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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 

where ρ$_{i,}$$_{0}$ is the density of ions i far from the surface. It is solved by an exponentially decaying function, 

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, 

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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$_{B}$ ) − $^{1}$. 

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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 

with a conservative force ⃗ F C ij 

(where r$_{ij}$ denotes the distance between the centers of particles i and j ), a dissipative force ⃗ F D ij 

with the friction coefficient γ DPD , and a random force ⃗ F R ij 

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, 

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, 

where r$_{c}$ denotes the cut-off radius. 

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 

with a dissipative contribution 

## III. SIMULATION METHOD 

## A. Dissipative Particle Dynamics 

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## B. Tunable slip boundaries 

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