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 $F$

$\ F=RTV(\,c_{1}\ln \ c_{1}+c_{2}\ln \ c_{2}+B_{11}{c_{1}}^{2}+B_{22}{c_{2}}^{2}+2B_{12}{c_{1}}{c_{2}})\,$

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.