Isomorph invariance of dynamics of sheared glassy systems

We study hidden scale invariance in the glassy phase of the Kob-Andersen binary Lennard-Jones system. After cooling below the glass transition, we generate a so-called isomorph from the fluctuations of potential energy and virial in the $NVT$ ensemble: a set of density, temperature pairs for which structure and dynamics are identical when expressed in appropriate reduced units. To access dynamical features, we shear the system using the SLLOD algorithm coupled with Lees-Edwards boundary conditions and study the statistics of stress fluctuations and the particle displacements transverse to the shearing direction. We find good collapse of the statistical data, showing that isomorph theory works well in this regime. The analysis of stress fluctuations, in particular the distribution of stress changes over a given strain interval, allows us to identify a clear signature of avalanche behavior in the form of an exponential tail on the negative side. This feature is also isomorph invariant. The implications of isomorphs for theories of plasticity are discussed briefly.

Figure 1

Black symbols indicate an isomorph in the supercooled liquid which includes the point $(\rho =1.2,T=0.44)$. We use this as a guide to locating the glass transition; its relaxation time is about 2700 in reduced units, corresponding to 3850 in LJ units at the lowest density 1.2. The inset shows the intermediate scattering function for the different state points, lying almost on top of each other. The red and green symbols indicate isomorphs generated in the glass which we use for studying deformation, referred to as those starting at temperatures $T=0.55$ and 0.1, respectively. Note that the starting densities are not the same, since these are taken from a cooling run at fixed pressure $P=10$.

Figure 2

Radial distribution function (RDF) for the large ($A$) particles in reduced units along glassy isomorphs starting at (a) $T=0.55$ and (b) $T=0.1$. Each figure shows ten curves, where the density is increased by 1% for each state point, giving a 9.4% change in density overall; the temperature increases by 54% overall. The insets show close-ups of (a) the first peak and (b) the second peak, where very some small deviations are discernible.

Figure 3

(a) Section of the stress-strain curve for the lowest-density state point on the higher-temperature isomorph $(\rho =1.265,T=0.550)$ at the lowest nominal strain rate ${10}^{-5}$. (b) Section of the stress-strain curve for a state point on the lower-temperature isomorph $(\rho =1.324,T=0.100)$ at the lowest strain rate ${10}^{-5}$. The abrupt drops can be identified with avalanches of plastic activity. The difference in vertical scale between (a) and (b) can be attributed partly to the definition of reduced units for stress.

Figure 4

Flow stress and standard deviation during the steady-state regime as a function of density along high-temperature isomorphs (left panels) and low-temperature isomorphs (right panels) for different strain rates. The legend indicates the nominal strain rates, that is, the real strain rates at the first point on each isomorph; for other state points in each data set, the reduced unit strain rate is the same. Error bars have been calculated as the error of the mean and of the standard deviation, assuming the data are normally distributed [see Eq. (4.14) and Appendix E of Ref. 40]. We choose the effective number of data points to be the total steady-state strain divided by 0.05; we believe data points are statistically independent for strain increments of 0.05. The horizontal lines indicate the mean value for the isomorph.

Figure 5

Normalized shear stress autocorrelation functions along the (a) high- and (b) low-temperature isomorphs for three different strain rates. Curves have been shifted for clarity. The dashed lines indicated fits using a compressed exponential function for the first curve in each set (lowest density and temperature); the parameters can be seen in Fig. 6.

Figure 6

Fits of shear stress autocorrelation to Eq. (6) shown as functions of density along the (a) high- and (b) low-temperature isomorphs. The characteristic strain over which decays occurs, ${\varepsilon }_c$, decreases approximately linearly as density increases.

Figure 7

Histograms of stress changes of intervals as indicated for the high-temperature isomorph for different strain rates. For each strain rate and $\mathrm{\Delta }\varepsilon$, distributions from the ten members of the isomorph are plotted in the same color. The fact that they appear as one curve for each color, apart from broadening due to statistical noise at the lowest strain rates, indicates a high degree of collapse. The distributions are essentially Gaussian for all strain rates and strain intervals $\mathrm{\Delta }\varepsilon$, and their widths are relative insensitive to $\mathrm{\Delta }\varepsilon$ even at the lowest strain rates, indicating that most of the fluctuations are thermal rather than strain driven. The inset in (d) shows an alternative way of exhibiting isomorph invariance for $\mathrm{\Delta }\varepsilon =0.000512$, by coloring different members of the isomorph differently, and on a linear scale.

Figure 8

Histograms of stress changes of intervals as indicated for the low-temperature isomorph for different strain rates. As in Fig. 7, distributions for a given strain rate and strain interval, but different members of the isomorph, are plotted in the same color. They are Gaussian for the largest strain intervals $\mathrm{\Delta }\varepsilon$ as well as for the shortest $\mathrm{\Delta }\varepsilon$ at the slowest strain rate, where the contribution of strain to the fluctuations is negligible compared to the thermal contribution. For larger $\mathrm{\Delta }\varepsilon$ at the slowest strain rate, an exponential tail on the negative side is a clear indication of plastic events organizing into avalanches. For even larger $\mathrm{\Delta }\varepsilon$ and at the larger strain rates, mixing of thermal and mechanic noise, and multiple avalanches lead to more disorganized histograms. The inset of (d) shows, on a linear scale, distributions of the second smallest strain interval with the different members of the isomorph represented with different colors as an alternative check of the invariance.

Figure 9

Fisher-Pearson skewness $S_{\text{FP}}$ of (reduced) stress drop distributions as a function of strain interval for different strain rates for the low-temperature isomorph. Different curves of the same color correspond to different points on the isomorph.

Figure 10

Histograms of (reduced) stress changes over strain intervals ${\varepsilon }_s$ chosen to minimize skewness for each strain rate, on the low-temperature isomorph. Data for different points on the isomorph are plotted in the same color for each nominal strain rate. In order of decreasing strain, the minimum-skew strain intervals, as judged by eye from Fig. 9, are 0.02, 0.008, 0.004, and 0.003.

Figure 11

Self-part of the intermediate scattering function for larger ($A$) particles based on particle displacements transverse to the shearing direction for (a) the high-temperature isomorph and (b) the low-temperature isomorph, for different strain rates.

Figure 12

Mean-square transverse displacement plotted in reduced units for (a) the high-temperature isomorph and (b) the low-temperature isomorph. The horizontal arrows indicate a factor of 10 in the time axis and can be used to judge by what factor the curves can be shifted onto each other in time.

Figure 13

The MSD curves from Figs. 12 and 12 plotted together, though without the short-time parts. At low strain rates the MSD appears to become independent of the isomorph, as well as which point on the isomorph. The definition of reduced units means that the curves for the different isomorphs are plotted in terms of different timescales, so caution is required when drawing conclusions from the apparent collapse.