## 2 Materials and Methods
### Capacitive Deionization Experiment  
The MCDI unit cell consisted of two parallel carbon electrodes. To prevent short-circuiting, a spacer with a thickness of 100 \(\upmu\)m (100 mesh, Nylon) was used to separate each pair of electrodes. In addition, the spacer was used as a flow channel for the influent water within the MCDI cell. The schematic of the single-pass MCDI experiment and cell configuration is depicted in Figure 1. The influent water flowed in from the marginality of the electrode and flowed out from the center of the electrode. A unit cell was assembled by placing plexiglas on both sides of the electrodes.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type** & **Sample** & **S\({}_{\text{BET}}\) (m\({}^{2}\) g\({}^{-1}\))** & **V\({}_{\text{total}}\) (cm\({}^{3}\) g\({}^{-1}\))** & **d\({}_{\text{BJH}}\) (nm)** \\ \hline Powder & Activated carbon & 1250.27 & 0.55 & 1.77 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Physical properties of activated carbon powder.  
Figure 1: Schematic diagram of the MCDI experiment and cell configuration.  
Deionization experiments were performed by the single-pass method, where the influent was passed through the MCDI cell. The concentration, pH, and flow rate of the feed solution varied from 50-200 mg L\({}^{-1}\) to 2-4 and 10-30 mL min\({}^{-1}\), respectively. An adsorption experiment was conducted at different cell potentials (0.8, 1.0, 1.2, 1.4 V) through a potentiostat (WPG100, WonA tech Co., Seoul, Korea). A desorption test was conducted by changing the cell potential to 0 V immediately after the adsorption test. At each cell potential, the adsorption/desorption experiment was repeated three times in succession to ensure equilibrium. A conductivity and pH measurement system (Orion, Versa Star Pro, Thermo Fisher Scientific, Waltham, MA, USA) was installed at the outlet of the cell and continuously monitored at intervals of 5 s. The concentration of Cs\({}^{+}\) ions was measured using inductively coupled plasma optical emission spectroscopy (ICP-OES, Perkin Elmer, Midland, ON, Canada).  
The specific adsorption capacity (SAC) of the electrode was calculated using the following equation:  
\[\text{SAC}(\text{mg}/\text{g})=\frac{V\big{(}C_{0}-C_{avg}\big{)}}{m} \tag{1}\]  
where \(V\) is the total volume of influent water supplied to the cell during the adsorption process; \(C_{0}\) and \(C_{avg}\) are the initial influent concentration and average effluent concentration, respectively; and \(m\) is the mass of the carbon electrode. The specific adsorption rate (SAR) was obtained by dividing the SAC value with the adsorption time.  
The charge efficiency (\(\Lambda\)) is the ratio of the salt adsorption amounts over the charge in a CDI, which was calculated by the following equation:  
\[\Lambda\ =\frac{V\cdot F\cdot\big{(}C_{0}-C_{avg}\big{)}}{M_{w}\int Idt} \tag{2}\]  
where \(F\) is Faraday's constant, \(M_{w}\) is the molar mass of CsNO\({}_{3}\) and \(I\) is the current density.  
The removal efficiency was calculated by the following equation:  
\[\text{Removal efficiency}\ (\%)\ =\frac{\big{(}C_{0}-C_{avg}\big{)}}{C_{0}}\ \times\ 100 \tag{3}\]