| This thesis contains two separate bodies of work, which are partitioned into two chapters.; The first chapter concerns two-dimensional random tilings that comprise squares and equilateral triangles. Such tilings provide a basis for modeling quasicrystals with twelve-fold symmetry. A (phason) elastic theory for random square-triangle tilings is constructed, whose order parameter is the phason field, and whose entropy density includes terms up to third order in the phason strain; this theory contains four unknown parameters. An adequate update move for square-triangle tilings rearranges a closed, nonlocal, one-dimensional chain of squares and triangles. Such update moves are called "zippers". By implementing this update move, measuring phason-mode fluctuations, introducing a pseudo-Hamiltonian, and using the Ferrenberg-Swendsen histogram method, estimates of these four parameters are obtained.; The second chapter concerns three-dimensional tilings that comprise canonical cells. These tilings provide a basis for modeling quasicrystals that exhibit icosahedral symmetry (e.g. i-AlMn(Si)). The space symmetries of various cubic canonical-cell approximants are determined. A rational methodology is described for decorating canonical-cell tilings with atoms. By mapping canonical-cell tilings onto packings that contain large triacontahedra and prolate rhombohedra, a decoration prescription that uses this methodology is formulated for AlMn(Si)-type quasicrystals. The whole approach is tested by generating model structures containing Al and Mn atoms, then relaxing these structures subject to interatomic forces defined by pair potentials; displacement coefficients are calculated, which quantify how, in terms of the decoration prescription, atoms move upon relaxation. |
