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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 ${\displaystyle F}$

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

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.

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 ${\displaystyle N}$, with ${\displaystyle N}$ greater or equal than 2.

References

1. Edmond, E.; Ogston, A.G. (1968). "An approach to the study of phase separation in ternary aqueous systems". Biochemical Journal. 109 (4): 569–576. doi:10.1042/bj1090569. PMC 1186942. PMID 5683507.
2. Bot, A.; Dewi, B.P.C.; Venema, P. (2021). "Phase-separating binary polymer mixtures: the degeneracy of the virial coefficients and their extraction from phase diagrams". ACS Omega. 6 (11): 7862–7878. doi:10.1021/acsomega.1c00450. PMC 7992149. PMID 33778298.
3. Clark, A.H. (2000). "Direct analysis of experimental tie line data (two polymer-one solvent systems) using Flory-Huggins theory". Carbohydrate Polymers. 42 (4): 337–351. doi:10.1016/S0144-8617(99)00180-0.
4. Bot, A.; van der Linden, E.; Venema, P. (2024). "Phase separation in complex mixtures with many components: analytical expressions for spinodal manifolds". ACS Omega. 9 (21): 22677–22690. doi:10.1021/acsomega.4c00339. PMC 11137696. PMID 38826518.

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