Experimental test of a predicted dynamics–structure–thermodynamics connection in molecularly complex glass-forming liquids

Significance

The physical origin of cooperative activated relaxation in glass-forming liquids, which underlies the huge growth of viscosity and suppression of diffusion with cooling, remains elusive. Understanding this problem is of fundamental scientific interest and can assist in the design of new materials. We combine experimental relaxation time and equation-of-state data of molecular liquids to provide strong evidence for the recent theoretical prediction that activated relaxation, though causally driven by dynamically coupled cage-scale hopping and nonlocal collective elasticity quantified by local structural correlations, is well captured based on a thermodynamic description with dimensionless compressibility as the key variable. A mechanistic deduction is the crossover from fragile to strong (e.g., inorganic networks) relaxation behavior arises from the irrelevance of long-range elasticity effects.

Abstract

Understanding in a unified manner the generic and chemically specific aspects of activated dynamics in diverse glass-forming liquids over 14 or more decades in time is a grand challenge in condensed matter physics, physical chemistry, and materials science and engineering. Large families of conceptually distinct models have postulated a causal connection with qualitatively different “order parameters” including various measures of structure, free volume, thermodynamic properties, short or intermediate time dynamics, and mechanical properties. Construction of a predictive theory that covers both the noncooperative and cooperative activated relaxation regimes remains elusive. Here, we test using solely experimental data a recent microscopic dynamical theory prediction that although activated relaxation is a spatially coupled local–nonlocal event with barriers quantified by local pair structure, it can also be understood based on the dimensionless compressibility via an equilibrium statistical mechanics connection between thermodynamics and structure. This prediction is found to be consistent with observations on diverse fragile molecular liquids under isobaric and isochoric conditions and provides a different conceptual view of the global relaxation map. As a corollary, a theoretical basis is established for the structural relaxation time scale growing exponentially with inverse temperature to a high power, consistent with experiments in the deeply supercooled regime. A criterion for the irrelevance of collective elasticity effects is deduced and shown to be consistent with viscous flow in low-fragility inorganic network-forming melts. Finally, implications for relaxation in the equilibrated deep glass state are briefly considered.

An enormous number of seemingly orthogonal proposals exist for a fundamental connection between a (typically scalar) structural or excess (configurational) thermodynamic quantity and activated relaxation in supercooled liquids (1–12). High chemical complexity for fragile glass formers which exhibit strongly non-Arrhenius relaxation greatly complicates the formulation of predictive theories. A common generic view (1, 3, 8) is that the structural or alpha relaxation time (and viscosity, inverse diffusivity) evolves with cooling as shown in Fig. 1A. Different dynamical mechanisms in the high-, intermediate-, and low-temperature regimes are often envisioned: noncooperative Arrhenius (∼1 ps to 100 ps), critical power law (∼0.1 ns to 100 ns), and cooperative non-Arrhenius (∼0.1 $\mu$s to 100 s or beyond), respectively. Typically a causal connection is postulated between the logarithm of the alpha time (an effective barrier in thermal energy units) and a specific “order parameter”: 1) in the structural class (6, 7, 13–17), the intensity of the cage peak of the structure factor S(k), local aspect(s) of the radial distribution function g(r), or specific packing motifs; 2) in the thermodynamics class, various measures of free volume (18, 19), excess entropy (20), configurational entropy (21–25), internal energy and enthalpy (26), or with some arguing for an equilibrium phase transition at an inaccessibly low (high) temperature (density) (23, 27–29); 3) in the short time class, the high-frequency shear modulus (2, 30–32), Debye–Waller factor (33), or amplitude of special vibrational modes (33–35); and 4) in the intermediate time class, the concentration of dilute mobile excitations [e.g., strings (36, 37) or facilitating defects (38)]. Many of the proposed order parameters are hard or impossible to uniquely define and/or experimentally measure. The diverse models often claim to capture relaxation data over limited time windows typically based on fitting but usually fail at low and/or high enough temperature (5).

