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Computer Simulations, Scaling and the Prediction 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
(supersaturated) 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);
Estimating the nucleation rate, J, from the molecular dynamics simulation at T = 80.3K. Supersaturation
ratio = P/Po = 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-
vol. = ( 60 x 60 x 60) 3;
Nucleation is a non-equilibrium process!
●There is no “first principles” theory from which to determine the nucleation rate. ● Most models attempt to treat nucleation as the decay of a near-equilibrium metastable (supersaturated) state.
● 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
(vapor-to-liquid)
Jclassical = [N1 v 4rn*2/3 ] · Nn*
= [Monomer flux] · [# Critical Clusters/Vol.]
Nn / N1 = exp [–Work of formation / kT] Work of formation of cluster from vapor:
W(n) = 4 rn2 - n kT ln S
S = P/Po
Estimating Nn
n* = critical sized cluster
has equal probability of growing or decaying
n* = critical sized cluster
at n = n*: dW(n) / dn = 0 Let
W(n) = An2/3 -nlnS
where A = [36]1/3 liq-2/3 /kT
liq= liquid number density
Volume / Surface in W(n*)
d/dn [ An2/3 - nlnS]n* = 0
(2/3)A n*-1/3 = lnS
n* = [2A/ 3lnS]3
W(n*) /kT = ½ n* lnS
= [16/3] [/(liq2/3 kT) ]3 / [lnS]2
liq= liquid number density
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.
1
T
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 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 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 Simulations
Ensemble A: (n -1) cluster plus monomerprobe interactions turned off
Ensemble B: n cluster withprobe interactions
normal
Calculate f(n) =[F(n)-F(n-1)]/kT
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?
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