FloryRehner

Flory–Rehner theory is a foundational framework in polymer science for describing the swelling behavior of cross-linked polymer networks, such as gels, in solvents. Named after Paul Flory and Rehner, the theory combines Flory–Huggins solution theory for polymer–solvent mixing with rubber elasticity theory to explain how networks take up solvent and reach equilibrium.

At its core, the theory envisions swelling as a balance between two opposing contributions to the free energy. The mixing term, governed by solvent quality and the polymer–solvent interaction parameter, favors uptake of solvent, while the elastic term, arising from the crosslinked network’s elasticity, resists deformation and limits swelling. The result is an equilibrium swollen state described by the polymer volume fraction in the swollen gel and its dependence on parameters such as the crosslink density and the interaction parameter chi. In practice, the Flory–Rehner relation is used to relate measurable swelling data to the network’s crosslink density.

Applications of the theory include determining crosslink density from swelling measurements, characterizing hydrogel materials, and guiding the design of soft, stretchable polymers for biomedical devices, drug delivery, and tissue engineering. It provides a practical tool for predicting how changes in solvent, temperature, or crosslinking will affect swelling.

Limitations and extensions are important. The original theory assumes a homogeneous, isotropic network with affine deformations and Gaussian chain elasticity, neglecting entanglements and complex microstructures. It also treats mixing without ionic effects. For ionic gels, nonneutral Donnan equilibrium and non-Gaussian elasticity are incorporated in extended models to improve accuracy. Despite limitations, Flory–Rehner remains a widely used framework for analyzing gel swelling.