Computer Simulations,
Nucleation Rate Predictions and
Scaling
Barbara Hale
Physics Department and Cloud and Aerosol
Sciences Laboratory,
University of Missouri – Rolla
Rolla, MO 65401 USA
Computer simulation techniques
and predictions of vapor to liquid
nucleation rates
• Molecular Dynamics: time dependent
simulation of nucleation process in a (usually
dense) vapor.
• Monte Carlo: statistical mechanical
calculations of small n-cluster free energies of
formation from the vapor , Gn
Monte Carlo Simulations
Randomly generate molecular equilibrium configurations

use “importance”, sampling in statistical mechanical ensemble:
U = Ufinal –Uinitial = change in potential energy
Canonical (N,T,V)  exp -[ U ] /kT
Gibbs
(N,T,P)  exp –[ U +PV] /kT
Grand Can. (,T,V)  exp –[ U -  N ] /kT
Metropolis method:
exp[ –U / kT] > Random #
;
 accept
Smart Monte Carlo: umbrella sampling, …
0<R<1
Nucleation rate via Monte Carlo

=
Calculate: < Gn > = < Gn – nG1 >
free energy of formation (schemes for doing this depend on n-cluster
definition, interaction potential and simulation efficiency….
must calculate free energy differences.)
 Nn/N1 = exp - < Gn>/kT
= n-cluster size distribution
or

Simulate: Pn = <Nn/N>
(constrained) system

in “larger” supersaturated
Gn = - kT ln <Nn/N>
Nucleation rate via Monte Carlo
From simulation:
Jn = monomer flux ·N1 e -G(n)/kT
Predict Nucleation rate, J:
•
•
J  [N1v1 4rn*2 ]· N1
J
e -G(n*)/kT
( often a classical model form is used)
-1
= [n Jn' ]-1
(steady-state nucleation rate summation)
Computer simulation study of gas-liquid nucleation
in a Lennard-Jones system.
P. R. ten Wolde and D. Frenkel
J. Chem. Phys. 109, 9901 (1998)
We report a computer-simulation study of homogeneous gasliquid nucleation in a Lennard-Jones system. Using umbrella
sampling, we compute the free energy of a cluster as a function of
its size. A thermodynamic integration scheme is employed to
determine the height of the nucleation barrier as a function of
supersaturation. Our simulations illustrate that the mechanical and
the thermodynamical surfaces of tension and surface tension differ
significantly. In particular, we show that the mechanical definition
of the surface tension cannot be used to compute this barrier height.
We find that the relations recently proposed by McGraw and
Laaksonen J. Chem. Phys. 106, 5284 (1997) for the height of the
barrier and for the size of the critical nucleus are obeyed.
Numerical calculation of the rate of homogeneousgas–liquid
nucleation in a Lennard-Jones system
P. R. ten Wolde and D. Frenkel
J. Chem. Phys. 110, 1591 (1999)
We report a computer-simulation study of the absolute
rate of homogeneous gas–liquid nucleation in a LennardJones system. The height of the barrier has been computed
using umbrella sampling, whereas the kinetic prefactor is
calculated using molecular dynamics simulations. The
simulations show that the nucleation process is highly
diffusive. We find that the kinetic prefactor is a factor of 10
larger than predicted by classical nucleation theory.
P. R. ten Wolde and D. Frenkel
J. Chem. Phys. 110, 1591 (1999)
•
•
•
•
•
(N,P,T) Monte Carlo, N = 864 particles
S = 1.53;
G/kT = 59.4
T = 80.9K = 0.741 ;
JMC/MD = 4.5 x 105 cm-3sec-1
Tc = 130K = 1.085 
(LJ truncated 2.5, shifted)
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 in supersaturated system.
 Measure J  rate at which clusters form
Molecular dynamics of homogeneous nucleation in
the vapor phase. I. Lennard-Jones fluid
K. Yasuoka and M. Matsumoto
J. Chem. Phys. 109, 8451 (1998)
Molecular dynamics computer simulation was carried out to
investigate the dynamics of vapor phase homogeneous nucleation at
the triple point temperature under supersaturation ratio 6.8 for a
Lennard-Jones fluid. To control the system temperature, the 5000
target particles were mixed with 5000 soft-core carrier gas particles.
The observed nucleation rate is seven orders of magnitude larger
than prediction of a classical nucleation theory. The kinetically
defined critical nucleus size, at which the growth and decay rates
are balanced, is 30–40, as large as the thermodynamically defined
value of 25.4 estimated with the classical theory. From the cluster
size distribution in the steady state region, the free energy of cluster
formation is estimated, which diminishes the difference between the
theoretical prediction and the simulational result concerning the
nucleation rate.
