Saddle-splay Elasticity Research Articles

Twist-bend nematic drops as colloidal particles: Structural features.

The mesogen CB7CB [1″,7″-bis(4-cyanobiphenyl-4'-yl)heptane], mixed with a small quantity of a long chain amphiphile, is examined for the structural features of twist-bend nematic (N_{TB}) drops acting as colloidal inclusions in the isotropic and nematic environments. In the isotropic phase, the drops nucleating in the radial (splay) geometry develop toward escaped radial, off-centered structures, involving both splay and bend distortions. With further growth, they transform into low-birefringence (near-homeotropic) objects, within which remarkably well-organized networks of parabolic focal conic defects evolve in time. In electrically reoriented near-homeotropic N_{TB} drops, the pseudolayers develop an undulatory boundary possibly attributable to saddle-splay elasticity. In the matrix of the planar nematic phase, N_{TB} droplets appearing as radial hedgehogs attain stability in the dipolar geometry, through their association with hyperbolic hedgehogs. With growth, on transformation of the hyperbolic defect into its topologically equivalent Saturn ring around the N_{TB} drop, the geometry turns quadrupolar. Significantly, dipoles are stable in smaller drops, while quadrupoles are stable in larger ones. The dipole-quadrupole transformation is reversible, but is hysteretic with respect to drop size. Importantly, this transformation is often mediated by nucleation of two loop disclinations, one appearing at a marginally lower temperature than the other. The existence of a metastable state with partial formation of a Saturn ring and persistence of the hyperbolic hedgehog raises a question relating to the conservation of topological charge. In twisted nematics, this state features in the formation of a giant unknot that binds all N_{TB} drops together.

Spontaneous mirror symmetry breaking and chiral segregation in the achiral ferronematic compound DIO.

An achiral compound, DIO, known to exhibit three nematic phases namely N, NX and NF, is studied by polarizing microscopy and electro-optics for different surface conditions in confinement. The high temperature N phase assigned initially as a conventional nematic phase, shows two additional unusual features: the optical activity and the linear electro-optic response related to the polar nature of this phase. An appearance of chiral domains is explained by the spontaneous symmetry breaking arising from the saddle-splay elasticity and followed by the formation of helical domains of the opposite chirality. This is the first example of helical segregation observed in calamitic non-chiral molecules in the nematic phase. As reported previously, the ferronematic NF shows strong polar azimuthal surface interaction energy which stabilizes a homogeneous structure in planar aligned LC cells rubbed parallel and exhibits a twisted structure in cells with antiparallel buffing. The transmission spectra are simulated using Berreman's 4 × 4 matrix method. The observed agreement between the experimental and the simulated spectra quantitatively confirms the presence of twisted structures in antiparallel rubbed cells.

Stable Assemblies of Topological Defects in Nematic Orientational Order.

We considered general mechanisms enabling the stabilization of localized assemblies of topological defects (TDs). There is growing evidence that physical fields represent fundamental natural entities, and therefore these features are of interest to all branches of physics. In general, cores of TDs are energetically costly, and consequently, assemblies of TDs are unfavorable. Owing to the richness of universalities in the physics of TDs, it is of interest to identify systems where they are easily experimentally accessible, enabling detailed and well-controlled analysis of their universal behavior, and cross-fertilizing knowledge in different areas of physics. In this respect, thermotropic nematic liquid crystals (NLCs) represent an ideal experiment testbed for such studies. In addition, TDs in NLCs could be exploited in several applications. We present examples that emphasize the importance of curvature imposed on the phase component of the relevant order parameter field. In NLCs, it is represented by the nematic tensor order parameter. Using a simple Landau-type approach, we show how the coupling between chirality and saddle splay elasticity, which can be expressed as a Gaussian curvature contribution, can stabilize Meron TDs. The latter have numerous analogs in other branches of physics. TDs in 2D curved manifolds reveal that the Gaussian curvature dominantly impacts the assembling and stabilization of TDs. Furthermore, a strong enough curvature that serves as an attractor for TDs is a respective field that could be imposed in a fast enough phase transition. Assemblies of created TDs created in such a disordered environment could be stabilized by appropriate impurities.

