混合溶剂中的高分子凝胶理论推导

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

高分子凝胶的自由能\(F\)有两项，混合项\(F_{mix}\)和弹性项\(F_{el}\)：

\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}