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 (in reduced units) 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 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 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. Polymer blends Synthetic polymers rarely consist of chains of uniform length in solvent. The Flory-Huggins free energy density can be generalized to an N-component mixture of polymers with lengths by For a binary polymer blend, where one species consists of monomers and the other 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. For large polymers of and these terms are negligibly small. This implies that for a stable mixture to exist , so for polymers A and B to blend their segments must attract one another. Limitations Flory-Huggins theory tends to agree well with experiments in the semi-dilute