Accurate Solutions to Non-Linear PDEs Underlying a Propulsion of Catalytic Microswimmers

Evgeny S. Asmolov, Tatiana V. Nizkaya and Olga I. Vinogradova
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Evgeny S. Asmolov: Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Science, 31 Leninsky Prospect, 119071 Moscow, Russia
Tatiana V. Nizkaya: Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Science, 31 Leninsky Prospect, 119071 Moscow, Russia
Olga I. Vinogradova: Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Science, 31 Leninsky Prospect, 119071 Moscow, Russia

Mathematics, 2022, vol. 10, issue 9, 1-10

Abstract: Catalytic swimmers self-propel in electrolyte solutions thanks to an inhomogeneous ion release from their surface. Here, we consider the experimentally relevant limit of thin electrostatic diffuse layers, where the method of matched asymptotic expansions can be employed. While the analytical solution for ion concentration and electric potential in the inner region is known, the electrostatic problem in the outer region was previously solved but only for a linear case. Additionally, only main geometries such as a sphere or cylinder have been favoured. Here, we derive a non-linear outer solution for the electric field and concentrations for swimmers of any shape with given ion surface fluxes that then allow us to find the velocity of particle self-propulsion. The power of our formalism is to include the complicated effects of the anisotropy and inhomogeneity of surface ion fluxes under relevant boundary conditions. This is demonstrated by exact solutions for electric potential profiles in some particular cases with the consequent calculations of self-propulsion velocities.

Keywords: self-propulsion; Nernst–Planck–Poisson equations; matched asymptotic expansions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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