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A domain decomposition method for the directional contact angle hysteresis interval on doubly periodic rough surfaces

Zijie Lin, William M Feldman, Braxton Osting

arXiv:2609.17811v1math.APcs.NAmath.NAphysics.comp-ph

TL;DR

Roughness can pin contact lines, allowing stationary droplets to exhibit apparent angles that differ from the local Young angle. The paper formulates directional CAH and introduces a two-scale method that combines near-region MBO updates with far-region minimal-surface updates. Computations show a strongly directional interval with a sharp change near diagonal orientations, constraining the wetted shapes the surface can support.

  • Problem

    Roughness can pin contact lines so stationary droplets exhibit a range of apparent angles despite locally satisfying Young’s angle.

    Source: §1 · p. 1

  • Method

    The two-scale alternating method couples an MBO diffusion-generated update near the contact line with a linearized minimal-surface update in the far region.

    Source: §1 · p. 2

  • Results

    The computed CAH interval is direction-dependent, widest along the diagonal, and changes sharply near that orientation.

    Source: §1.3 · p. 6, §6 · p. 27

  • Takeaways & Limitations

    Because a stationary droplet’s apparent angle must lie inside the directional interval along its contact line, the interval constrains supported wetted shapes; a diagonal-sided square-like set is admissible for the studied surface.

    Source: §6 · p. 27

    Computations use approximate far-field boundary conditions, one doubly periodic surface, and finitely many rational orientations, so observed directional changes are numerical evidence rather than a general proof of discontinuity.

    Source: §6 · p. 27

Abstract

from arXiv · show

We study wetting on a doubly periodic rough surface in three dimensions. Although the liquid-vapor interface meets the solid at the local Young's angle, microscale roughness can cause the macroscopic apparent angle to differ substantially from this value. We formulate the directional contact angle hysteresis (CAH) interval in terms of apparent angles associated with pinned microscopic configurations. To approximate its receding and advancing endpoints, we evolve capillary mean curvature flow (CMCF) toward extremal stationary states. Computing these states is difficult because the pinning that produces hysteresis is generated at the scale of the roughness, whereas the apparent angle is only meaningful at the macroscopic scale, so a single uniform grid must resolve both. We therefore introduce a two-scale alternating (TSA) method based on a Schwarz decomposition: a Merriman-Bence-Osher (MBO) diffusion-generated scheme resolves the contact-line near region, while a linearized minimal-surface problem updates the far region. For an idealized reference iteration, we prove decay of an approximate interfacial energy. Numerical experiments on a representative doubly periodic surface show a strongly anisotropic CAH interval whose width varies by more than a factor of three with contact-line orientation and changes sharply near the diagonal directions. Because a stationary droplet must meet the solid at an apparent angle inside this interval, the computed anisotropy constrains which macroscopic wetted regions the surface can support; it is consistent with a square-like stationary droplet whose sides align with the diagonal directions.

1. Introduction

The introduction formulates directional contact angle hysteresis (CAH) for periodic rough surfaces and presents a two-scale method for approximating its endpoints. It motivates the method by the mismatch between microscale pinning and macroscopic apparent angles.

1. Introduction

  • Pinned contact lines let stationary drops exhibit a range of apparent angles, a phenomenon relevant to imbibition, surface design, printing, bio-microarrays, and evaporation.CAH has also been linked to anisotropic macroscopic droplet shapes on micropatterned surfaces.Source: §1 · pp. 1–2
  • Estimating CAH endpoints is computationally difficult because roughness-scale pinning and macroscopic apparent angles require resolving different spatial scales.A uniform grid fine enough for the contact line must cover the whole domain, becoming prohibitive as ε → 0.Source: §1 · p. 2

1.1. Surface energy and capillary problems

  • For doubly periodic rough surfaces, the directional CAH interval is the set of apparent angles of stationary microscopic configurations and may depend on contact-line direction.The formulation replaces the classical contact-angle condition with an interval constraint.Source: §1.1 · pp. 3–4

1.2. The contact angle hysteresis (CAH) interval

  • The receding and advancing endpoints are approximated by evolving capillary mean curvature flow toward minimal and maximal stationary states.The method initializes lower and upper angle barriers; these states are expected to attain the smallest and largest apparent angles, respectively.Source: §1.3 · p. 6, §1.2 · p. 6

1.3. Overview of results

  • The two-scale alternating method couples an MBO update near the contact line with a linearized minimal-surface update in the far region, and a reference iteration has an approximate interfacial-energy decay property.The far-region update uses a Fourier-series solution based on boundary data from the near-region update.Source: §1.3 · p. 6
  • On the tested doubly periodic surface, the computed CAH is anisotropic and changes sharply near π/4; a diagonal-sided square-like droplet has contact angles inside the interval.The droplet is therefore stationary on the micropatterned surface.Source: §1.2 · p. 5, §1.3 · p. 6

2. Computation of the contact angle hysteresis (CAH) interval

The section develops two ways to approximate contact-angle hysteresis endpoints, first in a periodic toy model and then for capillary mean curvature flow on rough surfaces. In the capillary problem, stationary states approached from lower and upper angle barriers estimate the receding and advancing endpoints.

