Does a growing static length scale control the glass transition?

TL;DR

This paper challenges the idea that a growing static length scale solely controls the glass transition, showing that local dynamical slowdown plays a larger role in increasing relaxation times.

Contribution

The study provides evidence from simulations that static length scale growth has a limited effect on the glass transition, emphasizing the importance of local dynamical slowdown.

Findings

Static length scale growth contributes modestly to relaxation time increase.

Local dynamical slowdown is the dominant factor in the glass transition.

Strong coupling between particle diffusion and density fluctuations supports the findings.

Abstract

Several theories of the glass transition propose that the structural relaxation time {\tau}{\alpha} is controlled by a growing static length scale {\xi} that is determined by the free energy landscape but not by the local dynamical rules governing its exploration. We argue, based on recent simulations using particle-radius-swap dynamics, that only a modest factor in the increase in {\tau}{\alpha} on approach to the glass transition may stem from the growth of a static length, with a vastly larger contribution attributable instead to a slowdown of local dynamics. This reinforces arguments that we base on the observed strong coupling of particle diffusion and density fluctuations in real glasses

Full text

When a liquid is cooled sufficiently rapidly to avoid crystallization, its viscosity $\eta$ increases. In general, $\eta\simeq G\tau_{\alpha}$, where $G$ is the (plateau) shear modulus and $\tau_{\alpha}$ characterizes the relaxation time of density fluctuations. As the temperature $T$ is lowered, $G$ evolves mildly but $\tau_{\alpha}$ increases by about $15$ order of magnitude Ediger et al. (1996) until, at the transition temperature $T_{G}$, it becomes too large to measure experimentally. The liquid then becomes a glass, and falls out of equilibrium. Near the glass transition, the diffusion of a tagged particle also becomes very slow. The characteristic time $\tau_{D}$ over which a particle diffuses its own radius increases, albeit not as much as $\tau_{\alpha}$. The decoupling of these two quantities (referred to as the Stokes-Einstein breakdown) is significant, but comparatively mild: the ratio ${\cal S}=\tau_{\alpha}/\tau_{D}$ is increased by only a few orders of magnitude at $T_{G}$ Ediger (2000); Swallen et al. (2009).

The dependence of $\tau_{\alpha}$ on $T$ is used to classify glassy liquids Angell (1991). Writing $\tau_{\alpha}=\tau_{0}\exp(E/k_{B}T)$ where $\tau_{0}$ is a vibrational time scale (in the picosecond range) and $E$ some activation energy, one finds that $E$ is constant in some liquids, called strong, but increases up to a factor $5$ under cooling in other liquids, called fragile. Fragility is best shown in the ‘Angell plot’ of $\log(\tau_{\alpha})$ vs. $T_{G}/T$ Angell (1991), which is linear for strong liquids but highly curved for fragile ones. Quantitatively, fragility is defined as $m=d\log(\tau_{\alpha})/d\log(T_{G}/T)$ evaluated at $T_{G}$ itself. Strong liquids have $m\approx 25$; very fragile ones have $m\approx 120$.

There are competing explanations of the increase in the activation energy $E$ in fragile liquids, which occurs with no obvious change of static structure Berthier and Biroli (2011). Several theories, including the Adam-Gibbs scenario Adam and Gibbs (1965), Random First Order Theory (RFOT) Lubchenko and Wolynes (2007); Bouchaud and Biroli (2004); Kirkpatrick and Thirumalai (2015), and those involving locally favored structures Kivelson et al. (1995); Tanaka (2012), posit that the increase in $E$ stems from the growth of a purely static length scale $\xi$, characterizing some ‘hidden order’ in the many-body free-energy landscape that is not captured by traditional probes of static structure such as pair correlations.

