<s>
Classical	B-Algorithm
nucleation	I-Algorithm
theory	I-Algorithm
(	O
CNT	O
)	O
is	O
the	O
most	O
common	O
theoretical	O
model	O
used	O
to	O
quantitatively	O
study	O
the	O
kinetics	O
of	O
nucleation	B-Algorithm
.	O
</s>
<s>
Nucleation	B-Algorithm
is	O
the	O
first	O
step	O
in	O
the	O
spontaneous	O
formation	O
of	O
a	O
new	O
thermodynamic	O
phase	O
or	O
a	O
new	O
structure	O
,	O
starting	O
from	O
a	O
state	O
of	O
metastability	O
.	O
</s>
<s>
The	O
kinetics	O
of	O
formation	O
of	O
the	O
new	O
phase	O
is	O
frequently	O
dominated	O
by	O
nucleation	B-Algorithm
,	O
such	O
that	O
the	O
time	O
to	O
nucleate	B-Algorithm
determines	O
how	O
long	O
it	O
will	O
take	O
for	O
the	O
new	O
phase	O
to	O
appear	O
.	O
</s>
<s>
The	O
time	O
to	O
nucleate	B-Algorithm
can	O
vary	O
by	O
orders	O
of	O
magnitude	O
,	O
from	O
negligible	O
to	O
exceedingly	O
large	O
,	O
far	O
beyond	O
reach	O
of	O
experimental	O
timescales	O
.	O
</s>
<s>
One	O
of	O
the	O
key	O
achievements	O
of	O
classical	B-Algorithm
nucleation	I-Algorithm
theory	I-Algorithm
is	O
to	O
explain	O
and	O
quantify	O
this	O
immense	O
variation	O
.	O
</s>
<s>
The	O
central	O
result	O
of	O
classical	B-Algorithm
nucleation	I-Algorithm
theory	I-Algorithm
is	O
a	O
prediction	O
for	O
the	O
rate	O
of	O
nucleation	B-Algorithm
,	O
in	O
units	O
of	O
(	O
number	O
of	O
events	O
)	O
/	O
( volume·time	O
)	O
.	O
</s>
<s>
For	O
instance	O
,	O
a	O
rate	O
in	O
a	O
supersaturated	O
vapor	O
would	O
correspond	O
to	O
an	O
average	O
of	O
1000	O
droplets	O
nucleating	B-Algorithm
in	O
a	O
volume	O
of	O
1	O
cubic	O
meter	O
in	O
1	O
second	O
.	O
</s>
<s>
is	O
the	O
free	O
energy	O
cost	O
of	O
the	O
nucleus	O
at	O
the	O
top	O
of	O
the	O
nucleation	B-Algorithm
barrier	O
,	O
and	O
is	O
the	O
average	O
thermal	O
energy	O
with	O
the	O
absolute	O
temperature	O
and	O
the	O
Boltzmann	O
constant	O
.	O
</s>
<s>
is	O
the	O
number	O
of	O
nucleation	B-Algorithm
sites	I-Algorithm
.	O
</s>
<s>
This	O
expression	O
for	O
the	O
rate	O
can	O
be	O
thought	O
of	O
as	O
a	O
product	O
of	O
two	O
factors	O
:	O
the	O
first	O
,	O
,	O
is	O
the	O
number	O
of	O
nucleation	B-Algorithm
sites	I-Algorithm
multiplied	O
by	O
the	O
probability	O
that	O
a	O
nucleus	O
of	O
critical	O
size	O
has	O
grown	O
around	O
it	O
.	O
</s>
<s>
It	O
can	O
be	O
interpreted	O
as	O
the	O
average	O
,	O
instantaneous	O
number	O
of	O
nuclei	O
at	O
the	O
top	O
of	O
the	O
nucleation	B-Algorithm
barrier	O
.	O
</s>
<s>
So	O
if	O
is	O
large	O
and	O
positive	O
the	O
probability	O
of	O
forming	O
a	O
nucleus	O
is	O
very	O
low	O
and	O
nucleation	B-Algorithm
will	O
be	O
slow	O
.	O
</s>
<s>
have	O
used	O
computer	O
simulation	O
to	O
estimate	O
all	O
the	O
quantities	O
in	O
the	O
above	O
equation	O
,	O
for	O
the	O
nucleation	B-Algorithm
of	O
ice	O
in	O
liquid	O
water	O
.	O
</s>
<s>
At	O
a	O
supercooling	O
of	O
19.5	O
°C	O
,	O
i.e.	O
,	O
19.5	O
°C	O
below	O
the	O
freezing	O
point	O
of	O
water	O
in	O
their	O
model	O
,	O
they	O
estimate	O
a	O
free	O
energy	O
barrier	O
to	O
nucleation	B-Algorithm
of	O
ice	O
