Orientational Correlations in Active and Passive Nematic Defects

D. J. G. Pearce, J. Nambisan, P. W. Ellis, A. Fernandez-Nieves, and L. Giomi

Phys. Rev. Lett. 127, 197801 – Published 2 November 2021

Abstract

We investigate the emergence of orientational order among $+ 1 / 2$ disclinations in active nematic liquid crystals. Using a combination of theoretical and experimental methods, we show that $+ 1 / 2$ disclinations have short-range antiferromagnetic alignment, as a consequence of the elastic torques originating from their polar structure. The presence of intermediate $- 1 / 2$ disclinations, however, turns this interaction from antialigning to aligning at scales that are smaller than the typical distance between like-sign defects. No long-range orientational order is observed. Strikingly, these effects are insensitive to material properties and qualitatively similar to what is found for defects in passive nematic liquid crystals.

Received 10 March 2021
Accepted 10 September 2021

DOI:https://doi.org/10.1103/PhysRevLett.127.197801

Physics Subject Headings (PhySH)

Active defects Active nematics Liquid crystals Living matter & active matter

Polymers & Soft Matter

Authors & Affiliations

D. J. G. Pearce^1,2, J. Nambisan ^3,4, P. W. Ellis⁵, A. Fernandez-Nieves^3,4,6,*, and L. Giomi ^7,†

¹Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02142, USA
²Departments of Biochemistry and Theoretical Physics, Université de Genéve, 1205 Genéve, Switzerland
³Department of Condensed Matter Physics, University of Barcelona, 08028 Barcelona, Spain
⁴School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
⁵John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
⁶ICREA-Institucio Catalana de Recerca i Estudis Avancats, 08010 Barcelona, Spain
⁷Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, Netherlands

^*a.fernandeznieves@ub.edu
^†giomi@lorentz.leidenuniv.nl

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Vol. 127, Iss. 19 — 5 November 2021

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Images

Figure 1
(a) Schematic representation of a planar nematic featuring a $+ 1 / 2$ disclination with polarity $p$ . (b) Typical configuration of a microtubule-kinesin-based active nematic. $+ 1 / 2$ and $- 1 / 2$ disclinations are indicated with red and blue arrows.
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Figure 2
Numerical Results. (a) Number of defects $N_{d}$ versus time in coarsening passive nematics. The horizontal lines mark configurations featuring $N_{d} = 300$ , 500, 700, 900 defects. Inset: snapshot of a typical configuration. (b) Scale-dependent defect nematic order parameter, showing the characteristic scaling behavior of systems with no long- or quasi-long-ranged order: $S_{d} (ℓ) \sim ℓ^{- 1}$ . Inset: polar histogram of the orientation of $N_{d} = 100$ , $+ 1 / 2$ defects. (c) Polar correlation function $C_{p p} (r) = ⟨ p (r) \cdot p (0) ⟩$ . Inset: the same function versus $r / r_{0}$ , with $r_{0} \sim 1 / \sqrt{N_{d}}$ the average defect spacing. The data collapse on the same master curve. (d) Topological charge density correlation functions for $C_{+ -}$ ( $C_{+ +}$ ), showing a local depletion (accumulation) of like-sign (opposite-sign) defects at short distances. Inset: the same functions versus $r / r_{0}$ . (e)–(h) Same quantities as in (a)–(d) but for an active nematic whose dynamics are governed by Eqs. (3). The prominently different behaviors of $C_{+ -}$ in the passive (d) and active (h) cases $r \approx 0$ , originate from the random creation of defect pairs and subsequent self-propulsion of $+ 1 / 2$ defects in active nematics, which temporarily results in $\pm 1 / 2$ pairs at short distances. In all numerical simulations we have set $λ = 0.1$ , $ρ = 10^{- 2} kg m^{- 2}$ , $η = 10^{- 4} kg s^{- 1}$ and $α = {- 6.25, - 12.50, - 25.00, - 50.00} \times 10^{- 2} kg s^{- 2}$ [28].
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Figure 3
Experimental results. (a) Number of defects versus time in a microtuble-kinesin-based active nematic. Each color represents a different experimental sample, grouped into two sets with different $N_{d}$ . (b) Scale-dependent nematic order parameter of the $+ 1 / 2$ defects. Inset: polar histogram of the orientation of the $+ 1 / 2$ defects. (c) Polar correlation function $C_{p p} = ⟨ p (r) \cdot p (0) ⟩$ . (d) Correlation functions $C_{+ -}$ and $C_{+ +}$ . Insets: same functions versus $r / r_{0}$ .
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Figure 4
(a),(b) Snapshots illustrating the (a) antiferromagnetic alignment between $+ 1 / 2$ defects, and (b) ferromagnetic alignment between $+ 1 / 2$ defects in the presence of an intermediate $- 1 / 2$ defect. These effects determine the orientational correlation, for $r < r_{0}$ , and anticorrelation, for $r > r_{0}$ , in Figs. 2 and 3. (c),(d) Example of configurations consisting of two $+ 1 / 2$ disclinations in the absence (c) and in the presence (d) of an intermediate $- 1 / 2$ disclination. (e) Elastic free energy of the configurations displayed in panels (c) and (d) as a function of the angular separation $Δ ψ$ between $+ 1 / 2$ defects. These are energetically favored to be antialigned while separated by a defect free patch (red curve) and aligned in the presence of an intermediate $- 1 / 2$ defect (blue curve).
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