arXiv subjects

Gareth P. Alexander

Publications and source records attributed to Gareth P. Alexander.

At least 37 records · Page 2Linked to original sources

arXiv2018

Stable and Unstable Vortex Knots in Excitable Media

We study the dynamics of knotted vortices in a bulk excitable medium using the FitzHugh-Nagumo model. From a systematic survey of all knots of at most eight crossings we establish that the generic behaviour is of unsteady, irregular dynamics, with prolonged periods of expansion of parts of the vortex. The mechanism for the length expansion is a long-range `wave slapping' interaction, analogous to that responsible for the annihilation of small vortex rings by larger ones. We also show that there are stable vortex geometries for certain knots; in addition to the unknot, trefoil and figure eight knots reported previously, we have found stable examples of the Whitehead link and $6_2$ knot. We give a thorough characterisation of their geometry and steady state motion. For the unknot, trefoil and figure eight knots we greatly expand previous evidence that FitzHugh-Nagumo dynamics untangles initially complex geometries while preserving topology.

nlin.PS↗

arXiv2018

Maxwell's Theory of Solid Angle and the Construction of Knotted Fields

We provide a systematic description of the solid angle function as a means of constructing a knotted field for any curve or link in $\mathbb{R}^3$. This is a purely geometric construction in which all of the properties of the entire knotted field derive from the geometry of the curve, and from projective and spherical geometry. We emphasise a fundamental homotopy formula as unifying different formulae for computing the solid angle. The solid angle induces a natural framing of the curve, which we show is related to its writhe and use to characterise the local structure in a neighborhood of the knot. Finally, we discuss computational implementation of the formulae derived, with C code provided, and give illustrations for how the solid angle may be used to give explicit constructions of knotted scroll waves in excitable media and knotted director fields around disclination lines in nematic liquid crystals.

math-ph↗

arXiv2018

Point Defects, Chirality and Singularity Theory in Cholesteric Liquid Crystal Droplets

We develop a theory of point defects in cholesterics and textures in spherical droplets with normal anchoring. The local structure of chiral defects is described by singularity theory and a smectic-like gradient field establishing a nexus between cholesterics and smectics mediated by their defects. We identify the defects of degree $-2$ and $-3$ observed experimentally with the singularities $D_4^{-}$ and $T_{4,4,4}$, respectively. Radial point defects typical of nematics cannot be perturbed into chiral structures with a single handedness by general topological considerations. For the same reasons, the spherical surface frustrates the chirality in a surface boundary layer containing regions of both handedness.

cond-mat.soft↗

arXiv2018

Woven Nematic Defects, Skyrmions and the Abelian Sandpile Model

We show that a fixed set of woven defect lines in a nematic liquid crystal supports a set of non-singular topological states which can be mapped on to recurrent stable configurations in the Abelian sandpile model or chip-firing game. The physical correspondence between local Skyrmion flux and sandpile height is made between the two models. Using a toy model of the elastic energy, we examine the structure of energy minima as a function of topological class and show that the system admits domain wall Skyrmion solitons.

cond-mat.soft↗

arXiv2017

Polyomino Models of Surface Supramolecular Assembly: Design Constraints and Structural Selectivity

We examine emergent properties of 2D supramolecular networks, using enumeration of configurations formed by interacting dominoes on square lattices as a simple model system. Possible ground states are identified using a convex hull construction in the interaction parameters for nearest-neighbour bonds. We demonstrate how this construction can be used to design interaction parameters which lead to networks with specific properties, including chirality and highly degenerate ground states. We then introduce kinetics as simple local rearrangements. By partitioning the configuration space into smaller sets which satisfy different topological constraints, we can design configurations which are kinetically trapped. By considering heat capacity curves along directions through the convex hull, we also demonstrate design of interacting domino configurations to create tilings robust against temperature induced phase transitions. We discuss extension of this design construction to more complex molecular shapes.

cond-mat.soft↗

arXiv2017

Vortex formation and dynamics of defects in shells of active nematics

We present a hydrodynamic model for a thin spherical shell of active nematic liquid crystal with an arbitrary configuration of defects. The active flows generated by defects in the director lead to the formation of stable vortices, analogous to those seen in confined systems in flat geometries, which generate an effective dynamics for four +1/2 defects that reproduces the tetrahedral to planar oscillations observed in experiments. As the activity is increased and the vortices become stronger, the defects are drawn more tightly into pairs, rotating about antipodal points. We extend this situation to also describe the dynamics of other configurations of defects. For example, two +1 defects are found to attract or repel according to the local geometric character of the director field around them, while additional pairs of opposite charge defects can give rise to flow states containing more than two vortices. Finally, we describe the generic relationship between defects in the orientation and singular points of the flow, and suggest implications for the three-dimensional nature of the flow and deformation in the shape of the shell.

