Self-Consistent Approach to Global Charge Neutrality in Electrokinetics: A Surface Potential Trap Model

Abstract

In this work, we treat the Poisson-Nernst-Planck (PNP) equations as the basis for a consistent framework of the electrokinetic effects. The static limit of the PNP equations is shown to be the charge-conserving Poisson-Boltzmann (CCPB) equation, with guaranteed charge neutrality within the computational domain. We propose a surface potential trap model that attributes an energy cost to the interfacial charge dissociation. In conjunction with the CCPB, the surface potential trap can cause a surface-specific adsorbed charge layer σ. By defining a chemical potential μ that arises from the charge neutrality constraint, a reformulated CCPB can be reduced to the form of the Poisson-Boltzmann equation, whose prediction of the Debye screening layer profile is in excellent agreement with that of the Poisson-Boltzmann equation when the channel width is much larger than the Debye length. However, important differences emerge when the channel width is small, so the Debye screening layers from the opposite sides of the channel overlap with each other. In particular, the theory automatically yields a variation of σ that is generally known as the "charge regulation" behavior, attendant with predictions of force variation as a function of nanoscale separation between two charged surfaces that are in good agreement with the experiments, with no adjustable or additional parameters. We give a generalized definition of the ζ potential that reflects the strength of the electrokinetic effect; its variations with the concentration of surface-specific and surfacenonspecific salt ions are shown to be in good agreement with the experiments. To delineate the behavior of the electro-osmotic (EO) effect, the coupled PNP and Navier-Stokes equations are solved numerically under an applied electric field tangential to the fluid-solid interface. The EO effect is shown to exhibit an intrinsic time dependence that is noninertial in its origin. Under a step-function applied electric field, a pulse of fluid flow is followed by relaxation to a new ion distribution, owing to the diffusive counter current. We have numerically evaluated the Onsager coefficients associated with the EO effect, L 21 , and its reverse streaming potential effect, L 12 , and show that L 12 ¼ L 21 in accordance with the Onsager relation. We conclude by noting some of the challenges ahead.

Thin layer analysis of a non-local model for the double layer structure

“…To describe clearly the curvature effects on the thin EDL, it suffices to establish pointwise descriptions for boundary layer solutions near the boundary. Based upon [28,29,38,48,52] for boundary layer solutions of the CCPB equations, in this work we focus mainly on an electrolyte environment involving the mixture of one anion species with the charge valence −pe 0 and one cation species with the charge valence +qe 0 (p, q > 0 and e 0 is the elementary charge). Let Ω be a bounded domain in R N (N ≥ 2), where the boundary ∂Ω associated with the charged surface is smooth.…”

“…The Neumann boundary condition (1.2) is considered for a given surface charge distribution C bd at the charged surface; the Robin boundary condition (1.5) is related to the capacitance effect of the EDL, where η is related to the thickness of the Stern layer [2]. We refer the reader to [25,37,40,48,52] for the more details of the physical background information of the model (1.7)-(1.8) and boundary conditions (1.2) and (1.5).…”

Modeling and simulation of dielectrophoretic collective dynamics in a suspension of polarizable particles under the action of a gradient AC electric field

“…Dielectrophoresis (DEP) is defined as the motion imparted on polarized particles in suspension by exposure to a gradient AC electric field; along with other electro‐mechanical effects , DEP is a longstanding issue both in pure physics and a variety of scientific and technological fields. The past few decades have seen rapid growth in the use of DEP for collecting, manipulating, and discriminating particles or biological cells suspended in liquids, due to improvements in micro‐fabrication techniques .…”

Diffusion Limit of Kinetic Equations for Multiple Species Charged Particles

“…To increase the efficiency of numerical simulations, one may use the (multi-species) Poisson-Nernst-Planck (PNP) system [19,27,28,54,62], which is a macroscopic model to describe multispecies ion transport. Conventionally, the PNP system consists of coupled diffusion-convection equations and the Poisson equation being represented as follows:…”