Critical radius ratios

Figure 9 reports the effect of the pore-body to pore-throat radius ratio, R /R, on the critical capillary pressure for... 

Figure 1.37 Critical radius ratios for various coordination numbers. Adapted, from W. Callister, Materials Science and Engineering An Introduction, 5th ed., p. 384. Copyright 2000 by John Wiley Sons, Inc.

Fig. 14.4. Theoretical critical velocity as a function of separation distance for different values of solute radius/pore radius ratio (A). Solute radius (a) = 5 nm ko= 1.0 (k is the Debye parameter) membrane and solute surface potential = 50 mV. Values of A (a) 0.8 ...

FIGURE 11.4 Passage of bacterial cells through membrane pores. Cell damage occurs at critical pore radius/cell radius ratio. (From Goosen, M.F.A., Sablani, S.S., Al-Hinai, H., Al-Obeidani, S., Al-Belushi, R., and Jackson, D., Sep. Sci. TechnoL, 39, 2261, 2004.)... 

However, surfactants incorporated into the electrolyte solution at concentrations below their critical micelle concentration (CMC) may act as hydrophobic selectors to modulate the electrophoretic selectivity of hydrophobic peptides and proteins. The binding of ionic or zwitterionic surfactant molecules to peptides and proteins alters both the hydrodynamic (Stokes) radius and the effective charges of these analytes. This causes a variation in the electrophoretic mobility, which is directly proportional to the effective charge and inversely proportional to the Stokes radius. Variations of the charge-to-hydrodynamic radius ratios are also induced by the binding of nonionic surfactants to peptide or protein molecules. The binding of the surfactant molecules to peptides and proteins may vary with the surfactant species and its concentration, and it is influenced by the experimental conditions such as pH, ionic strength, and temperature of the electrolyte solution. Surfactants may bind to samples, either to the... 

Critical radius Predicted Ion Radius ratio coordination... 

Table 8 Critical cation environments to anion radius ratios for stability various coordination...

Fig. 3.07. Section through a unit cell of the caesium chloride structure on a vertical diagonal plane. The solid circles represent the cations. In (a) the ions are shown in their correct relative sizes for CsCl (b) corresponds to the critical radius ratio for anion-anion contact.

Thus the variation of the energy of the structure as a function of radius ratio will be of the form represented by curve (a) in fig. 3.08, with a discontinuity at this critical value of r+jr. ... 

We can continue this argument by considering the zincblende structure. In this case the critical radius ratio, corresponding to anion-anion contact, can be readily shown to be given by... 

A second feature of silicate crystal chemistry is peculiar to silicates alone, and arises accidentally from the particular value of the radius of the aluminium ion. The A1 0 radius ratio of 0 36 is so close to the critical value of 0 3 for transition from 6- to 4-co-ordination that this ion can occur in both conditions, sometimes in the same structure. When 4-co-ordinated the aluminium ion replaces silicon, and such replacement is purely random and may be of indefinite extent. For every aluminium ion so introduced a corresponding substitution of Ca2+ for Na+, Al3+ for Mg2+ or Fe3+ for Fe2+ must simultaneously occur else-... 

The crystal chemistry of the iron-carbon system is especially complex on account of the relatively small size of the iron atom, resulting in a carbon iron radius ratio of about o 6o, which is so close to the critical value 0 59 discussed above that both interstitial structures and structures of greater complexity may be expected. Added to this is the further complication that iron is dimorphous. Below about 910 °C, and from about 1400 °C to the melting point, the structure is cubic body centred, and is known as a iron. Between these two temperatures a cubic close-packed structure, termed y iron, is formed. The ferromagnetism of iron... 

A survey of all of the available data on the stability of halide complexes shows that generally the stability decreases in the series F>Cl>Br>I, but with some metal ions the order is the opposite, namely, Ftheoretical explanation for either sequence or for the existence of the two classes of acceptors relative to the halide ions has been given. It is likely that charge/radius ratio, polarizability, and the ability to use empty outer d orbitals for back-bonding are significant factors. From the available results it appears that for complexes where the replacement stability order is Clbond strength is Cl >Br >1, so that ionic size and polarizability appear to be the critical factors. 

