Brian Kirby

Species and Charge Transport

This chapter describes a general framework for species and charge transport, which assists us in understanding how electric fields couple to fluid flow in nonequilibrium systems. The following sections first describe the basic sources of species fluxes. These constitutive relations include the diffusivity, electrophoretic mobility, and viscous mobility. The species fluxes, when applied to a control volume, lead to the basic conservation equations for species, the Nernst–Planck equations . We then consider the sources of charge fluxes, which lead to constitutive relations for the charge fluxes and definitions of parameters such as the conductivity and molar conductivity. Because charge in an electrolyte solution is carried by ionic species (in contrast to electrons, as is the case for metal conductors), the charge transport and species transport equations are closely related – in fact, the charge transport equation is just a sum of species transport equations weighted by the ion valence and multiplied by the Faraday constant. We show in this chapter that the transport parameters D , µ EP , µ i , σ, and ∧ are all closely related, and we write equations such as the Nernst–Einstein relation to link these parameters. These issues affect microfluidic devices because ion transport couples to and affects fluid flow in microfluidic systems. Further, many microfluidic systems are designed to manipulate and control the distribution of dissolved analytes for concentration, chemical separation, or other purposes.

Units and Fundamental Constants

This book endeavors to maintain strict adherence to the SI System of Units. Quantities must be in SI units when inserted into the equations, and when used properly, the equations will yield values in SI units. This means, for example, that wavelengths are to be expressed in m, wave numbers in m−1, intensity iλ in Wm−3sr−1, intensity i0η in Wm−1sr−1, angles in radians or steradians, photon energy in J, and temperatures in K. The only exception to this rule is in gas pressure, which is sometimes expressed in atmospheres.

AC Electrokinetics and the Dynamics of Diffuse Charge

Equilibrium models of the EDL (Chapter 9) assume that the ion distribution is in equilibrium and use a Boltzmann statistical description to predict ion distributions. The equilibrium assumption is appropriate for the EDL at an electrically insulating surface such as glass or most polymers, because the ion distribution processes are typically fast relative to the phenomena that change the boundary condition ϕ 0 (e.g., surface adsorption or changes in electrolyte concentration or pH). In this chapter, we address the dynamics of diffuse charge. We focus primarily on the formation of thin double layers at electrodes with attention to the dynamics of doublelayer formation and equilibration. Unlike for the double layer formed at the surface of an insulator, the double-layer equilibration at an electrode (owing to the potential applied at that electrode) is not necessarily fast compared with the variation of the voltage at the electrode – high-frequency voltage sources can vary rapidly compared with double-layer equilibration. Thus the dynamic aspects of double-layer equilibration are critically pertinent. A full continuum description of these phenomena comes from the Poisson, Nernst–Planck, and Navier–Stokes equations combined with boundary conditions describing electrode kinetics. Because such analysis is daunting, we approximate the problem as that of predicting surface electroosmosis with time-dependent electrokinetic potentials, and use 1D models of the EDL to form equivalent circuits that can be used to model the temporal response of ϕ 0 .