Packing fraction calculation

Eq. (64), together with Eq. (58), indicates that does not depend on the size but instead the size distribution breadth expressed by Og. Experimental results show that the maximum packing fraction calculated from Eq. (58) is relatively higher than that determined with the rheological measurement [31]. The discrepancy is believed to be a result of the constant value of a, which should be a function of the particle size and particle packing... 

Figure A3.6.13. Density dependence of die photolytic cage effect of iodine in compressed liquid n-pentane (circles), n-hexane (triangles), and n-heptane (squares) [38], The solid curves represent calculations using the diffusion model [37], the dotted and dashed curves are from static caging models using Camahan-Starling packing fractions and calculated radial distribution fiinctions, respectively [38],...

In many tabulations of nuclear properties, such as that in Appendix B, the quantity that is tabulated is the mass excess or mass defect rather than the mass. The mass excess, A, is defined as M(A, Z) — A, usually given in units of the energy equivalent of mass. Since in most, if not all calculations, the number of nucleons will remain constant, the use of mass excesses in the calculations will introduce an arithmetic simplification. Another term that is sometimes used is the mass excess per nucleon or the packing fraction [=(M — A)/A]. 

The hard sphere diameters were then used to calculate the theoretical Enskog coefficients at each density and temperature. The results are shown in Figure 3 as plots of the ratio of the experimental to calculated coefficients vs. the packing fraction, along with the molecular dynamics results (24) for comparison. The agreement between the calculated ratios and the molecular dynamics results is excellent at the intermediate densities, especially for those ratios calculated with diameters determined from PVT data. Discrepancies at the intermediate densities can be easily accounted for by errors in measured diffusion coefficients and calculated diameters. The corrected Enskog theory of hard spheres gives an accurate description of the self-diffusion in dense supercritical ethylene. 

Figure 3. The ratio D/D as a function of packing fraction for supercritical ethylene. A indicates ratios calculated using hard sphere diameters determined from diffusion data. 0 indicates ratios calculated using hard sphere diameters determined from compressibility data. The solid lines are the molecular dynamics results, extrapolated to infinite systems, of Alder, Gass and Wainwright (Ref. 24).

Tc. The two power-law exponents are not independent but depend on a single parameter, the so-called critical exponent X, which is specific for a given interaction potential (e.g., hard spheres). Actually, the interaction potential enters the MCT equations only indirectly via the structure factor S(q), which fixes the nonlinear coupling in the generalized oscillator equation. It is important to note that the MCT exponents are not universal in contrast to those of second-order phase transitions. In the case of hard spheres, for example, S(q) can be calculated via the Percus-Yevick approximation [26], and the full time and -dependence of < >(q. f) were obtained. As an example, Fig. 10 shows the susceptibility spectra of the hard-sphere system at a particular q. Note that temperature cannot be defined in the hard-sphere system instead, the packing fraction cp is used as a parameter. Above the critical packing fraction 0), which corresponds to T < Tc in systems where T exists, the a-process is absent (frozen) and only the fast dynamics is present. At cp < tpc the a-peak and the concomitant susceptibility minimum shift to lower frequencies with increasing cp, so that the closer cp is to the critical value fast dynamics can be identified (curve c in Fig. 10). 

The concept of the free volume of disperse systems can also be correlated with the change in the structure of the composite of the type solid particles — liquid — gas during its compaction. In that case the value of the maximum packing fraction of filler (p in Eq. (80b) remains valid also for systems containing air inclusions, and instead of the value of the volume fraction of filler, characteristic for a solid particles — liquid dispersion-system solid particles — liquid — gas should be substituted. This value can be calculated as follows the ratio of concentrations Cs x g/Cs, to the first approximation can be substituted by the ratio of the densities of uncompacted and compacted composites, i.e. by parameter Kp. Then Eq. (80b) in view of Eq. (88), for uncompacted composites acquires the form ... 