K. Yasuoka and M. Matsumoto, LJ MD
J. Chem. Phys. 109, 8451 (1998) =2.15ps
K. Yasuoka and M. Matsumoto, LJ MD
J. Chem. Phys. 109, 8451 (1998)
=2.15ps
K. Yasuoka and M. Matsumoto
J. Chem. Phys. 109, 8463 (1998)
MD Lennard-Jones
• S = 6.8 (monomer depletion occurs)
• T = 80.3K = 0.67 
• JMD  1027 cm-3sec-1  107Jclassical LJ
• Tc LJ = 161.7K = 1.35 
• LJ surface tension: LJ > experiment
• LJ potentials truncated at 12;
( rcutoff < 5 can alter Tc and LJ )
Molecular dynamics of homogeneous
nucleation in the vapor phase. II. Water.
K. Yasuoka and M. Matsumoto
J. Chem. Phys. 109, 8463 (1998)
Homogeneous nucleation process in the vapor phase of water
is investigated with a molecular dynamics computer simulation at
350 K under supersaturation ratio 7.3. Using a method similar to
Lennard-Jones fluid (Part I), the nucleation rate is three orders of
magnitude smaller than prediction of a classical nucleation theory.
The kinetically defined critical nucleus size is 30–45, much larger
than the thermodynamically defined value of 1.0 estimated with
the classical theory. Free energy of cluster formation is estimated
from the cluster size distribution in steady state time region. The
predicted nucleation rate with this free energy agrees with the
simulation result. It is concluded that considering the cluster size
dependence of surface tension is very important.
K. Yasuoka and M. Matsumoto
J. Chem. Phys. 109, 8463 (1998)
MD TIP4P water
• S = 7.3 (monomer depletion occurs)
• T = 350K;
TIP4P =39.29 erg cm-2
• JMD  1027 cm-3sec-1  10-2 JClassical TIP4P
• Tc model = 563K ?
• TIP4P surface tension: TIP4P  0.65 experiment
Difficulties in comparing with
experimental data
 Simulation results depend on potential model.
The classical atom-atom interaction potentials can
have different surface tension, coexistence vapor
pressure, and critical temperature.
 Results can also depend on the definition of a
cluster and the simulation technique.
 How can the simulation results be evaluated
and compared with experiment?
Proposal: Use Scaled Supersaturation
B. N. Hale, Phys. Rev. A 33, 4156 (1986)
Scaled nucleation rate:
ln[Jscaled /1026 ]cgs  -(16/3) 3 [lnS/(Tc/T -1)3/2 ]-2
scaled supersaturation (similar to Binder’s):
lnSscaled = lnS/(Tc/T -1)3/2
  2 ( argon excess surface entropy/molecule)
  1.5 (water excess surface entropy/molecule)
Toluene (C7H8) nucleation data of Schmitt et al plotted
vs. scaled supersaturation, Co = [Tc /240-1]3/2 ; Tc = 591.8K
4
4
a)
Schmitt et al. toluene data
Schmitt et al. toluene data
3
log(J / cm-3s-1)
3
log(J / cm-3s-1)
b)
2
1
2
1
0
0
-1
-1
2
3
lnS
4
2
3
4
Co lnS/[Tc/T-1]3/2
Nonane (C9H20) nucleation data of Adams et al. plotted
vs. scaled supersaturation ; Co = [Tc/240-1]3/2 ; Tc = 594.6K
6
6
b)
Adams et al. nonane data
a)
Adams et al. nonane data
5
log(J / cm-3s-1)
log(J / cm-3s-1)
5
4
3
4
3
2
2
1
1
2
3
4
lnS
5
2
3
4
Co lnS/[Tc/T-1]3/2
5
Water nucleation rate data of Wölk and Strey plotted vs.
