Shearing small glass-forming systems: A potential energy landscape perspective

Understanding the yielding of glass-forming systems upon shearing is notoriously difficult since it is a strong nonequilibrium effect. Here we show that the concept of the potential energy landscape (PEL), developed for the quiescent state, can be extended to shearing. When introducing an appropriate coarse graining of the extended PEL for sheared systems, one can distinguish two fundamentally different types of plastic events, namely, inherent structure (IS) vs minimized structure (MS) transitions. We apply these concepts to noncyclic and cyclic shearings of small systems, which allow us to characterize the properties of elementary plastic events. Whereas the general properties of the stress-strain curves are similar to larger systems, a closer analysis reveals significantly different properties. This allows one to identify the impact of the elastic coupling in larger systems. The concept of MS enables us to relate the stress overshoot of a single trajectory to the emergence of a MS transition. Furthermore, the occurrence of limit cycles can be characterized in great detail for the small systems, and connections to the properties of the PEL can be formulated. Possible implications of our small-system results for the macroscopic limit are discussed.

Figure 1

Illustration of the minimization routine for a one-dimensional potential, shown for different strain values. Displayed are the contours of a specifically designed sixth order polynomial. The solid lines show the taken paths where the inherent structures would evolve by increasing or decreasing the strain. The vertical lines mark the strain values for which the potential is plotted in the inset.

Figure 2

(a) Lowest three eigenvalues of the Hessian matrix at the start of a shear simulation, starting at $\epsilon \approx -4.65$ with $T=0.01,N=130$. The dashed and solid vertical lines mark the detected transitions between ISs and MSs. (b) Energy and (c) stress of the inherent structures for the same simulation.

Figure 3

The average number of MS and $D_{\min }$ transitions since the start of the simulation. $D_{\min }$ transitions are used to detect IS transitions. The solid line denotes the strain at the overshoot maximum ${\gamma }_{\mathrm{peak}}$.

Figure 4

IS and MB activation energies for $N=130$ particles calculated from $\langle \tau |\epsilon ;T\rangle$. The activation energy for jumps to a higher or lower energetic IS is denoted by “up” and “down,” respectively. We also show the difference between the start energy and energy at the stress peak $\mathrm{\Delta }\epsilon ={\epsilon }_{\mathrm{peak}}-{\epsilon }_{\mathrm{start}}$. The dashed lines represent linear fits in the range of ${\epsilon }_{\mathrm{MB}},{\epsilon }_{\mathrm{start}}\in [-4.6,-4.52]$. The slope for $V\left(\epsilon \right)/N$ is $-0.275\left(19\right)$, and the slope for $\mathrm{\Delta }\epsilon$ is $-0.478\left(52\right)$.

Figure 5

Ensemble averaged (a) stress and (b) energy. The solid lines represent IS energies and stresses, and the dotted lines represent MS energies and stresses. The vertical lines show the position of the maximum of the stress overshoot for the respective starting energy. The value of ${\epsilon }_{\mathrm{start}}$ increases in direction of the arrow.

Figure 6

Probability distribution of the first MS transition ${\gamma }_{\mathrm{MS}}^1$. The dashed lines represent fits with Gaussian distributions having a fixed width of $\sigma =0.0675$ and a cutoff at ${\gamma }_{\mathrm{MS}}^1=0$. The value of ${\epsilon }_{\mathrm{start}}$ increases in direction of the arrow.

Figure 7

Change in MS properties after the first MS transition as a function of ${\gamma }_{\mathrm{MS}}^1$, i.e., the strain, where this transition occurs. Included are linear fits for ${\gamma }_{\mathrm{MS}}^1\le 0.05$ and ${\gamma }_{\mathrm{MS}}^1\in [0.1,0.2]$. (a) Change in MS energy. Intersection point: ${\gamma }_{\mathrm{MS}}^1=0.08$ for $N=130$. (b) Change in MS strain. Intersection point: ${\gamma }_{\mathrm{MS}}^1=0.10$ for $N=130$. (c) Euclidean distance between the first and the second MSs. Average intersection point: ${\gamma }_{\mathrm{MS}}^1=0.10$.

Figure 8

MS energy after the first MS transition with respect to the start energy ${\epsilon }_{\mathrm{start}}$. The vertical line marks the position of the average crossover at $\gamma =0.0765\left(46\right)$.

