SHARP-INTERFACE APPROACH FOR SIMULATING SOLID-STATE ... · Sharp-interface approach for simulating solid-state dewetting in three dimensions 3 Fig. 1.1. γ-plot, 1/γ-plot and ξ-plot

SHARP-INTERFACE APPROACH FOR SIMULATING SOLID-STATE

DEWETTING IN THREE DIMENSIONS

WEI JIANG∗, QUAN ZHAO† , AND WEIZHU BAO‡

Abstract. The problem of simulating solid-state dewetting of thin films in three dimensions(3D) by using a sharp-interface approach is considered in this paper. Based on the thermodynamicvariation, a speed method is used for calculating the first variation to the total surface energyfunctional. The speed method shares more advantages than the traditional use of parameterizedcurves (or surfaces), e.g., it is more intrinsic and its variational structure (related with Cahn-Hoffmanξ-vector) is clearer and more direct. By making use of the first variation, necessary conditions for theequilibrium shape of the solid-state dewetting problem is given, and a kinetic sharp-interface modelwhich includes the surface energy anisotropy is also proposed. This sharp-interface model describesthe interface evolution in 3D which occurs through surface diffusion and contact line migration. Bysolving the proposed model, we perform lots of numerical simulations to investigate the evolutionof patterned films, e.g., the evolution of a short cuboid and pinch-off of a long cuboid. Numericalsimulations in 3D demonstrate the accuracy and efficacy of the sharp-interface approach to capturemany of the complexities observed in solid-state dewetting experiments.

Key words. Solid-state dewetting, surface diffusion, Cahn-Hoffman ξ-vector, shape derivative,thermodynamic variation.

AMS subject classifications. 74G65, 74G15, 74H15, 49Q10

1. Introduction. Driven by capillarity effects, solid thin films sitting on a sub-strate are rarely stable and could exhibit complex morphological changes, e.g. thefaceting [28, 29, 56], edge retraction [54, 14, 57, 58, 32], pinch-off [23, 31], fingeringinstabilities [30, 11, 18] and so on. This phenomenon, known as solid-state dewet-ting [51], has been widely observed across many thin film/substrate systems in mate-rials science [28, 29, 32, 46]. Solid-state dewetting can be deleterious by fabricatingthe thin film structures, e.g., microelectronic and optoelectronic devices, thus de-stroying the reliability of the devices. On the other hand, it is advantageous andcan be positively used to produce the well-controlled formation of an array of micro-/nanoscale particles, e.g., used in sensors [36, 2] and as catalysts for carbon [43]and semiconductor nanowire growth [45]. Recently, the solid-state dewetting has at-tracted considerable interest, and has been studied extensively from the experimental(e.g., [56, 57, 58, 1, 42, 20, 39, 38, 33]) and theoretical (e.g., [23, 48, 49, 52, 25, 24,27, 6, 5, 14, 54, 32, 31, 30, 62]) points of view by many research groups. The under-standing of its equilibrium patterns and kinetic morphology evolution characteristicscould provide some important knowledge to develop new experimental methods tocontrol solid-state dewetting [34], and enhance its potential applications in thin filmtechnologies.

Modeling solid-state dewetting has been one of active research areas and become

∗School of Mathematics and Statistics & Computational Science Hubei Key Laboratory, WuhanUniversity, Wuhan 430072, P.R. China ([email protected]). This author’s research was sup-ported by the National Natural Science Foundation of China Nos. 11871384 and 91630313, and theNatural Science Foundation of Hubei Province Grant No. 2018CFB466.

†Corresponding author: Department of Mathematics, National University of Singapore, Singapore119076 ([email protected]). This author’s research was supported by the Ministry of Educationof Singapore grant R-146-000-247-114.

‡Department of Mathematics, National University of Singapore, Singapore 119076 ([email protected], URL: http://www.math.nus.edu.sg/˜bao/). This author’s research was sup-ported by the Ministry of Education of Singapore grant R-146-000-247-114 and the National NaturalScience Foundation of China No.91630207.

