Polyelectrolyte Electrophoresis in Nanochannels: A Dissipative Particle Dynamics Simulation Jens Smiatek 1, ∗ and Friederike Schmid 2, † $^{1}$Institut f¨ ur Physikalische Chemie, Universit¨ at M¨ unster, D-48149 M¨ unster, Germany 2 Institut f¨ ur Physik, Johannes Gutenberg-Universit¨ at, Staudinger Weg 7, D-55099 Mainz, Germany (Dated: August 1, 2018) We present mesoscopic DPD-simulations of polyelectrolyte electrophoresis in confined nanogeome- tries, for varying salt concentration and surface slip conditions. Special attention is given to the influence of electroosmotic flow (EOF) on the migration of the polyelectrolyte. The effective poly- electrolyte mobility is found to depend strongly on the boundary properties, i.e. , the slip length and the width of the electric double layer. Analytic expressions for the electroosmotic mobility and the total mobility are derived which are in good agreement with the numerical results. The relevant quantity characterizing the effect of slippage is found to be the dimensionless quantity κ δ$_{B}$ , where δ$_{B}$ is the slip length, and κ − 1 an effective electrostatic screening length at the channel boundaries. PACS numbers: 82.35.Rs 47.57.jd 47.61.-k 82.45+z 83.50.Lh In recent years there is growing interest in techniques for manipulating single nanoparticles or macromolecules in micro- and nanochannel systems. The flow profiles in these channels and the motion of the macromolecules can be controlled on the nanoscale by pressure gradients and electric fields, and by exploiting smart channel geome- tries. This explains the great potential and the broad applicability of nanochannel devices, e.g. , for analyzing tiny DNA or protein samples by electrophoresis $^{1–11}$. Such systems represent a challenge for theory and com- puter simulation due to their high complexity. In biotech- nological applications, the molecules of interest are often charged and dissolved in buffer solutions with high salt concentrations. Thus electrostatic and hydrodynamic ef- fects compete with each other, resulting, among other, in a remarkable ’electrohydrodynamic screening’ effect: In free solution electrophoresis ( i.e. , in a constant electric field), the counterion layer surrounding a charged par- ticle not only screens the electrostatic interactions, but also the dominant contribution to the hydrodynamic in- teractions, i.e. , those that are generated by the applied field $^{12,13}$. Therefore, simulations of electrophoresis that neglect the electrostatic and hydrodynamic interactions altogether often give results that are in good semiquan- titative agreement with experiments $^{1,14,15}$. More sophis- ticated approaches that still allow to avoid the explicit representation of charges have been devised as well $^{16,17}$. Nevertheless, it is clear that such simplified treatments disregard important physics, especially in confined ge- ometries where electrostatic interactions compete with the regular steric interactions with the confining walls $^{18}$. Simulations that take full account of electrostatic and hydrodynamic interactions are clearly desirable. Such simulations have recently been carried out for polyelec- trolyte electrophoresis in free solutions $^{19,20}$, but, to the best knowledge of the present authors, not yet for mi- crochannels. Now if the diffuse layer is thin compared to the chan- nel dimensions, Eq. (1) can be regarded as an effective boundary condition for a steady-state EOF velocity field v$_{EOF}$ ( r ) inside the channel. (We note that in a steady- state situation, the external field in the vicinity to a wall is necessary parallel to the wall.) Cummings et al. 21 have shown that for laminar incompressible flow, this bound- ary condition effectively defines the flow inside the chan- nel. Provided Eq. (1) also holds at the inlet and outlet boundaries of the channel, it becomes valid everywhere in the channel $^{22}$. Due to this remarkable similitude be- tween the steady-state velocity field of an EOF and the outside of the diffuse layer, with the so-called electroos- motic mobility µ$_{EOF}$ . The effective migration speed v P of particles in microchannels in response to the exter- nal fields then results from two contributions $^{15}$: The bare electrophoretic mobility µ$_{e}$ of the polyelectrolyte in a fluid at rest, and the background EOF velocity, v P = v$_{EOF}$ + µ$_{e}$ E ( ext $^{)}$. Experimentally, it has been found that the former may even dominate over the latter, such that the polyelectrolyte effectively migrates in the direc- tion opposite to the applied field $^{4}$. In many microchannels, an additional effect comes into play, which significantly modifies the effective elec- trophoretic response of particles to electric fields: The same electric field that drives the polyelectrolyte may also induce a total net fluid flow in the microchannel, the ’electroosmotic flow’ (EOF). Many materials commonly used in microtechnology like PDMS (Polydimethylsilox- ane) acquire charges if brought in contact with water, either by the ionization or dissociation of surface groups or the adsorption of ions from solution. To screen the charges on the channel walls, a diffuse layer of oppositely charged ions forms in front of the walls. In the presence of an external electric field E ( ext $^{)}$, these ions are pulled along, dragging the surrounding fluid with them. One gets a characteristic ’plug’ flow profile which saturates at a fluid velocity I. INTRODUCTION arXiv:1001.0885v1 [cond-mat.soft] 6 Jan 2010 v$_{EOF}$ = µ$_{EOF}$ E ( ext ) (1)