Memory-Dependent Dielectric Permittivity Modeling of Thermo-Responsive Hydrogels Using a Resonant Microwave Sensor Under Dynamic Thermal Conditions

Thermo-responsive materials have promising potential for applications in sensing, actuation, and biomedical imaging due to the unique tunability they offer under dynamic thermal conditions. Despite the growing interest in thermo-responsive hydrogels, their microwave dielectric behavior under dynamic thermal conditions remains largely unexplored, especially concerning hysteresis and thermal memory effects. In this study, we first perform broadband dielectric characterization of gelatin-based hydrogels at three different concentrations to establish a baseline and calibrate the resonance-based measurements. We then characterize the real-time microwave dielectric response of these hydrogels during a heating–cooling sequence from 24 °C to 41°C. A resonance-based microwave sensor is employed to track distinct heating and cooling trajectories via shifts in resonance frequency f<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> and amplitude A<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub>. The shifts translate into hysteresis-like behavior in the real permittivity ε’ and loss tangent tan δ. A memory-polynomial model of nonlinearity order N = 4 and memory depth M = 6 accurately captures this path-dependent dielectric response, confirming the hydrogel’s intrinsic thermal memory-dependent behavior.