Generalized nematostatics

The generalized equations of bulk and interfacial nematostatics in terms of the tensor order parameter are derived using calculus of variations, taking into account long and short range nematic bulk free energies as well as anchoring and saddle-splay surface free energies. A general expression for the surface stress tensor order parameter for a nematic liquid crystal/isotropic fluid (NLC/I) interface has been derived, and found to represent normal, shear, and bending stresses. It is shown that the surface stress tensor is asymmetric. It is also found that anchoring energy contributes to bending and normal stresses, while saddle-splay energy contributes to normal and shear stresses. The rotational identifies governing the bulk and surface stress tensors are derived and used to show that the equations of nematostatics are fully consistent with the general balance equations of polar fluids. The equations presented provide a theoretical framework for solving interfacial problems involving NLCs that is applicable to cases where variations in liquid crystalline order and saddle-splay energy play significant roles.     

Thermodynamics of soft anisotropic interfaces

The Gibbs-Duhem equation for interfaces between nematic liquid crystals and isotropic fluids is formulated and shown to be a generic equation for soft anisotropic surfaces. The one-to-one correspondence between the nematic and crystalline surface Gibbs-Duhem equations is established. Consistency between the surface Gibbs-Duhem equation and the classical equations of interfacial nematostatics is shown. Using a phase space that takes into account thermodynamics, liquid crystalline order, and geometric variables, the generalized nematic surface Gibbs-Duhem equation reveals the presence of couplings between shape, adsorption, temperature, and average molecular orientation. Merging the thermodynamic analysis with nematostatics results in a model for morphactancy, that is, adsorption-induced interfacial shape selection. The specific roles of gradient bulk Frank elasticity, interfacial tension, and anchoring energy are elucidated by analyzing particular paths in the thermodynamic-geometric phase space.     

Viscoelastic theory for nematic polymer interfaces

A macroscopic theory for the dynamics of compressible nematic polymer‐viscous fluid interfaces is developed from first principles. The theory is used to define and characterize the basic interfacial viscoelastic material properties of the ordered interfaces. The theory is based on a decomposition of the kinematic fields and nematic tensor order parameter that takes into account the symmetry breaking of the interface. The interfacial rate of entropy production used to identify the interfacial viscoelastic modes is given in terms of surface rate of deformation tensor and the surface Jaumann derivative of the tangential component nematic tensor order parameter. The derived surface viscous stress tensor is asymmetric and thus describes surface flow‐induced changes in the tensor order parameter. Consistency with the Boussinesq surface fluid appropriate for Newtonian interfaces is established. The interfacial material functions are identified as the dynamic surface tension, the interfacial dilational viscosities, and the interfacial shear viscosities. The interfacial material functions depend on the surface tensor order parameter and as a consequence anisotropy is their characteristic feature. Two characteristic interfacial tensions and two dilational viscosities are predicted depending on the director orientation. In addition six interfacial shear viscosities arise as the directors sample the velocity, velocity gradient, and vorticity directions. Finally the theory provides for the necessary theoretical tools needed to describe the interfacial dynamics of nematic polymer interfaces, such as capillary instabilities, Marangoni flows, and wetting phenomena.     

Dynamic behaviour at a nematic liquid crystal-rubbednylon interface using evanescent wave photon correlation spectroscopy

Photon correlation spectroscopy of light scattered by director fluctuations from an evanescent optical wave propagating in the nematic liquid crystal 5CB is used to study the interfacial dynamic behaviour of the liquid crystal. The intensity correlation function of light scattered by interfacial orientation fluctuations is measured by illuminating to give a short optical penetration depth within the nematic. These surface scattering correlation functions strongly differ from the bulk correlation function and are interpreted in terms of a nematic surface orientation mode arising from the coupling between the director field and the fluid velocity. It is shown that the analysis of the surface mode gives a method for measuring anchoring energies in liquid crystals. The anchoring energy obtained for rotation of the director away from the rubbing direction about an axis normal to the surface for 5CB at a rubbed nylon surface is 7.14±0.7 × 10-² ergcm-².     

