^{1}, Amit Kumar Bhattacharjee

^{2,a)}, Jürgen Horbach

^{2,b)}, Matthias Fuchs

^{1,c)}and Thomas Voigtmann

^{1,2,3}

### Abstract

We study the nonlinear rheology of a glass-forming binary mixture under the reversal of shear flow using molecular dynamics simulations and a schematic model of the mode-coupling theory of the glass transition (MCT). Memory effects lead to a history-dependent response, as exemplified by the vanishing of a stress-overshoot phenomenon in the stress–strain curves of the sheared liquid, and a change in the apparent elastic coefficients around states with zero stress. We investigate the various retarded contributions to the stress response at a given time schematically within MCT. The connection of this macroscopic response to single-particle motion is demonstrated using molecular-dynamics simulation.

We thank for funding by the Deutsche Forschungsgemeinschaft through Research Unit FOR 1394, projects P3 and P8. A.K.B. acknowledges funding through the German Academic Exchange Service, DAAD-DLR programme. T.V. is funded through the Helmholtz Gesellschaft (HGF, VH-NG 406) and the Zukunftskolleg of the University of Konstanz.

I. INTRODUCTION

II. THEORY

A. Schematic MCT model

B. Bauschinger effect

III. SIMULATION

IV. RESULTS AND DISCUSSION

A. Comparison of simulation and theory

B. Interpretation in a generalized Maxwell model

C. Discussion of the history dependence in the schematic model

D. Simulation results on average particle motion

V. CONCLUSIONS

### Key Topics

- Elasticity
- 35.0
- Correlation functions
- 28.0
- Elastic moduli
- 18.0
- Glass transitions
- 12.0
- Shear flows
- 11.0

## Figures

Linear plot of the stress–strain relation σ(γ) at fixed density and shear rate for various flow histories: start-up flow with constant shear rate (red), flow reversal in the steady state (s, purple), from the elastic regime (el, blue), and from the point of the stress overshoot (max, green). The linear increase with the quiescent elastic modulus *G* _{∞} in start-up, and with *G* _{eff} after shear-reversal are indicated by dotted lines. Characteristic stress–strain points, whose transient modulus is discussed in the text, are marked by symbols: The red square gives the stationary stress. It corresponds to an integration along the line from A to B in Fig. 7 . Triangles mark stresses after flow reversal in the linear regime and correspond to an integration along the line C' to E'. Discs mark stresses after flow reversal in the overshoot regime and are obtained via integrations along the line C to E in Fig. 7 .

Linear plot of the stress–strain relation σ(γ) at fixed density and shear rate for various flow histories: start-up flow with constant shear rate (red), flow reversal in the steady state (s, purple), from the elastic regime (el, blue), and from the point of the stress overshoot (max, green). The linear increase with the quiescent elastic modulus *G* _{∞} in start-up, and with *G* _{eff} after shear-reversal are indicated by dotted lines. Characteristic stress–strain points, whose transient modulus is discussed in the text, are marked by symbols: The red square gives the stationary stress. It corresponds to an integration along the line from A to B in Fig. 7 . Triangles mark stresses after flow reversal in the linear regime and correspond to an integration along the line C' to E'. Discs mark stresses after flow reversal in the overshoot regime and are obtained via integrations along the line C to E in Fig. 7 .

Plot of the stress–strain curves σ(γ) at fixed temperature for various shear rates. The dashed lines show the MCT results for Γ = 100, ε = −10^{−4}, γ_{ c } = 0.75, , and γ_{**}/γ_{*} = 1.33. γ_{*} is 10^{−1}, 7 × 10^{−2}, 6 × 10^{−2}, 4.5 × 10^{−2}, 3.5 × 10^{−2}, 1.5 × 10^{−2} for decreasing shear rates .

Plot of the stress–strain curves σ(γ) at fixed temperature for various shear rates. The dashed lines show the MCT results for Γ = 100, ε = −10^{−4}, γ_{ c } = 0.75, , and γ_{**}/γ_{*} = 1.33. γ_{*} is 10^{−1}, 7 × 10^{−2}, 6 × 10^{−2}, 4.5 × 10^{−2}, 3.5 × 10^{−2}, 1.5 × 10^{−2} for decreasing shear rates .

