Domain models, spherical

Figure 4.10 AX4, AX3E, and AX2E2 molecules (a) tangent sphere models or domain models with spherical domains B is a bonding pair and E is a lone pair and (b) conventional bond line structures.

It is a small step from van der Waals, electron-domain models of the C—H bonds of, e.g., biphenyl, cyclohexane, or methane (Figs. 1—3), to molecular models in which to a first, and useful, approximation each valence-shell electron-pair is represented by a spherical, van der Waals-like domain 7h (Non-spherical domains may be useful for describing, e.g., lone pairs about atoms with large atomic cores, -electrons, and the electron-pairs of multiple bonds vide infra.)... 

Spherical-domain models of three-center bonds in localized-molecular-orbital models of a nonclassical carbonium ion, B4CI4, and TaeClfJ have been described 49,52) a drawing of a spherical-domain model of the methyl lithium tetramer, (LiCH, is shown in Fig. 31. Large, outer circles represent domains of electron-pairs of C—H bonds. Solid circles represent domains of Li+ ions. Shaded circles represent 4-center lithium-lithium-lithium-carbon bonds — i.e., electron-pair domains that touch, simultaneously, three lithium ions and the kernel of a carbon atom. The... 

At the beginning of this section we enumerated four ways in which actual polymer molecules deviate from the model for perfectly flexible chains. The three sources of deviation which we have discussed so far all lead to the prediction of larger coil dimensions than would be the case for perfect flexibility. The fourth source of discrepancy, solvent interaction, can have either an expansion or a contraction effect on the coil dimensions. To see how this comes about, we consider enclosing the spherical domain occupied by the polymer molecule by a hypothetical boundary as indicated by the broken line in Fig. 1.9. Only a portion of this domain is actually occupied by chain segments, and the remaining sites are occupied by solvent molecules which we have assumed to be totally indifferent as far as coil dimensions are concerned. The region enclosed by this hypothetical boundary may be viewed as a solution, an we next consider the tendency of solvent molecules to cross in or out of the domain of the polymer molecule. 

FIG. 4 Onion model of spherical water-containing reversed micelles. Solvent molecules are not represented. A, surfactant alkyl chain domain B, head group plus hydration water domain C, hulk water domain. (For water-containing AOT-reversed micelles, the approximate thickness of layer A is 1.5 nm, of layer B is 0.4 nm, whereas the radius of C is given hy the equation r = 0.17R nm.)... 

For AX molecules with no lone pairs in the valence shell of A, both the VSEPR model and the LCP model predict the same geometries, namely AX2 linear, AX3 equilateral triangular, AX4 tetrahedral, AX5 trigonal bipyramidal, and AX octahedral. Indeed Bent s tangent sphere model can be used equally as a model of the packing of spherical electron pair domains and as a model of the close packing of spherical ligands around the core of the central atom. 

Modeling the initial structure by spherical domains in a bcc-lattice1, the theoretical intensity along the ellipsoidal ridge as a function of the angle y1 between fiber axis and the direction of the radial beam is... 

Several models of polyelectrolyte thermodynamics have been proposed and have been recently reviewed [84], Historically, two general approaches [84,85] have been used to model polyelectrolyte thermodynamics, spherical and chain models. In the former approach the coiled polyions are treated as spherical domains with charge density distributed continuously within the sphere... 

A model for wet scrubbing in a cross-flow is illustrated in Fig. 7.21. Consider a rectangular scrubbing domain of length L, height H, and width of unity in Cartesian coordinates. Assume that the gas-solid suspension flow is moving horizontally, and that the solid particles are spherical and of uniform size. The particle concentration across any plane perpendicular to the flow is assumed to be uniform. The water droplets fall vertically and are uniformly distributed in the flow system. 

A spherical model was used in Ref. [15] in order to obtain the shape of the domains, reversed under the fdb conditions. This model was widely applied for studies of different processes that take place in the field of afm tip (see Ref. [65]), including ferroelectric polarization reversal [66-69], In this model the field of the tip apex is supposed to coincide with a field of a metallic sphere, the radius of which is equal to the radius of curvature of the tip apex. Using a simple approximation it may be supposed that the tip charge is concentrated in the center of the sphere [15,64-69], We will take into account a more general model and check the accuracy of the simple spherical model application to the ferroelectric domain breakdown condition. 

When domain dimensions are much larger than the tip radius R, Equations (11), (12) and (13) become identical to the corresponding equations obtained in the framework of the theory [15,64] developed using the simple spherical model. In this model the energy of the long domain W r,l), created within the condition of fdb, equals to the summation of the energies from Equation (10.2) and Equation (10.11)... 

Sidgwick s discussion raises an important question What are the effective sizes and shapes of atoms in molecules From the viewpoint of the electride ion model of electronic structure, Sigdwick s circles for the fluoride ions in the first column of Fig. 15 are the wrong shape, if nearly the right overall size. In the electride-ion model a fluoride ion is composed of (approximately) spherical domains, but is not itself spherical, in the field of a cation, Fig. 16. Fig. 17 illustrates, correspondingly, the implied suggestion that, on the assumption that non-bonded interactions are not limiting, the covalency limits of an atom will be determined by the radius of the atom s core and by the effective radii, not of the overall van der Waals envelopes of the coordinated ions but, rather, by the radii of the individual, shared electron-pairs. 

Martin et al. [68] have modeled together solubility of 14 polycyclic aromatic hydrocarbons (PAHs) and fullerene in octanol and heptane utilizing descriptors calculated with the CODESS A package [69]. Also in this case, the applicability domain has not been validated. The structural difference between planar PAHs and spherical fullerene is probably too large for making reliable predictions. Moreover, the experimental solubilities of fullerene in both solvents (log S = —4.09 in heptane and log S>4.18 in octanol) are significantly lower than the solubilities of PAHs (—3.80[Pg.211]

---