For amorphous solids, it has been intensely debated whether the traditional view on solids, in terms of the ground state and harmonic low energy excitations on top of it, such as phonons, is still valid. Recent theoretical developments of amorphous solids revealed the possibility of unexpectedly complex free-energy landscapes where the simple harmonic picture breaks down. Here we demonstrate that standard rheological techniques can be used as powerful tools to examine nontrivial consequences of such complex free-energy landscapes. By extensive numerical simulations on a hard sphere glass under quasistatic shear at finite temperatures, we show that above the so-called Gardner transition density, the elasticity breaks down, the stress relaxation exhibits slow, and ageing dynamics and the apparent shear modulus becomes protocol-dependent. Being designed to be reproducible in laboratories, our approach may trigger explorations of the complex free-energy landscapes of a large variety of amorphous materials.
Amorphous and crystalline solids have very different behaviours under external perturbations, especially rheological properties under shear deformations1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18. It is well known that by increasing the shear strain, a crystal displays a linear elastic response, followed by plastic deformation and yielding. However, experiments and numerical simulations show that this picture breaks down for amorphous solids, such as glasses2,5,6,7,8,13,16,18,19, granular matter3,11,15 and foams4, where the elastic behaviour is mixed with plastic events. Such plastic events cause sudden drops in stress–strain curves, and are sometimes referred to as crackling noise20, due to their similarities to avalanches in earthquakes. An apparent shear modulus or rigidity μ, which is the ratio between the stress and strain, can be nevertheless defined and measured. Experiments on glassy emulsion systems21,22 show that μ scales linearly μ∼P with the pressure P both below and above the jamming density, while harmonic treatments predict μ∼P1.5 (below)23 and μ∼P0.5 (above)24, respectively. These contradictions reveal that amorphous solids can be strikingly softer than purely harmonic solids like crystals, even at sufficiently low temperatures where the harmonic expansion was conventionally expected to be valid.
On the theoretical side, the mean-field theory based on the exact solution in the large dimensional limit of the hard sphere glass has brought a more accurate and comprehensive picture beyond the harmonic description10,14,25,26,27,28,29,30. The main outcome is the prediction of a Gardner transition (see Fig. 1a) that divides the classical amorphous phase into two: in the stable phase (or normal phase), the state is confined in one of the simple smooth basins on the free-energy landscape; once the system is compressed above the Gardner transition density G (or is cooled down below the Gardner transition temperature TG), the simple glass basin splits into a fractal hierarchy of subbasins and the glass state becomes marginally stable. Although similar ideas of complex energy landscapes have been conceived phenomenologically in earlier works (see ref. 31 and references therein), the mean-field theory gives a firmer first principle ground for such a picture, with falsifiable predictions. In particular, the theory predicts that the elastic anomalies and nontrivial rheology should only appear in the marginally stable phase (or Gardner phase)10,14 that lies deep inside the glassy phase. However, the mean-field theory is exact only in the large dimensional limit, and its relevance in real systems is far from obvious. Here we test the theoretical proposal of the nontrivial rheology in physically relevant dimensions d=2 and 3 and compare quantitatively the theoretical predictions with our numerical data.
We design laboratory-reproducible rheological protocols to examine the signatures of the intriguing complex free-energy landscape. Our protocols are applied on densely packed hard spheres, a simple and representative glass-forming model. Our result shows the anticipated anomalous rheology emerging at the Gardner transition that turns out to be strikingly similar to the dynamical responses of spin glasses to an external magnetic field32,33. The evidence of a complex free-energy landscape in the Gardner phase is consistent with a previous numerical study34 where particles’ vibrational dynamics is analysed. That approach has been used in a recent experiment of an agitated granular system35. However, generalizing the method to other systems, such as molecular glasses, may not be easy due to the difficulty of tracking trajectories of individual particles. The approach proposed in this study overcomes this problem, since it requires no microscopic information, but only the standard macroscopic rheological measurements (the shear stress and strain) that are well accessible in many experimental systems, including molecular and metallic glasses, polymers and colloids. In the present paper, however, we do not attempt to judge whether the Gardner transition survives in finite dimensional systems as a sharp phase transition or becomes a crossover (in the thermodynamic limit), but rather we aim to explore the possibilities to observe its nontrivial signatures in experimentally feasible length/timescales.
Preparation of stable glasses
To avoid crystallization, we work on a polydisperse mixture of hard spheres whose diameters are distributed according to a probability distribution P(D)∼D−3, for Dmin≤D<Dmin/0.45 (refs 34, 36) (see Supplementary Note 1). A glass is typically obtained by a slow compression (or cooling) annealing from a dilute state, where it falls out of equilibrium at the compression (or cooling) rate-dependent glass transition density g (or glass transition temperature Tg). Since we choose hard spheres as our working system, the density is the control parameter.
