Universal Hydrodynamic Mechanisms for Crystallization in Active Colloidal Suspensions

Rajesh Singh and R. Adhikari

Phys. Rev. Lett. 117, 228002 – Published 23 November 2016

Abstract

The lack of detailed balance in active colloidal suspensions allows dissipation to determine stationary states. Here we show that slow viscous flow produced by polar or apolar active colloids near plane walls mediates attractive hydrodynamic forces that drive crystallization. Hydrodynamically mediated torques tend to destabilize the crystal but stability can be regained through critical amounts of bottom heaviness or chiral activity. Numerical simulations show that crystallization is not nucleational, as in equilibrium, but is preceded by a spinodal-like instability. Harmonic excitations of the active crystal relax diffusively but the normal modes are distinct from an equilibrium colloidal crystal. The hydrodynamic mechanisms presented here are universal and rationalize recent experiments on the crystallization of active colloids.

Received 17 June 2016

DOI:https://doi.org/10.1103/PhysRevLett.117.228002

Physics Subject Headings (PhySH)

Clustering Crystallization Emergence of patterns Microfluidics

Colloidal crystal Living matter & active matter Self-propelled particles

Condensed Matter, Materials & Applied PhysicsPolymers & Soft MatterStatistical Physics & Thermodynamics

Authors & Affiliations

Rajesh Singh^* and R. Adhikari^†

The Institute of Mathematical Sciences-HBNI, CIT Campus, Chennai 600113, India

^*rsingh@imsc.res.in
^†rjoy@imsc.res.in

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Vol. 117, Iss. 22 — 25 November 2016

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Images

Figure 1
Panels (a)–(c) are instantaneous configurations during the crystallization of 1024 active colloids of radius $b$ at a plane wall. The colloids are colored by their initial positions. Panels (d)–(f) show the structure factor $S (k)$ at corresponding instants. Wave numbers are scaled by the modulus of the reciprocal lattice vector $k_{0}$ and the contribution from $k = 0$ is discarded. Panel (g) shows the variation of the angular velocity $Ω_{c}$ of a crystallite with the number $N$ of colloids in it. A typical configuration is shown in the inset. Panel (h) is the state diagram for orientational stability in terms of the measure of chirality $V_{0}^{(3 a)}$ and bottom heaviness $T_{0}$ (see text). Each dot represents one simulation. Here, $v_{s}$ is the self-propulsion speed of an isolated colloid, $τ = b / v_{s}$ , and $ε$ is the scale of the repulsive steric potential.
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Figure 2
Distortion of the flow produced by leading polar $(l σ = 1 s)$ and apolar $(l σ = 2 s)$ slip terms in Eq. (2) as an active colloid of radius $b$ , shown in green, approaches a plane wall. Tracer colloids are show in white. The streamlines of fluid flow have been overlaid on a pseudocolor plot of logarithm of the magnitude of the local flow normalized by its maximum. The flow in (c) results when the colloid is brought to rest near the wall. Hydrodynamic forces attract nearby colloids, as shown by thick white arrows, leading to crystallization. Hydrodynamic torques tend to reorient the colloids as shown by curved red arrows. The remaining graphs show quantitative variation of active forces and torques from modes in Eq. (2) scaled by $F_{A} = 6 π η b v_{s}$ and $T_{A} = 8 π η b^{2} v_{s}$ respectively as a function of height $h$ of the colloid from the wall and distance, $r_{i j} = R_{i} - R_{j}$ , from other colloids. Solid and dotted lines represent analytical and numerical results, respectively (see text). Here, $∥$ and $⊥$ indicate directions parallel and perpendicular to the wall at $z = 0$ .
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Figure 3
Branches of the dispersion relation for the two planar normal modes of relaxation of a hexagonal active crystal. The curves in the upper panel show the dispersion along high symmetry directions of the Brillouin zone (first inset). The surfaces in the second and third insets show the dispersion over the entire Brillouin zone. Polar plots in the lower panel have comparisons of the full numerical solution of Eq. (6) with the approximate solution at small $k$ of Eq. (7) for $k = 0.01 k_{0}$ (left panel) and $k = 0.3 k_{0}$ (right panel).
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