Johari-Goldstein Relaxation Far Below $T_g$: Experimental Evidence for the Gardner Transition in Structural Glasses?

Experimental evidence for the Gardner transition, theoretically predicted to arise deep in the glassy state of matter, is scarce. At this transition, the energy landscape sensed by the particles forming the glass is expected to become more complex. In the present Letter, we report the dielectric response of two typical glass formers with well-pronounced Johari-Goldstein $\beta$ relaxation, following this response down to unprecedented low temperatures, far below the glass transition. As the Johari-Goldstein process is believed to arise from the local structure of the energy landscape, its investigation seems an ideal tool to seek evidence for the Gardner transition. Indeed, we find an unusual broadening of the $\beta$ relaxation below about 110 K for sorbitol and 100 K for xylitol, in excess of the expected broadening arising from a distribution of energy barriers. These results are well consistent with the presence of the Gardner transition in canonical structural glass formers.

Figure 1

(a–d) Schematic loss spectra of glass-forming materials in different temperature regimes: (a) The liquid state above $T_c$. (b) The supercooled liquid, $T_g<T<T_c$. (c) The glass below $T_g$. (d) The glass far below $T_g$ and below the Gardner transition, $T<T_G<T_g$. The different contributing dynamic processes are indicated by color: the $\alpha$ relaxation (green), the JG $\beta$ relaxation (blue), the fast process (gray), and the boson peak (yellow) [38, 39]. The red area in (d) indicates the suggested additional contribution arising from the sub-basins in the energy landscape induced by the Gardner transition. (e) Typical temperature dependence of the $\alpha$- and $\beta$-relaxation times in an Arrhenius plot with a possible additional Gardner relaxation arising below $T_G$. (f) Schematic view of the energy-landscape scenario assumed to explain the occurrence of the JG $\beta$ relaxation [28, 29, 40]. (Inset) Possible modification of the local $\beta$-relaxation basins by the Gardner transition, leading to a fractal roughening of the landscape.

Figure 2

Spectra of the dielectric constant (a) and loss (b) of sorbitol for various temperatures. The lines are fits by a combination of power laws and the CC function as described in the text. For clarity reasons, in (a), at low temperatures, curves are shown for less temperatures than in (b). The inset shows a magnified view of the loss at the lowest temperature, in the region of the minimum.

Figure 3

Circles in (a) and (b): temperature dependence of the width parameter $\alpha$ as obtained from fits of the dielectric spectra of sorbitol and xylitol, respectively. Circles in (c) and (d): Corresponding half-widths of the relaxation-time distribution. The insets show the same data multiplied by $T$; the arrows indicate the anomaly that may arise from the Gardner transition. The lines in (c) and (d) (including the insets) are $W_{\tau }\propto 1/T$ laws with the proportionality factor adapted to match the experimental data at high temperatures. From these, the lines in (a) and (b) were calculated. The colored regions indicate the additional broadening ascribed to the roughening of the energy landscape arising at the Gardner transition.