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Papers by Michael Snarski

Mathematical Models and Methods in Applied Sciences, 2016

Physical Review E, 2014

The topology and the geometry of a surface play a fundamental role in determining the equilibrium... more The topology and the geometry of a surface play a fundamental role in determining the equilibrium configurations of thin films of liquid crystals. We propose here a theoretical analysis of a recently introduced surface Frank energy, in the case of two-dimensional nematic liquid crystals coating a toroidal particle. Our aim is to show how a different modeling of the effect of extrinsic curvature acts as a selection principle among equilibria of the classical energy, and how new configurations emerge. In particular, our analysis predicts the existence of new stable equilibria with complex windings. PACS numbers: 61.30.Dk, 68.35.Md Due to their special optical properties and their controllability through electric and magnetic fields, liquid crystals have proven to be fundamental in many scientific and technological applications. Their properties have been deeply investigated for over a century and nowadays increasing emphasis is being placed on so-called nematic shells. These are microscopic particles coated by a thin film of liquid crystals, which develop defects with a topological charge, and thus have a tendency to selfassemble into metamaterials which may have new optical properties and a high potential for technological applications (see, e.g., ). The form of the elastic energy for nematics is well established, both in the framework of director theory, based on the works of Oseen, Zocher, and Frank, and in the framework of the order-tensor theory introduced by de Gennes (see, e.g., ). In contrast, there is no universal agreement on the two-dimensional free energy to model nematic shells. Different ways to take into account the distorsion effect of the substrate were proposed in [6-8] and recently by Napoli and Vergori ([9, 10]). Indeed, as observed in [11] and [12], the liquid crystal ground state (and all its stable configurations, in general) is the result of the competition between two driving principles: on one hand the minimization of the "curvature of the texture" penalized by the elastic energy, and on the other the frustration due to constraints of geometrical and topological nature, imposed by anchoring the nematic to the surface of the underlying particle. The new energy model ([9, 10]) affects these two aspects, focusing on the effects of the extrinsic geometry of the substrate on the elastic energy of the nematics. With the present Letter we aim at exploring the full consequences of the new model so that a detailed comparison with the classical one can be established. More precisely, we study the two-dimensional Napoli-Vergori director theory for nematic shells on a genus one surface: a) we analyze the dependence of the new energy on the mechanical parameters (splay, twist and bend moduli) and on the geometrical parameters (the radii of the torus); b) we highlight in which cases the new energy acts as a selection principle among the minimizers of the classical one, and in which cases new different states emerge; c) we predict the existence of stable equilibrium states carrying a higher energy than the ground state, in correspondence with the homotopy classes of the torus. Our analysis, in particular, agrees and makes more precise the statement of [10], according to which the new energy "promotes the alignment of the flux lines of the nematic director towards geodesics and/ or lines of curvature of the surface". The aspect of high energy equilibrium states is present in the classical energy as well, but it was neglected in previous research on genus one surfaces ([8]).

Periodic orbits and associated bifurcations of singularly perturbed state-dependent delay differe... more Periodic orbits and associated bifurcations of singularly perturbed state-dependent delay differential equations (DDEs) are studied when the profiles of the periodic orbits contain jump discontinuities in the singular limit. A definition of singular solution is introduced which is based on a continuous parametrisation of the possibly discontinuous limiting solution. This reduces the construction of the limiting profiles to an algebraic problem. A model two state-dependent delay differential equation is studied in detail and periodic singular solutions are constructed with one and two local maxima per period. A complete character-isation of the conditions on the parameters for these singular solutions to exist facilitates an investigation of bifurcation structures in the singular case revealing folds and possible cusp bifurcations. Sophisticated boundary value techniques are used to numerically compute the bifurcation diagram of the state-dependent DDE when the perturbation parameter is close to zero. This confirms that the solutions and bifurcations constructed in the singular case persist when the perturbation parameter is nonzero, and hence demonstrates that the solutions constructed using our singular solution definition are useful and relevant to the singularly perturbed problem. Fold and cusp bifurcations are found very close to the parameter values predicted by the singular solution theory, and we also find period-doubling bifurcations as well as periodic orbits with more than two local maxima per period, and explain the alignment between the folds on different bifurcation branches.

