Model Derivation
The mean-field Flory-Huggins-Rhener theory enables predicting the phase
behavior of gel matrices, providing an intuitive picture of the
underlying statistical thermodynamics of a solution
system8. In this theory, the thermodynamic mechanism
of gel volume phase transition (swelling and deswelling) is described as
an interplay between mixing free energy of the binary mixture (i.e.,
polymer and solvent) and elastic free energy of the solid-like polymer
network9.
The free energy of mixing is hypothesized to be quantified using a
liquid lattice model of n unit cells11. The
total Gibbs free energy (ΔF ) of the system is given by the
summation of mixing free energy (ΔFmix ) and
elastic free energy (ΔFel ):
\begin{equation}
F=F_{\text{mix}}+F_{\text{el}}.\nonumber \\
\end{equation}Typically, the mixing free energy of a binary system has been considred
to derive the well-known equation as below9:
\begin{equation}
F_{\text{mix}}=nk_{B}T\left[\left(1-\phi\right)\ln\left(1-\phi\right)+\chi^{\text{Flory}}\phi\left(1-\phi\right)\right],\nonumber \\
\end{equation}where φ is the total polymer volume fraction andχFlory is the Flory-Huggins interaction
parameters between the polymer and solvent, T is the temperature,
and kB is the Boltzmann constant.
The elastic free energy of the crosslinked polymer network is given by
in the simplest form as follows9:
\begin{equation}
\Delta F_{\text{el}}=\frac{3k_{B}T}{2N_{x}}\left[\left(\frac{\phi_{0}}{\phi}\right)^{\frac{2}{3}}-1-ln\left(\frac{\phi_{0}}{\phi}\right)\right],\nonumber \\
\end{equation}where φ0 represents the polymer volume fraction
at a reference state, and Nx is the degree of
polymerization between crosslinks.
In our model, we modify the mixing free energy term as that of a ternary
mixture, which means that the system comprises two types of polymers
(i.e., tissue-derived biopolymer and thermoresponsive pNIPAM) and a
common solvent (i.e., water). We assume that the elastic free energy is
attributed to the elasticity of the crosslinked biopolymer network, and
generalize the expression by introducing a dimensionless parameter,α > 0. Thus, the ternary mixing free energy and the
elastic free energy are given by the following:
\begin{equation}
F_{\text{mix}}=nk_{B}T\left[\frac{\phi_{\text{NIPA}}}{N_{\text{NIPA}}}\ln\phi_{\text{NIPA}}+\left(1-\phi_{\text{bio}}-\phi_{\text{NIPA}}\right)\ln\left(1-\phi_{\text{bio}}-\phi_{\text{NIPA}}\right)+\chi_{\text{bio}}\phi_{\text{bio}}\left(1-\phi_{\text{bio}}-\phi_{\text{bio}}\right)+\frac{\phi_{\text{bio}}}{N_{\text{bio}}}\ln\phi_{\text{bio}}+\chi_{\text{NIPA}}\phi_{\text{NIPA}}\left(1-\phi_{\text{bio}}-\phi_{\text{NIPA}}\right)+\chi_{bio-NIPA}{\phi_{\text{bio}}\phi}_{\text{NIPA}}\right]\approx k_{B}T\left[\frac{V}{v_{w}}\left(1-\frac{V_{\text{bio}}}{V}-\frac{V_{\text{NIPA}}}{V}\right)\ln\left(1-\frac{V_{\text{bio}}}{V}-\frac{V_{\text{NIPA}}}{V}\right)+\left(\chi_{\text{bio}}n_{\text{bio}}+\chi_{\text{NIPA}}n_{\text{NIPA}}\right)\left(1-\frac{V_{\text{bio}}}{V}-\frac{V_{\text{NIPA}}}{V}\right)+\chi_{bio-NIPA}n_{\text{NIPA}}\frac{V_{\text{bio}}}{V}\right],\nonumber \\
\end{equation}where\(n_{\text{NIPA}}=n\phi_{\text{NIPA}}=n\frac{V_{\text{NIPA}}}{V}\),\(n_{\text{NIPA}}v_{w}=V_{\text{NIPA}}\),\(n_{\text{bio}}v_{w}=V_{\text{bio}}\), \(nv_{w}=V\),
and
\begin{equation}
\Delta F_{\text{el}}=\frac{k_{B}T}{N_{x}}\left[\frac{3}{2}\left\{\left(\frac{\phi_{bio\_i}}{\phi_{\text{bio}}}\right)^{\frac{2}{3}}-1\right\}-\alpha ln\left(\frac{\phi_{bio\_i}}{\phi_{\text{bio}}}\right)\right]=\frac{k_{B}T}{N_{x}}\left[\frac{3}{2}\left\{\left(\frac{V}{V_{i}}\right)^{\frac{2}{3}}-1\right\}-\alpha ln\left(\frac{V}{V_{i}}\right)\right].\nonumber \\
\end{equation}Here, we denote n and vw as the total
number of lattice sites and the volume of a lattice (unit cell),
respectively. In addition, φbio andφNIPA are volume fraction of biopolymer and
pNIPAM, respectively. Similarly, Vbio andVNIPA are defined as the volume occupied by
biopolymer and by pNIPAM, respectively. The χbio ,χNIPA , and χbio-NIPA are
Flory-Huggins interaction parameters of biopolymer-water, pNIPAM-water,
and biopolymer-pNIPAM, respectively. The initial biopolymer volume
fraction φbio_i at the initial gel volumeVi and the biopolymer volume fractionφbio at a given gel volume V are defined
as \(\phi_{bio\_i}=\frac{V_{\text{bio}}}{V_{i}}\) and\(\phi_{\text{bio}}=\frac{V_{\text{bio}}}{V}\), respectively, whereVbio is the volume occupied by biopolymer chains.
