Microscopic understanding of the Johari-Goldstein $\beta$ relaxation gained from nuclear $\gamma$-resonance time-domain-interferometry experiments

Traditionally the study of dynamics of glass-forming materials has been focused on the structural α relaxation. However, in recent years experimental evidence has revealed that a secondary β relaxation belonging to a special class, called the Johari-Goldstein (JG) β relaxation, has properties strongly linked to the primary α relaxation. By invoking the principle of causality, the relation implies the JG β relaxation is fundamental and indispensable for generating the α relaxation, and the properties of the latter are inherited from the former. The JG β relaxation is observed together with the α relaxation mostly by dielectric spectroscopy. The macroscopic nature of the data allows the use of arbitrary or unproven procedures to analyze the data. Thus the results characterizing the JG β relaxation and the relation of its relaxation time ${\tau }_{\beta }$ to the α-relaxation time ${\tau }_{\alpha }$ obtained can be equivocal and controversial. Coming to the rescue is the nuclear resonance time-domain-interferometry (TDI) technique covering a wide time range (${10}^{-9}–{10}^{-5}\mathrm{s}$) and a scattering vector $q$ range ($9.6–40\mathrm{n}{\mathrm{m}}^{-1}$). TDI experiments have been carried out on four glass formers, ortho-terphenyl [M. Saito et al., Phys. Rev. Lett. 109, 115705 (2012)], polybutadiene [T. Kanaya et al., J. Chem. Phys. 140, 144906 (2014)], 5-methyl-2-hexanol [F. Caporaletti et al., Sci. Rep. 9, 14319 (2019)], and 1-propanol [F. Caporaletti et al., Nat. Commun. 12, 1867 (2021)]. In this paper the TDI data are reexamined in conjunction with dielectric and neutron scattering data. The results show the JG β relaxation observed by dielectric spectroscopy is heterogeneous and comprises processes with different length scales. A process with a longer length scale has a longer relaxation time. TDI data also prove the primitive relaxation time ${\tau }_0$ of the coupling model falls within the distribution of the TDI $q$-dependent JG β-relaxation times. This important finding explains why the experimental dielectric JG β-relaxation times ${\tau }_{\beta }\left(T,P\right)$ is approximately equal to ${\tau }_0\left(T,P\right)$ as found in many glass formers at various temperature $T$ and pressure $P$. The result, ${\tau }_{\beta }\left(T,P\right)\approx {\tau }_0\left(T,P\right)$, in turn explains why the ratio ${\tau }_{\alpha }\left(T,P\right)/{\tau }_{\beta }\left(T,P\right)$ is invariant to changes of $T$ and pressure $P$ at constant ${\tau }_{\alpha }\left(T,P\right)$, the α-relaxation time.

Figure 1

(a) Inverse temperature dependence of the relaxation time measured by TDI at different $q$ values: 14 ${\mathrm{nm}}^{-1}$ [40] (gray circles), $18\mathrm{n}{\mathrm{m}}^{–1}$ [41] (green circles), 23 ${\mathrm{nm}}^{-1}$ [40] (blue diamonds), and 29 ${\mathrm{nm}}^{-1}$ [41] (yellow right-pointing triangles). The solid lines of the corresponding color show the best fitting curves using the $T$ dependence for the α and JG β relaxation as reported in [40, 41]. The characteristic times of the JG β relaxation as measured by dielectric spectroscopy (DS); larger brown ovals from Ref. [45] and red squares from Ref. [4] are also reported for the sake of comparison. The red dash-dotted line is the Arrhenius dependence of the DS data from Ref. [4]. Inset: Static structure factor $S$($q$) of OTP. (b) Dielectric loss spectra of $o$-terphenyl versus log ($f$/Hz) at temperatures $T$ = 178, 202, 226, 242, 246, 250, 254, and 258 K, in the order of increasing peak frequency [45]. For the lines and arrow see text. Part of panel (a) is reproduced from Ref. [40] by permission from APS.