of	O
.	O
</s>
<s>
This	O
is	O
a	O
rate	O
of	O
homogeneous	O
nucleation	B-Algorithm
estimated	O
for	O
a	O
model	O
of	O
water	O
,	O
not	O
real	O
water	O
—	O
in	O
experiments	O
one	O
cannot	O
grow	O
nuclei	O
of	O
water	O
and	O
so	O
cannot	O
directly	O
determine	O
the	O
values	O
of	O
the	O
barrier	O
,	O
or	O
the	O
dynamic	O
parameters	O
such	O
as	O
,	O
for	O
real	O
water	O
.	O
</s>
<s>
However	O
,	O
it	O
may	O
be	O
that	O
indeed	O
the	O
homogeneous	O
nucleation	B-Algorithm
of	O
ice	O
at	O
temperatures	O
near	O
-20	O
°C	O
and	O
above	O
is	O
extremely	O
slow	O
and	O
so	O
that	O
whenever	O
water	O
freezes	O
at	O
temperatures	O
of	O
-20	O
°C	O
and	O
above	O
this	O
is	O
due	O
to	O
heterogeneous	O
nucleation	B-Algorithm
,	O
i.e.	O
,	O
the	O
ice	O
nucleates	B-Algorithm
in	O
contact	O
with	O
a	O
surface	O
.	O
</s>
<s>
Homogeneous	O
nucleation	B-Algorithm
is	O
much	O
rarer	O
than	O
heterogeneous	O
nucleation	B-Algorithm
.	O
</s>
<s>
However	O
,	O
homogeneous	O
nucleation	B-Algorithm
is	O
simpler	O
and	O
easier	O
to	O
understand	O
than	O
heterogeneous	O
nucleation	B-Algorithm
,	O
so	O
the	O
easiest	O
way	O
to	O
understand	O
heterogeneous	O
nucleation	B-Algorithm
is	O
to	O
start	O
with	O
homogeneous	O
nucleation	B-Algorithm
.	O
</s>
<s>
So	O
we	O
will	O
outline	O
the	O
CNT	O
calculation	O
for	O
the	O
homogeneous	O
nucleation	B-Algorithm
barrier	O
.	O
</s>
<s>
To	O
understand	O
if	O
nucleation	B-Algorithm
is	O
fast	O
or	O
slow	O
,	O
,	O
the	O
Gibbs	O
free	O
energy	O
change	O
as	O
a	O
function	O
of	O
the	O
size	O
of	O
the	O
nucleus	O
,	O
needs	O
to	O
be	O
calculated	O
.	O
</s>
<s>
is	O
the	O
difference	O
in	O
free	O
energy	O
per	O
unit	O
of	O
volume	O
between	O
the	O
phase	O
that	O
nucleates	B-Algorithm
and	O
the	O
thermodynamic	O
phase	O
nucleation	B-Algorithm
is	O
occurring	O
in	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
water	O
is	O
nucleating	B-Algorithm
in	O
supersaturated	O
air	O
,	O
then	O
is	O
the	O
free	O
energy	O
per	O
unit	O
of	O
volume	O
of	O
water	O
minus	O
that	O
of	O
supersaturated	O
air	O
at	O
the	O
same	O
pressure	O
.	O
</s>
<s>
As	O
nucleation	B-Algorithm
only	O
occurs	O
when	O
the	O
air	O
is	O
supersaturated	O
,	O
is	O
always	O
negative	O
.	O
</s>
<s>
The	O
rate	O
at	O
which	O
nucleation	B-Algorithm
occurs	O
is	O
then	O
limited	O
by	O
,	O
i.e.	O
,	O
determined	O
by	O
the	O
probability	O
,	O
of	O
forming	O
the	O
critical	O
nucleus	O
.	O
</s>
<s>
This	O
explains	O
why	O
in	O
homogeneous	O
nucleation	B-Algorithm
usually	O
only	O
spherical	O
nuclei	O
are	O
taken	O
into	O
account	O
.	O
</s>
<s>
Unlike	O
homogeneous	O
nucleation	B-Algorithm
,	O
heterogeneous	O
nucleation	B-Algorithm
occurs	O
on	O
a	O
surface	O
or	O
impurity	O
.	O
</s>
<s>
It	O
is	O
much	O
more	O
common	O
than	O
homogeneous	O
nucleation	B-Algorithm
.	O
</s>
<s>
This	O
is	O
because	O
the	O
nucleation	B-Algorithm
barrier	O
for	O
heterogeneous	O
nucleation	B-Algorithm
is	O
much	O
lower	O
than	O
for	O
homogeneous	O
nucleation	B-Algorithm
.	O
</s>
<s>
To	O
see	O
this	O
,	O
