The statistical mechanics of polymerized material

It is argued that a solid consisting of fully entangled long chain molecules will show elastic properties similar to those of cross linked, i.e. vulcanized material, provided that elastic moduli are studied in periods short compared with creep times. An appropriate formula for the free energy is deduced, and calculated for a model in which each chain is confined to an environment formed by the others. Under deformation this environment changes and the elastic energy can be deduced. If 1, 2, 3 are the principal strains, that part of the free energy due to the polymer entanglements is shown to be ΔF = - BκT(12 + 22 + 32)L(σl/m)1/2 for small strains, where B is a numerical constant, L is the total length of polymer present, σ the density of polymer, and l, m the effective length and mass of one of the submolecules from which the polymer is made. An expression is also given for the case of a strain of arbitrary size.