arXiv subjects

Apala Majumdar

Publications and source records attributed to Apala Majumdar.

63 records · Page 4Linked to original sources

arXiv2015

Free energy pathways of a Multistable Liquid Crystal Device

The planar bistable device [Tsakonas \textit{et al., Appl. Phys. Lett.}, 2007, {\textbf{90}}, 111913] is known to have two distinct classes of stable equilibria: the diagonal and rotated solutions. We model this device within the two-dimensional Landau-de Gennes theory, with a surface potential and without any external fields. We systematically compute a special class of transition pathways, referred to as minimum energy pathways, between the stable equilibria that provide new information about how the equilibria are connected in the Landau-de Gennes free energy landscape. These transition pathways exhibit an intermediate transition state, which is a saddle point of the Landau-de Gennes free energy. We numerically compute the structural details of the transition states, the optimal transition pathways and the free energy barriers between the equilibria, as a function of the surface anchoring strength. For strong anchoring, the transition pathways are mediated by defects whereas we get defect-free transition pathways for moderate and weak anchoring. In the weak anchoring limit, we recover a cusp catastrophe situation for which the rotated state acts as a transition state connecting two different diagonal states.

cond-mat.soft↗

arXiv2015

Nematic equilibria on a two-dimensional annulus: defects and energies

We study planar nematic equilibria on a two-dimensional annulus with strong and weak tangent anchoring, within the Oseen-Frank and Landau-de Gennes theories for nematic liquid crystals. We analyse the defect-free state in the Oseen-Frank framework and obtain analytic stability criteria in terms of the elastic anisotropy, annular aspect ratio and anchoring strength. We consider radial and azimuthal perturbations of the defect-free state separately, which yields a complete stability diagram for the defect-free state. We construct nematic equilibria with an arbitrary number of defects on a two-dimensional annulus with strong tangent anchoring and compute their energies; these equilibria are generalizations of the diagonal and rotated states observed in a square. This gives novel insights into the correlation between preferred numbers of defects, their locations and the geometry. In the Landau-de Gennes framework, we adapt Mironescu's powerful stability result in the Ginzburg-Landau framework (P. Mironescu, On the stability of radial solutions of the Ginzburg-Landau equation, 1995) to compute quantitative criteria for the local stability of the defect-free state in terms of the temperature and geometry.

math.AP↗

arXiv2013

Twisted rods, helices and buckling solutions in three dimensions

The study of slender elastic structures is an archetypical problem in continuum mechanics, dynamical systems and bifurcation theory, with a rich history dating back to Euler's seminal work in the 18th century. These filamentary elastic structures have widespread applications in engineering and biology, examples of which include cables, textile industry, DNA experiments, collagen modelling etc. One is typically interested in the equilibrium configurations of these rod-like structures, their stability and dynamic evolution and all three questions have been extensively addressed in the literature. However, it is generally recognized that there are still several open non-trivial questions related to three-dimensional analysis of rod equilibria, inclusion of topological and positional constraints and different kinds of boundary conditions.

math-ph↗

arXiv2011

Molecular Tilt on Monolayer-Protected Nanoparticles

The structure of the tilted phase of monolayer-protected nanoparticles is investigated by means of a simple Ginzburg-Landau model. The theory contains two dimensionless parameters representing the preferential tilt angle and the ratio epsilon between the energy cost due to spatial variations in the tilt of the coating molecules and that of the van der Waals interactions which favors uniform tilt. We analyze the model for both spherical and octahedral particles. On spherical particles, we find a transition from a tilted phase, at small epsilon, to a phase where the molecules spontaneously align along the surface normal and tilt disappears. Octahedral particles have an additional phase at small epsilon characterized by the presence of six topological defects. These defective configurations provide preferred sites for the chemical functionalization of monolayer-protected nanoparticles via place-exchange reactions and their consequent linking to form molecules and bulk materials.

cond-mat.soft↗

arXiv2011

Symmetry of uniaxial global Landau-de Gennes minimizers in the theory of nematic liquid crystals

