Uniform distortions and generalized elasticity of liquid crystals

Virga, Epifanio G.

doi:10.1103/physreve.100.052701

Cited by 39 publications

(83 citation statements)

References 42 publications

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“…Building upon the decomposition of ∇n in (3), it is natural to define a uniform nematic distortion as one for which all distortion characteristics (S, T, b 1 , b 2 , q) are constant in space [16]. Inside a uniform distortion, it is as if, sitting in a place in space, one sees the same orientation landscape against the local distortion frame (n 1 , n 2 , n) (which is intrinsically defined), irrespective of the place.…”

Section: Quasi-uniform Distortionsmentioning

confidence: 99%

“…Inside a uniform distortion, it is as if, sitting in a place in space, one sees the same orientation landscape against the local distortion frame (n 1 , n 2 , n) (which is intrinsically defined), irrespective of the place. It was shown in [16] that the family of all uniform distortions is characterized by having S = 0 and T = ±2q with, correspondingly, b 1 = ±b 2 . 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].…”

Section: Quasi-uniform Distortionsmentioning

confidence: 99%

“…In general, Φ changes from point to point. In one case, it is everywhere the same, that is, when all distortion characteristics are constant in space, as is the case for the uniform distortions, which have been characterized in [16]. If one allows Φ to be differently inflated at different places in space, while keeping everywhere the same shape, one defines a new class of quasi-uniform distortions.…”

Section: Spiralsmentioning

confidence: 99%

“…3 Since b · n ≡ 0, we can represent b as b = b 1 n 1 + b 2 n 2 . We shall call (n 1 , n 2 , n) the distortion frame and (S, T, b 1 , b 2 , q) the distortion characteristics of the director field n [16]. In terms of these, (3) can also be written as…”

Section: Introductionmentioning

confidence: 99%

“…IV we discuss the octupolar potential and its geometric representation for special director distortions. These include the uniform distortions, for which all distortion characteristics are constant in space [16], and new types of distortions, which we shall call quasi-uniform. In the last Sec.…”

Section: Introductionmentioning

confidence: 99%

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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: 99%

Section: Quasi-uniform Distortionsmentioning

confidence: 99%

Section: Spiralsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Liquid crystal distortions revealed by an octupolar tensor

Pedrini

Virga

2020

Phys. Rev. E

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Rotations with Constant $$\mathbf {{\text {curl }}}$$ are Constant

Ginster

Acharya

2022

Arch Rational Mech Anal

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We address a problem that extends a fundamental classical result of continuum mechanics from the time of its inception, as well as answers a fundamental question in the recent, modern nonlinear elastic theory of dislocations. Interestingly, the implication of our result in the latter case is qualitatively different from its well-established analog in the linear elastic theory of dislocations. It is a classical result that if $$u\in C^2({\mathbb {R}}^n;{\mathbb {R}}^n)$$ u ∈ C 2 ( R n ; R n ) and $$\nabla u \in SO(n)$$ ∇ u ∈ S O ( n ) , it follows that u is rigid. In this article this result is generalized to matrix fields with non-vanishing $${\text {curl }}$$ curl . It is shown that every matrix field $$R\in C^2(\varOmega ;SO(3))$$ R ∈ C 2 ( Ω ; S O ( 3 ) ) such that $${\text {curl }}R = constant$$ curl R = c o n s t a n t is necessarily constant. Moreover, it is proved in arbitrary dimensions that a measurable rotation field is as regular as its distributional $${\text {curl }}$$ curl allows. In particular, a measurable matrix field $$R: \varOmega \rightarrow SO(n)$$ R : Ω → S O ( n ) , whose $${\text {curl }}$$ curl in the sense of distributions is smooth, is also smooth.

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Stability Against the Odds: The Case of Chromonic Liquid Crystals

Paparini

Virga

2022

J Nonlinear Sci

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The ground state of chromonic liquid crystals, as revealed by a number of recent experiments, is quite different from that of ordinary nematic liquid crystals: it is twisted instead of uniform. The common explanation provided for this state within the classical elastic theory of Frank demands that one Ericksen’s inequality is violated. Since in general such a violation makes Frank’s elastic free-energy functional unbounded below, the question arises as to whether the twisted ground state can be locally stable. We answer this question in the affirmative. In reaching this conclusion, a central role is played by the specific boundary conditions imposed in the experiments on the boundary of rigid containers and by a general formula that we derive here for the second variation in Frank’s elastic free energy.

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Uniform distortions and generalized elasticity of liquid crystals

Cited by 39 publications

References 42 publications

Liquid crystal distortions revealed by an octupolar tensor

Liquid crystal distortions revealed by an octupolar tensor

Rotations with Constant $$\mathbf {{\text {curl }}}$$ are Constant

Stability Against the Odds: The Case of Chromonic Liquid Crystals

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