A sample of polyisoprene (C5H8) with Mw = 100,000 and Mw/Mn = 2.0 is cross-linked with sulfur. According to the classical theory, what fraction of monomers need to be in cross-links in order to reach the gel point? Given that the entanglement molecular weight for polyisoprene is 5400, how many cross-links per chain are required so that the shear modulus of the resulting rubber is equal to the plateau modulus of the polymer before cross-linking (see Chapter 11)? The rubber from part (b) is swollen to equilibrium with toluene, a good solvent. The volume fraction of polymer in the swollen gel is e. The surrounding excess solvent is then replaced with a toluene solution of the initial uncross-linked polyisoprene polymer, also with concentration e. Assuming the polymer does not diffuse into the gel, will there be a net solvent flow from gel to solution, from solution to gel, or will there be no solvent transfer? Explain your reasoning.