We study uniaxial energy-minimizers within the Landau-de Gennes theory for nematic liquid crystals on a three-dimensional spherical droplet subject to homeotropic boundary conditions. We work in the low-temperature regime and show that uniaxial energy-minimizers necessarily have the structure of the well-studied radial-hedgehog solution in the low-temperature limit. An immediate consequence of this result is that Landau-de Gennes energy minimizers cannot be purely uniaxial for sufficiently low temperatures.

math.AP↗

arXiv2010

The Radial-Hedgehog Solution in Landau--de Gennes' theory

We study the radial-hedgehog solution in a three-dimensional spherical droplet, with homeotropic boundary conditions, within the Landau-de Gennes theory for nematic liquid crystals. The radial-hedgehog solution is a candidate for a globally stable configuration in this model framework and is also a prototype configuration for studying isolated point defects in condensed matter physics. The static properties of the radial-hedgehog solution are governed by a nonlinear singular ordinary differential equation. We consider two different limits separately - the \emph{vanishing core limit} and \emph{low-temperature limit} respectively. We use a combination of Ginzburg-Landau techniques, perturbation methods and stability analysis to study the qualitative properties of the radial-hedgehog solution, both in the vicinity of and away from the defect core. We establish the instability of the radial-hedgehog solution with respect to biaxial perturbations in certain parameter regimes and demonstrate the stability of the radial-hedgehog solution in other parameter regimes. Our results complement previous work in the field, are rigorous in nature and give information about the role of geometry and temperature on the properties of the radial-hedgehog solution and the associated biaxial instabilities.

math.AP↗

arXiv2010

Uniaxiality in the Landau-de Gennes theory of nematic liquid crystals

We study uniaxial energy minimizers within the Landau-de Gennes theory for nematic liquid crystals, subject to dirichlet boundary conditions. Topological defects in such minimizers correspond to the zeros of the corresponding equilibrium field. We consider two-dimensional and three-dimensional domains separately and study the correspondence between Landau-de Gennes theory and Ginzburg-Landau theory for superconductors. We obtain results for the location and dimensionality of the defect set, the minimizer profile near the defect set and study the qualitative properties of uniaxial energy minimizers away from the defect set, in the physically relevant case of vanishing elastic constant. In the three-dimensional case, we establish the $C^{1,α}$-convergence of uniaxial minimizers to a limiting harmonic map, away from the defect set, for some $0<α<1$. Some generalizations for biaxial minimizers are also discussed. This work is motivated by the study of defects in liquid crystalline systems and their applications.

math.AP↗

arXiv2009

Equilibrium order parameters of liquid crystals in the Landau-De Gennes theory

We study nematic liquid crystal configurations in confined geometries within the continuum Landau--De Gennes theory. These nematic configurations are mathematically described by symmetric, traceless two-tensor fields, known as $\Qvec$-tensor order parameter fields. We obtain explicit upper bounds for the order parameters of equilibrium liquid crystal configurations in terms of the temperature, material constants, boundary conditions and the domain geometry. These bounds are compared with the bounds predicted by the statistical mechanics definition of the $\Qvec$-tensor order parameter. They give quantitative information about the temperature regimes for which the Landau-De Gennes definition and the statistical mechanics definition of the $\Qvec$-tensor order parameter agree and the temperature regimes for which the two definitions fail to agree. For the temperature regimes where the two definitions do not agree, we discuss possible alternatives.

math.AP↗

arXiv2008

Landau-De Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond

We study global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in $W^{1,2}$, to a global minimizer predicted by the Oseen-Frank theory for uniaxial nematic liquid crystals with constant order parameter. Moreover, the convergence is uniform in the interior of the domain, away from the singularities of the limiting Oseen-Frank global minimizer. We obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the Landau-De Gennes global minimizer. We also study the interplay between biaxiality and uniaxiality in Landau-De Gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions.

math.AP↗