Isomorphs in sheared binary Lennard-Jones glass: Transient response

We have studied shear deformation of binary Lennard-Jones glasses to investigate the extent to which the transient part of the stress strain curves is invariant when the thermodynamic state point is varied along an isomorph. Shear deformations were carried out on glass samples of varying stability, determined by cooling rate, and at varying strain rates, at state points deep in the glass. Density changes up to and exceeding a factor of two were made. We investigated several different methods for generating isomorphs but none of the previously developed methods could generate sufficiently precise isomorphs given the large density changes and nonequilibrium situation. Instead, the temperatures for these higher densities were chosen to give state points isomorphic to the starting state point by requiring the steady-state flow stress for isomorphic state points to be invariant in reduced units. In contrast to the steady-state flow stress, we find that the peak stress on the stress strain curve is not invariant. The peak stress decreases by a few percent for each ten percent increase in density, although the differences decrease with increasing density. Analysis of strain profiles and nonaffine motion during the transient phase suggests that the root of the changes in peak stress is a varying tendency to form shear bands, with the largest tendency occurring at the lowest densities. We suggest that this reflects the effective steepness of the potential; a higher effective steepness gives a greater tendency to form shear bands.

Figure 1

Examples of reduced-unit shear stress (top panels) and potential energy per particle (bottom panels) versus strain, from shear deformation of a binary Lennard-Jones glass with density 1.183 at temperature 0.3. The glass was prepared by cooling at the rate ${10}^{-7}$ (LJ units), while the deformation was carried out at reduced strain rate ${10}^{-5}$, meaning the real strain rate was ${10}^{-5}{\rho }^{1/3}{\left(T\right)}^{1/2}\simeq 5.8\times {10}^{-6}$. The left panels show curves from a single run, while the right panels show the average of 40 independent runs.

Figure 2

The five methods to obtain the temperature ratio $T_f/T_i$ used to identify the state point ${\rho }_f,T_f$ isomorphic to state point ${\rho }_i,T_i$ from simulations at the latter. The example here uses reference density and temperature ${\rho }_i=1.183$ and $T_i=0.3$ and a starting configuration cooled at $R_{c,0}={10}^{-5}$. We consider a new density ${\rho }_f=1.1{\rho }_i=1.301$. Black indicates results from NVT simulations and red is for the steady state from a shear simulation with the highest (reduced) strain rate $\widetilde{\stackrel{̇}{\epsilon }}={10}^{-3}$. For panels (a), (c), and (d) the system was sampled at regular intervals during the simulation and configurations uniformly scaled to ${\rho }_f$; the potential energy, forces, and shear stress were calculated on the scaled configurations, denoted with subscript $f$. (a) Temperature ratio given by the force method FM (blue), Eq. (2), and modified force method (green), Eq. (3) from the same shear simulation. The dotted horizontal lines indicate the corresponding temperature ratios from the NVT simulation. (b) Scatter-plot of the virial $W$ versus potential energy $U$. The slopes (correlation coefficient) of the two fits are 5.014 (0.869) and 5.143 (0.859), respectively, where the slopes can be considered estimates of the density scaling exponent $\gamma$, which yields the temperature factor via Eqs. (C4) and (C6). (c) Scatter-plot of $U_f$ against $U_i$ (DIC-pe). The slopes (correlation coefficient) are 1.593 (0.975) and 1.609 (0.972), respectively. Here the slopes correspond directly to the temperature ratios. (d) Scatter-plot of ${\sigma }_f$ against ${\sigma }_i$. The slopes (correlation coefficient) are 1.755 (1.00) and 1.757 (0.999). Here the temperature ratio is the slope divided by the density ratio (1.1). Table 1 gives the results of the different methods.

Figure 3

(a) temperature ratios (TR) from different methods against density for starting density 1.183. Solid lines are for NVT simulation (NVT) ${10}^7$ steps and dashed lines are for steady state (SSS). Inset zooms in for the largest two densities. (b) the ratio between the highest and lowest temperature ratios of the SSS from the top panel versus density. Solid line starts at $\rho =1.183$, and dashed line starts at $\rho =1.301$.

