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arxiv: 2604.25510 · v1 · submitted 2026-04-28 · 🧮 math.NA · cs.NA

A sharp-interface model for solid-state dewetting with wetting potential

Weijie Huang , Xinran Ruan This is my paper

Pith reviewed 2026-05-07 15:34 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords solid-state dewettingsharp-interface modelwetting potentialsurface diffusionfinite element methodthin filmsmorphological evolutionnumerical simulation
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The pith

A sharp-interface model incorporates wetting via thickness-dependent surface energy for solid-state dewetting.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes a sharp-interface model for the morphological evolution of thin solid films on substrates during dewetting, where wetting interactions are included by letting surface energy depend on local film thickness. The evolution follows surface diffusion along the film-vapor interface, with contact-line conditions that arise naturally from varying the energy. An efficient semi-implicit finite element method is constructed by applying a Taylor expansion to the wetting-potential term at each time step. Two-dimensional simulations reproduce multiple dewetting behaviors including retraction and island formation, while three-dimensional runs handle more intricate shapes. The computations also show that the model recovers an earlier thickness-independent sharp-interface version when the spatial range of the wetting potential is driven toward zero.

Core claim

The central claim is that wetting effects can be incorporated into a sharp-interface description of solid-state dewetting by replacing constant surface energy with a thickness-dependent form. The resulting system is governed by surface diffusion together with natural boundary conditions at the three-phase contact line. A semi-implicit finite element discretization is obtained by Taylor-expanding the wetting term, and this scheme is shown to be practical for both two- and three-dimensional computations that capture observed morphological changes while recovering the thickness-independent limit as the potential range vanishes.

What carries the argument

Thickness-dependent surface energy inside the sharp-interface energy functional, evolved by surface diffusion, together with the semi-implicit finite element scheme obtained from a Taylor expansion of the wetting-potential term.

Load-bearing premise

The wetting effect is adequately captured by a thickness-dependent surface energy whose range parameter can be taken to zero to recover the prior model, and that the Taylor expansion of the wetting term yields a stable and accurate semi-implicit scheme.

What would settle it

If two- or three-dimensional simulations with successively smaller values of the wetting-potential range fail to approach the morphological evolution produced by the thickness-independent model of reference [1], the recovery claim would be refuted.

Figures

Figures reproduced from arXiv: 2604.25510 by Weijie Huang, Xinran Ruan.

Figure 1
Figure 1. Figure 1: FIG. 1. A schematic illustration of the evolution from an ini view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Wetting potential as a function of the film thickness. view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. A schematic illustration of the wetting-potential view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison of the numerical equilibrium shapes view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Thickness of the wetting layer in equilibrium. view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Snapshots of the evolution of an initially small isla view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Evolution of (a) the normalized area view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Snapshots of the evolution of a long island film view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Snapshots of the evolution of a long island film with view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Number of agglomerates formed from a high-aspect view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Snapshots of the evolution of a long island film with view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Evolution of a semi-infinite film with view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Schematic of the fitting procedure used to define view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Evolution of the minimum valley thickness view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Effective contact point position versus time during view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. Snapshots of the evolution of an initially (30 view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. Cross-sections of the island film along the view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24. Snapshots of the evolution of an initially (60 view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25. Cross-sections of the island film along the view at source ↗
Figure 26
Figure 26. Figure 26: FIG. 26. Snapshots of the evolution of an initially (1 view at source ↗
Figure 31
Figure 31. Figure 31: FIG. 31. Snapshots of the evolution of an initially square view at source ↗
Figure 29
Figure 29. Figure 29: FIG. 29. Snapshots of the evolution of an initially square-r view at source ↗
Figure 30
Figure 30. Figure 30: FIG. 30. Cross-sections of the island geometry for the exam view at source ↗
Figure 35
Figure 35. Figure 35: FIG. 35. Snapshots of the evolution of an initially cross view at source ↗
Figure 36
Figure 36. Figure 36: FIG. 36. Snapshots of the evolution of an initially cross view at source ↗
read the original abstract

We propose a sharp-interface model for solid-state dewetting of thin films with wetting potential, where the wetting effect is incorporated through a thickness-dependent surface energy. The model is governed by surface diffusion together with natural boundary conditions, and describes the morphological evolution of the film-vapor interface. For its numerical approximation, we develop an efficient semi-implicit finite element method based on a Taylor expansion of the wetting-potential term. Numerical simulations in two dimensions show that the proposed model and method can capture various dewetting phenomena. They also indicate that, as the range of the wetting potential tends to zero, the proposed model approaches the sharp-interface model with thickness-independent surface energy proposed in [1]. The model and numerical method are further extended to three dimensions, where the computations capture complex morphological evolution in solid-state dewetting.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The paper proposes a sharp-interface model for solid-state dewetting of thin films that incorporates wetting effects through a thickness-dependent surface energy. The evolution is governed by surface diffusion with natural boundary conditions at the contact line. A semi-implicit finite-element scheme is derived by applying a first-order Taylor expansion to the wetting-potential contribution in the variational formulation. Two-dimensional simulations are presented to illustrate various dewetting morphologies, and the model is shown to approach the thickness-independent sharp-interface model of reference [1] as the range of the wetting potential tends to zero. The formulation and method are extended to three dimensions.

Significance. If the numerical method can be shown to be stable and consistent, the work would supply a practical sharp-interface framework that embeds wetting physics while recovering a known limit model, together with an efficient time-stepping scheme that enables both 2-D and 3-D computations of complex morphologies.

major comments (3)
  1. [Numerical scheme] Numerical scheme section: the semi-implicit discretization is obtained by replacing the wetting-potential term with its first-order Taylor expansion inside the variational form. No unconditional stability estimate, consistency analysis, or comparison of energy-dissipation rates between the linearized scheme and the fully nonlinear problem is supplied. Because the central claim that the simulations are reliable and recover the limit model rests on this approximation, the absence of such analysis is load-bearing.
  2. [Two-dimensional simulations] Two-dimensional simulations section: the statement that the model approaches the sharp-interface model of [1] as the range parameter tends to zero is supported only by visual inspection of interface shapes. No quantitative diagnostics (interface profiles, contact-angle errors, or energy-decay curves) are reported to measure the rate or accuracy of this limit, leaving the convergence claim unverifiable.
  3. [Abstract and numerical results] Abstract and numerical results: the manuscript asserts that the simulations capture dewetting phenomena and recover the limit case, yet supplies neither error analysis, convergence rates under mesh or time-step refinement, nor quantitative comparisons against a fully implicit solver on identical meshes. These omissions directly affect the credibility of the reported morphologies and the zero-range limit.
minor comments (1)
  1. [Model formulation] Notation for the range parameter of the wetting potential should be introduced once and used consistently in both the model derivation and the numerical experiments.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address each major comment below and indicate the revisions we intend to make to strengthen the numerical analysis and validation.

read point-by-point responses

  1. Referee: [Numerical scheme] Numerical scheme section: the semi-implicit discretization is obtained by replacing the wetting-potential term with its first-order Taylor expansion inside the variational form. No unconditional stability estimate, consistency analysis, or comparison of energy-dissipation rates between the linearized scheme and the fully nonlinear problem is supplied. Because the central claim that the simulations are reliable and recover the limit model rests on this approximation, the absence of such analysis is load-bearing.

    Authors: We agree that a more detailed analysis of the semi-implicit scheme is warranted. The first-order Taylor expansion is employed specifically to linearize the nonlinear wetting-potential contribution and thereby obtain an efficient scheme that does not require solving a nonlinear algebraic system at each step. While the resulting scheme is not unconditionally stable (its stability is conditional on the time-step size relative to the mesh size and the range parameter), we will add a consistency analysis of the linearization error together with a direct comparison of the discrete energy-dissipation rate produced by the linearized scheme versus the fully nonlinear variational formulation. These additions will appear in a new subsection of the numerical-method section. revision: yes

  2. Referee: [Two-dimensional simulations] Two-dimensional simulations section: the statement that the model approaches the sharp-interface model of [1] as the range parameter tends to zero is supported only by visual inspection of interface shapes. No quantitative diagnostics (interface profiles, contact-angle errors, or energy-decay curves) are reported to measure the rate or accuracy of this limit, leaving the convergence claim unverifiable.

