Distribution of free volume

In reality, the data on isothermal contraction for many polymers6 treated according to the free-volume theory show that quantitatively the kinetics of the process does not correspond to the simplified model of a polymer with one average relaxation time. It is therefore necessary to consider the relaxation spectra and relaxation time distribution. Kastner72 made an attempt to link this distribution with the distribution of free-volume. Covacs6 concluded in this connection that, when considering the macroscopic properties of polymers (complex moduli, volume, etc.), the free-volume concept has to be coordinated with changes in molecular mobility and the different types of molecular motion. These processes include the broad distribution of the retardation times, which may be associated with the local distribution of the holes. 

Kakizaki, M. and Hideshima, T., Effect of distribution of free volume on concentration dependence of dielectric relaxation in water mixtures with poly(ethylene glycol) and glucose, Jpn. J. Appl. Phys., Part 1, 1998, 37, 900. 

The distribution of free volume in amorphous polymers is of paramount importance for the respective material s transport behavior towards small and medium-sized penetrants. 

Apparently, there is less free volume in the aged polymer network. Any water coming into the aged network would tend to swell the polymer because there are simply less vacant sites in the distribution of free volume. In short, there is more polymer-solvent interactions as water diffuses into an aged epoxy network. 

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]. 

Conformer sequence probabilities Radial distribution functions Scattering functions Orientation correlation functions Mechanical properties Distribution of free volume... 

The complex decoloration kinetics were attributed to nonhomogeneous distribution of free volume and not to the isomerism of the merocyanine form. 

Recently" we have presented a quasi-chemical equation-of-state model that accounts for the nonrandom distribution of free volume in pure nonpolar fluids as well as in their mixtures. The model has proved successful in describing the phase equilibria of these systems, especially at the near-critical region. The essentials of this model follow. 

Using a Monte Carlo technique, Shah, Stern, and Ludovice evaluated the total free volume and the distribution of free volume available to spherical penetrants of different sizes in model structures of poly(propylene) and poly(vinyl chloride) at temperatures just below their glass transitions. Investigation of the... 

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. 

Our result for v>v is essentially identical to that derived earlier by Cohen and Turnbull for the most probable distribution of free volume X, P x) = y/Vfexp( — yx/Vf), where Vy is the free volume averaged over all cells, Vy —pvy, and y is a numerical factor between and 1 introduced to correct for overlap of the volume between neighboring cells. Comparing the exponent yx/vy =(y/p)x/vy whh the exponent in (6.10b), (v-v )/vy, we see that the two distributions are identical if x is taken as (c —u ) and y is taken as p, which would be close to j in the temperature region considered in Ref. 88. 

It is -interesting to observe here that less clear-cut separation in slow and fast isomers can occur in the kinetic scheme of the decoloration process. If the rate of decoloration depends to a considerable degree on the viscosity of the polymer matrix, it is also a very well known fact that the distribution of free volume in the glassy state can be Inhomogeneous, giving rise to a broad spectrum of relaxation times. This leads to a more complex mechanism of one-step chemical reactions in glassy polymers (111-114). 

Polymers designed for use as electrolytes in batteries are used at temperatures well above Tg. In this region a different model for conduction is more applicable, namely one based on the fluctuations in the distribution of free volume that occur above Tg due to the localised movements of the polymer chain segments. It is assumed that any ion sits in a region of free volume and can move into an adjacent space only when a sufficiently large element of free volume becomes available immediately adjacent to it. 

FIGURE 4.14 Distribution of free volume in PVAc immediately after the temperature was stepped to 30°C after equilibrium had been established at 40°C (wq) and after equilibrium is attained at 30°C (ILco)- Shown is the relative fraction of states having the free volume indicated by the abscissa (x). 

Merkel et al. [2002, 2003] carried out studies of gas and vapor permeability and PALS free volume in a poly(4-methyl-2-pentyne) (PMP)/fumed silica (FS) nanocomposite. It was observed that gas and vapor uptake remained essentially unaltered in nanocomposites containing up to 40 wt% FS, whereas penetrant diffusivity increased systematically with the spherical nanofiller content. The increased diffusivity dictates a corresponding increase in permeability, and it was further established that the permeability of large penetrants was enhanced more than that of small penetrants. PALS analysis indicated two o-Ps annihilation components, interpreted as indicative of a bimodal distribution of free-volume nanoholes. The shorter o-Ps lifetime remained unchanged at a value T3 2.3 to 2.6 ns, with an increase in filler content. In contrast, the longer lifetime, T4, attributed to large, possibly interconnected nanoholes, increased substantially from 7.6 ns to 9.5 ns as FS content increased up to 40 wt%. 

Thus, the distribution of free volume, or the LL environments, and the distributed material property affected, such as the local fluidity or relaxation behavior, reflects the variations in the local atomic packing discussed in Section 1.3. Such property variations have long been of interest (Scherer 1990). For the case presented above, in which the viscosity at T2 needs a certain relaxation time from that of Ti, the change in the time-dependent property, p (e.g., viscosity), is given by a relaxation function Mp(t),... 

Fig. 2.9 Calculated distribution of free volume evaluated on the basis of the size of spheres that can be inserted into empty sites (Shah et al. (1989) courtesy of the ACS).

Reaction of a positron with an electron gives a metastable positronium (Ps) particle, which may have antiparallel spins (para-positronium, p-Ps) or parallel spins (ort/jo-positronium, o-Ps). Within a polymer, the longer lifetimes of o-Ps may be related to the size, concentration and distribution of free volume elements. There have been a number of studies of PIM-1 by positron annihilation lifetime spectroscopy (PALS) [33-36]. 

One can discern for PTMSN the two broad minima in the curve AHjy that correspond to the values of Vc of 426cmVmol (u-Ct) and 754cmVmol (n-Cia). The presence of the two minima is rather unexpected because in most cases only one minimum was observed in the curves AHj[Vc). The only exception so far was amorphous Teflon AF2400, where also two minima, though flat and broad, could be noted [19], Here we have apparently another manifestation of bimodal size distribution of free volume in glassy polymers that attracts now a keen interest [33], The error bars shown in Figure 3.6 seem to support the assumption of bimodal size distribution in this polymer. 

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