## 4 Discussion
### Theoretical Consideration
The results in this study indicate the influence of cationic (amine) groups immobilized in the thermo-responsive polymer on the solution pH for the cross-linked gels. The hydroxide ions in the liquid phase in the slurry will be electrostatically attracted to the protonated amine groups on the surface of the gels; consequently, hydroxide ions that exist to neutralize the surface charge affect the pH of the slurry (pH in the liquid phase in the slurry, pH\({}_{\mathrm{L}}\)). In the model, the influence of surface charge on the gel was introduced as a surface charge contribution factor \(\beta\) = (amount of surface charge affecting the pH in the slurry)/(total amount of charge in the gels). The details of the definition of \(\beta\) in the model are shown in the Appendix A. Regarding the equilibrium, the absorption amount of CO\({}_{2}\) can be calculated as a function of the temperature by solving the above-mentioned equations of the system. According to the experimental conditions used in this study, the amount of dissolved CO\({}_{2}\) [H\({}_{2}\)CO\({}_{3}\)*] that is in equilibrium with CO\({}_{2}\) in the gas phase, was calculated at first, based on [PERSON]'s law. Subsequently, the composition in the liquid phase in the slurry and pH in the liquid phase (pH\({}_{L}\)) were calculated, in which the mass balance of CO\({}_{2}\), dissociation equilibrium of carbonate acid and its dissociated ion, and electric charge balance were satisfied. Subsequently, the composition in the gel phase was calculated using the composition in the liquid phase. Based on the Donnan equilibrium between the gel and the liquid phases in the slurry, the composition of the ions in the gel was calculated to satisfy the charge balance in Equation (18) and dissociation equilibrium of carbonate, fixed charge and water, respectively. The total amount of CO\({}_{2}\) adsorbed in the gel-slurry system (\(Q\)) is the sum of the amount of H\({}_{2}\)CO\({}_{3}\)*, HCO\({}_{3}\)*, and CO\({}_{3}\)2* in the slurry. The cycle capacity can be calculated by Equation (19) on the equilibrium model basis. Where \(T_{L}\) and \(T_{H}\) are operation low temperature for CO\({}_{2}\) adsorption and operation high temperature for CO\({}_{2}\) desorption, respectively. \[cycle\ capacity=Q_{T_{L}}-Q_{T_{H}} \tag{19}\]  
Figure 7a shows the temperature dependency of the amount of CO\({}_{2}\) absorption, \(Q\) (total amount absorbed CO\({}_{2}\) in the slurry), and Donnan ratio as functions of p\(K_{a,g}\). Depending on the p\(K_{a,g}\) of the amine groups in the gels, the amount of the absorption decreases with increasing temperature. The Donnan ratio tends to increase with the temperature, indicating a decrease in the difference of concentration of ions between the liquid and gel phases. When p\(K_{a,g}\) decreases (it will occur with the collapse of the gel) at a high-temperature region, the CO\({}_{2}\) that was adsorbed at the low temperature desorbs at the high-temperature region, depending on the temperature dependence of p\(K_{a,g}\) (refer the route of the arrow in the figure). The volume change and the p\(K_{a}\) change of the amine groups in the gel will be linked with each other. The p\(K_{a}\) affects the number of protonated fixed-charge sites (the number of electrostatic adsorption sites); thus, the \(Q\) depends on p\(K_{a,g}\) significantly. Figure 7b shows the dependence of the pH in the gel (pH\({}_{g}\)) and the pH in the liquid phase of the slurry (pH\({}_{L}\)) on the p\(K_{a,g}\) of the amine groups in the gel, as a function of temperature. For an aqueous solution system without the gels, the solution pH tends to increase with the temperature, as shown by dotted lines in the figure. The pH in the gel (pH\({}_{g}\)) becomes higher than the pH\({}_{L}\) in the liquid phase due to the Donnan equilibrium, and the pH in the gel increases with increasing temperature, as is the case with the aqueous solution system when the p\(K_{a}\) of the amine (p\(K_{a,g}\)) is constant.