Many physical systems spontaneously self-organize in regular patterns. Without external control, they evolve into almost crystalline states showing stripes or spots aligned in somewhat regular fashions. Examples range from the self-organizational processes in early developmental states of the embryo to phase separation dynamics in manufacturing processes such as dip-coating. In order to control those processes and possibly harvest the self-organizational capabilities for the manufacturing of micro-structured materials, one needs to understand how the patterning is influenced by parameters and system geometry. The investigator focuses on a particularly relevant situation in which patterns arise through a directional quenching process where the patterned region expands in space, either in an externally controlled fashion, or in a self-organized growth process. It turns out that the result of patterning is very rigid, across many physical and biological contexts, in the sense that the final pattern is robust against imperfections and reproducible from many different initial states of the experiment. The final pattern does however depend sensitively on growth rates and quenching geometry. The investigator and his collaborators develop analytic and numerical tools that enable systematic prediction and control of resulting patterns, with the ultimate goal of designing processes that result in a desired, pre-specified pattern. Graduate students are engaged in the research of the project.
The investigator and his collaborators analyze prototypical systems such as the Swift-Hohenberg or the Cahn-Hilliard equation in situations where the pattern-forming region expands in time at a prescribed rate. In the simplest case, these systems develop striped patterns with a fixed wavelength and orientation relative to the direction of growth. Numerical tools developed for this scenario allow for a systematic exploration of the relation between parameters and resulting orientations and wavelengths. Analytic tools can guide the numerics by exhibiting universal mechanisms such as pinning and detachment of structures. Analysis also complements numerical studies in limiting regimes where computational cost is prohibitively high. Both analysis and numerics focus on the study of coherent structures, which are the simplest pattern-forming dynamic states of the system. They are typically stationary or time-periodic in a frame moving with the quenching interface, and asymptotic to a selected pattern in the pattern-forming region. The first part of the project focuses on the formation of stripes, or lamellar crystals, in simple model problems, where the growth process roughly selects an orientation angle of stripes relative to the boundary. The second part broadens the scope by including more realistic models, different growth laws, and different preferred crystalline states. Graduate 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.
