Local versus Global Stretched Mechanical Response in a Supercooled Liquid near the Glass Transition

Amorphous materials have a rich relaxation spectrum, which is usually described in terms of a hierarchy of relaxation mechanisms. In this work, we investigate the local dynamic modulus spectra in a model glass just above the glass transition temperature by performing a mechanical spectroscopy analysis with molecular dynamics simulations. We find that the spectra, at the local as well as on the global scale, can be well described by the Cole-Davidson formula in the frequency range explored with simulations. Surprisingly, the Cole-Davidson stretching exponent does not change with the size of the local region that is probed. The local relaxation time displays a broad distribution, as expected based on dynamic heterogeneity concepts, but the stretching is obtained independently of this distribution. We find that the size dependence of the local relaxation time and moduli can be well explained by the elastic shoving model.

Figure 1

Local mechanical spectroscopy analysis. (a) Inherent structure energy vs temperature during the quench process. The solid lines from top to bottom represent three different quenching rates: $4.35\times {10}^{-3}$, $4.35\times {10}^{-4}$, and $4.35\times {10}^{-5}$. Solid points represent the quenching rate $1.67\times {10}^{-6}$. The state used for the mechanical spectroscopy analysis in this work is shown by the box, at temperature $T=0.435$, close to the mode coupling temperature of the model [20]. (b) Schematic representation of the local mechanical spectroscopy analysis for a selected region ($r_c$) with sinusoidal strain loading. (c) Local stress response of a region of size $r_c=7$ for two different frequencies shifted by one for clarity (upper, $\omega =2\pi /8000$; lower, $\omega =2\pi /2000$) in the first five cycles.

Figure 2

Probability distribution of local dynamic moduli for $r_c=7$. (a),(b) Probability distribution of local storage modulus $G^{\prime }(\omega )$ and local loss modulus $G^{\prime \prime }(\omega )$ for different frequencies. (c) Probability distribution of loss and storage modulus ratio $[G^{\prime \prime }(\omega )/G^{\prime }(\omega )]$; (d) collapse of the probability distribution in (c) by rescaling the data with ${\omega }^{\beta }$ for different loading frequencies (see the text and Fig. 4 for values of $\beta$). The solid line in (a) is a Gaussian distribution; the solid line in (b),(c) and the dashed line in (d) are Gumbel distributions [24]. Note that there is no particular physical reason to choose those distributions except to capture the trend of the data with the loading frequency.

Figure 3

Dynamic moduli spectra for different sizes. The upper panel shows the storage moduli and the lower panel the loss moduli, and the solid lines are a fit to the modified Cole-Davidson formula. Error bars are of the order of the symbol size and are obtained by averaging over different numbers of local regions, for $r_c=5$, 512 regions; $r_c=6$, 7, 10, 216 regions; and $r_c=15$, 64 regions, and for the bulk sample, ten independent samples.

Figure 4

Scale dependence of the heterogeneous relaxation parameters. (a) Cole-Davidson stretch exponent $\beta$ for different local radii (the dashed line represents the bulk value). (b),(c) Local $G_{\infty }$ and local relaxation time $\tau (r_c)$ vs inverse confinement radius $1/r_c$. The solid curve is a best fit of the data with the formula $(A/r_c)+(B/r_c^3)$ (see the text). (d) Correlation between local relaxation time $\tau$ and local $G_{\infty }$. Error bars are obtained from fitting the data in Fig. 3 within a 95% confidence interval.