<s>
In	O
thermodynamics	O
,	O
nucleation	B-Algorithm
is	O
the	O
first	O
step	O
in	O
the	O
formation	O
of	O
either	O
a	O
new	O
thermodynamic	O
phase	O
or	O
structure	O
via	O
self-assembly	O
or	O
self-organization	O
within	O
a	O
substance	O
or	O
mixture	O
.	O
</s>
<s>
Nucleation	B-Algorithm
is	O
typically	O
defined	O
to	O
be	O
the	O
process	O
that	O
determines	O
how	O
long	O
an	O
observer	O
has	O
to	O
wait	O
before	O
the	O
new	O
phase	O
or	O
self-organized	O
structure	O
appears	O
.	O
</s>
<s>
At	O
these	O
conditions	O
,	O
nucleation	B-Algorithm
of	O
ice	O
is	O
either	O
slow	O
or	O
does	O
not	O
occur	O
at	O
all	O
.	O
</s>
<s>
However	O
,	O
at	O
lower	O
temperatures	O
nucleation	B-Algorithm
is	O
fast	O
,	O
and	O
ice	O
crystals	O
appear	O
after	O
little	O
or	O
no	O
delay	O
.	O
</s>
<s>
Nucleation	B-Algorithm
is	O
a	O
common	O
mechanism	O
which	O
generates	O
first-order	O
phase	O
transitions	O
,	O
and	O
it	O
is	O
the	O
start	O
of	O
the	O
process	O
of	O
forming	O
a	O
new	O
thermodynamic	O
phase	O
.	O
</s>
<s>
Nucleation	B-Algorithm
is	O
often	O
very	O
sensitive	O
to	O
impurities	O
in	O
the	O
system	O
.	O
</s>
<s>
These	O
impurities	O
may	O
be	O
too	O
small	O
to	O
be	O
seen	O
by	O
the	O
naked	O
eye	O
,	O
but	O
still	O
can	O
control	O
the	O
rate	O
of	O
nucleation	B-Algorithm
.	O
</s>
<s>
Because	O
of	O
this	O
,	O
it	O
is	O
often	O
important	O
to	O
distinguish	O
between	O
heterogeneous	O
nucleation	B-Algorithm
and	O
homogeneous	O
nucleation	B-Algorithm
.	O
</s>
<s>
Heterogeneous	O
nucleation	B-Algorithm
occurs	O
at	O
nucleation	B-Algorithm
sites	I-Algorithm
on	O
surfaces	O
in	O
the	O
system	O
.	O
</s>
<s>
Homogeneous	O
nucleation	B-Algorithm
occurs	O
away	O
from	O
a	O
surface	O
.	O
</s>
<s>
Nucleation	B-Algorithm
is	O
usually	O
a	O
stochastic	O
(	O
random	O
)	O
process	O
,	O
so	O
even	O
in	O
two	O
identical	O
systems	O
nucleation	B-Algorithm
will	O
occur	O
at	O
different	O
times	O
.	O
</s>
<s>
This	O
shows	O
nucleation	B-Algorithm
of	O
a	O
new	O
phase	O
(	O
shown	O
in	O
red	O
)	O
in	O
an	O
existing	O
phase	O
(	O
white	O
)	O
.	O
</s>
<s>
The	O
standard	O
theory	O
that	O
describes	O
this	O
behaviour	O
for	O
the	O
nucleation	B-Algorithm
of	O
a	O
new	O
thermodynamic	O
phase	O
is	O
called	O
classical	B-Algorithm
nucleation	I-Algorithm
theory	I-Algorithm
.	O
</s>
<s>
However	O
,	O
the	O
CNT	O
fails	O
in	O
describing	O
experimental	O
results	O
of	O
vapour	O
to	O
liquid	O
nucleation	B-Algorithm
even	O
for	O
model	O
substances	O
like	O
argon	O
by	O
several	O
orders	O
of	O
magnitude	O
.	O
</s>
<s>
For	O
nucleation	B-Algorithm
of	O
a	O
new	O
thermodynamic	O
phase	O
,	O
such	O
as	O
the	O
formation	O
of	O
ice	O
in	O
water	O
below	O
0°C	O
,	O
if	O
the	O
system	O
is	O
not	O
evolving	O
with	O
time	O
and	O
nucleation	B-Algorithm
occurs	O
in	O
one	O
step	O
,	O
then	O
the	O
probability	O
that	O
nucleation	B-Algorithm
has	O
not	O
occurred	O
should	O
undergo	O
exponential	O
decay	O
.	O
</s>
<s>
This	O
is	O
seen	O
for	O
example	O
in	O
