Average hole volume

This is the distribution function which was said to be the goal at the beginning of Eq. (5.17). From it, the average hole volume and radius will shortly be seen to be obtainable (see Fig. 5.22). 

Fig. 5.47. Plot of the free volume per ion against the average hole volume ( ) LiCI, (a) NaCI, ( ) KCI, (a) CsCI, (+) NaBr, (v) KBr, ( ) CsBr, ( ) Nal, (o) Kl.

FIGURE 10.5 Specific volume V versus the average hole volume u (in spherical approximation) in the PS fractions. 

It is apparent that only the average value corresponds to the universal value in WLF theory, and that the deviations from this value can be very considerable. In spite of this, we may believe that free-volume is determined mainly by the hole volume. At the same time it follows from Eq. (63) that the SB constant is a very complex value and a function of p and a. Plotting the experimental data in coordinates... 

On the molecular level, the ratio vo/vf) is a measure of the average volume of the polymer relative to that of the holes. Thus, when vq > Vf, i.e., the polymer chain is larger than the average hole size, the viscosity will be correspondingly high, and when vq < Vf, the viscosity will be low. 

Kilburn et al. [2002] carried out a PALS study of free volume in semicrystalline poly(ethylene-co-l-octene) (PO) copolymers as well as high-density polyethylene (HDPF). The degree of crystallinity was characterized by DSC and WAXD analyses. A method was proposed to estimate the fractions of the RAF and MAF phases based on the observation that the mean thermal expansivity of free-volume holes, ea = d vh)/dt, varies as a function of Xc, which implies that the individual mean expansivities of holes in RAF and MAF phases are different. Thus, the thermal expansivity of the mean hole volume (i.e., averaged over the entire amorphous phase) may be expressed above To by... 

VpH average hole free volume per gram of mixture. 

The specific critical hole free volume, is estimated as the specific volume at 0 K, which in turn is obtained from group contribution methods. The ratio of the molar volumes of the jumping units of components i andj, is computed using the values at 0 K. The pre-exponential factor. Do, and the critical energy needed by a molecule to overcome the attractive force, E, are obtained by fitting the Dullien equation for the self-diffusion coefficient to viscosity versus temperature data. Finally, the average hole-free volume per gram of the mixture, VfH/y, can be estimated from those of the individual species ... 

An increase in the average free volume holes can be clearly monitored by positron annihilation. Positron annihilation is one of the useful techniques to investigate relaxation characteristics of polymers. Positrons emitted from Na induce radiation effect on polymer samples, and then resultant electrons are trapped in a shallow potential, which are formed at low temperature far below the glass temperature (T ). The increase in the number of these trapped electrons is observed as an increase in the intensity (1 ) of the long-lived component of ori/jo-positronium (o-Ps). 

Utiu et al. investigated the PFSA/Si02 composites by Xe NMR spectroscopy and relaxometry. The average free volume hole size, estimated by extrapolation of Xe chemical shift to zero pressure, reaches a maximum at... 

A simplified theory was proposed by Brandt, Berko and Walker [104] in which the positron of Ps wave function in the field of the electron was replaced by the wave function of the Ps atom. The Ps wave function was then calculated for different lattice structures in the Wigner-Seitz approximation. This approximation is generally referred to as the free volume model, since the free volume is used as one of the parameters in the calculation. This model relates o-Ps lifetime to the average free volume hole size of the medium, and results construed that the o-Ps lifetime would measure the lattice-Ps interaction. Later, Tabata et al. [105] and Ogata and Tao [106] each adopted similar - but different - approaches by considering a unit cell and Ps located at the center instead of the center of the molecule, as used by Brandt et al. [104]. 

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