In particular, in modern interpretations of RFOT Lubchenko and Wolynes (2007); Bouchaud and Biroli (2004), $\xi$ is a ‘point-to-set’ correlation length set by the minimum scale on which alternative packings are available to a patch of fluid whose environment is held frozen. Shorter scale motions do cross local barriers, but cannot discover a new pattern for the density which keeps returning to its initial state. In this view, regardless of how rapidly these local moves can permute the particles within the patch, $\tau_{\alpha}$ is controlled by the fact that the system can relax fully its density fluctuations only via collective rearrangements on the scale $\xi$ that ‘break’ the hidden order. The resulting collective activation energy is $E_{\rm coll}\simeq c_{0}(T)\xi(T)^{\psi}$, where $c_{0}(T)\simeq c_{0}(T_{G})$ is non-singular, and $\psi$ is some exponent. It can be expressed in terms of thermodynamic quantities alone and is thus independent of the details of the dynamics 111This statement holds true if the order parameter $Q$ on a correlation volume $\xi^{d}$ evolves smoothly at each numerical step, as it ensures that configurations with $Q=Q^{*}$ for which the free energy barrier is maximum are visited. This must be true for any Monte-Carlo where moves involves $n_{0}$ particles with $n_{0}<<\xi^{d}$. (see more on that below).

Although there is clear empirical evidence that a growing static length scale $\xi(T)$ exists (see e.g. Albert et al. (2016) and references therein), its role in the dynamics is debated. Alternative theories instead propose that the increase of $\tau_{\alpha}$ stems from growing barriers to the elementary rearrangements required to explore the landscape Dyre (2006); Torchinsky et al. (2009); Mirigian and Schweizer (2014); Garrahan et al. (2011); Lemaître (2014). Such barriers effectively add a term to the activation energy {\color[rgb]{0,0,0}E_{\rm el}}=c_{1}(T). Unlike $E_{\rm coll}$, these kinetic barriers can depend greatly on the details of the dynamical rules governing the system. While the definition of ‘elementary rearrangements’ can be multi-particle (see more on that below), {\color[rgb]{0,0,0}E_{\rm el}} describes local physics and cannot diverge. However it might grow strongly enough near $T_{G}$ to control the glass transition via \tau_{\alpha}\propto{\color[rgb]{0,0,0}\tau_{\rm el}}\simeq\tau_{0}\exp[{\color[rgb]{0,0,0}E_{\rm el}}/k_{B}T]. For example in elastic models Dyre (2006), rearrangements require a certain strain, giving $E(T)\sim G_{\infty}(T)$ where $G_{\infty}$ is the high-frequency (vibrational) shear modulus. Empirically these quantities are indeed strongly correlated Dyre (2006); Torchinsky et al. (2009), as seen by plotting $\log(\tau_{\alpha})$ vs. $(T_{G}G_{\infty}(T))/(TG_{\infty}(T_{G}))$, which now appears almost perfectly linear even for fragile liquids Torchinsky et al. (2009); Hecksher and Dyre (2015).

Combining the two activation terms gives $\ln(\tau_{\alpha}/\tau_{0})\simeq a_{0}\xi^{\psi}+a_{1}$, with $a_{0}=c_{0}/k_{B}T$ and $a_{1}=c_{1}/k_{B}T$. This estimate coincides in form with an upper bound on the relaxation time Montanari and Semerjian (2006); this bound shows that a diverging $\xi$ is rigorously required for an ideal glass transition ($\tau_{\alpha}/\tau_{0}\to\infty$). It is then tempting to conclude that growth of $\xi$ controls the real glass transition ($\tau_{\alpha}/\tau_{0}\to 10^{15}$).

In this work we present arguments for the contrary view, proposing instead that most of the increase in $\tau_{\alpha}$ stems from an increase upon cooling of {\color[rgb]{0,0,0}E_{\rm el}}. Our main argument is based on recent numerical observations Berthier et al. (2016); Gutiérrez et al. (2015); Ninarello et al. (2017); Berthier et al. (2017) in which a judicious choice of dynamics is shown to equilibrate systems far deeper inside the glass ($T\ll T_{G}$) than previously achievable. Indeed this dynamics almost abolishes the glass transition. We shall infer from this outcome that {\color[rgb]{0,0,0}E_{\rm el}} rather than $E_{\rm coll}$ is dominant in controlling $\tau_{\alpha}$. This finding applies, strictly speaking, only to the polydisperse systems studied numerically in Berthier et al. (2016); Gutiérrez et al. (2015); Ninarello et al. (2017); Berthier et al. (2017). We then give a more general and complementary argument, to the effect that the Stokes-Einstein factor should obey {\cal S}\geq\exp[E_{\rm coll}/k_{B}T]=\tau_{\alpha}/{\color[rgb]{0,0,0}\tau_{\rm el}}. It follows that the growth of $\xi$ contributes only a few decades at most to the 15 decades increase in $\tau_{\alpha}$ on the approach to $T_{G}$. Our findings are depicted schematically in Fig.1.

