A simulation method for the wetting dynamics of liquid droplets on deformable membranes

TL;DR

This work addresses the challenge of simulating wetting dynamics of liquid droplets on deformable membranes by formulating a thermodynamically consistent model that couples Navier–Stokes hydrodynamics with a diffuse-interface phase-field for the droplet and an explicit, elastic membrane energy (bending, tension, and stretch). An ALE-based fitted finite-element discretization tracks two moving subdomains separated by the membrane, enabling accurate treatment of high membrane curvature and pressure discontinuities, while a diffuse interface regularizes the moving three-phase contact line. The authors derive the governing equations from energy variations, implement a monolithic IMEX time-stepping scheme, and employ remeshing to handle large deformations; water permeability across the membrane is included to model slow fluxes. Validation against analytical shape equations and a range of axisymmetric 2D/3D tests demonstrates the method’s ability to reproduce adhesion, lens-like and wrapping configurations, membrane-mediated droplet interactions (inverted Cheerios effect), inverted endocytosis, and phase separation dynamics around membranes, highlighting its potential to study condensate–membrane interactions in biology and materials science.

Abstract

Biological cells utilize membranes and liquid-like droplets, known as biomolecular condensates, to structure their interior. The interaction of droplets and membranes, despite being involved in several key biological processes, is so far little understood. Here, we present a first numerical method to simulate the continuum dynamics of droplets interacting with deformable membranes via wetting. The method combines the advantages of the phase-field method for multi-phase flow simulation and the arbitrary Lagrangian-Eulerian (ALE) method for an explicit description of the elastic surface. The model is thermodynamically consistent, coupling bulk hydrodynamics with capillary forces, as well as bending, tension, and stretching of a thin membrane. The method is validated by comparing simulations for single droplets to theoretical results of shape equations, and its capabilities are illustrated in 2D and 3D axisymmetric scenarios.

Figures (9)

• Figure 1: Illustration a droplet wetting a vesicle. A deformable closed membrane $\Gamma$ separates the fluid domain $\Omega_{\text{in}}$ and a two-phase fluid domain $\Omega_{\text{out}}$. The two fluids in $\Omega_{\text{out}}$ are indicated by the value of the phase-field function $\phi$. The fluid-fluid interface is diffuse with a thickness $\varepsilon$ and surface tension $\sigma_{\text{f}}$. The membrane $\Gamma$ has two distinct surface tensions $\sigma_0$ and $\sigma_1$ depending on whether it is adjacent to the dilute or dense phase, respectively (see inset). Two different contact angles can be considered, where $\alpha$ is the macroscopic Neumann angle, that can be measured in experiments, while $\theta$ is the local Young's angle between the droplet interface and the locally flat membrane.
• Figure 1: Illustration of the numerical mesh. The membrane triangulation $T_{\Gamma}$ (red) is fitted to the triangulations $T_{\text{in}}$ (gray) and $T_{\text{out}}$. The phase field $\phi$ representing the droplet is defined in $T_{\text{out}}$ and controls the adaptive refinement. The white line indicates the fluid-fluid interface as 0.5-level set of $\phi$.
• Figure 1: Comparison of stationary shapes obtained by simulations (dashed lines) and theoretical model (solid lines) for a single droplet (blue) on an initially flat membrane (red). Compelling agreement between the two solutions is even visible in the close-up around the three-phase contact point. Parameters: Initial droplet shape is spherical (radius $50$nm) in the theoretical model, and half-spherical cap (radius $70.71$nm) in the simulations. The used surface tensions can be determined from \ref{['eq:theta']} with $\sigma_{\text{f}}=15\mu$N/m. Further, $K_A=0, K_B=8\cdot 10^{-20}$Nm, $P=0$.
• Figure 2: Time evolution for three different parameter configurations exhibit different categories of droplet-membrane interaction: adhesion (top row), lens shape (middle row), and wrapped/endocytosis (bottom row). The single top snapshot shows the initial configuration of droplet (blue) and membrane (red). Oriented streamlines illustrate fluid velocity in the $xy$-plane colored by magnitude in units of $10\,\mu$m/s. All snapshots have been zoomed in for visibility and therefore do not show all of $\Omega$. Parameters: Initially spherical membrane of radius of $5.4\mu$m and half-spherical cap droplet of radius $2.5\mu$m, centered on the membrane. Other parameters are $\varepsilon=0.02\mu$m, $K_B=8\cdot 10^{-19}\,\text{Nm}$, $K_A = 5\cdot 10^{-3}$ N/m, $P=10^{-7}$m$^2$s/kg, $\eta = 1\,\text{Pa}\cdot\text{s}$, $\rho = 10^3\,\text{kg/m}^3$. Axisymmetric simulations.
• Figure 3: Inverted endocytosis: A small spherical vesicle (red) is absorbed by a larger drop (blue). Time evolution of axisymmetric simulation. Vesicle can change shape due to imposed permeability. Parameters: $\sigma_{\text{f}} = 30\,\mu\text{N/m}, ~\sigma_0 = 31\,\mu\text{N/m}, ~\sigma_1 = 1\,\mu\text{N/m}, K_B=8\cdot 10^{-17}\,\text{Nm}, K_A = 5\cdot 10^{-3}\,\text{N/m}, P=10^{-7}$m$^2$s/kg, initial radii $1.25\,\mu$m (vesicle), $5\,\mu$m (drop). Axisymmetric simulation.