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Gareth P. Alexander

Publications and source records attributed to Gareth P. Alexander.

At least 19 recordsLinked to original sources
arXiv2025

Screw Symmetry, Chiral Hydrodynamics and Odd Instability in Active Cholesterics

Active cholesterics are chiral in both their structure, which has continuous screw symmetry, and their active stresses, which include contributions from torque dipoles. Both expressions of chirality give rise to curl forces in the hydrodynamics, which we derive from the active Ericksen-Leslie equations using a geometric approach. This clarifies the hydrodynamics of continuous screw symmetry and provides an example of generalised odd elastic forces that originate from an equilibrium free energy. We discuss also the nonlinear structure of the active hydrodynamics in terms of the Eulerian displacement field of the cholesteric pseudolayers. For the active instability, screw symmetry generates a contribution of chiral activity to the linearised pseudolayer hydrodynamics that is absent in materials with chiral activity but achiral structure. When the two forms are sufficiently antagonistic, this term produces a new active instability with threshold and characteristic wavevector distinct from those of the active Helfrich-Hurault instability in chiral active smectics. Finally, we comment on the isotropic chiral hydrodynamics of materials with three-dimensional screw symmetry.

cond-mat.soft↗
arXiv2025

Scallop Theorem for Swimming in Anisotropic Fluids

In isotropic fluids like water, micrometer-scale swimmers have evolved swim strokes to translate despite their tiny size. As described by Purcell in his Scallop Theorem, reciprocal motions, like those performed by a scallop, cannot drive swimming when inertial effects are absent, as is typical at micrometer length scales. Thus, microswimmers have evolved complex structures that can perform non-reciprocal swim strokes or body displacements to generate motion. Microswimmer dynamics in structured fluids differ fundamentally from those in isotropic fluids because of their inherent asymmetry. The orientation of elongated constituents and the topological defects that spontaneously form near microswimmers provide broken symmetries, even atequilibrium. This is sufficient for the dynamic disturbance of even the simplest isotropic swimmers to generate propulsion. We combine experiments on magnetically rotated colloids in nematic liquid crystals with analytic non-equilibrium solutions to formulate propulsion strategies for microswimmers in nematic fluids and determine how swimming velocity depends on the rotation rate, materials parameters, and forcing regimes. For example, we find that micro-scale spherical colloids swim effectively under continuous rotation and under reciprocal forcing.Thus, swim strokes that are ineffective in isotropic fluids are highly effective in nematic liquid crystals. In light of these observations, the Scallop Theorem is extended for structured fluids.

cond-mat.soft↗
arXiv2024

Chirality and odd mechanics in active columnar phases

Chiral active materials display odd dynamical effects in both their elastic and viscous responses. We show that the most symmetric mesophase with two-dimensional odd elasticity in three dimensions is chiral, polar and columnar, with two-dimensional translational order in the plane perpendicular to the columns and no elastic restoring force for their relative sliding. We derive its hydrodynamic equations from those of a chiral active variant of model H. The most striking prediction of the odd dynamics is two distinct types of column oscillation whose frequencies do not vanish at zero wavenumber. In addition, activity leads to a buckling instability coming from the generic force-dipole active stress analogous to the mechanical Helfrich-Hurault instability in passive materials, while the chiral torque-dipole active stress fundamentally modifies the instability by the selection of helical column undulations.

cond-mat.soft↗
arXiv2024

Escape into the Third Dimension in Cholesteric Liquid Crystals

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.

cond-mat.soft↗
arXiv2023

Three-Dimensional Spontaneous Flow Transition in a Homeotropic Active Nematic

We study the three-dimensional spontaneous flow transition of an active nematic in an infinite slab geometry using a combination of numerics and analytics. We show that it is determined by the interplay of two eigenmodes -- called S- and D-mode -- that are unstable at the same activity threshold and spontaneously breaks both rotational symmetry and chiral symmetry. The onset of the unstable modes is described by a non-Hermitian integro-differential operator, which we determine their exponential growth rates from using perturbation theory. The S-mode is the fastest growing. After it reaches a finite amplitude, the growth of the D-mode is anisotropic, being promoted perpendicular to the S-mode and suppressed parallel to it, forming a steady state with a full three-dimensional director field and a well-defined chirality. Lastly, we derive a model of the leading-order time evolution of the system close to the activity threshold.

