arXiv subjects

Epifanio G. Virga

Publications and source records attributed to Epifanio G. Virga.

At least 19 recordsLinked to original sources

arXiv2025

Metric Implications in the Kinematics of Surfaces

In the direct approach to continua in reduced space dimensions, a thin shell is described as a mathematical surface in three-dimensional space. An exploratory kinematic study of such surfaces could be very valuable, especially if conducted with no use of coordinates. Three energy contents have been identified in a thin shell, which refer to three independent deformation modes: stretching, drilling, and bending. We analyze the consequences for the three energy contents produced by metric restrictions imposed on the admissible deformations. Would the latter stem from physical constraints, the elastic response of a shell could be hindered in ways that might not be readily expected.

math.DG↗

arXiv2025

Neutral Deformation Modes of Minimal Surfaces

Stretching, drilling, and bending are the independent deformation modes of a thin shell, each of which has an individual energy content. When the energy content of a mode vanishes, that mode is neutral. We characterize all neutral modes of deformation of minimal surfaces into minimal surfaces. A hierarchy is found among these: a stretching neutral mode (which is an isometry) is also drilling neutral, and a drilling neutral mode is also bending neutral. Thus, all isometries of a minimal surface are globally neutral and give rise to soft elasticity. More generally, all minimal surfaces can be classified relative to a reference one in terms of three energy contents, which can be given in closed form.

math.DG↗

arXiv2025

Surface Nematic Quasi-Uniformity

Line fields on surfaces are a means to describe the nematic order that may pattern them. The least distorted nematic fields are called uniform, but they can only exist on surfaces with negative constant Gaussian curvature. To identify the least distorted nematic fields on a generic surface, we relax the notion of uniformity into that of quasi-uniformity and prove that all such fields are parallel transported (in Levi-Civita's sense) by the geodesics of the surface. Both global and local constructions of quasi-uniform fields are presented to illustrate both richness and significance of the proposed notion.

cond-mat.soft↗

arXiv2025

Singular Damped Twist Waves in Chromonic Liquid Crystals

Chromonics are special classes of nematic liquid crystals, for which a quartic elastic theory seems to be more appropriate than the classical quadratic Oseen-Frank theory. The relaxation dynamics of twist director profiles are known to develop a shock wave in finite time in the inviscid case, where dissipation is neglected. This paper studies the dissipative case. We give a sufficient criterion for the formation of shocks in the presence of dissipation and we estimate the critical time at which these singularities develop. Both criterion and estimate depend on the initial director profile. We put our theory to the test on a class of initial kink profiles and we show how accurate our estimates are by comparing them to the outcomes of numerical solutions.

math.AP↗

arXiv2025

Surface Nematic Uniformity

An ant-like observer confined to a two-dimensional surface traversed by stripes would wonder whether this striped landscape could be devised in such a way as to appear to be the same wherever they go. Differently stated, this is the problem studied in this paper. In a more technical jargon, we determine all possible uniform nematic fields on a smooth surface. It was already known that for such a field to exist, the surface must have constant negative Gaussian curvature. Here, we show that all uniform nematic fields on such a surface are parallel transported (in Levi-Civita's sense) by special systems of geodesics, which (with scant inventiveness) are termed uniform. We prove that, for every geodesic on the surface, there are two systems of uniform geodesics that include it; they are conventionally called right and left, to allude at a possible intrinsic definition of handedness. We found explicitly all uniform fields for Beltrami's pseudosphere. Since both geodesics and uniformity are preserved under isometries, by a classical theorem of Minding, the solution for the pseudosphere carries over all other admissible surfaces, thus providing a general solution to the problem (at least in principle).

cond-mat.soft↗

arXiv2025

A Variational Theory for Soft Shells

Three general modes are distinguished in the deformation of a thin shell; these are stretching, drilling, and bending. Of these, the drilling mode is the one more likely to emerge in a soft matter shell (as compared to a hard, structural one), as it is ignited by a swerve of material fibers about the local normal. We propose a hyperelastic theory for soft shells, based on a separation criterion that envisages the strain-energy density as the sum of three independent pure measures of stretching, drilling, and bending. Each individual measure is prescribed to vanish on all other companion modes. The result is a direct, second-grade theory featuring a bending energy quartic in an invariant strain descriptor that stems from the polar rotation hidden in the deformation gradient (although quadratic energies are also appropriate in special cases). The proposed energy functional has a multi-well character, which fosters cases of soft elasticity (with a continuum of ground states) related to minimal surfaces.