Linear plot of the stress–strain relation |σ(|γ − γ_{0}|)| at fixed temperature and shear rate for various flow histories: starting from equilibrium (EQ, red line), after flow reversal in the steady state (S, purple), from the elastic regime (el, blue), and from the point of the stress overshoot (max, green). Reversal at is in the steady state as seen from comparing with reversal at (black data). Smooth lines result from the schematic model.

Linear plot of the stress–strain relation |σ(|γ − γ_{0}|)| at fixed temperature and shear rate for various flow histories: starting from equilibrium (EQ, red line), after flow reversal in the steady state (S, purple), from the elastic regime (el, blue), and from the point of the stress overshoot (max, green). Reversal at is in the steady state as seen from comparing with reversal at (black data). Smooth lines result from the schematic model.

Velocity profiles *v* _{ x }(*y*) as a function of position along the gradient direction *y* for various times after application of a new flow direction. (a) Starting from a quiescent configuration (EQ), shear in +*x* direction. (b) Starting from the resulting steady state (S) at , shear in −*x* direction. Insets mark the points for which profiles are shown along the stress–strain curves. A total 100 independent configurations are averaged within a small time window to obtain the depicted graphics.

Velocity profiles *v* _{ x }(*y*) as a function of position along the gradient direction *y* for various times after application of a new flow direction. (a) Starting from a quiescent configuration (EQ), shear in +*x* direction. (b) Starting from the resulting steady state (S) at , shear in −*x* direction. Insets mark the points for which profiles are shown along the stress–strain curves. A total 100 independent configurations are averaged within a small time window to obtain the depicted graphics.

Bauschinger effect as illustrated by a generalized Maxwell model (see text). Solid lines are σ(*t*) as a function of normalized time, , for the case γ_{ w } = 0.1 corresponding to and γ_{ w } = 0.8 corresponding to . Dashed lines display the contributions σ_{I}(*t*) (upper, red) and σ_{II}(*t*) (lower, blue).

Bauschinger effect as illustrated by a generalized Maxwell model (see text). Solid lines are σ(*t*) as a function of normalized time, , for the case γ_{ w } = 0.1 corresponding to and γ_{ w } = 0.8 corresponding to . Dashed lines display the contributions σ_{I}(*t*) (upper, red) and σ_{II}(*t*) (lower, blue).

Linear stress responses from simulation, fits of the schematic model, and Maxwell model as labeled for various waiting strains γ_{ w }. The inset gives the strain γ_{ w } − γ_{0} required for the stress to reduce to zero after shear reversal. To include the simulation point for , this value has been divided by 200, using that the precise value of γ_{ w } is irrelevant in the steady state.

Linear stress responses from simulation, fits of the schematic model, and Maxwell model as labeled for various waiting strains γ_{ w }. The inset gives the strain γ_{ w } − γ_{0} required for the stress to reduce to zero after shear reversal. To include the simulation point for , this value has been divided by 200, using that the precise value of γ_{ w } is irrelevant in the steady state.

View on the *t* − *t* ^{′}-plane, with the three different time regimes for the density correlator Φ. As the *h*-function depends only on the absolute value of , the one-time-solutions and are equal.

View on the *t* − *t* ^{′}-plane, with the three different time regimes for the density correlator Φ. As the *h*-function depends only on the absolute value of , the one-time-solutions and are equal.

(a) Shear moduli *G*(*t*, *t* ^{′}) after shear reversal at in the steady state, shown for three different fixed times *t* (respectively, strains γ), as functions of *t* − *t* ^{′}. Curves marked with violet triangle, disc, and diamond symbols correspond to points marked with the same symbol in Fig. 1 . For times , all curves collapse by construction onto the start-up result *G*(*t*, 0) shown in red (marked with a square). The inset shows the corresponding correlators Φ(*t*, *t* ^{′}) (they overlap on the resolution of the figure). (b) Corresponding stress integrands on a linear *t* − *t* ^{′} axis. Curves after shear reversal which would be negative for *t* − *t* ^{′} → 0 are mirrored to positive initial decay, where they overlap with the start-up curve.

