Temperature and Concentration
Dependence of Interaction Parameter

Many thermodynamic properties of polymer solutions such as solubility / miscibility, swelling equilibria, and other properties that depend on the mixture composition can be expressed in terms of the polymer-solvent interaction parameter χ. This unitless quantity was first introduced by Paul Flory¹ and Maurice Huggins² independently as an exchange interaction parameter in their lattice model of polymer solutions. For this reason, this parameter is often called Flory-Huggins Parameter.

In early models and studies, the Flory-Huggins Parameter (χ) was assumed to be independent of concentration. However, this is often not the case; χ is usually a function of both temperature and composition (polymer concentration).

The temperature dependence of the interaction parameter is quite obvious from the definition of the χ parameter:

χ(T) = -z/2 (ε_pp + ε_ss - 2 ε_ps) / kT = B / T

where B describes the temperature dependence of χ. According to this equation, χ is a linear function of the inverse temperature. For the case B > 0 and (very) low temperatures, the free energy curve is concave and homogenous mixtures are unstable because the second derivative of the free energy of mixing is negative and the contribution of the entropy is small.

Empirically, the temperature dependence of χ is often written as a sum of two terms:

χ(T) = A + B/T

This equation is often valid for systems with an upper critical solution temperature (UCST). In some other cases, a noticeable nonlinearity is observed when χ is plotted versus 1/T. In such cases the data can often be fitted to a quadratic function in 1/T,

χ(T) = A + B/T + C/T²

For many polymer solutions, χ increases noticeably with polymer concentration, particularly for poor solvents.The concentration dependence of the Flory-Huggins parameter is given by

χ(φ_p) = a + c(1 - b) / (1 - bφ_p)²

where a, and b are constants within a certain temperature interval and the constant c depends on the temperature.

c = c₀ + c₁ / T

The concentration dependence of the Flory-Huggins parameter can be also approximated by a power series:

χ = ∑_i χ_i φ_pⁱ

i_max = 2 gives generally acceptable fits:

χ = χ₀ + χ₁ φ_p + χ₂ 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 experimental findings, three characterist behaviors are observed:

• 1. In some cases, χ increases greatly with polymer concentration. This is often the case for poor solvents.

• 2. In some other cases, χ is nearly independent of composition, as predicted by the original Flory-Huggins theory, which is often the case for good solvents.

• 3. In a very few cases, χ decreases with increasing polymer concentration. This behavior is sometimes observed for polymer-solvent systems that are highly exothermal.

The table below lists selected values of polymer-solvent interaction parameters at infinit dilution, χ_∞. For many solvent-polymer systems at low polymer concentrations and ambient temperatures, the χ_∞ parameter has a value between 0.45 and 0.50.