Effect of the topology on the gaussian elasticity of a network. The example of a fractal topology

By considering fractal topologies of polymer networks, we show that one can obtain much smaller values of the elastic modulus than predicted by classical models for the same density of crosslinks. In a regular fractal such as the Sierpinski gasket, one can replace the segments joining the modes by Gaussian chains. This allows one to draw some micronetworks of rank k = 1, 2 etc., with no dangling ends except for the few external points (3 for any rank). Within the frame of the phantom network model, we calculate the elastic modulus of the micronetwork. It varies as p^k−1 for p < 1, which leads, even for rather small orders k such as 2, 3, to values lower than the classical James and Guth modulus, which actually corresponds to the connectivity of a tree. Thus there is an influence of the topology. Fluctuations of the positions of crosslinks and of the distance between two crosslinks are estimated and are found to correspond to a lower deformation of the chains. Whether these topologies are more than an example and could explain the anomalies observed experimentally is a postponed discussion.