Stability of a split-core configuration induced by saddle-splay elasticity in a submicron nematic liquid crystal spherical droplet

ABSTRACT We use Landau–de Gennes theory to investigate the configurations of liquid crystals confined to a submicron spherical droplet subject to weak homeotropic anchoring. The results show that in a one-constant approximation, there are three equilibrium configurations: radial hedgehog, ring disclination and uniform, where the radial hedgehog configuration is a stable state. With increasing elastic anisotropy, which is essentially an increase in the splay elastic constant, the radial hedgehog configuration transforms into a split-core configuration, which is metastable. The split-core configuration becomes the lowest-energy state induced by saddle-splay elasticity within appropriate ranges of the radius and elastic anisotropy.

Saddle-splay elasticity induced chiral structures in cylindrical nematic liquid crystal shell

ABSTRACT Based on Landau-de Gennes theory and differential iteration method, in a cylindrical shell with degenerate planar anchoring at both interfaces, the double-twist director configuration caused by large saddle-splay elasticity is investigated. The combined effects of saddle-splay elasticity and the curvature on the spontaneous chiral structure are analysed. Between opposite-handed domains, we find two domain wall structures and a new 1/2 defect ring structure. The stabilities of these structures are analysed by an energy comparison.

Saddle-splay-induced periodic edge undulations in smectic-A disks immersed in a nematic medium.

We report experimental studies on the phase behavior of binary mixtures of 1″,7″-bis(4-cyanobiphenyl-4'-yl)heptane (CB7CB) and 4,4-diheptyloxyazoxybenzene, which exhibit, apart from the nematic (N) and twist-bend nematic (N_{TB}) phases, the induced smectic-A (Sm-A) phase for weight fraction of CB7CB between 0.05 and 0.70. In planar nematic layers, the N_{TB} phase separates as droplets of tactoidlike planform; the chirality of droplets manifests in the optical dissimilarity between their opposite angular ends. Our main result is that, in the appropriate two phase region, Sm-A nuclei with positive dielectric anisotropy change over to disks immersed in the nematic above some electric field, their edges decorated by periodic bright spots, a result which was earlier reported in another binary system exhibiting the induced Sm-A phase [R. Pratibha and N. V. Madhusudana, Physica A 224, 9 (1996)10.1016/0378-4371(95)00311-8]. We develop a simple theory for the threshold of this distortion, which is a periodic undulation of the edge of the disk, demonstrating that it arises from saddle-splay elasticity of Sm-A, the low Sm-A-N interfacial tension unable to suppress the distortion. The observed increases in the number of bright spots with field, and with the radius of the disk at a given field, in both the experimental systems are also accounted for by the model. The distortion, which results in the most direct visualization of saddle splay in Sm-A, is also exhibited by disks nucleating on surfaces treated for homeotropic anchoring.

Saddle–splay elasticity induced chiral structures in achiral liquid crystals within cylindrical geometry

ABSTRACTOn the basis of Landau–de Gennes theory and the finite-difference iterative method, spontaneous chiral structures in cylinders with degenerate planar boundary conditions are investigated. A double-twist director configuration can be achieved by the saddle–splay contribution, corresponding to the L24 term in Landau–de Gennes theory. The twist angle increases as the radius R of the cylinder is reduced because the curvature of the cylindrical surface becomes larger. Moreover, we find fine structures of the domain walls and point defects between opposite-handed domains. An energy comparison shows that domain walls are the stable state for K24/K> 0.6 (K11 = K33 = K, K11, K33 and K24 are the splay, bend, saddle–splay elastic constants in Frank theory), whereas point defects are the stable state for K24/K< 0.6.

Programming emergent symmetries with saddle-splay elasticity

The director field adopted by a confined liquid crystal is controlled by a balance between the externally imposed interactions and the liquid’s internal orientational elasticity. While the latter is usually considered to resist all deformations, liquid crystals actually have an intrinsic propensity to adopt saddle-splay arrangements, characterised by the elastic constant {K}_{24}. In most realisations, dominant surface anchoring treatments suppress such deformations, rendering {K}_{24} immeasurable. Here we identify regimes where more subtle, patterned surfaces enable saddle-splay effects to be both observed and exploited. Utilising theory and continuum calculations, we determine experimental regimes where generic, achiral liquid crystals exhibit spontaneously broken surface symmetries. These provide a new route to measuring {K}_{24}. We further demonstrate a multistable device in which weak, but directional, fields switch between saddle-splay-motivated, spontaneously-polar surface states. Generalising beyond simple confinement, our highly scalable approach offers exciting opportunities for low-field, fast-switching optoelectronic devices which go beyond current technologies.

In Silico Measurement of Elastic Moduli of Nematic Liquid Crystals.

