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Benno
Liebchen
*^{a},
Michael E.
Cates
^{b} and
Davide
Marenduzzo
^{a}
^{a}SUPA, School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, UK. E-mail: Benno.Liebchen@staffmail.ed.ac.uk
^{b}DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, UK

Received
19th May 2016
, Accepted 6th August 2016

First published on 8th August 2016

We demonstrate that active rotations in chemically signalling particles, such as autochemotactic E. coli close to walls, create a route for pattern formation based on a nonlinear yet deterministic instability mechanism. For slow rotations, we find a transient persistence of the uniform state, followed by a sudden formation of clusters contingent on locking of the average propulsion direction by chemotaxis. These clusters coarsen, which results in phase separation into a dense and a dilute region. Faster rotations arrest phase separation leading to a global travelling wave of rotors with synchronized roto-translational motion. Our results elucidate the physics resulting from the competition of two generic paradigms in active matter, chemotaxis and active rotations, and show that the latter provides a tool to design programmable self-assembly of active matter, for example to control coarsening.

Various self-propelled microorganisms, like the bacterium E. coli

Currently, the KS instability is attracting renewed attention in active synthetic Janus colloids. These particles catalyse reactions within a chemical bath, yielding gradients which drive self-propulsion via diffusiophoresis or a similar mechanism. Importantly, other colloids may feel rotational torques caused by the same long-ranged gradients and respond to them by adapting their swimming direction, thereby providing a synthetic analog of chemotactic signalling.^{9–11} The same KS equations can therefore be transferred from the microbiological world to phoretic colloids.^{10,12–14} Here, either the KS instability, or instabilities based on chemorepulsion^{15} may help explain the still elusive dynamic clustering observed in experiments.^{10,16,17}

In all these cases chemotactic instability relies on the ability of weak chemical fluctuations around the uniform density to align microswimmers up (or down) chemical gradients. However, under many circumstances microswimmers vary their swimming direction autonomously of chemical cues: this occurs, e.g., for bacteria swimming clockwise close to a glass wall,^{18} or anti-clockwise near an oil–water interface^{19} (see ESI† for a discussion on parameters). Synthetic examples of active signalling rotors with self-propulsion (sometimes called ‘circle swimmers’) may be realised with L-shaped phoretic swimmers,^{20,21} or with active particles with dipole moments^{16,22,23} which will track the rotation of applied external fields^{16,24} and can interact via self-produced phoretic fields.^{16}

We should expect that, close to uniform states, finite active torques as caused by intrinsic rotations will generally outcompete chemotactic torques which are proportional to the chemical gradient, when it comes to determining swimming direction: hence, they might generically lead to a breakdown of the linear KS instability. Does this rule out the possibility of phase separation and patterning in signalling rotors? This could have profound consequences for systems like thin (bio)films of chemotactic bacteria^{25} or actuated phoretic particles.

Here, we propose a generic model to study active signalling rotors. As a key result we find that even weak active rotations suppress the linear KS instability. However, we identify an instability mechanism which is distinct from the KS mechanism and creates a nonlinear, but deterministic, route to pattern formation.

For small rotation frequency, this route generates macroscopic phase separation: a fraction of the rotors condenses into a dense phase, separated from a dilute gas by a quasi-stationary interface where chemotaxis suppresses rotations and ‘locks’ the average swimming direction onto the upward density gradient. Away from the interfaces, rotations persist in both phases and lead to dynamic patterns, such as spots or moving stripes and spirals (Fig. 2h and i) This combination of phase separation with pattern formation within both phases represents a novel type of hierarchical structure formation. When increasing the rotation frequency above a certain threshold, strikingly, the growing clusters do not coarsen any more but form a global pattern of travelling stripes with self-limiting size, suggesting that active rotations can be used to control coarsening in suspensions of self-propelled particles (Fig. 2l).

We describe active particles which self-propel with velocity v_{0} and rotate with frequencies ω_{i} > 0 at a coarse grained level in two-dimensions. A smooth ρ(x,t) represents the active particle density and p(x,t) is the local average of the unit vector describing the direction of self-propulsion. We focus on polarized rotors with identical frequencies, which applies to synthetic colloidal rotors locked to a rotating field, but also to bacterial rotors where short-ranged alignment interactions can synchronize the individual rotations locally (see ESI† and final paragraph). Thus, in absence of signalling, p rotates with a collective frequency ω. Generally however, our rotors produce signalling molecules with a local rate k_{0}ρ; hence p also responds, via chemotaxis, to gradients of the resulting chemical field c(x,t), which is degraded with a rate k_{d} (compare ref. 5 and 26). This yields a competition of intrinsic active rotations and chemotactic alignment which determines the behaviour of the signalling rotors at large scales. Allowing for finite diffusion of the chemical and colloidal density fields with coefficients D_{c} and D_{ρ}, we describe signalling rotors phenomenologically by:

