Kinetic free volume fraction

Table 5 lists the composition, PP CE molar ratio, density, and segregation level determined by SAXS and the Tg determined by DSC of a PTE series where the ratio PP CE was systematically varied over extreme ranges, and DRS and TSDC were employed to investigate the molecular dynamics and to draw conclusions on the microphase separation. Figure 9 shows representative e" f) plots in the region of the a relaxation and Table 6 displays results obtained by analyzing the DRS data for some of the samples listed in Table 5. The kinetic free volume fraction at Tg, fg, was calculated by [57]...

Table 6. Results of DRS measurements for the a relaxation relaxation strength, shape parameter, (both at 263K), VTF parameters B and T, dielectric glass transition temperature, Tg dieb 4 kinetic free volume fraction at for selected samples of Table 5 [55]...

Privalko V P (1999) Glass transition in polymers dependence of the Kohlrausch stretching exponent of kinetic free volume fraction, J Non-Cryst Solids 255 259-263. Tant M R, Mauritz K A and Wilkes G L (Eds.) (1997) lonomers. Chapman and Hall, London. 

It is known that glassy polymer membranes can have a considerable size-sieving character, reflected mainly in the diffusive term of the transport equation. Many studies have therefore attempted to correlate the diffusion coefficient and the membrane permeability with the size of the penetrant molecules, for instance expressed in terms of the kinetic diameter, Lennard-Jones diameter or critical volume [40]. Since the transport takes place through the available free volume in the material, a correlation between the free volume fraction and transport properties should also exist. Through the years, authors have proposed different equations to correlate transport and FFV, starting with the historical model of Cohen and Turnbull for self diffusion [41], later adapted by Fujita for polymer systans [42]. Park and Paul adopted a somewhat simpler form of this equation to correlate the permeability coefficient with fractional free volume [43] ... 

Recent theoretical studies on the molecular structure of organic/C02 EFL mixtures show that the fraction of the total volume occupied by the van der Waals volumes of the mixture constituents decreases monotonically with addition of CO2 to conditions that correspond to 60% of those of the original liquids. Therefore, as mentioned earlier, significant free volume is present in these mixtures [17]. Large free volumes result in significantly decreased viscosities as compared to those of common liquids. For example, experimental (Figure 9.1) [8] and theoretical studies [17] on methanol/C02 mixtures show an approximate monotonic drop in viscosity with addition of CO2. For example, when 60 mol% CO2 is added to methanol, the viscosity drops by 67%. The low viscosities are expected to be favorable for fast kinetics. 

Equation (7.1.16) is asymptotically (cto — oo) exact. It shows that the accumulation kinetics is defined by (i) a fraction of AB pairs, 1 — u>, created at relative distances r > r0, (ii) recombination of defects created inside the recombination volume of another-kind defects. The co-factor (1 - <5a - <5b ) in equation (7.1.16) gives just a fraction of free folume available for new defect creation. Two quantities 5a and <5b characterizing, in their turn, the whole volume fraction forbidden for creation of another kind defects are defined entirely by quite specific many-point densities pmfl and po,m > he., by the relative distribution of similar defects only (see equation (7.1.17)). 

Equation (3) is the most widely used in analyzing experimental curves, since its form is intuitively clear the rate of the defect accumulation is determined by the fraction of free volume of the crystal not occupied by previously created defects, without taking account of the overlap of the annihilation volumes of similar defects. Evidently it is applicable only in the initial stage of accumulation kinetics at relatively low concentrations of defects, nvo superposition approximation corresponds to the first two terms of expansion (2) in powers of nvo-... 

After losing their kinetic energy the penetrated positrons may either directly annihilate with surrounding electrons into two gamma rays, or combine with an electron to form a Ps atom. Although both positrons and Ps are known to localize within the free volumes, a certain fraction of them may diffuse back to the surface and escape to the vacuum. The probability of positrons and Ps annihilating in the polymer depends on their diffusion coefficients. 

Kmilarly, Sung et al. recently reported the use of azo chromophoric labels as a molecular probe of physical aging in amorphous polymers . By measuring the kinetics of trans cis photoisomerization of azo duomophores covalently bonded to amorphous polyurethanes, they defined a parameter a which corresponds to the fraction of the free volume above a critical size at a given temperature and time of aging. 

Figure 12-15 A molecular interpretation of deviations from ideal behavior, (a) A sample of gas at a low temperature. Each sphere represents a molecule. Because of their low kinetic energies, attractive forces between molecules can now cause a few molecules to stick together. (b) A sample of gas under high pressure. The molecules are quite close together. The free volume is now a much smaller fraction of the total volume.

The kinetics of spiropyran and azobenzene photoisomerization deviate from first order when these dyes are entrapped in a solid matrix below Tg.24-34 This behavior has been attributed to the presence of a distribution of free volume within the matrix, as shown in Table 3.11 .35 When the probe is located in sites of free volume Vf greater than the critical volume for isomerization Vfc, the reaction proceeds at the same rate as in solution. For sites of Vf < Vfc, the reaction is retarded, since it becomes controlled by the matrix molecular motions. At low temperature, the local molecular motions are frozen and fluctuations of local free volume become increasingly small as the temperature decreases. Consequently the fraction of sites where Vf < Vk increases. 

The radicals desorbed from the surface should diffuse along the pores to the outer surface of catalyst grains and further through the catalyst bed with high probability to be trapped and terminated by the solid material, for instance the surface of the support. Hence only a small fraction of the radicals which left the surface eventually comes to the reactor free volume. Besides the variations of the relative sizes of reactors, kinetic evidence of the heterogeneous -homogeneous nature of hydrocarbons oxidation was provided by also varying the ratio of an inert material to the reactor free volume. 

Robertson et al. [1984] developed a stochastic model for predicting the kinetics of physical aging of polymer glasses. The equilibrium volume at a given temperature, the hole fraction, and the fluctuations in free volume were derived from the S-S cell-hole theory. The rate of volume changes was assumed to be related to the local free volume content thus, it varied from one region to the next according to a probability function. The model predictions compared favorably with the results from Kovacs laboratory. Its evolution and recent advances are discussed by Simha and Robertson in Chapter 4. 

This analysis leads us to the conclusion that ionic surfactants in salt-free solutions undergo kinetically limited adsorption. Indeed, dynamic surface tension curves of such solutions do not exhibit the diffusive asymptotic time dependence of non-ionic surfactants, depicted in Fig. 1. The scheme of Section 2, focusing on the diffusive transport inside the solution, is no longer valid. Instead, the diffusive relaxation in the bulk solution is practically immediate and we should concentrate on the interfacial kinetics, Eq. (21). In this case the subsurface volume fraction, t, obeys the Boltzmann distribution, not the Davies adsorption isotherm (15), and the electric potential is given by the Poisson-Boltzmann theory. By these observations Eq. (21) can be expressed as a function of the surface... 

The basic assumptions of the kinetic-molecular theory give us insight into why real gases deviate from ideal behavior. The molecules of an ideal gas are assumed to occupy no space and have no attractions for one another. Real molecules, however, do have finite volumes, and they do attract one another. As shown in Figure 10.25 , the free, unoccupied space in which molecules can move is somewhat less than the container volume. At relatively low pressures the volume of the gas molecules is negligible, compared with the container volume. Thus, the free volume available to the molecules is essentially the entire volume of the container. As the pressure increases, however, the free space in which the molecules can move becomes a smaller fraction of the container volume. Under these conditions, therefore, gas volumes tend to be slightly greater than those predicted by the ideal-gas equation. 

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