Spectral Graph Theory of Polymer Dynamics

This book presents familiar models—such as the Hookean dumbbell, the Rouse chain, ring polymers, trimers, stars, and more general bead–spring networks—within a unified framework. Classical polymer dynamics often treats each molecular architecture separately. A dumbbell gives one relaxation time, a Rouse chain gives the Rouse spectrum, a ring polymer gives a different set of degenerate modes, and branched polymers require still other calculations. The idea of this book is that these examples are not separate phenomena, but manifestations of one underlying structure: polymer topology: ?graph Laplacian -?spectrum -?relaxation -?rheology. The mathematical background is developed carefully, but it is not presented as an abstract text in graph theory. The emphasis is always on polymer modelling and dynamics. Graphs, incidence matrices, Laplacians, eigenvalues, eigenvectors, and modal decompositions are introduced because they are useful for understanding how polymer architecture controls relaxation and stress. The goal is not to turn students in polymer physics/engineering into graph theorists, but to provide them with enough spectral graph theory to use it effectively as a modelling tool.