# Brownian motion in liquids : theory and experiment

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Since the theoretical work of Einstein [1905] and von Smoluckowski [1906], and the experiments of Perrin [1909], Brownian motion at long time-scales has been extensively studied for over a century. Short time-scale aspects of Brownian motion are however becoming increasingly relevant, as technology attempts to make smaller and faster devices. The subject matter of this dissertation is the study of short time-scale (typically ~µs) aspects of Brownian motion of microscopic particles in liquids, where the dynamics of the fluid medium are significant.

We detail two recent experiments probing this regime: an experiment [Kheifets et. al., 2014] that measured the hydrodynamic instantaneous velocity of a dielectric particle in liquid medium and confirmed theories of Brownian motion based on hydrodynamics up to sub-microsecond time-scales, and a subsequent experiment [Mo et. al., 2015a] that verified the Maxwell-Boltzmann distribution of velocities well into the tails.

In a liquid medium, the presence of a boundary near a particle has a significant impact on the characteristics of its Brownian motion, owing to the hydrodynamic coupling between the bounding walls and the particle. However, exact solutions to the hydrodynamic equations are not known even for the common situation of a flat wall. An approximate theory was developed by Felderhof [2005], using a point-particle approximation. Despite agreement with previous experiments, that work results in a drag coefficient with a spurious dependence on the particle's density. In this work, we describe a modification [Simha et. al., 2017] to the point-particle approximation that resolves this inconsistency. Moreover, Felderhof's approximation scheme neglects the size of the particle not only in comparison to the distance of the bounding wall, but also to the skin-depth of rotational flow it generates in the fluid. Since this skin-depth depends on the time-scale of the motion, it is not obvious that such an approximation scheme works at all time-scales. We use the formalism of boundary integral equations to set up a perturbative framework, and obtain the point-particle framework through a series of systematic approximations. This derivation explains why the theory works so well at all time-scales. An alternative calculation for a simple case of a no-slip sphere near a full-slip wall is presented, with results indicating that the point-particle approximation may not capture all non-perturbative terms.

We then discuss an experiment [Mo et. al., 2015b] that probed the effects of a boundary on Brownian motion at short time-scales. The experiment agrees very well with the point-particle theory, demonstrating that the boundary significantly impacts Brownian motion down to a certain time-scale, and that the effects are diminished at shorter time-scales. Such effects allude to the possibility of using Brownian motion as a probe of the local environment.