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Dmitry Golovaty

Publications and source records attributed to Dmitry Golovaty.

45 records · Page 3Linked to original sources
arXiv2016

A non-traditional view on the modeling of nematic disclination dynamics

Nonsingular disclination dynamics in a uniaxial nematic liquid crystal is modeled within a mathematical framework where the kinematics is a direct extension of the classical way of identifying these line defects with singularities of a unit vector field representing the nematic director. It is well known that the universally accepted Oseen-Frank energy is infinite for configurations that contain disclination line defects. We devise a natural augmentation of the Oseen-Frank energy to account for physical situations where, under certain conditions, infinite director gradients have zero associated energy cost, as would be necessary for modeling half-integer strength disclinations within the framework of the director theory. Equilibria and dynamics (in the absence of flow) of line defects are studied within the proposed model. Using appropriate initial/boundary data, the gradient-flow dynamics of this energy leads to non-singular, line defect equilibrium solutions, including those of half-integer strength. However, we demonstrate that the gradient flow dynamics for this energy is not able to adequately describe defect evolution. Motivated by similarity with dislocation dynamics in solids, a novel 2D-model of disclination dynamics in nematics is proposed. The model is based on the extended Oseen-Frank energy and takes into account thermodynamics and the kinematics of conservation of defect topological charge. We validate this model through computations of disclination equilibria, annihilation, repulsion, and splitting. We show that the energy function we devise, suitably interpreted, can serve as well for the modeling of equilibria and dynamics of dislocation line defects in solids making the conclusions of this paper relevant to mechanics of both solids and liquid crystals.

cond-mat.soft↗
arXiv2015

Dimension reduction for the Landau-de Gennes model in planar 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 in the limit of vanishing thickness. In this asymptotic regime, surface energy plays a greater role and we take particular care in understanding its influence on the structure of the minimizers of the derived two-dimensional energy. We assume general weak anchoring conditions on the top and the bottom surfaces of the film and the strong Dirichlet boundary conditions on the lateral boundary of the film. The constants in the weak anchoring conditions are chosen so as to enforce that a surface-energy-minimizing nematic $Q$-tensor has the normal to the film as one of its eigenvectors. We establish a general convergence result and then discuss the limiting problem in several parameter regimes.

math.AP↗
arXiv2015

Numerical analysis of the vertex models for simulating grain boundary networks

Polycrystalline materials undergoing coarsening can be represented as evolving networks of grain boundaries, whose statistical characteristics determine macroscopic materials properties. The process of formation of various statistical distributions is extremely complex and is strongly influenced by topological changes in the network. This work is an attempt to elucidate the role of these changes by conducting a thorough numerical investigation of one of the simplest types of grain growth simulation models, called vertex models. While having obvious limitations in terms of its ability to represent realistic systems, the vertex model enables full control over topological transitions and retains essential geometric features of the network. We formulate a self-consistent vertex model and investigate the role of microscopic parameters on the mesoscale network behavior. This study sheds light onto several important questions, such as how statistics are affected by the choice of temporal and spatial resolution and rules governing topological changes. Statistical analysis of the data produced by the simulation is performed for both isotropic and anisotropic grain boundary energy.

cond-mat.mtrl-sci↗
arXiv2014

On Minimizers of the Landau-de Gennes Energy Functional on Planar Domains

We study tensor-valued minimizers of the Landau-de Gennes energy functional on a simply-connected planar domain $Ω$ with non-contractible boundary data. Here the tensorial field represents the second moment of a local orientational distribution of rod-like molecules of a nematic liquid crystal. Under the assumption that the energy depends on a single parameter---a dimensionless elastic constant $\eps>0$---we establish that, as $\eps\to0$, the minimizers converge to a projection-valued map that minimizes the Dirichlet integral away from a single point in $Ω$. We also provide a description of the limiting map.

math.AP↗
arXiv2013

Effective models for nematic liquid crystals composites with ferromagnetic inclusions

Molecules of a nematic liquid crystal respond to an applied magnetic field by reorienting themselves in the direction of the field. Since the dielectric anisotropy of a nematic is small, it takes relatively large fields to elicit a significant liquid crystal response. The interaction may be enhanced in colloidal suspensions of ferromagnetic particles in a liquid crystalline matrix---ferronematics--- as proposed by Brochard and de Gennes in 1970. The ability of these particles to align with the field and, simultaneously, cause reorientation of the nematic molecules, greatly increases the magnetic response of the mixture. Essentially the particles provide an easy axis of magnetization that interacts with the liquid crystal via surface anchoring. We derive an expression for the effective energy of ferronematic in the dilute limit, that is, when the number of particles tends to infinity while their total volume fraction tends to zero. The total energy of the mixture is assumed to be the sum of the bulk elastic liquid crystal contribution, the anchoring energy of the liquid crystal on the surfaces of the particles, and the magnetic energy of interaction between the particles and the applied magnetic field. The homogenized limiting ferronematic energy is obtained rigorously using a variational approach. It generalizes formal expressions previously reported in a physical literature.

math.AP↗
arXiv2010

Radially symmetric minimizers for a $p$-Ginzburg Landau type energy in $\R^2$

We consider the minimization of a p-Ginzburg-Landau energy functional over the class of radially symmetric functions of degree one. We prove the existence of a unique minimizer in this class, and show that its modulus is monotone increasing and concave. We also study the asymptotic limit of the minimizers as p \rightarrow \infty. Finally, we prove that the radially symmetric solution is locally stable for $p$ in the interval $(2,4]$.

math.AP↗
arXiv2007

Polygonization of carbon nanotubes

We use a multiscale procedure to derive a simple continuum model of multiwalled carbon nanotubes that takes into account both strong covalent bonds within graphene layers and weak bonds between atoms in different layers. The model predicts polygonization of crossections of large multiwalled nanotubes as a consequence of their curvature-induced turbostratic structure.

cond-mat.mtrl-sci↗
arXiv2006

Capacity of a multiply-connected domain and nonexistence of Ginzburg-Landau minimizers with prescribed degrees on the boundary

Suppose that $ω\subsetΩ\subset R^2$. In the annular domain $A=Ω\setminus\barω$ we consider the class $J$ of complex valued maps having degree 1 on $\partial Ω$ and on $\partialω$. It was conjectured by Berlyand and Mironescu ('04), that he existence of minimizers of the Ginzburg-Landau energy $E_κ$ in $J$ is completely determined by the value of the $H^1$-capacity $cap(A)$ of the domain and the value of the Ginzburg-Landau parameter $κ$. The existence of minimizers of $E_κ$ for all $κ$ when $cap(A)\geqπ$ (domain $A$ is ``thin'') and for small $κ$ when $cap(A)<π$ (domain $A$ is ``thick'') was established by Berlyand and Mironescu ('04). Here we provide the answer for the remaining case of large $κ$ when $cap(A)<π$. We prove that, when $cap(A)<π$, there exists a finite threshold value $κ_1$ of the Ginzburg-Landau parameter $κ$ such that the minimum of the Ginzburg-Landau energy $E_κ$ is not attained in $J$ when $κ>κ_1$ while it is attained when $κ<κ_1$.

math.AP↗