Abstract
Supercooled liquids exhibit a pronounced slowdown of their dynamics on cooling^{1} without showing any obvious structural or thermodynamic changes^{2}. Several theories relate this slowdown to increasing spatial correlations^{3,4,5,6}. However, no sign of this is seen in standard static correlation functions, despite indirect evidence from considering specific heat^{7} and linear dielectric susceptibility^{8}. Whereas the dynamic correlation function progressively becomes more nonexponential as the temperature is reduced, so far no similar signature has been found in static correlations that can distinguish qualitatively between a hightemperature and a deeply supercooled glassforming liquid in equilibrium. Here, we show evidence of a qualitative thermodynamic signature that differentiates between the two. We show by numerical simulations with fixed boundary conditions that the influence of the boundary propagates into the bulk over increasing length scales on cooling. With the increase of this static correlation length, the influence of the boundary decays nonexponentially. Such longrange susceptibility to boundary conditions is expected within the random firstorder theory^{4,9,10} (RFOT) of the glass transition. However, a quantitative account of our numerical results requires a generalization of RFOT, taking into account surface tension fluctuations between states.
Main
Inspired by critical phenomena, it is natural to expect that the slowing down of the dynamics is related to the vicinity of a thermodynamic phase transition, where some kind of longrange order would set in^{11}. This is the spirit of different recent theories^{4,9,12,13,14}, but seems at odds with others^{5,15}, at least at first sight. In particular, the crucial physical mechanism at the root of random firstorder theory^{4} (RFOT) is the emergence of longrange amorphous order, the precise definition and quantitative characterization of which is however far from obvious. Dynamic heterogeneities^{16} do show a growing dynamic correlation length accompanying the glass transition, both experimentally^{17} and numerically^{18}. This is certainly a first important step, but not sufficient to prune down—even at a qualitative level—different theories of the glass transition. In particular, it is not clear whether this phenomenon is due to an underlying static or purely dynamic phase transition.
The approach followed here is based on the very definition of a thermodynamic phase transition, where the effect of boundary conditions becomes longranged. The problem is that for glasses there are no natural boundary conditions, because these should be as ‘random’ as the bulk amorphous states that they favour. A possible solution is to use equilibrium liquid configurations to define the boundary^{19}. In the context of RFOT, this was suggested in ref. 9 (and further discussed in ref. 11), but the scope and some conclusions of this Gedankenexperiment are more general^{19,20,21}. Starting from a given equilibrium configuration, we freeze the motion of all particles outside a cavity of radius R and study the thermodynamics of the mobile particles with boundary conditions imposed by the frozen ones. Defining a suitable overlap (inside the cavity) between the original equilibrated configuration and the configurations equilibrated with the boundary pinning field, the existence of ‘order’ on a scale ξ implies a large overlap (or ‘pointtoset’ correlation) for R≪ξ and a small one for R≫ξ.
Such correlation length arises in RFOT^{9}, if we assume (or find by approximate calculations^{10,22}) that there are many amorphous metastable states, inside the cavity (Σ(T) is the configurational entropy, or complexity). The particles in the cavity can be either in the original state α in which the outside particles have been frozen or in any of the other states. The probability to flip to a different state γ is determined by the balance between the freeenergy loss due to mismatch between α and γ, (T)R^{θ}, and the gain from the entropic term T R^{d}Σ(T) ( is a generalized surface tension and θ≤d−1). A crossover between a boundarydominated regime (large overlap) and an entropydominated regime (small overlap) is expected for R=ξ∝(/T Σ)^{(1/d−θ)}. Note that this length diverges at the Kauzmann temperature T_{K} where Σ(T_{K}^{+})→0.
This Gedankenexperiment was realized numerically by three of us in ref. 23. This first study indeed suggested a growing static length, but this length was very small, and no sharp transition between high and low overlap was observed, in contrast with expectations based on RFOT. Is RFOT possibly in contradiction with numerical experiments? What is the physical meaning of the increasingly longrange susceptibility to boundary conditions if it is characterized by such a small length? To answer these crucial questions, we have carried out new simulations that measure the local overlap at the centre of the cavity. In fact, a major difficulty in interpreting the results of ref. 23 is that the overlap was computed as an average over the whole cavity. As a result, the overlap drop due to the (putative) change of state of the cavity is mingled with the decay of the overlap expected from a trivial weakening of the surface pinning field for larger spheres. This effect is indeed present even in the singlestate case^{21} at all temperatures. As we shall show, analysing the behaviour of the local overlap at the centre of the cavity yields sharper results that allow us to answer, at least partially, the above questions.
