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.
Although it makes simplifying assumptions, it generates useful results for interpreting experiments.
The result obtained by Flory[1] and Huggins[2] is The right-hand side is a function of the number of moles
to take account of the energy of interdispersing polymer and solvent molecules.
For a small solute, the mole fractions would appear instead, and this modification is the innovation due to Flory and Huggins.
[1][2] 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.
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.
[4] For a random walk on a lattice[3] we can calculate the entropy change (the increase in spatial uncertainty) as a result of mixing solute and solvent.
These are also the probabilities that a given lattice site, chosen at random, is occupied by a solvent molecule or a polymer segment, respectively.
that any such site is occupied by a solvent molecule,[6] we obtain the total number of polymer-solvent molecular interactions.
[1][2] 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.
The osmotic pressure (in reduced units) is The polymer solution is stable with respect to small fluctuations when the second derivative of this free energy is positive.
A little algebra then shows that the polymer solution first becomes unstable at a critical point at This means that for all values of
, 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
This is peculiar to polymers, a mixture of small molecules can be approximated using the Flory–Huggins expression with
Synthetic polymers rarely consist of chains of uniform length in solvent.
The Flory–Huggins free energy density can be generalized[5] to an N-component mixture of polymers with lengths
monomers this simplifies to As in the case for dilute polymer solutions, the first two terms on the right-hand side represent the entropy of mixing.
The theory qualitatively predicts phase separation, the tendency for high molecular weight species to be immiscible, the
interaction-temperature dependence and other features commonly observed in polymer mixtures.
However, unmodified Flory–Huggins theory fails to predict the lower critical solution temperature observed in some polymer blends and the lack of dependence of the critical temperature
[7] Additionally, it can be shown that for a binary blend of polymer species with equal chain lengths
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[4][6] effect on interaction energies neglected by unmodified Flory–Huggins theory is chain correlation.
As the polymer concentration increases, chains tend to overlap and the effect becomes less important.
In fact, the demarcation between dilute and semi-dilute solutions is commonly defined by the concentration where polymers begin to overlap
