Polymer free-volume models

Generally, the values of the scaling exponent are smaller for polymers than for molecular liquids, for which 3.2 < y < 8.5. A larger y, or steeper repulsive potential, implies greater influence of jamming on the dynamics. The smaller exponent found for polymers in comparison with small-molecule liquids means that volume effects are weaker for polymers, which is ironic given their central role in the historical development of free-volume models. The reason why y is smaller... 

In polymer electrolytes (even prevailingly crystalline), most of ions are transported via the mobile amorphous regions. The ion conduction should therefore be related to viscoelastic properties of the polymeric host and described by models analogous to that for ion transport in liquids. These include either the free volume model or the configurational entropy model . The former is based on the assumption that thermal fluctuations of the polymer skeleton open occasionally free volumes into which the ionic (or other) species can migrate. For classical liquid electrolytes, the free volume per molecule, vf, is defined as ... 

Larry Duda and Jim Vrentas were the first to systematically study the diffusion of small molecules in molten polymers, formulate a free volume-based theoretical model, and elucidate the sharp dependence of the diffusion coefficient on temperature and concentration.2 Figure 8.8 shows diffusivities of toluene in polystyrene as a function of concentration and temperature. The values were computed using the Vrentas and Duda (17) free volume model and, as shown, coincide well with available data. 

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. 

Among the popular methods for interpreting the diffusion of small penetrants in polymers are the so called free-volume models (6,11,13,51-54). The basic assumption of these models is that the mobility of both polymer segments and penetrant molecules is primarly determined by the available free-volume in the penetrant polymer system. The free-volume of the polymer is regarded as an empty volume between the chains of the polymer. Similarly the free-volume of the penetrant can be regarded as the volume not occupied between the molecules of the penetrant. 

Most free-volume models for diffusion in polymers follow the phenomenological basis set in (55) where the self-diffusion of an ideal liquid of hard spheres ( molecules ) has been analysed. These molecules are confined - for most of the time - in a cage formed by their immediate neighbours. A local fluctuation in density may open a hole within a cage, large enough to permit a considerable displacement of the sphere contained by it. This displacement gives rise to diffusion only if another sphere jumps into the hole before the first sphere returns to its initial position. Diffusion occurs not as a result of an activation process in the ordinary sense but rather as a result of the redistribution of the free-volume within the liquid of hard spheres. 

In the last two decades Vrentas, Duda and their co-workers have published a substantial number of papers (61-67) on the free-volume model of diffusion in polymer-solvent systems they developed in the late 70 s (68-72). This model, which is often cited and used in the literature, underwent a number of modifications over the years and appears to apply well to the diffusion of organic solvents in rubbery and glassy polymers. 

In order to develop a consistent free-volume diffusion model, there are some issues which must be addressed, namely i) how the currently available free-volume for the diffusion process is defined, ii) how this free-volume is distributed among the polymer segments and the penetrant molecules and iii) how much energy is required for the redistribution of the free-volume. Any valid free-volume diffusion model addresses these issues both from the phenomenologic and quantitative points of view such that the diffusion process is described adequately down to the microscopic level. Vrentas and Duda stated that their free-volume model addresses these three issues in a more detailed form than previous diffusion models of the same type. Moreover, it was stated that the model allows the calculation of the absolute value of the diffusion coefficient and the activation energy of diffusion mainly from parameters which have physical significance, i.e. so-called first principles . In the framework of this model the derivation of the relation for the calculation of the self-diffusion coefficient of the sol-... 

Local density fluctuations occur in penetrant polymer systems both above and below Tg. It is then reasonable to expect that a free-volume diffusion model should also provide an adequate description of the diffusion of small penetrants in glassy polymers. To reach this goal the free-volume model for diffusion of small penetrants in rubbery polymers, second part of Section 5.1.1, was modified to include transport below Tg (64,65,72,91-93). 

Finally, in some of the most widely used classical models - the free-volume models of Fujita, Vrentas and Duda and their alternatives (171-175) - more than a dozen structural and physical parameters are needed to calculate the free-volume in the penetrant polymer system and subsequently the D. This might prove to be a relatively simple task for simple gases and some organic vapors, but not for the non-volatile organic substances (rest-monomers, additives, stabilizers, fillers, plasticizers) which are typical for polymers used in the packaging sector. As suggested indirectly in (17) sometimes in the future it will maybe possible to calculate all the free-volume parameters of a classical model by using MD computer simulations of the penetrant polymer system. 

Wilkens, J. B., and F. A. Long A free-volume model for diffusion of small molecules in polymers. Trans. Faraday Soc. 53, 1146 (1957). 

Miyamoto, T. and Shibayama, W. D. Free volume model for ionic conductivity in polymers. J. Appl. Phys. 44 5372, 1973. 

