Preface

This Commemorative Issue of Liquid Crystals Reviews honors the memory of Maurice Kleman, a founder of the modern understanding of topological defects in ordered media and a prominent member of the liquid crystal community, who passed away on 29 January 2021, in Paris, France. Kleman contributed greatly to many branches of condensed matter physics, ranging from liquid crystals, magnetic systems, quasicrystals, as well as amorphous media and biological tissues. Through the variety of themes, the guiding idea of his research was the concept of topological defects, which he explored most fully for liquid crystalline phases, summarizing his results in scientific monographs [1,2], an autobiography [3], and multiple reviews [4–10]. His remarkable achievements include the elucidation of the fine structure and properties of line defects in chiral, uniaxial, twist-bend, and biaxial nematics, thermotropic and lyotropic smectics, columnar, blue, and twist-grain-boundary phases, descriptions of focal conic and developable domains, and the homotopy classification of defects in ordered media.

The articles in this issue, presented in two parts in volume 10 and volume 11, connect to Maurice Kleman in a variety of ways, by exploring research areas to which he made significant contributions, describing ideas inspired or influenced by him, and presenting historical reminiscences related to his research, mentorship, and life. Below we give a very brief outline of the Kleman’s contributions to the topic of defects in liquid crystals as a background which helps to introduce the contributed papers.

Maurice started his exploration of liquid crystals in 1969 at the Laboratoire de Physique des Solides, where he and his scientific mentor and friend Jacques Friedel [11] gave a geometrical description of line defects in cholesterics, stressing their disclination-dislocation duality and classifying them into the so-called $\tau$, $\lambda ,$ and $\chi$-lines, depending on how the Burgers vector is aligned with respect to the helicoidal axis and whether the core of the defect is singular in molecular orientation.

The classification of cholesteric defects continued by the demonstration of a profound difference between ‘thick’ and ‘thin’ disclinations in a nematic. Kleman and a postdoctoral visitor to Orsay, Patricia Cladis, demonstrated theoretically that the ‘thick’ lines with an alleged strength 1 or –1 (i.e. in which the director rotates by 360° around the defect axis) are not really defects: the director realigns along the axis so that the entire structure smoothly transforms into a defect-free state [12], a process called by Robert Meyer ‘escape into the third dimension’ [13]. If the rotation is 180°, the director could not escape and forms a singular core.

The difference between 360° (thick) and 180° (thin) lines manifested a deep connection between the type of ordering and the nature of defects in a medium, which eventually led to the homotopy classification of defects in ordered media by Kleman and Gérard Toulouse in 1976 [14]. To establish which defects are permissible in a medium, one needs to know the ‘degeneracy space’ of the order parameter, also called the ‘order parameter space’ and ‘ground state manifold’ and defined as the manifold of all possible values of the order parameter that do not alter the thermodynamic potentials of the system. For instance, in a recently discovered ferroelectric nematic, the degeneracy space is a sphere, each point of which corresponds to a possible orientation of the spontaneous electric polarization in space. In a conventional quadrupolar nematic, it is a sphere with antipodal points identified, because of the apolar character of the director. Generalizing the Burgers circuit in the construction of dislocations in solids to mappings of Burgers ‘enclosures’ around different defects in ordered media onto the degeneracy space, homotopy theory classifies various defects (such as points, lines, walls, and 3D textures) into separate classes that correspond to the elements of the homotopy groups of the ground state manifold. The stability of each class of defects, their behavior during merging, splitting, mutual entanglement, and crossing is described by topological charges, associated with the elements of these groups. The apolar character of the director in a conventional nematic permits topological stability of 180° disclinations, since a loop encircling such a disclination in real space is mapped onto the degeneracy space as a contour that connects diametrically antipodal points. Such a contour cannot be transformed into a point by any smooth transformation, guaranteeing the topological stability of the disclination and assigning it a nonzero topological charge. In contrast, there are no topologically stable disclination lines in a ferroelectric nematic, as a loop encircling a suspect line in real space is mapped onto a closed loop on the sphere of the ferroelectric degeneracy space; such a loop can always be shrunk into a point, which corresponds to a rearrangement of polarization into a uniformly aligned state. The homotopy classification uncovered a deep connection between liquid crystal defects to numerous topological formations in other types of condensed matter, such as solid crystals, superconductors and superfluids, ferromagnets and ferroelectrics, incompressible fluids, and objects such as magnetic monopoles, instantons, solitons, and cosmic strings.

