Flory—Huggins theory polymer-solvent mixtures

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... 

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 deficiencies of the Flory-Huggins theory result from the limitations both of the model and of the assumptions employed in its derivation. Thus, the use of a single type of lattice for pure solvent, pure polymer and their mixtures is clearly unrealistic since it requires that there is no volume change upon mixing. The method used in the model to calculate the total number of possible conformations of a polymer molecule in the lattice is also unrealistic since it does not exclude self-intersections of the chain. Moreover, the use of a mean-field approximation to facilitate this calculation, whereby it is assumed that the segments of the previously added polymer molecules are distributed uniformly in the lattice, is satisfactory only when the volume fraction (f>2 of polymer is high, as in relatively concentrated polymer solutions. 

In dilute polymer solutions, the polymer molecules are isolated from each other by regions of pure solvent, i.e., the polymer segments are not uniformly distributed in the lattice. In view of this, the Flory-Huggins theory is least satisfactory for dilute polymer solutions and only applies to concentrated solutions or mixtures. Furthermore, the interaction param-... 

The concepts derived from polymer solutions have been extended to considerations of polymer/polymer miscibility Scott [18] discussed the case of two homopolymers dissolved in a common solvent forming an equilibrium mixture and derived expressions for the free energy of mixing, using the Flory-Huggins theory... 

The assumption of unperturbed chain statistics. Implicit in Flory-Huggins theory is the assumption that the long-range chain statistics of polymer chains are ideal random walks. This is not to be expected a polymer chain in a solvent collapses as conditions are changed to bring about phase separation between the polymer and the solvent (Grosberg and Khokhlov 1994). One would expect a polymer chain in a mixture to do the same as the conditions for phase separation were approached (Sariban and Binder 1987). 

The Flory-Huggins theory, originally developed for polymer/solvent systems, was extended to polymer blends and other multicomponent mixtures by R. L. Scott, J. Polym. Sci. 9,423 (1952). 

In this section the basic principles of membrane formation by phase inversion will be described in greater detail. All phase inversion processes are based on the same thermodynamic principles, since the starting point in all cases is a thermodynamically stable solution which is subjected to demixing. Special attention will be paid to the immersion precipitation process with the basic charaaeristic that at least three components are used a polymer, a solvent and a nonsolvent where the solvent and nonsolvent must be miscible with each other. In fact, most of the commercial phase inversion membranes are prepared from multi-component mixtures, but in order to understand the basic principles only three component systems will be considered. An introduction to the thermodynamics of. polymer solutions is first given, a qualitatively useful approach for describing polymer solubility or polymer-penetrant interaction is the solubility parameter theory. A more quantitative description is provided by the Flory-Huggins theory. Other more sophisticated theories have been developed but they will not be considered here. 

Fortunately these complications do not exist when both Na and Nb are large the mean field (Flory-Huggins) theory is qualitatively correct for polymer mixtures without solvent. This can be shown from a detailed study of fluctuation effects.We can summarize the results by the following statement, which is used often in this chapter. If we focus our attention on the species (A) with the longest chains (A/a > Nb), each chain has a size Roa = o/Va and spans a volume / oa /Va . We now define a para-... 

In this section we consider a general model that has broad applicability to phase transitions in soft materials the Landau theory, which is based on an expansion of the free energy in a power series of an order parameter. The Landau theory describes the ordering at the mesoscopic, not molecular, level. Molecular mean field theories include the Maier-Saupe model, discussed in detail in Section 5.5.2. This describes the orientation of an arbitrary molecule surrounded by all others (Fig. 1.5), which set up an average anisotropic interaction potential, which is the mean field in this case. In polymer physics, the Flory-Huggins theory is a powerful mean field model for a polymer-solvent or polymer-polymer mixture. It is outlined in Section 2.5.6. 

We shall in section 4.2 deal with regular solutions of small-molecule substances. The construction of phase diagrams from the derived equations is demonstrated. The Flory—Huggins mean-field theory derived for mixtures of polymers and small-molecule solvents is dealt with in section 4.3. It turns out that the simple Flory—Huggins theory is inadequate in many cases. The scaling laws for dilute and semi-dilute solutions are briefly presented. The inadequacy of the Flory-Huggins approach has led to the development of the equation-of-state theories which is the fourth topic (section 4.6) Polymer-polymer mixtures are particularly complex and they are dealt with in section 4.7. 

The most relevant theory for modeling the free energy of binary polymer mixtures is the Flory-Huggins theory, initially employed for solvent-solvent and polymer-solvent mixtures. This theory was independently derived by Flory [4, 5] and Huggins [6, 7]. The key equation (combined from discussions earlier in this chapter on entropy and enthalpy of mixing) is ... 

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