## 3 Results and Discussion
### Theory and Model
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**AEM**} & \multicolumn{2}{c}{**Manning’s Models**} & \multicolumn{2}{c}{**Manning’s Models + \(\alpha\)**} & \multicolumn{2}{c}{**Manning’s Models + \(\alpha\), \(\beta\)**} \\ \cline{2-7}  & \(R^{2}\) & **SSE** & \(R^{2}\) & **SSE** & \(R^{2}\) & **SSE** \\ \hline ACLP-2\% & 0.6821 & 0.10170 & 0.8516 & 0.02820 & 0.9569 & 0.008770 \\ ACLP-3\% & 0.6984 & 0.06063 & 0.8950 & 0.01177 & 0.9955 & 0.000565 \\ ACLP-4\% & 0.6912 & 0.05767 & 0.8596 & 0.01383 & 0.9906 & 0.000785 \\ \hline \end{tabular}
\end{table} 
Table 4: The fitting constants of the three models predict the ion activity coefficients in the ACLP membrane equilibrated in NaBr solution.—the Manning model, the Manning model with the modified parameter \(\alpha\) (Manning + \(\alpha\)), and the Manning model with both modified parameters \(\alpha\) and \(\beta\) (Manning + \(\alpha\), \(\beta\))—were compared. In this context, \(R^{2}\) represents the correlation coefficient, while SST denotes the Sum of Squares Total Variance. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**AEM**} & \multicolumn{2}{c}{**Manning’s Models**} & \multicolumn{2}{c}{**Manning’s Models + \(\alpha\)**} & \multicolumn{2}{c}{**Manning’s Models + \(\alpha\), \(\beta\)**} \\ \cline{2-7}  & \(R^{2}\) & **SSE** & \(R^{2}\) & **SSE** & \(R^{2}\) & **SSE** \\ \hline ACLP-2\% & 0.6593 & 0.09758 & 0.8249 & 0.02734 & 0.9422 & 0.009565 \\ ACLP-3\% & 0.6594 & 0.05246 & 0.8378 & 0.01306 & 0.9931 & 0.000620 \\ ACLP-4\% & 0.6698 & 0.04871 & 0.8151 & 0.01433 & 0.9510 & 0.004272 \\ \hline \end{tabular}
\end{table} 
Table 3: The fitting constants of the three models predict the ion activity coefficients in the ACLP membrane equilibrated in NaCl solution.—the Manning model, the Manning model with the modified parameter \(\alpha\) (Manning + \(\alpha\)), and the Manning model with both modified parameters \(\alpha\) and \(\beta\) (Manning + \(\alpha\), \(\beta\))—were compared. In this context, \(R^{2}\) represents the correlation coefficient, while SST denotes the Sum of Squares Total Variance. Based on this, the modified Manning model expression can be expressed as follows:  
\[\left[\gamma_{+}^{\text{m}}\gamma_{-}^{\text{m}}\right]_{\text{mod}}=\frac{1}{ \beta}\left(\frac{\frac{m_{\text{s}}^{\text{m}}}{m_{\text{s}}^{\text{m}}}+1}{ \frac{m_{\text{s}}^{\text{m}}}{m_{\text{s}}^{\text{m}}}+1}\right)\exp\left[ \frac{-\frac{m_{\text{s}}^{\text{m}}}{m_{\text{s}}^{\text{m}}}}{\frac{m_{ \text{s}}^{\text{m}}}{m_{\text{s}}^{\text{m}}}+2\alpha\zeta}\right] \tag{26}\]  
where \(\alpha\) is the monomer projection length correction parameter, with a reasonable value of \(\alpha=\sqrt{3}\). \(\beta\) is the linear electric field superposition correction coefficient, which is calculated using Equation (25). The modified Manning model employs all the same parameters as the original model. Based on the above discussion and improvement, the modified model is given by Equation (26). In this way, the calculation process of the ion activity coefficient in the membrane phase can be summarized as follows:  
1. Substitute the monomer components in the polymer into Equation (14) to calculate the spacing \(b\) between fixed groups;  
2. Substitute the values of membrane moisture content and density into Equation (16) to calculate the volume fraction of water \(V_{\text{w}}\) and polymer \(V_{\text{p}}\) in the swelling polymer;  
3. Substitute \(V_{\text{w}}\) and \(V_{\text{p}}\) into Equation (15) to calculate the dielectric constant \(\varepsilon\) in the membrane;  
4. The values of \(b\) and \(\varepsilon\) are substituted into Equation (11) to calculate the line charge density \(\zeta\). The parameter \(\alpha\) is introduced and \(\zeta\) is corrected to \(\alpha\zeta\). 5. Calculate the empirical parameter \(\beta\) according to Equation (25);  
6.