Thermal fractional free volume

Table 14.2 Values of WLF Cj0 and C2° Parameters, Fractional Free Volumes f0 and/ and Thermal Expansion Coefficient af for PAMAM, PPI and PBzE dendrimers ... 

According to Ferry12 the free-volume per cm of substance, i. e. the fractional free-volume /, is hard to define exactly and should be regarded as merely a useful semi quantitative concept. Specifically, the thermal expansion coefficients of liquids for the most part reflect the increase in fractional free-volume only a small part is connected with the anharmonic dependence of potential energy or interatomic and intermoleeular distances. 

It is generally assumed that fractional free-volume changes with temperature according to Eq. (16), where the thermal expansion coefficient is expressed by Eq. (19). 

The factors (298 Tg) and (Tg 298) are the "thermal distances" of Tg from room temperature for rubbers and glasses respectively. The influence of these "thermal distances" is probably connected with the fractional free volume of the polymer in rubbery amorphous polymers this f.f.v. increases with decreasing Tg, in glassy amorphous polymers the f.f.v. increases with increasing Tg (increasing formation of micro-voids), hence lowering of the activation energy. 

Alternative ways of determining the free volume fraction without using I3 have also been proposed by Dlubek et al [28], as well as, Brandzuch et al [29], Dlubek et al used the coefficient of thermal expansion of the amorphous regions and hole volume determined from positron data to determine the number density of the free volume holes. Brandzuch et. al. used the coefficient of thermal expansion just above and just below the Tg to estimate the fractional free volumes. This model is based on the assumption that the expansion of the holes of the free volume, as seen by positrons, reflects the expansion of the total volume of the material. 

Fractional Free Volume and Coefficient of Thermal Expansion. The shift constants c and C2 from the WLF equation are not only fitting parameters that describe the frequency-temperature relation of a given polymer, but they are also related to chemical structure. Ferry has shown (6) that these constants can be related to the fractional free volume and coefficient of thermal expansion of the free volume, which have physical meaning in terms of the polymer structure. One can define the free volume at the glass transition divided by the total volume as fg and the coefficient of thermal expansion of... 

Table IV. WLF Shift Constants, Fractional Free Volume, and Thermal Expansion Coefficient...

This choice relies on the assumption that a constant Tg+T corresponds to a constant free volume state. Such an approximation presumes that the thermal expansion of free volume, Uf, as well as the fractional free volume, fg, at Tg, are independent of the hlend composition. In the case of binary blends with short chains, may depend on the composition of the blend, so that a normalisation of experimental times with this parameter would be misleading. 

Eisenbach also discussed the significance of the WLF-parameters with respect to the fractional free volume at Tg and to the difference of thermal dilation coefficients of rubbery and glassy states. 

There is no fundamental qualitative difference in mechanisms of low molecular weight (MW) penetrant diffusion in polymers above and below glass transition temperature, Tg, of the polymers [5,6]. The difference lies only in the fact that the movement of structural units of the macromolecule that are responsible for the transfer of penetrant molecules takes place at different supermolecular levels of the polymer matrix. At T > Tg the process of diffusion takes place in a medium with equilibrium or near-equUibrium packing of chains, and the fractional free volume, P(, in the polymer is equal to the fractional free volume in the polymer determined by thermal mobUity of strucmral units of macromolecules V((T), i e., V(= vut). At r< Tg the process of diffusion comes about under nonequihbrium packing conditions, although there exists a quasi-equilibrium structural organization of the matrix, where Vf> It is assumed that in this case Vf= where is the fractional free volume... 

Table 12. Thermal expansion behavior and fractional free volume at Tg for epoxy resin systems before and after crosslinking...

The concept of free volume has been of more limited use in the prediction of solubility coefficients although, Peterlin (H) has suggested that the solubility coefficient is directly proportional to the free volume available in the polymer matrix. In many respects, the free volume expressions closely resemble the relationships developed in the activated state approach. In fact for the case of diffusivity, the two models can be shown to be mathematically equivalent by incorporating thermal expansion models such as the one proposed by Fox and Flory (12). The usefulness of the free volume model however, lies in the accessibility of the fractional free volume, through the use of group contribution methods developed by Bondi (12.) and Sugden (li), for correlation of barrier properties of polymers of different structure as demonstrated by Lee (15.). ... 

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