2 externally applied electric field, the net electrophoretic velocity of nanoparticles or polyelectrolytes can be writ- ten approximately as with the effective total mobility µ$_{t}$ = µ$_{EOF}$ + µ$_{e}$ . We note that this expression relies on two assumptions which are both not obvious: The nanoparticles or polyelectrolytes stay well outside the diffuse layer covering the wall, and they themselves do not influence the EOF. The EOF amplitude at planar walls with no-slip boundary conditions has been calculated a long time ago by Smoluchowski $^{23}$. On the nanoscale, however, the no- slip boundary condition does not necessarily apply. Ex- periments have indicated $^{24–26}$, that the velocity profile is not strictly continuous at walls, i.e. , fluids exhibit a certain amoung of slippage. This effect can be enhanced significantly by using superhydrophobic walls which are covered by a thin gas layer. It can also be tuned to some extent by designing nanopatterned surfaces with alter- nating hydrophobic and hydrophilic sections $^{27}$. In all of these cases, the appropriate mesoscopic boundary condi- tion is the ’partial slip’ boundary condition, where z$_{B}$ is the position of the hydrodynamic boundary (which is usually close to the physical boundary, but not necessarily identical), and the ’slip length’ δ$_{B}$ character- izes the amount of slippage. No-slip boundaries corre- spond to δ$_{B}$ = 0, full-slip to δ$_{B}$ → ∞ . From Eq. (3), one would expect that the EOF amplitude is enhanced in the presence of slippage $^{27}$, and this is indeed found in simulations $^{28,29}$. The effect of slippage on EOF has been calculated within the linearized Poisson-Boltzmann the- ory, the Debye-H¨ uckel approximation, by Joly et al. $^{28}$. Below, we will derive a general expression which is also valid beyond the Poisson-Boltzmann theory. In this paper we present Dissipative Particle Dynam- ics (DPD)-simulations of polyelectrolyte electrophoresis in microchannels with varying slip lengths, at varying salt concentrations. We treat the solvent and all ions ex- plicitly, and all charges interact via unscreened Coulomb interactions. This allows to investigate the interplay of solvent and polyelectrolyte, of electrostatic and hydro- dynamic interactions, of electrophoresis and EOF in full detail, with almost no approximation. (Our only approx- imation is to neglect image charge effects, i.e. , the di- electric constant is taken to constant everywhere.) Our results indicate that the hydrodynamic boundary condi- tions strongly influence the total mobility of the polyelec- trolyte, and the total mobility can be tuned from positive to negative by varying the slip length. The simulation data are compared with a simple analytical expression, which is derived based on the assumption that the flow profile in the channel follows the Stokes equation. The numerical results are in very good agreement with the theory. and µ 0 EOF is the well-known Smoluchowski result 23 for the electroosmotic mobility at sticky walls, where we have defined the inverse ’surface screening length’ where we have used ψ (0) = 0 and identified the fluid velocity at the center of the channel with the EOF veloc- ity, v$_{x}$ (0) = v$_{EOF}$ . We further define ψ$_{B}$ := ψ ( ± z$_{B}$ ) (for no-slip boundaries, ψ$_{B}$ is the so-called Zeta-Potential $^{23}$). Inserting the partial-slip boundary condition for the flow, Eq. (3), we finally obtain the following simple expression for the electroosmotic mobility, (with the shear viscosity η$_{s}$ ), one finds immediately ∂$_{zz}$v$_{x}$ ( z ) = ∂$_{zz}$ψ ( z ) ( ϵ$_{r}$ E$_{x}$/η$_{s}$ ). For symmetry reasons, the profiles v$_{x}$ and ψ must satisfy the boundary condi- tion ∂$_{z}$v$_{x}$ |$_{z}$$_{=0}$ = ∂$_{z}$ψ |$_{z}$$_{=0}$ = 0 at the center of the channel. This gives the relation (where ϵ$_{r}$ is the dielectric constant), with the Stokes equation 30 We consider for simplicity a planar slit channel with identical walls at z = ± L/ 2, exposed to an external elec- tric field E$_{x}$ in the x direction. The electrostatic poten- tial Φ( x, y, z ) then takes the general form Φ( x, y, z ) = ψ ( z ) + E$_{x}$ x + const. where we can set ψ (0) = 0 for sim- plicity. The electrolyte in the channel is taken to contain n different ion species i with local number density ρ$_{i}$ ( z ) and valency Z$_{i}$ , which results in a net charge density ρ ( z ) = ∑ n i $_{=1}$( Z$_{i}$e ) ρ$_{i}$ ( z ). The electric field then generates a force density f$_{x}$ ( z ) = ρ ( z ) E$_{x}$ in the fluid. Comparing the Poisson equation for the electrostatic potential ψ , The paper is organized as follows. The theory is pre- sented in section 2. Section 3 focuses on the simulation method and the parameters used in our simulations. The numerical results will be shown in section 4. We conclude with a brief summary in section 5. II. THEORETICAL CONSIDERATIONS: EOF IN THE PRESENCE OF SLIPPAGE v P = ( µ$_{EOF}$ + µ$_{e}$ ) E ( ext ) =: µ$_{t}$ E ( ext ) (2) ∂ $^{2}$ψ ( z ) ∂z 2 = − ρ ( z ) ϵ$_{r}$ (4) v$_{x}$ ( ± z$_{B}$ ) = ∓ δ$_{B}$ ∂ ∂z v$_{x}$ ( z ) |$_{z}$$_{=}$$_{±}$$_{z}$$_{B}$ , (3) η$_{s}$ ∂ $^{2}$v$_{x}$ ( z ) ∂z 2 = − f$_{x}$ ( z ) = − ρ ( z ) E$_{x}$ (5) v$_{x}$ ( z ) = ϵ$_{r}$ E$_{x}$ η$_{s}$ ψ ( z ) + v$_{EOF}$, (6) µ$_{EOF}$ = v$_{EOF}$/E$_{x}$ = µ 0 EOF (1 + κ δ$_{B}$ ) , (7) κ := ∓ ∂ z ψ ψ | z = ± z $_{B}$, (8) µ 0 EOF = − ϵ$_{r}$ ψ$_{B}$/η$_{s}$. (9)