Fractional free volume Fractionation

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... 

Void fraction of packing under operating conditions = void fraction of dry packing minus the total hold-up (not the free volume of dry-packing)... 

The glass transition temperature of a dilute system, according to the free volume changes, is determined by the diluent volume fraction Vd, and changes of the thermal expansion coefficient, a, at Tg by using ... 

The free volume model seems to be more adequate to describe the plasticization behaviour of the systems of lower amine content. According to Eq. (5), the higher is the change of the expansion coefficient the lower is the influence of the diluent volume fraction. The three TGDDM-DDS mixtures cured with 20, 30 and 50 PHR of hardener were characterized l2) by changes of the expansion coefficient at the glass transition, respectively, of 0.63, 1.08 and 2.94x 10 3 °C l. The more dense and stiffer resin crosslinked with 50 PHR of DDS should be, in principle, the less... 

The sizes and concentration of the free-volume cells in a polyimide film can be measured by PALS. The positrons injected into polymeric material combine with electrons to form positroniums. The lifetime (nanoseconds) of the trapped positronium in the film is related to the free-volume radius (few angstroms) and the free-volume fraction in the polyimide can be calculated.136 This technique allows a calculation of the dielectric constant in good agreement with the experimental value.137 An interesting correlation was found between the lifetime of the positronium and the diffusion coefficient of gas in polyimide.138,139 High permeabilities are associated with high intensities and long lifetime for positron annihilation. 

We define a nucleation overpotential rjN EN E0 (Fig. 36) required to make the N0 oxidation nuclei appear. The nucleation overpotential is related to the degree of closure (compaction) of the polymeric entanglement ( ), expressed as the fraction of interchain free volume destroyed after polarization at a given potential Ec, compared with the amount of free volume present at Es. 

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]... 

An obvious refinement suggested by this second derivation would consist in ascribing different free volume fractions Vf to the two pure... 

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

Numerous models have been proposed to interpret pore diffusion through polymer networks. The most successful and most widely used model has been that of Yasuda and coworkers [191,192], This theory has its roots in the free volume theory of Cohen and Turnbull [193] for the diffusion of hard spheres in a liquid. According to Yasuda and coworkers, the diffusion coefficient is proportional to exp(-Vj/Vf), where Vs is the characteristic volume of the solute and Vf is the free volume within the gel. Since Vf is assumed to be linearly related to the volume fraction of solvent inside the gel, the following expression is derived ... 

The catalytic test of propane ODH reaction was performed in the 350-600°C range in a quartz fixed bed flow reactor with on line GC analysis. The free volume of the reactor after the catalyst bed was filled with quartz particles to minimize the homogeneous reactions. All the testing set was placed in a thermostat with heated lines to the gas chromatographs at about 100°C to prevent water condensation. The feed gas composition was C3H8/02/N2 = 20/10/70 vol.% at total gas flow 50 cm3 min-1. Catalyst fractions of 0.2-0.315 mm particle size and of 80 mg weight were loaded into the reactor. Before the reaction, the catalyst samples in the reactor were kept under airflow at 600°C for lh. 

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... 

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