{"id":528074,"date":"2026-06-10T13:04:22","date_gmt":"2026-06-10T13:04:22","guid":{"rendered":"https:\/\/www.europesays.com\/ie\/528074\/"},"modified":"2026-06-10T13:04:22","modified_gmt":"2026-06-10T13:04:22","slug":"crossover-of-quasi-localized-dynamics-and-diffusion-in-supercooled-liquids","status":"publish","type":"post","link":"https:\/\/www.europesays.com\/ie\/528074\/","title":{"rendered":"Crossover of quasi-localized dynamics and diffusion in supercooled liquids"},"content":{"rendered":"

A liquid cooled sufficiently fast below the melting temperature does not crystallize and can be deeply undercooled until it eventually reaches the glass state across the glass-transition temperature Tg (ref. 1<\/a>). Upon cooling, the molecular dynamics become slower and slower, and around a temperature often identified with the mode-coupling transition temperature Tc \u2248 1.2Tg, a plateau develops in the time dependence of the mean-squared molecular displacement2<\/a>,3<\/a>. This plateau is a signature of caged dynamics: the molecules remain trapped in the cage formed by their first neighbours (the plateau in the mean-squared displacement) before escaping and diffusing away. Tc also marks the temperature below which a secondary relaxation process, known as Johari\u2013Goldstein or \u03b2JG relaxation4<\/a>, branches off the slower, main relaxation process, known as \u03b1 or structural relaxation1<\/a>. The \u03b2JG relaxation is still under investigation and, despite appearing in the temperature range below Tc crucial for a detailed understanding of the glass-transition, its role in the vitrification process remains under debate5<\/a>,6<\/a>,7<\/a>.<\/p>\n

Johari and Goldstein were the first to report evidence of \u03b2JG relaxation in the dielectric spectra of glass formers composed of rigid molecules4<\/a>. A large literature of dielectric spectroscopy studies of mainly organic glass formers has been accumulated since then8<\/a>,9<\/a>, and criteria have been proposed to distinguish \u2018genuine\u2019 \u03b2JG relaxations from processes related to internal degrees of freedom10<\/a>. Although the \u03b2JG relaxation was initially considered purely rotational in nature due to its appearance in dielectric spectroscopy spectra11<\/a>, its observation by techniques probing density fluctuations12<\/a> and in mechanical spectroscopy studies of metallic glasses13<\/a>, where no orientation degrees of freedom are present, suggests a substantial translational component. An important outcome of the available studies has been a comprehensive understanding of the role of the \u03b2JG relaxation in a number of relevant mechanical properties such as, for example, the plastic response of the material13<\/a>. More recently, calorimetry has also been used to characterize the \u03b2JG process, adding an additional perspective14<\/a>,15<\/a>.<\/p>\n

Different conceptual frameworks, not clearly compatible with each other, have been used to describe the \u03b2JG relaxation. In a real-space picture, it is usually interpreted as a quasi-localized process occurring in loosely connected regions of the glass structure4<\/a>. This description is supported by confocal microscopy experiments, where clusters of fast-moving particles could be directly observed in colloidal supercooled fluids and glasses16<\/a>. Although real-space imaging of bulk liquids with atomic resolution is still very difficult, consistent results were obtained in experiments probing the structure of metallic glasses17<\/a>. One- and two-dimensional 2H-nuclear magnetic resonance (NMR) studies provided evidence that the secondary relaxation is associated with small-angle reorientational jumps, which involve essentially all molecules over time18<\/a>,19<\/a>. In the coupling model, the \u03b2JG relaxation is associated to the dynamics of particles that escape from their cage and is considered to be the precursor of the \u03b1 relaxation20<\/a>. In the potential energy landscape framework21<\/a>,22<\/a>, the \u03b2JG relaxation is associated with transitions between neighbouring minima with small energy barriers, which corrugate larger \u2018metabasins\u2019 explored during the \u03b1 relaxation. This topographic hierarchy in the potential energy landscape has been supported by a recent numerical simulation of an asymmetric dimer system23<\/a>. However, a hierarchical separation between basins of different heights does not seem to be necessarily required. Within random first-order transition theory, for example, the \u03b2JG relaxation is associated to a low free-energy tail of the activation-barrier distribution, which emerges because of the differences in the geometry of the reconfiguring regions related to the \u03b1 and \u03b2JG relaxations24<\/a>. Numerical simulations covering exceptionally long times instead support a scenario in which the \u03b2JG relaxation originates from a heterogeneous activated dynamics with dynamic facilitation, a phenomenon in which microscopic motion induces further motion nearby25<\/a>,26<\/a>.<\/p>\n

