Free-volume concept

It is possible to derive an expression equivalent to Eq. (4.67) starting from entropy rather than free volume concepts. We have emphasized the latter approach, since it is easier to visualize and hence to use for qualitative predictions about Tg. 

Kandyrin, L. B. and Kuleznev, V. N. The Dependence of Viscosity on the Composition of Concentrated Dispersions and the Free Volume Concept of Disperse Systems. Vol. 103, pp. 103-148. 

Cohen and Turnbull [20,21] laid down the foundation for the free volume concept in modeling self-diffusion in simple van der Waals liquids. They considered that the volume in a liquid is composed of two parts, the actual volume occupied by the liquid molecules and the free volume surrounding these molecules opened up by thermal fluctuation. Increasing temperature increases only the free volume and not the occupied volume. The average free volume per molecule, vf, can be defined as... 

It was shown, that the conception of reactive medium heterogeneity is connected with free volume representations, that it was to be expected for diffusion-controlled sohd phase reactions. If free volume microvoids were not connected with one another, then medium is heterogeneous, and in case of formation of percolation network of such microvoids - homogeneous. To obtain such definition is possible only within the framework of the fractal free volume conception. 

Generally, the VTF behaviour of all transport properties may be understood from the free volume concept introduced by Doolittle (1951) and further developed by Cohen and Turnbull (1959). Essentially, any diffusing species is depicted as encaged by the nearest atoms in a cell... 

The concept of free-volume appeared to be very useful and was applied for the theoretical description of many processes in liquids, including polymeric solutions and melts. Taking the free-volume concept as a basis, theories were developed for the diffusion of low-molecular-weight compounds into polymers14,1S, thermal conductivity16, solution and solubility of polymers17, etc. 

The free-volume concept was applied most widely in the theory of viscoelastic properties of polymers developed by Williams, Landel and Ferry (WLF theory), presented in detail in12. According to WLF theory, the changes in liquid viscosity with frequency and temperature from glass temperature T% to T may be plotted on a single master curve by using the reduction factor... 

The theory of viscoelastic properties based on the free-volume concept was developed in18, the main idea being that some structural elements are displaced when the free-volume fraction exceeds a critical value /c. The frequency v of the determining molecular process may be obtained from the equation... 

Simha and Boyer22 applied the free-volume concept to describe glass transitions in amorphous polymers. By analogy with Eq. (12), it was assumed that the fractional free-volume/g at glass-transition temperature Tg is... 

K i being a constant independent of other material properties (iso-free-volume concept of Tg). 

Difficulties arise, however, concerning the interpretation of T0 in terms of the free-volume concept. In order to overcome these difficulties, Miller31 suggested that the occupied volume could vary with temperature. 

A very interesting development of the free-volume concept has been made in some papers by Miller29,36 His main idea is that all excess thermodynamic functions... 

The expressions in brackets are the expansivities above and below Tg. The constant K3 is a function of bond type in chains and is really constant for every class of polymers. The physical interpretation of this equation may be consistent with the iso-free-volume concept. However, we believe that the introduction of this equality is in practise a denial of the concept. There are also other arguments against this concept. Kastner56 found, for example, that dielectric losses diminish during the isothermal volume contraction, which indicates a dependence of relaxation times on free-volume. However, if we assume that relaxation time depends exclusively on free-volume, the calculated reduction factor differs from the experimental one. 

A very interesting approach to the theoretical prediction of the limit value A a Tg was developed in 49. As a result of a thermodynamic consideration of mixing macromolecules with holes and taking into account the free-volume concept, it was shown that... 

Other cases of deviation from the behavior predicted within the framework of the free-volume concept have been reported 65 66. We have already mentioned Kanig s conclusion about the dependence of/g on chain flexibility45 and our own conclusion46,47 that it depends on packing density. In connection with this it is worth discussing whether there is really any dependence of the fractional free-volume at Tg on the molecular parameters of chains. 

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. 

From the point of view of the applicability of the iso-free-volume concept, it is of great interest to test it for some systems more complicated than simple polymeric liquids, despite the fact that, as we have already shown, this concept has failed in many cases even for simple polymers. The more complicated cases considered are compositions and polymeric blends and alloys. 

In addition, in polymeric compositions, i.e. in filled polymers, as a rule a broadening of the relaxation spectra takes place81. All these characteristics are closely associated with molecular parameters and free-volume, therefore it is interesting to consider how structural changes, induced in the same polymer by the action of the solid surface, may influence the applicability of the free-volume concept and the approach to such systems via the iso-free-volume state. In 82 the applicability of the... 

Let us now consider the applicability of the free-volume concept to the glass transition in heterogeneous polymeric systems consisting of two polymers. We think88 ... 

Everything discussed in the present paper shows that the free-volume concept, although very useful from the qualitative point of view, cannot be used for the quantitative description of many properties of polymer systems. This is especially clear when we consider glass-transition phenomena using the idea of the iso-free-volume state. Many experimental data, discussed above, show that this concept cannot be applied even to polymer materials having the same chemical nature but a different physical structure. From the experimental results Goldstein104 had already concluded that the concept of free-volume cannot be correct. These conclusions were carefully discussed later105. ... 

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