受限离子液体



BSK自由能

\begin{equation} \begin{split} G=& \int d\vec{r}g(\vec{r})\\ =&\int d\vec{r}\left \{ -\frac{\varepsilon}{2}[(\nabla \psi)^2+l_c^2(\nabla^2 \psi)^2]+ez(n_+-n_-)\psi \right \}\\ &+\frac{k_BT}{v}\int d\vec{r}\left [ vn_+\ln(vn_+)+vn_-\ln(vn_-)+(1-vn_+-vn_-)]\ln (1-vn_+-vn_-) \right ]\\ &-k_BT\int d\vec{r}\left ( \mu_+n_++\mu_-n_- \right )\\ =& \int d\vec{r}g(\psi,\nabla \psi,\nabla^2 \psi,n_+,n_-) \end{split} \label{BSK} \end{equation}

其中$l_c$为静电关联长度,$z$为离子价。

变分,由$\frac{\partial g}{\partial \psi}-\nabla\cdot \frac{\partial g}{\partial \nabla\psi}+\nabla^2 \frac{\partial g}{\partial \nabla^2\psi}=0$ 得

\begin{equation} ez(n_+-n_-)+\varepsilon \nabla^2 \psi-\varepsilon l_c^2\nabla^4 \psi=0 \label{Poisson4} \end{equation}

由 $\frac{\partial g}{\partial n_{\pm}}=0$得

\begin{equation} \begin{split} &\ln\frac{n_{\pm}v}{1-v(n_++n_-)}\pm \frac{ez\psi}{k_BT} =\mu_{\pm}=\ln\frac{n_0v}{1-2vn_0}\\\\ &\frac{n_{\pm}v}{1-v(n_++n_-)}=\frac{n_0v}{1-2vn_0}\exp(\mp\frac{ez\psi}{k_BT} )\\\\ &n_{\pm}=\frac{n_0}{1-\gamma+\gamma\cosh(\frac{ez\psi}{k_BT})}\exp(\mp\frac{ez\psi}{k_BT})=\frac{n_0}{1+2\gamma\sinh^2(\frac{ez\psi}{2k_BT})}\exp(\mp\frac{ez\psi}{k_BT}) \end{split} \label{Fermi} \end{equation}

其中 $\gamma=2vn_0$。将\eqref{Fermi} 式代入 \eqref{Poisson4} 式。

\begin{equation} \nabla^2 \psi- l_c^2\nabla^4 \psi=\frac{2n_0ez}{\varepsilon}\frac{\sinh(\frac{ez\psi}{k_BT})}{1+2\gamma\sinh^2(\frac{ez\psi}{2k_BT})} \label{Poisson-Fermi} \end{equation}

渗透压推导参考文献J. Phys. Chem. B2009,113,6001–6011 的附录A。为符号简单,泛函\eqref{BSK} 只考虑一维$z$情形。欧拉-拉格朗日方程

\begin{equation} \frac{\partial g}{\partial \psi}-\frac{d}{dz} \frac{\partial g}{\partial \psi'}+\frac{d^2}{dz^2}\frac{\partial g}{\partial \psi''}=0 \label{ELeq} \end{equation}

对自由能密度求导

\begin{equation} \begin{split} \frac{dg}{dz}&=\frac{\partial g}{\partial \psi}\psi'+\frac{\partial g}{\partial \psi'}\psi''+\frac{\partial g}{\partial \psi''}\psi''' \\ =&\left(\frac{d}{dz} \frac{\partial g}{\partial \psi'}-\frac{d^2}{dz^2}\frac{\partial g}{\partial \psi''}\right)\psi'+\frac{\partial g}{\partial \psi'}\psi''+\frac{\partial g}{\partial \psi''}\psi'''\\ =&\psi'\frac{d}{dz} \frac{\partial g}{\partial \psi'}+\frac{\partial g}{\partial \psi'}\psi''+\frac{\partial g}{\partial \psi''}\psi'''-\psi'\frac{d^2}{dz^2}\frac{\partial g}{\partial \psi''}\\ =&\frac{d}{dz}\left( \psi'\frac{\partial g}{\partial \psi'}\right)+\frac{d^2}{dz^2}\left (\psi' \frac{\partial g}{\partial \psi''}\right)-2\psi''\frac{d}{dz}\frac{\partial g}{\partial \psi''}-2\psi'\frac{d^2}{dz^2}\frac{\partial g}{\partial \psi''}\\ =&\frac{d}{dz}\left( \psi'\frac{\partial g}{\partial \psi'}\right)-2\frac{d}{dz}\left(\psi'\frac{d}{dz}\frac{\partial g}{\partial \psi''}\right)+\frac{d^2}{dz^2}\left (\psi' \frac{\partial g}{\partial \psi''}\right)\\ =&\frac{d}{dz}\left( \psi'\frac{\partial g}{\partial \psi'}\right)-\frac{d}{dz}\left(\psi'\frac{d}{dz}\frac{\partial g}{\partial \psi''}\right)+\frac{d}{dz}\left (\psi'' \frac{\partial g}{\partial \psi''}\right) \end{split} \label{dgdz} \end{equation}

所以渗透压为

\begin{equation} \begin{split} \Pi=&\psi'\frac{\partial g}{\partial \psi'}-\psi'\frac{d}{dz}\frac{\partial g}{\partial \psi''}+\psi'' \frac{\partial g}{\partial \psi''}-g \\ =&-\frac{\varepsilon}{2}\psi'^2+\varepsilon l_c^2\psi'\psi'''-\varepsilon l_c^2\psi''^2-\frac{k_BT}{v}\ln (1-vn_+-vn_-) \end{split} \label{osmoticpressure} \end{equation}

标签: 渗透压, 离子液体

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