All the fibres of a fibre bundle are, by definition, diffeomorphic to each other. In the absence of additional structure, however, there is no canonical way to single out a particular diffeomorphism between fibres. In the case of a product bundle, for example, such a special choice is indeed available because of the existence of the second projection map onto the typical fibre. In this extreme case, we may say that we are in the presence of a canonical distant parallelism in the fibre bundle. An equivalent way to describe this situation is by saying that we have a canonical family of non-intersecting smooth cross sections such that each point in the fibre bundle belongs to one, and only one, of them. In a general fibre bundle we can only afford this luxury noncanonically and locally. A connection on a fibre bundle is, roughly speaking, an additional structure defined on the bundle that permits to establish intrinsic fibre diffeomorphisms for fibres lying along curves in the base manifold. In other words, a connection can be described as a curve-dependent parallelism. Given a connection, it may so happen that the induced fibre parallelisms turn out to be curve-independent. A quantitative measure of this property or the lack thereof is provided by the vanishing, or otherwise, of the curvature of the connection.

Microsystems use several techniques to actuate particles beyond the electrophoresis discussed in Chapter 13. Two physical phenomena are described in this chapter – dielectrophoresis and magnetophoresis – which are commonly used in microdevices to manipulate particles or droplets in suspension. This chapter also discusses digital microfluidics, which is not a physical phenomenon, but rather a system concept for manipulating fluid droplets using AC electric fields.

DIELECTROPHORESIS

Dielectrophoresis (DEP) is often used in microsystems as a mechanism for manipulating particles. It is appealing because the dielectrophoretic force on a particle scales with the characteristic length scale of the system to the –3 power, and dielectrophoretic forces are quite large when small devices are used. Further, particle response varies based on the frequency and phase of the applied field. Because the user can change particle response by changing a setting on a function generator, DEP measurements afford great flexibility to the user. Because of this, DEP has been used for many applications, with one example shown in Fig. 17.1.

The term dielectrophoresis refers to the Coulomb response of an electrically polarized object in a nonuniform electric field. In contrast to linear electrophoresis, it (a) does not require that the object have a net charge and (b) has a nonzero time-averaged effect even if AC electric fields are used.

Consider, as an example, a spherical, uncharged, uniform, ideal dielectric particle with a finite polarizability, expressed using its electrical permittivity εp, suspended in empty space.

Our interest in flows of electrolyte solutions in microdevices requires that we keep track of electrolyte solutions themselves, as well as acid–base chemistry at surfaces and in buffers. The coupling of electric fields and fluid mechanics that is common in microfluidics leads also to coupling between fluid mechanics and chemistry, because acid–base chemistry describes most interfacial charge and the charge state of many common analytes. Acid–base reactions at interfaces dictate the interfacial charge and, in turn, the electroosmotic mobility of the interface. Acid–base reactions, for example, between water and DNA or between water and proteins dictate the electrophoretic mobility of proteins and DNA in solution.

Because of this, this appendix provides a description of the properties of water, electrolyte solutions, and the associated acid–base chemistry. It defines solution terminology, derives the Henderson–Hasselbach equation for dissociation equilibrium, shows how the Henderson–Hasselbach equation relates reaction p K a to acid dissociation, and shows how the water dissociation equation leads to a simple relation between pH and pOH for water at room temperature. This basic understanding explains, for example, the pH dependence of the electroosmotic mobility.

FUNDAMENTAL PROPERTIES OF WATER

Water is a unique molecule with several properties that are unusual compared with other liquids. A number of important fundamental properties of water are summarized in Table B.1.

In mathematics and the sciences, just as in the arts, there is no substitute for the masterpieces. Fortunately, both in Differential Geometry and in Continuum Mechanics, we possess a veritable treasure trove of fundamental masterpieces, classical as well as modern. Reading them may elicit a pleasure, even an emotion, comparable to that aroused by playing a Mozart piano sonata, the same sensation of perfection and beauty, the sweetness of drowning in such an ocean. This comparison is fair also in a different sense, namely, that for common mortals to achieve these spiritual heights requires a considerable amount of study and work, without which we must resign ourselves to the intuitive feelings evoked by the senses and to the assurances of the critics or the arbiters of taste. The aim of this book is to provide some familiarity with the basic ideas of Differential Geometry as they become actualized in the context of Continuum Mechanics so that the reader can feel more at home with the masters.

