arXiv subjects

Dmitry Golovaty

Publications and source records attributed to Dmitry Golovaty.

At least 37 records · Page 2Linked to original sources

arXiv2020

A One-Dimensional Variational Problem for Cholesteric Liquid Crystals with Disparate Elastic Constants

We consider a one-dimensional variational problem arising in connection with a model for cholesteric liquid crystals. The principal feature of our study is the assumption that the twist deformation of the nematic director incurs much higher energy penalty than other modes of deformation. The appropriate ratio of the elastic constants then gives a small parameter $\varepsilon$ entering an Allen-Cahn-type energy functional augmented by a twist term. We consider the behavior of the energy as $\varepsilon$ tends to zero. We demonstrate existence of the local energy minimizers classified by their overall twist, find the $Γ$-limit of the relaxed energies and show that it consists of the twist and jump terms. Further, we extend our results to include the situation when the cholesteric pitch vanishes along with $\varepsilon$.

math.AP↗

arXiv2020

A variational method for generating $n$-cross fields using higher-order $Q$-tensors

An $n$-cross field is a locally-defined orthogonal coordinate system invariant with respect to the cubic symmetry group. Cross fields are finding wide-spread use in mesh generation, computer graphics, and materials science among many applications. It was recently by other authors that $3$-cross fields can be embedded into the set of symmetric $4$th-order tensors. Another concurrent work further develops a relaxation of this tensor field via a certain set of varieties. In this paper, we consider the problem of generating an arbitrary $n$-cross field using a fourth-order $Q$-tensor theory that is constructed out of tensored projection matrices. We establish that by a Ginzburg-Landau relaxation towards a global projection, one can reliably generate an $n$-cross field on arbitrary Lipschitz domains. Our work provides a rigorous approach that offers several new results including porting the tensor framework to arbitrary dimensions, providing a new relaxation method that embeds the problem into a global steepest descent, and offering a relaxation scheme for aligning the cross field with the boundary. Our approach is designed to fit within the classical Ginzburg-Landau PDE theory, offering a concrete road map for the future careful study of singularities of energy minimizers.

math.AP↗

arXiv2020

Saturn ring defect around a spherical particle immersed in nematic liquid crystal

We consider a nematic liquid crystal occupying the three-dimensional domain in the exterior of a spherical colloid particle. The nematic is subject to Dirichlet boundary conditions that enforce orthogonal attachment of nematic molecules to the surface of the particle. Our main interest is to understand the behavior of energy-critical configurations of the Landau-de Gennes $Q$-tensor model in the limit of vanishing correlation length. We demonstrate existence of configurations with a single Saturn-ring defect approaching the equator of the particle and no other line or point defects. We show this by analyzing asymptotics of energy minimizers under two symmetry constraints: rotational equivariance around the vertical axis and reflection across the horizontal plane. Energy blow-up at the ring defect is a significant obstacle to constructing well-behaved comparison maps needed to eliminate the possibility of point defects. The boundary estimates we develop to address this issue are new and should be applicable to a wider class of problems.

math.AP↗

arXiv2020

Thin Film Liquid Crystals with Oblique Anchoring and Boojums

We study a two-dimensional variational problem which arises as a thin-film limit of the Landau-de Gennes energy of nematic liquid crystals. We impose an oblique angle condition for the nematic director on the boundary, via boundary penalization (weak anchoring.) We show that for strong anchoring strength (relative to the usual Ginzburg-Landau length scale parameter,) defects will occur in the interior, as in the case of strong (Dirichlet) anchoring, but for weaker anchoring strength all defects will occur on the boundary. These defects will each carry a fractional winding number; such boundary defects are known as "boojums". The boojums will occur in ordered pairs along the boundary; for nematic director with angle oblique to the normal vector, they serve to reduce the winding of the phase in two steps, in order to avoid the formation of interior defects. We determine the number and location of the defects via a Renormalized Energy and numerical simulations.

