Primal-dual splitting methods for phase-field surfactant model with moving contact lines

TL;DR

This work addresses the numerical simulation of a phase-field model for surfactant-laden droplets with moving contact lines by formulating a structure-preserving variational scheme based on the minimizing movement (JKO) principle for Wasserstein gradient flows with degenerate mobilities. The coupled φ–ψ–boundary dynamics are recast as convex minimizations and efficiently solved via a preconditioned primal-dual splitting method with FFT-based accelerations and adaptive time stepping. The scheme inherits original energy dissipation, bound preservation for the surfactant, and mass conservation at the discrete level, enabling stable long-time simulations and adaptive time stepping. Numerical experiments demonstrate accurate interface dynamics, controllable wetting/dewetting through surfactants, and substantial computational efficiency gains compared with traditional energy-stable schemes, highlighting practical impact for simulating complex droplet dynamics in engineering applications.

Abstract

Surfactants have important effects on the dynamics of droplets on solid surfaces, which has inspired many industrial applications. Phase-field surfactant model with moving contact lines (PFS-MCL) has been employed to investigate the complex droplet dynamics with surfactants, while its numerical simulation remains challenging due to the coupling of gradient flows with respect to transport distances involving nonlinear and degenerate mobilities. We propose a novel structure-preserving variational scheme for PFS-MCL model with the dynamic boundary condition based on the minimizing movement scheme and optimal transport theory for Wasserstein gradient flows. The proposed scheme consists of a series of convex minimization problems and can be efficiently solved by our proposed primal-dual splitting method and its accelerated versions. By respecting the underlying PDE's variational structure with respect to the transport distance, the proposed scheme is proved to inherits the desirable properties including original energy dissipation, bound-preserving, and mass conservation. Through a suite of numerical simulations, we validate the performance of the proposed scheme and investigate the effects of surfactants on the droplet dynamics.

Figures (21)

• Figure 1: Phase-field description of the diffused interface between two fluids ($\phi=\pm 1$) in contact with a solid substrate $\Gamma$ with a contact angle $\theta_s$. $\gamma_{1}$ and $\gamma_{2}$ are the fluid-solid substrate interfacial tensions with Fluid I (with $\phi=+1$) and Fluid II (with $\phi=-1$), respectively, and $\gamma_{12}$ is the interfacial tension for the fluid-fluid interface. Young’s angle $\theta_{s}$ satisfies the Young–Dupré equation $\gamma_{12}\mathrm{cos}(\theta_{s})+\gamma_{1}=\gamma_{2}$.
• Figure 2: Illustration of the variational approach for the PFS-MCL model.
• Figure 3: The equilibrium states of droplets and surfactants at $T = 200$ for $\mathrm{Cn}=0.025$ and $\theta_s=120^\circ$, with illustration of energy dissipation and mass conservation.
• Figure 4: First-order accuracy in time of $\phi$ and $\psi$ at $t=0.1$ with fixed $\Delta x=\Delta y=0.005$ and different time steps $\Delta t$, where the reference solution is obtained with $\Delta t=10^{-5}$.
• Figure 5: First-order accuracy for $\phi$ and $\psi$.

Theorems & Definitions (8)

• Remark 3.1
• Theorem 3.1
• Remark 3.2
• Remark 3.3
• Remark 4.1
• Remark 4.2
• Remark 4.3
• Remark 4.4