Fig. 1.

Global relaxation map and theoretical picture and key predictions. (A) Three-regime relaxation map (curves) for the alpha time with Arrhenius and strongly non-Arrhenius behaviors separated by a crossover regime perhaps of a critical power law (6) form. The proposed two-regime scenario of ECNLE theory (39–42) is based solely on noncooperative and cooperative activated dynamics (slightly overlapping orange and green regions) with the inverse dimensionless compressibility ($S_0^{-1}$) as the relevant thermodynamic quantity. The approximately five to six decade range that simulations can probe is indicated. (B) Dynamic free energy for a metastable hard sphere (diameter $\sigma$) fluid (42) as a function of particle displacement at a high packing fraction of $\varphi$ = 0.58. Relevant length and energy scales are indicated. (Inset) Schematic of the core physical idea for the alpha relaxation: hopping on the cage scale coupled with a collective elastic displacement of all particles outside the cage. (C) Main: local cage barrier as a function of inverse dimensionless compressibility for 0.44<$\varphi$ <0.61 corresponding to a 16 decade increase of the alpha time (39, 41, 42). The metastable regime begins at $\varphi$ ∼ 0.5 where the total barrier is ∼1.5 k_BT. (Inset) Total barrier as a function of $S_0^{-3}$ normalized by its $\varphi$ = 0.5 value. The elastic barrier is 1 k_BT at $\varphi$ ∼ 0.55. Packing fractions are given along the top x-axis.

Here we present, using only experimental data, a test of a relationship between activated relaxation, local pair structure, and a specific thermodynamic property predicted by the Elastically Collective Nonlinear Langevin Equation (ECNLE) theory (39–41). The results provide support for the following: 1) the coupled local–nonlocal nature of relaxation deeply connected with collective elasticity, 2) the dimensionless amplitude of thermal density fluctuations, S₀, as the relevant (nonexcess) thermodynamic property, 3) a roadmap for organizing relaxation data in S₀, not in temperature, space, 4) irrelevance of collective elasticity as the origin for the crossover from fragile to strong glass formers, and 5) an explicit demonstration that a dynamics–thermodynamics correlation can be a noncausal consequence of the causal relation between local pair structure and S₀.

Central Theoretical Prediction

The physical picture and key concepts of ECNLE theory are illustrated in Fig. 1B, all well documented in the literature (39–43). The entire pattern of relaxation in Fig. 1A arises from activated dynamics, in which the key slow variable is dynamic pair density fluctuations. Single-particle trajectories are related to local caging constraints embedded in a “dynamic free energy,” F_dyn(r), in which r is the particle displacement from its initial position, quantified by local features of g(r) or its Fourier space analog, S(k). A local cage barrier (F_B) for large amplitude hopping is predicted, which is intimately coupled to longer range, small amplitude, collective elastic displacements of all particles outside the cage. Construction of the scale-free elastic displacement field is physically motivated by the phenomenological shoving model (2, 30) but differs in four ways: 1) a particle-level Einstein glass framework is adopted (not continuum mechanics) to compute the elastic barrier (F_el), 2) F_el is not determined solely by the dynamic shear modulus, 3) the displacement field amplitude nucleated at the cage scale follows from the predicted thermodynamic state–dependent particle jump length ($\mathrm{\Delta }r$), and 4) F_el is not dominant relative to F_B even in the deeply supercooled regime. The alpha process total barrier is the sum of the two related barriers, (F_B + F_el), which both grow with cooling or densification. The elastic barrier grows more strongly with densification/cooling than F_B, and its relative importance increases with fragility (39, 41). No divergences are predicted at finite temperature nor below random close packing (RCP). Other germane aspects of ECNLE theory are recalled in SI Appendix.

