Close packed spheres, volume

A rough measure for the overlap concentration 4>ov is that volume fraction of polymer at which close-packed spheres with radius rg just touch. Then 4> =0.74 rl /(4Trr /3). Taking r — A((J> - 0) ... 

Close-packed spheres occupy 74.04% of a total volume, hence the hard-sphere radius of I" in these 2 1 salts in 2.03 A. Correction for the electrostatic attraction alone would give a monovalent iodide radius of about 2.24, an opposite repulsion-correction for the different co-ordination number would reduce this to about 2.10 A for the monovalent sodium-chloride type (see Appendix). Such values are consistent with our earlier estimates, but incompatible with the electron-density minimum value (4) of 1.94 A. 

An important quantity, which characterizes a macroemulsion, is the volume fraction of the disperse phase 4>a (inner phase volume fraction). Intuitively one would assume that the volume fraction should be significantly below 50%. In reality much higher volume fractions are reached. If the inner phase consists of spherical drops all of the same size, then the maximal volume fraction is that of closed packed spheres (fa = 0.74). It is possible to prepare macroemulsions with even higher volume fractions volume fractions of more than 99% have been achieved. Such emulsions are also called high internal phase emulsions (HIPE). Two effects can occur. First, the droplet size distribution is usually inhomogeneous, so that small drops fill the free volume between large drops (see Fig. 12.9). Second, the drops can deform, so that in the end only a thin film of the continuous phase remains between neighboring droplets. 

Hydraulic number for close-packed spheres. We consider a volume V which contains n spheres. Here, n is assumed to be a large number so that it is practically continuous. The volume filled by particles is V) = n 4/3 nR3. The total surface area of all particles is A = n 4ttR2. The free ( void ) volume is Vv = 0.26 V = 0.26 n 4/3 ttR3/0.74. Thus the hydraulic number is... 

The second term in Eq. (1.36) accounts for the non-isomorphic change of cell shape in the process of its transformation into a polyhedron. The value of this term does not exceed 2.8% of the main equation term and equals zero when volume fraction in close-packed spheres). 

The surface porosity is equal to the ratio of the pore area to membrane area multiplied by the number of pores. In most cases volume flux through ceramic membranes can be best described by the Kozeny-Carman relationship, which corresponds to a system of close packed spheres (see Figure 6.8a) ... 

Small volume of dispersed phase—this reduces the frequency of collisions and aggregation. Higher volumes are possible (for close-packed spheres the dispersed-phase volume fraction would be 0.74), but in practice the fraction can even be higher. 

The manufacture of molded articles is usually carried out with mixtures of aluminum oxides with different particle size distributions. This is particularly important when pore-free end-products are required, because this enables a higher volume concentration of aluminum oxide to be obtained than the 74% of ideally cubic close packed spheres by filling the gaps with smaller particles. The particle size distributions used in practice are usually determined using empirically determined approximate formulae (Andreasen or Fuller distribution curves) which take into account the morphology of the individual particles. 

Note that the volume density of closed pack spheres is about rfcpK, 0.74. The choice of t] = 0.45, which is about 6/10 of the maximum density, was chosen for convenience. In fact even at these densities converging of the Percus-Yevick equation is quite slow (see also Appendix E). 

Figure 2.40, presented earlier, is an example of concentration of whole yeast cells. Though the concentration of cells at the end of the run was only 10%, the data imply that cells may be concentrated up to the close packed sphere density of 75 volume percent. Indeed, yeast, and E. colt concentrations of 60% and 37% respectively have been harvested. 

Further increases in pressure will increase the solute concentration at the surface of the membrane (Cs) to a limiting concentration. Proteins begin to form a semisolid gel on the membrane surface with a gel concentration (Cg) between 20 and 50 wt %. Colloidal suspensions form a densely packed layer of close-packed spheres usually between 70 and 80 volume %. Under these conditions (see Figure 3.26), the membrane is said to be "gel-polarized", and further increases in pressure will not increase Cs. Therefore, once the membrane is "gel-polarized, the retention should be independent of pressure (see Equation 7). In Figure 3.23, the rejection of Dextran-80 is beginning to level out at the higher pressures as the membrane becomes gel-polarized. 

