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Apala Majumdar

Publications and source records attributed to Apala Majumdar.

At least 55 records · Page 3Linked to original sources

Dynamic tuning of the director field in liquid crystal shells using block copolymers

When a nematic liquid crystal (LC) is confined on a self-closing spherical shell, topological constraints arise with intriguing consequences that depend critically on how the LC is aligned in the shell. We demonstrate reversible dynamic tuning of the alignment, and thereby the topology, of nematic LC shells stabilized by the nonionic amphiphilic block copolymer Pluronic F127. Deep in the nematic phase, the director is tangential to the interface, but upon approaching the temperature TNI of the nematic-isotropic transition, the director realigns to normal. We link this to a delicate interplay between an interfacial tension that is nearly independent of director orientation, and the configuration-dependent elastic deformation energy of an LC confined in a shell. The process is primarily triggered by the heating-induced reduction of the nematic order parameter, hence realignment temperatures differ by several tens of degrees between LCs with high and low TNI, respectively. The temperature of realignment is always lower on the positive-curved shell outside than at the negative-curved inside, yielding a complex topological reconfiguration on heating. Complementing experimental investigations with mathematical modeling and computer simulations, we identify and investigate three different trajectories, distinguished by their configurations of topological defects in the initial tangential-aligned shell. Our results uncover a new aspect of the complex response of LCs to curved confinement, demonstrating that the order of the LC itself can influence the alignment and thereby the topology of the system. They also reveal the potential of amphiphilic block copolymer stabilizers for enabling continuous tunability of LC shell configuration, opening doors for in-depth studies of topological dynamics as well as novel applications in, e.g., sensing and programmed soft actuators.

cond-mat.soft↗

Discontinuous Galerkin Finite Element Methods for the Landau-de Gennes Minimization Problem of Liquid Crystals

We consider a system of second order non-linear elliptic partial differential equations that models the equilibrium configurations of a two dimensional planar bistable nematic liquid crystal device. Discontinuous Galerkin finite element methods are used to approximate the solutions of this nonlinear problem with non-homogeneous Dirichlet boundary conditions. A discrete inf-sup condition demonstrates the stability of the discontinuous Galerkin discretization of a well-posed linear problem. We then establish the existence and local uniqueness of the discrete solution of the non-linear problem. An a priori error estimate in the energy norm is derived and a best approximation property is demonstrated. Further, we prove the quadratic convergence of Newton's iterates along with complementary numerical experiments.

math.NA↗

Tailored Morphologies in 2D Ferronematic Wells

We focus on a dilute uniform suspension of magnetic nanoparticles in a nematic-filled micron-sized shallow well with tangent boundary conditions, as a paradigm system with two coupled order parameters. This system exhibits spontaneous magnetization without magnetic fields. We numerically obtain the stable nematic and associated magnetization morphologies, induced purely by the geometry, boundary conditions and the coupling between the magnetic nanoparticles and the host nematic medium. Our most striking observations pertain to domain walls in the magnetization profile whose location can be manipulated by the coupling and material properties, and stable interior and boundary nematic defects, whose location and multiplicity can be tailored by the coupling too. These novel morphologies are not accessible in uncoupled systems and can be used for new multistable systems with singularities and stable interfaces.

cond-mat.soft↗

Solution Landscapes in the Landau-de Gennes Theory on Rectangles

We study nematic equilibria on rectangular domains, in a reduced two-dimensional Landau-de Gennes framework. These reduced equilibria carry over to the three-dimensional framework at a special temperature. There is one essential model variable---$ε$ which is a geometry-dependent and material-dependent variable. We compute the limiting profiles exactly in two distinguished limits---the $ε\to 0$ limit relevant for macroscopic domains and the $ε\to \infty$ limit relevant for nano-scale domains. The limiting profile has line defects near the shorter edges in the $ε\to \infty$ limit whereas we observe fractional point defects in the $ε\to 0$ limit. The analytical studies are complemented by bifurcation diagrams for these reduced equilibria as a function of $ε$ and the rectangular aspect ratio. We also introduce the concept of `non-trivial' topologies and the relaxation of non-trivial topologies to trivial topologies mediated via point and line defects, with potential consequences for non-equilibrium phenomena and switching dynamics.

