Introduction

Because colloidal particles generally reside in a viscous fluid, the behavior of a dispersion is strongly influenced by hydrodynamic forces generated by the relative particle–fluid motion. Although many hydrodynamic effects can be deduced from the behavior of an isolated particle, the disturbance it causes decays so slowly with distance that interparticle effects are seldom negligible. Consequently, hydrodynamic forces transmitted from one particle to another through a viscous fluid must be understood. Interactions, as well as the behavior of isolated particles, are discussed here. The presentation is not meant to be a scaled-down text on hydrodynamics, but is intended to provide tools to deal with phenomena encountered in colloidal systems.

The next section presents the basic differential equations governing the behavior of an incompressible Newtonian fluid and an analysis of the relative importance of viscous and inertial effects. The analysis of two simple flows illustrates some basic principles about the kinematics of fluid motion. Then we turn our attention to flows for which inertial effects are negligible, Stokes flows. Special emphasis is given to singular solutions resulting from forces applied at points in the fluid. Subsequent sections deal with isolated spheres and two interacting spheres, first in a quiescent fluid and then in fluid undergoing laminar shear flow.

Introduction

Everyone has empirical knowledge of electrostatic and electromagnetic phenomena based on experiences such as the buildup of static charge on a comb or nature's grand displays of lightning and the auroras borealis and australis. Less obvious but no less familiar are the stabilizing effects of electrostatic forces in colloidal suspensions. Clay particles and silt carried in suspensions by rivers coagulate upon encountering the higher salt concentration of the sea to form huge deltas. Electrostatic stabilization is also responsible for the long shelf-life of certain latex paints. Needless to say, electrostatic forces play central roles in the behaviour of biological systems. Despite such diversity, electromagnetic and electrostatic phenomena can be understood in terms of the elegant theory embodied in Maxwell's equations. Here we take these equations as axioms and proceed deductively.

The presentation is organized as follows. First the equations governing quasi-static electric fields are set out. Starting with the balance laws and conditions prescribed at boundaries where electrical properties change abruptly, we are led to discuss dielectrics, polarization, free charge, and the electrical stress embodied in Maxwell's stress tensor. Then emphasis shifts to the electrical double layer and mathematical models describing its behavior. Here layers of charge are immobilized on a surface, while ions in the adjacent solution move freely under the influence of electrical forces and Brownian motion. After studying matters near a single surface, we turn our attention to the region between two surfaces and electrostatic forces between macroscopic particles in solutions containing dissolved ions.

Colloid science has its roots in nineteenth- and early twentieth-century discoveries concerning the behavior of minute particles. Its early development was stimulated by controversies regarding the very existence of molecules. Scientific interest, along with technological and biological applications, fostered several definitive monographs and textbooks in the 1930s and 1940s. However, interest in the field declined within many academic circles after the Second World War, especially in the United States, despite continued and widespread industrial applications. The resurgence of interest that began in the early 1960s arose from mutually reinforcing events. New technological problems appeared in, for example, the manufacture of synthetic dispersions for coatings, enhanced oil recovery, the development of new fuels, environmental pollution, ceramics fabrication, corrosion phenomena, biotechnology, and separations processes. In addition, monodisperse suspensions of colloidal particles of diverse sorts became readily available and advances in our understanding of fluid mechanics on the colloidal scale burgeoned almost simultaneously. Further stimuli were provided by the appreciation by colloid scientists of advances in the theory of interparticle forces coupled with the development of several new experimental techniques. Forces and particle properties have long been difficult to measure accurately on the colloidal scale and numerical values were often the result of a long uncertain chain of inference. The new techniques made possible direct, accurate measurements of size, shape, and concentration, as well as the attractive and repulsive forces between surfaces separated by a few nanometers.

