Scaled Nucleationin Lennard-Jones System

Barbara Hale and Tom MahlerPhysics Department

Missouri University of Science & Technology

Jerry KieferPhysics Department

St. Bonaventure University

Motivation

To understand how scaling of the nucleation rate is related to the microscopic energies of formation

of small clusters.

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)

Schmitt et al Toluene (C7H8) data

Co = [Tc /240-1]3/2 ; Tc = 591.8K

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

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 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)

Kinetic Nucleation Rate Formalism

1/J = n=1,M 1/Jn ; M large

Jn = n (N1S)2 j=2,n S[ N1j-1/j]

growth/decay rate constants S = Nexp

1 /N1 P/Po

Growth/Decay Rate Constants

Detailed balance:

n-1 Nn-1N1= n Nn

from Monte Carlo:ln[Qn/(Qn-1 Q1 n)]= ln[Nn/(Nn-1N1)]

= ln(n-1 /n) = - fn

Monte Carlo Simulations

Ensemble A: (n -1) cluster plus monomer

probe interactions turned off

Ensemble B: n cluster with normal probe interactions

Calculate fn = [Fn – Fn-1 ]/kT

LJ n-cluster Free Energy Differences

0

2

4

6

8

10

12

14

16

18

0 0.2 0.4 0.6 0.8 1n-1/3

-f n

T* = 0.50 60K)

T* = 0.32 (40K)

T* = 0.42 (50K)

LJ n-cluster Free Energy Differences

0

2

4

6

8

0 0.2 0.4 0.6 0.8 1

n-1/3

- f

n / [

Tc

/T-1

]

T* = 0.50 60K)

T* = 0.32 (40K)

T* = 0.42 (50K)

Scaling of free energy differences for small Lennard-Jones clusters

LJ n-cluster Free Energy Differences

0

2

4

6

8

10

12

14

16

18

0 0.2 0.4 0.6 0.8 1n-1/3

-f n

T* = 0.50 60K)

T* = 0.32 (40K)

T* = 0.42 (50K)

LJ n-cluster Free Energy Differences

0

2

4

6

8

0 0.2 0.4 0.6 0.8 1

n-1/3

- f

n / [T

c /T

-1]

T* = 0.50 60K)

T* = 0.32 (40K)

T* = 0.42 (50K)

LJ (argon) nucleation rates

0

2

4

6

8

10

12

0 2 4 6 8 10

lnS

log

(J/s

-1cm

-3)

T* = 0.34 (40K)

T* = 0.42 (50K)

T* = 0.50 (60K)

LJ (argon) nucleation rates

0

2

4

6

8

10

12

0 2 4 6 8 10

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

log

(J/s

-1c

m-3

)

T* = 0.34 (40K)

T* = 0.42 (50K)

T* = 0.50 (60K)

Comments & Conclusions

• Experimental data J (lnS/[Tc/T-1]3/2).

• Source of scaling?

• Monte Carlo LJ small cluster simulations scaled energies of formation.

• Scaling appears to emerge from [Tc/T-1] dependence of the fn .

LJ (argon) nucleation rates

0

2

4

6

8

10

12

0 2 4 6 8 10

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

log(

J/s

-1cm

-3)

T* = 0.34 (40K)

T* = 0.42 (50K)

T* = 0.50 (60K)

LJ (argon) nucleation rates

0

2

4

6

8

10

12

0 2 4 6 8 10

lnS

log(

J/s

-1cm

-3)

T* = 0.34 (40K)

T* = 0.42 (50K)

T* = 0.50 (60K)

LJ n-cluster Free Energy Differences

Scaled: Tc =1.313 (157K Argon)

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1 1.2

n-1/3

-W

n/ (T

c/T

-1)

T*= 0.50 (60K)

T*= 0.32 (40K)

T* = 0.42 (50K)

Classical Model

Model Lennard-Jones System Law of mass action

dilute vapor system with volume, V;

non-interacting mixture of ideal gases;

each n-cluster size is ideal gas of Nn particles;

full atom-atom LJ interaction potential;

separable classical Hamiltonian

Study of Scaling in LJ System

calculate rate constants for growth and decay of model Lennard-Jones clusters at three temperatures; determine model nucleation rates, J, from kinetic nucleation rate formalism; compare logJ vs lnS and

logJ vs lnS/[Tc/T-1]3/2

Law of Mass Action

Nn/[Nn-1N1] = Q(n)/[Q(n-1)Q(1)n]

Q(n) = n-cluster canonical

configurational

partition function

The nucleation rate can be calculated for a range of

supersaturation ratios, S.

1/J = n=1,M 1/Jn ; M large

Jn = (n) (N1S)2j=2,n [N1S (j-1)/(j)]

S = N1exp/N1

Free Energy Differences

- f(n) = ln [Q(n)/(Q(n-1)Q(1))]calculated

= ln [ (ρoliq/ρo

vap) (j-1)/(j) ]

Use Monte Carlo Bennett technique.