We design a numerical protocol to mimic a simple shear experiment of deeply annealed glasses (see Fig. 1). Our protocol includes three steps. We first use the swap algorithm35,36 to prepare a well-equilibrated, supercooled-liquid configuration at various densities g (see Supplementary Methods and Supplementary Fig. 1). The algorithm combines the Lubachevsky–Stillinger algorithm37 that consists of standard event-driven molecular dynamics (MD) and slow compression, with Monte Carlo swaps of particle diameters. The MD time is expressed in units of , where the particle mass m and mean diameter , as well as the inverse temperature β, are all set to unity. In other words, a particle travels over a distance of the order of the diameter within a unit MD time. From the thermodynamic point of view, the system is still in the liquid but we work at density g sufficiently above the mode-coupling theory (MCT) crossover density d. Then, once we switch off the particle swapping and return to the natural dynamics simulated by MD, the α-relaxation time has become much larger than our MD simulation timescales so that the system behaves essentially as a solid. This glass is thus ultrastable, in a sense similar to those obtained by vapour deposition experiments38,39,40. At a given density g, we prepare many of such equilibrated configurations that are statistically independent from each other, and we call them samples in the following.
Second, subsequently the equilibrated configuration is compressed up to a target density with a compression rate δg=10−3. From a single sample that is a starting equilibrated configuration at g, we generate an ensemble of compressed glasses at , obtained by choosing statistically independent initial particle velocities drawn from the Maxwell–Boltzmann distribution. We call each of such compressed glasses as a realization in the following. These realizations are out of equilibrium, since they no longer follow the liquid equation of state (EOS), but we consider that they remain in restricted equilibrium29 for <G, that is, they are equilibrated within the given glass state determined by the sample. The MD preserves the kinetic energy so that the system remains at the unit temperature throughout our simulations. The typical scale of the vibrations of the particles within the glass states depends on . For instance, it varies from 10−1 to 10−2 for =0.645 to =0.688 (for g=0.643, see Fig. 2 of ref. 34), so that particles make 10 to 102 collisions within a unit MD time.
Third, for a given realization, a simple shear is applied. The simple shear is modelled by an affine deformation of the x-coordinates of all particles, xi→xi+γzi, under the Lees–Edwards boundary condition41 with fixed system volume. The shear strain is increased quasistatically with a small constant shear rate , such that the shear rate dependence is negligible in the regime <G (see Supplementary Fig. 6 for a discussion on the dependence), and the shear stress Σ is measured at different γ. The shear stress Σ and the pressure P are both calculated from interparticle interactions due to collisions between hard sphere particles. For convenience, we introduce reduced pressure p=βP/ρ and reduced stress σ=βΣ/ρ, where ρ is the number density of the particles (see Supplementary Note 1). Note that as the pressure, the shear stress is entirely due to momentum exchanges between the particles so that the rigidity is purely entropic in hard sphere systems. Furthermore, because shear stress and pressure have the same physical dimension, it is convenient to introduce a rescaled stress .
Breakdown of elasticity
Figure 1 shows the phase diagram for our polydisperse hard sphere model, and typical stress–strain curves of individual realizations in different density regimes. In the stable glass phase g<<G (Fig. 1b), the stress–strain curve shows a smooth linear (harmonic) response regime at small γ, followed by a sharp drop of the stress σ, signalling the yielding of the system. At yielding, a system-wide shear band emerges (see Fig. 1c), and the system is driven out of a free-energy metastable glass basin. After yielding, the system enters a steady flow state, similar to those observed in athermal amorphous solids under quasistatic shear6,42. In the Gardner phase G<<J, where J is the jamming density, the harmonic response is punctuated by mesoscopic plastic events (MPEs) that can happen at very small γ (see Fig. 1d). These MPEs correspond to sudden avalanche-like heterogeneous rearrangements of particle positions without formation of band-like patterns (see Fig. 1e). Similar MPEs have been observed in quasistatic shear simulations at zero temperatures2,6, but our simulations are performed at finite temperatures. Note that the details of the plastic events, including the locations of yielding, jamming and MPEs, depend on the samples (see Supplementary Fig. 8) and realizations (see Supplementary Fig. 7). For the behaviour of the stress–strain curves averaged over many realizations and samples, see Supplementary Figs 2–4, as well as Supplementary Notes 2 and 3.
