有限伸长的弱聚电解质刷



传统的聚电解质刷的强拉伸自洽场理论里,聚电解质链是可以无限伸长的。文章The Journal of Chemical Physics 146, 214901 (2017)考虑了链的有限伸长性,改进了强拉伸自洽场理论。此文的方法还可以推广至弱聚电解质刷。

设弱聚电解质为弱酸,链长为$N$,库恩长度为$a$,带电分率分布为$\alpha(x)$,电离平衡关系为:

\begin{equation}
\begin{split}
\frac{\alpha(x)}{1-\alpha(x)}=&\frac{K_a}{c_{H^+}(x)}=\frac{K_a}{C_{H^+}}\exp[\psi(x)]\
=&\frac{\alpha_b}{1-\alpha_b}\exp[\psi_1(x)]
\end{split}
\label{disequil}
\end{equation}

其中,$K_a$为电离平衡常数,$C_{H^+}$为本体溶液氢离子浓度,$\psi_1(x)$为刷内电势分布。

参考Langmuir, 2011, 27 (17), 10615–10633Macromol. Theory Simul. 2003, 12, 223–228,刷内分子场为

\begin{equation}
\begin{split}
U(x)=&\ln[1-\alpha(x)]\
=&\Lambda+3\ln\cos\frac{\pi x}{2Na}=\Lambda+3\ln\cos\frac{\pi x}{2L}
\end{split}
\label{potential}
\end{equation}

这里$\Lambda$为待定常数。

设$\alpha(x=H)=\alpha_H$,代入\eqref{potential}式,得$\lambda=\ln(1-\alpha_H)-3\ln\cos\frac{\pi H}{2L}$,代入\eqref{potential}式,得

\begin{equation}
\alpha(x)=1-(1-\alpha_H)\left ( \frac{\cos\frac{\pi x}{2L}}{\cos\frac{\pi H}{2L}} \right )^3
\label{chgfrac}
\end{equation}

代入\eqref{disequil}式,得电势为

\begin{equation} \begin{split} \psi_1(x)=&\ln\left (\frac{\alpha(x)}{1-\alpha(x)}\frac{1-\alpha_b}{\alpha_b} \right ) \\ =&\ln \left[\frac{1- (1-\alpha_H)\left ( \frac{\cos\frac{\pi x}{2L}}{\cos\frac{\pi H}{2L}} \right )^3}{(1-\alpha_H)\left ( \frac{\cos\frac{\pi x}{2L}}{\cos\frac{\pi H}{2L}} \right )^3}\frac{1-\alpha_b}{\alpha_b}\right]\\ =& \ln \left[ \frac{1-\alpha_b}{\alpha_b(1-\alpha_H)}\left ( \frac{\cos\frac{\pi H}{2L}}{\cos\frac{\pi x}{2L}} \right )^3-\frac{1-\alpha_b}{\alpha_b}\right]\\ =&\ln\left[\left ( \frac{\cos\frac{\pi H}{2L}}{\cos\frac{\pi x}{2L}} \right )^3-(1-\alpha_H) \right] +\ln\frac{1-\alpha_b}{\alpha_b(1-\alpha_H)} \end{split} \label{psi1} \end{equation}

刷内单位面积上过量电荷

\begin{equation}
Q_1=\frac{1}{4\pi l_B}\frac{\mathrm d\psi_1(x)}{\mathrm dx}\Bigg |_{x=H}=\frac{3}{8Ll_B\alpha_H }\tan\frac{\pi H}{2L}
\label{Q1}
\end{equation}

对应的古依-查普曼(Gouy-Chapman)长度为

\begin{equation}
\Lambda_1=\frac{s}{2\pi l_BQ_1}=\frac{4\alpha_H sL}{3\pi}\cot\frac{\pi H}{2L}
\label{Gouy-Chapman}
\end{equation}

Langmuir, 2011, 27 (17), 10615–10633,知刷外电势分布为

\begin{equation}
\psi_2(x)=-2\ln\left[\frac{\left (\kappa \Lambda_1+\sqrt{(\kappa \Lambda_1)^2+1}-1 \right )+\left (\kappa \Lambda_1-\sqrt{(\kappa \Lambda_1)^2+1}+1 \right )e^{-\kappa(x-H)}}{\left (\kappa \Lambda_1+\sqrt{(\kappa \Lambda_1)^2+1}-1 \right )-\left (\kappa \Lambda_1-\sqrt{(\kappa \Lambda_1)^2+1}+1 \right )e^{-\kappa(x-H)}}\right ]
\label{psi2}
\end{equation}

其中$\kappa^{-1}$为德拜长度。

刷边缘处电势连续,

\begin{equation}
\psi_1(H)=\ln \left [ \frac{\alpha_H(1-\alpha_b)}{\alpha_b(1-\alpha_H)} \right ]=\psi_2(H)=-2\ln \left [ \frac{\kappa \Lambda_1}{\sqrt{(\kappa \Lambda_1)^2+1}-1} \right ]
\label{psiH}
\end{equation}

于是,可得

\begin{equation}
\frac{\alpha_H(1-\alpha_b)}{\alpha_b(1-\alpha_H)} = \left ( \frac{\kappa \Lambda_1}{\sqrt{(\kappa \Lambda_1)^2+1}-1} \right )^{-2}
\label{alphaH}
\end{equation}

刷内正负离子密度分布分别为

\begin{equation}
c_+(x)=c_{\mathrm s}e^{-\psi(x)}
\label{c+}
\end{equation}

\begin{equation}
c_-(x)=c_{\mathrm s}e^{\psi(x)}
\label{c-}
\end{equation}

高分子密度分布

\begin{equation} \begin{split} c_{\mathrm p}(x)=&\frac{1}{\alpha(x)}\left (c_+(x)-c_-(x)+\frac{1}{4\pi l_B}\frac{\mathrm d^2\psi_1(x)}{\mathrm dx^2} \right )\\ =&\frac{1}{\alpha(x)}\left (C_se^{-\psi_1(x)}-C_se^{\psi_1(x)}+\frac{1}{4\pi l_B}\frac{\mathrm d^2\psi_1(x)}{\mathrm dx^2} \right )\\ =&\frac{\kappa^2}{8\pi l_B\alpha(x)}\left (e^{-\psi_1(x)}-e^{\psi_1(x)} \right )+\frac{1}{4\pi l_B\alpha(x)}\frac{\mathrm d^2\psi_1(x)}{\mathrm dx^2} \end{split} \label{cp} \end{equation}

具体表达式比较冗繁,可以由Mathematica 等符号计算软件给出给出。上式中$C_s$为本体盐浓度,$\kappa^{-1}=(8\pi l_BC_s)^{-1/2}$为德拜屏蔽长度。

刷的厚度由下式给出

\begin{equation}
\int_0^Hc_{\mathrm p}(x)\mathrm dx=\frac{N}{s}
\label{cpcons}
\end{equation}

标签: 强拉伸理论, sst, 聚电解质刷

添加新评论

captcha
请输入验证码