The production of strings (disclination lines and loops) has been observed by means of the Kibble mechanism of domain (bubble) formation in the isotropic-nematic phase transition of the uniaxial nematic liquid crystal 4-cyano-4'-n-pentylbiphenyl. The number of strings formed per bubble is about 0.6. This value is in reasonable agreement with a numerical simulation of the experiment in which the Kibble mechanism is used for the order parameter space of a uniaxial nematic liquid crystal.

The Cosmological Kibble Mechanism in the Laboratory: String Formation in Liquid Crystals

Life at Low Reynolds Number

Concentrated suspensions of swimming microorganisms and other forms of active matter are known to display complex, self-organized spatiotemporal patterns on scales that are large compared with those of the individual motile units. Despite intensive experimental and theoretical study, it has remained unclear the extent to which the hydrodynamic flows generated by swimming cells, rather than purely steric interactions between them, drive the self-organization. Here we use the recent discovery of a spiral-vortex state in confined suspensions of Bacillus subtilis to study this issue in detail. Those experiments showed that if the radius of confinement in a thin cylindrical chamber is below a critical value, the suspension will spontaneously form a steady single-vortex state encircled by a counter-rotating cell boundary layer, with spiral cell orientation within the vortex. Left unclear, however, was the flagellar orientation, and hence the cell swimming direction, within the spiral vortex. Here, using a fast simulation method that captures oriented cell–cell and cell–fluid interactions in a minimal model of discrete particle systems, we predict the striking, counterintuitive result that in the presence of collectively generated fluid motion, the cells within the spiral vortex actually swim upstream against those flows. This prediction is then confirmed by the experiments reported here, which include measurements of flagella bundle orientation and cell tracking in the self-organized state. These results highlight the complex interplay between cell orientation and hydrodynamic flows in concentrated suspensions of microorganisms.

https://absimage.aps.org/image/MAR16/MWS_MAR16-2015-005717.pdf

Fluid flows created by swimming bacteria drive self-organization in confined suspensions

On computable numbers

Active matter encompasses different nonequilibrium systems in which individual constituents convert energy into non-conservative forces or motion at the microscale. This review provides an elementary introduction to the role of topology in active matter through experimentally relevant examples. Here, the focus lies on topological defects and topologically protected edge modes with an emphasis on the distinctive properties they acquire in active media. These paradigmatic examples represent two physically distinct classes of phenomena whose robustness can be traced to a common mathematical origin: the presence of topological invariants. These invariants are typically integer numbers that cannot be changed by continuous deformations of the relevant order parameters or physical parameters of the underlying medium. We first explain the mechanisms whereby topological defects self propel and proliferate in active nematics, leading to collective states which can be manipulated by geometry and patterning. Possible implications for active microfluidics and biological tissues are presented. We then illustrate how the propagation of waves in active fluids and solids is affected by the presence of topological invariants characterizing their dispersion relations. We discuss the relevance of these ideas for the design of robotic metamaterials and the properties of active granular and colloidal systems. Open theoretical and experimental challenges are presented as future research prospects.

Topological active matter

The formation of emulsions from multiple immiscible fluids is governed by classical concepts such as surface tension, differential chemical affinity and viscosity, and the action of surface-active agents. Much less is known about emulsification when one of the components is active and thus inherently not constrained by the laws of thermodynamic equilibrium. We demonstrate one such realization consisting in the encapsulation of an active liquid crystal (LC)–like gel, based on microtubules and kinesin molecular motors, into a thermotropic LC. These active nematic emulsions exhibit a variety of dynamic behaviors that arise from the cross-talk between topological defects separately residing in the active and passive components. Using numerical simulations, we show a feedback mechanism by which active flows continuously drive the passive defects that, in response, resolve the otherwise degenerated trajectories of the active defects. Our experiments show that the choice of surfactant, which stabilizes the active/passive interface, allows tuning the regularity of the self-sustained dynamic events. The hybrid active-passive system demonstrated here provides new perspectives for dynamic self-assembly driven by an active material but regulated by the equilibrium properties of the passive component.