For large , the stress σ grows dramatically at large γ, and appears to diverge (see Fig. 1d). This shear jamming phenomenon is due to the dilatancy effect of hard sphere glasses under shear: the pressure p increases with γ when the system volume is fixed. Note that if p is kept as a constant when γ is increased, then the volume expands due to the dilatancy effect. In that case, shear jamming does not appear and shear yielding is recovered (see Supplementary Fig. 5). While the switching from shear yielding to shear jamming with increasing is not a consequence of the Gardner transition, it implies that the system is trapped more deeply in the metastable basin, and that the activated barrier crossing between metastable basins becomes forbidden. However, the emergence of subbasins in the Gardner phase28,34 implies that even though the usual relaxation (α-relaxation) is frozen, an additional slow dynamics may appear. This aspect is explored below.
Ageing and slow dynamics
We next show that in the Gardner phase, the relaxation of shear stress becomes complicated, accompanied by ageing and a slow dynamics. Due to the similarity between the Gardner transition and the spin glass transition, here it is very useful to first recall what happens in spin glasses that are essentially disordered and highly frustrated magnets43,44. The mean-field spin glass theory has suggested complex free-energy landscapes of spin glasses manifested as continuous replica symmetry breaking45, much as what happens in the Gardner phase of hard sphere glasses27,28. Remarkably, this feature is predicted to have a reflection in the dynamics, resulting in nontrivial dynamical responses to external magnetic field, and ageing effects in the relaxation of magnetization46,47,48. In experiments, the simplest approach to examine the intriguing features of the dynamics is a combination of the so-called zero-field cooling (zfc) and field cooling (fc) protocols. In the zfc protocol, one cools a spin glass sample from a high temperature in the paramagnetic phase down to a target temperature T, where a magnetic field h is switched on and one measures the increase of the magnetization. In the fc protocol, one first switches on the magnetic field h, and then subsequently cools the system down to the target temperature T and measures the remanent magnetization. The key point is that, in the two protocols, the order of cooling and switching on of the magnetic field is reversed. In such experiments49,50, the magnetizations observed in the zfc/fc protocols are the same if the working temperature T is higher than the spin glass transition temperature, while the fc magnetization becomes larger than the zfc magnetization if T is lower than the spin glass transition temperature. The anomaly, that is, the difference between the zfc and fc magnetizations, is naturally explained by the mean-field theory45. Furthermore, examinations of the ageing effects by these protocols give detailed information about the complex free-energy landscape32,33,46,47,48.
It has been pointed out theoretically that the shear on structural glasses plays a very similar role as the magnetic field on spin glasses9,51, and that the relaxation of the shear stress should also reflect the complex free-energy landscape encoded by the continuous replica symmetry breaking solution in the Gardner phase (see Supplementary Fig. 9, and Fig. 2 of ref. 10). The shear strain and stress in structural glasses correspond to the magnetic field and magnetization in spin glasses, respectively. Furthermore, apparently compression in hard sphere glasses corresponds to cooling in spin glasses. Therefore, inspired by the zfc/fc experiments in spin glasses, we design two distinct protocols that are combinations of compression and shear exerted in reversed orders. In the zero-field compression (ZFC) protocol, we first compress the configuration from g to , and set the time to zero. We then wait for time tw before a shear strain γ is applied instantaneously (see Supplementary Methods), and measure the relaxation of the stress σZFC(t, tw) as a function of the time τ=t−tw elapsed after switching on the strain. On the other hand, in the field compression (FC) protocol, we first apply an instantaneous increment of shear strain at the initial density g, compress the configuration to and set the time to zero. Then, we measure the relaxation of the stress σFC(t) as the function of the elapsed time t.
For <G, no ageing effect is observed, and the dynamics is fast. The σZFC(t, tw) is stationary or time translationally invariant, that is, σZFC(τ, tw)=σZFC(τ), depending only on the time difference τ=t−tw but not on the waiting time tw (see Fig. 2). After a timescale τb corresponding to the ballistic motions of particles34, the ZFC stress σZFC(τ, tw) converges quickly to σFC(t) that is almost a constant in time.
In contrast, for >G, σZFC(τ, tw) displays strong tw-dependent ageing effects manifesting the out-of-equilibrium nature of the system, as well as a slow dynamics. In such a situation, different large time limits can emerge depending on the order of τ→∞ and tw→∞ (ref. 52). An important feature that can be seen in Fig. 2 is that σZFC(τ, tw) exhibits a plateau suggesting the existence of a large time limit σZFC≡limτ→∞ →∞ σ ZFC(τ, tw) where tw→∞ is taken before τ→∞. On the other hand, σFC(t) is again essentially constant in time t (for t>τb) and we shall denote it as σFC. In the reversed order of the large time limits, we expect that the ZFC shear stress decays to the FC one, →∞ limτ→∞σ ZFC(τ, tw)=σ FC. However, the convergence becomes slower as tw increases, and its corresponding timescale could be beyond the simulation time window, as shown in the case of Fig. 2.
