Capture–release dynamics of nanoparticles in a viscoelastic hydrogel under an oscillating magnetic field

A study was conducted to understand the effect of superparamagnetic nanoparticles on a viscoelastic hydrogel subject to an oscillating magnetic field directed tangent to the hydrogel surface. A capture–release kinetic model was formulated for the dynamic transition of nanoparticles between a freely oscillating state in the Newtonian solvent and a hydrogel-bound captured state. The viscoelastic hydrogel is modeled as an Oldroyd-B fluid. A parametric analysis was conducted to understand regimes of system behavior in terms of key dimensionless parameters. Increasing the viscous-to-thermal work ratio enhances detachment, while higher adhesion energy promotes particle retention by the hydrogel. The magnetic field amplitude controls both uptake rate and oscillatory strength, and the relative matrix-solvent velocity shapes the harmonic content of the response. A bifurcation was observed at a critical value of the ratio of retardation to relaxation times. For values of this ratio less than the critical value, the system exhibits multi-periodic elastic-regime responses, whereas for values of the ratio above the critical ratio, the system collapses into mono-periodic viscous-regime behavior. The capture rate coefficient includes an absolute value, so its product with sinusoidal magnetic forcing yields only odd harmonics in the response. Elastic waves propagate from the no-slip base, and height-averaged particle counts oscillate at twice the driving frequency. These findings elucidate how nonlinear capture–release kinetics and viscoelastic memory produce multi-periodic and odd-harmonic-dominated dynamics, informing the design of switchable smart materials for on-demand drug delivery and biofilm control.