Fractional free volume coefficient

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 idea that the fractional free-volume at glass temperature as found experimentally depends on the mode of molecular motions was put forward in 196746 47 as a result of calculating/g from data obtained from isothermal volume relaxation for some polymer systems. By estimating average relaxation time at different temperatures it was possible to find the fractional free-volume/g at Te according to WLF theory. If we accept the validity of the theory as regards the universal dependence of the reduction factor aT on (T - Tg), then on the basis of data on Aa and theoretical values aT calculated from universal values of the coefficients C and C, it is possible to make an estimate of/g. In this case the value found corresponds to the universal one. If, however, we use the experimental values aT, the fractional free-... 

Figure 2.24 Correlation of the oxygen permeability coefficient for a family of related polysulfones with inverse fractional free volume (calculated using the Bondi method) [33]. Reprinted with permission from C.L. Aitken, W.J. Koros and D.R. Paul, Effect of Structural Symmetry on Gas Transport Properties of Polysulfones, Macromolecules 25, 3424. Copyright 1992, American Chemical Society...

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...

The results have indicated that the monomeric friction coefficient is sensitive to the presence of the short chains, and is smaller, at a given temperature, than the homopolymer. This observed reduction in has been shown to be only attributed to an increase in the fractional free volume. Indeed, the temperature dependence of the monomeric friction coefficient is described by a WLF-type equation ... 

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. 

The models most frequently used to describe the concentration dependence of diffusion and permeability coefficients of gases and vapors, including hydrocarbons, are transport model of dual-mode sorption (which is usually used to describe diffusion and permeation in polymer glasses) as well as its various modifications molecular models analyzing the relation of diffusion coefficients to the movement of penetrant molecules and the effect of intermolecular forces on these processes and free volume models describing the relation of diffusion coefficients and fractional free volume of the system. Molecular models and free volume models are commonly used to describe diffusion in rubbery polymers. However, some versions of these models that fall into both classification groups have been used for both mbbery and glassy polymers. These are the models by Pace-Datyner and Duda-Vrentas [7,29,30]. 

Fujita (ljD related the thermodynamic diffusion coefficient, D, to the fractional free volume, v, by... 

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.). ... 

The fractional free volume / reaches a constant value, fg, at Tg and increases linearly above Tg with the coefficient of expansion Of. Following Chapter 4, substitution of equation (4-16) into equation (5-1) with Tg as the reference temperature yields the WLF equation where the constants C, and C2 are given by... 

The plot of frequency of e" maximum versus reciprocal temperature for the a transition showed typical WLF behavior in Figure 11. All these data were fit to a single WLF equation with the fractional free volume at Tg equal to. 0225 and the expansion coefficient of free volume above Tg equal to 5.3 X 10. ... 

The reduction in formation of new taut tie molecules per amorphous layer manifests itself in the almost constant sorption and diffusion coefficient above A. The constancy of sorption means that there is a constant fraction of almost unperturbed locations in the amorphous component where the molecules of the sorbate can be accommodated. The constancy of diffusion coefficient means that there is a constant fraction of taut tie molecules which, by their better packing, reduce the fractional free volume and which, by their tautness, increase the energy needed for hole formation, i.e. the activation energy of diffusion. 

Molecular Meaning of aT. The effect of temperature on viscosity is related to its effect on the friction coefficient, which, in turn, depends on the fractional free volume according to the equation ... 

Free-volume theories of the glass transition assume that, if conformational changes of the backbone are to take place, there must be space available for molecular segments to move into. The total amount of free space per unit volume of the polymer is called the fractional free volume Vf. As the temperature is lowered from a temperature well above Jg, the volume of the polymer falls because the molecules are able to rearrange locally to reduce the free volume. When the temperature approaches Tg the molecular motions become so slow (see e.g. fig. 5.27) that the molecules cannot rearrange within the time-scale of the experiment and the volume of the material then contracts like that of a solid, with a coefficient of expansion that is generally about half that observed above Tg. 

Fractional free volume/was then calculated according to the previous definition [Eq. (10.15)]. Its variation with temperature is shown in Figure 10.10, together with the theoretical free-volume fraction fi values of/(obtained assuming spherical holes, plotted as circles in Figure 10.10) are systematically lower than h for all the structures. Furthermore, the expansion coefficients of/are higher than the corresponding values deduced from the theory. 

---