math-ph↗

The Well Order Reconstruction Solution for Three-Dimensional Wells, in the Landau-de Gennes theory

We study nematic equilibria on three-dimensional square wells, with emphasis on Well Order Reconstruction Solutions (WORS) as a function of the well size, characterized by $λ$, and the well height denoted by $ε$. The WORS are distinctive equilibria reported in [10] for square domains, without taking the third dimension into account, which have two mutually perpendicular defect lines running along the square diagonals, intersecting at the square centre. We prove the existence of WORS on three-dimensional wells for arbitrary well heights, with (i) natural boundary conditions and (ii) realistic surface energies on the top and bottom well surfaces, along with Dirichlet conditions on the lateral surfaces. Moreover, the WORS is globally stable for $λ$ small enough in both cases and unstable as $λ$ increases. We numerically compute novel mixed 3D solutions for large $λ$ and $ε$ followed by a numerical investigation of the effects of surface anchoring on the WORS, exemplifying the relevance of the WORS solution in a 3D context.

math.AP↗

Magnetic Nanoparticles in a Nematic Channel: A One-Dimensional Study

We study a ferromagnetic suspension or a suspension of magnetic nanoparticles in an anisotropic nematic medium, in three different one-dimensional variational settings, ordered in terms of increasing complexity. The three models are featured by a nematic energy, a magnetic energy and a magneto-nematic coupling energy and the experimentally observed patterns are modelled as local or global energy minimizers. We numerically observe polydomains with distinct states of magnetization for weak to moderate magneto-nematic coupling in our models. We demonstrate that these polydomains are stabilised by lowering the temperature (as in Mertelj et al., 2013) and that the polydomain structures lose stability as the magneto-nematic coupling increases. Some exact solutions for prototypical situations are also obtained.

cond-mat.soft↗

Minimizers of a Landau-de Gennes Energy with a Subquadratic Elastic Energy

We study a modified Landau-de Gennes model for nematic liquid crystals, where the elastic term is assumed to be of subquadratic growth in the gradient. We analyze the behaviour of global minimizers in two- and three-dimensional domains, subject to uniaxial boundary conditions, in the asymptotic regime where the length scale of the defect cores is small compared to the length scale of the domain. We obtain uniform convergence of the minimizers and of their gradients, away from the singularities of the limiting uniaxial map. We also demonstrate the presence of maximally biaxial cores in minimizers on two-dimensional domains, when the temperature is sufficiently low.

math.AP↗

Theoretical Remarks on Cybotactic Clusters of Bent-Core Nematic Liquid Crystals in 1D Settings

The bent-core liquid crystals (LCs) are highly regarded as the next-generation materials for electro-optic devices. The nematic (N) phase of these LCs possesses highly ordered smectic-like cybotactic clusters which are promising in terms of ferroelectric-like behaviour in the N phase itself. We have studied a one-dimensional (1D) Landau-deGennes model of spatially inhomogeneous order parameters for the N phase of bent-core LCs. We investigate the effects of spatial confinement and coupling (between these clusters and the surrounding LC molecules) on the order parameters to model cluster formation in recently reported experiments. The coupling is found to increase the cluster order parameter significantly, suggesting an enhancement in the cluster formation and could also predict a possible transition to a phase with weak nematic-like ordering in the vicinity of nematic-isotropic transition upon appreciable increase of the coupling parameter γ.

cond-mat.soft↗

Order Reconstruction for Nematics on Squares with Isotropic Inclusions: A Landau-de Gennes Study

