Distribution free volume/hole

In practice the lifetime distributions are usually obtained using a computer program such as the MELT [21] or CONTIN [22, 23] programs. The reliablity of these programs for measurring the o-PS lifetime distribution in polymers was shown by Cao et al [24]. A detailed description of these methods of data analysis is presented in Chapter 4. The advantage of the continuous lifetime analysis is that one can obtain free volume hole distributions rather that the average values obtained in the finite analysis. 

The affect of polymer stereoregularity in the chains on the PAL data has also been studied. Hamielec et al [56] found what appears to be an increased lifetime (hole size) with increased randomness of the chain configuration in a series of polyvinlychloride (PVC) polymers, despite the large degree of scatter in the sample (probably due to the fact that a series of commercially available products were used.). They however found little correlation with tacticity in polypropylene. More recently a PAL study on a series of very well characterized polystyrene and poly(p-methlystyrene) samples of differing tacticity [57] was performed. In addition to finding that the polystyrene samples have smaller free volume holes than the poly(p-methylstyrene) samples, they found that the syndiotactic samples had broader hole distributions than the attactic samples. 

Since both positrons and Ps could, be localized in free-volume holes, the data of positron lifetime (t2) and o-Ps bulk lifetime (t3) provide information about the size and distribution of free-volume size as a function of the depth near the surface. Figure 11.6 shows the variation of positron lifetime and o-Ps bulk lifetime vs the depth. A significant increase of lifetimes near the surface shows a larger size of free volumes near the surface than in the bulk. Similar variations vs the positron energy indicates that both positrons and Ps are localized in free volumes and holes. Figure 11.7 shows the distributions of hole size in the polymer from the data of o-Ps lifetime distribution. Near the surface, not only the size is larger than the bulk, the distribution is significantly wider [10]. 

The larger free volume and distribution also indicates a larger fraction of free volume near the surface than in the bulk. According to the WFL theory [43], a larger free volume leads to a lower Tg. Indeed a significant Tg depression (as much as 70 °C) has been reported in the surface of polystyrene by using PAL method [10]. Other studies of polymer surfaces have shown that the size of the free volume holes near the surface of polyethylene [47] and polypropylene [48] are larger than the bulk. 

It can be observed that an infinite channel with a square cross section is obtained from Eq. (10.9) when m oo. Prisms and layered cavities have been used, for example, in structural studies of clays [Joshi et al., 1998 Consolati et al 2002], A discussion of pore-size distributions in low-dielectric thin films [Gidley et al 2000] was based on cubic structures. Of course, other geometries are possible For example, ellipsoidal holes were also considered [Jean and Shi, 1994], in an interesting attempt to frame free-volume holes in semicrystalline polymers subjected to tensile deformation. 

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. 

Additional information on the material under study can be obtained from the measurement of the e+ -e momentum distribution as mirrored in Doppler broadening of annihilation radiation (DEAR) and the angular correlation of annihilation radiation (ACAR). DEAR can be applied to study the chemical surroundings of free-volume holes [Dlubek et al., 2000a Bamford et al., 2006b], while ACAR is able to measure the anisotropy of the hole shape, as observed for highly crystalline fibers, for example [Jean et al., 1996 Bamford et al., 2001b]. 

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). 

Jean, Y. C., and Deng, Q., Direct measurements of free-volume hole distributions in polymers by using a positronium probe, J. Polym. Sci. B, 30, 1359-1364 (1992). 

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... 

In recent years, positron annihilation lifetime (PAL) spectroscopy has been demonstrated to be a special sub-nanometer probe to determine the free-volume hole size, fraction and distribution in a variety of polymers (4-9). In this technique, measured lifetimes and relative intensities of the positron and positronium, Ps (a bound atom which consists of an electron and a positron), are related to the size and fraction of sub-nanometer holes in polymeric materials. Because of the positive-charge nature, the positron and Ps are repelled by the ion core of polymer molecules and trapped in open spaces, such as holes, free volumes, and voids. The observed... 

Free-Voliime Hfree-volume holes in polymers have a distribution, a PAL spectrum can be expressed in a continuous form ... 

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). 

The decrease in volume that accompanies the physical aging process can be related to a change of the distribution of free volume holes. Since the size and concentration of free volume holes in amorphous polymers is closely linked to the thermal expansion, direct measurements of the free volume can be used to monitor the aging processes. 

Positron Annihilation Lifetime Spectroscopy (PALS) provides a measure of free volume holes or voids, free volume, and free volume distribution, at an atomic scale. The technique exploits the fact that the positively charged positron (e" ), the antiparticle to the electron, preferentially samples regions of low positive charge density. When injected in a polymer matrix, thermalized positrons can combine with an electron to form a bound state, known as positronium (Ps). This species can only exist in a void and it rapidly annihilates on contact with the electron cloud of a molecule. For polymer studies using PALS, it is ortho-positronium (oPs, a triplet state) which is of interest. The oPs spin exchanges with electrons of opposite spin on the walls of the cavity and it is annihilated. Thus, the oPs lifetime, 13, gives a measure of the mean free volume cavity radius, whereas the relative intensity of... 

With more and more data/experiments having been collected, positron lifetime spectroscopy might become a unique characterization method for polymers, that can even give qualitative results on the free-volume hole distribution of polymers. 

Fig. 39. Free-volume hole distributions of various polycarbonates. Dashed lines are the molecular volumes of common penetrants. Reprinted from Ref 181. This material is used by permission of John Wiley Sons, Inc.

Using PALS, Dammert et al. (92) and Yu et al. (93) examined the hole volume of a series of polystyrenes of different tacticity. They arrived at a free-volume hole size distribution maximum of around 110 at room temperature see Rgure 8.24 (92). This corresponds to an effective spherical hole radius of approximately 3 A. While this radius is somewhat larger than the theoretical value of 1.5 A found above, if the holes are actually irregular in shape the values are seen to agree quite well. 

Figure 8.24 PALS study of free volume hole size distribution in polystyrenes and poly(p-methylstyrenes) calculated from lifetime distributions of o-positronium.

Figure 27.7 (a) Free volume hole size distributions in PS/TMPC blends (b) Calculated inter-... 

Figure 2. Free-volume hole radius distribution R pdf(R) (relative units) of PTMSP (pdf(R) is probability density function) obtained from the two right-hand peaks in Fig. 1. The dashed line shows the calculated dependence (3) of annihilation rate, (in ns ) versus radius of FV elements used in confuting the size distribution.

A study of free volume in immiscible PS/PC and miscible PS/TMPC blends using PAL spectroscopy showed negative deviation in hnearity in the free volume of the miscible blend [390]. The free volume hole distribution was additive in the miscible blend, but broadened in the immiscible blend. Similar results were noted in comparison of SMA/SAN polymer blends [391]. The values of 13 were close to linearity with composition, whereas I3 showed negative deviation from linearity for the miscible blends. Immiscible blends were foimd to exhibit tmsymmetrical and broader oPs lifetime distributions than the miscible blends. 

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