Kleman's longtime fascination was with focal conic domains in liquid crystals such as smectics and cholesterics. According to his autobiography [3], these domains, appearing as beautiful pairings of ellipses and hyperbolae, were the first textures of liquid crystals he observed under a polarizing microscope. In 1922, prior to X-ray studies, these domains prompted Georges Friedel (the grandfather of Jacques Friedel) to discover that smectics are stacks of fluid layers that could easily bend but could not change their thickness. Maurice developed an analytical description of focal conics [15,16] and their associations, demonstrating how these are related to grain boundaries and dislocations [17,18]. With this brief background, we now introduce the contributions to this special issue.

Pawel Pieranski [19] returns to the historical roots of the experiments on the defects in cholesterics that were first performed in 1921 by François Grandjean [20] and then expanded in Orsay by Georges Durand and Madeleine Veyssié [21]. Grandjean used cleaved mica sheets to confine a cholesteric in a wedge-like geometry. It turns out that mica sheets impose a good planar alignment on the director, while their flexibility allows one to construct not only ‘linear’ wedges but also ‘curvilinear’ wedges between two crossed cylindrical sheets [19]. The samples reveal beautiful arrays of dislocations, straight and curved, with Burgers vector of both half-pitch and full pitch, thus substantiating the theoretical insights of Kleman and Friedel [11]. The crossed curved sheets produce dislocation loops that nucleate and change their radius each time the separation distance of the two sheets is changed. When the loop is not strictly perpendicular to the far-field helicoidal axis, it shows a kinked shape, with one or two cusps signaling a transition from one level along the helicoid to the next; Kleman and Friedel alluded to these shapes in their pioneering work [11]. Pawel stresses that the ‘cholesteric between mica’ experiments are well adapted for undergraduate labs since they do not require complex preparation and instrumentation, and yet yield a rich plethora of physical effects.

Jin-Sheng Wu and Ivan Smalyukh [22] continue the theme of defects in cholesterics by presenting a comprehensive review on the three-dimensional structure and visualization of defects by modern microscopy techniques such as fluorescence confocal polarizing microscopy (FCPM). FCPM permitts the visualization of the core structure of dislocations in cholesterics as pairs of $\tau$ and $\lambda$ disclinations, in complete agreement with the insight of Kleman and Friedel [23]. The review discusses deep analogies between the elastic and topological properties of cholesterics, smectics, biaxial nematics, and ferromagnets and describes many intriguing topological objects that the cholesterics offer under careful observation. Among these, besides lines of dislocation-disclination dual characters, are skyrmions, hopfions, heliknotons, torons, and twistions; to this list, one can also add a monopole, resembling structurally the Dirac’s magnetic charge. It is stressed that because of a rich structure of their degeneracy space, and the possibility to chemically ‘engineer’ the elastic constants to vary in a broad range, the cholesterics represent an excellent model system to explore nontrivial topologies of many other systems, the access to which is hindered either by small/large length scales or high energies. The authors describe $\tau$, $\lambda ,$and $\chi$-disclinations in terms of the group of unit quaternions thus linking the original classification of Kleman and Friedel to the later developed homotopy classification.

