Glassy aging with modified Kohlrausch-Williams-Watts form

In this paper, we address the question of whether aging in the nonequilibrium glassy state is controlled by the equilibrium $\alpha$-relaxation process, which occurs at temperatures above $T_g$. Recently, Lunkenheimer et al. [Phys. Rev. Lett. 95, 055702 (2005)] proposed a model for the glassy aging data of dielectric relaxation using a modified Kohlrausch-Williams-Watts form $\text{exp}\left[-{\left(t_{\text{age}}/{\tau }_{\text{age}}\right)}^{{\beta }_{\text{age}}}\right]$. The aging time $t_{\text{age}}$ dependence of the relaxation time ${\tau }_{\text{age}}$ is defined by these authors through a functional relation involving the corresponding frequency $\nu \left(t_{\text{age}}\right)=1/\left(2\pi {\tau }_{\text{age}}\right)$, but the stretching exponent ${\beta }_{\text{age}}$ is the same as ${\beta }_{\alpha }$, the $\alpha$-relaxation stretching exponent. We present here an alternative functional form for ${\tau }_{\text{age}}\left(t_{\text{age}}\right)$ directly involving the relaxation time itself. The proposed model fits the data of Lunkenheimer et al. perfectly with a stretching exponent ${\beta }_{\text{age}}$ different from ${\beta }_{\alpha }$.

Figure 1

(Color online) ${ϵ}^{″}\left(t_{\text{age}}\right)$ data for glycerol at $179\text{K}$ vs $t_{\text{age}}$ (s) at the frequencies displayed in the inset. Solid lines correspond to the best fits using the modified KWW form with ${\tau }_{\text{age}}\left(t_{\text{age}}\right)$ determined from Eq. (4).

Figure 2

(Color online) Best-fit values of ${ϵ}_{\text{st}}^{″}$ for the curves in Fig. 1 vs frequency $\nu$ (Hz) as obtained with the present work (solid line) and the corresponding results of Ref. [5] (dashed). The inset shows the best-fit ${ϵ}_{\text{eq}}^{″}$ with frequency $\nu$ respectively from the present work and Ref. [5].

Figure 3

(Color online) Best-fit $\tau \left(t_{\text{age}}\right)$ vs $t_{\text{age}}$ (s) for glycerol using ansatz (4) of the present work. Result of Ref. [5] for $\tau \left(t_{\text{age}}\right)$ as inset.

Figure 4

(Color online) Scaling of the dielectric data for glycerol at $179\text{K}$ corresponding to different frequencies. Fits obtained with ${\tau }_{\text{age}}\left(t_{\text{age}}\right)$ in the modified KWW ansatz determined from (a) Eq. (4) of the present work (solid line) and (b) Ref. [5] (dashed line). Also shown are best fits obtained taking ${\tau }_{\text{age}}={\tau }_{\text{st}}$ (short dashed) and ${\tau }_{\text{age}}={\tau }_{\text{eq}}$ (long dashed), respectively, with the stretching exponent being fixed at ${\beta }_{\alpha }$.

Figure 5

(Color online) (a) ${ϵ}^{″}\left(t_{\text{age}}\right)$ data for PC [6, 15] at $152\text{K}$ vs $t_{\text{age}}$ (s) at the frequencies displayed in the inset, (b) ${ϵ}^{″}\left(t_{\text{age}}\right)$ data for PG [8, 16] at $157\text{K}$(s) at the frequencies displayed in the inset, (c) ${ϵ}^{″}\left(t_{\text{age}}\right)$ data for xylitol [16] at $243\text{K}$ vs $t_{\text{age}}$ (s) at the frequencies displayed in the inset, and (d) $M^{″}\left(t_{\text{age}}\right)$ data for CKN [18] at $329\text{K}$ vs $t_{\text{age}}$ (s) at the frequencies displayed in the inset. In all panels, the solid lines are obtained using our scheme.

Figure 6

(Color online) $\tau \left(t_{\text{age}}\right)$ vs $t_{\text{age}}$ (s) for different materials using ansatz (4) in the present work.

Figure 7

(Color online) $\alpha$-relaxation time ${\tau }_{\alpha }$ vs inverse of temperature for CKN [6], xylitol [15], glycerol [16], PG [17], and PC [18]. The ${\tau }_{\text{eq}}$ obtained from the fitting of the corresponding data is shown by filled circles for the fitting of Ref. [5] and by filled triangles for the present fitting scheme.