Solid-State Dewetting Fundamentals

• Solid-state dewetting is the capillarity-driven evolution of solid films on substrates via surface diffusion and contact line migration, resulting in morphological changes like island formation and hole growth.
• The process is governed by anisotropic surface energies and kinetic mechanisms, where equilibrium shapes are determined by conditions like the Young–Herring relation and contact line dynamics.
• Advanced numerical models, such as sharp-interface PFEM and phase-field methods, capture SSD dynamics to inform applications in nanofabrication and materials design.

Solid-state dewetting (SSD) is the capillarity-driven morphological evolution of a solid thin film on a substrate upon annealing, often below the melting point. The film remains solid, and mass transport is dominated by surface diffusion along the film/vapor interface, together with migration of contact points or contact lines where film, vapor, and substrate meet. The resulting dynamics include edge retraction, hole growth, island formation, pinch-off, coarsening, and relaxation toward equilibrium shapes such as circular arcs in isotropic settings and Wulff/Winterbottom-type shapes in anisotropic settings (Zhao et al., 2020, Bao et al., 2020).

1. Physical basis and characteristic morphologies

SSD is governed by surface-diffusion-limited kinetics rather than viscous flow. In two dimensions, if $\mu$ denotes the chemical potential along the open curve $\Gamma(t)$, the surface mass flux and normal velocity satisfy

$$J=-\partial_s\mu,\qquad V_n=-\partial_s J=\partial_{ss}\mu,$$

so interface motion is driven by curvature-induced gradients of chemical potential rather than by bulk transport (Zhao et al., 2020). In three dimensions, the corresponding evolution law is

$V_n=\Delta_s\mu,$

with $\Delta_s$ the Laplace–Beltrami operator on the film/vapor surface (Bao et al., 2020).

This kinetic structure produces a characteristic sequence of morphologies. Reported phenomena include film edge retraction, hole nucleation and growth, ridge–valley formation, pinch-off events, island formation, coarsening, and eventual approach to equilibrium. In isotropic cases the equilibrium profile is a circular arc in 2D or a spherical-cap-like shape in 3D, whereas anisotropy introduces faceting, sharp corners, missing orientations, and orientation-dependent breakup pathways (Bao et al., 2016, Zhao, 2017).

A common misconception is to treat SSD as a straightforward solid analogue of liquid dewetting. The analogy is limited. The relevant kinetics are Mullins-type surface diffusion, mass is conserved along the evolving surface, and contact-line motion is coupled to interfacial energetics and finite mobility. In crystalline films, anisotropic surface energy can alter not only the equilibrium morphology but also whether particular dynamical pathways such as valley formation, saw-tooth ridges, or directional pinch-off are favored (Jiang et al., 2018).

2. Energetics, wetting, and equilibrium constructions

The central thermodynamic object is the total free energy. For a 2D open curve on a flat substrate with anisotropic film/vapor energy density $\gamma(\theta)$, a standard dimensionless form is

$$W(t)=\int_{\Gamma(t)}\gamma(\theta)\,ds-\sigma\bigl[x_c^r(t)-x_c^l(t)\bigr],$$

where $\sigma=(\gamma_{SV}-\gamma_{SL})/\gamma_0$ is the substrate-energy contrast normalized by a reference surface energy $\gamma_0$ (Bao et al., 2016). In the isotropic limit, this reduces to interface length minus a wetted-length contribution (Zhao et al., 2020).

The corresponding equilibrium contact condition is the anisotropic Young–Herring relation

$$\gamma(\theta)\cos\theta-\gamma'(\theta)\sin\theta=\sigma.$$

For isotropy, $\Gamma(t)$0, this becomes the classical Young equation

$\Gamma(t)$1

The surface stiffness

$\Gamma(t)$2

separates weak anisotropy, for which $\Gamma(t)$3, from strong anisotropy, for which $\Gamma(t)$4 for some orientations and the unregularized dynamics become ill-posed (Bao et al., 2016).

Equilibrium-shape constructions follow from this energetics. For weak anisotropy, equilibrium islands are described by Wulff or Winterbottom constructions, truncated by the substrate. For strong anisotropy, however, the classical Winterbottom construction can fail: multiple admissible equilibrium shapes may exist for the same material parameters, including metastable shapes that are not obtained by the standard “cut-and-keep” rule. A repaired Winterbottom construction was proposed precisely to account for these multiple equilibria and their dynamical accessibility (Jiang et al., 2015).

On curved substrates, the contact geometry must distinguish intrinsic and extrinsic angles. If $\Gamma(t)$5 are the film-interface angles and $\Gamma(t)$6 are the substrate tangent angles, then

$\Gamma(t)$7

and the equilibrium condition becomes

$\Gamma(t)$8

This setting admits curvature-biased equilibria and migration phenomena absent on flat substrates (Jiang et al., 2018).

