We present a generalized connectedness percolation theory reduced to a compact form for a large class of anisotropic particle mixtures with variable degrees of connectivity. Even though allowing for an infinite number of components, we derive a compact yet exact expression for the mean cluster size of connected particles. We apply our theory to rodlike particles taken as a model for carbon nanotubes and find that the percolation threshold is sensitive to polydispersity in length, diameter, and the level of connectivity, which may explain large variations in the experimental values for the electrical percolation threshold in carbon-nanotube composites. The calculated connectedness percolation threshold depends only on a few moments of the full distribution function. If the distribution function factorizes, then the percolation threshold is raised by the presence of thicker rods, whereas it is lowered by any length polydispersity relative to the one with the same average length and diameter. We show that for a given average length, a length distribution that is strongly skewed to shorter lengths produces the lowest threshold relative to the equivalent monodisperse one. However, if the lengths and diameters of the particles are linearly correlated, polydispersity raises the percolation threshold and more so for a more skewed distribution toward smaller lengths. The effect of connectivity polydispersity is studied by considering nonadditive mixtures of conductive and insulating particles, and we present tentative predictions for the percolation threshold of graphene sheets modeled as perfectly rigid, disklike particles.

We show that a generalized connectedness percolation theory can be made tractable for a large class of anisotropic particle mixtures that potentially contain an infinite number of components. By applying our methodology to carbon-nanotube composites, we explain the huge variations found in the onset of electrical conduction in terms of a percolation threshold that turns out to be sensitive to polydispersity in particle length and diameter. The theory also allows us to model the influence of the presence of nonconductive species in the mixture, such as is the case for single-walled nanotubes, showing that these raise the percolation threshold proportionally to their abundance.

We studied, by means of polarized light microscopy, the shape and director field of nematic tactoids as a function of their size in dispersions of colloidal gibbsite platelets in polar and apolar solvents. Because of the homeotropic anchoring of the platelets to the interface, we found large tactoids to be spherical with a radial director field, whereas small tactoids turn out to have an oblate shape and a homogeneous director field, in accordance with theoretical predictions. The transition from a radial to a homogeneous director field seems to proceed via two different routes depending in our case on the solvent. In one route, the what presumably is a hedgehog point defect in the center of the tactoid transforms into a ring defect with a radius that presumably goes to infinity with decreasing drop size. In the other route, the hedgehog defect is displaced from the center to the edge of the tactoid, where it becomes virtual again going to infinity with decreasing drop size. Furthermore, quantitative analysis of the tactoid properties provides us with useful information on the ratio of the splay elastic constant and the anchoring strength and the ratio of the anchoring strength and the surface tension.