The Free Volume Theory of Cohen and Turnbull

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 ... 

A well-known and simple theory for describing molecular transport in a liquid is the free-volume theory of Cohen and Turnbull [1959, 1970]. Employing statistical mechanics, these authors showed that the most probable size distribution of the free volume per molecule in a hard sphere liquid may be described by an exponential decreasing function. It was assumed that diffusion of the hard-spheres can only take place when, due to thermal fluctuations, holes are formed whose size is greater than a critical volume. When applying this theory to a structural relaxation process in a liquid, its (circular) frequency o) = r = 2jtv is expressed by... 

Physically, an elastomer is more liquid-like than solid-like, and therefore the mobility theory for a liquid seems the appropriate choice for ion-conducting polymer systems. The macroscopic viscosity of an elastomer cannot, however, be used in the Stokes equation. This is because the macroscopic viscosity is greatly enhanced by chain entanglement, which does not directly resist ion motion. However, the mobility in a liquid or elastomer can be derived according to the free-volume theory of Cohen and Turnbull, outlined below. Because it involves the glass transition temperature (Tg) as the major parameter, it is particularly applicable to amorphous polymers for which Jg is easily measured. 

The surface concentration dependence of the lateral mobility of Fig. 7 was analyzed in terms of the free-volume theory of hard sphere liquids of Cohen and Turnbull [55, 56], as well as in view of the Enskog theory of dense gases [57] extended by Alder s molecular dynamics calculations to liquid densities [58]. The latter approach was particularly successful. It revealed that the lateral diffusion constant of the Fc amphiphiles does follow the expected linear dependence on the relative free area, Af/Ao, where Af = A — Ao, A = MMA, and Aq is the molecular area of a surfactant molecule. It also revealed that the slope of this dependence which is expected to inversely depend on the molecular mass of a diffusing particle, was more than 3 orders of magnitude smaller [54]. Clearly, this discrepancy is due to the effect of the viscous drag of the polar head groups in water, a factor not included in the Enskog theory. 

The physical basis of equation 29 can be understood from the theory of Cohen and Turnbull,which treats the self-diffusion of spherical molecules with the assumption that motion of a molecule can occur only when a void exceeding some critical volume v , is available for it to move into. If no energy is required for free volume redistribution at constant volume, a calculation of fluctuations gives a result which is equivalent to expressing the translational friction coefficient f as follows ... 

The behavior of the WLF parameters can be explained using the free-volume concept. The two WLF parameters are described in the following form according to the free-volume theory developed by Cohen and Turnbull [105,106] ... 

The free-volume models reviewed here and in a later section are based on Cohen and Turnbull s theory (18) for diffusion in a hard-sphere liquid. These investigators argue that the total free volume is a sum of two contributions. One arises from molecular vibrations and cannot be redistributed without a large energy change, and the second is in the form of discontinuous voids. Diffusion in such a liquid is not due to a thermal activation process, as it is taken to be in the molecular models, but is assumed to result from a redistribution of free-volume voids caused by random fluctuations in local density. 

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... 

Theory must account for these properties of fluids, their temperature dependence, the glass transition, and the properties of the amorphous solid. The two most widely accepted theories are the entropy theory, as formulated by Gibbs and Di Marzio (1958), Gibbs (1960), and Adam and Gibbs (1965), and the free-volume theory as developed by Eyring (1936), Fox and Flory (1950), Williams, Landel and Ferry (1955), Cohen and Turnbull (1959), and Turnbull and Cohen (1961, 1970). 

Free-volume theories have been very successful in explaining concentration-dependent diffusion behavior of organic vapors in amorphous polymers, especially in cases where the penetrant is a good swelling agent for the polymer. The most significant free-volume theory is based on the works of Cohen and Turnbull (1959),... 

