Flory–Rehner equation
 | This article may require cleanup to meet Wikipedia's quality standards. The specific problem is: The current state does not provide insights into the model assumptions. It is not clear what the equations mean. (November 2019) (Learn how and when to remove this template message) |
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:
- [math]\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) }[/math]
where, [math]\displaystyle{ \nu_2 }[/math] is the volume fraction of polymer in the swollen mass, [math]\displaystyle{ V_1 }[/math] the molar volume of the solvent, [math]\displaystyle{ n }[/math] is the number of network chain segments bounded on both ends by crosslinks, and [math]\displaystyle{ \chi_1 }[/math] is the Flory solvent-polymer interaction term.[2]
In its full form, the Flory–Rehner equation is written as:[3]
- [math]\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) }[/math]
where, [math]\displaystyle{ \bar{\nu} }[/math] is the specific volume of the polymer, [math]\displaystyle{ M }[/math] is the primary molecular mass, and [math]\displaystyle{ M_c }[/math] is the average molecular mass between crosslinks or the network parameter.[3]
Flory–Rehner theory
The Flory–Rehner theory gives the change of free energy upon swelling of the polymer gel similar to the Flory–Huggins solution theory:
- [math]\displaystyle{ \Delta F = \Delta F_\mathrm{mix} + \Delta F_\mathrm{elastic} }[/math].
The theory considers forces arising from three sources:[2]
- The entropy change [math]\displaystyle{ \Delta S_\mathrm{mix} }[/math] caused by mixing of polymer and solvent
- The heat of mixing of polymer and solvent [math]\displaystyle{ \Delta U_\mathrm{mix} }[/math], which may be positive, negative, or zero so, that [math]\displaystyle{ \Delta F_\mathrm{mix}=\Delta U_\mathrm{mix}-T\Delta S_\mathrm{mix} }[/math]
- The entropy change caused by reduction in numbers of possible chain conformations via swelling [math]\displaystyle{ \Delta F_\mathrm{elastic} }[/math]
The Flory–Rehner equation was used to model the cooking of steaks in a journal article in 2020[4]
References
- ↑ Flory & Rehner 1943
- ↑ 2.0 2.1 Sperling 2006, p. 472
- ↑ 3.0 3.1 Alger 1997, p. 202
- ↑ Nelson, H.; Deyo, S.; Granzier-Nakajima, S.; Puente, P.; Tully, K.; Webb, J. (2020). "A mathematical model for meat cooking" (in en). The European Physical Journal Plus 135 (3): 322. doi:10.1140/epjp/s13360-020-00311-0. ISSN 2190-5444. Bibcode: 2020EPJP..135..322N. http://link.springer.com/10.1140/epjp/s13360-020-00311-0.
Bibliography
- Flory, Paul; Rehner, John (1943). "Statistical mechanics of cross-linked polymer networks II. Swelling". J. Chem. Phys. 11 (11): 521–526. doi:10.1063/1.1723792. Bibcode: 1943JChPh..11..521F.
- 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.
 |  |
