Size ratio

At equilibrium, in order to achieve equality of chemical potentials, not only tire colloid but also tire polymer concentrations in tire different phases are different. We focus here on a theory tliat allows for tliis polymer partitioning [99]. Predictions for two polymer/colloid size ratios are shown in figure C2.6.10. A liquid phase is predicted to occur only when tire range of attractions is not too small compared to tire particle size, 5/a > 0.3. Under tliese conditions a phase behaviour is obtained tliat is similar to tliat of simple liquids, such as argon. Because of tire polymer partitioning, however, tliere is a tliree-phase triangle (ratlier tlian a triple point). For smaller polymer (narrower attractions), tire gas-liquid transition becomes metastable witli respect to tire fluid-crystal transition. These predictions were confinned experimentally [100]. The phase boundaries were predicted semi-quantitatively. 

As shown in section C2.6.6.2, hard-sphere suspensions already show a rich phase behaviour. This is even more the case when binary mixtures of hard spheres are considered. First, we will mention tire case of moderate size ratios, around 0.6. At low concentrations tliese fonn a mixed fluid phase. On increasing tire overall concentration of mixtures, however, binary crystals of type AB2 and AB were observed (where A represents tire larger spheres), in addition to pure A or B crystals [105, 106]. An example of an AB2 stmcture is shown in figure C2.6.11. Computer simulations confinned tire tliennodynamic stability of tire stmctures tliat were observed [107, 1081. 

A second case to be considered is that of mixtures witli a small size ratio, <0.2. For a long time it was believed tliat such mixtures would not show any instability in tire fluid phase, but such an instability was predicted by Biben and Flansen [109]. This can be understood to be as a result of depletion interactions, exerted on the large spheres by tire small spheres (see section C2.6.4.3). Experimentally, such mixtures were indeed found to display an instability [110]. The gas-liquid transition does, however, seem to be metastable witli respect to tire fluid-crystal transition [111, 112]. This was confinned by computer simulations [113]. 

Imhof A and Dhont J K G 1995 Experimental phase diagram of a binary oolloidal hard-sphere mixture with a large size ratio Phys. Rev. Lett. 75 1662-5... 

Table 3 is arranged by crystal class (14). The crystal class of a given pigment is determined almost solely by the ratio of the ionic sizes of the cation and the anion and thek respective valences. Hence, for any given stoichiometry and ionic size ratio, only one or two stmctures will be possible. In some classes (spkiel, zkcon), a wide range of colors is possible within the confines of that class. Pigments within a given class usually have excellent chemical and... 

Wollastonite with an aspect ratio of 15 1 is useful as a replacement for asbestos and as a high-strength filler for plastics. The feed material with dgo of 45 [Lm was similarly ground. Beads of 0.3 mm gave faster grinding than 0.8 mm beads, and these corresponded to a bead-particle-size ratio of 19, confirming other results. 

Frame size Ratio oj copper Tarns pel coil Tarn insuhttion (Table 1.2) to insalalion to collage... 

Pores of the activated carbon exist throughout the particle in a manner illustrated in Figure 2. The pore structure of activated carbon affects the large surface-to-size ratio. The macropores do not add appreciably to the surface area of the carbon but provide a passageway to the particle interior and the micropores. The micropores are developed primarily during carbon activation and result in the large surface areas for adsorption to occur. 

Seleet size ratios preferably the same as the standard values (Table 7-2) to avoid experimentation or based on small-seale smdies. 

Clearly, the inclusion of the liquid phase (xi) will tend to reduce the number of fines in the system. Thus, a diminution in particle size must be effected to provide an equivalent particle population in the fluid-particle assembly. The particle size ratio normally used in practical systems tends to be somewhat lower than the one computed from Eq. (8). 

Table 27.7 Appropriate boiier size ratios assuming turndown to 30% of fuii ioad...

Installed boiler size ratios Lowest load (%) Load if largest fails (%) Installed boiler size ratios Lowest load (%) Load if largest fails (%) ... 

