Throughout this letter, by ‘areal density’ we refer to quantities normalized using the nominal area of the inner wall (2Π r 0 d x for a differential slice) and not the cross section of the channel. Also, for simplicity, we consider all impurities equal among them (subsequent click here generalization to multiple chemical species should be easy). The average radius of the impurities is noted ρ 0. The impurity concentration in the fluid is considered to be moderate enough as to not significantly

affect its viscosity and as for the impurities in the fluid to be noninteracting with each other (specially when colliding with the channel wall). Figure 1 Representation of a nanostructured channel filter as modelled in the present letter. The nominal shape of the channel is supposed to be cylindrical with length L, and the figure shows only the differential slice with axial coordinate from x to x + dx.

The radiuses r 0 and ρ 0 correspond to the average dimensions of the bare learn more channel and impurities. The effective radiuses r e and ρ e vary as trapped impurities cover the inner wall, via their dependences on, respectively, the areal density n of trapped impurities and on the areal density z e of effective charge of the inner wall. This z e reflects that exposed charges in a nanostructured surface attract the impurities in the fluid and also constitute binding anchors for those impurities. It is expected to diminish as impurities cover the surface, for which we assume the simple z e(n) dependence given by Equation 1 of the main text. Effective-charge density of the inner wall, z e We now introduce the important concept of a phenomenological ‘effective charge’ of the inner wall of the channel. We quantify this effective charge via its areal density PJ34 HCl z e , and as already commented on in the introduction, it reflects the fact that the nanostructured walls expose charges that induce both electrostatic and van der Waals attractions over the

components of the impurities in the fluid. Indeed, z e will depend on the areal density of already trapped impurities n (which will screen out the wall) and also on the chemistry specifics of the wall and impurities. Let us focus on the mutual interplays between n and z e and in obtaining an equation for their evolution with time as flow passes through the channel. In particular, the interdependence z e (n) may be naturally expected to be continuously decreasing when n increases, to take a finite value z 0 at n = 0 (clean filter), and to saturate to zero when n reaches some critical value n sat at which all active centers of the wall become well covered by impurities. We thus postulate the simplest z e (n) dependence fulfilling such conditions: (1) where the notation ∥…∥ stands for min1,…. Obviously, other sensible choices for z e (n) are possible such as, e.g.