Mass diffusion and bending in dynamic wetting by phase-field and sharp-interface models

TL;DR

This work addresses the challenge of dynamic wetting at moving contact lines by connecting phase-field diffusion with sharp-interface modeling. A toy model derived from the Cahn–Hilliard framework links diffusive mass transport to wall curvature, introducing a fictitious boundary at distance $\delta$ and a dynamic apparent angle $\theta_a$ to implement a Navier boundary condition within a sharp-interface solver. The authors validate the approach by comparing VOF simulations using the toy-boundary to full phase-field results in two test cases, finding strong agreement in interface shapes, curvature, and contact-angle evolution. The resulting methodology offers a cost-effective route to capture phase-field–level dynamics with sharp-interface, inputting parameters from PF to enable broader CFD applicability. The framework has potential extensions to single-component fluids with phase change and provides open data for reproducibility and further exploration.

Abstract

Dynamic wetting poses a well-known challenge in classical sharp-interface formulation as the no-slip wall condition leads to a contact line singularity that is typically regularized with a Navier boundary condition, often requiring empirical fitting for the slip length. On the other hand, this paradox does not appear in phase-field models as the contact line moves through diffusive mass transport. In this work, we present a toy model that accounts for mass diffusion at the contact line within a sharp-interface framework. This model is based on a theoretical relation derived from the Cahn-Hilliard equations, which links the total diffusive mass transport to the curvature at the wall. From an estimate of the chemical potential on a curved interface, we obtain an expression for the width of the highly curved region $δ$ and the apparent angle. In the sharp-interface model, we then introduce a fictitious boundary, displaced by a distance $δ$ into the domain, where a Navier boundary condition is applied along a dynamic apparent contact angle that accounts for the local interface bending. The robustness of the model is assessed by comparing the toy model results with standard phase-field ones on two cases: the steady state profiles of a liquid bridge between two plates moving in opposite directions and the transient behaviors of a drop spreading on a solid with a prescribed equilibrium angle. This offers a practical and efficient alternative to solve contact line problems at lower cost in a sharp-interface framework with input parameters from phase-field models.

Figures (10)

• Figure 1: Sketch of the region of high curvature close to a moving contact line.
• Figure 2: Correlation between the radius of curvature $R$ and \ref{['eq:Rab']}. (a) Each simulation corresponds to a unique set of Peclet ($\operatorname{Pe}$) and Cahn ($\operatorname{Cn}$) numbers while maintaining a constant capillary number ($\operatorname{Ca}$). The toy model parameters are set to $a = 2.5$ and $b = 0.2170$, and the fitted line demonstrates a clear one-to-one relationship between these two quantities. (b) Steady-state solutions of phase-field (PF) simulations for cases highlighted in red in the left panel.
• Figure 3: Comparison between the phase-field left interface (black points) and the volume-of-fluid interface (red line) for six distinct sets of Peclet ($\operatorname{Pe}$) and Cahn ($\operatorname{Cn}$) numbers. Panels (a) to (f) correspond to the following parameter sets: ($\operatorname{Pe} = 30, \: \operatorname{Cn} = 0.02$), ($\operatorname{Pe} = 30, \: \operatorname{Cn} = 0.04$), ($\operatorname{Pe} = 100, \: \operatorname{Cn} = 0.02$), ($\operatorname{Pe} = 160, \: \operatorname{Cn} = 0.04$), ($\operatorname{Pe} = 300, \: \operatorname{Cn} = 0.02$), and ($\operatorname{Pe} = 300, \: \operatorname{Cn} = 0.04$).
• Figure 4: Comparison between the phase-field angle along the left interface (black points) and the volume-of-fluid angle (red symbols) for six distinct sets of Peclet ($\operatorname{Pe}$) and Cahn ($\operatorname{Cn}$) numbers.
• Figure 5: Comparison of interface shapes at t = 0.1 for phase-field simulations (black lines) and volume-of-fluid toy model (red lines). (a)-(c): Effect of the variation of the geometrical factor $a$ for a fixed value of $b$. (d)-(e) Effect of the variation of $b$ for a fixed value of $a$.