Investigating the formation of three-dimensional symmetry broken quasicrystals

Abstract

Periodic crystals form ordered arrangements of atoms or molecules with rotational and transla- tional symmetry, while quasicrystals (QCs) lack translational symmetries. QCs can be quasiperiodic in all three dimensions with icosahedral symmetry. Recently, quasicrystals have also been observed in soft matter systems, where the first observation of soft matter quasicrystals was of two-dimensional dodecagonal quasicrystals that were columnar along the third direction. Analysis of soft matter crystallization employs numerical simulation and numerical continuations methods. These methods are highly sensitive to chosen initial conditions. Reliable initial guesses are usually created based on symmetry considerations. This implies that patterns with full symmetry are preferentially investigated. In this thesis, we consider different reduced symmetry subspaces of icosahedral symmetry quasicrystals and thereby expand the repertoire of potential solutions that can be explored in order to obtain parameter ranges where the related solutions in the partial differential equation (PDE) model are stable. The investigation involves modelling the pattern formation of three-dimensional soft matter quasicrystals using a PDE model that describes pattern formation at two lengthscales. We implemented linear stability and weakly nonlinear analysis. From the weakly nonlinear analysis, we derived the amplitude equation for different patterns. During weakly non-linear analysis, we chose different symmetry subgroups, which can be divided into simple and complex patterns. In the case of simple patterns, we have only one amplitude equation to solve for thier equilibria, which we can solve analytically. In the case of complex patterns where we have more than one amplitude equation to solve for its equilibria, homotopy continuation is used to solve polynomial equations of the corresponding complex pattern amplitude equations. In order to see if the solutions of the amplitude equation of each pattern have corresponding solutions for the PDE model, we use these solutions as the initial condition of the numerical simulation. Therefore, we are able to see if icosahedral symmetry has broken symmetry subgroups and identify the stable solutions of each pattern.