Dynamics of Reversible Plasticity in an Amorphous Solid

TL;DR

The study investigates how oscillatory shear couples to local plastic rearrangements in a two-dimensional colloidal amorphous solid. By tracking non-affine particle motion and collapsing the dynamics onto an overdamped one-dimensional model with a cubic force, the authors reveal a frequency-independent effective potential that typically takes a quartic double-well form and captures rearrangement cycles across a broad frequency range. They show that apparent rearrangement timescales depend on driving frequency, with a quasi-static limit far below the experimental range and a high-frequency regime where rearrangements can be suppressed or skipped, indicating latent dynamics at play. The work connects microscopic rearrangements to energy landscapes and rheology in glasses, highlights the limits of quasistatic models for driven soft solids, and suggests rich dynamical memory phenomena tied to the full dynamical trajectories rather than turning points alone.

Abstract

Local rearrangements are the elements of plastic deformation in an amorphous solid. In oscillatory shear, they can switch reversibly between two distinct configurations. While these repeating relaxations are typically considered in the limit of slow driving, their dynamics is less well understood. We perform experiments on a colloidal amorphous solid at an oil-water interface. The rearrangement timescales we observe span at least 1 decade, with no apparent upper bound. As frequency is increased, individual rearrangements appear faster and more hysteretic, but may disappear entirely above a crossover frequency -- suggesting that in practical experiments, the slowest rearrangements may be latent. We show how to find the effective potential energy that reproduces a particle's frequency-dependent motion. In rare cases, this potential energy has only one minimum. Our results have implications for the energy landscapes and rheology of amorphous or glassy solids, for sound propagation in nonlinear media, and for mechanical memory and history-dependence.

Figures (6)

• Figure 1: At the microscopic scale, the response of an amorphous solid to oscillatory shear can be plastic or elastic, depending on frequency. (A) Select particle trajectories over a 258-second shear cycle (with maximum shear strain = 0.06; see Fig. \ref{['fig:theory']} for waveform) overlaid on experimental micrograph of part of the monolayer. Background image is from the first video frame. Both rearranging (looped trajectories) and non-rearranging (linear trajectories) can be seen. All particles return to the original position after strain returns to 0. Color indicates passage of time. (B) Particle displacements during first half of the cycle in (A) with global affine motion subtracted. Here the displacement vectors shown are 2$\times$ their actual sizes for emphasis purposes. Background image faded for clarity. (C) Same particle trajectories over an 8.06-second cycle. In this area, rearranging trajectories are replaced by non-rearranging trajectories.
• Figure 2: (A) Plot of $D_{\mathrm{min}}$ of one particle against shear strain $\gamma$ during one cycle. $D_{\mathrm{min}}$ measures the non-affine displacement of a particle compared to its initial placement among its nearest neighbors. Arrows mark the direction of travel, starting at lower left. In this example, the local structure is bistable in the range of shear strain $0.014<\gamma<0.038$, which means its state in this range depends on its history. (B) Distribution of rearrangement timescales $(\tau^+$, $\tau^-)$ (measured in seconds), for 4161 rearranging particles at $f=0.004$ Hz. Central plot: Darker color indicates greater frequency. Dashed diagonal line indicates $\tau^+=\tau^-$. Histograms on axes: sums along each row/column. (C) Spatial map of the rearranging particles in (B), colored by rearrangement timescale $\tau^+$. Each dot marks the location of one particle (not indicative of size). Particles in the same rearranging area tend to have similar relaxation times.
• Figure 3: Observed rearrangement dynamics depend on frequency. (A) Frequency sweep protocol, increasing by factors of 2 from $f=f_0 = 0.004$ Hz to $f=32 f_0$. (B)$D_{\mathrm{min}}$ trajectory for a single particle during the frequency sweep. The three cycles at each frequency use the color bar above (A). Repetitions have the same color but diminished intensity. (C) Using Eq. \ref{['eqn:force2']}, we obtained the $D_{\min}$-dependent $\hat{F}_\text{rep}$ curve, and show that it is independent of driving frequency and direction. Loops at the same frequencies are averaged over. Dashed line is the best fit cubic force model. Inset: interpreting the $\hat{F}_\text{rep}$ curve as the quasi-static limit of the rearrangement; stable portions of the curve are followed up to the limits of stability. (D) Model output using identical driving waveform and force model. (E) Extracted timescales from (B), compared with model output with the same force model and variable frequency. The model is extended by over 3 decades in the low frequency range, showing the transition from power-law to quasi-static regime. Orange dashed line shows an empirical fit (see text). $1/(2f)$ (purple dashed line) bounds $\tau^+$ from above.
• Figure 4: Example of rearrangement-like motion without static hysteresis. (A) Frequency-dependent $D_{\min}$ loops. (B) Inferred $\hat{F}_\text{rep}$ curve; showing positive $\hat{B}$ term. As such, no hysteresis exists in the low frequency limit. (C) Model output from integrating Eq. \ref{['eqn:force2']}. (D) Particle non-affine displacements in half a cycle, similar to those of Fig. \ref{['fig:traj']}(B), for the rearrangement shown in (A--C) corresponding to the lowest frequency; circle marks the particle whose $D_{\min}$ is measured. Arrows are twice the actual displacement for clarity.
• Figure 5: Because of their dynamics, some rearrangements can be skipped at high frequencies. (A)$D_{\mathrm{min}}$ loops for a central particle in the rearrangement of Fig. \ref{['fig:traj']}(A--C), showing a sharp decrease in the non-affine motion as driving frequency increases. (B) The same curves as in (A) but with frequency dependent parts removed, as before. Notice that the rearrangement is stalled at the turning point for the highest two frequencies. (C) Model output with identical driving waveform and force model. (D) The peak $D_\mathrm{min}$ drops quickly at a threshold frequency $\sim 0.03$ s$^{-1}$, which is verified with model output. (E--G) Plot of $D_{\min}$ in a representative region, for different driving frequencies.