## 3 Results and Discussion
### Ion Concentration and Activity Coefficient in AEMs  
The values for the concentrations of ions in the membranes equilibrated with NaCl and NaBr solutions as a function of external electrolyte concentration are presented in Figure 4. Following the conventions of solution thermodynamics, these concentrations are expressed in terms of molality (mol\(\cdot\)kg\({}^{-1}\) H\({}_{2}\)O). As previously reported for commercial ion-exchange membranes (IEMs) [10; 11; 12], the co-ion concentration in the membrane, \(m_{\text{Na}^{+}}^{\text{m}}\), is significantly lower than the counter-ion concentration \(m_{\text{X}^{-}}^{\text{m}}\). The co-ion concentration remains at a low level, and the magnitude of the difference increases sharply as the concentration of the external electrolyte decreases. In contrast, the counter-ion concentration remains relatively stable and ultimately converges to the value of the fixed group concentration. From the perspective of the chemical environment of ions within the film, co-ions primarily originate from the absorption of the solution during the swelling of the dry film. The electrolyte in this chemical environment is termed the mobile electrolyte concentration, which is equivalent to the concentration of co-ions. The Gibbs-Donnan effect prevents the values of \(m_{\text{s}}^{\text{m}}\) from exceeding those of \(m_{\text{Na}^{+}}^{\text{s}}\), particularly when the external salt concentration is low (below 0.03 mol\(\cdot\)kg\({}^{-1}\)) [5; 45]. The concentration of \(m_{\text{Na}^{+}}^{\text{m}}\) in the AEM can be so low that the co-ion concentration in the solution during the desorption experimental step drops below 80 \(\mu\)g\(\cdot\)L\({}^{-1}\), leading to increased measurement errors. In contrast to co-ions, counter-ions exist in two distinct chemical environments. Some are found as ion pairs with fixed groups, maintaining the charge balance within the membrane before the equilibration experiment, and their values should equal \(m_{\text{A}}^{\text{m}}\). The other portion of the counter-ions, similar to co-ions, is absorbed from the mobile electrolyte during the swelling process. Therefore, when the mobile electrolyte concentration is below 0.1 mol\(\cdot\)kg\({}^{-1}\), \(m_{\text{A}}^{\text{m}}\) is already two to three orders of magnitude higher than \(m_{\text{s}}^{\text{m}}\), and the concentrations of \(m_{\text{Na}^{+}}^{\text{m}}\) and \(m_{\text{X}^{-}}^{\text{m}}\) do not vary by an order of magnitude. As the external electrolyte concentration rises above 0.1 mol\(\cdot\)kg\({}^{-1}\), the proportion of mobile electrolytes \(m_{\text{s}}^{\text{m}}\) increases. Concurrently, the deswelling effect leads to a reduction in the solvent absorbed by the AEM, which causes \(m_{\text{A}}^{\text{m}}\) to increase; these two factors together facilitate a rapid increase in \(m_{\text{X}^{-}}^{\text{m}}\) within the high concentration range [10; 46]. Figure 3: The influence of external solution molality and cross-linking on the water uptake of ACLP AEMs (excluding the internal PP fiber) is significant. Water uptake decreases with increasing external solution molality and cross-linking. The water uptake values are reported as the mass of water per unit mass of the ion-exchange polymer in the membrane. Accordingly, the ion activity coefficient can be derived from the aforementioned concentrations in AEMs using the Donnan Equilibrium [26; 27]. At ion sorption equilibrium, the electrochemical potential of ion \(i\) within the IEM is equal to that in the external solution [50]:  
\[\mu_{i}^{\odot,\text{m}}\left(T\right)+RT\text{ In }\gamma_{i}^{\text{m}}m_{i}^{\text{m}}+z_{i}F \varphi^{\text{m}}=\mu_{i}^{\odot,s}(T)+RT\text{ In}\gamma_{i}^{s}m_{i}^{s}+z_{i}F \varphi^{s} \tag{6}\]  
where **a**, \(\mu_{i}^{\odot,\text{m}}\), \(\mu_{i}^{\odot,s}\) are the standard chemical potential of ion \(i\) in the membrane and solution at temperature \(T\), respectively.