Chirp-Z analysis for sol–gel transition monitoring

Gelation is a complex reaction that transforms a liquid medium into a solid one: the gel. In gel state, some gel materials (DMAP) have the singular property to ring in an audible frequency range when a pulse is applied. Before the gelation point, there is no transmission of slow waves observed; after the gelation point, the speed of sound in the gel rapidly increases from 0.1 to 10 m/s. The time evolution of the speed of sound can be measured, in frequency domain, by following the frequency spacing of the resonance peaks from the Synchronous Detection (SD) measurement method. Unfortunately, due to a constant frequency sampling rate, the relative error for low speeds (0.1 m/s) is 100%. In order to maintain a low constant relative error, in the whole speed time evolution range, Chirp-Z Transform (CZT) is used. This operation transforms a time variant signal to a time invariant one using only a time dependant stretching factor (S). In the frequency domain, the CZT enables us to stretch each collected spectrum from time signals. The blind identification of the S factor gives us the complete time evolution law of the speed of sound. Moreover, this method proves that the frequency bandwidth follows the same time law. These results point out that the minimum wavelength stays constant and that it only depends on the gel.