Resonances arising from hydrodynamic memory in Brownian motion

Abstract

Observation of the Brownian motion of a small probe interacting with its environment provides one of the main strategies for characterizing soft matter. Essentially, two counteracting forces govern the motion of the Brownian particle. First, the particle is driven by rapid collisions with the surrounding solvent molecules, referred to as thermal noise. Second, the friction between the particle and the viscous solvent damps its motion. Conventionally, the thermal force is assumed to be random and characterized by a Gaussian white noise spectrum. The friction is assumed to be given by the Stokes drag, suggesting that motion is overdamped at long times in particle tracking experiments, when inertia becomes negligible. However, as the particle receives momentum from the fluctuating fluid molecules, it also displaces the fluid in its immediate vicinity. The entrained fluid acts back on the particle and gives rise to long-range correlations. This hydrodynamic 'memory' translates to thermal forces, which have a coloured, that is, non-white, noise spectrum. One hundred years after Perrin's pioneering experiments on Brownian motion, direct experimental observation of this colour is still elusive. Here we measure the spectrum of thermal noise by confining the Brownian fluctuations of a microsphere in a strong optical trap. We show that hydrodynamic correlations result in a resonant peak in the power spectral density of the sphere's positional fluctuations, in strong contrast to overdamped systems. Furthermore, we demonstrate different strategies to achieve peak amplification. By analogy with microcantilever-based sensors, our results reveal that the particle-fluid-trap system can be considered a nanomechanical resonator in which the intrinsic hydrodynamic backflow enhances resonance. Therefore, instead of being treated as a disturbance, details in thermal noise could be exploited for the development of new types of sensor and particle-based assay in lab-on-a-chip applications.

Superdiffusive trajectories in Brownian motion

“…The present observation of the super-diffusive regime in Brownian motion thus offers the first direct evidence of the random short-time correlated force Langevin had imagined, that force solely directing the motion of a particle initially at rest in a thermal bath. The interpretation of this new observation relies solely on the original formulation of the Langevin model with a δ-correlated force, irrespective of any other effect causing possibly a correlation of the random force itself, such as hydrodynamic memory effects [15,16]. The presence of these effects is well-documented in the case where the fluid density is comparable to the particle density, such as in water, but is negligible in the present case of a rarefied gas.…”

“…(1) correctly predicts the diffusive motion of Brownian particles immersed in a thermal bath at all time scales. In the case of a thermal distribution of initial conditions, the long-and short-time predictions have been confirmed experimentally [5,[13][14][15]. In order to observe super-diffusion in standard Brownian motion, it is necessary to perform conditional statistics on Brownian trajectories with fixed (or narrowly distributed) initial velocity, so that dispersion statistics can be calculated for sets of trajectories with zero (or very small) relative initial velocity.…”

Brownian Motion of Boomerang Colloidal Particles

“…Our experimental observations show that the mean displacements are biased towards the center of hydrodynamic stress (CoH), and that the mean-square displacements exhibit a crossover from short time faster to long time slower diffusion with the short-time diffusion coefficients dependent on the points used for tracking. A model based on Langevin theory elucidates that these behaviors are ascribed to a superposition of two diffusive modes: the ellipsoidal motion of the CoH and the rotational motion of the tracking point with respect to the CoH.Brownian motion as a general phenomenon of the diffusion processes has inspired extensive research [1][2][3][4][5][6][7][8][9][10][11][12] due to both its interesting physics and practical applications such as in microrheology [13][14][15][16], selfpropelled microswimmers [17] and particle and molecular separation [18][19][20]. Inspired by the diverse geometric shapes of biological macromolecules, Brenner and others have extended the hydrodynamic theory of Brownian motion to particles with irregular shapes [21][22][23][24][25][26].…”

Molecular Hydrodynamics from Memory Kernels

“…These non-Markovian effects arise because of momentum conservation, leading to slow hydrodynamic modes that manifest themselves as long-time tails in the velocity autocorrelation function (VACF) [3][4][5][6]. Recent experiments have demonstrated that the force exerted by the bath includes a deterministic component [7], well described for large colloidal spheres by the Basset-Boussinesq (BB) hydrodynamic force [8,9]:…”