Nematic Membranes Research Articles

Mechanics of nematic membranes: Euler–Lagrange equations, Noether charges, stress, torque and boundary conditions of the surface Frank’s nematic field

The mechanics of a flexible membrane decorated with a nematic liquid-crystal texture is considered in a variational framework. The variations on the splay, twist and the bend energy are obtained from the local deformations leading to changes in the shape membrane. The Euler–Lagrange derivatives and the Noether charges are identified from the variational equations. The nematic stress tensor is obtained as a consequence of translational invariance. Likewise, the rotational invariance implies the torque nematic tensor. The corresponding boundary conditions are obtained for the free-edges in the open-membrane configuration. These results constitute the basis of a generalized theory of elasticity for anisotropic nematic membranes. Some relevant consequences of the presence of nematic ordering are visualized at revolution surfaces with axial symmetry.

Stresses in curved nematic membranes.

Ordering configurations of a director field on a curved membrane induces stress. In this work, we present a theoretical framework to calculate the stress tensor and the torque as a consequence of the nematic ordering; we use the variational principle and invariance of the energy under Euclidean motions. Euler-Lagrange equations of the membrane as well as the corresponding boundary conditions also appear as natural results. The stress tensor found includes attraction-repulsion forces between defects; likewise, defects are attracted to patches with the same sign in Gaussian curvature. These forces are mediated by the Green function of the Laplace-Beltrami operator of the surface. In addition, we find nonisotropic forces that involve derivatives of the Green function and the Gaussian curvature, even in the normal direction to the membrane. We examine the case of axial membranes to analyze the spherical one. For spherical vesicles we find the modified Young-Laplace law as a consequence of the nematic texture. In the case of spherical cap with defect at the north pole, we find that the force is repulsive with respect to the north pole, indicating that it is an unstable equilibrium point.

Morphology change in nematic membranes induced by defects

The cell membrane is one of the most important structures of living organisms. This is due to the many functions attributed to it such as permeable selectivity, protection, anchoring to the cytoskeleton and so many others. Any change in the shape of the cell membrane may affect directly the properties and abilities. In this article, we study how defects in the liquid crystalline organization of a membrane can change its shape. For performing this, we consider a membrane with orientational order, i.e., a nematic membrane, which can happen in biological membranes, nematic films and other systems and study how a defect in this order can change the shape of the membrane when the bending rigidity is considered. We find that depending on the ratio of rigidity and elastic constant, buckling of this membrane may happens and turn it into pseudo-spheres.

Equilibrium of nematic vesicles

A variational scheme is proposed which allows the derivation of a concise and elegant formulation of the equilibrium equations for closed fluid membranes, endowed with a nematic microstructure. The nematic order is described by an in-plane nematic director and a degree of orientation, as customary in the theory of uniaxial nematics. The only constitutive ingredient in this scheme is a free-energy density which depends on the vesicle geometry and order parameters. The stress and the couple stress tensors related to this free-energy density are provided. As an application of the proposed scheme, a certain number of special theories are deduced: soap bubbles, lipid vesicles, chiral and achiral nematic membranes, and nematics on curved substrates.

Defects in nematic membranes can buckle into pseudospheres

Many factors influence the shapes of living and manufactured membranes. In addition to boundary conditions, surface tension, and curvature, the ordering of particles embedded in or attached to a membrane can strongly influence its equilibrium shape. As a simple model of such ordering, we consider rodlike particles that align to form a so-called nematic phase in the plane of the membrane. We call any sheet with such embedded orientational order a nematic membrane. Nematic membranes can occur in biological cells, liquid crystal films, manufactured materials, and other soft matter systems. By formulating the free energy of nematic films using tensor contractions from differential geometry, we elucidate the elastic terms allowed by symmetry, and indicate differences from hexatic membranes. We find that topological defects in the orientation field can cause the membrane to buckle over a size set by the competition between surface tension and in-plane elasticity. In the absence of bending rigidity the resulting shape is universal, known as a parabolic pseudosphere or a revolved tractrix. This buckling is the two-dimensional analog of the bent cores of line defects that are frequently observed in bulk nematic liquid crystals. Bending costs oppose such buckling and modify the shape in a predictable manner. In particular, the anisotropic rigidities of nematic membranes lead to different shapes for aster and vortex defects, in principle enabling measurement of couplings specific to nematic membranes.

Nematic membranes: Shape instabilities of closed achiral vesicles

We consider the coupling between the local curvature tensor of a membrane and the local two-dimensional nematic order parameter, deriving it from a quasi-microscopic argument. This coupling makes the nematic director aligned along the lowest curvature eigenvector in a local metric. Local bending of a membrane may then generate nematic ordering. Alternatively, emerging nematic order leads to shape instabilities of closed vesicles. The theory is applied to a spherical isotropic vesicle, which turns into a prolate shape with two +1 disclinations on its poles as the nematic order sets in the membrane, described within the Landau-de Gennes continuum model.

Higher-order assembly of microtubules by counterions: from hexagonal bundles to living necklaces.

Cellular factors tightly regulate the architecture of bundles of filamentous cytoskeletal proteins, giving rise to assemblies with distinct morphologies and physical properties, and a similar control of the supramolecular organization of nanotubes and nanorods in synthetic materials is highly desirable. However, it is unknown what principles determine how macromolecular interactions lead to assemblies with defined morphologies. In particular, electrostatic interactions between highly charged polyelectrolytes, which are ubiquitous in biological and synthetic self-assembled structures, are poorly understood. We have used a model system consisting of microtubules (MTs) and multivalent cations to examine how microscopic interactions can give rise to distinct bundle phases in biological polyelectrolytes. The structure of these supramolecular assemblies was elucidated on length scales from subnanometer to micrometer with synchrotron x-ray diffraction, transmission electron microscopy, and differential interference contrast microscopy. Tightly packed hexagonal bundles with controllable diameters were observed for large trivalent, tetravalent, and pentavalent counterions. Unexpectedly, in the presence of small divalent cations, we have discovered a living necklace bundle phase, comprised of 2D dynamic assemblies of MTs with linear, branched, and loop topologies. This new bundle phase is an experimental example of nematic membranes. The morphologically distinct MT assemblies give insight into general features of bundle formation and may be used as templates for miniaturized materials with applications in nanotechnology and biotechnology.

Bilayer membranes with 2D-nematic order of the surfactant polar heads

In this paper we review and discuss the possibility of having a 2D-nematic ordering of the surfactant polar heads in bilayer membranes. The coupling between the nematic order parameters and the curvature can yield instabilities in the shape of the membrane, generally leading to tubules with mesoscopic radiuses. The nematic directors in the two monolayers should present coupled disclinations with peculiar characteristics. Para-nematic membranes, i.e., nematic membranes that are in the isotropic phase when flat, should behave as the nematic membranes in the curved state.

Fluctuating Membranes with Tilt Order

Thermal fluctuations are important for amphiphilic bilayer membranes since typical bending stiffnesses can be a few $k_B T$. The rod-like constituent molecules are generically tilted with respect to the local normal for packing reasons. We study the effects of fluctuations on membranes with nematic order, a simplified idealization with the same novel features as realistic tilt order. We find that nematic membranes lie in the same universality class as hexatic membranes, {\it i.e.} the couplings that distinguish nematic from hexatic order are marginally irrelevant. Our calculation also illustrates the advantages of conformal gauge, which brings great conceptual and technical simplifications compared to the more popular Monge gauge.