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Effects of Alcohol Chain Length and Salt on Phase Behavior and Critical Phenomena in SDS Microemulsions
Published in Promod Kumar, K. L. Mittal, Handbook of Microemulsion Science and Technology, 2018
where C1 and C2 are the local principal curvatures of the surfactant layer and C0 the spontaneous curvature [59,60]. The contribution of Fb to the total free energy is crucial in determining the type and characteristic size of the structure. The first term, which characterizes the rigidity, represents the energy required to bend a unit area of interface by a unit amount of curvature. The second term plays an important role in the change of membrane topology and consequently in the phase transition where the genus of the surface is changed. This is a direct consequence of the Gauss-Bonnet theorem, which states that the integral of the Gaussian curvature over a given structure depends on its topological type. C0 describes the tendency of the surfactant film to bend either toward the water or toward the oil. It arises from the competition in the packing of the polar heads and hydrocarbon tails of the surfactant molecules [61]. Roughly, one can say that if Co is sufficiently large, a droplet structure is formed, whereas when Co vanishes, a lamellar phase and/or a random bicontinuous structure is obtained.
Geometry of the Middle Surface
Published in Eduard Ventsel, Theodor Krauthammer, Thin Plates and Shells, 2001
Eduard Ventsel, Theodor Krauthammer
The first of these quantities is called the Gaussian curvature and the second is the mean curvature of a surface. The concept of Gaussian curvature is fundamental in the theory of surfaces. If the coordinate curves are lines of curvature, then the expression for Γ, Eq. (11.28), is reduced to the following equation (because a12 = b12 = 0) : () Γ=k1k2=b11b22A2B2,
Industrial statistics and manifold data
Published in Quality Engineering, 2020
Enrique del Castillo, Xueqi Zhao
In Zhao and del Castillo (2019), rather than solving registration problems via ICP or similar algorithms for on-line control, we propose instead to do SPC on properties of the surface (a 2-dimensional manifold) of an object that are not only invariant to rigid transformations (as the shape of an object is invariant) but also that are intrinsic properties. A geometric property of a manifold is intrinsic if it is computed without any use of the coordinates or other information from the space it is embedded in, and only uses coordinates (or information, in general) defined on the manifold. For the surface of an object, surface coordinates and geodesic distances are intrinsic properties, so is the Gaussian curvature. They are all also invariant properties. In contrast, Euclidean distances between points on an object are invariant but not intrinsic. By only using intrinsic properties computed from each part, it is possible to avoid the computationally expensive registration step (ICP registration could still be used for post alarm diagnostics, helping to locate the defect on the part, but not for on-line monitoring).
Finetuning Discrete Architectural Surfaces by use of Circle Packing
Published in Journal of Asian Architecture and Building Engineering, 2023
Among the design targets involving geometric property of a surface, the Gaussian curvature serves as one of the most important indices. In the past decades, double curvature structural envelopes (that is, surfaces with non-vanishing Gaussian curvature) have become increasingly popular in architectural design. Human preference for curvature is biologically determined (Gómez-Puerto, Munar, and Nadal 2016) and the impact of curvature on the impression of architectural facades is investigated in Ruta et al. (2019). The Guggenheim Museum makes use of the ever-present curvature of the various geometries, not only for aesthetic purposes but also to significantly enhance stiffness of the structure against lateral loads (Iyengar et al. 1998).