    Authors: We accept that visual inspection alone does not constitute rigorous verification of the zero-range limit. In the revised manuscript we will supplement the existing figures with quantitative diagnostics: (i) L²-norm differences between the computed interface positions and those of the thickness-independent model for a sequence of decreasing range parameters, (ii) tabulated contact-angle errors relative to the limit value, and (iii) overlaid energy-decay curves that demonstrate the convergence of the dissipation rate. These data will be presented in a dedicated convergence subsection. revision: yes

  3. Referee: [Abstract and numerical results] Abstract and numerical results: the manuscript asserts that the simulations capture dewetting phenomena and recover the limit case, yet supplies neither error analysis, convergence rates under mesh or time-step refinement, nor quantitative comparisons against a fully implicit solver on identical meshes. These omissions directly affect the credibility of the reported morphologies and the zero-range limit.

    Authors: The manuscript is primarily devoted to model derivation and the demonstration of morphological capabilities rather than a comprehensive numerical-analysis study. Nevertheless, to bolster credibility we will include (a) spatial and temporal convergence studies under successive mesh and time-step refinement for a representative dewetting configuration, reporting observed orders of accuracy, and (b) a side-by-side quantitative comparison, on identical meshes, between the semi-implicit scheme and a fully implicit nonlinear solver for at least one benchmark problem. These results will be added to the numerical-results section and referenced in the abstract. revision: yes

Circularity Check

0 steps flagged

Independent modeling choice with non-load-bearing consistency check

full rationale

The derivation introduces a thickness-dependent surface energy as an explicit modeling choice to capture wetting, independent of prior results. The semi-implicit FEM is constructed via a first-order Taylor expansion of the wetting term inside the variational form, presented as a numerical approximation rather than an exact reduction. The statement that the model approaches the thickness-independent model of [1] as the range parameter tends to zero is reported solely as a numerical observation from simulations, serving as a consistency verification and not as a load-bearing step in deriving the main model or scheme. No equations reduce to their inputs by construction, no fitted parameters are relabeled as predictions, and the central claims rest on the new formulation and its direct numerical implementation.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The model rests on the standard assumption that morphological evolution is driven by surface diffusion and on the modeling choice that wetting is represented by a thickness-dependent surface energy; no new particles or forces are postulated.

free parameters (1)
  • range of wetting potential
    A tunable length scale whose zero limit is used to recover the earlier model; its specific value is chosen for simulation studies.
axioms (2)
  • domain assumption Surface diffusion governs the morphological evolution of the film-vapor interface
    Invoked in the governing equations and boundary conditions stated in the abstract.
  • domain assumption Wetting effects can be incorporated via thickness-dependent surface energy
    Central modeling hypothesis that defines the new term in the energy.

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Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

  1. [1]

    The total free energy of the system is given by W ε(h) = ∫ Γ γ ε(h)ds = ∫ b a γ ε(h) √ 1 + (∂xh)2dx, (2) where Γ = Γ( t) denotes the moving surface profile, namely the film-vapor interface, and s denotes the arc length along the interface. The chemical potential is defined as the variational derivative of the free energy with respect to the height function h...

    work page

  2. [2]

    5 and x2 = 2

    with x1 = − 2. 5 and x2 = 2. 5, Fig. 4 shows that the computed equilibrium shapes approach the theoretical equilibrium shape of the h-independent model predicted by the generalized Win- terbottom construction [ 17] as ε decreases. We also examine the thickness of the wetting layer in equilibrium. We denote by h∗ the nearly uniform film thickness away from ...

    work page

  3. [3]

    01 and perform a series of numerical simulations for large islands with dif- ferent aspect ratios and different values of θi

    To examine this depen- dence more systematically, we fix ε = 0. 01 and perform a series of numerical simulations for large islands with dif- ferent aspect ratios and different values of θi. The results are summarized in Fig

    work page

  4. [4]

    For comparison, the corresponding results for the h-independent model reported by Dornel [ 18] are also shown

    We observe clear boundaries separating the regions with 1, 2, and 3 (or more) ag- glomerates. For comparison, the corresponding results for the h-independent model reported by Dornel [ 18] are also shown. IV.4. Dynamics of semi-infinite films In this subsection, we consider the evolution of ini- tially semi-infinite films described by ( 16) with x1 = 0 and x2...

