Chemistry:Edmond–Ogston model

The Edmond–Ogston model is a thermodynamic model proposed by Elizabeth Edmond and Alexander George Ogston in 1968 to describe phase separation of two-component polymer mixtures in a common solvent. At the core of the model is an expression for the Helmholtz free energy [math]\displaystyle{ F }[/math]

[math]\displaystyle{ \ F = RTV(\,c_1\ln\ c_1 + c_2\ln\ c_2 + B_{11} {c_1}^2 + B_{22} {c_2}^2 + 2 B_{12} {c_1} {c_2}) \, }[/math]

that takes into account terms in the concentration of the polymers up to second order, and needs three virial coefficients [math]\displaystyle{ B_{11}, B_{12} }[/math] and [math]\displaystyle{ B_{22} }[/math] as input. Here [math]\displaystyle{ c_i }[/math] is the molar concentration of polymer [math]\displaystyle{ i }[/math], [math]\displaystyle{ R }[/math] is the universal gas constant, [math]\displaystyle{ T }[/math] is the absolute temperature, [math]\displaystyle{ V }[/math] is the system volume. It is possible to obtain explicit solutions for the coordinates of the critical point

[math]\displaystyle{ (c_{1,c},c_{2,c}) = (\frac{1}{2(B_{12} \cdot S_c-B_{11})} \,,\frac{1}{2(B_{12}/S_c-B_{22})} \,) }[/math],

where [math]\displaystyle{ -S_c }[/math] represents the slope of the binodal and spinodal in the critical point. Its value can be obtained by solving a third order polynomial in [math]\displaystyle{ \sqrt{S_c} }[/math],

[math]\displaystyle{ \ B_{22} {\sqrt{S_c}}^3 + B_{12} {\sqrt{S_c}}^2 - B_{12} {\sqrt{S_c}} - B_{11} = 0 \, }[/math],

which can be done analytically using Cardano's method and choosing the solution for which both [math]\displaystyle{ c_{1,c} }[/math] and [math]\displaystyle{ c_{2,c} }[/math] are positive.

The spinodal can be expressed analytically too, and the Lambert W function has a central role to express the coordinates of binodal and tie-lines.

The model is closely related to the Flory–Huggins model.