Introduction

The timescale, τα, of structural rearrangements in a typical liquid is in the ps range. If the liquid is supercooled below the melting temperature, τα gets longer the lower is the temperature, until the dynamics become so slow that we speak of a solid rather than of a liquid: a glass has been obtained. Conventionally, the glass transition temperature, Tg, is defined as the temperature where this characteristic time is 100 s. However, this is not the only characteristic timescale for the molecular dynamics of a strongly supercooled liquid. At temperatures somewhat higher than Tg, a second characteristic timescale associated with a secondary relaxation process appears. This secondary process, known as Johari-Goldstein or βJG1, can be considered a precursor of the glass-transition in two ways: time and temperature. In time: it anticipates the structural (or α) relaxation that freezes in at Tg by many decades, being typically in the ms-μs time range at Tg and remaining active also in the glass2. In temperature: it decouples from the α process in the supercooled liquid phase at temperatures lower than Tαβ ≈ 1.2 Tg, thus anticipating the glass-transition by a stretch. On the role of the βJG-relaxation in the glass-transition at least three generations of scientists have been debating since its discovery in the early seventies1. While its phenomenology has been largely explored, the concomitant effects of: (i) the lack of a firm microscopic theory of the glass-transition and (ii) the difficulty of experimental and numerical techniques to probe the spatial and the temporal properties of this process in the relevant time-range and temperature-range, make this debate still unsettled.

From the glass-transition perspective, it is becoming more and more clear that the vitrification kinetics is not only driven by the structural relaxation, as traditionally believed, but other, faster processes must play a role to reduce the effectiveness of the structural relaxation to lock the atomic dynamics3. This is consistent with recent theoretical and numerical work pointing out that the free energy landscape relevant for the glassy state is much rougher than classically thought, with intra-state barriers associated with secondary relaxations4. In the context of these findings, it is clear that direct investigations of the βJG-relaxation are of crucial importance for a better understanding of the liquid-to-glass transition.

Many basic properties of the βJG process are known, mostly thanks to a large body of dielectric spectroscopy (DS)2,5,6 and nuclear magnetic resonance7 data. It is e.g., nowadays understood that the βJG-relaxation is strongly connected to the α-process2,5,6, and thus is sensitive to the glass-transition. This conclusion is in apparent contrast with the restricted re-orientational dynamics of the process7 and with its mild Arrhenius T-dependence, usually interpreted as the signature of a hindered, simply activated process2. The few available neutron and X-ray scattering experiments also describe it, above Tg, as an essentially restricted if not localized molecular motion occurring mainly within the transient cages formed by the nearest-neighbors8,9,10,11,12. Sub-Tg experiments13 and atomistic simulations14,15 support instead of the idea that the βJG-relaxation is rather cooperative and characterized by string-like excitations occurring via hopping between adjacent atomic sites.

Several phenomenological models of the βJG process are also available, based on an intuition of the main physical mechanism giving rise to it. For instance, according to the coupling model, the βJG-relaxation consists of a distribution of elementary processes evolving with time by increasing the number of participating molecular units and culminating in the cooperative α-relaxation: the α-relaxation and βJG-relaxation thus share many relevant properties2. In another model based on the hypothesis of strong dynamical heterogeneity of the supercooled liquid, the βJG-relaxation originates from the population exchange between regions of tightly confined molecules and of loosely confined ones11. The random first-order transition theory identifies instead the βJG-relaxation with cooperative rearrangements of string-like shaped regions of molecules which become more prominent with increasing temperature16. While these ideas are not necessarily alternative to each other, unambiguous experimental input is required as a guide among these rather different views. Combining new X-ray scattering data on 1-propanol with data available in the literature8,9,10,12 and here re-analyzed, evidence shall now be provided of two striking features related to the molecules participating in the βJG-relaxation: (i) their mean-squared displacement satisfies the Lindemann criterion for structural instability and (ii) their number matches the threshold for site percolation. They, therefore, form a mobile percolating, infinite cluster pervading the whole sample. This cluster evolves on the timescale of the βJG-process, and careful experimental design is required to study it.

