Relation between some secondary relaxations and the α relaxations in glass-forming materials according to the coupling model

Some secondary or β relaxations in glass-forming materials involve molecular motions that bear strong resemblance to the primitive α relaxations of the coupling model, although the two are not identical. For these β relaxations, at the glass transition temperature Tg the relaxation time τβ(Tg) is expected to be shorter than but not too different in order of magnitude from τ0(Tg), the primitive α-relaxation time at Tg. The latter can be calculated by the coupling model from the relaxation time τα(Tg), the exponent (1−n) of the Kohlrausch–Williams–Watts (KWW) correlation function exp[−(t/τα)1−n], and the experimental crossover time, tc≈2 ps, of the α relaxation. From experimental data of β and α relaxations in a variety of glass-forming materials, it is found that τβ(Tg) and τ0(Tg) are close to each other in order of magnitude as anticipated. The results indicate these β relaxations indeed bear some close relation to the corresponding primitive α relaxation, although they are not the same process. Since the relaxation times of the majority of these β relaxations have the Arrhenius temperature dependence, τβ(T)=τβ∞ exp(Eβ/RT), where τβ∞ is of the order of 10−13–10−16 s, knowing, approximately, the value of τβ(T) at one temperature Tg means the location of the β relaxation in the relaxation map can be roughly determined from the α relaxation. The findings can be restated as the empirical result: there exists a strong correlation between the value of log[τβ(Tg)] and the KWW exponent (1−n) of the α relaxation in many glass-formers. A smaller KWW exponent of the α relaxation corresponds to shorter τβ(Tg) or smaller log[τβ(Tg)]. This remarkable cross correlation between the α relaxation and the β relaxation should be of interest for any model or theory of molecular dynamics of glass formers.