Three-Dimensional Active Defect Loops

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Three-Dimensional Active Defect Loops

Jack Binysh, Žiga Kos, Simon Čopar, Miha Ravnik, and Gareth P. Alexander

Phys. Rev. Lett. 124, 088001 – Published 25 February 2020

Abstract

We describe the flows and morphological dynamics of topological defect lines and loops in three-dimensional active nematics and show, using theory and numerical modeling, that they are governed by the local profile of the orientational order surrounding the defects. Analyzing a continuous span of defect loop profiles, ranging from radial and tangential twist to wedge $\pm 1 / 2$ profiles, we show that the distinct geometries can drive material flow perpendicular or along the local defect loop segment, whose variation around a closed loop can lead to net loop motion, elongation, or compression of shape, or buckling of the loops. We demonstrate a correlation between local curvature and the local orientational profile of the defect loop, indicating dynamic coupling between geometry and topology. To address the general formation of defect loops in three dimensions, we show their creation via bend instability from different initial elastic distortions.

Received 16 September 2019
Accepted 23 January 2020

DOI:https://doi.org/10.1103/PhysRevLett.124.088001

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)

Biomimetic & bio-inspired materials Line defects Topological defects

Active nematics Living matter & active matter

Polymers & Soft Matter

Authors & Affiliations

Jack Binysh ¹, Žiga Kos², Simon Čopar ², Miha Ravnik^2,3,*, and Gareth P. Alexander ^4,†

¹Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom
²Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
³Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia
⁴Department of Physics and Centre for Complexity Science, University of Warwick, Coventry CV4 7AL, United Kingdom

^*miha.ravnik@fmf.uni-lj.si
^†G.P.Alexander@warwick.ac.uk

Article Text

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Supplemental Material

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References

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Issue

Vol. 124, Iss. 8 — 28 February 2020

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Images

Figure 1
Three-dimensional local profiles of defect lines and their flows. (a) A local director profile (green cylinders) corresponds to a path on the unit sphere connecting antipodal points. We consider minimal distortion profiles corresponding to half a great circle (thickened green line)—these are specified by a twist angle $β$ and a phase offset angle $α$ . (b) Selected examples of defect profiles of $+ 1 / 2$ and “pure twist” type. (c) The active flows (blue arrows) generated by the corresponding nematic defect profiles. We show the case of extensile activity ( $ζ > 0$ ); for contractile activity the direction of the flows reverses.
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Figure 2
Flows and dynamics of different active nematic defect loops with zero topological charge. (a) Defect loop surrounding a cylindrical splay domain (director shown in green). The top panel shows a schematic with the analytically predicted self-propulsion velocity (blue arrows), the middle panel shows the numerically calculated flows in the cell midplane, and the bottom panel shows the dynamics of the defect loop position and shape. (b) Defect loop surrounding a cylindrical bend domain with the same comparison between analytic predictions and full simulation as in (a). (c) Defect loop surrounding a cylindrical twist domain. Here the profile is of twist type ( $β = π / 2$ ) everywhere and $α$ makes two full rotations around the loop. The self-propulsion velocities contract the loop along $x$ , expand it along $y$ , and buckle it on its diagonals. Color on the middle panel velocity gives its $z$ component ( $+ z$ red, $- z$ blue). Time in all panels is given in units of the active timescale $τ_{ζ}$ (see Supplemental Material [28]).
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Figure 3
Statistics of defect loop profiles. (a) Typical snapshot of a spherical droplet in the turbulent regime, showing three defect loops with cross sections colored by twist angle $β$ . Insets show the variation of profiles for each loop in an equal-area projection of the ( $α$ , $β$ ) sphere. (b) Cumulative statistics of defect loop profiles (zero topological charge loops only). The majority are of twist type ( $β \approx π / 2$ ), with a maximum at tangential twist profiles ( $α \approx \pm π / 2$ ). (c) Correlation between local defect line curvature and twist angle $β$ (units are inverse active length scale $ξ_{ζ}^{- 1}$ ; see Supplemental Material [28]).
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Figure 4
Nucleation of zero topological charge active nematic defect loops. (a) A localized bend distortion in the director (inset, green) seeds a defect loop (red) via the generic bend instability. The nucleated loop has the same structure as that surrounding the splay domain shown in Fig. 2. (b) Seeds of different elastic distortions generate defect loops of the same type, emerging from regions of maximal bend by the same generic instability.
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