The Flory-Huggins Treatment of Polymer Mixtures

Flory and Huggins devised a general scheme that enables one to deal with the mixing properties of a pair of poljuners. It provides a basic understanding of the occurrence of different types of phase diagrams, in dependence on temperature and the molar masses. 

The decomposition of A mix in these two contributions points to the two main aspects of the mixing process, but this alone would not be of much value. What is needed for actual use are explicit expressions for A St and A ioc, so that the sum of the two contributions can be calculated. The Flory-Fluggins treatment is based on approximate equations for both parts. We formulate them first and then discuss their origins and the implications. The equations have the following forms  

Introducing the volmne fractions (j A and of the two components in the mixture, given by 

It includes two parameters. The less important one is Vq, denoting the (molar) volume of a reference unit common to both polymers. Principally it can be chosen arbitrarily, but usually it is identified with the volume occupied by one of the monomeric units. The decisive factor is the Flory-Huggins parameter y. It is dimensionless and determines in empirical manner the change in the local free energy per reference unit. 

the effective coordination number Zes gives the number of nearest neighbors (in reference units) on other chains, and a division by 2 is necessary to avoid a double count of the pair contacts. An increase in the local Gibbs free energy only results if an AB-pair is formed and this occurs with a probability 

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

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. 

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... 

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... 

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. 

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... 

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). 

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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