3. Sharp-interface formulations

A widely used isotropic 2D sharp-interface model parameterizes the film/vapor interface as an open curve $\Gamma(t)$9 and writes the evolution as

$$J=-\partial_s\mu,\qquad V_n=-\partial_s J=\partial_{ss}\mu,$$0

for $$J=-\partial_s\mu,\qquad V_n=-\partial_s J=\partial_{ss}\mu,$$1, together with the substrate-attachment conditions

$$J=-\partial_s\mu,\qquad V_n=-\partial_s J=\partial_{ss}\mu,$$2

the no-flux conditions

$$J=-\partial_s\mu,\qquad V_n=-\partial_s J=\partial_{ss}\mu,$$3

and the relaxed contact-angle laws

$$J=-\partial_s\mu,\qquad V_n=-\partial_s J=\partial_{ss}\mu,$$4

The crucial feature is that the relaxed angle condition can be rewritten as a Robin-type boundary condition and imposed as a natural boundary term in the weak formulation, so the interface and contact points evolve simultaneously (Zhao et al., 2020).

For anisotropic 2D SSD, the Cahn–Hoffman $$J=-\partial_s\mu,\qquad V_n=-\partial_s J=\partial_{ss}\mu,$$5-vector formulation is especially useful. With unit tangent $$J=-\partial_s\mu,\qquad V_n=-\partial_s J=\partial_{ss}\mu,$$6, unit normal $$J=-\partial_s\mu,\qquad V_n=-\partial_s J=\partial_{ss}\mu,$$7, and orientation angle $$J=-\partial_s\mu,\qquad V_n=-\partial_s J=\partial_{ss}\mu,$$8, one has

$$J=-\partial_s\mu,\qquad V_n=-\partial_s J=\partial_{ss}\mu,$$9

and the chemical potential may be written as

$V_n=\Delta_s\mu,$0

The interface then evolves by

$V_n=\Delta_s\mu,$1

Under strong anisotropy, regularization by a Willmore term modifies the chemical potential to

$V_n=\Delta_s\mu,$2

together with additional endpoint conditions such as $V_n=\Delta_s\mu,$3 at the contact points (Jiang et al., 2018).

In three dimensions, the sharp-interface model treats the film/vapor interface as a moving open surface $V_n=\Delta_s\mu,$4 with boundary $V_n=\Delta_s\mu,$5 on the substrate. In the isotropic case,

$V_n=\Delta_s\mu,$6

and the relaxed contact-angle law is

$V_n=\Delta_s\mu,$7

with zero mass flux

$V_n=\Delta_s\mu,$8

For anisotropic energies in Riemannian metric form, the Cahn–Hoffman vector becomes

$V_n=\Delta_s\mu,$9

and the contact-angle law is modified by the anisotropic conormal vector $\Delta_s$0 (Bao et al., 2020).

4. Variational structure, conservation laws, and regularization

The sharp-interface SSD equations are naturally expressed as gradient flows of interfacial energy under mass conservation. In 2D isotropic SSD on a flat substrate, the continuous weak formulation yields exact area conservation

$\Delta_s$1

and energy dissipation

$\Delta_s$2

The contact-point dissipation appears explicitly and is tied to the Robin-type boundary contribution in the weak form (Zhao et al., 2020).

The 3D isotropic model has the analogous structure:

$\Delta_s$3

and

$\Delta_s$4

In the anisotropic extension, $\Delta_s$5 is replaced by the anisotropic chemical potential $\Delta_s$6, and the same mass-conserving, energy-dissipative structure persists (Bao et al., 2020).

Curved substrates require additional geometric care. A structure-preserving formulation for 2D anisotropic SSD on curved substrates uses an arclength parameterization of the substrate and a symmetrized surface-energy matrix

$\Delta_s$7

which yields

$\Delta_s$8

This formulation gives an unconditional discrete energy law, and a localized correction of the discrete normals restores exact area preservation for general curved substrates (Bao et al., 2024).

Axisymmetric SSD leads to related structure-preserving formulations. For anisotropic surfaces of revolution, the chemical potential contains both meridional and azimuthal curvature contributions, and weak formulations based on anisotropic matrices $\Delta_s$9 produce provable volume conservation and energy stability (Li et al., 2024). In the strongly anisotropic axisymmetric setting, Willmore regularization adds curvature-squared energy and restores well-posedness when the stiffness becomes negative, while retaining discrete volume conservation and energy dissipation (Li et al., 8 Jan 2025).

