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.

The question of whether the dramatic slowing down of the dynamics of glass-forming liquids near the structural glass transition is caused by the growth of one or more correlation lengths has received much attention in recent years. Several proposals have been made for both static and dynamic length scales that may be responsible for the growth of timescales as the glass transition is approached. These proposals are critically examined with emphasis on the dynamic length scale associated with spatial heterogeneity of local dynamics and the static point-to-set or mosaic length scale of the random first-order transition theory of equilibrium glass transition. Available results for these length scales, obtained mostly from simulations, are summarized, and the relation of the growth of timescales near the glass transition with the growth of these length scales is examined. Some of the outstanding questions about length scales in glass-forming liquids are discussed, and studies in which these questions may be addressed are suggested.

The effect of finite temperature T and finite strain rateγ on the statistical physics of plastic deformations in amorphous solids made of N particles is investigated. We recognize three regimes of temperature where the statistics are qualitatively different. In the first regime the temperature is very low, T < Tcross(N ), and the strain is quasi-static. In this regime the elasto-plastic steady state exhibits highly correlated plastic events whose statistics are characterized by anomalous exponents. In the second regime Tcross(N ) < T < Tmax(γ) the system-size dependence of the stress fluctuations becomes normal, but the variance depends on the strain rate. The physical mechanism of the crossover is different for increasing temperature and increasing strain rate, since the plastic events are still dominated by the mechanical instabilities (seen as an eigenvalue of the Hessian matrix going to zero), and the effect of temperature is only to facilitate the transition. A third regime occurs above the second cross-over temperature Tmax(γ) where stress fluctuations become dominated by thermal noise. Throughout the paper we demonstrate that scaling concepts are highly relevant for the problem at hand, and finally we present a scaling theory that is able to collapse the data for all the values of temperatures and strain rates, providing us with a high degree of predictability.

Characterizing the glass state remains elusive since its distinction from a liquid state is not obvious. Glasses are liquids whose viscosity has increased so much that they cannot flow. Accordingly there have been many attempts to define a static length-scale associated with the dramatic slowing down of supercooled liquid with decreasing temperature. Here we present a simple method to extract the desired length-scale which is highly accessible both for experiments and for numerical simulations. The fundamental idea is that low lying vibrational frequencies come in two types, those related to elastic response and those determined by plastic instabilities. The minimal observed frequency is determined by one or the other, crossing at a typical length-scale which is growing with the approach of the glass transition. This length-scale characterizes the correlated disorder in the system, where on longer length-scales the details of the disorder become irrelevant, dominated by the Debye model of elastic modes.The phenomenon of the glass transition in which supercooled liquids exhibit a dramatic slowdown in their dynamics upon cooling is attracting a tremendous amount of effort. One of the thorny issues has to do with attempts to define and measure a static length-scale that will grow upon the approach to the glass transition. Several such attempts were published in the recent literature, using delicate measures of higher-order correlation functions [1,2], of the effects of boundary conditions, [3], point to set correlations [4], the scaling of the non affine displacement field [5] and patch correlation scale [6]. In this Letter we propose a different approach which yields a very natural static length that appears to fit the bill. An advantage of this approach is that it can be immediately and easily applied to both experimental [10] and numerical data (see below).Our starting point is the fact that at lower frequency the density of state (DOS) for a disorder system reflects the access of plastic modes which are not exhibited by the density of states of the corresponding crystalline materials [11,12]. This excess of modes is sometime referred to as the 'Boson Peak' [13]. It is most natural to discuss the 'modes' in terms of the eigenfunctions of the Hessianwhere U (r 1 , r 2 , . . . r N ) is the total energy of the system consisting of N particles, and {r k } N k=1 are their coordinates. Being real and symmetric, the Hessian is diagonalizable, and in all discussions below we refer to this matrix with the Goldstone modes (zero modes due to symmetries) being pruned out. Recently we had progress in understanding the analytic form of the density of states of generic glassy systems [14]. The eigenvalues appear in two distinct families, one corresponding to the Debye model of an elastic body, while the density of access plastic modes could be approximated aswhere the pre-factor B(T ) being strongly dependent on temperature and the exponent θ being weakly dependent on temperature. This dependence is a partial measure of the de...