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

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.

Figures (18)

• Figure 1: The geometric setting involving the partition of $\Omega$ into two disjoint time dependent open sets $\Omega^+(t)$ and $\Omega^-(t)$ that are separated by an interface $\Sigma(t)$. The unit normal ${\boldsymbol \nu}$ of $\Sigma(t)$ points into $\Omega^+(t)$.
• Figure 2: The geometric setting for the flat multilayered solutions. The unit normals point into $\Omega_+(t)$.
• Figure 3: The geometric setting for the radial multilayered solutions.
• Figure 4: ($\Omega = (0,8)^2$) Moving front for $\alpha=0.1$, $m_\pm=1$, $\rho_\pm = 1$, $S_\mp=\pm1$, $S_I=0$ with $q(0)=0.3$ (left) and $q(0)=7.7$ (right). We plot $q(t)$ and $q_h(t)$ over time.
• Figure 5: ($\Omega = (0,8)^2$) $\alpha=0.2$, $m_\pm=0.1$, $S_\mp=\pm1$, $\rho_\pm=0.1$, $S_I=0$. Evolution for a perturbed flat interface at position ${q(0) = 4}$. We show the solution at times $t=0,1,\ldots,10$, and separately at time $t=10$.

Theorems & Definitions (16)

• Proposition 2.1: Dissipative inequality in two space dimensions
• proof
• Remark 2.1
• Lemma 3.1
• proof
• Remark 3.1
• Proposition 4.1
• proof
• Proposition 4.2
• proof