Fractional free volume calculation

Later Kanig took into account the temperature dependence of the fractional free-volume, calculated according to his equations. Below Tg, as temperature decreases, sPfl increases as a result of a decrease in expansion volume at frozen hole volume. Above Tg, pn increases due to the sharp rise in hole volume. At Te the value ip i is at its minimum. From the condition d [Pg.75]

Figure 2.24 Correlation of the oxygen permeability coefficient for a family of related polysulfones with inverse fractional free volume (calculated using the Bondi method) [33]. Reprinted with permission from C.L. Aitken, W.J. Koros and D.R. Paul, Effect of Structural Symmetry on Gas Transport Properties of Polysulfones, Macromolecules 25, 3424. Copyright 1992, American Chemical Society...

The relative fractional free volume (/), calculated variously as the product Ij, vh)h, or C vh)h, where C is an empirical constant, did not exhibit regular behavior when plotted as a function of the blend composition, behaving essentially similar to I3. 

FIG. 12-8. Diffusion coefficient of n-hexadccane at 2S°C through random styrene-butadiene copolymers plotted against reciprocal of fractional free volume calculated from equation 8. Open circles, uncross-linked samples black circles, cross-linked. Lines drawn with slope of — 1/2.3Q3 as specified by equation 7. (Rhee and Ferry. ) Reproduced, by permission, from the Journal of Applied Polymer Science. 

Molar masses, glass transition temperatures, and fractional free volume (calculated from densities and Paul and Park s series of increments) of the polymers are summarized in Tables I and II (6-8). Fractional free volume (Vf) calculated by the method of Park and Paul (P) differs for different gases, depending on the diameters of the gas molecules. Here, the values for O2 and N2 are given as examples. As expected, the values for Vf increase with decreasing kinetic diameter of the gas molecules. 

The fractional free volume,/, calculated as a function of aging time using the Struik model can then be used in the following correlation to give gas permeability ... 

Also, in cases where the dimensions of a regular particle vary throughout a bed of such particles or are not known, but where the fractional free volume and specific surface can be measured or calculated, the shape factor can be calculated and the equivalent diameter of the regular particle determined from Figure 2. 

Values of the fractional free volumes at the glass temperature were calculated for PBzEs and PAMAMs from fg = /ref + (Tg — 7 ref)/2.303C1° [53], and the results obtained are listed in Table 14.2 and shown (for PAMAMs) as C in Figure 14.12. It can be seen from these data that fg appears to be independent of... 

The idea that the fractional free-volume at glass temperature as found experimentally depends on the mode of molecular motions was put forward in 196746 47 as a result of calculating/g from data obtained from isothermal volume relaxation for some polymer systems. By estimating average relaxation time at different temperatures it was possible to find the fractional free-volume/g at Te according to WLF theory. If we accept the validity of the theory as regards the universal dependence of the reduction factor aT on (T - Tg), then on the basis of data on Aa and theoretical values aT calculated from universal values of the coefficients C and C, it is possible to make an estimate of/g. In this case the value found corresponds to the universal one. If, however, we use the experimental values aT, the fractional free-... 

In 60 the fractional free-volume was calculated according to the relationship ... 

In some reports83,84) the change in the fractional free-volume was calculated at temperatures above Tg for epoxy resin filled with polystyrene particles on the basis of the experimental value of the reduction factor aT and the universal value fg according to the equation... 

It was found that the total fraction of the free-volume in the system increases with increasing concentration of the polymeric filler. The temperature dependence of fg for the epoxy matrix was calculated on the supposition that free-volume is an additive value of the constituent components and using the temperature dependence of the fractional free-volume of polystyrene. It was found that with increasing filler concentration the fractional free-volume becomes greater than for pure epoxy resin. Since the fraction of the free-volume increases with increasing total surface area of the filler, it may be supposed that this effect is associated with the surface layers of polymer. It was found that the rate of free-volume expansion in a filled system is higher than in an unfilled one, which means that the expansivity of the free-volume... 

