Edmond–Ogston model

Template:Use dmy dates 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 $F$

$F=RTV(c_1\ln c_1+c_2\ln c_2+B_{11}{c_1}^2+B_{22}{c_2}^2+2B_{12}{c}_1{c}_2)$

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

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

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

$B_{22}{\sqrt{S_c}}^3+B_{12}{\sqrt{S_c}}^2-B_{12}\sqrt{S_c}-B_{11}=0$,

which can be done analytically using Cardano's method and choosing the solution for which both $c_{1,c}$ and $c_{2,c}$ 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.

The model and its solutions have been generalized to mixtures with an arbitrary number of components $N$, with $N$ greater or equal than 2.