Analyses of glass transition phenomena by solving differential equation with delay effect

A linear differential equation for the analyses of glass transition phenomena has been proposed by taking into account the delay effect due to the change in transportation of atoms near the glass transition temperature (Tg). Under the condition maintaining the order of the differential equation as the second, the non-linear differential equation proposed by Van Den Beukel and Sietsma is modified to obtain the analytic solution for a linear equation by introducing the following points: the delay effect which is described with a term of Mackey-Glass model, a concept of effective free volume (xfeeff) and its concentration expression (Cfeeff) which correspond to the equilibrium, and an additional term associated with Cfeeff. In analyzing the linear equation, Doyle's p-function was used for the integral of reaction rate with respect to temperature (T). It is found that the linear equation proposed in the present study can describe the changes in free volume (x) with increasing temperature in the dx/dT − T chart, the sharp increase in free volume at Tg, and over shooting phenomena of free volume slightly above the Tg, as experimentally in thermal analyses for metallic glasses. The linear solution obtained in the present study is of great importance for the analyses of the glass transition because the change in free volume with increasing temperature on heating is described with fundamental functions.