## 1 Introduction  
Capacitive deionization (CDI), often referred to as electrosorption, is the process of driving counter ions toward an electrode by applying an electrical potential between two electrodes [1, 2, 3, 4]. Ions such as Cl-, F-, HCO3-, SO42-, NO3-, ClO4-, K+, Na+, Mg2+, Ca2+, Cd2+, Pb2+, and Cu2+ are held at the electrode-solution interface by an electrical double layer (EDL) formed under the applied potential [5, 6]. CDI is easily operated under ambient conditions and low voltages (1.2 V) without requiring high-pressure pumps, thermal heaters, distillation columns, or membranes [7, 8]. Therefore, electrosorption has attracted widespread attention in the adsorption processes of brackish water desalination [9, 10, 11, 12], wastewater treatment [13, 14], and the purification of supplied water [6, 15, 16].  
More importantly, in CDI based on EDL theory, contaminants from the electrodes can be easily removed by shorting or reversing the electrodes in the absence of chemicals [17]. However, this scenario is true only in ideal CDI operations, when the amount of charge during adsorption equals that during desorption. In fact, electrodesorption by shorting or reversing the electrodes cannot completely desorb the contaminants. [PERSON] et al. [18] reported a 40-46% desorption efficiency at 10 mA/cm2. [PERSON] et al. [19] studied different regeneration modes--open-circuit potential (OCP), short circuit (SC), and reverse voltage (RV)--in a CDI system equipped with asymmetric electrodes. The highest regeneration recovery in their study was 64%. Many recent research studies have reached similar conclusions [20; 21; 22]. Electrodesorption is a widely acknowledged problem in CDI and has defied solutions because the key factors of the existing regeneration technology remain unclear. The incomplete regeneration of electrode materials has increased the cost of electrosorption and inevitably introduced secondary pollution.  
This study employs a three-dimensional electrode reactor with the characteristics of many small particles, which shortens the transfer distance between the reactant and the electrode and hence increases the specific surface area of the working electrode [23; 24; 25; 26]. However, recycling smaller particles from a solution in a three-dimensional electrode reactor is a difficult task and risks forming a new environmental pollutant [24]. Porous carbons such as activated carbon [25; 27], activated carbon fibers [28], and carbon aerogels [29] are commonly used as electrosorption electrode materials. Activated carbon is a popular choice, especially for particle electrodes in three-dimensional electrode reactors, owing to its high electrical conductivity, good mechanical strength, and low cost [30; 31; 32; 33]. Under an electric field, the particles in a particle electrode can carry the same charge as the adjacent electrode. Hence, electrode matching is another key requirement of three-dimensional electrode reactors. If the electrodes are well matched, the overall performance of the electrodesorption process is improved. Poorly matched electrodes can degrade the performance or cause electrode damage [34; 35].  
The present study on the electrodeposition process is based on our previous three-dimensional electrode system but focuses on electrode matching to improve the cathode potential. According to theoretical electrochemistry, the cathode potential plays a definitive role. Here, the complete desorption potential is predicted in a theoretical electrochemistry analysis, and the extended cathode potential after matching the electrodes is discussed in detail. After performing electrosorption and desorption cycles, the electrode configuration and the predicted complete desorption potential are applied to the electrodesorption of SO\({}_{4}\)\({}^{2-}\). The effects of ionic strengths and circulation velocities on the desorption efficiency are evaluated in electrodesorption batch tests.