Confinement and polydispersity in granular systems and liquid crystals

In chapter 2 we presented results obtained through experiments on a vibrated monolayer of granular metallic rods, focusing on the effects that introducing a circular obstacle to the centre of the cavity (thus creating an annulus) would have on the steady states of the system. Although topological arguments suggested that defects found on systems with no obstacle would not appear, we observed a separation of the system into regions consisting of smectic and tetratic ordering, the latter containing completely unordered point-like defects. We argue that the topological approach based on a continuum argument is rendered invalid by the size of the elements involved, in particular the ratio of the central obstacle size to particle lengths. The tetratic structures can be viewed as domain walls separating the smectic regions, which were invariably equal to four in our experiments. In chapter 3 we used the SPT to study two different systems of two-dimensional particles. In the first one, composed of particles shaped like hard isosceles triangles, we found, for the one component fluid, 1st and 2nd order Isotropic-Nematic transitions and for symmetric binary mixtures a 1st order Nematic-Nematic transition ending in a critical point and a Landau point where two first-order Nematic-Triatic transitions and a Nematic-Nematic demixing transition coalesce. We also found that for these particular mixtures a Triatic phase can be stabilized even though both species involved do no exhibit a stable Triatic phase on the one-component limit. To the best of our knowledge this is the first example of a Nematic-Nematic transition on one-component hard particle systems. For the second system, of hard rectangular particles, we found that polydispersity has an important effect on the phase diagram, strengthening the 1st order Isotropic-Nematic transition and decreasing the stability of the Tetratic phase. Also we found that polydispersity induces strong fractionation between the coexisting phases. Finally in chapter 4 we studied the effects of nano-confinement on a system of hardboard like particles. Following the restricted orientation (Zwanzig) approach we obtained a DF using the FMT to approximate the excess part of the free-energy of the system by means of numerical minimization. We considered a particular set of particle aspect ratios as well as slightly biaxial particles. We hoped to determine whether homeotropic configurations could be stabilized by the presence of two hard walls. We obtained that, for particular particle aspect ratios, homeotropic phases do stabilize and also that allowing for particle biaxiality greatly enriches the phase diagram. The results obtained can be explained by considering how entropy is maximized as the system approaches close packing. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Luis Mederos Martín.- Secretario: Daniel de las Heras Díaz-Plaza.- Vocal: Giorgo Cinacchi