Eur. Phys. J. E (2016) 39: 51
https://doi.org/10.1140/epje/i2016-16051-4

Regular Article

Mode instabilities and dynamic patterns in a colony of self-propelled surfactant particles covering a thin liquid layer

¹ Department of Mathematics, Faculty of Science Engineering and Technology, Swinburne University of Technology, 3122, Hawthorn, Victoria, Australia
² Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm Klemm Str. 9, 48149, Münster, Germany
³ Center of Nonlinear Science (CeNoS), Westfälische Wilhelms Universität Münster, Corrensstr. 2, 48149, Münster, Germany
⁴ Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623, Berlin, Germany

^* e-mail: u.thiele@uni-muenster.de
^** e-mail: Holger.Stark@tu-berlin.de

Received: 21 December 2015
Accepted: 25 March 2016
Published online: 6 May 2016

Abstract

We consider a colony of point-like self-propelled surfactant particles (swimmers) without direct interactions that cover a thin liquid layer on a solid support. The particles predominantly swim normal to the free film surface with only a small component parallel to the film surface. The coupled dynamics of the swimmer density and film height profile is captured in a long-wave model allowing for diffusive and convective transport of the swimmers (including rotational diffusion). The dynamics of the film height profile is determined by i) the upward pushing force of the swimmers onto the liquid-gas interface, ii) the solutal Marangoni force due to gradients in the swimmer concentration, and iii) the rotational diffusion of the swimmers together with the in-plane active motion. After reviewing and extending the analysis of the linear stability of the uniform state, we analyse the fully nonlinear dynamic equations and show that point-like swimmers, which only interact via long-wave deformations of the liquid film, self-organise in highly regular (standing, travelling, and modulated waves) and various irregular patterns.