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.[1] At the core of the model is an expression for the Helmholtz free energy ${\displaystyle F}$

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

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

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

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

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

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

The model is closely related to the Flory–Huggins model.[3]