M. Carme Calderer

Liquid crystal flow: dynamic and static configurations

We consider a system modeling the flow of nematic liquid crystals with variable degrees of orientation. Although the set of constitutive equations involves many different parameters, the flow behavior is determined mostly by three nondimensional parameters, i.e., the Ericksen number, the Reynolds number, and the interface number. We establish a dissipative relation of the system in general domains. In the case of plane Poiseuille flow, we prove existence and regularity of solutions. Moreover, we discuss the stationary configurations with a large number of defects due to the large Ericksen number of the flow.

Chevron patterns in liquid crystal flows

We study plane shear and Poiseuille flows of uniaxially nematic liquid crystals for large values of the Ericksen number, e. This is a non-dimensional quantity that renders rigorous notions of fast flow regimes. (The former corresponds to large values of e.) The model employed is that due to Ericksen for liquid crystals with variable degree of orientation (Ericksen, 1991). The constitutive equations derived by Kuzuu and Doi (1984) and Marrucci (1981), play an essential role in the analysis. Variables of the problem include the velocity field, v, the pressure, p, the scalar order parameter, s, and the director, n. Recent experimental work involving, both, low molecular weight liquid crystals as well as polymers, has pointed out that the positivity of the ratio α2α3 of the Leslie coefficients does not necessarily guarantee uniform molecular alignment in shear flow regimes, as previously believed. In particular, for fast regimes, the flow breaks up into stripped configurations, chevrons parallel to the velocity field. Taking such observations into account, here, we consider regimes that fulfill the alignment condition, and show that for large values of e, solutions exhibiting disclinations in the flow region can be found. Moreover we point out that across singular lines, the director experiences jump discontinuities of approximately ±45° to the direction of flow. We construct solutions that present domain structures in the form of stripes parallel to the flow direction, on the plane of shear. The interface between two such stripes corresponds to a nearly isotropic layer.

Incipient Dynamics of Swelling of Gels

In this article, we analyze a model of the incipient dynamics of gel swelling and perform numerical simulations. The governing system consists of balance laws for a mixture of nonlinear elastic solid and solvent yielding effective equations for the gel. We discuss the multiscale nature of the problem and identify physically realistic regimes. The mixing mechanism is based on the Flory–Huggins energy. We consider the case that the dissipation mechanism is the solid-solvent friction force. This leads to a system of weakly dissipative nonlinear hyperbolic equations. After addressing the Cauchy problem, we propose physically realistic boundary conditions describing the motion of the swelling boundary. We study the linearized version of the free boundary problem. Numerical simulations of solutions are presented too.

Poiseuille flow of nematic liquid crystals

Abstract We study flow phenomena of nematic liquid crystals with variable degree of orientation. There are three non-dimensional parameters, the Ericksen number, the Reynolds number and the Interface number that are very relevant in determining the flow behavior. We establish a dissipative relation for the governing system that holds for general flow domains. In the case of plane Poiseuille flow, we study the well-posedness of the governing system of differential equations. We discuss stationary configurations with many defects that are due to the large Ericksen number of the flow. We also present numerical simulations of the boundary value problem for the stationary system.

On Poiseuille flow of liquid crystals

We consider Poiseuille flow of polymeric liquid crystals corresponding to large values of the velocity gradient. The model employed [Ericksen, 1991] proposes governing equations for the velocity field, v, the pressure p, the director n, and the order parameter s. The constitutive functions for the Leslie coefficients alphai derived from the molecular theory of Doi [1981] play a crucial role in the modelling. In addition to the Ericksen number, E, the present model exhibits a new non-dimensional parameter I, that represents the contribution of the elastic free energy of non-gradient type with respect to Frank-Oseens elasticity. One of the goals of the analysis was to examine the role of s in describing singularities as well as in obtaining regimes which are not predicted by the previous Leslie-Ericksen model. In particular, solutions are obtained that correspond to domain structures parallel to the flow. Such domains are separated by singular lines across which the director experiences jumps of, approxima...

Stability of shear flows of polymeric liquid crystals

Abstract This article deals with flow phenomena in liquid crystal materials. The governing equations are a generalization of the Leslie-Ericksen equations for liquid crystals with variable degree of orientation. The goal of the analysis is to study the role of the order parameter in the model. We examine the multiplicity of solutions, phases , that occur in shear flow regimes and analyze their stability under physically realistic conditions (such conditions turn out to be compatible with the high viscosities of polymeric materials). In particular, we observe that ellipticity of the linearized system of governing equations is a consequence of the Clausius-Duhem inequality.

Analysis of Nonlocal Electrostatic Effects in Chiral Smectic C Liquid Crystals

We present modeling and analysis of smectic C phases of liquid crystals capable of sustaining spontaneous polarization. The layered liquid crystals are also assumed to be chiral. We study minimization of the total energy subject to electrostatic constraints. In order to determine mathematically and physically relevant boundary conditions, we appeal to the analogy between the current problem and the vorticity in fluids. We place a special emphasis on the nonlocal and self‐energy effects arising from spontaneous polarization. We discuss examples pertaining to the electric field created by the liquid crystal in dielectric medium, and also to the possible role of a domain shape as an energy reduction mechanism.

Radial configurations of smectic A materials and focal conics

Abstract Radial configurations of smectic A liquid crystals in a disk are considered. Focal conics solutions in the disk are constructed from the original radial solution, and the stability of radial solution is shown.

Poiseuille flow of liquid crystals: highly oscillatory regimes

In this article we discuss numerical solutions of the boundary value problem of stationary, plane Poiseuille flow of nematic liquid crystals with variable degree of orientation. The role of the Ericksen number in determining the structure of defects and oscillations of the solutions is analyzed. We find that, at the limit of very large Ericksen number, the very large density of defects render the flow effectively isotropic. This follows from the dependence of the Leslie coefficients on the order parameter; it is a property of the model, and it does not depend on the particular sets of data used in the numerical simulations. From physical point of view, this work intends to address flow phenomena of polymeric liquid crystals. One relevant feature of such a polymeric flow is the textured appearance that results from very large defect densities. This article provides a quantitative, one-dimensional model of such defect phenomena. Moreover, it shows that, for large Ericksen number, a flow that satisfies the alignment condition may actually behave as non-aligning.

Mathematical developments in the study of smectic A liquid crystals

Abstract We consider free energy functionals to model equilibrium smectic A liquid crystal configurations in the neighborhood of the nematic phase transition. We first look at the functional proposed by de Gennes based on the Ginzburg–Landau model for superconductivity, and consider its covariant formulations. We construct the focal conic solution and study its stability. We also study the role of chirality in the model of smectic A * . In the final part of the paper, we use an argument of symmetry breaking to model smectic phases as periodic solutions of an extend nematic theory.