Glasses are amorphous solids whose constituent particles are caged by their neighbours and thus cannot flow. This sluggishness is often ascribed to the free energy landscape containing multiple minima (basins) separated by high barriers. Here we show, using theory and numerical simulation, that the landscape is much rougher than is classically assumed. Deep in the glass, it undergoes a 'roughness transition' to fractal basins, which brings about isostaticity and marginal stability on approaching jamming. Critical exponents for the basin width, the weak force distribution and the spatial spread of quasi-contacts near jamming can be analytically determined. Their value is found to be compatible with numerical observations. This advance incorporates the jamming transition of granular materials into the framework of glass theory. Because temperature and pressure control what features of the landscape are experienced, glass mechanics and transport are expected to reflect the features of the topology we discuss here.

Despite decades of work, gaining a first-principle understanding of amorphous materials remains an extremely challenging problem. However, recent theoretical breakthroughs have led to the formulation of an exact solution of a microscopic model in the mean-field limit of infinite spatial dimension, and numerical simulations have remarkably confirmed the dimensional robustness of some of the predictions. This review describes these latest advances. More specifically, we consider the dynamical and thermodynamic descriptions of hard spheres around the dynamical, Gardner and jamming transitions. Comparing meanfield predictions with the finite-dimensional simulations, we identify robust aspects of the theory and uncover its more sensitive features. We conclude with a brief overview of ongoing research.

This article describes how the dimensions of nanowires affect the transmittance and sheet resistance of a random nanowire network. Silver nanowires with independently controlled lengths and diameters were synthesized with a gram-scale polyol synthesis by controlling the reaction temperature and time. Characterization of films composed of nanowires of different lengths but the same diameter enabled the quantification of the effect of length on the conductance and transmittance of silver nanowire films. Finite-difference time-domain calculations were used to determine the effect of nanowire diameter, overlap, and hole size on the transmittance of a nanowire network. For individual nanowires with diameters greater than 50 nm, increasing diameter increases the electrical conductance to optical extinction ratio, but the opposite is true for nanowires with diameters less than this size. Calculations and experimental data show that for a random network of nanowires, decreasing nanowire diameter increases the number density of nanowires at a given transmittance, leading to improved connectivity and conductivity at high transmittance (>90%). This information will facilitate the design of transparent, conducting nanowire films for flexible displays, organic light emitting diodes and thin-film solar cells.Indium tin oxide (ITO) is the material of choice for transparent conducting films in flat-panel displays, organic solar cells, and organic light emitting diodes because, with a sheet resistance of 10 U sq À1 at a transmittance of 90% (l ¼ 550 nm), it is highly conductive and transparent. However, indium is a scarce element, ITO is brittle, and ITO film is expensive because it is produced with a vapor-phase coating process that is 1000 times slower than newspaper printing.1,2 These problems have motivated a search for alternatives to ITO that are flexible and can be deposited from liquids at high coating rates.3-9 As discussed in recent reviews, promising solution-processed alternatives to ITO include poly(3,4-ethylenedioxythiophene)poly(styrenesulfonate), carbon nanotubes, graphene, ITO nanowires, and metal nanowires.10-15 Of these alternatives, films of silver nanowires currently have the highest conductance and transmittance. 4,16-18For example, Leem et al. have recently reported obtaining silver nanowire films with a sheet resistance of 10 U sq À1 at a transmittance of 89.3% (l ¼ 550 nm), nearly matching ITO. Although silver ($1000 kg À1) is more expensive than indium ($800 kg À1 ), the fact that silver nanowire films can be produced with highthroughput wet-coating processes allows them to achieve lower costs. 19,20Here we report a simple polyol synthesis that enables control over the length and diameter of silver nanowires, as well as their production on the gram scale. By measuring the properties of films composed of nanowires with distinct ranges of dimensions, we have obtained the first quantitative confirmation of theoretical predictions for the effect of nanowire length and number density on the conductance of 2D...

In this set of lecture notes we review the mode-coupling theory of the glass transition from several perspectives. First, we derive mode-coupling equations for the description of density fluctuations from microscopic considerations with the use the Mori-Zwanzig projection operator technique. We also derive schematic mode-coupling equations of a similar form from a field-theoretic perspective. We review the successes and failures of modecoupling theory, and discuss recent advances in the applications of the theory.

