4 coupling to the relative velocity ( v $_{i}$− v $_{wall}$) of the particle with respect to the wall, and a stochastic force Table I: Slip lengths δ$_{B}$ for different layer friction coefficients γ$_{L}$ , compared with theoretical value δ T B according to Ref. 38. which satisfies the fluctuation-dissipation relation and thus ensures that the local equilibrium distribution is again a Boltzmann distribution. Here α is α = x, y, z and χ$_{i,α}$ is a Gaussian distributed random variable with mean zero and variance one: 〈 χ$_{i,α}$ 〉 = 0, 〈 χ$_{i,α}$χ$_{j,β}$ 〉 = δ$_{ij}$δ$_{αβ}$ . The viscous coupling between fluid and wall is achieved by the locally varying viscosity γ$_{L}$ω$_{L}$ ( z ) with ω$_{L}$ ( z ) = 1 − z/z$_{c}$ up to a cut-off distance z$_{c}$ . The pref- actor γ$_{L}$ can be used to tune the strength of the friction force and hence the value of the slip length. Within this approach it is possible to tune the slip length δ$_{B}$ system- atically from full-slip to no-slip, and to derive an analytic expression for the slip length as a function of the model parameters $^{38}$. We have studied the electrophoresis of charged poly- mers of length N = 20 in electrolyte solutions, confined by a planar slit channel with charged walls. All particles, polymer, solvent and ions, are modeled explicitly. We use a simulation box of size (12 σ × 12 σ × 10 σ ) which is peri- odic in x - and y -direction and confined by impermeable walls in the z -direction. The walls repel the particles via a soft repulsive WCA potential 39 of range σ and am- plitude ϵ . (Hence the accessible channel width for the particles is actually L$_{z}$ = 8 σ ). Ions and monomers repel each other with the same WCA potential. In addition, chain monomers are connected by harmonic springs with the spring constant k = 25 ϵ/σ 2 and r$_{0}$ = 1 . 0 σ . Neu- tral solvent particles have no conservative interactions except with the walls. The wall contains immobilized, negatively charged par- ticles at random positions. Every second monomer on the polyelectrolyte carries a negative charge. The solvent contains the positive counterions for the walls and the polyelectrolyte, and additional (positive and negative) salt ions. All charges are monovalent, and the system as a whole is electroneutral. In addition to their other interac- tions, charged particles interact via a Coulomb potential with the Bjerrum length λ$_{B}$ = e $^{2}$/ 4 πϵ$_{r}$k$_{B}$T = 1 . 0 σ , and they are exposed to an external field E$_{x}$ = − 1 . 0 ϵ/eσ . Specifically, we have studied systems with a surface charge density of σ$_{A}$ = − 0 . 208 eσ − $^{2}$. The total counte- rion density was ρ = 0 . 06 σ − 3 and the salt density varied between ρ$_{s}$ =0.05625, 0.0375, 0.03, 0.025, and 0 . 015 σ − $^{3}$. We use DPD simulations with a friction coefficient γ DPD = 5 . 0 σ − $^{1}$( mϵ ) 1 / $^{2}$. The density of the solvent par- ticles was ρ = 3 . 75 σ − $^{3}$, and the temperature of the sys- tem was T = 1 . 0 ϵ/k$_{B}$ . For these parameters, the shear to the ionic difference Δ ρ = ρ$_{c}$ − ρ$_{a}$ . The exponential fit describes the data very well (black solid line in Fig. 1). The fit parameters for κ are listed in Table II, along with Fig. 1 shows the average ionic distributions of an- ions ρ$_{a}$ and cations ρ$_{c}$ for the salt concentration ρ$_{s}$ = 0 . 05625 σ − $^{3}$. Here the ’cations’ include the positively charged salt ions and the counterions of the wall and the polyelectrolyte, and the ’anions’ only the negatively charged salt ions. Due to the presence of the polyelec- trolyte in the middle of the channel, the average cation density there is slightly increased. To determine the in- verse effective screening length κ , we have thus fitted the following function The electrostatics were calculated by P3M 40 and the ELC (electrostatic layer correction)-algorithm 41 for 2 D + h slabwise geometries. All simulations have been carried out with the freely available software package ESPResSo 42,43 Tunable-slip boundary conditions were used with fric- tion coefficients γ$_{L}$ =0.1, 0.25, 0.5, 0.75, 1.0, and 6 . 1 σ − $^{1}$( mϵ ) 1 / $^{2}$. The range of the viscous layer was z$_{c}$ = 2 . 0 σ . Only the solvent particles interact with the tunable-slip boundaries. By performing Plane Poiseuille and Plane Couette flow simulations with the above given parameters, the slip length δ$_{B}$ and the hy- drodynamic boundary positions z$_{B}$ can be determined independently $^{38}$. The hydrodynamic boundary position is found at | z$_{B}$ | = (3 . 866 ± 0 . 266) σ in all simulations. The corresponding slip lengths are presented in Table I together with the theoretical values predicted by the an- alytic expression in Ref. 38. The comparison shows that the simulated results are in good agreement with the the- ory. viscosity of the DPD fluid – as determined by fitting the amplitude of Plane Poiseuille flows 38 – is given by η$_{s}$ = (1 . 334 ± 0 . 003) σ − $^{2}$( mϵ ) 1 / $^{2}$. The DPD timestep was δt = 0 . 01 σ ( m/ϵ ) 1 / $^{2}$. F R i,α = √ 2 γ$_{L}$ k$_{B}$T ω$_{L}$ ( z ) χ$_{i,α}$ (22) U$_{harmonic}$ = 1 2 k ( r$_{ij}$ − r$_{0}$ ) 2 (23) C. Simulation details IV. NUMERICAL RESULTS Δ ρ = Δ ρ$_{0}$ ( e − κz + e $^{κz}$) + c (24) γ$_{L}$ [ σ − $^{1}$( mϵ ) 1 / $^{2}$] δ$_{B}$ [ σ ] ± δ$_{B}$ [ σ ] δ T $_{B}$[ σ ] 0 . 1 14 . 977 1 . 879 14 . 000 0 . 25 5 . 664 0 . 783 5 . 458 0 . 5 2 . 626 0 . 521 2 . 613 0 . 75 1 . 765 0 . 409 1 . 664 1 . 0 1 . 292 0 . 423 1 . 190 6 . 1 0 . 000 0 . 197 0 . 000