arXiv subjects

Apala Majumdar

Publications and source records attributed to Apala Majumdar.

At least 37 records · Page 2Linked to original sources

arXiv2023

Stochastic effects on solution landscapes for nematic liquid crystals

We study the effects of additive and multiplicative noise on the solution landscape of nematic liquid crystals confined to a square domain within the Landau-de Gennes framework, as well as the impact of additive noise on the symmetric radial hedgehog solution for nematic droplets. The introduction of random noise can be used to capture material uncertainties and imperfections, which are always present in physical systems. We implement random noise in our framework by introducing a Q-Wiener stochastic process to the governing differential equations. On the square, the solution landscape for the deterministic problem is well understood, enabling us to compare and contrast the deterministic predictions and the stochastic predictions, while we demonstrate that the symmetry of the radial hedgehog solution can be violated by noise. This approach of introducing noise to deterministic equations can be used to test the robustness and validity of predictions from deterministic liquid crystal models, which essentially capture idealised situations.

cond-mat.soft↗

arXiv2023

Two-dimensional Ferronematics, Canonical Harmonic Maps and Minimal Connections

We study a variational model for ferronematics in two-dimensional domains, in the "super-dilute" regime. The free energy functional consists of a reduced Landau-de Gennes energy for the nematic order parameter, a Ginzburg-Landau type energy for the spontaneous magnetisation, and a coupling term that favours the co-alignment of the nematic director and the magnetisation. In a suitable asymptotic regime, we prove that the nematic order parameter converges to a canonical harmonic map with non-orientable point defects, while the magnetisation converges to a singular vector field, with line defects that connect the non-orientable point defects in pairs, along a minimal connection.

math.AP↗

arXiv2023

The Radial Hedgehog Solution in the Landau-de Gennes Theory: Effects of the Bulk Potentials

We study equilibrium configurations in spherical droplets of nematic liquid crystal with strong radial anchoring, within the Landau-de Gennes theory with a sixth-order bulk potential. The sixth-order potential predicts a bulk biaxial phase for sufficiently low temperatures, which the conventional fourth-order potential cannot predict. We prove the existence of a radial hedgehog solution, which is a uniaxial solution with a single isotropic point defect at the droplet centre, for all temperatures and droplet sizes, and prove that there is a unique radial hedgehog solution for moderately low temperatures, but not deep in the nematic phase. We numerically compute critical points of the Landau-de Gennes free energy with the sixth order bulk potential, with rotational and mirror symmetry, and find at least two competing stable critical points: the biaxial torus and split core solutions, which have biaxial regions around the centre, for low temperatures. The size of the biaxial regions increases with decreasing temperature. We also compare the properties of the radial hedgehog solution with the fourth-order and sixth-order potentials respectively, in terms of the Morse indices as a function of the temperature and droplet radius; the role of the radial hedgehog solution as a transition state in switching processes; and compare the bifurcation plots with temperature, with the fourth- and sixth-order potentials. Overall, the sixth-order potential has a stabilising effect on biaxial critical points and a de-stabilising effect on uniaxial critical points and we discover an altogether novel bulk biaxial critical point of the Landau-de Gennes energy with the sixth-order potential, for which the bulk biaxiality is driven by the sixth-order potential.

cond-mat.soft↗

arXiv2023

A Multi-Faceted Study of Nematic Order Reconstruction in Microfluidic Channels

We study order reconstruction (OR) solutions in the Beris-Edwards framework for nematodynamics, for both passive and active nematic flows in a microfluidic channel. OR solutions exhibit polydomains and domain walls, and as such, are of physical interest. We show that OR solutions exist for passive flows with constant velocity and pressure, but only for specific boundary conditions. We prove the existence of unique, symmetric and non-singular nematic profiles, for boundary conditions that do not allow for OR solutions. We compute asymptotic expansions for OR-type solutions for passive flows with non-constant velocity and pressure, and active flows, which shed light into the internal structure of domain walls. The asymptotics are complemented by extensive numerical studies that demonstrate the universality of OR-type structures in static and dynamic scenarios.

math.AP↗

arXiv2023

Hierarchies of Critical Points of a Landau-de Gennes Free Energy on Three-Dimensional Cuboids