The concept of free volume in a polymer is an extension of the ideas of Cohen and Turnbull [141], first used to describe the self-diffusion in a liquid of hard spheres. Such theories suggest that the permeant diffuses by a cooperative movement between the permeant and the polymer segments, from one hole to the other within the polymer. The creation of a hole is caused by fluctuations of local density due to thermal motion. Based on the concept of the redistribution of free volume to represent the thermodynamic diffusion coefficient [142], and the standard reference state for free volume [143], Stem and Fang [144] interpreted their permeability data for nonporous membranes, and Fang, Stem, and Frisch [ 145] extended the theory to include the case of permeation of gas and liquid mixtures. 

These relations for the oligomers agreed with the results of Sasabe et al. s work [24] on amorphous polymers, such as PVC and PVDC. The meaning of m for the oligomer was explained according to both the WLF equation and the free-volume theory developed by Cohen and Turnbull [25,26] ... 

Free volume present in nanocomposite systems plays a major role in determining the overall performance of the membranes. Positron annihilation lifetime spectroscopy (PALS) is an efficient technique used for the analysis of free volume. The diffusion of permeant through polymeric membranes can be described by two theories, namely, molecular and free-volume theories. According to the free-volume theory, the diffusion is not a thermally activated process as in the molecular model, but it is assumed to be the result of random redistributions of free-volume voids within a polymer matrix. Cohen and Turnbull developed the free-volume models that describe the diffusion process when a molecule moves into a void larger than a critical size, Vc- Voids are formed during the statistical redistribution of free volume within the polymer. It is found that the relative fractional free volume of unfilled polymer decreases on the addition of layered silicates. The decrease is attributed to the interaction between layered silicate and polymer because of the platelet structure and high aspect ratio of layered silicates. The decrease is explained to the restricted mobility of the chain segments in the presence of layered silicates. This results in reduced free-volume concentration or relative fractional free volume [49]. 

Mathematical treatment of molten salts that supercool was first carried out by Cohen and Turnbull. The principal idea of the hole theory—that diffusion involves ions that wait for a void to turn up before jumping into it—is maintained. However, Cohen and Turnbull introduced into their model a property called thefree volume, Vf. What is meant by this free volume It is the amount of space in addition to that, Vq, filled by matter in a closely packed liquid. Cohen and Turnbull proposed that the free volume is linearly related to temperature... 

As seen in Table 1, the decrease in permeability can be directly attributed to a dramatic reduction in the effective diffusion coefficient, while there is a much smaller effect on the apparent solubility. A similar dependence of the solubility and diffusion coefficients on the draw ratio has been observed in other uniaxially oriented polymers (35-37). Because the glass transition and density of the polystyrene samples were found independent of the draw ratio, they concluded that the reduction in diffusivity was due to anisotropic redistribution of the free volume during drawing. Using an expansion coefficient related to draw ratio, the polystyrene data were successfully correlated using the Cohen-Turnbull free volume theory. However, the situation was found to be more complex for PVC (i ) ... 

Cohen and Turnbull s critical free-volume fluctuations picture of selfdiffusion in dense liquids is similar to the vacancy model of self-diffusion in crystals. However, in crystals individual vacancies exist and retain their identity over long periods of time, whereas in liquids the corresponding voids are ephemeral. The free volume is distributed statistically so that at any given instance there is a certain concentration of molecule-sized voids in the liquid. However, each such void is short-lived, being created and dying in continual free-volume fluctuations. The Frenkel hole theory of liquids ignores this ephemeral, statistical character of the free volume. 

The temperature dependence of the dynamic viscosity t of a liquid close to its glass temperature Tg can be described by the Vogel-Tammann-Fulcher (VTF) equation [43-45] or by the Theory of free volume introduced by Doolittle [46 8], Cohen and Turnbull [49, 50]. An exponential dependence from the reciprocal temperature 1/T is found (see (8.8)). 

On cooling the liquid, flow becomes impossible because of unavailable free volume for atomic jump. The main theories accounting for this phenomenon are the fi-ee-volume model (Cohen and Turnbull, 1959 Turnbull and Cohen, 1961) and the configurational entropy model (Adams and Gibbs, 1965). 

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