1

2 W. Bao, W. Jiang and Q. Zhao

increasingly urgent in decades. Surface diffusion flow and contact line migration arerecognised as the two main kinetic features for the evolution of solid-state dewetting.The first sharp-interface model dated back to 1986, when Srolovitz and Safran [49]proposed a model to study the hole growth in solid-state dewetting under the threeassumptions, i.e., isotropic surface energy, small slope profile and cylindrical symme-try. The phase field approach has also been utilized to model solid-state dewetting,and the derived models can naturally capture the complex topology change duringthe evolution [23, 3]. Moreover, many other approaches have been proposed and dis-cussed in order to include the anisotropic surface energy effects, such as the discretemodel [14], a kinetic Monte Carlo model [41, 15], a crystalline model [10, 61] andcontinuum models based on partial differential equations (PDEs) [52, 25, 6, 17].

Under isothermal conditions, the equilibrium shape for a free-standing solid par-ticle can be formulated by minimizing the interfacial energy subject to the constraintof a constant volume:

(1.1) minΩW :=W (S) =

∫

S

γ(n) dS s.t. |Ω| = const,

where Ω is the enclosed domain by a closed surface S, and γ(n) is the surface en-ergy (density) with n = (n1, n2, n3)

T representing the crystallographic orientation.Based on the γ-plot, the equilibrium shape can be geometrically constructed via thewell-known Wulff (Gibbs-Wulff) construction [55]. The resulted Wulff shape, is theinner convex region bounded by all planes that are perpendicular to orientation n

and at a distance of γ(n) from the origin. The Winterbottom construction [53] wassubsequently proposed to treat the substrate case by truncating the Wulff shape witha flat plane, and the distance of the plan depends on the wettability of the substrate(substrate energy). On the other hand, many theories [8, 9] demonstrate that thederivatives of γ(n) play an important role in investigating the dynamics and equilib-rium for anisotropic interface problems. In 1972, Cahn and Hoffman developed thetheory of ξ-vector [22, 8] to describe the surface energy density of the solid materials.It is defined based on a homogeneous extension of γ:

(1.2) ξ(n) = ∇γ(p)∣∣∣p=n

, with γ(p) = |p|γ

(p

|p|

)

, ∀p ∈ R3\0,

where |u| :=√

u21 + u22 + u23 for u = (u1, u2, u3)T ∈ R

3. Under this extension, γsatisfies

(1.3) γ(λp) = |λ|γ(p), ∇γ(p) · p = γ(p), ∀λ 6= 0,p ∈ R3\0.

Compared to the traditional use of scalar function γ (or γ-plot), ξ-vector formula-tion has some advantages in the description of equilibrium shapes and thermodynamicevolution for crystalline interfaces. Because the magnitude of the normal componentfor ξ equals to γ(n) (By (1.3), we have ξ · n = γ(n)), ξ-plot shares similar geome-try with the Wulff shape [8, 40]. The ξ-plot can be regarded as the mathematicalrepresentation of the equilibrium shape for weakly anisotropic case (when 1/γ-plotis convex). Fig. 1.1 depicts the γ-plot,1/γ-plot and ξ-plot for four different types ofanisotropies: (a) the isotropic surface energy, i.e., γ(n) ≡ 1; (b) cubic surface energyγ(n) = 1+a[n4

1+n42+n

43] with a representing the degree of anisotropy; (c) ellipsoidal

surface energy γ(n) =√

a21n21 + a22n

22 + a23n

23; (d) “cusped” surface energy defined as

Sharp-interface approach for simulating solid-state dewetting in three dimensions 3

Fig. 1.1. γ-plot, 1/γ-plot and ξ-plot for different anisotropies: (a) isotropic surface energy;(b) cubic anisotropic surface energy defined as γ(n) = 1 + 0.3(n4

1+ n4

2+ n4

3); (c) ellipsoidal surface

energy γ(n) =√

4n2

1+ n2

2+ n2

3; (d) “cusped” surface energy defined as γ(n) = |n1|+ |n2|+ |n3|.