the	O
nucleation	B-Algorithm
of	O
ice	O
in	O
supercooled	O
small	O
water	O
droplets	O
.	O
</s>
<s>
The	O
decay	O
rate	O
of	O
the	O
exponential	O
gives	O
the	O
nucleation	B-Algorithm
rate	O
.	O
</s>
<s>
Classical	B-Algorithm
nucleation	I-Algorithm
theory	I-Algorithm
is	O
a	O
widely	O
used	O
approximate	O
theory	O
for	O
estimating	O
these	O
rates	O
,	O
and	O
how	O
they	O
vary	O
with	O
variables	O
such	O
as	O
temperature	O
.	O
</s>
<s>
It	O
correctly	O
predicts	O
that	O
the	O
time	O
you	O
have	O
to	O
wait	O
for	O
nucleation	B-Algorithm
decreases	O
extremely	O
rapidly	O
when	O
supersaturated	O
.	O
</s>
<s>
It	O
is	O
not	O
just	O
new	O
phases	O
such	O
as	O
liquids	O
and	O
crystals	O
that	O
form	O
via	O
nucleation	B-Algorithm
followed	O
by	O
growth	O
.	O
</s>
<s>
The	O
self-assembly	O
process	O
that	O
forms	O
objects	O
like	O
the	O
amyloid	O
aggregates	O
associated	O
with	O
Alzheimer	O
's	O
disease	O
also	O
starts	O
with	O
nucleation	B-Algorithm
.	O
</s>
<s>
Energy	O
consuming	O
self-organising	O
systems	O
such	O
as	O
the	O
microtubules	O
in	O
cells	O
also	O
show	O
nucleation	B-Algorithm
and	O
growth	O
.	O
</s>
<s>
Heterogeneous	O
nucleation	B-Algorithm
,	O
nucleation	B-Algorithm
with	O
the	O
nucleus	O
at	O
a	O
surface	O
,	O
is	O
much	O
more	O
common	O
than	O
homogeneous	O
nucleation	B-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
in	O
the	O
nucleation	B-Algorithm
of	O
ice	O
from	O
supercooled	O
water	O
droplets	O
,	O
purifying	O
the	O
water	O
to	O
remove	O
all	O
or	O
almost	O
all	O
impurities	O
results	O
in	O
water	O
droplets	O
that	O
freeze	O
below	O
around	O
−35°C	O
,	O
whereas	O
water	O
that	O
contains	O
impurities	O
may	O
freeze	O
at	O
−5°C	O
or	O
warmer	O
.	O
</s>
<s>
This	O
observation	O
that	O
heterogeneous	O
nucleation	B-Algorithm
can	O
occur	O
when	O
the	O
rate	O
of	O
homogeneous	O
nucleation	B-Algorithm
is	O
essentially	O
zero	O
,	O
is	O
often	O
understood	O
using	O
classical	B-Algorithm
nucleation	I-Algorithm
theory	I-Algorithm
.	O
</s>
<s>
This	O
predicts	O
that	O
the	O
nucleation	B-Algorithm
slows	O
exponentially	O
with	O
the	O
height	O
of	O
a	O
free	O
energy	O
barrier	O
G*	O
.	O
</s>
<s>
For	O
homogeneous	O
nucleation	B-Algorithm
the	O
nucleus	O
is	O
approximated	O
by	O
a	O
sphere	O
,	O
but	O
as	O
we	O
can	O
see	O
in	O
the	O
schematic	O
of	O
macroscopic	O
droplets	O
to	O
the	O
right	O
,	O
droplets	O
on	O
surfaces	O
are	O
not	O
complete	O
spheres	O
and	O
so	O
the	O
area	O
of	O
the	O
interface	O
between	O
the	O
droplet	O
and	O
the	O
surrounding	O
fluid	O
is	O
less	O
than	O
a	O
sphere	O
's	O
.	O
</s>
<s>
This	O
reduction	O
in	O
surface	O
area	O
of	O
the	O
nucleus	O
reduces	O
the	O
height	O
of	O
the	O
barrier	O
to	O
nucleation	B-Algorithm
and	O
so	O
speeds	O
nucleation	B-Algorithm
up	O
exponentially	O
.	O
</s>
<s>
Nucleation	B-Algorithm
can	O
also	O
start	O
at	O
the	O
surface	O
of	O
a	O
liquid	O
.	O
</s>
<s>
For	O
example	O
,	O
computer	O
simulations	O
of	O
gold	O
nanoparticles	O
show	O
