1,354,345 research outputs found
Polydispersity and surface energy strength in nematic colloids
No full textWe consider a Landau-de Gennes model for a polydisperse, inhomogeneous suspension of colloidal inclusions in a nematic host, in the dilute regime. We study the homogenised limit and compute the effective free energy of the composite material. By suitably choosing the shape of the inclusions and imposing a quadratic, Rapini-Papoular type surface anchoring energy density, we obtain an effective free energy functional with an additional linear term, which may be interpreted as an "effective field" induced by the inclusions. Moreover, we compute the effective free energy in a regime of "very strong anchoring", that is, when the surface energy effects dominate over the volume free energy
On a sharp Poincare-type inequality on the 2-sphere and its application in micromagnetics
No full textThe main aim of this note is to prove a sharp Poincare-type inequality for vector-valued functions on S2 that naturally emerges in the context of micromagnetics of spherical thin films
Refined approximation for minimizers of a Landau-de Gennes energy functional
No full textWe study minimizers of a Landau-de Gennes energy functional in the asymptotic regime of small dimensionless elastic constant L > 0. The results on the convergence to a minimizer of the limit Oseen-Frank functional in Majumdar and Zarnescu (Arch Ration Mech Anal 196:227–280, 2010) are revisited and improved, which in effect lead to a sharp rate of convergence. The equation for the first-order correction term is derived: it has a “normal component” given by an algebraic relation and a “tangential component” given by a linear system
Design of effective bulk potentials for nematic liquid crystals via colloidal homogenisation
No full textWe consider a Landau–de Gennes model for a suspension of small colloidal inclusions in a nematic host. We impose suitable anchoring conditions at the boundary of the inclusions, and we work in the dilute regime — i.e. the size of the inclusions is much smaller than the typical separation distance between them, so that the total volume occupied by the inclusions is small. By studying the homogenised limit, and proving rigorous convergence results for local minimisers, we compute the effective free energy for the doped material. In particular, we show that not only the phase transition temperature, but also any coefficient of the quartic Landau–de Gennes bulk potential can be tuned, by suitably choosing the surface anchoring energy density
Nonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy
No full textIn this paper, we prove the existence of global-in-time weak solutions for an evolutionary PDE system modelling nonisothermal Landau-de Gennes nematic liquid crystal (LC) flows in three dimensions of space. In our model, the incompressible Navier-Stokes system for the macroscopic velocity is coupled to a nonlinear convective parabolic equation describing the evolution of the Q-tensor Q, namely a tensor-valued variable representing the normalized second-order moments of the probability distribution function of the LC molecules. The effects of the (absolute) temperature are prescribed in the form of an energy balance identity complemented with a global entropy production inequality. Compared to previous contributions, we can consider here the physically realistic singular configuration potential introduced by Ball and Majumdar. This potential gives rise to severe mathematical difficulties since it introduces, in the Q-tensor equation, a term that is at the same time singular in Q and degenerate in v. To treat it, a careful analysis of the properties of , particularly of its blow-up rate, is carried out
Evolution of non-isothermal Landau–de Gennes nematic liquid crystals flows with singular potential
No full textWe discuss a 3D model describing the time evolution of nematic liquid crystals in the framework of Landau-de Gennes theory, where the natural physical constraints are enforced by a singular free energy bulk potential proposed by J.M. Ball and A. Majumdar. The thermal effects are present through the component of the free energy that accounts for intermolecular interactions. The model is consistent with the general principles of thermodynamics and mathematically tractable. We identify the a priori estimates for the associated system of evolutionary partial differential equations and construct global-in-time weak solutions for arbitrary physically relevant initial data
Weak sequential stability for a nonlinear model of nematic electrolytes
No full textIn this article we study a system of nonlinear PDEs modelling the electrokinetics of a nematic electrolyte material consisting of various ions species contained in a nematic liquid crystal. The evolution is described by a system coupling a Nernst-Planck system for the ions concentrations with a Maxwell's equation of electrostatics governing the evolution of the electrostatic potential, a Navier-Stokes equation for the velocityfield, and a non-smooth Allen-Cahn type equation for the nematic directorfield. We focus on the two-species case and prove apriori estimates that provide a weak sequential stability result, the main step towards proving the existence of weak solutions
Half-Integer Point Defects in the Q-Tensor Theory of Nematic Liquid Crystals
No full textWe investigate prototypical profiles of point defects in two-dimensional liquid crystals within the framework of Landau–de Gennes theory. Using boundary conditions characteristic of defects of index k/2, we find a critical point of the Landau–de Gennes energy that is characterised by a system of ordinary differential equations. In the deep nematic regime, b2 small, we prove that this critical point is the unique global minimiser of the Landau–de Gennes energy. For the case b2=0, we investigate in greater detail the regime of vanishing elastic constant L→0, where we obtain three explicit point defect profiles, including the global minimiser
Landau-de Gennes Corrections to the Oseen-Frank Theory of Nematic Liquid Crystals
No full textWe study the asymptotic behavior of the minimisers of the Landau-de Gennes model for nematic liquid crystals in a two-dimensional domain in the regime of small elastic constant. At leading order in the elasticity constant, the minimum-energy configurations can be described by the simpler Oseen-Frank theory. Using a refined notion of Γ -development we recover Landau-de Gennes corrections to the Oseen-Frank energy. We provide an explicit characterisation of minimizing Q-tensors at this order in terms of optimal Oseen-Frank directors and observe the emerging biaxiality. We apply our results to distinguish between optimal configurations in the class of conformal director fields of fixed topological degree saturating the lower bound for the Oseen-Frank energy
Anina from Mining for Coal to Mining for Ideas. A Possible Path for Regeneration / Anina, de la Mina de carbuni la Mina de idei. O posibilă renastere
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