math.AP↗

arXiv2019

A Novel Landau-de Gennes Model with Quartic Elastic Terms

Within the framework of the generalized Landau-de Gennes theory, we identify a $Q$-tensor-based energy that reduces to the four-constant Oseen-Frank energy when it is considered over orientable uniaxial nematic states. Although the commonly considered version of the Landau-de Gennes theory has an elastic contribution that is at most cubic in components of the $Q$-tensor and their derivatives, the alternative offered here is quartic in these variables. One clear advantage of our approach over the cubic theory is that the associated minimization problem is well-posed for a significantly wider choice of elastic constants. In particular, quartic energy can be used to model nematic-to-isotropic phase transitions for highly disparate elastic constants. In addition to proving well-posedness of the proposed version of the Landau-de Gennes theory, we establish a rigorous connection between this theory and its Oseen-Frank counterpart via a $Γ$-convergence argument in the limit of vanishing nematic correlation length. We also prove strong convergence of the associated minimizers.

cond-mat.soft↗

arXiv2019

Phase Transitions in Nematics: Textures with Tactoids and Disclinations

We demonstrate that a first order isotropic-to-nematic phase transition in liquid crystals can be succesfully modeled within the generalized Landau-de Gennes theory by selecting an appropriate combination of elastic constants. The numerical simulations of the model established in this paper qualitatively reproduce the experimentally observed configurations that include interfaces and topological defects in the nematic phase.

cond-mat.soft↗

arXiv2019

Existence of superconducting solutions for a reduced Ginzburg-Landau model in the presence of strong electric currents

In this work we consider a reduced Ginzburg-Landau model in which the magnetic field is neglected and the magnitude of the current density is significantly stronger than that considered in a recent work by the same authors. We prove the existence of a solution which can be obtained by solving a non-convex minimization problem away from the boundary of the domain. Near the boundary, we show that this solution is essentially one-dimensional. We also establish some linear stability results for a simplified, one-dimensional version of the original problem.

math-ph↗

arXiv2018

A Model Problem for Nematic-Isotropic Transitions with Highly Disparate Elastic Constants

We analyze a model problem based on highly disparate elastic constants that we propose in order to understand corners and cusps that form on the boundary between the nematic and isotropic phases in a liquid crystal. For a bounded planar domain $Ω$ we investigate the $\varepsilon \to 0$ asymptotics of the variational problem \[\inf \frac{1}{2}\int_Ω\left( \frac{1}{\varepsilon} W(u)+\varepsilon |\nabla u|^2 + L_\varepsilon(\mathrm{div}\, u)^2 \right) \,dx\] within various parameter regimes for $L_\varepsilon > 0.$ Here $u:Ω\to\mathbb{R}^2$ and $W$ is a potential vanishing on the unit circle and at the origin. When $\varepsilon\ll L_\varepsilon\to 0$, we show that these functionals $Γ-$converge to a constant multiple of the perimeter of the phase boundary and the divergence penalty is not felt. However, when $L_\varepsilon \equiv L > 0$, we find that a tangency requirement along the phase boundary for competitors in the conjectured $Γ$-limit becomes a mechanism for development of singularities. We establish criticality conditions for this limit and under a non-degeneracy assumption on the potential we prove compactness of energy bounded sequences in $L^2$. The role played by this tangency condition on the formation of interfacial singularities is investigated through several examples: each of these examples involves analytically rigorous reasoning motivated by numerical experiments. We argue that generically, "wall" singularities between $\mathbb{S}^1$-valued states are expected near the defects along the phase boundary.

