Brownian Motion of Arbitrarily Shaped Particles in Two-Dimensions
  Here we implement microfabricated boomerang particles with unequal arm lengths as a model for non-symmetry particles and study their Brownian motion in a quasi-two dimensional geometry by using high precision single particle motion tracking. We show that due to the coupling between translation and rotation, the mean squared displacements of a single asymmetric boomerang particle exhibit a non-linear crossover from short time faster to long time slower diffusion, and the mean displacements for fixed initial orientation are non-zero and saturate out at long time. The measured anisotropic diffusion coefficients versus the tracking point position indicate that there exists one unique point, i.e. the center of hydrodynamic stress (CoH), at which all coupled diffusion coefficients vanish. This implies that in contrast to motion in 3D where the CoH only exists for high symmetry particles, the CoH always exists for Brownian motion in 2D. We develop an analytical model based on Langevin theory to explain the experimental results and show that among the 6 anisotropic diffusion coefficients only 5 are independent because the translation-translation coupling originates from the translation-rotation coupling. Finally we classify the behavior of 2D Brownian motion of arbitrarily shaped particles into four groups based on the particle shape symmetry group. 
Ayan Chakrabarty, Andrew Konya, Feng Wang, Jonathan V. Selinger, Kai   Sun and Qi-Huo Wei
cond-mat.soft