Experiments on confined droplets of the nematic liquid crystal 5CB have questioned long-established bounds imposed on the elastic free energy of nematic systems. This elasticity, which derives from molecular alignment within nematic systems, is quantified through a set of moduli which can be difficult to measure experimentally and, in some cases, can only be probed indirectly. This is particularly true of the surfacelike saddle-splay elastic term, for which the available experimental data indicate values on the cusp of stability, often with large uncertainties. Here, we demonstrate that all nematic elastic moduli, including the saddle-splay elastic constant k_{24}, may be calculated directly from atomistic molecular simulations. Importantly, results obtained through in silico measurements of the 5CB elastic properties demonstrate unambiguously that saddle-splay elasticity alone is unable to describe the observed confined morphologies.

Cylindrical nematic liquid crystal shell: effect of saddle-splay elasticity.

This study introduces cylindrical nematic liquid crystal (LC) shells. Shells as confinement can provide soft matter with intriguing topology and geometry. Indeed, in spherical shells of LCs, rich defect structures have been reported. Avoiding the inherent Plateau-Rayleigh instability of cylindrical liquid-liquid interfaces, we realize the cylindrical nematic LC shell by two different methods: the phase separation in the nematic-isotropic coexistence phase and a cylindrical cavity with a glass rod suspended in the middle. Specifically, the director configurations of lyotropic chromonic LCs (LCLCs) in the cylindrical shell and their energetics are investigated theoretically and experimentally. Unusual elastic properties of LCLCs, i.e., a large saddle-splay modulus, and a shell geometry with both concave and convex curvatures, result in a double-twist director configuration.

Relevance of saddle-splay elasticity in complex nematic geometries.

We demonstrate the relevance of saddle-splay elasticity in nematic liquid crystalline fluids in the context of complex surface anchoring conditions and the complex geometrical confinement. Specifically, nematic cells with patterns of surface anchoring and colloidal knots are shown as examples where saddle-splay free energy contribution can have a notable role which originates from nonhomogeneous surface anchoring and the varying surface curvature. Patterned nematic cells are shown to exhibit various (meta)stable configurations of nematic field, with relative (meta)stability depending on the saddle-splay. We show that for high enough values of saddle-splay elastic constant K24 a previously unstable conformation can be stabilised, more generally indicating that the saddle-splay can reverse or change the (meta)stability of various nematic structures affecting their phase diagrams. Furthermore, we investigate saddle-splay elasticity in the geometry of highly curved boundaries - the colloidal particle knots in nematic - where the local curvature of the particles induces complex spatial variations of the saddle-splay contributions. Finally, a nematic order parameter tensor based saddle-splay invariant is shown, which allows for the direct calculation of saddle-splay free energy from the Q-tensor, a possibility very relevant for multiple mesoscopic modelling approaches, such as Landau-de Gennes free energy modelling.

Erratum: Chiral structures and defects of lyotropic chromonic liquid crystals induced by saddle-splay elasticity [Phys. Rev. E91, 050501(R) (2015)

Erratum: Chiral structures and defects of lyotropic chromonic liquid crystals induced by saddle-splay elasticity [Phys. Rev. E<b>91</b>, 050501(R) (2015)

Chiral structures and defects of lyotropic chromonic liquid crystals induced by saddle-splay elasticity.

An experimental and theoretical study of lyotropic chromonic liquid crystals (LCLCs) confined in cylinders with degenerate planar boundary conditions elucidates LCLC director configurations. When the Frank saddle-splay modulus is more than twice the twist modulus, the ground state adopts an inhomogeneous escaped-twisted configuration. Analysis of the configuration yields a large saddle-splay modulus, which violates Ericksen inequalities but not thermodynamic stability. Lastly, we observe point defects between opposite-handed domains, and we explain a preference for point defects over domain walls.

Nanoparticle-induced twist-grain boundary phase.

By means of high-resolution ac calorimetry and polarizing optical microscopy, it is demonstrated that surface-functionalized spherical CdSSe nanoparticles induce a twist-grain boundary phase when dispersed in a chiral liquid crystal. These nanoparticles can effectively stabilize the one-dimensional lattice of screw dislocations, thus establishing the twist-grain boundary order between the cholesteric and the smectic-A phases. A Landau-de Gennes-Ginzburg model is used to analyze the impact of nanoparticles on widening the temperature range of molecular organizations possessing a lattice of screw dislocations. We show that in addition to the defect-core-replacement mechanism, the saddle-splay elasticity may also play a significant role.

