Shape, Frustration, and Elasticity in Colloidal Liquid Crystals: A Particle Based Simulation Study

Liquid crystals are intermediate phases between fluids and solids: they flow like liquids while retaining orientational or partial positional order. This duality makes them sensitive to geometry, confinement, external fields, and microscopic particle shape. In colloidal liquid crystals, where the constituent particles can be directly designed and resolved, shape provides a particularly powerful route to controlling collective order. This thesis investigates how particle geometry encodes elastic frustration and thereby drives the emergence of complex liquid-crystalline phases. Using particle-based simulations, complemented where possible by experiments and coarse-grained modelling, we show that simple anisotropic building blocks can self-assemble into modulated, chiral, topological, and networked structures through minimal physical mechanisms. The conceptual framework of the thesis is provided by Frank–Oseen elasticity. The four fundamental director deformations—splay, twist, bend, and saddle-splay—can each be associated with a characteristic local packing geometry. However, these ideal deformation modes are generally incompatible with uniform tiling of three-dimensional Euclidean space. As a result, particle shapes that locally favour a given elastic mode generate geometric frustration. This frustration may be resolved by coupling to other elastic deformations, by developing density modulations, or by introducing defects and topological textures. The central question addressed here is how such local geometric preferences translate into macroscopic liquid-crystalline order. We first examine bend frustration in systems of achiral hard banana-shaped particles. Despite the absence of intrinsic chirality or attractive interactions, these particles spontaneously stabilise chiral and topological structures. Under confinement, they form double-twist skyrmions arranged in hexagonal lattices; in bulk, they assemble into blue-phase-like networks of skyrmion filaments. This demonstrates that curvature alone, through excluded-volume interactions, can generate complex chiral order. We then extend this picture to mixtures of bent and straight colloidal rods, showing that particle-shape dispersity can stabilise twist–bend nematic order and striped skyrmion phases. These results reveal that combining different local deformation tendencies broadens the landscape of accessible modulated phases. The thesis next turns to saddle-splay frustration using distorted tetrahedral particles. Simulations show that these simple hard polyhedra self-assemble hierarchically into a rich variety of mesophases, including spontaneous cholesterics, biaxial nematics, disordered splay-nematic structures, twisted lamellae, hexagonal columnar phases, and gyroids. These findings establish saddle-splay as a productive design principle for generating complex soft-matter architectures from entropic interactions alone. Beyond hard-particle models, we develop a machine-learning framework for coarse-grained anisotropic interaction potentials. By expressing orientation-dependent interactions through particle-centred descriptors and spherical-tensor-inspired expansions, this approach enables accurate and efficient simulations of particles with complex shapes, surface patterns, and depletion interactions. Finally, we explore field-driven colloidal rod networks, where gravity, electric fields, confinement, and electrohydrodynamic interactions cooperate to form reversible percolating structures with tunable connectivity. This thesis establishes a geometric design perspective for colloidal liquid crystals. It shows that microscopic shapeand local packing frustration can be systematically connected to emergent elasticity, topology, and self-assembled structure. By linking minimal particle models, continuum elastic intuition, machine-learned potentials, and experimental realisations, the work provides a framework for designing soft materials in which complexity arises not from chemical specificity, but from geometry itself.