## 3 Results and Discussion
### Theory and Model  
[PERSON]'s Counterion Condensation Theory [28; 29] is one of the most widely accepted theoretical models used to describe the complexities of polyelectrolyte solutions [10; 11; 12; 13]. It models a polymer chain with a fixed group as a linear charge chain, assuming that the solute and solvent can provide sufficient shielding to allow the linear charge to ignore interactions between wires while focusing on the long-range electrostatic force between line and point charges [28], as well as counter-ion condensation. As the Figure 6a, the ion-exchange membrane can be regarded as a polyelectrolyte confined within a specific geometry in its chemical structure. Therefore, the long-range forces that influence the nonideal behavior of polyelectrolytes may similarly affect ion-exchange membranes. This perspective is valid [36; 37; 38]. To accurately assess the improvements in membrane properties discussed in this paper, we will first continue to employ the original model. In the assumptions of [PERSON]'s theory, a polymer chain modeled as a linear charge generates an additional electric field energy effect on nearby ions. This effect is closely related to the line charge density, \(\xi\), and its critical value, \(\xi_{\text{crit}}\), which are defined as follows [28]:  
\[\xi\ =\frac{\text{e}^{2}}{4\pi\epsilon\varepsilon_{0}kTb}=\frac{\lambda_{ \text{b}}}{b} \tag{11}\]  
\[\xi_{\text{crit}}=\frac{1}{\left|z_{\text{j}}z_{\text{p}}\right|} \tag{12}\]  
where \(e\) is the value of elementary charge, \(\varepsilon_{0}\) is the vacuum permittivity, \(\varepsilon\) is the dielectric constant of solvent, \(k\) is Boltzmann's constant, \(T\) is the Kelvin temperature, \(b\) is the average distance between two charges on the polyelectrolyte chain, and \(\lambda_{\text{b}}\) is the Bjerrum length (At this length, the work required to separate a counter-ion from a fixed charge is four times the average kinetic energy per degree of freedom [57; 58]). \(z_{\text{i}}\) is the amount of charge carried by each counter-ion and \(z_{\text{p}}\) is by the fixed group. The values of \(\xi\) and \(\xi_{\text{crit}}\) significantly influence the discussion of ion activity coefficients [28]. Considering the case where both the ion and the fixed group are monovalent, if \(\xi<\xi_{\text{crit}}\), then \(\lambda_{\text{b}}<b\); in this scenario, counterions can overcome the electrostatic attraction energy of the immobilized group, although this energy increases as they approach. Conversely, if \(\xi>\xi_{\text{crit}}\), an electrostatic overlap region exists within the system, making it difficult for counterions to overcome the electrostatic attraction energy. Consequently, they tend to remain in this region, leading to counter-ion condensation. The condensed ions will shield the electric field of the fixed group until \(\xi\) decreases to 1 [28]. Previous studies have shown that reasonable values of \(\xi\) for most cross-linked ion-exchange membranes are greater than 1 [59]. The interaction between the uncondensed mobile ions and the line  
Figure 6: (**a**) illustrates the scenarios described by [PERSON]’s theory, specifically for the cases where \(b>\lambda_{\text{b}}\) and \(b<\lambda_{\text{b}}\). In the case where \(b<\lambda_{\text{b}}\), the overlap of the Bjerrum length creates a localized energy minimum region where counterions are likely to reside, leading to “counterion condensation” [28]. (**b**) mathematically presents the left and right limits predicted by the [PERSON] model under this assumption, as expressed by Equation (13).