arXiv subjects

Epifanio G. Virga

Publications and source records attributed to Epifanio G. Virga.

At least 37 records · Page 2Linked to original sources

arXiv2022

Analytical Thermodynamics

This paper proposes a theory that bridges classical analytical mechanics and nonequilibrium thermodynamics. Its intent is to derive the evolution equations of a system from a stationarity principle for a suitably augmented Lagrangian action. This aim is attained for homogeneous systems, described by a finite number of state variables depending on time only. In particular, it is shown that away from equilibrium free energy and entropy are independent constitutive functions.

cond-mat.stat-mech↗

arXiv2022

Model for Photoresponsive Nematic Elastomers

We study the equilibria of a photoresponsive nematic elastomer ribbon within a continuum theory that builds upon the statistical mechanics model put forward by Corbett and Warner [Phys. Rev. E 78, 061701 (2008)]. We prove that the spontaneous deformation induced by illumination is not monotonically dependent on the intensity $I$. The ribbon's deflection first increases with increasing $I$, as expected, but then decreases and abruptly ceases altogether at a critical value $I_\mathrm{e}$ of $I$. $I_\mathrm{e}$, which is enclosed within a hysteresis loop, marks a first-order shape transition. Finally, we find that there is a critical value of the ribbon's length, depending only on the degree of cross-linking in the material, below which no deflection can be induced in the ribbon, no matter how intense is the light shone on it.

cond-mat.soft↗

arXiv2022

Paradoxes for Chromonic Liquid Crystal Droplets

Chromonic liquid crystals constitute a novel lyotropic phase, whose elastic properties have so far been modeled within the classical Oseen-Frank theory, provided that the twist constant is assumed to be considerably smaller than the saddle-splay constant, in violation of one Ericksen inequality. This paper shows that paradoxical consequences follow from such a violation for droplets of these materials surrounded by an isotropic fluid. For example, tactoids with a degenerate planar anchoring simply disintegrate indefinitely in myriads of smaller ones.

cond-mat.soft↗

arXiv2022

A ribbon model for nematic polymer networks

We present a theory of deformation of ribbons made of nematic polymer networks (NPNs). These materials exhibit properties of rubber and nematic liquid crystals, and can be activated by external stimuli of heat and light. A two-dimensional energy for a sheet of such a material has already been derived from the celebrated neo-classical energy of nematic elastomers in three space dimensions. Here, we use a dimension reduction method to obtain the appropriate energy for a ribbon from the aforementioned sheet energy. We also present an illustrative example of a rectangular NPN ribbon that undergoes in-plane serpentine deformations upon activation under an appropriate set of boundary conditions.

cond-mat.soft↗

arXiv2022

Stability Against the Odds: the Case of Chromonic Liquid Crystals

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 of Frank's elastic free energy.

cond-mat.soft↗

arXiv2021

Cavity volume and free energy in many-body systems

Within this work we derive and analyse an expression for the free energy of a single-species system in the thermodynamic limit in terms of a generalised cavity volume, that is exact in general, and in principle applicable to systems across their entire range of density, as well as to particles within a general coordinate space. This provides a universal equation of state, and can thus relate the cavity volume to classical results, such as Mayer's cluster expansions. Through this we are able to provide some insight into the connections between cavity volume and free energy density, as well as their consequences. We use examples which permit explicit computations to further probe these results, reclaiming the exact results for a classical Tonks gas and providing a novel derivation of Onsager's free energy for a single species, isotropic system. Given the complexity of the problem we also provide a local lattice ansatz, exact in one dimension, with which we may approximate the cavity volume for hard sphere systems to provide an accurate equation of state in the cases of hard disks and spheres in both dilute regimes as well as beyond the freezing transition.

cond-mat.stat-mech↗

arXiv2021

Shape Bistability in Squeezed Chromonic Droplets

We study droplets of chromonic liquid crystals squeezed between parallel plates inducing degenerate tangential anchoring on the nematic director. In the coexistence regime, where droplets in the nematic phase are at equilibrium with the surrounding melt, our two-dimensional theoretical model predicts a regime of shape bistability for sufficiently large bipolar droplets, where tactoids (pointed shapes) and discoids (smooth shapes) coexist in equilibrium. This phenomenon has not yet been observed. Two-dimensional droplets of disodium cromoglycate (DSCG) have been the object of a thorough experimental study [Y.-K. Kim et al., J. Phys.: Condens. Matter 25, 404202 (2013)]. We show that our theory is in good quantitative agreement with these data and extract from them what promises to be a more accurate estimate for the isotropic surface tension at the nematic/melt interface of DSCG.

