Elastic free energy in swollen polymer networks

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Abstract
Polymer networks with sparse chemical crosslinks are composed in general of deeply interpenetrating coils. The microscopic statistical features of such gels are defined by three characteristic lengths: R t , the separation between the end (junction) points of each component coil; R s , the distance between two spatially nearest neighbor junction points; and ξ, the polymer–polymer correlation length. These distances are related through R 2 t =A R 3 s /ξ, where A is a numerical constant that depends upon the network structure. While R t varies extremely weakly with polymer volume fraction φ, and ξ depends strongly on φ as predicted by scaling theory, only R s deforms affinely (i.e., as φ− 1 / 3) upon swelling. The swelling pressure in such a system consists of two parts, mixing and elastic. The mixing part for each coil contains a factor (R t0/R t )2 to allow for correlations in the concentration, where R t0 is the unperturbed end‐to‐end distance. This situation is the same as for polymer solutions. For the elastic part of the swelling pressure, experimental observations are described that prove the following equivalences, in lightly crosslinked polyvinyl acetate gels: (1) the shear modulus is equal to the volume elastic modulus; (2) the ratio of the shear modulus of a gel swollen in a theta diluent to that of the same gel swollen to the same degree in a good solvent is a constant equal to the ratio of the corresponding temperatures, T/θ. It is concluded that the individual component coils make no significant contribution to the network elasticity, the latter being provided only by the mutual folding of the interpenetrating coils.