Free volume, definition fractional

Covacs6 proposed another definition of the fractional free-volume. As the extrapolated volume of supercooled liquids reaches the crystalline state, vc, at temperature rc > 0 K, it is possible to compare this critical temperature with rg , the limit glass temperature at infinitely slow cooling. Free-volume in the crystalline state is assumed to be zero. In this case the value /g according to Doolittle may be compared with /g, c, which characterizes the excess free-volume of glass as compared with the crystalline state ... 

Here the proportionality factor Ad is considered to be dependent primarily upon the size and shape of the diluent molecule and hence may be independent of temperature and diluent concentration. From its definition the parameter Bd may also be treated as independent of these factors. Thus we see that the mobility md of a given diluent is primarily determined by the magnitude of the average fractional free volume / of the system. 

The quantity f(0,T), which is the value of / at zero penetrant concentration, represents the average fractional free volume in the pure polymer. As will be expected from its definition, the quantity y(T) may be compared with the fractional free volume of the diluent. A similar conclusion has been deduced by Ferry and Stratton (1960). 

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. 

The definitions of the free-volume fraction discussed above are dependent on taking the volume as an initial marking-off state, i.e. on the manner of calculation. It is already apparent here that there is some uncertainty regarding the physical meaning of free-volume. 

Some very interesting ideas concerning the relationship between free-volume formation and the energy of one mole of hole formation were developed in detail by Kanig42. Kanig introduced some improvements to the definition of free-volume, On the basis of Frenkel s ideas43 he divided the free-volume into two parts, one of which is determined only by the thermal vibrations of atoms in the lattice of a real crystal while the other is connected with inherent free-volume, i.e. voids and holes. It is the latter that makes possible the exchange of particles, i.e. the very existence of the liquid state. He introduced some new definitions of fractions of free-volume ... 

It is, therefore, not surprising that there exists a definite relationship between Aand the enthalpy of vaporization, Av H, the former constituting a fraction between 0.2 and 0.3 of the latter, as is readily obtained from the data in Tables 3.1 and 3.9. The pressure dependence of the viscosity is also closely related to the free volume of the solvent. The fluidity (O = l/r ) is proportional to the ratio between the free and the occupied volume, the former, as mentioned above, being the difference between the actual molar volume and the intrinsic molar volume (Tables 3.1 and 3. 4) (Hildebrand 1978). In fact, the logarithm of the viscosity of liquids was found (Marcus 1998) to be described well for some 300 liquids by the empirical relationship ... 

To predict the viscosity reduction, the change in the free volume fraction caused by CO2 dissolution has to be calculated. Extending the definition of free volume fraction, the free volume fraction is given as a function of temperature, pressure, and dissolution of gas ... 

Figure 12.4 presents the density decrease and the total liactional free volume (FFV) increase by addition of PEG into the Pebax . The experimental values of density, which are lower than those calculated by the additive model, suggest that the total free volume is increased. The FFV was estimated from a definition of the volume fraction of the filler by using density values [76], and by solving the following equations ... 

Because B is an arbitrary parameter, the free-volume fraction cannot be determined directly from these equations. As noted previously, Doolittle " " was concerned with the definition of free volume and in Figure 20 is shown a possible definition for the free-volume fraction based on the concept of an occupied volume. When one considers B to be unity, the value of the free-volume fraction at falls between 0.013 and 0.034 for many polymers, and the universal value originally proposed by Williams, Landel and Ferrywas taken to be 0.025 (see equation 23 also the discussion about the free-volume fraction derived from the Gibbs-DiMarzio " lattice model). 

The following symbols are used in the definitions of the dimensionless quantities mass (m), time (t), volume (V area (A density (p), speed (u), length (/), viscosity (rj), pressure (p), acceleration of free fall (p), cubic expansion coefficient (a), temperature (T surface tension (y), speed of sound (c), mean free path (X), frequency (/), thermal diffusivity (a), coefficient of heat transfer (/i), thermal conductivity (/c), specific heat capacity at constant pressure (cp), diffusion coefficient (D), mole fraction (x), mass transfer coefficient (fcd), permeability (p), electric conductivity (k and magnetic flux density ( B) ... 

These ideas can be formalised in terms of statistical mechanics to some extent, and an outline of the main ideas is given in the following section. We remark parenthetically that there are profound difficulties confronting the definition of an aggregate. The nature of the hydrophobic free energy of transfer of a hydrocarbon from water to the hydrophobic core of a micelle can be measured, but its temperature dependence is not understood because it depends on water, an unknown quantity. For the same reasons, solution theory, does not even tell us whether mole fractions or mole volumes are the correct ratios to use to determine entropy. However, provided certain assumptions are allowed [62-65], then simple rules emerge. The rules are if v is the hydrocarbon chain volume, a the head-group area, and 1 of an optimal... 

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