math.AP↗

arXiv2018

A Ginzburg-Landau type problem for highly anisotropic nematic liquid crystals

We carry out an asymptotic analysis of a thin nematic liquid crystal in which one elastic constant dominates over the others, namely \begin{align} \label{energyab} \inf E_\varepsilon(u)\quad\mbox{where}\quad E_\varepsilon(u) := \frac{1}{2}\int_Ω\left\{\varepsilon\,|\nabla u|^2 + \frac{1}{\varepsilon} \,(|u|^2 - 1)^2 + L \,(\mathrm{div}\,u)^2\right\} \,dx. \end{align} Here $u: Ω\to \mathbb R^2$ is a vector field, $0 < \varepsilon \ll 1 $ is a small parameter, and $L > 0$ is a fixed constant, independent of $\varepsilon$. We derive the $Γ$-limit $E_0$, which is a sum of a bulk term penalizing divergence and an Aviles-Giga type wall energy involving the cube of the jump in the tangential component of the $\mathbb{S}^1$-valued order parameter. We then derive criticality conditions for $E_0$ and analyze minimization of $E_0$ both rigorously and numerically for various domains $Ω$ and a variety of Dirichlet boundary conditions.

math.AP↗

arXiv2017

A Discrete-to-Continuum Model of Weakly Interacting Incommensurate Chains

In this paper we use a formal discrete-to-continuum procedure to derive a continuum variational model for two chains of atoms with slightly incommensurate lattices. The chains represent a cross-section of a three-dimensional system consisting of a graphene sheet suspended over a substrate. The continuum model recovers both qualitatively and quantitatively the behavior observed in the corresponding discrete model. The numerical solutions for both models demonstrate the presence of large commensurate regions separated by localized incommensurate domain walls.

cond-mat.mtrl-sci↗

arXiv2017

A Discrete-to-Continuum Model of Weakly Interacting Incommensurate Two-Dimensional Lattices

In this paper we propose a continuum variational model for a two-dimensional deformable lattice of atoms interacting with a two-dimensional rigid lattice. The two lattices have slightly different lattice parameters and there is a small relative rotation between them. This is a prototypical example of a three-dimensional system consisting of a graphene sheet suspended over a substrate. The continuum model recovers both qualitatively and quantitatively the behavior observed in the corresponding discrete model. The continuum model predicts that the deformable lattice develops a network of domain walls characterized by large shearing, stretching, and bending deformation that accommodate the misalignment and/or mismatch between the deformable and rigid lattices. Two integer-valued parameters that can be identified with the components of a Burgers vector, describe the mismatch between the lattices and determine the geometry and the detail of deformation associated with the domain walls.

cond-mat.mtrl-sci↗

arXiv2017

Dimension reduction for the Landau-de Gennes model on curved nematic thin films

We use the method of $Γ$-convergence to study the behavior of the Landau-de Gennes model for a nematic liquid crystalline film attached to a general fixed surface in the limit of vanishing thickness. This paper generalizes the approach that we used previously to study a similar problem for a planar surface. Since the anchoring energy dominates when the thickness of the film is small, it is essential to understand its influence on the structure of the minimizers of the limiting energy. In particular, the anchoring energy dictates the class of admissible competitors and the structure of the limiting problem. We assume general weak anchoring conditions on the top and the bottom surfaces of the film and strong Dirichlet boundary conditions on the lateral boundary of the film when the surface is not closed. We establish a general convergence result to an energy defined on the surface that involves a somewhat surprising remnant of the normal component of the tensor gradient. Then we exhibit one effect of curvature through an analysis of the behavior of minimizers to the limiting problem when the substrate is a frustrum.

math.AP↗

arXiv2017

Q-tensor model for electrokinetics in nematic liquid crystals

We use a variational principle to derive a mathematical model for a nematic electrolyte in which the liquid crystalline component is described in terms of a second-rank order tensor. The model extends the previously developed director-based theory and accounts for presence of disclinations and possible biaxiality. We verify the model by considering a simple but illustrative example of liquid crystal-enabled electro-osmotic flow (LCEO) around a stationary dielectric spherical particle placed at the center of a large cylindrical container filled with a nematic electrolyte. Assuming homeotropic anchoring of the nematic on the surface of the particle and uniform distribution of the director on the surface of the container, we consider two configurations with a disclination equatorial ring and with a hyperbolic hedgehog, respectively. The computed electro-osmotic flows show a strong dependence on the director configurations and on the anisotropies of dielectric permittivity and electric conductivity of the nematic characteristic of liquid crystal-enabled electrokinetics. Further, the simulations demonstrate space charge separation around the dielectric sphere, even in the case of isotropic permittivity and conductivity. This is in agreement with the induced-charge electro-osmotic effect described for isotropic electrolytes surrounding dielectric spheres.