Figure 9

Average stress overshoot peak position for all trajectories with the MS transition at a given strain. The solid black line reflects the diagonal.

Figure 10

Probability that after $n_{\mathrm{cycle}}$ cycles the system has visited a limit cycle. The value of ${\gamma }_{\max }$ increases in direction of the arrow.

Figure 11

(a) The red points: Average value of $n_{\mathrm{cycle}}$ where a limit cycle is entered in cyclic AQS simulations at $N=130$ as a function of ${\gamma }_{\max }$. The dark blue line denotes a power-law fit of the form ${\gamma }_{\mathrm{start}}=c{({\gamma }_{\text{nl}}-{\gamma }_{\max })}^{\delta }$ with the fit parameters $c=1.44\times {10}^{-6},\delta =-5.49$, and ${\gamma }_{\text{nl}}=0.231$. The blue points: The value of $n_{\mathrm{cycle}}$ where 90% of all simulations have entered a limit cycle. (b) Probability to be already in a limit cycle at the start of a cyclic AQS simulation at $N=130$. The solid red line represents an approximation using MS statistics (see the text). For comparison the dashed red line shows $P({\gamma }_{\mathrm{MS}}>{\gamma }_{\max })$.

Figure 12

Probability to visit the starting MS by AQS shear in the reverse direction (up to $\gamma =-0.3$) after the first MS transition in noncyclic simulations. The solid lines represent linear fits to the data, and for $N=130$, it has a slope of $-2.64\left(20\right)$ and an intercept of 0.74(3). By extrapolation, $p_{\mathrm{return}}$ is zero for ${\gamma }_{\mathrm{MS}}\ge 0.279\left(10\right)$ at $N=130$.

Figure 13

(a) The average MS energy after $n_{\mathrm{cycle}}$ cycles for different values of ${\gamma }_{\max }$. The value of ${\gamma }_{\max }$ increases in direction of the arrow. (b) The final MS energy after 100 AQS cyclic shear simulations together with the final MS energy under the condition that the system is in a limit cycle after 100 cycles. Additionally the MS energy from FIG. 5 is included.

Figure 14

Probability density $p\left({\sigma }_{\mathrm{next}}\right|{\sigma }_{\mathrm{end}})$. The stress before the MS and the non-MS transition is called ${\sigma }_{\mathrm{end}}$, and the stress after the transition is called ${\sigma }_{\mathrm{next}}$. To calculate the probabilities, we used all transitions with $\gamma \le 1.0$, including both the overshoot region and the flow regime. The value of ${\sigma }_{\mathrm{end}}$ increases in direction of the arrow.

Figure 15

Distribution of energy drops at MS and non-MS transitions. The straight black line reflects a power-law behavior with a slope of $-\frac{5}{4}$.

Figure 16

Parametrization of the ${\gamma }_{\mathrm{MS}}^1$ distribution, which has a cutoff Gaussian shape. The blue line shows the fit parameter $\mu$. The black straight line represents a fit to these values with a slope of $-0.953\left(28\right)$ and an intercept of $-4.26\left(13\right)$. The red line shows the average value of ${\gamma }_{\mathrm{MS}}$, calculated directly from the distribution. From the linear fit of $\mu$, we also calculated the average ${\gamma }_{\mathrm{MS}}$, which is shown as the black dashed line. For all cutoff Gaussian distributions, we used a fixed $\sigma =0.0675$, which we determined as the mean fit parameter for the lowest ${\epsilon }_{\mathrm{start}}$.

Figure 17

Scatter plot of the position of the first MS transitions. The color shows if the starting structure can be found to shear in the reverse direction until $\gamma =-0.3$ immediately after the MS transition.

Figure 18

(a) Stress of subensembles with a given strain ${\gamma }_{\mathrm{MS}}<0.09$. Shown are trajectories for temperatures $T\in \{1\times {10}^{-4},5\times {10}^{-4}\}$. (b) The same for ${\gamma }_{\mathrm{MS}}\in (0.09,0.2)$ and $T={10}^{-4}$. (c) For ${\gamma }_{\mathrm{MS}}>0.21$ we plot all available values for $T\in [{10}^{-4},{10}^{-2}]$.