We assume that the Flory-Huggins interaction parameter between
biopolymer and pNIPAM can be approximated as a decoupled form as
follows:
\begin{equation}
\chi_{bio-NIPA}\approx\chi_{\text{bio}}+\chi_{\text{NIPA}}-2\sqrt{\chi_{\text{bio}}}\sqrt{\chi_{\text{NIPA}}}.\nonumber \\
\end{equation}Then, the partial derivative of the mixing free energy with volume can
be approximated as:
\begin{equation}
\frac{\partial F_{\text{mix}}}{\partial V}\approx\frac{k_{B}T}{v_{w}}\left[\ln\left(1-\frac{V_{\text{bio}}+V_{\text{NIPA}}}{V}\right)+\frac{V_{\text{bio}}+V_{\text{NIPA}}}{V}+\chi_{\text{bio}}\frac{V_{\text{bio}}^{2}}{V^{2}}+\chi_{\text{NIPA}}\frac{V_{\text{NIPA}}^{2}}{V^{2}}+2\sqrt{\chi_{\text{bio}}\chi_{\text{NIPA}}}\frac{V_{\text{bio}}V_{\text{NIPA}}}{V^{2}}\right]\approx\frac{k_{B}T}{v_{w}}\left(\frac{V_{i}}{V}\right)^{2}\left(\sqrt{\chi_{\text{bio}}}\frac{V_{\text{bio}}}{V_{i}}+\sqrt{\chi_{\text{NIPA}}}\frac{V_{\text{NIPA}}}{V_{i}}\right)^{2}.\nonumber \\
\end{equation}Additionally, the time derivative of the elastic free energy becomes:
\begin{equation}
\frac{\partial F_{\text{el}}}{\partial V}=\frac{k_{B}T}{{N_{x}V}_{i}}\left[\left(\frac{V_{i}}{V}\right)^{\frac{1}{3}}-\alpha\left(\frac{V_{i}}{V}\right)\right].\nonumber \\
\end{equation}Equilibrium swelling and deswelling are calculated by assuming the
balance of elastic and mixing contributions to the osmotic pressure in
the gel matrix9,12. Then, the osmotic pressure is the
negative time-derivative of the total free energy and equals zero when
the contributions from mixing and elastic free energies are in
equilibrium:
\begin{equation}
\Pi=-\frac{\partial F}{\partial V}=-\left(\frac{\partial F_{\text{mix}}}{\partial V}+\frac{\partial F_{\text{el}}}{\partial V}\right)=-\left[\frac{k_{B}T}{v_{w}}\left(\frac{V_{i}}{V}\right)^{2}\left(\sqrt{\chi_{\text{bio}}}\frac{V_{\text{bio}}}{V_{i}}+\sqrt{\chi_{\text{NIPA}}}\frac{V_{\text{NIPA}}}{V_{i}}\right)^{2}+\frac{k_{B}T}{{N_{x}V}_{i}}\left\{\left(\frac{V_{i}}{V}\right)^{\frac{1}{3}}-\alpha\left(\frac{V_{i}}{V}\right)\right\}\right]=0.\nonumber \\
\end{equation}This equation gives the final formula as follows:
\begin{equation}
\Lambda+\Gamma\left\{\left(\frac{V}{V_{i}}\right)^{\frac{5}{3}}-\alpha\left(\frac{V}{V_{i}}\right)\right\}=0,\nonumber \\
\end{equation}where the coefficients Λ > 0 and Γ> 0 are defined as
\begin{equation}
\Lambda=\frac{1}{v_{w}}\left(\sqrt{\chi_{\text{bio}}}\frac{V_{\text{bio}}}{V_{i}}+\sqrt{\chi_{\text{NIPA}}}\frac{V_{\text{NIPA}}}{V_{i}}\right)^{2}\nonumber \\
\end{equation}and
\begin{equation}
\Gamma=\frac{1}{{N_{x}V}_{i}}.\nonumber \\
\end{equation}Therefore, in isobaric conditions, an analytical relation between the
gel deswelling parameter V/Vi and the variablesNx and χNIPA is obtained.
If we substitute χNIPA(T) as an explicit function
of only temperature in a conventional expression12,
\begin{equation}
\chi_{\text{NIPA}}\left(T\right)=\frac{1}{2}-A\left(1-\frac{\Theta}{T}\right),\nonumber \\
\end{equation}the analytical relation between T , Nx , andV/Vi is finally derived as follows:
\begin{equation}
\frac{T}{\Theta}=\left[1-\frac{1}{2A}+\frac{1}{A}\left[\frac{V_{i}}{V_{\text{NIPA}}}\sqrt{\frac{1}{N_{x}}\frac{v_{w}}{V_{i}}\left\{\alpha\left(\frac{V}{V_{i}}\right)-\left(\frac{V}{V_{i}}\right)^{\frac{5}{3}}\right\}}-\frac{\sqrt{\chi_{\text{bio}}}}{r_{\text{polym}}}\right]^{2}\right]^{-1},\nonumber \\
\end{equation}where Θ is the Flory’s Θ temperature of pNIPAM, A is a
factor which can be experimentally determined. In addition, the doping
ratio of thermoresponsive polymer to tissue-derived biopolymer is
defined as
\begin{equation}
r_{\text{polym}}=\frac{V_{\text{NIPA}}}{V_{\text{bio}}}.\nonumber \\
\end{equation}Here, 0 < rpolym ≤ 1, and we assume
that Vi/VNIPA = 20 for all cases.
The parameter values and plotting variables are summarized in Tables 1
and 2, which are used for the numerical graphs in Figs. 2 to 5.