Solid elasticity and liquid-like behaviour in randomly crosslinked polymer networks

We use the results of exact replica calculations on randomly cross-linked networks of Gaussian phantom chains in order to study the effect of quenched fluctuations of local crosslink density on the elastic response of the network on all length scales. We verify Alexander's conjecture about the existence of length scales on which rigidity and affinity are established and show that they depend on the local crosslink density as well as on the state of deformation of the whole sample. The non-affine character of the deformation in the intermediate range between these length scales is discussed. We calculate the fractions of elastically effective and affinely deformed chains in the network and find that while the former increases, the latter decreases as a function of elongation. We consider the limit of applicability of linear elasticity and estimate the stress-carrying fraction of the chains at the onset the non-linear regime. Experimental tests of our predictions are proposed.