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.Comment: 11 pages, 8 figure

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.Comment: 10 pages, 4 figure

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.Comment: 6 pages, 6 figure

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: $\lambda$ 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 Poincar\'{e} 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.Comment: 14 pages, 9 figure

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.Comment: 20 pages, 9 figure

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.Comment: 9 pages, 3 figures, submitted to J. Interface Focu

The Sonic hedgehog and Wnt signalling pathways in interstitial lung disease and CD4⁺T cell activation

The Sonic Hedgehog (Shh) signalling pathway plays an important role in lung development where it promotes branching morphogenesis through epithelialmesenchymal interactions. Increased Shh expression promotes epithelial and mesenchymal proliferation in vitro and in vivo. TGFß is also expressed in embryonic lung where it acts to inhibit branching morphogenesis. TGFß overexpression results in lung hypoplasia, a similar phenotype to that seen in Shh⁻/⁻ mutants; suggesting that Shh and TGFß have opposing roles. Evidence to date would suggest that although TGFß and Shh may not directly interact in lung development, they probably have common targets and may function in a shared regulatory circuit.Interstitial lung disease (ILD) is the end result of a multiplicity of pathological processes. It has been recently proposed that the commonest form, Idiopathic Pulmonary Fibrosis (IPF) or Usual Interstitial Pneumonia (UIP), is due to abnormal wound healing in the lung, characterized by epithelial-fibroblast interactions; a process similar to foetal lung branching and epithelialization. TGFß has been strongly linked with ELD in both animal models and human disease. Based on the link between TGFß and Shh in lung morphogenesis, the initial aim of this thesis was to determine whether or not Shh signalling was upregulated in ILD. The work presented confirms that Shh and TGFß expression are increased in the airway epithelium of fibrotic but not non-fibrotic lung both in the murine FITC model of ILD and in biopsy sections from patients with IPF. Expression of Patched (Ptc), the Shh receptor, is unchanged in epithelial cells. Notably, Ptc is present both in alveolar macrophages and lymphocytic infiltrates. However, there is no discernible difference in the fibrotic response in the lungs of mice containing heterozygous mutations of Gli2 and 3, used as models of dysregulated Shh signalling, nor in mice treated with intra-tracheal SPC-Shh cDNA.The Shh pathway has recently been shown to play a role in thymocyte development. The findings presented demonstrate that both Shh and Ptc are expressed in human T cells. In addition, upregulation of Shh signalling enhances and blocking of endogenous Shh inhibits T cell receptor mediated T cell activation, respectively, as determined by proliferation, cytokine production and CD25 and CD69 expressionWnt signalling is also thought to play a role in lung branching morphogenesis. It is known to interact both with Shh signalling and TGFß. However, using presently available antibodies, there is no evidence of upregulation of Wnt signalling in ILD. In an attempt to drive the Wnt Pathway, a replication-deficient adenovirus expressing Dvll (Ad5-MCMV-Dvll) was successfully rescued. Although the virus drives Dvll mRNA and protein expression in vitro and in vivo, it does not consistently mimic Wnt signalling, nor does it appear to affect Shh or TGFß signalling. Furthermore such Dvll overexpression has little effect on cell proliferation either in vitro or in vivo, and does not cause lung fibrosis in mice.Thus Shh signalling appears to be upregulated in fibrotic lung in mice and humans. However the work presented does not define whether or not the pathway plays a specific role in the pathogenesis of ILD. There is also no evidence relating Wnt signalling or Dvll upregulation to ILD. Finally, Shh is shown to influence TCR mediated signalling and clonal expansion. It may be that damaged epithelial cells and the immune system communicate via this pathway