Topological Methods for Polymeric Materials: Characterizing the Relationship Between Polymer Entanglement and Viscoelasticity

W e draw on mathematical results from topology to develop quantitative methods for polymeric materials to characterize the relationship between polymer chain entanglement and bulk viscoelastic responses. We generalize the mathematical notion of the Linking Number and Writhe to be applicable to open (linear) chains. We show how our results can be used in practice by performing fully three-dimensional computational simulations of polymeric chains entangled in weaves of a few distinct topologies and with varying levels of chain densities. We investigate relationships between our topological characteristics for chain entanglement and viscoelastic responses by performing Lees-Edwards simulations of the rheology over a broad range of frequencies. Our topological measures of entanglement indicate the global topology is the dominant factor in characterizing mechanical properties. We find an almost linear relation between the mean absolute Writhe Wr and the loss tangent and an almost inverse linear relation between the mean absolute Periodic Linking Number LKP and the loss tangent. These results indicate the potential of our topological methods in providing a characterization of the level of chain entanglement useful in understanding the origins of mechanical responses in polymeric materials.