A regularized model for wetting/dewetting problems: asymptotic analysis and $\Gamma$-convergence

TL;DR

This paper introduces a novel regularized variational model for wetting/dewetting problems that incorporates height-dependent surface energy, enabling fixed domain simulations, automatic topological changes, and convergence to classical models.

Contribution

The paper develops a new regularized model with a precursor layer, facilitating analysis, simulations, and convergence to sharp-interface models without explicit contact line tracking.

Findings

Precursor layer covers the substrate with height depending on regularization parameter.

Model allows simulation in fixed domain and captures topological changes.

Numerical results confirm theoretical convergence and efficiency.

Abstract

By introducing height dependency in the surface energy density, we propose a novel regularized variational model to simulate wetting/dewetting problems. The regularized model leads to the appearance of a precursor layer which covers the bare substrate, with the precursor height depending on the regularization parameter $\varepsilon$. The new model enjoys lots of advantages in analysis and imulations. With the help of the precursor layer, the regularized model is naturally extended to a larger domain than that of the classical sharp-interface model, and thus can be solved in a fixed domain. There is no need to explicitly track the contact line motion, and difficulties arising from free boundary problems can be avoided. In addition, topological change events can be automatically captured. Under some mild and physically meaningful conditions, we show the positivity-preserving property of the minimizers of the new model. By using asymptotic analysis and $\Gamma$-convergence, we investigate the convergence relations between the new regularized model and the classical sharp-interface model. Finally, numerical results are provided to validate our theoretical analysis, as well as the accuracy and efficiency of the new regularized model.