Scaled Nucleation in Lennard-Jones System Barbara Hale and Tom Mahler Physics Department Missouri...
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Transcript of Scaled Nucleation in Lennard-Jones System Barbara Hale and Tom Mahler Physics Department Missouri...
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?