Design and Control of Quasiperiodic Patterns of Particles with Standing Acoustic Waves

Abstract. We propose a method to design tunable quasiperiodic structures of particles suspended in a fluid by creating and controlling standing acoustic waves. The method can be applied to any linear wave-like phenomena and is presented here in the framework of ultrasound directed self-assembly, which allows the fabrication of 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 is used to characterize the quasiperiodic designs based on local minima of the acoustic radiation potential and to explain the influence of the controls on the particle patterns. We propose geodesics for the smooth transformation of the quasiperiodic patterns by continuous transitions between different controls. The developed approach unlocks a route to creating tunable quasiperiodic and moiré structures known for their unconventional superconductivity and other extraordinary properties. Several examples of constructing quasiperiodic structures, including in two and three dimensions, are given.