Truncated sphere

Li et al. [189] assumed that a pair of deformable droplets has the shape of truncated sphere separated by a planar film and used the improved Carnahan-Starling equation to describe the repulsion term as ... 

The third discharge stage is accompanied by current stoppage, separation of shock wave from the gas bubble and continuation of its expansion by its own momentum. The shock wave is suppressed by the surrounding liquid, and the pressure inside the bubble falls abruptly. Carbon structures take under such conditions the shape of a tore, truncated sphere, etc. 

Figure 3.12. A liquid drop is spherical in shape (dashed line) while the equilibrium shape of a solid monocrystalline particle of cubic structure metal is a truncated sphere (full line).

Another complication arises when strong attractive forces operate between the drops or bubbles. This may lead to a finite contact angle, 6, between the intervening film (of reduced tension) and the adjaeent bulk interfaces (21, 24-26). Under those conditions, droplets will spontaneously deform into truncated spheres upon contact and can thus pack to much higher densities. For monodisperse drops, the ideal close-packed density, consistent with minimization of the system s surface free energy, is given (21) by... 

Buzza and Gates (102) also addressed the question whether disorder or the increased dimensionality from two to three dimensions is responsible for the observed experimental behavior of the shear modulus. In particular, they explored the lack of the sudden jump in G from zero to a finite value at 0 = 0Q that is predicted by the perfectly ordered 2-D model. We have seen above that disorder appears to remove that abrupt jump in two dimensions (90). For drops on a simple cubic lattice, Buzza and Cates analyzed the drop deformation in uniaxial strain close to 0 = 0q, first using the model of truncated spheres . (For reasons given above, we believe this to be a very poor model.) They showed that this model did not eliminate the discontinuous jump in G. An exact model, based on a theory by Morse and Witten (103) for weakly deformed drops, led to G a 1/ In (0 - 0q), which eliminates the discontinuity, but still shows an unrealistically sharp rise at 0 = 0q and is qualitatively very different from the experimentally observed linear dependence of G on (0 - 0q). Similar conclusions were reached by Lacasse and coworkers (49, 104). A simulation of a disordered 3-D model (104) indicated that the droplet coordination number increased from 6 at to 10 at 0 = 0.84, qualitatively similar to what is seen in disordered 2-D systems (90). Combined with a suitable (anharmonic) interdroplet force potential, the results of the simulation were in close agreement with experimental shear modulus and osmotic pressure data. It therefore appears again that disor-... 

Which of the following diagrams represent elements and which represent compounds Each color sphere (or truncated sphere) represents an atom. 

Truncated sphere Athermal increasing with time 2-3... 

In this method the field on a dipole in the simulation consists of two parts the first a short-range contribution from molecules situated within a truncation sphere the second from molecules outside the sphere which are considered to form a dielectric continuum (e ) producing an Onsager reaction field at the centre of the cavity. 

There are a number of different geometries convenient for immersion microlenses in photodetection. Probably, the most well known and widely used form is hemisphere. The use of microsystem technologies allows the fabrication of various discrete or arrayed microlenses, with spherical surfaces (calottes, hemispheres and truncated spheres, full spheres), aspheric (ellipsoids, paraboloids, cylinders, cones), toroid, as well as various nonmonotonic surfaces consisting of two or more monotonous segments. Most of the microlenses convenient to increase the incident flux to the detector are plano-convex ones. 

Fig. 2.4 Some spherical and aspheric discrete microlenses for optical concentrators in MWIR and LWIR range that may be fabricated by microsystem technologies, a calotte b hemisphere c hyperhemisphere d ball lens e truncated sphere f bulb g hemi-cylinder h cylinder i curvilinear cone j concave concentrator k gradient-index lens (GRIN) I complex two-element lens (sphere/GRIN). The grid corresponds to the homogeneity of refractive index, i.e., describes its gradient...

This section outlines the approach for calculation of the pair interaction energy between two equally sized deformable fluid droplets (the generalisation to droplets of different size is often straightforward). The fluid particles are assumed to acquire the shape of truncated spheres when the distance between them is small enough (see Figure 10.1). The accuracy of this idealisation is discussed at the end of this section. 

The formation of a doublet of droplets (as shown in Figure 10.1) is accompanied by a shape change from spherical to that of a truncated sphere. This deviation from the optimal (spherical) shape also contributes to the pair interdroplet energy. The... 

All the results described and used above are based on the assumption that the interacting deformable droplets acquire the shape of truncated spheres when sufficiently strong attraction is present. This is certainly an approximation and needs to be compared with results based on the actual shape of the droplets. The latter could be obtained by solving the augmented (including interactions) Laplace equations for the fluid droplet interfaces... 

The comparison of the results based on the Laplace equations (described above) to those obtained assuming the model shape of truncated spheres, showed that both approaches are in very good agreement within 7% see also Table 10.1. This (exact) approach allows a thermodynamic treatment of doublets of droplets by an analogy with macroscopic liquid foam and emulsion films, involving the concept of line and transversal tensions, thermod5mamic film radius and contact angle. Such an analysis is presented in ref. 49. 

Table 10.1 Comparison of the results obtained by the approach based on (i) the numerical integration of the augmented Laplace equation and (ii) the model approach (shape of truncated spheres)...

Consider two bubbles pressed into contact by body forces F. The force F represents the sum of body forces, such as gravity, and the net force imposed by other contacting bubbles. We assume that the deviation from a spherical shape is small and that the bubble s deformed shape is a truncated sphere [16] see Figure 12.7a. Deformed bubbles do not, in fact, assume this shape the sharp corner at the edge of the thin film, in particular, is unrealistic. Nevertheless, the anszatz suffices to give an approximate idea of the elastic interaction between bubbles of similar size. 

Given the truncated sphere of Figure 12.7, the Young-Laplace law can be used to obtain an approximate relation between the imposed force and the resulting deformation. For each bubble, the body force F must be balanced by a force with magnitude Ap nrj) applied by the fluid in the thin film, where ry V2R6 is the radius of the facet and 8 = 8(F) is the dilference between the sphere radius and the distance from the bubble center to the center of the facet. The radius of the... 

Figure 12.7 (a) A contact formed between two bubbles in a wet foam subject to body forces F. The bubbles shapes are approximated as truncated spheres. A thin film of liquid separates two circular facets, (b) Each facet has a radius ry and an effective overlap 5. A disjoining pressure in the thin film applies a force on the facet. 

Equation 12.7 resembles the force law of a simple linear spring. The result is only approximate due to the truncated sphere assumption. More accurate calculations that do not presume the deformed shape a priori are available in the literature [16,19]. These show that Equation 12.8 overestimates the magnitude of fceff and misses logarithmic corrections that cause the stiffness to vanish as f 0. Nevertheless, Equation 12.8 establishes useful intuition. 

This could be explained by the fact that the model of Park assumes that the droplet has the form of a truncated sphere rather than a cylindrical disk, that is, a droplet on a substrate with a certain (small) contact angle. Only for a value of 23 6 kDa did the calculated maximum spreading factor deviate strongly from the experimental results. 

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