Consider the model picture showing closely packed spheres with diameters of d shown at the right bottom in Figure 14.11. The lined rhombus (a layer) contains four spheres that occupy a volume of (2d)(v d)(v A )- The second layer spheres occupy the space at the center of a triangle of spheres. This leads to a space volume of S j /7. for one sphere. The actual volume of a sphere is iTd /6. Thus, S2 ttI6 = 0.74) of the space is filled with close-packing spheres. 74%. 

P20.9 As demonstrated in Justijkation 20.3 of the text, close-packed spheres fill 0.7404 of the total volume of the crystal. Therefore 1 cm- of clase-packed carbon atoms would contain... 

What are the virtues of these emerging photoelectrode materials The first is related to their enormous surface area. Consider that the 3D structure is built up of close-packed spheres of radius, r. Then ignoring the void space, the specific area. As (area/volume) is given by 3/r [205]. For r = 10 nm, Ag is on the order of 10 cm , and for a 1 cm film of 1 pm thickness, this value corresponds to an internal sxtrface area of 100 cm (i.e. a surface roughness factor of 100). Clearly, this becomes important if we want the electrolyte redox species to be adsorbed on the electrode surface (see following). Alternatively, a large amount of sensitization dye can be adsorbed onto the support semiconductor although this dye sensitization approach is not considered... 

The phase transition appeared suddenly, as the close-packed sphere stracture was being diluted by shrinking each sphere diameter, when the spheres occupied 49% of the volume, i.e. at 0.49 packing fraction. The pressure was calculated by working out the impacts of the spheres on one square meter of the cell in one second. As the volume of the spheres was reduced during the computation, the... 

It is noted that it is easy to compress particles to a state of approximating cubic packing where the body has about 50% by volume of pores. However, at 100 tons in.- the porosity drops to 0.204, which is even less than that of perfect close, which is 0.255. Under this pressure some of the particles may have become flattened together in which case the formulas are not applicable. 

This factor is near the value, 1.159, that is geometrically required for close-packed spheres of arbitrary but comparable sizes. These intrinsic volumes for the monatomic cations dealt with in this chapter and anions with radii <0.2 nm are shown as in Table 2.8. 

Figure 5 Phase diagram of diblock copolymers with equal segmental lengths and segmental volumes of both block components. % Flory-Huggins-Staverman interaction parameter, N degree of polymerization, ( ) volume fraction, D disordered phase, CPS close packed spheres, BCC body-centered cubic spheres, H hexagonally packed cylinders, G gyroid, L lamellae. (From Ref. 110, Copyright 1996 American Chemical Society.)...

Discontinuous 3D cubic structures commonly observed in BCP SA are spherical morphologies as shown in Fig. 2. In this case, the minority block of the BCP forms spheres and the spatial arrangement of the spheres varies from body-centered cubic (Im3 m, Q229 space group) to closed packed spheres as the volume fraction of the minority block and/or the product iflV decrease [18]. Furthermore, the so-called A15 phase with Pm3 n symmetry was observed in amphiphilic di-BCPs containing a linear block and a dendron block [17]. 

The relationship between the ionic radii and the ionic volumes vi leads directly to Vi = (4 r/3)(ri) and a set of such volumes, denoted as vm is reported in [17]. The ionic volumes obtained from the formula unit volumes in crystals [28], denoted as vj, is also shown in [ 17], with vj = (1.258 0.01 6)vm having been established with a correlation coefficient of 0.995 for 55 cations. The larger size of vj takes into account the void spaces between the (more or less spherical) ions. Another way to express this, based on the suggestion of Mukerjee [29] regarding such ions in solution, is to use the factor =1.213 multiplying the radius of univalent monatomic ions vi = (4jr/3)( ri). This factor is near the value, 1.159, that is geometrically required for close-packed spheres of arbitrary but comparable sizes. 

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