Role of topological defects in the two-stage melting and elastic behavior of active Brownian particles

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Role of topological defects in the two-stage melting and elastic behavior of active Brownian particles

Siddharth Paliwal and Marjolein Dijkstra

Phys. Rev. Research 2, 012013(R) – Published 14 January 2020

Abstract

We find that crystalline states of repulsive active Brownian particles at high activity melt into a hexatic phase, but this transition is not driven by an unbinding of bound dislocation pairs as suggested by the Kosterlitz-Thouless-Halperin-Nelson-Young theory. Upon reducing the density, the crystalline state melts into a high-density hexatic state devoid of any defects. Decreasing the density further, the dislocations proliferate and introduce plasticity in the system, nevertheless maintaining the hexatic state, but eventually melting into a fluid state. Remarkably, the elastic constants of active solids are equal to those of their passive counterparts, as the swim contribution to the stress tensor is negligible in the solid state. The sole effect of activity is that the stable solid regime shifts to higher densities. Furthermore, discontinuities in the elastic constants as a function of density correspond to changes in the defect concentrations rather than to the solid-hexatic transition.

Received 1 July 2019
Revised 11 October 2019

DOI:https://doi.org/10.1103/PhysRevResearch.2.012013

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Phase diagrams Phase transitions

Active Brownian particles

Brownian dynamics

Statistical Physics & Thermodynamics

Authors & Affiliations

Siddharth Paliwal ^* and Marjolein Dijkstra ^†

Soft Condensed Matter, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 1, 3584 CC Utrecht, The Netherlands

^*s.paliwal@uu.nl
^†m.dijkstra@uu.nl

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Vol. 2, Iss. 1 — January - March 2020

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Images

Figure 1
State diagram in the Pe- $ρ σ^{2}$ representation exhibiting fluid (circles, red area), fluid-hexatic coexistence (diamonds, yellow area), hexatic (triangles, green area), and crystal (squares, blue area) states. The symbols denote state points used in the simulations, the background colors denote the boundaries of the labeled regions, and the black dashed line indicates the densities beyond which the concentration of topological defects vanishes in simulations of $N = 72 \times 10^{3}$ particles. The inset is a magnification of the low-Pe regime.
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Figure 2
Typical sections of active Brownian particle configurations ( $80 σ \times 80 σ$ ) for $Pe = 0.0$ , 4.8, and 71.5 at labeled states, showing fivefold (blue) and sevenfold (red) defects, other defects (black), and particles with $N_{b} = 6$ (gray). Some vacancies are indicated by arrows in the hexatic and solid states. The configurations at $Pe = 4.8$ are similar to the corresponding passive states. For $Pe = 71.5$ we find that with increasing density the hexatic states show a decrease in the number fraction of defects, with a complete absence of defects at a density $ρ σ^{2} ≳ 1.560$ . The positional correlations become quasi-long-range around a density $ρ σ^{2} ≃ 1.900$ for $Pe = 71.5$ as shown in Fig. 1. The insets in the top right and bottom right corners show the same configurations as the main panels but colored according to the hexatic and positional order parameters $ψ_{6}$ and $ψ_{T}$ , respectively, following the color mapping shown at the top.
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Figure 3
(a) Number fraction of specific defects as a function of density $ρ σ^{2}$ for various Pe. The background colors mark the boundaries as indicated in the state diagram (Fig. 1) and the region devoid of any defects is crosshatched. The symbols on the axis indicate the densities where the defects become absent ( $\times$ ) and the hexatic-solid transition densities ( $★$ ). (b) Difference in the number fraction of unpaired dislocations and bound dislocation pairs $(N_{pair} - N_{quart}) / N$ (squares) and difference in the number fraction of unpaired dislocations and free disclinations $(N_{pair} - N_{free}) / N$ (circles) as a function of $ρ σ^{2}$ for various Pe as labeled in the legend.
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Figure 4
(a) Young's modulus $K$ as a function of density $ρ σ^{2}$ collapses onto a single master curve $K \propto exp (a ρ^{3} + b ρ^{2} + c ρ + d)$ for $0.0 \leq Pe \leq 71.5$ . The plus ( $+$ ) markers denote the discontinuous jump in $K$ for the corresponding activity, cross ( $\times$ ) markers show the densities where the defects disappear for a large system size of $N = 72 \times 10^{3}$ particles, and star ( $★$ ) markers denote the transition densities from the decay of $ψ_{T}$ , all offset vertically for clarity. (b) Bare Young's modulus $K$ (open circles) obtained directly from the Lamé coefficients and the corresponding renormalized values $K_{R}$ (triangles) for $Pe = 0.0$ , 0.8, 1.6, and 2.4. The vertical lines mark the hexatic-solid transitions identified from the decay of $ψ_{T}$ for $N = 72 \times 10^{3}$ particles.
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