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 ϕ0 (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 ϕ0.

Chemical separations are a critical component of analytical and synthetic chemistry. In microchip applications, a sample comprising multiple chemical species is separated spatially into individual components by inducing the components of a sample to move at differing velocities in a microchannel. This is shown schematically in Fig. 12.1 and a sample experimental result is shown in Fig. 12.2. Separations are achieved by inserting a sample fluid bolus into a microchannel, inducing motion of these species with velocities that differ from species to species, and detecting the concentration of species as a function of time as these species elute (i.e., arrive) at the location of the detector (Fig. 12.1). Many microfluidic separations are modified from capillary or column-based techniques, and draw advantages from more optimal fluid transport, thermal dissipation, or system integration. One example of a chemical separation is an electrophoresis separation, which can be used to separate species that have different electrophoretic mobilities. In this case, species motion is induced by an electric field aligned along the axis of the microchannel, which induces electroosmosis and electrophoresis. Because this technique requires only that electric fields be applied, it integrates easily into microsystem designs, and a large fraction of the microchip analyses developed since 1995 use microchip electrophoresis (see one example in Fig. 12.3). This is true for both protein analysis (Section 12.5) and DNA analysis and sequencing (Chapter 14).

Most of this text uses continuum, ideal-solution theory to describe transport in microand nanoscale systems. Molecule-scale interactions, however, require that the interactions of the solvent, dissolved electrolytes, and macromolecules be treated in more detail. The first example of this in the text is the steric modification of the Poisson–Boltzmann description of the EDL presented in Chapter 9. Other examples of this include the intermolecular potentials used in atomistic simulations, excluded-volume modeling for predictions of DNA conformation in nanochannels, and colloidal simulations. This appendix focuses on the general concepts of interaction potentials and distribution functions, with a focus on atomistic modeling. As is often done in the atomistic/molecular dynamics literature, we use the term atom in this appendix in the original sense of an indivisible unit rather than the modern chemical sense of a nucleus surrounded by an electron cloud. Thus the term atom in this appendix refers in general to any entity that we model based on its potential energy – e.g., solvent molecules or dissolved ions or even boundaries.

THERMODYNAMICS OF INTERMOLECULAR POTENTIALS

Atoms that interact rarely are straightforward to model thermodynamically. That these atoms rarely interact is the basic premise of the ideal gas law and the ideal solution approximation. For example, the Boltzmann approximation for ions in an electrolyte consists of treating the ions as if they do not interact with each other, interacting only with a continuum electrical field.

To this point, we have considered flow in channels whose dimension was large relative to the Debye length or the size of any molecules or particles suspended in the flow. When we use channels with shallow (e.g., nanoscale) depths d, we cannot separate the EDL from the bulk fluid by using boundary-layer theory; instead, we must account for the presence of net charge density in the bulk flow field. Even if the double layers remain thin, the perturbative effects of double layers (for example, surface conductance) are more significant as the channel becomes small. Because these phenomena typically coincide with transport through nanoscale channels, the term nanofluidics is often used to refer to flows with small d * or flows with molecules or particles comparable to the size of the channel, though the scale need not be nanoscopic for these phenomena to be important, and these phenomena are unimportant in some nanoscale flows. Despite this, some authors use the term nanofluidics to refer specifically to flows in nanoscale channels with no reference to molecular size or λD. Because our interest is the interplay of electrokinetic effects with channels and molecular-scale confinement, our focus is on channels with molecular scale or of a size comparable to λD, and we pay only cursory attention to the absolute dimension of the channel.

For unidirectional flow in infinitely long, uniform-cross-section channels, thick-EDL effects are observed primarily through changes in the elements of the electrokinetic coupling matrix.