Chemical complexity can be handled by mapping (39, 41) a thermal liquid to a temperature-dependent effective hard sphere fluid that exactly reproduces its experimental dimensionless compressibility, $S\left(k\to 0\right)\equiv S_0=\rho k_{\text{B}}T{\kappa }_T\propto ⟨{\left(\delta \rho \right)}^2⟩$. This quantity scales as the amplitude of long wavelength density fluctuations and is thus the natural thermodynamic variable for the dynamical ECNLE theory. Note that S₀ is a product of three T-dependent factors, is not an excess thermodynamic property, quantifies density fluctuations (second derivative of free energy), and never vanishes above zero Kelvin or below RCP. These features are in contrast to thermodynamic models of dynamics built on configurational entropy, internal energy, or enthalpy. Microscopically, S₀ encodes in average manner information about molecular size, shape, and repulsive and attractive forces. Quantitative calculations capture well the alpha time growth of hard sphere fluids and colloidal suspensions (nonpolar molecular liquids using the mapping) over approximately six (∼14) decades (39, 41).

However, assessment of the core dynamical ideas of ECNLE theory for structurally complex viscous liquids involves two uncertainties: 1) the accuracy of the required pair structure information as computed using approximate integral equation theory, and 2) validity of the mapping. This recently motivated our combined theoretical and simulation analysis (42, 44) which established specific links between dynamics, pair structure, and S₀ predicted by ECNLE theory for the metastable hard sphere fluid for which 1) is minimized and 2) is eliminated. A key prediction (42) is that the total barrier (logarithm of the alpha time in units of a short elementary time ${\tau }_0$) scales inversely with S₀ to an integer power equal to unity in the noncooperative regime ($\text{log}\left({\tau }_{\alpha }/{\tau }_0\right)\propto \beta F_{\text{B}}\propto S_0^{-1}$) in which collective elasticity effects are negligible and three in the deeply supercooled cooperative regime, $\text{log}\left({\tau }_{\alpha }/{\tau }_0\right)\propto \beta \left(F_{\mathrm{B}}+F_{\mathrm{el}}\right)\propto S_0^{-3}$ (Fig. 1C). The latter scaling reflects a subtle compensation between the cage and elastic barrier contributions (42) and is not conceptually identical to the shoving model (2, 30) (SI Appendix). The location of the crossover between the two regimes is known for hard spheres (per Fig. 1C) but varies due to chemical complexity reflecting the relative importance of elastic to cage physics (42, 45). However, the predicted thermodynamics–dynamics connection stated above generically holds (42) over a wide range of fragilities. Importantly, it is not literally causal in the sense that thermodynamics determines dynamics. Rather it is a predicted consequence of fundamental connections between pair structure and thermodynamics in equilibrium statistical mechanics.

Here, we provide direct support for our theoretically derived dynamically noncausal connection between the alpha time and S₀ for molecular and inorganic liquids. We also construct a master curve in S₀ space, predict and verify its consequences in temperature space, identify and test a mechanism for the fragile-to-strong crossover, and briefly discuss what ECNLE theory predicts in the equilibrated “deep glass” regime.

Experimental Test of Theoretical Predictions

We first consider the isobaric, 1 atm relaxation of six well-studied fragile glass-forming liquids of diverse sizes, shapes, and attractive interactions (inset of Fig. 2A): toluene (TOL), salol (SAL), ortho-terphenyl (OTP), trisnapthylbenzene (TNB), sorbitol (SORB), and glycerol (GLY). SI Appendix, Table S1 (in which data sources are cited) shows that their T_g values vary widely from 126 to 344 K, and their fragility values, defined as $m\equiv d\log \left({\tau }_{\alpha }\right)/d{\left(T_{\mathrm{g}}/T\right)}_{T_{\mathrm{g}}}$, vary from 53 (GLY) to 115 (TOL). The alpha time (${\tau }_{\alpha }$) data are plotted in Fig. 2A in the standard Angell representation and were obtained primarily from dielectric spectroscopy, supplemented for a few systems with viscosity, NMR, dynamic light scattering, and/or mechanical spectroscopy data (SI Appendix). Relaxation times span ∼14.4, 15.1, 12.7, 14.3, 13.0, and 11.7 decades for TOL, SORB, SAL, OTP, TNB, and GLY, respectively. At high temperatures, a very narrow (∼0.5 to 1.5 decades) Arrhenius behavior is observed. The nonpolar molecule data collapse over many decades, while the lowest-fragility hydrogen-bonding system (GLY) exhibits a large deviation. We also analyzed two high-pressure states of OTP (SI Appendix, Fig. S4A) and SAL at a constant volume state with the same T_g as at 1 atm. The alpha time data for the latter nearly overlaps that of GLY in Fig. 2A, reflecting the much smaller isochoric fragility of SAL compared to its isobaric analog (m ∼36 versus 73).

