Developments of Flory-Huggins Theory

Having derived an expression for A5mfac, the second stage in the development of Flory-Huggins theory is to derive an expression for the effects of intermolecular interactions. In the original theory, this was considered only in terms of an enthalpy change, AJJ ix- Calculation of for polymer... 

Prausnitz and coworkers [36] have developed a theory based on hydrogen bonds in a lattice that permits holes and thus compressibility. This overcomes the deficiencies of Flory-Huggins theory however, Guassian chains are still assumed, and therefore it is difficult to fit experimental results with a single set of parameters over the entire range of swelling. However, the theory... 

Further improvements of Flory-Huggins theory (the third approximation) were possible. after development of experimental methods to determine the phase separation region, the spinodal, the critical point, and Flory-Huggins interaction parameter. 

We concluded the last section with the observation that a polymer solution is expected to be nonideal on the grounds of entropy considerations alone. A nonzero value for AH would exacerbate the situation even further. We therefore begin our discussion of this problem by assuming a polymer-solvent system which shows athermal mixing. In the next section we shall extend the theory to include systems for which AH 9 0. The theory we shall examine in the next few sections was developed independently by Flory and Huggins and is known as the Flory-Huggins theory. 

The lattice model that served as the basis for calculating ASj in the last section continues to characterize the Flory-Huggins theory in the development of an expression for AHj . Specifically, we are concerned with the change in enthalpy which occurs when one species is replaced by another in adjacent lattice sites. The situation can be represented in the notation of a chemical reaction ... 

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... 

The corresponding-states theory of polymer solution thermodynamics, developed principally by Prigogine and Flory, has provided a reliable predictive tool requiring only minimal information. We have seen here several examples of the use of the corresponding-states theory. We have also seen that the corresponding-states theory is a considerable improvement over the older Flory-Huggins theory. 

Flory-Huggins Theory. The simplest quantitative model for AGmx that includes the most essential elements needed for polymer blends is the Flory-Huggins theory, originally developed for polymer solutions (3,4). It assumes the only contribution to the entropy of mixing is combinatorial in origin and is given by equation 3, for a unit volume of a mixture of polymers A. and B. Here, pt and... 

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 ... 

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... 

To calculate AWm (the enthalpy of mixing) the polymer solution is approximated by a mixture of solvent molecules and polymer segments, and AW is estimated from the number of 1,2 contacts, as in Section 12.2.1. The terminology is somewhat different in the Flory-Huggins theory, however. A site in the liquid lattice is assumed to have z nearest neighbors and a line of reasoning similar to that developed above for the solubility parameter model leads to the expression... 

The solution properties of copolymers are much more compHcated. This is due mainly to the fact that the two copolymer components A and B behave differently in different solvents, and only when the two components are soluble in the same solvent will they exhibit similar solution properties. This is the case, for example for a nonpolar copolymer in a nonpolar solvent. It should also be emphasised that the Flory-Huggins theory was developed for ideal Hnear polymers. Indeed, with branched polymers with a high monomer density (e.g. star-branched polymers), the 0-temperature will depend on the length of the arms, and is in general lower than that of a linear polymer with the same molecular weight. 

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. 

The first stage in the development of the Flory-Huggins theory is to derive an expression for A mix when Alfniix = 0- mixing... 

Thus it is clear that the term "clustering" can be used in a variety of situations, and is not precisely defined. In the present case, it will be used in the context of the statistical thermodynamic treatment of binary solutions developed by Zlmm ( ) and Zimm and Lundberg ( ), which provides a calculation of a cluster integral, and which can be extended to specify a cluster size for each component. In describing the role of the "clustering" theory in relationship to previously developed solution theories, such as the widely used Flory-Huggins theory ( ), Zimm and Lundberg point out that "Our considerations are not intended as a replacement for the previous theories, but as... 

Another useful and simpler theory is the Lattice-Fluid (LF) Theory developed by Sanchez and Lacombe w-23-241. This theory has much in common with the Flory-Huggins theory but differs in one important respect in that it allows the lattice to have some vacant sites and to be compressible. Thus the compressible lattice theory is capable of describing volume changes on mixing as well as LCST and UCST behaviours. As with the theory of Flory and his co-workers, X12 (which is proportional to the change in energy that accompanies the formation of a 1 2 contact from a 1-1 and a 2-2 contact) is obtainable from experimental values of heats of mixing. 

In this section, we mention very briefly some recent theoretical developments, which go far beyond the simple Flory-Huggins theory. As was emphasized above, the Flory-Huggins theory suffers from two basic defects (i) Using a lattice model where polymers are represented as self-avoiding walks is a crude approximation, which neglects the disparity in size and shape of subunits of the two types of chain in a polymer blend, as well as packing constraints, specific interactions etc. (ii) Even within the realm of a lattice model, the statistical mechanics (involving approximations beyond the mean field approximation) is far too crude. 

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