Flory–Huggins solution theory


Flory–Huggins solution theory is a lattice 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 for mixing a polymer with a solvent. Although it makes simplifying assumptions, it generates useful results for interpreting experiments.

Theory

The thermodynamic equation for the Gibbs energy change accompanying mixing at constant temperature and pressure is
A change, denoted by, is the value of a variable for a solution or mixture minus the values for the pure components considered separately. The objective is to find explicit formulas for and, the enthalpy and entropy increments associated with the mixing process.
The result obtained by Flory and Huggins is
The right-hand side is a function of the number of moles and volume fraction of solvent, the number of moles and volume fraction of polymer, with the introduction of a parameter to take account of the energy of interdispersing polymer and solvent molecules. is the gas constant and is the absolute temperature. The volume fraction is analogous to the mole fraction, but is weighted to take account of the relative sizes of the molecules. For a small solute, the mole fractions would appear instead, and this modification is the innovation due to Flory and Huggins. In the most general case the mixing parameter,, is a free energy parameter, thus including an entropic component.

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. 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. 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
is the number of solvent molecules and is the number of polymer molecules, each of which has segments.
For a random walk on a lattice we can calculate the entropy change as a result of mixing solute and solvent.
where is Boltzmann's constant. Define the lattice volume fractions and
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, equals one, the volume fractions reduce to molecular or mole fractions, and we recover the usual entropy of mixing.
In addition to the entropic effect, we can expect an enthalpy change. There are three molecular interactions to consider: solvent-solvent, monomer-monomer , and monomer-solvent. Each of the last occurs at the expense of the average of the other two, so the energy increment per monomer-solvent contact is
The total number of such contacts is
where is the coordination number, the number of nearest neighbors for a lattice site, each one occupied either by one chain segment or a solvent molecule. That is, is the total number of polymer segments in the solution, so is the number of nearest-neighbor sites to all the polymer segments. Multiplying by the probability that any such site is occupied by a solvent molecule, we obtain the total number of polymer-solvent molecular interactions. An approximation following mean field theory is made by following this procedure, thereby reducing the complex problem of many interactions to a simpler problem of one interaction.
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 and to moles and by transferring the Avogadro constant to the gas constant.
The value of the interaction parameter can be estimated from the Hildebrand solubility parameters and
where is the actual volume of a polymer segment.
In the most general case the interaction and the ensuing mixing parameter,, is a free energy parameter, thus including an entropic component. This means that aside to the regular mixing entropy there is another entropic contribution from the interaction between solvent and monomer. This contribution is sometimes very important in order to make quantitative predictions of thermodynamic properties.
More advanced solution theories exist, such as the Flory–Krigbaum theory.

Liquid-liquid phase separation

Polymers can separate out from the solvent, and do so in a characteristic way. The Flory-Huggins free energy per unit volume, for a polymer with monomers, can be written in a simple dimensionless form
for the volume fraction of monomers, and. The osmotic pressure is.
The polymer solution is stable with respect to small fluctuations when the second derivative of this free energy is positive. This second derivative is
and solution first becomes unstable then this and the third derivative are both equal to zero. A little algebra then shows that the polymer solution first becomes unstable at a critical point at
This means that for all values of the monomer-solvent effective interaction is weakly repulsive, but this is too weak to cause liquid/liquid separation. However, when, there is separation into two coexisting phases, one richer in polymer but poorer in solvent, than the other.
The unusual feature of the liquid/liquid phase separation is that it is highly asymmetric: the volume fraction of monomers at the critical point is approximately, which is very small for large polymers. The amount of polymer in the solvent-rich/polymer-poor coexisting phase is extremely small for long polymers. The solvent-rich phase is close to pure solvent. This is peculiar to polymers, a mixture of small molecules can be approximated using the Flory-Huggins expression with, and then and both coexisting phases are far from pure.

Footnotes

  1. "Thermodynamics of High Polymer Solutions," Paul J. Flory Journal of Chemical Physics, August 1941, Volume 9, Issue 8, p. 660 . Flory suggested that Huggins' name ought to be first since he had published several months earlier: Flory, P.J., "Thermodynamics of high polymer solutions," J. Chem. Phys. 10:51-61
  2. "Solutions of Long Chain Compounds," Maurice L. Huggins Journal of Chemical Physics, May 1941 Volume 9, Issue 5, p. 440
  3. We are ignoring the free volume due to molecular disorder in liquids and amorphous solids as compared to crystals. This, and the assumption that monomers and solute molecules are really the same size, are the main geometric approximations in this model.
  4. For a real synthetic polymer, there is a statistical distribution of chain lengths, so would be an average.
  5. The enthalpy is the internal energy corrected for any pressure-volume work at constant . We are not making any distinction here. This allows the approximation of Helmholtz free energy, which is the natural form of free energy from the Flory–Huggins lattice theory, to Gibbs free energy.
  6. In fact, two of the sites adjacent to a polymer segment are occupied by other polymer segments since it is part of a chain; and one more, making three, for branching sites, but only one for terminals.