## 2 Materials and Methods  
### System Model  
Figure 1 presents the architecture of the proposed desalination device, built by an array of \(N\) cells. Each cell resembles a nanochannel, where water molecules and Na\({}^{+}\) and Cl\({}^{-}\) ions flow between two planar, periodic, carbon surfaces, theoretically analogous to Poiseuille flow (see the inset of Figure 1). In a previous work, we have investigated the effect of the electric field strength, \(E\), and the distance between the two wall surfaces, \(h\), on desalination performance in a stand-alone nanochannel [30]. In this work, we consider a parallel network of such nanochannels and extend the investigation over the whole network. In each cell, water/ion solution is inserted on the left, while water and ions are separated at the outlet, as the solution flows from left to right, driven by a piston. This is achieved by the application of an external electric field, \(E\), acting perpendicular to the walls, which is expected to lead Na\({}^{+}\) near the upper wall, Cl\({}^{-}\) ions near the lower wall, and clean water in the region around the cell centerline. During the simulations, the solution density is constant, \(\rho=1078\) kg/m\({}^{3}\), and temperature remains constant at \(T=300\) K with the application of Nose-Hoover thermostats. Cells of various heights (i.e., the distance between the walls) are simulated and, along with the number of cells, \(N\), inside the device, the flow rate of clean water is estimated in Section 3. Simulation parameters for all atoms in the system are described by the Lennard-Jones (LJ) potential. For two \(i\) and \(j\) species, the potential is given by [31]  
\[u\left(\mathbf{r}_{ij}\right)=\left\{ \begin{array}{cc}4\varepsilon\left[\left( \frac{\varepsilon}{r_{ij}}\right)^{12}-\left(\frac{\varepsilon}{r_{ij}}\right) ^{6}\right]&\mathbf{r}_{ij}<\mathbf{r}_{c}\\ \\ 0&\mathbf{r}_{ij}\geq\mathbf{r}_{c}\end{array} \right. \tag{1}\]  
where parameters \(\varepsilon\) and \(\sigma\) are the energy and size parameters, respectively, and \(\mathbf{r}_{c}=9\) A is the cut-off radius. Moreover, coulombic interaction between hydrogen, oxygen, and ions is given by \(V_{c}=\frac{C_{\mathrm{eff}}\mu_{ij}}{\varepsilon_{0}r_{ij}}\), where \(C\) the energy conversion constant, \(q_{i}\) and \(q_{j}\) the charges of interacting atoms, and \(\varepsilon_{0}\) is the dielectric constant. For two different species \(i\) and \(j\) we consider \(\sigma_{ij}=\left(\sigma_{i}+\sigma_{j}\right)/2\), \(\varepsilon_{ij}=\sqrt{\left(\varepsilon_{i}\varepsilon_{j}\right)}\). When \(i\) and \(j\) correspond to wall (w) and fluid (f) atoms, the ratio \(\varepsilon_{mf}/\varepsilon_{ff}\) represents whether the wall is treated as hydrophobic (\(\varepsilon_{mf}/\varepsilon_{ff}=0.1\) and \(0.2\)) or hydrophilic (\(\varepsilon_{mf}/\varepsilon_{ff}=0.5\) and \(1.0\)). Water molecules comply to the SPC/E (extended simple point charge) pair potential, which has been found to reproduce adequately the structural and dynamic properties of water [32], especially under confinement [33]. Molecular Dynamics is a common method for interpretation and investigation of nanoscale phenomena. This does not exclude investigation with Navier-Stokes equations or quasi-continuum approaches, provided that one considers the appropriate scale factors and coefficients [34]. For example, shear viscosity across a nanochannel is not constant  
Figure 1: The proposed desalination device, consisted by an array of \(N\) cells. The inset figure presents a detailed view of each cell. [PERSON] and red circles are Na\({}^{+}\) and Cl\({}^{-}\) ions, respectively. The external electric field \(E_{z}\) is applied perpendicular to the piston-driven flow. Clean water is gathered from each cell to the common outlet, while the remaining water/ion solution (brine) is recirculated to further separate water from ions. and this has to be addressed when applying the classical N-S theory [35].