Cluster formation in water-in-oil microemulsions at percolation: Evaluation of the electrical properties

We study water-in-oil microemulsion systems in the droplet phase and in the vicinity of a percolation transition in the non-percolating region. We focus on the electrical conductivity and permittivity, quantities that show large variations when approaching the percolation threshold. The accepted model for the interpretation of the increasing conductivity-very large compared to that of the bathing oil phase-is related to clustering of the microemulsion droplets and migration of charges within the aggregates. Power laws have been used to interpret the behaviour of the static dielectric properties and scaling functions proposed for the frequency-dependent conductivity and permittivity. We review some relevant experiments in this field and the proposed interpretations, and formulate a phenomenological model of conduction. It is based on the physical picture of cluster formation due to attractive interactions among the constituent water droplets, anomalous diffusion in the bulk of fractal aggregates and polydispersity of the clusters. The model gives quantitative expressions for both conductivity and permittivity over the entire frequency range of the percolative relaxation phenomena, including the static behaviour. A closed expression is derived for the scaling function of a scaling variable which involves frequency, the cut-off cluster size and the parameters of the bulk components. The results are also expressed in the time domain in terms of the polarization time correlation function. The latter exhibits a rather interesting behaviour, since it gradually evolves from an exponential decay to a power-law decay and to a stretched exponential as time increases. The time-scales of the different stages are obtained from the typical decay times of the single droplet and the largest cluster. We have analysed many different sets of data obtained for different microemulsion systems as functions of the composition of the dispersed phase, the temperature and the frequency of the applied field, with a very good agreement with the model in all cases.