Anisotropic velocity statistics of topological defects under shear flow

Active smectics on a sphere

The dynamics of active smectic liquid crystals confined on a spherical surface is explored through an active phase field crystal model. Starting from an initially randomly perturbed isotropic phase, several types of topological defects are spontaneously formed, and then annihilate during a coarsening process until a steady state is achieved. The coarsening process is highly complex involving several scaling laws of defect densities as a function of time where different dynamical exponents can be identified. In general the exponent for the final stage towards the steady state is significantly larger than that in the passive and in the planar case, i.e. the coarsening is getting accelerated both by activity and by the topological and geometrical properties of the sphere. A defect type characteristic for this active system is a rotating spiral of evolving smectic layering lines. On a sphere this defect type also determines the steady state. Our results can in principle be confirmed by dense systems of synthetic or biological active particles.

Inelastic rotations and pseudoturbulent plastic avalanches in crystals

Plastic deformations in crystals produce microstructures with randomly oriented patches of unstressed lattice forming complex textures. We use a mesoscopic Landau-type tensorial model of crystal plasticity to show that in such textures rotations can originate from crystallographically exact microslips which self organize in the form of laminates of a pseudotwin type. The formation of such laminates can be viewed as an effective internal "wrinkling" of the crystal lattice. While such "wrinkling" disguises itself as an elastically neutral rotation, behind it is inherently dissipative, dislocation-mediated process. Our numerical experiments reveal pseudoturbulent effective rotations with power-law distributed spatial correlations which suggests that the process of dislocational self-organization is inherently unstable and points toward the necessity of a probabilistic description of crystal plasticity.

Robustness of topological defects in discrete domains

Topological defects are singular points in vector fields, important in applications ranging from fingerprint detection to liquid crystals to biomedical imaging. In discretized vector fields, topological defects and their topological charge are identified by finite differences or finite-step paths around the tentative defect. As the topological charge is (half) integer, it cannot depend continuously on each input vector in a discrete domain. Instead, it switches discontinuously when vectors change beyond a certain amount, making the analysis of topological defects error prone in noisy data. We improve existing methods for the identification of topological defects by proposing a robustness measure for (i) the location of a defect, (ii) the existence of topological defects and the total topological charge within a given area, (iii) the annihilation of a defect pair, and (iv) the formation of a defect pair. Based on the proposed robustness measure, we show that topological defects in discrete domains can be identified with optimal trade-off between localization precision and robustness. The proposed robustness measure enables uncertainty quantification for topological defects in noisy discretized nematic fields (orientation fields) and polar fields (vector fields).

System size dependence of the structure and rheology in a sheared lamellar liquid crystalline medium

The structural and rheological evolution of an initially disordered lamellar phase system under a shear flow is examined using a mesoscale model based on a free energy functional for the concentration field, which is the scaled difference in the concentration between the hydrophilic and hydrophobic components. The dimensionless numbers which affect the shear evolution are the Reynolds number (γ˙¯L²/ν), the Schmidt number (ν/D), a dimensionless parameter Σ=(Aλ²/ρν²), a parameter μ_r which represents the viscosity contrast between the hydrophilic and hydrophobic components, and (L/λ), the ratio of system size and layer spacing. Here, ρ, ν, and D are the density, kinematic viscosity (ratio of viscosity and density), and the mass diffusivity, and A is the energy density in the free energy functional which is proportional to the compression modulus. Two distinct modes of structural evolution are observed for moderate values of the parameter Σ depending only on the combination ScΣ and independent of system size. For ScΣ less than about 10, the layers tend to form before they are deformed by the mean shear, and layered but misaligned domains are initially formed, and these are deformed and rotated by the flow. In this case, the excess viscosity (difference between the viscosity and that for an aligned state) does not decrease to zero even after 1000 strain units, but appears to plateau to a steady state value. For ScΣ greater than about 10, layers are deformed by the mean shear before they are fully formed, and a well aligned lamellar phase with edge dislocation orders completely due to the cancellation of dislocations. The excess viscosity scales as t^-1 in the long time limit. The maximum macroscopic viscosity (ratio of total stress and average strain rate over the entire sample) during the alignment process increases with the system size proportional to (L/λ)^3/2. For large values of Σ, there is localisation of shear at the walls, and the bulk of the sample moves as a block. The thickness of the shearing region appears to be invariant with the system size, leading to an increase of viscosity proportional to L. The time for structural evolution is found to be the inverse of the strain rate γ˙^-1. In the case of a significant viscosity contrast between the hydrophilic and hydrophobic parts, the average viscosity increases by 1-2 orders of magnitude due to the defect pinning mechanism, where the regions between defects move as a block, and shear localisation at the wall.

