Biomedical Engineering Reference
In-Depth Information
is the density of the pure solvent. The time derivative here is the “material
derivative”—the derivative of states associated with particles rather than position.
The velocity of the solvent is that of the gel plus the differential velocity due to
diffusion. The diffusion is driven by osmotic pressures, which are functions of two
internal coordinates or states: solvent concentration,
ρ
s
c
, and the mass of
H
ions per
+
unit volume of swollen polymer,
. Equivalent measures of these quantities are the
mass fractions of gel, which are solvent and hydrogen ions, respectively. These two
quantities as well as the displacement components of the gel are used as primary
variables in this exposition. The solvent concentration can be represented in terms
of a mass fraction by:
H
ρξ
ps
c
=
(6.2)
ξ
ρ
(
1
−
s
(
ρρ
−
))
s
p
s
where
is the solvent mass fraction.
For a system of three components (polymer, solvent, and protons), there are two
independent diffusion equations, each depending on the gradients of, at most, two
of the components. The isothermal diffusion equation describing the evolution of
the solvent mass fraction
ρ
is the density of pure polymer and
ξ
p
s
ξ
is
s
ρ
D
Dt
ξ
(
s
)
s
=∇
.
D
(
ξξ
)
∇
ξ
+
D
(
ξξ
)
∇
ξ
(6.3)
sp
s
,
H
s
H
sH
,
H
ρ
−
c
s
s
where the terms
D
are diffusivities,
ξ
is the mass fraction of the
H
ions, and
∇
+
ij
H
represents spatial gradient. Because the preceding evolution equation is written in a
frame moving with the gel, the convective term appears differently than it would in
a Eulerian formulation. Derivation of this equation requires exploitations of the
continuity equation (6.1). The transport of
H
+
is similar to that of solvent, but also
involves a source term:
D
Dt
ξ
D
D
ξ
∂∂
−
c
/
∼
(
)
∇
H
s
s
=∇⋅
D
ξξ
ξ
+
ξ
+
ξ
(6.4)
Hs
,
H
H
H
ρ
c
t
s
ξ
∼
ξ
∼
where
. Note that is a
tunable parameter that may be varied through electrical or chemical means. Also
note the modification to the convective terms resulting from formulation of the
equations in a frame that convects with the gel.
The diffusion relationships are taken from Flory-Huggins theory. The hydrogen
diffusion coefficient is represented as
accounts for the rate of creation or neutralization of
H
12
/
1
1
ξ
ξ
−
+
T
p
p
−
52
DD
H
=
≈
10
ξ
(6.5)
H
s