Differential Geometry is a rather sophisticated blend of Algebra, Topology, and Analysis. My selection of topics as a nonmathematician is rather haphazard and the depth and rigor of the treatment vary from topic to topic. As befits human nature, I am understandably forgiving of myself for this lack of consistency, but the reader may not be so kind, in which case I will not take it as a personal offence.

Although the Navier–Stokes solutions cannot be solved analytically in the general case, we can still obtain solutions that guide engineering analysis of fluid systems. If we make certain geometric simplifications, specifically that the flow is unidirectional through a channel of infinite extent, the Navier–Stokes equations can be simplified and solved by direct integration. The key simplification enabled by this assumption is that the convective term of the Navier–Stokes equations can be neglected, because the fluid velocity and the velocity gradients are orthogonal. The solutions in this limit include laminar flow between two moving plates (Couette flow) and pressure-driven laminar flow in a pipe (Poiseuille flow). These flows are simultaneously the simplest solutions of the Navier–Stokes equations and the most common types of flows observed in long, narrow channels. Many microchannel flows are described by these solutions, their superposition, or a small perturbation of these flows. This chapter presents these solutions and interprets these solutions in terms of flow kinematics, viscous stresses, and Reynolds number.

STEADY PRESSURE- AND BOUNDARY-DRIVEN FLOW THROUGH LONG CHANNELS

The Navier–Stokes equations can be simplified when flow proceeds through an infinitely long channel of uniform cross section, owing to the geometric elimination of the nonlinear term in the equation. Couette and Poiseuille flows, in turn, describe flow driven by boundary motion or pressure gradients.

Microdevices for analyzing deoxyribonucleic acid (DNA) are ubiquitous in biological analysis, and techniques for analyzing DNA in microchips pervade the analytical chemistry literature. Use of nanochannels to study polymer physics has also become common. Owing to DNA's huge biological importance, its chemical properties have been thoroughly studied, and the experimental tools available for chemical analysis of DNA are numerous. The ubiquity and convenience of DNA has also led to extensive study of its physical properties. DNA is therefore an excellent example of how microscale systems facilitate analysis, as well as a model system for examining the effect of nanostructured devices on molecular transport of linear polyelectrolytes. Because the chemistry for fluorescently labeling DNA is relatively inexpensive and available commercially, fluorescence microscopy of DNA is a widely used means for visualizing DNA. It is quite routine to fluorescently label and observe the gross morphology of a single DNA molecule with 1-µm resolution, and thus straightforward experiments can be brought to bear on questions of molecular configuration.

DNA (and other idealized linear polymers) behave physically somewhere in between small molecules (which behave like idealized points) and particles (which behave like rigid continuous solid phases). The behavior observed (and the models that describe this behavior) incorporates aspects of point and particle behavior, and these behaviors are different depending on the type of transport.

This appendix outlines the role of several key dimensional and nondimensional parameters in micro- and nanoscale fluid mechanics that come from nondimensionalization of governing equations. A key advantage of nondimensionalization is that it leads to a compact description of flow parameters (i.e., Re) and thus leads to generalization. Nondimensionalization can be a powerful tool, but it is useful only if implemented with insight into the physics of the problems. Our stress here is the process of nondimensionalization, rather than a listing of nondimensional parameters, and we focus on only a few examples.

BUCKINGHAM Π THEOREM

The Buckingham Π theorem is a theorem in dimensional analysis that quantifies how many nondimensional parameters are required for specifying a problem. It also provides a process by which these nondimensional parameters can be determined. The Buckingham Π theorem states that a system with n independent physical variables that are a function of m fundamental physical quantities can be written as a function of n – m nondimensional quantities. As an example, the steady Navier–Stokes equations have four parameters: a characteristic length ℓ, a characteristic velocity U, the viscosity η, and the fluid density ρ. These are a function of three fundamental physical quantities: mass, length, and time. Thus the system can be described in terms of 4 − 3 = 1 nondimensional quantity, and it can be shown that the nondimensional quantity must be proportional to ρUℓ/η to some power.

This chapter discusses the physical relevance of potential fluid flow to flows in microfluidic devices and describes analytical tools for creating potential flow solutions. This chapter also focuses on the use of complex mathematics for 2D potential problems with plane symmetry. These plane-symmetric flows are relevant for microsystems, because microchannels are often shallower than they are wide and thus depth-averaged properties are often well approximated by 2D analysis.

