A Monte Carlo discrete A Monte Carlo discrete sum (MCDS) approach to sum (MCDS) approach to energies of formation energies of formation for small methanol for small methanol clusters clusters Srivatsan Raman*, Barbara Srivatsan Raman*, Barbara Hale and Gerald Wilemski 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
A Monte Carlo discrete sum (MCDS) approach to energies of formation for small methanol clusters Srivatsan Raman*, Barbara Hale and Gerald Wilemski Physics.
Dec 22, 2015
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A Monte Carlo discrete sum A Monte Carlo discrete sum (MCDS) approach to energies (MCDS) approach to energies
of formation for small of formation for small methanol clustersmethanol clusters
Srivatsan Raman*, Barbara Hale and Srivatsan Raman*, Barbara Hale and Gerald WilemskiGerald Wilemski
Physics Department, *Chemical Engineering Department
University of Missouri-Rolla, Rolla, MO – 65409, USA
Supported by the Engineering Physics Program, U.S. DOE
log J CLASSICAL x 10-7(cm-3sec-1)
0 2 4 6 8 10 12 14
log
J E
XP (
cm-3
sec-1
)
0
2
4
6
8
10
12
14 Methanol Nucleation Rates
Strey, Wagner and Schmeling JCP 1986
272 K
255 K 240 K 230 K
Experimental Nucleation rates JEXP vs Classical Theory Predictions JCLASSICAL
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:MCDS nucleation rate model:
(n*) (n*) = critical size cluster = critical size cluster concentration from Monte Carlo concentration from Monte Carlo
Nucleation rate, Nucleation rate,
JJMCDSMCDS = classical steady state form = classical steady state form
J J MCDS MCDS = = JJo clo cl (n*)(n*) Monte CarloMonte Carlo
JJo clo cl = = monomer flux factor times Zeldovich factormonomer flux factor times Zeldovich factor
Use sum of Monte Carlo free energy differences.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
CH3
Methyl group
Hydrogen++
Atom/Func grp
O 86.5 3.030 -0.700
CH3 105.2 3.740 +0.265
H 0.0 0.0 +0.435rCO 1.4246 Å
rOH 0.9451 Å108.53o
Uαβ = ULJ + UCOULOMB
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
‘S1’ 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 -δFn
Methanol Clusters
n-1/3
0.0 0.5 1.0
-f n
0
10
20
T = 240K
experiment
vs
vs
Methanol Clusters
n-1/3
0.0 0.5 1.0
-f n
0
10
20
T = 260K
experiment
vs
Methanol Clusters
n-1/3
0.0 0.5 1.0
-f n
0
10
20
T = 280K
experiment
MOTIVATION FOR SCALING OF FREE ENERGY DIFFERENCES WITH (TC/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’
Scaled Free Energy Differences
n-1/3
0.0 0.5 1.0
-f n
/ [ T
c /T
-1
]
0
10
20
T = 260K T = 240K
T = 280K
experiment
Average of Scaled -fn
n-1/3
0.0 0.5 1.0
-f n
/ [
Tc
/T -
1 ]
0
10
20
= 1.2
- - - 10.75 - 3.87 n-1/3
experiment
log J CLASSICAL x 10-7(cm-3sec-1)
0 2 4 6 8 10 12 14
log J
EX
P (cm
-3se
c-1)
0
2
4
6
8
10
12
14 Methanol Nucleation Rates
Strey, Wagner and Schmeling JCP 1986
272 K
255 K 240 K 230 K
Experimental Nucleation rates JEXP vs Classical Theory Predictions JCLASSICAL
Experimental Nucleation rates JEXP vs Monte Carlo calculated Nucleation rates JMCDS
log JMCDSx 10-7(cm-3sec-1)
0 2 4 6 8 10 12 14
log J
EX
P (cm
-3se
c-1)
0
2
4
6
8
10
12
14 Methanol Nucleation Rates
Strey, Wagner and Schmeling JCP 1986
272 K
255 K
240 K 230 K
Results and Discussion
• Potential model and free energy difference results:
-- slope agrees with σbulk 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?
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