lnS / [Tc/T-1]3/2 ; Co = [Tc/240-1]3/2 ; Tc = 647 K
b)
a)
260 K
240 K 230 K
220 K
log J /(cm-3sec-1)
Wolk and Strey H2O data
10
250 K
8
6
log [ J / cm-3 / sec-1 ]
10
8
6
4
4
1.8
2.0
2.2
2.4
2.6
lnS
2.8
3.0
3.2
1.8
2.0
2.2
2.4
2.6
2.8
Co lnS / [Tc/T -1]3/2
3.0
3.2
Wolk and Strey H2O data
log [ J / cm-3 / sec-1 ]
10
8
6
4
1.8
2.0
2.2
2.4
2.6
2.8
Co lnS / [Tc/T -1]3/2
3.0
3.2
Example 1: using scaled supersaturation to
compare experimental data and computer
simulation predictions for Jwater
Plot
-log[J/1026]cgs units vs. Co' [lnS/(Tc/T -1)3/2 ]-2
Co' = 23.1 = -(16/3) 3 / ln(10) ;
Tc= 647 K ;
 = 1.47
Sexp or Smodel
- log [ J / 10 26 cm-3s-1 ]
20
H2O: Wolk and Strey
0
0
10
20
23.1 [Tc/T -1]3/ (lnS)2
30
- log [ J / 10 26 cm-3s-1 ]
H2O: Miller et al.
20
H2O: Wolk and Strey
0
0
10
20
23.1 [Tc/T -1]3/ (lnS)2
30
- log [ J / 10 26 cm-3s-1 ]
H2O: Miller et al.
20
H2O: Wolk and Strey
D2O, H2O
Wyslouzil et al.
0
0
10
20
23.1 [Tc/T -1]3/ (lnS)2
30
- log [ J / 10 26 cm-3s-1 ]
H2O: Miller et al.
20
H2O: Wolk and Strey
D2O, H2O
Wyslouzil et al.
0
MD TIP4P: Yasuoka et al.
T = 350K, S = 7.3
0
10
20
23.1 [Tc/T -1]3/ (lnS)2
30
- log [ J / 10 26 cm-3s-1 ]
H2O: Miller et al.
20
H2O: Wolk and Strey
MC TIP4P
Vehkamaki
D2O, H2O
Wyslouzil et al.
0
MD TIP4P: Yasuoka et al.
T = 350K, S = 7.3
0
10
20
23.1 [Tc/T -1]3/ (lnS)2
30
- log [ J / 10 26 cm-3s-1 ]
H2O: Miller et al.
20
H2O: Wolk and Strey
MC TIP4P
Hale, DiMattio
Vehkamaki
D2O, H2O
Wyslouzil et al.
0
MD TIP4P: Yasuoka et al.
T = 350K, S = 7.3
0
10
20
23.1 [Tc/T -1]3/ (lnS)2
30
- log [ J / 10 26 cm-3s-1 ]
Miller et al.
20
Wolk and Strey
Wyslouzil
0
MC TIP4P
Hale, DiMattio
Vehkamaki
MD TIP4P: Yasuoka et al.
T = 350K, S = 7.3
0
10
20
23.1 [Tc/T -1]3/ (lnS)2
30
Comments on water data and
predictions for J
• Predicted rates using TIP4P are about 4 orders
of magnitude too large, but appear to have
correct “scaled supersaturation” dependence.
• TIP4P critical temperature < 647K
• MD and MC show similar results.
• Significance (if any) to “shifted” scaled
supersaturation for TIP4P?
Example 2: using the scaled supersaturation to
compare experimental data and computer
simulation predictions for Jargon.
Plot
-log[J/1026]cgs units vs. Co' [lnS/(Tc/T -1)3/2 ]-2
Co' = 24.6 = -(16/3) 3 ;
 = 1.5
(Tc= 150 K, Sexp) for data;
(Tc model , Smodel ) for simulation results.
Argon experimental rates
Assume “onset” rates:
shock tube:  10102 cm-3 s-1
nozzle:
 1016 2 cm-3 s-1
Fast expansion chamber:  107 2 cm-3 s-1
+ Fladerer
Argon Rate Data / LJ Simulations
- log [ J / 10 26 cm-3s-1 ]
50
30
10
-10
0
10
20
30
40
24.6 [Tc/T -1]3/ (lnS)2
50
+ Fladerer
 Zahoransky
Argon Rate Data / LJ Simulations
- log [ J / 10 26 cm-3s-1 ]
50
30
10
-10
0
10
20
30
40
24.6 [Tc/T -1]3/ (lnS)2
50
+ Fladerer
 Zahoransky
Argon Rate Data / LJ Simulations
○ Stein
- log [ J / 10 26 cm-3s-1 ]
50
30
10
-10
0
10
20
30
40
24.6 [Tc/T -1]3/ (lnS)2
50
+ Fladerer
 Zahoransky
Argon Rate Data / LJ Simulations
○ Stein
- log [ J / 10 26 cm-3s-1 ]
50
 Matthew et al.
30
10
-10
0
10
20
30
40
24.6 [Tc/T -1]3/ (lnS)2
50
+ Fladerer
 Zahoransky
Argon Rate Data / LJ Simulations
○ Stein
- log [ J / 10 26 cm-3s-1 ]
50
 Matthew et al.