We investigate critical points of a Landau-de Gennes (LdG) free energy in three-dimensional (3D) cuboids, that model nematic equilibria. We develop a hybrid saddle dynamics-based algorithm to efficiently compute solution landscapes of these 3D systems. Our main results concern (a) the construction of 3D LdG critical points from a database of 2D LdG critical points and (b) studies of the effects of cross-section size and cuboid height on solution landscapes. In doing so, we discover multiple-layer 3D LdG critical points constructed by stacking 3D critical points on top of each other, novel pathways between distinct energy minima mediated by 3D LdG critical points and novel metastable escaped solutions, all of which can be tuned for tailor-made static and dynamic properties of confined nematic liquid crystal systems in 3D.

cond-mat.soft↗

arXiv2022

Multistability for a Reduced Nematic Liquid Crystal Model in the Exterior of 2D Polygons

We study nematic equilibria in an unbounded domain, with a two-dimensional regular polygonal hole with $K$ edges, in a reduced Landau-de Gennes framework. This complements our previous work on the "interior problem" for nematic equilibria confined inside regular polygons (SIAM Journal on Applied Mathematics, 80(4):1678-1703, 2020). The two essential dimensionless model parameters are $λ$--the ratio of the edge length of polygon hole to the nematic correlation length, and an additional degree of freedom, $γ^*$--the nematic director at infinity. In the $λ\to 0$ limit, the limiting profile has two interior point defects outside a generic polygon hole, except for a triangle and a square. For a square hole, the limiting profile has either no interior defects or two line defects depending on $γ^*$, and for a triangular hole, there is a unique interior point defect outside the hole. In the $λ\to\infty$ limit, there are at least $\binom{K}{2}$ stable states %differentiated by the location of two bend vertices and the multistability is enhanced by $γ^*$, compared to the interior problem. Our work offers new insights into how to tune the existence, location, and dimensionality of defects.

math-ph↗

arXiv2022

Uniaxial versus Biaxial Pathways in One-Dimensional Cholesteric Liquid Crystals

Cholesteric liquid crystals exhibit great morphological richness of static metastable states. Understanding the transitions between such states is key for the development of switchable devices. We show, using a quasi-one-dimensional model, that cholesterics exhibit distinct uniaxial and biaxial pathways between distinct minima. We study transitions between different layer numbers and prove, and show, that transition states are distinguished either through splay-mediated untwisting, understood through contact topology, or the presence of biaxiality. Furthermore we characterise a menagerie of additional saddle points that dictate the connectivity of the solution landscape.

cond-mat.soft↗

arXiv2022

Weak solutions to the equations of stationary compressible flows in active liquid crystals

The equations of stationary compressible flows of active liquid crystals are considered in a bounded three-dimensional domain. The system consists of the stationary Navier-Stokes equations coupled with the equation of Q-tensors and the equation of the active particles. The existence of weak solutions to the stationary problem is established through a two-level approximation scheme, compactness estimates and weak convergence arguments. Novel techniques are developed to overcome the difficulties due to the lower regularity of stationary solutions, a Moser-type iteration is used to deal with the strong coupling of active particles and fluids, and some weighted estimates on the energy functions are achieved so that the weak solutions can be constructed for all values of the adiabatic exponent $γ>1$.

math.AP↗

arXiv2021

One-dimensional ferronematics in a channel: order reconstruction, bifurcations and multistability

We study a model system with nematic and magnetic orders, within a channel geometry modelled by an interval, $[-D, D]$. The system is characterised by a tensor-valued nematic order parameter $\mathbf{Q}$ and a vector-valued magnetisation $\mathbf{M}$, and the observable states are modelled as stable critical points of an appropriately defined free energy. In particular, the full energy includes a nemato-magnetic coupling term characterised by a parameter $c$. We (i) derive $L^\infty$ bounds for $\mathbf{Q}$ and $\mathbf{M}$; (ii) prove a uniqueness result in parameter regimes defined by $c$, $D$ and material- and temperature-dependent correlation lengths; (iii) analyse order reconstruction solutions, possessing domain walls, and their stabilities as a function of $D$ and $c$ and (iv) perform numerical studies that elucidate the interplay of $c$ and $D$ for multistability.

math.AP↗

arXiv2021

Pattern Formation for Nematic Liquid Crystals-Modelling, Analysis, and Applications