5. Numerical methodologies and accuracy

Parametric finite element methods (PFEMs) are the dominant sharp-interface discretization strategy in the SSD literature. An early anisotropic PFEM for open curves used piecewise linear elements and semi-implicit time stepping, treated the contact lines explicitly, preserved area to machine precision, and exhibited numerical stability with $\gamma(\theta)$0 rather than the much more restrictive $\gamma(\theta)$1 typical of explicit marker-particle schemes. For open curves, however, its spatial convergence was approximately first order, attributed in part to the forward Euler update of the contact lines (Bao et al., 2016).

The energy-stable PFEM (ES-PFEM) for 2D isotropic SSD modified this picture by incorporating the relaxed contact-angle condition as a natural Robin boundary term. With backward Euler in time and geometric coefficients frozen at the previous step, the fully discrete problem is linear at each time step, requires the solution of a single linear system, is unconditionally energy-stable, and achieves second-order spatial convergence measured by a manifold distance between curves. The same formulation also exhibits long-time equal mesh distribution, quantified by the mesh-ratio indicator $\gamma(\theta)$2, and convergence to constant-curvature equilibria, quantified by the curvature-variation indicator $\gamma(\theta)$3 (Zhao et al., 2020).

The 3D counterpart uses continuous piecewise linear elements on triangulated surfaces and preserves the same structural philosophy: simultaneous evolution of the interface and its contact line, backward Euler time stepping, and unconditional energy stability. Convergence is measured by a manifold distance between triangulated surfaces, and simulations of an initially cuboid island show approximately second-order spatial convergence, while discrete volume loss remains small, about $\gamma(\theta)$4–$\gamma(\theta)$5. The linear systems are solved efficiently by a Schur-complement strategy or by GMRES with ILU preconditioning (Bao et al., 2020).

Diffuse-interface and phase-field methods occupy the complementary end of the methodological spectrum. A diffuse-interface approach based on an anisotropic Cahn–Hilliard equation with degenerate mobility recovers, in the sharp-interface limit, anisotropic surface diffusion together with anisotropic Young’s law and a zero-flux condition at the contact line. Weak solutions were proved for both smooth and double-obstacle potentials, and finite element computations showed excellent agreement with the sharp-interface limit in 2D and 3D (Garcke et al., 2022). A related phase-field framework for anisotropic SSD on patterned substrates uses two order parameters, one for the film and one for the substrate, and is particularly effective when topology changes and arbitrary non-planar geometries are central (Radice et al., 2024).

A recurrent methodological distinction follows from these developments. Sharp-interface PFEMs directly resolve curvature, normal velocity, and boundary conditions with high geometric accuracy and strong structure-preservation properties, whereas phase-field and diffuse-interface formulations handle pinch-off, hole nucleation, reconnection, and complex substrate geometries more naturally. The literature presents these as complementary rather than mutually exclusive descriptions (Bao et al., 2020).

6. Materials systems, substrates, and current extensions

Experimental SSD spans metals, alloys, nanocrystalline films, and structured substrates. In nanostructured Ag films of thickness $\gamma(\theta)$6 on amorphous silica, annealing in air produced an induction regime at $\gamma(\theta)$7, hole formation at $\gamma(\theta)$8, and a fully dewetted island state at $\gamma(\theta)$9. The RMS roughness increased from about $$W(t)=\int_{\Gamma(t)}\gamma(\theta)\,ds-\sigma\bigl[x_c^r(t)-x_c^l(t)\bigr],$$0 in the as-deposited film to $$W(t)=\int_{\Gamma(t)}\gamma(\theta)\,ds-\sigma\bigl[x_c^r(t)-x_c^l(t)\bigr],$$1, $$W(t)=\int_{\Gamma(t)}\gamma(\theta)\,ds-\sigma\bigl[x_c^r(t)-x_c^l(t)\bigr],$$2, and $$W(t)=\int_{\Gamma(t)}\gamma(\theta)\,ds-\sigma\bigl[x_c^r(t)-x_c^l(t)\bigr],$$3, while substrate exposure rose from less than $$W(t)=\int_{\Gamma(t)}\gamma(\theta)\,ds-\sigma\bigl[x_c^r(t)-x_c^l(t)\bigr],$$4 to $$W(t)=\int_{\Gamma(t)}\gamma(\theta)\,ds-\sigma\bigl[x_c^r(t)-x_c^l(t)\bigr],$$5 and $$W(t)=\int_{\Gamma(t)}\gamma(\theta)\,ds-\sigma\bigl[x_c^r(t)-x_c^l(t)\bigr],$$6. Power spectral density analysis extracted characteristic wavelengths $$W(t)=\int_{\Gamma(t)}\gamma(\theta)\,ds-\sigma\bigl[x_c^r(t)-x_c^l(t)\bigr],$$7 at $$W(t)=\int_{\Gamma(t)}\gamma(\theta)\,ds-\sigma\bigl[x_c^r(t)-x_c^l(t)\bigr],$$8, $$W(t)=\int_{\Gamma(t)}\gamma(\theta)\,ds-\sigma\bigl[x_c^r(t)-x_c^l(t)\bigr],$$9 at $\sigma=(\gamma_{SV}-\gamma_{SL})/\gamma_0$0, and $\sigma=(\gamma_{SV}-\gamma_{SL})/\gamma_0$1 at $\sigma=(\gamma_{SV}-\gamma_{SL})/\gamma_0$2, showing the transition from hillock formation to hole growth and island separation (Berni et al., 2020).

