## 4 Modelling the Structure-Property Relationships
### \"Pore-flow\" Models  
The \"pore-flow\" models consider the transport of species as occurring within the membrane pores forming ion and water conducting channels [85; 86]. This approach originates from the description of ion and water transport inside microcapillaries with charged walls [177; 178; 90; 179; 180]. Namely, the main goal of these models is the description of electrokinetic phenomena, that is, coupled transfer of a liquid relative to a charged solid (or liquid) surface or the transfer a charged solid particles relative to a liquid [181; 182]. The main role in these phenomena is played by the EDL on charged surfaces. The term \"space-charge models\" [183; 184; 185; 186; 187; 188; 189; 190] is often applied in the literature for this kind of models. In these models, the ion and water transport is described on the basis of the Nernst-Planck-Poisson-Navier-Stokes equations. The main parameters characterizing the membrane are pore size and wall charge density [191]. The basic version of a space-charge model deals with a pore (schematically shown in Figure 7). It involves the extended Nernst-Planck equation containing the convective term, Equation (22) written in 2D geometry. The diffusion coefficients are usually assumed to be the same as in free solution. The cross-sectional distribution of local concentration, \(\widetilde{c}_{i}\), is described by the Poisson-Boltzman equation. The fluid flow (assumed to occur only in the axial direction) is described by the Navier-Stokes equation, where the axial pressure gradient and the electrical body force provided by the tangential electric field applied to the space charge in EDL are taking into account [177; 183]. The most studies dealt with the pores whose walls are uniformly charged, such as shown in Figure 7, bottom. The impact of a non-uniform distribution of the pore wall charge on NF performance has been investigated by the group of [PERSON] in Refs. [186; 192; 193]. The research field making accent on the fluid transfer in narrow channels was only recently termed as nanofluidics but it has a quite rich history, in particular, in membrane science [194]. Nano- and microfluidics research as well as the applications in this field have experienced extensive growth in recent years [195; 196; 197; 198; 199]. 100 nm enables the occurrence of the phenomena, which are impossible at bigger length scales. Figure 6: Proton conductivity in water \(\kappa\) (**a**) and diffusion permeability \(P\) (from 0.1 M HCl solution to water) (**b**) of a Nafion® membrane doped with caesium acid salt of phosphotungstic heteropolyacid, as functions of nanoparticle volume fraction, \(f_{p}\). Diamond icons show experimental results [170]; the curves are calculated by applying the model [174] for different particle diameters shown in nm near the curves. The best fitting is obtained, when assuming that the nanoparticle size is 16 nm. Reproduced with permission from [174], Elsevier, 2016. The \"pore-flow\" models involve the effects, which are specific for interfaces: change of the dielectric constant as a function of the distance from the pore wall [183], finite ion sizes, ion hydration effects [200], adsorption [185] and other [196]. These effects are taken into account when computing the streaming potential [201], diffusion permeability [186] and permselectivity [202], pore conductivity [180,203]. This type of models has been shown as very useful for describing the ion and fluid transport in and around nanometre-sized objects [204]. [PERSON] et al. [182,205] paid attention to the steric effects near a non-permeable wall at high voltages taking into account that solvated counter-ions are crowded there. [PERSON] and [PERSON] [206] and later on [PERSON] et al. [207] have calculated using space-charge models the macroscopic Onsager's \(L_{ij}\) coefficients for a Nafion membrane as functions of the membrane nanostructure parameters (which are the pore radius, pore wall charge density and the tortuosity factor).