Emergence of Zero Modes in Disordered Solids Under Periodic Tiling

Abstract

In computational models of particle packings with periodic boundary conditions, it is assumed that the packing is attached to exact copies of itself in all possible directions. The periodicity of the boundary then requires that all of the particles’ images move together. An infinitely repeated structure, on the other hand, does not necessarily have this constraint. As a consequence, a jammed packing (or a rigid elastic network) under periodic boundary conditions may have a corresponding infinitely repeated lattice representation that is not rigid or indeed may not even be at a local energy minimum. In this manuscript, we prove this claim and discuss ways in which periodic boundary conditions succeed in capturing the physics of repeated structures and where they fall short. :contentReference[oaicite:0]{index=0}

Publication Details

• Authors: R. Cameron Dennis, Varda F. Hagh, and Eric I. Corwin
• Journal: Physical Review E
• Volume: 106
• Issue: 4
• Article Number: 044901
• Publication Date: October 2022
• DOI: 10.1103/PhysRevE.106.044901

Key Findings

• Periodic Boundary Conditions Limitations: Demonstrated that while periodic boundary conditions (PBCs) are commonly used in simulations to mimic infinite systems, they impose constraints causing all particle images to move collectively. This collective movement may not accurately represent the behavior of an infinitely repeated structure, potentially leading to discrepancies in mechanical stability assessments.
• Emergence of Zero Modes: Proved that a jammed packing or rigid elastic network under PBCs might correspond to an infinitely repeated lattice that is not rigid or even at a local energy minimum. This finding indicates that such systems can exhibit zero modes—deformations that do not change the system’s energy—when analyzed beyond the constraints of PBCs.
• Implications for Material Modeling: Highlighted the necessity for caution when using PBCs in modeling disordered solids. The study suggests that while PBCs are useful, they may not always capture the true mechanical behavior of materials, especially concerning rigidity and stability in infinitely repeated structures.

This study provides critical insights into the limitations of periodic boundary conditions in modeling the mechanical stability of disordered solids, emphasizing the potential emergence of zero modes in such systems.

url_pdf: “https://arxiv.org/pdf/2201.05711.pdf"