cond-mat.soft↗

arXiv2021

Leaky Cell Model of Hard Spheres

We study packings of hard spheres on lattices. The partition function, and therefore the pressure, may be written solely in terms of the accessible free volume, i.e. the volume of space that a sphere can explore without touching another sphere. We compute these free volumes using a leaky cell model, in which the accessible space accounts for the possibility that spheres may escape from the local cage of lattice neighbors. We describe how elementary geometry may be used to calculate the free volume exactly for this leaky cell model in two- and three-dimensional lattice packings and compare the results to the well-known Carnahan-Starling and Percus-Yevick liquid models. We provide formulas for the free volumes of various lattices and use the common tangent construction to identify several phase transitions between them in the leaky cell regime, indicating the possibility of coexistence in crystalline materials.

cond-mat.soft↗

arXiv2021

Ridge approximation for thin nematic polymer networks

Nematic polymer networks (NPNs) are nematic elastomers within which the nematic director is enslaved to the elastic deformation. The elastic free energy of a NPN sheet of thickness $h$ has both stretching and bending components (the former scaling like h, the latter scaling like $h^3$). NPN sheets bear a director field $\mathbf{m}$ imprinted in them (usually, uniformly throughout their thickness); they can be activated by changing the nematic order (e.g. by illumination or heating). This paper illustrates an attempt to compute the bending energy of a NPN sheet and to show which role it can play in determining the activated shape. Our approach is approximate: the activated surface consists of flat sectors connected by ridges, where the unit normal jumps and the bending energy is concentrated. By increasing the number of ridges, we should get closer to the real situation, where the activated surface is smooth and the bending energy is distributed on it. The method is applied to a disk with imprinted a spiraling hedgehog. It is shown that upon activation the disk, like a tiny hand, is able to grab a rigid lamina.

cond-mat.soft↗

arXiv2020

Ridge energy for thin nematic polymer networks

Minimizing the elastic free energy of a thin sheet of nematic polymer network among smooth isometric immersions is the strategy purported by the mainstream theory. In this paper, we broaden the class of admissible spontaneous deformations: we consider ridged isometric immersions, which can cause a sharp ridge in the immersed surfaces. We propose a model to compute the extra energy distributed along such ridges. This energy comes from bending; it is shown under what circumstances it scales quadratically with the sheet's thickness, falling just in between stretching and bending energies. We put our theory to the test by studying the spontaneous deformation of a disk on which a radial hedgehog was imprinted at the time of crosslinking. We predict the number of folds that develop in terms of the degree of order induced in the material by external agents (such as heat and illumination).

cond-mat.soft↗

arXiv2020

Nematic Tactoid Population

Tactoids are pointed, spindle-like droplets of nematic liquid crystal in an isotropic fluid. They have long been observed in inorganic and organic nematics, in thermotropic phases as well as lyotropic colloidal aggregates. The variational problem of determining the optimal shape of a nematic droplet is formidable and has only been attacked in selected classes of shapes and director fields. Here, by considering a novel class of admissible solutions for a bipolar droplet, we study the prevalence in the population of all equilibrium shapes of each of the three that may be optimal (tactoids primarily among them). We show how the prevalence of a shape is affected by the drop's volume $V_0$ and the saddle-splay constant $K_{24}$ of the material. Tactoids, in particular, prevail for small $V_0$ and small $K_{24}$ (appropriately scaled). Our class of shapes (and director fields) is sufficiently different from those employed so far to unveil a rather different role of $K_{24}$.

cond-mat.soft↗

arXiv2020

On the Kirchhoff-Love hypothesis (revised and vindicated)

The Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This hypothesis has a long history checkered with the vicissitudes of life: even its attribution has been questioned, and recent rigorous dimension-reduction tools (based on $Γ$-convergence) have proven to be incompatible with it. We find that an appropriately revised version of the Kirchhoff-Love hypothesis is a valuable means to derive a two-dimensional variational model for elastic plates from a three-dimensional nonlinear free-energy functional. The bending energies thus obtained for a number of materials also show to contain measures of stretching of the plate's mid surface (alongside the expected measures of bending). The incompatibility with $Γ$-convergence also appears to be removed in the cases where contact with that method and ours can be made.

