参考资料：THE JOURNAL OF CHEMICAL PHYSICS 137, 024902 (2012)

高分子凝胶的自由能

有两项，混合项

和弹性项

：

\begin{equation}
F=F_{mix}+F_{el}
\label{Fen}
\end{equation}

两种混合溶剂中的高分子凝胶，混合自由能为

\begin{equation}
F_{mix}=\frac{k_BT}{v_0}\left[ V_g f(\phi_{1g},\phi_{2g},\phi_3)+V_s f(\phi_{1s},\phi_{2s},0) \right ]
\label{Fmix}
\end{equation}

其中，$V_g$ 和 $V_s$ 分别为凝胶和凝胶外部溶液的体积，$v_0$为单体和溶剂分子的体积，$f(\phi_{1},\phi_{2},\phi_3)$为Flory-Huggins混合自由能：

\begin{equation}
f(\phi_{1},\phi_{2},\phi_3)=\phi_{1}\ln\phi_{1}+\phi_{2}\ln\phi_{2}+\sum_{i\lt j}\chi_{ij}\phi_i\phi_j
\label{FH}
\end{equation}

其中，$\phi_{1}$和$\phi_{2}$分别为溶剂分子的体积分数，下标$\phi_{ig}$和$\phi_{is}$分别表示第$i$（$i=1,2$）种组分凝胶内外的体积分数，$\phi_{3}$表示高分子网络的体积分数，$\chi_{ij}$为$i$和$j$两种组分的相互作用参数。

高分子网络的熵弹性能为

\begin{equation}
F_{el}=\frac{1}{2}k_BT\nu V_{g0}\left[ 3\left ( \frac{\phi_{30}}{\phi_3}\right )^{2/3}-2B\ln\left ( \frac{\phi_{30}}{\phi_3}\right ) \right ]
\label{Fel}
\end{equation}

其中，$V_{g0}$ 为处于参考态的凝胶的体积，$\nu$ 为交联点密度，$B$ 为非线性弹性系数，$\phi_{30}$ 为处于参考态的凝胶的体积分数。

凝胶内外满足不可压缩性条件：

\begin{equation}
\phi_{1g}+\phi_{2g}+\phi_3=1
\label{Incom1}
\end{equation}

\begin{equation}
\phi_{1s}+\phi_{2s}=1
\label{Incom2}
\end{equation}

可定义如下巨势：

\begin{equation}
\Omega=F_{mix}+F_{el}-\mu_2(V_g\phi_{2g}+V_s\phi_{2s})+\kappa (V_g+V_s)
\label{Grand}
\end{equation}

其中，$\mu_2$和$\kappa$ 分别为保证第二种组分和总体积不变的拉格朗日乘子。对巨势求极小可得平衡态：

\begin{equation}
\frac{\partial \Omega}{\partial \phi_{2g}}=\frac{\partial \Omega}{\partial \phi_{2s}}=0
\label{Mini1}
\end{equation}

\begin{equation}
\frac{\partial \Omega}{\partial V_g}=\frac{\partial \Omega}{\partial V_s}=0
\label{Mini2}
\end{equation}

由

\begin{equation*}
\frac{\partial \Omega}{\partial \phi_{2g}}=\frac{\partial F_{mix}}{\partial \phi_{2g}}-\mu_2V_g=\frac{k_BT}{v_0}V_g\frac{\partial f}{\partial \phi_{2g}}-\mu_2V_g=0
\end{equation*}

于是得

\begin{equation*}
\tilde{\mu}(\phi_{2g},\phi_3)=\frac{\partial f}{\partial \phi_{2g}}=\frac{v_0\mu_2 }{k_BT}
\end{equation*}

同理，由$\frac{\partial \Omega}{\partial \phi_{2s}}=0$可得

\begin{equation*}
\tilde{\mu}(\phi_{2s},0)=\frac{\partial f}{\partial \phi_{2s}}=\frac{v_0\mu_2 }{k_BT}
\end{equation*}

由方程\eqref{Incom1}和\eqref{Incom2}，可将\eqref{FH}化为：

\begin{equation*}
\begin{split}
f(\phi_{2},\phi_3)=&(1-\phi_{2}-\phi_3)\ln(1-\phi_{2}-\phi_3)+\phi_{2}\ln\phi_{2}\\
&+\chi_{12}(1-\phi_{2}-\phi_3)\phi_2 +\chi_{13}(1-\phi_{2}-\phi_3)\phi_3+\chi_{23}\phi_2\phi_3
\end{split}
\end{equation*}

于是得

\begin{equation}
\tilde{\mu}(\phi_{2},\phi_{3})=\ln\frac{\phi_2}{1-\phi_2-\phi_3}+\chi_{12}(1-2\phi_2-\phi_3)+(\chi_{23}-\chi_{13})\phi_3
\label{ChemPot}
\end{equation}