Critical Reynolds Number For concentric annular ducts, the critical Reynolds number at which turbulent flow occurs varies with the radius ratio. Hanks [109] has determined the lower limit of Recrit for concentric annular ducts from a theoretical perspective for the case of a uniform flow at the duct inlet. This is shown in Fig. 5.16. The critical Reynolds number is within 3 percent of the selected measurements for air and water [109]. 

Three factors are critical in determining the structure of ceramic compounds crystal stoichiometry, the radius ratio, and the propensity for covalency and tetrahedral coordination. 

Since cations are usually smaller than anions, the crystal structure is usually determined by the maximum number of anions that it is possible to pack around the cations, which, for a given anion size, will increase as the size of the cation increases. Geometrically, this can be expressed in terms of the radius ratio rjr, where r. and r are the cation and anion radii, respectively. The critical radius ratios for various coordination numbers are shown in Fig. 3.3. Even the smallest cation can be surrounded by two anions and results in a linear arrangement (not shown in Fig. 3.3). As the size of the cation increases, i.e., as increases, the number of anions that can be accommodated around a given cation increases to 3 and a triangular arrangement becomes stable (top of Fig. 3.3). r. /i, > 0.225, the tetrahedral arrangement becomes stable, and so forth. 

Figure 3.2. Stability criteria used to determine critical radius ratios.

Figure 3.3. Critical radius ratios for various coordination numbers. The most stable structure is usually the one with the maximum coordination allowed by the radius ratio.

Derive the critical radius ratio for the tetrahedral arrangement (second from top in Fig.. 3.3). 

The easiest way to derive this ratio is to appreciate that when the radius ratio is critical, the cations just touch the anions, while the latter in turn are just touching one another (i.e., the anions are closely packed). Since the coordinates of the tetrahedral position in a close-packed arrangement (Fig. 3.46) are 1/4,1/4,1/4, it follows that the distance between anion and cation centers is... 

Ceramic structures can be quite complicated and diverse, and for the most part depend on the type of bonding present. For ionically bonded ceramics, the stoichiometry and the radius ratio of the cations to the anions are critical determinants of structure. The former narrows the possible structures, and the latter determines the local arrangement of the anions around the cations. The structures can be best visualized by focusing first on the anion arrangement which, for the vast majority of ceramics, is FCC, HCP, or simple cubic. Once the anion sublattice is established, the structures that arise will depend on the fractional cationic occupancy of the various interstitial sites defined by the anion sublattice. 

Pressure ratio Critical pressure ratio Knuckle radius Mean radius of pipe usii nominal wall thickness T Specific surface area Fluid head loss Specific energy loss Speed... 

Obviously, thermal runaway occurs in the previous example if the flow rate ratio is unity. However, it is possible to control a double-pipe reactor with = I by decreasing the radius ratio. This is illustrated in Table 4-4 for conditions described in the previous example. Thermal runaway occurs when k > /Ccriticai and the critical radius ratio lies somewhere between 0.10 and 0.15. 

Hence, the parametric sensitivity analysis outlined in this section identifies critical values, or a range of critical values, for (1) the outside wall temperature, (2) the heat transfer time constant, (3) the flow rate ratio xj/, and (4) the radius ratio k, which delineates the boundary between well-behaved reactor performance and thermal runaway. Other parameters that exhibit critical values and... 

Table 6.1. The radius-ratio values given in Table 6.1 are consistent with a CN of 6 based on the critical radius ratios given earlier in Table 5.4. The interstitial atoms are located either in an octahedral site or in the center of a trigonal prism. For the transition metals, the tetrahedral interstices in the close-packed structures are too small for C or N.

Figure 16.13. Calculated (Eq. (16.17)) and measured critical grain radius ratio, a APp)ja(0), with excess internal gas pressure, APpP ...

Core No. DtO Boron Cone Cone (Mole%) (g B/1) No. of Fuel Rods Critical Radius (cm) Critical Height (cm) Critical Buckling (xl0 cm" ) Thermal DisadvanU e Factor Cadmium Ratio ... 

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