    work page

  5. [5]

    In the following computations, we choose hc = 0

    More precisely, it is defined as the intersection of the substrate y = 0 and a quadratic fitting of the film profile in a prescribed region. In the following computations, we choose hc = 0. 2 and α = 0. 1. fitting region FIG. 18. Schematic of the fitting procedure used to define the effective contact point position xc. The solid black curve denotes the film profil...

    work page

  6. [6]

    In particular, for both ε = 0

    over a wide time interval. In particular, for both ε = 0. 05 and ε = 0. 025, the curves correspond- ing to different values of σ follow the same overall trend predicted by this law. This suggests that, during the first mass-shedding cycle, the motion of the effective contact point in the present model is still well captured by the fitting law from the h-indep...

    work page

  7. [7]

    The inward shrinkage of such toroidal structures has also been reported in [ 23, 24]

    As the evolution proceeds, the resulting ring-like structure becomes nearly axisymmet- ric, shrinks inward, and eventually merges into a single island. The inward shrinkage of such toroidal structures has also been reported in [ 23, 24]. The corresponding cross-sections are shown in Fig. 23. For an even larger square island, namely (60 , 60, 1), the film a...

    work page

  8. [8]

    As the cuboid becomes longer, pinch-off occurs and the film breaks into several isolated particles, as shown in Figs

    As the evolution proceeds, the accumulated ma- terial moves toward the center, and the island eventually relaxes to a single cap-shaped equilibrium. As the cuboid becomes longer, pinch-off occurs and the film breaks into several isolated particles, as shown in Figs. 27 and 28. The breakup in the transverse direction 10 -40 -20 0 20 40 0 4 8 (a) -40 -20 0 20...

    work page

  9. [9]

    When the length is further increased to (1 , 24, 1), more particles are formed, and a clear coarsening process is observed, in which larger particles gradually absorb smaller ones, as shown in Fig. 28. These examples show a clear aspect-ratio effect in the evolution of elongated cuboid islands. As the length in- creases, the morphology changes from retract...

    work page

  10. [10]

    The corresponding cross-sections are shown in Fig. 30. As the size of the square-ring island increases, more complicated topological changes occur. For c = 11, pinch-off takes place and the film breaks into four iso- lated islands, which then evolve toward their equilibrium shapes, as shown in Fig

    work page

  11. [11]

    When the size is further in- creased to c = 17, more isolated islands are formed and a clear coarsening process is observed, as shown in Fig. 33. To better illustrate these evolutions, the corresponding cross-sections are plotted in Fig. 32 and Fig. 34. FIG. 29. Snapshots of the evolution of an initially square-r ing island obtained from a (6 , 6, 1) cubo...

    work page

  12. [12]

    As time evolves, the largest particle in the center gradually absorbs the smaller ones, indicating a coarsening process, as shown in Fig

    When the limb length is increased to c = 9, the island breaks into five isolated particles. As time evolves, the largest particle in the center gradually absorbs the smaller ones, indicating a coarsening process, as shown in Fig. 36. These examples indicate that the proposed model can also handle complex initial geometries and capture rich three-dimensiona...

    work page

  13. [13]

    with respect to the height function h(x). For any g ∈ H 1(I) with I = (a,b ), inte- gration by parts yields d dαW ε(h +αg ) ⏐ ⏐ ⏐ ⏐ α =0 = ∫ b a [ (γ ε)′(h) √ 1 + (∂xh)2g + γ ε(h)∂xh∂ xg√ 1 + (∂xh)2 ] dx = ∫ b a [ (γ ε)′(h) √ 1 + (∂xh)2 − ∂x ( γ ε(h)∂xh√ 1 + (∂xh)2 )] gdx + γ ε(h)∂xh√ 1 + (∂xh)2 g ⏐ ⏐ ⏐ x=b x=a = ∫ b a [ (γ ε)′(h) √ 1 + (∂xh)2 − γ ε(h)∂xx...

    work page

  14. [14]

    Y. Wang, W. Jiang, W. Bao, and D. J. Srolovitz, Phys. Rev. B 91, 045303 (2015)

    work page 2015

  15. [15]