that	O
the	O
crystal	O
phase	O
nucleates	B-Algorithm
at	O
the	O
liquid-gold	O
surface	O
.	O
</s>
<s>
Classical	B-Algorithm
nucleation	I-Algorithm
theory	I-Algorithm
makes	O
a	O
number	O
of	O
assumptions	O
,	O
for	O
example	O
it	O
treats	O
a	O
microscopic	O
nucleus	O
as	O
if	O
it	O
is	O
a	O
macroscopic	O
droplet	O
with	O
a	O
well-defined	O
surface	O
whose	O
free	O
energy	O
is	O
estimated	O
using	O
an	O
equilibrium	O
property	O
:	O
the	O
interfacial	O
tension	O
.	O
</s>
<s>
Also	O
nucleation	B-Algorithm
is	O
an	O
inherently	O
out	O
of	O
thermodynamic	O
equilibrium	O
phenomenon	O
so	O
it	O
is	O
not	O
always	O
obvious	O
that	O
its	O
rate	O
can	O
be	O
estimated	O
using	O
equilibrium	O
properties	O
.	O
</s>
<s>
However	O
,	O
modern	O
computers	O
are	O
powerful	O
enough	O
to	O
calculate	O
essentially	O
exact	O
nucleation	B-Algorithm
rates	O
for	O
simple	O
models	O
.	O
</s>
<s>
These	O
have	O
been	O
compared	O
with	O
the	O
classical	O
theory	O
,	O
for	O
example	O
for	O
the	O
case	O
of	O
nucleation	B-Algorithm
of	O
the	O
crystal	O
phase	O
in	O
the	O
model	O
of	O
hard	O
spheres	O
.	O
</s>
<s>
So	O
for	O
the	O
simple	O
models	O
we	O
can	O
study	O
,	O
classical	B-Algorithm
nucleation	I-Algorithm
theory	I-Algorithm
works	O
quite	O
well	O
,	O
but	O
we	O
do	O
not	O
know	O
if	O
it	O
works	O
equally	O
well	O
for	O
(	O
say	O
)	O
complex	O
molecules	O
crystallising	O
out	O
of	O
solution	O
.	O
</s>
<s>
Nucleation	B-Algorithm
of	O
the	O
crystal	O
is	O
then	O
being	O
prevented	O
by	O
a	O
substantial	O
barrier	O
.	O
</s>
<s>
In	O
small	O
volumes	O
,	O
such	O
as	O
in	O
small	O
droplets	O
,	O
only	O
one	O
nucleation	B-Algorithm
event	O
may	O
be	O
needed	O
for	O
crystallisation	O
.	O
</s>
<s>
In	O
these	O
small	O
volumes	O
,	O
the	O
time	O
until	O
the	O
first	O
crystal	O
appears	O
is	O
usually	O
defined	O
to	O
be	O
the	O
nucleation	B-Algorithm
time	O
.	O
</s>
<s>
Visualization	O
of	O
the	O
initial	O
stage	O
of	O
crystal	O
nucleation	B-Algorithm
of	O
sodium	O
chloride	O
was	O
achieved	O
through	O
atomic-resolution	O
real-time	O
video	O
imaging	O
.	O
</s>
<s>
Calcium	O
carbonate	O
crystal	O
nucleation	B-Algorithm
depends	O
not	O
only	O
on	O
degree	O
of	O
supersaturation	O
but	O
also	O
the	O
ratio	O
of	O
calcium	O
to	O
carbonate	O
ions	O
in	O
aqueous	O
solutions	O
.	O
</s>
<s>
In	O
larger	O
volumes	O
many	O
nucleation	B-Algorithm
events	O
will	O
occur	O
.	O
</s>
<s>
A	O
simple	O
model	O
for	O
crystallisation	O
in	O
that	O
case	O
,	O
that	O
combines	O
nucleation	B-Algorithm
and	O
growth	O
is	O
the	O
KJMA	O
or	O
Avrami	O
model	O
.	O
</s>
<s>
The	O
time	O
until	O
the	O
appearance	O
of	O
the	O
first	O
crystal	O
is	O
also	O
called	O
primary	O
nucleation	B-Algorithm
time	O
,	O
to	O
distinguish	O
it	O
from	O
secondary	O
nucleation	B-Algorithm
times	O
.	O
</s>
<s>
Primary	O
nucleation	B-Algorithm
describes	O
the	O
transition	O
to	O
a	O
new	O
phase	O
that	O
does	O
not	O
rely	O
on	O
the	O
new	O
phase	O
already	O
being	O
present	O
,	O
either	O
