## Polymer solubility

### Quick Start

Put a polymer into a solvent and everything seems fine - till things separate into a dilute solution of polymer in solvent and a solution of solvent in polymer. This split takes place if the Flory-Huggins χ parameter is large (~0.5) and if the polymer MWt is large and if the solvent is large.

To start, take χ just over 0.5 and slide the polymer and solvent MWt sliders. If the graph always goes negative, that means a stable solution. As soon as it starts to flatline, you're in trouble, as the test tube will show if you set the polymer concentration, φ2. to a fairly low value!

### Flory-Huggins

Flory-Huggins χ
Polymer MWt
Solvent MWt
φ2max
φ2 Concentration
φ2l   :  φ2s  :  D

The solubility behaviour of a polymer in a solvent depends on a balance of entropic and enthalpic effects. These are captured in the Flory-Huggins equation showing the dependence of the chemical potential of the polymer/solvent combination, Δμs, on the solvent and polymer volume fractions φ1, φ2:

Δμ=RT[ln(φ_1)+φ_2(1-1/x)+χ.φ_2²]

The factor x is the ratio of the MWts of the polymer and the solvent and is large as the polymer MWt becomes very large so that 1-1/x approaches 1 so that increasing φ2 increases the sum of the two entropic effect, reducing solubility.

For a low χ value and for low MWt polymer and solvent, the curve just shows a gentle fall-off. But at a critical χ, which is lower for high MWt polymers and solvents, the system becomes unstable and splits into a "spinodal" system with one phase containing a very low concentration of polymer in solvent, φ2l and a phase of polymer swollen with solvent φ2s. The two values are shown graphically and numerically, along with the HSP Distance, D, discussed below.

If your real φ2 value (shown by the green dashed line) is between those two values then the two phases always have those values but the volume of those phases change. At low φ2 you mostly have a tube of low concentration polymer solution with a small amount of swollen polymer at the bottom, at high φ2 you mostly have swollen polymer with a small amount of low concentration polymer solution at the top. Move the φ2 slider to get a feel for this.

This lessons are.

1. Low MWt polymers are more easily solubilised.
2. Low MWt solvents are more effective.
3. Once there's a phase separation, as MWt or Chi increases, the concentration of polymer in solution decreases.
4. And as MWt or Chi increase the concentration of polymer in the second phase increases or, to put it more intuitively, the concentration of solvent in the swollen polymer (1-φ2) decreases.

For those familiar with Hansen Solubility Parameters and the idea of a Distance, D, between polymer and solvent, χ is given by, at temperature T and gas constant R:

χ=(MVol.D^2)/(4RT)

So a small HSP Distance gives better solubility and, if there is phase separation, a larger Distance not only means that the polymer is basically insoluble in the solvent, but that the solvent swells the polymer to a smaller and smaller extent.

For those interested in technicalities, the phase separation is binodal at the minima and maxima of the graph but in reality you get a spinodal split at points somewhat lower and higher than the binodal values. The spinodal points are where the second derivative of the curve equals zero:

-1/(1-φ_2)+1-1/x+2φ_2χ = 0