微凝胶的泊松-玻尔兹曼-弗洛里理论



半巨势:

\begin{equation} \begin{split} \Omega=&F_{\mathrm{ela}}+F_{\mathrm{exc}}+\Omega_{\mathrm{ele}} \\ =&\frac{3N_{\mathrm c}}{2}\left [\left (\frac{a}{a_0}\right )^{2}-\frac{1}{3}\ln\frac{a}{a_0} -1 \right ]k_BT+k_BT v b^3\frac{N^2}{a^3} \\ & -\frac{\epsilon}{2}\int_V (\nabla\phi)^2dV + \int_V n_{\mathrm e} \phi dV + \\ & k_BT\int_V \left [ n_+\ln \frac{n_+}{n_0}+n_-\ln \frac{n_-}{n_0} -(n_++n_--2n_0) \right ] dV \end{split} \label{Omega} \end{equation}

其中,$N_{\mathrm c}$为凝胶内部链数目,$a_0=(N_{\mathrm c}N_{\mathrm m})^{1/3}b=N^{1/3}b$为凝胶参考态大小,令参考态中链数密度和高分子链节数密度分别为$\nu_{\mathrm c}=3N_{\mathrm c}/(4\pi a_0^3)$和$\rho_{\mathrm c}=3N/(4\pi a_0^3)$,$n_{\mathrm e}$为体系中电荷密度,为

\begin{equation} n_{\mathrm e} = n_+-n_-+n_{\mathrm m} = n_+-n_-+\frac{3fN_{\mathrm c}N_{\mathrm m}}{4\pi a^3}\Theta(a-r) = n_+-n_-+\frac{3Z}{4\pi a^3}\Theta(a-r) \label{rho} \end{equation}

将半巨势变分:

\begin{equation} \frac{\delta \Omega}{\delta \phi} = \epsilon \nabla^2\phi + e n_{\mathrm e} = 0 \label{deltaphi} \end{equation}

这正是泊松方程。

\begin{equation} \frac{\delta \Omega}{\delta n_j} = q_j\phi + k_BT\ln n_j - \mu_j = 0 \label{deltanj} \end{equation}

得离子分布:

\begin{equation} n_{\pm} = n_{0}e^{\mp e\phi/k_BT} \label{n+-} \end{equation}

令$\psi=e\phi/(k_BT)$,$l_{\mathrm B}=e^2/(4\pi \epsilon k_BT)$,$\kappa^2=8\pi l_{\mathrm B} n_0$

\begin{equation} \nabla^2\psi =-\frac{e^2}{\epsilon k_BT} (n_+-n_-+\rho_{\mathrm m})=\kappa^2\sinh \psi - \frac{3Zl_{\mathrm B}}{4\pi a^3}\Theta(a-r) \label{deltapsi} \end{equation}

将\eqref{rho}、\eqref{n+-}三式代入\eqref{Omega},得

\begin{equation} \frac{\Omega_{\mathrm{ele}}}{k_BT} =-\int_V \left [\frac{1}{8\pi l_{\mathrm B}}(\nabla\psi)^2 - n_{\mathrm m} \psi + 2n_0\cosh \psi \right ] dV \label{Omelen} \end{equation}

凝胶大小由$\frac{\partial \Omega}{\partial a}=0$(略去弹性能中对数项和排除体积作用能)得到:

\begin{equation} \frac{3N_{\mathrm c}a}{a_0^2}-\frac{9Z}{a^4}\int_0^a \psi(r) r^2dr+\frac{3Z}{a}\psi(a)=0 \label{a} \end{equation}

用$a_0$标记长度,泊松-玻尔兹曼方程\eqref{deltapsi}化为:

\begin{equation} \frac{d^2\psi}{dr^2}+\frac{2}{r}\frac{d\psi}{dr}=\kappa^2\sinh \psi - \frac{3Z l_{\mathrm B}}{a^3}\Theta(a-r) \label{DLPoisson} \end{equation}

这里,$\kappa \rightarrow \kappa a_0$,$a\rightarrow a/a_0$,$l_{\mathrm B}\rightarrow l_{\mathrm B}/a_0$
凝胶大小方程:

\begin{equation} \nu_{\mathrm c}a-\frac{3f\rho_{\mathrm c}}{a^4}\int_0^a r^2 \psi(r) dr+\frac{f\rho_{\mathrm c}}{a}\psi(a)=0 \label{a'} \end{equation}

边界条件:

\begin{equation} \begin{split} & \frac{d\psi}{dr}\bigg |_{r=0}=0 \\ & \frac{d\psi}{dr}\bigg |_{r=R}=0 \\ & \psi(a^-)=\psi(a^+) \\ & \frac{d\psi}{dr}\bigg |_{y=a^-}=\frac{d\psi}{dr}\bigg |_{y=a^+} \end{split} \label{bc} \end{equation}

德拜-休克尔近似

如果$|\psi|\ll 1$,方程\eqref{DLPoisson}化为

\begin{equation} \frac{d^2\psi}{dr^2}+\frac{2}{r}\frac{d\psi}{dr}=\kappa^2 \psi -\frac{3Z l_{\mathrm B}}{a^3}\Theta(a-r) \label{DH} \end{equation}

解此方程,得

\begin{equation} \psi(r)= \begin{cases} & \frac{3Z l_{\mathrm B} }{a^3 \kappa ^2}-C_1\frac{ e^{\kappa r} -e^{-\kappa r}}{\kappa r} , 0\lt r\le a \\ & C_2\frac{e^{-\kappa r}}{\kappa r}+C_2\frac{e^{\kappa (r+2R)}}{\kappa r}\frac{\kappa R+1}{\kappa R-1}, a\le r\le R \end{cases} \label{psi} \end{equation}

其中,

\begin{equation} C_1=-\frac{3Z l_{\mathrm B}}{2\kappa ^2 a^3}\frac{ e^{\kappa a} (\kappa a -1) (\kappa R+1)-e^{ \kappa (2 R-a)}(\kappa a +1)(\kappa R-1)}{ e^{2 \kappa R} (\kappa R-1)+\kappa R+1 } \label{C1} \end{equation}

\begin{equation} C_2= \frac{3Z l_{\mathrm B}}{2\kappa ^2 a^3} \frac{ \left[e^{2 a \kappa } (\kappa a -1)+\kappa a +1\right] (\kappa R-1) e^{\kappa (2 R-a)}}{e^{2 \kappa R} (\kappa R-1)+\kappa R+1 } \label{C2} \end{equation}

将\eqref{psi}代入\eqref{a'},可得聚电解质大小$a$满足如下方程:

\begin{equation} \begin{split} & \nu_{\mathrm c}a+\frac{f\rho_{\mathrm c}}{a}\psi(a)= \\ &\frac{3f\rho_{\mathrm c}Z l_{\mathrm B}}{2 \kappa ^4 a^7}\frac{[(2 a-3) (\kappa a)^2+3] (\kappa R-1) e^{2 \kappa R}+[(2 a+3) (\kappa a)^2-3] (\kappa R+1)+3 e^{2 \kappa a} (\kappa a -1)^2 (\kappa R+1)-3 e^{2 \kappa (R-a)} (\kappa a +1)^2(\kappa R-1)}{e^{2 \kappa R} (\kappa R-1)+\kappa R+1} \end{split} \label{ares} \end{equation}

标签: 泊松-玻尔兹曼-弗洛里理论, 凝胶

添加新评论

captcha
请输入验证码