For reliable application of the free volume concept of disperse systems one must have dependable methods of determination of the maximum packing fraction of the filler tpmax. Unfortunately, the possibility of a reliable theoretical calculation of its value, even for narrow filler fractions, seems to be problematic since there are practically no methods available for calculations for filler particles of arbitrary shape. The most reliable data are those obtained by computer simulation of the maximum packing fraction for spherical particles which give the value associated with possible particle aggregation, so that they are probable for fractions of small particle size. Deviations of particle shape is nearly cubic. At present the most reliable method of determination of [Pg.142]

At the optimum point corresponding to the maximum packing fraction, e1 = e11, since the porosity value at this point does not depend on whether fine or coarse particles form the mixture skeleton. The Equation (90) permit calculation of the porosity values for a mixture of fractions of any composition. A similar procedure can be used for calculating the porosity of mixtures of three or more fractions. A laboratory check of these Equations (90) confirmed good agreement between the calculation forecast and the experimental determination of porosity coefficients and wide variety of combinations of narrow filler fractions. 

F.q. (16-23). Subsequently, it became clear that a theoretical form for S (q)S(q) given by Percus and Yevick (1958) and Pcrcus (1962) was more convenient, and probably as accurate as the experiment for the resistivity calculation. This approach was used by Ashcroft and Lckner (1966) for an extensive study of the resistivity of all the simple liquid metals. The form due to Pcrcus and Yevick depends only upon two parameters, a hard-sphere diameter and a packing fraction these lead to a simple form in terms of elementary functions Ashcroft and Lckner discuss the choice of parameters. This form is presumably just as appropriate for other elemental liquids. 

A mixture of 231 g. of the above barium salt and 300 ml. of sulfuric acid (sp. gr. 1.67) is distilled with superheated steam (inlet steam temperature at 200-220° and exit steam at 150-160°). The oily layer of the distillate is separated, washed with water, dried, and distilled twice through a 1-m. packed fractionating column to give 40 g. of o-xylene and 149 g. of l-bromo-2,3-dimethylbenzene, b.p. 210.5-212.5°. This is a 42% yield calculated on the barium salt. 

The KBIs" were calculated at constant packing fraction... 

Fig. 16 Calculated star phase diagram (inverse functionality f again packing fraction t)) [178],...

Predictions of the (idealized) MCT equations for the potential part of the equilibrium, time-dependent shear modulus ( ) of hard spheres for various packing fractions 0 are shown in Fig. 3 and calculated from the limit of (lid) for vanishing shear rate ... 

Fig, 6 The reduced storage (diamonds and solid lines ) and loss (squares and broken lines) modulus for a fluid state at effective packing fraction fef = 0.540 from [32]. The vertical bars mark the minimal rescaled frequency above which the influence of crystallisation can be neglected. Parameters in the MCT calculation given as blue lines = -0.01, = (). 5, and t) = 0.3 kgT/(Dijlln ) -. 

Fig. 21 Steady state incoherent intermediate scattering functions d> (r) as functions of accumulated strain yt for various shear rates y the data were obtained in a col loidal hard sphere dispersion at packing fraction

Table 1 Packing fraction 0eff and parameters V(,r,Yc, and ri ofthe fit using the schematic Fj2-model for the measurements shown in Fig. 23 from [33], The parameters , short time diffusion coefficient Ds/Dty, and rescaling factor Cy from the microscopic linear response calculation using...

Fig. 2.8 The pair correlation function g r) for a fluid composed of hard spheres at a packing fraction of = 0.49 calculated as a function of distance, r, using the Ornstein-Zernike equation with the direct correlation function given by equations (2.6.6) and (2.6.10). The data shown as ( ) are from a Monte Carlo calculation.

Estimate the packing fraction for a hard-sphere liquid with a density of 21.25 atoms nm and a hard-sphere diameter of 350 pm. Use this result to calculate the Percus-Yevick product for the system at 85 K using the Carnahan-Starling equation of state (equation (2.9.11)). 

In order to estimate the hard-sphere contribution the packing fraction is calculated first ... 

The maximum packing fraction cp. can be easily calculated for monodisperse rigid spheres. For an hexagonal packing

random packing (Pp = 0.64. The maximum packing fraction increases with polydisperse suspensions for example, for a bimodal particle size distribution (with a ratio of 10 1), 0.8. 

Gupta and Seshadri (1986) discussed calculations of the maximum loading level of filled liquids. They found that a geometric method for calculating the maximum packing fraction works well for polydisperse spherical suspensions. 

Figure 2.1.6 Trial plot of the resulting crystal packing as calculated from the published fractional coordinates and space group based on a 0.3-0.7 mm long crystal of a-cinnamic acid 1 after tail irradiation to give a claimed [25] 100% conversion to a-truxillic acid 2.

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