Flory–Huggins Phase Behaviour

The one equation, and the one consequence

Flory–Huggins writes the free energy of mixing per lattice site as an entropy term for each component and one enthalpy term for the contact between them, with φ the volume fraction of component 1 and N the number of segments per chain:

The two entropy terms are what drive mixing, and both carry a 1/N. That is the whole story of polymer thermodynamics in one place: a chain of a thousand segments gains a thousandth of the mixing entropy that a thousand free segments would, because the segments are tied together and cannot spread out independently. The enthalpy term carries no 1/N at all. So as chains get longer, the thing opposing mixing stays the same size while the thing favouring it vanishes.

Setting the second and third derivatives to zero gives the critical point in closed form, and it makes the consequence quantitative:

For two chains of equal length that is χ_c = 2/N. For a polymer in a small-molecule solvent (N₁ = 1) it tends to ½ from above as the chain grows, which is exactly what a theta solvent means: χ = 0.5 is the borderline where a very long chain is on the edge of precipitating. And for two small molecules it gives χ_c = 2, the regular-solution result. One expression, three familiar limits.

Phase diagram

Why two polymers almost never mix

χ_c = 2/N for a symmetric blend is a brutal criterion. Put numbers on it:

Typical χ between two chemically distinct polymers runs from about 0.01 to well above 0.1. Compare that with the 0.002 a pair of N = 1000 chains is allowed, and the rule follows: polymer blends are immiscible unless the two polymers are nearly the same molecule, or something other than contact enthalpy is paying for the mixing – hydrogen bonding, a specific interaction, or a copolymer engineered to sit between them. This is why compatibilisers exist as an industry, and why "alloy" in a polymer datasheet usually means a controlled two-phase morphology rather than a solution.

It also runs the other way, and that is the useful direction: the same 1/N is why a polymer dissolves at all. A solvent has N = 1 and keeps its full mixing entropy, so it can tolerate χ up to about ½. Solvency is not a property of the polymer, it is the solvent's entropy doing the work.

The same two blocks, joined

Tie the two polymers together at one end and they cannot separate macroscopically – the junction will not let a chain leave its partner. What is left is separation on the scale of the chain itself, and mean-field theory puts the order–disorder transition for a symmetric diblock at χN = 10.495, where N is the total degree of polymerization.

The contrast with the blend is the point. A symmetric blend of the same two polymers, each of length N/2, demixes once χN exceeds 4 – that is χ_c = 2/(N/2). The diblock needs χN > 10.495, a factor of 2.6× more incompatibility, and even then it orders into domains of tens of nanometres instead of separating into two layers. The covalent bond is worth that much frustration, and it is the entire reason block copolymers are useful: they are held permanently at a length scale that a blend would race past on its way to two macroscopic phases.

Two cautions. 10.495 is the mean-field value for a symmetric diblock; composition asymmetry raises it, and fluctuation corrections raise it further for short chains, so a real transition sits above this number rather than on it. And χ here must be defined on the same reference volume as the N you typed – a χ quoted per repeat unit and an N counted in repeat units of a different size do not multiply.

Reading the diagram

• Outside the binodal the mixture is stable: one phase, and it stays one phase.
• Between binodal and spinodal it is metastable. It wants to separate but has to nucleate a droplet of the other phase first, and the barrier can hold for a long time. Separation here is nucleation and growth: discrete droplets that coarsen, with a well-defined size distribution.
• Inside the spinodal there is no barrier at all – the free energy curves the wrong way, so any composition fluctuation grows rather than decays. That is spinodal decomposition, and it produces the characteristic co-continuous, interpenetrating morphology instead of droplets. If you have ever seen a blend with two interlocking phases rather than spheres in a matrix, it was quenched into this region.
• The tie-line ends are the compositions of the two phases you actually get, and they do not depend on where you started between them. Only the amounts change, by the lever rule, which the calculator reports.

What the theory is missing

• χ is not a constant. It is treated here as one number, but a measured χ varies with composition and with molecular weight, and the temperature dependence is rarely exactly A + B/T. A χ taken from one composition and applied across the whole diagram is an approximation the diagram cannot show you the error of.
• No volume change on mixing, and no equation of state. The lattice is incompressible by construction, which is why plain Flory–Huggins cannot produce LCST behaviour from an entirely enthalpic χ. Real polymer solutions that phase separate on heating need free volume, which needs a compressible theory.
• Mean field. Concentration fluctuations are ignored, which matters most near the critical point and for short chains – exactly where the numbers look most precise.
• Monodisperse chains. A real N is a distribution, and polydispersity broadens the coexistence region and shifts the cloud point away from the binodal computed here.

Related tools

• Crosslink density – uses the same χ in the Flory–Rehner swelling equation, and shows how much the value you pick is worth.
• Block copolymers – the chemistry of the chains this page treats as two numbers.
• Chain dimensions – how big a coil is, which sets the domain spacing a diblock orders into.