Maurice Kleman

Defects in liquid crystals

Defects are local breakings of symmetry in an ordered medium. The physics of defects has long been reduced to the study of dislocations in solids, and to the main physical phenomena they are responsible for, like plastic deformation. Dislocations break translational symmetries. Disclinations break rotational symmetries and are the basic defects of media with continuous symmetries, like liquid crystals. In this review, it is stressed how their study has contributed to a renewal of the physics of defects. Static and dynamic properties in the nematic, cholesteric, blue, smectic and columnar phases of liquid crystals are described in detail, up to the most recent results, for small-molecule thermotropic liquid crystals as well as for lyotropic and polymeric liquid crystals. The authors also discuss the homotopy classification of defects, including point defects, and compare it with the Volterra classification; finally they present the curved-crystal description of frustration in the light of the relevant situation in liquid crystals. The usefulness of the concepts introduced for liquid crystals for the study of other systems (structural problems in biology, dissipative structures, magnetic domains, etc.) is also emphasised.

Topological point defects in nematic liquid crystals

Point defects in nematics, also called hedgehogs, are topological entities that have no equivalent in ordered atomic solids, despite the homonymy. They have been the subject of intense experimental and, above all, theoretical (analytical and computational) investigations in the last thirty years. They are present in bulk specimens and at the specimen boundaries. This review article stresses the importance of the core structure of the defect, the possibility of it splitting into a disclination loop, and boundary conditions, as well as taking stock of the recent advances on point defects in nematic colloidal suspensions. An important topic is the formation of strings between opposite hedgehogs (radial and hyperbolic), and their role in the dynamic properties of nematics.

Defect structures in lyotropic smectic phases revealed by freeze-fracture electron microscopy

Abstract Freeze-fracture electron microscopy has been used to examine defect structures such as confocal domains, screw dislocations, disclinations, and walls in lamellar phospholipid phases containing known amounts of water (from about 5 to 65% by weight). In a concurrent study (Costello and Gulik-Krzywicki 1976) all samples were examined by X-ray diffraction before and after freeze-quenching to ensure that the freezing process had not induced a phase transition. Most of the prominent defect structures, especially confocal domains, have been observed in all samples. Electron micrographs showing the lamellar arrangements of the defects near their cores are presented. The most important observation about the frequency of occurrence of defects is the proliference of screw dislocations and the absence of edge dislocations. This lends support to the commonly proposed hypothesis that the deformation of smectic layers tends to occur at constant layer thickness.

Textural analysis of a mesophase with banana shaped molecules

Abstract.Observed under the polarizing microscope, the

Imperfections in focal conic domains: the role of dislocations

B7_{\rm bis}

Liquid crystal helical ribbons as isometric textures

phase in the banana compound D14F3 [J.P. Bedel et al., Liq. Cryst. 27, 1411 (2000)] displays two types of textures of defects, namely (a): helical ribbons, that nucleate in large quantities when the samples are quenched from a sufficiently high temperature in the isotropic phase (b)- shapes with no helicity having the structure of developable domains much akin to those observed in columnar phases, either resulting from the annealing of the helical ribbons or nucleating under slow cooling processes. The existence of these two kinds of defects points toward the complex nature of the structure of the B7 phase, which is at the same time a columnar and a smectic phase. Our observations fit the model [M. Kleman, J. Phys. France 46, 1193 (1985)] according to which the geometry of a helical ribbon is that one of the central region of a screw dislocation with a giant Burgers vector, split into two helical disclination lines of strength k = 1/2 which bound the ribbon. Textures and defects, already partly documented, and growth features and annealing processes, not yet reported in the literature, are analyzed. We conclude that the helical ribbons and the developable domains with no helicity are textures of two different B7 states, namely a metastable state and the ground state respectively. Comparative textural analysis is performed for two other banana compounds exhibiting B2 phases.

Screw dislocations in the smectic C phase of liquid crystals

It is usual to think of focal conic domains (FCDs) as perfect geometric constructions in which the layers are folded into Dupin cyclides about an ellipse and a hyperbola that are conjugate. This ideal picture is often far from reality. We have investigated in detail the FCDs in several materials that have a transition from a smectic A (SmA) to a nematic phase (N). The ellipse and the hyperbola are seldom perfect, and the FCD textures also suffer large transformations (in shape and/or in nature) when approaching the transition to the nematic phase, or appear imperfect on cooling from the nematic phase. We interpret these imperfections as due to the interaction of FCDs with dislocations. We analyze theoretically the general principles subtending the interaction mechanisms between FCDs and finite Burgers vector dislocations, namely the formation of kinks on disclinations, to which dislocations are attached, and we present models relating to some experimental results. Whereas the principles of the interactions are very general, their realizations can differ widely in function of the boundary conditions.

Imperfect focal conic domains in A smectics: a textural analysis

Abstract.Deformations that conserve the parallelism and the distances – between layers, in smectic phases; between columns, in columnar phases – are commonplace in liquid crystals. The resulting isometric deformed textures display specific geometric features. The corresponding order parameter singularities extend over rather large, macroscopic, distances, e.g., cofocal conics in smectics. This well-known picture is modified when, superimposed to the 1D or 2D periodicities, the structure is helical. However isometry can be preserved. This paper discusses the case of a medium whose structure is made of 1D modulated layers (a lamello-columnar phase), assuming that the modulations rotate helically from one layer to the next. The price to pay is that any isometric texture is necessarily frustrated; it consists of layers folded into a set of parallel helicoids, in the manner of a screw dislocation (of macroscopic Burgers vector), the modulations being along the helices, i.e. double-twisted. The singularity set is made of two helical disclination lines. We complete this geometric analysis by a crude calculation of the energy of a helical ribbon. It is suggested that the helical ribbons observed in the B7 phase of banana-like molecules are such isometric textures. As a side result, let us mention that the description of double-twist, traditionally made in terms of a partition of the director field into nested cylinders, could more than often be profitably tested against a partition into nested helicoids.

Dislocations observed by transmission electron microscopy in the quasicrystalline phase T2 of Al-Li-Cu after deformation

The displacement field of the screw dislocation, the interaction energy of two parallel screw dislocations and their self-energy is calculated for a smectic C liquid crystal in the approximation of...

Effect of Frustration in Liquid Crystals and Polymers

We have analysed optical microscopy observations of focal conic domains (FCDs) with imperfect ellipses on the basis of the theoretical concepts developed by Kleman et al. (Philos. Mag. 2006, 86, 4439). Two types of imperfect ellipses are observed: with kinks (elementary imperfections resulting from a topological interaction of a disclination with a dislocation, at their point of junction) being either in the ellipse plane (‘mouse’) or perpendicular to it (‘turtle’). The experimental conditions for the observation of imperfect FCDs of both types are described. A model describing the shape of a mouse‐type ellipse is compared with observations. Two experimental observations in nematogenic smectics (i.e. which have a smectic → nematic transition) emphasise the predominant role of kinks in dynamical phenomena involving dislocations and FCDs: (i) a reversible and sudden temperature‐induced transformation between two FCD textures, i.e. FCDs with fewer kink‐carrying distortions (called isometric textures) and the higher temperature FCDs with a proliferation of kinks (called non‐isometric); (ii) a shrinking of the ellipses upon heating, up to their sudden and total disappearance at a temperature well below the phase transition. In contrast, in a non‐nematogenic smectic, the ellipse size does not vary upon heating until the appearance of the isotropic phase.