cond-mat.soft↗

arXiv2024

Singular Twist Waves in Chromonic Liquid Crystals

Chromonic liquid crystals are lyotropic nematic phases whose applications span from food to drug industries. It has recently been suggested that the elastic energy density governing the equilibrium distortions of these materials may be quartic in the measure of twist. Here we show that the non-linear twist-wave equation associated with such an energy has smooth solutions that break down in a finite time, giving rise to the formation of a shock wave, under rather generic assumptions on the initial profile. The critical time at which smooth solutions become singular is estimated analytically with an accuracy that numerical calculations for a number of exemplary cases prove to be satisfactory.

cond-mat.soft↗

arXiv2024

Bending-Neutral Deformations of Minimal Surfaces

Minimal surfaces are ubiquitous in nature. Here they are considered as geometric objects that bear a deformation content. By refining the resolution of the surface deformation gradient afforded by the polar decomposition theorem, we identify a bending content and a class of deformations that leave it unchanged. These are the bending-neutral deformations, fully characterized by an integrability condition; they preserve normals. We prove that (1) every minimal surface is transformed into a minimal surface by a bending-neutral deformation, (2) given two minimal surfaces with the same system of normals, there is a bending-neutral deformation that maps one into the other, and (3) all minimal surfaces have indeed a universal bending content.

math.DG↗

arXiv2024

What a twist cell experiment tells about a quartic twist theory for chromonics

The elastic theory of chromonic liquid crystals is not completely established. We know, for example, that for anomalously low twist constants (needed for chromonics) the classical Oseen-Frank theory may entail paradoxical consequences when applied to describe the equilibrium shapes of droplets surrounded by an isotropic phase: contrary to experimental evidence, they are predicted to dissolve in a plethora of unstable smaller droplets. We proposed a quartic twist theory that prevents such an instability from happening. Here, we apply this theory to the data of two experiments devised to measure the planar anchoring strength at the plates bounding a twist cell filled with a chromonic liquid crystal; these data had before been interpreted within the Oseen-Frank theory. We show that the quartic twist theory affords a better agreement with the experimental data, while delivering in one case a larger value for the anchoring strength.

cond-mat.soft↗

arXiv2024

Inversion Ring in Chromonic Twisted Hedgehogs: Theory and Experiment

Twisted hedgehogs are defects in spherical cavities with homeotropic anchoring for the nematic director that arise when twist distortions are sufficiently less energetic than splay (and bend) distortions. They bear a characteristic inversion ring, where the director texture changes the sense it spirals about the center of the cavity. This paper applies a quartic twist theory recently proposed to describe the elasticity of chromonics to explain a series of inversion rings observed in aqueous solutions of SSY at two different concentrations. The theory features a phenomenological length a, whose measure is extracted from the data and shown to be fairly independent of the cavity radius, as expected for a material constant.

cond-mat.soft↗

arXiv2023

Pure measure of bending for soft plates

This paper, originally motivated by a question raised by Wood and Hanna [Soft Matter, 15, 2411 (2019)], shows that pure measures of bending for soft plates can be defined by introducing the class of bending-neutral deformations, finite incremental changes of the plate's shape bearing no further bending. This class of deformations is subject to a geometric compatibility condition, which is fully characterized. A tensorial pure measure of bending, which is accordingly invariant under bending-neutral deformations, is described in details. As shown by an illustrative class of examples, the general notion of pure measure of bending could be of use to formulate direct theories for soft plates, where stretching and bending energies are kept separate.

cond-mat.soft↗

arXiv2023

A confined rod: mean field theory for hard rod-like particles

In this paper, we model the configurations of a system of hard rods by viewing each rod in a cell formed by its neighbors. By minimizing the free energy in the model and performing molecular dynamics, where, in both cases, the shape of the cell is a free parameter, we obtain the equilibrium orientational order parameter, free energy and pressure of the system. Our model enables the calculation of anisotropic stresses exerted on the walls of the cell due to shape change of the rod in photoisomerization. These results are a key step towards understanding molecular shape change effects in photomechanical systems under illumination.

cond-mat.soft↗

arXiv2023

A geometric method to determine chromonics' planar anchoring strength

Chromonic nematics are lyotropic liquid crystals that have already been known for half a century, but have only recently raised interest for their potential applications in life sciences. Determining elastic constants and anchoring strengths for rigid substrates has thus become a priority in the characterization of these materials. Here, we present a method to determine chromonics' planar anchoring strength. We call it geometric as it is based on recognition and fitting of the stable equilibrium shapes of droplets surrounded by the isotropic phase in a thin cell with plates enforcing parallel alignments of the nematic director. We apply our method to shapes observed in experiments; they resemble elongated rods with round ends, which are called batonnets. Our theory also predicts other droplets' equilibrium shapes, which are either slender and round, called discoids, or slender and pointed, called tactoids. In particular, sufficiently small droplets are expected to display shape bistability, with two equilibrium shapes, one tactoid and one discoid, exchanging roles as stable and metastable shapes upon varying their common area.

