## 2 Materials and Methods
### Mathematical Formulation  
Shear viscosity, \(\mu\), is given by the Stokes-Einstein's relation [37; 38] as  
\[\mu=\frac{k_{B}T}{3\alpha\pi D_{ch}} \tag{2}\]  
where constant \(\alpha\) = 1.7 A is the effective hydrodynamic diameter of the water molecule, and \(k_{B}\) the Boltzmann constant.  
The channel diffusion coefficient, \(D_{ch}\), is obtained from the time-averaging of the mean square displacement of \(N\) fluid particles [39] as  
\[MSD=\frac{1}{N}\left\langle\sum_{j=1}^{N}\left[r_{j}(t)-r_{j}(0)\right]^{2}\right\rangle \tag{3}\]  
and  
\[D_{ch}=\underset{t\to\infty}{lim}\frac{1}{6t}MSD \tag{4}\]  
where \(r_{j}\) is the position vector of the \(j\)th atom. The brackets indicate time average. We distinguish a parallel, \(D_{P}\), and a perpendicular to the wall direction, \(D_{T}\), component, as  
\[D_{P}=\underset{t\to\infty}{lim}\frac{1}{4t}\frac{1}{N}\left\langle\sum_{j=1 }^{N}\left[r_{j}^{xy}(t)-r_{j}^{xy}(0)\right]^{2}\right\rangle \tag{5}\]  
\[D_{T}=\underset{t\to\infty}{lim}\frac{1}{2t}\frac{1}{N}\left\langle\sum_{j=1 }^{N}\left[r_{j}^{z}(t)-r_{j}^{z}(0)\right]^{2}\right\rangle \tag{6}\]  
where, \(r_{j}^{xy}\) is the position vector in \(x\)- and \(y\)-dimensions, and \(r_{j}^{z}\) the position vector in \(z\)-dimension of the \(j\)th atom.  
To calculate shear viscosity in direction parallel to the cell walls, \(\mu_{P}\), or perpendicular, \(\mu_{T}\), we, respectively, substitute in Equation (2) the diffusion coefficients \(D_{P}\) and \(D_{T}\).  
Diffusion coefficient calculations from the above equations, characterizing the mechanism of mass transfer at the lower scales, improve our understanding on the microscopic effects taking place at these scales. This would be of interest for problems that have to do with electric fields, since the system behavior is investigated at the lower possible level and all hidden effects are revealed [40]. Equation (2) provides calculations on shear viscosity, bypassing a significant amount of time-dependent statistical mechanics simulations needed to extract shear viscosity by other methods, such as the Green-Kubo formalism, and has been widely considered in nanoscale systems [37; 38]. It shows that the size effect of \(D_{ch}\) is directly influenced by the size effect of the shear viscosity \(\mu\). As \(D_{ch}\) is decreasing with the channel width, \(\mu\) is increasing. Shear viscosity is connected to friction, i.e., a macroscopic property, and it becomes clear that it also affects the ion and clean water flow rate for the proposed device.  
In order to argue on the achievable performance of the proposed desalination method, the volumetric flow rate in every channel is calculated, as  
\[Q=\overline{v}\cdot A=\overline{v}\cdot\left(h_{o}\cdot L_{y}\right) \tag{7}\]  
where \(A\) is the cross-sectional channel area, \(h_{o}\) the outlet height, \(\overline{v}\) the average fluid velocity and \(L_{y}\) is the channel \(y\)-dimension. To disjoin calculations from \(L_{y}\), since the simulation system is periodic in the \(y\)-direction, we consider the more general volumetric flow rate per unit length to be  
\[q=\frac{Q}{L_{y}}=\overline{v}\cdot h_{o} \tag{8}\]