Reversibility, Chaos, and Attractors in Periodically Sheared Elastic Filaments

TL;DR

The study investigates how Brownian, inextensible elastic filaments behave under strong oscillatory shear, revealing a regime where buckling and thermal fluctuations drive irreversible, non-reciprocal dynamics. By combining microfluidic actin experiments, Brownian dynamics simulations, and a reduced-order elastica model, it uncovers two coexisting, time-glide-symmetric attractors that lead to intermittent switching between quasi-reversible and chaotic states. A stroboscopic framework shows that fluctuations seed buckling and that the system exhibits stochastic symmetry breaking in a minimal nonequilibrium soft-matter setting. These findings offer insights into controlling the rheology of soft materials under time-dependent flows and suggest broader implications for synchronization and intermittency in driven elastic systems.

Abstract

The dynamics of filaments in flow are central to understanding a wide range of biological and soft-matter systems, yet their behavior under time-dependent forcing remains poorly understood. Here, we investigate the long-time dynamics of Brownian inextensible elastic filaments subjected to strong uniform oscillatory shear by combining microfluidic experiments on actin filaments with numerical simulations based on a fluctuating Euler-Bernoulli elastica model in a viscous fluid. As the oscillation period increases, irreversibility emerges from the interplay of flow-induced deformations and thermal noise. This leads to a departure from reversible, deterministic rigid-body dynamics: in this regime, the filaments cycle between nearly straight, flow-aligned conformations at full periods and buckled shapes at half periods. Owing to the time-glide symmetry of the system, two such attracting states in fact coexist with a phase shift of half a period. The system spontaneously selects one, but occasionally switches between them as a result of noise, producing intermittent transitions between apparent order and disorder. This system constitutes an experimentally accessible realization of stochastic symmetry breaking, attractor hopping, and intermittency in a minimal nonequilibrium soft-matter system, with novel implications for the design and control of soft matter systems under time-dependent flows.

Figures (6)

• Figure 1: (a) Evolution of the time-periodic shear flow, with shear rate $\dot{\gamma}(t)/\dot{\gamma}_m =\sin{(2\pi t/T)}$, over the course of one period $T$ of oscillation. The plot compares experimental microfluidic shear rate measurements with a sinusoidal wave. (b) Schematic of a filament with arclength $s$, whose shape is described by the tangent vector $\hat{\mathbf{t}}(s,t)$, or equivalently by the tangent angle $\theta(s,t)$ measured from the negative $x$-direction. The positive $x$-direction is defined as the flow direction at $t=0$. The background flow lines illustrate the underlying straining flow, which is extensional in the first and third quadrants and compressional in the second and fourth quadrants during the first half period when $\dot{\gamma}>0$. (c) Snapshots of representative filament trajectories over the course of one period, from experiments (E) and simulations (S) with $\ell_p/L = 1.3$, $\bar{\mu}_m \approx 1 \times 10^4$. The dimensionless period is $\rho=6$ in (i) and $\rho=13$ in (ii)-(v). Cases (ii)-(v) differ by the initial orientation of the filament. The blue and red vertical lines label the deformation state of the filament at the start and end of each half cycle: straight (blue) or deformed (red). (d) Orientational dynamics in the (${\theta_0,\dot{\theta}_0}$) phase plane, for the same five cases as in (c). In each case, the experimental trajectory (E) is compared to the deterministic non-Brownian rigid-rod trajectory (RR) governed by Jeffery's equation Eq. (\ref{['eq:Jeffery']}). Deviations from Jeffery orbits coincide with periods of strong filament deformation.
• Figure 2: Mode dynamics from simulations and modeling over a single oscillation period for three filaments with different initial orientations: (a) $\theta_0^i = 10^{\circ}$, (b) $30^{\circ}$, and (c) $90^{\circ}$. Top row: Filament conformations from Brownian flexible simulations, where the conformations at the start and end of each half-cycle are highlighted using the same color code as in Fig. \ref{['fig:1']}(c). Middle row: Time evolution of the mean filament orientation $\theta_0(t)$. Brownian simulation results (orange circles) are compared with non-Brownian simulations (purple squares), the predictions from the semi-analytical reduced-order model (black solid line; see SM and Fig. S2 Supplemental_Info), and the Jeffery orbit for a rigid rod (RR, blue dashed line). The colored background highlights regions of compressional and extensional flow, following the same color scheme as in Fig. \ref{['fig:1']}(d). Bottom row: Time evolution of the filament deformation $\|\boldsymbol{\Psi}(t)\|$, computed using the first five normal modes. The legend matches that of the middle row. All cases use a dimensionless period $\rho = 14$ and an elastoviscous number $\bar{\mu}_m = 10^4$. Brownian simulations are performed with $\ell_p/L = 50$.
• Figure 3: Ensemble-averaged stroboscopic changes in (a) mean filament orientation, $\langle | \delta \theta_0 | \rangle = \langle | \theta_0(T) - \theta_0(0) | \rangle$, and (b) filament deformation, $\langle \| \delta\boldsymbol{\Psi} \| \rangle = \langle \| \boldsymbol{\Psi}(T) - \boldsymbol{\Psi}(0) \| \rangle$, as functions of the dimensionless period $\rho$ and initial orientation $\theta_0^i$. Averages were computed from 50 simulations per $(\rho, \theta_0^i)$ pair, each with a different random seed. Simulations used $\ell_p/L = 50$ and $\bar{\mu}_m = 10^4$. Black squares indicate the three examples shown in Fig. \ref{['fig:2']}.
• Figure 4: (a) Top panel: Evolution of the two stroboscopic sequences $\mathcal{P}_0(n)$ (black) and $\mathcal{P}_{T/2}(n)$ (red) over 1500 periods in a Brownian simulation with $\rho = 20$ and $\bar{\mu}_m = 1 \times 10^4$. The orange dashed line indicates the switching angle $\theta_S$ between the two time series, and the green dotted line marks the mean attracting angle $\theta_A$ for stable dynamics. The phase chart from Fig. \ref{['fig:3']}(a), showing the ensemble-averaged stroboscopic change $\langle | \delta \theta_0 | \rangle$, is overlaid on the right. Purple arrows highlight crossings between the two sequences. Bottom panel: Time evolution of the order parameter $S$ that characterizes filament deformation states (see main text for definition). (b) Zoom-in on the first 100 periods from (a), highlighting three distinct dynamical regimes using blue, green, and orange interrogation windows. (c) Filament morphologies corresponding to the three windows in (b). The top and bottom rows depict the two attracting states, where one sequence is quasi-reversible and the other chaotic, resulting in $S = 0.5$. The middle row shows a switching event, in which both sequences are reversible and the filament adopts symmetric straight conformations, yielding $S = 0$. Also see Supplemental Videos 1 to 3 Supplemental_Info for movies showing the dynamics.
• Figure 5: (a) Switching angle $\theta_S$ marking the transition between stable and chaotic dynamics, and attracting angle $\theta_A$ during stable dynamics, shown as functions of the dimensionless period $\rho$. Symbols represent simulation results; dashed lines indicate theoretical predictions. (b) Mean residence time $N_A$ in a stable attracting state, and the normalized switching frequency $N_S/N_T$, where $N_S$ is the number of transitions between stable and chaotic dynamics and $N_T = 2500$ is the total number of periods. Error bars indicate standard deviations. Averages were computed from 12 long Brownian simulations with $\bar{\mu}_m = 1 \times 10^4$ and different initial orientations. All results correspond to Brownian filaments with $\ell_p/L = 50$.