## Generalized Flory-Huggins Model

The classical thermodynamics of polymer solutions was
developed by Paul Flory^{1} and Maurice Huggins^{2}.It is widely
used to predict phase-separation phenomena in binary and ternary polymer
solutions and mixtures. Its popularity results from its computational
simplicity. But this theory has also some shortcomings; in the original version
of the FH theory, the interaction parameter is assumed to be independent of the
mixture composition. However, this assumptions is not valid for most mixtures.
Much research has been done to improve the original FH theory, for example by
including corrections for the concentration and temperature dependence of the FH
interaction paramater. Some of these theories will be discussed below.

To improve the predictions, the FH interaction parameter must be replaced by a temperature and composition-dependent interaction parameter. The parameter is
often written as the sum of two parts: one part describes the enthalpic effect, due

to the energy change upon mixing, and the other part describes the entropic effect due to
noncombinatorial entropy:

*χ* = *χ _{H}* +

*χ*

_{S}A number of expressions have been suggested by various researchers to adequately describe the temperature and concentration dependence. Often, the composition dependence of the FH interaction parameter is described with a polynomial function of the mixture composition,

*χ* = *χ _{0}*(

*T*) +

*χ*(

_{1}*T*)

*φ*+

_{p}*χ*

_{2}(

*T*)

*φ*

_{p}^{2}

where each coefficient of the polynomial, *χ _{i}*, is assumed to be a function of temperature, which, in its general form, can consist of inverse, linear, and
logarithmic terms of temperature. According to the original FH lattice theory, the coefficients are only a function of the inverse temperature, but this is an oversimplification, and for actual
systems, complementary terms must be considered.

In several publications, empirical functions for the heat
of mixing and the FH interaction parameter have been suggested. For
example, Taimoori, et al.^{3} suggested a combined function of temperature and mixture composition:

*χ = χ_{0} + χ_{1} T + χ_{2} φ+ χ_{3} T φ*

where the values of *χ*_{0 } - *χ*_{3} are assumed to be constants, that
is, they are independent of temperature and mixture composition. Substituting for *χ* in

Δ*h _{mix}* /

*kT*≈

*χ*

_{ps}φ_{s}φ_{p}gives the heat of mixing as a third-order polynomial with respect to the volume fraction, *φ,* with temperature-dependent coefficients:

Δ*h _{mix}* /

*kT*≈ (

*χ*

_{0}+

*χ*

_{1}

*T*)

*φ*+ [(

*χ*

_{2}-

*χ*

_{0}) + (

*χ*

_{3}-

*χ*

_{1})

*T*]

*φ*

^{2}- (

*χ*+

_{2}*χ*

_{3}*T*)

*φ*

^{3}

Another empirical relation has been suggested by Taimoori, Modarrress and Mansoori (2000):^{3}

Δ*h _{mix}* /

*kT*≈ (

*χ*

_{1}+

*χ*

_{3}

*φ*) (

*1/T*) - (

*χ*

_{2}+

*χ*

_{4}

*φ*) ln

*T*+

*χ*

_{5}The authors of this equation have shown that their model can predict all types of heat-of-mixing curves including exothermic, endothermic, and sigmoidal types. It also predicts all typs of occuring spinodal phase curves, including the upper and lower critical solution temperatures, and closed-loop miscibility regions.

Nedoma and Robertson (2008) investigated
the concentration and temperature dependence of the FH *χ*
parameter at the critical point for the system PIB/dPBD^{4}. They examined several expression and found that the following expression gives the best fit

*χ*_{c} = *A _{c}*(

*T*) +

*B*(

_{c}*T*) (2

*φ*- 1) /

*N*

_{AVE}where *N _{AVE}* is defined as

*N _{AVE}* = 4 (

*r*

^{ -½}+

*s*

^{-½})

^{-2}

and *r* and *s* are the
number of repeat unit per polymer chain and

*χ _{c}* = 2 /

*N*

_{AVE}Both *A _{c}*(

*T*) and

*B*(

_{c}*T*) were assumed to be quadratic functions of 1/

*T*.

*A _{c}*(

*T*) =

*χ*+

_{A0}*χ*/

_{A1}*T*+

*χ*/

_{A2}*T*

^{2}

*B _{c}*(

*T*) =

*χ*+

_{B0}*χ*/

_{B1}*T*+

*χ*/

_{B2}*T*

^{2}

Nedoma et al. showed that the same equations also apply to
non critical values of *χ*. They found

*A*(*T*) = *A _{c}*(

*T*)

*B*(*T*) = *B _{c}*(

*T*) / 3

##### References

- 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 - M. Taimoori, H. Modarrress and
G.A. Mansoori ,
*J. Appl. Poly. Sci.*, Vol. 78, 1328–1340 (2000) - A.J. Nedoma, M.L. Robertson, N.S. Wanuakule, and N.P. Balsara,
*Macromolecules*, vol. 41, 15, 5773 - 5779 (2008)