We combine an analytically solvable mean-field elasto-plastic model with molecular dynamics simulations of a generic glass former to demonstrate that, depending on their preparation protocol, amorphous materials can yield in two qualitatively distinct ways. We show that well-annealed systems yield in a discontinuous brittle way, as metallic and molecular glasses do. Yielding corresponds in this case to a first-order nonequilibrium phase transition. As the degree of annealing decreases, the first-order character becomes weaker and the transition terminates in a second-order critical point in the universality class of an Ising model in a random field. For even more poorly annealed systems, yielding becomes a smooth crossover, representative of the ductile rheological behavior generically observed in foams, emulsions, and colloidal glasses. Our results show that the variety of yielding behaviors found in amorphous materials does not necessarily result from the diversity of particle interactions or microscopic dynamics but is instead unified by carefully considering the role of the initial stability of the system.

We implement and optimize a particle-swap Monte-Carlo algorithm that allows us to thermalize a polydisperse system of hard spheres up to unprecedentedly-large volume fractions, where previous algorithms and experiments fail to equilibrate. We show that no glass singularity intervenes before the jamming density, which we independently determine through two distinct non-equilibrium protocols. We demonstrate that equilibrium fluid and non-equilibrium jammed states can have the same density, showing that the jamming transition cannot be the end-point of the fluid branch.PACS numbers: 05.20.Jj,We clarify the behavior of non-crystalline states of hard spheres at very large densities where both a glass transition (in colloidal systems) and a jamming transition (in non-Brownian systems) are observed [1][2][3]. Glass and jamming transitions are usually studied through distinct protocols, and understanding the relation between these two broad classes of phase transformations and the resulting amorphous arrested states is an important research goal [1][2][3][4][5]. These questions impact a wide range of fields, from the rheological properties of soft materials to optimization problems in computer science [1,6].Let us first consider Brownian hard spheres. When size polydispersity is introduced, crystallization can be prevented and the thermodynamic properties of the fluid studied at increasing density until a glass transition takes place, where particle diffusivity becomes very small [7]. Upon further compression, the pressure of the glass increases until a jamming transition occurs, where particles come at close contact and the pressure diverges [8]. Because the laboratory glass transition arises from using a finite observation timescale, two scenarios were proposed to describe the hypothetical situation where thermalization is no longer an issue [9]. A first possibility is that slower compressions reveal an ideal glass transition density, ϕ 0 , above which the equilibrium state is a glass, not a fluid [2]. Jamming would then be observed upon further non-equilibrium compression of these glass states [10]. Alternatively, it may be that slower compressions continuously shift the kinetic glass transition to higher densities. In this view, it is plausible that jamming becomes the end-point of the equilibrium fluid branch [11,12].Distinguishing between these two scenarios by direct numerical measurements is challenging. For a wellstudied binary mixture of hard spheres, for instance, thermalization can be achieved up to ϕ max ≈ 0.60 [9,13]. The location of the glass transition must be extrapolated using empirical fits based on activated relaxation. Values in the range ϕ 0 = 0.615 − 0.635 were obtained [9], depending on the fitting function. Fitting the relaxation times to a power law yields ϕ mct ≈ 0.59 < ϕ max , so that the associated mode-coupling transition [14] corresponds to an avoided singularity. For the same system, jamming transitions were located in the range ϕ J = 0.648 − 0.662 depending on the chosen protocol [15][1...