2.1. A toy model for the 2D hysteresis

  • The toy model pins precisely for forces in [a,b], and serves as a reduced account of contact-line pinning and hysteresis.Its scalar coordinate represents contact-line displacement, while the forcing plays the role of apparent angle.Source: §2.1 · pp. 7–8

2.1.1. Approach I: Bisection Method

  • Bisection approximates both endpoint forces by evolving the system at trial forces and classifying whether motion traverses several periods.The bracketing interval converges linearly; the method uses motion because the periodic force cannot be evaluated explicitly.Source: §2.1.1 · p. 8

2.1.2. Approach II: Forced Method

  • As ε tends to zero, the forced steady state approaches a maximum of the periodic force, and the two endpoints are approximated with O(ε) error.The method replaces constant forcing with a decreasing function of position; initializing on opposite sides of the force range targets the two extrema.Source: §2.1.2 · p. 8

2.2. Methods to approximate the CAH interval

  • The capillary approaches mirror the toy methods: half-space bisection varies the prescribed far-field angle, while a finite channel uses a fictitious-boundary condition.The channel’s Dirichlet condition supplies the decreasing forcing analogue.Source: §2.2 · p. 9

2.2.1. Approach I: Bisection method in a half-space domain

  • In the half-space, lower and upper angle barriers bracket stationary apparent angles, and bisection classifies trial flows as advancing, receding, or pinned.The planar initial configurations need not themselves satisfy the Young-angle condition or be stationary solutions.Source: §2.2.1 · pp. 9–10

2.2.2. Approach II: CMCF in a finite height channel

  • In the finite channel, lower and upper barrier starts converge to extremal stationary states that approximate θrec and θadv; this approach was observed to cost less than half-space bisection.The apparent angle is insensitive to the artificial height R once R is large compared with the roughness period.Source: §2.2.2 · p. 10, Figure 5 · p. 11
  • For general rough surfaces, there is no theoretical guarantee that the angle barriers produce advancing and receding solutions; in practice, they are chosen far from the observed pinning interval.The interfaces were empirically observed to evolve monotonically across several roughness periods before converging.Source: §2.2.2 · p. 10

3. A two-scale alternating (TSA) method for capillary mean curvature flow (CMCF) (10)

The two-scale alternating method handles the different spatial scales of capillary mean curvature flow by updating overlapping near and far regions in turn. It combines a fine-scale MBO update near the contact line with a linearized minimal-surface update farther away.

3. A two-scale alternating (TSA) method for capillary mean curvature flow (CMCF) (10)

  • TSA alternates updates across overlapping regions: MBO resolves contact-line dynamics nearby, while a linearized minimal-surface problem updates the macroscopic far region.The decomposition avoids carrying a uniformly fine discretization throughout the domain.Source: §3 · p. 11, Figure 6 · p. 12

3.1. Near-region: MBO diffusion-generated method

  • The near-region MBO update decreases the approximate interfacial energy for τ1 > 0.The method uses a variant without a volume constraint; its time-step choices preserve the Young angle at the contact line.Source: §3.1 · pp. 12–14

3.2. Far-region: minimal surface approximation

  • The far-region interface is updated as a stationary minimal surface using the previous near-region interface as boundary data, then linearized about its mean slope.The linearization uses that the far-region interface remains close to a plane.Source: §3.2 · pp. 14–15

This linear problem admits the following Fourier series solution

The macroscopic slope links interfacial energy to apparent angle: on either monotone energy branch, the slope evolves until pinning determines the stationary angle. The TSA energy analysis introduces an intermediate reference functional and proves that its iterations decrease energy.

