On a Mullins-Sekerka model for the growth of active droplets modelling protocells: Stability analysis and numerical computations

TL;DR

This paper investigates the stability and dynamics of chemically active Mullins-Sekerka models for droplet growth, providing theoretical analysis and a numerical method to simulate complex behaviors like splitting and shell formation.

Contribution

It offers the first detailed stability analysis of active Mullins-Sekerka models and introduces a numerical approach capable of capturing topological changes in droplet evolution.

Findings

Radially symmetric solutions exist and are analyzed for stability.

Numerical simulations confirm theoretical stability and reveal complex behaviors.

Droplet splitting and shell formation are demonstrated through simulations.

Abstract

Mullins-Sekerka models with chemical reactions can lead to scenarios where droplets grow, become unstable, split, grow and undergo further division. These grow and division cycles have been proposed as a model for protocells and are believed to play a fundamental role in living systems by providing chemical compartments which are important in the organization of living systems. This paper analyses chemically active Mullins-Sekerka models. Existence of radially symmetric solutions is shown and a detailed stability analysis in radial as well as planar situations is given. In particular, we also analyze multilayered solutions leading to shell-type situations. Finally, we introduce a numerical method based on a parametric finite element approach that explicitly accounts for topological changes, thereby allowing for droplet splitting and merging. Several numerical simulations verify the findings of the theoretical stability analysis and show complex dynamical behavior, including multiple instabilities, splittings of droplets and appearance of shell-type solutions.