Figure 4

The reduced flow stress $\widetilde{{\sigma }_f}$ and reduced peak stress $\widetilde{{\sigma }_p}$ against temperature near the point of matching stress at ${\rho }_1=1.301$ (left two panels) and ${\rho }_9=2.789$ (right two panels). Glasses cooled (at lowest density) at rate $R_{c,0}={10}^{-5}$ and sheared at reduced strain rate $\dot{\stackrel{̃}{\epsilon }}={10}^{-3}$. Solid black lines are linear fits. Gray horizontal lines indicate $\widetilde{{\sigma }_f}$ in panels (a) and (b), and $\widetilde{{\sigma }_p}$ in panels (c) and (d) at the reference density with shaded region indicating error. Arrows in panels (a) and (b) point to $T$ estimated using: (i) DIC stress method; (ii) DIC PE method; (iii) $WU$/analytic; (iv) FMMOD; (v) FM.

Figure 5

The isomorph determined by matching flow stress again on glasses cooled (at lowest density) at rate $R_{c,0}={10}^{-5}$ and sheared at $\dot{\stackrel{̃}{\epsilon }}={10}^{-3}$. The large black diamond indicates the reference state point. The two black squares represent points whose temperatures were identified by matching the reduced flow stress to that of the reference, using the linear fits in Fig. 4, and the red diamonds are points whose temperatures have been determined by interpolation between the black squares, using Eq. (4).

Figure 6

Reduced stress-strain curves along isomorph determined by the procedure illustrated in Fig. 5. Glasses cooled (at lowest density) at rate $R_{c,0}={10}^{-5}$ and sheared at reduced strain rate $\dot{\stackrel{̃}{\epsilon }}={10}^{-3}$.

Figure 7

(a) $\widetilde{{\sigma }_f}$ and (b) $\widetilde{{\sigma }_p}$ against density along the isomorph for $R_{c,0}={10}^{-5},{10}^{-6},{10}^{-7}$ (black, red, blue), and $\widetilde{\stackrel{̇}{\epsilon }}={10}^{-3},{10}^{-4},{10}^{-5}$ (circle, square, triangle). Each point in panel (a) is an average of 40 shear simulations (on 40 individual configurations, or 30 for $\widetilde{\stackrel{̇}{\epsilon }}={10}^{-5}$) and the uncertainties are computed as described in Sec. 3b. For the peak stress [panel (b)], we first divide the 40 shear runs into five groups and obtain five averaged stress and strain curves. We then fit the data for 10% strain around the maximum stress using a fourth-degree polynomial and identify the maximum of the fit as the $\widetilde{{\sigma }_p}$. The error is the standard deviation of the five values divided by $\sqrt{5}$. The three families in panel (a) correspond to $\widetilde{\stackrel{̇}{\epsilon }}={10}^{-3},{10}^{-4},{10}^{-5}$ from top to bottom respectively. The errors are all smaller than the marker size. The position of the gray bars in panel (a) are the reference $\widetilde{{\sigma }_f}$ (at ${\rho }_1$ cooled with $R_{c,0}={10}^{-5}$ and sheared with $\widetilde{\stackrel{̇}{\epsilon }}={10}^{-3}$) and the width of the bar indicates the reference $\widetilde{{\sigma }_f}$ plus or minus the uncertainty.

Figure 8

Comparison between stress-strain curves obtained from shearing at the lowest density (blue), computing the stress from the same configurations scaled to the highest density (green), and actually shearing at the highest density (orange) with $\dot{\stackrel{̃}{\epsilon }}={10}^{-3}$ and $R_{c,0}={10}^{-7}$.

Figure 9

Collapse of (a) $\langle \mathrm{\Delta }{\tilde{r}}_{yz}^2\left(\tilde{t}\right)\rangle$ and (b) $g\left(\tilde{r}\right)$ (for AA pairs) for steady state only along an isomorph containing 10 points in the phase diagram calculated using the method described in Sec. 4b. The starting configuration is at ${\rho }_0=1.183$ and $T_0=0.3$ and was cooled with $R_{c,0}={10}^{-5}$. Shearing for all state points was with reduced strain rate $\dot{\stackrel{̃}{\epsilon }}={10}^{-4}$. The inset in panel (b) shows a close-up of the first peak.

