Lattice theories free volume theory

Consolati, G., Quasso, R, Simha, R., and Olson, G. B., On the relation betwen positron annihilation lifetime spectroscopy and lattice-hole-theory free volume, J. Polym. Sci. B, 43, 2225-2229 (2005). 

Several theories have been developed to describe glass transition, such as the thermodynamic theory [24, 25], free volume theory [26], or kinetic theory [27]. The former employs the lattice model to establish the partition function and the entropy of polymer can be calculated through this partition function. The latter takes the volume changes during the glass transition stage into account. 

Nine different equations-of-state, EOS theories are described including Flory Orwoll Vrij (FOV) Prigogine Square Well cell model, and the Sanchez Lacombe free volume theory. When the mathematical complexity of the EOS theories increases it is prudent to watch for spurious results such as negative pressure and negative volume expansivity. Although mathematically correct these have little physical meaning in polymer science. The large molecule effects are explicitly accounted for by the lattice fluid EOS theories. The current textbooks on thermodynamics discuss... 

Free-volume theory is an improved cell or lattice model for the liquid state by introduction of vacancies in the lattice. In the free-volume theory, the Simha-Somcynsky equation of state of a polymer system is written as (10)... 

Other free volume theories such as those of Sanchez and Lacombe and Simha and Somcynski are based on a lattice model and all or part of the free volume arises from vacancies on the lattice, unlike the Flory theory where free volume arises from an overall increase in molecular separations. Such theories are discussed in the chapter on polymer solutions (Volume 2, Chapter 3) and have not been much used in relation to polymer mixtures. Their use may -well prove to be valuable since, especially using the theory of Simha and Somcynski, they much better describe the properties of the pure components. 

The model of Marchetti et al. is based on the compressible lattice theory which Sanchez and Lacombe developed to apply to polymer-solvent systems which have variable levels of free volume [138-141], This theory is a ternary version of classic Flory-Huggins theory, with the third component in the polymer-solvent system being vacant lattice sites or holes . The key parameters in this theory which affect the polymer-solvent phase diagram are ... 

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). 

Later Hildebrand [10] obtained the same result assuming that free volume available to the molecules per unit volume of liquid is the same for the polymer as for the solvent. The heat of mixing is defined as the difference between the total interaction energy in the mixture compared with that of pure components. Based on their lattice theory model, Flory [7,8,9] and Huggins [11,12] obtained the following expression for the heat of mixing ... 

The existing theories adopt, more or less, a successive approximate method in which one first considers segments to be independent of each other and then introduces their connectivity in some way or the other. The first step has been taken not only by the lattice theories but also by the free volume theories (43). However, it is the second step which is not simple and important. While this second step was taken in our theory discussed in the previous section, one must still try to develop an improved treatment of the chain connectivity, avoiding, for example, the product assumption such as Eq. (6.7). 

The lattice fluid equation-of-state theory for polymers, polymer solutions, and polymer mixtures is a useful tool which can provide information on equa-tion-of-state properties, and also allows prediction of surface tension of polymers, phase stability of polymer blends, etc. [17-20]. The theory uses empty lattice sites to account for free volume, and therefore one may treat volume changes upon mixing, which are not possible in the Flory-Huggins theory. As a result, lower critical solution temperature (LCST) behaviors can, in principle, be described in polymer systems which interact chiefly through dispersion forces [17]. The equation-of-state theory involves characteristic parameters, p, v, and T, which have to be determined from experimental data. The least-squares fitting of density data as a function of temperature and pressure yields a set of parameters which best represent the data over the temperature and pressure ranges considered [21]. The method,however,requires tedious experiments to deter-... 

C. Lattice Theories (Hole, Cell and Free Volume Theories).. 238... 

Another drawback of the H-F theory was the initial assumption that all lattice cells are occupied by either solvent molecules or polymeric segments that are of equal size. As a consequence the free volume contribution was neglected. Maron [1959] pointed out that dissolution of polymer is associated with volume changes — his modification of... 

We want to pursue the subject by starting with a brief review for the present purposes of the essentials of lattice-hole theory, then follow with a consideration of free-volume mobility connections, and continue with some comparisons of experiment versus theory. Finally, we propose and sketch modifications of the theory. These may open the way to generalizations and more insightful relations to empirical formulations, such as the KAHR model [Kovacs et al., 1977, 1979] for volume relaxation. 

The Simha and Somcynsky (S-S) [1969] cell-hole theory is based on the lattice-hole model. The molecular segments of an -mer occupy ay-fraction of the lattice sites, while the remaining randomly distributed sites, /i = 7 — y, are left as empty holes. The fraction /i is a measure of the free-volume content. The goal was to provide improved description of fluids, ranging from low-molecular-weight spherical molecules (such as argon) to macromolecular chains. The S-S configurational partition function is... 

Much of the work stems from Simha-Somcynsky (S-S) [1969] hole theory, developed originally to describe polymers in the liquid state. They introduced the free volume by using the formalism of vacant cells or holes in a lattice and developed an equation of state that could be used to calculate the fraction of sites occupied and hence the fractional free volume. As discussed in Chapter 6, the concept has been developed further by Simha and his co-workers. 

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