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Median Filters for Anisotropic Wetting / Dewetting Problems

Jiajia Guo, Selim Esedoglu

TL;DR

This work addresses simulating area-preserving, anisotropic three-phase wetting/dewetting dynamics in 2D with a stationary solid phase. It develops vectorial median filter level-set schemes, derived from threshold dynamics, using anisotropic convolution kernels (constructed from a two-circle kernel) to encode surface tensions and mobilities, ensuring correct limiting dynamics and indirect enforcement of Herring angle conditions at junctions. The authors establish convergence evidence and subgrid accuracy across prescribed, fully anisotropic, and topology-changing scenarios, with time-discretization errors aligning with the expected rate near 1/2. Numerical experiments compare against front-tracking benchmarks and demonstrate robust topology handling, suggesting the method as a practical tool for complex mesoscale interfacial problems and a path toward grain-boundary network simulations. The approach combines variational threshold-dynamics concepts with level-set regularity to achieve a flexible, subgrid-accurate, topologically robust framework for anisotropic multiphase flows.

Abstract

We present new level set methods for multiphase, anisotropic (weighted) motion by mean curvature of networks, focusing on wetting-dewetting problems where one out of three phases is stationary -- a good testbed for checking whether complicated junction conditions are correctly enforced. The new schemes are vectorial median filters: The level set values at the next time step are determined by a sorting procedure performed on the most recent level set values. Detailed numerical convergence studies are presented, showing that the correct angle conditions at triple junctions (which include torque terms due to anisotropy) are indeed indirectly and automatically attained. Other standard benefits of level set methods, such as subgrid accuracy on uniform grids via interpolation and seamless treatment of topological changes, remain intact.

Median Filters for Anisotropic Wetting / Dewetting Problems

TL;DR

This work addresses simulating area-preserving, anisotropic three-phase wetting/dewetting dynamics in 2D with a stationary solid phase. It develops vectorial median filter level-set schemes, derived from threshold dynamics, using anisotropic convolution kernels (constructed from a two-circle kernel) to encode surface tensions and mobilities, ensuring correct limiting dynamics and indirect enforcement of Herring angle conditions at junctions. The authors establish convergence evidence and subgrid accuracy across prescribed, fully anisotropic, and topology-changing scenarios, with time-discretization errors aligning with the expected rate near 1/2. Numerical experiments compare against front-tracking benchmarks and demonstrate robust topology handling, suggesting the method as a practical tool for complex mesoscale interfacial problems and a path toward grain-boundary network simulations. The approach combines variational threshold-dynamics concepts with level-set regularity to achieve a flexible, subgrid-accurate, topologically robust framework for anisotropic multiphase flows.

Abstract

We present new level set methods for multiphase, anisotropic (weighted) motion by mean curvature of networks, focusing on wetting-dewetting problems where one out of three phases is stationary -- a good testbed for checking whether complicated junction conditions are correctly enforced. The new schemes are vectorial median filters: The level set values at the next time step are determined by a sorting procedure performed on the most recent level set values. Detailed numerical convergence studies are presented, showing that the correct angle conditions at triple junctions (which include torque terms due to anisotropy) are indeed indirectly and automatically attained. Other standard benefits of level set methods, such as subgrid accuracy on uniform grids via interpolation and seamless treatment of topological changes, remain intact.

Paper Structure

This paper contains 11 sections, 46 equations, 31 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction and Background
  2. The Vapor-Liquid-Substrate Model in 2D
  3. Threshold Dynamics for the VLS Model
  4. Median Filter for the VLS Model
  5. Anisotropic Kernels for Median Filters
  6. Numerical Experiments
  7. Prescribed anisotropic $\sigma_{VL}$
  8. Prescribed anisotropic $\sigma_{VL}$ and $\textcolor{black}{m}_{VL}$
  9. Fully anisotropic case
  10. Topological Changes
  11. Conclusion

Figures (31)

  • Figure 1: A triple junction in two dimensions
  • Figure 1: Diagram of the dewetting/wetting setting. The substrate $S$ need not be flat.
  • Figure 1: Dynamics of the anisotropic droplet using the sorting algorithm \ref{['alg:pvs_mf_sorting1']}. The filled component is the liquid droplet at different times $T$, the red line is the theoretic Wulff shape associated with the surface tension, and the black lines are interfaces.
  • Figure 1: Left: Polar plot for the surface tension $\sigma_{VL}$ and its Wulff shape. The blue line represents the plot for surface tension, and the orange line depicts the Wulff shape. The region of the Wulff shape above the dashed line (where $\sigma_{VS} - \sigma_{LS} = -0.5$) corresponds to the equilibrium state of this droplet. Right: The weight function associated with the surface tension $\sigma_{VL}$ and $\textcolor{black}{m_{VL}}$.
  • Figure 1: Error table for the dynamic convergence test of a droplet with anisotropic surface tension $\sigma_{VL}(\theta) = \sqrt{1+\cos^2(\theta-\frac{\pi}{3})}$ and mobility $\textcolor{black}{m}_{VL}(\theta) = \frac{(3 - \cos(\frac{\pi}{3} + 2\theta))^{3/2}}{4\sqrt{2}}$ on a curved solid substrate. Final time $T = 0.008$.
  • ...and 26 more figures

Theorems & Definitions (5)

  • Claim 4.1
  • Proof 1
  • Remark 4.2
  • Claim 5.1
  • Proof 2