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Relationship between Solubility and Interaction Parameters

Many thermodynamic properties of polymer solutions and polymer blends such as solubility, swelling, osmotic pressure, and many other properties can be understood in terms of polymer-solvent and polymer-polymer interaction parameters χ. This dimensionless quantity was originally introduced by P. J. Flory1 and M. L. Huggins2 as an exchange interaction parameter in their lattice theory of polymer solutions.

The Flory-Huggins interaction parameter, χ, can be expressed as the sum of an entropic and enthalpic contribution:

χ = χH + χS

where χH is the enthalpic component and χS is the entropic component. The parameter χS is usually assigned a value between 0.3 and 0.4 for nonpolar systems. Often a value of about χS = 0.34 is chosen.

The enthalpic component of the Flory-Huggins interaction parameter,  χH, describes the energy part of the monomer-solvent interaction. It can be derived from the cohesive energy of the pair interactions:

kTχH = z/2 · (εpp - 2εps + εss) ≈ z/2 · {εpp - 2ppεss)1/2 + εss}

where subscripts s and p stand for solvent and polymer, respectively, and εij is the interaction energy between two segments of type i and j. The expression ij/2 is the total cohesive energy of a segment that is surrounded by z molecules.3

If one divides the interaction energy εii by the volume of the repeat unit vm,i, one gets an expression for the cohesive energy density. The square root is the so called solubility parameter:

δi = (Ecoh,i / Vm,i)1/2 = (z eij / 2vm,i)1/2

where Ecoh,i is the molar cohesive energy and Vm,i the molar volume of the molecule i. If we assume that the cross-interaction energy is the geometric average of the interaction energies of the pure components, the equation above can be written as

χH = {vpδp2 - 2(vpvs)1/2δpδs + vsδs2} / RT

or

χH ≈ (vpvs)1/2(δp - δs)2 / RT

This is the so called Hildebrand-Scott equation which can be used to estimate interaction parameters.

Combining the two expression for χH and χS yields

χ ≈ (vpss)1/2(δp - δs)2 / RT + β

The parameter β is sometimes called the lattice constant. For polymer-solvent blends, a value of around 0.34 is often chosen. However, a value of zero yields sometimes better correlations, as it is the case for polymer blends.

A criterion for good solubility is δp = δs. This means, the enthalpic part of the interaction parameter should be close to zero to ensure miscibility of polymers.

References and Notes
1. P. J. Flory, J. Chem. Phys. 9, 660 (1941); 10, 51 (1942)
2. M. L. Huggins, J. Phys. Chem. 46, 151 (1942); J. Am. Chem. Soc. 64, 1712 (1942)
3. The factor 1/2 takes into account that the interaction energy has to be equally divided between the adjacent molecules.
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Summary

Flory & Huggins

interaction parameters can be expressed as the sum of an entropic and enthalpic contribution.

• Hildebrand & Scott

relation can be used to estimate interaction parameters from solubility parameters.

• Polymer Blends

Miscibiltiy can only be achieved if the enthalpic part of the Flory-Huggins interaction parameter is close to zero.

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