The glass transition, whereby liquids transform into amorphous solids at low temperatures, is a subject of intense research despite decades of investigation. Explaining the enormous increase in relaxation times of a liquid upon supercooling is essential for understanding the glass transition. Although many theories, such as the Adam-Gibbs theory, have sought to relate growing relaxation times to length scales associated with spatial correlations in liquid structure or motion of molecules, the role of length scales in glassy dynamics is not well established. Recent studies of spatially correlated rearrangements of molecules leading to structural relaxation, termed ''spatially heterogeneous dynamics,'' provide fresh impetus in this direction. A powerful approach to extract length scales in critical phenomena is finite-size scaling, wherein a system is studied for sizes traversing the length scales of interest. We perform finite-size scaling for a realistic glass-former, using computer simulations, to evaluate the length scale associated with spatially heterogeneous dynamics, which grows as temperature decreases. However, relaxation times that also grow with decreasing temperature do not exhibit standard finite-size scaling with this length. We show that relaxation times are instead determined, for all studied system sizes and temperatures, by configurational entropy, in accordance with the Adam-Gibbs relation, but in disagreement with theoretical expectations based on spin-glass models that configurational entropy is not relevant at temperatures substantially above the critical temperature of mode-coupling theory. Our results provide new insights into the dynamics of glass-forming liquids and pose serious challenges to existing theoretical descriptions.correlation length ͉ dynamic heterogeneity ͉ finite-size scaling ͉ glass transition ͉ relaxation time M ost approaches to understanding the glass transition and slow dynamics in glass formers (1-10) are based on the intuitive picture that the movement of their constituent particles (atoms, molecules, polymers) requires progressively more cooperative rearrangement of groups of particles as temperature decreases (or density increases). Structural relaxation becomes slow because the concerted motion of many particles is infrequent. Intuitively, the size of such ''cooperatively rearranging regions'' (CRR) is expected to increase with decreasing temperature. Thus, the above picture naturally involves the notion of a growing length scale, albeit implicitly in most descriptions. The notion of such a length scale, related to the configurational entropy S c (see Methods), forms the basis of rationalizing (1, 6, 7) the celebrated Adam-Gibbs (AG) relation (1) between the relaxation time and S c .More recently, a number of theoretical approaches have explored the relevance of a growing length scale to dynamical slow down (5,7,9). A specific motivation for some of these approaches arises from the study of heterogeneous dynamics in glass formers (11)(12)(13)(14). In particular, c...

The art of making structural, polymeric and metallic glasses is rapidly developing with many applications. A limitation to their use is their mechanical stability: under increasing external strain all amorphous solids respond elastically to small strains but have a finite yield stress which cannot be exceeded without effecting a plastic response which typically leads to mechanical failure. Understanding this is crucial for assessing the risk of failure of glassy materials under mechanical loads. Here we show that the statistics of the energy barriers ∆E that need to be surmounted changes from a probability distribution function (pdf) that goes smoothly to zero to a pdf which is finite at ∆E = 0. This fundamental change implies a dramatic transition in the mechanical stability properties with respect to external strain. We derive exact results for the scaling exponents that characterize the magnitudes of average energy and stress drops in plastic events as a function of system size.In this Letter we focus on the statistical physics of the yielding transition at very low temperatures and quasistatic external straining conditions, (the so-called athermal quasi-static or AQS limit) where very precise simulation results exist for the dependence of energy and stress drops in plastic events as a function of system size [1]. Consider Fig. 1 which demonstrates the nature of the yielding transition. We plot here the conditional mean energy drop in a plastic event as a function of the external strain γ for two-dimensional systems (see below) consisting of N particles, with N ranging between 484 and 20164. To read this figure properly, one should understand that in some realizations there are no plastic events at all at a given external stain. What is measured here is the size of the mean energy drop if such a drop happened at an external strain value between γ and γ + dγ, averaged over numerous realizations of the random structure of the system (see below for details). We see that in the early stages of the loading, the plastic events are localized and the amount of energy released in events is system-size independent. This is followed by a smooth rise in these curves, showing an increasingly sharper transition to the plastic flow state in which the plastic events become non-localized avalanches whose total energy release increases with the system size. This very interesting system size dependence will be quantified below. We note in passing that the stress itself cannot be a proper order parameter; states with the same stress level (shown for example in Fig. 1 as two magenta circles) have very different conditional mean plastic energy drops. Here we explore the statistical physics that is responsible for the difference between these iso-stress states, which also have very similar potential energy and pressure. We point out that the precise nature of this strain-induced transition from the solid-like jammed state to the steady flow state, where the plastic flow events resemble liquidlike dynamics, is still unclear. Althoug...

The breakdown of the Stokes-Einstein (SE) relation between diffusivity and viscosity at low temperatures is considered to be one of the hallmarks of glassy dynamics in liquids. Theoretical analyses relate this breakdown with the presence of heterogeneous dynamics, and by extension, with the fragility of glass formers. We perform an investigation of the breakdown of the SE relation in 2, 3, and 4 dimensions in order to understand these interrelations. Results from simulations of model glass formers show that the degree of the breakdown of the SE relation decreases with increasing spatial dimensionality. The breakdown itself can be rationalized via the difference between the activation free energies for diffusivity and viscosity (or relaxation times) in the Adam-Gibbs relation in three and four dimensions. The behavior in two dimensions also can be understood in terms of a generalized Adam-Gibbs relation that is observed in previous work. We calculate various measures of heterogeneity of dynamics and find that the degree of the SE breakdown and measures of heterogeneity of dynamics are generally well correlated but with some exceptions. The two-dimensional systems we study show deviations from the pattern of behavior of the three- and four-dimensional systems both at high and low temperatures. The fragility of the studied liquids is found to increase with spatial dimensionality, contrary to the expectation based on the association of fragility with heterogeneous dynamics.

We derive expressions for the lowest nonlinear elastic constants of amorphous solids in athermal conditions (up to third order), in terms of the interaction potential between the constituent particles. The effect of these constants cannot be disregarded when amorphous solids undergo instabilities such as plastic flow or fracture in the athermal limit; in such situations the elastic response increases enormously, bringing the system much beyond the linear regime. We demonstrate that the existing theory of thermal nonlinear elastic constants converges to our expressions in the limit of zero temperature. We motivate the calculation by discussing two examples in which these nonlinear elastic constants play a crucial role in the context of elastoplasticity of amorphous solids. The first example is the plasticity-induced memory that is typical to amorphous solids (giving rise to the Bauschinger effect). The second example is how to predict the next plastic event from knowledge of the nonlinear elastic constants. Using the results of our calculations we derive a simple differential equation for the lowest eigenvalue of the Hessian matrix in the external strain near mechanical instabilities; this equation predicts how the eigenvalue vanishes at the mechanical instability and the value of the strain where the mechanical instability takes place.

We study the elastic theory of amorphous solids made of particles with finite range interactions in the thermodynamic limit. For the elastic theory to exist, one requires all the elastic coefficients, linear and nonlinear, to attain a finite thermodynamic limit. We show that for such systems the existence of nonaffine mechanical responses results in anomalous fluctuations of all the nonlinear coefficients of the elastic theory. While the shear modulus exists, the first nonlinear coefficient B(2) has anomalous fluctuations and the second nonlinear coefficient B(3) and all the higher order coefficients (which are nonzero by symmetry) diverge in the thermodynamic limit. These results call into question the existence of elasticity (or solidity) of amorphous solids at finite strains, even at zero temperature. We discuss the physical meaning of these results and propose that in these systems elasticity can never be decoupled from plasticity: the nonlinear response must be very substantially plastic.