because	O
it	O
is	O
the	O
very	O
first	O
nucleus	O
of	O
that	O
phase	O
to	O
form	O
,	O
or	O
because	O
the	O
nucleus	O
forms	O
far	O
from	O
any	O
pre-existing	O
piece	O
of	O
the	O
new	O
phase	O
.	O
</s>
<s>
Particularly	O
in	O
the	O
study	O
of	O
crystallisation	O
,	O
secondary	O
nucleation	B-Algorithm
can	O
be	O
important	O
.	O
</s>
<s>
So	O
both	O
primary	O
and	O
secondary	O
nucleation	B-Algorithm
increase	O
the	O
number	O
of	O
crystals	O
in	O
the	O
system	O
but	O
their	O
mechanisms	O
are	O
very	O
different	O
,	O
and	O
secondary	O
nucleation	B-Algorithm
relies	O
on	O
crystals	O
already	O
being	O
present	O
.	O
</s>
<s>
It	O
is	O
typically	O
difficult	O
to	O
experimentally	O
study	O
the	O
nucleation	B-Algorithm
of	O
crystals	O
.	O
</s>
<s>
In	O
large	O
liquid	O
volumes	O
there	O
are	O
typically	O
multiple	O
nucleation	B-Algorithm
events	O
,	O
and	O
it	O
is	O
difficult	O
to	O
disentangle	O
the	O
effects	O
of	O
nucleation	B-Algorithm
from	O
those	O
of	O
growth	O
of	O
the	O
nucleated	O
phase	O
.	O
</s>
<s>
As	O
nucleation	B-Algorithm
is	O
stochastic	O
,	O
many	O
droplets	O
are	O
needed	O
so	O
that	O
statistics	O
for	O
the	O
nucleation	B-Algorithm
events	O
can	O
be	O
obtained	O
.	O
</s>
<s>
To	O
the	O
right	O
is	O
shown	O
an	O
example	O
set	O
of	O
nucleation	B-Algorithm
data	O
.	O
</s>
<s>
It	O
is	O
for	O
the	O
nucleation	B-Algorithm
at	O
constant	O
temperature	O
and	O
hence	O
supersaturation	O
of	O
the	O
crystal	O
phase	O
in	O
small	O
droplets	O
of	O
supercooled	O
liquid	O
tin	O
;	O
this	O
is	O
the	O
work	O
of	O
Pound	O
and	O
La	O
Mer	O
.	O
</s>
<s>
Nucleation	B-Algorithm
occurs	O
in	O
different	O
droplets	O
at	O
different	O
times	O
,	O
hence	O
the	O
fraction	O
is	O
not	O
a	O
simple	O
step	O
function	O
that	O
drops	O
sharply	O
from	O
one	O
to	O
zero	O
at	O
one	O
particular	O
time	O
.	O
</s>
<s>
The	O
model	O
assumes	O
that	O
nucleation	B-Algorithm
occurs	O
due	O
to	O
impurity	O
particles	O
in	O
the	O
liquid	O
tin	O
droplets	O
,	O
and	O
it	O
makes	O
the	O
simplifying	O
assumption	O
that	O
all	O
impurity	O
particles	O
produce	O
nucleation	B-Algorithm
at	O
the	O
same	O
rate	O
.	O
</s>
<s>
The	O
fit	O
values	O
are	O
that	O
the	O
nucleation	B-Algorithm
rate	O
due	O
to	O
a	O
single	O
impurity	O
particle	O
is	O
0.02/s	O
,	O
and	O
the	O
average	O
number	O
of	O
impurity	O
particles	O
per	O
droplet	O
is	O
1.2	O
.	O
</s>
<s>
Within	O
the	O
model	O
this	O
is	O
assumed	O
to	O
be	O
because	O
,	O
by	O
chance	O
,	O
these	O
droplets	O
do	O
not	O
have	O
even	O
one	O
impurity	O
particle	O
and	O
so	O
there	O
is	O
no	O
heterogeneous	O
nucleation	B-Algorithm
.	O
</s>
<s>
Homogeneous	O
nucleation	B-Algorithm
is	O
assumed	O
to	O
be	O
negligible	O
on	O
the	O
timescale	O
of	O
this	O
experiment	O
.	O
</s>
<s>
These	O
data	O
are	O
just	O
one	O
example	O
,	O
but	O
they	O
illustrate	O
common	O
features	O
of	O
the	O
nucleation	B-Algorithm
of	O
crystals	O
in	O
that	O
there	O
is	O
clear	O
evidence	O
for	O
heterogeneous	O
nucleation	B-Algorithm