Results

Nuclear γ-resonance time-domain-interferometry (TDI) is one of the few techniques able to probe directly microscopic density fluctuations at the timescale where the βJG-process decouples from the structural one. TDI is based on an interferometric analysis of the X-rays scattered by the sample and, as such, provides information that is both time and q (and therefore space) resolved (q being the momentum exchanged in the scattering process)8,12. TDI has been here employed to study 1-propanol, a model glass-former with a genuine βJG-process17 rather coupled to the α-relaxation. Some examples of the measured TDI beating patterns are reported in Fig. 1 (red squares) along with the fitting curves obtained using a model that accounts for one relaxation process (black lines, see “Methods” section). The beating pattern contrast function \({\phi }^{\prime}(q,t)\) (blue solid line in Fig. 1) is proportional to the density correlation function and allows extracting its main relaxation parameters, in particular the relaxation time, τ, at different q values. TDI is sensitive to the fastest relaxation process active in the explored time window: given its dynamical range of fewer than two decades in time, it was not possible to detect more than one relaxation process per experimental curve even when both the α and the βJG process were expected to be active.

Fig. 1: Temperature and scattering vector dependence of time-domain interferograms.
figure 1

Time-domain interferometry data as a function of time (squares with ±1 SD errorbars) at different temperatures and at a fixed exchanged wave-vector (q = 15 nm−1, roughly corresponding to the inter-molecular distance) a and at the same temperature T = 122.5 K in the supercooled liquid state and at different q values b. The raw data have been averaged over a time range ±0.7 and ±0.9 ns, respectively, depending on the collected statistics, in order to improve the figure readability. The black solid lines are the model curves obtained from the fitting procedure. The blue solid lines are the calculated contrast functions, \({\phi }^{\prime}\), along with the 68% confidence interval (gray area). \({\phi }^{\prime}\) is proportional to the density correlation function.

Of particular interest is the T-dependence of τ, which was investigated at two q-values, 15 and 25 nm−1, corresponding to the inter-molecular and to an intra-molecular length-scale, respectively. The measured τ values are shown in Fig. 2 along with the ones obtained by DS and relative to the reorientational dynamics (see “Methods” section and Supplementary Note 1).

Fig. 2: Relaxation map from dielectric spectroscopy and time-domain interferometry measurements.
figure 2

Inverse temperature dependence of the relaxation time (τ) measured by dielectric spectroscopy (gray and green squares with ± 1 SD errorbars) and time-domain interferometry at two different q-values: 15 (blue circles with ±1 SD errorbars) and 25 nm−1 (red squares with ± 1 SD errorbars). The gray and green dash-dotted lines are fit to the dielectric data. The α-relaxation is described using the Vogel-Fulcher-Tamman equation \(\tau ={\tau }_{0}\exp (D{T}_{0}/(T-{T}_{0}))\) (D = 37(2), τ0 = 10−15.4(2)s, T0 = 51(1) K) (gray dash-dotted line) while the βJG relaxation using the Arrhenius T-dependence with a reduced activation energy \({E}_{{\beta }_{JG}}/{k}_{B}=4.04(4)\times 1{0}^{3}\) K (green dash-dotted line). The blue and red solid lines are the model results fitted to the dielectric spectroscopy data and scaled to match the time-domain interferometry data. The blue dashed line is the best match of the βJG T-dependence to the whole set of time-domain interferometry data at 15 nm−1. The gray and green areas delimited by dashed lines at 1000/T = 7.5 and 7.8 K−1 are the distributions of relaxation times, \(G(\mathrm{ln}\,\tau )\), associated with the α-process and βJG-process, respectively, and extracted from dielectric spectroscopy investigations. The base widths of the two areas correspond to the FWHM of the two distributions. Inset: diffuse scattering pattern of 1-propanol at T = 122.5 K, with the indication of the q values and of the corresponding ranges covered in the time-domain interferometry measurements reported in the main figure.

The DS data provide a precise estimation of the T-dependence of the α-process and βJG-process characteristic times (gray and green dash-dotted lines in Fig. 2). These T-dependencies were then scaled onto the TDI data. This procedure, along with the study of the q-dependence of the relaxation time (see below), allowed us to unambiguously associate the process appearing in the density fluctuations to the α-relaxation or βJG-relaxation, see also Supplementary Note 2 for additional support to this interpretation.

