Hole volume distributions

In this chapter we demonstrate the potential of PALS for the study of the size distribution of subnanometer-size local free volumes (holes) in amorphous polymers. We employ the routine LifeTime in its version 9.0 (LT9.0 [Kansy, 1996, 2002]) for the analysis of lifetime spectra and discuss its advantage for analyzing < -Ps lifetime distributions. From these distributions the hole radius and hole volume distributions are calculated. The assumptions underlying this type of analysis, present-day understanding, and possible complications (e.g., tunneling, weighting) are discussed briefly. 

Since Xpo follows a distribution and the relation between A.po and the hole radius r is nonlinear [Eq. (11.3)], we prefer to estimate the mean hole-volume as the mean of the number-weighted hole volume distribution. The radius distribution [the probability density function (pdf)], n(rfc), can be calculated from n (rh) = - 3 (A.) (dx/drh) [Gregory, 1991 Deng et al., 1992b] ... 

It is interesting to compare the hole sizes for polymers of different Tg values and different chemical structure. Such a comparison is made in Figure 11.5, showing plots of the temperature-dependent mean hole volume, ( ua ), and the standard deviation, <7ft, of the hole volume distribution for a large collection of polymers with Tg values between 200 and 500 K. We have grouped the polymers under discussion into... 

Baugher, A. H., Kossler, W. J., and Petzinger, K. G., Does quantum mechanical tunneling affect the validity of hole volume distributions obtained from positron annihilation lifetime measurements Macromolecules, 29, 7280-7283 (1996). 

Figure 9. Hole volume distributions for Avicel samples.

From the point of view of the ideas discussed above concerning the variability on the free-volume fraction at Tg, even for the same modes of molecular motion in different polymers, there is great interest in some new concepts about the free-volume distribution, in the system, first proposed in 24. The starting point is the suggestion that all molecular motions, like transfer phenomena, can take place only when the size of the voids or holes in the system exceeds a critical value v. This critical volume appears as a result of redistribution of the free-volume within the system. 

Lattice coordination numbers (2) and the cell volumes (vjj) for both the pure components and mixture lattices are assumed to have the same value. The partition function for this ensemble can be formulated following Equation 2. It is assumed now that the partition function, far from the binary critical point can be approximated by its largest term. Since molecule segments and holes can distribute themselves non-randomly, the partition function must incorporate terms to account for this effect. The nonrandomness correction rjj... 

Kluin, J.E., Yu, Z., Vleeshouwers, S., McGervey, J.D., Jamieson, A.M., Simha, R., Sommer, K. (1993) Ortho-positronium lifetime studies of free volume in polycarbonates of different structures Influence of hole size distributions . Macromolecules. 26, 1853. 

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). 

A change In the nonequilibrium substructures of the material with different packing densities may Influence the complex rate of these volume changes. A simplified model might Involve an alteration of the hole structure and the clustering of the molecules with a smaller overall free volume distributed Into larger microscopic voids. 

Experimental data from our laboratories will be shown for an extensive series of amorphous polymers with glass transitions between Tg = 200 and 500 K. We discuss the temperature dependence of the hole-size distribution characterized by its mean and width and compare these dependencies with the hole fraction calculated from the equation of state of the Simha-Somcynsky lattice-hole theory from pressure-volume-temperature PVT) experiments [Simha and Somcynsky, 1969 Simha and Wilson, 1973 Robertson, 1992 Utracki and Simha, 2001]. The same is done for the pressure dependence of the hole free-volume. The free-volume recovery in densified, and gas-exposed polymers are discussed briefly. It is shown that the holes detected by the o-Ps probe can be considered as multivacancies of the S-S lattice. This gives us a chance to estimate reasonable values for the o-Ps hole density. Reasons for its... 

We assume that the o-Ps hole-size distribution above Tg directly mirrors the thermal density fiuctuation. This allows us to extract information on the length scale of the dynamic heterogeneity in polymers. Using a fluctuation approach, the temperature dependency of the volume of the smallest representative freely fluctuating subsystem can be estimated. Limits of this interpretation for polymers with a high structural disorder, which already appears in the glass, are discussed. 

FIGURE 11.6 Mean volume, (Vhg), and the mean dispersion,, of the hole-size distribution... 

This picture seems to be true only for polymers with flexible chains, which are characterized by a low Tg value. For high-Tg polymers, not only do the volume parameters h, Vf, and (u/j) exhibit rather large values at Tg (see Sections 11.3 and 11.5) but the hole-size distribution is broad (has a large an. Figure 11.5b). This leads to an unusually small value for (Vsv) at a of 1 to 3 nm, (Tsv) explained this by an inherent topologic disorder due to the bulky units (mers) and inflexible chains, respectively, of these polymers, which increases a above the value that would be caused by thermal fluctuations [Dlubek et al., 2007c]. 

Jean, Y. C., Comments on the paper Can positron annihilation lifetime spectroscopy measure the free-volume hole size distribution in amorphous polymers Macromolecules, 29,5756-5757 (1996). 

These observations were explained in terms of a free-volume treatment that adopts the Grest-Cohen model, in which a system consists of free-volume cells, each having a total hole volume vh. These free-volume cells can be classified as solidlike (n < v c) or liquidlike (w > Vhc), where Vhc is a critical hole volume. Moreover, it is assumed that the free volume associated with a liquidlike cell of the amorphous phase consists of free-volume holes whose size distribution is given by a normal frequency distribution, H vk). This leads to a cumulative distribution function of free-volume hole sizes, r vh), given by... 

The resulting strain dependence of (w ) and the width of the hole-size distribution computed from F(vh) are both in good agreement with the experimental data, assuming a value p = 0.65, indicative that the above-described free-volume model provides a satisfactory description of the PALS data. Computation of the fractional free volume from the experimental g(vh). 

G. Dlubek, A. P. Clarke, H. M. FretweU, S. B. Dugdale, M. A. Alam, Positron lifetime studies of free volume hole size distribution in glassy polycarbonate and polystyrene, Phys. Stat. Sol. A, 157, 351 (1996). 

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