Free volume fractional

Next we assume that the state designated by the subscript 2 in Eq. (4.61) corresponds to Tg, we designate the fraction free volume at Tg by fg. Likewise,... 

The time-temperature superpositioning principle was applied f to the maximum in dielectric loss factors measured on poly(vinyl acetate). Data collected at different temperatures were shifted to match at Tg = 28 C. The shift factors for the frequency (in hertz) at the maximum were found to obey the WLF equation in the following form log co + 6.9 = [ 19.6(T -28)]/[42 (T - 28)]. Estimate the fractional free volume at Tg and a. for the free volume from these data. Recalling from Chap. 3 that the loss factor for the mechanical properties occurs at cor = 1, estimate the relaxation time for poly(vinyl acetate) at 40 and 28.5 C. 

Polyisobutylene and similar copolymers appear to "pack" well (density of 0.917 g/cm ) (86) and have fractional free volumes of 0.026 (vs 0.071 for polydimethylsiloxane). The efficient packing in PIB is attributed to the unoccupied volume in the system being largely at the intermolecular interfaces, and thus a polymer chain surface phenomenon. The thicker cross section of PIB chains results in less surface area per carbon atom. 

Limiting flow rates are hsted in Table 23-16. The residence times of the combined fluids are figured for 50 atm (735 psi), 400°C (752°F), and a fraction free volume between particles of 0.4. In a 20-m (66-ft) depth, accordingly, the contact times range from 6.9 to 960 s in commercial units. In pilot units the packing depth is reduced to make the contact times about the same. 

These two nomographs provide a convenient means of estimating the equivalent diameter of almost any type of particle Figure 1 of regular particles from their dimensions, and Figure 2 of irregular particles from fractional free volume, specific surface, and shape. 

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. 

What is the equivalent diameter of crushed glass (t ) = 0.65) with a fractional free volume of 0.55 and a specific... 

If we, however, consider that viscosity is inversely related to the fractional free volume, which increases from a small value at the glass transition temperature Tg linearly with temperature above this figure, then it is possible to derive an equation. 

Now it has been argued that the viscosity is related to the fractional free volume by an expression of the form... 

According to free-volume interpretations, the rate of molecular motions is governed entirely by the available unoccupied space ( free volume ). Early studies of molecular liquids led to the Doolittle equation, relating the viscosity to the fractional free volume, / [23,24]... 

B0 Static magnetic field (flux density) (2) Fractional free-volume ... 

A gas phase reaction has a zero order rate equation in the concentration range of interest. Given the additional data following, find the space velocity, cuft of feed/(hr)(cuft of catalyst bed), needed for 95% conversion. C0 = 0.005 lbmol/cuft, inlet concentration k = 5 lbmol/(hr)(cuft of catalyst), specific rate D = 0.1 ft2/hr, diffusivity c = 0.40, fractional free volume... 

A packed bed of particles of diameter dp and fractional free volume c is modelled as a group of parallel capilaries with a perimeter... 

In attempts to better understand dendrimer intramolecular morphology, considerable attention was devoted to the fractional free volume near the glass temperature [40, 49, 50], Because all of the studies were performed within the WLF temperature range, the data were analyzed using the equation... 

Figure 14.12 Dependencies of glass temperature (Tg) (B), fractional free volume (fo/B) (A) and fractional free volume at glass temperature (fg/B) (C) on dendrimer molecular weight (i.e. generation) for the first seven generations of PAMAM dendrimers (according to ref. [40])...

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... 

Table 14.2 Values of WLF Cj0 and C2° Parameters, Fractional Free Volumes f0 and/ and Thermal Expansion Coefficient af for PAMAM, PPI and PBzE dendrimers ... 

Figure 14.13 Dependencies of the fractional free volume (f098a/B) on molecular weight for PPI (A), PAMAM (B) and PBzE (C) dendrimers. Data are from ref. [40], [50] and [49], respectively (see Table 14.2)...

Hougham et al. reported that, on average, the decrease in dielectric constant in a polyimide series that was attributable to the change in fractional free volume that accompanied fluorine substitution was greater than 50% Recently, these authors have further refined these estimates. ... 

The three broad categories of ways in which fluorine can affect the dielectric constant are by changing (1) the polarizability, (2) the hydrophobicity, and (3) the fractional free volume. 

Recently, the same series of six polyimides was studied by positron annihilation spectroscopy to determine die fractional free volume directly. In all three H/F analogue pairs, the increased free volume of the fluorinated polymer accounted for around 50% of the observed decrease in refractive index and dielectric constant. This result confims an astonishingly large free volume contribution predicted by our earlier estimates.Future work will investigate the generality of this result to other polymer systems. 

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