Free-volume theory requirements

Miyamoto and Shibayama (1973) proposed a model which is essentially an extension to free volume theory, allowing explicitly for the energy requirements of ion motion relative to counter ions and polymer host. This has been elaborated (Cheradame and Le Nest, 1987) to describe ionic conductivity in cross-linked polyether based networks. The conductivity was expressed in the form... 

A modified version of the free-volume theory is used to calculate the viscoelastic scaling factor or the Newtonian viscosity reduction where the fractional free volumes of pure polymer and polymer-SCF mixtures are determined from thermodynamic data and equation-of-state models. The significance of the combined EOS and free-volume theory is that the viscoelastic scaling factor can be predicted accurately without requiring any mixture rheological data. 

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). 

The real reason for this serious limitation of the free volume theory is as yet not clarified. Kishimoto, Maekawa and Fujita (1960) have considered that diffusion of small molecules such as water requires for their jumping only a very local cooperation of the solid-like vibrations of two or three monomers and, therefore, their rate of diffusion would not depend on physical factors through the average free volume of the system. It would simply increase with temperature as a result of enhanced thermal agitations of individual monomer units as well as the penetrant molecules themselves. However, an explanation like this is as yet no... 

A different approach was made by Dyre et al. (1996) to account for the experimental viscosity variations with temperature as an alternative to VTF and AG models. They considered the flow in viscous liquids to arise from sudden events involving motion and reorganization of several molecules. From the viewpoint of mechanism, the energy required for such flow is minimized if the surrounding liquid is shoved aside to create the necessary volume for rearrangement. This volume is fundamentally different from the volume of the free volume theory and is, in principle, an activation volume. The free energy involved may be written as... 

While the WLF equation is based on the free-volume theory of glass transition which is concerned with the introduction of free volume as a requirement for coordinated molecular motion, the equation also serves to introduce some kinetic aspects into the quantitative theory of glass transition. 

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. 

The free-volume theory of diffusion was developed by Vrentas and Duda. This theory is based on the assumption that movement of a small molecule (e.g., solvent) is accompanied by a movement in the solid matrix to fill the free volume (hole) left by a displaced solvent molecule. Several important conditions must be described to model the process. These include the time scales of solvent movement and the movement of solid matrix (e.g. polymer segments, called jumping units), the size of holes which may fit both solvent molecules and jumping units, and the energy required for the diffusion to occur. 

In this chapter we discuss the basics of the phase behaviour of hard spheres plus depletants. Phase transitions are the result of physical properties of a collection of particles depending on many-body interactions. In Chap. 2 we focused on two-body interactions. As we shall see, depletion elfects are commonly not pair-wise additive. Therefore, the prediction of phase transitions of particles with depletion interaction is not straightforward. As a starting point a description is required for the thermodynamic properties of the pure colloidal dispersion. Here the colloid-atom analogy, recognized by Einstein and exploited by Perrin in his classical experiments, is very useful. Subsequently, we explain the basics of the free volume theory for the phase behaviour of colloids -I- depletants. In this chapter we treat only simplest type of depletant, the penetrable hard sphere. 

The free-volume theory of the glass transition, as developed in Section 8.6.1, is concerned with the introduction of free volume as a requirement for coordinated molecular motion, leading to reptation.The WLF equation also serves to introduce some kinetic aspects. For example, if the time frame of an experiment is decreased by a factor of 10 near Tg, equations (8.47) and (8.48) indicate that the glass transition temperature should be raised by about 3°C ... 

The free-volume theory introduces free volume in the form of segment-size voids as a requirement for the onset of coordinated molecular motion. This theory provides relationships between coefficients of expansion below and above Tg and yields equations relating viscoelastic motion to the variables of time and temperature. 

To recapitulate, the Flory version of the Prigogine free-volume or corresponding-states polymer solution theory requires three pure-component parameters (p, v, T ) for each component of the solution and one binary parameter (p ) for each pair of components. 

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