Spontaneous chiral symmetry breaking in collective active motion

Rebekka E. Breier, Robin L. B. Selinger, Giovanni Ciccotti, Stephan Herminghaus, and Marco G. Mazza

Phys. Rev. E 93, 022410 – Published 18 February 2016

Abstract

Chiral symmetry breaking is ubiquitous in biological systems, from DNA to bacterial suspensions. A key unresolved problem is how chiral structures may spontaneously emerge from achiral interactions. We study a simple model of active swimmers in three dimensions that effectively incorporates hydrodynamic interactions. We perform large-scale molecular dynamics simulations (up to $10^{6}$ particles) and find long-lived metastable collective states that exhibit chiral organization although the interactions are achiral. We elucidate under which conditions these chiral states will emerge and grow to large scales. To explore the complex phase space available to the system, we perform nonequilibrium quenches on a one-dimensional Lebwohl-Lasher model with periodic boundary conditions to study the likelihood of formation of chiral structures.

Received 18 November 2014
Revised 8 January 2016

DOI:https://doi.org/10.1103/PhysRevE.93.022410

Physics Subject Headings (PhySH)

Low Reynolds number swimmers

Chiral symmetry Theories of collective dynamics & active matter

Physics of Living Systems

Authors & Affiliations

Rebekka E. Breier¹, Robin L. B. Selinger², Giovanni Ciccotti^3,4, Stephan Herminghaus¹, and Marco G. Mazza^1,*

¹Max Planck Institute for Dynamics and Self-Organization (MPIDS), Am Fassberg 17, 37077 Göttingen, Germany
²Liquid Crystal Institute, Kent State University, Kent, Ohio 44242, USA
³Department of Physics, University of Rome “La Sapienza”, Piazzale A. Moro 5, 00185 Rome, Italy
⁴School of Physics, University College Dublin, Belfield, Dublin 4, Ireland

^*Corresponding author: marco.mazza@ds.mpg.de

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Vol. 93, Iss. 2 — February 2016

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Images

Figure 1
Phase diagram of the system of self-propelled particles in terms of Péclet number $P$ and density $ρ$ . The color indicates the global nematic order parameter and shows the regions of the nematic and the isotropic phases (denoted by $•$ ). The symbol $★$ indicates a chiral pattern which developed in one specific simulation run and can arise everywhere in the nematic region of the phase diagram.
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Figure 2
Cross sections of a typical chiral configuration ( $ρ = 1.625, P = 0.08$ ). The color code represents the local nematic order parameter and the arrows represent the SPP. The ribbon in the bottom shows the helicoidal behavior of ${\hat{d}}^{} loc$ in different cross sections.
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Figure 3
(a) Components of ${\hat{d}}^{loc}$ along the helical axis ( $\hat{x}$ ) of the chiral pattern (see Fig. 2). The symbols are the components of ${\hat{d}}^{} loc$ (in slices normal to the helical axis) and the solid lines are sinusoidal least square fits $d_{y, z}^{loc} = cos (π x / L + ϕ_{y, z})$ . (b) Time evolution of the chiral order parameter for the same simulation. The transient evolution before reaching the steady state can extend up to $10^{4}$ time steps.
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Figure 4
Ideal splay (a), bend (b), and twist (c) deformations of the director field. The reference particle ( $★$ ) does not experience any torque from the neighboring particles ( $•$ ).
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Figure 5
Formation of the chiral pattern. (a) Sketch of two competing layers (each nematically aligned). Here, the angle between the two ${\hat{d}}^{} loc$ is $π / 2$ . (b) and (c) Example from the same simulation as in Figs. 2 and 3 at time $t = 2000$ time steps. The system is divided into $10^{3}$ boxes in each of which the fraction $f_{box}^{{y, z}}$ of particles with $|e_{y}| > 0.9$ (b) and $|e_{z}| > 0.9$ (c) is determined. Only boxes with $f_{box}^{{y, z}} > 0.15$ are plotted. $f_{box}^{x}$ can be defined accordingly. (d) and (e) Evolution of the fraction $f^{{x, y, z}}$ of boxes with $f_{box}^{{x, y, z}} > 0.15$ for the same chiral simulation (d) and a nematic example (e). The chiral structure shows two growing domains of similar size and different orientations, while only one orientation dominates in the nematic case.
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Figure 6
Inset: When annealed, the 1D Lebwohl-Lasher model can evolve into the untwisted ground state or a chiral metastable state with $\pm n$ twists. Main panel: Frequency of final states has a maximum value at $|n| = 1$ for a long chain with $N = 800$ (green $▿$ , fit as green dashed line), and at $|n| = 0$ for a short chain with $N = 100$ (blue $◊$ , fit as blue solid line). The lines are guides for the eye.
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