MasonWeaver-analytic: Numerical evaluation of analytic solutions of the Mason-Weaver equation

MasonWeaver-analytic: Numerical evaluation of analytic solutions of the Mason-Weaver equation

L. Barthe and M. H. V. Werts, 2022

ENS Rennes / CNRS

The Mason-Weaver equation (MWE) is a partial differential equation that describes the sedimentation of small particles in a fluid with the particles being subject to Brownian motion.[1] Understanding the evolution of the vertical concentration profile of an initially homogeneous solution of nanoparticles undergoing sedimentation requires solving the MWE. Previously, we used a numerical finite-difference scheme to find concentration profiles obeying the Mason-Weaver equation.[2][3]

We have now developed Python code that evaluates directly the analytic solutions given  in the original publication by Mason and Weaver.[1] The task of numerically evaluating the analytic expressions was not as straight-forward as initially expected (as  described in the accompanying technical note [4]), but we have finally arrived at an efficient and robust program for obtaining sedimentation concentration profiles from the analytic solutions of the Mason-Weaver equation.

[1] Mason, M.; Weaver, W. "The Settling of Small Particles in a Fluid". Phys. Rev. 1924, 23, 412. doi:10.1103/PhysRev.23.412

[2] Midelet, J.; El-Sagheer, A. H.; Brown, T.; Kanaras, A. G.; Werts, M. H. V. "The Sedimentation of Colloidal Nanoparticles in Solution and Its Study Using Quantitative Digital Photography". Part. & Part. Syst. Charact. 2017, 34, 1700095. doi:10.1002/ppsc.201700095

[3] MasonWeaver-finite-diff

[4] Barthe, L.; Werts, M. H. V. "Sedimentation of colloidal nanoparticles in fluids: efficient and robust numerical evaluation of analytic solutions of the Mason-Weaver equation". ChemRXiv 2022. doi:10.26434/chemrxiv-2022-91vrq

v1.0

Initial release. Calculations are done in the module masonweaver_analytic.py which has been tested in a variety of situations and parameter settings. Accurate, physically sound results are obtained for the time-evolution of the concentration profiles of Brownian particles subject to sedimentation.