3 The remaining task is to determine the screening pa- rameter κ . If the surface charges are very small and the ions in the liquid are uncorrelated, it can be calculated analytically within the linearized Debye-H¨ uckel theory $^{31}$. The Debye-H¨ uckel equation for the evolution of the po- tential ψ in an electrolyte solution reads ∂$_{zz}$ψ = κ 2 $_{D}$ψ with the inverse Debye-H¨ uckel screening length Newtonian fluids are effectively modeled by the Dis- sipative Particle Dynamics (DPD) method $^{36,37}$. DPD is a coarse-grained, momentum-conserving simulation tech- nique which creates a well-defined canonical ensemble. The basic DPD equations are defined in terms of the forces on one particle, which involve two-particle interac- tions where ρ$_{i,}$$_{0}$ is the density of ions i far from the surface. It is solved by an exponentially decaying function, with a conservative force ⃗ F C ij Inserting that in Eq. (8), one finds κ = κ$_{D}$ , i.e. , the sur- face screening length is identical with the Debye screening length. Eq. (9) with κ = κ$_{D}$ basically corresponds to the result of Joly et al. $^{28}$. Unfortunately, the range of validity of the Debye- H¨ uckel theory is limited, it breaks down already for mod- erate surface potentials ψ$_{B}$ and/or for highly concen- trated ion solutions. Nevertheless, the exponential be- havior often persists even in systems where the Debye- H¨ uckel approximation is not valid. For high ion con- centrations, detailed studies based on integral equa- tions have lead to the conclusion that the Debye-H¨ uckel approximation can still be used in a wide parameter range, if κ$_{D}$ is replaced by a modified effective screening length $^{32–34}$. For high surface charges, analytical solu- tions are again available in the so-called ’strong coupling limit’, where the profiles are predicted to decay expo- nentially with the Guy-Chapman length $^{35}$. This limit is very special and rarely encountered. At intermediate coupling regimes, the decay length must be obtained em- pirically, e.g. , by fitting the charge distribution ρ ( z ) to an exponential behavior, which is characterized by the same exponential behavior than ψ ( z ) by virtue of the Poisson equation, Putting everything together, the total net electrophoretic mobility µ$_{t}$ (Eq. 2) of a polyelectrolyte in the channel can be expressed in terms of the electroosmotic mobility µ$_{EOF}$ as where the ratio µ$_{e}$/µ 0 EOF depends only weakly on the ionic strength of the electrolyte and the slip length of the sur- face. The main effect of slippage is incorporated in the factor (1 + κ δ$_{B}$ ) − $^{1}$. with a dissipative contribution The hydrodynamic boundary condition at the walls is realized with a recently developed method 38 that allows to implement arbitrary partial-slip boundary conditions: We introduce an additional coordinate-dependent viscous force that mimicks the wall/fluid friction where r$_{c}$ denotes the cut-off radius. which ensures that an equilibrium simulation samples a canonical ensemble $^{36,37}$. Otherwise, the weight function ω DPD is arbitrary and will be chosen linear here as often in the literature, Here ˇ ζ$_{ij}$ = ˇ ζ$_{ji}$ is a symmetric random number with zero mean and unit variance, and the weighting functions of the dissipative and the stochastic force are related by a fluctuation-dissipation theorem, with the friction coefficient γ DPD , and a random force ⃗ F R ij (where r$_{ij}$ denotes the distance between the centers of particles i and j ), a dissipative force ⃗ F D ij III. SIMULATION METHOD A. Dissipative Particle Dynamics B. Tunable slip boundaries n ∑ i = q ( Z$_{i}$e ) ρ$_{i}$ ( z ) ∝ ∂ $^{2}$ψ ( z ) ∂z 2 ∝ ( e κz + e − $^{κz}$) . (12) µ$_{t}$ µ$_{EOF}$ = 1 + µ$_{e}$ µ 0 $_{EOF}$(1 + κ δ$_{B}$ ) , (13) F L i = F D i + F R i (20) F D i = − γ$_{L}$ ω$_{L}$ ( z ) ( v i − v $_{wall}$) (21) ω DPD ( r$_{ij}$ ) = { 1 − r$_{ij}$ r$_{c}$ : r$_{ij}$ < r$_{c}$ 0 : r$_{ij}$ ≥ r$_{c}$ (19) ω$_{D}$ ( r$_{ij}$ ) = [ ω$_{R}$ ( r$_{ij}$ )] 2 ≡ ω DPD ( r$_{ij}$ ) , (18) ⃗ F R ij = √ 2 γ DPD k$_{B}$T ω$_{R}$ ( r$_{ij}$ ) ˇ ζ$_{ij}$ ˆ r$_{ij}$ . (17) ⃗ F D ij = − γ$_{DPD}$ω$_{D}$ ( r$_{ij}$ )(ˆ r$_{ij}$ · ⃗ v$_{ij}$ )ˆ r$_{ij}$ (16) ⃗ F C ij = − ⃗ ∇$_{ij}$ U$_{ij}$ ( r$_{ij}$ ) . (15) ⃗ F DPD i = ∑ i = j ⃗ F C ij + ⃗ F D ij + ⃗ F R ij (14) κ$_{D}$ = √ ∑ n i $_{=1}$( Z$_{i}$e ) $^{2}$ρ$_{i,}$$_{0}$ ϵ$_{r}$ k$_{B}$ T , (10) ψ ( z ) ∝ ( e κ$_{D}$z + e − κ$_{D}$z − 2) . (11)