Umbilic Lines in Orientational Order

Machon, Thomas; Alexander, Gareth P.

doi:10.1103/physrevx.6.011033

Cited by 54 publications

(87 citation statements)

References 115 publications

(144 reference statements)

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“…Moreover, this two-fold family is exhausted by the heliconical director fields first envisioned by Meyer [22] and which have recently been identified experimentally with the ground state of twist-bend nematic phases [23]. 11 Our "octupolar eye" does not see T and, as already remarked, it is also insensitive to rescaling (by a constant) all distortion characteristics. 12 The octupolar potential (and its graphical representation) is thus especially suited to describe uniform distortions.…”

Section: Quasi-uniform Distortionsmentioning

confidence: 98%

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Liquid crystal distortions revealed by an octupolar tensor

Pedrini

Virga

2020

Phys. Rev. E

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The classical theory of liquid crystal elasticity as formulated by Oseen and Frank describes the (orientable) optic axis of these soft materials by a director n. The ground state is attained when n is uniform in space; all other states, which have a non-vanishing gradient ∇n, are distorted. This paper proposes an algebraic (and geometric) way to describe the local distortion of a liquid crystal by constructing from n and ∇n a third-rank, symmetric and traceless tensor A (the octupolar tensor). The (nonlinear) eigenvectors of A associated with the local maxima of its cubic form Φ on the unit sphere (its octupolar potential ) designate the directions of distortion concentration. The octupolar potential is illustrated geometrically and its symmetries are charted in the space of distortion characteristics, so as to educate the eye to capture the dominating elastic modes. Special distortions are studied, which have everywhere either the same octupolar potential or one with the same shape, but differently inflated. * andrea.pedrini@unipv.it † eg.virga@unipv.it arXiv:1911.03333v1 [cond-mat.soft]

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Section: Quasi-uniform Distortionsmentioning

confidence: 98%

“…Greek counterpart ∆ used in both [11] and [10]. The only other point where our notation differs from that of [10] is in calling b the bend vector, which there was B.…”

Section: Introductionmentioning

confidence: 99%

Liquid crystal distortions revealed by an octupolar tensor

Pedrini

Virga

2020

Phys. Rev. E

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“…), in its nonzero 2D block, following [48,49]. The modes of zero H, then, have both locally isotropic gradients of t (vanishing biaxial splay), and constant cross-sectional area per filament (vanishing splay).…”

Section: Local Metric and Convective Flow Tensormentioning

confidence: 99%

Constant spacing in filament bundles

2019

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Assemblies of one-dimensional filaments appear in a wide range of physical systems: from biopolymer bundles, columnar liquid crystals, and superconductor vortex arrays; to familiar macroscopic materials, like ropes, cables, and textiles. Interactions between the constituent filaments in such systems are most sensitive to the distance of closest approach between the central curves which approximate their configuration, subjecting these distinct assemblies to common geometric constraints. In this paper, we consider two distinct notions of constant spacing in multi-filament packings in 3  : equidistance, where the distance of closest approach is constant along the length of filament pairs; and isometry, where the distances of closest approach between all neighboring filaments are constant and equal. We show that, although any smooth curve in 3  permits one dimensional families of collinear equidistant curves belonging to a ruled surface, there are only two families of tangent fields with mutually equidistant integral curves in 3  . The relative shapes and configurations of curves in these families are highly constrained: they must be either (isometric) developable domains, which can bend, but not twist; or (non-isometric) constant-pitch helical bundles, which can twist, but not bend. Thus, filament textures that are simultaneously bent and twisted, such as twisted toroids of condensed DNA plasmids or wire ropes, are doubly frustrated: twist frustrates constant neighbor spacing in the cross-section, while nonequidistance requires additional longitudinal variations of spacing along the filaments. To illustrate the consequences of the failure of equidistance, we compare spacing in three 'almost equidistant' ansatzes for twisted toroidal bundles and use our formulation of equidistance to construct upper bounds on the growth of longitudinal variations of spacing with bundle thickness.

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“…Finally, we note for completeness that it should also be possible (though we do not attempt it here) to construct tb g ( ) and establish a similar result using the natural Frenet-Serret apparatus that one may associate to the pitch axis [22], so that tb g ( ) becomes a generalisation of a Frenet-Serret self-linking number, preserved as long as Tr 0 c ¹ ( ) , where n n ij ljk k i l  c = ¶ is the chirality tensor [21,22,47].…”

Section: Layer Number Invariantsmentioning

confidence: 99%

“…A characterisation of defects in these structures can be given in terms of pitch-axis descriptions of the cholesteric phase [19,20]. The pitch axis is defined as the local direction along which the cholesteric twists, and its defects can be formalised in terms of singularities or degeneracies, such as umbilic lines, in the gradient tensor of the director field [21,22]. These defects, which are found at the centre of double-twist cylinders or where the layer structure of a cholesteric changes, can be locally characterised in this way.…”

Section: Introductionmentioning

confidence: 99%

Contact topology and the structure and dynamics of cholesterics

Machon

2017

New J. Phys.

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Using tools and concepts from contact topology we show that non-vanishing twist implies conservation of the layer structure in cholesteric liquid crystals. This leads to a number of additional topological invariants for cholesteric configurations, such as layer numbers, that are not captured by traditional descriptions, characterises the nature and size of the chiral energy barriers between metastable configurations, and gives a geometric characterisation of cholesteric dynamics in any context, including active systems, those in confined geometries or under the influence of an external field.

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Umbilic Lines in Orientational Order

Cited by 54 publications

References 115 publications

Liquid crystal distortions revealed by an octupolar tensor

Liquid crystal distortions revealed by an octupolar tensor

Constant spacing in filament bundles

Contact topology and the structure and dynamics of cholesterics

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