Slow dynamics in glass-forming materials at α- and β-relaxation stages based on time-convolutionless mode-coupling theory

The slow dynamics of glass-forming materials in α and β stages are analyzed based on the time-convolutionless mode-coupling theory recently proposed by the present author. Similarly to the analyses used in the mode-coupling theory, the two-step relaxation processes, so called critical decay with a time exponent a and von Schweidler decay with a time exponent b, are first discussed in β stage. In α stage, the Kohlrausch–Williams–Watts type function, often called stretched exponential with a time exponent β, is also investigated in the manner similar to that in β stage. The exponents a, b, and β are then shown to satisfy the relations Γ[1−a]2Γ[1−2a]=Γ[1+b]2Γ[1+2b]=Γ[2β−1]Γ[2β+1]Γ[β+1]Γ[−β−1]=λ,12a+12b=γ,where λ is an exponent parameter, γ a power-law exponent of the α-relaxation time, and Γ[x] an usual Γ-function. Thus, the numerical values of those exponents calculated under a given value of γ are compared with the simulation results for three types of glass-forming liquids with three different values of γ and are shown to describe them well within error.