Apparently σZFC is larger than σFC when >G, implying the ergodicity breaking. The ageing effect and the slowing down of dynamics show the similarities between the Gardner transition and the liquid–glass transition, demonstrating that the Gardner transition could be considered as a ‘glass transition within the glass phase’ (see also Supplementary Note 3). In a sharp contrast, because the Gardner transition is absent in a crystal, its shear stress relaxes faster when increases, and no ageing is present.
Protocol-dependent shear modulus
The above observation suggests that the linear shear moduli measured by the two protocols should be distinct in the Gardner phase. We determine the apparent shear modulus μ as μZFC=(σZFC−σ0)/γ and μFC=(σFC−σ0)/γ, where σ0 is the remanent shear stress at before γ is applied. The shear strain is increased quasistatically with rate up to a predetermined small target γ. The shear stress is measured at τ=1 after waiting for tw=10. Details on the time and g dependences of the shear modulus are discussed in Supplementary Figs 10, 13 and 14.
Figure 3 shows that while μZFC and μFC are indistinguishable in the stable glass phase <G (or p<pG), they become clearly distinct in the Gardner phase >G (or p>pG). For a similar result of a two-dimensional bidisperse hard disk model, see Supplementary Fig. 16. This behaviour of shear modulus is a consequence of the time dynamics of the shear stress illustrated in Fig. 2: at the timescales used to measure the shear modulus (τ=1 and tw=10), the two shear stresses σZFC and σFC have converged to the same value for <G, but remain different for >G. The bifurcation point determines the Gardner transition threshold G (or pG). Within the numerical accuracy, the G determined from this approach is fully consistent with the previous estimate based on particles’ vibrational motions and caging order parameters34. To further test this result, we perform detailed analysis on its dependence on the number of particles N and the shear strain γ, as discussed below.
We find no appreciable finite size effects for μFC (see Fig. 3a) that is in contrast to the observation in nonequilibrated systems, where μFC decreases to zero in the thermodynamic limit13. It suggests that preparing deeply equilibrium configurations is the key to observe the nonvanishing μFC. While the shear moduli measured around the Gardner transition, and therefore the determination of G, are N independent, stronger finite size effects are observed for μZFC at large p near the jamming limit: μZFC is lower in larger systems, suggesting a stronger nonlinear effect. Nevertheless, the data of μZFC(p), with a fixed γ, appear to converge for N≳2,000, confirming that μZFC(p) and μFC(p) remain distinguishable in the thermodynamic limit, for >G.
Regarding the γ-dependence, Fig. 3b shows that, within the numerical accuracy, μFC is independent of γ, as long as γ is sufficiently small. On the other hand, for >G and a given N, μZFC slightly increases with decreasing γ. This result shows that in the Gardner phase, the nonlinear effect on μZFC remains even for very small γ, consistent with the observation of elasticity breakdown in Fig. 1. Such nonlinear effects are observed for any N studied (see Supplementary Fig. 12), and we expect that in the thermodynamic limit N→∞, a pure linear behaviour of μZFC can only exist in the limit γ→0 (ref. 53). The vanishing of the pure elastic regime distinguishes the Gardner phase from the normal glass and crystalline phases. For a more detailed discussion on how the shear moduli depend on the strain γ, the particle number N, the initial density g and the waiting time tw, see Supplementary Note 4.
For >g, the mean-field theory predicts two power-law scalings in the large p limit10: with κ=1.41574..., and μFC∼p. The first scaling has also been derived semiempirically by an independent approach54. We find good agreement between the theory and simulation on the scaling of μFC (see Fig. 3). For μZFC, a noticeable discrepancy is observed in the limit of large N for a fixed finite γ (Fig. 3a), but the discrepancy decreases when γ→0 for a fixed N (Fig. 3b), or when N is decreased for a fixed γ (Fig. 3a). This is because the mean-field μZFC is obtained in the pure linear response limit γ→0, while the nonlinear effect caused by MPEs would increase with γ and N, as discussed above. The scaling μFC∼p is consistent with the experimental observation in emulsions21,22. Considering the experimental system is possibly not deeply equilibrated, we expect that the relaxation of experimental μ(t) is sufficiently fast, and the measurement was performed in the long time limit μ(t→∞)→μFC (see the discussion of Fig. 2).
