Defect dynamics in active nematics

Abstract

Topological defects are distinctive signatures of liquid crystals. They profoundly affect the viscoelastic behaviour of the fluid by constraining the orientational structure in a way that inevitably requires global changes not achievable with any set of local deformations. In active nematic liquid crystals, topological defects not only dictate the global structure of the director, but also act as local sources of motion, behaving as self-propelled particles. In this article, we present a detailed analytical and numerical study of the mechanics of topological defects in active nematic liquid crystals.

Keywords: active liquid crystals; chaotic dynamics; self-propelled particles; topological defects.

Figure 1.

Schematic of the region where an active nematic is linearly unstable to splay (yellow/light) and to bend (orange/dark) fluctuations in the plane of the alignment parameter λ and the activity . The unstable regions are bounded by the critical activity given in equation (2.5). Flow tumbling extensile nematic with |λ|<1 are unstable to bend when active stresses are extensile (α<0) and to splay when active stresses are contractile (α>0). Conversely, strongly flow aligning (|λ|≫1) is unstable to splay when active stresses are extensile (α<0) and to bend when active stresses are contractile (α>0). (Online version in colour.)

Figure 2.

Example of a (a) and (b) disclination. The solid red lines are tangent to the director field with θ_d=kϕ and . The background shows the active backflow associated with the disclinations and obtained by solving the Stokes equation (3.6) with no-slip boundary conditions on a circle (dashed black line). The intensity of the background colour is proportional to the magnitude of the flow velocity. The white streamlines are given by equation (3.9). (Online version in colour.)

Figure 3.

Snapshots of a disclination pair shortly after the beginning of relaxation. (top) Director field (black lines) superimposed on a heat map of the nematic order parameter and (bottom) flow field (arrows) superimposed on a heat map of the concentration for a contractile system with α=0.8 (a,c) and an extensile system with α=−0.8 (b,d). In the top panel, the colour denotes the magnitude of the nematic-order parameter S relative to its equilibrium value . In the bottom panel, the colour denotes the magnitude of the concentration c relative to the average value c₀=2c^⋆. Depending on the sign of α, the backflow tends to speed up (α>0) or slow down (α<0) the annihilation process by increasing or decreasing the velocity of the disclination. For α negative and sufficiently large in magnitude, the defect reverses its direction of motion (d) and escapes annihilation. (Online version in colour.)

Figure 4.

Schematic of the effective attractive/repulsive interaction promoted by the active backflow. Depending on the sign of the active stress α, disclinations self-propel in the direction of their ‘tail’ (contractile) or ‘head’ (extensile). Based on the mutual orientation of the defects, this can lead to an attractive or repulsive interaction. (Online version in colour.)

Figure 5.

Defect pair production in an active suspension of microtubules and kinesin (top panels) and the same phenomenon observed in our numerical simulation of an extensile nematic fluid with γ=100 and α=−2. The experimental pictures are reprinted with permission from T. Sanchez et al., Nature (London) 491, 431 (2012). Copyright • 2012, Macmillan. (Online version in colour.)

Figure 6.

Defect trajectories and annihilation times obtained from a numerical integration of equations (2.1) for various γ and α values. The initial condition is a pair of defects with the relative orientation shown in figure 3 and separation L/2. (a) Defect trajectories for γ=5 and various α values (indicated in the plot). The upper (red) and lower (blue) curves correspond to the positive and negative disclination, respectively. The defects annihilate where the two curves merge. (b) The same plot for γ=10. Slowing down the relaxational dynamics of the nematic phase increases the annihilation time and for α=−0.8 reverses the direction of motion of the disclination. (c) Defect separation as a function of time for α=0.8 and various γ values. (d) Annihilation time normalized by the corresponding annihilation time obtained at α=0 (i.e. ). The line is a fit to with t_a given by equation (4.6). As described in the text this expression only depends on the dimensionless parameter . The fit is obtained with . (Online version in colour.)

Figure 7.

Phase diagram showing the various flow regimes of an active nematic obtained by varying activity α and rotational viscosity γ for both contractile (α>0) and extensile (α<0) systems. The dashed lines bounding the region where the homogeneous ordered state (H) is stable are the boundaries of linear stability given in equation (2.5). With increasing activity, the system exhibits relaxation oscillations (O), non-periodic oscillations characterized by the formation and unzipping of walls (W), and turbulence (T). (Online version in colour.)

Figure 8.

Dynamical states obtained from a numerical integration of equations (2.1) with γ=20 and various values of activity for an extensile system. (a) Average nematic-order parameter versus time. The dashed line for α=−0.3 identifies the relaxation oscillations regime with the labels (b), (c) and (d) marking the times corresponding to the snapshots on the top-left panel. The solid red line for α=−0.8 indicates the non-periodic oscillatory regimes characterized by the formation of walls (e) and the unzipping of walls through the unbinding of defect pairs: (f) and (g). The symbols filled circles and open triangles mark the positions of and disclinations respectively. The dotted blue line for α=−1.2 corresponds to the turbulent regime in which defects proliferate: (h), (i) and (j). In all the snapshots, the background colours are set by the magnitude of the vorticity ω and the order parameter S rescaled by the equilibrium value , whereas the solid lines indicate velocity (top) and director field (bottom). Movies displaying the time evolution of each state are included as the electronic supplementary material. (Online version in colour.)

Figure 9.

A magnification of the snapshot of figure 8f, showing the creation of a (filled circle) and a (open triangle) defect pair along a wall or α=−0.8 and γ=20. The black arrows indicate the flow velocity, while the background colour is related with the local vorticity. The wall is also the boundary between a pair of vortices of opposite circulation. The flow field of opposite-signed vortices adds at the wall, yielding a region of high shear that promotes defect unbinding. (Online version in colour.)

Figure 10.

Defects area fraction Nπa²/L² as a function of the active Ericksen number Er_α=αγL²/(ηK) for contractile (symbols in red tones) and extensile (symbols in blue tones) systems. In both cases, the area fraction saturates when the activity increase is compensated by a drop of the order parameter which effectively reduces the injected active stress. (Online version in colour.)