A clear change in the T-dependence of τ measured by TDI occurs around Tαβ 131 K at q = 25 nm−1. At q = 15 nm−1 the crossover from the α to the βJG-process is instead too weak to be appreciated. Interestingly, at q = 15 nm−1 the T-dependence of τ can also be described accounting only for that of the βJG-relaxation (blue dashed line in Fig. 2): a similar match cannot be obtained instead considering only the α-process. It is then evident that, at least for T ≤ 131.4 K, TDI is sensitive to the βJG-process both at the inter and intra-molecular lengthscales. This provides broader validity to similar observations for another monoalcohol12.

Further insights can be gained from the q-dependence of the relaxation parameters which was investigated at two temperatures: 131.4 and 122.5 K (see Fig. 3a). At the former one, slightly higher than the decoupling temperature Tαβ, the α-relaxation is expected to be dominant, whereas at the latter one Fig. 2 shows that TDI is sensitive only to the βJG-process. In order to facilitate the comparison to the DS results, the TDI characteristic times have been converted to those, τp, corresponding to the peak of the susceptibility function (see “Methods” section). The obtained τp values are plotted in Fig. 3b along with the curves obtained from fitting to them a power-law τpqn. The power-law exponents, n, obtained from the fits, namely n = 2.0(7) at T = 131.4 K and n = 3.9(9) at T = 122.5 K (see Fig. 3c), strongly support that for T > Tαβ the α-relaxation, with a quadratic q-dependence of the characteristic time, dominates the microscopic dynamics, whereas below Tαβ the characteristic super-quadratic q-dependence of τp confirms that the βJG-relaxation is rather observed.

Fig. 3: Scattering vector dependence of the relaxation time of density fluctuations.
figure 3

a temperature dependence of the relaxation time at q = 25 nm−1, as in Fig. 2. The green and blue circles show the two temperatures at which the q-dependencies in the main figure are reported. b wave-number (q) dependence of the characteristic time (τp) of density fluctuations identified by the peak position of the susceptibility function at T = 131.4 K (α-relaxation, blue circles with ± 1 SD errorbars) and 122.5 K (βJG-relaxation, green squares with ±1 SD errorbars). The solid-lines are obtained fitting a power-law τpqn to the experimental data. This simple model can account for the q-dependence of the βJG-relaxation at T = 122.5 K in the whole explored q-range. At T = 131.4 K, where the α-relaxation is the dominant process, τp shows an oscillation close to the peak of the S(q) that can be associated with the well-known de Gennes narrowing effect42. A power-law is indeed able to well reproduce the experimental data only starting from q = 18 nm−1. The blue and green boxes indicate the values of the corresponding relaxation times measured by dielectric spectroscopy. Their widths show the corresponding uncertainty (±1 SD). For T = 122.5 K the reported value of τ is the mean of our dielectric spectroscopy result and of that from ref. 36. The open diamond with horizontal ± 1 SD errorbar is the q-value, qDS, at which the TDI and DS relaxation times match at T = 122.5 K. The diffuse scattering pattern at 122.5 K (gray line) is rescaled and reported on the same plot for the sake of comparison. c the n values extracted from the power-law fits are plotted as a function of the inverse temperature. The errorbars correspond to ±1 SD.

Discussion

The α-process in 1-propanol is still diffusive around Tαβ while the microscopic dynamics associated with the βJG-process is clearly restricted (n > 2) already at a relatively high T (122.5 K  1.25 Tg), giving broader significance to similar observations reported for the other two glass-formers studied using TDI: 5M2H12 and OTP8,9.

There are then some interesting common features for the three glass-formers investigated by TDI: (i) density fluctuations are coupled to the βJG relaxation at/close to both the inter and intra-molecular length-scale; (ii) the characteristic times of the βJG relaxation appearing in the density fluctuations show a strong q-dependence, indicative of a restricted dynamics, and (iii) cross the dielectric \({\tau }_{{\beta }_{{\rm{JG}}}}^{{\rm{DS}}}\) at a q-value, qDS, somewhat larger than qmax. The length \(\frac{1}{{q}_{{\mathrm{DS}}}}\) identifies the characteristic scale for the center-of-mass motion within the βJG-relaxation12. More in detail, it is possible to relate qDS to the molecular root-mean-squared displacement within an anomalous diffusion model18 (see discussion in the Supplementary Note 3) as:

$${{\Delta }}{r}_{{\rm{JG}}}=\sqrt{\big\langle {r}^{2}\big({\tau }_{{\beta }_{{\rm{JG}}}}^{{\rm{DS}}}\big)\big\rangle }=\frac{\sqrt{6}}{{q}_{{\rm{DS}}}}\ .$$
(1)