When electric fields are applied across capillaries or microchannels, bulk fluid motion is observed. The velocity of this motion is linearly proportional to the applied electric field, and dependent on both (a) the material used to construct the microchannel and (b) the solution in contact with the channel wall. This motion is referred to as electroosmosis and stems from electrical forces on ions in the electrical double layer or EDL, a thin layer of ions that is located near a wall exposed to an aqueous solution. If the fluid velocity is interrogated at micrometer resolution, for example, by observation of the fluid flow with a light microscope, the fluid flow in a channel of uniform cross section appears to be uniform. If the fluid velocity is interrogated with nanometer resolution (which is experimentally difficult), we find that the fluid velocity is uniform far from the wall but decays to zero at the wall over a length scale λD ranging from approximately 0.5 to 200 nm. Figure 6.1 illustrates the velocity profile in an electroosmotic flow.

This fluid flow can be immensely useful in microfluidic systems, because it is often much more straightforward experimentally to address voltage signals sent to electrodes rather than to implement and control a miniaturized mechanical pressure pump. This flow, however, comes with its own complications: Its velocity distribution is different from pressure-driven flow, it is sensitive to chemical features at the interface, and the act of applying electric fields can also move particles relative to the fluid or cause Joule heating (i.e., resistive heating) throughout the fluid.

Previous chapters assert that a potential drop occurs over an EDL, consistent with the fact that chemical reactions occur at the surface to induce ionization of wall species. We now return to this subject in greater detail. Our goal is to be able to predict the equilibrium surface potential at microfluidic device interfaces as a function of the device material and solution conditions. This chapter frames the problem, describes associated parameters, and lists several models that can be used to attack this problem and interpret experimental data. We start by clarifying notation and terminology. We then discuss the chemical origins of surface charge for both Nernstian and non-Nernstian surfaces, discuss techniques for measuring and modifying electrokinetic potentials, and summarize observed zeta potentials for microfluidic substrates. Finally, we discuss how EDL theory is related to interpretation of zeta potential data and the relation between ζ and ϕ0.

DEFINITIONS AND NOTATION

Here we must define the distinct meanings of several terms, namely the zeta potential, the electrokinetic potential, the interfacial potential, the double-layer potential, and the surface potential. These terms have different meanings and are used differently by various authors. Further, some of these terms become equivalent if specific models are used to describe the interface, but have different meanings if other models are used.

Surface potential (or, equivalently, interfacial potential or double-layer potential) typically implies the difference between the potential in a bulk, electroneutral solution and the potential at the wall.

Micro- and nanofabricated devices have led to revolutionary changes in our ability to manipulate tiny volumes of fluid or micro- and nanoparticles contained therein. This has led to countless applications for chemical and particulate separation and analysis, biological characterization, sensors, cell capture and counting, micropumps and actuators, high-throughput design and parallelization, and system integration, to name a few areas. Because biological and chemical analysis is typically concerned with molecules and bioparticles with small dimensions (some examples are shown in Fig. 0.1), the tools used to manipulate these objects are naturally of a similar scale, and the developments in micro- and nanofabrication in recent decades has brought engineering tools to a scale that easily matches these objects.

From a fluid-mechanical standpoint, our ability to manufacture micro- and nanoscale devices creates a number of challenges and provides matching opportunities, some of which are denoted schematically in Fig. 0.2. If we focus on liquid-phase devices, which have dominated most bioanalytical applications, shrinking the length scales makes interfacial phenomena and electrokinetic phenomena much more important, and reduces the importance of gravity and pressure. The no-slip boundary condition, safely assumed for macroscopic flows, can be inaccurate when the length scale is small. Although the low-Reynolds-number characteristic of most of these flows eliminates the challenges of nonlinearity in the convective term and the associated difficulty in modeling turbulent flows, we are instead forced to consider the nonlinearity of the source term in the Poisson–Boltzmann equation, nonlinearity of the coupling of electrodynamics with fluid flow, and uncertainty in predicting electroosmotic boundary conditions.

Many microfluidic systems are used to manipulate the distribution of chemical species. Chemical separations, for example, physically separate components of a multispecies mixture so that the quantities of each component can be analyzed or so that useful species can be concentrated or purified from a mixture. Many biochemical assays, for example DNA microarrays, require that a reagent be brought into contact with the entirety of a functionalized surface, i.e., that the reagents in the system be well mixed. Studies of homogeneous kinetics in solution require that a system become well mixed on a time scale faster than the kinetics of the reaction. In contrast to these, extracting functionality from a spatial variation of surface chemistry often depends on the ability to pattern surface chemistry with flow techniques, which requires that components of the solution remain unmixed.