Fig. 2.

Alpha time data and test of the predicted relation between dynamics and thermodynamics. (A) Experimental alpha time versus reduced inverse temperature at 1 atm, plus one constant volume (0.7896 cm³/g) result for SAL; inset shows the molecular structures. (B and C) Experimental tests at 1 atm of the predicted two-regime linear variation of the logarithm of the alpha time with scaled inverse dimensionless compressibility raised to the (B) first and (C) third powers; data are shifted vertically for clarity. Two lines are drawn through the TNB data in C corresponding to just the low temperature regime (dashed) and all the data (solid). (D) Same as C but contrasts isochoric (squares) and isobaric (circles) SAL data; inset shows the very different temperature dependences of 1/S₀.

To test the theoretical prediction requires dimensionless compressibility data which are also given in SI Appendix. Fig. 2 B and C present the key cross plots at 1 atm: alpha time versus normalized inverse dimensionless compressibility in the two representations motivated by the theory. Overall, the plots are consistent with the predicted two activated regimes in S₀ space. Fig. 2B tests the idea of a “renormalized” noncooperative regime. Consistent with Fig. 1C, renormalized Arrhenius behavior applies over ∼1.5 to 3 decades, roughly double or more the range that classic Arrhenius behavior applies. This widening of the noncooperative regime is a consequence of S₀ not being a linear function of temperature for fragile liquids which have many configurational packing structures, the importance of which for relaxation changes with cooling (7, 41). Molecules with similar fragilities (SAL, OTP, and TNB) follow the renormalized Arrhenius behavior over nearly the same number of decades (∼2), consistent with the fragility index correlating with the relative importance of the noncooperative cage versus cooperative elastic physics (42, 45). As expected, the least fragile GLY exhibits the widest renormalized Arrhenius regime.

Fig. 2C shows the corresponding alpha time plot versus the scaled inverse cube of S₀. Drawing one line through all the data might seem to contradict the two activated regimes idea. However, Fig. 1C shows $S_0^{-3}$ scaling emerges at a relatively low total barrier in which the (narrow) $S_0^{-1}$ scaling also works in the crossover regime. Overall, the inverse cube law works well over ∼14.2, 15.1, 12.1, 14.3, 13 (8 for dashed line), and 7 decades for TOL, SORB, SAL, OTP, TNB, and GLY, respectively. Nearly identical behavior is found for OTP at high pressures (SI Appendix, Fig. S4B).

Fig. 2D contrasts SAL relaxation behavior under isochoric and isobaric conditions. In the $S_0^{-3}$ representation, SAL appears more fragile at constant volume. This is a dramatic consequence of the much-reduced growth with cooling of $S_0^{-1}$ (see inset) due to the isothermal compressibility increasing, not decreasing, upon isochoric cooling. Nevertheless, the isochoric alpha time data are well straightened over ∼10 decades, only slightly less than at constant pressure. Especially given the practical uncertainties of using the Tait equation of state (EOS) (SI Appendix), Fig. 2D supports S₀ as the key thermodynamic variable beyond isobaric conditions.

