^{1}, Nicoletta Gnan

^{1}, Francesco Sciortino

^{1,2}and Emanuela Zaccarelli

^{1,2}

### Abstract

We performed extensive molecular dynamics (MD) simulations, supplemented by Mode Coupling Theory (MCT) calculations, for the square shoulder model, a purely repulsive potential where the hardcore is complemented by a finite shoulder. For the one-component version of this model, MCT predicted [Sperl et al. , Phys. Rev. Lett.104, 145701 (Year: 2010)10.1103/PhysRevLett.104.145701] the presence of diffusion anomalies both upon cooling and upon compression and the occurrence of glass-glass transitions. In the simulations, we focus on a non-crystallising binary mixture, which, at the investigated shoulder width, shows a non-monotonic behaviour of the diffusion upon cooling but not upon isothermal compression. In addition, we find the presence of a disconnected glass-glass line in the phase diagram, ending in two higher order singularities. These points generate a logarithmic dependence of the density correlators as well as a subdiffusive behaviour of the mean squared displacement, although with the interference of the nearby liquid-glass transition. We also perform novel MCT calculations using as input the partial structure factors obtained within MD, confirming the simulation results. The presence of two hard sphere glasses, differing only in their hardcore length, is revealed, showing that the simple competition between the two is sufficient for creating a rather complex dynamical behaviour.

We acknowledge helpful discussions with Matthias Sperl, Sergey V. Buldyrev, Pradeep Kumar, and H. Eugene Stanley. This work was supported by EU through ITN-234810-COMPLOIDS and ERC-226207-PATCHYCOLLOIDS, and by MIUR-FIRB ANISOFT (RBFR125H0M).

I. INTRODUCTION

II. METHODS: SIMULATIONS AND THEORY

III. RESULTS

A. Iso-diffusivity lines and ideal liquid-glass line from simulations

B. Comparison with old and new MCT results: Role of the input structure factors and mapping to simulations

C. Searching for higher order singularities

D. Non-ergodicity parameters from MCT and simulations

IV. DISCUSSION AND CONCLUSIONS

### Key Topics

- Glasses
- 48.0
- Glass transitions
- 40.0
- Phase diagrams
- 14.0
- Diffusion
- 10.0
- Molecular dynamics
- 9.0

##### F25

## Figures

Square shoulder potential for a generic additive binary mixture of species *i*, *j*. Here, σ_{ ij } = (1/2)(σ_{ i } + σ_{ j }) are the hardcores, Δσ_{ ij } are the shoulder widths, and *u* _{0} = 1 is the shoulder height.

Square shoulder potential for a generic additive binary mixture of species *i*, *j*. Here, σ_{ ij } = (1/2)(σ_{ i } + σ_{ j }) are the hardcores, Δσ_{ ij } are the shoulder widths, and *u* _{0} = 1 is the shoulder height.

Normalized diffusion coefficient *D* _{ A }/*D* _{0} as a function of (a) ϕ for several isotherms, as reported in the labels. At all investigated *T* data show a monotonic decrease with increasing ϕ, which clearly indicates the absence of diffusion anomalies associated with compression/expansion. A crossing of the data at high ϕ however signals the presence of a diffusivity maximum associated with cooling; (b) *T* for several isochores, as reported in the labels.

Normalized diffusion coefficient *D* _{ A }/*D* _{0} as a function of (a) ϕ for several isotherms, as reported in the labels. At all investigated *T* data show a monotonic decrease with increasing ϕ, which clearly indicates the absence of diffusion anomalies associated with compression/expansion. A crossing of the data at high ϕ however signals the presence of a diffusivity maximum associated with cooling; (b) *T* for several isochores, as reported in the labels.

(a) Isodiffusivity lines for *D* _{ A }/*D* _{0} = 1.0 × 10^{−3}, 1.0 × 10^{−4}, and 1.1 × 10^{−5}, as well as the extrapolated arrest (*D* _{ A } = 0) lines from the fits *D* _{ A } ∼ |ϕ − ϕ_{ g }(*T*)|^{γ(T)} along isotherms and *D* _{ A } ∼ |*T* − *T* _{ g }ϕ|^{γ(ϕ)} along isochores. The data display a reentrance in *T* (inset), while no reentrance in ϕ is observed; (b) Power law fits along isotherms (left) and isochores (right).

(a) Isodiffusivity lines for *D* _{ A }/*D* _{0} = 1.0 × 10^{−3}, 1.0 × 10^{−4}, and 1.1 × 10^{−5}, as well as the extrapolated arrest (*D* _{ A } = 0) lines from the fits *D* _{ A } ∼ |ϕ − ϕ_{ g }(*T*)|^{γ(T)} along isotherms and *D* _{ A } ∼ |*T* − *T* _{ g }ϕ|^{γ(ϕ)} along isochores. The data display a reentrance in *T* (inset), while no reentrance in ϕ is observed; (b) Power law fits along isotherms (left) and isochores (right).

Static structure factors for a monodisperse SS system at *T* = 0.5, ϕ = 0.45 calculated by MD simulations as well as solving the Ornstein-Zernike equation within Rogers-Young (RY) and Percus-Yevick (PY) closures. (Inset) MCT results for the liquid-glass and glass-glass lines using PY and RY.

Static structure factors for a monodisperse SS system at *T* = 0.5, ϕ = 0.45 calculated by MD simulations as well as solving the Ornstein-Zernike equation within Rogers-Young (RY) and Percus-Yevick (PY) closures. (Inset) MCT results for the liquid-glass and glass-glass lines using PY and RY.

