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B. Hale, G. Wilemski and A. Viets Physics Department Missouri University of Science & Technology

Monte Carlo Simulations of Growth/Decay Rate Constant Ratios for Small Methanol Clusters: Application to Nucleation Data Analysis. B. Hale, G. Wilemski and A. Viets Physics Department Missouri University of Science & Technology. Outline.

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B. Hale, G. Wilemski and A. Viets Physics Department Missouri University of Science & Technology

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  1. Monte Carlo Simulations of Growth/Decay Rate Constant Ratios for Small Methanol Clusters: Application to Nucleation Data Analysis B. Hale, G. Wilemski and A. Viets Physics Department Missouri University of Science & Technology

  2. Outline • Scaling of nucleation rates / non scaling of methanol experimental rates • Monte Carlo methanol cluster simulations: small cluster growth/decay rates  subcritical cluster heats of formation • Prediction of higher experimental T, lower S. • Summary of results.

  3. Scaling Analysis of the Nucleation Rate* at T << Tc*B. Hale, Phys. Rev A 33, 4156 (1986); B. Hale, J. Chem. Phys. 122, 204509 (2005); B. Hale & M. Thomason, Phys. Rev. Letters 105, 046101 (2010) the scaling function J(S, T) = J( lnS/[Tc/T-1]3/2) Scaling Plot: -log(J/1026) = Co [Tc/T - 1]3/[ln(S)2] Substances demonstrating scaling include water, toluene, nonane, n-octane, n-decane, n-pentanol, n-butanol, and n-propanol. C0 = (16π/3)Ω3/ln(10) Ω ~ 2.1 for normal liquids ~1.5 for substances with dipole moment

  4. Scaling of toluene data Schmitt et al.,J. Chem. Phys. 79, 4496 (1983);Hale & Thomason, Phys. Rev. Letters 105, 046101 (2010) Tc = 591.8K C7H8 Co = [Tc/240-1]3/2 lnS C0lnS/[Tc/T-1]3/2

  5. Scaling plot for water data Ω = 1.47

  6. The SSW methanol data do not scale Strey, Schmeling, and Wagner, JCP 85, 6192 (1986) (SSW) Rates have been corrected by SSW for heats of formation for n = 2 – 4. Subcritical cluster heats of formation lead to an increase in the final T, and a lowering of S.

  7. Monte Carlo simulations are used to estimate small n-cluster heats of formation (including n > 4) in the adiabatic expansion. • 3- site pair potential of M. E. van Leeuwen and B. Smit, J. PhysChem, 99,1831 (1995); • Bennett Monte Carlo technique  free energy differences. δfn= - ln[Qn/(Qn-1Q1)] n ranges from 2 to 96 (for formalism see B. Hale & M. Thomason, PRL 105, 046101 (2010))

  8. Monte Carlo results for van Leeuwen / Smit methanol potential -δfnscale at low T with [Tc/T-1]. Tc for model (no cutoff) ~ 532K

  9. The averaged scaled free energy differences are used to calculate -δfnat variable T. Average of MC Scaled Free Energy differences

  10. Equilibrium constants and -δfn • Equilibrium constant for the n-mer: Kn= Nn/(N1)n = (kTnliquid)1-nexp(Σj=2…n-δfn) • Kn= Kn-1 exp(-δfn) for n = 3 and n > 4 • K2 and K4 normalized to experimental values (Renner, Kucera & Blander, JCP 66, 177 (1977)) • Tetramer molar dissociation enthalpy and entropy: ΔH = 23.9 kcal/mol, ΔS = 81.0 cal/mol/K

  11. Experimental methanol rates with MC corrected T and S The supersonic nozzle data of Laksmono, Tanamura and Wyslouzil et al. [JPC 135, 07435 (2011)] are as published. Strey, Wagner & Schmelling, JCP 84, 2325 (1986) data have corrected T, S from Monte Carlo results presented here. ΔH4= 23.9 kcal/mol ΔS4= 81.0 cal/mol/K

  12. Monte Carlo results for the methanol nucleation rate from the van Leeuwen &Smit model potential MC nucleation rates are calculated using kinetic steady-state nucleation rate formalism with T’c/T’ = Tc/T. and T’c = 532 K for model potential.* *Dunikov et al. JCP 115, 6623 (2001)

  13. Old with n = 2 - 4 New with MC n = 2-12

  14. Summary & Conclusions • MC free energy differences for n = 2-12 applied to estimates of subcritical heats of formation can improve scaling properties of the methanol ratedata. • Questions remain regarding potential model Tc and the thermodynamic properties of the tetramer. • Use of [Tc/T ]model = [Tc/T]exp in MC potential model predictions for J gives good results.

  15. Thank you!

  16. data from Renner, Kucera& Blander, JCP 66, 177 (1977)

  17. Final S and T depend on largest cluster size, n, used in expansion calculations

  18. The heat of association (and sub-critical cluster formation) can greatly affect the final thermodynamic ( T, S) state of the gas. T and S are plotted vs. the pressure drop of expansion (heats of formation included for n = 2-4). From Strey, Schmeling, and Wagner, JCP 1986.

  19. Small n-mer concentrations formed during expansion from T=278 K, pM=5.53 kPa

  20. Nucleation rates can be calculated from the Monte Carlo results using the kinetic steady state 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 constant ratio S = Nexp1 /N1  P/Po

  21. Growth/Decay Rate Constant Ratios Calculated from Monte Carlo Detailed balance: n-1 Nn-1N1= nNn ln(N1n-1 /n) =ln[Nn/Nn-1] = - fn– ln[liquid/1] from Monte Carlo

  22. MC results must be calibrated using experimental dimer and tetramer data K(n)=K(n-1) exp(–δfn), n>5 K(5)=KEX(4)exp(–δf5)

  23. Dimer and tetramer equilibration rates were normalized to experimental values

  24. Scaling:Wölk and Strey Water DataCo = [Tc/240-1]3/2 ; Tc = 647.3 K B. Hale, J. Chem. Phys. 122, 204509 (2005)

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