Despite a large amount of work that has already been devoted to this subject, pinpointing the microscopic characteristics of the \u03b2JG relaxation process, for example, the rotational versus translational molecular displacements involved in it and their distribution in space and time, remains an extraordinary theoretical, numerical and experimental challenge. We here contribute to this matter by reporting an experimental study of the wavenumber-resolved dynamical response of the \u03b2JG relaxation in a model hydrogen-bonded liquid, 5-methyl-2-hexanol (5M2H), in the temperature range where it separates from the \u03b1 relaxation. The data collected, analysed using a new inversion approach, put us in a position to provide a clear picture of the \u03b2JG relaxation: it corresponds to the critical rattling of the molecules in the cage formed by their first neighbours, triggering the onset of diffusion. This work thus identifies the microscopic motions that signal the onset of glassy behaviour.<\/p>\n

In the nuclear \u03b3-resonance time-domain experiment that we discuss here (see Supplementary Fig. 1<\/a> for an example of raw data), it is possible to extract the (normalized) intermediate scattering function, f(q<\/b>, t), that is the q-component of the (normalized) density correlation function at time t (ref. 27<\/a>) (see Methods<\/a> and Supplementary Text 1<\/a> for more details). This technique is used here to study the microscopic density fluctuations in deeply supercooled 5M2H, a model hydrogen-bonded glass former characterized by a genuine \u03b2JG process28<\/a> and with a characteristic timescale ideal for the dynamic range of time-domain interferograms (TDI)29<\/a>. Figure 1a<\/a> reports the intermediate scattering function of 5M2H measured at 176.9\u2009K as a function of time for three q-values (see Fig. 1b<\/a> for the corresponding total scattering profile). The decay with time of density fluctuations is well described by a stretched exponential, also called the Kohlrausch-Williams-Watts (KWW) function, as in previous studies12<\/a>,29<\/a>,30<\/a><\/p>\n

$$f({\\bf{q}},t)={f}_{{\\bf{q}}}(T)\\exp \\left[-{\\left(\\frac{t}{\\tau ({\\bf{q}},T)}\\right)}^{{\\beta }_{\\mathrm{KWW}}}\\right].$$<\/p>\n

\n (1)\n <\/p>\n

By fitting equation (1<\/a>) to the experimental data of f(q<\/b>, t) for different temperatures and wavenumbers, as shown in the examples reported in Fig. 1a<\/a>, the relaxation time, \u03c4(q, T), and the relaxation strength, fq(T), have been extracted. The shape parameter, \u03b2KWW, has been fixed to 0.5 for all temperatures and q to reduce the number of free fitting parameters. However, the results of this study are not affected by variations of \u03b2KWW in the \u00b10.2 range (see Supplementary Text 2<\/a>, Supplementary Fig. 2<\/a> and Supplementary Table 1<\/a> for justification and more details).<\/p>\n

Fig. 1: Nuclear \u03b3-resonance time-domain experimental results.<\/b>\"Fig.The alternative text for this image may have been generated using AI.<\/p>\n

a<\/b>, Intermediate scattering functions extracted from \u03b3-resonance TDI measured at 176.9\u2009K at the indicated wavenumbers. The solid lines show best fits to the data using the KWW function (equation (1<\/a>)). The error bars denote \u00b11\u2009s.d., computed after fitting and reduction of the TDI beating patterns (ref. 56<\/a> and Supplementary Text 1<\/a>) and subsequent logarithmic binning on the time axis. Each bin contains n = 11 time delays for t < 60\u2009ns, increasing to n = 22 for 60 < t < 120\u2009ns and to n > 100 at longer times owing to reduced count rates. The s.d. combines the within-bin variance with propagated uncertainties from the data reduction procedure. Each beating pattern has a total integrated intensity >1 \u00d7 105 counts, with an average of >50 counts per time delay (Supplementary Fig. 1<\/a>). Shaded regions indicate the 68% confidence intervals obtained from the fitting procedure. b<\/b>, Total scattered intensity measured at 176.9\u2009K. The covered q-interval ranges from 9 to 40\u2009nm\u22121 and is associated to the shown range of colours. The horizontal bar indicates the typical wavenumber resolution in the experiment. c<\/b>, The diamonds represent the temperature dependence of the relaxation time, \u03c4, measured at different wavenumbers identified according to the colours in b<\/b>. In the plot, we also report data obtained in a previous experiment (circles)29<\/a>. Error bars are \u00b11\u2009s.d., obtained from the weighted nonlinear fitting of the experimental data57<\/a>.<\/p>\n

