Flory–Rehner equation

In polymer science Flory–Rehner equation is an equation that describes the mixing of polymer and liquid molecules as predicted by the equilibrium swelling theory of Flory and Rehner.^[1] It describes the equilibrium swelling of a lightly crosslinked polymer in terms of crosslink density and the quality of the solvent. The theory considers forces arising from three sources:^[2]

The entropy change caused by mixing of polymer and solvent.
The entropy change caused by reduction in numbers of possible chain conformations via swelling.
The heat of mixing of polymer and solvent, which may be positive, negative, or zero.

The equation is written as —

-\left[\ln {\left(1-\nu _{2}\right)}+\nu _{2}+\chi _{1}\nu _{2}^{2}\right]=V_{1}n\left(\nu _{2}^{\frac {1}{3}}-{\frac {\nu _{2}}{2}}\right)

where, $\nu_2$ is the volume fraction of polymer in the swollen mass, $V_{1}$ the molar volume of the solvent, $n$ is the number of network chain segments bounded on both ends by crosslinks, and $\chi_1$ is the Flory solvent-polymer interaction term.^[2]

In its full form, the equation is written as^[3] —

-\left[\ln {\left(1-\nu _{2}\right)}+\nu _{2}+\chi _{1}\nu _{2}^{2}\right]={\frac {V_{1}}{{\bar {\nu }}M_{c}}}\left(1-{\frac {2M_{c}}{M}}\right)\left(\nu _{2}^{\frac {1}{3}}-{\frac {\nu _{2}}{2}}\right)

where, ${\bar {\nu }}$ is the specific volume of the polymer, $M$ is the primary molecular mass, and $M_c$ is the average molecular mass between crosslinks or the network parameter.^[3]

References

Bibliography

Flory, Paul; Rehner, John (1943). "Statistical mechanics of cross-linked polymer networks II. Swelling". J. Chem. Phys. 11 (11): 521–526. Bibcode:1943JChPh..11..521F. doi:10.1063/1.1723792.
Sperling, Leslie H. (2006). Introduction to Physical Polymer Science (4th ed.). Bethlehem, PA: John Wiley & Sons. ISBN 978-0-471-70606-9.
Alger, Mark (1997). Polymer Science Dictionary (2nd ed.). London: Chapman & Hall. ISBN 978-0-412-60870-4.

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[Flory43-1] Flory and Rehner 1943

[Sperling06-2] 1 2 Sperling 2006, p. 472

[Alger97-3] 1 2 Alger 1997, p. 202