Experimental points show no tendency to scale.

IntroductionNucleation rates for methanol, an associating vapor, are made with nonclassical gradient theory (GT) for T = 230K to 275K. The SAFT-0 EOS is used to account for molecular association effects . [Chapman, Gubbins, Jackson, Radosz, Ind. Eng. Chem. Res. 29, 1709 (1990)] Calculated rates are compared to experimental values of Strey, Wagner, Schmeling [J. Chem. Phys. 84, 2325 (1986)]Monte Carlo results are used in a hybrid scheme to correct experimental S and T values for the effects of the heat of associationHale plots are used to elucidate and assess the observed S and T behavior seen in the data and calculated rates. For more background see Obeidat, Gharaibeh, Ghanem, Hrahsheh, Al-Zoubi, Wilemski, ChemPhysChem 11, 3987 (2010)

AcknowledgmentsThis work is supported by a grant from the National Science foundation. We thank Barbara Wyslouzil for very helpful discussions.

Free energy as a functional of fluid density ρ(r):

20

[ ( )] [ ( )]

( [ ( )] ( / 2)[ ( )] )

F r dr f r

dr f r c r

Euler-Lagrange equation:2

0 ( ) ( )c r

Work of Formation:

2( / 2)W w c dV

( ) ( )ew w w 0( )w f

ρe = equilibrium density of bulk phase

f and f0 are the respective Helmholtz free energy densities of the inhomogeneous and homogeneous fluids; c is the influence parameter evaluated here as a function of temperature by forcing agreement between calculated and experimental values of the bulk surface tension

Nucleation rate:

22

expl P WJ

m S kT kT

GT points cluster in a tight band with Ω = 1.27 in surprising agreement with T dependence found “universally” for water, many other alcohols, and hydrocarbons (with different values of Ω ).

The new GT points (color) scale in a manner similar to the experimental rates with corrected S and T values.

GT results are now in remarkably good agreement with experimental rates. The theoretical and experimental S and T dependence of the nucleation rates are now mutually consistent.

Jscaled = J0exp(-Wscaled), J0=1026 cm-3 s-1

Wscaled=(16π/3) Ω3[(TC/T) –1]3/[lnS]2, C0 = (16π/3) Ω3/ln(10)

Ω = excess surface entropy per molecule (~1.5 for polar species; >2 for nonpolar)

• Original data show anomalous S-T dependence and no tendency to scale.

• Corrected data show greatly improved scaling behavior.

• GT rates are in excellent accord with corrected data and scale well.

• GT rates scale better at high S than at low S.• The methanol Ω value 1.27 is consistent with

the systematic variation seen for higher molecular weight alcohols: ethanol (1.5), propanol (1.74), butanol (1.83), pentanol (1.94)

Scaling behavior and gradient theory of Methanol nucleation Rates

G. Wilemski, B. Hale, and F. Hrahsheh Missouri University of Science and Technology

Rolla, Missouri , USA

1. SAFT-0 EOS for Methanol

2. Gradient Theory (GT)

3. Scaling behavior of methanol nucleation rates using original T and S of Strey, Wagner, Schmeling

4. Small cluster equilibrium constants K(n) from Monte Carlo plus experiment*- hybrid scheme

5. Better treatment of heat releasefrom small cluster formation leads to improved scaling behavior

6. Conclusions

Experimental points (color) show no tendency to scale.

K(n) values from MC free energy values (see poster by Hale, et al.) are too small, so we use experimental K(2) and K(4) values to adjust K(n) for n=3 and n>4.

*W. Weltner, K.S. Pitzer, J. Am. Chem. Soc. 73, 2606 (1951), T. A. Renner, G.H. Kucera, M. Blander, J. Chem. Phys. 66, 177 (1977)

These modified K(n) values are used in a code that models the cluster concentrations and heat release in methanol vapor undergoing an isentropic expansion. To get fully corrected S and T values, cluster sizes up to n=12 must be included.

6. With corrected S and T values,GT rates scale well and agree nicely with experimental values

• The corrected S values are smaller than the original values of SWS while the final temperatures are slightly higher (left).

• With the corrected S and T values, the experimental rates show much improved scaling behavior (right).

GT scales better at high S. As S decreases, the different temperature rate curves tend to grow farther apart.