The temperature dependence of \u03c4, probed at different q, is shown in the relaxation map reported in Fig. 1c<\/a>. As already observed in ref. 29<\/a>, a change in the activation energy of the relaxation time can be observed around T\u03b1\u03b2\u2009=\u2009181\u2009K, with the activation energy matching that of the \u03b1 relaxation above T\u03b1\u03b2 and that of the \u03b2JG relaxation below T\u03b1\u03b2 (ref. 12<\/a>). In the present work, we extend our previous investigation reported in ref. 29<\/a> to a larger q-range and to a denser q-grid, as shown in Fig. 2<\/a>, for the q-dependence of the relaxation time.<\/p>\n

Fig. 2: The \u03b1-to-\u03b2JG transition observed as a function of q.<\/b>\"Fig.The alternative text for this image may have been generated using AI.<\/p>\n

a<\/b>\u2013c<\/b>, Wavenumber dependence of the relaxation time at 188 K (a<\/b>), 181 K (b<\/b>) and 176.9 K (c<\/b>) across T\u03b1\u03b2 \u2248 181\u2009K (ref. 29<\/a>), as reported in the legends. The dashed and dashed-dotted lines through the data for \u03c4 at T\u2009=\u2009181\u2009K (b<\/b>) and T\u2009=\u2009176.9\u2009K (c<\/b>) show, at q \u2248 16\u2009nm\u22121, a transition from a q\u22122 regime at low q to a q\u22124 regime at high q. The full lines through the data correspond to equation (2<\/a>). The data for \u03c4 at T\u2009=\u2009188\u2009K (a<\/b>) only show the q\u22122 regime. The error bars denote \u00b11\u2009s.d. from the nonlinear fitting procedure of the beating patterns.<\/p>\n

Above T\u03b1\u03b2, we observe in Fig. 2a<\/a> an inversely quadratic wavenumber dependence for \u03c4, compatible with the diffusive behaviour expected in this region, as also reported for other glass formers2b,c<\/a>). In more detail, at both 181\u2009K and 179.6\u2009K and at scattering vectors larger than ~16\u2009nm\u22121, \u03c4 displays, in agreement with previous observations, a super-quadratic q-dependence, \u03c4 \u221d q\u2212n with n > 2, which is a hallmark of restricted subdiffusive dynamics. In particular, the value of the exponent n averaged over the data at 181\u2009K and 176.9\u2009K is \u3008n\u3009 = 4.3(3). Instead, at smaller q-values, an inversely quadratic q-dependence is observed, indicating that the diffusive dynamics is recovered at small scattering vectors (that is, large distances and long times). As already reported in our previous study29<\/a> and here even more clearly demonstrated, the wavenumber at which the q-dependence changes from diffusive to subdiffusive, qDS \u2248 16 nm\u22121, corresponds to a root mean-squared displacement \\(\\sqrt{6}\/{q}_{\\mathrm{DS}}\\simeq 1.5\\,\\mathring{\\rm A} \\), which is ~20% of the average intermolecular distance. This value is compatible with Lindemann\u2019s criterion for structural instability and with the observation that, in simulated metallic glasses, the \u03b1 relaxation appears when molecular displacements reach roughly 20% of the distance of nearest neighbours32<\/a>. From a different perspective, we also observe that the visibility of the \u03b2JG relaxation in the density correlation function at high q might be explained by thinking that this process has a strong rotational component that dominates the density correlation function at high q, as the rotational\u2013translational coupling becomes strong there, consistent with the results of a molecular dynamics study of undercooled water33<\/a>. The transition between the two different regimes for \u03c4 is smooth and can be described using the simple expression<\/p>\n

$$\\tau (q)=\\frac{1}{{D}_{{\\rm{\\alpha }}}{q}^{2}+{\\widetilde{D}}^{2}{q}^{4}},$$<\/p>\n

