Exact integrated equations to describe diffusion kinetics

Crank's solutions for Fick's second law remain the foundation of diffusion kinetics models, yet many studies use simplified forms of these solutions for fitting experimental data. Here, we derive and summarize the exact diffusion equations that can be used to model the analyte uptake and release in permeable films, with emphasis on two cases: a free-standing film and a film mounted on an impermeable substrate. For both cases, we present the analytical expressions and their integrated forms. The integrations consider two different experimental scenarios that are common when measuring the kinetics of analyte uptake and desorption: (1) the average analyte concentration is determined across the entire film and (2) the average concentration of analyte is determined in a localized region of interest. While the former is relevant to, e.g. gravimetric measurements, the latter is particularly relevant to plasmonic sensing applications and evanescent field interactions, where the measurable signal is dependent on the analyte concentration near the film interface. We provide a comprehensive framework for fitting experimental diffusion curves to physically meaningful models, enabling a more accurate determination of diffusion coefficients across a range of polymer-analyte systems.