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 . By defining a chemical potential 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 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 potential that reflects the strength of the electrokinetic effect; its variations with the concentration of surface-specific and surface-nonspecific 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, , and its reverse streaming potential effect, , and show that in accordance with the Onsager relation. We conclude by noting some of the challenges ahead.
- Received 8 April 2013
DOI:https://doi.org/10.1103/PhysRevX.4.011042
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Published by the American Physical Society
Popular Summary
When a channel made of silica, for example, is filled with water, the channel’s surface becomes ionized by shedding small ions that become mobile in water. An interesting situation appears: A diffuse layer of the small mobile ions forms very close to the oppositely charged channel wall, as a result of a fine balance between thermal Brownian motion and Coulomb attraction. How does one describe this electric double layer quantitatively? And, how does it influence the flow of the solution if an electric field is applied, as the mobile ions will be forced by the field to move in a particular direction? These questions are in fact almost two centuries old, central to the science of electrokinetics as electric double layers are ubiquitous in electrochemistry, fuel-cell science, biology, and chemical analysis. It turns out, however, that the existing approaches to answering this question are more empirically effective than fundamentally rigorous, and they face challenges in contemporary contexts where the channel dimensions are in the nanoscales.
In this work, we present an approach that is rigorous in terms of both the fundamental physics and the mathematical formulation and that, going beyond just making an academic point, leads to new predictions of nanoscale and time-dependent electrokinetic phenomena.
The crucial new element in our approach is the introduction of a surface potential trap, which attributes an energy cost to the charge separation process at the fluid-solid interface. The fundamental constraint of global charge neutrality, that the total number of charges carried by the small mobile ions should cancel the oppositely charged ions at the fluid-solid interface, is enforced rigorously at the initial mathematical formulation of the problem. The end result is a set of integral-differential equations that describe the electrical double layer and its dynamics under an externally applied electric field, applicable to both macroscopic and nanoscale systems, and also to both steady-state and time-dependent processes. We show its much broader applicability with a number of results that quantitatively explain some of the existing experimental data without the need of additional parameters, and with new predictions about time dependence of the electro-osmotic effects—an area rarely explored so far.
We expect our approach, and its further adaptations to a broad range of electrokinetic contexts, to become a fundamental extension to the theoretical toolset for the cross-disciplinary science of electrokinetics.