This linear problem admits the following Fourier series solution

  • Linearizing the far-region minimal-surface equation about mean slope m yields u_zz + (1 + m^2)u_yy = 0, with periodic boundary data handled by Fourier modes.The Fourier series is truncated in practice; its truncation parameter is chosen according to the mode damping.Source: This linear problem admits the… · p. 16

3.3. Energy-slope relation of the macroscopic interface

  • The channel energy has a leading O(R) planar-interface contribution and an O(1) correction from microscopic roughness, assuming the interface stays nearly planar above height r.The effective solid energy densities are treated as constants in this heuristic relation.Source: §3.3 · p. 16
  • On either monotone branch of F(m), the slope moves until pinning, and its stopping value determines the apparent angle of the extremal stationary state.The leading O(R) energy term has a unique minimum at m = −cot θ̄Y; the O(1) roughness contribution may produce small oscillations around the trend.Source: §3.3 · pp. 16–17

3.4. Reference formulation of the two-scale alternating (TSA) method

  • The localized and global energies have matching first variations with respect to the interface perturbation.Source: §3.4 · p. 17
  • The reference TSA iteration uses an intermediate energy functional to connect the localized and global energies and make an energy analysis possible.For small τ1, the intermediate energy approximates the global energy; the localized energy is expected to approximate it at first variation.Source: §3.4 · p. 18
  • The reference iterations (23) and (24) satisfy an energy-decreasing property, although the implemented alternating updates minimize different energies.The distinct regional energies make an energy principle for the complete iteration non-immediate.Source: §3.4 · pp. 17–18

4. Implementation of the TSA method

The TSA method approximates extremal stationary states by alternating a contact-line-near MBO update with a far-region linearized minimal-surface update. A two-level stopping procedure refines the time step before the stationary apparent angles are used as CAH endpoints.

4. Implementation of the TSA method

  • The channel method approximates receding and advancing CAH endpoints by evolving from lower and upper angle barriers toward minimal and maximal stationary states.The stationary states are computed for rational directions of the doubly periodic rough surface.Source: §4 · p. 19, §4.2 · p. 20

4.1. Overview of the TSA method

  • The near-region MBO step diffuses and thresholds the liquid and vapor regions, while the far-region reconstructs the interface by solving a linearized minimal-surface problem from near-region trace data.The analytical reference problem uses Dirichlet data, whereas the numerical MBO implementation uses a Neumann-like condition suited to convolution.Source: §4.1 · p. 19

4.5. Implementation details for Step 4: two-level stopping criterion

  • The two-level stopping criterion first checks convergence at a fixed time step, then halves the step and stops when further refinement no longer changes the computed configuration on the grid.The resulting stationary states supply the apparent receding and advancing angles; the apparent angle is recorded as π − θi.Source: §4.1 · p. 19, §4.5 · p. 21

4.2. Implementation details for Step 1: initialization

  • Initialization places the interface in a centered near-region and uses a reflected copy in the x-direction to support FFT computation without boundary artifacts.The computation samples rational directions chosen to preserve the surface’s Z2-periodicity.Source: §4.2 · p. 20

4.3. Implementation details for Step 2: near-region update (MBO)

  • The MBO update uses FFT-based convolution with periodicity in y and reflection in x, while time steps enforce the prescribed Young and imaginary contact angles.Source: §4.3 · p. 21

4.4. Implementation details for Step 3: far-region update (minimal surface)

  • The far-region interface is represented as a graph and reconstructed from Fourier coefficients, with the truncated series updating only the near/far transition region.Source: §4.4 · p. 21

5. Numerical experiments

The sampled CAH intervals are anisotropic, with the widest interval at the diagonal and sharp endpoint changes near that direction. The computed slope and contact-line structures are consistent with localized transitions and a square-like wetted region aligned with the diagonals.

5.1. Directional CAH over rational directions

  • Every sampled orientation has distinct receding and advancing angles; the interval is widest at the diagonal, with sharp endpoint changes nearby that do not numerically establish a discontinuity.Source: §5.1 · p. 22
  • 0.0130 and 0.0222 radians are the endpoint differences at ±arctan(1/3), while 0.0088 and 0.0545 radians are the differences at ±π/4.The computed endpoints reproduce the rough surface’s directional symmetry to four decimal places at the sampled orientations described in the passage.Source: §5.1 · p. 22
  • Whether finite-channel computations at opposite orientations preserve symmetry depends on the parity of the direction integers, because some transformations change the phase fixed by the upper boundary.Source: §5.1 · p. 23

5.2. Large-scale monotonicity of the slope m

  • For the two representative orientations, advancing and receding slope trajectories follow the predicted large-scale monotone trend, with small variations near their final stages.Source: §5.2 · p. 23

5.3. Interface and contact-line structures near the CAH discontinuity

  • The interfaces become planar-like with height, while nearby stationary contact lines share long pattern segments and differ through localized switches to neighboring patterns.The proposed typewriter-like rearrangement is a geometric interpretation of stationary configurations, not a model of transient physical dynamics.Source: §5.3 · pp. 24, 26, Figure 15 · p. 26

5.4. Revisiting the anisotropy effect

  • The wider diagonal pinning range and localized contact-line transitions are consistent with faceted, square-like wetted regions whose sides align with diagonal directions.The connection to faceting is less clear in this volume-constrained setting than in related settings without a volume constraint.Source: §5.4 · p. 26

6. Discussion

The TSA method combines a near-contact-line MBO update with a far-region minimal-surface approximation, and its reference iteration has a proved approximate-energy decay property. Computations show directional CAH anisotropy that constrains admissible macroscopic droplet shapes, while the reported conclusions remain limited to a single surface and finite rational orientations.