cond-mat.soft↗

arXiv2017

Strongly nonlinear asymptotic model of cellular instabilities in premixed flames with stepwise ignition temperature kinetics

In this paper we consider ignition-temperature, first-order reaction model of thermo-diffusive combustion that describes dynamics of thick flames arising in a theory of combustion of hydrogen-oxygen and ethylene-oxygen mixtures. These flames often assume the shape of propagating curved interfaces that correspond to level sets of constant temperature. We derive a fully nonlinear equation that governs dynamics of these level sets under a single assumption of small curvature. We study this equation for various asymptotic parameter regimes and discuss the ranges of validity of the corresponding simplified models. Our theoretical findings are supported by numerical simulations.

nlin.CD↗

arXiv2017

On approximation of Ginzburg-Landau minimizers by $\mathbb S^1$-valued maps in domains with vanishingly small holes

We consider a two-dimensional Ginzburg-Landau problem on an arbitrary domain with a finite number of vanishingly small circular holes. A special choice of scaling relation between the material and geometric parameters (Ginzburg-Landau parameter vs hole radius) is motivated by a recently dsicovered phenomenon of vortex phase separation in superconducting composites. We show that, for each hole, the degrees of minimizers of the Ginzburg-Landau problems in the classes of $\mathbb S^1$-valued and $\mathbb C$-valued maps, respectively, are the same. The presence of two parameters that are widely separated on a logarithmic scale constitutes the principal difficulty of the analysis that is based on energy decomposition techniques.

math.AP↗

arXiv2016

Electro-osmosis in nematic liquid crystals

We derive a mathematical model of a nematic electrolyte based on a variational formulation of nematodynamics. Extending our previous work, we consider a general setup which incorporates dielectric anisotropy of the liquid-crystalline matrix and the full set of nematic viscosities. We verify the model by comparing its predictions to the results of the experiments on the substrate-controlled liquid-crystal-enabled electrokinetics. In the experiments a nematic liquid crystal confined to a thin planar cell with surface-patterned anchoring conditions exhibit electro-osmotic flows along the "guiding rails" imposed by the spatially varying director.

cond-mat.soft↗

arXiv2016

Modeling of nematic electrolyte and nonlinear electroosmosis

We derive a mathematical model of a nematic electrolyte based on the Leslie-Ericksen theory of liquid crystal flow. Our goal is to investigate the nonlinear electrokinetic effects that occur because the nematic matrix is anisotropic, in particular, transport of ions in a direction perpendicular to the electric field as well as quadratic dependence of the induced flow velocity on the electric field. The latter effect makes it possible to generate sustained flows in the nematic electrolyte that do not reverse their direction when the polarity of the applied electric field is reversed. From a practical perspective, this enables the design of AC-driven electrophoretic and electroosmotic devices. Our study of a special flow in a thin nematic film shows a very good agreement with laboratory experiments.

cond-mat.soft↗

arXiv2016

Euler elastica as a $Γ$-Limit of discrete bending energies of one-dimensional chains of atoms

This work is motivated by discrete-to-continuum modeling of the mechanics of a graphene sheet, which is a single-atom thick macromolecule of carbon atoms covalently bonded to form a hexagonal lattice. The strong covalent bonding makes the sheet essentially inextensible and gives the sheet a resistance to bending. We study a one-dimensional atomistic model that describes the cross-section of a graphene sheet as a collection of rigid links connected by torsional springs. $Γ$-convergence is used to rigorously justify an upscaling procedure for the discrete bending energy of the atomistic model. Our result establishes that as the bond length in the atomistic model goes to 0, the bending energies $Γ$-converge to Euler's elastica.

math-ph↗