External degree of freedom parameter

Here C., is an external degree of freedom parameter for the solvent. 

Cj = an external degree of freedom parameter for solvents = 1.1 v j = the reduced volume of solvent i, given by Equation (3C-14) vm = the reduced volume of the mixture, given by Equation (3C-15) The reduced volume of solvent i, v j, is given by the following equation. 

Step 8 Obtain the external degree of freedom parameters, CTo k, CT k, and Ck°, for each group k from Table 3D-1. 

To get a predictive model, the reduced volumes and the external degree of freedom parameter are not calculated from Flory s equation of state, Equation [4.4.72], but from some simple approximations as given by the following relations ... 

C external degree of freedom parameter of the solvent I, usually fixed = 1.1... 

The molar hard-core is calculated from the pure component hard-core molar volumes using a linear mixing mle [Eq. (57)]. The same type of mixing rule is used for the number of external degrees of freedom parameter C [Eq. (58)]. 

In the UNIFAC-FV model as suggested by Oishi and Prausnitz the parameters C (3c is the number of external degrees of freedom) and b are set to constant values (c, = 1.1 and b = 1.28). The performance is rather satisfactory, as shown by many investigators,for a large variety of polymer-solvent systems. When referring to UNIFAC-FV in the rest of this section, the original version with constant b and c values is employed. 

The spinodal, binodal and critical point Equations derived on the basis of this theory will be discussed later. When the theory has been tested it has been found to describe the properties of polymer blends much better than the classical lattice theories 17 1B). It is more successful in interpreting the excess properties of mixtures with dispersion or weak attraction forces. In the case of mixtures with a strong specific interaction it suffers from the results of the random mixing assumption. The excess volumes observed by Shih and Flory, 9, for C6H6-PDMS mixtures are considerably different from those predicted by the theory and this cannot be resolved by reasonable alterations of any adjustable parameter. Hamada et al.20), however, have shown that the theory of Flory and his co-workers can be largely improved by using the number of external degrees of freedom for the mixture as ... 

Eqs 2.23 and 2.24 provide a corresponding states description of PVT behavior of any liquid. Once the four characteristic parameters P, V, T and 3c/s are known, the specific volume and all its derivatives are known in the full range of P and T. For linear molecules the external degrees of freedom are proportional to number of segments 3Cj = Sj -I- 3. Thus, for linear polymers, where Sj 3, the external degree of freedom 3c./s. = 1, i.e., for polymers, only three parameters P, V, and T, are required. Typical values of P, V, and T for selected polymers are listed in Table 2.3. 

The most important of the new parameters is the so-called structural effect, which is related to the number of degrees of freedom 3c that a molecule possesses, divided by the number of extonal contacts q. This stmctural factor dq) is a measure of the number of external degrees of freedom per segment and changes with the length of the component. Thus, the ratio decreases as a liquid becomes increasingly polymeric. 

Historically, the MNSJ equations (6.33) and (6.34) were evaluated using x-ray data on the crystalline specific volume. Thus, the isobars of linear polyethylene (LPE), poly(viny lidene fluoride) (PVDF), and poly(chlorotrifluoroethylene) (PCIF3), and an isotherm at atmospheric pressure of LPE were fitted to the theoretical dependencies [Simha and Jain, 1978 Jain and Simha, 1979a,b]. The fitting requires five characteristic parameters P T V, c/s, and 60, the last two having universal values. While in the melt, the macromolecular external degrees of freedom 3c/s 1 in the crystalline polymers, c/s l. For different crystalline species the characteristic reduced quantum temperature value adopted at 0 (K) is o = (hpvo/ksT ) 0.022, where... 

In order to avoid the uncertainty of the parameter c which characterizes the decrease in the external degrees of freedom, a much simpler mathematical form of the equation of state was proposed on the basis of the Ising (lattice) fluid model (88,89)... 

We now consider mixtures of Na molecules A occupying sites of the quasi lattice and characterized by Sc i external degrees of freedom with Nb molecules B diaracterized by ra and 3cb. The intermolecolar eneigies between the elements of A and B are represented by the two parameter interaction law (2.4.2)... 

The invariant hole size and energy of hole formation in the Lacombe-Sanchez model implies a dependance of the internal energy on the density that is strictly of the van der Waals form, as in Flory s theory. Another, somewhat more complicated, model developed by Simha and his collaborators is similar to that of Sanchez and Lacombe in its use of a liquid lattice with vacant sites, but it also retains features of Prigogine s earlier cell model in the c parameter for external degrees of freedom and a lattice energy with a density dependence based on an effective (6-12) pair potential. Like the other theories, this one has been successful in correlating equation of state data both for neat polymer liquids and for nondilute solutions. 

A phase is defined as the part of the system that has uniquely distinguishing properties from the other part of the system. That property can be, for example, density (e.g., water-ice-water vapor) or different crystallographic forms (e.g., a — Pd//3 - Pd). The coexistence and number of different phases p depends on the number of components c, and on external physical parameters called degrees of freedom /. These are most typically pressure and temperature. The governing relationship is the Gibbs phase rule. 

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