Measuring collective diffusion properties by counting particles in boxes

Abstract: The collective diffusion coefficient $D_\mathrm{coll}$ is a key quantity for describing the macroscopic transport properties of soft matter systems. However, measuring $D_\mathrm{coll}$ is a fundamental experimental and numerical challenge, as it either relies on nonequilibrium techniques that are hard to interpret or, at equilibrium, on Fourier-based approaches which are fraught with difficulties associated with Fourier transforms. In this work, we investigate the equilibrium diffusive dynamics of a 2D colloidal suspension experimentally and numerically. We use a "Countoscope" technique, which analyses the statistics of particle number counts $N(t)$ in virtual observation boxes of a series of microscopy images at equilibrium, to measure $D_\mathrm{coll}$ for the first time. We validate our results against Fourier-based approaches and establish best practices for measuring $D_\mathrm{coll}$ using fluctuating counts. We show that Fourier techniques yield inaccurate long-range collective measurements because of the non-periodic nature of an experimental image, yet counting exploits this property by using finite observation windows. Finally, we discuss the potential of our method to advance our understanding of collective properties in suspensions, particularly the role of hydrodynamic interactions.