Active nematics are out-of-equilibrium fluids composed of rod-like subunits, which can generate large-scale, self-driven flows. We examine a microtubule-kinesin-based active nematic confined to two dimensions, exhibiting chaotic flows with moving topological defects. Applying tools from chaos theory, we investigate self-driven advection and mixing on different length scales. Local fluid stretching is quantified by the Lyapunov exponent. Global mixing is quantified by the topological entropy, calculated from both defect braiding and curve extension rates. We find excellent agreement between these independent measures of chaos, demonstrating that the extensile stretching between microtubules directly translates into macroscopic braiding of positive defects. Remarkably, increasing extensile activity (through ATP concentration) does not increase the dimensionless topological entropy. This study represents an application of chaotic advection to the emerging field of active nematics and quantification of the collective motion of an ensemble of defects (through topological entropy) in a liquid crystal.
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The data that support the findings of this study are available from the corresponding authors on reasonable request.
The MATLAB code for computing nematic director fields and topological defects is available on request from K.A.M. The E-tec code is available in Python from S.A.S. (firstname.lastname@example.org) on request.
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Our group is grateful to Z. Dogic for the generous contribution of microtubules and molecular motors and to L. Lemma for sample preparation. We also acknowledge useful discussions with S. Sindi. The authors acknowledge generous funding from the National Science Foundation, through several awards (DMR-1808926), NSF-CREST: Center for Cellular and Biomolecular Machines at UC Merced (HRD-1547848), and from the Brandeis Biomaterials facility MRSEC-1420382.
The authors declare no competing interests.
Peer review information: Nature Physics thanks Jordi Ignes-Mullol, Idan Tuval and Julia Yeomans for their contribution to the peer review of this work.
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Supplementary material and Supplementary Figs. 1–6.
Fluorescence microscopy video of the microtubule/kinesin active nematic at 50 µM ATP concentration.
Fluorescence microscopy video of the active nematic with topological defects tracked at 50 µM ATP concentration. +1/2 defects are shown by white circles and –1/2 defects are shown by yellow triangles.
Optical microscopy bright-field video of the active nematic with tracked beads marked in blue and numbered. 50 µM ATP concentration.
Optical microscopy bright-field video of the active nematic at 50 µM ATP concentration with tracked beads marked and numbered. A nematic contour growing between a pair of tracked beads is shown in blue.
Fluorescence microscopy video of the active nematic at 50 µM ATP concentration with topological defects marked and tracked. +1/2 defects are shown by white circles and –1/2 defects are shown by yellow triangles. A nematic contour growing between a pair of tracked defects is shown in blue.
Braiding motion of the tracked +1/2 defects with E-tec triangulation showing the growth of the stretched mesh (red lines). 50 µM ATP concentration.
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Tan, A.J., Roberts, E., Smith, S.A. et al. Topological chaos in active nematics. Nat. Phys. 15, 1033–1039 (2019). https://doi.org/10.1038/s41567-019-0600-y
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