Finite-size effects on sound damping in stable computer glasses

TL;DR

This paper discusses how finite-size effects influence sound damping in stable computer glasses, providing insights into the limitations and implications of recent simulation results.

Contribution

It offers a critical analysis of finite-size effects on sound damping, highlighting their impact on interpreting simulation data in stable glasses.

Findings

Finite-size effects significantly affect sound damping measurements.

Analysis clarifies limitations of previous simulation results.

Provides guidelines for future computational studies.

Abstract

In this brief note we comment on the recent results presented in arXiv:1812.08736v1

Figures

b of Wang *et al.* ([2018](#bib.bib6)).

Equations

$${\Gamma }_{T,L}\sim \Delta \omega (k)\sim \frac{k\sqrt{n_q(k)}}{\sqrt{N}}fork<\frac{{\omega }_{†}(N)}{c_T}.$$

Full text

Finite-size effects on sound damping in stable computer glasses

Corrado Rainone1, Avraham Moriel2, Geert Kapteijns1, Eran Bouchbinder2, and Edan Lerner1

1Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands

2Chemical and Biological Physics Department, Weizmann Institute of Science, Rehovot 7610001, Israel

It is well known that disorder gives rise to attenuation of low frequency elastic waves in amorphous solids, even if dynamics are strictly confined to the harmonic regime Buchenau et al. (1992); Ganter and Schirmacher (2010); Marruzzo et al. (2013); Gelin et al. (2016); Mizuno and Ikeda (2018). In a recent preprint Wang et al. (2018), Szamel and coworkers present results regarding sound attenuation rates measured within the harmonic regime in very stable three-dimensional (3D) computer glasses; the key result announced in Wang et al. (2018) is the discovery of a new scenario of sound damping in which the shear (transverse) wave attenuation rate follows $\Gamma_{T}\!\sim\!k^{2}$ in the low wavenumber $k$ regime. This scenario has been asserted to be experimentally relevant.

In this brief note we show that the key result of Wang et al. (2018) is in fact a finite-size effect, which has been recently discussed and fully explained theoretically in Bouchbinder and Lerner (2018). Consequently, the sound damping scenario reported on in Wang et al. (2018) is neither new nor experimentally relevant; it will disappear in the thermodynamic limit of macroscopic glasses. Central to understanding the observations of Wang et al. (2018) is the following theoretical prediction for $\Gamma_{T}$ and $\Gamma_{L}$ (the sound/longitudinal wave attenuation rate) presented in Bouchbinder and Lerner (2018)

[TABLE]

Here $\omega_{\dagger}$ is a crossover frequency Bouchbinder and Lerner (2018), divided by the shear (transverse) wave-speed $c_{T}$, below which phonons cluster into discrete bands of disorder-induced width $\Delta\omega(k)$ and degeneracy $n_{q}(k)$ 111$n_{q}$ is the number of different solutions to the integer sum of squares problem $q\!=\!n_{x}^{2}\!+n_{y}^{2}\!+n_{z}^{2}$, and $k\!=\!2\pi\sqrt{q}/L$ in finite-size systems composed of $N$ particles, as demonstrated in Fig. 1. Finally, note that it has been shown that $\omega_{\dagger}(N)\!\sim\!N^{-2/(2{\,\mathchar 22\relax\mkern-12.0mud}+{\,\mathchar 22\relax\mkern-12.0mud}^{2})}$ in ${\,\mathchar 22\relax\mkern-12.0mud}$ dimensions Bouchbinder and Lerner (2018).

To support our main assertion that the low wavenumber regime reported on in Wang et al. (2018) is described by the finite-size theory prediction in Eq. (1), we present in Fig. 2a measurements of phonon band frequency widths $\Delta\omega(k)$, extracted (as described in Bouchbinder and Lerner (2018)) from the vibrational modes of between $50\!-\!100$ independent stable computer glasses footnote1 in ${\,\mathchar 22\relax\mkern-12.0mud}\!=\!3$, for $N\!=16\hbox{K},64\hbox{K},256\hbox{K}$, and plotted against the phonon band wavenumber $k$. We find that the widths approximately follow $\Delta\omega(k)\!\sim\!k^{2}$ (dashed line in Fig. 2a), which corresponds to the observations in Wang et al. (2018). This scaling, however, is an apparent one; the data in fact follow the finite-size theory prediction in Eq. (1), as shown next.

In Fig. 2b we plot the phonon band widths against the rescaled wavenumber $k\sqrt{n_{q}(k)}/\sqrt{N}$. These data establish that the phonon band widths follow the finite-size scaling as given by Eq. (1), even for very stable glasses, and consequently that the $k^{2}$ scaling reported on in Wang et al. (2018) will not persist in the thermodynamic limit $N\!\to\!\infty$, for which $\omega_{\dagger}(N)\!\to\!0$. Note, though, that the crossover wavenumber $k_{\dagger}\!\equiv\!\omega_{\dagger}/c_{T}$ depends very weakly on system size ($\sim\!N^{-2/15}$ in ${\,\mathchar 22\relax\mkern-12.0mud}\!=\!3$, consistent with our estimations shown in Fig. 1); varying $N$ by a factor of $4$, as done in Wang et al. (2018), changes $k_{\dagger}$ by merely $20\%$, hence explaining the apparent $N$-independent crossover to the finite-size regime observed in Fig. 2b of Wang et al. (2018).

Finally, an approximate quartic scaling $\Gamma_{L}\!\sim\!k^{4}$ of the sound (longitudinal) wave attenuation rate is also reported on in Wang et al. (2018), exhibiting no finite-size effects as predicted in Eq. (1). We assert that for the system sizes employed in Wang et al. (2018), (all but the very lowest-$k$) sound waves reside above $\omega_{\dagger}$ (see Fig. 1a-b, where sound waves are marked with arrows), explaining why sound attenuation rates are devoid of finite-size effects in this case. In very stable 2D glasses several sound waves do reside below $\omega_{\dagger}$, and consequently attenuation rates also follow the predicted finite-size scaling Eq. (1), as demonstrated in Fig. 1c.