Credit: Pierfrancesco Urbani

How can an amorphous material be rigid? Glass – the prototypical and ubiquitous amorphous solid – inhabits an incredibly ramified and complex energy landscape, which presumably underlies its rigidity. But how? Dealing with so many relevant energy minima and the ensuing far-from-equilibrium dynamics has emerged as one of the central problems in statistical physics. Tackling it requires new tools and concepts.

This collaboration, addressing such fundamental issues as disorder, nonlinear response and far-from-equilibrium dynamics, builds upon three powerful approaches: the study of marginal stability at jamming, the mean-field theory of glasses in infinite dimension, and the dynamics of systems in complex landscapes. The convergence of recent breakthroughs in these areas generates a unique opportunity to come to grips with three outstanding and intimately related challenges. This is our strategy for cracking the glass problem.  Tackling the prodigious intellectual challenges of this Collaboration requires assembling an international group of scientists with expertise in an array of techniques and topics.  In order to take full advantage of this diverse talent, we have agreed on the free exchange of ideas and results.  We work together on topics of mutual interest in a spirit of cooperation rather than competition.

1. Marginal stability

Considerations of marginal stability first emerged in the study of athermal granular materials. At their jamming point, these systems are marginally stable; the number of contacts between particles is precisely the minimal number required for mechanical stability. This observation has important consequences. In particular, near the jamming transition soft system-spanning excitations appear, and these give rise to materials with properties vastly different from those of conventional solids, such as crystals.

Marginal stability also independently (and surprisingly) emerged in the mean-field theory of glasses in infinite dimensions. In that context, marginality is associated with the emergence of flat directions in phase space, an infinite correlation length, and a diverging susceptibility. These features are characteristic of a second-order phase transition, which in this context is known as a Gardner transition. An important prediction of mean-field theory is that marginal stability persists away from jamming, and thus gives rise to a Gardner glassy phase with exotic properties, such as diverging fluctuations of linear and nonlinear elastic moduli.

The main goals of the collaboration on this theme are: to relate the two notions of marginal stability; to understand to what extent marginal stability describes conventional glasses, which are far from any jamming transition; and to describe how the soft excitations associated with marginal stability affect macroscopic properties, such as glassy rheology.

Figure (from Brito and Wyart, J. Chem. Phys. 2009): snapshot of the mechanical force network in a particle system at jamming, showing the extreme heterogeneity that accompanies marginal stability.
2. Dynamics and algorithms

The glass transition occurs when a liquid spontaneously forms an amorphous solid upon cooling. Characterizing the complex relaxation pathways that glass formers use to explore their rough energy landscape is fundamental to understanding the breaking of ergodicity that accompanies this transition. A key goal of the collaboration is to unify the amorphous states at the landscape minima with the dynamics that connects them. We will thus combine the description of glassy dynamics in large-dimensional systems with a detailed description of particle motion within model liquids. This will give rise to a general theory of glassy dynamics that explains universal phenomena, such as the super-Arrhenius dynamical slowdown of glass formers as well as glass plasticity and aging. Both novel analytical and numerical techniques are needed to address this problem.

The collaboration has developed a numerical scheme that takes advantage of the flexibility offered by computer simulations for sampling configuration space to bypass and overtake experimental dynamics. We are now in a position to study glass configurations that would have taken thousands or more years to prepare in standard experiments. These system will allow us to gain unprecedented insight into the structure and dynamics of amorphous materials.

We are also developing a theory of activated dynamical processes in glasses. In contrast to simpler cases of activated dynamics – such as nucleation at first-order phase transitions, tunneling between degenerate vacua in quantum field theory, and the Freidlin-Wentzell theory – glasses have an exponential (in the size N of the system) number of paths connecting an exponential number of metastable states. Identifying the dynamically relevant set of paths requires fully grasping the balance between the entropy of paths and their energetic cost. The collaboration has thus embarked on a journey to elucidate the perturbative and nonperturbative corrections to mean-field glassy dynamics. We will identify the typical energy barriers overcome during the dynamics and characterize their real-space counterparts.

Figure (credit: Chiara Cammarota): A schematic rugged energy landscape with a multitude of energy minima, maxima, and saddles. Arrows denote some of the possible relaxation pathways.
3. Rheology

Glassy rheology, which describes how amorphous solids respond to mechanical perturbations, is surprisingly far from theoretical control. This severely limits our ability to design disordered materials and to predict the behavior of systems as varied as polymer melts and earthquake faults. Recent progress by the collaboration in obtaining analytical solutions for mean-field models of glassy materials under external shear and in developing numerical methods that identify the structural excitations that facilitate flow, however, suggest that predictive rheological descriptions are within reach.

To achieve this goal, the collaboration is focusing on three main directions. The first goal is to specify the nature and universality class of the yielding transition. Yielding is reminiscent both of a depinning transition in systems with quenched disorder and of a system approaching a spinodal. Using analytical tools we have recently developed, we will study the spinodal of disordered models with long-range interactions. Using simulations, we will also quantify how less and more stable regions interact in amorphous solids.

The second goal is to study the nature of shear-induced excitations. There presumably exists a crossover from mean-field-like excitations near jamming to localized shear transformation zones away from it. But how is this transition reflected in the excitation modes of the system? Can schematic models predict the anomalous scaling of the density of states? The similarity between the modes excited by shear with those that give rise to the low-temperature calorimetric anomalies of glasses will also be examined.

The third goal is to develop new continuum models that quantitatively predict the rheology of glasses from first principles. Existing models are phenomenological with no first-principle motivation or explicit connection to microscopic glass properties, but ongoing theoretical advances should provide better theoretical control. We will use these improved models to relate shear-activated excitations to strain localization, and predict the statistics of avalanches. Scaling arguments for the length scales that characterize these processes are also examined.

Figure (from Widmer-Cooper et al., Nature Physics, 2008): Correlations between the mobility of particles in the liquid (circles) and their participation in localised soft modes at jamming (red zones) suggest the presence of structure-dynamics coupling.