Numerical analysis for a nematic droplet in polymer dispersed liquid crystals

Starting from the Landau-de Gennes free energy expression, the author has numerically analysed the director pattern in a nematic droplet of polymer dispersed liquid crystals. The nematic director has been understood as the eigenvector, which corresponds to the largest eigenvalue of the tensor order parameter. To investigate the droplet structure influence, all equations have been treated on the curvilinear coordinate system which is generated along the droplet boundary. In the case of spherical and spheroidal droplets with normal strong anchoring, the director exhibits an axial configuration and a disclination ring. The ring radius and the capactiance of the system change without hysteresis with the applied voltage.     

Interfacial forces and Marangoni flow on a nematic drop retracting in an isotropic fluid

Nematic-isotropic interfaces exhibit novel dynamics due to anchoring of the liquid crystal molecules on the interface. The objective of this study is to demonstrate the consequences of such dynamics in the flow field created by an elongated nematic drop retracting in an isotropic matrix. This is accomplished by two-dimensional flow simulations using a diffuse-interface model. By exploring the coupling among bulk liquid crystal orientation, surface anchoring and the flow field, we show that the anchoring energy plays a fundamental role in the interfacial dynamics of nematic liquids. In particular, it gives rise to a dynamic interfacial tension that depends on the bulk orientation. Tangential gradient of the interfacial tension drives a Marangoni flow near the nematic-isotropic interface. Besides, the anchoring energy produces an additional normal force on the interface that, together with the interfacial tension, determines the movement of the interface. Consequently, a nematic drop with planar anchoring retracts more slowly than a Newtonian drop, while one with homeotropic anchoring retracts faster than a Newtonian drop. The numerical results are consistent with prior theories for interfacial rheology and experimental observations.     

Recent advances in nematic and smectic A anchoring on amorphous solid surfaces [1]

New anchoring properties of liquid crystals on amorphous solid surfaces are presented. In nematics (N), angular anchoring is usually described in terms of the Rapini-Papoular form, assuming constant surface order parameter. We generalize this expression, predicting a decrease of surface order for strong surface disorientation. Recent experiments on anchorings of varying strength confirm these predictions. Conjectures for the angular anchoring of smectic A on a solid amorphous surface explain the two easy layer orientations, normal to the surface or parallel, faceting inside a small critical angle. Roughness-induced surface transitions are discussed. For antagonistic nematic and smectic anchorings, we expect, below the N-S_A transition, a bent nematic surface boundary layer, recently observed by smectization under an electric field. Finally, the positional anchoring strength of smectics is introduced in terms of shear induced surface melting, and confirmed by a recent observation of oscillating shear stresses at the layer period.     

Influence of K24 on the structure of nematic liquid crystal droplets

A phenomenological free energy is used to describe the stable ordering of nematic liquid crystals confined to supramicron spherical cavities. In particular the effects of the saddle splay elastic constant, K₂₄, on the equilibrium structures and phase diagram of droplets with homeotropic surface anchoring are discussed. Some structures are illustrated by the corresponding simulated polarization microscope textures. Possibilities for an experimental determination of the saddle-splay elastic constant and surface anchoring strength by studying the radial-axial structural transition in such droplets are analysed. It is shown that the K₂₄ term in the elastic free energy stabilizes a deformed droplet structure even in the limit of the zero anchoring strength.     

Nemato-capillarity theory and the orientation-induced Marangoni flow

The macroscopic equations of nemato-capillarity, including the interfacial linear momentum balance equation and the interfacial director torque balance equation, are presented. The interfacial linear momentum balance equation for isotropic fluid-nematic liquid crystals involves the surface divergence of the surface stress tensor. It is shown that the surface stress tensor for isotropic fluid-nematic interfaces is, in most cases of interest, dominated by elastic modes. It is found that the anisotropic elastic contribution to the surface stress tensor gives rise to bending stresses, not observed in interfaces between isotropic fluids. In addition it is found that the anisotropic contribution to the surface elasticity also gives rise to tangential forces. Thus when the director orientation deviates from the easy axis of an isotropic fluidnematic interface and the deviation has surface gradients, an orientation-driven Marangoni flow can exist. The strength of this novel effect is proportional to the anchoring energy of the interface, and the direction of flow is from low energy regions towards high energy regions, that is, from regions where the director is aligned along the easy axis towards regions where the director deviates from the easy axis.     