γ(n) = |n1|+ |n2|+ |n3|. In the application of materials science, the surface energy isusually piecewise smooth and has some “cusped” points, where it is not differentiable[40, 19]. A typical example is the “cusped” surface energy defined above. For thesecases, we can regularize the surface energy with a small parameter 0 < ε ≪ 1 toensure the use of sharp-interface kinetic model in this paper, i.e.,

(1.4) γ(n) =√

ε2 + (1− ε2)n21 +

√

ε2 + (1− ε2)n22 +

√

ε2 + (1− ε2)n23.

The Cahn-Hoffman ξ-vector has been recently utilized to describe the solid-statedewetting problem in two dimensions (2D) [26]. Based on the thermodynamic varia-tion, the authors derived sharp-interface models via the Cahn-Hoffman ξ-vector for-mulation for describing the evolution of solid-state dewetting. In this scenario, themoving interface is described as a parameterization over a time-independent domain,and the variation is performed by considering an infinitesimal perturbation with re-spect to an open curve. However, performing the thermodynamic variation for the3D problem by using the approach of parameterized surfaces could be very differ-ent. Firstly, the calculations of the energy variation via surface parameterization aretedious and awkward. Secondly, for the solid-state dewetting problem, the infinites-imal perturbation to a surface in the tangential direction plays an important rolein investigating the contact line migration along the substrate [25, 5]. These diffi-culties eliminate the possibility of treating the problem with the classic approach ofparameterized surface or simply doing a normal perturbation. In the literature, theshape optimization problem is popular in the design and construction of industrialstructures, and the speed method and shape derivatives have been widely utilized toperform the shape sensitivity analysis of such shape optimization problems [47, 21, 13].This approach avoids the parameterization of a surface on a fixed reference domain


and is able to deal with perturbations along arbitrary directions.Therefore, based on the ξ-vector formulation and the speed method, the objec-

tives of this paper are as follows: (i) to calculate the first variation of the energyfunctional for solid-state dewetting in 3D; (ii) to provide a rigorous derivation of thethermodynamic description of the equilibrium shape for solid-state dewetting in 3D;(iii) to develop a kinetic sharp-interface model which includes the surface diffusion andcontact line migration for simulating morphology evolution in 3D; and (iv) to presentnumerical simulations to investigate important characteristics of the morphologicalevolution for solid-state dewetting observed in experiments.

The rest of the paper is organized as follows. In Section 2, we briefly introducethe speed method and sharp derivatives, and then apply them for calculating the firstvariation of the total free energy functional. In Section 3, we rigorously derive thenecessary conditions for the equilibrium shape and explicitly give an expression forthe equilibrium shape by using a parameterized formula. In Section 4, based on ther-modynamic variation, a sharp-interface model is proposed for simulating solid-statedewetting of thin films in 3D. Subsequently, we perform some numerical simulationsto demonstrate the accuracy and efficacy of our proposed model in Section 5. Finally,we draw some conclusions in Section 6.

2. Thermodynamic variation. The solid-state dewetting problem can be il-lustrated as Fig. 2.1, where a solid thin film (in blue) can dewet or agglomerate on aflat rigid substrate (in gray) due to capillarity effects. The total interfacial free energyof the system can be written as [5, 26]

(2.1) W =

∫

SFV

γFV dSFV +

∫

SFS

γFS dSFS +

∫

SV S

γV S dSV S

︸︷︷︸

Substrate energy

,

where SFV := S, SFS and SV S represent the film/vapor, film/substrate and and

Substrate

Film

Vaporx

yz

γV S

γF V

= γ (n)

S

cΓ

nΓ

n

Γ

γF S

Lx

τΓ

Ly

Ssu b

Fig. 2.1. A schematic illustration of the solid-state dewetting of a solid thin film (in blue) ona flat, rigid substrate (in gray) in three dimensions.

vapor/substrate interfaces, respectively, and γFV , γFS and γV S represent the corre-sponding surface energy densities. In solid-state dewetting problems, we often as-sume that γFS, γV S are two constants, and γFS is a function of the orientation of the


film/vapor interface, i.e., γFV := γ(n) with n representing the unit normal vector ofthe film/vapor interface, which points outwards to the vapor phase. The film/vaporinterface is here described by an open two-dimensional surface S with boundary Γ(i.e., the contact line), which is a closed plane curve in the flat substrate Ssub.

