Biomedical Engineering Reference
In-Depth Information
The average p
K
a
of the hydrogel can be estimated by plotting swelling vs. pH and take the
pH of the inflection point. Also,
α
is the fraction of charged monomers.
For amine ionizable groups,
α
α
pH p
−
K
+
log
10
=
0
(5.137)
a
1
+
Recent progress has enabled researchers to model a more complicated case for the
purpose of controlling drug release rate. Shang et al. [70] proposed a method to tailor the
chemical potential equilibrium at the donor-membrane interface. Their method works as
follows: a certain amount of the solute molecule is first immobilized in the membrane.
The immobilized solute could contribute to the overall chemical potential of solute in
the membrane; hence, the amount of free solute molecules from donor that could parti-
from donor that could parti-
tion into the membrane is reduced. It is because that less amount of solute molecules is
needed to balance its chemical potential in the donor because of the contribution of immo-
bilized ones. In conventional thermodynamics, all components in a system are considered
independent (Equation 5.65). However, in practice, the amount of one component could
affect the amount of the other, which means dependence shall be considered. The infi ni-
s from donor that could parti-
The inini-
tesimal expression of Gibb's free energy equation (Equation 5.63) is reconfigured in the
following way to handle the “dependent” case: in a general scenario, two dependent
substances (A and B) in the membrane are grouped together, represented by a combined
chemical potential (
μ
M,grouping
), and the total concentration of solute molecules in the mem-
in the mem-
brane (
C
M,grouping
=
C
M,A
+
C
M,B
), where
C
M,A
and
C
M,B
are the concentrations of A and B in
the membrane, and they are dependent; subscript M denotes membrane. Mathematically,
the above description writes
the total concentration of solute molecules in the mem-
total concentration of solute molecules in the mem-
of solute molecules in the mem-
∑
d
G
= −
S T V P
d
+
d
+
(
d
C
+
d
C
)
+
d
C
(5.138)
M,A
M,A
M,B
M,B
i
i
i
=
3
and
μ
M,grouping
d
C
M,grouping
=
μ
A
d
C
A
+
μ
B
d
C
B
(5.139)
where
S
is entropy.
Hence, the combined chemical potential is
C
C
d
C
C
d
M,A
M,B
=
+
(5.140)
M,grouping
M,A
M,B
d
d
grouping
grouping
T P n
i
,
,
T P n
i
>
,
,
2
>
2
and let
d
C
d
C
M,A
M,B
=
and
=
(5.141)
α
β
d
C
d
C
M,grouping
M,
grouping
T P n
i
,
,
T P n
i
,
,
2
>
2
>
At chemical potential equilibrium, it is assumed that the combined chemical potential in
the membrane is balanced by the chemical potential of substance A in the donor:
