^{1}, J. R. Morris

^{2}and T. Egami

^{3}

### Abstract

The atomic level origin of viscosity and of various relaxation times is of primary interest in the field of supercooled liquids and the glass transition. Previously, by starting from the Green-Kubo expression for viscosity and by decomposing it into correlation functions between local atomic level stresses, we showed that there is a connection between shear stress waves and viscosity, and that the range of propagation of shear waves is also the range that is relevant for viscosity. Here, the behavior of the atomic level stress correlation function at different temperatures is discussed in more detail. The comparison of different time scales of the system shows that the long time decay of the stress correlation function (τ_{ S }) is approximately three times shorter than the long time decay of the intermediate self-scattering function (τ_{α}), while the the Maxwell relaxation time (τ_{ M }) is approximately five times shorter than τ_{α}. It is demonstrated how different timescales of the stress correlation function contribute to the Maxwell relaxation time. Finally, we discuss the non-trivial role of periodic boundary conditions.

The authors wish to thank K. Trachenko, V. N. Novikov, and K. A. Lokshin for useful discussions. This research was supported by the (U.S.) Department of Energy (DOE), Basic Energy Sciences, Materials Sciences and Engineering Division.

I. INTRODUCTION

II. THE IDEA OF THE APPROACH

III. SIMULATION DETAILS

IV. RESULTS

A. Relative importance of the auto and the cross terms of the sscf

B. Time and distance dependence of the cross term

C. Results for the microscopic viscosity

D. Connection with diffusion, intermediate self-scattering function, and relevant timescales

E. An insight into the Maxwellrelaxation time

V. CONCLUSION

### Key Topics

- Viscosity
- 61.0
- Correlation functions
- 21.0
- Shear rate dependent viscosity
- 16.0
- Relaxation times
- 15.0
- Glass transitions
- 9.0

## Figures

Stress correlation functions for a system of 5488 particles at 2000 K. The black solid line shows correlation function for the total stress. The red dashed line shows autocorrelation correlation for atomic level stress. The blue dotted-dashed line shows the difference between the total and the autocorrelation functions. The error in the correlation function of the total stress is comparable with the line thickness. The error in the autocorrelation function is much smaller than the line thickness.

Stress correlation functions for a system of 5488 particles at 2000 K. The black solid line shows correlation function for the total stress. The red dashed line shows autocorrelation correlation for atomic level stress. The blue dotted-dashed line shows the difference between the total and the autocorrelation functions. The error in the correlation function of the total stress is comparable with the line thickness. The error in the autocorrelation function is much smaller than the line thickness.

Dependencies of the total and autocorrelation viscosities on the upper cutoff in integration time at 2000 K. Obtained from the data in Fig. 1 .

Dependencies of the total and autocorrelation viscosities on the upper cutoff in integration time at 2000 K. Obtained from the data in Fig. 1 .

Dependencies on temperature of the final contributions to viscosity from the auto and the cross terms.

Dependencies on temperature of the final contributions to viscosity from the auto and the cross terms.

Normalized microscopic sscf's *F*(*r*, *t*)/*F*(*r* = 0, *t* = 0) for (a) 5488 particles system (*L*/2 ∼ 20.60 Å) at 1500 K, (b) 43 904 particles system (*L*/2 ∼ 41.21 Å) at 1500 K, and (c) 43 904 particles system at 2000 K. The values of the stress autocorrelation functions at zero time are ∼0.41 eV at 1500 K, ∼0.55 eV at 2000 K. Comparison of the panels (a) and (b) clearly shows that finite size of the system and the PBC affect the stress field. Comparison of panels (b) and (c) shows temperature dependence of the stress field in a moderately supercooled system.

Normalized microscopic sscf's *F*(*r*, *t*)/*F*(*r* = 0, *t* = 0) for (a) 5488 particles system (*L*/2 ∼ 20.60 Å) at 1500 K, (b) 43 904 particles system (*L*/2 ∼ 41.21 Å) at 1500 K, and (c) 43 904 particles system at 2000 K. The values of the stress autocorrelation functions at zero time are ∼0.41 eV at 1500 K, ∼0.55 eV at 2000 K. Comparison of the panels (a) and (b) clearly shows that finite size of the system and the PBC affect the stress field. Comparison of panels (b) and (c) shows temperature dependence of the stress field in a moderately supercooled system.

(a)–(d) Normalized microscopic sscf's *F*(*r*, *t*)/*F*(*r* = 0, *t* = 0) for the system of 5488 particles at temperatures 3000 K, 5000 K, 7000 K, 10 000 K correspondingly. The values of the stress autocorrelation functions at zero time are ∼0.86 eV at 3000 K, ∼1.52 eV at 5000 K, ∼2.18 eV at 7000 K, ∼3.13 eV at 10 000 K. (e) Normalized sscf at temperature 10 000 K for the large system of 43 904 particles. The comparison of panels (d) and (e) shows that PBC introduce some distortion into the sscf even at such a high temperature as 10 000 K.

