A Diffusion-Controlled Model for Swelling Behavior in Polymeric Hydrogels

Abstract

Hydrogels are cross-linked polymer networks capable of absorbing large amounts of water, supporting diverse biomedical, pharmaceutical, and agricultural applications. Their swelling behavior is governed by water diffusion within the polymer matrix and is commonly described by the classical Crank solution for Fickian diffusion in a plane sheet, expressed in terms of the normalized fractional water uptake (Mt/M∞). Therefore, an additional conversion is required to relate it to the swelling degree S(t).This study presents the one-dimensional Fickian diffusion solution for a hydrogel slab of thickness L, with surfaces held at a constant equilibrium concentration, and recasts it as a closed-form expression for the time-dependent swelling degree, S(t), directly in terms of the effective diffusion coefficient (D) and the equilibrium swelling degree (S_eq). The resulting solution reproduces characteristic Fickian relationship at short times, and predicts the scaling law . Parametric analysis further shows that governs the swelling rate, whereas  determines the equilibrium swelling capacity. The predicted trends are qualitatively consistent with reported hydrogel swelling behavior. Whereas the classical Crank model focuses on relative water uptake, the proposed formulation translates diffusion kinetics into the swelling degree, S(t), enabling direct interpretation of measurable swelling behavior through a simple and physically meaningful analytical framework.