po) Jclassical= [ 1 v 4rn*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rn2- nkTln(P/Po)] = 0 d/dn [ An2/3 -nlnS] = 0 ………………………………………….. A = [36]1/3  /[liq2/3kT ] ; S = P/Po liq=n/[4 rn3/3]

  8. Volume / Surface in ∆G(n*) d/dn [ An2/3 - nlnS]n* = 0 (2/3)A n*-1/3 - lnS = 0 n* = [2A/ 3lnS]3 ∆G(n*) /kT = (1/2) [2A/ 3lnS]3lnS ∆G(n*) /kT= [16/3] [/(liq2/3 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 << 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.

  12. Scaling of the surface tension at T << Tc Assume a scaled form for :  = o’ [Tc- T]  with  =1for 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 << 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

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

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

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

  17. Comparison of Jscaled with water data from different experimental techniques: plot log[J/J0,scaled] vs. J0,scaled [2mkTc/h2]3/2 s-1  1026 cm-3 s-1 for most materials (corresponding states)

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

  19. 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)

  20. Nucleation rate via Monte Carlo Calculation of Nucleation rate from Monte Carlo free energy differences, -f(n) : Jn = [1v1 4rn2 ]· 1 exp 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.

  21. Monte Carlo TIP4P nucleation rate resultsfor experimental water data points (Si,Ti)

  22. 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?

  23. 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. • MeasureJ  rate at which clusters form

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