Temperature-responsive Polymer - Thermodynamics

Thermodynamics

Polymers dissolve in a solvent when the Gibbs energy of the system decreases, i.e., the change of Gibbs energy (ΔG) is negative. From the known Legendre transformation of the Gibbs-Helmholtz equation it follows that ΔG is determined by the enthalpy of mixing (ΔH) and entropy of mixing (ΔS).

Without interactions between the compounds there would be no enthalpy of mixing and the entropy of mixing would be ideal. The ideal entropy of mixing of multiple pure compounds is always positive (the term -T∙ΔS is negative) and ΔG would be negative for all compositions, causing complete miscibility. Therefore, the fact that miscibility gaps are observed can only be explained by interaction. In the case of polymer solutions, polymer-polymer, solvent-solvent and polymer-solvent interactions have to be taken into account. A model for the phenomenological description of polymer phase diagrams was developed by Flory and Huiggens (see Flory-Huggins solution theory). The resulting equation for the change of Gibbs energy consists of a term for the entropy of mixing for polymers and an interaction parameter that describes the sum of all interactions.

where

• R = universal gas constant
• m = number of occupied lattice sites per molecule (for polymer solutions m₁ is approximately equal to the degree of polymerization and m₂=1)
• φ = volume fraction of the polymer and the solvent, respectively
• χ = interaction parameter

A consequence of the Flory-Huiggens theory is, for instance, that the UCST (if it exists) increases and shifts into the solvent-rich region when the molar mass of the polymer increases. Whether a polymer shows LCST and/or UCST behavior can be derived from the temperature-dependence of the interaction parameter (see figure). It has to be noted that the interaction parameter not only comprises enthalpic contributions but also the non-ideal entropy of mixing, which again consists of many individual contributions (e.g., the strong hydrophobic effect in aqueous solutions). For these reasons, classical Flory Huiggens theory cannot provide much insight into the molecular origin of miscibility gaps.