Swap algorithms. Recent Monte-Carlo schemes have managed to equilibrate liquids at temperatures where traditional algorithms would need at least a $10^{10}$ speed-up to allow equilibration Grigera and Parisi (2001); Fernández et al. (2006); Berthier et al. (2016); Gutiérrez et al. (2015); Ninarello et al. (2017); Berthier et al. (2017). These swap algorithms consider polydisperse particles. In addition to the usual translational moves, they allow for moves in which two particles of different radius exchange places. This can also be viewed as a radius-swap at fixed particle positions (in which case there is no direct contribution to tagged-particle diffusivity). One then has particles whose sizes are an extra set of dynamical variables, with update rules that preserve detailed balance. Accordingly the free energy landscape $F(\{{\bf r}\})$ as a function of the particle positions $\{{\bf r}\}$ is, after integration over sizes, well defined and independent of the chosen dynamical rules. The swap rules can involve either well-separated particles as reported in Ninarello et al. (2017); Berthier et al. (2017), or be local, for instance restricted to nearest neighbour swaps Berthier (2017). For the dynamical rules of Ninarello et al. (2017); Berthier et al. (2017), the time scale $\tau_{\alpha}^{+}$ on which density fluctuations relax with swaps 555 Note that in Berthier et al. (2017), $\tau_{D}$ and $\tau_{\alpha}$ are both treated as one-particle properties. Here we use $\tau_{\alpha}$ to describe collective density relaxations. can be measured and compared to the value $\tau_{\alpha}^{-}$ without swaps allowed. The latter case behaves like molecular dynamics simulations, and display the standard glass transition. Remarkably though, with the swap dynamics switched on the glass transition is essentially gone: $\tau_{\alpha}^{+}(T_{G})/\tau_{0}$ is only around $10^{2}-10^{3}$. This contrasts with $\tau_{\alpha}^{-}/\tau_{0}\simeq 10^{15}$ for the standard dynamics at $T_{G}$.

Implications of these observations. Local and long-range swap dynamics do appear to give very similar results for $\tau_{\alpha}$ Berthier et al. (2017), as we justify in S.I. This suggest that either type of swap dynamics can mollify local barriers, leading to much faster relaxation than the conventional dynamics of real glasses. It follows that the latter is dominated by {\color[rgb]{0,0,0}E_{\rm el}}. Indeed, switching from standard to swap dynamics simply enhances the local move set: swapping is a specific local move involving two particles only 333A different instance of local dynamical rules profoundly affecting the glass transition is discussed in Jacquin et al. (2015).. The underlying free-energy landscape $F(\{{\bf r}\})$ is unaffected. As such, so long as one first equilibrates the system, any static (i.e., thermodynamic) length $\xi$ must be identical with both dynamics. Indeed, a major selling-point of the swap algorithms is that they do allow full equilibration over an unprecedentedly wide temperature range. We further consider (see proof below) that for large enough $\tau_{\alpha}$ the collective barriers $E_{\rm coll}$ must be identical for swap and non-swap dynamics. We can now compare the dynamics with ($+$) and without ($-$) swaps. Since \tau_{\alpha}^{\mp}={\color[rgb]{0,0,0}\tau_{\rm el}}^{\mp}\exp[E_{\rm coll}(\xi)/k_{B}T], if glass physics is dominated by the growing length scale $\xi$, then $\tau_{\alpha}^{-}/\tau_{\alpha}^{+}$ should depend only weakly on $T$: indeed the only such dependence is through the factor {\color[rgb]{0,0,0}\tau_{\rm el}}^{-}/{\color[rgb]{0,0,0}\tau_{\rm el}}^{+}. But in the simulations, $\tau_{\alpha}^{-}/\tau_{\alpha}^{+}$ increases without apparent limit as $T$ is lowered: this increase can be tracked for about 4 decades before it becomes too large to measure Ninarello et al. (2017); Berthier et al. (2017). In contrast, the growth of $\tau_{\alpha}^{+}$, which bounds above the slowing down caused by the growth of $\xi$, only shows a 2 or 3 decade increase at $T_{G}$. However, it continues to grow by several more decades as $T$ is further decreased.