Concerning “straightening” of alpha time data, Medvedev and Caruthers (26) recently showed this can be impressively achieved for molecules based on the ansatz that the barrier scales as the inverse configurational internal energy. For the systems we study, the number of decades of straightening is nearly the same as in Fig. 2C. However, use of configurational internal energy or enthalpy (26, 46) has no theoretical basis nor clear underlying physical picture. Moreover, no mechanism for the significant high temperature deviations sometimes found were proposed, isochoric relaxation was not analyzed, the approach [as noted (26)] is not consistent with confinement effects on glassy relaxation, and use of an excess thermodynamic property may result in a finite temperature divergence. None of these features apply to ECNLE theory, which predicts, not assumes, a (dynamically noncausal) connection of relaxation time to a thermodynamic property.

Fig. 3 explores the idea motivated by Fig. 1C that a linear master curve can be constructed using the reduced variable $\overline{\mathrm{X}}$ = S₀(T_g)/S₀(T) based on one adjustable dimensionless parameter, w (varies in the range 0 to 1), that captures the noncooperative-to-cooperative crossover: $\log \left({\tau }_{\alpha }/{\tau }_0\right)\propto w\overline{\mathrm{X}}+\left(1-w\right){\overline{\mathrm{X}}}^3$. Physically, (1−w)/w quantifies the chemically specific relative importance of collective elasticity. Fig. 3 shows an overall good collapse for the 1 atm data (no vertical shifts), with the largest (but weak) deviations occurring at high temperatures; the analogous vertically shifted plots are shown in SI Appendix, Fig. S5. An equally good linear master curve is found for OTP at high pressures (SI Appendix, Fig. S4B).

Fig. 3.

Master curve for the alpha time and implications for the connection of fragility and collective elasticity. Linearization and collapse of the 1 atm alpha time data using the reduced variable $\overline{\mathrm{X}}$ = S₀(T_g)/S₀(T) based on one adjustable parameter, w, that describes the nonuniversal crossover from noncooperative to cooperative relaxation. (Inset) Deduced exponential relation between w and fragility index, in which m is determined from experimental data (see SI Appendix, Table S1 for values). Two high-pressure (78.5 and 125 MPa) results for OTP (SI Appendix) are included.

The inset of Fig. 3 plots the extracted w values versus fragility index, m. Based on the two-barrier idea of ECNLE theory (39–42), one a priori expects w⁻¹ grows monotonically with fragility (39, 40), as seen in Fig. 3. By fitting only the 1 atm data, an exponential relation is found, $w^{-1}=e^{\left(m-41.5\right)/20}$. The high-pressure OTP results fall on this master curve to within uncertainties. The precise w versus m relationship is not a priori predicted since small-length–scale nonuniversalities affect the jump distance which enters quantification of the absolute magnitude of the elastic barrier but not its temperature nor density dependences (42, 45). Taking the master curve seriously, a small extrapolation leads to the prediction that collective elasticity becomes irrelevant (w→1) when m < 41.5. Hence, a generalized (in S₀ space) fragile-to-strong crossover associated with the dominance of structural relaxation by local caging is deduced.

Strong Glass Formers.

The idea that when m < 41.5, the alpha time obeys the generalized noncooperative scaling $\log \left({\tau }_{\alpha }/{\tau }_0\right)\propto S_0^{-1}$ is tested using two very different “strong” inorganic network forming liquids, silica (SiO₂; m∼18) and boron oxide (B₂O₃; m∼37). Using experimental EOS data (47–49), the main frame of Fig. 4A shows their dimensionless compressibilities are qualitatively different, in magnitude and temperature dependence, from that of fragile molecular liquids (SI Appendix, Fig. S2) and also from each other. This reflects the organized structure of network liquids with strong directional interactions and a microstructural change in the B₂O₃ melt (48, 49). The silica S₀(T) decreases linearly with temperature, akin to a harmonic crystal or amorphous solid (47). On the other hand, B₂O₃ shows very complex behavior since the density grows with cooling, but the isothermal compressibility, ${\kappa }_{\mathrm{T}}\left(T\right)$, displays a nonmonotonic evolution (insets of Fig. 4A), resulting in the shown unusual $S_0\left(T\right)=\rho \left(T\right)\cdot k_{\mathrm{B}}T\cdot {\kappa }_{\mathrm{T}}\left(T\right)$.

