Free-volume approach

Matthews-Akgerman The free-volume approach of Hildebrand was shown to be valid for binary, dilute liquid paraffin mixtures (as well as self-diffusion), consisting of solutes from Cg to Cig and solvents of Cg and C o- The term they referred to as the diffusion volume was simply correlated with the critical volume, as = 0.308 V. We can infer from Table 5-15 that this is approximately related to the volume at the melting point as = 0.945 V, . Their correlation was vahd for diffusion of linear alkanes at temperatures up to 300°C and pressures up to 3.45 MPa. Matthews et al. and Erkey and Akger-man completea similar studies of diffusion of alkanes, restricted to /1-hexadecane and /i-octane, respectively, as the solvents. 

To account for the variation of the dynamics with pressure, the free volume is allowed to compress with P, but differently than the total compressibility of the material [22]. One consequent problem is that fitting data can lead to the unphysical result that the free volume is less compressible than the occupied volume [42]. The CG model has been modified with an additional parameter to describe t(P) [34,35] however, the resulting expression does not accurately fit data obtained at high pressure [41,43,44]. Beyond describing experimental results, the CG fit parameters yield free volumes that are inconsistent with the unoccupied volume deduced from cell models [41]. More generally, a free-volume approach to dynamics is at odds with the experimental result that relaxation in polymers is to a significant degree a thermally activated process [14,15,45]. 

Various theories have been proposed to describe the transport in all of these types of polymer membranes. Theories for macroporous and microporous membranes have been based on hydrodynamic and frictional considerations while those for nonporous gels have been based on Eyring s theory and use a free volume approach to describe the movement of solute through the mesh of the polymer. 

Free volume approach to polystyrene melt viscosity. J. Appl. Phys- 29, 1395-1398 (1958). 

The Oishi-Prausnitz model cannot be defined strictly as a lattice model. The combinatorial and residual terms in the original UNIFAC and UNIQUAC models can be derived from lattice statistics arguments similar to those used in deriving the other models discussed in this section. On the other hand, the free volume contribution to the Oishi-Prausnitz model is derived from the Flory equation of state discussed in the next section. Thus, the Oishi-Prausnitz model is a hybrid of the lattice-fluid and free volume approaches. 

The free volume approach has been an increasingly popular method to relate polymer structure to gas transport properties. The basic premise of this technique is that a polymer with an open, poorly packed structure will have a large unoccupied free volume through which a gas can diffuse with ease. In a typical model, set forth by Lee (d2.) a specific free volume, SFV, is derived from the difference between the molar volume, Vm, (determined from the experimental density of a polymer) and the occupied volume, Vo, (calculated using a group additive method, in this case, that of Bondi) (4(1). ... 

A detailed analysis of the 3, 3 -disubstituted bisphenol containing polycarbonates provided an explanation for their observed oxygen permeabilities. However such an approach is not of general utility. Using an existing free-volume approach, a qualitative... 

Kreiiuss and Frisch also have tmated the effect of crystallinity on transport in tubhety polymers using the so-called free-volume approach, which is discussed helow. In essence, their ideas suggest that intro-daction of crystallinity reduces the freedom of motion of amorphous chain segments between crystals. By... 

LIQUIDS, GLASSES, AND THE GLASS TRANSITION A FREE-VOLUME APPROACH... 

Liquids, Glasses, and the Glass Transition A Free-Volume Approach... 

Free volume approach to the combinatorial entropy The combinatorial entropy of mixing can be more readily derived by a free volume approach which renders the assumptions inherent in the Flory-Huggins theory more transparently obvious. Anticipating what is to be presented in Section 3.3, vis-d-vis the equation-of-state theory, we present a brief account of this alternative derivation. 

The preceding derivation makes clear, in the context of the free volume approach, what assumptions are necessary to derive the Flory-Huggins combinatorial entropy of mixing. Specifically, these are that the solvent and polymer possess identical free volume fractions and that the total free volume is conserved on mixing. 

The term is the communal entropy according to this free volume approach. 

We stress that the foregoing thermodynamic notions for aqueous dispersions are quite speculative. A better understanding awaits the formulation of theories of polymer solution thermodynamics that comprehend aqueous systems. These may well include factors that are absent from the free volume approach. 

Finally, we stress that the free volume approach is only applicable to nonpolar systems. Aqueous dispersions fall outside its scope. This is vividly illustrated by the data of Evans et al. (1975), who determined experimentally that d(UCFT)/d7 = — 1 x 10 KPa for latex particles sterically stabilized by poly(oxyethylene) in aqueous 0-43 molal magnesium sulphate solutions. Both the sign and magnitude of this quantity is different from that predicted by free volume theory for the UCFT of non aqueous dispersions. Paradoxically, it falls in line with the predictions, both in sign and magnitude, published by Croucher and Hair (1979) for the pressure dependence of the LCFT of poly(a-methylstyrene) in -butyl chloride. This may be merely coincidental, but the very small pressure dependence exhibited by the UCFT of aqueous sterically stabilized dispersions emphasizes the major differences between the origins of flocculation at the UCI T for aqueous and nonaqueous dispersions. The small pressure dependence observed for aqueous systems is scarcely surprising since the UCFT of an aqueous dispersion occurs far from the critical point of water whereas that for nonaqueous dispersions is quite close to the critical point of the dispersion medium. 

Grest, G. S., and Cohen, M. H., Liquids, glasses, and the glass transition a free-volume approach, Arfv. Chem. Phys., 48, 455-525 (1981). 

Various types of coupled non-linear Fickian diffusion processes were numerically simulated using the free-volume approach given by equation [12.8], as well as non-Fickian transport. The non-Fickian transport was modeled as a stress-induced mass flux that typically occurs in the presence of non-uniform stress fields normally present in complex structures. The coupled diffusion and viscoelasticity boundary value problems were solved numerically using the finite element code NOVA-3D. Details of the non-hnear and non-Fickian diffusion model have been described elsewhere [14]. A benchmark verification of the linear Fickian diffusion model defined by equations [12.3]-[12.5] under a complex hygrothermal loading is presented in Section 12.6. 

Raj, J. M., Kumaraswamy, G. N., Ranganathaiah, C., Interfacial stabilization of binary polymer blends through radiation treatment A free volume approach. Physica Status Solidi (C) 2009,6,2404—2406. 

Cohen, M.H. and Crest, G.S. (1979) Liquid-glass transition, a free-volume approach. Phys. Rev. B, 20, 1077-1098. 

The entropy of mixing was originally calculated by Flory using a lattice approach, but it can also be derived by using a free volume approach. The entropy of mixing, A5, can be denoted by the following equation (1) ... 

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