A Monte Carlo discrete sum (MCDS) approach to energies of formation for small methanol clusters Srivatsan Raman*, Barbara Hale and Gerald Wilemski Physics Department, *Chemical Engineering Department University of Missouri-Rolla, Rolla, MO – 65409, USA Supported by the Engineering Physics Program, U.S. DOE

Experimental Nucleation rates J EXP vs Classical Theory Predictions J CLASSICAL

MOTIVATION FOR THIS WORK Classical Nucleation Model – Poor temperature dependence To apply a molecular treatment of small Methanol clusters – Avoid use of bulk surface tension and describe cluster as discrete set of molecules To build a foundation for treating binary systems comprising methanol and water

MCDS nucleation rate model:  (n*) = critical size cluster concentration from Monte Carlo Nucleation rate, J MCDS = classical steady state form J MCDS = J o cl  (n*) Monte Carlo J o cl = monomer flux factor times Zeldovich factor Use sum of Monte Carlo free energy differences.

THREE-SITE INTERMOLECULAR PAIR POTENTIAL FOR METHANOL* * Monica E. van Leeuwen and Berend Smit, J. Phys Chem, 99,1831 (1995) Oxygen CH 3 Methyl group Hydrogen + + Atom/Func grp O CH H r CO Å r OH Å o U αβ = U LJ + U COULOMB

STATISTICAL MECHANICAL FORMALISM Law of Mass Action (Assuming non-interacting mixture of ideal gases with each cluster constituting an ideal gas system) Separating the kinetic energy contribution from the canonical partition function, Z, we have…. ‘Q’ is the configurational partition function

After algebraic manipulations, we have ‘S 1 ’ is the monomer supersaturation ratio where,

The Two Canonical Ensembles Ensemble A n molecules Ensemble B (n-1) molecules in cluster + one monomer

We plot vs or, ANALYSIS OF -δF n

vs

MOTIVATION FOR SCALING OF FREE ENERGY DIFFERENCES WITH (T C /T – 1) is the cluster excess surface entropy per molecule is a nearly universal constant. It is about ‘2’ for most substances, but for associated liquids, it is approximately ‘1.5’

Experimental Nucleation rates J EXP vs Classical Theory Predictions J CLASSICAL

Experimental Nucleation rates J EXP vs Monte Carlo calculated Nucleation rates J MCDS

Results and Discussion Potential model and free energy difference results: -- slope agrees with σ bulk in the limit of large cluster sizes -- intercept indicates about the right vapor pressure -- the free energies scale with [Tc/T -1] and permit predictions of J over range of T Prediction of nucleation rate: -- no improvement over classical model in terms of magnitude -- improved temperature dependence for 255 K and 272 K data Large discrepancy in magnitude of J: Experimental data are corrected for small n-mer formation (Strey et al). Can present model provide improved estimate of heat of association effect on final temperature?