math-ph↗

arXiv2019

Liquid Crystal Distortions Revealed by an Octupolar Tensor

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 $\mathbf{n}$. The ground state is attained when $\mathbf{n}$ is uniform in space; all other states, which have a non-vanishing gradient $\nabla\mathbf{n}$, are distorted. This paper proposes an algebraic (and geometric) way to describe the local distortion of a liquid crystal by constructing from $\mathbf{n}$ and $\nabla\mathbf{n}$ a third-rank, symmetric and traceless tensor $\mathbf{A}$ (the octupolar tensor). The (nonlinear) eigenvectors of $\mathbf{A}$ 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.

cond-mat.soft↗

arXiv2019

Uniform distortions and generalized elasticity of liquid crystals

Ordinary nematic liquid crystals are characterized by having a uniform director field as ground state. In such a state, the director is the same everywhere and no distortion is to be seen at all. We give a definition of uniform distortion which makes precise the intuitive notion of seeing everywhere the same director landscape. We characterize all such distortions and prove that they fall into two families, each described by two scalar parameters. Uniform distortions exhaust R. Meyer's heliconical structures, which, as it has recently been recognized, include the ground state of twist-bend nematics. The generalized elasticity of these new phases is treated with a simple free-energy density, which can be minimized by both uniform and non-uniform distortions, the latter injecting a germ of elastic frustration.

cond-mat.soft↗

arXiv2018

The symmetries of octupolar tensors

Octupolar tensors are third order, completely symmetric and traceless tensors. Whereas in 2D an octupolar tensor has the same symmetries as an equilateral triangle and can ultimately be identified with a vector in the plane, the symmetries that it enjoys in 3D are quite different, and only exceptionally reduce to those of a regular tetrahedron. By use of the octupolar potential that is, the cubic form associated on the unit sphere with an octupolar tensor, we shall classify all inequivalent octupolar symmetries. This is a mathematical study which also reviews and incorporates some previous, less systematic attempts.

math-ph↗

arXiv2018

Lily-like twist distribution in toroidal nematics

Toroidal nematics are droplets of nematic liquid crystals in the form of a circular torus. When the nematic director is subject to planar degenerate boundary conditions, the bend-only director field with vector lines along the parallels of all internal torodial shells is an equilibrium solution for all values of the elastic constants. Local stability analyses have shown that an instability is expected to occur for sufficiently small values of the twist elastic constant. It is natural to conjecture that in this regime the global equilibrium would be characterized by a maximum twist deflection on the boundary of the torus, with a twist distribution over the torus' cross-section represented by a fennel-like surface. We prove that surprisingly the stable twist distribution is instead represented by a lily-like surface. Thus the overall maximum twist deflection falls well within the torus. To cope with the complexity of the elastic free-energy functional in the fully non-linear setting, we developed an ad hoc deep-learning optimization method, which here is also duly validated and documented for it promises to be applicable to other similar problems, equally intractable analytically.

cond-mat.soft↗

arXiv2018

Contributions of Repulsive and Attractive Interactions to Nematic Order

Both repulsive and attractive molecular interactions can be used to explain the onset of nematic order. The object of this paper is to combine these two nematogenic molecular interactions in a unified theory. This attempt is not unprecedented; what is perhaps new is the focus on the understanding of nematics in the high density limit. There, the orientational probability distribution is shown to exhibit a unique feature: it has compact support on configuration space. As attractive interactions are turned on, the behavior changes, and at a critical attractive interaction strength, thermotropic behavior of the Maier-Saupe type is attained.

cond-mat.soft↗

arXiv2018

Instability of toroidal nematics

Toroidal nematics are nematic liquid crystals confined within a circular torus and subject to planar degenerate anchoring on the boundary of the torus. They may be droplets floating in an isotropic environment or cavities carved out of a solid substrate. A universal solution of Frank's elastic free energy is an equilibrium configuration for the nematic director field, irrespective of the values of the elastic constants, whose vector lines are the coaxial parallels of the torus. We explore the local stability of this configuration and identify a range of parameters where the main drive towards instability does not come from the surface-like elastic constant $K_{24}$ being large, but from the the ratio $K_2/K_3$ of the twist to bend elastic constants being small, which also makes our study relevant to chromonic liquid crystals.

cond-mat.soft↗