We prove the existence of a well order reconstruction solution (WORS) type Landau-de Gennes critical point on a square domain with an isotropic concentric square inclusion, with tangent boundary conditions on the outer square edges. There are two geometrical parameters - the outer square edge length $λ$, and the aspect ratio $ρ$, which is the ratio of the inner and outer square edge lengths. The WORS exists for all geometrical parameters and for all temperatures; we prove that the WORS is globally stable for either $λ$ small enough or for $ρ$ sufficiently close to unity. We study three different types of critical points in this model setting: critical points with the minimal two degrees of freedom consistent with the imposed boundary conditions, critical points with three degrees of freedom and critical points with five degrees of freedom. In the two-dimensional case, we use $Γ$-convergence techniques to identify the energy-minimizing competitors. We decompose the second variation of the Landau-de Gennes energy into three separate components to study the effects of different types of perturbations on the WORS solution and find that it is most susceptible to in-plane perturbations. In the three-dimensional setting, we numerically find up to $28$ critical points for moderately large values of $ρ$ and we find two critical points with the full five degrees of freedom for very small values of $ρ$, with an escaped profile around the isotropic square inclusion.

math.AP↗

Global Weak Solutions for the Compressible Active Liquid Crystal System

We study the hydrodynamics of compressible flows of active liquid crystals in the Beris-Edwards hydrodynamics framework, using the Landau-de Gennes $Q$-tensor order parameter to describe liquid crystalline ordering. We prove the existence of global weak solutions for this active system in three space dimensions by the three-level approximations and weak convergence argument. New techniques and estimates are developed to overcome the difficulties caused by the active terms.

math.AP↗

Remarks on Uniaxial Solutions in the Landau-de Gennes Theory

We study uniaxial solutions of the Euler-Lagrange equations for a Landau-de Gennes free energy for nematic liquid crystals, with a fourth order bulk potential, with and without elastic anisotropy. In the elastic isotropic case, we show that (i) all uniaxial solutions of the Euler-Lagrange equations, with a director field of certain symmetry, necessarily have the radial-hedgehog structure modulo an orthogonal transformation, (ii) the "escape into third dimension" director cannot correspond to a purely uniaxial solution of the Landau-de Gennes Euler-Lagrange equations and we do not use artificial assumptions on the scalar order parameter and (iii) there are no non-trivial uniaxial solutions that have $\mathbf{e}_z$ as an eigenvector. In the elastic anisotropic case, we prove that all uniaxial solutions of the corresponding Euler-Lagrange equations, with a certain symmetry, are strictly of the radial-hedgehog type, i.e. the elastic anisotropic case enforces the radial-hedgehog structure (or the degree $+1$-vortex structure) more strongly than the elastic isotropic case and the associated partial differential equations are technically far more difficult than in the elastic isotropic case.

math.AP↗

Nematohydrodynamics for Colloidal Self-Assembly and Transport Phenomena

We study colloidal particles in a nematic-liquid-crystal-filled microfluidic channel and show how elastic interactions between the particle and the channel wall lead to different particle dynamics compared with conventional microfluidics. For a static particle, in the absence of a flow field, the director orientation on the particle surface undergoes a rapid transition from uniform to homeotropic anchoring as a function of the anchoring strength and the particle size. In the presence of a flow field, in addition to fluid viscous stresses on the particle, nematic-induced elastic stresses are exerted as a result of the anchoring conditions. The resulting forces are shown to offer the possibility of size-based separation of individual particles. For multi-particle systems, the nematic forces induce inter-particle attraction induces particle attraction, following which the particles aggregate and reorientate. These results illustrate how coupled nematic-hydrodynamic effects can affect the mobility and spatial reorganization of colloidal particles in microfluidic applications.

cond-mat.soft↗

Global Existence and Regularity for the Active Liquid Crystal System

In this paper, we study the active hydrodynamics, described in the Q-tensor liquid crystal framework. We prove the existence of global weak solutions in dimension two and three, with suitable initial datas. By using Littlewood-Paley decomposition, we also get the higher regularity of the weak solutions and the uniqueness of weak-strong solutions in dimension two.

math.AP↗

Solution landscapes in nematic microfluidics

We study the static equilibria of a simplified Leslie--Ericksen model for a unidirectional uniaxial nematic flow in a prototype microfluidic channel, as a function of the pressure gradient $\mathcal{G}$ and inverse anchoring strength, $\mathcal{B}$. We numerically find multiple static equilibria for admissible pairs $(\mathcal{G}, \mathcal{B})$ and classify them according to their winding numbers and stability. The case $\mathcal{G}=0$ is analytically tractable and we numerically study how the solution landscape is transformed as $\mathcal{G}$ increases. We study the one-dimensional dynamical model, the sensitivity of the dynamic solutions to initial conditions and the rate of change of $\mathcal{G}$ and $\mathcal{B}$. We provide a physically interesting example of how the time delay between the applications of $\mathcal{G}$ and $\mathcal{B}$ can determine the selection of the final steady state.

