Kirkwood-Buff and the polymer χ parameter

Many polymer scientists are familiar with the Flory-Huggins χ (chi) parameter (which captures the dis-similarity between solvent and polymer) and the fact that when it reaches a value of 0.5 the solvent and polymer separate into two phases: a solvent phase containing a (low) saturated concentration of polymer and a polymer phase containing lots of solvent. This split into two phases is called the spinodal point.

An app showing the classic Flory-Huggins calculation is based on standard lattice theory with its mixture of entropic and enthalpic effects.

Here we show the same effect via the assumption-free molecular thermodynamics of Kirkwood-Buff where we are interested in three molecular parameters:

1. G₁₁ which is the self-liking of the solvent
2. G₁₂ which is the like/dislike of the solvent and the polymer
3. G₂₂ which is the self-liking of the polymer

The only inputs required are the MVols, V₁ and V₂ of the solvent and polymer, and the χ parameter which is specified as χ = a + bφ + cφ². The standard assumption is that b and c are zero, i.e. a constant χ but some real-world examples with different b and c are discussed below. For the G_ij values, self-liking should give a positive value and dis-liking gives a negative value. However, the large size of the polymer means that it excludes solvent from itself and excludes itself even more, so even when the polymer is happy with itself and the solvent, G₂₂ remains strongly negative. Excluded volume effects, which are usually not taken into account because they seem so dull, are of super-importance!

The theory behind the app is described below. But the key is to see what happens to the chemical potential μ₂ as you approach then exceed χ=0.5. It becomes rather obvious that bad things happen at that point. The G_ij graphs show the whole range of volume fraction φ₂ but the chemical potential graph covers a smaller range to show the interesting bits.

The theory comes from an excellent paper by Dr Arturo Horta at UNED in Madrid. I have simplified the nomenclature for ease of reading on the web. Where i=1 for solvent and 2 for polymer, for volume fractions φ_i, molar volumes V_i and ratio r=V₂/V₁,the starting point is the link between the chi parameter and the zero angle scattering factor, S.

`1/S = 1/φ_1+1/rφ_2-2χ`

We then go straight to the G_ij values via:

`G_(11) = V_1/φ_1(S/φ_1-1)`

`G_(12) = (-S.V_2)/(r.φ_1*φ_2)`

`G_(22) = V_2/φ_2(S/(r.φ_2)-1)`

Finally we can derive the derivative of the chemical potential versus concentration via:

`(δμ_2)/(δc_2) = 1/(c_2(1+c_2(G_(22)-G_(12)))`

And with some minor arithmetic that becomes δμ₂/δφ₂ from which μ₂ can be calculated and becomes unstable once we hit the spinodal point. Technically, KB cannot handle such a phase separation so the numbers in this region are not significant - it is the trend that is important.

What does it mean?

The formulae are rather easy, but what do they mean and why do things go wrong when χ reaches 0.5? The key is the balance between G₂₂ and G₁₂. G₁₂ doesn't do anything very interesting, it is G₂₂ which changes rapidly as χ exceed 0.5. This is because the polymer becomes balanced between two phenomena - the large negative value from the excluded volume effect and the desire to escape from the solvent which is especially large when there is little polymer and much solvent. You can see how G₂₂ starts to curve up as things start to go wrong, and can see how G₂₂-G₁₂ becomes positive at the spinodal.

This approach to understanding the spinodal phase separation of polymers in bad solvents seems to me to be much more powerful and intuitive than the one we have all been taught which has some entropic and enthalpic effects in some lattice approximation to reality.

Examples

In a follow-up paper² Dr Horta provided some examples which are interesting to play with:

• Polyisobutylene/Cyclohexane V₁=109, r=350, a=0.45, b=0.1, c=0
• Polystyrene/MEK V₁=90, r=700, a=0.46, b=0.5, c=0
• Poly(4-hydroxystyrene)/Acetone V₁=74, r=20, a=0.3, b=-0.5, c=-1.7