cond-mat.soft↗

arXiv2023

A Review on Octupolar Tensors

In its most restrictive definition, an octupolar tensor is a fully symmetric traceless third-rank tensor in three space dimensions. So great a body of works have been devoted to this specific class of tensors and their physical applications that a review would perhaps be welcome by a number of students. Here, we endeavour to place octupolar tensors into a broader perspective, considering non-vanishing traces and non-fully symmetric tensors as well. A number of general concepts are recalled and applied to either octupolar and higher-rank tensors. As a tool to navigate the diversity of scenarios we envision, we introduce the octupolar potential, a scalar-valued function which can easily be given an instructive geometrical representation. Physical applications are plenty; those to liquid crystal science play a major role here, as they were the original motivation for our interest in the topic of this review.

math-ph↗

arXiv2023

Relieving nematic geometric frustration in the plane

Frustration in nematic-ordered media (endowed with a director field) is treated in a purely geometric fashion in a flat, two-dimensional space. We recall the definition of quasi-uniform distortions and envision these as viable ways to relieve director fields prescribed on either a straight line or the unit circle. We prove that using a planar spiral is the only way to fill the whole plane with a quasi-uniform distortion. Apart from that, all relieving quasi-uniform distortions can at most be defined in a half-plane; however, in a generic sense, they are all asymptotically spirals.

math-ph↗

arXiv2023

Bending and Stretching in a Narrow Ribbon of Nematic Polymer Networks

We study the spontaneous out-of-plane bending of a planar untwisted ribbon composed of nematic polymer networks activated by a change in temperature. Our theory accounts for both stretching and bending energies, which compete to establish equilibrium. We show that when equilibrium is attained these energy components obey a \emph{complementarity} relation: one is maximum where the other is minimum. Moreover, we identify a \emph{bleaching} regime: for sufficiently large values of an activation parameter (which measures the mismatch between the degrees of order in polymer organization in the reference and current configurations), the ribbon's deformation is essentially independent of its thickness.

cond-mat.soft↗

arXiv2023

Spiraling Defect Cores in Chromonic Hedgehogs

An elastic quartic twist theory has recently been proposed for chromonic liquid crystals, intended to overcome the paradoxical conclusions encountered by the classical Oseen-Frank theory when applied to droplets submerged in an isotropic fluid environment. However, available experimental data for chromonics confined to cylindrical cavities with degenerate planar anchoring on their lateral boundary can be explained equally well by both competing theories. This paper identifies a means to differentiate these theories both qualitatively and quantitatively. They are shown to predict quite different core defects for the twisted hedgehogs that chromonics generate when confined to a fixed spherical cavity with homeotropic anchoring. In the quartic twist theory, the defect core is estimated to be nearly one order of magnitude larger (tens of microns) than in the other and, correspondingly, the director field lines describe Archimedean spirals instead of logarithmic ones.

cond-mat.soft↗

arXiv2022

An Elastic Quartic Twist Theory for Chromonic Liquid Crystals

Chromonic liquid crystals are lyotropic materials which are attracting growing interest for their adapatbility to living systems. To describe their elastic properties, the classical Oseen-Frank theory requires anomalously small twist constants and (comparatively) large saddle-splay constants, so large as to violate one of Ericksen's inequalities, which guarantee that the Oseen-Frank stored-energy density is bounded below. While such a violation does not prevent the existence and stability of equilibrium distortions in problems with fixed geometric confinement, the study of free-boundary problems for droplets has revealed a number of paradoxical consequences. Minimizing sequences driving the total energy to negative infinity have been constructed by employing ever growing needle-shaped tactoids incorporating a diverging twist [Phy. Rev. E 106, 044703 (2022)]. To overcome these difficulties, we propose here a novel elastic theory that extends for chromonics the classical Oseen-Frank stored energy by adding a quartic twist term. We show that the total energy of droplets is bounded below in the quartic twist theory, so that the known paradoxes are ruled out. The quartic term introduces a phenomenological length $a$ in the theory; this affects the equilibrium of chromonics confined within capillary tubes. Use of published experimental data allows us to estimate $a$.

cond-mat.soft↗