cond-mat.soft↗
arXiv2023

Colloids in Two-Dimensional Active Nematics: Conformal Cogs and Controllable Spontaneous Rotation

A major challenge in the study of active systems is to harness their non-equilibrium dynamics into useful work. We address this by showing how to design colloids with controllable spontaneous propulsion or rotation when immersed in active nematics. This is illustrated for discs with tilted anchoring and chiral cogs, for which we determine the nematic director through conformal mappings. Our analysis identifies two regimes of behaviour for chiral cogs: orientation-dependent handedness and persistent active rotation. Finally, we provide design principles for active nematic colloids to achieve desired rotational dynamics.

cond-mat.soft↗
arXiv2023

A Twist On Active Membranes: Odd Mechanics, Spontaneous Flows and Shape Instabilities

Living systems are chiral on multiple scales, from constituent biopolymers to large scale morphology, and their active mechanics is both driven by chiral components and serves to generate chiral morphologies. We describe the mechanics of active fluid membranes in coordinate-free form, with focus on chiral contributions to the stress. These generate geometric `odd elastic' forces in response to mean curvature gradients but directed perpendicularly. As a result, they induce tangential membrane flows that circulate around maxima and minima of membrane curvature. When the normal viscous force amplifies perturbations the membrane shape can become linearly unstable giving rise to shape instabilities controlled by an active Scriven-Love number. We describe examples for spheroids, membranes tubes and helicoids, discussing the relevance and predictions such examples make for a variety of biological systems from the sub-cellular to tissue level.

cond-mat.soft↗
arXiv2023

Dynamics of Ordered Active Columns: Flows, Twists, and Waves

We formulate the hydrodynamics of active columnar phases, with two-dimensional translational order in the plane perpendicular to the columns and no elastic restoring force for relative sliding of the columns, using the general formalism of an active model H$^*$. Our predictions include: two-dimensional odd elasticity coming from three-dimensional plasmon-like oscillations of the columns in chiral polar phases with a frequency that is independent of wavenumber and non-analytic; a buckling instability coming from the generic force-dipole active stress analogous to the mechanical Helfrich-Hurault instability in passive materials; the selection of helical column undulations by apolar chiral activity.

cond-mat.soft↗
arXiv2023

Active Nematic Multipoles: Flow Responses and the Dynamics of Defects and Colloids

We introduce a general description of localised distortions in active nematics using the framework of active nematic multipoles. We give the Stokesian flows for arbitrary multipoles in terms of differentiation of a fundamental flow response and describe them explicitly up to quadrupole order. We also present the response in terms of the net active force and torque associated to the multipole. This allows the identification of the dipolar and quadrupolar distortions that generate self-propulsion and self-rotation respectively and serves as a guide for the design of arbitrary flow responses. Our results can be applied to both defect loops in three-dimensional active nematics and to systems with colloidal inclusions. They reveal the geometry-dependence of the self-dynamics of defect loops and provide insights into how colloids might be designed to achieve propulsive or rotational dynamics, and more generally for the extraction of work from active nematics. Finally, we extend our analysis also to two dimensions and to systems with chiral active stresses.

cond-mat.soft↗
arXiv2022

Contact Topology and the Classification of Disclination Lines in Cholesteric Liquid Crystals

We give a 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.

cond-mat.soft↗
arXiv2021

Defect Loops in Three-Dimensional Active Nematics as Active Multipoles

We develop a description of defect loops in three-dimensional active nematics based on a multipole expansion of the far-field director and show how this leads to a self-dynamics dependent on the loop's geometric type. The dipole term leads to active stresses that generate a global self-propulsion for splay and bend loops. The quadrupole moment is non-zero only for non-planar loops and generates a net `active torque', such that defect loops are both self-motile and self-orienting. Our analysis identifies right- and left-handed twist loops as the only force and torque free geometries, suggesting a mechanism for generating an excess of twist loops. Finally, we determine the Stokesian flows created by defect loops and describe qualitatively their hydrodynamics.

