Reversible to Irreversible Transitions in Pattern-Forming Systems with Cyclic Interactions

Abstract

Transitions from reversible to irreversible or fluctuating states above a critical density and shear amplitude have been extensively studied in non-thermal cyclically sheared suspensions and amorphous solids. Here, we propose that the same type of reversible to irreversible transition occurs for a system of particles with competing short-range attraction and long-range repulsion, which can form crystals, stripes, and bubbles as the ratio of attraction to repulsion varies. By oscillating the strength of the attractive part of the potential, we find that the system can organize into either time-periodic states consisting of nondiffusive complex closed orbits, or into a diffusive fluctuating state. A critical point separates these states as a function of the maximum strength of the attraction, oscillation frequency, and particle density. We also find a re-entrant behavior of the reversible state as a function of the strength of the attraction and the oscillation frequency.

Figures (4)

• Figure 1: (a) The SALR interaction potential $V(r_{ij},t)=1/r_{ij} - B(t)\exp(-r_{ij})$ for $B_{\rm max}=2.9$ and $\omega=1.25\times 10^{-4}$ at two different times, $t=\pi n/\omega$ (black) where $B(t)=B_{\rm max}$ and $t=\pi(n+1/2)/\omega$ (red) where $B(t)=0$ and the interaction is purely repulsive. (b) The corresponding short range interaction strength $B(t) = B_{\max}|\cos(\omega t)|$ plotted against cycle number $n$. (c-f) Particle configurations for a system with $B_{\rm max}=2.9$, $\rho=0.5$ and $\omega=1.25\times 10^{-4}$ during cycle $n=20$. (c) Uniform state at $B/B_{\rm max} = 0.0$. (d) Beginning of bubble formation at $B/B_{\rm max} = 0.75$ for increasing $B$. (e) Fully formed bubbles at $B/B_{\rm max} = 1.0$. (f) The bubble structure expands back toward a uniform state at $B/B_{\rm max} = 0.5$ for decreasing $B$.
• Figure 2: (a) Cumulative displacements $d(n)$ vs $n$ for a system with $\rho=0.5$ and $\omega=1.25\times 10^{-4}$ at $B_{\rm max} = 2.0$, 2.4, 2.6, 2.7, 2.8, 2.9, 3.0, 3.2, and $4.0$, from bottom to top. (b) The corresponding $R(n)$ vs $n$. The behavior is reversible for $B_{\rm max} < 2.7$ and irreversible for $B_{\rm max} \geq 2.7$. (c) $R$ after $n=200$ cycles vs $B_{\rm max}$ for the same system, showing a critical transition near $B_{\rm max} = 2.65$. The dashed line is a fit to $R \propto (B_{\rm max} - B_{\rm max}^c)^\beta$ with $B_{\rm max}^c = 2.69$ and $\beta = 1/2$. (d) $R$ after $n=200$ cycles vs $B_\text{max}$ for $\rho = 0.2$, 0.225, 0.25, 0.275, 0.3, 0.35, 0.4, 0.5, 0.6, and $0.8$, from bottom to top. The lowest three values of $\rho$ give $R=0$ for all $B_{\rm max}$.
• Figure 3: Particle configurations (red) and trajectories (blue) during $n=5$ cycles in the steady state. (a) Disordered flow in the irreversible state at $\rho = 0.5$, $B_{\rm max} = 2.9$, and $\omega=1.25\times 10^{-4}$. (b) Reversible state at $\rho=0.15$, $B_{\rm max}=2.9$, and $\omega=1.25\times 10^{-4}$. (c) Reentrant reversible state at $\rho = 0.25$, $B_{\rm max}=4.0$, and $\omega=1.25\times 10^{-4}$. (d) A stripe-like reversible state at $\rho = 0.5$ and $B_{\rm max} = 3.2$ for a higher frequency of $\omega = 8.0\times 10^{-4}$.
• Figure 4: (a) Boundary between reversible (blue) and irreversible (red) regimes as a function of $\omega$ vs $B_{\rm max}$ for $\rho=0.5$. When $B_{\rm max} < 2.0$ and/or at high $\omega$, the system is in a reversible uniform state. As a function of $B_{\rm max}$ there is a reentrant transition to the reversible state, and the irreversible state reaches its greatest extent near $B_{\rm max} = 3.2$. (b) A heat map of $R$ after $n=200$ cycles as a function of $\rho$ vs $B_{\rm max}$ in a system with $\omega=1.25\times 10^{-4}$ showing reversible (blue) and irreversible (red) regimes.