Theory of pair connectedness in templated quenched-annealed systems

Recently, several families of promising porous materials have been proposed where the porous matrix forms in the presence of additional molecules or templates. These materials find applications in separations, sensing, catalysis, and other technologies. For these systems, it is important to understand the connectedness of the matrix species and the porous space. In the first case, this would characterize the integrity of the porous material, whereas the second property is directly related to the accessibility of the interior porous space and thus to the function of the material. Here, we propose an integral equation theory which describes cluster population and percolation phenomena for matrix and template species at the stage of the templated material formation. We also extend this theory to provide structural characterization of the fluid confined in a templated structure. The predictions of the theory are tested for the case of rigid molecular species made of hard sphere interaction sites and compared with computer simulations. We discuss the effect of the system density, species structure, and other parameters on the average cluster size and percolation threshold for the components of the system.