Diffusivity liquids, free-volume theory

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... 

FREE VOLUME THEORY FOR SELF-DIFFUSIVITY OF SIMPLE NON-POLAR LIQUIDS. 

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. 

Cohen and Crest [91] extended the free volume theory by introducing the concept of percolation for particle diffusion in the liquid by focusing on the random distribution of free volume. The singularity at Ty in (10.24) represents the singularity induced by the percolation threshold. The free volume regions do not percolate below Ty, so the particle diffusion is limited as expected in the glassy region. Above Ty, the percolated network of free volume allows the particle diffusion to occur over the entire volume, which makes the system behave like a fluid. 

Application of this theory to both polymeric and small molecule liquids has given jump distances 6 which correspond roughly to the dimensions of small molecules (i.e., from 2 to 20 A). Typical jump frequencies range from 10 to 10 per second. These and other free volume theories of diffusion are described in more detail in Crank and Park [4]. The important concept, common to all free volume theories, is that diffusion occurs in polymers through free volume obtained by minor displacements of side groups or segments of the chain but without net translational movements of the centre of mass of the polymer. 

To evaluate the time-dependent function, X(t), a simple model of diffusion is proposed. Starting from Langmuir adsorption theory, we consider that liquid molecules having diffused into the elastomer are localized on discrete sites (which might be free volume domains). In these conditions, we can deduce the rate of occupation of these sites by TCP with time. Only the filhng of the first layer of the sites situated below the liquid/solid interface at a distance of the order of the length of intermolecular interaction, i.e., a few nanometers, needs to be considered to estimate X(t). 

These authors were the first FGSE workers to make extensive use of the concept of free volume 42,44) and its effect on transport in polymer systems. That theory asserts that amorphous materials (liquids, polymers) above their glass transition temperature T contain unoccupied volume randomly distributed and in parcels of sufficient size to permit jumps of small molecules — and of polymer jumping segments — to take place. Since liquids have a fractional free volume fdil typically greater than that, f, of polymers, the diffusion rate both of diluent molecules and (uncrosslinked and unentangled) polymer molecules should increase with increasing diluent volume fraction vdi,. The Fujita-Doolittle expression 43) describes this effect quantitatively for the diluent diffusion ... 

The concept of free-volume appeared to be very useful and was applied for the theoretical description of many processes in liquids, including polymeric solutions and melts. Taking the free-volume concept as a basis, theories were developed for the diffusion of low-molecular-weight compounds into polymers14,1S, thermal conductivity16, solution and solubility of polymers17, etc. 

Another advance in the concepts of liquid-phase diffusion was provided by Hildebrand [Science, 174, 490 (1971)] who adapted a theory of viscosity to self-diffusivity. He postulated that DAA = B(V-V )/V , where DAA is the self-diffusion coefficient, V is the molar volume, and V is the molar volume at which fluidity is zero (i.e., the molar volume of the solid phase at the melting temperature). The difference (V -V ) can be thought of as the free volume, which increases with temperature and B is a proportionality constant. 

Ertl and Dullien (ibid.) found that Hildebrand s equation could not fit their data with B as a constant. They modified it by applying an empirical exponent n (a constant greater than unity) to the volumetric ratio. The new equation is not generally useful, however, since there is no means for predicting n. The theory does identify the free volume as an important physical variable, since n > 1 for most liquids implies that diffusion is more strongly dependent on free volume than is viscosity. 

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... 

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. 

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. 

It is expected that the jump distance 8 will be relatively constant from one polymer to the next, and that variations in the diffusion constant are caused primarily by differences in the jump frequency c ). This fiequency depends on the probability of collecting a sufficient number of packets of free volume to give a hole of volume V of sufficient size that the diffusing molecule can move into it. The creation of a hole in the polymer requires the expenditure of a certain quantity of energy E, and the probability of forming such a hole can be calculated from Boltzmann statistics to be proportional to exp(—Application of this theory to both polymeric and small-molecule liquids has given jump distances 8 which conespond roughly to the dimensions of small molecules (i.e., 2-20... 

For diffusion of liquid through rubbery polymer composites, Fickian and non-Fickian diffusion theories are frequently used to describe the mechanism of transport, but for gas or vapour, other models have been developed to fit experimental data of diffusion profiles. The models of gas transport include Maxwell s model," free volume increase mechanism," solubility increase mechanism," nanogap hypothesis," Nielsen model, " " Bharadwaj model, ° Cussler model " " and Gusev and Lusti model, " etc. 

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. 

Equations 29 and 30 imply that free volume is the sole parameter in determining the rate of molecular rearrangements and transport phenomena such as diffusion and viscosity which depend on them. In older theories of liquids, - the temperature dependence of viscosity is determined by an energy barrier for hole formation. This leads to a viscosity proportional to 6 p AH RT), where A//, is the activation energy for flow, independent of temperature—an Arrhenius form. It will be shown in Section 6 that the latter type of temperature dependence is applicable at temperatures very far above Tg.- whereas equation 29 is applicable for 100 or so above Tg, and hybrid expressions may also be useful over a more extended range. 

According to the kinetic theory of gases, the self-diffusivity of a hard-sphere gas is given by DG = (2/5)(u)L, where (u) is the average velocity and L is the mean free path [4]. Because the mean free path of a confined particle in the liquid is about equal to the diameter of its confining volume, the contribution of the confined particle to the self-diffusivity of the liquid may be written... 

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