Hole radius distributions

Figure 8. Hole radius distributions for Avicel samples.

Figure 2. Free-volume hole radius distribution R pdf(R) (relative units) of PTMSP (pdf(R) is probability density function) obtained from the two right-hand peaks in Fig. 1. The dashed line shows the calculated dependence (3) of annihilation rate, (in ns ) versus radius of FV elements used in confuting the size distribution.

Since the desired distribution function only concerns the radii (or sizes) of holes, it is sufficient to have the probability that the hole radius is between r and r + dr irrespective of the location and the translational and breathing momentum of the hole. This probability Pr dr of the hole s radius being between r and r + dr is obtained from Eq. (5.16) by integrating over all possible values of the location, and of the translational and breathing momentum of the hole, i.e.,... 

In this chapter we demonstrate the potential of PALS for the study of the size distribution of subnanometer-size local free volumes (holes) in amorphous polymers. We employ the routine LifeTime in its version 9.0 (LT9.0 [Kansy, 1996, 2002]) for the analysis of lifetime spectra and discuss its advantage for analyzing < -Ps lifetime distributions. From these distributions the hole radius and hole volume distributions are calculated. The assumptions underlying this type of analysis, present-day understanding, and possible complications (e.g., tunneling, weighting) are discussed briefly. 

Since Xpo follows a distribution and the relation between A.po and the hole radius r is nonlinear [Eq. (11.3)], we prefer to estimate the mean hole-volume as the mean of the number-weighted hole volume distribution. The radius distribution [the probability density function (pdf)], n(rfc), can be calculated from n (rh) = - 3 (A.) (dx/drh) [Gregory, 1991 Deng et al., 1992b] ... 

Using PALS, Dammert et al. (92) and Yu et al. (93) examined the hole volume of a series of polystyrenes of different tacticity. They arrived at a free-volume hole size distribution maximum of around 110 at room temperature see Rgure 8.24 (92). This corresponds to an effective spherical hole radius of approximately 3 A. While this radius is somewhat larger than the theoretical value of 1.5 A found above, if the holes are actually irregular in shape the values are seen to agree quite well. 

The Lewis dot formalism shows any halogen in a molecule surrounded by three electron lone pairs. An unfortunate consequence of this perspective is that it is natural to assume that these electrons are equivalent and symmetrically distributed (i.e., that the iodine is sp3 hybridized). Even simple quantum mechanical calculations, however, show that this is not the case [148]. Consider the diiodine molecule in the gas phase (Fig. 3). There is a region directly opposite the I-I sigma bond where the nucleus is poorly shielded by the atoms electron cloud. Allen described this as polar flattening , where the effective atomic radius is shorter at this point than it is perpendicular to the I-I bond [149]. Politzer and coworkers simply call it a sigma hole [150,151]. This area of positive electrostatic potential also coincides with the LUMO of the molecule (Fig. 4). 

We may wonder what effect these gigantic explosions and accompanying radiation may have on the galactic environment of the hypernova. Such an energy release could only produce holes, hollow shells in the distribution of interstellar matter. After 1 million years, these cavities reach a radius of 150 to 500 light-years, according to calculation. The initial excesses are followed by a calmer expansion at speeds below 10 km s ... 

Tu, as a function of Newtonian wall shear rate, F, for a number of common polymers. Consider the melt spinning of PS at a volumetric flow rate of 4.06 x 10 cm /s through a spinneret that contains 100 identical holes of radius 1.73 X 10 cm and length 3.46 x 10 cm. Assume that the molecular weight distribution is broad. 

The structure of crystalline FeO belongs to the NaCl type. When iron(II) oxide is prepared under normal conditions, the composition of the product (wustite) is always Fei 50. In order to retain overall electric neutrality, part of the Fe2+ is oxidized to Fe3+, and the chemical formula becomes Fc Fe. O. Since the radius of Fe3+ is small, the Fe3+ cations tend to occupy the tetrahedral holes to form a short-range ordered Fe40io cluster, which is called the Koch cluster of Fei 0, as shown in Fig. 10.1.3. The Koch clusters are distributed randomly in the crystal structure. To satisfy charge neutrality, the formation of a Koch cluster must be accompanied by the presence of six Fe2+ vacancies, one of which is located at the center of the cluster, and the remaining five are distributed randomly at the centers of the edges of the cubic unit cell. 

According to the normal equations of classical statistical mechanics, which are used to express velocities and momenta distributed in three dimensions, the probability that the location of a hole is between x and x + dx, y and y + dy, z and z + dz that its translational momenta lie between and + dp, Py and Py + dpy, p and p + dp that its breathing momentum is between p and p + dp/, and, finally, that its radius is between r and r+ dr, is proportional to the Boltzmann probability factor,... 

This is the distribution function which was said to be the goal at the beginning of Eq. (5.17). From it, the average hole volume and radius will shortly be seen to be obtainable (see Fig. 5.22). 

Using the distribution function, make a plot of probability that a hole has a radius in molten sodium chloride at 1170 K. The surface tension of molten sodium... 

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