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 x

_{1}and x

_{2}, 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

*φ*and_{1}*φ*, the free energy of mixing per site would be:_{2}
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

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.

*V*is the molar volume of the solvent and δ_{0}_{1}and δ_{2}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.

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