Introduction

Optical microscopic observations of small particles dispersed in water reveal a constant state of random motion. The discovery of this phenomenon is now attributed to Robert Brown, a botanist, although other publications predate his descriptions of 1828 and 1829. While Brown correctly attributed the motion to the molecular nature of matter, controversy persisted until the experiments of Gouy in 1888 ruled out extraneous causes such as mechanical vibrations, convection currents, and illumination and focused attention on molecular agitation. As Perrin (1910) concluded, the particles seem to move independently with no effect of density or composition, although the amplitude of the motion is greater for smaller particles, with less viscous fluids, and at higher temperatures. The displacements are significant; for example, 0.2-μm spheres in water wander 10 μm from their starting point in a bit over 30 seconds. Gouy and Perrin both attributed the motion of the particle to incessant impacts of fluid molecules which impart kinetic energy equal to 3/2 kT, partitioned equally among the three translational degrees of freedom. The irregularity of the translational motion and the rapid damping of the random fluctuations by the viscous fluid, however, confounded early attempts to measure this kinetic energy by calculating the instantaneous velocity from the observed trajectory. This failure to verify directly the origin of Brownian motion led to theoretical treatments appropriate for the longer diffusion time scale.

Introduction

Electrokinetic processes derive from interactions between macroscopic motion and diffuse electric charge. Here the emphasis is on phenomena used to characterize the electrical properties of particles in aqueous suspensions, and to introduce the subject some simple models will be described. The detailed discussions begin with electrophoresis, the motion of individual particles due to an external field. This technique is widely used to measure particle charge. The electrical conductivity of a suspension also reflects electrokinetic processes and is addressed next. The theory of conductivity is based on the same basic model as electrophoresis, extended to cover contributions from many particles. Measurements of mobility and conductivity are complementary, since one reflects events around an individual particle directly while the other averages over the particle population. The final topic concerns the response of colloidal dispersions to unsteady electric fields. Double layers are polarized by external fields and an unsteady field engenders relaxation processes at relatively low frequencies. Thus, studies of dielectric relaxation provide additional insight into electrokinetic behavior.

Examples of electrokinetic phenomena

Suppose that charged colloidal particles suspended in an ionic solution are somehow confined to a channel connecting two reservoirs. Flow of the solution sweeps part of the diffuse charge from one reservoir to the other, producing a streaming current. When there is no external connection between the reservoirs, charge accumulation produces a difference in electrical potential known as the streaming potential.

Colloidal phenomena

Colloidal particles dispersed in liquids exhibit astonishing properties. Dispersions such as the colloidal gold sol prepared by Faraday (1791 – 1867) over a century ago can persist almost indefinitely, yet the addition of salt would cause rapid, irreversible flocculation. In fact, for many dispersions the physical state, i.e. the stability or phase behavior, can be altered dramatically by modest changes in composition. This complex behavior stems from the different forces that act among the particles, determining their spatial distribution and governing the dynamics. Brownian motion and dispersion forces (arising from London–van der Waals attraction) would flocculate Faraday's gold sol were it not for electrostatic repulsion between the particles. The addition of salt increases the concentration of ions screening the surface charge, suppressing repulsion and allowing flocculation. Doublets and more complicated structures formed during flocculation have long lifetimes, since Brownian motion is too weak to overcome the strong attractive force between particles near contact. Indeed, removal of the salt does not usually lead to spontaneous redispersion, so mechanical means must be used.

Another type of transformation occurs when ions are removed from electrostatically stabilized systems. Polymer latices in an electrolyte solution are milky-white fluids, but dialysis eliminates the ions and leads to iridescence owing to Bragg diffraction of visible light from an ordered structure (Fig. 1.1). Here the absence of screening allows long-range electrostatic repulsion to induce a disorder-order phase transition.

Introduction

As suggested in Chapter 6, the adsorption or anchoring of polymer onto the surface of colloidal particles provides an alternate means of imparting stability. Indeed, polymeric stabilization was exploited by the ancient Egyptians as early as 2500 bc (Napper, 1983, §2.1). They formulated inks by dispersing carbon black particles in a solution of naturally occurring polymer such as casein or gum arabic. Adsorption of the polymer onto the carbon black maintained the dispersion and also allowed redispersion after drying.