Here ΔrJG has to be regarded as the most probable displacement of the molecules participating in the βJG-relaxation at \({\tau }_{{\beta }_{{\rm{JG}}}}^{{\rm{DS}}}\). Figure 4a shows ΔrJG for 1-propanol (red symbol), 5M2H (blue symbols), and OTP (green symbols, see Supplementary Note 4) estimated from the q-dependencies reported in Fig. 3 and in refs. 8,9,12 and normalized to the corresponding average inter-molecular distance (center-of-mass to center-of-mass), \({r}_{{{p}}}={v}_{{\rm{mol}}}^{\frac{1}{3}}\), where vmol is the molecular volume. Figure 4b reports the corresponding relaxation times of the three samples probed by DS and depolarized dynamic light-scattering as a function of Tg/T and scaled by the value of \({\tau }_{{\beta }_{{\rm{JG}}}}^{{\rm{DS}}}\) at Tg. Figure 4a shows that ΔrJG is larger for 1-propanol (28%) than for 5M2H and OTP (12%). This difference can be related to the higher normalized temperature at which the dynamics of 1-propanol were investigated, i.e., T/Tg 1.25. In all cases, ΔrJG amounts to a non-negligible fraction of the average inter-molecular distance, a result evocative of the Lindemann criterion for the stability of crystalline solids19. In fact, all the estimated values for ΔrJG/rp fall inside the gray band in Fig. 4a which enlightens the range of typical values for the Lindemann criterion in crystals (see ref. 20 and references therein) expressed in terms of root-mean-squared displacement. This suggests that the restricted molecular motion associated with the βJG-relaxation corresponds, in average, to locally unstable cages: the molecules participating in the βJG-process can be seen as ’uncaged’ at \({{\tau }_{{\beta }_{{\rm{JG}}}}^{DS}}\) and are characterized by higher-than-average mobility as they can eventually sub-diffuse to longer distances prior the onset of the α-relaxation.

Fig. 4: Root mean squared displacement for the molecules participating in the \({\beta }_{{\rm{JG}}}\) process at \({\tau }_{{\beta }_{{\rm{JG}}}}^{{\rm{DS}}}\).
figure 4

a Characteristic center-of-mass root-mean-squared displacement at \({\tau }_{{\beta }_{{\rm{JG}}}}^{{\rm{DS}}}\) for the molecules participating in the βJG-relaxation rescaled to the corresponding inter-molecular distance (center-of-mass to center-of-mass). Red circles: 1-propanol; blue circles: 5M2H; green circles: OTP. The gray area shows the range of typical values for the Lindemann criterion in crystals (see ref. 20 and references therein) expressed in terms of root-mean-squared displacement. b Tg-rescaled inverse T-dependence of the α-relaxation and βJG-relaxation times (τ) measured by dielectric spectroscopy and/or depolarized dynamic light-scattering for 5M2H (blue diamonds,12), 1-propanol (red left-pointing triangles and squares,36; red circles, this work) and OTP (green squares,43; green circles,1; green diamonds, this work). To facilitate the comparison the relaxation times have been scaled to the value of the corresponding \({\tau }_{{\beta }_{{\rm{JG}}}}^{{\rm{DS}}}\) at Tg. All errorbars shown in panels a and b correspond to ±1 SD.

Further insights on the structural properties of the βJG-relaxation are provided by its microscopic relaxation strength, \({f}_{q}^{{\rm{JG}}}\). This information is hard to obtain even for the dynamic range of TDI and the only available estimate is for 5M2H12 where it was found that around qDS (31 nm−1 < q < 40 nm−1) and above Tg (1.07 Tg < T < 1.21 Tg) \({f}_{q}^{{\rm{JG}}}\simeq 0.25\): one molecule out of four, on average, participates to the βJG-process at \({\tau }_{{\beta }_{{\rm{JG}}}}^{{\rm{DS}}}\).