\n (2)\n <\/p>\n

where D\u03b1 is the diffusion coefficient, and \\(\\widetilde{D}\\) is an anomalous diffusion coefficient associated with the subdiffusive dynamics within the \u03b2JG relaxation29<\/a>,34<\/a>. The full lines through the experimental data at 181\u2009K and 176.9\u2009K in Fig. 2<\/a> are the results of a fit using equation (2<\/a>) and describe the data rather well. At first glance, this result could be interpreted, following a study on dielectric spectroscopy and neutron scattering experiments34<\/a>,35<\/a>, assuming that density fluctuations around the decoupling temperature T\u03b1\u03b2 relax via two independent relaxation channels, the \u03b1 and the \u03b2JG ones, which exist and are active at all q. The intermediate scattering function could then be written as the product of the two relaxation processes: f(q, t) = f\u03b1(q, t)f\u03b2(q, t). This implies that the density relaxation time would be determined by the faster process, namely the \u03b1 relaxation for q < 16\u2009nm\u22121 and the \u03b2JG relaxation for q > 16\u2009nm\u22121.<\/p>\n

To test this hypothesis, we consider the q-dependence of the intermediate scattering function f(q, t)\/fq probed at different times, as reported in Fig. 3<\/a>, for 188, 181 and 176.9\u2009K at three different times.<\/p>\n

Fig. 3: Wavenumber dependence of the intermediate scattering function.<\/b>\"Fig.The alternative text for this image may have been generated using AI.<\/p>\n

Top: the experimental data for f(q, t)\/fq are reported as a function of q at three different times (columns) and at three different temperatures (rows) across T\u03b1\u03b2. The full red, blue and black lines correspond to the shape expected for the \u03b1 relaxation and the \u03b2JG relaxation and their product, respectively. The first two contributions describe the data well at low- and high q, respectively, but their product fails to describe the overall dataset. At T = 188\u2009K > T\u03b1\u03b2 the contribution of the \u03b1 relaxation is sufficient to describe the q-dependence of f(q, t) at each probed time. The error bars correspond to \u00b11\u2009s.d., obtained after data reduction and logarithmic binning, with uncertainty propagation. The data at 176.9 K combine the results of two independent measuring runs.<\/p>\n

Following the previous discussion, we model f(q, t) by (1) considering the measured q-dependence of the relaxation time and (2) assuming that f(q, t) is well described by a stretched exponential (equation (1<\/a>)) in the time window that is here probed, with \u03b2KWW = 0.5 at all q. In particular, consistent with the q\u22122 dependence of the relaxation time in the \u03b1 relaxation regime, we can assume for the \u03b1 relaxation the simple expression<\/p>\n

$$\\frac{{f}_{{\\rm{\\alpha }}}({\\bf{q}},t)}{{f}_{q}^{\\,{\\rm{\\alpha }}}}=\\exp \\left[-{({D}_{{\\rm{\\alpha }}}{q}^{2}t)}^{{\\beta }_{\\mathrm{KWW}}}\\right]\\approx \\exp [-q\\sqrt{{D}_{{\\rm{\\alpha }}}t}],$$<\/p>\n

\n (3)\n <\/p>\n

Here \\({f}_{q}^{\\,{\\rm{\\alpha }}}\\) is the strength of the process, and D\u03b1 is the diffusion coefficient that can be obtained from the quadratic q-dependence at high temperatures\/small scattering vectors (Fig. 2<\/a>). Equation (3<\/a>) is reported in Fig. 3<\/a>, red lines, and well describes the f(q, t) data at low q, in the \u03b1 relaxation regime. The super-quadratic q-dependence of the inverse relaxation time of the \u03b2JG relaxation can instead be described in terms of an anomalous diffusion model. By combining the approximately quartic q-dependence of the \u03b2JG inverse relaxation time with equation (1<\/a>) for \u03b2KWW\u2009=\u20090.5, it is possible to write34<\/a>,36<\/a><\/p>\n

$$\\frac{{f}_{\\mathrm{JG}}({\\bf{q}},t)}{{f}_{q}^{\\,\\mathrm{JG}}}\\approx \\exp \\left[-\\widetilde{D}{q}^{2}{t}^{{\\beta }_{\\mathrm{KWW}}}\\right]=\\exp \\left[-\\frac{\\langle {r}^{2}(t)\\rangle {q}^{2}}{6}\\right],$$<\/p>\n