Mathematical Models and Methods in Applied Sciences, 2016

Physical Review E, 2014

The topology and the geometry of a surface play a fundamental role in determining the equilibrium... more The topology and the geometry of a surface play a fundamental role in determining the equilibrium configurations of thin films of liquid crystals. We propose here a theoretical analysis of a recently introduced surface Frank energy, in the case of two-dimensional nematic liquid crystals coating a toroidal particle. Our aim is to show how a different modeling of the effect of extrinsic curvature acts as a selection principle among equilibria of the classical energy, and how new configurations emerge. In particular, our analysis predicts the existence of new stable equilibria with complex windings. PACS numbers: 61.30.Dk, 68.35.Md Due to their special optical properties and their controllability through electric and magnetic fields, liquid crystals have proven to be fundamental in many scientific and technological applications. Their properties have been deeply investigated for over a century and nowadays increasing emphasis is being placed on so-called nematic shells. These are microscopic particles coated by a thin film of liquid crystals, which develop defects with a topological charge, and thus have a tendency to selfassemble into metamaterials which may have new optical properties and a high potential for technological applications (see, e.g., ). The form of the elastic energy for nematics is well established, both in the framework of director theory, based on the works of Oseen, Zocher, and Frank, and in the framework of the order-tensor theory introduced by de Gennes (see, e.g., ). In contrast, there is no universal agreement on the two-dimensional free energy to model nematic shells. Different ways to take into account the distorsion effect of the substrate were proposed in [6-8] and recently by Napoli and Vergori ([9, 10]). Indeed, as observed in [11] and [12], the liquid crystal ground state (and all its stable configurations, in general) is the result of the competition between two driving principles: on one hand the minimization of the "curvature of the texture" penalized by the elastic energy, and on the other the frustration due to constraints of geometrical and topological nature, imposed by anchoring the nematic to the surface of the underlying particle. The new energy model ([9, 10]) affects these two aspects, focusing on the effects of the extrinsic geometry of the substrate on the elastic energy of the nematics. With the present Letter we aim at exploring the full consequences of the new model so that a detailed comparison with the classical one can be established. More precisely, we study the two-dimensional Napoli-Vergori director theory for nematic shells on a genus one surface: a) we analyze the dependence of the new energy on the mechanical parameters (splay, twist and bend moduli) and on the geometrical parameters (the radii of the torus); b) we highlight in which cases the new energy acts as a selection principle among the minimizers of the classical one, and in which cases new different states emerge; c) we predict the existence of stable equilibrium states carrying a higher energy than the ground state, in correspondence with the homotopy classes of the torus. Our analysis, in particular, agrees and makes more precise the statement of [10], according to which the new energy "promotes the alignment of the flux lines of the nematic director towards geodesics and/ or lines of curvature of the surface". The aspect of high energy equilibrium states is present in the classical energy as well, but it was neglected in previous research on genus one surfaces ([8]).

Periodic orbits and associated bifurcations of singularly perturbed state-dependent delay differe... more Periodic orbits and associated bifurcations of singularly perturbed state-dependent delay differential equations (DDEs) are studied when the profiles of the periodic orbits contain jump discontinuities in the singular limit. A definition of singular solution is introduced which is based on a continuous parametrisation of the possibly discontinuous limiting solution. This reduces the construction of the limiting profiles to an algebraic problem. A model two state-dependent delay differential equation is studied in detail and periodic singular solutions are constructed with one and two local maxima per period. A complete character-isation of the conditions on the parameters for these singular solutions to exist facilitates an investigation of bifurcation structures in the singular case revealing folds and possible cusp bifurcations. Sophisticated boundary value techniques are used to numerically compute the bifurcation diagram of the state-dependent DDE when the perturbation parameter is close to zero. This confirms that the solutions and bifurcations constructed in the singular case persist when the perturbation parameter is nonzero, and hence demonstrates that the solutions constructed using our singular solution definition are useful and relevant to the singularly perturbed problem. Fold and cusp bifurcations are found very close to the parameter values predicted by the singular solution theory, and we also find period-doubling bifurcations as well as periodic orbits with more than two local maxima per period, and explain the alignment between the folds on different bifurcation branches.