The ultimate goal of materials design is to start by a specifying a set of desirable properties and then to follow-up by fabricating a material that meets these specifications optimally. By controlling and patterning the micro- and nano- structures, this dream is growing ever closer to reality. Materials that can be uniquely targeted to specific applications have the potential to make an enormous technological impact. In this talk, we present mathematical theory that can be used for controlling the shapes of growing crystals at the microscale and controlling the spatial orientation of nanostructures (quantum dots) during epitaxial growth of thin films.
At the microscale, we demonstrate that there exist critical conditions of growth such that the Mullins-Sekerka instability may be suppressed and instead universal limiting shapes exist. That is, we find that the morphologies of the nonlinearly evolving crystals tend to limiting shapes that evolve self-similarly and depend only on the far-field conditions. We then design protocols by which the compact growth of crystals with desired symmetries can be achieved. We present both 2D and3D results using adaptive boundary integral methods. Preliminary experimental results are presented that suggest the confirmation of the theory.
The theory at the microscale is then extended to nanoscale studies of monolayer, epitaxially growing islands. Here, the control variables are the deposition flux and a far-field flux that can be manipulated so as to control the shape of the island. We conclude with a study of strained epitaxial thin films. In this case, the relaxation of strain provides a mechanism for influencing the self-organization of quantum dot structures. Using newly-developed, adaptive phase-field methods, we demonstrate that strain patterning, as well as control of the deposition flux, may result in ordered self-organized arrays of nanostructures (quantum dots).