\n (4)\n <\/p>\n

where \\({f}_{q}^{\\,\\mathrm{JG}}\\) is the relaxation strength of the \u03b2JG relaxation. Within this approximation, the normalized intermediate scattering function associated with the \u03b2JG relaxation is approximately Gaussian in shape. This allows us to express f(q, t) in equation (4<\/a>) in terms of the mean-squared molecular displacement, \u3008r2(t)\u3009, which is then related to the anomalous diffusion coefficient \\(\\widetilde{D}\\), obtained from fitting the q-dependence of the data in Fig. 2<\/a>, by the relation<\/p>\n

$$\\langle {r}^{2}(t)\\rangle =\\langle | {\\bf{r}}(t)-{\\bf{r}}(0){| }^{2}\\rangle \\simeq 6\\widetilde{D}{t}^{{\\beta }_{\\mathrm{KWW}}}.$$<\/p>\n

\n (5)\n <\/p>\n

Equation (4<\/a>) is reported in all panels of Fig. 3<\/a>, blue lines, and describes well the f(q, t) data at high q, in the \u03b2JG relaxation regime.<\/p>\n

We are now in the position to test the previously discussed scenario for the \u03b1 and \u03b2JG relaxation processes. The black solid lines reported in Fig. 3<\/a> have been calculated as the product of equations (3<\/a>) and (4<\/a>), and it is clear that the assumption that the f(q, t) is described by the product of two independent contributions does not match well our experimental data. We also underline that above the crossover temperature (T\u03b1\u03b2\u2009=\u2009181\u2009K), where only the \u03b1 relaxation is present, equation (3<\/a>) accounts well for the q-dependence of f(q, t) (Fig. 3<\/a>, top). We can then conclude this section underlining that the transition from diffusive to subdiffusive dynamics seen in the q-dependence of the relaxation time is also clearly reflected in the q-dependence of the shape of the intermediate scattering function. Around the crossover temperature T\u03b1\u03b2, the \u03b1 and \u03b2JG relaxations cannot be considered as two independent relaxation processes: instead, the relaxation process that affects the density fluctuations changes in nature from what we call the \u03b2JG process to the \u03b1 process as the length-scale (and therefore the timescale) of observation is increased across the Lindemann length. Interestingly, our findings (see also hereafter) provide a microscopic picture of the results of previous NMR investigations establishing a strong correlation between the \u03b1 and \u03b2JG processes37<\/a>.<\/p>\n

The observation that the \u03b1-relaxation-controlled regime is characterized by a q\u22122 dependence of the relaxation time encourages us to estimate a diffusion coefficient at all probed temperatures, including those where only few q-points have been collected. Figure 4a<\/a> reports the Tg-scaled temperature dependence of the extracted diffusion coefficient (Fig.\u20094a<\/a>, blue diamonds, left axis).<\/p>\n

Fig. 4: A real-space view of the secondary relaxation in 5M2H.<\/b>\"Fig.The alternative text for this image may have been generated using AI.<\/p>\n

a<\/b>, Left axis: diffusion coefficients extracted from the q-dependence of the relaxation time for 5M2H (full blue diamonds, this work) and OTP (full red diamonds, extracted from the data reported in refs. 12<\/a>,39<\/a>). For OTP, direct measurements of the diffusion coefficient from ref. 46<\/a> (NMR, red circles) and ref. 40<\/a> (isothermal desorption, red squares) are reported as well. Right axis: Stokes\u2013Einstein estimation of the temperature dependence of the diffusion coefficient based on literature data for the \u03b1 relaxation time for 5M2H29<\/a> (dashed blue line) and OTP4<\/a>,41<\/a>,42<\/a>,43<\/a>,44<\/a> (dashed red line). b<\/b>, Anomalous diffusion coefficient of 5M2H (full triangles) as a function of the Tg-scaled inverse temperature. The slope of the data is consistent with the expected value of \\({E}_{{\\rm{JG}}}\/(2{k}_{{\\rm{B}}}{T}_{{\\rm{g}}}{\\rm{ln}}(10))\\) (dashed line), where EJG is the activation energy of the \u03b2JG relaxation29<\/a>. c<\/b>, Time dependence of the mean-squared displacement at T = 181\u2009K computed from both the \u03b1 and \u03b2JG regimes. The full and dashed lines are the power laws expected in the \u03b1 and \u03b2JG regimes, respectively. d<\/b>, As in c<\/b> but for T = 176.9\u2009K. Right axis: the dash-dotted red line refers to numerical simulation data for a three-dimensional size-polydisperse mixture of soft repulsive spheres from ref. 26<\/a>. The blue area in d<\/b> delimited by a dashed line shows the distribution of relaxation times, G(log(\u03c4)), associated to the \u03b2JG relaxation of 5M2H as extracted from dielectric spectroscopy data. The base width correspond to the FWHM of the distribution. The coloured rectangles in c<\/b> and d<\/b> show the mean-squared molecular displacement estimated from the q-dependence of the relaxation strength fq at large scattering vectors (Supplementary Information<\/a>). The vertical error bars in a<\/b> (some are smaller than the marker size) and b<\/b> and the horizontal error bars in c<\/b> and d<\/b> correspond to \u00b11\u2009s.d. as obtained from the weighted nonlinear fitting of the experimental data described in the main text. In c<\/b> and d<\/b>, the vertical error bars represent \u00b11\u2009s.d. as obtained by propagating the uncertainties in the diffusion coefficients (\\(\\widetilde{D}\\) and D\u03b1).<\/p>\n

