Frequency Selection for Dynamic Impedimetric Analysis
When performing impedimetry of dynamically hydrating hydrogels in order to determine their responses during swelling, i.e. swelling kinetics, one is generally focused on the hydrogel‘s “membrane resistance”, RM, according to an appropriate equivalent circuit model. The frequency at which impedimetry is conducted - or better, the “impedimetric frequency” (ωi) - should be carefully selected based upon the frequencies that capture RM best at various degrees of hydration. For simplicity, frequencies will only be considered for “dehydrated” (<50% hydration) and “hydrated” (>50% hydration) states. To calculate ωi, first, the breakpoint frequencies (the frequency at which Bode magnitude transitions from resistive to capacitive and crosses phase of 45° 1 ,2) corresponding to the dehydrated and hydrated states, ω1d and ω1h, respectively, must be determined. Breakpoint frequencies can be calculated with the calculator provided here using inputs (RCT and RM) from equivalent circuit models of the electrical impedance spectra (EIS) and the characteristic frequencies (ωc) related to hydrogel interfacial behavior (RCTQDL) from both dehydrated and hydrated hydrogel states. Characteristic frequencies are defined as those frequencies at which phase angles are highest in Bode plots or, in other words, those that correspond to the topmost points of appropriate semi-circles in Nyquist plots 3 ,4. The average of these two values may be suitable for impedimetry. However, by assigning weights based on the total hydration time that hydrogels will spend in each state, if they were to follow the equation for swelling by Fickian diffusion and polymer relaxation (Mt/Minf = k1t + k2t0.5), then a more appropriate impedimetric frequency will be returned. Thus, ω1d and ω1h, are weighted by 0.175 and 0.825, respectively, to return the ωi at which impedimetry of your hydrogel should be performed.
RCT-Dehydrated (Ω) RCT-Hydrated (Ω)
ωc-Dehydrated (Hz) ωc-Hydrated (Hz)