Chemistry Reference
In-Depth Information
A particle charge of 1 has been assumed here. Three regimes for this equation are
possible. First, if there is no current
½
j
ð
r
;
t
Þ¼
0
, Eq. [52] along with the Poisson
equation [which determines the potential
] lead to the PB equation; this is
the equilibrium situation. Second, if the particle densities are not changing
[
f
ð
r
Þ
r
ð
r
;
t
Þ¼
constant], the steady-state (Nernst-Planck) case is obtained:
r
j
ð
r
;
t
Þ¼
0
½
53
By insertion of Eq. [52b] for
j
ð
r
;
into this equation, it is easy to see that a
Poisson-type equation is obtained (actually a Laplace equation, with the rhs
equal to zero). If the diffusion coefficient is a constant over the domain,
then that variable drops out. Notice that, in Eq. [52b], an effective ''dielectric
constant'' [exp
t
Þ
] appears, and this expression can vary over orders of
magnitude due to large variations in the potential. Finally, if we seek a solu-
tion to the time-dependent problem, we must solve the full Smoluchowski
equation iteratively. The particle density then depends explicitly on time.
The well-known PNP transport equations correspond to the steady-state
case. Any transients are assumed to have died out, so the particle number den-
sities are constant; transport still occurs in a way that maintains those constant
particle densities. The PNP-type equations have found wide application in
semiconductor physics
152
and more recently in ion channel biophysics. A
recent study has moved beyond the steady-state regime to examine real-time
transport related to enzyme kinetics.
153
At the PNP level, our group has developed an efficient FAS-MG solver
for the coupled equations, and we found that the choice of relaxation and
interpolation schemes plays a crucial role in stability and efficiency. The Pois-
son part of the problem is standard and requires no special considerations. The
Nernst-Planck part, however, contains strongly varying functions (as dis-
cussed above), and this is where focus on the relaxation and interpolation
operations is required.
The Laplace equation for the diffusion part of the PNP equations (for
one of the ionic species) can be written as
ð
bf
ð
r
ÞÞ
r½
y
ð
r
Þr
c
ð
r
Þ¼
0
½
54
where
y
ð
r
Þ¼
exp
½
b
z
f
ð
r
Þ
½
55
and
c
ð
r
Þ¼
exp
½
b
z
f
ð
r
Þ
r
ð
r
;
t
Þ
½
56
