Sedimentation of colloidal nanoparticles in fluids: efficient and robust numerical evaluation of analytic solutions of the Mason-Weaver equation

The Mason-Weaver equation (MWE) is an advection-diffusion equation that describes the time-evolution of the concentration profile of a solution of nanoparticles undergoing sedimentation and Brownian motion from an initially homogeneous state towards sedimentation equilibrium. In spite of the availability of analytic solutions, recent work has used numerical schemes to obtain practical solutions for the MWE. Here, the numerical evaluation of analytic solutions of the MWE is investigated using standard floating-point computations. It was found that the numerical evaluation is not always straightforward as this involves summing over an infinite series of terms which is sometimes automatically truncated due to limitations in floating-point computation. By combining several analytic expressions, each having its own range of validity in the MWE parameter space, robust and computationally efficient numerical evaluation of the solution is finally achieved. The expressions and the numerical procedure have been coded into a computer program enabling practical calculation of nanoparticle sedimentation profiles.