Kinetic modelling and equilibrium behaviour of methylene blue adsorption in sequentially modified PVDF/NC/TiO₂/carbon...

Hybrid polymeric membranes are often evaluated in terms of flux, rejection, and antifouling performance, whereas their adsorption kinetics and equilibrium behaviour are commonly treated as secondary considerations. This separation limits mechanistic understanding because dye removal by functional membranes is governed by the coupling between convective transport, pore-mediated retention, surface adsorption, filler dispersion and foulant-membrane interaction. This study aims to evaluate the kinetic and equilibrium behaviour of methylene blue (MB) adsorption by sequentially modified polyvinylidene fluoride (PVDF)-based hybrid membranes incorporating nanocellulose (NC), titanium dioxide (TiO₂), graphene (GR) and graphene oxide-derived modified hybrid material (GO-mHM). Correspondingly, this study evaluates adsorption behaviour across the full sequence of membrane modification, enabling a comparative assessment of the contribution of each material component to adsorption kinetics and equilibrium performance, rather than limiting the analysis to the final optimised membrane formulation. Adsorption experiments were conducted at 25 °C using MB concentrations of 1-5 ppm, with adsorption capacity normalised to membrane area. The pseudo second order (PSO) model provided the most consistent kinetic fit with R ² values generally ranging from 0.96 to 0.98. At 5 ppm MB, the equilibrium adsorption capacity increased from 1049.79 mg m^-2 for pristine PVDF to 2080.69 mg m^-2 for PVDF-4% NCM, followed by further improvement to 2408.14, 2442.68, and 2462.38 mg m^-2 for PVDF/NC/3.0% TiO₂, PVDF/NC/TiO₂/0.2% GR, and PVDF/NC/TiO₂/0.3% GO-mHM, respectively. Isotherm equilibrium modelling showed that the Freundlich model best described MB adsorption which indicates adsorption on a heterogeneous membrane surface with non-uniform adsorption sites. PVDF/NC/TiO₂/0.3% GO-mHM exhibited the strongest adsorption response, recording a Freundlich K _F of 75 501.71 mg m^-2. These findings are consistent with MB adsorption occurring across chemically and structurally heterogeneous membrane domains, with filler composition, dispersion and interfacial accessibility influencing the observed adsorption response. The specific contributions of individual molecular interactions were not directly resolved and are therefore interpreted mechanistically rather than assigned as definitive adsorption pathways.