The dependence of f% on filler content was carefully investigated for filled polystyrene87 and the values for/g were calculated in different ways, using various values for the occupied volume v0. The results of these calculations have shown that the values for /g do not coincide when calculated in different ways. Nor are these values constant for the different amounts of filler incorporated. This shows once more that the glass temperature is not a temperature of constant fractional free-volume. 

We calculated the fractional free-volumes/gl and/g2 according to the method developed by Covacs from the curves of the isothermal contraction82. The values obtained are not really the fractional free-volumes of the components at the corresponding temperatures, since in the calculation from contraction curves it is impossible to exdude the contributions of both components to the total free-volume. 

From experimental data on viscoelastic properties the fractional free-volume was calculated according to the WLF equation for pure hardened resin, filled with different amounts of polymeric filler obtained from the same hardened resin. By using special methods for preparation of filled specimens, it was possible to obtain... 

A modified version of the free-volume theory is used to calculate the viscoelastic scaling factor or the Newtonian viscosity reduction where the fractional free volumes of pure polymer and polymer-SCF mixtures are determined from thermodynamic data and equation-of-state models. The significance of the combined EOS and free-volume theory is that the viscoelastic scaling factor can be predicted accurately without requiring any mixture rheological data. 

Table 2.2 Calculated fractional free volume for representative membrane materials at ambient temperatures (Bondi method)...

The other parameter, C2, has been used for the fg determination of the crosslinked DGEBA network. Miyamoto and Shibayama [ 118,124] calculated the fractional free-volume of the DGEBA network using the C2 parameter based on the ionic conduction. The DC conduction measurement is the method investigating the same kind of moving unit, the ionic charge carrier, both in the un-crosslinked oligomer and in the crosslinked network. Table 12 summarizes the fg values that are obtained from the DC conduction measurement for three... 

In most cases, the fractional free volume amounts to 0.025 — 0.030. a value which is considered as a normal one for amorphous polymers. For the photochromic polyesters inadminissibly high fg values were however found (corrected fg 0.13 to 0.197 instead of 0.27 to 0.45 as indicated erroneously). On the basis of the calculation of Sevens49 fg is 0.048 which is much more reasonable when taking into account the bulkiness of the bisphotochrome and the disturbing effects on its direct vicinity. 

Much of the work stems from Simha-Somcynsky (S-S) [1969] hole theory, developed originally to describe polymers in the liquid state. They introduced the free volume by using the formalism of vacant cells or holes in a lattice and developed an equation of state that could be used to calculate the fraction of sites occupied and hence the fractional free volume. As discussed in Chapter 6, the concept has been developed further by Simha and his co-workers. 

Fractional free volume/was then calculated according to the previous definition [Eq. (10.15)]. Its variation with temperature is shown in Figure 10.10, together with the theoretical free-volume fraction fi values of/(obtained assuming spherical holes, plotted as circles in Figure 10.10) are systematically lower than h for all the structures. Furthermore, the expansion coefficients of/are higher than the corresponding values deduced from the theory. 

FIGURE 11.9 Specific total, V, free Vf = hV, and occupied, Vqcc = (1 — h)V, volume of PC as a function of T at ambient P. h is the hole fraction calculated from V using the S-S equation of state. Open symbols, experimental data dots, S-S equation of state fits to the volume in the temperature range T>Tg-, stars, free volume calculated from Vf=N v,), where (Vf,) is the mean hole volume from PALS and is the specific hole density, assumed to be constant at N f = 0.67 X 10 g (corresponding to 0.81 nm at 300 K). (Adapted from Dlubek et al., [2007d].)... 

Figure 11.15b shows a comparison of various fractional free volumes in ER6. The total free volume is defined as the difference between the total, V, and the van der Waals specific volume, Vvdw, which was calculated from the group contributions given by van Krevelen [1993]. In the temperature range from Tg (PVT) = 325 to 470 K,/f varies between 0.316 and 0.373. Its value for a hexagonal close-packed (hep) structure is 0.26. The Bondi free volume [Bondi, 1968] assumes a occupied volume of 1.3Vvdw-/Bondi varies between 0.111 and 0.185. It is distinctly larger than the hole free volume determined from the S-S equation of state of PVT (and from PALS) experiments, h = Vf IV, which increases from 0.0576 to 0.134. 

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