Understanding Plastic Deformation in Thermal Glasses from Single-Soft-Spot Dynamics

By considering the low-frequency vibrational modes of amorphous solids, Manning and Liu [Phys. Rev. Lett. 107, 108302 (2011)] showed that a population of “soft spots” can be identified that are intimately related to plasticity at zero temperature under quasistatic shear. In this work, we track individual soft spots with time in a two-dimensional sheared thermal Lennard Jones glass at temperatures ranging from deep in the glassy regime to above the glass transition temperature. We show that the lifetimes of individual soft spots are correlated with the time scale for structural relaxation. We additionally calculate the number of rearrangements required to destroy soft spots and show that most soft spots can survive many rearrangements. Finally, we show that soft spots are robust predictors of rearrangements at temperatures well into the supercooled regime. Altogether, these results pave the way for mesoscopic theories of plasticity of amorphous solids based on dynamical behavior of individual soft spots.

Popular Summary

All around us, solids flow under deformation: Forks are bent, glass is shattered, and butter is spread. This process has long been understood for crystalline materials such as the metals that we use to build our planes, trains, and automobiles. Almost all of the atoms in a crystal have environments that look indistinguishable from one another. However, certain places in crystals, called defects, have different environments from the rest of the system and typically feature extra or missing atoms in their neighborhoods. When a crystal deforms, atoms preferentially rearrange near these defects, allowing the material to assume its new shape. We show that the deformation of an amorphous solid can be understood by considering regions of the material called “soft spots” that are analogous to defects in crystals.

In disordered solids such as glass and sand, constituent atoms or particles are arranged at random and no two locations look the same. Since the particles do not follow any particular pattern, it is extremely difficult to identify special places in the material—the so-called soft spots—where particles should rearrange when the material is deformed. Previous studies have used sound waves to acoustically probe the positions of soft spots, overcoming the inherent difficulty in locating soft spots before atomic rearrangement occurs. We use numerical simulations of a rapidly deformed glass to show that particle rearrangements occur preferentially at soft spots over a range of temperatures, from well below the glass transition to above it. Furthermore, we find that the lifetimes of individual soft spots are related directly to the rate at which the material relaxes in response to an applied stress.

An understanding of crystalline defects has helped scientists and engineers to develop crystalline materials with more desirable material properties. Likewise, our analysis builds toward an understanding of amorphous solids in terms of soft spots that could enable the development of disordered solids with improved properties.

Figure 1

An example configuration of the system at $T=0.1$ and $\dot{\gamma }=10^{-4}$. Particles are colored according to their $D_{\min }^2$ value (see text). Particles outlined in black are members of the soft spots for this configuration. The soft spots have been generated using $N_m=430$ and $N_p=20$. Inset: A single soft spot coinciding with a rearrangement.

Figure 2

The probability of a particle residing in a soft spot as a function of its $D_{\min }^2$ value. Panel (a) shows a comparison of the temperatures studied from $T=0.1$ (in blue) to $T=0.4$ (in red) at a strain rate of $\dot{\gamma }=10^{-4}$, and panel (b) shows a comparison of strain rates studied from $\dot{\gamma }=10^{-5}$ (in dark blue) to $\dot{\gamma }=10^{-3}$ (in green) at a temperature of $T=0.1$. In all cases, we see that the probability increases with $D_{\min }^2$ until some threshold $D_{\mathrm{th}}^2\simeq 15T$ (vertical dashed line) at which point the probability reaches some plateau value $P^{*}$. The soft-spot density ${\rho }_{SS}$ is marked by a horizontal dashed line.

Figure 3

The difference in probability, $\mathrm{\Delta }{P}^{*}=P^{*}(T,\dot{\gamma })-{\rho }_{SS}$, as a function of $N_m$ and $N_p$ for a temperature of $T=0.1$ and strain rate $\dot{\gamma }=10^{-4}$. We see a broad plateau over which $\mathrm{\Delta }{P}^{*}$ is largely independent of $N_m$ and $N_p$ with a weak maximum occurring at $N_m^{\star }=430$ and $N_p^{\star }=20$ (marked by a star.) The behavior of $\mathrm{\Delta }{P}^{*}$ is largely independent of temperature and strain rate.

