Nucleation
Don H. Rasmussen
Box 5705
Clarkson University
rasmu@clarkson.edu
Homogeneous Nucleation

Fluctuations in composition and structure which are small in
extent but large in degree result in small new phase nuclei
which are in local equilibrium but unstable to growth in an
undercooled or supersaturated parent phase.
How does Homogeneous Nucleation
Occur?





Stable clusters form when their formation decreases total free
energy.
Growth of small clusters is limited because new particle surface
costs more free energy than the bulk free energy reduction. Only
large clusters are stable.
Clusters grow and decay by monomer addition/evaporation and
there is in a metastable cluster size distribution.
The larger the supersaturation or undercooling, the greater the
number and maximum size of the existing clusters and the
smaller the necessary critical cluster size for continued growth.
For clusters just larger than critical, the growth rate increases
along with the size in an autocatalytic fashion.
Numerical Model of Homogeneous
Nucleation

Nucleation Rate, J, is the product of
 q,
the net probability of an atom jumping
across interface and into the critical cluster
(per unit surface area)
 t, jump frequency of monomer is fluid
 Ac, the surface area of the critical cluster
 nc, the concentration of critical clusters per
unit volume
J  q t Ac nc
q is the net rate of diffusion across the surface of a cluster
dC
 ci  co 
q  D
  D

dx
 l

where Do is the diffusion coefficient in the liquid.
  Qd 
D  Do exp

 RT 
and ci and co are the concentration of crystallizing atoms on the
two sides of the interface of thickness, l.
The surface area of a spherical cluster is
Ac  4 r
2
The concentration of critical clusters is
  G f 
nc  no exp 

 RT 
Nucleation Rate per Unit Volume, J
 G f 
(co  ci )
 Qd 
J  qt Acnc  no t Ac Do
exp 
exp
 kT 

l
kT




The pre-exponential factors are almost constant and
approximately 1035 nuclei/cm3sec.
  G f   nuclei 
  Qd 

J  10 exp
exp
  3

 kT 
 kT   cm sec 
35
Effect of Temperature on Bulk
Free Energy Change
Free energy

T
Gv  H  TS  H 1 
 TM
Gv L



S
GS
GL
T
T
TE
Temperature
Free Energy of a Cluster as a
Function of Size
4
G f (r )  4 r    r 3 Gv
3
2
9
2 10
9
2 10
9
1 10
dA( r )
F ( r )
0
VdP ( r )
9
1 10
9
2 10 2 109
0
0
20
40
60
r
80
100
100
Influence of increasing
Undercooling or Supersaturation
From 1 to 5 the supersaturation or undercooling increases which results
in a decrease in both the critical cluster size and the barrier to nucleation.
9
1 10
F 1 ( r )
8
5 10
0
F 2 ( r )
F 3 ( r )
F 4 ( r )
F 5 ( r )
8
5 10
9
1 10
9
1.5 10
9
2 10
0
20
40
60
r
80
100
Conditions for critical cluster or
nucleus
d [G f (r )]
dr
 0  8 r  4 r 2 Gv
 2
r 
Gv
*
16 
16  T
G 

2
3 Gv
3 H 2 T 2
3
*
3
2
M
Critical Cluster Size and Free Energy
Barrier versus Undercooling
Free Energy Barrier
Radius (nm)
100
50
0
0
50
Undercooling
100
5
1 10
4
5 10
0
0
50
Undercooling
100
Temperature Dependence of
Nucleation Rate
  G f   nuclei 
  Qd 

J  10 exp
exp
  3

 kT 
 kT   cm sec 
35
33
1 10
J(T)
32
1 10
31
1 10
30
1 10
0
100
T
200
Time Temperature Transformation
Curves
The delay time is related to the reciprocal of the nucleation rate and here the
delay time is plotted as a function of the undercooling.
0
T
200
1
10
100
Time (Seconds)
3
1 10
Heterogeneous Nucleation
Nucleation at the surface of an impurity
particle or on the walls of the container.
 “Catalysts for Nucleation” are surfaces
which significantly lower the barrier to
new phase formation.
 Heterogeneous nucleation occurs at low
undercooling and at high rates.

Nucleation on a substrate takes
less material
liquid
nl
sl
q nucleus
sn
substrate
Fraction of the critical cluster which
must form at any specific undercooling
Exposed Volume
1
2
f (q )  2  cos(q ) 1  cos(q ) 
4
1
0.5
0
0
2
4
Contact Angle (radians)
Free Energy of Formation of the Nucleus versus
Contact Angle at Fixed Undercooling
3
16


1
*
2
G 
(2  cos(q ) )(1  cos(q ))
2
3 Gv 4
Barrier Height
5
1.5 10
5
1 10
4
5 10
0
0
1
2
3
Contact An gle