= −v_{0}∇·(ρp) + D_{ρ}∇^{2}ρ + K∇^{2}ρ^{3} | (1) |

= ω + β × ∇c; p = (cosϕ, sinϕ)^{T} | (2) |

ċ = k_{0}ρ − k_{d}c + D_{c}∇^{2}c + ε(c_{0} − c)^{3}. | (3) |

We now explore the competition of active rotations and chemotaxis by solving eqn (1)–(3) for different Δ on a square lattice with L_{x} × L_{y} grid points and periodic boundary conditions using finite difference methods. As an initial state, we choose a small and random perturbation of the spatially uniform and coherently rotating state (ρ,ϕ,c) = (ρ_{0},ωt,(k_{0}/k_{d})ρ_{0}) which solves eqn (1)–(3). In absence of rotations (Δ = 0) we observe clusters growing out of the uniform state (Fig. 2a), which colocalise with chemical density maxima and coarsen at long times (Fig. 2b), yielding one dense macroscopic cluster coexisting with a dilute gas (Fig. 2c). Here, instability of the uniform state is expected due to the positive feedback loop of particle aggregation and chemical production we discussed in the introduction and in ref. 15.

For Δ = 5 × 10^{−3} this picture changes dramatically. Now the uniform state persists for a certain duration (see Videos 1 and 2 in ESI†): we call this the initial lag regime. Then, almost suddenly, fluctuations ‘awake’ and grow to create clusters (Fig. 2d). These clusters coarsen into a relatively dense phase separated from a dilute rotor gas (Fig. 2f) by a slowly-moving interface where rotations are suppressed and particles swim, on average, up the chemical gradient (Video 2, ESI†). Away from the interface, all colloids perform a ‘stop-and-go’ rotation, which is associated with dynamic short-lived density dips and peaks (white spots and red dips in yellow region in Fig. 2e and f). Choosing stronger rotations (Δ = 0.024) we observe a similar suppression of rotations at the interfaces, but within both phases stripe and spiral patterns form (Fig. 2g–i and Video 2, ESI†).

For Δ = 0.044 complexity suddenly breaks down: after a long initial lag regime and an intermediate regime where short-lived dynamic clusters continuously emerge and decay, we find a travelling wave made of straight parallel stripes (Fig. 2k and l, Video 4, ESI†) with self-limiting wavelength. This sudden emergence of a length scale upon increasing Δ allows control of coarsening via active rotations. The length scale of the stripe pattern which decreases as ω increases and thus can be also controlled.

= ω + β|∇c|sin(ϕ + δ); δ = arg(−∂_{x}c + i∂_{y}c) | (4) |

This raises the question why we could still observe cluster growth in our simulations. The answer is that chemotaxis can abduct the rotors into the nonlinear regime before they complete a full rotation, and this in turn can lead to a nonlinear instability. To understand the underlying physical mechanism, let us reconsider our coherently rotating uniform initial state, where all rotors are in phase (Fig. 1c). While ρ, c remain approximately uniform in the course of the early-stage dynamics, weak fluctuations of c continuously dephase the orientation field p due to chemotaxis (Fig. 1d). Once the rotors are sufficiently out of phase, they can form, temporarily, aster-like ‘seed’ configurations (Fig. 1e), where all particles in the vicinity of a positive fluctuation of c swim up the chemical gradient. This configuration promotes, temporarily, a growth of the fluctuation via the standard KS feedback loop. If this growth generates a chemical gradient surpassing the locking-threshold (β|∇c| > ω) before the seed decays, then p remains locked in its vicinity – leading to a stable cluster as just discussed.^{35} Hence, we call our nonlinear instability the ‘locking instability’. Whether the locking instability is effective or not in practice is a matter of competing timescales, between the instantaneous growth rate of chemical gradients and ω – the former needs to dominate to create locking. Video 5 in the ESI† shows the early stage dynamics of c and p, reflecting that c and its gradients can indeed grow beyond the locking threshold on timescales where p rotates only slightly.

This process of forming short lived clusters which prevent their decay by merging and moving into a collective direction (spontaneous symmetry breaking) constitutes a second nonlinear instability mechanism of the uniform state that is independent of the locking mechanism. In contrast with the latter, this mechanism works at all frequencies, but is comparatively slow since it involves the coordination of several clusters. Hence this second route to structure formation in signalling rotors is particularly relevant in parameter regimes where the locking mechanism is ineffective, i.e., for large ω, but applies similarly to the moving stripes formed within the dense and the dilute phase in Fig. 2g–i.

As its most remarkable feature and contrasting the locking instability, this route to pattern formation introduces a length scale in steady state. Within a large region of parameter space, this length scale increases approximately as v_{0}/ω. Hence, for synthetic rotors this length scale can be tuned via the frequency of an applied rotating field. We emphasize that this length scale is determined purely dynamically and cannot be calculated via the standard tools for linear (supercritical) instabilities like amplitude equations.^{32}

(5) |

In conclusion, although even weak active rotations linearly stabilise the uniform phase in ensembles of auto-chemotactic particles, they generate a nonlinear route to structure formation. This route creates novel patterns including hierarchically organized states combining phase separation and pattern formation. It also allows for features which would be impossible to achieve in linear instability scenarios, such as a delayed onset of patterning whose lag time can be programmed via the initial conditions. More generally, we showed that rotations provide a versatile new tool to design self-assembly and collective behaviour of active matter, for example to control coarsening.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6sm01162d |

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