Figure 4

The plateau probability $P^{*}$ for a particle with high $D_{\min }^2$ to reside in a soft spot, normalized by the soft-spot density, $P_{SS}$. This represents how much more likely rearrangements are to be found at soft spots than if the soft-spot map were completely uncorrelated with rearrangements. A value of 1 (dashed line) represents the uncorrelated value. The ratio is 0 when the soft-spot map is anticorrelated and so describes no plastic activity. The value of 5.2 represents the maximum possible value of $P^{*}$, $1/{\rho }_{SS}$, which occurs if all of the plastic activity resides in soft spots. Data are shown for strain rates $\dot{\gamma }=10^{-5}$ (in dark blue) to $\dot{\gamma }=10^{-3}$ (in green).

Figure 5

Correlation functions of the $D_{\min }^2$ field and the soft-spot population. On the left are comparisons of the temperatures considered from $T=0.1$ (in blue) to $T=0.4$ (in red) at a strain rate of $\dot{\gamma }=10^{-4}$. In the middle column are comparisons of the strain rates considered from $\dot{\gamma }=10^{-5}$ (in dark blue) to $\dot{\gamma }=10^{-3}$ (in green) scaled by the strain rate, $t/\tau \to \dot{\gamma }t$ at a temperature $T=0.1$. On the right are comparisons of the strain rates considered from $\dot{\gamma }=10^{-5}$ (in brown) to $\dot{\gamma }=10^{-3}$ (in yellow) scaled by the strain rate, $t/\tau \to \dot{\gamma }t$ at a temperature $T=0.4$. Figures (a)–(c) show the self-intermediate scattering function $F(q_{\max },t)$ evaluated at $\mathit{q}=2\pi /{\sigma }_{AA}\widehat{\mathit{y}}$. Figures (d)–(f) show the autocorrelation function for $D_{\min }^2$. Figures (g)–(i) show the autocorrelation function for the soft-spot population. In (g), at each temperature, comparisons are made with the cumulative probability density function for individual soft-spot lifetimes, introduced in Sec. 5, overlaid in dashed lines. In (h), a single comparison is made to $P({\tau }_L\ge \delta t)$, shown using a dashed black line, for lifetimes aggregated from lifetimes collected at all three different strain rates. In (i), predictions from the cumulative probability density function for individual soft-spot lifetimes are shown for the two faster strain rates overlaid in a dashed line. Figures (j)–(l) show the cross correlation between $D_{\min }^2$ and the soft-spot population. The plots in both the middle and left columns feature two vertical dashed lines to serve as guides to the eye. The earlier line occurs at a time ${\tau }^{*}$ when the self-intermediate scattering function first drops below the plateau. The later line occurs at the relaxation time ${\tau }_{\alpha }$ defined so that $F(q_{\max },{\tau }_{\alpha })\sim e^{-1}$.

Figure 6

(a)–(c) Probability distributions for single-soft-spot lifetimes at different temperatures and strain rates, respectively. In (a), temperatures of $T=0.1$ (blue) to $T=0.4$ (red) are shown at a strain rate of $\dot{\gamma }=10^{-4}$. Overlaid in dashed lines are the predictions of the discrete model. In (b), we show lifetime distributions at strain rates of $\dot{\gamma }=10^{-5}$ (dark blue) to $\dot{\gamma }=10^{-3}$ (green) at a temperature of $T=0.1$. Again, predictions from the discrete model are shown in dashed lines for strain rates of $\dot{\gamma }=10^{-4}$ and $\dot{\gamma }=10^{-3}$. In (c), we show lifetime distributions at strain rates of $\dot{\gamma }=10^{-5}$ (brown) to $\dot{\gamma }=10^{-3}$ (yellow) at a temperature of $T=0.4$. Predictions from the discrete model are overlaid in dashed lines. Figures (e)–(f) show the probability distributions for the number of rearrangements needed to destroy a single soft spot using the same color scheme as in (a)–(c). In all cases, we see that the number of rearrangements appears to be power-law distributed with an exponential tail. For each distribution, the mean number of rearrangements needed to destroy a soft spot is overlaid in a dashed line.