A simplified nonlinear memory function for the dynamics of glass-forming materials based on time-convolutionless mode-coupling theory

A simplified nonlinear memory function is proposed in the ideal time-convolutionless mode-coupling theory equation to study the dynamics of glass-forming liquids. The numerical solutions are then compared with the simulation results performed on fragile liquids and strong liquids. They are shown to recover the simulation results in a supercooled state well within error, except at a $\beta$-relaxation stage because of the ideal equation. A temperature dependence of the nonlinearity $\mu$ in the memory function then suggests that the supercooled state must be clearly separated into two substates, a weakly supercooled state in which $\mu$ increases rapidly as $T$ decreases and a deeply supercooled state in which $\mu$ becomes constant up to the glass transition as $T$ decreases. On the other hand, it is shown that in a glass state $\mu$ increases rapidly as $T$ decreases, while it is constant in a liquid state. Thus, it is emphasized that the new model for the simplified memory function is much more reasonable than the conventional one proposed earlier by the present author not only qualitatively but also quantitatively.