MCT results for the binary mixture under study using the static structure factors calculated from simulations as input, labeled as liquid glass (filled squares) and glass-glass (open squares). Arrest curve drawn from ϕ_{ g } (filled circles) and *T* _{ g } (filled diamonds) obtained from power-law fits of *D* _{ A } as in Fig. 3 . Mapped MCT lines onto the arrest curve: liquid-glass (filled triangles) and glass-glass (open triangles). Stars are the two predicted higher order singularities and .

MCT results for the binary mixture under study using the static structure factors calculated from simulations as input, labeled as liquid glass (filled squares) and glass-glass (open squares). Arrest curve drawn from ϕ_{ g } (filled circles) and *T* _{ g } (filled diamonds) obtained from power-law fits of *D* _{ A } as in Fig. 3 . Mapped MCT lines onto the arrest curve: liquid-glass (filled triangles) and glass-glass (open triangles). Stars are the two predicted higher order singularities and .

MSD for a particles as a function of scaled time *tD* _{0} for ϕ = 0.525 as a function of *T*, indicated in the labels. The vertical dotted lines indicate as guides to the eye the regime of subdiffusive behaviour, which is highlighted by the dashed line (∝ *t* ^{0.5}).

MSD for a particles as a function of scaled time *tD* _{0} for ϕ = 0.525 as a function of *T*, indicated in the labels. The vertical dotted lines indicate as guides to the eye the regime of subdiffusive behaviour, which is highlighted by the dashed line (∝ *t* ^{0.5}).

The density autocorrelation functions for ϕ = 0.525 as a function of time for (a) several wave vectors at *T* = 0.375. From top to bottom, *q*σ_{ AA } = 1.88, 2.81, 4.68, 5.63, 7.5, 10.32, 13.12, 17.82, 28.13. A concave-convex shape transition is observed around *q**σ_{ AA } ≈ 7.0, where the decay of is almost purely logarithmic; (b) several *T* at fixed wave vector *q* = *q**. From left to right, temperatures are *T* = 1.0, 0.6, 0.5, 0.4, 0.375, 0.35, 0.3, 0.29.

The density autocorrelation functions for ϕ = 0.525 as a function of time for (a) several wave vectors at *T* = 0.375. From top to bottom, *q*σ_{ AA } = 1.88, 2.81, 4.68, 5.63, 7.5, 10.32, 13.12, 17.82, 28.13. A concave-convex shape transition is observed around *q**σ_{ AA } ≈ 7.0, where the decay of is almost purely logarithmic; (b) several *T* at fixed wave vector *q* = *q**. From left to right, temperatures are *T* = 1.0, 0.6, 0.5, 0.4, 0.375, 0.35, 0.3, 0.29.

Critical non-ergodicity parameters for the A species calculated within MCT along the liquid-glass (curves labeled from 1 to 5) and along the glass-glass (curves labeled from 6 to 8) lines. The corresponding state points and their position on the MCT lines are reported in the inset: a non-monotonic behaviour with increasing ϕ is observed for both sets of data.

Critical non-ergodicity parameters for the A species calculated within MCT along the liquid-glass (curves labeled from 1 to 5) and along the glass-glass (curves labeled from 6 to 8) lines. The corresponding state points and their position on the MCT lines are reported in the inset: a non-monotonic behaviour with increasing ϕ is observed for both sets of data.

(a): Non-ergodicity parameters obtained from simulations, fitting the density auto-correlation functions with stretched exponentials, for the state points reported in the upper inset. The behaviour along the liquid-glass line is strikingly similar to that of MCT predictions, reported in Fig. 8 . (Lower inset) for low ϕ (and low *T*) (state point 1) is identical to that of high ϕ (state point 6) upon a rescaling by the effective diameter σ + Δ; (b) Stretching exponents β^{ AA } obtained from the fits as a function of wave vector for the same state points considered in (a).

(a): Non-ergodicity parameters obtained from simulations, fitting the density auto-correlation functions with stretched exponentials, for the state points reported in the upper inset. The behaviour along the liquid-glass line is strikingly similar to that of MCT predictions, reported in Fig. 8 . (Lower inset) for low ϕ (and low *T*) (state point 1) is identical to that of high ϕ (state point 6) upon a rescaling by the effective diameter σ + Δ; (b) Stretching exponents β^{ AA } obtained from the fits as a function of wave vector for the same state points considered in (a).

## Tables

Extrapolated values of γ(*T*), ϕ_{ g }, γ(ϕ), and *T* _{ g } obtained from fitting data of Figs. 3(a) and 3(b) with MCT predictions of Eqs. (3) and (4) for the diffusion coefficient *D* _{ A }. Error bars of the fit parameters typically amount to a few percent for the values of ϕ_{ g } and *T* _{ g }, while the γ exponents can vary systematically over different fit intervals, so they should be taken with caution.

Extrapolated values of γ(*T*), ϕ_{ g }, γ(ϕ), and *T* _{ g } obtained from fitting data of Figs. 3(a) and 3(b) with MCT predictions of Eqs. (3) and (4) for the diffusion coefficient *D* _{ A }. Error bars of the fit parameters typically amount to a few percent for the values of ϕ_{ g } and *T* _{ g }, while the γ exponents can vary systematically over different fit intervals, so they should be taken with caution.

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