We study a softsphere model^{24} that we can equilibrate below the modecoupling transition temperature T_{MC}=0.226 (ref. 25) and for large systems (see the Methods section). After equilibration, several independent reference configurations are chosen as starting points for runs with all but M particles frozen. These mobile particles are confined inside a sphere of radius R such that the density inside equals the bulk density. After the confined runs reach equilibrium, the local overlap at the centre q_{c}(R) is measured. To define q_{c}(R), we partition the simulation box into many small cubic boxes of side ℓ, such that the probability of finding more than one particle in a single box is negligible. Let n_{i} equal the number of particles in box i, then
where the sum runs over all boxes within a small volume v at the centre of the sphere, N_{i} is the number of boxes and 〈⋯〉 means thermal average. To minimize statistical uncertainty without losing the local nature of q_{c}(R), we choose N_{i}=v/ℓ^{3}=125. Normalization is such that the overlap of two identical configurations is 1 on average, whereas for totally uncorrelated configurations q_{c}=q_{0}≡ℓ^{3}=0.062876.
We show q_{c}(R)−q_{0} for several temperatures in Fig. 1. The decay becomes slower at lower temperature: the effect of boundary conditions propagates on larger length scales. This clearly shows the growth of static order that expands into the bulk in the deeply supercooled phase. The most striking result, however, is that at low temperatures the decay is no longer the simple exponential that prevails in a standard liquid state^{21}. The relaxation can be fitted by a ‘compressed exponential’,
where ζ increases at low temperatures (Fig. 2a) above its hightemperature liquid value ζ=1 (Fig. 2b). Larger values of ζ mean a sharper crossover between large and small overlaps. The best fit parameters are given in Table 1. Interestingly, the value of the length scale ξ found here is significantly larger than those in ref. 23. This is partly due to the fact that ζ>1 at low temperatures (see Supplementary Information for further discussion). The overlap q_{c}(R) therefore appears as a thermodynamic quantity able to single out the deeply supercooled liquid in terms of (1) a large correlation length ξ and (2) an anomalous nonexponential relaxation, characterized by the exponent ζ>1.
The growth of ξ strongly suggests that the liquid state should be thought of as a mosaic of regions (transiently) ordered on a length scale ξ. Furthermore, following suggestions based on RFOT^{4,9,12}, it is natural to conjecture that we are probing the growth of positional amorphous order. Interestingly, in this scenario, all finitepoint static correlations remain featureless, whereas the pointtoset correlations^{11,26}, captured by q_{c}(R), grow. The simplest version of RFOT predicts at low temperature a sharp crossover of q_{c}(R) for R=ξ (ref. 27). Our numerical results show that this is not what happens: the crossover is much smoother than expected (see also the recent work^{28} on a 1−d Kac model).
