The first section below on single-chain conformations addresses the case of individual polymers in solution or in interaction with confining surfaces. Many basic concepts of polymer science have arisen in the study of these single-and few-chain systems, and dilute solution experiments are still essential in characterizing the properties of the polymers, so that polymeric materials can be made reproducibly and theories of polymer properties can be tested against carefully controlled experiments. The next section considers protein properties and scientific questions relevant to applications in biotechnology that are associated with predicting the conformation, folding, and long-time dynamics of proteins.
Subsequent sections discuss the flow and equilibrium properties of polymers in the liquid state, the state from which many polymeric materials are fabricated by extrusion, molding, and so forth. Descriptions of the viscoelastic properties of molten polymers often rely on ad hoc phenomenological models. Although these display remarkable agreement with a wide range of experiments, a firm theoretical underpinning for these models is lacking. Mathematical issues arise from this viscoelasticity because the responses typically associated with viscous liquids and elastic solids are combined; the stress at any position depends on the entire deformation history of the material element located at that position. The formulation of viscoelastic constitutive equations that accurately portray material behavior remains a subject of research. Fundamental issues of existence, uniqueness, and qualitative behavior have not been adequately addressed. Computational schemes for flows in complex geometries often fail to converge under conditions of processing interest. Instabilities occur that are absent in low-molecular-weight liquids. Many of the issues discussed here have been reviewed previously (Denn, 1990).
Many theoretical and mathematical sciences issues are involved in the construction of and solutions to models of polymer fluids, which are being developed along three distinct lines. One of the oldest involves the use of an underlying lattice to greatly simplify the mathematics at some concomitant expense in representation of reality. Recent developments in polymer theory build on successes made over the past few decades in the theory of simple molecular fluids, especially in integral equations and density functional methods. Density functional methods are discussed in this chapter in relation to studies of interfaces in polymer systems, followed by a description of problems in block copolymers and in stiff polymers and liquid crystals, as well as mention of other areas; see also Chapter 8.
Problems in polymer science are almost as diverse as those of all of materials science, and are associated with issues of fracture, thermal stability, ageing, transport, adhesion, and so on. Polymers may be used as organic conductors and nonlinear optical materials. Some of the issues are addressed in other chapters in this report. The omission of specific areas is indicative merely of the limited nature of this report.
Very interesting mathematics arises naturally in the study of polymeric materials. The most basic model in this discipline is the Wiener path model of single polymer chains. This picture represents the shapes of polymer chains as being much like the dynamical trajectories swept out by particles undergoing Brownian motion. At the next level of study,