Free volume distribution curve

In paper [82] the dependence of a on free volume microvoid radius r was obtained, which has the sense of a free volume distribution curve. A similar curve is adduced in Figure 5.22, where the value of r was calculated according to the free volume microvoid volume (in assumption of its spherical shape), estimated according to the following relationship [80] ... 

In the case of an isotropic Xe distribution (large cavities), the slope d8/dn of the straight section of the d-curve is inversely proportional to the free volume of the cavities accessible to the xenon atoms. 

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. 

Positronium lifetime spectroscopy is particularly well suited for stud)hng defects in crystals and structural fluctuations in amorphous materials and can give an estimate of free volumes in condensed matter [116]. It is a useful technique to estimate the free volume of polymeric membranes [117]. In a study on silica gels, the decay lifetime has been found (Fig. 4.16) to be proportional to the pore diameters (measured by N2 adsorption) between 30 and 100 A [118]. Information on pore size distribution and surface area may also be obtained by means of calibration curves. 

The excellent combination of efficiency, reversibility, and selectivity in extraction by ABC extractants provides for saving on capital cost. In many cases, stoichiometric extraction results in about 1 mole of extracted acid per mole of amine in the loaded extractant (organic-to-aqueous volume ratio of about 3 in treating an effluent comprising 30% free sulfuric acid). Well-adjusted distribution curves achieve nearly complete extraction and a quite concentrated back-extract by three or four theoretical stages in extraction and a similar number of theoretical stages in back-extraction. Phase mixing and phase separation are satisfactory too. 

Figure 2.14 gives a schematic representation of the distribution function of the free volume fraction F(f) vs. / for blends of various degrees of compatibility. When the components are mixed to a heterogeneity limit of 10-20 A, the curve of F f) will approach the curve for a molecularly mixed system. 

Figure 2.14. Schematic representation of the distribution function of free volume fraction F f) for various blend systems. Curves 1 and 2 show the shapes of the functions of components 1 and 2, respectively. Curves S., P.M., M.H., and P.F.M. indicate two separated phases, partially miscible, microheterogeneous, and perfectly miscible systems, respectively. The volume fraction of each component is 50% for all these blends. The broken line shows the F(f) of the components in the perfectly miscible system. (Manabe et al, 1971.)...

One of the most important conclusions associated with the Takayanagi model is that the distribution function of the free volume fraction Fj f) is evaluated by the following equation based on the specific volume-temperature curve ... 

One can discern for PTMSN the two broad minima in the curve AHjy that correspond to the values of Vc of 426cmVmol (u-Ct) and 754cmVmol (n-Cia). The presence of the two minima is rather unexpected because in most cases only one minimum was observed in the curves AHj[Vc). The only exception so far was amorphous Teflon AF2400, where also two minima, though flat and broad, could be noted [19], Here we have apparently another manifestation of bimodal size distribution of free volume in glassy polymers that attracts now a keen interest [33], The error bars shown in Figure 3.6 seem to support the assumption of bimodal size distribution in this polymer. 

If free volume is a distribution rather than being single valued then the dynamic, time-dependent, behaviour would conform to a single relaxation process and have an ideal form. However, if there is a distribution of sites at which relaxation can occur with different values of the free volume, then there will be a distribution of relaxation times. This is what is observed experimentally and produces the form of the curves shown in Figure 7.13 and 7.14. 

With respect to the shape of the pore size distribution curves, figure 3 shows two types of pores the pores around 30-45A in diameter which are the normal size of pores when no additive is used, and the pores around 120 A in diameter, which are the result of the change in structure promoted by the interaction of the NH4 and CO ions with the aluminium in the mixed oxide. The higher diameter pores found in the pure titanium sample may be the result of the generation of CO2 gas fix>m the decomposition, during caldnation, of the free ionic 3 found in this sample, which promotes the formation of a few large pores of more than 200A in diameter and wtuch contribute little to the total surface area (42 mVg) and the cumulative pore volume (0,102 cm /g). 

The shape of the kinetic decay curves can be used to determine the distribution of free volume in the polymer matrix. There will be a small number of large holes and a larger number of small holes, giving a typical distribution curve as shown in Figure 13.13. 

Positron annihilation rate distributions for a typical immiscible polymer blend PVC/PS of (80/20) composition and pure PVC and PS are shown in Figure 27.4a, and the radius and free volume cell size distribution PDFs are shown in Figure 27.4b. These PDFs are obtained from CONTIN-PALS2 analysis of the measured lifetime spectra. It is clear from the distribution curves that the width of the curve for PVC is wider compared to PS, while the blend has a distribution in between the constituent polymer distribution, thus demonstrating the ability of the computer routine CONTIN-PALS2 (from the present authors studies). 

---