Its temperature dependence agrees well with that expected from the Stokes\u2013Einstein equation D\u03b1 \u221d T\/\u03c4\u03b1 (ref. 38<\/a>) (Fig.\u20094d<\/a>, dashed line, right axis), where we have used literature data for the \u03b1 relaxation time, \u03c4\u03b1 (ref. 29<\/a>). This is interesting, as the diffusion coefficient is known to become q-dependent at high q (ref. 38<\/a>). We expect that this q-dependence should not affect the temperature dependence of D\u03b1, as the q-range used to extract it from the experimental data is basically the same at all temperatures. To test this assumption, because we are not aware of other independently measured diffusion coefficient data for 5M2H, we perform the same analysis in the \u03b1-dominated regime using literature TDI data for the intermediate scattering function of o-terphenyl (OTP) (Fig.\u20094a<\/a>, full red diamonds, left axis)12<\/a>,39<\/a>. OTP has in fact been studied in detail, and direct measurements are available of the diffusion coefficient (Fig.\u20094a<\/a>, full red circles and squares, left axis)40<\/a> and of the timescale of the \u03b1 relaxation from several techniques4<\/a>,41<\/a>,42<\/a>,43<\/a>,44<\/a>, which we again use to estimate the temperature dependence of D\u03b1 from the Stokes\u2013Einstein expression (Fig.\u20094a<\/a>, dashed line, right axis). We note that the same y-scales have been used for 5M2H and OTP. We observe that (1) the Stokes\u2013Einstein expression describes the temperature dependence of the diffusion coefficient of OTP well, down to a certain temperature in the deeply supercooled liquid, below which it is violated, as well documented in the literature45<\/a>,46<\/a>; (2) the direct measurements of the diffusion coefficient in OTP agree rather well (within ~30%) with the estimates obtained from the intermediate scattering function measured at high q. We can conclude that it is possible to extract reasonably consistent diffusion coefficients from our data, which gives us confidence in the results of the model used to describe them. Similarly, we can compute the anomalous diffusion coefficient \\(\\widetilde{D}\\) associated with the \u03b2JG relaxation from the regime where the inverse relaxation time shows an almost quartic dependence on q (\\(\\tau ={\\widetilde{D}}^{-2}{q}^{-4}\\)). The data obtained are reported in Fig. 4b<\/a> as a function of the Tg-scaled inverse temperature. From the relation between \\(\\widetilde{D}\\) and the relaxation time of the JG relaxation, which can be worked out from equation (4<\/a>), we can expect that the slope of \\({\\log }_{10}(\\widetilde{D})\\) versus Tg\/T is \\({E}_{{\\rm{JG}}}\/(2{k}_{{\\rm{B}}}{T}_{{\\rm{g}}}{\\rm{ln}}(10))\\), where EJG = 0.28 \u00b1 0.02\u2009eV is the activation energy of the \u03b2JG relaxation36<\/a> and Tg = 155\u2009K is the glass-transition temperature of 5M2H36<\/a>. This prediction is indeed verified in Fig. 4b<\/a>, sign of an overall internal consistency of the data analysis.<\/p>\n

