Hydrodynamic synchronization of colloidal oscillators

Abstract

Two colloidal spheres are maintained in oscillation by switching the position of an optical trap when a sphere reaches a limit position, leading to oscillations that are bounded in amplitude but free in phase and period. The interaction between the oscillators is only through the hydrodynamic flow induced by their motion. We prove that in the absence of stochastic noise the antiphase dynamical state is stable, and we show how the period depends on coupling strength. Both features are observed experimentally. As the natural frequencies of the oscillators are made progressively different, the coordination is quickly lost. These results help one to understand the origin of hydrodynamic synchronization and how the dynamics can be tuned. Cilia and flagella are biological systems coupled hydrodynamically, exhibiting dramatic collective motions. We propose that weakly correlated phase fluctuations, with one of the oscillators typically precessing the other, are characteristic of hydrodynamically coupled systems in the presence of thermal noise.colloidal particles | hydrodynamic interaction | metachronal wave | optical tweezers | nonlinear dynamical system

Dynamics of a linear magnetic “microswimmer molecule”

“…Aspects of our results may further be important for the behavior of interacting magnetotactic bacteria [41,42]. More artificially, spring-like interactions between the constituents could be mimicked by caging them in comoving optical laser traps using feedback control loops [43].…”

Krylov subspace methods for computing hydrodynamic interactions in Brownian dynamics simulations

“…1 On the other hand, it is essential to include HI to correctly capture the dynamics of colloidal spheres, macromolecules, and swimming bacteria at a low Reynolds number; in particular, their collective, intermolecular motions can give rise to qualitatively different dynamic behavior. [2][3][4] Hydrodynamic interactions are modeled by a configuration-dependent diffusion matrix, D, of size 3N × 3N for a system of N particles. This matrix is dense, owing to the long-ranged nature of HI.…”

“…7 In this approximation technique, an approximate correlated vector is computed as p(D)z, where p(D) is a polynomial in D that approximates the principal square root of D. This square root corresponds to the factorization in Eq. (4), where B = B T . The technique is based on Chebyshev polynomials and requires estimates of the extreme eigenvalues of D. The matrix p(D) itself is not necessary and is never formed explicitly, and thus O(N 3 ) matrix-matrix multiplications are avoided.…”

Stochastic Hydrodynamic Synchronization in Rotating Energy Landscapes

“…The phases of these basic units are not constrained by the applied forces, but are free to change in response to hydrodynamic couplings with nearby particles. This requirement of a free phase can be obtained in linear oscillators using a geometric switch, that is an external potential that switches between two shapes whenever some geometric trigger is activated based on particle instantaneous positions [7,8]. When a nonlinear potential is used, the obtained oscillations are asymmetric under time reversal and therefore can give rise to a quick synchronization of nearby oscillators [9].…”

“…Fig.3 reports the average lifetimes measured from experimental trajectories plotted against their theoretical expected values according to Eq. (8).…”