5  0  0.1  0.2  0.3  0.4 -4 -3 -2 -1  0  1  2  3  4  [ -3 ] z [ ]  0  0.1  0.2  0.3  0.4 -4 -3 -2 -1 0  1  2  3  4 [ -3 ] [ ] −1 Exponential fit -2.5 -2 -1.5 -1 -0.5  0  0.5 -4 -3 -2 -1  0  1  2  3  4 v x (z) [ (m/ ) -1/2 ] z [ ]  0  10  20  30  40  50  0  10  20  30  40  50 $_{µ}$EOF  /  EOF $_{s}$ [ $^{-3}$] B B -2 -1 0  0  0.02 0.04 0.06  B  [k B T/e] Figure 1: Distribution of the ionic difference Δ ρ = ρ$_{c}$ − ρ$_{a}$ between cations and anions in the solution (not counting the polyelectrolyte) for an exemplary salt concentration of ρ$_{s}$ = 0 . 05625 σ − 3 and the surface charge density σ$_{A}$ = − 0 . 208 eσ − $^{2}$. The black line corresponds to an exponential fit (Eq. (24)) with an effective inverse screening length of κ = 2 . 305 ± 0 . 025 σ − $^{1}$. Inset: Distribution of cations (cir- cles) and anions (triangles) for the same system. Figure 2: Exemplary flow profiles for a salt concentration ρ$_{s}$ = 0 . 05625 σ − 3 for varying slip lengths (from bottom to top: δ$_{B}$ = (14 . 98 , 5 . 66 , 2 . 63 , 1 . 77 , 1 . 29 , 0 . 00) σ .) The black lines are the theoretical predictions obtained by integrating the Stokes equation (Eq. (5)) with a fitted inverse screening length of κ = 2 . 305 σ − $^{1}$. Figure 3: Ratio µ$_{EOF}$ /µ 0 EOF plotted against δ$_{B}$ κ for the differ- ent salt concentrations and screening lengths given in Table I and II. The black line is the theoretical prediction of Eq. (7) with slope 1+ κδ$_{B}$ . Inset: Surface potential as obtained from µ 0 EOF using Eq. (9) (circles) and indepently by a test charge method (triangles) as a function of the salt concentration ρ$_{s}$ . the values for the Debye-H¨ uckel screening parameter κ$_{D}$ (Eq. (10)). The decay lengths are overall very different from those predicted by the Debye-H¨ uckel theory. We conclude that the system is outside the validity region of the linearized Poisson-Boltzmann approximation. This is perhaps not surprising, given that the individual ion profiles (inset of Fig. 1) at the walls deviate strongly from their bulk value, i.e. , these deviations can hardly be con- sidered as small perturbations. The surface charge is too high. On the other hand, the electrostatic coupling con- stant Ξ = 2 πZ $^{3}$λ3 $_{B}$σ$_{A}$ ∼ 0 . 2 ( Z = 1 is the valency of the cations) is still much smaller than unity, hence we are still in a ’weak coupling’ regime. This is also evi- dent from the fact that the effective screening param- eter κ differs strongly from the Guy Chapman length, µ − 1 = 2 πλ$_{B}$Zσ$_{A}$ = 1 . 31 σ . The EOF profiles for the same salt concentration ( ρ$_{s}$ = 0 . 05625 σ − $^{3}$) are shown in Fig. 2. The different curves correspond to different hydrodynamic boundary condi- tions (slip lengths). As expected, the flow velocity in- creases drastically for larger slip lengths. All curves are in good agreement with the theoretical predictions, which were obtained by integrating the Stokes equation (24) numerically with the correct partial-slip boundary con- ditions. In agreement with our earlier studies at zero salt concentration $^{29}$, we thus find that a description based on the Stokes equation – a continuum equation – remains valid even for very narrow channels. The flow velocity in the middle of the channel gives the EOF mobility. Fig. 3 compares our numerical re- sults for all salt concentrations and slip lengths with the theoretical prediction of Eq. (7), where µ 0 EOF has been determined by a linear regression for each salt concen- tration independently. We find good agreement between simulation data and theory. This confirms the validity of our theoretical result, Eq. (7). It also demonstrates that the polyelectrolyte, which was present in all simulations, does not perturb the EOF even in very narrow channels. Table II: Fitted inverse screening lengths κ and Debye-H¨ uckel screening parameter κ$_{D}$ for different salt concentrations ρ$_{s}$ and the fixed counterion density of ρ = 0 . 06 σ − $^{3}$. ρ$_{s}$ [ σ − $^{3}$] κ [ σ − $^{1}$] ± κ [ σ − $^{1}$] κ$_{D}$ [ σ − $^{1}$] 0.015 1.996 0.041 0.98 0.0225 2.011 0.049 1.02 0.03 1.983 0.041 1.07 0.0375 2.182 0.047 1.11 0.05625 2.305 0.025 1.21