/pdf/active-nematic-emulsions-36dw4h4vf6.pdf

Active nematic emulsions

Topological defects are singularities in material fields that play a vital role across a range of systems: from cosmic microwave background polarization to superconductors and biological materials. Although topological defects and their mutual interactions have been extensively studied, little is known about the interplay between defects in different fields-especially when they coevolve-within the same physical system. Here, using nematic microfluidics, we study the cross-talk of topological defects in two different material fields-the velocity field and the molecular orientational field. Specifically, we generate hydrodynamic stagnation points of different topological charges at the center of star-shaped microfluidic junctions, which then interact with emergent topological defects in the orientational field of the nematic director. We combine experiments and analytical and numerical calculations to show that a hydrodynamic singularity of a given topological charge can nucleate a nematic defect of equal topological charge and corroborate this by creating [Formula: see text], [Formula: see text], and [Formula: see text] topological defects in four-, six-, and eight-arm junctions. Our work is an attempt toward understanding materials that are governed by distinctly multifield topology, where disparate topology-carrying fields are coupled and concertedly determine the material properties and response.

Cross-talk between topological defects in different fields revealed by nematic microfluidics.

We describe the flows and morphological dynamics of topological defect lines and loops in three-dimensional active nematics and show, using theory and numerical modeling, that they are governed by the local profile of the orientational order surrounding the defects. Analyzing a continuous span of defect loop profiles, ranging from radial and tangential twist to wedge 
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 profiles, we show that the distinct geometries can drive material flow perpendicular or along the local defect loop segment, whose variation around a closed loop can lead to net loop motion, elongation, or compression of shape, or buckling of the loops. We demonstrate a correlation between local curvature and the local orientational profile of the defect loop, indicating dynamic coupling between geometry and topology. To address the general formation of defect loops in three dimensions, we show their creation via bend instability from different initial elastic distortions.

Three-dimensional active defect loops

Active nematics contain topological defects that (under sufficient activity) move, create, and annihilate in a chaotic quasi-steady state, called active turbulence. However, understanding active defects under confinement is an open challenge, especially in three dimensions. Here, we demonstrate the topology of three-dimensional active nematic turbulence under the spherical confinement, using numerical modeling. In such spherical droplets, we show the three-dimensional structure of the topological defects, which, because of closed confinement, emerge in the form of closed loops or surface-to-surface spanning line segments. In the turbulent regime, the defects are shown to be strongly spatially and time varying, with ongoing transformations between positive winding, negative winding, and twisted profiles, and with defect loops of zero and nonzero topological charge. The timeline of the active turbulence is characterized by four types of bulk topology-linked events—breakup, annihilation, coalescence, and crossover of the defects—which we note could be used for the analysis of the active turbulence in different three-dimensional geometries. The turbulent regime is separated by a first-order structural transition from a low activity regime of a steady-state vortex structure and an offset single point defect. We also demonstrate coupling of surface and bulk topological defect dynamics by changing from strong perpendicular to in-plane surface alignment. More generally, this work aims to provide insight into three-dimensional active turbulence, distinctly from the perspective of the topology of the emergent three-dimensional topological defects.

Topology of Three-Dimensional Active Nematic Turbulence Confined to Droplets

We demonstrate the relevance of saddle-splay elasticity in nematic liquid crystalline fluids in the context of complex surface anchoring conditions and the complex geometrical confinement. Specifically, nematic cells with patterns of surface anchoring and colloidal knots are shown as examples where saddle-splay free energy contribution can have a notable role which originates from nonhomogeneous surface anchoring and the varying surface curvature. Patterned nematic cells are shown to exhibit various (meta)stable configurations of nematic field, with relative (meta)stability depending on the saddle-splay. We show that for high enough values of saddle-splay elastic constant K24 a previously unstable conformation can be stabilised, more generally indicating that the saddle-splay can reverse or change the (meta)stability of various nematic structures affecting their phase diagrams. Furthermore, we investigate saddle-splay elasticity in the geometry of highly curved boundaries – the colloidal particle knots in nematic – where the local curvature of the particles induces complex spatial variations of the saddle-splay contributions. Finally, a nematic order parameter tensor based saddle-splay invariant is shown, which allows for the direct calculation of saddle-splay free energy from the Q-tensor, a possibility very relevant for multiple mesoscopic modelling approaches, such as Landau-de Gennes free energy modelling.

/pdf/relevance-of-saddle-splay-elasticity-in-complex-nematic-4phi77flnt.pdf

Žiga Kos

Papers

Active nematic emulsions

Cross-talk between topological defects in different fields revealed by nematic microfluidics.

Three-dimensional active defect loops

Topology of Three-Dimensional Active Nematic Turbulence Confined to Droplets

Relevance of saddle-splay elasticity in complex nematic geometries