(a)–(d) Normalized microscopic sscf's *F*(*r*, *t*)/*F*(*r* = 0, *t* = 0) for the system of 5488 particles at temperatures 3000 K, 5000 K, 7000 K, 10 000 K correspondingly. The values of the stress autocorrelation functions at zero time are ∼0.86 eV at 3000 K, ∼1.52 eV at 5000 K, ∼2.18 eV at 7000 K, ∼3.13 eV at 10 000 K. (e) Normalized sscf at temperature 10 000 K for the large system of 43 904 particles. The comparison of panels (d) and (e) shows that PBC introduce some distortion into the sscf even at such a high temperature as 10 000 K.

Constant time cuts of the (non-normalized) sscf from Fig. 4(b) . The legends show the cut times in femtoseconds.

Constant time cuts of the (non-normalized) sscf from Fig. 4(b) . The legends show the cut times in femtoseconds.

The black curves show zero-time cuts of the sscf's normalized to the zero-time value of the corresponding autocorrelation functions on the system of 5488 particles. The red curves show corresponding PDFs scaled in such a way that the heights of the first peaks in the scaled PDFs would match the heights of the first peaks in the sscf's. Note that the splitting of the second peak in PDF at 1500 K is also reflected in the sscf.

The black curves show zero-time cuts of the sscf's normalized to the zero-time value of the corresponding autocorrelation functions on the system of 5488 particles. The red curves show corresponding PDFs scaled in such a way that the heights of the first peaks in the scaled PDFs would match the heights of the first peaks in the sscf's. Note that the splitting of the second peak in PDF at 1500 K is also reflected in the sscf.

Dependencies of the microscopic viscosities on inclusion distance *R* _{ max } for the intermediate (5488) and the large (43 904) systems at 1500 K for some values of *t* _{ max }. For all *t* _{ max }, the red curves correspond to the intermediate size system, while the black curves correspond to the large system.

Dependencies of the microscopic viscosities on inclusion distance *R* _{ max } for the intermediate (5488) and the large (43 904) systems at 1500 K for some values of *t* _{ max }. For all *t* _{ max }, the red curves correspond to the intermediate size system, while the black curves correspond to the large system.

Dependencies of the microscopic viscosities on inclusion distance *R* _{ max } for the systems of different sizes at 2000 K for some values of *t* _{ max }. For all *t* _{ max }, the blue curves correspond to the smallest system, the red curves correspond to the intermediate size system and the black curves correspond to the large system.

Dependencies of the microscopic viscosities on inclusion distance *R* _{ max } for the systems of different sizes at 2000 K for some values of *t* _{ max }. For all *t* _{ max }, the blue curves correspond to the smallest system, the red curves correspond to the intermediate size system and the black curves correspond to the large system.

Dependencies of the particles mean square displacement on time at different temperatures. The results were obtained on a system of 5488 particles.

Dependencies of the particles mean square displacement on time at different temperatures. The results were obtained on a system of 5488 particles.

Fourier transform of the pair distribution function , which is related to the normalized scattering intensity *S*(*q*). Note that the first peak occurs at *q* ≈ 3.05 Å^{−1}. The inset shows the pair potential used in our MD simulations.

Fourier transform of the pair distribution function , which is related to the normalized scattering intensity *S*(*q*). Note that the first peak occurs at *q* ≈ 3.05 Å^{−1}. The inset shows the pair potential used in our MD simulations.

Dependencies of the intermediate self-scattering function on time for several temperatures. The data from MD simulations (solid curves) were fitted with the dashed curves. The error bars for the MD curves are smaller than the widths of the curves. The results were obtained on a system of 5488 particles. The legends show the temperatures and the parameters of the stretched exponential *A* exp [−(*t*/τ)^{β}] fits, i.e., *A*, τ, β.

Dependencies of the intermediate self-scattering function on time for several temperatures. The data from MD simulations (solid curves) were fitted with the dashed curves. The error bars for the MD curves are smaller than the widths of the curves. The results were obtained on a system of 5488 particles. The legends show the temperatures and the parameters of the stretched exponential *A* exp [−(*t*/τ)^{β}] fits, i.e., *A*, τ, β.

