A (very simplified) review of Flory-Huggins theory

This post is a simple and qualitative review of regular solution theory and how it extends into the Flory-Huggins theory. Regular solution theory is a statistical model that gives an expression for the free energy of mixing of a binary system. Though simple, this theory provides a useful picture for the phase behavior between two components and lays the framework for the Flory-Huggins theory.

Regular solution theory considers two contributions to the free energy of mixing; namely, the entropy of mixing and the enthalpy of mixing. For a binary system with components 1 and 2 having mole fractions x₁ and x₂, Boltzmann’s definition of entropy gives us the following expression for entropy of mixing per mole of lattice sites (I will omit the math here, but this comes from applying Stirling's equation to Boltmann's entropy):

In other words, the entropy of mixing describes the number of ways in which objects 1 and 2 can be arranged on a lattice and is thus related to intermolecular interactions. On the other hand, the enthalpy of mixing considers interactional contributions - that is, the exchange energies between pairwise nearest neighbors. By defining the interaction parameter χ as the exchange energy per molecule normalized by kT, we obtain the following expression for enthalpy of mixing per mole of lattice sites:

Combining these two expressions using ΔG_m = ΔH_m – TΔS_m, we obtain the well-known expression for the free energy of mixing per site:

By extending this, we can understand the Flory-Huggins theory, which describes the interactions between a polymer and solvent. In this model, we consider each lattice site to be the volume of one solvent molecule and we assume that each polymer occupies N lattice sites (such that the volume of one solvent molecule = the volume of one polymer segment = the volume of one lattice site). For volume fractions φ₁ and φ₂, the free energy of mixing per site would be:

In the above equation, the entropic terms favor mixing (note that as N increases, entropy decreases) and the enthalpic term opposes mixing (when chi is positive). The Flory-Huggins interaction parameter chi can be determined by the Hildebrand solubility parameters using:

where V₀ is the molar volume of the solvent and δ₁ and δ₂ are the solubility parameters of the polymer and solvent. 

Finally, it is worthwhile to mention some of the simplifying assumptions made in this theory. For example, it is a mean-field theory (treats local interactions the same way as bulk interactions), which is inaccurate for polymer solutions because crosslinking inevitably causes non-uniformity in the solution. This model also assumes that there is no change in free volume during mixing and that entropic contributions only come from translational configurations.