Biomedical Engineering Reference
In-Depth Information
αβ β
2
=+
2
B
(1.23)
N
)
(
)
+=
(
)
++ −
(
)
(
0
2
−+
2
ln
1
−
φφ
1
τφ
2
f
1 2
αβ
1
β
0
(1.24)
φ
0
where
F
/
kT
is the reduced temperature.
Therefore, the anisotropy of deformation of the disk is uniquely determined by
B
. Namely, for positive
B
, the disk is compressed more in its length than in diameter
(
τ
= 1 - 2
∆
α
>
β
) and, for negative
B
, the compression in the radial direction is more than
axial (
α
<
β
). By plotting
α
(radial expansion) versus
β
(axial expansion) of a rod-
shaped gel, various values for
B
and
can be obtained.
De Rossi and coworkers (1985) of the University of Pisa in Italy explained the
dynamic behavior of these gels from a thermodynamic point of view according to
the general expression:
τ
dU
=− ++ +
TdS
PdV
Fdl
µ
dn
ε
dq
(1.25)
i
i
where mechanical, chemical, and electrical energy terms are present and
Fdl
is the
mechanical work performed by a polymer fiber muscle during uniaxial elongation
dl
, and keeping
S, V, n
i
, and
q
constant.
ε
dq
is the electrical work term and
µ
i
dn
i
is
the chemical energy term.
They further proposed an analytical model describing mechanical parameters
governing the kinetics of thermally cross-linked PVA-PAA polyelectrolyte gel. For
a thin film of the specimen, they found a relation between swelling rate and linear
dimension and diffusion coefficient of the material. To analyze and describe the
transient mechanical behavior of a polyelectrolyte gel element in response to proton
and salt concentration gradients generated by electrode reactions and delivered
within the gel by electrochemical potential differences, several rate processes are
taken into account. These can include ion diffusion, diffusion-limited chemical
reaction, ion migration, and consequent electrical and mechanical realignment of
the polymer network.
Typically, proton diffusion reaction and mechanical realignment of the polymer
network are slow and a system of coupled differential equations is formulated to
determine the kinetics of gel de-swelling under electromechanical stimuli. A com-
parison among proton diffusion and gel-swelling characteristic time constants might
prove very useful in decoupling the chemical and mechanical problems, particularly
in the case of specific gel systems in which these limiting time constants are
considerably different.
De Rossi and colleagues analyzed the kinetics of free swelling of a partially
dehydrated gel in the case of spherical and thin film samples. The only problem
with their formulations is that they consider constitutive equations used in linear
infinitesimal elasticity formulations that are not appropriate for rather large swelling
and de-swelling deformation of ionic polymeric gels.
Osada and Hasebe (1985) of Ibaraki University in Japan were pioneers in
attempting to synthesize various polyelectrolytes and describing their dynamic