Interpretation of results
The Gardner transition is a consequence of the split of glass basins in the phase space28, and the split of particle cages in the real space (see Fig. 4). The schematic plot of the free-energy F as a function of γ in Fig. 4 illustrates how a glass basin splits into many subbasins once the system is compressed above G. Here we interpret our results based on this free-energy landscape viewpoint. First, in the ZFC protocol, the system intends to remain in one of the subbasins after compression (note that different realizations may end up in different subbasins), but as γ increases in a quasistatic shear procedure (Fig. 1), it may become unstable where the shear stress drops abruptly, resulting in a MPE. The MPE could be interpreted as shear-induced barrier crossing between subbasins, analogous to the barrier crossing between basins in a yielding event. Second, if γ is fixed, the shear stress relaxes with time, and according to the Arrhenius law, the emergence of barriers between subbasins would result in a slowing down of the relaxation dynamics with (Fig. 2). The appearance of ageing further reveals the emergence of complex structures within a basin, similar to the mechanism of ageing in the glass transition52. Third, because in the FC protocol, the system can overcome the subbasin barriers, the μFC always corresponds to the second-order curvature of basins, rather than that of subbasins as in the μZFC case. This results in μFC<μZFC for the regime >G as observed in Fig. 3. Note that according to Fig. 4e, one should obtain a shear modulus close to μZFC if an additional strain is applied after FC, as confirmed in Supplementary Fig. 15. On the other hand, no basin split occurs and therefore two protocols are equivalent in the stable phase <G. A previous study13 has shown that μZFC=μFC for crystals. The similarity between crystals and stable glasses further confirms that their free-energy basins are similarly structureless.
We wish to stress that our data cannot exclude that the Gardner transition becomes just a crossover in finite dimensions, such that no real phase transition exists. Yet, irrespective of the sharpness of the Gardner transition, we rationalize here in a unified framework all the observations obtained on the rheological behaviour of the simple hard sphere glass, and find quantitatively reasonable agreement between the theory and simulations. Thus, even if the Gardner transition is not sharp in the thermodynamic limit, for accessible sizes in numerical simulations, and likely for those in experiments as well35, a behaviour reminiscent of the transition can be clearly observed.
Finally, we make remarks on experimental consequences. It is an intriguing question to clarify whether the phase diagram presented in Fig. 1a is generic in a wide range of amorphous solids, ranging from different kinds of glasses to soft matter such as colloids (one can choose to change the temperature or pressure as the control parameter depending on specific systems). The crucial point is to keep track of the dynamical effects that might have been overlooked in some previous experiments, for the following two reasons. First, in reasonably stabilized dense systems, the liquid EOS (green line in Fig. 1a) and the Gardner line (red line) becomes separated enough, so that the liquid dynamics (α-relaxation) and the intriguing internal glassy dynamics (β-relaxation induced by the Gardner transition) can be well separated in timescales. In this respect, recently developed experimental techniques, such as the vapour deposition38,39,40 and the high pressure path55, or the use of sufficiently old natural glasses56, would provide ideal settings. If such an ideal setting is not possible, one could freeze the α-relaxation out of the experimental time window, by working at sufficiently low temperatures or high densities. The second reason is that by experimentally studying the ageing effects due to the internal dynamics of the amorphous solids, the complexity of the free-energy landscape could become manifested as we demonstrated in the present paper.
The data that support the findings of this study are available in Osaka University Knowledge Archive (OUKA) with the identifier. http://hdl.handle.net/11094/59688.
How to cite this article: Jin, Y. & Yoshino, H. Exploring the complex free-energy landscape of the simplest glass by rheology. Nat. Commun. 8, 14935 doi: 10.1038/ncomms14935 (2017).
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We thank Giulio Biroli, Ludovic Berthier, Patrick Charbonneau, Olivier Dauchot, Anaël Lematre, Corrado Rainone and Pierfrancesco Urbani for useful discussions, and especially Francesco Zamponi, Daijyu Nakayama, and Satoshi Okamura for many stimulating interactions. This work was supported by KAKENHI (No. 25103005 ‘Fluctuation & Structure’) from MEXT, Japan. We also wish to thank financial support from Cybermedia Center, Osaka University. The computations were performed using Research Center for Computational Science, Okazaki, Japan, and the computing facilities in the Cybermedia Center, Osaka Univerisity.
The authors declare no competing financial interests.
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Jin, Y., Yoshino, H. Exploring the complex free-energy landscape of the simplest glass by rheology. Nat Commun 8, 14935 (2017). https://doi.org/10.1038/ncomms14935