cond-mat.soft↗

arXiv2017

Hydrodynamic instabilities in active cholesteric liquid crystals

We describe the basic properties and consequences of introducing active stresses, with principal direction along the local director, in cholesteric liquid crystals. The helical ground state is found to be linearly unstable to extensile stresses, without threshold in the limit of infinite system size, whereas contractile stresses are hydrodynamically screened by the cholesteric elasticity to give a finite threshold. This is confirmed numerically and the non-linear consequences of instability, in both extensile and contractile cases, are studied. We also consider the stresses associated to defects in the cholesteric pitch ($λ$ lines) and show how the geometry near to the defect generates threshold-less flows reminiscent of those for defects in active nematics. At large extensile activity $λ$ lines are spontaneously created and can form steady state patterns sustained by constant active flows.

cond-mat.soft↗

arXiv2016

Exact solutions for hydrodynamic interactions of two squirming spheres

We provide exact solutions of the Stokes equations for a squirming sphere close to a no-slip surface, both planar and spherical, and for the interactions between two squirmers, in three dimensions. These allow the hydrodynamic interactions of swimming microscopic organisms with confining boundaries, or each other, to be determined for arbitrary separation and, in particular, in the close proximity regime where approximate methods based on point singularity descriptions cease to be valid. We give a detailed description of the circular motion of an arbitrary squirmer moving parallel to a no-slip spherical boundary or flat free surface at close separation, finding that the circling generically has opposite sense at free surfaces and at solid boundaries. While the asymptotic interaction is symmetric under head-tail reversal of the swimmer, in the near field microscopic structure can result in significant asymmetry. We also find the translational velocity towards the surface for a simple model with only the lowest two squirming modes. By comparing these to asymptotic approximations of the interaction we find that the transition from near- to far-field behaviour occurs at a separation of about two swimmer diameters. These solutions are for the rotational velocity about the wall normal, or common diameter of two spheres, and the translational speed along that same direction, and are obtained using the Lorentz reciprocal theorem for Stokes flows in conjunction with known solutions for the conjugate Stokes drag problems, the derivations of which are demonstrated here for completeness. The analogous motions in the perpendicular directions, i.e. parallel to the wall, currently cannot be calculated exactly since the relevant Stokes drag solutions needed for the reciprocal theorem are not available.

physics.flu-dyn↗

arXiv2016

Straight Round the Twist: Frustration and Chirality in Smectics-A

Frustration is a powerful mechanism in condensed matter systems, driving both order and co plexity. In smectics, the frustration between macroscopic chirality and equally spaced layers generates textures characterised by a proliferation of defects. In this article, we study several different ground states of the chiral Landau-de Gennes free energy for a smectic liquid crystal. The standard theory finds the twist grain boundary (TGB) phase to be the ground state for chiral type II smectics. However, for very highly chiral systems, the hierarchical helical nanofilament (HN) phase can form and is stable over the TGB.

cond-mat.soft↗

arXiv2016

Global Defect Topology in Nematic Liquid Crystals

We give the global homotopy classification of nematic textures for a general domain with weak anchoring boundary conditions and arbitrary defect set in terms of twisted cohomology, and give an explicit computation for the case of knotted and linked defects in $\mathbb{R}^3$, showing that the distinct homotopy classes have a 1-1 correspondence with the first homology group of the branched double cover, branched over the disclination loops. We show further that the subset of those classes corresponding to elements of order 2 in this group have representatives that are planar and characterise the obstruction for other classes in terms of merons. The planar textures are a feature of the global defect topology that is not reflected in any local characterisation. Finally, we describe how the global classification relates to recent experiments on nematic droplets and how elements of order 4 relate to the presence of $τ$ lines in cholesterics.

cond-mat.soft↗

arXiv2016

Instabilities and Solitons in Minimal Strips

We show that highly twisted minimal strips can undergo a non-singular transition, unlike the singular transitions seen in the Möbius strip and the catenoid. If the strip is non-orientable this transition is topologically frustrated, and the resulting surface contains a helical defect. Through a controlled analytic approximation the system can be mapped onto a scalar $ϕ^4$ theory on a non-orientable line bundle over the circle, where the defect becomes a topologically protected kink soliton or domain wall, thus establishing their existence in minimal surfaces. Experimental studies of soap films confirm these results and demonstrate how the position of the defect can be controlled through boundary deformation.