6. Discussion

  • The TSA method couples near-contact-line MBO updates with a Fourier-based far-region minimal-surface approximation, and an idealized reference iteration has approximate interfacial-energy decay.The energy result, together with a heuristic energy-slope relation, explains the observed large-scale monotone behavior of the macroscopic slope m.Source: §6 · p. 26
  • The computed CAH interval is widest along the diagonal, with differences between stationary contact lines spatially localized near that orientation.The localized transition gives a typewriter-like geometric interpretation; the other diagonal directions follow by surface symmetry.Source: §6 · p. 27
  • A stationary droplet’s apparent angle must lie inside the directional interval along its contact line, making the interval a necessary constraint on supported shapes.For the studied surface, a square-like wetted set with diagonal sides is admissible; Surface Evolver contact angles lie inside the computed interval around the contact line.Source: §6 · p. 27
  • The computations use approximate far-field boundary conditions, one doubly periodic surface, and finitely many rational orientations, so the directional changes are numerical evidence rather than proof of discontinuity.Additional roughness geometries, nonperiodic surfaces, and broader parameter ranges are identified as needed to assess robustness.Source: §6 · p. 27
  • The formulation permits spatially varying Young’s angles, and the authors expect the decomposition to extend to chemically patterned surfaces.Directly incorporating a volume constraint to compute stationary droplets remains challenging; the closed-interval conjecture in the non-graphical setting and whether the sharp change at π/4 is a discontinuity remain open.Source: §6 · p. 27

Appendix A. Derivation of the MBO scheme for a capillary problem without volume constraint

The appendix derives the capillary MBO threshold choice and analyzes an iterative update obtained by minimizing a linearized energy. It also establishes energy decrease for the capillary MBO scheme under the stated positive-parameter condition.

Appendix A. Derivation of the MBO scheme for a capillary problem without volume constraint

  • The derivation arrives at a threshold choice of 0 for the capillary MBO scheme.The appendix frames this as deriving the threshold choice for a variation of MBO on a rough surface.Source: Appendix A · p. 30
  • At iteration k, the method linearizes the near-region energy about the current approximation and takes the minimizer of that linearized energy as the next update.The update is expressed through indicator functions of the resulting liquid and vapor sets.Source: Appendix A · p. 30
  • The next liquid and vapor sets are selected by the sign of φ, with φ nonnegative on one set and negative on the other.The proof uses these sign conditions to compare the updated sets with an arbitrary admissible pair.Source: Appendix A · p. 30
  • The capillary MBO scheme’s energy Eτ decreases for all τ1 > 0.The appendix presents this as stability of the scheme.Source: Appendix A · p. 30

Appendix B. Justification of the linearization far from the boundary

Under the stated trapping and periodicity assumptions, the appendix justifies representing the sufficiently high portion of a minimizing interface as a smooth graph. Its deviation from a tilted reference plane satisfies a periodic elliptic equation and decays exponentially toward a plane at large height.

Appendix B. Justification of the linearization far from the boundary

  • Above a height z∗, a surface-area minimizer trapped between parallel planes is a globally smooth subgraph in the (y,z) variables.The local regularity argument applies once the rescaled flatness is below the De Giorgi threshold; the trapping assumption bounds the graph’s deviation from the reference plane.Source: Appendix B · pp. 31–32
  • The proof outline first establishes graphicality using De Giorgi ε-regularity, then applies an elliptic boundary-layer result to the deviation from the reference plane.The graphicality step yields a smooth solution of the graphical minimal-surface equation before the deviation equation is considered.Source: Appendix B · pp. 31–32
  • The interface approaches a plane exponentially with height: |X(y,z)−(mz+s)| ≤ Ce^−c(z−z∗)/T.The stated bound follows from applying the periodic half-space elliptic result to the deviation equation.Source: Appendix B · pp. 32–33
  • The graph X(y,z) is periodic in y and solves the graphical minimal-surface equation; its deviation v = X − mz satisfies a linear divergence-form equation with periodic coefficients.The appendix uses this equation to connect the graph representation to a periodic elliptic boundary-layer result.Source: Appendix B · pp. 32–33
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