Fig. 4.

Test of the predicted connection between thermodynamics and dynamics for strong inorganic liquids. (A) Dimensionless compressibility of Silica and Boron Oxide at 1 atm; the black line for SiO₂ is S₀(T) = 5.4*10⁻⁵ T (Kelvin). Density and isothermal compressibility versus temperature for B₂O₃ are shown in the upper and lower insets, respectively. (B) Logarithm of the viscosity of SiO₂ and B₂O₃ melts as a function of the scaled inverse dimensionless compressibility (main) and scaled inverse temperature (Inset).

The inset of Fig. 4B plots the experimental shear viscosities in the Angell representation. Silica is essentially Arrhenius, while boron oxide is not. The main frame of Fig. 4B shows $\log \left({\tau }_{\alpha }/{\tau }_0\right)\propto S_0^{-1}$ applies to silica, as expected since $S_0\left(T\right)\propto T$. More remarkably, the B₂O₃ data are also linearized as predicted, despite the complexity of S₀(T). These findings support our idea that collective elasticity is unimportant for such low-fragility melts, presumably because the length scale associated with “breaking” strong directional attractive interactions is small.

Implications in Temperature Space.

Although our main goal is to test the theoretical prediction of a noncausal connection between the alpha time and S₀, its implications in temperature space are also of interest. This aspect is studied in detail in SI Appendix with the key results summarized below.

Our thesis is that S₀(T) is the key thermodynamic variable, but its temperature dependence is not universal, even within a single-material class such as molecules. However, for nonpolar molecules at constant pressure, over a nontrivial range of temperatures, 1/S₀(T)∼1/T² is reasonably well obeyed (SI Appendix, Fig. S2). If so, then ECNLE theory predicts that a log-linear plot of the alpha time versus (T_g/T)⁶ should be linear in the deeply supercooled regime. Fig. 5 verifies this holds over ∼10, 12.4, 11, 13.6, and 7 decades for TOL, SORB, SAL, OTP, and TNB, respectively. Note that, as theoretically expected, the $S_0^{-3}$ correlation in Fig. 2C is better since 1/S₀∼1/T² is not exact. Although the idea that the effective barrier scales as inverse temperature to a power (∼T⁻ⁿ with a variable exponent n) has been discussed in conceptually distinct approaches (5, 8, 10, 23, 38), our prediction n = 6 for fragile liquids is distinct. Such a large exponent is far beyond the dimension of space and does not seem consistent with theories that relate the barrier to a growing static amorphous order or dynamic heterogeneity length scale (8, 10–12, 23, 38).

Fig. 5.

Consequences of the predicted relation between thermodynamics and dynamics in temperature space. Logarithm of the 1 atm alpha time data of Fig. 2 versus scaled inverse temperature to the sixth power; data are shifted vertically for clarity. The short horizontal pink lines indicate 1 $\mu$s.

To buttress our findings, we have further tested the T⁻⁶ scaling against the deeply supercooled molecular liquid relaxation data (40 systems) in ref. 5. (SI Appendix, Fig. S8). With the exception of three hydrogen-bonding molecules, the agreement with the predicted ∼T⁻⁶ behavior is overall very good. Notably, ECNLE theory predicts the T⁻⁶ scaling should fail if 1/S₀∼1/T² does not hold. This is the case for hydrogen-bonding GLY in which 1/S₀∼T^−1.45 (SI Appendix, Fig. S2), and thus, a T^−4.35 barrier scaling is predicted, which experimentally holds over approximately nine decades (SI Appendix, Fig. S9). Since 1/S₀(T) is not an inverse power law in T under isochoric conditions (Fig. 2D, inset), power scaling of the barrier with temperature is neither expected nor found for SAL at constant volume (Fig. 2D).

