Measuring collective diffusion coefficients by counting particles in boxes

The collective diffusion coefficient Dcoll is a key quantity for describing the macroscopic transport properties of soft matter systems. However, measuring Dcoll 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 Dcoll for the first time. We validate our results against Fourier-based approaches and establish best practices for measuring Dcoll 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.