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 = χ_H + χ_S

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

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

kTχ_H = z/2 · (ε_pp - 2ε_ps + ε_ss) ≈ z/2 · {ε_pp - 2(ε_ppε_ss)^1/2 + ε_ss}

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

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

δ_i = (E_coh,i / V_m,i)^1/2 = (z e_ij / 2v_m,i)^1/2

where E_coh,i is the molar cohesive energy and V_m,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

χ_H = {v_pδ_p² - 2(v_pv_s)^1/2δ_pδ_s + v_sδ_s²} / RT

or

χ_H ≈ (v_pv_s)^1/2(δ_p - δ_s)² / RT

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

Combining the two expression for χ_H and χ_S yields

χ ≈ (v_ps_s)^1/2(δ_p - δ_s)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 δ_p = δ_s. This means, the enthalpic part of the interaction parameter should be close to zero to ensure miscibility of polymers.

References & 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.