To account for our numerical results, we now propose a natural generalization of RFOT, based on the idea that the effective interface tension, , is in fact state dependent. Repeating the argument of ref. 9 with this extra assumption, the probability that the cavity is found in a state γ different from the outside pinning (frozen) state α, is:
where β=1/T, f^{*} and Σ^{*}=Σ(f^{*}) are respectively the free energy and configurational entropy of the equilibrium states and p_{α}(f) is the distribution of surface tension for state α with other states of free energy f (see the Methods section). As our simulations average over many realizations of the external state α, we in fact compute p_{out}^{(α)}(R) averaged over all possible pinning states:
where w_{α} is the thermodynamic weight of each state. The leading contribution of the integral over is shown in the Methods section to be e^{−βy}_{α}R^{θ}, where y_{α} is the lower edge of p_{α}(f) and where the exponent θ is possibly renormalized (this might explain why the effective value of θ can be less than the naively expected value d−1). Introducing , we find
The simple mosaic result of ref. 9 is recovered setting p_{α}(f)=δ(−_{0}) and P(Y)=δ(Y −Y_{0}). Expression (4) can be simplified further by noticing that the approximation (1+e^{R}^{d}Σ^{*}−β Y R^{θ})^{−1}≈Θ(Y −T Σ^{*}R^{d−θ}) holds even for rather small values of R (Θ is the step function). Hence,
The overlap at the centre of the cavity is given by q_{c}(R)=p_{in}(R)q_{1}+p_{out}(R)q_{0}, where q_{1} is the selfoverlap of the cavity (which is not necessarily 1 at finite temperatures). Using equation (5), we finally obtain
At this point, some assumption about P(Y) needs to be made. An interesting parameterization is
(ν>0), which describes either a stiff distribution (small values of Y suppressed) at large ν or a soft one (small values of Y enhanced) for small ν. Y_{c} is the typical value of . The standard mosaic picture corresponds to . Equation (6) now reads, explicitly:
where the usual RFOT relation between ξ and Σ is recovered, ξ=(Y_{c}/T Σ^{*})^{1/(d−θ)}. As in the Adam–Gibbs treatment^{29}, the growth of the length is caused by the decreasing of the configurational entropy at low temperature. The expression (7) is the compressed exponential form (1) used above to fit our numerical data. Within this framework, the thermodynamic anomaly ζ=ν(d−θ) is directly related (at low temperatures) to the exponent ν describing the surface tension distribution.
A nontrivial P(Y) could arise for two main reasons. One is strong preasymptotic corrections to standard RFOT: although the surface tension between two typical states could be selfaveraging in the limit, fluctuations can be state dependent and important at finite R. This is often what happens for interfaces in random media (see the Supplementary Information). The second possibility is that the effective interface tension fluctuates even in the largeR limit. If the exponent θ turns out to be less than d−1, as suggested by phenomenological arguments in ref. 4, then the interface should be a highly fluctuating object. We find that ζ, and therefore ν, increases when T decreases, indicating that the crossover of q_{c}(R) becomes sharper at larger sizes (Fig. 1). This is compatible with a finitesizeeffect scenario. This behaviour is also expected within the RFOT scenario that predicts a vanishing surface tension at the modecoupling transition T_{MC}, which behaves as a spinodal point. Coherent amorphous order droplets should therefore be fractal around T_{MC} and compact below^{30}, which suggests an increase of the effective value of ν as T decreases. A firstprinciples RFOT computation of q_{c}(R) for the model we simulated would be very instrumental to clarify this issue.
We have developed a theoretical framework, based on a generalization of RFOT, that explains these results as a onestate to multistate transition governed by the surface tension distribution. The sharpening of the decay at low temperature corresponds to more and more regions developing a large surface tension. From a more general perspective, our numerical results strongly support a mosaic picture where the supercooled liquid is characterized by a ‘hidden’ static order on an increasingly larger scale on cooling. Although this is a natural consequence of RFOT, other theoretical approaches may also account for these phenomena at least on a qualitative level: in particular the frustrationlimited domain theory^{14}, the effective potential approach^{31} and, perhaps surprisingly, some kinetically constrained models (see ref. 20). The main difference is the physical origin of the growing static length scale: within RFOT and at variance with other approaches, it is closely linked to the decrease of the configurational entropy. Our work opens the way to a quantitative study of this issue and, hence, to a clearcut test of RFOT as a valid theory of the glass transition. From a more theoretical point of view, several crucial questions remain elusive: can an RFOTlike transition exist outside meanfield? How precisely can amorphous metastable states be defined? Does the notion of effective surface tension between these states make sense? A definitive test of the mosaic scenario requires finding a way to measure directly this surface tension and its distribution in the deeply supercooled phase. Work in this direction is in progress.
