The Excluded Region

4 Since the water framework is formed by a space-filling arrangement of face-sharing polyhedra, 12 (i.e., 24/2) of the 44 protons in the [20H2O+4OH]+ form disordered bonds with the adjacent polyhedra. 

This concept promises to be as useful for analyzing water structure in macro-molecular crystals as were the Ramachrandran plots for limiting the conformation possibilities in polypeptides. 

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Use of the concept of the Excluded Region to limit the orientation of -OH groups and both orientation and translations of water molecules (48). 

Figure 16.7. Experimental constraints on the density of axions in the galactic halo near the Sun as a function of the axion mass (upper scale) and cavity frequency (lower scale). The regions above the curves marked DFSZ and KSVZ are excluded fro the respective axion models. The currently accepted value for the local dark halo density is 0.45 GeV/cm3, which is approximately the extension of the excluded region for the KSVZ axion. (Figure from Asztalos et al.(2004).)...

Let us now turn to the problem of minimizing equation (5) with respect to T and a, subject to both equations (9) and (10). It is usual in practice simply to perform a sequence of calculations optimizing E against T for discrete sets of exponents aj., v.% interpolation procedure. This means that the constraint of equation (10) causes no trouble at all, since we can simply choose all our discrete sets to obey it, and forbid interpolation into the excluded regions. However, it is also easy to see that a simple transformation... 

Figure 2.10 shows a plot of the excluded region for the Ow-H - - O bonds discussed in Part IV, Chap. 22. 

As the O-ft - O angle in a three-center hydrogen bond approaches 90°, the O - - O van der Waals repulsive force comes into play which gives rise to the excluded region described in Part I. The bond length of the minor component tends to increase. For an H O hydrogen bond of 2.5 A, a covalent O-H bond of... 

The mixing of surfactant and polymer in the porous medium occurs due to both dispersion and the excluded volume effect for the flow of polymer molecules in porous media, which in turn could lead to the phase separation. Figure 16 illustrates the schematic explanation of the surfactant-polymer incompatibility and concomittant phase separation. We propose that around each micelle there is a region of solvent that is excluded to polymer molecules. However, when these micelles approach each other, there is overlapping of this excluded region. Therefore, if all micelles separate out then the excluded region diminishes due to the overlap of the shell and more solvent becomes available for the polymer molecules. This effect is very similar to the polymer depletion stabilization (55). Therefore, this is similar to osmotic effect where the polymer molecule tends to maximize the solvent for all possible configurations. ... 

Figure 4.10 also shows the excluded volume for this pair of molecules. The dashed line is roughly the closest distance that the center of a water molecule can approach. Thus, if we replace the pair of methane molecules at this specific configuration by an ethane molecule, we expect that the geometry of the excluded region will not be changed. 

A special case of (G.5) is for a hard-sphere (HS) solute. In this case, the interaction between s and any other molecule is either zero or infinity according to whether the molecule is away from or within the excluded region produced by. The function expl— Biis] is accordingly either zero or one. Therefore, the average in (G.5) may be written as... 

Figure 5.22 The volume excluded by a spherical particle of diameter ri is a larger sphere of radius d. The volume of the excluded region is 4Trd l3.

Figure 1.5 Schematic representation of the situation giving rise to depletion flocculation. Polymer molecules are excluded from the space between particles causing an osmotic pressure differential between the excluded region and the continuous phase and giving rise to a net attractive depletion force.

Fig. 3.8 Dependence of the exclusion length on injected current density in an n v exclusion photoconductor for various concentrations of donors in the excluded region. T = 300 K, material is HgCdTe with a composition of 0.165...

Equality is forbidden since it yields an infinite curvature the excluded region near by is the core region. 

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