Hashmi, Bahaudin A., authorShipman, Patrick D., advisorLiu, Jiangguo, advisorThompson, Stephen, committee memberBates, Daniel J., committee member2017-09-142017-09-142017https://hdl.handle.net/10217/183970We have analyzed pattern formation in two different systems: (1) Vapor-to-particle reaction diffusion systems and (2) Highly ordered square arrays in ion bombardment. The vapor-to-particle reaction exhibits oscillatory behavior and produces a spatial pattern called Liesegang rings. In this thesis, we develop a finite element scheme to model the hydrogen chloride and ammonia vapor-to-particle reaction. In our simulations, we develop parametric regions for different patterns and corroborate data obtained from experiments of this reaction. For the ion bombardment of a planar surface, we add the Ehrlich-Schwoebel barrier terms to the Bradley-Shipman equations of motion and see the impact of ion bombardment at normal incidence on a binary crystalline material. A weakly nonlinear stability analysis was conducted and regions were determined where highly ordered square pyramids formed.born digitaldoctoral dissertationsengCopyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright.ion bombardmentthreshold kineticsnucleation and growthfinite element methodsText