In particular, we want to retain perspective on the engineering importance of these flows as well as on the relative importance of analysis versus numerics. The Laplace equation is rather straightforward to solve numerically, and therefore numerical simulation is a suitable approach for most Laplace equation systems. For example, simulation of the electroosmotic flow within a microdevice with a complicated geometry would be simulated, because analytical solution would be impossible. Despite the importance of numerics, the analytical solutions are important because they lend physical insight and because simple analytical solutions for important cases (for example, the potential flow around a sphere) facilitate expedient solutions of more complicated problems. For example, the study of electrophoresis of a suspension of charged spheres is typically analyzed with techniques informed by the analytical solution for potential flow around a sphere and not with detailed and extensive numerical solutions of the Laplace equation.

If Mathematics is the language of Physics, then the case for the use of Differential Geometry in Mechanics needs hardly any advocacy. The very arena of mechanical phenomena is the space-time continuum, and a continuum is another word for a differentiable manifold. Roughly speaking, this foundational notion of Differential Geometry entails an entity that can support smooth fields, the physical nature of which is a matter of context. In Continuum Mechanics, as opposed to Classical Particle Mechanics, there is another continuum at play, namely, the material body. This continuous collection of particles, known also as the body manifold, supports fields such as temperature, velocity and stress, which interact with each other according to the physical laws governing the various phenomena of interest. Thus, we can appreciate how Differential Geometry provides us with the proper mathematical framework to describe the two fundamental entities of our discourse: the space-time manifold and the body manifold. But there is much more.

When Lagrange published his treatise on analytical mechanics, he was in fact creating, or at least laying the foundations of, a Geometrical Mechanics. A classical mechanical system, such as the plane double pendulum shown in Figure 1.4, has a finite number of degrees of freedom. In this example, because of the constraints imposed by the constancy of the lengths of the links, this number is 2.

The Norwegian mathematician Sophus Lie (1842–1899) is rightly credited with the creation of one of the most fertile paradigms in mathematical physics. Some of the material discussed in the previous chapter, in particular the relation between brackets of vector fields and commutativity of flows, is directly traceable to Lie's doctoral dissertation. Twentieth-century Physics owes a great deal to Lie's ideas, and so does Differential Geometry.

Introduction

We will revisit some of the ideas introduced in Example 4.6 from a more general point of view. Just as the velocity field of a fluid presupposes an underlying flow of matter, so can any vector field be regarded as the velocity field of the steady motion of a fluid, thereby leading to the mathematical notion of the flow of a vector field. Moreover, were one to attach a marker to each of two neighbouring fluid particles, representing the tail and the tip of a vector, as time goes on the flow would carry them along, thus yielding a rate of change of the vector they define. The rigorous mathematical counterpart of this idea is the Lie derivative.

Let V : M → TM be a (smooth) vector field. A (parametrized) curve γ : I → M is called an integral curve of the vector field if its tangent at each point coincides with the vector field at that point.

This text describes liquid flow in microsystems, primarily flow of water and aqueous solutions. To this end, this chapter describes basic relations suitable for describing the flow of water. For flows in microfluidic devices, liquids are well approximated as incompressible, i.e., having approximately uniform density, so this text describes incompressible flow exclusively.

This chapter describes the kinematics of flow fields, which describes the motion and deformation of fluids. As part of this process, key concepts are introduced, such as streamlines, pathlines, streaklines, the stream function, vorticity, circulation, and strain rate and rotation rate tensors. These concepts provide the language used throughout the text to communicate the modes of fluid motion and deformation. We discuss conservation of mass and momentum for incompressible flows of Newtonian fluids. Finally, we discuss boundary conditions for the governing equations, including solid and free interfaces with surface tension, and in particular we give attention to the no-slip condition and its applicability in micro- and nanoscale devices. This chapter assumes familiarity with vector calculus, which is reviewed in Appendix C. Importantly, Appendix C also covers the notation and coordinate systems used throughout.

We define a fluid as a material that deforms continuously when experiencing a nonuniform stress of any magnitude. We are primarily interested in a continuum description of the fluid flow, meaning that we are interested in the macroscopic manifestation of the motions of the individual molecules that make up the fluid, i.e., the velocity and the pressure of the fluid as a function of time and space.

This text is primarily concerned with the behavior of fluids in micro- and nanofabricated systems; however, the ubiquity of diffuse charge in solution and the routine use of applied electric fields requires that the electrodynamic equations be solved simultaneously with the fluid equations, leading to a body force term that modifies the fluid velocity fields. This chapter summarizes the fundamental equations of electrostatics and electrodynamics with specific focus on aqueous solutions with boundary conditions typical of microfluidic devices. A description of electrical circuits, which describe the electrostatics and electrodynamics of discretized elements, is also presented.

