Lifetime free volume distribution

Kristiak, J., Kristiakova, K., Sausa, O., Bandzuch, P., Bartos, J. (1993) Temperature dependence of free volume distributions in polymers studied by position lifetime spectroscopy . Journal De Physique IV. 265. 

Hofmann, D., Enhialgo-Castano, M., Lebret, A., Heuchel, M., and Yampolsldi, Y., Molecular modeling investigation of free volume distributions in stiff chain polymers with conventional and ultrahigh free volume comparison between molecular modeling and positron lifetime studies. Macromolecules, 36, 8528-8538 (2003). 

D. Hofmann, M. Heuchel, Y. Yampolskii, V. Khotimskii, V. Shantarovich, Free volume distributions in ultrahigh and lower free volume polymers Comparison between molecular modeling and positron lifetime smdies. Macromolecules, 35, 2129-2140 (2002). 

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... 

Hofmaim, D., et al.. Molecular Modeling Investigation of Free Volume Distributions in Stiff Chain Polymers with Conventional and Ultrahigh Free Volume Comparison Between Molecular Modeling and Positron Lifetime Studies. Macromolecules, 2003, 36(22), 8528-8538. 

When dealing with the free volume or interstices in dense membranes, small angle X-ray spectroscopy (SAXS) is very helpful, along with positron annihilation lifetime spectroscopy (PALS) [30], or even ellipsometry [31, 32], which measures density locally giving a first insight on the space free volume distribution in depth. 

Free Volume Distribution-Lifetime Analysis by Laplace Transform Method... 

Positron annihilation lifetime spectroscopy (PALS) is a more recent tool used to probe free volume and free volume distribution in polymers (38, 59). PALS uses orthoPositronium (oPs) as a probe of free volume in the polymer matrix. oPs resides in regions of reduced electron density, such as free volume elements between and along chains and at chain ends (38). The lifetime of oPs in a polymer matrix reflects the mean size of free volume elements accessible to oPs. The intensity of oPs annihilations in a polymer sample reflects the concentration of accessible free volume elements. The oPs lifetime in a polymer sample is finite (on the order of several nanoseconds), so PALS probes the availability of free volume elements on nanosecond timescales (40). The minimum free volume cavity diameter required by oPs for localization is 3.SA (41), which is equal to the kinetic diameter of methane (42). Thus, PALS probes the dynamic availability of free volume elements similar in size to those important for gas separations applications. Several recent studies demonstrate the strong correlation of PALS parameters and transport properties in polymers (34, 38, 43-45). The chapter by Yampol skii and Shantarovich in this book describes the use of PALS to characterize free volume distribution in membrane polymers. 

A fascinating insight into the impact that modelling can make in polymer science is provided in an article by Miiller-Plathe and co-workers [136]. They summarise work in two areas of experimental study, the first involves positron annihilation studies as a technique for the measurement of free volume in polymers, and the second is the use of MD as a tool for aiding the interpretation of NMR data. In the first example they show how the previous assumptions about spherical cavities representing free volume must be questioned. Indeed, they show that the assumptions of a spherical cavity lead to a systematic underestimate of the volume for a given lifetime, and that it is unable to account for the distribution of lifetimes observed for a given volume of cavity. The NMR example is a wonderful illustration of the impact of a simple model with the correct physics. 

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]. 

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. 

Abdel-Hady, E.E., El-Sayed, A.M.A. (1995) Free volume home distributions of polymers via the positron lifetime techniques . Polymer degradation and stability. 47, 369. 

This technique, firstly applied to metals and ceramics, has become a popular tool in polymers science for the determination of free volume [4,6-8] and starts to be applied to carbonaceous materials [9-12], Positron studies of porous materials have been predominantly oriented towards the chemical interaction of positrons with gases filling the porosity or with molecular layers adsorbed on the pore surface. Few studies have focused in the relation between annihilation characteristics with pore size and pore size distribution. Only in same cases, the annihilation time and the pore size have been directly related, and most of these studies have been carried out with silica gels [5,13,14], although other materials like porous resins (XADS) [15] have also been studied. In all these studies, it has been observed that the lifetime of positrons (t) increases with pore width. 

Positronium in condensed matter can exist only in the regions of a low electron density, in various kinds of free volume in defects of vacancy type, voids sometimes natural free spaces in a perfect crystal structure are sufficient to accommodate a Ps atom. The pick-off probability depends on overlapping the positronium wavefunction with wavefunctions of the surrounding electrons, thus the size of free volume in which o-Ps is trapped strongly influences its lifetime. The relation between the free volume size and o-Ps lifetime is widely used for determination of the sub-nanovoid distribution in polymers [3]. It is assumed that the Ps atom is trapped in a spherical void of a radius R the void represents a rectangular potential well. The depth of the well is related to the Ps work function, however, in the commonly used model [4] a simplified approach is applied the potential barrier is assumed infinite, but its radius is increased by AR. The value of AR is chosen to reproduce the overlap of the Ps wavefunction with the electron cloud outside R. Thus,... 

Positronium lifetime spectroscopy is particularly well suited for stud)hng defects in crystals and structural fluctuations in amorphous materials and can give an estimate of free volumes in condensed matter [116]. It is a useful technique to estimate the free volume of polymeric membranes [117]. In a study on silica gels, the decay lifetime has been found (Fig. 4.16) to be proportional to the pore diameters (measured by N2 adsorption) between 30 and 100 A [118]. Information on pore size distribution and surface area may also be obtained by means of calibration curves. 

As for the volumes of the atoms, the thermal expansion and compressibility is composed of two main terms, the cavity and the hydration. An estimate of the contribution of each factor relies on assumptions that are not easy to check. An estimate of the expansion or compression of the cavities should be possible with positron annihilation lifetime spectroscopy. This approach has proven to be a useful tool for determining the size of cavities and pores in polymers and materials. The lifetime is sensitive to the size of the cavity in which it is localized. A number of empirical relations correlate the distribution of the lifetime and the free volume [33]. Data on the pressure effect on the lifetime are only available for polymers. The results suggest that there may be a considerable contribution of the reduction in cavity size to the compressibility of a protein. 

The photoinduced susceptibility shown in Equation 11.14a is the sum of two terms one with exp(-2Dt) (relaxation of the first-order parameter Ai) decay and the second with exp(-12Dt) (relaxation of the third-order parameter A3) decay. Hence, the first very rapid decay may contain the fast exp(-12Df) contribution. Howeveg as can be seen from Figure 11,14, the relative magnitude of this initial very fast decay does not depend on the optimization of the intensity ratio between the writing beams. So, this first rapid decay may not be due to the decay of the third-order parameter A3. In addition, because the hyperpolarizability P of DRl is different in the cis and in the traits state, the first very rapid decay also contains a contribution connected with the lifetime of the metastable cis form, which is due to molecules coming back to the trans form without any net orientation. A better model would have to account for a distribution of diffusion constants for molecules embedded with various free volumes, which may explain the multiexponential behavior of the decay. ... 

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