We offer the following interpretation for these results. The relaxation time $\tau_{\alpha}^{+}$ describing the with-swap dynamics, for which local barriers are mollified, may ultimately be controlled by the growth of $\xi$, although other effects are plausible  333333One may expect a transition for swap dynamics at some temperature $T_{MCT}^{+}$ where local minima appear in $F(\{{\bf r}\})$ with $T_{MCT}^{-}>T_{MCT}^{+}$. Indeed for the swap dynamics stability is more demanding since radii can smoothly evolve (if poly-dispersity is continuous).. However, with the swap-free dynamics relevant to real glasses, the resulting contribution to $\tau_{\alpha}^{-}/\tau_{0}$ is smaller than 3 decades at the glass transition. Thus the lion’s share of the growth in $\tau_{\alpha}$ stems not from a growing static length but from growing local barriers. Only these barriers are affected by introducing local swaps, so that the resulting collapse of the glass transition points directly to their dynamical importance.

Stokes-Einstein violation. The arguments above are restricted to polydisperse systems, as simulated by radius-swap algorithms. However for conventional swap-free dynamics the relaxation in these systems is no different from classical numerical models of structural glasses Ninarello et al. (2017); Berthier et al. (2017). This suggests that polydispersity is immaterial to the issue of whether a growing static length controls the glass transition. To support this view, we now present a polydispersity-free argument that again points toward control predominantly by local barriers.

For concreteness we suppose RFOT to correctly identify the static length scale $\xi$. Recall that in this approach, each configuration is a mosaic of states, whose characteristic size $\xi$ results from a competition between configurational entropy and “surface tension” between states. The activation barrier to nucleate a new density configuration on a length scale $r$ then varies as $c_{0}r^{\psi}$: the dynamics is fast on short length scales, but slow on long length scales. However for $r<\xi$, a single state is thermodynamically favored. Local rearrangements are possible, but these will not fully relax density fluctuations. Processes that can relax density fluctuations occur on a scale $\xi$, the smallest on which alternative density patterns can appear Bouchaud and Biroli (2004).

We believe that this picture contradicts the observation that in real glasses the Stokes Einstein (SE) factor ${\cal S}=\tau_{\alpha}/\tau_{D}$ at $T_{G}$ is of order $10^{3}$ Swallen and Ediger (2011); Swallen et al. (2009), rather than a much larger value. In practice, much of the violation is thought to arise from dynamic heterogeneity (DH), in which diffusive and structural relaxation times are dominated by the most liquid and most solid regions respectively Tarjus and Kivelson (1995). However, DH can only increase ${\cal S}$: any other mechanism found to contribute to the SE violation therefore gives a lower bound on it. We may therefore write ${\cal S}={\cal S}_{1}{\cal S}_{2}\geq{\cal S}_{1}$, where ${\cal S}_{2}$ accounts for DH, and ${\cal S}_{1}$ accounts for the possibility that particles can exchange positions while leaving density fluctuations unchanged Charbonneau et al. (2014a). We now argue that ${\cal S}_{1}\sim\exp[a_{1}\xi^{\psi}]$, by noting first that, even for $r<\xi$, there is always an exponentially large number of states available for conventional dynamics, corresponding to permutations of the particles within a fixed density pattern. Any such state can be reached by a series of local permutations, which can each be generated by local rearrangements only. Without polydispersity there cannot be any thermodynamic reason for a particular permutation to be preferred, so a tagged particle can move diffusively with \tau_{D}\simeq\tau_{0}\exp[{\color[rgb]{0,0,0}E_{\rm el}}/k_{B}T]=\tau_{\alpha}/{\cal S}_{1}.

