Hyperbolic Geometry

Millman, Richard S.; Parker, George D.

doi:10.1007/978-1-4684-0130-1_8

Cited by 3 publications

(3 citation statements)

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“…Using the first and second fundamental forms 27 associated with the parametrization in eq 11, one can write the mean curvature and differential area for ellipsoids as follows:…”

Section: = − −mentioning

confidence: 99%

Are Ellipsoids Feasible Micelle Shapes? An Answer Based on a Molecular-Thermodynamic Model of Nonionic Surfactant Micelles

Iyer

Blankschtein

2012

J. Phys. Chem. B

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The existence of ellipsoidal micelles in aqueous solution has been debated in the literature. Although a number of experimental studies suggest that certain surfactants form ellipsoidal micelles, many theoretical studies have claimed that micelles with an ellipsoidal shape cannot exist. To shed light on this topic, in this paper, we develop a curvature-corrected, molecular-thermodynamic model for the free energy of micellization of nonionic surfactant biaxial ellipsoidal micelles. We subsequently use this model to evaluate the feasibility of forming ellipsoidal micelles, compared to forming spherical, spherocylindrical, and discoidal micelles, and conclude that ellipsoidal micelles can exist in solution. Utilizing the model developed here, we also establish theoretical limits on the size of the ellipsoidal micelles. These limits depend solely on the chemical structure of the surfactant molecule.

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“…Using the first and second fundamental forms 27 associated with the parametrization in eq 11, one can write the mean curvature and differential area for ellipsoids as follows:…”

Section: = − −mentioning

confidence: 99%

Are Ellipsoids Feasible Micelle Shapes? An Answer Based on a Molecular-Thermodynamic Model of Nonionic Surfactant Micelles

Iyer

Blankschtein

2012

J. Phys. Chem. B

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“…Consider the quadratic surface in . By standard surface theory (see e.g., [ 45 ]), the principal curvatures of are given by The principal directions of are counterclockwise rotations of the x and y axes by , where . Here, the direction of (which is either or ) is called the principal angle of .…”

Section: Proof Of Main Theorems In the Discrete Settingmentioning

confidence: 99%

Bakry–Émery Ricci Curvature Bounds for Doubly Warped Products of Weighted Spaces

Fathi

Lakzian

2022

J Geom Anal

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We introduce a notion of doubly warped product of weighted graphs that is consistent with the doubly warped product in the Riemannian setting. We establish various discrete Bakry–Émery Ricci curvature-dimension bounds for such warped products in terms of the curvature of the constituent graphs. This requires deliberate analysis of the quadratic forms involved, prompting the introduction of some crucial notions such as curvature saturation at a vertex. In the spirit of being thorough and to provide a frame of reference, we also introduce the -doubly warped products of smooth measure spaces and establish -Bakry–Émery Ricci curvature (lower) bounds thereof in terms of those of the factors. At the end of these notes, we present examples and demonstrate applications of warped products with some toy models.

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“…The second term gives the integrated Gaussian curvature, which is constant according to the Gauss-Bonnet theorem. 47 For a given surface tension and volume, the equilibrium shape will correspond to an energy minimum determined by the shape equation δF ) 0. The structures are more readily solved by first rescaling the free energy of the model to become dimensionless, such that where and S 0 is the area of the base.…”

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confidence: 99%

Epitaxially Driven Formation of Intricate Supported Gold Nanostructures on a Lattice-Matched Oxide Substrate

Devenyi

Hughes

et al. 2009

Nano Lett.

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A new class of gold nanostructures has been fabricated on the (100), (111), and (110) surfaces of lattice-matched MgAl(2)O(4) substrates. The nanostructures were fabricated through a synthesis route where a thin gold film dewets, liquefies, and then slowly self-assembles. The supported nanostructures are intricately shaped, crystalline, and epitaxially aligned. Simulations based on a continuum elastic theory indicate that the self-assembly is driven by strained epitaxy and minimization of the surface free energy.

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Hyperbolic Geometry

Cited by 3 publications

References 0 publications

Are Ellipsoids Feasible Micelle Shapes? An Answer Based on a Molecular-Thermodynamic Model of Nonionic Surfactant Micelles

Are Ellipsoids Feasible Micelle Shapes? An Answer Based on a Molecular-Thermodynamic Model of Nonionic Surfactant Micelles

Bakry–Émery Ricci Curvature Bounds for Doubly Warped Products of Weighted Spaces

Epitaxially Driven Formation of Intricate Supported Gold Nanostructures on a Lattice-Matched Oxide Substrate

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