^{1}and J. Horbach

^{2}

### Abstract

We present extensive molecular dynamics computer simulations of a glass-forming Yukawa mixture, investigating the nonlinear response of a single particle that is pulled through the system by a constant force. Structural changes around the pulled particle are analyzed by pair correlation functions, measured in the deeply supercooled state of the system. A regime of intermediate force strengths is found where the structural changes around the pulled particle are small, although its steady-state velocity shows a strong nonlinear response. This nonlinear response regime is characterized by a force-temperature superposition principle of a Peclet number and anisotropic diffusive behavior. In the direction parallel to the force, mean-square displacements show anomalous superdiffusion in the long time limit. We analyze this superdiffusive behavior by means of the van Hove correlation function of the pulled particle. Perpendicular to the force, the driven particle shows diffusive behavior for all considered force strengths and temperatures. We discuss the dynamics perpendicular and parallel to the force in terms of effective temperatures.

We thank Kurt Binder, Pinaki Chaudhuri, Matthias Fuchs, Christian Harrer, Andreas Heuer, Robert Jack, Antonio Puertas, Carsten Schroer, and Thomas Voigtmann for useful discussions. The authors acknowledge financial support by the German Deutsche Forschungsgemeinschaft (DFG), Project Nos. SFB TR 6/A5 and FOR 1394/P8 (J.H.). Computer time at the NIC Jülich is gratefully acknowledged.

I. INTRODUCTION

II. DETAILS OF THE SIMULATION

III. RESULTS

A. Structural changes

B. Dynamics in force direction

C. Dynamics perpendicular to force direction

IV. CONCLUSIONS

### Key Topics

- Diffusion
- 17.0
- Correlation functions
- 11.0
- Self diffusion
- 11.0
- Friction
- 8.0
- Molecular dynamics
- 7.0

## Figures

Pair correlation functions around a pulled A particle at *T* = 0.14 for different forces *f* and at equilibrium (dashed lines). The solid lines in the plots correspond to the forces *f* = 2.5, 6.0, 8.0, 10.0, and 20.0 (from top to bottom). Note that the latter curves are shifted with respect to the equilibrium curves by multiples of −0.5 on the ordinate. *g* _{AA}(*x*) in (a) and *g* _{AB}(*x*) in (b) are the pair correlation functions in force direction, *g* _{AA}(*r*) in (c) and *g* _{AB}(*r*) in (d) those perpendicular to the force.

Pair correlation functions around a pulled A particle at *T* = 0.14 for different forces *f* and at equilibrium (dashed lines). The solid lines in the plots correspond to the forces *f* = 2.5, 6.0, 8.0, 10.0, and 20.0 (from top to bottom). Note that the latter curves are shifted with respect to the equilibrium curves by multiples of −0.5 on the ordinate. *g* _{AA}(*x*) in (a) and *g* _{AB}(*x*) in (b) are the pair correlation functions in force direction, *g* _{AA}(*r*) in (c) and *g* _{AB}(*r*) in (d) those perpendicular to the force.

The same as Fig. 1 , but now for a pulled B particle. *g* _{BA}(*x*) in (a) and *g* _{BB}(*x*) in (b) are the pair correlation functions in force direction, *g* _{BA}(*r*) in (c) and *g* _{BB}(*r*) in (d) those perpendicular to the force.

The same as Fig. 1 , but now for a pulled B particle. *g* _{BA}(*x*) in (a) and *g* _{BB}(*x*) in (b) are the pair correlation functions in force direction, *g* _{BA}(*r*) in (c) and *g* _{BB}(*r*) in (d) those perpendicular to the force.

Time dependence of displacement Δ*x* of a pulled A particle for the forces *f* = 0.5. 1.0, 1.5, 2.0, 2.5, and 5.0 (from right to left) at temperature *T* = 0.14. Solid lines correspond to the transient state from switch-on of the force at *t* = 0 to the steady-state at long times. Dashed lines correspond to the steady state where Δ*x* displays a linear time dependence.

Time dependence of displacement Δ*x* of a pulled A particle for the forces *f* = 0.5. 1.0, 1.5, 2.0, 2.5, and 5.0 (from right to left) at temperature *T* = 0.14. Solid lines correspond to the transient state from switch-on of the force at *t* = 0 to the steady-state at long times. Dashed lines correspond to the steady state where Δ*x* displays a linear time dependence.

(a) Steady state velocity of the pulled A particle as a function of *f* for different temperatures *T*, as indicated. The bold dashed line is a linear fit to the data for *T* = 1.0. (b) Peclet number Pe* as a function of (see text).

(a) Steady state velocity of the pulled A particle as a function of *f* for different temperatures *T*, as indicated. The bold dashed line is a linear fit to the data for *T* = 1.0. (b) Peclet number Pe* as a function of (see text).

