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This flux is a combination of the bulk convection ucvk, the bulk diffusion Dkcvk∇lncvk, the bulk conduction Dkzkcv∇ψ ̃v, as well as the surface convection ueffδk −ucv, the surface diffusion Dk(δk −1)cv∇lncv, kkkkk and the surface conduction Dkzk(δk − 1)cvk∇ψ ̃v. Then we can integrate the mass conservation equation (Eq.(4)(5)), water dissociation equation (Eq.(13)), continuity equation (Eq.(8)), and PB equation (Eq.(15)) in the z direction, and arrive at and ∂(hck)+∇·(hJk)=0 {k=1,2,···,Ns}, (25) ∂t ∂(hcw) + ∇ · (hJw) = 0, (26) ∂t cNs+1 cNs+1 = KwδNs+1δNs+2, (27) N ∇·huP+uEO+ uDO =0, (28) k k=1 N zkck −cs =0. (29) k=1 where cs = −σ/hF as the volume-averaged negative surface charge density. This last equation is called the electroneutrality equation. Now we have N + 2 equations (Eq.(25)-(29)) in terms of N + 2 independent variables ck, pv, and ψv, which can be solved with proper boundary conditions and known coefficients δk, αEO, αDO, βP , βEO, βDO. In fact, these coefficients are largely determined by how much of the channel k k k kl is occupied by EDLs. Please see Appendix Appendix A for how we calculate the coefficients. In the shock ED experiments [8, 17], typically I = 0.01–100mM, which corresponds to λD = 1–100nm, and the pore diameter (i.e., 4h = 4Vp/ap for cylindrical pores) is around 1 μm for the macroporous material and around 1 nm for the membrane. So the EDL can be thin, comparable, or thick compared with these two pore sizes. Next, we will discuss the limits of thin and thick EDLs in the depth-averaged model. If the EDL is much thinner than the channel depth (λD ≪ h), δ ,αEO,βP,βEO,βDO ∼1, (30a) (30b) Therefore, the pore-scale details are no longer important, and the depth-averaged model is reduced to the homogenized model [13], where kEO = −ǫζ/μ, kDO = 0, and ueff = u. kk If the EDL is much thicker than the channel depth (λD ≫ h), we can assume φ ≈ ζ, |ζ ̃| ≫ 1, and k kkkl αDO ∼ 0. k σ ≈−N zkcvexp(−zkζ ̃)[33].Therefore, hF k=1k δk ∼ exp(−zkζ ̃), αDO∼ 1exp(−zkζ ̃) ifzkζ ̃<0 (31a) (31b) (31c) (31d) We can further investigate the scales of the velocities. kP ∼ O(h2), and Eq.(31c) indicates that the kEO ∼ 0. Assuming constant σ, we have maxk exp(−zkζ ̃) ∼ O(h−1), and so maxk kDO ∼ O(h). Therefore, the k pressure-driven flow, electroosmosis, and diffusioosmosis all decrease as h becomes smaller, while diffusion and electromigration are not much influenced by the pore size. To conclude, the flow can be neglected at the thick EDL limit and extremely small h. As a result, we can apply these scalings to membranes. In fact, under such extreme confinement, the electroneutrality may break down [34] and the ion transport may be correlated [35]. However, for typical ion concentrations (∼ mM) and membrane charge (charge concentration ∼ M) in this work, the above effects should not be very important. 7 k31 ̃ −3 if zkζ > 0 αEO ∼0, βP,βEO,βDO ∼1. k k klPDF Image | Theory of shock electrodialysis
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