Memorial Article in honor of Maurice Kleman

Liquid Crystals Reviews 10(1-2) (2022) 6-33. 

The initial aim of this memorial article was to illustrate with color pictures the very first article of Maurice Kleman and Jacques Friedel devoted to dislocations in cholesterics : 

Color pictures of cholesteric dislocations in a wedge made of mica sheets cholesterics : 

Nets of dislocations in Grandjean wedges produced by a partial cleavage of mica sheets and filled with 5CB/CB15 cholesteric mixtures. (a–c) Single dislocations lines (SL) located in the thinnest areas of wedges. (d) Coexistence of single (SL) and double (DL) dislocations lines.  In agreement with a commonly believed paradigm, the single and double lines are located respectively in thinner and thicker areas of wedges.  

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Memorial Article in honor of Gérard Toulouse

Comptes Rendus Physique, 25 (2024) 367-388

Abstract. Classification of defects in ordered systems, based on the homotopy theory, conceived by Gérard Toulouse and Maurice Kléman has a very wide range of applications. We illustrate its modus operandi with three experimental examples. We deal first with dislocations in a dissipative periodic pattern of convection rolls in a shear flow instability in nematics. As the second example we chose the captive disclination loops threaded on polymer fibers immersed in nematics. Third, we focus on objects with double topological character defined for the first time in the generic article [1] coauthored by Gérard Toulouse: the “double et tripple anneau” Hopf links made of interlaced dislocation loops in cholesterics. Finally, we report on the recent discovery of their generalised, beads necklace version made of many minimal dislocation loops threaded, like pearls, on a string-like dislocation loops.

[1]  : Y. Bouligand, B. Derrida, V. Poénaru, Y. Pomeau and G. Toulouse, “Distorsions with double topological character:

the case of cholesterics”, J. Phys. France 39 (1978), pp. 863–867.

“The triple ring structure” predicted in ref. [1]. a) Reproduction of  Figure 1 from the ref. [1]. b) The three components Hopf link observed in our experiments. c) Perspective view showing three kinks on the cargo loop carrying the two minimal loops.

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Decay of skeins of dislocations in cholesterics: 

rewiring Conway's tangles into necklaces of bangles

Abstract: Knotted and linked skeins of vortices and disclinations generated, respectively, by symmetry-breaking normal - superfluid and isotropic - nematic phase transitions are known to untie, by rewiring of their crossings, into independent unknots that finally shrink and collapse until the defect-free ground state is reached. We demonstrate that the decay of skeins of dislocations, generated by the isotropic - cholesteric phase transition within a cylinder/cylinder gap, leads to stable necklace-like states made of numerous minimal loops, called bangles, tethered to kinks of much larger loops called cargo. We analyze the topological decay of skeins of dislocations in terms of the Conway–Kauffman theory of knots, showing that the necklace state results from rewiring of crossings triggered by collisions of tangles with their numerator closure. We point out that, in general, for symmetry reasons, kinks on edge dislocations are chiral. Their handedness, right or left, directly depends on the sign of kinks on which they are localized. In cholesterics with intrinsic chirality, the energy of kinks bearing bangles depends on their handedness. For this reason, within necklaces, all bangles are tethered to kinks of the same sign.

Jun-Yong Lee et al. , Soft Matter 21 (2025) 8205

The necklace state

(a) Geometry of experiments with cholesteric droplets (5CB/CB15 mixtures) confined between cylindrical mica sheets. The upper mica sheet can be moved in the x, y and z directions. (b) Microscopy image of a necklace composed of a circular dislocation called a cargo loop bearing five small circular dislocations called minimal loops or bangles. N is the number of cholesteric pitches p E 5 mm located between the mica sheets. (c) Close-up image of a bangle tethered to a kink of the cargo loop. The radius of the bangle is about three times smaller than the cholesteric pitch: rb E p/3. It varies with the concentration cCB15 of the chiral compound CB15 in the nematic 5CB. Here, cCB15 = 0.86% and p E 25 mm. (d) Perspective view of the necklace state. As all minimal loops are tethered exclusively on the +p kinks of the cargo loop, this configuration is chiral (for details see Fig. 10).

Bangle tethered to a kink of a dislocation. 

(a) Bright-field image of the bangle structure. (b) Cross-sectional fluorescence confocal polarizing microscopy (FCPM) images along the dotted lines (1) and (2) in (a), showing vertical deviation of the dislocation with pitch p. The polarization direction is along the x-direction. (c) and (d) Cross-sectional FCPM images in planes (c) perpendicular and (d) parallel to the edge dislocation line far from the kink. For each slice, fluorescence intensities acquired under four linearly polarized excitations (0°, 45°, 90°, and 135°) are summed. Hence, bright regions correspond to in-plane alignment of the director, whereas dark regions indicate vertical alignment with the director, roughly along the z-axis. Images i–vi correspond to cross-sections along the lines i–vi marked in (a). (e) In-plane intensity-summed FCPM images at different heights. (f) 3D reconstruction of the dislocation kink and surrounding bangle based on experimental observations. (g) Plot of the relative height (DZd) versus the relative lateral positions (DXd) of the dislocation. Both positions are extracted from the experimental images like the ones shown in (d). For the sake of clarity, the dihedral angle of the wedge represented in the inset is exaggerated. (ZLI-3412/CB15 mixture with the cholesteric pitch p = 5 microns.)

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New concept

Topological metadefects: tangles of dislocations

Abstract: The concept of topological defects is universal. In condensed matter, it applies to disclinations, dislocations or vortices that are fingerprints of symmetry breaking during phase transitions. Using as a generic example the tangles of dislocations, we introduce the concept of topological metadefects i.e. defects made of defects.  We show that in cholesterics, dextrogyre and levogyre primary tangles are generated through the D2 -> C2  symmetry breaking from the coplanar dislocation pair called Lehmann cluster submitted to a high enough tensile strain. The primary tangles can be wound up individually into double-helices. They can also annihilate in pairs or associate into tangles of higher orders following simple algebraic rules.

Published in Physical Review Letters 131 (2023) 128101 

Generation of the dextrogyre primary tangle, driven by a tensile strain, from the Lehmann cluster: a) variation of the Peach-Koehler force with the gap thickness, b-e) overlapping instability of the Lehmann cluster, f) geometry of the Lehmann cluster, g-k) detailed views of the overlapping instability leading to the generation of the dextrogyre tangle. 

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The Geilo School 2022

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Publication de la version française du livre collectif sur les cristaux liquides

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Collective book on liquid crystals 

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Article published in Communications Materials