The Flory–Huggins solution theory' is a mathematical model of the thermodynamics of polymer solutions which takes account of the great dissimilarity in molecular sizes in adapting the usual expression for the entropy of mixing. The result is an equation for the Gibbs free energy change
Contents
Theory
The thermodynamic equation for the Gibbs energy change accompanying mixing at constant temperature and (external) pressure is
A change, denoted by
The result obtained by Flory[1] and Huggins[2] is
The right-hand side is a function of the number of moles
Derivation
We first calculate the entropy of mixing, the increase in the uncertainty about the locations of the molecules when they are interspersed. In the pure condensed phases — solvent and polymer — everywhere we look we find a molecule.[3] Of course, any notion of "finding" a molecule in a given location is a thought experiment since we can't actually examine spatial locations the size of molecules. The expression for the entropy of mixing of small molecules in terms of mole fractions is no longer reasonable when the solute is a macromolecular chain. We take account of this dissymmetry in molecular sizes by assuming that individual polymer segments and individual solvent molecules occupy sites on a lattice. Each site is occupied by exactly one molecule of the solvent or by one monomer of the polymer chain, so the total number of sites is
From statistical mechanics we can calculate the entropy change, the increase in spatial uncertainty, as a result of mixing solute and solvent.
where
These are also the probabilities that a given lattice site, chosen at random, is occupied by a solvent molecule or a polymer segment, respectively. Thus
For a small solute whose molecules occupy just one lattice site,
In addition to the entropic effect, we can expect an enthalpy change.[5] There are three molecular interactions to consider: solvent-solvent
The total number of such contacts is
where
The enthalpy change is equal to the energy change per polymer monomer-solvent interaction multiplied by the number of such interactions
The polymer-solvent interaction parameter chi is defined as
It depends on the nature of both the solvent and the solute, and is the only material-specific parameter in the model. The enthalpy change becomes
Assembling terms, the total free energy change is
where we have converted the expression from molecules
The value of the interaction parameter can be estimated from the Hildebrand solubility parameters
where
This treatment does not attempt to calculate the conformational entropy of folding for polymer chains. (See the random coil discussion.) The conformations of even amorphous polymers will change when they go into solution, and most thermoplastic polymers also have lamellar crystalline regions which do not persist in solution as the chains separate. These events are accompanied by additional entropy and energy changes.
It should be noted that in the most general case the interaction
More advanced solution theories exist, such as the Flory-Krigbaum theory.