Modeling enthalpy relaxation using the Mittag-Leffler function

Understanding the structural relaxation plays an essential role in the application of amorphous solids. Most of the current models use the fictive temperature concept to describe structural relaxation. The fictive temperature is assumed to evolve with temperature through the Kohlrausch-Williams-Watt (KWW) function. In this work, we use the Mittag-Leffler (ML) function to replace the KWW function. We will demonstrate the similarity and difference of the two models. Both the continuous model in the integral form and the discrete model with multiple relaxation processes in differential form are presented. We provide a numerical algorithm to obtain the relaxation spectra of the discrete models. It is shown that the spectrum of the model with the KWW kernel is asymmetric while the spectrum of the model with the ML kernel is symmetric. We further demonstrate that the continuous model and the discrete model are numerically equivalent. The performances of the models with the KWW kernel and the ML kernel are compared in details. It is showed that both models can capture the same trend on the effects of the cooling rate and annealing time on the enthalpy response. However, the magnitude and location of the enthalpy overshoot predicted by the two models are different. Thus, this work provides another choice for the relaxation function.