Abstract
We prove lower semicontinuity and lower bounds for a Chen-Lubensky energy describing nematic/smectic liquid crystals with physically realistic boundary conditions. The Chen-Lubensky energy captures stable phases of the liquid crystal material, ranging from purely nematic or smectic states to coexisting nematic/smectic states. By including appropriate additional terms, the model includes the effects of applied electric or magnetic fields, and/or electrical self-interactions in the case of polarized liquid crystals. As a consequence of our results, we establish existence of minimizers with weak or strong anchoring of the director field (describing molecular orientation) at the boundary, and Dirichlet or Neumann boundary conditions on the smectic order parameter for the liquid crystal material.
Highlights
1.1 Background on Liquid CrystalsLiquid crystals are intermediate states between liquid and solid states that occur in a certain class of anisotropic materials, and are typically made up of elongated ”rod-like” molecules
The molecules still tend to align in layers but their long axes have a preferred tilt angle with the layer normal; this is called the smectic C phase. (See Figure 1.) Stable states of liquid crystals need not be uniform
A smectic phase can locally melt into a different smectic phase or a less ordered phase. This can occur if the liquid crystal is subjected to external stresses thereby introducing defects into the layer structure or locally altering the tilt angle
Summary
Liquid crystals are intermediate states between liquid and solid (crystal) states that occur in a certain class of anisotropic materials, and are typically made up of elongated ”rod-like” molecules. The de Gennes energy for nematic/smectic A liquid crystals combined an Oseen-Frank energy term for a unit vector field, n(x), representing the average local orientation of the liquid crystal molecule, and a GinzburgLandau term involving the covariant derivative of a complex-valued order parameter, ψ(x), related to the local smectic layers. It was shown in [2] by Bauman, Calderer, Liu, and Phillips that de Gennes’ energy is coercive and lower semicontinuous among admissible families of functions in an appropriate Sobolev space with physically realistic boundary conditions. The boundary conditions include weak or strong anchoring at the boundary, and boundary values that naturally result in mixedstate nematic/smectic minimizers
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