## 3 Theoretical Model Description
### Model Implementation  
The proposed numerical simulation method was implemented in COMSOL Multiphysics 5.5 and developed based on finite element analysis and Newtonian particle dynamics. Figure 6 shows the schematic loop diagram of the numerical solution procedure of the developed method. The velocities of particles are initialized by Maxwell distribution at the first time step of the simulation--in the following steps, Newton's equations are solved to obtain particle movement This loop is carried through all the particles individually at each time increment. To calculate the charge quantity, the loop is also performed on all the contact SPs within the interaction range.  
Figure 5: Model geometry design with two channels separated by a membrane. (**a**) Circular channel, (**b**) box-shaped with the linear current collector, and (**c**) box-shaped with the extended current collector.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Geometry Design Parameters**} \\ \hline
**Parameter** & **Symbol** & **Value** & **Reference** \\ \hline Diameter of flow channel & \(D_{fc}\) & 5 mm & [49; 54] \\ \hline Length of flow channel & \(L\) & 120 mm & [49] \\ \hline Volume of flow channel & \(V\) & 2.3562 \(\times\) 10\({}^{-6}\) m\({}^{3}\) & - \\ \hline \multicolumn{4}{c}{Carbon particles parameters} \\ \hline Parameter & Symbol & Value & Reference \\ \hline Diameter of carbon particle & \(d_{p}\) & 50 nm & [49; 54] \\ \hline Volume of carbon particle & \(V_{p}\) & 6.545 \(\times\) 10\({}^{-23}\) m\({}^{3}\) & Calculated \\ \hline Number of carbon particles in 1 g & \(n_{c}\) & 6 \(\times\) 10\({}^{18}\) & Calculated \\ \hline Flow velocity of the slurry & \(u_{0}\) & 1.69 mm/s & [54] \\ \hline Total no. of carbon particles in the slurry & & 1.8 \(\times\) 10\({}^{14}\)/mm\({}^{3}\) & Calculated [49] \\ \hline Total no. of electrolytic ions in the slurry & & 1.8066 \(\times\) 10\({}^{14}\)/mm\({}^{3}\) & Calculated [49; 54] \\ \hline Mass of single carbon particle & \(m_{cp}\) & 1.67 \(\times\) 10\({}^{-19}\) g & Calculated \\ \hline Number of Na ions per carbon particle & \(\beta_{Na}\) & 1 & Needs to be optimized \\ \hline \multicolumn{4}{c}{SP parameters} \\ \hline Parameter & Symbol & Value & Reference \\ \hline Diameter of an SP & \(D_{sp}\) & 100 μm & Needs to be optimized \\ \hline Volume of an SP & \(V_{sp}\) & 5.236 \(\times\) 10\({}^{-13}\) m\({}^{3}\) & \\ \hline Average number of carbon particles in an SP & \(n_{NPS}\) & 9.4248 \(\times\) 10\({}^{10}\) & Calculated \\ \hline Number of SPs needed & \(needed_{NSP}\) & 2.0281 \(\times\) 10\({}^{5}\) & Calculated \\ \hline Number of carbon particles in an SP & \(NPS\) & 1.885 \(\times\) 10\({}^{11}\) & - \\ \hline Number of carbon particles should be in an SP & \(NC_{box}\) & 7.0686 \(\times\) 10\({}^{14}\) & - \\ \hline Total volume of SPs & \(V_{SP}\) & 3.927 \(\times\) 10\({}^{-9}\) m\({}^{3}\) & - \\ \hline Interaction diameter of SPs & \(D_{IA}\) & \(D_{SP}\) & - \\ \hline Number of SPs actually used in each channel & \(actual_{NSP}\) & 2000 & Needs to be optimized \\ \hline Ratio of needed to actual SPs & \(NSP_{ratio}\) & \(\frac{need_{NSE}}{actual_{NSP}}\) & - \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters for the particle-based stochastic model.