Sound wave propagation and existence of a two step relaxation process in a glass-former melt

The velocity and damping of longitudinal and transverse sound waves of a prototypical glass-forming melt, $m$-tricresylphosphate, are studied over a very wide frequency span by an ultrasonic and a light scattering experiment ${(10\mathrm{}}^7$ and ${10}^{11}$ rad/s). In the Brillouin light scattering measurements, the central quasielastic line has been also analyzed. Unpublished data on velocity and damping (frequency $\approx {10}^9$ rad/s) are also included in the present analysis. A quantitative analysis of the data gives good agreement for individual spectra as well as for the general temperature trends. The analysis is based on the existence of fast $\left(\beta \right)$ and slow $\left(\alpha \right)$ relaxation processes in the liquid. A phenomenological description of a two step relaxation function is proposed, and the phenomenological parameters [relaxation times, ${\tau }_{\alpha }$ and ${\tau }_{\beta },$ relative weight, $\tilde{f}\mathrm{}\left(T\right)\mathrm{}$, etc.] of the two relaxation processes are derived. As a consequence of the two step relaxation process, we propose to revise the concept of “infinite frequency” ${(c}_{\infty })$ sound velocity, introducing two characteristic sound velocities, one which applies if $\omega {\tau }_{\beta }\gg 1,$ $c_{\infty },$ and one, $c_{\infty \alpha },$ which describes the elastic response of the liquid at frequency higher than $1/{\tau }_{\alpha }$ but smaller than $1/{\tau }_{\beta }.$ Within the framework of a two step relaxation process, the meaning of “nonergodicity” parameter of mode-coupling theory and its relation with $\tilde{f}\mathrm{}\left(T\right)\mathrm{}$ are also discussed. The weight $\tilde{f}\mathrm{}\left(T\right)\mathrm{}$ is found to decrease monotonically from near 1 to near 0.4 between $T_g$ and ${1.5T}_g.$ The comparison of our findings with the predictions of the mode-coupling theory shows qualitative accord in several important points.