Jun-Ichi Fukuda [24] presents further exploration of exotic structures such as arrays of skyrmions, whirl-like localized topological formations that, according to the presented Landau-de Gennes theory, should occur in cholesteric materials with a very short pitch (<300 nm), so that the bulk ordering tends to form blue phases with double-twisted helicoidal director elements and disclination lines relieving structural frustrations. The surface anchoring is assumed to be sufficiently strong to prevent an ideal bulk structure of various cubic symmetries and to produce arrays of skyrmion-like structures as a compromise between the bulk and surface interactions. Note here that Kleman developed the first analytical model of double-twist in blue phases and in chromosomes of dinoflagellates, stressing the important role of the saddle splay elastic modulus K₂₄ in their stabilization and the possibility of coexistence of cholesteric and two-dimensional periodic translational order [25].

The topological theme is continued by Gareth Alexander and Randall Kamien [26] who show a deep connection between two subtle effects predicted by homotopy theory. One is the nontrivial entangled configuration of two closed linear defects, which in certain cases must be connected by a third defect. Kleman predicted this effect in 1977 [27], immediately after the homotopy theory had been presented, specifying that the extra defect could be observed in a biaxial nematic. Toulouse came to a similar conclusion, predicting the emergence of an extra defect that prevents crossing and disentanglement of two disclinations in a biaxial nematic [28]. The corresponding configurations of the biaxial director fields are prohibitively complex to draw, thus it is hard to imagine that such a profound ‘quark-interaction-like’ effect could be predicted by means other than the homotopy groups classification. The second effect underscoring the power of the homotopy classification is the ambiguity in the result of merging of two point defects-hedgehogs of charge 1 described by Volovik and Mineev [29]: In the presence of a disclination line, the two could either annihilate or produce a defect of a charge 2. Alexander and Kamien [26] demonstrate that the two effects arise from the same mathematical structure, related to the Whitehead products of homotopy groups. The authors apply the concept to classify defect structures introduced by two colloidal particles with homeotropic anchoring, each of which mimics a radial point defect-hedgehog. The work extends Kleman’s insight into linked disclinations to higher-dimension textures, which are nonsingular and yet topologically stable, interpreting the most famous example, the Hopf texture, in terms of the Whitehead product.

Saša Harkai, Charles Rosenblatt, Samo Kralj and their colleagues [30] review both the theoretical and experimental aspects of topological defects in liquid crystals, with an emphasis on how one could create long-lived defect states. One of the approaches to stabilize a nontrivial topological charge and avoid annihilation of opposite charges is via surface interactions in sandwich-type cells, in which the bounding plates are predesigned with a non-uniform director field. Desired director patterns could be mechanically nano-inscribed into polymer coatings by a stylus of an atomic force microscope (AFM). The seeded point defects at these substrates serve as the ends of disclinations once the sample with two parallel patterned substrates is filled with a liquid crystal, which allows one to perform a number of interesting experiments on the fine structure of the defect cores, switching one defect type into another by external fields, and motion of defects in electrohydrodynamic instability. The authors also describe other approaches to defect production and stabilization. For example, transient defects could be created during an isotropic-nematic phase transition in ‘bubbles’ of the nematic which must carry nontrivial charge because of the spherical topology and non-vanishing surface anchoring, thanks to the analogy to the beautiful Kibble mechanism of cosmic string formations in early stages of the Universe and the appearance of defects during the isotropic-to-nematic phase transition. Finally, the special case of stable disclination networks, the blue phases, could be extended by adding nanoparticles that accumulate at the disclination cores.

