Monte Carlo Simulations of Growth/Decay Rate Constant Ratios

for Small Methanol Clusters: Application to Nucleation Data Analysis

B. Hale, G. Wilemski and A. VietsPhysics Department

Missouri University of Science & Technology

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.

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)

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.

the scaling function

C0 = (16π/3)Ω3/ln(10) Ω ~ 2.1 for normal liquids

~1.5 for substances with dipole moment

Scaling of toluene data

Schmitt et al., J. Chem. Phys. 79, 4496 (1983); Hale & Thomason, Phys. Rev. Letters 105, 046101 (2010)

Co lnS/[Tc/T-1]3/2

2 3 4lo

g(J

/ cm

-3s-1

)1-

0

1

2

3

4

259K

217K

233K

Jexp (O) Jscaled (+)

Schmitt et al. toluene data b)

lnS

2 3 4

log(

J / c

m-3

s-1)

1-

0

1

2

3

4

259K

217K

233K

Jexp (O) Jscaled (+)

Schmitt et al. toluene data a)

Co = [Tc /240-1]3/2

Tc = 591.8K

C7H8

C0 lnS/[Tc/T-1]3/2lnS

Scaling plot for water data

Ω = 1.47

The SSW methanol data do not scale

Strey, Schmeling, and Wagner, JCP 85, 6192 (1986) (SSW)

Rates have beencorrected by SSW for

heats of formationfor n = 2 – 4.

Subcritical cluster heats of formationlead to an increase in the final T, and

a lowering of S.

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. Phys Chem, 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))

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

20.00

f(x) = − 4.28783133333334 x + 10.4548919743521R² = 1

f(x) = − 2.02049154003061 ln(x) + 6.8212830784748R² = 0.35135659250292

T = 220 KT = 240 KT = 260 KLinear (T = 260 K)Logarithmic (T = 260 K)

n-1/3

df

n /

[ T

c /

T-1

]Tc = 532K

W

Scaled MC free energy differences-δfn scale at low T

with [Tc/T-1].

Tc for model(no cutoff)

~ 532K

Monte Carlo results for van Leeuwen / Smit methanol potential

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

20.00

n-1/3

-,

dfscaven

Methanol Potential of van Leeuwen & Smit Tc = 532K = 1.33W

Average of MC Scaled Free Energy differences

The averaged scaled free energy differences are used to calculate -δfn at variable T.

Equilibrium constants and -δfn

• Equilibrium constant for the n-mer:

Kn = Nn /(N1)n

= (kTnliquid)1-n exp(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

Experimental methanol rates with MC corrected T and S

The supersonic nozzledata of Laksmono,

Tanamura and Wyslouzil et al.

[JPC 135, 07435 (2011)]are as published.

Strey, Wagner & Schmelling,JCP 84, 2325 (1986) datahave corrected T, S from

Monte Carlo resultspresented here.

ΔH4 = 23.9 kcal/mol ΔS4 = 81.0 cal/mol/K

MC nucleation ratesare calculated usingkinetic steady-state

nucleation rateformalism with

T’c/T’ = Tc/T.and T’c = 532 K

for model potential.*

*Dunikov et al. JCP 115, 6623 (2001)

Monte Carlo results for the methanol nucleation ratefrom the van Leeuwen & Smit model potential

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

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 rate data.

• 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.

Thank you!

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

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

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.

10-5

10-4

10-3

10-2

Cn

/C1

280 270 260 250 240T (K)

S=1 S=2 S=3 S=4

C2

C3

C4 C5

C6

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

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

Growth/Decay Rate Constant Ratios Calculated from Monte Carlo

Detailed balance:

n-1 Nn-1N1= n Nn

ln(N1n-1 /n) = ln[Nn/Nn-1]

= - fn – ln [liquid/1] from Monte Carlo

MC results must be calibrated using experimental dimer and tetramer data

K(n)=K(n-1) exp(–δfn), n>5K(5)=KEX(4)exp(–δf5)

Dimer and tetramer equilibration rates were normalized to experimental values

Scaling: Wölk and Strey Water Data Co = [Tc/240-1]3/2 ; Tc = 647.3 K

lnS

1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2

log

J /(

cm-3

sec-1

)

4

6

8

10

a)

260 K 250 K

240 K 230 K 220 K

Co lnS / [Tc/T -1]3/2

1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2

log

[ J

/ cm

-3 /

sec-1

]

4

6

8

10 Wolk and Strey H2O data

b)

255 K

240 K 230 K

B. Hale, J. Chem. Phys. 122, 204509 (2005)

-0.399999999999999 0.100000000000001 0.6000000000000010.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

20.00

f(x) = − 3.95481947639095 x + 8.68853347303938R² = 0.968809052065814

f(x) = − 4.46630876230872 x + 9.86454262667881R² = 0.961615539487383

f(x) = − 4.55738464474925 x + 10.9580150800744R² = 0.939061852919328

f(x) = − 5.32906817435849 x + 12.7505419052869R² = 0.934551910933465

f(x) = − 6.80505320487129 x + 14.9127759884948R² = 0.938197672250135

f(x) = − 7.56305171860654 x + 17.3698917557686R² = 0.931140973539482

-δfn for n > 5T = 200 K Linear (T = 200 K) T = 220 K

Linear (T = 220 K) T = 240 K Linear (T = 240 K)

T = 260 K Linear (T = 260 K) Linear (T = 260 K)

T = 280 K Linear (T = 280 K) T = 300 K

Linear (T = 300 K) int240 int260

int280 int300 int220

240 small n 260 small n 280 small n

300 small n 220 small n 200 small n

int200