## 3 Theoretical Model Description  
The modeling approach adopted for FE charging in EFC geometry takes the physical properties of carbon electrodes into account, such as size/shape, particle volume fraction, and concentration. Approximately 1.8 \(\times\) 10\({}^{17}\) carbon particles were in 1 cm\({}^{3}\) of the slurry electrode. Solving the differential equations simultaneously for time, three-dimensional position, and distribution in such a dense system of nano-particles would be costly. For a more efficient simulation, the concept of a superparticle representing many real particles weakly interacting at overlapping distances was applied. The building block of the model is not of single particles but rather of collective clouds: each computational particle (defined  
Figure 2: (**a**) Photograph of flow capacitor cell and experimental set-up for charging [49]. (**b**) Potentiostatic measurements at constant voltages of 0.25, 0.5, and 0.75 V and (inset) at constant currents of 5, 10, 20, and 50 \(\upmu\)A. (**c**) Nyquist plot for carbon slurry in 0.1 M Na\({}_{2}\)SO\({}_{4}\). as a superparticle) represents a group of particles that can be visualized as a small piece of phase space. This particle-based model was developed to study the dense particle concentration by following each particle's trajectory using Newton's law of motion [51]. In addition, this method allows the rescaling of the number of particles because the Lorentz force's acceleration depends only on the charge-to-mass ratio so that each superparticle follows the same trajectory as a real carbon particle. The SP diameter should be smaller than the width of the flow channel and several orders larger than the size of carbon particles. Figure 3 shows the main idea, depicting the lumping of several real particles into a distribution of superparticles. Due to stochastic nature of the model, the main advantage of such an approach is its capacity to add arbitrary random and fluctuating side reactions or forces to the simulation. The method solves second-order ordinary differential equations for the components of each superparticle position. The conservation of linear momentum of each superparticle (SP) takes the familiar form of Newton's second law of motion as  
\[F_{t}=\frac{d\left(m_{sp}v\right)}{dt} \tag{1}\]  
where \(m_{sp}\) is the mass of the individual SP and \(v\) is its velocity. Generally, the total force \(F_{t}\) acting on the particles can be divided into two categories, those due to external fields (electric field or magnetic field) and due to interactions between particles. It can be a combination of drag (subjected to the velocity of the particle), dielectrophoretic (particle under the influence of electric field), and Brownian (imbalanced collision of particles) forces, among others, depending on the requirement of the system under study. The Brownian movement at the microscale is negligible [52], and the drag and dielectrophoretic forces for spherical particles located in an infinite medium without any neighboring particle are of the order of \(10^{-16}\), which is considered ineffective for such systems [53]. Thus, momentum Equation (1) gives the force on each particle to track the motion of computational superparticles. Using the finite element method, the flow channel's continuous domain was divided into a mesh of discrete elements. The position of an SP released in the channel is given by \(q\left(q_{x},q_{y},q_{z}\right)\), and the respective velocity at a specific location is given as \(v\left(v_{x},v_{y},v_{z}\right)\). The force \(F_{t}\) on the SP in the background fluid, e.g., water is given by Equation (1). The trajectories of individual particles are always solved in the time domain. At each time step taken by the solver, the force acting on each particle is queried at the current particle position. During each time step, the particles may interact with boundaries in the geometry, or they may be subjected to other phenomena that can discontinuously change the particle velocity.