the scaling of nucleation rates barbara hale physics department and cloud and aerosol sciences...
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![Page 1: The Scaling of Nucleation Rates Barbara Hale Physics Department and Cloud and Aerosol Sciences Laboratory University of Missouri – Rolla Rolla, MO 65401](https://reader036.vdocument.in/reader036/viewer/2022062409/56649d585503460f94a36d71/html5/thumbnails/1.jpg)
The Scaling of Nucleation Rates
Barbara HalePhysics Department and
Cloud and Aerosol Sciences LaboratoryUniversity of Missouri – Rolla
Rolla, MO 65401 USA
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Nucleation : formation of embryos of the new phase from the metastable parent phase
K. Yasuoka and M.
Matsumoto, J. Chem. Phys. 109,
8451 (1998)
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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
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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-
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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.
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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)
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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]
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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
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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 ;
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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.
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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.
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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)
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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
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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
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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)
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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)
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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
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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
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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
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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
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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.
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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.
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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
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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?
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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