We then have ${\cal S}_{1}=\exp[a_{1}\xi^{\psi}]<{\cal S}\simeq 10^{3}$, implying that the growth of $\xi$ cannot account for more than about a 3 decade increase in $\tau_{\alpha}$ at $T_{G}$. Were the glass transition controlled purely by growth of $\xi$, a much stronger decoupling of $\tau_{D}$ and $\tau_{\alpha}$ could be expected. This is incompatible with the fact that the fragility defined from $\tau_{D}$ is almost as large as that defined from $\tau_{\alpha}$ (only 13-25% 343434This change of fragility can be deduced from the power law fit reported between $\tau_{D}$ and $\tau_{\alpha}$ in Swallen and Ediger (2011); Swallen et al. (2009). smaller for the liquids studied in Swallen and Ediger (2011); Swallen et al. (2009)). In practice therefore, diffusive and density relaxations are much more strongly coupled than RFOT seems to imply, and this points to a major role for {\color[rgb]{0,0,0}E_{\rm el}} at the glass transition. The same arguments apply to any theory for which fast local moves are insufficient to break a hidden order on a growing scale $\xi$.

In the specific case of RFOT our argument can be restated as follows: for the nucleation picture to hold (whereby a new density pattern appears by an activated barrier crossing at scale $\xi$) local equilibrium in the landscape $F(\{{\bf r}\})$ should be reached, on all length scales shorter than $\xi$, at time scales much smaller than $\tau_{\alpha}$. But this local exploration of phase space permutes particles, leading to diffusion on the same fast timescale. Thus in any temperature regime where RFOT is dominant in driving the slowdown in $\tau_{\alpha}$, a severe Stokes Einstein breakdown should occur. This is not seen in practice.

Asymptotic equivalence of landscapes. We now return to the case of polydisperse particles and explain why, as we assumed above, the free energy landscape $F(\{{\bf r}\})$ is the same for dynamics with and without swaps, for large enough $\tau_{\alpha}$. This implies in turn that $E_{\rm coll}$ is the same (including its prefactor $c_{1}$) for the two types of dynamics, precisely in the regime in which a growing static length could come to dominate the dynamics. The argument is almost the same as the one just given for monodisperse particles: for large $\xi$, local rearrangements are rapid in comparison to $\tau_{\alpha}$, sampling with Boltzmann weight configurations where groups of particles are permuted locally. Thus traditional (non-swap) dynamics already performs local swaps on time scales much shorter than $\tau_{\alpha}$ in this limit. The collective free energy barriers, when large, must thus be described by the same $F(\{{\bf r}\})$ with and without swaps, which is what we assumed above.

Discussion. We are not suggesting that a growing length scale $\xi$ cannot contribute at all to the slowdown of dynamics in real glasses. But to play more than a supporting role, the resulting order must affect elementary rearrangements themselves, via {\color[rgb]{0,0,0}E_{\rm el}}. For example, in the spirit of elastic models, locally favored structures that grow under cooling Kivelson et al. (1995); Tanaka (2012) could increase $G_{\infty}$, which slows down local moves requiring finite strain. Similarly, the observation that radius-swap moves dramatically speed up the dynamics of glasses does not contradict dynamically-facilitated models Garrahan and Chandler (2002); Toninelli et al. (2007), since in these models the choice of local dynamics is important. However they do need to be consistent with the fact that particle diffusion and density relaxation dynamics are strongly coupled.

In our view, the proof that a static length scale must diverge if $\tau_{\alpha}$ does Montanari and Semerjian (2006) is of limited practical relevance for real liquids near their glass transition, since the proper signal for this divergence is $\tau_{\alpha}/\tau_{D}$ which increases relatively mildly within the experimental range. (Moreover, $\xi$ need only diverge as $(\ln(\tau_{\alpha}/\tau_{D}))^{1/\psi}$.) The same conclusion is reached if, instead of considering the dynamics in terms of the rescaled temperature $T/T_{G}$, one normalizes it by a natural energy scale $G_{\infty}(T)/G_{\infty}(T_{G})$ (which cannot be singular for smooth interactions). As stated earlier, the dynamics shows no sign of divergence if plotted in terms of $(T_{G}G_{\infty}(T))/(TG_{\infty}(T_{G}))$ Torchinsky et al. (2009); Hecksher and Dyre (2015).

On the other hand, with local barriers mollified by swap dynamics as in Ninarello et al. (2017); Berthier et al. (2017), the growth of $\xi$ could become paramount, and it is possible that RFOT is the correct theory of such dynamics. For realistic dynamics however theories of local barriers appear to be needed to describe the glass transition. One such theory is provided by elastic models Dyre (2006); Torchinsky et al. (2009), for which {\color[rgb]{0,0,0}E_{\rm el}} is proportional to $G_{\infty}$. However the spatial description of local moves remains very crude in these models. Importantly, they can involve finite collections of particles, for purely dynamical reasons. For example, it may require less strain energy to permute three particles forming a triangle than two neighboring particles.

