Propagating irreversibility fronts in cyclically-sheared suspensions

Abstract

The interface separating a liquid from its vapor phase is diffuse: the composition varies continuously from one phase to the other over a finite length. Recent experiments on dynamic jamming fronts in two dimensions [Waitukaitis et al., Europhysics Letters 102, 44001 (2013)] identified a diffuse interface between jammed and unjammed discs. In both cases, the thickness of the interface diverges as a critical transition is approached. We investigate the generality of this behavior using a third system: a model of cyclically-sheared non-Brownian suspensions. As we sediment the particles towards a boundary, we observe a diffuse traveling front that marks the interface between irreversible and reversible phases. We argue that the front width is linked to a diverging correlation lengthscale in the bulk, which we probe by studying avalanches near criticality. Our results show how diffuse interfaces may arise generally when an incompressible phase is brought to a critical point.

Figures (3)

• Figure 1: Self-organized compaction front.(a) Simplified model of a cyclically-sheared, sedimenting suspension after Ref. Corte09. In each cycle, a uniform sedimentation velocity $v_s$ is applied to all particles, and particles that overlap (red) are given random kicks. (b) Typical simulation showing a traveling front between a dense fluctuating region and a dilute reversible region. The front moves at constant speed $v_f$ until it reaches the top of the sediment and a fluctuating steady state begins. Here, $N=1273$, $\phi_0=0.2$, $\epsilon=0.5$, $W=50$, $H=100$, $v_s=2\times 10^{-5}$. (c) Scaled front velocity, $v_f/v_s$. The data over a wide range of parameters are well-described by Eq. (\ref{['eq:v_f']}), which assumes the two phases have uniform densities equal to $\phi_0$ and $\phi_c$. Here, $0.05 \leq \epsilon \leq 10$; $300 < N < 16300$; $10^{-6} \leq v_s \leq 4 \times 10^{-4}$; $0.05 \leq \phi_0 \leq 0.40$; $0.16 < \phi_c < 0.46$.
• Figure 2: Interface shape and thickness.(a) Density profile snapshots for $\epsilon=0.5$, $N=16297$, $v_s=1.7\times10^{-5}$, $W=100$, sampled at a regular period. Each curve is averaged over $200$ systems. The data plateau to the dashed lines at $\phi_0=0.2$, $\phi_c=0.376$. (b) Translating these 6 profiles atop one another shows that the front moves with fixed shape and width. The profile is consistent with a sigmoid [dashed line: Eq. (\ref{['eq:phi_y']})]. (c) Measured front width, $\Delta_f$, versus proximity to criticality of the sedimenting phase, $\phi_c - \phi_0$. Closed symbols: transient fronts. Open symbols: interface at the top of the system in the steady state (where $\phi_0 = 0$ above the sediment). The data are consistent with a power law with exponent $-1.15 \pm 0.18$ (dashed line), over a wide range of parameters ($0.05 \leq \epsilon \leq 10$; $300 < N < 16300$; $10^{-7} \leq v_s \leq 4 \times 10^{-4}$; $0.05 \leq \phi_0 \leq 0.40$; $0.16 < \phi_c < 0.46$). (d) The front width does not depend on $\epsilon$. Here we adjust $\phi_0$ so that $\phi_c - \phi_0 = 0.1$ is constant; all other parameters are fixed ($N=1730$, $v_s=10^{-6}$, $W=50$). Dashed line: value of the fit in panel (c). (e) Scaling the front width by the power law fit from panel (c), which shows that $\Delta_f$ does not depend strongly on the system width, $W$.
• Figure 3: Response to point perturbation.(a) Starting from a quiescent state, a perturbation may set off a chain reaction where many particles are activated before the system becomes quiescent again. Colored particles were active at some time during the avalanche, and the darker particles received more total kicks. (b) Histograms collected over many systems for the distance to the farthest activated particle, $\ell$, the number of activated particles, $n$, and the avalanche duration in cycles, $t$. Solid lines: Fits to Eq. (\ref{['eq:hist']}), where the measured exponent $\alpha$ is indicated in each panel. (c) The curves are approximately collapsed when scaled by the location of the exponential cutoff. (d) Value of the cutoff versus $\phi_c - \phi_0$. Each curve diverges as a power law, with an exponent that is distinct from $\alpha$ (see Table \ref{['tab:1']}). All systems have $\epsilon = 0.5$, $W=H=400$, and $\phi_c = 0.375$.