Graph theory of viscoelastic and configurational properties of Gaussian chains

The spring-and-bead model proposed by Rouse and Zimm can theoretically treat the viscoelastic behaviour of polymers. In this paper, we first point out that the Rouse and Zimm matrices in the molecular theory of polymer viscoelasticities are equivalent to the adjacency matrix and admittance (or Kirchhoff) matrix in graph theory, respectively. In order to solve the eigen-value problems of Rouse and Zimm matrices, the matrices are first represented by their corresponding eigen-graphs, which reflect the topological structure of the real chain. Starting from the eigen-graph, instead of tedious mathematics, the eigen-value problems are solved by a series of simple graphic operations, such as cutting-off the bonds, removing the closed pathways, etc. The eigen-polynomial of Rouse and Zimm matrices and the viscoelastic properties of the chain are obtained by using the theorems given in this paper. It is also shown that the eigen-polynomial of the chain can be greatly reduced if the chain graph has elements of symmetry. As the Rouse theory of viscoelasticity is closely related to the conformational statistics of Gaussian chains, it is demonstrated that the graph-theoretic approach developed here can also be applied to solve the configurational properties of Gaussian chains, such as the distribution function of the radius of gyration and its moments, the shape of a Gaussian chain, etc. We have also demonstrated that the graph-theoretic approach developed here is also applicable to copolymeric chains.