Tactoids are droplets of a nematic phase that under suitable conditions form in dispersions of elongated colloidal particles. We theoretically study the shape and the director-field configuration of such droplets for the case where a planar anchoring of the director field to the interface is favored. A minimum of four regimes can be identified in which the droplets have a different structure. Large droplets tend to be nearly spherical with a director field that is bipolar if the surface tension is strongly anisotropic and homogeneous if this is not so. Small droplets can become very elongated and spindlelike if the surface tension is sufficiently anisotropic. Depending on the anchoring strength, the director field is then either homogeneous or bipolar. We find that the more elongated the tactoid, the more strongly it resists the crossing over from a homogeneous to a bipolar structure. This should have implications for the nucleation rate of the nematic phase. Our calculations qualitatively describe the size dependence of the aspect ratio of tactoids found in recent experiments.

We apply continuum connectedness percolation theory to realistic carbon nanotube systems and predict how bending flexibility, length polydispersity, and attractive interactions between them influence the percolation threshold, demonstrating that it can be used as a predictive tool for designing nanotube-based composite materials. We argue that the host matrix in which the nanotubes are dispersed controls this threshold through the interactions it induces between them during processing and through the degree of connectedness that must be set by the tunneling distance of electrons, at least in the context of conductivity percolation. This provides routes to manipulate the percolation threshold and the level of conductivity in the final product. We find that the percolation threshold of carbon nanotubes is very sensitive to the degree of connectedness, to the presence of small quantities of longer rods, and to very weak attractive interactions between them. Bending flexibility or tortuosity, on the other hand, has only a fairly weak impact on the percolation threshold.

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.