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.

All structures are three-dimensional, and the exact analysis of stresses in them presents formidable difficulties. However, such precision is seldom needed, nor indeed justified, for the magnitude and distribution of the applied loading and the strength and stiffness of the structural material are not known accurately. For this reason it is adequate to analyse certain structures as if they are one- or two-dimensional. Thus the engineer's theory of beams is one-dimensional: the distribution of direct and shearing stresses across any section is assumed to depend only on the moment and shear at that section. By the same token, a plate, which is characterized by the fact that its thickness is small compared with its other linear dimensions, may be analysed in a two-dimensional manner. The simplest and most widely used plate theory is the classical small-deflexion theory which we will now consider.

The classical small-deflexion theory of plates, developed by Lagrange (1811), is based on the following assumptions:

1. (i) points which lie on a normal to the mid-plane of the undeflected plate lie on a normal to the mid-plane of the deflected plate;

2. (ii) the stresses normal to the mid-plane of the plate, arising from the applied loading, are negligible in comparison with the stresses in the plane of the plate;

3. (iii) the slope of the deflected plate in any direction is small so that its square may be neglected in comparison with unity;

4. […]

The exact large-deflexion analysis of plates generally presents considerable difficulties, but there are three classes of plate problems for which simplified theories are available for describing their behaviour under relatively high loading. These ‘asymptotic’ theories are membrane theory, tension field theory (sometimes called wrinkled membrane theory) and inextensional theory. All are described below. For a plate of perfectly elastic material, the error involved in using these theories tends to zero as the loading is increased or as the thickness is reduced. In any practical material, however, there is a limit to the elastic strain that may be developed, and this in turn limits the range of validity of these asymptotic theories to plates which are very thin. For steel and aluminium alloys, a typical limit to the elastic strain is 0.004, and this restricts the range of validity of the asymptotic theories as follows. For membrane theory and tension field theory the thickness must be less than about 0.001 of a typical planar dimension, while for inextensional theory the thickness must be less than about 0.01 of a typical planar dimension.

Membrane theory (considered by Föppl 1907)

When a thin plate is continuously supported along the boundaries in such a manner that restraint is afforded against movement in the plane of the plate, the load tends to be resisted to an increasing extent by middle-surface forces.

In the first edition of this book, I attempted to present a concise and unified introduction to elastic plate theory. Wherever possible, the approach was to give a clear physical picture of plate behaviour. The presentation was thus geared more towards engineers than towards mathematicians, particularly to structural engineers in aeronautical, civil and mechanical engineering and to structural research workers. These comments apply equally to this second edition. The main difference here is that I have included thermal stress effects, the behaviour of multi-layered composite plates and much additional material on plates in the largedeflexion régime. The objective throughout is to derive ‘continuum’ or analytical solutions rather than solutions based on numerical techniques such as finite elements which give little direct information on the significance of the structural design parameters; indeed, such solutions can become simply number-crunching exercises that mask the true physical behaviour.