It is interesting to read this value in the light of the number of nearest neighbors (z) for the samples here considered. z can be estimated from the radial pair distribution function, g(r): for 1-propanol and OTP the room temperature values are z 12 (D.T. Bowron, private communication) and 1521, respectively (see Supplementary Note 5). No g(r) data are instead available in the literature for 5M2H but we can expect its number of nearest neighbors to be similar to that of 1-propanol, given the similarity of their structure. The available values of z are close to that for an fcc lattice, and remarkably lattices with such connectivity present a threshold for site percolation (\({p}_{{\rm{c}}}^{{\rm{f.c.c}}}=\) 0.19822) which is comparable with \({f}_{q}^{{\rm{JG}}}\). The molecules participating in the βJG-relaxation must then be spatially connected in a percolating (or close to percolation) cluster pervading the whole sample, see Fig. 5. Clearly, this spatial connection does not imply dynamical correlations among all the molecules participating in the βJG-process. In fact, though the βJG-relaxation might be cooperative to some extent, as our results for 5M2H12 and 1-propanol also suggest, its strong sub-diffusivity clearly evidences that it is of prevalent local nature.

Fig. 5: Sketch of the spatial distribution of the molecules participating in the βJG-process at a given time.
figure 5

The molecules undergoing the βJG-relaxation (red spheres) are highly mobile and, after a time of the order of \({\tau }_{{\beta }_{{\rm{JG}}}}^{{\rm{DS}}}\), perform larger spatial excursions than the rest of the molecules (white spheres). These spatial excursions are, on average, of the order of those prescribed by the Lindemann criterion for crystal instability19. These "un-caged" molecules form a (close to) percolating cluster. The sketch is based on an experimentally measured configuration for a colloidal glass of hard spheres (volume fraction 0.61) reported in ref. 44 but has no direct connection to the discussion reported in that study. The cluster of mobile molecules was obtained by randomly selecting particles with a probability of 0.25, according to the here discussed estimate for the fraction of molecules participating in the βJG-relaxation at \({\tau }_{{\beta }_{{\rm{JG}}}}^{{\rm{DS}}}\) (relaxation strength).

The present results establish an interesting correlation between spatial and temporal properties of the molecules participating in the βJG process: the most mobile molecules (on a time scale that precedes the α-process) are spatially distributed as a percolating cluster. In other terms, the βJG process marks the development of a mosaic state in the undercooled phase with patches of less mobile molecules separated by an intermittent and ever-changing network of more mobile ones. These results, obtained in the deeply supercooled liquid phase, give then a new perspective to concepts used to imagine the dynamical heterogeneities which develop on lowering the temperature towards Tg23,24,25: while the size of the patches of the mosaic structure shows in the investigated cases a mild temperature dependence26,27,28, the development of a percolating, infinite cluster at temperatures below Tαβ ≈ 1.2 Tg signaled by the βJG-process is a very suggestive result supporting the idea that a dynamical transition is taking place there. This point is in line with the basic idea underlying the random phase order transition theory16,29.

Methods

Nuclear γ-resonance time-domain interferometry measurements

The nuclear γ-resonance TDI experiments were performed at the nuclear resonance beamline ID1830 of the European Synchrotron Radiation Facility (ESRF) in Grenoble (F) employing an optimized implementation of what is originally described in ref. 31.

TDI experiments are designed to detect X-ray scattering using an interferometer that allows probing density fluctuations in the 10 ns–10 μs time range8,10,12. Two single-line 57Fe-containing absorbers were therefore installed upstream of the sample, on the incoming beam, and downstream of the sample, on the scattered beam, to provide the probe and reference beams of a time-domain interferometer. In order to obtain different excitation energies, the probe absorber was mounted on a velocity transducer and driven at the constant velocity v 10 mm/s with a relative accuracy better than 0.1%. The resulting shift was of Ω = 105 Γ0, where Γ0 = 4.66 neV is the natural linewidth of the first excited state of 57Fe. The incident X-ray radiation used to excite the nuclear resonance of the absorbers was characterized by a bandwidth of 2.5 meV at the energy of the first nuclear transition of 57Fe at 14.412 keV. The photons quasi-elastically scattered by the sample at two different scattering vectors \(q=2{k}_{0}\sin (\theta /2)\), where θ is the scattering angle and k0 = 73 nm−1 is the wave-vector of the nuclear fluorescence from the first excited state of 57Fe, were simultaneously collected by two avalanche photodiode (APD) detectors. The setup was designed to span the q-range 9–42 nm−1.

