Measurement of the Brownian motion diffusion coefficient: Displacement-distribution versus mean-squared-displacement trajectory technique

Abstract

We investigate the possible difference in the diffusion coefficient D value when measured from the variance of the normal displacement-distribution (D_var), and from the slope of the linear mean-squared-displacement (MSD) against sampling time interval, (D_MSD). We considered the pure Brownian motion (BM) of solid 2μm spheres in water, with a viscosity of 0.89mPa-s at 25°C, observed over 60s every 0.1s. For each generated trajectory, D_var and D_MSD were calculated and compared against the theoretical D of 0.245 μm²/s. For the D_MSD measurements, the effect of the number of fitting points, f, used for plotting the MSD graph and the number of displacement steps, s, for computing each MSD per 𝜏 were studied. We have observed that D_MSD accuracy increases with f (D_MSD is within 5% accuracy at f > 30). We have also found that D_var is more accurate (to within one percent) and precise than D_MSD, regardless of the f and s parameters. Our results indicate that using the variance of the displacement distribution may lead to more accurate and precise D measurements of BM trajectories.