313 research outputs found

    Construction of exact minimal parking garages:nonlinear helical motifs in optimally packed lamellar structures

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    Minimal surfaces arise as energy minimizers for fluid membranes and are thus found in a variety of biological systems. The tight lamellar structures of the endoplasmic reticulum and plant thylakoids are comprised of such minimal surfaces in which right- and left-handed helical motifs are embedded in stoichiometry suggesting global pitch balance. So far, the analytical treatment of helical motifs in minimal surfaces was limited to the small-slope approximation where motifs are represented by the graph of harmonic functions. However, in most biologically and physically relevant regimes the inter- motif separation is comparable with its pitch, and thus this approximation fails. Here, we present a recipe for constructing exact minimal surfaces with an arbitrary distribution of helical motifs, showing that any harmonic graph can be deformed into a minimal surface by exploiting lateral displacements only. We analyse in detail pairs of motifs of the similar and of opposite handedness and also an infinite chain of identical motifs with similar or alternating handedness. Last, we study the second variation of the area functional for collections of helical motifs with asymptotic helicoidal structure and show that in this subclass of minimal surfaces stability requires that the collection of motifs is pitch balanced

    Moving frames and compatibility conditions for three-dimensional director fields

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    The geometry and topology of the region in which a director field is embedded impose limitations on the kind of supported orientational order. These limitations manifest as compatibility conditions that relate the quantities describing the director field to the geometry of the embedding space. For example, in two dimensions (2D) the splay and bend fields suffice to determine a director uniquely (up to rigid motions) and must comply with one relation linear in the Gaussian curvature of the embedding manifold. In 3D there are additional local fields describing the director, i.e. fields available to a local observer residing within the material, and a number of distinct ways to yield geometric frustration. So far it was unknown how many such local fields are required to uniquely describe a 3D director field, nor what are the compatibility relations they must satisfy. In this work, we address these questions directly. We employ the method of moving frames to show that a director field is fully determined by five local fields. These fields are shown to be related to each other and to the curvature of the embedding space through six differential relations. As an application of our method, we characterize all uniform distortion director fields, i.e., directors for which all the local characterizing fields are constant in space, in manifolds of constant curvature. The classification of such phases has been recently provided for directors in Euclidean space, where the textures correspond to foliations of space by parallel congruent helices. For non-vanishing curvature, we show that the pure twist phase is the only solution in positively curved space, while in the hyperbolic space uniform distortion fields correspond to foliations of space by (non-necessarily parallel) congruent helices. Further analysis is expected to allow to also construct of new non-uniform director fields

    Construction of exact minimal parking garages:nonlinear helical motifs in optimally packed lamellar structures

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    Minimal surfaces arise as energy minimizers for fluid membranes and are thus found in a variety of biological systems. The tight lamellar structures of the endoplasmic reticulum and plant thylakoids are comprised of such minimal surfaces in which right- and left-handed helical motifs are embedded in stoichiometry suggesting global pitch balance. So far, the analytical treatment of helical motifs in minimal surfaces was limited to the small-slope approximation where motifs are represented by the graph of harmonic functions. However, in most biologically and physically relevant regimes the inter- motif separation is comparable with its pitch, and thus this approximation fails. Here, we present a recipe for constructing exact minimal surfaces with an arbitrary distribution of helical motifs, showing that any harmonic graph can be deformed into a minimal surface by exploiting lateral displacements only. We analyse in detail pairs of motifs of the similar and of opposite handedness and also an infinite chain of identical motifs with similar or alternating handedness. Last, we study the second variation of the area functional for collections of helical motifs with asymptotic helicoidal structure and show that in this subclass of minimal surfaces stability requires that the collection of motifs is pitch balanced