Saddle-splay elasticity and field induced soliton in nematics

The symmetry breaking Fréedericksz transitions, when a uniformly aligned nematic state is replaced by a homogeneously or periodically distorted state, have been extensively studied before. Here we analyse the influence of the saddle-splay elasticity on the non-linear ground state of a nematic liquid crystal in the presence of a magnetic field above the Fréedericksz threshold. We identify the bifurcation point when the localized soliton-like state is linearly unstable with respect to the perturbations of the wavevector in the direction perpendicular to the initial plane of the soliton. This instability occurs only if the ratio of the saddle-splay elastic constant to the elastic modulus of nematics in the one-constant approximation is above the critical value .

Critical Reexamination of Berreman's Theory on Surface Anchoring

We critically reexamine well-known Berreman's theory [Phys. Rev. Lett. 28, 1683 (1972)] on the anchoring of a nematic liquid crystal due to its elastic distortions induced by a sinusoidally grooved surface. We put emphasis on the effect of azimuthal distortions of the director n and the contribution of saddle-splay surface elasticity characterized by K 24. We give a correct calculation of the anchoring energy and show that Berreman's theory gives a correct result only when K 1 = K 2 and K 24 = 0, where K 1 and K 2 are the splay and twist elastic constants, respectively. We also present our preliminary numerical attempts to evaluate the anchoring energy of a surface with square patterns and compare the anchoring energy calculated numerically with an analytical one obtained by a direct extension of our theoretical argument on one-dimensional parallel grooves.

Effect of Saddle-Splay Elasticity on Stability of Disclination Rings in Nematic Liquid Crystals

In this paper, the stability of disclination ring in nematic liquid crystals is studied. In the presence of saddle-splay elasticity (characterized by k24) the disclination ring has a universal equilibrium radius. Depending on the values of the saddle-splay constant k24, the universal equilibrium radius is altered. When k24 > 0.92k (m = 1/2) and k24 > 0.88k (m = −1/2), the disclination will be a point rather than a ring, where k is the Frank elastic constant in the one-constant approximation.

Stability of a hyperbolic disclination ring in a nematic liquid crystal.

The stability of a hyperbolic disclination ring in a nematic liquid crystal is considered by extending the argument for a radial disclination ring due to Mori and Nakanishi. The ring configuration is indeed stable in the presence of the saddle-splay elasticity (characterized by K24). The ring radius is estimated to be a approximately 2.9r(c), with r(c) being the core radius when K24 approximately K/2, where K is the Frank elastic constant in the one-constant approximation.

Saddle-splay elasticity and interfacial nematostatics.

This Brief Report drives the generalized force balance equations of interfacial statics between nematic liquid crystals (NLC) and isotropic fluids (I), using the classical equations of liquid crystal physics, taking into account an important class of gradient surface elasticity, known as saddle-splay elasticity. The objective is to identify the exact nature of the saddle-splay contributions to the fundamental interfacial force balance equations, known as the Laplace-Young equation and the Marangoni force equation. General expressions for the dynamic generalization of these two equations were given by Shih, Mann, and Brown [Mol. Cryst. Liq. Cryst. 98, 47 (1983)], but the specific form of the static terms appearing in these two equations were missing in the literature, and are now given in this paper. It is found that the tensorial order and functional form of the contributions of saddle-splay elasticity to the two force balance equations are congruent with those arising from the interfacial tension. Therefore, to generalize the interfacial equations of nematostatics by including saddle-splay energy, the interfacial tension must be renormalized with the saddle-splay energy contribution. In addition, saddle splay gives rise to distortion stresses, the two-dimensional analog to the bulk Ericksen stresses, which contribute to the tangential Marangoni force. Exact expressions for pressure jumps across NLC/I interfaces and for the tangential Marangoni force are derived and analyzed. These generalized results are expected to be useful in the characterization of nematocapillarity phenomena, such as wetting, spreading, and the mechanics of thin nematic films.

SADDLE-SPLAY ELASTICITY IN NEMATIC LIQUID CRYTALS

The effect of the saddle-splay elastic constant K24 on nematic director-fields is a topic of fundamental interest for the elastic theory of liquid crystals and plays a vital role in device applications where nematic liquid crystals are confined to a curved geometry. The first measurement of K24 came more than sixty years after the pioneering work of Oseen, who made first steps toward the elastic theory of liquid crystals. Today the study of saddle-splay elasticity has become an important topic in liquid crystal physics. In this review we summarize the various methods used to measure K24 in curved geometries and present the detailed theory behind the measurements.