Zhulina 的高分子刷理论

高分子刷的解析平均场理论有两种表述方式。一个是MWC理论(Macromolecules 1988, 21, 2610-2619),另外一个就是Zhulina和Birshtein这两位俄罗斯老太太的理论(Macromolecules 1991, 24, 140-149)。后者在物理上更直接,重新整理如下。

高分子刷的(平均一根链的)自由能$\Delta F$为链的熵弹性$\Delta F_{el}$与排除体积作用能$\Delta F_{conc}$之和:

\begin{equation} \Delta F=\Delta F_{el}+\Delta F_{conc} \tag{1}\label{eq:F} \end{equation}

刷内相互积作用能

\begin{equation} \Delta F_{conc}=\frac{\sigma}{a^3}\int f[\varphi(x)] \mathrm dx \tag{2}\label{eq:Fconc} \end{equation}

其中$\sigma$为平均一根链在接枝面上所占据的面积,$\varphi(x)$为高分子体积分数,$f[\varphi(x)]/a^3$为相互自由能密度。

接枝链的熵弹性:

\begin{equation*} \begin{split} \Delta F_{el}(x')&=\frac{3}{2a^2}\int_0^N\left (\frac{\mathrm dx}{\mathrm dn}\right )^2\mathrm dn=\frac{3}{2a^2}\int_0^N\frac{\mathrm dx}{\mathrm dn}\frac{\mathrm dx}{\mathrm dn}\mathrm dn\\ &=\frac{3}{2a^2}\int_0^{x'}\frac{\mathrm dx}{\mathrm dn}\mathrm dx =\frac{3}{2a^2}\int_0^{x'}\frac{\mathrm dx}{\mathrm dn}\mathrm dx\\ &=\frac{3}{2a^2}\int_0^{x'}E(x,x')\mathrm dx \end{split} \end{equation*}

其中$x'$为高分子链的末端所在位置,$H$为刷的高度,$E(x,x')=\frac{\mathrm dx}{\mathrm dn}$,并满足:

\begin{equation}\int_0^{x'} \frac{1}{E(x,x')}\mathrm dx=N \tag{3}\label{eq:Econs}\end{equation}

接枝链的末端的分布为$g'(x')$,$g'(x')\mathrm dx'$为$x'$处$A\mathrm dx'$体积范围内接枝链末端的数目,满足

\begin{equation*}A\int_0^H g'(x')\mathrm dx'=n_P\end{equation*}

其中$A$为接枝表面的总面积,$n_P$为接枝链的总数目。

平均一条链的熵弹性能为:

\begin{equation} \begin{split} \Delta F_{el}&=\frac{A}{n_P}\int_0^H \Delta F_{el}(x')g'(x')\mathrm dx'\\ &=\frac{3}{2a^2}\int_0^H g(x')\mathrm dx'\int_0^{x'}E(x,x')\mathrm dx \end{split}\tag{4}\label{eq:Fel}\end{equation}

其中,$g(x')=\frac{A}{n_P}g'(x')$,为$x'$处$\mathrm dx'$厚度范围内接枝链末端的数目,满足$\int_0^H g(x')\mathrm dx'=1$。

高分子体积分数$\varphi(x)$满足:

\begin{equation} \begin{split} \varphi(x)&=\frac{a^3}{\sigma}\int_0^H\frac{\mathrm dn}{\mathrm dx} g(x')\mathrm dx'\\ &=\frac{a^3}{\sigma}\int_0^H\frac{g(x')}{E(x,x')} \mathrm dx' \end{split}\tag{5}\label{eq:varphi}\end{equation}

\begin{equation}\sigma\int_0^{H} \varphi(x)\mathrm dx=Na^3\tag{6}\label{eq:varphicons}\end{equation}

要得到刷的结构,需要对如下泛函求变分:

\begin{equation}F'=\Delta F+\lambda_1 \int_0^{H} \varphi(x)\mathrm dx +\int_0^H \lambda_2(x')\mathrm dx'\int_0^{x'} \frac{1}{E(x,x')}\mathrm dx \tag{7}\label{eq:Fp}\end{equation}

其中$\lambda_1$和$\lambda_2(x')$分别为拉格朗日乘子。

对$F'$变分有:

\begin{equation} \begin{split} \delta F'=&\delta \Delta F_{el}+\delta \Delta F_{conc} + \lambda_1 \int_0^{H}\delta \varphi(x)\mathrm dx\\ &-\int_0^H \lambda_2(x')\mathrm dx'\int_0^{x'} \frac{\delta E(x,x')}{E^2(x,x')}\mathrm dx\\ =& \frac{3}{2a^2}\int_0^H \mathrm dx'\int_0^{x'}\left [g(x')\delta E(x,x') + E(x,x') \delta g(x')\right ]\mathrm dx \\ &+\frac{\sigma}{a^3}\int \frac{\delta f[\varphi(x)]}{\delta \varphi(x)} \delta \varphi(x) \mathrm dx + \lambda_1 \int_0^{H}\delta \varphi(x)\mathrm dx\\ &-\int_0^H \lambda_2(x')\mathrm dx'\int_0^{x'} \frac{\delta E(x,x')}{E^2(x,x')}\mathrm dx \end{split} \tag{8}\label{eq:var} \end{equation}

