Physical Aging and Heterogeneous Dynamics

Physical aging appears consistent with homogeneous models that are based upon a single “inner clock”, but incompatible with the established heterogeneous nature of relaxation in glass-forming materials and the concomitant dispersion of aging rates. This work demonstrates that aging follows the conceptually simpler model of heterogeneous dynamics that differs from the homogeneous case in the rate at which equilibrium is approached. However, the very fast modes within the relaxation time dispersion age according to a common inner clock, because their fictive temperatures are slaved to macroscopic softening. Evidence is provided for such a transition to homogeneous aging as the frequency is increased into the excess wing. The results explain why aging of glasses appears homogeneous and consistent with time aging-time superposition in cases where observations are based on the high frequency behavior.

Figure 1

Calculated aging behavior for HN type dynamics with ${\tau }_{\mathrm{HN}}=3370\mathrm{s}$ at the initial temperature $T_0$ ($\approx 182.5\mathrm{K}$) and ${\tau }_{\mathrm{HN}}=63500\mathrm{s}$ for steady state at $T_{\infty }=179\mathrm{K}$, and with temperature invariant exponents ${\alpha }_{\mathrm{HN}}=0.95$ and ${\gamma }_{\mathrm{HN}}=0.50$ (corresponding to $\beta =0.55$ and chosen to match the case of glycerol of Ref. 25). Solid lines represent the dielectric loss spectra at $t=0$ and $t=\infty$, as indicated. The dashed lines are for various aging times, indicated by the numbers in terms of ${log}_{10}(t/s)$. Panel (a) is for homogeneous dynamics based on Eq. (3), while panel (b) represents the heterogeneous case calculated from applying Eq. (2) separately to each domain.

Figure 2

Evolution of the logarithmic peak loss frequency, ${log}_{10}({\nu }_{\max }/\mathrm{Hz})$, versus aging time based on the same parameters leading to the data in Fig. 1. The two solid curves are the results for homogeneous and heterogeneous dynamics, as labeled, with the heterogeneous scenario approaching steady state conditions a factor of $\approx 10$ faster in time. The dashed lines are KWW fits to the “hom” and “het” curves, with $\tau =50000\mathrm{s}$, $\beta =0.42$ and $\tau =7500\mathrm{s}$, $\beta =0.55$, respectively. Circles indicate times for which spectra are shown. The inset shows deviations from time aging-time superposition for the heterogeneous case by redrawing the dashed curves of Fig. 1 on normalized scales, ${\epsilon }^{\prime \prime }/{\epsilon }_{\max }^{\prime \prime }$ versus $\nu /{\nu }_{\max }$. In arrow direction, the aging times are given by ${log}_{10}(t/s)=6$, 5, 2, 4, and 3.

Figure 3

Increase of fictive temperatures $T_f$ during energy absorption at different frequencies for MTHF at $T=96.1\mathrm{K}$. The two top panels show the time resolved $T_f(t)$ in terms of the relative change in dielectric loss factor, $\Delta \ln (\tan \delta )$. For the lower frequencies, $\nu \le 1\mathrm{kHz}$ (a), curves coincide on a $\nu t$ scale, indicating that the rates of the inner clocks scale with frequency. For the higher frequencies, $\nu \ge 2\mathrm{kHz}$ (b), curves follow a common inner clock as they coincide on an absolute $t$ scale. Panel (c) indicates the frequency positions within the loss spectrum ${\epsilon }^{\prime \prime }(\omega )$ at which fictive temperature relaxation behavior has been determined. Data from Ref. 30.