cond-mat.soft↗

arXiv2015

Motility of active fluid drops on surfaces

Drops of active liquid crystal have recently shown the ability to self-propel, which was associated with topological defects in the orientation of active filaments [Sanchez {\em et al.}, Nature {\bf 491}, 431 (2013)]. Here, we study the onset and different aspects of motility of a three-dimensional drop of active fluid on a planar surface. We analyse theoretically how motility is affected by orientation profiles with defects of various types and locations, by the shape of the drop, and by surface friction at the substrate. In the scope of a thin drop approximation, we derive exact expressions for the flow in the drop that is generated by a given orientation profile. The flow has a natural decomposition into terms that depend entirely on the geometrical properties of the orientation profile, i.e. its bend and splay, and a term coupling the orientation to the shape of the drop. We find that asymmetric splay or bend generates a directed bulk flow and enables the drop to move, with maximal speeds achieved when the splay or bend is induced by a topological defect in the interior of the drop. In motile drops the direction and speed of self-propulsion is controlled by friction at the substrate.

cond-mat.soft↗

arXiv2015

Umbilic Lines in Orientational Order

Three-dimensional orientational order in systems whose ground states possess non-zero, chiral gradients typically exhibits line-like structures or defects: $λ$ lines in cholesterics or Skyrmion tubes in ferromagnets for example. Here we show that such lines can be identified as a set of natural geometric singularities in a unit vector field, the generalisation of the umbilic points of a surface. We characterise these lines in terms of the natural vector bundles that the order defines and show that they give a way to localise and identify Skyrmion distortions in chiral materials -- in particular that they supply a natural representative of the Poincaré dual of the cocycle describing the topology. Their global structure leads to the definition of a self-linking number and helicity integral which relates the linking of umbilic lines to the Hopf invariant of the texture.

cond-mat.soft↗

arXiv2015

The many-body reciprocal theorem and swimmer hydrodynamics

We present a reinterpretation and extension of the reciprocal theorem for swimmers, extending its application from the motion of a single swimmer in an unbounded domain to the general setting, giving results for both swimmer interactions and general hydrodynamics. We illustrate the method for a squirmer near a planar surface, recovering standard literature results and extending them to a general squirming set, to motion in the presence of a ciliated surface, and expressions for the flow field throughout the domain. Finally, we present exact results for the hydrodynamics in two dimensions which shed light on the near-field behaviour.

cond-mat.soft↗

arXiv2014

Dynamics of Self-Threading Ring Polymers in a Gel

We study of the dynamics of ring polymers confined to diffuse in a background gel at low concentrations. We do this in order to probe the inter-play between topology and dynamics in ring polymers. We develop an algorithm that takes into account the possibility that the rings hinder their own motion by passing through themselves, i.e. "self-threading". Our results suggest that the number of self-threadings scales extensively with the length of the rings and that this is substantially independent of the details of the model. The slowing down of the rings' dynamics is found to be related to the fraction of segments that can contribute to the motion. Our results give a novel perspective on the motion of ring polymers in gel, for which a complete theory is still lacking, and may help us to understand the irreversible trapping of ring polymers in gel electrophoresis experiments.

cond-mat.soft↗

arXiv2014

The Geometry of the Cholesteric Phase

We propose a construction of a cholesteric pitch axis for an arbitrary nematic director field as an eigenvalue problem. Our definition leads to a Frenet-Serret description of an orthonormal triad determined by this axis, the director, and the mutually perpendicular direction. With this tool we are able to compare defect structures in cholesterics, biaxial nematics, and smectics. Though they all have similar ground state manifolds, the defect structures are different and cannot be, in general, translated from one phase to the other.

cond-mat.soft↗

arXiv2014

Knotted Defects in Nematic Liquid Crystals

We show that the number of distinct topological states associated to a given knotted defect, $L$, in a nematic liquid crystal is equal to the determinant of the link $L$. We give an interpretation of these states, demonstrate how they may be identified in experiments and describe the consequences for material behaviour and interactions between multiple knots. We show that stable knots can be created in a bulk cholesteric and illustrate the topology by classifying a simulated Hopf link. In addition we give a topological heuristic for the resolution of strand crossings in defect coarsening processes which allows us to distinguish topological classes of a given link and to make predictions about defect crossings in nematic liquid crystals.

cond-mat.soft↗

arXiv2013

Knotted Nematics

Knotted line defects in continuous fields entrain a complex arrangement of the material sur- rounding them. Recent experimental realisations in optics, fluids and nematic liquid crystals make it important to fully characterise these textures and to understand how their properties relate to the knot type. We characterise knotted nematics through an application of classical knot theory founded upon the Pontryagin-Thom construction for nematic textures and give explicit closed form constructions for knots possessing Milnor fibrations with general boundary conditions. For links we construct nematic textures corresponding to all possible assigments of linking numbers and discuss the relevance to recent, and classic, experiments on Hopf links.

cond-mat.soft↗