Polymer Solutions: a Geometric Introduction∗

Why should a physicist be interested in polymers? They do not hold the key to vast sources of energy as atomic nuclei do. They do not defy the intuition with ultrasmall dissipation as superconductors and superfluids do. They do not reveal subtle new nonabelian symmetries as do subatomic particles. Nor do they hold secrets about the origin or fate of the universe. Polymers are just ordinary matter—just insulating organic molecules. These molecules are merely larger than usual and are in the form of chains of small subunits. Yet these chain molecules have ways of interacting unmatched in other forms of matter. The study of polymers over the last few decades has forced us to broaden our notion of how matter can behave—how it can organize itself in space, how it can flow, and how it can transmit forces. These new behaviors arise from a few qualitative features of the polymer molecule’s structure. The potential for shaping these phenomena is just beginning to be realized as the power of synthetic chemists to control the molecular structure increases. The purpose of this introductory essay is to convey an understanding of how polymer liquids differ from other liquids and from other forms of condensed matter. Excellent treatments exist already (Jannink et al. 1992), notably Scaling Concepts in Polymer Physics by P. G. deGennes (DeGennes 1979). Here we adopt a geometric approach, exploiting the “fractal” structure of the polymers. This enables us to provide an economical and unified overview of the phenomena treated in greater depth in these books. As much as possible we shall attempt to account for the interaction of polymers with their surroundings by the mathematical laws describing the intersections between two fractal structures. We begin by recalling the scaling properties of any flexible chain of randomly-oriented links, noting that such a random-walk structure has the spatial scaling properties of a fractal object as defined by Mandelbrot (Mandelbrot 1982). We shall then survey the important ways in which fractal structures, including polymers, interact with their environments. This leads to a discussion of the thermodynamics and hydrodynamics of a polymer solution. With the fractal properties in mind we discuss the interaction of a random-walk polymer with itself, finding that these interactions change the spatial arrangement of the molecule considerably. Having described the behavior of individual polymer molecules, we can discuss the behavior of solutions, notably those where the polymer chains interpenetrate strongly. The spatial, energetic and dynamic behavior of these solutions can be understood in its major outlines from the fractal properties of isolated polymers treated earlier. We collect in Table I the main quantities used for this discussion.