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Wikipedia / 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 for mixing a polymer with a solvent. Although it makes simplifying assumptions, it generates useful results for interpreting experiments.

The theory is named after Paul Flory and Maurice Loyal Huggins who developed it independently in 1941.

Theory

The thermodynamic equation for the Gibbs energy change accompanying mixing at constant temperature and (external) pressure is

, 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, 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 of solvent (component of polymer (component to take account of the energy is the gas constant is the absolute temperature. The volume fraction is analogous to the mole fraction, 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 – a molecule exists for any arbitrarily small volume element. 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

segments.

For a random walk on a lattice we can calculate the entropy change (the increase in spatial uncertainty) as a result of mixing solute and solvent.

is the Boltzmann constant. Define the lattice volume fractions

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

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. (not the covalent bonding. 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

is the number of nearest-neighbor sites to all 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

by transferring the Avogadro constant to the gas constant.

The value of the interaction parameter can be estimated from the Hildebrand solubility parameters

is the actual volume of a polymer segment.

, is a free energy parameter, thus including an entropic component. The Flory–Huggins interaction parameter χ represents the energetic interaction between polymer segments and solvent molecules. It reflects the difference between polymer–solvent interactions and the average interactions of polymer–polymer and solvent–solvent pairs. Larger positive values of χ indicate unfavorable mixing and may lead to phase separation, whereas smaller values correspond to better miscibility. In many polymer–solvent systems the parameter is temperature dependent and often decreases as temperature increases. 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. monomers, can be written in a simple dimensionless form

. The osmotic pressure (in reduced units) is

  1. .

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 the solution first becomes unstable when 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

, there is separation into two coexisting phases, one richer in polymer but poorer in solvent, than the other.

and both coexisting phases are far from pure.

Polymer blends

Synthetic polymers rarely consist of chains of uniform length in solvent. The Flory–Huggins free energy density can be generalized by

For a binary polymer blend monomers this simplifies to

, so for polymers A and B to blend their segments must attract one another.

Limitations

interaction-temperature dependence and other features commonly observed in polymer mixtures. However, unmodified Flory–Huggins theory fails to predict the lower critical solution temperature. parameter complex dependence on temperature, blend composition, and chain length was discarded. Specifically, interactions beyond the nearest neighbor may be highly relevant to the behavior of the blend and the distribution of polymer segments is not necessarily uniform, so certain lattice sites may experience interaction energies disparate from that approximated by the mean-field theory.

One well-studied which can be estimated as

Here, m is the chain's radius of gyration.

Footnotes

  1. "Thermodynamics of High Polymer Solutionss", Paul J. Flory Journal of Chemical Physics, August 1941, Volume 9, Issue 8, p.660 Abstract. 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 (1942)

  2. "Solutions of Long Chain Compoundss", 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 would be an average.

  5. The enthalpy is the internal energy corrected for any pressure-volume work. 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 terminalss.

References