,	O
and	O
that	O
nucleation	B-Algorithm
is	O
clearly	O
stochastic	O
.	O
</s>
<s>
Clouds	O
form	O
when	O
wet	O
air	O
cools	O
(	O
often	O
because	O
the	O
air	O
rises	O
)	O
and	O
many	O
small	O
water	O
droplets	O
nucleate	B-Algorithm
from	O
the	O
supersaturated	O
air	O
.	O
</s>
<s>
The	O
excess	O
vapor	O
begins	O
to	O
nucleate	B-Algorithm
and	O
to	O
form	O
small	O
water	O
droplets	O
which	O
form	O
a	O
cloud	O
.	O
</s>
<s>
Nucleation	B-Algorithm
of	O
the	O
droplets	O
of	O
liquid	O
water	O
is	O
heterogeneous	O
,	O
occurring	O
on	O
particles	O
referred	O
to	O
as	O
cloud	O
condensation	O
nuclei	O
.	O
</s>
<s>
Bubbles	O
of	O
carbon	O
dioxide	O
nucleate	B-Algorithm
shortly	O
after	O
the	O
pressure	O
is	O
released	O
from	O
a	O
container	O
of	O
carbonated	O
liquid	O
.	O
</s>
<s>
Nucleation	B-Algorithm
in	O
boiling	O
can	O
occur	O
in	O
the	O
bulk	O
liquid	O
if	O
the	O
pressure	O
is	O
reduced	O
so	O
that	O
the	O
liquid	O
becomes	O
superheated	O
with	O
respect	O
to	O
the	O
pressure-dependent	O
boiling	O
point	O
.	O
</s>
<s>
More	O
often	O
,	O
nucleation	B-Algorithm
occurs	O
on	O
the	O
heating	O
surface	O
,	O
at	O
nucleation	B-Algorithm
sites	I-Algorithm
.	O
</s>
<s>
Typically	O
,	O
nucleation	B-Algorithm
sites	I-Algorithm
are	O
tiny	O
crevices	O
where	O
free	O
gas-liquid	O
surface	O
is	O
maintained	O
or	O
spots	O
on	O
the	O
heating	O
surface	O
with	O
lower	O
wetting	O
properties	O
.	O
</s>
<s>
Some	O
champagne	O
stirrers	O
operate	O
by	O
providing	O
many	O
nucleation	B-Algorithm
sites	I-Algorithm
via	O
high	O
surface-area	O
and	O
sharp	O
corners	O
,	O
speeding	O
the	O
release	O
of	O
bubbles	O
and	O
removing	O
carbonation	O
from	O
the	O
wine	O
.	O
</s>
<s>
The	O
surface	O
of	O
Mentos	O
candy	O
provides	O
nucleation	B-Algorithm
sites	I-Algorithm
for	O
the	O
formation	O
of	O
carbon-dioxide	O
bubbles	O
from	O
carbonated	O
soda	O
.	O
</s>
<s>
Both	O
the	O
bubble	B-Algorithm
chamber	I-Algorithm
and	O
the	O
cloud	O
chamber	O
rely	O
on	O
nucleation	B-Algorithm
,	O
of	O
bubbles	O
and	O
droplets	O
,	O
respectively	O
.	O
</s>
<s>
Liquid	O
water	O
does	O
not	O
freeze	O
at	O
0°C	O
unless	O
there	O
is	O
ice	O
already	O
present	O
;	O
cooling	O
significantly	O
below	O
0°C	O
is	O
required	O
to	O
nucleate	B-Algorithm
ice	O
and	O
so	O
for	O
the	O
water	O
to	O
freeze	O
.	O
</s>
<s>
crystalline	O
iron	O
made	O
from	O
liquid	O
iron	O
cast	O
into	O
a	O
mold	O
,	O
so	O
the	O
nucleation	B-Algorithm
of	O
crystalline	O
materials	O
is	O
widely	O
studied	O
in	O
industry	O
.	O
</s>
<s>
In	O
addition	O
to	O
the	O
nucleation	B-Algorithm
and	O
growth	O
of	O
crystals	O
e.g.	O
</s>
<s>
in	O
non-crystalline	O
glasses	O
,	O
the	O
nucleation	B-Algorithm
and	O
growth	O
of	O
impurity	O
precipitates	O
in	O
crystals	O
at	O
,	O
and	O
between	O
,	O
grain	O
boundaries	O
is	O
quite	O
important	O
industrially	O
.	O
</s>
<s>
For	O
example	O
in	O
metals	O
solid-state	O
nucleation	B-Algorithm
and	O
precipitate	O
growth	O
plays	O
an	O
important	O
role	O
e.g.	O
</s>