Nanocrystalline alloy films reveal a different balance between grain-size stabilization and structural stability. In sputtered Cu–Nb films annealed at $\sigma=(\gamma_{SV}-\gamma_{SL})/\gamma_0$3 for $\sigma=(\gamma_{SV}-\gamma_{SL})/\gamma_0$4, the dewetted area fraction increased systematically with Nb content: $\sigma=(\gamma_{SV}-\gamma_{SL})/\gamma_0$5 for Cu-4Nb, $\sigma=(\gamma_{SV}-\gamma_{SL})/\gamma_0$6 for Cu-8Nb, $\sigma=(\gamma_{SV}-\gamma_{SL})/\gamma_0$7 for Cu-21Nb, and $\sigma=(\gamma_{SV}-\gamma_{SL})/\gamma_0$8 for Cu-46Nb. In Cu-5Zr films, extensive SSD occurred despite thermodynamic stabilization in bulk nanocrystalline form; in a $\sigma=(\gamma_{SV}-\gamma_{SL})/\gamma_0$9-thick film annealed at $\gamma_0$0, the dewetted particle contained less than $\gamma_0$1 Zr, whereas the residual porous film beneath it contained $\gamma_0$2 Zr (Schuler et al., 2019). A continuum model for single-crystal binary alloy films complements these observations by coupling surface segregation, bulk phase separation, and morphology evolution, and predicts core–shell particles with a nanometric segregated shell atop a wetting layer that is modestly rich in the $\gamma_0$3 phase (Khenner, 2018).

Polycrystalline SSD introduces grain boundaries and triple junctions as distinct dewetting initiators. A recent three-dimensional multi-phase-field framework showed that hole formation begins preferentially at grain-boundary triple junctions, and derived refined onset criteria for the critical aspect ratio. In 2D,

$\gamma_0$4

where $\gamma_0$5 is the groove angle and $\gamma_0$6 is a diffuse-interface correction; in 3D regular $\gamma_0$7-gonal tilings,

$\gamma_0$8

These criteria matched the simulated onset of dewetting and clarified why breakup starts earlier at grain-boundary junctions than along free edges (Hoffrogge et al., 14 Jul 2025).

Substrate geometry profoundly changes SSD. For small 2D islands on curved substrates, the migration velocity obeys

$\gamma_0$9

so it is proportional to the substrate-curvature gradient $$\gamma(\theta)\cos\theta-\gamma'(\theta)\sin\theta=\sigma.$$0 and inversely proportional to $$\gamma(\theta)\cos\theta-\gamma'(\theta)\sin\theta=\sigma.$$1, while decreasing as the Young angle $$\gamma(\theta)\cos\theta-\gamma'(\theta)\sin\theta=\sigma.$$2 increases (Jiang et al., 2018). On non-planar substrates, phase-field simulations further show that crystalline particles lose the self-similarity characteristic of planar Winterbottom shapes and exhibit a volume-dependent apparent contact angle, with opposite trends for convex and concave profiles (Radice et al., 2024).

More recent generalizations extend SSD beyond the explicit moving-contact-line paradigm. A sharp-interface model with thickness-dependent surface energy

$$\gamma(\theta)\cos\theta-\gamma'(\theta)\sin\theta=\sigma.$$3

incorporates a wetting potential directly into the film/vapor energy, leaves a thin wetting layer on the substrate, and removes the explicit contact line; numerical results indicate that, as $$\gamma(\theta)\cos\theta-\gamma'(\theta)\sin\theta=\sigma.$$4, this model approaches the thickness-independent sharp-interface theory (Huang et al., 28 Apr 2026). Another extension treats a double-bubble thin film with three coupled open curves, a film–film interface, a shared triple junction, and three substrate contact points, and derives a structure-preserving PFEM that conserves area and dissipates energy (Li et al., 2 Mar 2025).

Two broader points follow from these developments. First, the classical Winterbottom construction is not universally exhaustive: strong anisotropy can admit multiple stable island morphologies (Jiang et al., 2015). Second, SSD is not a single model class but a family of energetically related formulations—sharp-interface, phase-field, diffuse-interface, polycrystalline, curved-substrate, axisymmetric, and wetting-potential models—whose differences mainly reflect which geometric, kinetic, or topological mechanisms are retained explicitly.