Biomedical Engineering Reference
In-Depth Information
where
is the density of the liquid solvent, which is assumed to be incompressible
v
is the three-dimensional liquid velocity vector
∇
ρ
is the gradient vector operator
∇
2
is the Laplacian operator
g
is the local gravitational acceleration
µ
is the solvent viscosity
p
is the hydrostatic pressure
E
is the imposed electric field vector
ρ
* is the charge density governed by the following Poisson's equation
(Lifson and Katchalsky, 1954):
2
ρ
*
=− ∇
D
ψ
(6.74)
where
D
is the dielectric constant of the liquid phase and
ψ
is the local electrostatic
potential. Furthermore,
is governed by the following Poisson-Boltzmann equation
(see Lifson and Katchalsky, 1954):
ψ
(
)
2
exp
(6.75)
∇=
ψπε
4
nD
/
−
εψ
/
kT
where
n
is the number density of counter-ions
ε
is their average charge
k
is the Boltzmann constant
T
is the absolute temperature
According to Lifson and Katchalsky (1954), the electrostatic potential in polyelec-
trolyte solutions for fully stretched macromolecules is given by the following equation,
which is an exact solution to the Poisson-Boltzmann equation in cylindrical coordinates:
{
}
(
)
()
=
2
2
2
2
−
1
ψ
rt
,
kT
/
ε
n
r
/
r
−
r
Sinh
β
n r
(
/
r
0
t n
β
(6.76)
0
i
)
−
where
β
is related to
λ
, which is a dimensionless parameter given by
(
)
λαε π
=
*
2
/
4
DbkT
,
(6.77)
where
* =
a
=
n
/
Z
, where
n
is the number of ions
and
Z
is the number of ionizable groups) and
b
is the distance between polyions in
the network. Furthermore,
n =
α
* is the degree of ionization (i.e.,
α
[α
*/
π
(
r
o
2
-
r
i
2
)
b
]
and betas are found from the following
equation:
.
0
0
2
0
0
2
1
−
β
λ
=
(6.78)
(
)
1
+
β
Coth
β
n r
/
r
i
0
