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Edmond–Ogston model

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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.[1] At the core of the model is an expression for the Helmholtz free energy

that takes into account terms in the concentration of the polymers up to second order, and needs three virial coefficients and as input. Here is the molar concentration of polymer , is the universal gas constant, is the absolute temperature, is the system volume. It is possible to obtain explicit solutions for the coordinates of the critical point

,

where represents the slope of the binodal and spinodal in the critical point. Its value can be obtained by solving a third order polynomial in ,

,

which can be done analytically using Cardano's method and choosing the solution for which both and 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]

References

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  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.