    Moving frames and compatibility conditions for three-dimensional director fields

    Get PDF
    The geometry and topology of the region in which a director field is embedded impose limitations on the kind of supported orientational order. These limitations manifest as compatibility conditions that relate the quantities describing the director field to the geometry of the embedding space. For example, in two dimensions (2D) the splay and bend fields suffice to determine a director uniquely (up to rigid motions) and must comply with one relation linear in the Gaussian curvature of the embedding manifold. In 3D there are additional local fields describing the director, i.e. fields available to a local observer residing within the material, and a number of distinct ways to yield geometric frustration. So far it was unknown how many such local fields are required to uniquely describe a 3D director field, nor what are the compatibility relations they must satisfy. In this work, we address these questions directly. We employ the method of moving frames to show that a director field is fully determined by five local fields. These fields are shown to be related to each other and to the curvature of the embedding space through six differential relations. As an application of our method, we characterize all uniform distortion director fields, i.e., directors for which all the local characterizing fields are constant in space, in manifolds of constant curvature. The classification of such phases has been recently provided for directors in Euclidean space, where the textures correspond to foliations of space by parallel congruent helices. For non-vanishing curvature, we show that the pure twist phase is the only solution in positively curved space, while in the hyperbolic space uniform distortion fields correspond to foliations of space by (non-necessarily parallel) congruent helices. Further analysis is expected to allow to also construct of new non-uniform director fields

    Compatible director fields in R3\mathbb{R}^3

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    The geometry and interactions between the constituents of a liquid crystal, which are responsible for inducing the partial order in the fluid, may locally favor an attempted phase that could not be realized in R3\mathbb{R}^3. While states that are incompatible with the geometry of R3\mathbb{R}^3 were identified more than 50 years ago, the collection of compatible states remained poorly understood and not well characterized. Recently, the compatibility conditions for three-dimensional director fields were derived using the method of moving frames. These compatibility conditions take the form of six differential relations in five scalar fields locally characterizing the director field. In this work, we rederive these equations using a more transparent approach employing vector calculus. We then use these equations to characterize a wide collection of compatible phases.Comment: 37 pages (32 in the published version), 5 figures. Keywords: Geometric frustration, Incompatibility, Liquid crystal, Frobenius. Note: This version is essentially the same as the published one. In addition, it is part of a Collection: Soft Matter Elasticity (https://link.springer.com/collections/ijhficbgii

    Shape selection in non-Euclidean plates

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    We investigate isometric immersions of disks with constant negative curvature into R3\mathbb{R}^3, and the minimizers for the bending energy, i.e. the L2L^2 norm of the principal curvatures over the class of W2,2W^{2,2} isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls in H2\mathbb{H}^2 into R3\mathbb{R}^3. In elucidating the connection between these immersions and the non-existence/singularity results of Hilbert and Amsler, we obtain a lower bound for the L∞L^\infty norm of the principal curvatures for such smooth isometric immersions. We also construct piecewise smooth isometric immersions that have a periodic profile, are globally W2,2W^{2,2}, and have a lower bending energy than their smooth counterparts. The number of periods in these configurations is set by the condition that the principal curvatures of the surface remain finite and grows approximately exponentially with the radius of the disc. We discuss the implications of our results on recent experiments on the mechanics of non-Euclidean plates

    Pathophysiology of Crohn’s disease inflammation and recurrence

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    Chron's Disease is a chronic inflammatory intestinal disease, first described at the beginning of the last century. The disease is characterized by the alternation of periods of flares and remissions influenced by a complex pathogenesis in which inflammation plays a key role. Crohn's disease evolution is mediated by a complex alteration of the inflammatory response which is characterized by alterations of the innate immunity of the intestinal mucosa barrier together with a remodeling of the extracellular matrix through the expression of metalloproteins and increased adhesion molecules expression, such as MAcCAM-1. This reshaped microenvironment enhances leucocytes migration in the sites of inflammation, promoting a TH1 response, through the production of cytokines such as IL-12 and TNF-α. IL-12 itself and IL-23 have been targeted for the medical treatment of CD. Giving the limited success of medical therapies, the treatment of the disease is invariably surgical. This review will highlight the role of inflammation in CD and describe the surgical approaches for the prevention of the almost inevitable recurrence
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