链状离子液体格气模型

本文将离子液体格气模型推广至链状离子液体。

假设正离子有一定链长,设链长为 $n$,如上图所示。

此时

\begin{equation} \Omega=\frac{N!}{(nN_+)!(N-nN_+)!}\frac{[(N-nN_+)/\xi]}{N_-!\{[(N-nN_+)/\xi]-N_-\}!} \label{distnumn} \end{equation}

\begin{equation} \begin{split} \ln\Omega=&N\ln N-(nN_+)\ln (nN_+)-(N-nN_+)\ln(N-nN_+)\\ &+\frac{N-nN_+}{\xi}\ln\frac{N-nN_+}{\xi}-N_-\ln N_-\\ &-\left (\frac{N-nN_+}{\xi}-N_- \right )\ln \left (\frac{N-nN_+}{\xi}-N_- \right ) \end{split} \label{lndistnumn} \end{equation}

\begin{equation} \begin{split} \frac{\mu_+}{kT}=&\frac{1}{kT}\frac{\partial F}{\partial N_+}=\frac{e\varphi}{kT}-\frac{\partial \ln\Omega}{\partial N_+}\\ =&\frac{e\varphi}{kT}-n\ln(N-nN_+)+n\ln N_++\frac{n}{\xi}\ln\left(\frac{N-nN_+}{\xi}\right)-\frac{n}{\xi}\ln\left(\frac{N-nN_+}{\xi}-N_- \right)\\ =&-n\ln(N-nN_0)+n\ln N_0+\frac{n}{\xi}\ln\left(\frac{N-nN_0}{\xi}\right)-\frac{n}{\xi}\ln\left(\frac{N-nN_0}{\xi}-N_0 \right) \end{split} \label{mun+} \end{equation}

\begin{equation} \begin{split} \frac{\mu_-}{kT}=&\frac{1}{kT}\frac{\partial F}{\partial N_-}=-\frac{e\varphi}{kT}-\frac{\partial \ln\Omega}{\partial N_-}\\ =&-\frac{e\varphi}{kT}+\ln N_--\ln\left(\frac{N-nN_+}{\xi}-N_- \right)\\ =&\ln N_0-\ln\left(\frac{N-nN_0}{\xi}-N_0 \right) \end{split} \label{mun-} \end{equation}

\begin{equation} \frac{e\varphi}{kT}-n\ln\frac{N-nN_+}{N-nN_0}+n\ln \frac{N_+}{N_0}+\frac{n}{\xi}\ln\frac{N-nN_+}{N-nN_0}-\frac{n}{\xi}\ln\frac{N-nN_+-\xi N_-}{N-nN_0-\xi N_0}=0 \label{mun+eq} \end{equation}

\begin{equation} -\frac{e\varphi}{kT}+\ln \frac{N_-}{N_0}-\ln\frac{N-nN_+-\xi N_-}{N-nN_0-\xi N_0}=0 \label{mun-eq} \end{equation}

此时孔隙率定义为 $\eta=2/\gamma-n-\xi$。$c_+/c_0$和$c_-/c_0$满足如下方程组

\begin{equation} -u=n\ln\frac{c_+}{c_0}-n\frac{\xi-1}{\xi}\ln\frac{\frac{2}{\gamma}-n\frac{c_+}{c_0}}{\xi+\eta}-\frac{1}{\xi}\ln\frac{\frac{2}{\gamma}-n\frac{c_+}{c_0}-\xi\frac{c_-}{c_0}}{\eta} \label{cn+eq} \end{equation}

\begin{equation} u=\ln\frac{c_-}{c_0}-\ln\frac{\frac{2}{\gamma}-n\frac{c_+}{c_0}-\xi\frac{c_-}{c_0}}{\eta} \label{cn-eq} \end{equation}

\begin{equation} e^{-u}=\left(\frac{c_+}{c_0}\right)^n \left(\frac{\frac{2}{\gamma}-n\frac{c_+}{c_0}}{\xi+\eta}\right)^{-n\frac{\xi-1}{\xi}}\left(\frac{\frac{2}{\gamma}-n\frac{c_+}{c_0}-\xi\frac{c_-}{c_0}}{\eta}\right)^{-\frac{n}{\xi}} \label{ecn+eq} \end{equation}

\begin{equation} e^u=\frac{\eta \frac{c_-}{c_0}}{\frac{2}{\gamma}-n\frac{c_+}{c_0}-\xi\frac{c_-}{c_0}} \label{ecn-eq} \end{equation}

没有办法得到$c_+/c_0$和$c_-/c_0$的显示表达式。

标签: 离子液体

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