Free volume theory, of liquids

Arakawa, K. Ondhe free volume theory of liquid. J. Phys. Soc. Japan 9, 647 (1954). 

Note that in the one-dimensional case, the canonical partition function has the form of Q = Vj1 /N A where is the free volume. In this case, the quantity Vf is indeed the volume unoccupied by particles. In the free volume theories of liquids, this form of the partition function was assumed to hold for a three-dimensional liquid. 

The free volume theory of liquids dates from the beginning of the 20th century. Two expressions for the free volume fraction, f, have been proposed, either f = (V - V )/V or less frequently used I d = (V - occy occ (Vocc is Ihe occupied volume). The theory was used to interpret the temperature (T) and pressure (P) dependencies of liquid viscosity [Batschinski, 1913]. The was defined as the specific volume at which the liquid viscosity... 

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 ... 

Detonation, Free Volume Theory of the Liquid State Developed by Eyring et al and by Lennard-Jones-Devonshire. The free volume theory of the liquid state developed by Eyring Hirshfelder (Ref 1) and by Lennard-Jones Devonshire (Ref 2) has provided a useful approximate description of the thermodynamic props of liquids in terms of intermolecular forces... 

See Detonation, Free Volume Theory of the Liquid State Developed by Eyring et al in this Volume... 

Detonation, free volume theory of the liquid state developed by Eyring et al and by Lennard-Jones-Devonshire 4 D349... 

The results of Collins and Raffel are obtained by making use of the uniform potential free-volume theory of Eyring and Hirsch-felder17 which gives the following equation of state for the liquid ... 

Kirkwood, J. G., Critique of the free volume theory of the liquid state, J. Chem. Phys., 18, 380-382(1950). 

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... 

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. 

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. 

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. 

Pratt, L. R., Hummer, G., and Garde, S. (1999). Theories of hydrophobic effects and the description of free volume in complex liquids. In New Approaches to Problems in Liquid State Theory (C. Caccamo, J.-P., Hansen, and G. Stell, eds.), vol. 529, pp. 407-420. Kluwer, Netherlands. NATO Science Series. 

It was found that Afi Tg and Aa Tg are not constant and therefore the SB equation has limited applicability. Hie results indicate an increase in Aa Te with increasing Tg. Therefore it is inadmissible to use the product A a Tg as a universal value in any theoretical discussion of the glass-transition phenomenon. At the same time, this conclusion in no way excludes the free-volume theory and the role of free-volume in the transition from the glassy to the liquid or rubberlike state. 

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