Biomedical Engineering Reference
In-Depth Information
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(a)
(b)
FIGURE 5.2
Deformation of PAMPS ionic gel cylinder due to an imposed radial voltage gradient.
mobile ions in the gel. Note that an electric field gradient may be imposed across
the body of the gel by charged surfactant molecules.
If the gel possesses a specific capacitance
C
and a specific resistance
R
, the
g
g
Kirchhoff's law can be written in the following form:
Qi
vCQRQ
g
=
−1
+
, where
=
(5.1)
g
where
v
is the voltage across the thickness of the gel and
Q
is the specific charge
Q
(charge per unit mass) accumulated in the gel,
through
the gel cylindrical sample across its radius. Equation (5.1) can be readily solved to yield
being the current density
i
QCv
=
1 xp(
−
−
tRC
)
,
(5.2)
g
g
g
assuming that at
= 0.
Equation (5.2) relates the voltage drop across the radius of the gel to the charge
accumulated, which eventually causes the gel to deform. Thus, equation (5.2) will
serve as a basis for the electrical control of gel deformations.
The imposed voltage gradient across the radius of the gel forces the internal
fixed and mobile ions to redistribute. A possible charge distribution of the gel is
shown in figure 5.3. In order to mathematically model the nonhomogeneous defor-
mation or bending forces at work in an ionic polyelectrolyte gel strip, a number of
simplifying assumptions are made.
The first assumption is that the polymer segments carrying fixed charges are
cylindrically distributed along a given polymer chain and independent of the cylin-
drical angle
t
= 0,
Q
. This assumption is not essential but greatly simplifies the analysis.
Consider the field of attraction and repulsion among neighboring rows of fixed or
mobile charges in an ionic gel. Let
θ
r
and
Z
be the cylindrical polar coordinates with
i
i
the
i
th row as an axis such that the origin is at a given polymer segment.
Let the spacing in the
i
th row be
b
, and let the forces exerted by the atoms be
i
central and of the form
= 10
represent the Coulomb, van der Waals, and short-range repulsive forces, respectively.
cr
-s
such that the particular cases of
s
= 2,
s
= 7, and
s
