Modulated structures in electroconvection in nematic liquid crystals

Abstract

Motivated by experiments in electroconvection in nematic liquid crystals with homeotropic alignment we study the coupled amplitude equations describing the formation of a stationary roll pattern in the presence of a weakly damped mode that breaks isotropy. The equations can be generalized to describe the planarly aligned case if the orienting effect of the boundaries is small, which can be achieved by a destabilizing magnetic field. The slow mode represents the in-plane director at the center of the cell. The simplest uniform states are normal rolls, which may undergo a pitchfork bifurcation to abnormal rolls with a misaligned in-plane director. We present a new class of defect-free solutions with spatial modulations perpendicular to the rolls. In a parameter range where the zigzag instability is not relevant these solutions are stable attractors, as observed in experiments. We also present two-dimensionally modulated states with and without defects which result from the destabilization of the one-dimensionally modulated structures. Finally, for no (or very small) damping, and away from the rotationally symmetric case, we find static chevrons made up of a periodic arrangement of defect chains (or bands of defects) separating homogeneous regions of oblique rolls with very small amplitude. These states may provide a model for a class of poorly understood stationary structures observed in various highly conducting materials ("prechevrons" or "broad domains").

Global behavior of solutions to chevron pattern equations

“…i.e., the semigroup S(t) : X 0 → X 0 , t ≥ 0, generated by the problem (1)-( 3) is a compact semigroup. Thanks to (12), this semigroup is also bounded dissipative. Therefore, the following theorem holds true (see, e.g., Ref.…”

“…In other words, this problem generates a continuous semigroup S(t), t ≥ 0, in the phase space X 0 ∶= L 2 (Ω) × L 2 (Ω). Moreover, (12) implies that this semigroup is bounded dissipative in the phase space X 0 . Remark II.2 (Ref.…”

Dynamics of laser-induced electroconvection pulses

“…[1][2][3] A wide range of EC investigations have been carried out using planar [4][5][6][7][8][9][10] and homeotropic [11][12][13][14] nematic alignment; in this work we are interested in the planar configuration. The EC process is in some respects the anisotropic counterpart of Rayleigh-Bénard convection (RBC), with the EC electric field replacing the vertical thermal gradient in RBC.…”

Generalized Maxwell-Higgs vortices in models with enhanced symmetry

“…Gauge symmetries are often used to conveniently account for extra degrees of freedom and other redundancies that one may wish to keep in their description of a system, and need not relate to any fundamental interaction. Applications of this kind include [41][42][43][44][45] and may be among the factors in the product group considered here. Being unrelated to electromagnetism, the degrees of freedom associated to these U(1) factors give rise to fields that transform independently of the gauge transformations of electromagnetism and possibly of each other which, in our formalism, translates to the fact that each scalar field of the theory is charged under only one U(1) subgroup.…”