Nematic braids: topological invariants and rewiring of disclinations

The conventional topological description given by the fundamental group of nematic order parameter does not adequately explain the entangled defect line structures that have been observed in nematic colloids. We introduce a new topological invariant, the self-linking number, that enables a complete classification of entangled defect line structures in general nematics, even without particles, and demonstrate our formalism using colloidal dimers, for which entangled structures have been previously observed. We also unveil a simple rewiring scheme for the orthogonal crossing of two -1/2 disclinations, based on a tetrahedral rotation of two relevant disclination segments, that allows us to predict possible nematic braids and calculate their self-linking numbers.Comment: 7 pages, 5 figures, accepted for publication in PRL, extended version with supplementary informatio

Reconfigurable knots and links in chiral nematic colloids

Tying knots and linking microscopic loops of polymers, macromolecules, or defect lines in complex materials is a challenging task for material scientists. We demonstrate the knotting of microscopic topological defect lines in chiral nematic liquid crystal colloids into knots and links of arbitrary complexity by using laser tweezers as a micromanipulation tool. All knots and links with up to six crossings, including the Hopf link, the Star of David and the Borromean rings are demonstrated, stabilizing colloidal particles into an unusual soft matter. The knots in chiral nematic colloids are classified by the quantized self-linking number, a direct measure of the geometric, or Berry's, phase. Forming arbitrary microscopic knots and links in chiral nematic colloids is a demonstration of how relevant the topology can be for the material engineering of soft matter.Comment: 6 pages, 3 figure

Singular Values, Nematic Disclinations, and Emergent Biaxiality

Both uniaxial and biaxial nematic liquid crystals are defined by orientational ordering of their building blocks. While uniaxial nematics only orient the long molecular axis, biaxial order implies local order along three axes. As the natural degree of biaxiality and the associated frame, that can be extracted from the tensorial description of the nematic order, vanishes in the uniaxial phase, we extend the nematic director to a full biaxial frame by making use of a singular value decomposition of the gradient of the director field instead. New defects and degrees of freedom are unveiled and the similarities and differences between the uniaxial and biaxial phase are analyzed by applying the algebraic rules of the quaternion group to the uniaxial phase.Comment: 5 pages, 1 figure, submitted to PR

Three-dimensional active defect loops

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 ± 1 / 2 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

Orientational properties of nematic disclinations

Topological defects play a pivotal role in the physics of liquid crystals and represent one of the most prominent and well studied aspects of mesophases. While in two-dimensional nematics, disclinations are traditionally treated as point-like objects, recent experimental studies on active nematics have suggested that half-strength disclinations might in fact possess a polar structure. In this article, we provide a precise definition of polarity for half-strength nematic disclinations, we introduce a simple and robust method to calculate this quantity from experimental and numerical data and we investigate how the orientational properties of half-strength disclinations affect their relaxational dynamics.Comment: 6 pages, 5 figures, supplementary movies at http://wwwhome.lorentz.leidenuniv.nl/~giomi/sup_mat/20150720

The Geometry of the Cholesteric Phase

We propose a construction of a cholesteric pitch axis for an arbitrary nematic director field as an eigenvalue problem. Our definition leads to a Frenet-Serret description of an orthonormal triad determined by this axis, the director, and the mutually perpendicular direction. With this tool we are able to compare defect structures in cholesterics, biaxial nematics, and smectics. Though they all have similar ground state manifolds, the defect structures are different and cannot be, in general, translated from one phase to the other.Comment: 5 pages, the full catastroph

Geometry of the cholesteric phase

We propose a construction of a cholesteric pitch axis for an arbitrary nematic director field as an eigenvalue problem. Our definition leads to a Frenet-Serret description of an orthonormal triad determined by this axis, the director, and the mutually perpendicular direction. With this tool, we are able to compare defect structures in cholesterics, biaxial nematics, and smectics. Though they all have similar ground state manifolds, the defect structures are different and cannot, in general, be translated from one phase to the other

Topological and geometric decomposition of nematic textures

Directional media, such as nematic liquid crystals and ferromagnets, are characterized by their topologically stabilized defects in directional order. In nematics, boundary conditions and surface-treated inclusions often create complex structures, which are difficult to classify. Topological charge of point defects in nematics has ambiguously defined sign and its additivity cannot be ensured when defects are observed separately. We demonstrate how the topological charge of complex defect structures can be determined by identifying and counting parts of the texture that satisfy simple geometric rules. We introduce a parameter called the defect rank and show that it corresponds to what is intuitively perceived as a point charge based on the properties of the director field. Finally, we discuss the role of free energy constraints in validity of the classification with the defect rank.Comment: 16 pages, 5 figure