Mechanical model for anisotropic curved interfaces with applications to surfactant-laden liquid-liquid crystal interfaces

A mechanical model for anisotropic curved interfaces, applicable to thermodynamically closed surfactant-laden liquid-liquid crystal interfaces is developed. The model takes into account the mechanical effects due to surface bending and surface tilting (anchoring) and incorporates liquid crystal anisotropy into classical fluid membrane mechanics. In the absence of the aligned liquid crystal, the model converges to the fluid membrane mechanical model, and in the absence of surfactant, it converges to the nematic interface mechanical model. Use of the well-known Helfrich-Rapini-Papoular surface energies leads to the Laplace equation for anisotropic curved interfaces, whose material limits are the vesicle shape equation and the liquid crystal Herring equation. Applications of the model to shape selection in liquid drops embedded in aligned nematic liquid crystals illustrates the competition between anchoring and bending and shows how anisotropic surface tension distorts the droplet and how bending tends to restore the spherical shape. This theory presented in this article shows that the interaction of interfacial anchoring and bending creates new regimes in classical fluid membrane mechanics.     

The stability diagram of a nematic liquid crystal confined to a cylindrical cavity

Different nematic structures confined to a long cylindrical cavity with homeotropic surface anchoring are studied using a numerical minimization of the free energy of the uniaxial nematic liquid crystal. The stability of escaped radial structures and planar polar structures (with and without line defects) is analysed in terms of the ratio of elastic constants K₂₄/K₁₁, K₃₃/K₁₁, anchoring strength and external magnetic field applied perpendicular to the symmetry axis of the cylinder. We draw the analogy between the stability diagram of the cylindrical structures and structures in a spherical droplet. In particular, a simple way extracting the value of the saddle-splay elastic constant K₂₄ from the stability studies is discussed.     

An extension of the Landau-Ginzburg-de Gennes theory for liquid crystals

Using angular momentum representation a method is proposed that allows the systematic construction of a generalized Landau-de Gennes elastic free energy of liquid crystals, in powers of a symmetric and traceless tensor order parameter, polarization field, of external fields and all respective derivatives. By this method all linearly independent elastic invariants and surface terms are constructed for nematics and cholesterics up to fourth order terms. In particular it is shown that up to fourth order in the tensor order parameter there are nineteen bulk elastic constants and four surface terms in the free energy of a general, biaxial nematic. In addition, the stability of this expansion is studied in detail. Some special cases of the elastic free energy of liquid crystals, already discussed in the literature, are reexamined and discrepancies with our results are emphasized. Finally, a thermo-dynamically correct way of establishing contact between the generalized de Gennes elastic free energy and other theories, like those of Oseen-Frank or Meyer, is proposed by applying fluctuation theory. Thus, the degeneracy of splay and bend elastic constants is removed even when these are calculated from the standard de Gennes free energy. Restrictions on higher order elastic constants are also obtained by comparing mean field relations and stability conditions with available experimental data.     

Interfacial Thermodynamics of Polymeric Mesophases

Summary: A complete mechanical‐thermodynamical formulation for multicomponent nematic polymer‐isotropic fluid interfaces is derived, validated, and used to derive the structure and shape equations for these soft anisotropic polymer interfaces. The fundamental role of liquid crystalline order and long range effects in coupling bulk and interfacial effects, and in coupling thermodynamical/liquid crystalline order/geometrical variables is demonstrated, discussed, and validated. The Gibbs‐Duhem nemato‐thermodynamics equation emerges from an interfacial tension γ = γ(Θ, μ, Q , ∇_s Q , k ) that depends on temperature (Θ), chemical potential (μ), nematic tensor order parameter Q , surface gradients of Q , and geometry k , and leads to new couplings in these enhanced phase spaces. The role of entropy and adsorption, and long range effects on interfacial shape and structure selection is revealed. For flat interfaces the preferred structure emerges from a competition between energy, entropy, and adsorption.     

