Generalized Flory-Huggins Model

The classical thermodynamics of polymer solutions was developed by Paul Flory¹ and Maurice Huggins².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 = χ₀(T) + χ₁(T) φ_p + χ₂(T) φ_p²

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 = χ₀ + χ₁ T + χ₂ φ+ χ₃ T φ

where the values of χ₀ - χ₃ are assumed to be constants, that is, they are independent of temperature and mixture composition. Substituting for χ in

Δh_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:

Δh_mix / kT ≈ (χ₀ + χ₁ T) φ + [(χ₂ - χ₀) + (χ₃ - χ₁) T ]φ² - (χ₂ + χ₃ T) φ³

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

Δh_mix / kT ≈ (χ₁ + χ₃ φ) (1/T) - (χ₂ + χ₄φ) ln T + χ₅

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/dPBD⁴. They examined several expression and found that the following expression gives the best fit

χ_c = A_c(T) + B_c(T) (2φ - 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/T²

B_c(T) = χ_B0 + χ_B1/T + χ_B2/T²

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

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