Design and control of quasiperiodic patterns of particles with standing acoustic waves

Abstract

We develop a method to design tunable quasiperiodic structures of particles suspended in a fluid by controlling standing acoustic waves. One application of our results is to ultrasound directed self-assembly, which allows fabricating composite materials with desired microstructures. Our approach is based on identifying the minima of a functional, termed the acoustic radiation potential, determining the locations of the particle clusters. This functional can be viewed as a two- or three-dimensional slice of a similar functional in higher dimensions as in the cut-and-project method of constructing quasiperiodic patterns. The higher dimensional representation allows for translations, rotations, and reflections of the patterns. Constrained optimization theory is used to characterize the quasiperiodic designs based on local minima of the acoustic radiation potential and to understand how changes to the controls affect particle patterns. We also show how to transition smoothly between different controls, producing smooth transformations of the quasiperiodic patterns. The developed approach unlocks a route to creating tunable quasiperiodic and moir\'e structures known for their unconventional superconductivity and other extraordinary properties. Several examples of constructing quasiperiodic structures, including in two and three dimensions, are given.