The estimation of the working surface area of solid electrodes is a difficult matter owing to irregularities at a submicroscopic level.10 15 20 24 32 63 64 67 68 73 74 218-224 Depending on the irregularity-to-probe size ratio, either the entire surface or only a fraction of it is accessible to a particular measurement. Only when the size of the molecule or ion used as a probe particle is smaller than the smallest surface irregularity... 

Tetrahedral and octahedral interstitial holes are formed by the vacancies left when anions pack in a ccp array, (a) Which hole can accommodate the larger ions (b) What is the size ratio of the largest metal cation that can occupy an octahedral hole to the largest that can occupy a tetrahedral hole while maintaining the close-packed nature of the anion lattice (c) If half the tetrahedral holes are occupied, what will be the empirical formula of the compound MVAV, where M represents the cations and A the anions ... 

Honda et al. disclosed that the size of particles and the size ratio of guest to host are important... 

The mesh expansion ratio, i.e. the size ratio of neighboring cells, should be kept small. Particularly in regions of large gradients, mesh size discontinuities should be avoided. 

Chromatographic use of monolithic silica columns has been attracting considerable attention because they can potentially provide higher overall performance than particle-packed columns based on the variable external porosity and through-pore size/skeleton size ratios. These subjects have been recently reviewed with particular interests in fundamental properties, applications, or chemical modifications (Tanaka et al., 2001 Siouffi, 2003 Cabrera, 2004 Eeltink et al., 2004 Rieux et al., 2005). Commercially available monolithic silica columns at this time include conventional size columns (4.6 mm i.d., 1-10 cm), capillary columns (50-200 pm i.d., 15-30 cm), and preparative scale columns (25 mm i.d., 10 cm). 

Variations in fluid density on reaction can have significant effects on the size ratio, but the effects are secondary when compared to the variations in reaction order. For positive values of the expansion parameter SA, the volume ratio is increased, for negative values of 8a, the volume ratio decreases. However, the fact that in practice CSTR s are used only for liquid phase reactions makes this point academic. 

As Levenspiel points out, the optimum size ratio is generally dependent on the form of the reaction rate expression and on the conversion task specified. For first-order kinetics (either irreversible or reversible with first-order kinetics in both directions) equal-sized reactors should be used. For orders above unity the smaller reactor should precede the larger for orders between zero and unity the larger reactor should precede the smaller. Szepe and Levenspiel (14) have presented charts showing the optimum size ratio for a cascade of two reactors as a function of the conversion level for various reaction orders. Their results indicate that the minimum in the total volume requirement is an extremely shallow one. For example, for a simple... 

The prototype FeCr sigma phase is of particular interest because the free atoms have very nearly the same size (ratio = 1.01), but they condense into a rather intricate structure. In the pure metals, the diameter of Cr is 2.50 A, while that of Fe is 2.48 A. (a difference of less than one percent), and both are bcc. Therefore, the existence of the sigma phase is determined by spd-hybridization of the electron orbitals. It is sometimes called a size-effect phase, but this is not really descriptive. 

When the atomic size ratio is near 1.2 some dense (i.e., close-packed) structures become possible in which tetrahedral sub-groups of one kind of atom share their vertices, sides or faces to from a network. This network contains holes into which the other kind of atoms are put. These are known as Laves phases. They have three kinds of symmetry cubic (related to diamond), hexagonal (related to wurtzite), and orthorhombic (a mixture of the other two). The prototype compounds are MgCu2, MgZn2, and MgNi2, respectively. Only the simplest cubic one will be discussed further here. See Laves (1956) or Raynor (1949) for more details. 

It may be expected that many intermetallic compounds will behave like metals during plastic deformation. However, some that contain covalent bonds will behave differently. In these, the size ratio tends to be 1.2 or greater. For NiAl the size ratio is 2.86/2.49 = 1.149. This may be compared with TIC (2.89/1.54 = 1.88), or TlB2 (2.89/1.72 = 1.68). The latter are clearly covalently bonded. 

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