Heterogeneous relaxation dynamics in amorphous materials under cyclic loading

Molecular dynamics simulations are performed to investigate heterogeneous dynamics in amorphous glassy materials under oscillatory shear strain. We consider three-dimensional binary Lennard-Jones mixture well below the glass transition temperature. The structural relaxation and dynamical heterogeneity are quantified by means of the self-overlap order parameter and the dynamic susceptibility. We found that at sufficiently small strain amplitudes, the mean square displacement exhibits a broad subdiffusive plateau and the system undergoes nearly reversible deformation over about ${10}^4$ cycles. Upon increasing strain amplitude, the transition to the diffusive regime occurs at shorter time intervals and the relaxation process involves intermittent bursts of large particle displacements. The detailed analysis of particle hopping dynamics and the dynamic susceptibility indicates that mobile particles aggregate into clusters whose sizes increase at larger strain amplitudes. Finally, the correlation between particle mobilities in consecutive time intervals demonstrates that dynamic facilitation becomes increasingly pronounced at larger strain amplitudes.

Figure 1

A snapshot of instantaneous positions of particles $A$ (large blue circles) and $B$ (small red circles) at mechanical equilibrium. The particle sizes are not drawn to scale. During the oscillatory motion, the periodic shear strain was applied in the $xz$ plane (indicated by the arrows).

Figure 2

The mean square displacement of $A$ and $B$ particles as a function of time for the oscillation frequency $\omega \tau =0.02$ and period $T=2\pi /\omega =314.16\tau$. The strain amplitudes from bottom to top are ${\gamma }_0=0.02,0.03,0.04,0.05,0.06,0.07,\text{and}0.08$. The dashed line with unit slope is plotted for reference.

Figure 3

The self-correlation function $Q_s(a,t)$ defined by Eq. (2) for the oscillation frequency $\omega \tau =0.02$ and period $T=2\pi /\omega =314.16\tau$. The probed length scale is $a=0.12\sigma$. The strain amplitudes from top to bottom are ${\gamma }_0=0.02,0.03,0.04,0.05,0.06,0.07,\text{and}0.08$.

Figure 4

The dynamic susceptibility ${\chi }_4(a,t)$ for the oscillation frequency $\omega \tau =0.02$ and $a=0.12\sigma$. The strain amplitudes from bottom to top are ${\gamma }_0=0.02,0.03,0.04,0.05,0.06$. The oscillation period is $T=314.16\tau$. The inset shows the dynamic correlation length ${\xi }_4={\left[{\chi }_4^{\text{max}}\left(t\right)\right]}^{1/3}$ as a function of the strain amplitude ${\gamma }_0$. The red line with a slope $0.9$ is the best fit to the data.

Figure 5

The number of particles undergoing cage jumps as a function of time for the oscillation frequency $\omega \tau =0.02$, period $T=2\pi /\omega =314.16\tau$, and strain amplitudes (a) ${\gamma }_0=0.02$, (b) ${\gamma }_0=0.04$, and (c) ${\gamma }_0=0.06$. Note that the vertical scale is different in panel (c).

Figure 6

Typical clusters of mobile particles $A$ (large blue circles) and $B$ (small red circles) for the oscillation frequency $\omega \tau =0.02$ and strain amplitudes (a) ${\gamma }_0=0.02$, (b) ${\gamma }_0=0.03$, (c) ${\gamma }_0=0.04$, and (d) ${\gamma }_0=0.05$.

Figure 7

The ratio of dynamically facilitated mobile particles and the total number of mobile particles at a given time step, provided that they were immobile during the preceding time interval $\Delta t$ (see text for details). The strain amplitudes from bottom to top are ${\gamma }_0=0.02,0.03,0.04,0.05,0.06$. The inset shows the ratio of dynamically facilitated particles and the total number of particles that become mobile during the time interval $\Delta t$, given that they were immobile during the previous time interval $\Delta t$. The color code for ${\gamma }_0$ is the same; the strain amplitude increases from bottom to top.