Understanding the geometry of such elementary rearrangements is a challenge. Recently there has been a convergence of different approaches to compute the properties of soft collective vibrational modes in hard sphere liquids: a real space description Brito and Wyart (2006, 2009), mean-field approximation DeGiuli et al. (2014a, b) and exact calculations in infinite dimensions Franz et al. (2015); Altieri et al. (2016); Maimbourg et al. (2016). (The discussion below may apply to other liquids by replacing the packing fraction $\phi$ by $T$.) It is found that some elastic modes become stable only above an onset packing fraction $\phi_{0}$ Brito and Wyart (2009); Charbonneau et al. (2014b); Maimbourg et al. (2016). For $\phi<\phi_{0}$, one expects flow to be fast as the system can relax without jumping over free energy barriers along unstable directions of phase space. By contrast, flow must be activated for $\phi>\phi_{0}$. This phenomenon is captured in part by the mode-coupling theory of liquids and by mean-field theory Rabochiy et al. (2013); Lubchenko and Wolynes (2007); Grigera et al. (2002); Franz and Montanari (2007); Andreanov et al. (2009). One expects a growing dynamical length scale at such intermediate $\phi$ because the last modes to remain unstable as $\phi$ increases are increasingly collective. For larger $\phi$, the simplest guess is that activated elementary rearrangements occurs along similar modes to those whose stability is exchanged at $\phi_{0}$. This is consistent with the observation that the dynamical length scale almost saturates (at the scale of collective motions of a few tens of particles) once $\tau_{\alpha}$ has increased by a few orders of magnitude Dalle-Ferrier et al. (2007).

This line of thought is consistent with the observations of swap algorithms. Indeed if elementary rearrangements become more collective because local moves are too costly, then dynamical correlations can disappear if effective local moves such as radius-swaps are allowed for. Alongside much faster relaxation, dynamical correlations near the mode coupling temperature should then be eliminated. This is indeed observed: dynamical correlations become very small when swaps are used Ninarello et al. (2017). Furthermore, dynamical correlations near $T_{G}$ also become very small. This observation undermines the RFOT result that dynamical heterogeneities near $T_{G}$ in liquids reflect $\xi$ Berthier and Biroli (2011), since $\xi$ is unaffected by the choice of dynamics. Instead it supports the view that for non-swap dynamics, elementary rearrangements are spatially more extended than $\xi$, further indicating that $T_{G}$ lies outside the temperature range where RFOT might apply.

Another point to clarify concerning elastic models is why $G_{\infty}$ increases under cooling. From that perspective, the exchange of stability of some elastic modes at $\phi_{0}$ can be shown to lead to an increase in shear modulus under cooling Lubchenko and Wolynes (2007); DeGiuli et al. (2014b); Nakayama et al. (2016), and is thus consistent with mean-field approaches. Other factors susceptible to stiffen the material may be very system-specific, including the previously mentioned growth of locally-favored structures Kivelson et al. (1995); Tanaka (2012).

A promising avenue that appears consistent with the swap-algorithm observations would thus be to combine elastic models with a more detailed description of collective dynamical modes beyond their linear regime. This could also account for other key facts of the dynamics in liquids, in particular the presence of growing dynamical length scales, the correlations between activation energy and high frequency shear modulus, and the correlations between entropy and dynamics Wyart (2010).

Conclusion. The recent observation that swap algorithms can essentially eliminate the glass transition without changing the free energy landscape casts doubt on theories in which a growing static length scale, determined solely by this landscape and setting a minimum scale for finding new density patterns, controls the slowdown of relaxations in glasses. Instead, these observations cap the contribution from this source at a few decades of growth in the structural relaxation time, with a much larger factor arising instead from the growing barriers to local rearrangement. This concurs with a similar cap derived from the Stokes Einstein violation for real glasses which is likewise only a few orders of magnitude, requiring tagged particle diffusion (asymptotically unaffected by the growth of $\xi$) and structural relaxation to remain strongly coupled near the glass point. Among theories of the local barrier physics, a combination of elastic models with an improved description of collective dynamic modes may offer a promising route forward.