math.AP↗

Order Reconstruction for Nematics on Squares and Regular Polygons: A Landau-de Gennes Study

We construct an order reconstruction (OR)-type Landau-de Gennes critical point on a square domain of edge length $λ$, motivated by the well order reconstruction solution numerically reported by Kralj and Majumdar. The OR critical point is distinguished by an uniaxial cross with negative scalar order parameter along the square diagonals. The OR critical point is defined in terms of a saddle-type critical point of an associated scalar variational problem. The OR-type critical point is globally stable for small $λ$ and undergoes a supercritical pitchfork bifurcation in the associated scalar variational setting. We consider generalizations of the OR-type critical point to a regular hexagon, accompanied by numerical estimates of stability criteria of such critical points on both a square and a hexagon in terms of material-dependent constants.

math.AP↗

Front Propagation at the Isotropic Nematic Transition Temperature

We study the gradient flow model for the Landau-de Gennes energy functional for nematic liquid crystals at the nematic-isotropic transition temperature on prototype geometries. We study the dynamic model on a three-dimensional droplet and on a disc with Dirichlet boundary conditions and different types of initial conditions. In the case of a droplet with radial boundary conditions, a large class of physically relevant initial conditions generate dynamic solutions with a well-defined isotropic-nematic interface which propagates according to mean curvature for small times. On a disc, we make a distinction between "planar" and "non-planar" initial conditions and "minimal" and "non-minimal" Dirichlet boundary conditions. Planar initial conditions generate solutions with an isotropic core for all times whereas non-planar initial conditions generate solutions which escape into the third dimension. Non-minimal boundary conditions generate solutions with boundary layers and these solutions can either have a largely ordered interior profile or an almost entirely disordered isotropic interior profile. Our examples suggest that whilst critical points of the Landau-de Gennes energy typically have highly localized disordered-ordered interfaces, the transient dynamics exhibit observable isotropic-nematic interfaces of potential experimental relevance.

math.AP↗

Radial Symmetry on 3D Shells in the Landau-de Gennes Theory

We study the stability of the radial-hedgehog solution on a three-dimensional (3D) spherical shell with radial boundary conditions, within the Landau-de Gennes theory for nematic liquid crystals. We show that the radial-hedgehog solution has no zeroes for a sufficiently narrow shell, for all temperatures below the nematic supercooling temperature. We prove that the radial-hedgehog solution is the unique global Landau-de Gennes energy minimizer for a sufficiently narrow 3D spherical shell, for all temperatures below the nematic supercooling temperature. We provide explicit geometry-dependent criteria for the global minimality of the radial-hedgehog solution in this temperature regime. In the low temperature limit, we prove the global minimality of the radial-hedgehog solution on a 3D spherical shell, for all values of the inner and outer radii.

math.AP↗

Uniaxial versus Biaxial Character of Nematic Equilibria in Three Dimensions

We study global minimizers of the Landau-de Gennes (LdG) energy functional for nematic liquid crystals, on arbitrary three-dimensional simply connected geometries with topologically non-trivial and physically relevant Dirichlet boundary conditions. Our results are specific to the low-temperature limit. We prove (i) that (re-scaled) global LdG minimizers converge uniformly to a (minimizing) limiting harmonic map, away from the singular set of the limiting map; (ii) there exist both a point of maximal biaxiality and a nonempty Lebesgue-null set of uniaxial points near each singular point of the limiting harmonic map (this improves the recent results of \cite{contreraslamy}); (iii) estimates for the size of "strongly biaxial" regions in terms of the reduced temperature $t$. We further show that global LdG minimizers in the restricted class of uniaxial $\Qvec$-tensors cannot be stable critical points of the LdG energy for low temperatures.

math.AP↗