Figure 10

Comparison of nonaffine particle motion over 12%-strain intervals between (a) transient state with $\epsilon \in [0,0.12]$ and (b) steady state with $\epsilon \in [3.65,3.77]$. Data is averaged over all particles and 40 independent runs. Color indicates the three components of the nonaffine motion squared and shown in the legend in panel (a).

Figure 11

Example of a displacement profile exhibiting shear-banding. Displacement is calculated in steady state between $\epsilon =3.65$ and $\epsilon =3.77$ from a shear simulation at $\dot{\stackrel{̃}{\epsilon }}={10}^{-4}$ from a configuration at $\rho =1.301$ originally cooled at $R_{c,0}={10}^{-5}$, by averaging the $x$ component of the particles' displacements over all particles within a bin defined by their $y$ coordinate. The straight line is the affine displacement following the applied strain. Dotted red lines are linear fits at different regions with two distinct slopes (treating the leftmost and rightmost regions together due to periodic boundary conditions). The norm, defined as the root mean-square deviation of the bin values from the linear profile, is 0.022. The inset shows the histogram of norms of all 40 configurations at the corresponding density, cooling rate, and shear rate. The vertical dashed line indicates the mean value of the norm.

Figure 12

Comparison of norm of nonaffine motion in velocity direction averaged of 40 configurations over 12%-strain intervals between transient state with $\epsilon \in [0,0.12]$ and steady state with $\epsilon \in [3.65,3.77]$. Errors are the standard deviation of 40 divided by square root of 40. The definition of norm is the mean-squared deviation of the displacement profile in Fig. 11.

Figure 13

Reduced peak stress ${\tilde{\sigma }}_p$ and flow stress ${\tilde{\sigma }}_f$ as function of reduced strain rate $\widetilde{\stackrel{̇}{\epsilon }}$ (a, c) and cooling rate $R_{c,0}$ (b, d). Black, red, and blue are for $R_{c,0}={10}^{-5},{10}^{-6},{10}^{-7}$; sphere, square, and up triangle are for $\widetilde{\stackrel{̇}{\epsilon }}={10}^{-3},{10}^{-4},{10}^{-5}$. The flow stress ${\tilde{\sigma }}_f$ is within errors independent of $R_{c,0}$ but increases with $\widetilde{\stackrel{̇}{\epsilon }}$. See the text for a discussion of the uncertainties of these measurements, which for these data are smaller than the symbol size. The three horizontal lines in panel (d) are the average of the corresponding three points of the same $\widetilde{\stackrel{̇}{\epsilon }}$.

Figure 14

Reduced potential energy against strain sheared with $\widetilde{\stackrel{̇}{\epsilon }}={10}^{-3},{10}^{-5}$ for the three cooling rates $R_{c,0}={10}^{-5}$ (blue and orange), ${10}^{-6}$ (green and red), ${10}^{-7}$ (purple and brown) at $\rho =1.183$ and $T=0.3$. Each curve is an average of 40 simulations.

Figure 15

Stress and strain curves of glasses cooled at three rates and sheared at reduced strain rate $\dot{\stackrel{̃}{\epsilon }}={10}^{-3}$. We only show strain up to 1 so that the peak is more visible; only four densities are shown for clarity. The black horizontal dashed lines indicate the flow stress. These curves are where the data of Fig. 7 are derived from.

Figure 16

Same as in Fig. 15 but now for glasses cooled at three rates and sheared at reduced strain rate $\dot{\stackrel{̃}{\epsilon }}={10}^{-4}$.

Figure 17

Same as in Fig. 15 but now for glasses cooled at three rates and sheared at reduced strain rate $\dot{\stackrel{̃}{\epsilon }}={10}^{-5}$.

Figure 18

Collapse of $g\left(\tilde{r}\right)$ for AB and BB pairs as complement to Fig. 9.