## 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. Flory^{1} and M. L. Huggins^{2} 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*is the entropic component. The parameter

_{S}*χ*is usually assigned a value between 0.3 and 0.4 for nonpolar systems. Often a value of about

_{S}*χ*= 0.34 is chosen.

_{S}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} - 2*ε*_{ps} + *ε*_{ss}) ≈
*z*/2 · {*ε*_{pp} - 2*(ε*_{pp}*ε*_{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*/ 2

_{ij}*v*)

_{m,i}^{1/2}

where *E _{coh,i}* is the molar cohesive energy and

*V*the molar volume of the molecule

_{m,i}*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*_{p}*δ _{p}*

^{2}- 2(

*v*

_{p}

*v*

_{s})

^{1/2}

*δ*

_{p}*δ*+

_{s}*v*

_{s}

*δ*

_{s}^{2}} /

*RT*

or

*χ*_{H} ≈ (*v*_{p}*v*_{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*yields

_{S}*χ* ≈ (*v*_{p}*s*_{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*. This means, the enthalpic part of the interaction parameter should be close to zero to ensure miscibility of polymers.

_{s}##### References and Notes

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