<s>
The	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
is	O
a	O
useful	O
equation	O
in	O
many	O
settings	O
,	O
whether	O
it	O
be	O
to	O
understand	O
physiological	O
interfaces	O
,	O
polymer	O
science	O
,	O
electron	O
interactions	O
in	O
a	O
semiconductor	O
,	O
or	O
more	O
.	O
</s>
<s>
The	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
is	O
derived	O
via	O
mean-field	O
assumptions	O
.	O
</s>
<s>
From	O
the	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
many	O
other	O
equations	O
have	O
been	O
derived	O
with	O
a	O
number	O
of	O
different	O
assumptions	O
.	O
</s>
<s>
The	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
describes	O
a	O
model	O
proposed	O
independently	O
by	O
Louis	O
Georges	O
Gouy	O
and	O
David	O
Leonard	O
Chapman	O
in	O
1910	O
and	O
1913	O
,	O
respectively	O
.	O
</s>
<s>
The	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
describes	O
the	O
electrochemical	O
potential	O
of	O
ions	O
in	O
the	O
diffuse	O
layer	O
.	O
</s>
<s>
is	O
the	O
temperature	O
in	O
kelvins	B-Operating_System
.	O
</s>
<s>
Finally	O
the	O
charge	O
density	O
can	O
be	O
substituted	O
into	O
the	O
Poisson	O
equation	O
to	O
produce	O
the	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
.	O
</s>
<s>
The	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
can	O
take	O
many	O
forms	O
throughout	O
various	O
scientific	O
fields	O
.	O
</s>
<s>
In	O
biophysics	O
and	O
certain	O
surface	O
chemistry	O
applications	O
,	O
it	O
is	O
known	O
simply	O
as	O
the	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
.	O
</s>
<s>
Only	O
minor	O
modifications	O
are	O
necessary	O
to	O
apply	O
the	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
to	O
various	O
interfacial	O
models	O
,	O
making	O
it	O
a	O
highly	O
useful	O
tool	O
in	O
determining	O
electrostatic	O
potential	O
at	O
surfaces	O
.	O
</s>
<s>
Because	O
the	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
is	O
a	O
partial	O
differential	O
of	O
the	O
second	O
order	O
,	O
it	O
is	O
commonly	O
solved	O
numerically	B-General_Concept
;	O
however	O
,	O
with	O
certain	O
geometries	O
,	O
it	O
can	O
be	O
solved	O
analytically	O
.	O
</s>
<s>
Below	O
is	O
the	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
solved	O
analytically	O
in	O
terms	O
of	O
a	O
second	O
order	O
derivative	O
with	O
respect	O
to	O
x	O
.	O
</s>
<s>
When	O
using	O
the	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
,	O
it	O
is	O
important	O
to	O
determine	O
if	O
the	O
specific	O
case	O
is	O
low	O
or	O
high	O
potential	O
.	O
</s>
<s>
In	O
the	O
low-potential	O
condition	O
,	O
the	O
linearized	O
version	O
of	O
the	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
(	O
shown	O
below	O
)	O
is	O
valid	O
,	O
and	O
it	O
is	O
commonly	O
used	O
as	O
it	O
is	O
more	O
simple	O
and	O
spans	O
a	O
wide	O
variety	O
of	O
cases	O
.	O
</s>
<s>
In	O
order	O
to	O
obtain	O
the	O
equation	O
,	O
the	O
general	O
solution	O
to	O
the	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
is	O
used	O
and	O
the	O
case	O
of	O
low	O
potentials	O
is	O
dropped	O
.	O
</s>
<s>
The	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
can	O
be	O
applied	O
in	O
a	O
variety	O
of	O
fields	O
mainly	O
as	O
a	O
modeling	O
tool	O
to	O
make	O
approximations	O
for	O
applications	O
such	O
as	O
charged	O
biomolecular	O
interactions	O
,	O
dynamics	O
of	O
electrons	O
in	O
semiconductors	O
or	O
plasma	O
,	O
etc	O
.	O
</s>
<s>
The	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
can	O
be	O
applied	O
to	O
biomolecular	O
systems	O
.	O
</s>
<s>
The	O
linearized	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
can	O
be	O
used	O
to	O
calculate	O
the	O
electrostatic	O
potential	O
and	O
free	O
energy	O
of	O
highly	O
charged	O
molecules	O
such	O
as	O
tRNA	O
in	O
an	O
ionic	O
solution	O
with	O
different	O
number	O
of	O
bound	O
ions	O
at	O
varying	O
physiological	O
ionic	O
strengths	O
.	O
</s>
<s>
Another	O
example	O
of	O
utilizing	O
the	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
is	O
the	O
determination	O
of	O
an	O
electric	O
potential	O
profile	O
at	O
points	O
perpendicular	O
to	O
the	O
phospholipid	O
bilayer	O
of	O
an	O
erythrocyte	O
.	O
</s>
<s>
The	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
can	O
also	O
be	O
used	O
to	O
calculate	O
the	O
electrostatic	O
free	O
energy	O
for	O
hypothetically	O
charging	O
a	O
sphere	O
using	O
the	O
following	O
charging	O
integral	O
:	O
</s>
<s>
The	O
following	O
expression	O
utilizes	O
chemical	O
potential	O
of	O
solute	O
molecules	O
and	O
implements	O
the	O
Poisson-Boltzmann	B-Algorithm
Equation	I-Algorithm
with	O
the	O
Euler-Lagrange	O
functional	O
:	O
</s>
<s>
An	O
analytical	O
solution	O
to	O
the	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
can	O
be	O
used	O
to	O
describe	O
an	O
electron-electron	O
interaction	O
in	O
a	O
metal-insulator	O
semiconductor	O
(	O
MIS	O
)	O
.	O
</s>
<s>
This	O
is	O
done	O
by	O
solving	O
the	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
analytically	O
in	O
the	O
three-dimensional	O
case	O
.	O
</s>
<s>
Therefore	O
,	O
it	O
is	O
essential	O
to	O
analytically	O
solve	O
the	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
in	O
order	O
to	O
obtain	O
the	O
analytical	O
quantities	O
in	O
the	O
MIS	O
tunneling	O
junctions	O
.	O
</s>
<s>
As	O
with	O
any	O
approximate	O
model	O
,	O
the	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
is	O
an	O
approximation	O
rather	O
than	O
an	O
exact	O
representation	O
.	O
</s>
<s>
The	O
Poisson	B-Algorithm
–	I-Algorithm
Boltzmann	I-Algorithm
equation	I-Algorithm
is	O
most	O
appropriate	O
for	O
approximating	O
the	O
electrostatic	O
potential	O
at	O
the	O
surface	O
for	O
aqueous	O
solutions	O
of	O
univalent	O
salts	O
at	O
concentrations	O
smaller	O
than	O
0.2	O
M	O
and	O
potentials	O
not	O
exceeding	O
50	O
–	O
80mV	O
.	O
</s>
<s>
In	O
the	O
limit	O
of	O
strong	O
electrostatic	O
interactions	O
,	O
a	O
strong	O
coupling	O
theory	O
is	O
more	O
applicable	O
than	O
the	O
weak	O
coupling	O
assumed	O
in	O
deriving	O
the	O
Poisson-Boltzmann	B-Algorithm
theory	O
.	O
</s>
