Lattice holes

Gitter-konstante, /. lattice constant grating constant, -loch, n. (Cryst.) lattice hole, lattice void, -masche, /. grid mesh, -span-nung, /. (Elec.) grid voltage, -spektrograph,... 

Nitrogen clathrate in j3-quinol was studied by Scott28) in spite of many experimental difficulties. Among the findings obtained in this study, we may mention the existence of a seven-line spectrum whose intensity depends upon the preparation, history and age of the sample a slow loss of nitrogen by the sample the fact that the line frequencies are independent of the factors which alter line intensities and that the structure may be due to a partial filling of the (3-quinol lattice holes. 

The GC-EOS based on a series expansion of the nonrandom lattice-hole theory is written by[l,6],... 

Hyper- and hypostoichiometric, 1102+. and UO2 j, respectively, are also known and have been analyzed by X-ray crystallography. In the case of the UO2+, extra oxygen atoms occupy central lattice holes in the normal UO2 structme. 

We interpret these results by assuming that the high values for the heats of adsorption pertain to specifically active centers, such as lattice holes (7). Dielectric relaxation measurements (4) and KMR studies (2) substantiate such conclusions. 

Figure 2.17. Surface tension of linear alkanes as a function of (a) temperature and (b) redueed temperature. Cp Cg, etc. refer to alkanes with 1, 2,. .. CH2 groups per chain. Lattice hole theory, redrawn after Schlangen et al. (J. Phys. Chem. 100 (1996) 3607.)...

A Schottky defect consists of an atom or ion vacancy in a crystal lattice, but the stoichiometry of a compound (and thus electrical neutrality) must be retained. In a metal lattice, a vacant atom site may be present. Examples of Schottky defects in ionic lattices are a vacant cation and a vacant anion site in an MX salt, or a vacant cation and two vacant anion sites in an MX2 salt. Figure 5.26 illustrates a Schottky defect in an NaCl lattice holes are present (Figure 5.26b) where ions are expected on the basis of the ideal lattice (Figure 5.26a). 

VOLUME RELAXATION AND THE LATTICE HOLE MODEL 4.1 INTRODUCTION... 

We want to pursue the subject by starting with a brief review for the present purposes of the essentials of lattice-hole theory, then follow with a consideration of free-volume mobility connections, and continue with some comparisons of experiment versus theory. Finally, we propose and sketch modifications of the theory. These may open the way to generalizations and more insightful relations to empirical formulations, such as the KAHR model [Kovacs et al., 1977, 1979] for volume relaxation. 

The literature offers empirical expressions that relate free volume to relaxation times. In particular, we refer to the Vogel and Williams-Landel-Ferry (WLF) relations derived from fluidity measurements. These macroscopically defined equations provide relaxation rates (i.e., reciprocal relaxation times, r) as functions of temperature. We can convert these to functions of free volume, /, or lattice-hole fraction, h. Due to the essentially linear dependence of h on T, the mathematical form of the original equation is preserved, and thus one has [Robertson, 1992]... 

Utracki, L. A., and Simha, R., Analytical representation of solutions to lattice-hole theory, Macrvmol. Theory Simul., 10, 17-24 (2001). 

The Simha and Somcynsky (S-S) [1969] cell-hole theory is based on the lattice-hole model. The molecular segments of an -mer occupy ay-fraction of the lattice sites, while the remaining randomly distributed sites, /i = 7 — y, are left as empty holes. The fraction /i is a measure of the free-volume content. The goal was to provide improved description of fluids, ranging from low-molecular-weight spherical molecules (such as argon) to macromolecular chains. The S-S configurational partition function is... 

Correlation between liquid behavior at thermodynamic equilibrium and that during flow follows from the mean-field approach, which assumes that liquids are structureless and that the dynamic behavior can be considered a semiequilibrium state. Evidently, this approach is unable to explain kinetic phenomena. The S-S lattice-hole mean-field theory does not consider polymeric chain structure, but its effects are reflected in the values of the characteristic reducing parameters, P, T, V, and tlie L-J interaction parameters. Characteristically, the PVT data rarely show secondary transformation temperatures at about 0.8r and 1.2r, which are evident in derivative properties (see Figures 6.1 and 6.2). By contrast, all flow models (e.g., reptation, cell structures, hole jumping) implicitly postulate that such configurational or conformational changes affect liquid dynamic behavior. 

Simha, R., and Xie, H., Applying lattice-hole theory to gas solubility in polymers, Polym. Bull., 40, 329-335 (1998). 

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. 

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