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Systems involving nematic liquid crystals subjected to magnetic fields or electric fields are modeled using the Oseen--Frank macroscopic continuum theory, and general criteria are developed to assess the local stability of equilibrium solutions. The criteria take into account the inhomogeneity of the electric field and the mutual influence of the electric field and the liquid-crystal director field on each other. The criteria show that formulas for the instability thresholds of electric-field Fréedericksz transitions cannot in all cases be obtained from those for the analogous magnetic-field transitions by simply replacing the magnetic parameters by the corresponding electric parameters, contrary to claims in standard references. A simple analytical test is provided to determine when an electric-field-induced instability can differ qualitatively from the analogous magnetic-field-induced instability. For the systems we study, it is found that taking into account the full coupling between the electric field and the director field can either elevate or leave unchanged an instability threshold (never lower it), compared to the threshold predicted by the magnetic-field analogy. The physical mechanism that underlies the effect of elevating an instability threshold is the increase in free energy associated with a first-order change in the ground-state electric field caused by a perturbation of the ground-state director field. Examples are given that involve classical Fréedericksz transitions and also periodic instabilities. The inclusion of flexoelectric terms in the theory is studied, and it is found that these terms are not capable of altering the instability thresholds of any of the classical Fréedericksz transitions. [ABSTRACT FROM AUTHOR]

Some recent analytical papers have explored limiting behaviors of Landau- de Gennes models for liquid crystals in certain extreme ranges of the model param- eters: limits of "vanishing elasticity" (in the language of some of these papers) and "low-temperature limits." We use simple scaling analysis to show that these limits are properly interpreted as limits in which geometric length scales (such as the size of the domain containing the liquid crystal material) become large compared to intrin- sic length scales (such as correlation lengths or coherence lengths, which determine defect core sizes). This represents the natural passage from a mesoscopic model to a macroscopic model and is analogous to a "London limit" in the Ginzburg-Landau theory of superconductivity or a "large-body limit" in the Landau-Lifshitz theory of ferromagnetism. Known relevant length scales in these parameter regimes (nematic correlation length, biaxial coherence length) can be seen to emerge via balances in equilibrium Euler-Lagrange equations associated with well-scaled Landau-de Gennes free-energy functionals. [ABSTRACT FROM AUTHOR]

We consider the nonlinear systems of equations that result from discretizations of a prototype variational model for the equilibrium director field characterizing the orientational properties of a liquid crystal material. In the presence of pointwise unit-vector constraints and coupled electric fields, the numerical solution of such equations by Lagrange-Newton methods leads to linear systems with a double saddle-point form, for which we have previously proposed a preconditioned nullspace method as an effective solver [A. Ramage and E. C. Gartland, Jr., SIAM J. Sci. Comput., 35 (2013), pp. B226-B247]. Here we propose and analyze a modified outer iteration ("renormalized Newton method") in which the orientation variables are normalized onto the constraint manifold at each iterative step. This scheme takes advantage of the special structure of these problems, and we prove that it is locally quadratically convergent. The renormalized Newton method bears some resemblance to the truncated Newton method of computational micromagnetics, and we compare and contrast the two. This brings to light some anomalies of the truncated Newton method. [ABSTRACT FROM AUTHOR]

We present a preconditioned nullspace method for the numerical solution of large sparse linear systems that arise from discretizations of continuum models for the orientational properties of liquid crystals. The approach effectively deals with pointwise unit-vector constraints, which are prevalent in such models. The indefinite, saddle-point nature of such problems, which can arise from either or both of two sources (pointwise unit-vector constraints, coupled electric fields), is illustrated. Both analytical and numerical results are given for a model problem. [ABSTRACT FROM AUTHOR]

The extension of Bogoliubov Jr.’s minimax principle to biaxial nematic liquid crystals has recently made it possible to derive the universal mean-field phase diagram for all quadrupolar interactions between nematogenic biaxial molecules, including those rendering the mean-field free energy $${\fancyscript{F}_0}$$ indefinite. To justify this extension, so far largely based on heuristic arguments, we prove here a minimum principle that also applies when $${\fancyscript{F}_0}$$ is indefinite. Bogoliubov Jr.’s minimax principle then emerges as a characterization of this minimum principle. The theory presented here is more general than its application to biaxial nematic liquid crystals could reveal. It essentially builds upon a concavity property of $${\fancyscript{F}_0}$$ in the order tensor associated with the repulsive component of the molecular interaction. [ABSTRACT FROM AUTHOR]

Motivated by a recent investigation of Millar and McKay [Director orientation of a twisted nematic under the influence of an in-plane magnetic field. Mol. Cryst. Liq. Cryst 435, 277/[937]-286/[946] (2005)], we study the magnetic field twist-Fréedericksz transition for a nematic liquid crystal of positive diamagnetic anisotropy with strong anchoring and pre-twist boundary conditions. Despite the pre-twist, the system still possesses ℤ2 symmetry and a symmetry-breaking pitchfork bifurcation, which occurs at a critical magnetic-field strength that, as we prove, is above the threshold for the classical twist-Fréedericksz transition (which has no pre-twist). It was observed numerically by Millar and McKay that this instability occurs precisely at the point at which the ground-state solution loses its monotonicity (with respect to the position coordinate across the cell gap). We explain this surprising observation using a rigorous phase-space analysis. [ABSTRACT FROM AUTHOR]

This paper describes the Landau-de Gennes free-energy minimization problem for computing equilibrium configurations of the tensor order parameter field that characterizes the molecular orientational properties of liquid crystal materials. Analytical and numerical issues are addressed. Conditions guaranteeing well-posedness (existence, regularity) of the problem are given, as is a nonlinear finite element convergence analysis. [ABSTRACT FROM AUTHOR]