## Flory-Huggins Parameter

Many thermodynamic properties of polymer solutions such as
solubility, phase equlibria as well as swelling equilibria of isolated polymer
coils and polymer networks are often expressed in terms of
the *polymer-solvent interaction parameter*. This parameter was first
introduced by Flory^{1} and Huggins^{2} independently to describe the
interaction between solvent and polymer molecules in their lattice
model of polymer solutions.

To calculate this parameter, we assume that the polymer segments and
the solvent molecules are randomly distributed in the coil volume
*R*^{3}, and that the heat of mixing is proportional to the volume fraction of polymer segments in
the coil volume, *R*^{3}. The overall system energy
is the sum of all interactions between neighboring molecules. For
simplicity, we further assume that the solvent and polymer segments are of
same size and that only nearest neighbor interaction have to be
included in the calculation. For this simple case, the energy if
mixing can be written as:

*E*_{mix} = <*N*_{pp}> *ε*_{pp} + <*N*_{ps}> *ε*_{ps} + <*N*_{ss}> *ε*_{ss}

where *ε _{ij}* is the interaction energy between two molecules, and <

*N*> is the average number of contacts between molecules

_{ij}*i*and

*j*. The probability

*P*

_{v}that a site is occupied by a molecule of type

*i*is simply proportional to its volume fraction:

*P*_{p} = *φ*_{p }
= 1 - *φ*_{s} = *N **l*^{3}/*R*^{3},
*P*_{s} = *φ*_{s }
= 1 - *φ*_{p} = 1 - *N **l*^{3}/*R*^{3}

where *l* is the effective length of a monomer. Hence, the average number of solvent-polymer,
polymer-polymer and solvent-solvent pairs is given by

<*N _{ps}*> =

*zN*(1 -

*φ*

_{p}),

<

*N*> = ½

_{pp}*zN*

*φ*

_{p},

<

*N*> =

_{ss}*N*-

_{0}*zN*[½

*φ*+ (1-

_{p}*φ*)],

_{p}where *N* is the number of repeat units
(segments) in the polymer chain, *N _{0}* is the pairs of neighboring solvent molecules in the absence of a polymer, and

*z*is the (average) number of contacts per molecule, also called

*lattice coordination number*(Flory-Huggins lattice model). Inserting the three expressions for the pair contacts into the equation for the energy of mixing gives

*E*_{mix} ≈ ½ *zNφ _{p}*
[

*ε*

_{pp}- 2

*ε*

_{ps}+

*ε*

_{ss}] + terms f ≠ (

*φ*)

Flory defined a dimensionless quantity which characterizes the interaction energy per solvent molecule divided by *kT*:

*χ*_{} = *-z*/2 (*ε*_{pp} + *ε*_{ss} - 2 *ε*_{ps}) / *kT*

This quantity is called polymer-solvent interaction parameter or *Flory-Huggins* parameter
because it was first introduced by P. J. Flory^{1} and M. L. Huggins^{2}
as an exchange interaction parameter in their lattice model of
polymer solutions. *zkTχ* is simply the difference in energy
between a solvent molecule immersed in pure polymer and in pure solvent. If we introduce this parameter, the energy
of mixing
can be written as

*E _{mix}* /

*kT*=

*χ N*(1 -

*φ*) =

_{p}*χ*(

*N*-

*N*

^{2}

*l*

^{3}/

*R*

^{3})

or per site

*e _{mix}* /

*kT*=

*χ φ*(1 -

_{p}*φ*)

_{p}Thus, the effect of solvent interactions can be expressed in terms of a single parameter.

##### References

- P. J. Flory,
*J. Chem. Phys.*9, 660 (1941); 10, 51 (1942); 17, 303 (1949) M. L. Huggins,

*J. Chem Phys.*9, 440 (1941);*J. Phys. Chem.*46, 151 (1942);

J. Am. Chem. Soc.-
Paul L. Flory,

*Principles of Polymer Chemistry*, Ithaca, New york, 1953