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Theory description
The swelling behavior of the hydrogels was analyzed within a framework of the
Flory–Rehner theory of swelling equilibrium, including the ideal Donnan equilibria.
According to the Flory–Rehner theory, the osmotic pressure
π
of a gel is the sum of three
contributions [59, 60].
𝜋 = 𝜋
𝑚𝑖𝑥
+ 𝜋
𝑒𝑙
+ 𝜋
𝑖𝑜𝑛
Equation 2
where
π
mix
,
π
el
, and
π
ion
are the osmotic pressures due to polymer–solvent mixing, due to
deformation of network chains to a more elongated state, and due to a non-uniform
distribution of mobile counter-ions between the gel and the solution, respectively. In our
study, nonionic monomers were employed in preparing our hydrogels. Additionally, to
consider if the IPN hydrogels are sensitive to salt (see Figure 7b), the assumptions were:
i) while Na+ and Cl- ions are small enough to pass through the porous hydrogel network,
at equilibrium state we assume the change in non-uniformity of counter-ions should be
small enough to be negligible, i.e.,
ion
0, and ii) added salt affected the swelling of
pure PAM network through changing
𝝌
1
, the polymer–solvent interaction parameter, to
make the PAM network shrink. Related to
ion
, the
𝝌
1
as a function of the ionic strength
185
of the solution is not readily expressed by solubility parameter. From this point of view
the polymer-solvent interaction parameter,
𝝌
1
, is assumed to change and accounts for
changes in SR ratio. The incorporation of the PVA network, if less affected in its
solubility, may constrain as well as sustain the SR of the PAM network in low and high
ionic strengths, respectively.
To simplify calculation, only
π
mix
and
π
el
were considered in studying the swelling
kinetic of the IPN hydrogels in 1 wt.-% NaCl. At equilibrium, the total osmotic pressure
(
π
), which is the sum of osmotic pressures from mixing (
π
mix
) and network elasticity (
π
mix
),
must be zero [60, 65] and
mix
= -
el
and will result in Equation 3, where
is the strand
density,
M
c
is average molecular weight between crosslinks,
V
1
is the molar volume of 1
wt.- % NaCl solution (18.42 cm
3
/mol), and
𝑣
2
is the average volume fraction of polymer
in the hydrogel when it reaches the equilibrium swelling state. We assumed a value of
𝝌
1
=0.48, which provided a good fit to the experimental swelling data of acrylamide-based
nonionic anionic, cationic, and ampholytic hydrogels of various compositions.[61-64]
The value of
2
is given by Equation 5, where
𝜌
𝑝
is the polymer density (1.13 g/cm
3
),
𝜌
𝑠
is
the density of 1 wt.-% NaCl solution (1.0053 g/cm
3
at 25 °C) and SR is the swelling ratio
of hydrogel..
𝜋
𝑚𝑖𝑥
+ 𝜋
𝑒𝑙
= 0
Equation 3
𝒗 =
𝝆
𝒑
𝑴
𝒄
=
𝒍𝒏(𝟏−𝒗
𝟐
)+𝒗
𝟐
+𝝌
𝟏
𝒗
𝟐
𝟐
𝑽
𝟏
(𝒗
𝟐
𝟏
𝟑
−
𝟏
𝟐
𝒗
𝟐
)
Equation 4
𝒗
𝟐
= (
𝟏
𝝆
𝒑
)/[(
𝑺𝑾
𝝆
𝒔
) + (
𝟏
𝝆
𝒑
)]
Equation 5
The inverse of molecular weight per crosslink (
e
) is defined as the crosslink
density (Equation 6). From these equations we observe that a measured crosslink density
186
is proportionally related to a change in
𝝌
1
,
2
2
, and/or inversely related to SR
2
(Equation
7).
𝑣
𝑒
=
1
𝑀
𝑐
Equation 6
𝑣
𝑒
∗=
1
𝑀
𝑐
∝ 𝜒
1
,
1
𝑀
𝑐
∝ 𝑣
2
2
∝ (
1
𝑠𝑤
)
2
Equation 7
Fig. 8
Dependence of the effective crosslinking density (
v
e
) on the concentration of
PVA
In this study, the PVA IPN was introduced as a secondary network. The PVA
could then be considered as a crosslinker of PAM network. Figure 8 shows the
relationship between the effective crosslinking density (
v
e
) and the concentration of PVA.
We observe the data of
v
e
had a linear trend versus the gel IPN concentration of PVA.
Here,
𝑣
𝑒
*is not considered as an absolute
𝑣
𝑒
, rather
𝑣
𝑒
* is determined by two factors:
crosslink density and
𝝌
1
, each having a proportionality relationship with
.
Therefore, to a certain extent, the loaded of PVA network could be considered as
the contributor of crosslinking in the experimental situation. Additionally, it is well
187
known that, network strength will increase with increasing crosslink density.[67,68] If so,
there should be a proportional relationship between the loading of PVA and the IPN gel
strength as well. The dependence of the complex modulus (G*) as a function of the
loading of PVA was plotted in Figure 9. We observe an approximately linear trend
between PVA content and elastic modulus. It could be concluded that, for PAM/PVA
IPN hydrogels, the strength could be easily tuned by adjusting the loading of the
secondary PVA network.
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