下面再看方程\eqref{Mini2}，

\begin{equation}
\frac{\partial \Omega}{\partial V_g}=\frac{\partial F_{mix}}{\partial V_g}+\frac{\partial F_{el}}{\partial V_g}+\kappa=0
\label{POV}
\end{equation}

方程\eqref{POV}中第一项

\begin{equation}
\frac{\partial F_{mix}}{\partial V_g}=\frac{k_BT}{v_0}\left [f(\phi_{2g},\phi_3)+V_g\frac{\partial f(\phi_{2g},\phi_3)}{\partial V_g} \right ]
\label{PFmVg1}
\end{equation}

其中，

\begin{equation*}
\begin{split}
V_g\frac{\partial f(\phi_{2g},\phi_3)}{\partial V_g}=& -\phi_{2g}\frac{\partial f(\phi_{2g},\phi_3)}{\partial \phi_{2g}}-\phi_{3}\frac{\partial f(\phi_{2g},\phi_3)}{\partial \phi_{3}} \\
=&-(\phi_{2g}+\phi_3)\ln(1-\phi_{2g}-\phi_3)-\phi_{2g}\ln\phi_{2g}\\
&-\chi_{12}(1-\phi_{2g}-\phi_3)\phi_{2g}+\chi_{12}\phi_{2g}^2+\chi_{13}\phi_{2g}\phi_3\\
&-\chi_{23}\phi_{2g}\phi_3+\phi_3+\chi_{12}\phi_{2g}\phi_3+\chi_{13}\phi_3^2\\
&-\chi_{13}(1-\phi_{2g}-\phi_3)\phi_3-\chi_{23}\phi_2\phi_3
\end{split}
\end{equation*}

代入方程\eqref{PFmVg1}，得

\begin{equation}
\begin{split}
\frac{\partial F_{mix}}{\partial V_g}\left (\frac{k_BT}{v_0}\right )^{-1}=& \ln(1-\phi_2-\phi_3)+\phi_3\\
&+\chi_{12}\phi_{2g}^2+\chi_{13}\phi_{3}^2\\
&-(\chi_{23}-\chi_{12}-\chi_{13})\phi_{2g}\phi_3
\end{split}
\label{PFmVg2}
\end{equation}

方程\eqref{POV}中第二项

\begin{equation}
\begin{split}
\frac{\partial F_{el}}{\partial V_g}=&\frac{\partial F_{el}}{\partial \phi_3}\frac{\partial \phi_3}{\partial V_g }=-\frac{\phi_3}{V_g}\frac{\partial F_{el}}{\partial \phi_3}\\
=&-k_BT\nu \phi_3 \frac{V_{g0}}{V_g}\left [-\left (\frac{\phi_{30}}{\phi_3}\right )^{2/3}\frac{1}{\phi_3}+\frac{B}{\phi_3} \right ]\\
=&k_BT\nu \left [\left (\frac{\phi_{3}}{\phi_{30}}\right )^{1/3}-B\frac{\phi_{3}}{\phi_{30}} \right ]
\end{split}
\label{PFelVg}
\end{equation}

方程\eqref{Mini2}中第二个偏导的结果为

\begin{equation}
\frac{\partial \Omega}{\partial V_s}=\frac{\partial F_{mix}}{\partial V_s}+\kappa=0
\label{POVs}
\end{equation}

其中

\begin{equation}
\begin{split}
\frac{\partial F_{mix}}{\partial V_s}=&\frac{k_BT}{v_0}\left [f(\phi_{2s},0)+V_s\frac{\partial f(\phi_{2s},0)}{\partial V_s} \right ]\\
=&\frac{k_BT}{v_0}\left [\ln(1-\phi_{2s})+\chi_{12}\phi_{2s}^2\right]
\end{split}
\label{PFmVs}
\end{equation}

综上，凝胶平衡态结构由如下方程给出：

\begin{equation}
\begin{split}
&\ln\frac{\phi_{2g}}{1-\phi_{2g}-\phi_3}+\chi_{12}(1-2\phi_{2g}-\phi_3)+(\chi_{23}-\chi_{13})\phi_3=\\
&\ln\frac{\phi_{2s}}{1-\phi_{2s}}+\chi_{12}(1-2\phi_{2s})
\end{split}
\label{Struc1}
\end{equation}

\begin{equation}
\begin{split}
&\ln(1-\phi_{2s})-\ln(1-\phi_{2g}-\phi_3)-\phi_3\\
&-\chi_{12}(\phi_{2s}^2-\phi_{2g}^2)-\chi_{13}\phi_3^2+(\chi_{23}-\chi_{12}-\chi_{13})\phi_{2g}\phi_3\\
&-\nu v_0 \left [\left (\frac{\phi_{3}}{\phi_{30}}\right )^{1/3}-B\frac{\phi_{3}}{\phi_{30}} \right ]=0
\end{split}
\label{Struc2}
\end{equation}