X Wu et al.
30
10
-10
0
10
20
30
40
24.6 [Tc/T -1]3/ (lnS)2
50
+ Fladerer
 Zahoransky
Argon Rate Data / LJ Simulations
○ Stein
- log [ J / 10 26 cm-3s-1 ]
50
 Matthew et al.
X Wu et al.
Calculations:
●
30
10
-10
0
10
20
30
40
24.6 [Tc/T -1]3/ (lnS)2
50
ten Wolde
+ Fladerer
 Zahoransky
Argon Rate Data / LJ Simulations
○ Stein
- log [ J / 10 26 cm-3s-1 ]
50
 Matthew et al.
X Wu et al.
Calculations:
●
ten Wolde
■ Yasuoka and
Matsumoto
30
10
-10
0
10
20
30
40
24.6 [Tc/T -1]3/ (lnS)2
50
+ Fladerer
 Zahoransky
Argon Rate Data / LJ Simulations
○ Stein
- log [ J / 10 26 cm-3s-1 ]
50
 Matthew et al.
X Wu et al.
Calculations:
●
ten Wolde
■ Yasuoka and
Matsumoto
▼ Senger, Reiss,
et al.
30
10
-10
0
10
20
30
40
24.6 [Tc/T -1]3/ (lnS)2
50
+ Fladerer
 Zahoransky
Argon Rate Data / LJ Simulations
○ Stein
- log [ J / 10 26 cm-3s-1 ]
50
 Matthew et al.
X Wu et al.
Calculations:
●
ten Wolde
■ Yasuoka and
Matsumoto
▼ Senger, Reiss,
et al.
▲ Oh and Zeng
30
10
-10
0
10
20
30
40
24.6 [Tc/T -1]3/ (lnS)2
50
+ Fladerer
 Zahoransky
Argon Rate Data / LJ Simulations
○ Stein
- log [ J / 10 26 cm-3s-1 ]
50
 Matthew et al.
X Wu et al.
Calculations:
●
ten Wolde
■ Yasuoka and
Matsumoto
▼ Senger, Reiss,
et al.
▲ Oh and Zeng
 Chen et al.
30
10
-10
0
10
20
30
40
24.6 [Tc/T -1]3/ (lnS)2
50
Argon Rate Data
+ Fladerer
 Zahoransky
LJ Simulations
○ Stein
- log [ J / 10 26 cm-3s-1 ]
50
 Matthew et al.
X Wu et al.
Calculations:
●
ten Wolde
■ Yasuoka and
Matsumoto
▼ Senger, Reiss,
et al.
▲ Oh and Zeng
Chen et al.
□ Hale
30
10
-10
0
10
20
30
40
24.6 [Tc/T -1]3/ (lnS)2
50
Argon Data /
+ Fladerer
 Zahoransky
LJ Simulations
○ Stein
- log [ J / 10 26 cm-3s-1 ]
50
 Matthew et al.
X Wu et al.
Calculations:
●
ten Wolde
■ Yasuoka and
Matsumoto
▼ Senger, Reiss,
et al.
▲ Oh and Zeng
Chen et al.
□ Hale, Kiefer
□ CNT
30
10
-10
0
10
20
30
40
24.6 [Tc/T -1]3/ (lnS)2
50
Argon Data /
LJ Simulations
50
- log [ J / 10 26 cm-3s-1 ]
+ Fladerer
 Zahoransky
 Stein
 Matthew et al.
X Wu et al.
Calculations:
 = 1.5
●
ten Wolde
■ Yasuoka and
Matsumoto
▼ Senger, Reiss,
et al.
▲ Oh and Zeng
Chen et al.
□ Hale, Kiefer
□ CNT
--- Scaled model,
 = 2.0
--- Scaled model,
 = 1.5
30
10
-10
0
10
20
30
40
24.6 [Tc/T -1]3/ (lnS)2
50
Comments on argon data and
predictions for J
• Limited experimental rate data for argon; no rate dependence on
temperature; rates are estimated from “onset” assumptions.
• Lennard-Jones MC and MD simulations at small scaled
supersaturations (high nucleation rates) appear to be consistent.
• Monte Carlo LJ simulation and CNT results at higher scaled
supersaturations (lower nucleation rates) are about 10-20 smaller
than experimental “onset” rates.
• ten Wolde’s LJ MC/MD predicted rate (105) is closer to Fladerer’s
experimental rate (1072) ; Tc model  130K and surface tension is
smaller than experimental value.