When shape is enough: from colloidal spheres to twisted polyhedra, from icosahedral to chiral order

In this thesis, we study entropy-driven phase transitions in suspensions of colloidal particles. Colloids are small particles, with typical sizes ranging from the nanometer to the micron, dispersed in a medium that is composed of much smaller particles (atoms or molecules). Because of this size difference, colloids experience Brownian motion that allows them to move around the medium, exploring the microscopic configurations available, and possibly arrange themselves in ordered structures. This process is called self-assembly. For example, colloids self-assemble in fluids (disordered arrangements of particles), liquid-crystals (partially ordered), and crystals (fully ordered). The stability and the transitions between these phases, i.e., the phase behaviour, depend both on the interactions between the colloids and on thermodynamic parameters like temperature or density. Here, we focus on systems composed of hard colloidal particles, that are particles that do not have any interactions except for the fact that cannot overlap with each other. By using computer simulations and theory, we show that the particle shape is enough for the formation of several thermodynamic phases. In other words, it is possible to obtain order in complete absence of any attraction in the system, just by increasing the density of the system. These transitions are (fully) entropy-driven since no change in energy is associated to the phase transition but only a change in entropy, that is a quantity related to the number of possible microscopic particle arrangements. In chapter 2 we show that when (tens of) thousands of hard spheres are compressed, while being confined in a spherical cavity, they do not self-assemble into an FCC crystal, that is the equilibrium bulk crystal structure, but rather they form stable icosahedral clusters. In chapter 3, we study hard spherocylinders forming liquid crystals. In chapter 4, we study liquid crystals formed by binary mixtures of colloidal rods and spheres, focusing in particular on the binary smectic phase, that consists in alternating layers of rods and spheres. In chapters 2, 3, and 4 our theoretical and simulation results are compared with experiments performed in our group. In chapter 5, we consider hard rod-like particles with a polyhedral shape and we identify the conditions to form prolate, oblate and biaxial nematic phases. In chapters 6, 7 and 8, we consider colloidal particles with a chiral shape forming chiral liquid-crystal phases. In particular, in chapter 6 we develop a theory to predict the equilibrium cholesteric pitch as a function of thermodynamic state and microscopic details. Applying the theory to hard helices, we observe both right- and left-handed cholesteric phases that depend on a subtle combination of particle geometry and system density. In chapter 7, we introduce particles with a twisted polyhedral shape and obtain, for the first time, a stable fully-entropy-driven cholesteric phase by computer simulations. Our results unveil how the competition between particle biaxiality and chirality is transmitted at a higher level into the nematic phases and new theoretical challenges on the self-assembly of chiral particles are addressed in chapter 8.