volatility and viscosity, the theoretical basis has been constructed (Wu, 1987); for nonequilibrium behavior of the fluid phase, such as rheology, and the optical and mechanical properties of the final solid, the basic theory is only partially in place.
A second example currently receiving considerable attention is the formulation and application of electrorheological fluids, which consist of polar particles in a nonpolar fluid. These are normally low-viscosity fluids, but a strong electric field induces dipoles in the particles and creates a volume-filling, particulate network with an elastic or pseudoplastic response to an applied stress. Here the issue is how to control or optimize the magnitude and dynamics of the response (Gast and Zukoski, 1990). The physical phenomena are reasonably well defined, but many-body interactions dominate and the behavior is highly nonlinear.
The fabrication of ceramics by colloidal processing aims for dense packing without large-scale inhomogeneities (Lange, 1989). Consolidation via centrifugation or filtration permits application of forces strong enough to overcome the short-range attractions inevitable between dense particles of 0.1-to 10.0-micron diameter in water or an organic liquid. Given accurate constitutive relations for the permeability and the stress supported by the particulate network, continuum theory describes the evolution of density profiles during the consolidation process (Buscall and White, 1987; Auzerais et al., 1990). The mathematical problem is to solve the highly nonlinear set of equations accurately enough to capture the propagation of several nearly discontinuous transitions in a one-, two-, or three-dimensional spatial domain.
For composite materials, the challenge is to relate the spatial structure and properties of the individual phases to macroscopic mechanical, optical, and electronic properties. Prediction of equilibrium structure via statistical mechanics is being developed for some classes of systems (Bates and Frederickson, 1990; Monovoukas and Gast, 1989). Considerable progress has been made in predicting linear responses through variational and bounding techniques (Torquato, 1990). However, nonlinear behaviors of considerable interest, such as yielding, fracture, and nonlinear optical properties, are still largely untouched.
One might define the challenge for mathematical theory as developing techniques for accurately handling strong many-body interactions. This challenge is complementary to the challenge to physics of describing interparticle forces, equilibrium phase behavior, and polymer thermodynamics. Many aspects of the subject have advanced to the point that sophisticated mathematical analysis and numerical solution procedures are needed to provide the quantitative predictions necessary to keep up with the physics and experimental investigations in this area. As is pointed out in Chapter 8, this is the case not just for this area but also in many other areas of materials science.
One can identify a number of relatively advanced approaches to this class of problems:
For colloidal problems, significant movement occurs only after many thermal