cond-mat.soft↗
arXiv2021

Intrinsic Geometry and Director Reconstruction for Three-Dimensional Liquid Crystals

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.

cond-mat.soft↗
arXiv2020

Active cholesterics: odder than odd elasticity

In equilibrium liquid crystals, chirality leads to a variety of spectacular three-dimensional structures, but chiral and achiral phases with the same broken continuous symmetries have identical long-time, large-scale dynamics. In this paper, we demonstrate that chirality qualitatively modifies the dynamics of layered liquid crystals in active systems in both two and three dimensions due to an active "odder" elasticity. In three dimensions, we demonstrate that the hydrodynamics of active cholesterics differs fundamentally from smectic-A liquid crystals, unlike their equilibrium counterpart. This distinction can be used to engineer a columnar array of vortices, with anti-ferromagnetic vorticity alignment, that can be switched on and off by external strain. A two-dimensional chiral layered state -- an array of lines on an incompressible, free-standing film of chiral active fluid with a preferred normal direction -- is generically unstable. However, this instability can be tuned in easily realisable experimental settings, when the film is either on a substrate or in an ambient fluid.

cond-mat.soft↗
arXiv2020

Geometry of Bend: Singular Lines and Defects in Twist-Bend Nematics

We describe the geometry of bend distortions in twist-bend nematic liquid crystals in terms of their fundamental degeneracies, which we call $β$ lines. These represent a new class of line-like topological defect. We use them to construct and characterise novel structures, including grain boundary and focal conic smectic-like defects, Skyrmions, and knotted merons. We analyse 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 new geometric perspective on the fractionalisation of Skyrmions.

cond-mat.soft↗
arXiv2020

Three-Dimensional Active Defect Loops

We describe the flows and morphological dynamics of topological defect lines and loops in three-dimensional active nematics and show, using theory and numerical modelling, that they are governed by the local profile of the orientational order surrounding the defects. Analysing a continuous span of defect loop profiles, ranging from radial and tangential twist to wedge $\pm 1/2$ profiles, we show that the distinct geometries can drive material flow perpendicular or along the local defect loop segment, whose variation around a closed loop can lead to net loop motion, elongation or compression of shape, or buckling of the loops. We demonstrate a correlation between local curvature and the local orientational profile of the defect loop, indicating dynamic coupling between geometry and topology. To address the general formation of defect loops in three dimensions, we show their creation via bend instability from different initial elastic distortions.

cond-mat.soft↗
arXiv2019

A Björling Representation for Jacobi Fields on Minimal Surfaces and Soap Film Instabilities

We develop a general framework for the description of instabilities on soap films using the Björling representation of minimal surfaces. The construction is naturally geometric and the instability has the interpretation as being specified by its amplitude and transverse gradient along any curve lying in the minimal surface. When the amplitude vanishes, the curve forms part of the boundary to a critically stable domain, while when the gradient vanishes the Jacobi field is maximal along the curve. In the latter case, we show that the Jacobi field is maximally localised if its amplitude is taken to be the lowest eigenfunction of a one-dimensional Schrödinger operator. We present examples for the helicoid, catenoid, circular helicoids and planar Enneper minimal surfaces, and emphasise that the geometric nature of the Björling representation allows direct connection with instabilities observed in soap films.

cond-mat.soft↗
arXiv2019

Singular Contact Geometry and Beltrami Fields in Cholesteric Liquid Crystals

The description of point defects in chiral liquid crystals via topological methods requires the introduction of singular contact structures, a generalisation of regular contact structures where the plane field may have singularities at isolated points. We characterise the class of singularities that may arise in such structures, as well as the subclass of singularities that can occur in a Beltrami field. We discuss questions of global existence, and prove that all singular contact structures with nonremovable singularities are overtwisted. To connect the theory to experiment we also discuss normal and tangential boundary conditions for singular contact structures, and show we can realise all desired boundary conditions except for normal anchoring on a sphere, where a theorem of Eliashberg and Thurston provides an obstruction to having a singular contact structure in the interior. By introducing a singular version of the Lutz twist we show that all contact structures are homotopic within the larger class of singular contact structures. We give applications of our results to the description of topological defects in chiral liquid crystals.

math.SG↗