For ideal systems (usually as in elastomers), the solubility wiU be independent of concentration and the sorption curve will follow Henry s law (Equation 4.6), i.e., gas concentration within the polymer is proportional to the applied pressure. For nonideal systems (usually as in glassy polymers), the sorption isotherm is generally curved and highly nonlinear. Such behavior can be described by free-volume models and Flory-Huggins thermodynamics—comprehensive discussions on this may be found elsewhere [1,25,26]. 

The models most frequently used to describe the concentration dependence of diffusion and permeability coefficients of gases and vapors, including hydrocarbons, are transport model of dual-mode sorption (which is usually used to describe diffusion and permeation in polymer glasses) as well as its various modifications molecular models analyzing the relation of diffusion coefficients to the movement of penetrant molecules and the effect of intermolecular forces on these processes and free volume models describing the relation of diffusion coefficients and fractional free volume of the system. Molecular models and free volume models are commonly used to describe diffusion in rubbery polymers. However, some versions of these models that fall into both classification groups have been used for both mbbery and glassy polymers. These are the models by Pace-Datyner and Duda-Vrentas [7,29,30]. 

The various models developed to describe the diffusion of small gas molecules in polymers generally fall into two categories (1) molecular models analyze specific penetrant and chain motions together with the pertinent intermolecular forces, and, (2) "free-volume" models attempt to elucidate the relationship between the diffusion coefficient and the free volume of the system, without consideration of a microscopic description. 

Stern, Frisch, and coworkers have extended Fujita s free-volume model to the permeation of light gases (31-33) (see Figure 3) and binary gas mixtures (34,25) (see Figure 4) through polymer membranes. The extended model was found to describe satisfactorily the dependence of permeability coefficients on pressure and temperature for a variety of light gases in polyethylene, as well the dependence on composition for several binary mixtures in the same polymer. The validity of the extended model is limited to total penetrant concentrations of up to 20-25 mol-%. 

Other free-volume models have been discussed by Frisch and Stern (2), by Kumins and Kwei <2 .) and by Rogers and Machin (3 9) Free-volume models which are applicable to both rubbery and glassy polymers are described in a following section of this review. 

The diffusion coefficient of small penetrants in glassy polymers can also be correlated with the polymer free volume. In view of the fact that experimental techniques for the determination of the free volume are inherently difficult, Shah, Stern, and Ludovice (Shah, V.M. Stern, S.A. Ludovice, P.J., submitted for publication in Macromolecules) have utilized the detailed atomistic modeling of relaxed polymer glasses developed by Theodorou and Suter ( 2, 3) to estimate the free volume available in polymer glasses for the diffusion of small molecules. 

MODELS FOR DIFFUSION IN BOTH RUBBERY AND IN GLASSY POLYMERS Some models are applicable to diffusion of small molecules in glassy as well as in rubbery polymers. These too fall into the general categories of molecular models and free-volume models. Recent molecular dynamics simulations of simple polymer/penetrant systems will also be discussed. 

The free-volume model proposed by Vrentas and Duda (67-69) is based on the models of Cohen and Turnbull and of Fujita, while utilizing Bearman s (7j0) relation between the mutual diffusion coefficient and the friction coefficient as well as the entanglement theory of Bueche (71) and Flory s (72) thermodynamic theory. The formulation of Vrentas and Duda relaxes the assumptions deemed responsible for the deficiencies of Fujita s model. Among the latter is the assumption that the molecular weight of that part of the polymer chain involved in a unit "jump" of a penetrant molecule is equal to the... 

A substantial number and variety of models of gas transport in polymers have been proposed during the last 20-30 years, in view of the great practical and scientific importance of this process. Molecular-type models are potentially most useful, since they relate diffusion coefficients to fundamental physicochemical properties of the polymers and penetrant molecules, in conjunction with the pertinent molecular interactions. However, the molecular models proposed up to now are overly simplified and contain one or more adjustable parameters. Phenomenological models, such as the dual-mode sorption model and some free-volume models, are very useful for the correlation and comparison of experimental data. 

The concept of free volume has been of more limited use in the prediction of solubility coefficients although, Peterlin (H) has suggested that the solubility coefficient is directly proportional to the free volume available in the polymer matrix. In many respects, the free volume expressions closely resemble the relationships developed in the activated state approach. In fact for the case of diffusivity, the two models can be shown to be mathematically equivalent by incorporating thermal expansion models such as the one proposed by Fox and Flory (12). The usefulness of the free volume model however, lies in the accessibility of the fractional free volume, through the use of group contribution methods developed by Bondi (12.) and Sugden (li), for correlation of barrier properties of polymers of different structure as demonstrated by Lee (15.). ... 

---