## 2 Materials and Methods
### Membrane Characterizations
After washing with distilled water, the sample was immersed in 0.5 NaCl solution for 3 h or more so that Na\({}^{+}\) was exchanged with H\({}^{+}\) present in the sample. The amount of H\({}^{+}\) existing in the solution was then determined by a traditional acid-base titration using a phenolphthalein indicator and a 0.01 M NaOH titration solution. Finally, the IEC value of the sample was calculated by substituting the measured parameters into the following Equation (5) [31]:  
\[\text{IEC}=\frac{C\cdot V_{s}}{W_{dry}}\left[\frac{\text{meq.}}{\text{g}_{ \text{dry memb}}}\right] \tag{5}\]  
where \(C\) is the normal concentration of ions measured through titration (meq./L), \(V_{s}\) is the solution volume (L), and \(W_{dry}\) is the dry membrane weight (g). The transport numbers (\(t_{-}\) for anion and \(t_{+}\) for cation) of the IEMs were determined by measuring the membrane potential using a pair of Ag/AgCl electrodes in a two-compartment diffusion cell and calculated by the following Equations (6) and (7) [32]:  
\[E_{m}=\frac{RT}{F}(2t_{+}-1)\ln\frac{C_{L}}{C_{H}} \tag{6}\]  
\[t_{+}+t_{-}=1 \tag{7}\]  
where \(E_{m}\) is the measured membrane potential, \(R\) is the gas constant, \(T\) is the absolute temperature, \(F\) is the Faraday constant, and \(C_{L}\) and \(C_{H}\) are NaCl concentrations of the compartments (1 and 5 mM, respectively). To evaluate the oxidation stability of the prepared IEMs, a membrane sample (2 \(\times\) 2 cm\({}^{2}\)) was immersed in an aqueous solution containing 3% H\({}_{2}\)O\({}_{2}\) and 3 ppm Fe\({}^{2+}\) (i.e., Fenton'sreagent), and then the reaction proceeded at 80 \({}^{\circ}\)C for 8 h. During the oxidation test, the membranes could be decomposed by free radicals (-OH and -COOH) formed by H\({}_{2}\)O\({}_{2}\) in the presence of Fe\({}^{2+}\) ions [33]. Therefore, each sample reacted for 0 (fresh), 4, and 8 h was washed several times with distilled water and dried in a dry oven at 80 \({}^{\circ}\)C for more than 6 h, and then the weight was measured to confirm the weight loss of the sample. Additionally, it was attempted to confirm the chemical stability of the IEMs in the VRFB system through a vanadium oxidation stability test. This is based on the principle that VO\({}_{2}\)+, which is a V(V) species, is reduced to VO\({}^{2+}\), which is a V(IV) species, by an oxidation reaction of a film immersed in a VO\({}_{2}\)+/H\({}_{2}\)SO\({}_{4}\) solution. For the measurement of vanadium oxidation stability, a membrane sample (2 \(\times\) 2 cm\({}^{2}\)) was immersed in 20.0 mL of 0.1 M V\({}_{2}\)SO\({}_{5}\) (in 5 M H\({}_{2}\)SO\({}_{4}\)) solution, and the temperature was maintained at 40 \({}^{\circ}\)C. The concentration of VO\({}^{2+}\) ions in the solution was determined by measuring the absorbance using UV-VIS spectroscopy (UV-2600, Shimadzu). A permeability test was also carried out to confirm the vanadium crossover through the IEMs. A membrane sample having a size of 5 \(\times\) 5 cm\({}^{2}\) was immersed in a 2 M H\({}_{2}\)SO\({}_{4}\) solution for 2 h or more to reach an equilibrium state, and then inserted in a lab-made two-compartment diffusion cell. Amounts of 2 M VOSO\({}_{4}\)/2 M H\({}_{2}\)SO\({}_{4}\) (feed) and 2 M MgSO\({}_{4}\)/2 M H\({}_{2}\)SO\({}_{4}\) (permeate) solutions were filled in each compartment.