自由能“作用量”



以文献Ions in Mixed Dielectric Solvents: Density Profiles and Osmotic Pressure between Charged Interfaces为例,说明将自由能看做作用量的处理手法。

体系如上图所示,电解质溶液受限于两带电平面,溶液中有A、B两种溶剂。

电解质为单价离子,正负离子数密度分别为$n_+$、$n_-$。A、B两种溶剂体积分数分别为$\phi_A$、$\phi_B$,介电常数分别为$\varepsilon_A$、$\varepsilon_B$。假设溶液局域介电常数为两溶剂介电常数的线性组合:

\begin{equation}\label{localdielectric} \varepsilon(\vec{r})=\varepsilon_A\phi_A(\vec{r})+\varepsilon_B\phi_B(\vec{r})=\varepsilon_0-\varepsilon_r\phi(\vec{r}) \end{equation}

其中已令$\varepsilon_0 \equiv \varepsilon_A$,$\varepsilon_r \equiv \varepsilon_A-\varepsilon_B$,$\phi(\vec{r}) \equiv \phi_B(\vec{r})$,假设体系是不可压缩的,$\phi_A(\vec{r})=1-\phi(\vec{r})$。

体系巨势为

\begin{equation} \begin{split} G=&\int d\vec{r}g(\vec{r})\\ =&\int d\vec{r}\left [ -\frac{\varepsilon(\vec{r})}{8\pi}(\nabla \psi)^2+e(n_+-n_-)\psi \right ]\\ &+k_BT\int d\vec{r}\left [ n_+(\ln (n_+a^3)-1)+n_-(\ln (n_-a^3)-1) \right ]\\ &+\frac{k_BT}{a^3}\int d\vec{r}\left [ \phi\ln \phi+(1-\phi)\ln (1-\phi)+\chi\phi(1-\phi) \right ]\\ &+k_BT\int d\vec{r}(\alpha_+n_++\alpha_-n_-)\phi\\ &-k_BT\int d\vec{r}\left [ \mu_+n_++\mu_-n_-+\mu_{\phi}\frac{\phi}{a^3} \right ]\\ =&\int d\vec{r}g(\psi,n_+,n_-,\phi) \end{split} \label{grandpotential} \end{equation}

倒数第二行为溶解能(solvation energy)。

巨势$G$可看做作用量,“广义坐标”$Q$为$\psi$、$n_+$、$n_-$、$\phi$,“广义动量”为$P=\nabla Q$。由$\frac{\partial g}{\partial Q}-\nabla\cdot\frac{\partial g}{\partial P}=0$得以下“运动方程”:

\begin{equation}\label{motioneqpsi} \nabla\cdot \left ( \frac{\varepsilon(\vec{r})}{4\pi}\nabla \psi \right) +e(n_+-n_-)\psi =0 \end{equation}

\begin{equation}\label{motioneqn} \pm \frac{e\psi}{k_BT} +\ln(n_{\pm}a^3)+\alpha_{\pm}\psi-\mu_{\pm} =0 \end{equation}

\begin{equation}\label{motioneqphi} \ln\frac{\phi}{1-\phi}+\chi(1-2\phi)+\frac{\varepsilon_ra^3}{8\pi k_BT}(\nabla \psi)^2+a^3(\alpha_+n_++\alpha_-n_-)-\mu_{\phi}=0 \end{equation}

从\eqref{grandpotential}式可看出,“作用量”密度$g$不显含$\vec{r}$,因此有“守恒量”

\begin{equation} \begin{split} \Pi=&\sum_{\nu}P_{\nu}\nabla Q_{\nu}-g=P_{\psi}\nabla \psi-g=\frac{\partial g}{\partial \nabla \psi}\nabla \psi-g \\ =&-\frac{\varepsilon(\vec{r})}{4\pi}(\nabla \psi)^2-g\\ =&-\frac{\varepsilon(\vec{r})}{4\pi}(\nabla \psi)^2-\left [ -\frac{\varepsilon(\vec{r})}{8\pi}(\nabla \psi)^2+e(n_+-n_-)\psi \right ]\\ &-k_BT \left [ n_+(\ln (n_+a^3)-1)+n_-(\ln (n_-a^3)-1) \right ]\\ &-\frac{k_BT}{a^3} \left [ \phi\ln \phi+(1-\phi)\ln (1-\phi)+\chi\phi(1-\phi) \right ]\\ &-k_BT (\alpha_+n_++\alpha_-n_-)\phi\\ &+k_BT \left [ \mu_+n_++\mu_-n_-+\mu_{\phi}\frac{\phi}{a^3} \right ]\\ =&-\frac{\varepsilon(\vec{r})}{8\pi}(\nabla \psi)^2\\ &-en_+\psi+k_BTn_+\ln (n_+a^3)-k_BT \alpha_+n_+\phi+k_BT \mu_+n_+\\ &+en_-\psi+k_BTn_-\ln (n_-a^3)-k_BT \alpha_-n_-\phi+k_BT \mu_-n_-\\ &+k_BT(n_++n_-)\\ &-\frac{k_BT}{a^3} \left [\ln (1-\phi)+ \phi\ln\frac{\phi}{1-\phi}+\chi\phi(1-\phi)-\mu_{\phi}\phi \right ]\\ =&-\frac{\varepsilon(\vec{r})}{8\pi}(\nabla \psi)^2+k_BT\left [n_++n_--\frac{\ln (1-\phi)}{a^3}\right ]\\ &+k_BT\left [(\alpha_+n_++\alpha_-n_-)\phi-\frac{\chi\phi^2}{a^3}\right ]+\frac{\varepsilon_r}{8\pi}(\nabla \psi)^2\\ =&-\frac{\varepsilon_0-2\varepsilon_r\phi}{8\pi}(\nabla \psi)^2+k_BT\left [n_++n_--\frac{\ln (1-\phi)}{a^3}\right ]\\ &+k_BT\left [(\alpha_+n_++\alpha_-n_-)\phi-\frac{\chi\phi^2}{a^3}\right ] \end{split} \label{PI} \end{equation}

上式推导中用到了“运动方程”。

\eqref{PI}式中“守恒量”压强减去本体压强$\Pi$ 正是体系的渗透压。

标签: 渗透压, 自由能, 作用量

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