Free-volume correction

Fig. 10-4. Left panel, raw viscosity data for ring and linear PS in bulk at 169.5°C. Right panel, free-volume corrected data at the same temperature. Both are due to Roovers [5].

McKenna et al. carried out sample preparation and viscosity measurements with admirable care and precision. Here in Figure 10-5, we quote as the most important of their findings the data of free-volume corrected 7(r) for the ring samples (presumably less knotted and less contaminated with linears), which were the l5est selected from Roovers as well as their own. We may note the following. 

Fig. 10-5. Solid line, fitting (free-volume corrected) rjit) for cyclic PS in the melt at 169.5°C ( , samples of McKenna et al. , Roovers samples). Dashed line, fitting the corresponding ry( ) for linear PS.

M.J. Misovich and E.A. Grulke, Prediction of solvent activities in polymer solutions using an empirical free volume correction, Ind. Eng. Chem. Res., 27 (1988) 1033-1041. 

The steady-flow viscosity t]q has repeatedly been found to be determined by M , in blends and distributions of widely varying composition, " provided the free volume correction mentioned above is made when necessary, to eliminate the complication that tjw friction coefficient depends on M whereas the effects of chain length depend on Mw. 

Enskog free volume correction function Energy ratio, x = (-)... 

The first correction leads to a free volume (v — b) instead of v ... 

A fascinating insight into the impact that modelling can make in polymer science is provided in an article by Miiller-Plathe and co-workers [136]. They summarise work in two areas of experimental study, the first involves positron annihilation studies as a technique for the measurement of free volume in polymers, and the second is the use of MD as a tool for aiding the interpretation of NMR data. In the first example they show how the previous assumptions about spherical cavities representing free volume must be questioned. Indeed, they show that the assumptions of a spherical cavity lead to a systematic underestimate of the volume for a given lifetime, and that it is unable to account for the distribution of lifetimes observed for a given volume of cavity. The NMR example is a wonderful illustration of the impact of a simple model with the correct physics. 

The configurational entropy model describes transport properties which are in agreement with VTF and WLF equations. It can, however, predict correctly the pressure dependences, for example, where the free volume models cannot. The advantages of this model over free volume interpretations of the VTF equation are numerous but it lacks the simplicity of the latter, and, bearing in mind that neither takes account of microscopic motion mechanisms, there are many arguments for using the simpler approach. 

Endless discussion exists regarding whether a theory based on the configurational entropy or the excess free volume 8v provides the more correct description of glass formation. Thus, this section briefly analyzes the relation... 

In 1873 van der Waais pointed out that real gases do not obey the ideal gas equation PV = RT and suggested that two correction terms should be included to give a more accurate representation, of the form (P + ali/) V - b) = RT. The term a/v corrects for the fact that there will be an attractive force between all gas molecules (both polar and nonpolar) and hence the observed pressure must be increased to that of an ideal, non-interacting gas. The second term (b) corrects for the fact that the molecules are finite in size and act like hard spheres on collision the actual free volume must then be less than the total measured volume of the gas. These correction terms are clearly to do with the interaction energy between molecules in the gas phase. 

Tager and co-workers (51) have invoked bundle structures to explain correlations between the viscosities of concentrated polymer solutions and the thermodynamic interactions between polymer and solvent. They note, for example, that solutions of polystyrene in decalin (a poor solvent) have higher viscosities than in ethyl benzene (a good solvent) at the same concentration, and quote a number of other examples. Such results are attributed to the ability of good solvents to break up the bundle structure the bundles presumably persist in poor solvents and give rise to a higher viscosity. It seems possible that such behavior could also be explained, at least in part, by the effects of solvent on free volume (see Section 5). Berry and Fox have found, for example, that concentrated solution data on polyvinyl acetate in solvents erf quite different thermodynamic interaction could be reduced satisfactorily by free volume considerations alone (16). Differences due to solvent which remain after correction for free volume... 

Fig. 28 shows a typical example of experimental data plotted according to Eq. (IV-25). From the slope a value of 1/30 is calculated for the quantity y>j3N (1 — ip). Since in this example N < 50, this would mean that the free volume 1 y> amounts to 0.16 which seems unreasonably high. A direct comparison of Eq. (IV-23) with DiMarzio s Eq. (IV-22a) shows that the free volume would amount to 1 — y> = 0.25. It seems, likely, therefore, that Jackson c.s. overestimate the effect of excluded volume obstruction, and that DiMarzio s and Khasanovich s treatments are more nearly correct. 

Figure 24. A relationship between free volume and the feasibility of a reaction in an organized media. Filled areas correspond to the shapes and sizes of reactants and products. Note the shape changes between the reactant and the product. Free volume around a reaction center is represented as unfilled regions. In all three cases shown here the total free volume present is much larger than needed for a reaction to occur, but it is not present at the correct location. Importance of location and directionality of free volume highlighted.

It is now generally accepted that the viscous flow of polymeric liquids is connected with chain segment rotation, i.e. with configurational entropy. From this point of view Miller concluded that the Simha-Boyer equation was not correct since the relative free-volume in SB theory equals zero at 0 K, not at T = T0. If the latter... 

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. ... 

The calculation of the solubility of gases and other small molecules in various polymers showed that for each polymer the relative solubility of the compounds was well predicted, but one polymer-specific correction had to be fitted. The results of this study are shown in Fig. 10.8. The correction constant may have to do with the missing combinatorial term, or with free-volume effects in polymers, or with structural differences of polymers, or with a combination of several such effects. The lack of a temperature dependence suggests that it is mainly of entropic nature. Unfortunately, no clear relationship between the value of... 

Thus measurements of the viscosity 9 (0,7) over a range of temperature allow determination of f(0,T) as a function of T, provided the value of /(0,T) at a certain temperature T is known from other source. For this purpose we may utilize the measurement of viscosity as a function of diluent concentration at the given T ] the substitution of such data into Eq. (40) may lead to the determination of the required f(0,T ). It is to be expected that, if the free volume theories of viscosity and diffusion developed above are at all correct, the values of /(0,7) thus derived from y data should agree with those obtained from ae data by application of Eq. (40) and also with those from DT data analyzed in terms of Eq. (36). 

In a search for the defining structural parameter of a composite, the free volume of disperse system proved to be the most sound one from the physical standpoint Presumably, for disperse systems the free volume is a measure of the mobility of filler particles, just as for liquids it is a measure of the mobility of molecules. But as applied to highly-loaded coarse systems of the type solid particles — liquid — gas this notion requires a certain correction. In characterizing the structure of such specific systems as highly-loaded coarse composites, it should be noted that to prevent their settling and separation into layers under the action of vibration, the concentration of the finest filler fraction with the largest specific surface in dispersion medium should be the maximum possible. Because of this and also because of the small size of particles (20-40 pm), the fine fraction suspended in the dispersion medium practically does not participate in the formation of the composite skeleton, which consists of coarser particles. Therefore... 

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