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 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]