J. Pollard and R.G. Morris, Preprint, arxiv.org:2408.01032, 2024

The topological understanding of nematic liquid crystals is traditionally centered on singularities, or defects, and their classification via homotopy theory. However, this approach has ultimately proved insufficient to properly capture a range of complex behaviours that have been reported in three dimensions. To address this, we argue that a finer understanding of topology is required, in which non-singular but non-trivial topological solitons - so-called merons - play a central role in mediating interactions between disclination lines. We present a comprehensive framework for capturing such behaviour that draws heavily on aspects of Morse theory; the key notion being that merons appear singular under projection onto a two-dimensional surface. This permits the use of singularity theory and dividing curves to characterise nematic textures via tomography, as well as an understanding of topological transitions via surgery theory. We use our ideas to understand and classify complex three-dimensional behaviours, such as the linking, rewiring and crossing of disclination lines, as well as to provide a new perspective on defect charge.

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J. Pollard, S.C. Al-Izzi, and R.G. Morris, Preprint (under review at J. Fluid Mech.), arxiv.org:2406.18014, 2024

Morphodynamic descriptions of fluid deformable surfaces are relevant for a range of biological and soft matter phenomena, spanning materials that can be passive or active, as well as ordered or topological. However, a principled, geometric formulation of the correct hydrodynamic equations has remained opaque, with objective rates proving a central, contentious issue.We argue that this is due to a conflation of several important notions that must be disambiguated when describing fluid deformable surfaces. These are the Eulerian and Lagrangian perspectives on fluid motion, and three different types of gauge freedom: in the ambient space; in the parameterisation of the surface, and; in the choice of frame field on the surface. We clarify these ideas, and show that objective rates in fluid deformable surfaces are time derivatives that are invariant under the first of these gauge freedoms, and which also preserve the structure of the ambient metric. The latter condition reduces a potentially infinite number of possible objective rates to only two: the material derivative and the Jaumann rate. The material derivative is invariant under the Galilean group, and therefore applies to velocities, whose rate captures the conservation of momentum. The Jaumann derivative is invariant under all time-dependent isometries, and therefore applies to local order parameters, or symmetry-broken variables, such as the nematic Q-tensor. We provide examples of material and Jaumann rates in two different frame fields that are pertinent to the current applications of the fluid mechanics of deformable surfaces.

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J. Pollard and G.P. Alexander, New. J. Phys. 26, 063027, 2024

Integer winding disclinations are unstable in a nematic and are removed by an `escape into the third dimension', resulting in a non-singular texture. This process is frustrated in a cholesteric material due to the requirement of maintaining a uniform handedness and instead results in the formation of strings of point defects, as well as complex three-dimensional solitons such as heliknotons that consist of linked dislocations. We give a complete description of this frustration using methods of contact topology. Furthermore, we describe how this frustration can be exploited to stabilise regions of the material where the handedness differs from the preferred handedness. These `twist solitons' are stable in numerical simulation and are a new form of topological defect in cholesteric materials that have not previously been studied.

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J. Pollard and G.P. Alexander, Phys. Rev. Lett. 130, 2023

We give a complete topological classification of defect lines in cholesteric liquid crystals using methods from contact topology. By focusing on the role played by the chirality of the material, we demonstrate a fundamental distinction between “tight” and “overtwisted” disclination lines not detected by standard homotopy theory arguments. The classification of overtwisted lines is the same as nematics, however, we show that tight disclinations possess a topological layer number that is conserved as long as the twist is nonvanishing. Finally, we observe that chirality frustrates the escape of removable defect lines, and explain how this frustration underlies the formation of several structures observed in experiments.