Capillary Thermodynamics of Nematic Polymer Interfaces

The Cahn‐Hoffman capillarity vector thermodynamics formalism for curved anisotropic interfaces is adapted to soft liquid crystalline polymer‐isotropic fluid interfaces. The Cahn‐Hoffman formalism in conjunction with the interfacial Landau‐de Gennes model is used to derive compact expressions for the capillary pressure, tangential Marangoni force, and interfacial torque. It is shown that the interfacial thermodynamics of nematic polymers can be analyzed in terms of three distinct modes: (i) area size change, (ii) area rotation, and (iii) tensor order parameter curvature. The formalism allows to clearly identify the nature and magnitude of these three contributions. Characteristic cases biaxial and uniaxial nematic ordering states are analyzed. Anisotropic liquid crystal surfaces display a number of novel interfacial effects: (a) capillary pressure even for flat surfaces, (b) tensor order parameter‐dependent renormalization of the tension coefficients due to anchoring energy, (c) tensor order parameter‐driven transitions between classical Laplace pressure and non‐classical behavior, (d) Laplace‐like capillary pressure due solely to orientation curvature, (e) generation of Marangoni forces through interfacial order and/or orientation gradients, and (f) surface torques generation.

Schematic of the geometry considered in this article.     

Confinement of a liquid crystal to small droplets and its effect on nuclear magnetic relaxation

Abstract

The spatial dependence of the orientation of the molecular director and of the nematic order parameter is obtained by minimization of the Landau–de Gennes free energy of the nematic liquid crystal confined in a spherical droplet. Special attention is given to the vicinity of the nematic–isotropic transition. The influence of the resulting nematic structure, large liquid crystal–polymer interface and restricted molecular diffusion on the nuclear magnetic relaxation is analysed. The translationally-induced molecular reorientation and the liquid crystal–polymer cross relaxation are discussed in particular. The possibility of an indirect study of the molecular anchoring on the polymer surface is demonstrated.     

Numerical analysis of the radial-axial structure transition with an applied field in a nematic droplet

In this paper the director configurations and the free energies of a nematic droplet with a surface normal anchoring condition are calculated numerically. For this surface anchoring, a transition occurs between the radial and axial structures with respect to an applied field. In the calculation of the director configurations, the position of a disclination has been fixed. Comparing the free energies for different disclinations, the stable position which gives the minimum free energy is found. In calculating the free energy of a droplet, it is assumed that the free energy density of the nematic phase does not exceed the isotropic free energy density, so that the large distortion in the vicinity of the disclination causes a nematic-isotropic transition and the free energy density of the disclination core becomes equal to the isotropic free energy density. The director configuration in a droplet is calculated as a function of an applied field for different isotropic free energy densities, elastic constant ratios and droplet shapes. The relation between the radial-axial structure transition and these factors are clarified.     

Influence of anchoring at a nematic cell surface on threshold spatially periodic reorientation of a director

We have analysed the influence of surface director anchoring in a planar flexoelectric nematic cell on the threshold spatially periodic reorientation of the director in an external dc electric field. By minimizing the free energy of the nematic cell we obtained the equations for a director and numerically solved them in the one elastic constant approximation. The dependences of the threshold electric field and the spatial period of director structure on the azimuthal and polar anchoring energy, as well as the flexoelectric parameters, are determined. It is shown that the domain of the flexoelectric parameter values, at which the spatially periodic reorientation of a director takes place, increases with decreasing azimuthal anchoring energy and increasing polar anchoring energy.     

Anchoring energies of liquid crystals measured on surfaces presenting oligopeptides

We report a methodology that permits quantitation of the azimuthal anchoring energy of the nematic liquid crystal 4-cyano-4'-pentyl-biphenyl on surfaces patterned with oligopeptides. The oligopeptide (IYGEFKKKC), an optimized substrate for the Src protein kinase, was covalently immobilized via the terminal cysteine to monolayers of amine-terminated tetra(ethylene glycol) formed on gold films. The measurements of anchoring energies, which were based on a torque-balance method, revealed a systematic decrease in anchoring energy from 3.7 +/- 0.6 microJ/m2 with increasing surface density of oligopeptide. We calculate that a mass density of oligopeptide of less than 1 ng/cm2 can lead to a measurable change in the anchoring energy of the nematic liquid crystal. These results suggest that measurements of anchoring energies of liquid crystals on surfaces may offer the basis of quantitative and label-free methods for detecting biomolecules on surfaces.