Flory–Rehner equation

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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 Flory–Rehner equation is written as:

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

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

In its full form, the Flory–Rehner equation is written as:^[3]

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

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

The Flory–Rehner theory gives the change of free energy upon swelling of the polymer gel similar to the Flory–Huggins solution theory:

${\displaystyle \Delta F=\Delta F_{\mathrm {mix} }+\Delta F_{\mathrm {elastic} }}$.

The theory considers forces arising from three sources:^[2]

1. The entropy change ${\displaystyle \Delta S_{\mathrm {mix} }}$ caused by mixing of polymer and solvent
2. The heat of mixing of polymer and solvent ${\displaystyle \Delta U_{\mathrm {mix} }}$, which may be positive, negative, or zero so, that ${\displaystyle \Delta F_{\mathrm {mix} }=\Delta U_{\mathrm {mix} }-T\Delta S_{\mathrm {mix} }}$
3. The entropy change caused by reduction in numbers of possible chain conformations via swelling ${\displaystyle \Delta F_{\mathrm {elastic} }}$

The Flory–Rehner equation was used to model the cooking of steaks in a journal article in 2020^[4]

• 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.