Biomedical Engineering Reference
In-Depth Information
C
1
,...,C
M
(Che et al.,
2008
)
M
ε
2
|∇
C
i
2
F
el
[
ϕ,C
1
,...,C
M
]=
−
ϕ
|
+
ρ(r)ϕ
+
RT
Ω
i
=
1
d
Ω,
M
M
C
i
ln
Λ
3
N
A
C
i
−
1
−
+
RT
μ
i
N
A
C
i
i
=
1
i
=
1
(2.9)
where
ε
is the dielectric permittivity of the solution,
N
A
is the Avogadro constant,
C
i
and
μ
i
are the bath concentration and chemical potential of the
i
th ionic species,
respectively, and
Λ
is the thermal de Broglie wavelength. The local charge density
ρ(r)
is given by
M
=
+
ρ(r)
ρ
f
(r)
Fz
i
C
i
(r),
(2.10)
i
=
1
where
ρ
f
is the fixed charge density from the GAG disaccharide units and
z
i
is the
valence number for the
i
th ionic species. In Eq. (
2.9
), the first two terms are the
internal electrostatic energy, the third term is the osmotic pressure from the mobile
ions, the fourth term constitutes the ideal gas entropy and the last term accounts for
the chemical potential. Setting the first variation of the free energy
G
with respect
to the concentration
C
i
to zero leads to
exp
,
Fϕ(r)
RT
C
i
C
i
(r)
=
−
z
i
(2.11)
which is the Boltzmann distribution for the concentrations at equilibrium. The vari-
ation of
F
el
with respect to the potential
ϕ
yields the Poisson equation
∇·
ε
∇
ϕ(r)
=−
ρ(r).
(2.12)
The Poisson-Boltzmann equation (PBE) is obtained by combining Eqs. (
2.10
)-
(
2.12
),
exp
.
M
−
z
i
Fϕ(r)
RT
2
ϕ(r)
=
ρ
f
(r)
+
Fz
i
C
i
−
ε
∇
(2.13)
i
=
1
Substituting Eqs. (
2.10
), (
2.11
)into(
2.9
), the free energy at equilibrium is obtained
as
1
d
Ω.
C
j
exp
M
ε
2
|∇
Fϕ(r)
RT
2
F
el
[
ϕ
]=
−
ϕ
|
+
ρ
f
ϕ
−
RT
−
z
j
−
Ω
j
=
1
(2.14)
