Tunneling conductivity in composites of attractive colloids

In conductor-insulator nanocomposites in which conducting fillers are dispersed in an insulating matrix the electrical connectedness is established by interparticle tunneling or hopping processes. These systems are intrinsically non-percolative and a coherent description of the functional dependence of the conductivity $\sigma$ on the filler properties, and in particular of the conductor-insulator transition, requires going beyond the usual continuum percolation approach by relaxing the constraint of a fixed connectivity distance. In this article we consider dispersions of conducting spherical particles which are connected to all others by tunneling conductances and which are subjected to an effective attractive square well potential. We show that the conductor-insulator transition at low contents $\phi$ of the conducting fillers does not determine the behavior of $\sigma$ at larger concentrations, in striking contrast to what is predicted by percolation theory. In particular, we find that at low $\phi$ the conductivity is governed almost entirely by the stickiness of the attraction, while at larger $\phi$ values $\sigma$ depends mainly on the depth of the potential well. As a consequence, by varying the range and depth of the potential while keeping the stickiness fixed, composites with similar conductor-insulator transitions may display conductivity variations of several orders of magnitude at intermediate and large $\phi$ values. By using a recently developed effective medium theory and the critical path approximation we explain this behavior in terms of dominant tunneling processes which involve interparticle distances spanning different regions of the square-well fluid structure as $\phi$ is varied. Our predictions could be tested in experiments by changing the potential profile with different depletants in polymer nanocomposites.