Having worked out and validated a description of our experimental data for the intermediate scattering function at both small and large q, we are now in position to extract the time dependence of the mean-squared displacement. In fact, we can associate a time t = 1\/(D\u03b1q2) to all q where the density fluctuations show a diffusive decay, and we can then evaluate the mean-squared displacement at that time using the simple random-walk relation: \u3008r2(t)\u3009 = 6D\u03b1t (ref. 38<\/a>). Similarly, we can associate from equation (4<\/a>) a time \\(t=1\/({\\widetilde{D}}^{2}{q}^{4})\\) to all q where the density fluctuations are subdiffusive (see also Supplementary Fig. 4<\/a> and Supplementary Text 3<\/a> for further analysis), and we can then evaluate the mean-squared displacement at that time using the relation: \\(\\langle {r}^{2}(t)\\rangle =6\\widetilde{D}{t}^{{\\beta }_{\\mathrm{KWW}}}\\). The two regimes cross at ~16\u2009nm\u22121. The values obtained for the mean-squared displacement, after normalization for the intermolecular distance rp = 0.76\u2009nm (evaluated from the molecular volume36<\/a>) are reported in Fig. 4c,d<\/a> for T = 181 and 176.9\u2009K, respectively. The shortest time reported here corresponds to the highest q that we have reached, whereas the longest time corresponds to the shortest investigated q. Moreover, we also report in Fig. 4c,d<\/a> the mean-squared displacement (coloured rectangles), computed from the q-dependence of the relaxation strength fq (see Supplementary Fig. 3<\/a> for details). The full and dashed lines in Fig.\u20094c,d<\/a> are the power laws expected in the \u03b1 (\u3008r2(t)\u3009 \u221d t) and \u03b2JG (\u3008r2(t)\u3009 \u221d t0.5) regimes, respectively, and are well consistent with the data. It is remarkable that the mean-squared displacements derived from two different regimes in q display such a continuous time dependence.<\/p>\n

At both temperatures, our data correspond to the portion of the time dependence of the mean-squared displacement comprised between the exit from the plateau (which we barely reach) and the start of the diffusion regime. They clearly show that, although the \u03b1 relaxation corresponds to the diffusion regime, as already known and expected, the \u03b2JG relaxation corresponds to a sublinear regime in the mean-squared displacement that follows the plateau and precedes the diffusion regime. In Fig. 4d<\/a>, we also report the distribution of relaxation times associated with the \u03b2JG process (blue area) as extracted from dielectric measurements29<\/a>. The overlap in time between this distribution and the sublinear regime that appears in the mean-squared displacement is noticeable. Therefore, in 5M2H, the \u03b2JG relaxation corresponds to the critical rattling of the molecules within the cage of their closest neighbours just before molecular diffusion, borrowing from the classical mode-coupling theory a terminology used for the von Schweidler\u2019s regime predicted above the critical temperature Tc (ref. 47<\/a>). We underline that the sublinear time dependence of the mean-squared displacement reported in Fig. 4<\/a> covers a wide range of two decades and more in time and reflects the width of the distribution of relaxation times for the \u03b2JG relaxation: in the framework presented here, it is the q-dependence of the \u03b2JG relaxation that translates a relaxation peak (as observed, for example, in dielectric spectroscopy) into a power law in time.<\/p>\n

In Fig. 4d<\/a> (right axis) we also report the simulation data for a three-dimensional size-polydisperse mixture of soft repulsive spheres from ref. 26<\/a> (dash-dotted red line, right axis). This dataset was selected because it corresponds to the same value of \u03c4\u03b1 (in physical units) as in our data. Specifically, we selected the mean-squared displacement at T = 0.085 (in reduced units), for which \u03c4\u03b1 \u2248 2\u2009\u03bcs (one simulation unit ~3 \u00d7 10\u221211\u2009s (ref. 26<\/a>)). The consistency between the two datasets is remarkable. Although these simulation data have not been analysed similarly to what we do here, the match between the experimental and simulation data shown in Fig. 4d<\/a> opens the possibility that the conclusions that we have reported here for 5M2H might be valid for a broad class of systems.<\/p>\n

Although from the available data for 5M2H we cannot accurately estimate the location of the mode-coupling temperature, Tc, it has often been reported that Tc \u2248 T\u03b1\u03b2 \u2248 1.2Tg (ref. 48<\/a>). This suggests that the sublinear diffusion process that we report here below T\u03b1\u03b2 might also be related to Tc. It is then interesting to observe that recent molecular dynamics simulations have revealed the emergence across Tc of a novel subdiffusive regime, responding to a different underlying mechanism than above Tc (ref. 49<\/a>), which might be related to the subdiffusive regime that we discuss here. This is a critical time and temperature range to advance our understanding of the mechanisms that lead to the glass transition. For instance, the onset of subdiffusion has recently been related to emergent facilitation and correlated dynamics in a two-dimensional lattice model of glassy behaviour50<\/a>, and facilitation is a mechanism that is becoming more and more the focus of recent investigations7<\/a>,26<\/a>. Combining these results with our observations, we can infer that the \u03b2JG process might be related to facilitation, a fascinating connection that will require more investigation.<\/p>\n