Structure-rheology relationship in a sheared lamellar fluid

The structure-rheology relationship in the shear alignment of a lamellar fluid is studied using a mesoscale model which provides access to the lamellar configurations and the rheology. Based on the equations and free energy functional, the complete set of dimensionless groups that characterize the system are the Reynolds number (ργL(2)/μ), the Schmidt number (μ/ρD), the Ericksen number (μγ/B), the interface sharpness parameter r, the ratio of the viscosities of the hydrophilic and hydrophobic parts μ(r), and the ratio of the system size and layer spacing (L/λ). Here, ρ and μ are the fluid density and average viscosity, γ is the applied strain rate, D is the coefficient of diffusion, B is the compression modulus, μ(r) is the maximum difference in the viscosity of the hydrophilic and hydrophobic parts divided by the average viscosity, and L is the system size in the cross-stream direction. The lattice Boltzmann method is used to solve the concentration and momentum equations for a two dimensional system of moderate size (L/λ=32) and for a low Reynolds number, and the other parameters are systematically varied to examine the qualitative features of the structure and viscosity evolution in different regimes. At low Schmidt numbers where mass diffusion is faster than momentum diffusion, there is fast local formation of randomly aligned domains with "grain boundaries," which are rotated by the shear flow to align along the extensional axis as time increases. This configuration offers a high resistance to flow, and the layers do not align in the flow direction even after 1000 strain units, resulting in a viscosity higher than that for an aligned lamellar phase. At high Schmidt numbers where momentum diffusion is fast, the shear flow disrupts layers before they are fully formed by diffusion, and alignment takes place by the breakage and reformation of layers by shear, resulting in defects (edge dislocations) embedded in a background of nearly aligned layers. At high Ericksen number where the viscous forces are large compared to the restoring forces due to layer compression and bending, shear tends to homogenize the concentration field, and the viscosity decreases significantly. At very high Ericksen number, shear even disrupts the layering of the lamellar phase. At low Ericksen number, shear results in the formation of well aligned layers with edge dislocations. However, these edge dislocations take a long time to anneal; the relatively small misalignment due to the defects results in a large increase in viscosity due to high layer stiffness and due to shear localization, because the layers between defects get pinned and move as a plug with no shear. An increase in the viscosity contrast between the hydrophilic and hydrophobic parts does not alter the structural characteristics during alignment. However, there is a significant increase in the viscosity, due to pinning of the layers between defects, which results in a plug flow between defects and a localization of the shear to a part of the domain.

Vortex clustering and universal scaling laws in two-dimensional quantum turbulence

We investigate numerically the statistics of quantized vortices in two-dimensional quantum turbulence using the Gross-Pitaevskii equation. We find that a universal -5/3 scaling law in the turbulent energy spectrum is intimately connected with the vortex statistics, such as number fluctuations and vortex velocity, which is also characterized by a similar scaling behavior. The -5/3 scaling law appearing in the power spectrum of vortex number fluctuations is consistent with the scenario of passive advection of isolated vortices by a turbulent superfluid velocity generated by like-signed vortex clusters. The velocity probability distribution of clustered vortices is also sensitive to spatial configurations, and exhibits a power-law tail distribution with a -5/3 exponent.

Intermittent dislocation density fluctuations in crystal plasticity from a phase-field crystal model

Plastic deformation mediated by collective dislocation dynamics is investigated in the two-dimensional phase-field crystal model of sheared single crystals. We find that intermittent fluctuations in the dislocation population number accompany bursts in the plastic strain-rate fluctuations. Dislocation number fluctuations exhibit a power-law spectral density 1/f2 at high frequencies f. The probability distribution of number fluctuations becomes bimodal at low driving rates corresponding to a scenario where low density of defects alternates at irregular times with high populations of defects. We propose a simple stochastic model of dislocation reaction kinetics that is able to capture these statistical properties of the dislocation density fluctuations as a function of shear rate.

Dislocation-mediated growth of bacterial cell walls

Recent experiments have illuminated a remarkable growth mechanism of rod-shaped bacteria: proteins associated with cell wall extension move at constant velocity in circles oriented approximately along the cell circumference [Garner EC, et al., (2011) Science 333:222-225], [Domínguez-Escobar J, et al. (2011) Science 333:225-228], [van Teeffelen S, et al. (2011) PNAS 108:15822-15827]. We view these as dislocations in the partially ordered peptidoglycan structure, activated by glycan strand extension machinery, and study theoretically the dynamics of these interacting defects on the surface of a cylinder. Generation and motion of these interacting defects lead to surprising effects arising from the cylindrical geometry, with important implications for growth. We also discuss how long range elastic interactions and turgor pressure affect the dynamics of the fraction of actively moving dislocations in the bacterial cell wall.