Micromechanical analysis of ionic clustering in Na®on per ̄uorinated membrane

The cluster morphology in a water-swollen Na®on per ̄uorinated membrane is studied using a micromechanics approach. The cluster size is determined from the minimization of the free energy as a function of the equivalent weight of Na®on, the volume fraction of water, and the temperature, taking into account the electrostatic dipole interaction energy, the elastic polymer chain reorganization energy, and the cluster surface energy, leading to results which are in accord with experimental observations. By minimizing the sum of: (1) the electro-elastic interaction energy between an ionic cluster and the ̄uorocarbon matrix, and (2) the cluster surface energy, it is concluded that the e€ective cluster shape is spherical in the absence of an electric ®eld, and becoming an oblate spheroid when an electric ®eld is applied. The e€ect of cluster morphology on the e€ective electro-elastic moduli and the e€ective ionic conductivity is then studied by a micromechanical multi-inclusion model. The result seems to describe the available empirical relation when a spherical cluster shape is assumed. It correctly predicts the insulator-to-conductor transition which occurs in Na®on, as the water volume fraction is increased. Ó 2000 Elsevier Science Ltd. All rights reserved.