Free-volume theories models

If the molecule moves without hindrance in a rigid-walled enclosure (the free enclosure ), as assumed in free volume theories, then rattling back and forth is a free vibration, which could be considered as coherent in such a cell. The transfer time between opposite sides of the cell t0 is roughly the inverse frequency of the vibration. The maximum in the free-path distribution was found theoretically in many cells of different shape [74]. In model distribution (1.121) it appears at a > 2 and shifts to t0 at a - oo (Fig. 1.18). At y — 1 coherent vibration in a cell turns into translational velocity oscillation as well as a molecular libration (Fig. 1.19). 

In the literature there is only one serious attempt to develop a detailed mechanistic model of free radical polymerization at high conversions (l. > ) This model after Cardenas and 0 Driscoll is discussed in some detail pointing out its important limitations. The present authors then describe the development of a semi-empirical model based on the free volume theory and show that this model adequately accounts for chain entanglements and glassy-state transition in bulk and solution polymerization of methyl methacrylate over wide ranges of temperature and solvent concentration. 

A useful model should account for a reduction of kt and kp with increase in polymer molecular weight and concentration and decrease in solvent concentration at polymerization temperatures both below and above the Tg of the polymer produced. For a mechanistic model this would involve many complex steps and a large number of adjustable parameters. It appears that the only realistic solution is to develop a semi-empirical model. In this context the free-volume theory appears to be a good starting point. 

Numerous models have been proposed to interpret pore diffusion through polymer networks. The most successful and most widely used model has been that of Yasuda and coworkers [191,192], This theory has its roots in the free volume theory of Cohen and Turnbull [193] for the diffusion of hard spheres in a liquid. According to Yasuda and coworkers, the diffusion coefficient is proportional to exp(-Vj/Vf), where Vs is the characteristic volume of the solute and Vf is the free volume within the gel. Since Vf is assumed to be linearly related to the volume fraction of solvent inside the gel, the following expression is derived ... 

Equation 39 has the structure proposed for the rate constants on the basis of the free volume theory (1,5,9). From this, it would be expected that the models developed from the free volume theory would be very successful in predicting both, the rate behaviour and the molecular properties at high conversions. The reason why these models have been only partially successful stems from the... 

The rate parameters follow similar conversion trajectories. Therefore, the rate constants and the initiator efficiency can be modelled with the same equation. An equation of the form of equation 39 is suggested. The theoretical Justification for the form of equation 39 stems from the free volume theory. 

Analysis of mixture models, established techniques, 61 Analysis of styrene suspension polymerization continuous models, 210-211 efficiency, 211,212f,213 free volume theory, 215,217 initiator conversion vs. 

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... 

This is a theoretical equation that was derived from free volume theory. If extruding materials at lower than normal temperatures, the higher sensitivity of the viscosity to temperature is an issue that needs to be considered. The engineering-based viscosity equation developed by Adams and Campbell [18] has been shown to hold for all nominal processing temperatures, from within a few degrees of Tg [26, 27] to conventional extruder melt temperatures. The Adams-Campbell model limiting shear temperature dependence is ... 

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. 

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. 

The Perez model comes from an approach in which the source of mobility is the existence of quasi-punctual defects characterized by positive or negative fluctuations of packing density, whereas classical free volume theories take into account only the domains of low packing density, e.g., the holes. The model leads to the following equation for the complex modulus ... 

A comprehensive model which is based on the free-volume theory and which accounts for the effect of molecular weight and solvent on chain entanglements and glassy-state transition has been recently developed by Marten and Hamielec (7 ). This model accounts for diffusion-controlled termination and propagation... 

In this system the a relaxation can be analyzed by the symmetric equation of Fuoss-Kikwood and a new model which is similar to Havriliak- Negami equation used in the analysis of dielectric spectroscopy. According to the Tg values calculated for these systems, the free volume can be appropriately described by the free volume theory. The analysis of these families of poly(methacrylate)s allow to understand in a good way the effect of the structure and nature of the side chain on the viscoleastic behavior of polymers [33],... 

Summing up, the two-phase model is physically consistent and may be applied for designing industrial systems, as demonstrated in recent studies [10, 11], Modeling the diffusion-controlled reactions in the polymer-rich phase becomes the most critical issue. The use of free-volume theory proposed by Xie et al. [6] has found a large consensus. We recall that the free volume designates the fraction of the free space between the molecules available for diffusion. Expressions of the rate constants for the initiation efficiency, dissociation and propagation are presented in Table 13.3, together with the equations of the free-volume model. 

Fig. 22. Extent of reaction at vitrification vs. reaction temperature for linear free-radical polymerization (styrene) for f = 0.5 and [II, = 0.10 mole/1. The solid line is for the results from the T,-mole-cular weight model [Eq. (21)] the dashed line is for the results from the free volume theory [Eq. (26)]. [Aronhime, M, T., Gillham, J. K. J. Coat. Tech. 56 (718), 35 (1984)]...

Figure 4.7 illustrates how the available free volume for transport increases with increasing temperature (Ff = Vj - V ), and the remarkable change when passing the Tg of a polymer [50]. According to the free volume diffusion model, the diffusion of molecules depends on the available free volume as well as sufficient energy to overcome polymer-polymer attractive forces. The specific volume at a particular temperamre can be obtained from the polymer density, whereas the volume occupied at 0 K can be estimated from group contributions. Details on this theory may be found in relevant handbooks, textbooks, and numerous publications [25,48-52]. 

The PERVAP simulator (tubular module) was developed by Alvarez (2005), using FORTRAN language (Compaq Visual Fortran Professional Edition 6.6.a). The mathematical model applied is based on the solution-diffusion mechanism. Activity coefficients of the components in the feed phase (jj) were determined using the UNIFAC method (Magnussen et al, 1981). The prediction of diffusion coefficient (Z) ) was carried out using the free-volume theory. 

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... 

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