Weighted mean curvature

The expression for weighted mean curvature for any surface in local equilibrium is simplified when the Wulff shape is completely faceted [10, 12], In this case,... 

The capillarity vector and the weighted mean curvature are discussed in more detail in Section C.3.2. 

J.E. Taylor. Overview No. 98. II—Mean curvature and weighted mean curvature. Acta Metall., 40(7) 1475-1485, 1992. 

There are two different varieties of the curvature of an interface which are convenient to use in capillarity studies mean curvature, denoted by k, and the weighted mean curvature, denoted by k7. 

Weighted Mean Curvature of an Interface. The weighted mean curvature, k7, has exactly the same geometrical properties as the mean curvature except that it is weighted by the possibly orientation-dependent magnitude of the interfacial tension. It is particularly useful for addressing capillarity problems when the interfacial energy is anisotropic, that is, dependent upon the interface orientation (Section C.3). 

The weighted mean curvature is the local rate of interfacial energy change with a local addition of volume. This establishes the connection to the work, 8W, to pass a small volume of material, 8V, through an interface. 8W/8V = /c7(f), in the limit of small volumes. 

Of all local motions, v(r), of an interface that pass the same amount of volume from one side to the other, the motion that is normal to the interface with magnitude proportional to the weighted mean curvature, v f) oc /c7n, increases the interfacial energy the fastest. However, fastest depends on how distance is measured. How this distance metric alters the variational principles that generate the kinetic equations is discussed elsewhere [14]. 

The weighted mean curvature is the interface divergence of the evaluated on the unit sphere. The interface divergence is defined within the interface, and if the interface is not differentiable, subgradients must be used. The convex portion of is equivalent to the the Wulff shape, so the interface divergence is operating from one interface onto another. This form can get very complicated. 

Fig. 60 Schematic illustration for formation of cylindrical morphology in a blend of slightly asymmetric lower molecular weight PS-b-PI (/3-chain) with large symmetric PS-fc-PI (a-chain). a Molecule of /S-chain with non-zero spontaneous curvature, b Cylindrical morphology formed by neat /3 chains shown in a. Here mean curvature of cylinder (solid line) is larger than spontaneous curvature of /3-chain (dashed lines). c Cylindrical morphology formed by binary blend of /3-chains shown in a and large symmetric copolymers (a-chain). In this case, mean curvature of cylinder closely fits to spontaneous curvature of /3-chain. From [180]. Copyright 2001 American Chemical Society...

There are static and dynamic methods. The static methods measure the tension of practically stationary surfaces which have been formed for an appreciable time, and depend on one of two principles. The most accurate depend on the pressure difference set up on the two sides of a curved surface possessing surface tension (Chap. I, 10), and are often only devices for the determination of hydrostatic pressure at a prescribed curvature of the liquid these include the capillary height method, with its numerous variants, the maximum bubble pressure method, the drop-weight method, and the method of sessile drops. The second principle, less accurate, but very often convenient because of its rapidity, is the formation of a film of the liquid and its extension by means of a support caused to adhere to the liquid temporarily methods in this class include the detachment of a ring or plate from the surface of any liquid, and the measurement of the tension of soap solutions by extending a film. 

Here, cu and C are the local mean curvatures of each of the two membrane monolayers, and nm denotes the bending rigidity of a single monolayer that is here assumed to be the same for each leaflet and for both lipid species. The spontaneous curvatures of the two leaflets, and cj are described as sums of the spontaneous curvatures of the pure lipid constituents weighted by their local compositions. This approximation has been previously validated [36,48]. 

Fig. 15.4. A. Some examples of cutoff values reported in the literature and their reliability (in this example, mean of negatives is 0.12 with an SD of 0.043). For example, a cut-off value of mean 2 SD approximates 95% confidence if 3 replicates are used to calculate the mean. B. The DI(-I-) and the DI(—) are inversely related and moving (as shown in frames l-III) the cut-off value to other absorbance levels (abscissae, frequency in ordinates) will increase one DI at the expense of the other. If no difference exists in the mean and distribution of the positive and negative sera (I), a straight line will be obtained in B. More curvature is indicative of more difference between the positive and negative sera distributions. The relative importance of DI(—) and DI(-l-) determines at which point on the curve the optimal combination is found (Section 15.2.3 at a both are equally important, whereas at b 3 x more weight is attached to DI(-l-)).

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