    D. J. Srolovitz and S. A. Safran, J. Appl. Phys. 60, 247 (1986)

    work page 1986

  16. [16]

    H. Wong, P. W. Voorhees, M. J. Miksis, and S. H. Davis, Acta Mater. 48, 1719 (2000)

    work page 2000

  17. [17]

    W. Bao, W. Jiang, Y. Wang, and Q. Zhao, J. Comput. Phys. 330, 380 (2017)

    work page 2017

  18. [18]

    Jiang, Q

    W. Jiang, Q. Zhao, and W. Bao, SIAM Journal on Ap- plied Mathematics 80 (2020)

    work page 2020

  19. [19]

    Q. Zhao, W. Jiang, and W. Bao, SIAM Journal on Sci- entific Computing 42 (2020)

    work page 2020

  20. [20]

    Bao and Q

    W. Bao and Q. Zhao, J. Comput. Math. 41, 771 (2023)

    work page 2023

  21. [21]

    Jiang, W

    W. Jiang, W. Bao, C. V. Thompson, and D. J. Srolovitz, Acta Mater. 60, 5578 (2012)

    work page 2012

  22. [22]

    Dziwnik, A

    M. Dziwnik, A. M¨ unch, and B. Wagner, Nonlinearity 30, 1465 (2017)

    work page 2017

  23. [23]

    Pierre-Louis, A

    O. Pierre-Louis, A. Chame, and Y. Saito, Phys. Rev. Lett. 103, 195501 (2009)

    work page 2009

  24. [24]

    Chiu and H

    C.-H. Chiu and H. Gao, Mater. Res. Soc. Symp. Proc. 356, 33 (1995)

    work page 1995

  25. [25]

    Peschka, S

    D. Peschka, S. Heafner, L. Marquant, K. Jacobs, A. M¨ unch, and B. Wagner, Proceedings of the Nati- nal Academy of Sciences of the United States of America 116.19, 9275 (2019)

    work page 2019

  26. [26]

    A. K. Tripathi and O. Pierre-Louis, Phys. Rev. E 97, 022801 (2018)

    work page 2018

  27. [27]

    A. K. Tripathi and O. Pierre-Louis, Physical Review E 101, 042802 (2020)

    work page 2020

  28. [28]

    Z. Zhou, W. Jiang, and Z. Zhang, Journal of Nonlinear Science 34 (2024)

    work page 2024

  29. [29]

    R. W. Balluffi, S. M. Allen, and W. C. Carter, Wiley, London (2005)

    work page 2005

  30. [30]

    W. Bao, W. Jiang, D. J. Srolovitz, and Y. Wang, SIAM J. Appl. Math. 77, 2093 (2017)

    work page 2093

  31. [31]

    Dornel, J

    E. Dornel, J. C. Barbe, F. DeCr´ ecy, G. Lacolle, and J. Eymery, Phys. Rev. B 73, 115427 (2006)

    work page 2006

  32. [32]

    C. V. Thompson, Annu. Rev. Mater. Res. 42, 399 (2012)

    work page 2012

  33. [33]

    Ye and C

    J. Ye and C. V. Thompson, Appl. Phys. Lett. 97, 071904 (2010)

    work page 2010

  34. [34]

    Ye and C

    J. Ye and C. V. Thompson, Adv. Mater. 23, 1567 (2011)

    work page 2011

  35. [35]

    Naffouti, R

    M. Naffouti, R. Backofen, M. Salvalaglio, T. Bottein, M. Lodari, A. Voigt, T. David, A. Benkouider, I. Fraj, L. Favre, et al. , Science Advances 3, 1472 (2017)

    work page 2017

  36. [36]

    Q. Zhao, J. Comput. Appl. Math. 361 (2019)

    work page 2019

  37. [37]

    Jiang, Q

    W. Jiang, Q. Zhao, T. Qian, D. J. Srolovitz, and W. Bao, Acta Materialia 163, 154 (2019)

    work page 2019

  38. [38]

    G. H. Kim and C. V. Thompson, Acta Mater. 84, 190 (2015)

    work page 2015

  39. [39]

    Rayleigh, Proc

    L. Rayleigh, Proc. Lond. Math. Soc 1, 4 (1878)

    work page