(a) Mean-squared displacement ⟨Δ*x* ^{2}(*t*)⟩ − ⟨Δ*x*(*t*)⟩^{2} for pulled A particle at *T* = 0.14. The curves correspond to the forces *f* = 0.0, 0.5, 1.0, 1.5, 2.5, 4.0, 6.0, and 10.0 (from right to left). (b) Effective exponents α as a function of *f* for different temperatures, as indicated.

(a) Mean-squared displacement ⟨Δ*x* ^{2}(*t*)⟩ − ⟨Δ*x*(*t*)⟩^{2} for pulled A particle at *T* = 0.14. The curves correspond to the forces *f* = 0.0, 0.5, 1.0, 1.5, 2.5, 4.0, 6.0, and 10.0 (from right to left). (b) Effective exponents α as a function of *f* for different temperatures, as indicated.

Mean-squared displacement ⟨Δ*x* ^{2}(*t*)⟩ − ⟨Δ*x*(*t*)⟩^{2} for pulled A particle at *T* = 0.14 and *f* = 2.0 for different thermostats (see text). The inset shows the corresponding mean-squared displacement displacements divided by *t*.

Mean-squared displacement ⟨Δ*x* ^{2}(*t*)⟩ − ⟨Δ*x*(*t*)⟩^{2} for pulled A particle at *T* = 0.14 and *f* = 2.0 for different thermostats (see text). The inset shows the corresponding mean-squared displacement displacements divided by *t*.

(a) Typical trajectories, *x*(*t*), of pulled A particles at *T* = 0.14 and *f* = 1.0. (b) Typical trajectories, *x*(*t*), of pulled A particles at *f* = 1.0 and different temperatures, as indicated.

(a) Typical trajectories, *x*(*t*), of pulled A particles at *T* = 0.14 and *f* = 1.0. (b) Typical trajectories, *x*(*t*), of pulled A particles at *f* = 1.0 and different temperatures, as indicated.

van Hove correlation function of pulled A particle at *T* = 0.14 for the indicated times, (a) *f* = 1.0, (b) *f* = 2.5.

van Hove correlation function of pulled A particle at *T* = 0.14 for the indicated times, (a) *f* = 1.0, (b) *f* = 2.5.

Dependence of self-diffusion constant on inverse temperature for pulled A particles for the forces *f* = 0.0 (equilibrium), 0.5, 1.0, 1.5, 2.0, 2.5, and 5.0. The insets show the same data, but now using the scaled effective temperatures *T* _{eff}(*f*) instead of *T*.

Dependence of self-diffusion constant on inverse temperature for pulled A particles for the forces *f* = 0.0 (equilibrium), 0.5, 1.0, 1.5, 2.0, 2.5, and 5.0. The insets show the same data, but now using the scaled effective temperatures *T* _{eff}(*f*) instead of *T*.

Time dependence of the incoherent scattering functions, *F* _{s}(*q*, *t*) in perpendicular direction for A particles at *f* = 1.0 and *q* = 6.0 for the temperatures *T* = 0.14, 0.15, 0.16, 0.17, 0.18, 0.21, 0.25, 0.30, and 0.34 (from right to left).

Time dependence of the incoherent scattering functions, *F* _{s}(*q*, *t*) in perpendicular direction for A particles at *f* = 1.0 and *q* = 6.0 for the temperatures *T* = 0.14, 0.15, 0.16, 0.17, 0.18, 0.21, 0.25, 0.30, and 0.34 (from right to left).

Dependence of (a) α relaxation time and (b) friction coefficient on inverse temperature for pulled A particles for the same forces as in Fig. 9 . The insets show the data as function of the effective temperatures *T* _{eff}(*f*).

Dependence of (a) α relaxation time and (b) friction coefficient on inverse temperature for pulled A particles for the same forces as in Fig. 9 . The insets show the data as function of the effective temperatures *T* _{eff}(*f*).

*T* _{eff}/*T* − 1 as function of *f*, as obtained from the scaling of the self-diffusion constant *D* _{orth} and the friction coefficient ξ. Results for A and B particles are shown, as indicated. The dotted and dashed lines are fits with *T* _{eff}/*T* − 1 = *C* × *f* ^{2} yielding the fit parameters *C* = 0.0326 and *C* = 0.0181 for *D* _{orth} and *C* = 0.0266 and *C* = 0.0134 for ξ of A and B particles, respectively.

*T* _{eff}/*T* − 1 as function of *f*, as obtained from the scaling of the self-diffusion constant *D* _{orth} and the friction coefficient ξ. Results for A and B particles are shown, as indicated. The dotted and dashed lines are fits with *T* _{eff}/*T* − 1 = *C* × *f* ^{2} yielding the fit parameters *C* = 0.0326 and *C* = 0.0181 for *D* _{orth} and *C* = 0.0266 and *C* = 0.0134 for ξ of A and B particles, respectively.

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