In the first part of this paper, we derive the general replica equations that describe infinitedimensional hard spheres at any level of replica symmetry breaking (RSB) and in particular in the fullRSB scheme. We show that these equations are formally very similar to the ones that have been derived for spin glass models, thus showing that the analogy between spin glasses and structural glasses conjectured by Kirkpatrick, Thirumalai, and Wolynes is realized in a strong sense in the mean field limit. We also suggest how the computation could be generalized in an approximate way to finite dimensional hard spheres. In the second part of the paper, we discuss the solution of these equations and we derive from it a number of physical predictions. We show that, below the Gardner transition where the 1RSB solution becomes unstable, a fullRSB phase exists and we locate the boundary of the fullRSB phase. Most importantly, we show that the fullRSB solution predicts correctly that jammed packings are isostatic, and allows one to compute analytically the critical exponents associated with the jamming transition, which are missed by the 1RSB solution. We show that these predictions compare very well with numerical results. Contents

Low-temperature properties of crystalline solids can be understood using harmonic perturbations around a perfect lattice, as in Debye's theory. Low-temperature properties of amorphous solids, however, strongly depart from such descriptions, displaying enhanced transport, activated slow dynamics across energy barriers, excess vibrational modes with respect to Debye's theory (i.e., a boson peak), and complex irreversible responses to small mechanical deformations. These experimental observations indirectly suggest that the dynamics of amorphous solids becomes anomalous at low temperatures. Here, we present direct numerical evidence that vibrations change nature at a well-defined location deep inside the glass phase of a simple glass former. We provide a real-space description of this transition and of the rapidly growing time-and lengthscales that accompany it. Our results provide the seed for a universal understanding of low-temperature glass anomalies within the theoretical framework of the recently discovered Gardner phase transition.glass transition | disordered solids | Gardner transition | computer simulations | hard spheres U nderstanding the nature of the glass transition, which describes the gradual transformation of a viscous liquid into an amorphous solid, remains an open challenge in condensed matter physics (1, 2). As a result, the glass phase itself is not well understood either. The main challenge is to connect the localized, or "caged," dynamics that characterizes the glass transition to the low-temperature anomalies that distinguish amorphous solids from their crystalline counterparts (3-7). Recent theoretical advances, building on the random first-order transition approach (8), have led to an exact mathematical description of both the glass transition and the amorphous phases of hard spheres in the mean-field limit of infinite-dimensional space (9). A surprising outcome has been the discovery of a novel phase transition inside the amorphous phase, separating the localized states produced at the glass transition from their inherent structures. This Gardner transition (10), which marks the emergence of a fractal hierarchy of marginally stable glass states, can be viewed as a glass transition deep within a glass, at which vibrational motion dramatically slows down and becomes spatially correlated (11). Although these theoretical findings promise to explain and unify the emergence of low-temperature anomalies in amorphous solids, the gap remains wide between mean-field calculations (9, 11) and experimental work. Here, we provide direct numerical evidence that vibrational motion in a simple 3D glass-former becomes anomalous at a well-defined location inside the glass phase. In particular, we report the rapid growth of a relaxation time related to cooperative vibrations, a nontrivial change in the probability distribution function of a global order parameter, and the rapid growth of a correlation length. We also relate these findings to observed anomalies in low-temperature laboratory glasses. These re...

The mechanical properties of jammed packings depend sensitively on their detailed local structure. Here we provide a complete characterization of the pair correlation close to contact and of the force distribution of jammed frictionless spheres. In particular we discover a set of new scaling relations that connect the behavior of particles bearing small forces and those bearing no force but that are almost in contact. By performing systematic investigations for spatial dimensions d ¼ 3-10, in a wide density range and using different preparation protocols, we show that these scalings are indeed universal. We therefore establish clear milestones for the emergence of a complete microscopic theory of jamming. This description is also crucial for high-precision force experiments in granular systems.

Recent theoretical advances offer an exact, first-principles theory of jamming criticality in infinite dimension as well as universal scaling relations between critical exponents in all dimensions. For packings of frictionless spheres near the jamming transition, these advances predict that nontrivial power-law exponents characterize the critical distribution of (i) small interparticle gaps and (ii) weak contact forces, both of which are crucial for mechanical stability. The scaling of the interparticle gaps is known to be constant in all spatial dimensions d-including the physically relevant d ¼ 2 and 3, but the value of the weak force exponent remains the object of debate and confusion. Here, we resolve this ambiguity by numerical simulations. We construct isostatic jammed packings with extremely high accuracy, and introduce a simple criterion to separate the contribution of particles that give rise to localized buckling excitations, i.e., bucklers, from the others. This analysis reveals the remarkable dimensional robustness of mean-field marginality and its associated criticality.