We summarise some recent results on solution landscapes for two-dimensional (2D) problems in the Landau--de Gennes theory for nematic liquid crystals. We study energy-minimizing and non energy-minimizing solutions of the Euler--Lagrange equations associated with a reduced Landau-de Gennes free energy on 2D domains with Dirichlet tangent boundary conditions. We review results on the multiplicity and regularity of solutions in distinguished asymptotic limits, using variational methods, methods from the theory of nonlinear partial differential equations, combinatorial arguments and scientific computation. The results beautifully canvass the competing effects of geometry (shape, size and symmetry), material anisotropy, and the symmetry of the model itself, illustrating the tremendous possibilities for exotic ordering transitions in 2D frameworks.

cond-mat.soft↗

arXiv2021

Parameter dependent finite element analysis for ferronematics solutions

This paper focuses on the analysis of a free energy functional, that models a dilute suspension of magnetic nanoparticles in a two-dimensional nematic well. The {\it first part} of the article is devoted to the asymptotic analysis of global energy minimizers in the limit of vanishing elastic constant, $\ell \rightarrow 0$ where the re-scaled elastic constant $\ell$ is inversely proportional to the domain area. The first results concern the strong $H^1$-convergence and a $\ell$-independent $H^2$-bound for the global minimizers on smooth bounded 2D domains, with smooth boundary and topologically trivial Dirichlet conditions. The {\it second part} focuses on the discrete approximation of regular solutions of the corresponding non-linear system of partial differential equations with cubic non-linearity and non-homogeneous Dirichlet boundary conditions. We establish (i) the existence and local uniqueness of the discrete solutions using fixed point argument, (ii) a best approximation result in energy norm, (iii) error estimates in the energy and $L^2$ norms with $\ell $- discretization parameter dependency for the conforming finite element method. Finally, the theoretical results are complemented by numerical experiments on the discrete solution profiles, the numerical convergence rates that corroborates the theoretical estimates, followed by plots that illustrate the dependence of the discretization parameter on $\ell$.

math.NA↗

arXiv2021

Solution Landscapes of the Simplified Ericksen--Leslie Model and its Comparison with the Reduced Landau--de Gennes Model

We investigate the solution landscapes of a simplified Ericksen--Leslie (sEL) vector model for nematic liquid crystals, confined in a two-dimensional square domain with tangent boundary conditions. An efficient numerical algorithm is developed to construct the solution landscapes by utilizing the symmetry properties of the model and the domain. Since the sEL model and the reduced Landau--de Gennes (rLdG) models can be viewed as Ginzburg--Landau functionals, we systematically compute the solution landscapes of the sEL model, for different domain sizes, and compare with the solution landscapes of the corresponding rLdG models. There are many similarities, including the stable diagonal and rotated states, bifurcation behaviors, and sub-solution landscapes with low-index saddle solutions. Significant disparities also exist between the two models. The sEL vector model exhibits the stable solution $C\pm$ with interior defects, high-index "fake defects" solutions, novel tessellating solutions, and certain types of distinctive dynamical pathways. The solution landscape approach provides a comprehensive and efficient way for model comparison and is applicable to a wide range of mathematical models in physics.

cond-mat.soft↗

arXiv2021

Elastic anisotropy of nematic liquid crystals in the two-dimensional Landau-de Gennes model

We study the effects of elastic anisotropy on the Landau-de Gennes critical points for nematic liquid crystals, in a square domain. The elastic anisotropy is captured by a parameter, $L_2$, and the critical points are described by three degrees of freedom. We analytically construct a symmetric critical point for all admissible values of $L_2$, which is necessarily globally stable for small domains i.e., when the square edge length, $λ$, is small enough. We perform asymptotic analyses and numerical studies to discover at least $5$ classes of these symmetric critical points - the $WORS$, $Ring^{\pm}$, $Constant$ and $pWORS$ solutions, of which the $WORS$, $Ring^+$ and $Constant$ solutions can be stable. Furthermore, we demonstrate that the novel $Constant$ solution is energetically preferable for large $λ$ and large $L_2$, and prove associated stability results that corroborate the stabilising effects of $L_2$ for reduced Landau-de Gennes critical points. We complement our analysis with numerically computed bifurcation diagrams for different values of $L_2$, which illustrate the interplay of elastic anisotropy and geometry for nematic solution landscapes, at low temperatures.