In conclusion, here we propose an experimentally extracted real-space picture of the \u03b2JG relaxation in the temperature range where it decouples from the structural process. Our results show that the \u03b1 and \u03b2JG relaxations cannot be treated as two independent processes: the microscopic mechanism affecting density fluctuations evolves from what we call the \u03b2JG process to the \u03b1 process as the length-scale (and therefore the timescale) of observation is increased. The \u03b2JG relaxation can be associated with the subdiffusive regime preceding the onset of the \u03b1 relaxation but not independent of it and possibly inducing or facilitating it. We also stress that, although the results presented here refer to a typical hydrogen-bonded glass former, 5M2H\u2014the sample where we first managed to present a detailed study in the time and temperature range where the \u03b1 and \u03b2JG relaxations decouple\u2014they are consistent with partial results available in the literature for different classes of glass formers12<\/a>,30<\/a>,36<\/a>,51<\/a> and are probably generic. We are, however, aware that still much has to be done to solve the many remaining inconsistencies. For instance, long simulations using the Kob\u2013Andersen Lennard\u2013Jones mixture, which is surely the most used model to study the glass transition, do not seem to show the transition in q between the \u03b1 and the \u03b2JG relaxation that we report here52<\/a>, though a qualitative change in the subdiffusive dynamics across Tc is observed49<\/a>. We might advocate that this model refers to a rather strong glass former, for which the \u03b2JG relaxation is known to be weak or rather merged with the \u03b1 relaxation53<\/a>,54<\/a>, though this issue requires further investigation. Moreover, recent numerical simulations23<\/a> of an asymmetric dimer system report that the self intermediate scattering function displays both the \u03b1 and \u03b2JG relaxations even above Tc, unlike our results, which show that these two processes correspond to two different q-ranges (length scales). Recent depolarized light-scattering experiments55<\/a> and a large corpus of dielectric spectroscopy data20<\/a>, which, however, refer to a different observable, also report evidence of both processes in the same spectrum. The distinction between universal and system-specific effects, as well as the comparison of results across different observables, remains a challenge which will require sustained, collective efforts from the scientific community studying the glass-transition. We believe that the results reported here contribute to this effort by clarifying the role of the \u03b2JG process as a precursor of the \u03b1 relaxation and thus its role in the glass transition.<\/p>\n","protected":false},"excerpt":{"rendered":"A liquid cooled sufficiently fast below the melting temperature does not crystallize and can be deeply undercooled until…\n","protected":false},"author":2,"featured_media":528075,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_share_on_mastodon":"0"},"categories":[271],"tags":[3554,3553,3557,914,18,910,56815,19,17,3552,3555,3556,452,133,61948,21730,3551],"class_list":["post-528074","post","type-post","status-publish","format-standard","has-post-thumbnail","category-physics","tag-atomic","tag-classical-and-continuum-physics","tag-complex-systems","tag-condensed-matter-physics","tag-eire","tag-general","tag-glasses","tag-ie","tag-ireland","tag-mathematical-and-computational-physics","tag-molecular","tag-optical-and-plasma-physics","tag-physics","tag-science","tag-statistical-physics","tag-structure-of-solids-and-liquids","tag-theoretical"],"share_on_mastodon":{"url":"https:\/\/pubeurope.com\/@ie\/116725958764987073","error":""},"_links":{"self":[{"href":"https:\/\/www.europesays.com\/ie\/wp-json\/wp\/v2\/posts\/528074","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.europesays.com\/ie\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.europesays.com\/ie\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/ie\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/ie\/wp-json\/wp\/v2\/comments?post=528074"}],"version-history":[{"count":0,"href":"https:\/\/www.europesays.com\/ie\/wp-json\/wp\/v2\/posts\/528074\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/ie\/wp-json\/wp\/v2\/media\/528075"}],"wp:attachment":[{"href":"https:\/\/www.europesays.com\/ie\/wp-json\/wp\/v2\/media?parent=528074"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.europesays.com\/ie\/wp-json\/wp\/v2\/categories?post=528074"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.europesays.com\/ie\/wp-json\/wp\/v2\/tags?post=528074"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}