参考文献:J. Chem. Phys. 141, 234902 (2014)
聚电解质微凝胶由\(N\)条链组成,每条链链长为\(m\),链带电分率为\(\alpha\),电离出反离子数目为\(Z=fNm\)。微凝胶半径为\(a\)。溶液中平均每个凝胶占据的体积为\(\frac{4\pi}{3}R^3\)
体系自由能(以\(k_BT\)约化)为:
\begin{equation} \begin{split} F=& F_{ela}+\int f(\vec{r})\mathrm d\vec{r}\\ =& F_{ela}+\int [f_{ele}(\vec{r})+f_{tr}(\vec{r})+f_{mix}(\vec{r})]\mathrm d\vec{r} \end{split} \label{Fenergy} \end{equation}
\eqref{Fenergy}式中\(F_{ela}\)为链的熵弹性能:
\begin{equation} F_{ela}=\frac{3N}{2}\left [\left (\frac{\phi_0}{\phi}\right )^{2/3}-\ln\left (\frac{\phi_0}{\phi}\right ) \right ] \label{Fela} \end{equation}
其中\(\phi_0\)、\(\phi\)分别为凝胶在参考态和当前态的链节体积分数。
\eqref{Fenergy}式中\(f_{ele}(\vec{r})\)为静电自由能密度:
\begin{equation} f_{ele}(\vec{r})=-\frac{1}{8\pi l_B}\mid \nabla \psi(\vec{r}) \mid^2+[n_+(\vec{r})-n_-(\vec{r})-\alpha\phi/v_0]\psi(\vec{r}) \label{fele} \end{equation}
其中\(\psi(\vec{r})\)为电势(以\(k_BT/e\)约化),\(n_{\pm}\)为小离子的数密度。
\eqref{Fenergy}式中\(f_{tr}(\vec{r})\)为小离子的平动熵:
\begin{equation} f_{tr}(\vec{r})=n_+(\vec{r})[\ln (n_+\lambda_B^3)-1]+n_-(\vec{r})[\ln (n_-\lambda_B^3)-1]-\mu_+n_+(\vec{r})-\mu_-n_-(\vec{r}) \label{ftr} \end{equation}
其中\(\lambda_B\)为德布罗意热波长,\(\mu_{\pm}\)为正负小离子的化学势。
\eqref{Fenergy}式中\(f_{mix}(\vec{r})\)为溶剂和凝胶的混合自由能:
\begin{equation} f_{tr}(\vec{r})=\frac{1}{v_0}[(1-\phi)\ln (1-\phi)+\chi \phi(1-\phi)] \label{fmix} \end{equation}
其中\(v_0\)为溶剂分子和凝胶链节的体积。
\eqref{Fenergy}式对\(\psi(\vec{r})\)变分,得
\begin{equation} \nabla^2 \psi(\vec{r})=-4\pi l_B[n_+(\vec{r})-n_-(\vec{r})-\alpha \phi/v_0] \label{PB} \end{equation}
\eqref{Fenergy}式对\(n_{\pm}(\vec{r})\)变分,并整理,得
\begin{equation} n_{\pm}(\vec{r})=\mu_{\pm}e^{\mp\psi(\vec{r})} \label{nden} \end{equation}
在本体溶液\(\psi=0\),\(n_{\pm}=c_s\),这里\(c_s\)为本体溶液盐浓度。所以上式可化为
\begin{equation} n_{\pm}(\vec{r})=c_se^{\mp\psi(\vec{r})} \label{ndis} \end{equation}
\eqref{Fela}式对\(V\)求导,得熵弹性力
\begin{equation} \Pi_{ela}=-\frac{\mathrm dF_{ela}}{\mathrm dV}=-\frac{N}{V_0}\left [\left ( \frac{\phi}{\phi_0}\right )^{1/3} -\frac{2}{3}\frac{\phi}{\phi_0} \right ] \label{Piela} \end{equation}
凝胶内小分子渗透压为
\begin{equation} \begin{split} \Pi_{g}=&n_+\frac{\partial f}{\partial n_+}+n_-\frac{\partial f}{\partial n_-}+\mid \nabla \psi\mid\frac{\partial f}{\partial \nabla \psi}+\phi\frac{\partial f}{\partial \phi}-f\\ =&-\frac{1}{8\pi l_B}\mid \nabla \psi(\vec{r}) \mid^2+n_+(\vec{r})+n_-(\vec{r}) \\ & -\frac{1}{v_0}\left [\ln(1-\phi)+\chi\phi^2+\phi\right ] \end{split} \label{Pig} \end{equation}
本体溶液小分子渗透压为
\begin{equation} \Pi_{b}=2c_s \label{Pib} \end{equation}
体系渗透压差为
\begin{equation}
\begin{split}
\Pi_{os}=\Pi_{g}-\Pi_{b}=&-\frac{1}{8\pi l_B}\mid \nabla \psi(\vec{r}) \mid^2+n_+(\vec{r})+n_-(\vec{r})-2c_s \
& -\frac{1}{v_0}\left [\ln(1-\phi)+\chi\phi^2+\phi\right ]
\end{split}
\label{Pios}
\end{equation}
渗透压差与熵弹性力二者平衡:
\begin{equation} \begin{split} \Pi_{ela}+\Pi_{os}=&-\frac{N}{V_0}\left [\left ( \frac{\phi}{\phi_0}\right )^{1/3} -\frac{2}{3}\frac{\phi}{\phi_0} \right ]\\ &-\frac{1}{8\pi l_B}\mid \nabla \psi(\vec{r}) \mid^2+n_+(\vec{r})+n_-(\vec{r})-2c_s \\ & -\frac{1}{v_0}\left [\ln(1-\phi)+\chi\phi^2+\phi\right ]\\ =&0 \end{split} \label{Pieq} \end{equation}
此式中\(\vec{r}\)可取凝胶内部任意一点,显然取\(r=0\)处最为方便。
体系状态由\eqref{PB}、\eqref{ndis}和\eqref{Pieq}组成的方程组给出。体系具有球对称性,以上方程在球坐标系求解。要求解的方程即为:
\begin{equation*} \begin{cases} \frac{\mathrm d^2}{\mathrm dr^2}\psi(r)+\frac{2}{r}\frac{\mathrm d}{\mathrm dr}\psi(r)=-4\pi l_B[n_+(r)-n_-(r)-\alpha\phi/v_0]\\ \\ n_{\pm}(r)=c_se^{\mp\psi(r)}\\ \\ -\frac{N}{V_0}\left [\left ( \frac{\phi}{\phi_0}\right )^{1/3} -\frac{2}{3}\frac{\phi}{\phi_0} \right ]+n_+(0)+n_-(0)-2c_s\\ -\frac{1}{v_0}\left [\ln(1-\phi)+\chi\phi^2+\phi\right ]=0 \\ \phi=\frac{mNv_0}{\frac{4\pi a_3}{3}} \Theta(a-r) \end{cases} \end{equation*}
其中\(\Theta(x)\)为阶跃函数。
边界条件:
\begin{equation*} \begin{cases} \frac{\mathrm d}{\mathrm dr}\psi(r)\Big|_{r=0}=0 \\ \psi(r=a^-)=\psi(r=a^+)\\ \frac{\mathrm d}{\mathrm dr}\psi(r)\Big|_{r=R}=0 \end{cases} \end{equation*}