## 3 Theoretical Model Description
The mesh boundaries not parallel to each another alter the velocity magnitude and direction. After the collision event, the velocity of the SP must be reinitialized by sampling  
Figure 3: Schematics of carbon particles and the SP system for parameters representation (not to scale). the velocity of the background fluid at random from the drifting Maxwellian distribution given by  
\[\mathrm{f}(v_{i})=\sqrt{\frac{m_{sp}}{2\pi\mathrm{k_{B}}T_{0}}}\mathrm{exp}\ \left(\frac{-m_{sp}{v_{i}}^{2}}{2\mathrm{k_{B}}T_{0}}\right) \tag{2}\]  
where \(k_{B}\) is the Boltzmann constant, and \(T_{0}=273\) K is the temperature in Kelvin. The total mass of superparticles was equal to the total mass of spherical carbon particles in the experimental slurry [49, 54]. The motion of superparticles was responsible for the physical charge redistribution, which occurred in two ways. First, a neutral superparticle would experience diffuse scattering by the outer wall of the flow channel and gain a unit charge. Later, this charge would be transferred/distributed homogenously to the neighboring SPs in the interaction range. The activated carbon black used as an electrode in the slurry had an average particle size of 50 nm and a bulk density of 96.11 kg/m\({}^{3}\). The slurry comprised 1.80 \(\times\) 10\({}^{14}\) carbon particles per mm\({}^{3}\) at an estimated distance apart of 17 nm and 1.80 \(\times\) 10\({}^{14}\) electrolytic ions available per mm\({}^{3}\), resulting in approximately one ion per carbon particle free to adsorb on the carbon surface. With the high number density of carbon, the COMSOL Multiphysics particle tracing module was customized to track particle motion under the influence of momentum force. The SP size was influenced by the physical properties of carbon particles and the volume \(V\) of the electrode flowing channel. The total number of SPs, \(N_{sp}\) needed in the flow channel can be approximated as  
\[N_{sp}=\frac{V\ D_{sp}}{L\ V_{sp}} \tag{3}\]  
where \(L\) is the length of the flow channel, \(V_{SP}\) is the volume of the SP, and \(D_{SP}\) is the SP diameter. Theoretically, the number of carbon particles in an SP \(n_{NPS}\) can be estimated by using the \(V_{sp}\) and volume of a single carbon particle \(v_{p}\):  
\[n_{NPS}=\frac{V_{sp}}{v_{p}} \tag{4}\]  
The total mass of the SP is dependent on the mass of carbon \(m_{cp}\) and \(n_{NPS}\):  
\[m_{sp}=\frac{m_{cp}\ n_{NPS}}{NSP_{ratio}} \tag{5}\]  
where \(NSP_{ratio}\) is the ratio of the total number of SPs needed in the system, \(N_{sp}\), to the number of SPs actually used in a flow channel of the model. Each SP, carrying thousands of carbon particles, moves around in the flow channel and obtains surface charge while scattering from the current collector (CC) [55]. After this particle-boundary wall interaction, the force caused by the momentum exchange between neutral SPs and the current-collector surface is large enough to move SPs away from the CC. Since the SP velocity vector is discretely changing in each time step crossing the boundary, it must be reinitialized to the velocity of SP in the previous boundary. This reinitialization condition is set for the size of the domain comparable to the diameter of SP itself. In the positive flow channel, when the neutral SPs scatter from the CC boundary, they obtain the charge, \(\beta_{Na}\) representing the sodium ion (Na\({}^{1+}\)) being adsorbed on the carbon particle. Similarly, in the negative flow channel, the carbon particle adsorbs the surface charge, \(\beta_{SO_{4}}\) representing sulfate ion (SO\({}_{4}\)\({}^{2-}\)). The SP charging scheme at the boundary and charge distribution within the interaction range is given in Figure 4.  
The variable \(Z_{p}\) is the initial charge on the SP, and \(\Delta q\) is the charge difference before and after the charge transfer to the SP. Since the initial SP release in the flow channel is random and diffuse scattering randomly interacts with other SPs, the charge transfer is also considered random.