Radius-ratio principle

Using cultured mammalian sarcoma cells, it has been found that transferrin is necessary in the growth medium for galllum-67 uptake to occur (95,96.97). A transferrin receptor on EMT-6 sarcoma cells for 25i iabeled transferrin was characterized by Scatchard analysis to have an average association constant K = 4.54 X 10 1/mole and approximately (with variation) 500,000 receptors per cell ( ). It was proposed that tumor accumulation of galllum-67 can occur only if the metal is complexed with transferrin so that it can interact with the receptors of tumor, as well as non-mallgnant cells (.33). The complex then enters the cell via an adsorptive endocytosis process (95.96.97.98.99) similar to the manner in which iron is taken up by reticulocytes and bone marrow cells (100.101). These transferrin receptors are saturable (that is, a plot of 125i transferrin uptake versus extracellular transferrin concentration reaches a peak ( at about 200 u g/ml) as more carrier transferrin is added to the medium) (95). Since uptake is also proportional to the fraction of 

ACS Symposium Series American Chemical Society Washington, DC, 1980. 

It has been further shown that a majority (approximately 60jt) of the extractable Qa (about 70 of the cellular gallium) from tumor and liver cells of the rat is associated with two macromolecular fractions of molar weight 1-1.2 x 105 Daltons and 4-5 X 10 Daltons (64) The 1-1.2 x 1o5 D band is found in both liver and tumor cells, whereas the 4-5 x 10 D band is found 

Lactoferrin, with a molecular weight of 8.5-9.0 x 10 and a structure similar to transferrin (112) has been suggested as an alternative intracellular gallium-binding agent (113). 

---

We focus attention on the fact that the crystal radii (CRs) for the various cations listed in table 1.11 are simply equivalent to the effective ionic radii (IRs) augmented by 0.14 A. Wittaker and Muntus (1970) observed that the CR radii of Shannon and Prewitt (1969) conform better than IR radii to the radius ratio principle and proposed a tabulation with intermediate values, consistent with the above principle (defined by the authors as ionic radii for geochemistry ), as particularly useful for sihcates. It was not considered necessary to reproduce the... 

The so-called radius ratio principle establishes that, for a cation/anion radius ratio lower than 0.414, the coordination of the complex is 4. The coordination numbers rise to 6 for ratios between 0.414 and 0.732 and to 8 for ratios higher than 0.732. Actually the various compounds conform to this principle only qualitatively. Tossell (1980) has shown that, if Ahrens s ionic radii are adopted, only 60% of compounds conform to the radius ratio principle. ... 

Solutions of Mo acidified to H /[Mo04] — 1.8 contain one or more very large polymolybdate structures. Earlier measurements based on ultracentrifugation and EMF studies, through to a recent structure report,are consistent with a fonnula [Mo360ii2(H20)i8] ". The reaction can be summarized as in equation (35). There are two seven-coordinate Mo atoms in this structure. In all isopoly and heteropoly structures the metal ion does not lie at the centre of its polyhedron, but is displaced towards the exterior of the structure and towards a vortex or edge of its own polyhedron. Structures appear to be governed by electrostatic and radius-ratio principles as observed for extended ionic lattices. The Mo tetrahedral radius is 0.55 A (0.56 A for W), the octahedral radius 0.73 A (0.74 A for W), and the radius of 0 is 1.40 A. 

Even if there are exceptions to the radius ratio rule, or if exact data are hard to come by, it is still a valid guiding principle. Cite three independent examples of pairs of compounds illustrating structural differences resulting from differences in ionic radii. 

We have seen in Chapter 4 that the coordination number of ions in lattices is related to the ratio of the radii of the ions. The same general principles apply to coordination compounds, especially when a single coordination number, such as 4. has two common geometries—tetrahedral and square planar, An extended list of radius ratios is given in Table 12,1. 

Although the ionic radius criterion of Goldschmidt continues to serve as a useful principle of crystal chemistry, attention has been drawn to limitations of it (Bums and Fyfe, 1967b Bums, 1973). As noted earlier, the magnitude of the ionic radius and the concept of radius ratio (i.e. cation radius/anion radius) has proven to be a valuable guide for determining whether an ion may occupy a specific coordination site in a crystal structure. However, subtle differences between ionic radii are often appealed to in interpretations of trace element distributions during mineral formation. 

Changes of coordination number A guiding principle of crystal chemistry is that the coordination number of a cation depends on the radius ratio, RJR, where Rc and / a are the ionic radii of the cation and anion, respectively. Octahedrally coordinated cations are predicted when 0.414 < 7 c// a < 0.732, while four-fold (tetrahedral) and eight- to twelvefold (cubic to dodecahedral) coordinations are favoured for radius ratios below 0.414 and above 0.732, respectively. The ionic radii summarized in Appendix 3... 

For simplicity we shall discuss complex oxides and complex oxy-salts, but the same principles apply to complex fluorides and ionic oxyfluorides. A complex oxide is an assembly of 0 ions and cations of various kinds which have radii ranging from about one-half to values rather larger than the radius of 0 . It is usual to mention in the present context some generalizations concerning the structures of complex ionic crystals which are often referred to as Pauling s rules . The first relates the c.n. of M"" to the radius ratio rg. The general increase of c.n. with increasing radius ratio is too well known to call for further discussion... 

It follows from the foregoing discussion that, at least in principle, it should be possible to predict the local arrangement of ions in a crystal if the ratio rjr is known. To illustrate the general validity of this statement, consider the oxides of group IVA elements. The results are summarized in Table 3.1, and in all cases the observed structures are what one would predict based on the radius ratios. 

The biological function of Group lA and IIA cations of the periodic table is reviewed against the background of their chemistry. Utilization of these cations arises from an ability to form different types of complex compounds, which is dependent upon the radius-ratio effect. If the details of their biochemistry are to be understood, new probe methods for following the cations in biological systems must be devised. Some possibilities based upon the principle of isomorphous replacement are described and tested. 

The neosilicates (orthosilicates) are composed of isolated 8104 tetrahedrons. The mineral olivine, Mg2Si04, is an example. The radius ratio rule for Mg " " and 0 predicts an octahedral coordination for Mg " " (65/140 = 0.46). Thus, the ebs for Mg is I /3, while that of Si + is I. In order to satisfy the principle of charge neutrality, each must coordinate to three Mg " " ions and one Si + ion —2=3(—l/3)-l-1(—I). The structure of olivine is shown in Figure 12.17. There are two types of distorted edge-sharing octahedrons for Mg " MI and Ml), shown in dark green and red, in addition to the isolated tetrahedrons containing Si, shown in blue. 

Although computational fluid dynamics (CFD) methods like FEM are in principle capable of dealing with arbitrarily shaped particles, the inherent limitation of SD necessitates the choice of spherical particles for a comparative study. Particles have small gaps between surfaces (gap-to-radius ratio of 0.0003), allowing for an easier mesh generation. Our method validation includes simple structures also chosen by Binder et al. [54]—a doublet of two spherical particles and a 7-particle star. 

The emulsification process in principle consists of the break-up of large droplets into smaller ones due to shear forces (10). The simplest form of shear is experienced in lamellar flow, and the droplet break-up may be visualized according to Figure 4. The phenomenon is governed by two forces, ie, the Laplace pressure, which preserves the droplet, and the stress from the velocity gradient, which causes the deformation. The ratio between the two is called the Weber number. We, where Tj is the viscosity of the continuous phase, G the velocity gradient, r the droplet radius, and y the interfacial tension. 

---