Brownian motion and the hydrodynamic friction tensor for colloidal particles of arbitrary shape Daniela J. Kraft, 1, * Raphael Wittkowski, 2 Borge ten Hagen, 3 Kazem V. Edmond, 1 David J. Pine, 1 and Hartmut L¨ owen 3 1 Center for Soft Matter Research, Department of Physics, New York University, New York, NY 10003 2 SUPA, School of Physics and Astronomy, University of Edinburgh, Edinburgh, EH9 3JZ, United Kingdom 3 Institut f¨ ur Theoretische Physik II, Weiche Materie, Heinrich-Heine-Universit¨ at D¨ usseldorf, D-40225 D¨ usseldorf, Germany (Dated: May 7, 2013) We synthesize colloidal particles with various anisotropic shapes and track their orientationally resolved Brownian trajectories using confocal microscopy. An analysis of appropriate short-time correlation functions provides direct access to the hydrodynamic friction tensor of the particles revealing nontrivial couplings between the translational and rotational degrees of freedom. The results are consistent with calculations of the hydrodynamic friction tensor in the low-Reynolds- number regime for the experimentally determined particle shapes. PACS numbers: 82.70.Dd, 05.40.Jc Spherical colloidal particles have served as a model system for investigating Brownian motion since the pi- oneering studies of A. Einstein [1] and M. J. Perrin [2]. Such particles are characterized by a translational dif- fusion coefficient that is linked to the Stokes friction coefficient through the well-known Stokes-Einstein rela- tion [1]. Particles encountered in nature and industry usually have complex non-spherical shapes, and describ- ing their Brownian motion raises fundamental questions about how translational and rotational diffusion are cou- pled. However, in most studies, translational and rota- tional diffusion are considered separately, which is valid only for certain highly symmetrical particle shapes. Recently, model colloids with well-characterized but complex shapes have become available [3–6], which per- mits the quantitative study of the hydrodynamic cou- pling between translational and rotational diffusion for nontrivial particle shapes for the first time. In general, the dynamics of a colloidal particle sus- pended in a liquid is described by a Langevin equation that equates the Stokes friction forces and torques with random thermal forces and torques on a particle. For an arbitrary colloidal particle suspended in a liquid, the friction forces and torques are described by a symmetric second-rank hydrodynamic friction tensor H [7, 8], which includes off-diagonal terms coupling the three transla- tional and three rotational degrees of freedom. In all, H has 21 independent elements. For spherical particles, H is diagonal, with two distinct entries corresponding to the inverse translational and rotational friction coefficients [9, 10]. For rod-like particles, both the translational and rotational entries involve two different coefficients, corre- sponding to parallel and perpendicular particle orienta- tion, but H remains diagonal, meaning that translation and rotation remain decoupled [11, 12]. For a general biaxial particle, H involves nonzero off- diagonal elements that couple translational and rota- tional motion. The corresponding Langevin equation involves intricate multiplicative noise terms due to this coupling, which makes a description of the Brownian dy- namics much more difficult. Although a first theoretical treatment dates back to P. F. Perrin [13, 14], it was not reconsidered until much later, and only by a few authors [7, 15–17] who never explicitly applied it to experiments for biaxial non-orthotropic particle shapes. In this Letter, we report experimental measurements and theoretical calculations of the hydrodynamic friction tensor for several anisotropic colloidal particles, including a general irregular biaxial shape with three fused spheres of different diameters. The particle shape and size are determined by confocal as well as scanning electron mi- croscopy (SEM). We track the Brownian trajectories of these anisotropic colloidal particles with full orientational resolution in real space by confocal microscopy. This 3D real-space technique allows for tracking the motion of arbitrarily complex colloidal particles, even in crowded environments. Based on the generalized Stokes-Einstein relation, we then propose a theoretical framework to ex- tract all independent hydrodynamic friction coefficients from the short-time limit of appropriate correlation func- tions. Our results are consistent with low-Reynolds- number hydrodynamic calculations of the friction ten- sor assuming stick boundary conditions of the solvent at the particle surface, where the experimentally deter- mined particle shapes are taken as an input. Since the full orientational resolution of the individual particle tra- jectories reveals the couplings between different degrees of freedom of Brownian motion, the information obtained by our analysis is much more basic and detailed than av- eraged quantities derived from dynamic light scattering [18] or sedimentation [19] experiments of biaxial colloidal particles. Our method can be used to analyze the Brow- nian dynamics of any rigid irregularly-shaped colloidal particles. arXiv:1305.1253v1 [cond-mat.soft] 6 May 2013