Localized Structures in Spatially Extended Systems: Fronts and Defects

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Spatially localized structures such as fronts, defects, spots or pulses are common in many continuum systems, and include pulses propagating along nerve fibers, dissipative solitons in optical and chemical systems, localized buckling of slender structures under compression, and oscillons in vibrating granular media. Examples from fluids include localized convection, vortices and drops. These diverse systems have two things in common: (i) they are dissipative systems driven by spatially uniform forcing, and (ii) there is range of forcing within which the application of different finite amplitude perturbations can lead to distinct localized states. The investigator seeks to extend existing theory in new directions, focusing on problems arising in materials science such as the nucleation and growth of crystals from a supercooled liquid, and the ordered and disordered structures that may result. These include structures with short-range order but no long-range order called quasicrystals. The type of structure that forms in turn determines the strength and other properties of the resulting material and this depends strongly on the speed of the crystallization process. The aim of the project is to provide a comprehensive understanding of the mechanisms behind the different types of growth that take place at different temperatures of the liquid both in this and in related systems. Two postdoctoral students are engaged in the research of the project.In this project the investigator and his colleagues study the properties of spatially localized structures in two and three dimensions. Both conserved and nonconserved systems are considered with a focus on phase field and dynamical density functional theory models of soft matter crystallization from a supercooled melt. Depending on the speed of this process, the resulting material may be crystalline, amorphous, or a quasicrystal. Localized structures of each type may serve as critical nuclei for the nucleation of the crystal and are therefore of particular interest. The project focuses on the properties of these structures, and the properties of the fronts that separate them from the melt. Both steady and propagating fronts are considered and the processes that determine the propagation speed are studied in detail. These include pinning of fronts to the microstructure behind them and the interaction with the conserved mass mode. The project also includes a parallel study of traveling pulses in nonconserved systems and a detailed study of different types of defects where pinning to microstructure is important. The techniques used by the investigator include bifurcation theory, coupled with numerical branch-following and direct numerical simulations of realistic systems. Two postdoctoral students are engaged in the research of the project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.