Stability of Texture and Shape of Circular Domains of Langmuir Monolayers

Finite domains of a Langmuir monolayer in a phase with tilted molecules can be modeled by a simple elastic free energy of an XY order parameter with isotropic and anisotropic line tension terms. The domains can and often do contain nontrivial textures, which in turn influence the shape of the domains. Herein we investigate the properties of a simplified isotropic model with a single elastic constant. For circular domains a first-order phase transition is found between two distinct textures: an exterior defect (or ``virtual boojum'') texture, and an interior defect texture. Starting with a circular domain and either of these two textures as a ground state, we find shape instabilities develop that depend on the elastic constants and line tensions in the simplified model. In both cases a necessary but not sufficient condition for the onset of shape instabilities is the possibility for a local negative effective line tension to develop from the anisotropic line tension term.Comment: 14 double column RevTex pages, 8 eps figure

Dynamics of smectic elastomers

We study the low-frequency, long-wavelength dynamics of liquid crystal elastomers, crosslinked in the smectic-$A$ phase, in their smectic-$A$, biaxial smectic and smectic-$C$ phases. Two different yet related formulations are employed. One formulation describes the pure hydrodynamics and does not explicitly involve the Frank director, which relaxes to its local equilibrium value in a non-hydrodynamic time. The other formulation explicitly treats the director and applies beyond the hydrodynamic limit. We compare the low-frequency, long-wavelength dynamics of smectic-$A$ elastomers to that of nematics and show that the two are closely related. For the biaxial smectic and the smectic-$C$ phases, we calculate sound velocities and the mode structure in certain symmetry directions. For the smectic-$C$ elastomers, in addition, we discuss in some detail their possible behavior in rheology experiments.Comment: 19 pages, 3 figure

Chiral Lyotropic Liquid Crystals: TGB Phases and Helicoidal Structures

The molecules in lyotropic membranes are typically aligned with the surface normal. When these molecules are chiral, there is a tendency for the molecular direction to twist. These competing effects can reach a compromise by producing helicoidal defects in the membranes. Unlike thermotropic smectics, the centers of these defects are hollow and thus their energy cost comes from the line energy of an exposed lamellar surface. We describe both the twist-grain-boundary phase of chiral lamellar phases as well as the isolated helicoidal defects.Comment: 10 pages, plain TeX, two included figures, revision corrects figures onl

Interactions between Membrane Inclusions on Fluctuating Membranes

We model membrane proteins as anisotropic objects characterized by symmetric-traceless tensors and determine the coupling between these order-parameters and membrane curvature. We consider the interactions between transmembrane proteins that respect up-down (reflection) symmetry of bilayer membranes and that have circular or non-circular cross-sectional areas in the tangent-plane of membranes. Using a field theoretic approach, we find non-entropic $1/R^{4}$ interactions between reflection-symmetry-breaking transmembrane proteins with circular cross-sectional area and entropic $1/R^{4}$ interactions between transmembrane proteins with circular cross-section that do not break up-down symmetry in agreement with previous calculations. We also find anisotropic $1/R^{4}$ interactions between reflection-symmetry-conserving transmembrane proteins with non-circular cross-section, anisotropic $1/R^{2}$ interactions between reflection-symmetry-breaking transmembrane proteins with non-circular cross-section, and non-entropic $1/R^{4}$ many-particle interactions among non-transmembrane proteins. For large $R$, these interactions might provide the dominant force inducing aggregation of the membrane proteins.Comment: REVTEX, 29 pages with 4 postscript figures compressed using uufiles. Introduction and Discussion sections revised. To appear in J. Phys. France I (September

Measure Factors, Tension, and Correlations of Fluid Membranes

We study two geometrical factors needed for the correct construction of statistical ensembles of surfaces. Such ensembles appear in the study of fluid bilayer membranes, though our results are more generally applicable. The naive functional measure over height fluctuations must be corrected by these factors in order to give correct, self-consistent formulas for the free energy and correlation functions of the height. While one of these corrections -- the Faddeev-Popov determinant -- has been studied extensively, our derivation proceeds from very simple geometrical ideas, which we hope removes some of its mystery. The other factor is similar to the Liouville correction in string theory. Since our formulas differ from those of previous authors, we include some explicit calculations of the effective frame tension and two-point function to show that our version indeed secures coordinate-invariance and consistency to lowest nontrivial order in a temperature expansion.Comment: 24 pp; plain Te