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

0 10 20 30 40 50

- lo

g [

J

/ 10 2

6 c

m-3

s-1 ]

10-

10

30

50 Model LJ System

T* = 0.335 T* = 0.418 T* = 0.503

0

10

20

30

40

50

60

70

80

90

100

0 2 4 6 8 10 12

colnS/[Tc/T-1]3/2

Lo

g(J

/J o)

T= 60K scaled

T= 50K scaled

T= 40K scaled

0

10

20

30

40

50

60

70

80

90

100

0 5 10

lnS

Lo

g(J

/Jo)

T = 60K

T = 50K

T = 40K

n-1/3

0.0 0.5 1.0

- f

n(n

) / [

Tc

/ T

- 1

]

0

1

2

3

4

5

6

7

8Lennard-Jones clusters

T*c = 1.31 T* = 0.335 T* = 0.419 T* = 0.503

192 20 6 2 n

Classical Nucleation Rate

2

liq

3

liq

22/12

oclassical

Sln

kT3

16exp

S

m

2

kT

PJ

(T) a – bT is the bulk liquid surface tension ;

Scaled Nucleation Rate at T << TcB. N. Hale, Phys. Rev A 33, 4156 (1986); J. Chem. Phys. 122, 204509 (2005)

2

3

c3

scaled,0scaledSln

1TT

3

16expJJ

J0,scaled [thermal (Tc)] -3 s-1

“scaled supersaturation” lnS/[Tc/T-1]3/2

Toluene (C7H8) nucleation data of Schmitt et al plotted

vs. scaled supersaturation, Co = [Tc /240-1]3/2 ; Tc = 591.8K

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

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 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)

Nonane (C9H20) nucleation data of Adams et al. plotted

vs. scaled supersaturation ; Co = [Tc/240-1]3/2 ; Tc = 594.6K

lnS

2 3 4 5

log(

J / c

m-3

s-1)

1

2

3

4

5

6

259K

217K

233K

Jexp (O) Jscaled (+)

Adams et al. nonane data a)

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

2 3 4 5

log(

J / c

m-3

s-1)

1

2

3

4

5

6

259K

217K

233K

Jexp (O) Jscaled (+)

Adams et al. nonane data b)

23.1 [Tc/T -1]3/ (lnS)2

0 10 20 30

- lo

g [

J /

10 2

6 c

m-3

s-1 ]

0

20

D2O, H2O

Wyslouzil et al.

H2O: Miller et al.

H2O: Wolk and Strey

Missing terms in the classical work of formation?

?..

Sln

kT3

16exp

S

m

2

kT

P2

liq

3

liq

22/12

oclassicalJ

2

3

c3

scaled,0scaledSln

1TT

3

16expJJ

Monte Carlo Helmholtz free energy differences for small water clusters: f(n) =[F(n)-F(n-1)]/kT

B.N. Hale and D. J. DiMattio, J. Phys. Chem. B 108, 19780 (2004)

n-1/3

0.0 0.5 1.0

- f

c(n

) / [

Tc /

T - 1

]

0

2

4

6

8

10

12H2O TIP4P clusters

Tc = 647 K exp. values 260 K

280 K300 K

192 20 6 2 n

Nucleation rate via Monte Carlo

Calculation of Nucleation rate from Monte Carlo -f(n) :

Jn = flux · Nn* Monte Carlo

= [N1v1 4rn2 ] · N1 exp 2,n(-f(n´) – ln[liq/1o]+lnS)

J -1 = [n Jn ]-1

The steady-state nucleation rate summation procedure requires no determination of n* as long as one sums over a sufficiently large number of n values.

Monte Carlo TIP4P nucleation rate resultsfor experimental water data points (Si,Ti)

log ( JMCDS TIP4P x 10-4 / cm-3 s-1 )

0 2 4 6 8 10 12

log ( J

/ cm

-3 s

-1 )

0

2

4

6

8

10

12

Wolk and Strey

Miller et al.

23.1 [Tc/T -1]3/ (lnS)2

0 10 20 30

- lo

g [

J /

10 2

6 c

m-3

s-1 ]

0

20

Wyslouzil MC TIP4P

Vehkamaki Hale, DiMattio

MD TIP4P: Yasuoka et al. T = 350K, S = 7.3

Miller et al.

Wolk and Strey

Comments & Conclusions

• Experimental data indicate that Jexp is a function of lnS/[Tc/T-1]3/2

• A “first principles” derivation of this scaling effect is not known;

• Monte Carlo simulations of f(n) for TIP4P water clusters show evidence of scaling;

• Temperature dependence in pre-factor of classical model can be partially cancelled when energy of formation is calculated from a discrete sum of f(n) over small cluster sizes.

• Can this be cast into more general formalism?