Polymer science presents an enormous array of other problems for which progress may benefit from improved mathematical approaches. The above discussions are illustrative of some of the ways in which mathematicians can contribute to polymer science. There are many more; see, for example, Kroschwitz (1990). Although these cannot all be discussed in detail here, it is perhaps worthwhile to give several examples.
Most of the polymeric materials ubiquitous around us are actually not pure, but rather are combinations of different types of molecules. For example, the reason that polymeric glasses do not shatter is that they are impact modified. That means that they are blended with rubbery materials. The phases do not truly mix, but rather form an intimate composite of small rubbery domains within the glassy matrix. Many other examples of polymeric composites exist, and frequently polymeric materials contain a large fraction of solid filler material. An important problem is to characterize the path of a fracture, and how the rubbery inclusions in a glass cause the fracture to branch and be terminated (Kausch, 1978; Brostow and Corneliussen, 1986). It is also important to be able to predict the properties of composites, particularly what type of averaging is necessary to describe effective properties (for example, effective viscosity, dielectric constant, or elastic constant). These subjects are discussed in Chapters 5 and 6.
The glassy state of matter is a form in which many polymers find use. There appear to he numerous mathematical issues to be sorted out in the development of a theory for this type of material (see Chapter 6). Many of these problems parallel ones associated with the description of protein dynamics and folding. Descriptions of relaxation phenomena in glasses require the solutions of multidimensional integro-(partial-)differential equations with a wide range of relevant time scales. Methods must be developed for isolating the subset of "relevant" degrees of freedom and for extracting the difficult-to-compute, long-time relaxation dynamics.