根据方程\eqref{eq:varphi},有:

\begin{equation} \delta \varphi(x)=\frac{a^3}{\sigma}\int_0^H\left [\frac{\delta g(x')}{E(x,x')}-\frac{g(x')}{E^2(x,x')}\delta E(x,x') \right ] \mathrm dx' \tag{9}\label{eq:varvarphi} \end{equation}

将方程\eqref{eq:varvarphi}带入方程\eqref{eq:var},得

\begin{equation} \begin{split} \delta F'= &\int_0^H \mathrm dx' \int_0^{x'} \mathrm dx \delta E(x,x')\\ &\left [\frac{3g(x')}{2a^2}-\frac{\lambda_2(x')}{E^2(x,x')}-\left (\lambda_1+\frac{\delta f[\varphi(x)]}{\delta \varphi(x)}\right )\frac{g(x')}{E^2(x,x')} \right ]\\ & \int_0^H \delta g(x') \mathrm dx' \int_0^{x'} \mathrm dx \\ &\left [\frac{3E(x,x')}{2a^2}+\frac{1}{E(x,x')}\left (\lambda_1+\frac{\delta f[\varphi(x)]}{\delta \varphi(x)}\right ) \right ] \end{split}\tag{10}\label{eq:varesult} \end{equation}

相应地我们可得如下两个变分方程:

\begin{equation} \frac{3g(x')}{2a^2}-\frac{\lambda_2(x')}{E^2(x,x')}-\left (\lambda_1+\frac{\delta f[\varphi(x)]}{\delta \varphi(x)}\right )\frac{g(x')}{E^2(x,x')} =0 \tag{11}\label{eq:var1} \end{equation}

\begin{equation}
\frac{3E(x,x')}{2a^2}+\frac{1}{E(x,x')}\left (\lambda_1+\frac{\delta f[\varphi(x)]}{\delta \varphi(x)}\right )=0
\tag{12}\label{eq:var2}
\end{equation}

由方程\eqref{eq:var1},

\begin{equation} E^2(x,x')=U_1(x')-U_2(x) \tag{13}\label{eq:EU12} \end{equation}

其中,

\begin{equation} U_1(x')=\frac{2a^2\lambda_2(x')}{3g(x')} \tag{14}\label{eq:U1} \end{equation} \begin{equation} U_2(x)=-\frac{2a^2}{3}\left ( \lambda_1+\frac{\delta f[\varphi(x)]}{\delta \varphi(x)}\right ) \tag{15}\label{eq:U2} \end{equation}

链的末端不受拉伸,则$E(x,x)=0$,于是 $U_1=U_2$,我们有

\begin{equation} E(x,x')=\sqrt{U(x')-U(x)} \tag{16}\label{eq:EU} \end{equation}

$U(x)$仍是未知函数,将方程\eqref{eq:EU}代入方程\eqref{eq:Econs},得

\begin{equation} U(x)=\frac{\pi^2 x^2}{4N^2} \tag{17}\label{eq:Ux} \end{equation}

将方程\eqref{eq:Ux}代入方程\eqref{eq:EU}得

\begin{equation} E(x,x')=\frac{\pi }{2N}\sqrt{x'^2-x^2}\tag{18}\label{eq:Ex} \end{equation}

将方程\eqref{eq:Ux}代入方程\eqref{eq:U2}得

\begin{equation}\lambda_1+\frac{\delta f[\varphi(x)]}{\delta \varphi(x)}=-\frac{3\pi^2x^2}{8N^2}\tag{19}\label{eq:varphix} \end{equation}

将方程\eqref{eq:Ux}代入方程\eqref{eq:varphicons},得如下积分方程:

\begin{equation} \varphi(x)=\frac{2Na^3}{\pi\sigma}\int_0^{x'}\frac{g(x')}{\sqrt{x'^2-x^2}}\mathrm dx \tag{20}\label{eq:inteq}\end{equation}

从方程\eqref{eq:varphix}到高分子体积分数$\varphi(x)$,解积分方程\eqref{eq:inteq}就可得高分子链末端的分布。积分方程的解可从积分方程手册中查到,在pp21。

标签: 高分子刷, 强拉伸理论, sst

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