ELECTROSTATICS IN MATTER

Electrostatics describes the effects caused by stationary source charges or static electric fields on other charges, termed test charges. The electrostatic limit applies when all charges are stationary and the current is zero. The equations and boundary conditions of electrostatics are all derived from Coulomb's law.

We study charges in matter, usually in an aqueous solution (we also call this an electrolyte solution) or in a metal conductor. In matter, it becomes unwieldy to keep track of all of the electrostatic interactions. To simplify things, we distinguish between free charge and bound charge and keep detailed track of only the free charge. Free charge implies a charge that is mobile over distances that are large relative to atomic length scales. Free charge typically comes from electrons (in a metal) or ions (in an aqueous solution).

The Laplace and Stokes equations are both linear in the fluid velocity and are both amenable to solution by superposition of Green's functions; further, each equation has a set of solutions whose superposition is termed the multipolar expansion. This appendix details these solutions.

LAPLACE EQUATION

The Laplace equation governs key electromagnetic solutions and, in the case of purely electroosmotic flow far from walls, the Laplace equation can govern fluid flow as well. Although numerical approaches are the norm for solving the governing equations in complex geometries, a great wealth of analytical solutions to the Laplace equation can be found by solving by separation of variables, especially for systems with symmetry. These solutions have mathematical importance because they lead to a convenient series expansion (the multipolar expansion) that approximates the correct solutions, and these solutions have physical importance because the building blocks of electrical systems (point charges), magnetic systems (point magnetic dipoles), and fluid systems (sources, sinks, vortexes) correspond to individual terms in this expansion.

This appendix describes solutions to the Laplace equation with both axial and plane symmetry. The solutions for plane-symmetric flows forecast the potential flows discussed in Chapter 7, and the solutions for axial symmetry forecast the use of the multipolar expansion for modeling the dielectrophoretic response of particles (Chapter 17).

Electrophoresis is the motion of a charged body proportional to an electric field. In contrast to Chapter 11, in which electrophoresis of molecules is described; this chapter discusses the motion of charged particles. Particle electrophoresis is a straightforward way to manipulate particles in microfluidic devices, both for positioning and for separation; it is ubiquitous if electric fields are used for any purpose.

Particles and molecules differ in that small molecules can be treated as point charges from the standpoint of how they perturb the surrounding electric field, and small molecules cannot support enough counterions to create a continuum electrical double layer. Particles, in contrast, have a large-enough charge that they are surrounded by a continuum EDL. For large particles, electrophoresis is described by use of analyses similar to electroosmosis, albeit with different boundary conditions. For smaller particles, we account for the breakdown of the thin-EDL assumption, leading to size-dependent electrophoretic velocity. Throughout, we discuss both the velocity distribution of the fluid and the velocity of the particle with respect to the bulk fluid.

INTRODUCTION TO ELECTROPHORESIS: ELECTROOSMOSIS WITH A MOVING BOUNDARY AND QUIESCENT BULK FLUID

Electrophoresis of particles and electroosmosis of fluid are both caused by the same physical phenomena – electrostatic forces on the wall and double layer counteracted at steady state by viscous forces in the fluid or elastic forces in the solid.

The previous chapters introduce the governing equations for fluid flow and provide solutions for simple unidirectional flows. Although we can only rarely use these simple solutions to exactly describe real flows, these unidirectional flows provide a framework for generating engineering estimates for many flows, and these estimation techniques reduce complex systems with large numbers of components into a relatively simple, approximate, linear description. This is relevant for flow in microdevices both (1) because microdevices can straightforwardly be made with large numbers of microchannels and (2) because those microchannels often have long, thin geometries that lead to largely unidirectional flows that are well approximated by the solutions presented in Chapter 2. Our approach, called hydraulic circuit analysis, involves assuming that the Poiseuille flow derived in Chapter 2 provides a sound engineering estimate of the pressure drops and flow rates through long, straight channels, even if the cross section is not exactly circular and the channels are neither perfectly straight nor infinite in extent. We thus write an approximate linear relation between pressure drops and flow rates through these channels – the Hagen – Poiseuille law. This law, combined with conservation of mass, approximately prescribes fluid flow through complex rigid networks by solution of sets of algebraic equations. By adding a hydraulic capacitance or compliance, this linearized analysis also allows prediction of unsteady flow through channels with a finite flexibility.