The TDI interferograms were analyzed as reported in the literature12,32,33,34. More precisely, in TDI experiments with identical single-line nuclear absorbers the time evolution of the intensity emerging from the interferometer is described by12,32,33:

$$I(q,t)=| R(t){| }^{2}\left[1+\cos ({{\Omega }}t){\phi }^{\prime}(q,t)\right] .$$
(2)

Here R(t) is the time response of the nuclear absorbers12,32,33 and \({\phi }^{\prime}(q,t)\) (blue solid line with gray area in Fig. 1) is the contrast function12. \({\phi }^{\prime}(q,t)\) is related to the density correlation function (also known as normalized intermediate scattering function), ϕ(q, t), via:

$${\phi }^{\prime}(q,t)=\phi (q,t)\frac{2}{1+{f}_{{{\Delta }}E}(q,T)},$$
(3)

where \({f}_{{{{\Delta }}}_{{\rm{E}}}}\) depends both on the experimental set-up and on sample properties32,34. \({\phi }^{\prime}\) has been modeled using the KWW function2 in order to introduce a minimum bias in the fitting procedure8,12:

$${\phi }^{\prime}(q,t)={f}_{q}^{\prime}\exp \left[-{\left(\frac{t}{\tau }\right)}^{{\beta }_{{\rm{KWW}}}}\right]\ .$$
(4)

Here \({f}_{q}^{\prime}\) is the initial beating pattern contrast (see Supplementary Note 6 for the obtained results), τ is the relaxation time of density fluctuations and βKWW is the stretching parameter. In the fitting procedure βKWW was fixed to an average value obtained from DS measurements (〈βKWW〉 = 0.66) for \(q\, > \, {q}_{\max }\), similarly to what reported in refs. 8,9,10,12. Concerning the data collected at \({q}_{\max }\), βKWW has been increased by 20% (〈βKWW〉 = 0.79) in order to take into account its q-dependence, consistently to what discussed in ref. 12. \({\phi }^{\prime}\) has also been modeled using the Mittag–Leffler function35, the Fourier transform of the Cole–Cole function well known to properly describe the βJG relaxation in DS data2. This approach is delicate given the fact that (i) the TDI signal covers less than two decades in time and (ii) the Fourier transform of the Cole–Cole function has a very stretched tail. In other terms, the measured TDI signal is in general not very sensitive to the parameters of the (Fourier transform of) the Cole–Cole function. However, for the data-sets where this happens to be the case, modeling the TDI signal within the βJG-relaxation using the Cole–Cole susceptibility provides results very close to those obtained using the KWW function for what concerns both the quality of the fit and the estimated parameters, which turn out to be in mutual agreement within one standard deviation.

1-propanol

The 1-propanol sample was purchased from Sigma Aldrich (anhydrous, 99.7% pure) and used as received. 1-propanol, differently from other glass-formers investigated by TDI8,12, is characterized, above Tg = 97 K, by rather coupled α and βJG processes36, thus offering the possibility to study the βJG-relaxation in a regime still unexplored by density fluctuations. The TDI experiment on 1-propanol has been carried out in the temperature interval from 1.44 Tg down to 1.16 Tg and at scattering vectors (q) ranging from the peak of the structure factor \({q}_{\max }=\)15 nm−1 up to 40 nm1. The temperature of the sample was controlled using a He-flow cryostat with ±0.1 K stability.

Dielectric spectroscopy measurements

The complex permittivity of the sample was measured in the range 10 mHz–10 MHz using a lumped impedance technique and the Novocontrol Alpha-Analyzer, whereas in the range 1–3 GHz using the coaxial reflectometric technique37 employing the Agilent 8753ES network analyzer. The dielectric cell consisted of a parallel plate capacitor with a silica spacer and filled with the sample in the liquid state. The temperature of the sample was controlled using a dry nitrogen-flow Quatro cryostat with a temperature accuracy better than 0.1 K.