math.AP↗

arXiv2021

Tailored Nematic and Magnetization Profiles on 2D Polygons

We study dilute suspensions of magnetic nanoparticles in a nematic host, on two-dimensional (2D) polygons. These systems are described by a nematic order parameter and a spontaneous magnetization, in the absence of any external fields. We study the stable states in terms of stable critical points of an appropriately defined free energy, with a nemato-magnetic coupling energy. We numerically study the interplay between the shape of the regular polygon, the size of the polygon and the strength of the nemato-magnetic coupling for the multistability of this prototype system. Our notable results include (i) the co-existence of stable states with domain walls and stable interior and boundary defects, (ii) the suppression of multistability for positive nemato-magnetic coupling, and (iii) the enhancement of multistability for negative nemato-magnetic coupling.

cond-mat.soft↗

arXiv2021

Quantum-Dots Dispersed Bent-core Nematic Liquid Crystals and Cybotactic Clusters: Experimental and Theoretical Insights

We study a quantum-dots (QDs) dispersed bent core liquid crystalline system in planar geometry and present experimental measurements of the order parameter, dielectric dispersion and absorption spectra, optical textures, with attention to variations with temperature. A bent core liquid crystal (LC) 14-2M-CH$_3$ is used as the host material and CdSe/ZnS core-shell type QDs are used as the dopant. The nematic (N) phase exhibited by the pristine (undoped) LC 14-2M-CH$_3$ contains cybotactic clusters, which are retained by its QDs incorporated LC nanocomposite. Our notable findings concern the reduction of the orientational order parameter of the QDs dispersed LC system compared to its pristine counterpart, at fixed temperatures, and a reduction of the size of the cybotactic clusters due to the incorporation of QDs. The reduced order parameter for the doped system is accompanied by reduced birefringence, increased activation energy and a qualitative reduction in the dielectric anisotropy. We complement the experiments with a novel Landau-de Gennes type free energy for a doped bent core LC system, that qualitatively captures the doping-induced reduced order parameter and its variation with temperature. The dependency of the mean order parameter on several other factors (e.g. cluster size, coupling parameter) are also analyzed.

cond-mat.soft↗

arXiv2020

Error analysis of Nitsche's and discontinuous Galerkin methods of a reduced Landau-de Gennes problem

We study a system of semi-linear elliptic partial differential equations with a lower order cubic nonlinear term, and inhomogeneous Dirichlet boundary conditions, relevant for two-dimensional bistable liquid crystal devices, within a reduced Landau-de Gennes framework. The main results are (i) a priori error estimates for the energy norm, within the Nitsche's and discontinuous Galerkin frameworks under milder regularity assumptions on the exact solution and (ii) a reliable and efficient {\it a posteriori} analysis for a sufficiently large penalization parameter and a sufficiently fine triangulation in both cases. Numerical examples that validate the theoretical results, are presented separately.

math.NA↗

arXiv2020

Solution landscape of a reduced Landau-de Gennes model on a hexagon

We investigate the solution landscape of a reduced Landau--de Gennes model for nematic liquid crystals on a two-dimensional hexagon at a fixed temperature, as a function of $λ$---the edge length. This is a generic example for reduced approaches on regular polygons. We apply the high-index optimization-based shrinking dimer method to systematically construct the solution landscape consisting of multiple defect solutions and relationships between them. We report a new stable T state with index-$0$ that has an interior $-1/2$ defect; new classes of high-index saddle points with multiple interior defects referred to as H class and TD class; changes in the Morse index of saddle points with $λ^2$ and novel pathways mediated by high-index saddle points that can control and steer dynamical pathways. The range of topological degrees, locations and multiplicity of defects offered by these saddle points can be used to navigate through complex solution landscapes of nematic liquid crystals and other related soft matter systems.

math-ph↗

arXiv2020

A Reduced Study for Nematic Equilibria on Two-Dimensional Polygons

We study reduced nematic equilibria on regular two-dimensional polygons with Dirichlet tangent boundary conditions, in a reduced two-dimensional Landau-de Gennes framework, discussing their relevance in the full three-dimensional framework too. We work at a fixed temperature and study the reduced stable equilibria in terms of the edge length, $λ$ of the regular polygon, $E_K$ with $K$ edges. We analytically compute a novel "ring solution" in the $λ\to 0$ limit, with a unique point defect at the centre of the polygon for $K \neq 4$. The ring solution is unique. For sufficiently large $λ$, we deduce the existence of at least $\left[K/2 \right]$ classes of stable equilibria and numerically compute bifurcation diagrams for reduced equilibria on a pentagon and hexagon, as a function of $λ^2$, thus illustrating the effects of geometry on the structure, locations and dimensionality of defects in this framework.

math-ph↗