Figure 2

(a) Temperature dependence of the TDI mean relaxation time $\langle {\tau }_{\mathrm{KWW}}\rangle$ and fitting curves given by symbols and lines, respectively, for PB at $Q=9.6$ (open circles, chain line), 15 (filled circles, thin solid line), 21 (open squares, dotted line), 27 (closed squares, dashed line), 32 (open triangles, two-dotted chain line), and 39 (closed triangles, thick solid line) ${\mathrm{nm}}^{-1}$. All are shown in black and reproduced from Ref. [42]. The dielectric ${\tau }_{\beta }$ (yellow filled squares) and ${\tau }_{\alpha }$ (green filled triangles), and the primitive relaxation times ${\tau }_0$ (red filled circles). The magenta arrow locates 1000/178 ${\mathrm{K}}^{-1}$. The purple line represents ${\tau }_{\beta }\left(q,T\right)$ from neutron spin echo at $Q=27.1 {\mathrm{nm}}^{-1}$. The green line represents the neutron spin echo ${\tau }_{\beta }\left(q,T\right)$ data at $Q=18.8 {\mathrm{nm}}^{-1}$ from Ref. [68]. (b) Structure factor $S\left(Q\right)$ of PB at 190 K and $Q$ dependence of the mean relaxation time $\langle {\tau }_{\mathrm{KWW}}\rangle$ at 220 K. (c) Scaled dielectric loss spectrum at 178 K. The dashed-dotted and the dotted lines represent the contributions of the α relaxation and the effective β relaxation, respectively, to the total loss [50]. The downward pointing arrows indicate ${\omega }_{\beta }\left(q,T\right)=1/{\tau }_{\beta }\left(q,T\right)$ at $q=39,32,27$, and 21 ${\mathrm{nm}}^{-1}$ and $T=178$ K from right to left. The longest downward pointing arrow indicates the Cole-Cole peak frequency ${\omega }_{\beta }$ = 1/${\tau }_{\beta }$ determined by the fit. The upward pointing arrow on the right is the primitive relaxation frequency ${\omega }_0$ = 1/${\tau }_0$ calculated by the CM Eq. (2). Part of panel (a) is reproduced from Ref. [42] by permission from AIP.

Figure 3

(a) $T$ dependence of the relaxation time measured by TDI at three different $q$ values: $q$ = 13 (blue stars), 24 (magenta triangles), and $37{\mathrm{nm}}^{-1}$ (green squares) and by dielectric spectroscopy (DS) (gray circles, red circles). The continuous lines are fits to the DS data (α and JG β relaxation). The solid lines are the same fits to the α and JG β relaxations rescaled to match the TDI data. The primitive relaxation times predicted by CM at two temperatures (164.7 and 176.6 K) are represented by the two yellow circles. Vertical dashed lines show their separation from α relaxation. Inset: diffuse scattering pattern of 5M2H at $T=187.6$ K, with the indication of the $q$ values and of the corresponding ranges covered in the TDI measurements reported in the main figure. Data taken from Ref. [43] and replotted with added information. (b) Direct comparison of dielectric (green squares) and DDLS data of 5M2H (black circles, and black line is fit by Gabriel et al. [36] at 157 K, demonstrating that α and β relaxations appear to be almost identical in both methods in their line shapes and time constants (DDLS, red circles). Fit of the Debye relaxation (light blue dashed line for dielectric data and green dashed line for DDLS data were done by Gabriel et al. The blue solid line is the KWW fit to α relaxation for both dielectric and DDSL data with $n$ = 0.44.

Figure 4

(a) Inverse temperature dependence of the relaxation time (τ) measured by dielectric spectroscopy (gray and green squares) and time-domain interferometry at two different $q$ values: 15 (blue circles with error bars) and 25 ${\mathrm{nm}}^{-1}$ (red squares with error bars). The gray line is the fit to ${\tau }_{\alpha }$ from dielectric data. The green arrow locates 1000/112.7 ${\mathrm{K}}^{-1}$. The purple closed triangles are the primitive relaxation times ${\tau }_0$. The inset is the static structure factor $S$($q$) of 1-propanol. (b) Susceptibility from DDLS experiment [35] at 112.7 K. The dashed line is the Kohlrausch fit with $n=0.40$. The lines are labeled.