The measured permittivity function ϵ(ν) was analyzed fitting simultaneously its real \({\epsilon }^{\prime}(\nu )\) and imaginary part ϵ(ν), where ν is the frequency. The function used for the fits is:

$$\epsilon (\nu ) = \frac{{{\Delta }}{\epsilon }_{{\rm{D}}}}{1+j2\pi \nu {\tau }_{{\rm{D}}}}+{{\Delta }}{\epsilon }_{\alpha }{L}_{j2\pi \nu }\left\{-\frac{d}{dt}\exp \left[-{\left(\frac{t}{{\tau }_{\alpha }}\right)}^{{\beta }_{{\rm{KWW}}}}\right]\right\}\\ +\frac{{{\Delta }}{\epsilon }_{{\beta }_{{\rm{JG}}}}}{1+{\left(j2\pi \nu {\tau }_{{\beta }_{{\rm{JG}}}^{{\rm{DS}}}}\right)}^{a}}+\frac{\sigma }{j2\pi \nu {\epsilon }_{0}}+{\epsilon }_{\infty },$$
(5)

where the first Lorentzian term accounts for the Debye relaxation; the second Kohlrausch-Williams-Watts (KWW) term for the α-relaxation; the third one is the Cole–Cole function that describes the βJG-relaxation; the fourth term accounts for the d.c. conductivity contribution; and the last one is for the induced polarization dielectric constant. \({{\Delta }}{\epsilon }_{{\rm{D}},\alpha ,{\beta }_{{\rm{JG}}}}\) are the dielectric relaxation strengths of the different processes. Lj2πν{\,} is the Laplace transform evaluated at j2πν.

Treating the α and βJG-processes as statistically independent is a common practice in the literature when the latter is not resolved. Nevertheless one has to be aware that the two processes are strongly connected in properties and not independent, and their cross-correlation terms could play a role. Our data are in agreement with those reported in ref. 36, as can be observed from Fig. 4 in the main text, and also with those in ref. 38. More details on the model used here can be found in ref. 39. An example of a dielectric spectrum measured for 1-propanol along with the fitting curve is shown in Supplementary Fig. 1.

Comparison between TDI and DS data

Since the TDI and the DS data were fitted employing different models, namely the KWW equation (Eq. (4)) for the TDI data and the Cole–Cole equation (third term of Eq. (5)) for the DS data, the comparison between the two timescales, required to estimate qDS, was performed using in both cases the characteristic time identified by the position of the maximum of the corresponding susceptibility, χ(ω). To this aim, starting from the τ values obtained from fitting Eq. (4) to the TDI signal, the corresponding susceptibility was numerically computed in order to obtain the characteristic time τp defined as:

$${\tau }_{{\rm{p}}}=\frac{1}{{\omega }_{{\rm{p}}}},$$
(6)

where ωp is the maximal loss angular frequency for that susceptibility. ωp is displaced to slightly lower frequencies with respect to 1/τ. The susceptibility corresponding to Eq. (4) was computed using the algorithm described in ref. 40 since the KWW equation has no analytical Fourier/Laplace transform.

In the case of the Cole–Cole expression, \({\tau }_{{\beta }_{{\rm{JG}}}}^{{\rm{DS}}}\) is already equal to \(\frac{1}{{\omega }_{{\rm{p}}}}\).

Mean square amplitude and mean-squared displacement

The values indicated in ref. 20 for the Lindemann criterion, i.e., molecular displacements comprised between 5% and 20% of the average intermolecular distance, refer to the mean-square amplitude (MSA):

$${\rm{MSA}}=\sqrt{\left \langle \left|\bf{r}-\left\langle {\bf{r}}\right\rangle \right|^{2}\right\rangle },$$
(7)

where r is the molecular position, 〈r〉 is the average position and 〈  〉 denotes the ensemble average. ΔrJG, as explained in the main text, is instead evaluated from the mean-squared displacement which is defined as:

$${\rm{MSD}}=\left\langle \left|{\bf{r}}(t)-{\bf{r}}(0)\right|^{2}\right\rangle .$$
(8)

Under the hypothesis that the molecular positions at times 0 and t are uncorrelated, the MSD and the MSA are related by41:

$${\rm{MSA}}=\frac{\sqrt{{\rm{MSD}}}}{\sqrt{2}}\ .$$
(9)

In order to perform the comparison shown in Fig. 4, the range of values reported in ref. 20 has therefore been multiplied by \(\sqrt{2}\).