(a) Shear moduli *G*(*t*, *t* ^{′}) after shear reversal at in the steady state, shown for three different fixed times *t* (respectively, strains γ), as functions of *t* − *t* ^{′}. Curves marked with violet triangle, disc, and diamond symbols correspond to points marked with the same symbol in Fig. 1 . For times , all curves collapse by construction onto the start-up result *G*(*t*, 0) shown in red (marked with a square). The inset shows the corresponding correlators Φ(*t*, *t* ^{′}) (they overlap on the resolution of the figure). (b) Corresponding stress integrands on a linear *t* − *t* ^{′} axis. Curves after shear reversal which would be negative for *t* − *t* ^{′} → 0 are mirrored to positive initial decay, where they overlap with the start-up curve.

Similar to Fig. 8 , but now for shear reversal at in the elastic regime. The curves end at a time , where the memory has not decayed to zero yet. This is as expected, because the system still responds elastic like before the early reversal at ; see the blue curve in Fig. 1 .

In Panel (a), linear stress response as function of waiting strain γ_{ w } in a fluid and a glass state; for the latter, two different shear rates are shown. The quiescent elastic shear constant *G* _{∞} is observed in start-up flow. In Panel (b), the corresponding curves are given for the relative stress overshoots σ_{ max }/σ_{ s } − 1.

In Panel (a), linear stress response as function of waiting strain γ_{ w } in a fluid and a glass state; for the latter, two different shear rates are shown. The quiescent elastic shear constant *G* _{∞} is observed in start-up flow. In Panel (b), the corresponding curves are given for the relative stress overshoots σ_{ max }/σ_{ s } − 1.

Simulation results of the self intermediate scattering functions *F* _{ s }(*q*, *t*) for the B-species in the two directions perpendicular to shear during start-up and after shear reversal in the stationary state, for wave vectors *q* = [π/σ_{ AA }, 2π/σ_{ AA }, 3π/σ_{ AA }, 4π/σ_{ AA }] in descending order. A total 200 independent simulational runs are averaged to obtain these graphics. The corresponding equilibrium functions are included (dashed lines, label EQ), shifted to coincide at short times.

Simulation results of the self intermediate scattering functions *F* _{ s }(*q*, *t*) for the B-species in the two directions perpendicular to shear during start-up and after shear reversal in the stationary state, for wave vectors *q* = [π/σ_{ AA }, 2π/σ_{ AA }, 3π/σ_{ AA }, 4π/σ_{ AA }] in descending order. A total 200 independent simulational runs are averaged to obtain these graphics. The corresponding equilibrium functions are included (dashed lines, label EQ), shifted to coincide at short times.

(a) Mean-squared displacement of B particles in the vorticity direction. Shown are the equilibrium curve for *T* = 0.4 (label EQ), and transient MSDs δ*z* ^{2}(*t*, *t* ^{′}) as a function of , where *t* ^{′} is the time of initial flow start-up from equilibrium (EQ), respectively, flow reversal according to the cases discussed in Fig. 2 (steady-state, s; elastic transient, el; strain corresponding to the overshoot, max). The equilibrium curve is shifted to coincide at short times. (b) Logarithmic derivative for the cases shown in (a).

(a) Mean-squared displacement of B particles in the vorticity direction. Shown are the equilibrium curve for *T* = 0.4 (label EQ), and transient MSDs δ*z* ^{2}(*t*, *t* ^{′}) as a function of , where *t* ^{′} is the time of initial flow start-up from equilibrium (EQ), respectively, flow reversal according to the cases discussed in Fig. 2 (steady-state, s; elastic transient, el; strain corresponding to the overshoot, max). The equilibrium curve is shifted to coincide at short times. (b) Logarithmic derivative for the cases shown in (a).

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