The Statistical Mechanics of Rubbers

Summary

The paper concerns the basis of the statistical mechanics of rubber. It is firstly shown that an extension of conventional (i.e. history independent) statistical mechanics can be made for systems with permanent constraints when the weighting of configurations is that of an equilibrium system without the constraints. A method of calculation is then given based on carrying out the usual kind of statistical mechanical calculations with all the coordinates of the ensemble retained i.e. to obtain the effective free energy of a constrained rubber one may average over the constraints of a system with (n+l) members, roughly speaking the rubber in 3(n+l) dimensions. This technique is used to calculate the free energy of rubber firstly in the simplest system, crosslinked interpenetrable chains, then to systems in which the topological constraints are added and finally (but rather incompletely) to the full case with internal energy.

It is possible to determine certain conditions in which a rigorous solution can be found, and for the interpenetrable network a rigorous solution can be obtained which can be shown to be microscopically affine. However the topological constraints can be shown to be not invariant under affine transformations, so that the general microscopic effect of a deformation cannot be affine. The final form of the free energy is a complicated function of the deformation since it does not appear possible to find a simple coupling constant which will lead to a simple answer. The simple Σλ ²_i law of the phantom chains is replaced by a more complex law containing all the elastic invariants, when the lack of interpenetrability of chains is allowed for.