Understanding the atomic-level Green-Kubo stress correlation function for a liquid through phonons in a model crystal

In order to gain insight into the connection between the vibrational dynamics and the atomic-level Green-Kubo stress correlation function in liquids, we consider this connection in a model crystal instead. Of course, vibrational dynamics in liquids and crystals are quite different and it is not expected that the results obtained on a model crystal should be valid for liquids. However, these considerations provide a benchmark to which the results of the previous molecular dynamics simulations can be compared. Thus, assuming that vibrations are plane waves, we derive analytical expressions for the atomic-level stress correlation functions in the classical limit and analyze them. These results provide, in particular, a recipe for analysis of the atomic-level stress correlation functions in Fourier space and extraction of the wave-vector and frequency-dependent information. We also evaluate the energies of the atomic-level stresses. The energies obtained are significantly smaller than the energies previously determined in molecular dynamics simulations of several model liquids. This result suggests that the average energies of the atomic-level stresses in liquids and glasses are largely determined by the structural disorder. We discuss this result in the context of equipartition of the atomic-level stress energies. Analysis of the previously published data suggests that it is possible to speak about configurational and vibrational contributions to the average energies of the atomic-level stresses in a glass state. However, this separation in a liquid state is problematic. We also introduce and briefly consider the atomic-level transverse current correlation function. Finally, we address the broadening of the peaks in the pair distribution function with increase of distance. We find that the peaks' broadening (by $\approx 40\%$) occurs due to the transverse vibrational modes, while contribution from the longitudinal modes does not change with distance.

Figure 1

Dispersion curves for the longitudinal and transverse waves in the continuous spherical approximation. Thick lines show the results obtained without the long-wavelength approximation. Thin lines show the results obtained with the long-wavelength approximation. Note that $\left(Q_{\max }a\right)\cong 3.9$.

Figure 2

Functions $H_L^p\left(qa\right)$, $H_L^{xy}\left(qa\right)$, and $H_T^{xy}\left(qa\right)$ from (44, 45, 46). Note that the $H_L^{xy}\left(qa\right)$ curve was scaled by 100, while the $H_T^{xy}\left(qa\right)$ curve was scaled by 50. Thus the contribution of the transverse waves to the average square of the shear stress is significantly larger than the contribution from the longitudinal waves. Also note that there are two polarizations of the transverse waves, while the figure shows the contribution from one polarization only.

Figure 3

Pressure-pressure correlation function without the long-wavelength approximation. The scaled time is ${\omega }_{o}t$.

Figure 4

Pressure-pressure correlation function with the long-wavelength approximation. The scaled time is ${\omega }_{o}t$.

Figure 5

Contribution from one polarization of the transverse waves to the shear-stress correlation function. The scaled time is ${\omega }_{o}t$. No damping of the waves is assumed.

Figure 6

Scaled shear-stress correlation function, $f(t,r)$ (66), due to the transverse waves with exponential damping in time and its Fourier transforms. (a) $r$-scaled shear-stress correlation function (62, 66) integrated over all $q$ with $E({\omega }_{\mathit{q}},t)$ given by (64). It was assumed that the numerical prefactor is equal to 1. Scaled time is ${\omega }_{o}t$. (b) Time to frequency Fourier transform of the scaled stress correlation function in the panel (a). Angular frequency is measured in units of ${\omega }_o$. (c) Distance to wave vector Fourier transform of the scaled stress correlation function in the panel (a). The unit of the wave vector is $1/a$. (d) Time to frequency and distance to wave-vector Fourier transforms of the scaled stress correlation function in the panel (a). Note that the center of the diagonal high-intensity region follows the dispersion curve for the transverse waves in Fig. 1. It is clear that the broadening of the dispersion curve is caused by the exponential damping (64).

Figure 7

Contributions to the peak's width in the pair distribution function from longitudinal and transverse waves as a function of distance. The blue curve represents contributions from all longitudinal waves. The red curve represents contributions from one polarization of all transverse waves.