Edmond–Ogston model explained

F=RTV(c_1ln c₁+c_2ln c₂+B₁₁

	2
{c
	1}

+B₂₂

	2
{c
	2}

+2B₁₂{c_1}{c_2})

(c_1,c,c_2,c)=(

	1	,
	2(B₁₂ ⋅ S_c-B₁₁)

	1
	2(B₁₂/S_c-B₂₂)

)

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}

B₂₂

	3
{\sqrt{S
	c}}

+B₁₂

	2
{\sqrt{S
	c}}

-B₁₂{\sqrt{S_c}}-B₁₁=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.^[2]

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

The model and its solutions have been generalized to mixtures with an arbitrary number of components

, with

greater or equal than 2.^[4]

Notes and References

Edmond. E.. Ogston. A.G.. 1968. An approach to the study of phase separation in ternary aqueous systems. Biochemical Journal. 109. 4. 569–576. 10.1042/bj1090569. 5683507. 1186942.
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. 10.1021/acsomega.1c00450. 33778298. 7992149. free.
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. 10.1016/S0144-8617(99)00180-0.
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. 10.1021/acsomega.4c00339. 38826518. 11137696. free.