Mean-field stability map of hard-sphere glasses

Ada Altieri and Francesco Zamponi

Phys. Rev. E 100, 032140 – Published 26 September 2019

Abstract

The response of amorphous solids to an applied shear deformation is an important problem, both in fundamental and in applied research. To tackle this problem, we focus on a system of hard spheres in infinite dimensions as a solvable model for colloidal systems and granular matter. The system is prepared above the dynamical glass transition density, and we discuss the phase diagram of the resulting glass under compression, decompression, and shear strain, expanding on previous results [Urbani and Zamponi, Phys. Rev. Lett. 118, 038001 (2017)]. We show that the solid region is bounded by a “shear jamming” line, at which the solid reaches close packing, and a “shear yielding” line, at which the solid undergoes a spinodal instability towards a liquid flowing phase. Furthermore, we characterize the evolution of these lines upon varying the glass preparation density. This paper aims to provide a general overview on yielding and jamming phenomena in hard-sphere systems by a systematic exploration of the phase diagram.

Received 12 July 2019

DOI:https://doi.org/10.1103/PhysRevE.100.032140

Physics Subject Headings (PhySH)

Flow instability Jamming Rheology Shear deformation

Disordered systems

Replica methods

Condensed Matter, Materials & Applied PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

Ada Altieri and Francesco Zamponi

Laboratoire de Physique de l'École Normale Supérieure, Université PSL, CNRS, Sorbonne Université, Université de Paris, 75005 Paris, France

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Issue

Vol. 100, Iss. 3 — September 2019

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Images

Figure 1
Mean-field phase diagrams of a hard-sphere glass upon adiabatic following in compression and decompression at density $\hat{φ}$ and in the presence of a quasistatic shear strain $γ$ for three different preparation densities ${\hat{φ}}_{g} = 5.5, {\hat{φ}}_{g} = 8$ , and ${\hat{φ}}_{g} = 12$ , respectively. The stability region of the glass is delimited by the shear jamming (blue) and shear yielding (red) lines. At sufficiently low ${\hat{φ}}_{g}$ , shear jamming disappears, and the system can only yield under an applied strain. The Gardner transition line, above which the RS solution is unstable, is shown in green. In the left panel, we expect that RSB effects change qualitatively the phase diagram in both gray-shaded regions.
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Figure 2
(a) Breaking point $λ$ and (b) inverse slope $1 / \dot{Δ} (λ)$ as a function of the applied shear strain $γ$ along the Gardner transition line for three different increasing values of ${\hat{φ}}_{g}$ .
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Figure 3
(a) Glass phase diagram on the $({\hat{φ}}_{g}, \hat{φ})$ plane. Glasses are prepared in equilibrium on the bisector line $\hat{φ} = {\hat{φ}}_{g} \geq {\hat{φ}}_{d} = 4.8067$ [dot-dashed line (black line)]. For a given ${\hat{φ}}_{g}$ , a glass can be followed in compression up to the jamming transition [blue line (upper curve) above which there are no states] or in decompression down to the melting transition (red dot-dashed line on the bottom). Above the Gardner transition [green line (light gray line)], the RS solution becomes unstable towards RSB. In this region for ${\hat{φ}}_{g} \leq 6.667$ , the RS solution predicts an unphysical spinodal (blue dashed line inside the full RSB region) at which the solution is lost before jamming is reached. (The continuation of the jamming line for ${\hat{φ}}_{g} \leq 6.667$ is based on full RSB data [29].) The critical density ${\hat{φ}}_{c}$ [red line (full gray line)] separates the phase diagram into two regions. Above it, the system jams under shear, whereas below it, it yields. (b) The same phase diagram on the plane of inverse reduced pressure $1 / \hat{p}$ and scaled packing fraction $\hat{φ}$ . Glasses are prepared on the equilibrium liquid line (dot-dashed black line) $\hat{p} = {\hat{φ}}_{g} / 2$ . The dynamical transition density ${\hat{φ}}_{d} ≃ 4.8067$ is marked by a full black circle. Upon compression or decompression, the glasses follow their equation of state (full black lines), reported for ${\hat{φ}}_{g} = 5.5, 7, 8, 9, 10, 12, 15, 17$ (bottom to top), which extends from the melting point [red line (light gray dot-dashed line)] up to jamming [blue line (full line)] for ${\hat{φ}}_{g} \geq 6.667$ or to the unphysical spinodal [dashed blue (gray line) merging with the critical jamming line] for ${\hat{φ}}_{g} \leq 6.667$ . Below the Gardner transition [green (light gray)] line, states are unstable towards RSB. The red full line separates the shear yielding and shear jamming regimes.
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