These topics all motivate discussion of the passive scalar transport equation. This convection–diffusion equation governs the transport of any conserved property that is carried along with a fluid flow, moves with the fluid, and does not affect that fluid flow. Chemical species and temperature are two examples of properties that can be handled in this way, as long as (1) the chemical concentration or temperature variations are low enough that transport properties such as density or viscosity can safely be assumed uniform, and (2) we neglect electric fields, which can cause migration of chemical species relative to the fluid.

The Navier–Stokes equations have not been solved analytically in the general case, and the only available analytical solutions arise from simple geometries (for example, the 1D flow geometries discussed in Chapter 2). Because of this, our analytical approach for solving fluid flow problems is often to solve a simpler equation that applies in a specific limit. Some examples of these simplified equations include the Stokes equations (applicable when the Reynolds number is low, as is usually the case in microfluidic devices) and the Laplace equation (applicable when the flow has no vorticity, as is the case for purely electrokinetic flows in certain limits). These simplified equations guide engineering analysis of fluid systems.

In this chapter, we discuss Stokes flow (equivalently termed creeping flow), in which case the Reynolds number is so low that viscous forces dominate over inertial forces. The approximation that leads from the Navier–Stokes equations to the Stokes equations is shown, and analytical results are discussed. The Stokes flow equations provide useful solutions to describe the fluid forces on small particles in micro- and nanofluidic systems, because these particles are often well approximated by simple geometries (for example, spheres) for which the Stokes flow equations can be solved analytically. The Stokes flow equations also lead to simple solutions (Hele-Shaw flows) for wide, shallow microchannels of uniform depths.

The principal aim of this book is to emphasize the geometric structure of Continuum Mechanics, but a reader not familiar with the by now standard presentation of this discipline is likely to miss the punch line. In part to avoid this unintended situation and in part to have a basic conceptual and terminological framework for the rest of the book, in this Appendix we provide a concise presentation of the subject as it can be found in more or less standard elementary textbooks. The level of mathematical sophistication is kept as low as possible: vectors are arrows, Pythagoras reigns supreme, everything is nice and smooth.

Bodies and Configurations

The passage from the classical mechanics of finite systems of particles and rigid bodies to the mechanics of deformable continua is not a trivial one. Already at the beginning of the subject we find ourselves confronted with the problem of defining the main concept: the body, or material continuum. Is it merely an infinite collection of particles? Because we must, at the very least, be able to define fields (temperature, velocity, and so on) over this entity, it is clear that we need a rigorous definition. This need is all the more pressing since, from our experience with centuries-old particular theories (hydrodynamics, linear elasticity, and so on), we know that soon enough temporal and spatial derivatives of these fields will enter the scene.

Integration of Forms in Affine Spaces

As a first step towards a theory of integration of differential forms on manifolds, we will present the particular case of integration on certain subsets of an affine space (without necessarily having an inner-product structure). In Section 2.8.3 we introduced the rigorous concept of an affine simplex and, later, in Section 3.4, we developed the idea of the multivector uniquely associated to an oriented affine simplex. Moreover, we have already advanced, on physical grounds, the notion that the evaluation of an r-form on an r-vector conveys the meaning of the calculation of the physical content of the quantity represented by the form within the volume represented by the multivector. For this idea to be of any practical use, we should be able to pursue it to the infinitesimal limit. Namely, given an r-dimensional domain D in an affine space A and, given at each point of this domain a (continuously varying, say) r-form ω, we would like to be able to subdivide the domain into small r-simplexes and define the total content as the limit of the sum of the evaluations of the r-forms on a point of each of the simplexes. In this way, we would have a generalization of the concept of Riemann integral.

Simplicial Complexes

A domain of integration within an n-dimensional affine space A may be a rather general set.