Mixtures Flory-Huggins treatment

Sanchez and Lacombe (1976) developed an equation of state for pure fluids that was later extended to mixtures (Lacombe and Sanchez, 1976). The Sanchez-Lacombe equation of state is based on hole theory and uses a random mixing expression for the attractive energy term. Random mixing means that the composition everywhere in the solution is equal to the overall composition, i.e., there are no local composition effects. Hole theory differs from the lattice model used in the Flory-Huggins theory because here the density of the mixture is allowed to vary by increasing the fraction of holes in the lattice. In the Flory-Huggins treatment every site is occupied by a solvent molecule or polymer segment. The Sanchez-Lacombe equation of state takes the form... 

Volume fraction of species a in Flory-Huggins treatment of polymer mixtures. 

Interaction parameter in Flory-Huggins treatment of polymer mixtures after normalization on a per monomer basis this becomes Xay also susceptibilities in discussion of density functional theories. Applied external field acting on species a in conformation X". Applied external field as a function of position acting on species a. Contribution of attractive interactions to the second virial coefficient for species pair ay also van der Waals coefficient. 

Although a modelling of a liquid polymer mixture on a lattice may first look rather artificial, it makes sense because it retains the important aspects of both the entropic and enthalpic part of A mix- In recent years, lattice models have gained a renewed importance as a concept which is suitable for computer simulations. Numerical investigations make it possible to check and assess the validity range of the Flory-Huggins treatment. In fact, limitations exist and, as analytical calculations are difficult, simulations are very helpful and important. We shall present one example in a later section. 

Af (PS) = 2 10, M(PVME) — 4.7 10. For molecular weights in this range the contribution of the translational entropy becomes very small indeed and mixing properties are mostly controlled by x- The curved appearance of the binodal, which contrasts with the result of the model calculation in Fig. 3.18 where we obtained for polymers with medium or high molecular weights a nearly horizontal line, is indicative of a pronounced compositional dependence of X- This represents a case where the Flory-Huggins treatment does not provide a comprehensive description. Interactions in this mixture are of a complex nature and apparently change with the sample composition, so that it becomes impossible to represent them by just one constant. 

Hence, we obtain an explicit expression for the collective response coefficient in terms of the known single chain response coefficients and a . Next, we turn to a non-athermal mixture. The difference in the interaction between like and unlike chains may be approximately accounted for in the spirit of the Flory-Huggins treatment, by introduction of the x-parameter. This is achieved by changing Eq. (A.92) into... 

The Flory-Huggins Treatment of Polymer Mixtures 111 and add up to unity,... 

Dealing with the osmotic pressure we first consider the case of an electrically neutral network in a good solvent. We refer here to the Flory-Huggins treatment of A/B-polymer mixtures (Sect. 4.1) and use it for a network(A)-solvent(B) system, by just setting... 

The well-known Flory treatment [50-52] of the en-thropic contribution to the Gibbs energy of mixing of polymers with solvents is still the simplest and most reliable theory developed. It is quite apparent, however, that the Flory-Huggins theory was established on the basis of the experimental behavior of only a few mixtures investigated over a very narrow range of temperature. Strict applications of the Flory-Huggins approach... 

Equation (12-23) suffers from the same limitations as the simple solubilty parameter model, because the expression for Wm is derived by assuming that in-termolecular forces are only nondirectional van der Waals interactions. Specific interactions like ionic or hydrogen bonds arc implicitly eliminated from the model. The solubility parameter treatment described to this point cannot take such inler-actions into account because each species is assigned a solubility parameter that is independent of the nature of the other ingredients in the mixture. The x parameter, on the other hand, refers to a pair of components and can include specific interactions even if they are not explicitly mentioned in the basic Flory-Huggins theory. Solubility parameters are more convenient to use because they can be assigned a priori to the components of a mixture, x values are more realistic, but have less predictive use because they must be determined by experiments with the actual mixture. 

Also, the original Hildebrand approach has been refined to take into account the contribution of polar groups and hydrogen bsolubility parameters. These mndifications of the Flory-Huggins theory and of the solubility parameter concept have made these methods an even more useful tool in the description of solutions, especially of mixtures containing polymer compounds. A comprehensive treatment of these extensions of Flory-Huggins and Hildebrand s theories, as well as the new equation of state approach of Flory (1965), bns re ntly been published (Shinoda, 1978 Olahisi et al 1979). 

Most theoretical procedures for deriving expressions for AG iix start with the construction of a model of the mixture. The model is then analyzed by the techniques of statistical thermodynamics. The nature and sophistication of different models vary depending on the level of the statistical mechanical approach and the seriousness of the mathematical approximations that are invariably introduced into the calculation. The immensely popular Flory-Huggins theory, which was developed in the early 1940s, is based on the pseudolattice model and a rather low-level statistical treatment with many approximations. The theory is remarkably simple, explains correctly (at least qualitatively) a large number of experimental observations, and serves as a starting point for many more sophisticated theories. 

Finally, segment-molar activity coefficients Ta and fa, are introduced to describe all deviations from the Flory-Huggins mixture (with x = 0). Within the continuous treatment the segment-mole fraction Vb, will be replaced by [29]... 

The melting temperature-composition relation for the conunon situation of two dissimilar polymers, only one of which crystallizes, was formulated by Nishi and Wang.(17) This relation is based on the free energy of mixing of two dissimilar polymers in the disordered state, as given by Scott (18), within the framework of the Flory-Huggins lattice treatment.(7) The chemical potentials of each species in the binary mixture can be expressed as... 

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