Model Flory-Huggins, polymer mixing

Taking into account the modes in which the water can be sorbed in the resin, different models should be considered to describe the overall process. First, the ordinary dissolution of a substance in the polymer may be described by the Flory-Huggins theory which treats the random mixing of an unoriented polymer and a solvent by using the liquid lattice approach. If as is the penetrant external activity, vp the polymer volume fraction and the solvent-polymer interaction parameter, the relationship relating these variables in the case of polymer of infinite molecular weight is as follows ... 

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... 

Later Hildebrand [10] obtained the same result assuming that free volume available to the molecules per unit volume of liquid is the same for the polymer as for the solvent. The heat of mixing is defined as the difference between the total interaction energy in the mixture compared with that of pure components. Based on their lattice theory model, Flory [7,8,9] and Huggins [11,12] obtained the following expression for the heat of mixing ... 

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... 

Our discussion here explores active connections between the potential distribution theorem (PDT) and the theory of polymer solutions. In Chapter 4 we have already derived the Flory-Huggins model in broad form, and discussed its basis in a van der Waals model of solution thermodynamics. That derivation highlighted the origins of composition, temperature, and pressure effects on the Flory-Huggins interaction parameter. We recall that this theory is based upon a van der Waals treatment of solutions with the additional assumptions of zero volume of mixing and more technical approximations such as Eq. (4.45), p. 81. Considering a system of a polymer (p) of polymerization index M dissolved in a solvent (s), the Rory-Huggins model is... 

The Flory-Huggins model uses a simple lattice representation for the polymer solution and calculates the total number of ways the lattice can be occupied by small molecules and by connected polymer segments. Each lattice site accounts for a solvent molecule or a polymer segment with the same volume as a solvent molecule. This analysis yields the following expression for A5m, the entropy of mixing A l moles of solvent with N2 moles of polymer. 

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 free energy of mixing (per mole of lattice sites) of a polymer solution (according to the Flory-Huggins model) is... 

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. 

Traditionally, the thermodynamics of polymer mixtures was developed in terms of a lattice model, with each monomer unit of the polymer chains occupying a single lattice site. The free energy of mixing of polymers in solution can be described by the Flory-Huggins equation ... 

The Flory-Huggins lattice model assumes that a polymer chain consists of a number of equivalent segments. The extension to polymer solvent interactions assumes that the polymer solution consists of a three-dimensional lattice and each lattice site is occupied either by a polymer segment or by a solvent molecule. Flory and Huggins calculated the entropy change of mixing as... 

Polymers dissolved in mixed solvents show the phenomenon of Preferential Adsorption. Experimentally, the preferential adsorption eoeffieient, X, is determined. Xis the volume of one of the liquids sorbed in exeess by the polymer (per unit mass of polymer). In general, the Flory-Huggins model of polymer solutions is used to deseribe the Preferential Adsorption. More reeently, equation of state theories have been applied. ... 

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