Spherical hole/cavity

Four volumetric defects are also included a spherical cavity, a sphere of a different material, a spheroidal cavity and a cylinderical cavity (a side-drilled hole). Except for the spheroid, the scattering problems are solved exactly by separation-of-variables. The spheroid (a cigar- or oblate-shaped defect) is solved by the null field approach and this limits the radio between the two axes to be smaller than five. 

The second noteworthy morphological feature is presented in Fig. 12b. This micrograph depicts the pre-crack front of 15-1500-70F, which had a value significantly above that of the control, as shown in Fig. 11 a. The holes may be examples of the dilatation effect observed in CTBN-modified epoxies l9,22> in which rubber particles dilate in mutually perpendicular directions under the application of a triaxial stress and then collapse into spherical cavities following fracture. Dilatation requires a mismatch in coefficients of thermal expansion of resin and rubber 11. This effect will therefore be most striking when the elastomeric phase is homogeneous, as is apparently the case here. 

Whereas macrocycles define a two-dimensional, circular hole, macrobicycles define a three-dimensional, spheroidal cavity, particularly well suited for binding the spherical alkali cations (AC) and alkaline-earth cations (AEC). 

A 1-m-diameter spherical cavity is maintained at a uniform temperature of 600 K. Now a 5-mm-diameter hole is drilled. Determine the maximum rate of radiation energy streaming through the hole. AVhal would your answer be if the diameter of the cavity were 3 m ... 

A properly folded chain molecule can form a cavity with a shape globally complementing the shape of an enclosed, quasi-spherical molecule. If, however, the enclosed molecule has a toroidal or more complicated topology, where the hole of the torus is small so the surrounding chain molecule cannot enter this hole, then an approximate global complementarity by a chain molecule is unlikely to reflect the correct topology of the enclosed molecule. 

For the seemingly simplest case of spherical metal ion complexation, the hole-size fit often, but not necessarily, holds. Fig. 2.5 illustrates the classical case where the cavity diameter of an ionophore determines the selectivity of cation complexation according to its radius [31]. As long as sufficient contact between the metal ion and the donor atom of the hgand is possible, the complexation free energy will be just a linear function of the number of such interactions and their donor quality... 

In the following we discuss various experimental data, with the aim of showing that use of nonspherical geometries for holes can in some cases supply better agreement between free-volume fractions /and h than can the spherical model. Furthermore, results in some polymers suggest a nonisotropic growth of the cavities with temperature. 

Yu et al. [1994] carried out PALS measurements on four PS fractions (4, 9, 25, and 400 kDa, respectively) versus temperature (Figure 10.4). They evaluated the free-volume fractions on the basis of the proportionality between the free-volume fraction as probed by o-Ps and the product of the o-Ps intensity I3 and the mean cavity volume assumed spherical, as sketched previously [Eq. (10.16)]. On this basis they observed agreement with the free-volume fraction predicted as given by the lattice-hole model [Simha and Somcynsky, 1969] over a range of temperatures above Tg, the proportionality constant C being a molar mass-dependent fitting parameter. 

It is worth pointing out that this result does not depend on the spherical shape adopted for the estimate of the hole size indeed, essentially identical discrepancies with the theory are obtained using cubic, prismatic, or cylindrical holes. All these models assume an isotropic expansion of holes that is, Vh characteristic dimension of the hole hosting Ps. The values of s, as evaluated from any model, are necessarily approximate estimates the irregular shape of real holes precludes the deduction of an exact value of the cavity size. Equations (10.6), and (10.8)-( 10.13) are all analogous in that they obtain from ts a characteristic hole dimension s, whose numerical value depends only slightly on the model adopted. This is a possible reason that discrepancy between/and h persists whatever shape is assumed for holes. 

Figure 5.1 Sketches of geometries for polymer translocation (a) a tiny hole in a thin planar membrane, (b) two spherical cavities connected by a bridge, (c) a cylindrical pore, and (d) an infinitely wide channel.

The above calculation can be readily implemented for the process depicted in Figure 5.9, where the chain undergoes translocation from one spherical cavity of radius to another spherical cavity of radius R2 through a hole. For this simation, Fq and Ff are given by Equation 5.10 with R = R and R = R2, respectively. F is again given by Equation 5.14 with R = R. The... 

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