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J. Pollard and S.M. Fielding, Phys. Rev. Research 4, 2022

Widespread processes in nature and technology are governed by the dynamical transition whereby a material in an initially solid-like state, whether soft or hard, then yields. Major unresolved questions concern whether any material will yield smoothly and gradually (“ductile” behavior) or fail abruptly and catastrophically (“brittle” behavior); the roles of sample annealing, disorder, and shear band formation in the onset of yielding and failure; and, most importantly from a practical viewpoint, whether any impending catastrophic failure can be predicted before it happens. We address these questions by studying theoretically the yielding of slowly sheared athermal amorphous materials, within a minimal mesoscopic lattice elastoplastic description. Our contributions are fourfold. First, we elucidate whether yielding will be ductile or brittle, for any given level of sample annealing prior to shear. For highly annealed samples, we find brittle yielding for all samples sizes. For poorly annealed samples we uncover an important dependence on the size of the sample of material being sheared, with ductile yielding for small samples, and brittle yielding only for large system sizes. Second, we show that yielding comprises two distinct stages: a prefailure stage, in which small levels of strain heterogeneity slowly accumulate within the material, followed by a catastrophic brittle failure event, in which a shear band quickly propagates across the sample via a cooperating line of (individually) localized plastic events. Third, we provide an exact expression for the slowly growing level of strain heterogeneity in the prefailure stage, expressed in terms of the macroscopically measured stress-strain curve and the sample size, and in excellent agreement with our simulation results. Fourth, we elucidate the basic mechanism via which a shear band then nucleates, in terms of the onset of cooperativity between plastic events. We furthermore provide an expression for the probability distribution of shear strains at which failure occurs, expressed in terms of the sample size and the disorder inherent in the sample, as determined by the degree of annealing prior to shear.

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J. Eun, J. Pollard, S-J. Kim, T. Machon, J. Jeong, PNAS 118, 2021

Our study of cholesteric lyotropic chromonic liquid crystals in cylindrical confinement reveals the topological aspects of cholesteric liquid crystals. The double-twist configurations we observe exhibit discontinuous layering transitions, domain formation, metastability, and chiral point defects as the concentration of chiral dopant is varied. We demonstrate that these distinct layer states can be distinguished by chiral topological invariants. We show that changes in the layer structure give rise to a chiral soliton similar to a toron, comprising a metastable pair of chiral point defects. Through the applicability of the invariants we describe to general systems, our work has broad relevance to the study of chiral materials.

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J. Pollard and G.P. Alexander, New. J. Phys. 23, 2021

We give a description of the intrinsic geometry of elastic distortions in three-dimensional nematic liquid crystals and establish necessary and sufficient conditions for a set of functions to represent these distortions by describing how they couple to the curvature tensor. We demonstrate that, in contrast to the situation in two dimensions, the first-order gradients of the director alone are not sufficient for full reconstruction of the director field from its intrinsic geometry: it is necessary to provide additional information about the second-order director gradients. We describe several different methods by which the director field may be reconstructed from its intrinsic geometry. Finally, we discuss the coupling between individual distortions and curvature from the perspective of Lie algebras and groups and describe homogeneous spaces on which pure modes of distortion can be realised.

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J.Binysh, J. Pollard and G.P. Alexander, Phys. Rev. Lett. 125, 2020

We describe the geometry of bend distortions in liquid crystals and their fundamental degeneracies, which we call β lines; these represent a new class of linelike topological defect in twist-bend nematics. We present constructions for smecticlike textures containing screw and edge dislocations and also for vortexlike structures of double twist and Skyrmions. We analyze their local geometry and global structure, showing that their intersection with any surface is twice the Skyrmion number. Finally, we demonstrate how arbitrary knots and links can be created and describe them in terms of merons, giving a geometric perspective on the fractionalization of Skyrmions.

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J. Pollard, G. Posnjak, S. Čopar, I. Muševič, and G.P. Alexander, Phys. Rev. X 9, 2019

We provide a characterization of point defects in droplets of cholesteric liquid crystal, using a combination of experiment, simulation, and theoretical analysis. These droplets display a range of structures including realizations of defects with high topological charge and arrangements of multiple defects in “topological molecules.” We show that there are certain defects that are incompatible with a uniform sense of chiral twisting for topological reasons. Furthermore, those defects that are compatible with twist of a single handedness are shown to have the structure of the gradient field of an isolated critical point and, hence, are described by singularity theory. We show that the mathematical tools of singularity theory reproduce, with excellent agreement, the experimental observations of high charge defects and topological molecules. Our results have implications beyond liquid crystal droplets in characterizing chiral materials and their topology in general.

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