The total sscf and the *auto* contribution to it at 1325 K for the system of 5488 particles. Both panels have the same data, but the upper panel has linear scale on *y*-axis, while the lower panel has log scale on *y*-axis (to better demonstrate quality of the fit at small values of sscf). The upper solid line (in both panels) corresponds to the total sscf, while the lower corresponds to the *auto* contribution. The dashed curves are the fits made with the sums of compressed and stretched exponential functions. The equation of the fit curve for the total sscf is 0.427 exp [−(*t*/39)^{2}] + 0.220 exp [−(*t*/2350)^{3/4}]. The equation of the fit curve for the *auto* term is 0.254 exp [−(*t*/34)^{2}] + 0.112 exp [−(*t*/2500)^{3/4}]. The size of the oscillations at very large times that could be observed in the total sscf gives some estimate of the errors. The errors at short times are smaller than the thickness of the curves.

The total sscf and the *auto* contribution to it at 1325 K for the system of 5488 particles. Both panels have the same data, but the upper panel has linear scale on *y*-axis, while the lower panel has log scale on *y*-axis (to better demonstrate quality of the fit at small values of sscf). The upper solid line (in both panels) corresponds to the total sscf, while the lower corresponds to the *auto* contribution. The dashed curves are the fits made with the sums of compressed and stretched exponential functions. The equation of the fit curve for the total sscf is 0.427 exp [−(*t*/39)^{2}] + 0.220 exp [−(*t*/2350)^{3/4}]. The equation of the fit curve for the *auto* term is 0.254 exp [−(*t*/34)^{2}] + 0.112 exp [−(*t*/2500)^{3/4}]. The size of the oscillations at very large times that could be observed in the total sscf gives some estimate of the errors. The errors at short times are smaller than the thickness of the curves.

Dependencies of the different relaxation times on temperature for the system of 5488 particles. The time τ_{α} is obtained by fitting the tails of the intermediate self-scattering function with the stretched exponential. The time τ_{2} is obtained by fitting with stretched exponential the tails of the total sscf. Note that τ_{α} ≈ 3τ_{2}. The time τ_{ S } was calculated from the behavior of total sscf as it was approximated with twice the value of the *auto* term of sscf and the *auto* term was fitted with the sum of the compressed and stretched exponential functions. Note that τ_{α} is longer than τ_{ S } by nearly a factor of five. The τ_{ M } curve was obtained by explicit integration of the total sscf and by calculating the values of the shear modulus *G* from the instant configurations. Note that there is a relatively good agreement between τ_{ s } and η/*G* curves, as it should be. The time τ_{1} corresponds to the compressed exponent that describes short time behavior of the sscf.

Dependencies of the different relaxation times on temperature for the system of 5488 particles. The time τ_{α} is obtained by fitting the tails of the intermediate self-scattering function with the stretched exponential. The time τ_{2} is obtained by fitting with stretched exponential the tails of the total sscf. Note that τ_{α} ≈ 3τ_{2}. The time τ_{ S } was calculated from the behavior of total sscf as it was approximated with twice the value of the *auto* term of sscf and the *auto* term was fitted with the sum of the compressed and stretched exponential functions. Note that τ_{α} is longer than τ_{ S } by nearly a factor of five. The τ_{ M } curve was obtained by explicit integration of the total sscf and by calculating the values of the shear modulus *G* from the instant configurations. Note that there is a relatively good agreement between τ_{ s } and η/*G* curves, as it should be. The time τ_{1} corresponds to the compressed exponent that describes short time behavior of the sscf.

The dependencies of the microscopic viscosities on inclusion distance from the intermediate (5488), the large (43 904) and the folded/mapped sscf at 1500 K and cutoff in time 2000 fs. The folded/mapped sscf function was produced by the mapping of the sscf obtained on the large system onto the small system as described in the text. We see that the mapped curve is rather similar to the curve obtained on the intermediate size system. The final saturated value of the mapped viscosity should be the same as the final value of the viscosity calculated on the large system (note that it is the case). The differences between the folded viscosity and the viscosity calculated on the intermediate size system directly could originate in statistical error or in the fact that our large system is not the infinite size system. However, these differences could also contain additional information about the nature of the atomic level stresses and the stress waves. Further investigation of these details might be interesting.

The dependencies of the microscopic viscosities on inclusion distance from the intermediate (5488), the large (43 904) and the folded/mapped sscf at 1500 K and cutoff in time 2000 fs. The folded/mapped sscf function was produced by the mapping of the sscf obtained on the large system onto the small system as described in the text. We see that the mapped curve is rather similar to the curve obtained on the intermediate size system. The final saturated value of the mapped viscosity should be the same as the final value of the viscosity calculated on the large system (note that it is the case). The differences between the folded viscosity and the viscosity calculated on the intermediate size system directly could originate in statistical error or in the fact that our large system is not the infinite size system. However, these differences could also contain additional information about the nature of the atomic level stresses and the stress waves. Further investigation of these details might be interesting.

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