Bruno Zappone and Emmanuelle Lacaze [31] discuss periodic patterns and defects in thin coatings of a smectic-A shaped by the balance of bulk elasticity, preferring the molecular layers to preserve their equidistance, and the external forces, such as hybrid alignment of the director that requires both splay and bend of the director. In a nematic, hybrid alignment introduces smooth splay-bend deformations along the normal to the film or, when the film is of a submicron thickness, large-period stripes incorporating also twist and saddle splay [32]. In a smectic-A director bend and twist are expelled from the structure because of the layers’ equidistance. As a result, thin submicron coatings placed onto a substrate with a unidirectional planar anchoring show a one-dimensional periodic array of semicylindrical domains, while thicker coatings develop two-dimensional arrays of focal conic domains. Interestingly, these domains are very different from the toroidal domains expected when the two bounding surfaces yield a degenerate in-plane anchoring: the elliptical bases are tilted with respect to the planar substrate, and large portions of the ellipse and hyperbola are virtual, extending beyond the physical limits of the sample. The periodic patterns of distorted smectic layers are effective in trapping and aligning nanoparticles and their arrays. Aligned arrays of gold nanorods show highly anisotropic absorption of light, suggesting practical applications.

Yuriy Nastishin and Claire Meyer [33] discuss the complex issue of imperfect defects in smectics-A, on which the authors worked in collaboration with Maurice [34]. The issue is a facet of a deep interconnection between curvature defects, disclinations, and translational defects-dislocations as outlined by Kleman and Friedel in Reviews of Modern Physics in 2008 [10]. In smectics, the ellipse and hyperbola framing a focal conic domain are essentially curved disclinations and thus could emit both disclinations and dislocations. A grain boundary separating two domains with differently tilted smectic layers could be relaxed by a wall of edge dislocations, when the misalignment angle is small, or by a chain of focal conic domains connected by short segment of dislocations, when the angle is not small. Another experimental realization of the dislocation-disclination emitting ability of focal conic domains are the so-called oily streaks in smectics and cholesterics, which represent bundles of disclinations, dislocations, and focal conic domains decorating these bundles [17]. Nastishin and Meyer discuss numerous geometries in which the emitting properties of defects in smectic-A appear explicitly under a polarizing microscope as macroscopic kinks in the ellipse-hyperbola and parabolic frames of focal conic domains and also in the structure of double-helicoidal domains, which are screw dislocations with a giant Burgers vector.

Antal Jakli, Yuriy Nastishin and Oleg Lavrentovich [35] describe how the change in the molecular shape of mesomorphic materials affects their ordering and the consequent topological defects. One of the long-standing problems in the physics of liquid crystals is the absence of solid experimental evidence of the existence of a biaxial nematic. The interconnected disclination loops in the biaxial nematics predicted by Kleman [27] and Toulose [28] might serve as an excellent test of biaxiality in the search of this elusive mesophase. Twist-bend nematics formed by flexible or rigid bent-cores show a dual character of defects, as their nanoscale director modulation yields smectic-like defects such as focal conic domains; Kleman wrote his last scientific papers on the structure of these defects, in collaboration with Krishnamurthy [36,37]. Finally, the review discusses mesophases in which the defects are not merely a departure from the ground state but an intrinsic feature of this ground state. A classic example are the blue phases of cholesterics with the network of disclinations and the twist-grain-boundary phases predicted by Renn and Lubensky [38]. More recent cases are the so-called helical nanofilament phase and ‘dark conglomerate’ phase, considered as a mesh of interconnected small focal conic domains [39].

Tim Sluckin [40] celebrates Kleman’s contribution to science by providing an account of his personal life, reviewing some of the research achievements and placing his work in a broad context of the topological defects theme, which extends from the mathematical discipline of topology to the science of living systems. One finds short excursion into Maurice’s early childhood, life under Nazi occupation, education, and his scientific career, intertwined closely with the members of the Friedel scientific family, both directly and indirectly. The paper describes the origin and development of classic ideas that shaped Maurice’s vision, from the Volterra process to dislocations in solid crystals, from algebraic topology to the Frank-Oseen model of nematic elasticity and defects. Maurice’s research is connected to the parallel work on superfluid anisotropic phases of He-3, cosmic strings in Kibble’s scenarios of the early Universe, elementary particles viewed as ‘kinks’, and, especially intriguing, defects in living tissues. The historical excursions nicely complement Kleman’s autobiography [3], while the subtle humor of the writing connects to the warmth of Maurice’s personality, which many miss deeply.