The Scaling of Nucleation Rates

Barbara HalePhysics Department and

Cloud and Aerosol Sciences LaboratoryUniversity of Missouri – Rolla

Rolla, MO 65401 USA

Nucleation : formation of embryos of the new phase from the metastable parent phase

K. Yasuoka and M.

Matsumoto, J. Chem. Phys. 109,

8451 (1998)

Molecular dynamics of homogeneous nucleation in the vapor phase: Lennard-Jones fluid,

K. Yasuoka and M. Matsumoto, J. Chem. Phys. 109, 8451 (1998); = 2.15ps; vol. = ( 60 x 60 x 60) 3; T = 80.3 K; S = 6.8

Estimating the nucleation rate, J, from the molecular

dynamics simulation for (argon) LJ at T = 80.3K; S =6.8

time volume

] formed embryos phase liquid of # [ J

)s cm 10 ~(J

s cm 10

s)10 (2.5 10 cm) 10 3.4 (60

embryos 30

-13-22classical

-13-29

-1238-

Nucleation is generally treated theoretically as the decay of a

metastable state – a non-equilibrium process.

●There is no “first principles” theory from which to determine the nucleation rate. ● Most models attempt to predict nucleation rates using properties of near-equilibrium metastable states.

● The classical nucleation theory (CNT) model was first developed in 1926 by Volmer and Weber, and by Becker and Döring in 1935 …. following a proposal by Gibbs.

● CNT treats nucleation as a fluctuation phenomenon in which small embryos of the new phase overcome free energy barriers and grow irreversibly to macroscopic size.

Classical Nucleation Theory

n (S, T) = 1 exp[- ∆G(n) /kT ]; S = P/Po

( includes effect of clusters near n*)

∆G(n) = (n) – n1 (free energy of formation) = G(n)surface + n liq – n1

= 4rn2 - nkTln(P/Po)

Jclassical = [ 1 v 4rn*2] · n* (S, T)

= [Monomer flux] · [# Critical Clusters/Vol.] (vapor-to-liquid nucleation rate)

n* = critical sized cluster

equal probability of growing or decaying

at n = n*:

d/dn[ 4rn2- nkTln(P/Po)] = 0

d/dn [ An2/3 - nlnS] = 0

…………………………………………..

A = [36]1/3 /[liq2/3

kT ] ; S = P/Po

liq= n/[4 rn3/3]

Volume / Surface in ∆G(n*)

d/dn [ An2/3 - nlnS]n* = 0

(2/3)A n*-1/3 - lnS = 0

n* = [2A/ 3lnS]3

∆G(n*) /kT = (1/2) [2A/ 3lnS]3 lnS

∆G(n*) /kT = [16/3] [/(liq2/3 kT) ]3 / [lnS]2

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 ;

Homogeneous Nucleation rate data for water:classical nucleation rate model has wrong T dependence

log ( Jclassical / 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.

Motivation for Scaling J at T << Tc

The CNT nucleation rate depends exponentially on (T)3 / [ln (P/Po(T))]2 . To obtain a physically realistic T dependence of J, a good starting point is to require functional forms for (T) and Po(T) which reflect “universal” properties of surface tension and vapor pressure.

Scaling of the surface tension at T << Tc

Assume a scaled form for :

= o’ [Tc- T]

with =1 for simplicity. Many substances fit this form and the critical exponent (corresponding to ) is close to 1.

1T

T1

T

T

k

'

kTcc

3/2.liq

03/2

.liq

= excess surface entropy per molecule / k 2 for normal liquids

1.5 for substances with dipole moment(a law of corresponding states result; Eötvös 1869)

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

Water nucleation rate data of Wölk and Strey plotted vs. lnS / [Tc/T-1]3/2 ; Co = [Tc/240-1]3/2 ; Tc = 647.3 K

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

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

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)

Comparison of Jscaled with water data from

different experimental techniques: plot log[J/J0,scaled] vs.

J0,scaled [2mkTc/h2]3/2 s-1

1026 cm-3 s-1

for most materials (corresponding states)

2

3

c3

Sln

1T

T

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 nucleation rate energy 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 free energy differences, -f(n) :

Jn = [1v1 4rn

2 ]· 1 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?

Molecular Dynamics Simulations

Solve Newton’s equations,

mi d2ri/dt2 = Fi = -i j≠i U(rj-ri),

iteratively for all i=1,2… n atoms;

Quench the system to temperature, T, and

monitor cluster formation.

Measure J rate at which clusters form