Methods
Simulation
We have studied the softsphere binary mixture^{24}, a fragile model glassformer. In addition to capturing the essential features of fragile glasses, this model can be thermalized below the modecoupling temperature with the swap Monte Carlo algorithm of ref. 32. Particles are of unit mass and belong to one of two species μ=1,2, present in equal amounts and interacting through a potential
where the radii σ_{μ} are fixed by the conditions σ_{2}/σ_{1}=1.2, (2σ_{1})^{3}+2(σ_{1}+σ_{2})^{3}+(2σ_{2})^{3}=4ℓ_{0}^{3} and ℓ_{0} is the unit of length. The particle density is ρ=N/V =l_{0}^{−3}. A smooth longrange cutoff is imposed setting v_{i j}(r)=B_{i j}(a−r)^{3}+C_{i j} for and v_{i j}(r)=C_{i j} for r>a, where a, B_{i j} and C_{i j} are fixed by requiring continuity up to the second derivative of v_{i j}(r). Temperature is measured in units of energy. To obtain the reference configurations, four to eight replicas of systems with N=2,048 or N=16,384 were equilibrated in a cubic box with periodic boundary conditions. Then the overlap was computed in systems with frozen boundaries and M mobile particles, with M=20, 30, 40, 50, 100, 150, 200, 300, 400, 800, 1,600, 3,200 mobile particles (1.684≤R≤9.142). The results are averaged over 8 to 32 outer states (reference configurations). Data were collected for at least 10 relaxation times (up to 10^{6} Monte Carlo steps), after discarding an initial portion of at least one relaxation time.
Overlap
To obtain equation (2), write the partition function for the mobile cavity surrounded by pinning state α,
so that
Introducing , the sum can be written
where in the last equality we have defined , and is the number of states with free energy f. Equation (2) follows approximating the integral over f with the saddlepoint method, which picks f=f^{*} as the dominant contribution. p_{α}(f) is the fraction of states γ (inside the cavity) with free energy f and effective interface tension , which we assume to be R independent (actually, some R dependence would not affect the result; the important requirement is that this dependence does not lead to values exponentially large or small in R).
Now the integral in equations (2) and (3) can be simplified using the saddlepoint method, which is a very good approximation even for rather small values of R. Because of the exponential term in R^{θ}, the integral is dominated by the lowest values of supported by the distribution p_{α}(f). There are two possibles cases. (1) If this function has a left edge, that is, it vanishes for <y_{α}, we find up to subleading terms: . Otherwise, (2) there are arbitrarily small effective tensions . Remarkably, in this case, depending on the form of p_{α}(f) at small , we can obtain a renormalization of θ. For instance, in the case p_{α}(f)≃exp(−c_{α}/^{a}), where a is a positive exponent, we find , where y′_{α} is a constant dependent on the outside state α and temperature and θ′=a θ/(a+1)<θ.
Comparison with the results of ref. 23
Following the same numeric protocol of this work, in ref. 23 the influence of the boundary conditions on the total overlap within the sphere was studied. It was observed that the decay of q_{tot}(R) is described sufficiently well within the singlestate framework and the standard RFOT scenario was ruled out. In this work, we show however that at low enough temperatures, neither the standard RFOT nor the singlestate scenario accounts for the local overlap data and a generalized RFOT theory is presented that is far more successful.
We might retrospectively wonder if the data in ref. 23 could have been used to discriminate between the onestate and the generalized RFOT scenarios. The answer is no. Allowing for an R dependence of q_{0} and q_{1} of the type suggested in ref. 23,
with x≡R/λ_{0;1}, the generalized RFOT prediction for q_{tot}(R) reads now:
Figure 3 shows that the generalized RFOT encoded in (8) and the onestate prediction given by the formula (6) of ref. 23 fit the total overlap data at T=0.203 at a comparable level of accuracy. Following Occam’s razor principle (‘entia non sunt multiplicanda praeter necessitatem’), the onestate theory should be chosen, which is the one with the smallest number of parameters.
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Acknowledgements
We thank C. Cammarota, L. A. Fernandez, G. Gradenigo, I. Giardina, A. Lefèvre, V. MartínMayor, A. Montanari, G. Parisi, D. Reichman, M. Tarzia and F. Zamponi for useful discussions. G.B. and J.P.B. are supported by ANR Grant DYNHET. T.S.G. thanks ECT^{*} and Dipartimento di Fisica, Universitá di Trento for hospitality and partial support and acknowledges partial support from CONICET and ANPCyT (Argentina) and ICTP (Trieste, Italy).
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Biroli, G., Bouchaud, JP., Cavagna, A. et al. Thermodynamic signature of growing amorphous order in glassforming liquids. Nature Phys 4, 771–775 (2008). https://doi.org/10.1038/nphys1050
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DOI: https://doi.org/10.1038/nphys1050
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