Activity Free volume contribution

The UNIFAC (Unified quasi chemical theory of liquid mixtures Functional-group Activity Coefficients) group-contribution method for the prediction of activity coefficients in non-electrolyte liquid mixtures was first introduced by Fredenslund et al. (1975). It is based on the Unified Quasi Chemical theory of liquid mixtures (UNIQUAC) (Abrams and Prausnitz, 1975), which is a statistical mechanical treatment derived from the quasi chemical lattice model (Guggenheim, 1952). UNIFAC has been extended to polymer solutions by Oishi and Prausnitz (1978) who added a free volume contribution term (UNIFAC-FV) taken from the polymer equation-of-state of Flory (1970). 

Mixtures of hydrocarbons are assumed to be athermal by UNIFAC, meaning there is no residual contribution to the activity coefficient. The free volume contribution is considered significant only for mixtures containing polymers and is equal to zero for liquid mixtures. The combinatorial activity coefficient contribution is calculated from the volume and surface area fractions of the molecule or polymer segment. The molecule structural parameters needed to do this are the van der Waals or hard core volumes and surface areas of the molecule relative to those of a standardized polyethylene methylene CH2 segment. UNIFAC for polymers (UNIFAC-FV) calculates in terms of activity (a,-) instead of the activity coefficient and uses weight fractions... 

Calculations using the UNIFAC-FV model are carried out as follows for a binary mixture of solute (a = 1) dissolved in a solid polymer (P = 2) (Goydan et al., 1989) The activity of the solute, aj, is separated into the three components the combinatorial contribution, a, the residual contribution, a, and the free-volume contribution,... 

YiC.Yir.Yifv components of activity coefficient represent combinatorial, residual and free-volume contributions 4.3.2... 

The free volume contribution to the activity coefficient fljfv for each component i is given by... 

The free volume contribution to the weight fraction activity coefficient, QjFV, is given by the following equation. 

As an example we will give here the expression for the activity coefficient of the solvent for a model which combines the p free-volume combinatorial term with a segment-based Wu-NRTL residual term [44]. The p free-volume combinatorial term combines combinatorial contributions and free-volume contributions [45]. The activity coefficient of the solvent is given by Eq. (44). 

Here, R is the gas constant, Mj is the solvent s molecular weight, and P 22 are the saturated vapour pressure and second virial coefficient of the pure solvent at temperature T, respectively. The required vapour pressures and virial coefficients can be calculated or estimated from known relationships [473-477]. Other treatments for the determination of y , which include a term to correct for the free volume contribution due to differences in the size of the solute and solvent, have also been used [478, 479]. The temperature dependence of the resulting solute activity coefficient is related to the infinite dilmion partial molar excess Gibbs energy (AG ) through Eq. (9). 

The activity coefficient is described in UNIFAC-FV and several other estimation models as being roughly composed of two or three different components. These components represent combinatorial contributions (yf) which are essentially due to differences in size and shape of the molecules in the mixture, residual contributions (yt) which are essentially due to energy interactions between molecules in the solution, and free volume (y[v) contributions which take into consideration differences between the free volumes of the mixture s components ... 

For systems containing polymers the free volume activity contribution must be calculated to account for the large differences between molecules. 

The procedure is based on the UNIFAC-Free Volume method developed by T. Oishi and J. M. Prausnitz, "Estimation of Solvent Activities in Polymer Solutions Using a Group-Contribution Method," Ind. Eng. Chem. Process Des. Dev., 17, 333 (1978). The UNIFAC-FV method is presented by Aa. Fredenslund, J. Gmehling, and P. Rasmussen, Vapor-Liquid Equilibria Using UNIFAC, Elsevier Scientific Publishing, New York (1977). The group... 

The Free-Volume Concept Group-Contribution Free-Volume Activity Coefficient Models... 

Many properties of pure polymers (and of polymer solutions) can be estimated with group contributions (GC). Examples of properties for which (GC) methods have been developed are the density, the solubility parameter, the melting and glass transition temperatures, as well as the surface tension. Phase equilibria for polymer solutions and blends can also be estimated with GC methods, as we discuss in Section 16.4 and 16.5. Here we review the GC principle, and in the following sections we discuss estimation methods for the density and the solubility parameter. These two properties are relevant for many thermodynamic models used for polymers, e.g., the Hansen and Flory-Hug-gins models discussed in Section 16.3 and the free-volume activity coefficient models discussed in Section 16.4. 

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 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.). ... 

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