Concluding Remarks and Future Outlook

The presented successful confrontation with experiment of a prediction by ECNLE theory of a dynamics–thermodynamics correlation significantly supports 1) its coupled local–nonlocal activated relaxation mechanism involving cage scale hopping and longer-range collective elasticity quantified by local structural constraints, 2) identification of S₀ as the key “thermodynamic order parameter” for all activated regimes, and 3) the generic two-regime map in Fig. 1A. Crucially, the theory does not predict activated relaxation is a literal consequence of thermodynamics or k = 0 density fluctuations. Rather, the success of the S₀–dynamic correlation reflects the strong causal connection between local pair structure and thermodynamics (or long wavelength structure) as a matter of equilibrium statistical mechanics. Predicted and verified consequences in temperature space include that the alpha time varies exponentially with T⁻⁶ in the deeply supercooled regime of nonpolar molecular liquids and a fragile-to-strong transition associated with the irrelevance of collective elasticity when m < 41.5. Our ideas and data analysis approach can be applied to other systems, including polymer melts (45).

Finally, what might our perspective imply modestly or far (“deep glass” regime) below T_g under equilibrated conditions? This issue is under intense debate (50, 51), with experimental evidence accumulating that the alpha time does not diverge at nonzero temperature (5, 52–54), which agrees with ECNLE theory. Experiments also suggest that increase of the alpha time with cooling strongly slows down below T_g, perhaps becoming Arrhenius, which challenges configurational thermodynamic models. Although a detailed analysis is beyond the scope of our article, we offer a few germane comments.

To address the deep glass regime in the present context would involve extrapolation of the $S_0^{-3}$ barrier growth law to very low (high) temperature (density). However, this incurs technical uncertainties associated with the integral equation theory employed to deduce the dynamics–thermodynamics connection (SI Appendix). Moreover, the aging to equilibrium of the fluctuation quantity S₀ is poorly established and may be exceptionally slow (55, 56). Thus, the most robust ECNLE theory prediction in the deep glass regime is based solely on dynamical properties, per the final proportionality in SI Appendix, Eq. S5. It represents a causal connection between the total (not elastic) barrier and two related measures of short time dynamics: the transient localization length (r_loc in Fig. 1B) and high frequency shear modulus (G′), $\log \left({\tau }_{\alpha }/{\tau }_0\right)\propto \left[F_{\mathrm{B}}\left(T\right)+F_{\mathrm{el}}\left(T\right)\right]/k_{\mathrm{B}}T\propto {\left[\sigma /r_{\mathrm{loc}}\left(T\right)\right]}^2\propto {\sigma }^{3}G\text{'}\left(T\right)/k_{\mathrm{B}}T$, where $\sigma$ is molecular size. At low temperatures as the bottom of the potential energy landscape is approached, one generically expects G′ approaches a constant, and the solid-state–like harmonic relation $r_{\mathrm{loc}}^2\sim T$ applies. The above relations then predict the alpha time crosses over to an Arrhenius form and with a sensible activation energy (SI Appendix). Work is in progress exploring this idea in detail, partially building on prior theoretical progress (7) below T_g.

Materials and Methods

Experimental data employed in this article are all from the literature. The data sources for the alpha time, T_g and fragility, and the S₀ analysis for the six fragile molecules (TOL, SORB, SAL, OTP, TNB, and GLY) under 1 atm, high pressure, and constant volume conditions are given in SI Appendix. The sources of the dynamic (thermodynamic S₀) data for silica and boron oxide are given in SI Appendix (main article). Germane theoretical background results are briefly recalled in SI Appendix, along with a modestly extended discussion of the equilibrated deep glass regime. SI Appendix also presents our analysis of experimental data of 40 deeply supercooled liquids to further test the predicted inverse in temperature power law scaling of the activation barrier.

Data Availability

All data employed in this work are included in the article and/or SI Appendix.