Simple model of the diffusive scattering law in glass-forming liquids

Abstract

We found that the decay rate, \ensuremath{\lambda}, of the correlation function, G, in glass-forming liquids can be expressed in terms of G itself. A three-parameter function of the form \ensuremath{\lambda}=${\ensuremath{\lambda}}_{0}$+${\ensuremath{\lambda}}_{1}$${\mathit{G}}^{\mathit{k}}$ leads to an analytic solution for the function G in the time domain. This assumption also allows one to express the function G as an infinite series of Lorentzians in the frequency domain. This model has been applied to M\"ossbauer absorption spectra of aqueous ${\mathrm{FeCl}}_{2}$ solutions in glycerol (14.4-keV line of $^{57}\mathrm{Fe}$) and higher temperature quasielastic (Rayleigh) scattering data obtained by the scattering of the $^{183}\mathrm{W}$ 46.5-keV M\"ossbauer line from pure glycerol. The model describes the M\"ossbauer data as well as the empirical Kohlrausch or Cole-Davidson (CD) laws. Also, the function \ensuremath{\lambda}(G) can be expressed in a two-parameter form as ${\ensuremath{\lambda}}_{0}$exp(\ensuremath{\alpha}G). It has been shown, however, that a closed-form expression for the function G does not exist in the time or frequency domain for this latter form. On the other hand, the exponential form gives a two-parameter fit to existing data, suggesting a physical basis to this form and implies that small changes in G are proportional to fractional changes in \ensuremath{\lambda}. It has been found that the parameter \ensuremath{\alpha} has some universal meaning as it remains constant over a significant low-temperature range accessible experimentally, decreasing to the zero value with increasing temperature. Our analysis suggests that the parameter \ensuremath{\alpha} may change in steps as sample temperature is increased. Such a behavior suggests that some processes (degrees of freedom) are ``freezing out'' at well-defined temperatures.