Generalized free-volume theory

We treat the extension of FVT and incorporate correct expressions for the (polymer concentration-dependent) depletion thickness and osmotic pressure, resulting in generalized free volume theory (GFVT). Expression (4.4) for the semigrand potential is still valid in GFVT it does not contain any assumption as yet on the physical properties of the depletants or the colloids. But now we need to... 

For q = 0.57 the composition of the colloidal liquid that coexists with a colloidal gas and crystal was determined by Moussaid et al. [26]. In Table 4.1 we compare these data with FVT and GFVT predictions. The experimental colloid volume fraction and polymer concentration clearly deviate significantly from FVT. Especially the polymer concentration of the coexisting colloidal liquid phase is about 30 times larger than the FVT prediction. Generalized free volume theory gives a much better prediction of the composition, especially if the polymers are assumed to be in a good solvent. 

Expression (6.46) may be regarded as a generalized free volume theory (GFVT) semi-grand potential for rods plus interacting polymers. Subsequently, we can specify the quantities a and P for interacting polymers. 

Fig. 6.10 Phase diagrams calculated using generalized free volume theory for spherocylinders with L/D = 20 plus interacting polymer chains in a good solvent dashed curves) for size ratios q = 1 left) and q = 2.5 right) in the reservoir representation. Full curves are FVT results for (noninteracting) penetrable hard spheres as depletants as identical to those in Fig. 6.7. As in Fig. 6.7 the Gaussian form for the ODF was used...

Similarly to Fig. 5-4 for other glassy polymer-solvent systems also the predictions of this free-volume theory are in general agreement with experimental data on the temperature dependence of D in the vicinity of Tg2. In particular, the theory predicts a step change in Ed at Tg2, and this is consistent with most experimental investigations of polymer-solvent diffusion at temperatures just above and below the glass transition temperature (6,11,15). 

The introduction forms Chapter 1 of this survey. Chapter 2 deals briefly with various approaches to the description of the concentration dependence of the viscosity of disperse systems, including the transition region from fluid to solid-like systems. Chapter 3 describes viscosity from the standpoint of the free volume theory and the specific features of the transition from mobile to glasslike systems. Chapter 4 presents the concept of the free volume of disperse systems developed by us as well as the results of experiments illustrating it Chapter 5 contains the pertinent generalizations and conclusions. 

Application to Polvmer-Solvent Systems. Fujita (231 was the first to use the free-volume theory of transport to derive a free-volume theory for self-diffusion in polymer-solvent systems. Berry and Fox (241 showed that, for the temperature intervals usually considered (smaller than 200°C), the theories that consider a redistribution energy for the voids gives results similar to those of the theories that assume a zero energy of redistribution for the free volume available for molecular transport. Vrentas and Duda (5.61 re-examined the free-volume theory of diffusion in polymer-solvent systems and proposed a more general version of the theory presented by Fujita. They concluded that the further restrictions needed for the theory of Fujita are responsible for the failures of the Fujita theory in describing the temperature and concentration dependence... 

In conclusion it can be stated from these general studies that photochromic processes in polymer matrices can be described by the free volume theory moreover, these processes are controlled by the kind of local environment around the photochromes,... 

General comparisons have been presented between diffusion coefficients and apparent activation energies derivable from the "global" free volume theory of Vrentas and Duda, and the statistical mechanical model of Pace and Datyner, as an initial step in the utilization of these theories. 

There is as yet no adequate theory to describe the thermodynamics of aqueous polymer solutions. Any discussion of the origins of the steric repulsion in aqueous dispersions must therefore be speculative. In what follows, the modifications to the free volume theory that may reasonably be required to account for the specific interactions between the water molecules and the stabilizing moieties will be enunciated. This general scheme seems pertinent to the stabilization imparted by the industrially important polymers poly(oxyethylene) and poly(vinyl alcohol). It is unlikely, however, to be relevant to the stabilization by poly(acrylic acid) at low pH or polyacrylamide. The latter stabilizers, as noted previously, appear to fall within the guidelines set out in Table 7.4. 

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. 

The temperature dependence of the conductivity in polymer electrolytes has often been taken as indicative of a particular type of conduction mechanism In particular, a distinction is generally made between systems which show an Arrhenius type of behaviour and those which present a curvature in log a or log aT inverse temperature plots. In the latter case, empirical equations derived from the free-volume theory have... 

The Prigogine simple cell model (P) considers each monomer in the system to be trapped in the cell created by its surroundings. The general cell potential, generated by the surroundings, is simplified to be athermal (cf. free volume theory), whereas the mean potential between the centers of different cells are described by the Leimard-Jones 6-12 potential. The P model EoS can be summarized as... 

Even with this modification, the resulting equation was no better than Fujita s original equation, which only correlates data at 0i < 0.2. These limitations seem to be absent in the free-volume theory of Vrentas and Duda, in which they obtained a general expression for the mutual diffusion coefficient, D, as... 

Other theories within the general framework of the free-volume concept have been advanced. They include the works of Kumins and Roteman (1961), Bueche (1953), Barrer (1957), DiBenedetto (1963), DiBenedetto and Paul (1964), Wilkens and Long (Wilkens, 1957 Wilkens and Long, 1957), and Vasenin (1960). Part of the reason that these theories are not so popular lies in the fact that their predictions of D c) were no better than those obtained by Fujita and Doolittle. In addition, most of these other theories concentrated on the temperature dependence of the intrinsic mobility, which is less important compared to its concentration dependence. In spite of the predictive limitations of the free-volume theory, it is applicable at the widest concentration range, and certainly it is the best theory to use for modeling diffusion limited behavior of polymerization systems. 

In order to determine the various diffusion-controlled rate coefficients, monomer and polymer diffusivities were estimated. In general, the free-volume theory is the accepted model for the determination of these quantities. In this model, diffusion is... 

We now incorporate the correct depletion thickness into free volume theory presented in Sect. 3.3. We consider the osmotic equilibrium between a polymer solution (reservoir) and the colloid-polymer mixture (system) of interest, see Fig. 4.6. The general expression for the semi-grand potential for Nc hard spheres plus interacting polymers as depletants, see (3.18), is... 

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