Abstract
Ensembles of particles rotating in a two-dimensional fluid can exhibit chaotic dynamics yet develop signatures of hidden order. Such rotors are found in the natural world spanning vastly disparate length scales — from the rotor proteins in cellular membranes to models of atmospheric dynamics. Here we show that an initially random distribution of either driven rotors in a viscous membrane, or ideal vortices with minute perturbations, spontaneously self assemble into a distinct arrangement. Despite arising from drastically different physics, these systems share a Hamiltonian structure that sets geometrical conservation laws resulting in prominent structural states. We find that the rotationally invariant interactions isotropically suppress long-wavelength fluctuations — a hallmark of a disordered hyperuniform material. With increasing area fraction, the system orders into a hexagonal lattice. In mixtures of two co-rotating populations, the stronger population will gain order from the other and both will become phase enriched. Finally, we show that classical 2D point vortex systems arise as exact limits of the experimentally accessible microscopic membrane rotors, yielding a new system through which to study topological defects.
Highlights
Ensembles of particles rotating in a two-dimensional fluid can exhibit chaotic dynamics yet develop signatures of hidden order
We have shown that driven rotors in a membrane or a soap film, like point vortices in an ideal 2D fluid, have geometrical conservation laws which limit their distribution
We suspect that a completely pure system of point vortices may never reach hyperuniformity due to a dynamical bottleneck, but have shown that hyperuniformity is robust to two forms of perturbations, whether arising due to numerical errors or steric interactions
Summary
Ensembles of particles rotating in a two-dimensional fluid can exhibit chaotic dynamics yet develop signatures of hidden order. Structurally identical Hamiltonian and moment constraints can arise in the microscopic, viscously dominated realm from a strict balance of dissipation with drive on immersed rotating objects These objects include models of interacting transmembrane ATP-synthase rotor proteins[15–17], and the planar interactions of rotors—microscopic particles driven to rotate by an external torque[18,19]. The long-wavelength configuration at steady state is characterized by an isotropically vanishing structure factor, S(q → 0) → 0, leading to an isotropic band-gap[27–29] To investigate this prediction, we numerically simulate assemblies of both BDD and point vortices and observe (see Fig. 1): (i) hyperuniformity for BDD systems; (ii) evidence that point vortex systems can become hyperuniform depending on how they are perturbed; (iii) phase enrichment (in both cases); and (iv) crystallization (for BDD). Our observations lead us to conclude that rotational dynamics provide a mechanism for the self-assembly of particles into a disordered hyperuniform 2D material
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