Several reasons persist for using polymeric, instead of electrostatic, stabilization. In some aqueous systems, electroviscous effects and the accompanying sensitivity to electrolyte concentration may be undesirable. In non-aqueous solvents with low dielectric constants, and surface charge densities typically one to two orders of magnitude smaller than in water, electrostatic repulsion frequently does not suffice. In addition, polymeric stabilization can be more robust than the electrostatic mode, providing stability for a longer time and at higher solids concentrations. When flocculation or phase separation does occur, it is normally reversible, i.e. a suitable change in the solvent conditions will redisperse the particles spontaneously.

Napper's (1983, §2.4) review of early studies indicates the evolution in the nature of the polymers used for this purpose. Work in the nineteenth century, and the early part of the twentieth, dealt with aqueous systems and employed biopolymers that were generally globular, crosslinked, or highly branched.

Introduction

As noted in Chapter 5, dispersion forces acting between similar particles suspended in a chemically different liquid are inevitably attractive, providing a driving force toward macroscopic phase separation. Hence, maintenance of a dispersed state requires an opposing interparticle repulsion, most commonly achieved through electrostatic forces in aqueous dispersions or the adsorption of soluble polymer in either aqueous or non-aqueous environments. Since all characteristics of colloidal systems change markedly in the transition from the dispersed to the aggregated state, the question of stability occupies a central position in colloid science (e.g. Verwey & Overbeek, 1948; Napper, 1983; Hunter, 1987).

Even among unstable or aggregated systems, the nature or degree of aggregation varies. Following the convention of La Mer (1964), many authors have attempted to distinguish between flocculation, referring to loose aggregation, with highly porous flocs and/or particles held relatively far apart, and coagulation, with more closely packed flocs of particles in contact. Unfortunately, floc structure has been quantified only recently, leaving the classification ambiguous in many cases.

In the following, we distinguish instead on the basis of the strength of the attractive potential responsible for aggregation. Then the criterion becomes whether the system attains equilibrium in the period of interest. For attractions strong relative to the thermal energy kT, Brownian motion eventually eliminates all individual particles, producing a non-equilibrium phase whose structure is governed by the range of the attractive potential and the mode of aggregation.

This chapter is devoted to three additional topics.

The equations we have discussed have been those for which the inverse scattering transform is applicable. However, given any evolution equation, it is natural to ask whether it can be solved by the inverse scattering transform (1ST); in other words, how do we decide if a given equation is completely integrable? This question is still open but a promising conjecture concerns the so-called Painlevé property. We shall describe how the Painlevé equations arise, what they are and the conjecture itself.

If the evolution equation cannot be solved by the 1ST, but is close to one which can be (by virtue of a small parameter), we may adopt the following procedure. The 1ST method is formulated in the conventional way but the time evolution of the scattering data now involves the small parameter. This parameter can be used as the basis for generating an asymptotic solution of the inverse scattering problem, and hence of the original equation. We shall outline the development of this argument.

Finally, if neither of the above methods is applicable, or if a graphical representation of the solution is required, then we may use a numerical solution. Indeed, the original motivation for the 1ST came from a study of numerical solutions of the KdV equation. We shall, in the final section of this chapter, present some numerical methods suited to the solution of the initial-value problem for evolution equations.

The theory of solitons is attractive and exciting; it brings together many branches of mathematics, some of which touch on deep ideas. Several of its aspects are amazing and beautiful; we shall present some of them in this book. The theory is, nevertheless, related to even more areas of mathematics, and has even more applications to the physical sciences, than the number which are included here. It has an interesting history and a promising future. Indeed, the work of Kruskal and his associates which gave us the ‘inverse scattering transform’ – a grand title for soliton theory – is a major achievement of twentieth-century mathematics. Their work was stimulated by a physical problem together with some surprising computational results. This is a classic example of how numerical results lead to the development of new mathematics, just as observational and experimental results have done since the time of Archimedes.

This book has grown out of Solitons written by one of us (PGD). That book originated from lectures given to final-year mathematics honours students at the University of Bristol. Much of the material in this version has also been used as the basis for an introductory course on inverse scattering theory given to MSc students at the University of Newcastle upon Tyne. In both courses the aim was to present the essence of inverse scattering clearly, rather than to develop the theory rigorously and completely. That is also the overall aim of this book.