note	O
that	O
the	O
nucleation	B-Algorithm
barrier	O
is	O
determined	O
by	O
the	O
positive	O
term	O
in	O
the	O
free	O
energy	O
,	O
which	O
is	O
proportional	O
to	O
the	O
total	O
exposed	O
surface	O
area	O
of	O
a	O
nucleus	O
.	O
</s>
<s>
For	O
homogeneous	O
nucleation	B-Algorithm
the	O
surface	O
area	O
is	O
simply	O
that	O
of	O
a	O
sphere	O
.	O
</s>
<s>
For	O
heterogeneous	O
nucleation	B-Algorithm
,	O
however	O
,	O
the	O
surface	O
area	O
is	O
smaller	O
since	O
part	O
of	O
the	O
nucleus	O
boundary	O
is	O
accommodated	O
by	O
the	O
surface	O
or	O
impurity	O
onto	O
which	O
it	O
is	O
nucleating	B-Algorithm
.	O
</s>
<s>
The	O
free	O
energy	O
needed	O
for	O
heterogeneous	O
nucleation	B-Algorithm
,	O
,	O
is	O
equal	O
to	O
the	O
product	O
of	O
homogeneous	O
nucleation	B-Algorithm
,	O
,	O
and	O
a	O
function	O
of	O
the	O
contact	O
angle	O
,	O
:.The	O
schematic	O
to	O
the	O
right	O
illustrates	O
the	O
decrease	O
in	O
the	O
exposed	O
surface	O
area	O
of	O
the	O
droplet	O
as	O
the	O
contact	O
angle	O
decreases	O
.	O
</s>
<s>
In	O
practice	O
,	O
this	O
means	O
that	O
nucleation	B-Algorithm
will	O
tend	O
to	O
occur	O
on	O
surface	O
imperfections	O
.	O
</s>
<s>
Classical	B-Algorithm
nucleation	I-Algorithm
theory	I-Algorithm
makes	O
a	O
number	O
of	O
assumptions	O
which	O
limit	O
its	O
applicability	O
.	O
</s>
<s>
In	O
addition	O
,	O
the	O
classical	O
theory	O
places	O
restrictions	O
on	O
the	O
kinetic	O
pathways	O
by	O
which	O
nucleation	B-Algorithm
occurs	O
,	O
assuming	O
clusters	O
grow	O
or	O
shrink	O
only	O
by	O
single	O
particle	O
adsorption/emission	O
.	O
</s>
<s>
One	O
difficulty	O
is	O
the	O
exquisite	O
sensitivity	O
of	O
the	O
nucleation	B-Algorithm
rate	O
to	O
the	O
free	O
energy	O
:	O
even	O
small	O
discrepancies	O
in	O
the	O
microscopic	O
parameters	O
can	O
lead	O
to	O
enormous	O
changes	O
in	O
the	O
predicted	O
nucleation	B-Algorithm
rate	O
.	O
</s>
<s>
For	O
simple	O
model	O
systems	O
,	O
modern	O
computers	O
are	O
powerful	O
enough	O
to	O
calculate	O
exact	O
nucleation	B-Algorithm
rates	O
numerically	O
.	O
</s>
<s>
An	O
example	O
is	O
the	O
nucleation	B-Algorithm
of	O
the	O
crystal	O
phase	O
in	O
a	O
system	O
of	O
hard	O
spheres	O
,	O
which	O
is	O
a	O
simple	O
model	O
of	O
colloids	O
consisting	O
of	O
perfectly	O
hard	O
spheres	O
in	O
thermal	O
motion	O
.	O
</s>
<s>
For	O
example	O
,	O
in	O
the	O
context	O
of	O
vapor	O
to	O
liquid	O
nucleation	B-Algorithm
,	O
the	O
CNT	O
predictions	O
for	O
the	O
nucleation	B-Algorithm
rate	O
are	O
incorrect	O
by	O
several	O
orders	O
of	O
magnitude	O
on	O
an	O
absolute	O
scale	O
—	O
that	O
is	O
,	O
without	O
renormalizing	O
with	O
respect	O
to	O
experimental	O
data	O
.	O
</s>
<s>
computationally	O
explored	O
the	O
nucleation	B-Algorithm
of	O
small	O
water	O
clusters	O
using	O
a	O
classical	O
water	O
model	O
.	O
</s>
<s>
It	O
was	O
found	O
that	O
CNT	O
could	O
describe	O
the	O
nucleation	B-Algorithm
of	O
clusters	O
of	O
8-50	O
water	O
molecules	O
well	O
,	O
but	O
failed	O
to	O
describe	O
smaller	O
clusters	O
.	O
</s>
