PHYSICAL REVIEW E 104, 044902 (2021) Classification of emerging patterns in self-assembled two-dimensional magnetic lattices Ehsan Norouzi, Audrey A. Watkins , and Osama R. Bilal * Wave Engineering through eXtreme & Intelligent matTEr Laboratory, Department of Mechanical Engineering, University of Connecticut, Storrs, Connecticut 06269, USA (Received 6 January 2021; revised 12 July 2021; accepted 5 August 2021; published 4 October 2021) Self-assembled granular materials can be utilized in many applications such as shock absorption and energy harvesting. Such materials are inherently discrete with an easy path to tunability through external applied forces such as stress or by adding more elements to the system. However, the self-assembly process is statistical in nature with no guarantee for repeatability, stability, or order of emergent final assemblies. Here we study both numerically and experimentally the two-dimensional self-assembly of free-floating disks with repulsive magnetic potentials confined to a boundary with embedded permanent magnets. Six different types of disks and seven boundary shapes are considered. An agent-based model is developed to predict the self-assembled patterns for any given disk type, boundary, and number of disks. The validity of the model is experimentally verified. While the model converges to a physical solution, these solutions are not always unique and depend on the initial position of the disks. The emerging patterns are classified into monostable patterns (i.e., stable patterns that emerge regardless of the initial conditions) and multistable patterns. We also characterize the emergent order and crystallinity of the emerging patterns. The developed model along with the self-assembly nature of the system can be key in creating re-programmable materials with exceptional nonlinear properties. DOI: 10.1103/PhysRevE.104.044902 I. INTRODUCTION Granular media have inspired an extensive number of stud- ies on the interplay between nonlinearity and discreteness [1] with many potential applications in shock absorption, acoustic sensing, switching, and energy harvesting [1–6]. In addition to their potential as a fascinating platform to study nonlinear dynamics [7–10], they can be easily prototyped from simple basic particles such as beads and disks. The inherent dis- creteness in such media makes them an ideal candidate for fabrication through self-assembly processes [10–17,17–21]. However, the statistical nature of the self-assembly process can hinder their practicality as the final assembly depends highly on the initial positions of the system elements [22,23]. Even for highly repeatable patterns, a certain amount of disor- der usually emerges within the assembly. The present study considers the interplay between the symmetry of granular elements, their confining boundary shape, and resulting final pattern. The repeatability, stability, and inherent order in the self-assembled emergent patterns are studied. In this paper we study the self-assembly of various types of free-floating magnetic disks in two dimensions confined to a fixed boundary with embedded magnets. In particular, the repeatability, stability, and crystallinity of these emerging pat- terns under different boundary conditions is considered. The free-floating disks are subject mainly to repulsive magnetic interactions among themselves and their confining boundary in two dimensions (i.e., with negligible friction). We develop a numerical model to predict the final assembly of the free- * osama.bilal@uconn.edu floating disks based on an agent-based model. We consider six different types of disks, with the number of embedded permanent magnets ranging from 1 to 6. We consider seven types of boundaries, with the number of sides varying from 3 (triangle) to 8 (octagon) in addition to a circular boundary (Fig. 1). The model is executed multiple times with different initial positions of the disks and their final assembly after convergence is recorded. The final assemblies are classified as monostable (the same pattern emerges regardless of the initial position of the disks) or multistable (where the initialization plays a crucial rule in the final pattern) patterns. To classify the stability of these emerging patterns as monostable or mul- tistable, we calculate the relative position between the disks (i.e., relative distance and angles between the disks) at their FIG. 1. Concept. The self-assembly of six disk types (left) with embedded identical magnets is considered. The disks are confined to different boundary shapes (middle). The boundaries also have identi- cal permanent magnets with the same polarity as the disks. The final self-assembly can be mono- or multistable, crystalline or amorphous depending on the number and type of disks and the boundary shape. 2470-0045/2021/104(4)/044902(8) 044902-1 ©2021 American Physical Society