Introduction

Analytical methods based on the work done by applied loads and the changes in the energy of a system that accompany a real or virtual crack advance have been of central importance in the development of fracture mechanics. These methods have provided a degree of unification of seemingly diverse ideas in fracture mechanics, and they have led to procedures of enormous practical significance for the characterization of the fracture behavior of materials. In addition, some of the most elegant theoretical analyses in the field have been those associated with energy methods. In this chapter, energy concepts that are particularly relevant to the study of dynamic fracture processes are considered.

The importance of the variation of energy measures during crack growth was recognized by Griffith (1920) in his pioneering discussion of brittle fracture, as outlined in Section 1.1.2. The extension of a crack requires the formation of new surface, he reasoned, with its associated surface energy. Consequently, a crack in a brittle solid should advance when the reduction of the total potential energy of the body during a small amount of crack advance equals the surface energy of the new surface thereby created. For an elastic body containing a crack, the negative of the rate of change of total potential energy with respect to crack dimension is called the energy release rate. This quantity, which is usually denoted by the symbol G, is a function of crack size, in general.

Introduction

The preceding chapters address interactions between colloidal particles dispersed in pure liquid or electrolyte solution. The hydrodynamic and dispersion forces depend only on the bulk properties of the individual phases, i.e. the viscosity and the dielectric permittivities. Electrostatic forces arising from the surface charges, however, are accompanied by free electrolyte. The associated electric fields distribute these additional species non-uniformly in the surrounding fluid, thereby producing a spatially varying osmotic pressure. Electrostatic interactions between particles alter these ion distributions, affecting the electric and pressure fields and generating an interparticle force.

We now consider another component commonly present in colloidal systems, soluble polymer. In many ways, the phenomena and the theoretical treatment resemble those for electrostatics. The interactions between polymer and particle generate non-uniform distributions of polymer throughout the solution. Particle–particle interactions alter this equilibrium distribution, producing a force whose sign and magnitude depend on the nature of the particle–polymer interaction. The major difference from the ionic solutions lies in the internal degrees of freedom of the polymer, which necessitate detailed consideration of the solution thermodynamics.

The reasons for adding soluble polymer to colloidal dispersions are several. The earliest known role, as stabilizer, aids or preserves the dispersion through adsorption of the macromolecule onto the surfaces of the particles to produce a strongly repulsive interaction. Homopolymers achieve this by adsorbing to particles non-specifically at multiple points along their backbone, while block or graft copolymers adsorb irreversibly at one end with the other remaining in solution (Napper, 1983).

Introduction

In the preceding chapters we have examined the response of colloidal particles to interactions with one another in a quiescent fluid, to interactions with large collectors while being convected by the fluid, and to imposed forces due to electric fields, gravity, or concentration gradients. In each case, equilibrium or non-equilibrium, static or dynamic, the interparticle forces and the resulting suspension microstructure play key roles. Now we consider the stresses and the non-equilibrium microstructure generated in a flowing suspension when the velocity varies spatially on a scale large with respect to the size of the particles.

A Newtonian incompressible liquid is characterized by a linear relation between the stress tensor and the rate-of-strain tensor, with the constant of proportionality being the viscosity. Polymeric liquids are well known for their non-Newtonian behavior including shear-rate-dependent viscosities, elasticity manifested in recoil upon the cessation of flow, solid-like fracture during extrusion, and a variety of secondary flow phenomena. Colloidal suspensions also depart from Newtonian behavior. They often behave as solids requiring a finite stress, the yield stress, before deforming continuously as a liquid. The contrast with the polymeric liquids reflects the fundamentally different microstructures. Both microstructures deform under stress, but macromolecular systems can recover from strains of several hundred per cent because the restoring force increases with the degree of deformation. The interparticle forces governing the microstructure in colloidal dispersions generally have a short range and the magnitude decreases with increasing separation, providing no mechanism for recovery beyond strains of a few per cent.

Introduction

Microscopic observations of colloidal particles in the nineteenth century revealed their tendency to form persistent aggregates through collisions induced by Brownian motion, clearly indicating an attractive interparticle force. Identification of its origin, however, awaited the quantitative descriptions of van der Waals forces between molecules developed in the 1920s (Israelachvili, 1985). This development prompted Kallman & Willstätter (1932) and Bradley (1932) to realize the summation over pairs of molecules in interacting particles would yield a long-range attraction.

Subsequently, de Boer (1936) and Hamaker (1937) performed explicit calculations of dispersion forces between colloidal particles by assuming the intermolecular forces to be strictly pairwise additive. Although approximate, this theory captures the essence of the phenomenon. The attraction arises because local fluctuations in the polarization within one particle induce, via the propagation of electromagnetic waves, a correlated response in the other. The associated free energy decreases with decreasing separation. Phase shifts introduced at large separations by the finite velocity of propagation reduce the degree of correlation, and, therefore, the magnitude of the attraction. Although the intermolecular potential decays rapidly on the molecular scale, the cumulative effect is a long-range interparticle potential that scales on the particle size.

Introduction

In the preceding chapters, fundamental aspects of colloid behavior have been emphasized. Now we are ready to apply this knowledge to processes involving suspensions. Here we investigate the capture of small particles by stationary collector units, one aspect of filtration technology.

Elementary considerations show that a strong attractive force is necessary if freely suspended particles are to come together, because at close separations viscous resistance increases dramatically. Since the interparticle force derives from the combination of electrostatic and dispersion forces, capture is particularly sensitive to the balance between colloidal and hydrodynamic forces. Several mechanisms contribute to particle capture and retention. Inertia is the dominant factor when fast-moving particles impact on a stationary object, whereas geometry and proximity govern the interception of slow-moving particles. The capture of submicron particles is influenced enormously by interparticle forces and Brownian motion. All these aspects are treated here, but technological issues are ignored. For example, a persistent problem encountered in the filtration of small particles is buildup of a deposit. Our treatment deals with the behavior of clean collector units to emphasize basic colloidal phenomena.

Aerosols have received the most study by a wide margin and many comprehensive reviews exist, e.g. Hidy & Brock (1970), Davies (1973), Friedlander (1977), and Kirsch & Stechkina (1978). Ives (1975) and Tien & Payatakes (1979) present broad reviews of liquid filtration; Spielman (1977) concentrates on small-scale processes in liquids.

Introduction

The sedimentation of colloidal particles is important both in technology and in the laboratory. Gravity settlers, thickeners, or clarifiers commonly remove particles from waste streams issuing from a variety of processes. These generally operate as continuous processes that split the feed into two product streams, one the clear fluid and the other a sludge. Successful design requires knowledge of the sedimentation velocity of the particles over the relevant range of volume fractions and the role of interparticle forces in determining the structure of the dense sludge. Centrifugation provides a means of enhancing the driving force for commercial-scale operations, as well as concentrating or analyzing dispersions in the laboratory.

Despite their longstanding use, much remains to be understood about the details of processes which convert dilute dispersions into dense sediments. The key issues appear to be

1. (i) the variation of the settling velocity with volume fraction and interparticle potential,

2. (ii) the role of forces transmitted by interparticle potentials, and

3. (iii) the formulation of macroscopic models to predict the evolution of volume fraction as a function of position and time.

As with other colloidal phenomena, the complexity arises from the importance of a variety of interparticle forces and the fact that many systems of interest tend to be flocculated.

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.