Assume that we consider a bounded domain with size Lx × Ly on the substrate(shown in Fig. 2.1), and we label the surface area enclosed by the contact line Γ asA(Γ), then the total interfacial free energy of the system can be calculated as

W =

∫

SFV

γFV

dSFV +

∫

SFS

γFS

dSFS

+

∫

SV S

γV S

dSV S

=

∫

S

γ(n) dS +A(Γ)γFS

+ (Lx Ly −A(Γ)) γV S

=

∫

S

γ(n) dS + (γFS

− γV S

)A(Γ) + Lx Ly γV S.(2.2)

By dropping off the constant term, we can simplify the total interfacial free energyas the following two parts, i.e., the film/vapor interface energy term Wint and thesubstrate energy term Wsub,

(2.3) W =Wint +Wsub =

∫

S

γ(n) dS + (γFS

− γV S

)A(Γ).

As shown in Fig. 2.1, we introduce three unit vectors nΓ, τ

Γand c

Γ, which are

defined along the boundary Γ. More precisely, nΓis the outer unit normal vector of

the plane curve Γ on the substrate Ssub; τ Γis the unit tangent vector of Γ on the

substrate surface Ssub, which points anticlockwise when looking from top to bottom;c

Γis called as the co-normal vector, which is normal to Γ and tangent to the surface S,

and points outwards to the vapor phase. For any point x ∈ S (with x = (x1, x2, x3)T

or (x, y, z)T ), if we label Tx S and Nx S as the tangent and normal vector spaces toS at x, respectively, then the following properties are valid,

τΓ(x) ∈ Tx S, τ

Γ(x) Ssub, τ

Γ(x) ⊥ c

Γ(x), ∀x ∈ Γ,

cΓ(x) ∈ Tx S, n

Γ(x) Ssub, n

Γ(x) ⊥ τ

Γ(x), ∀x ∈ Γ.(2.4)

2.1. Differential operators on manifolds. We start by introducing some ba-sic knowledge about surface calculus and differential geometry. For more details,please refer to [12, 16].

Definition 2.1. Suppose S ⊂ R3 is a two-dimensional smooth manifold, and a

function f is defined on S such that f ∈ C2(S). Let n = (n1, n2, n3)T be the unit

normal vector of S, and f be an extension of f in the neighbourhood of S such thatf is differentiable, then the surface gradient of f on S is defined as

(2.5) ∇Sf = ∇f − (∇f · n)n,

with ∇ denoting the usual gradient in R3. It is easy to show that ∇

Sf is independent

of the extension of f and only dependent on the value of f on S. If we denote ∇Sas

a vector operator

(2.6) ∇S= (D1, D2, D3)

T ,

then we can easily obtain

(2.7) Dixj = δij − ni nj , ∀1 ≤ i, j ≤ 3,


where δij is the Kronecker delta. The surface divergence of a vector-valued functiong = (g1, g2, g3)

T ∈ [C1(S)]3 is defined as

(2.8) ∇S· g =

3∑

i=1

Di gi.

Moreover, the Laplace-Beltrami operator on S can be expressed as

(2.9) ∆Sf = ∇

S· (∇

Sf) =

3∑

i=1

DiDif.

In the definition of surface gradient, since the normal component has been subtractedfrom ∇f , ∇

Sf can be viewed as the tangential component of ∇f , and thus we have

∇Sf(x) ∈ Tx S, ∀x ∈ S.On the other hand, the integration by parts on an open smooth surface S with

smooth boundary Γ reads as (see Theorem 2.10 in [16], and we omit the proof here)

(2.10)

∫

S

∇Sf dS =

∫

S

f Hn dS +

∫

Γ

f cΓdΓ,

where n and cΓare the normal and co-normal vectors (shown in Fig. 2.1), respectively,

and H is the mean curvature, which is defined as the surface divergence of the unitnormal vector, i.e., H = ∇

S·n. Furthermore, by using the product rule that∇

S(fg) =

g∇Sf + f ∇

Sg, we can obtain

(2.11)

∫

S

g∇Sf dS = −

∫

S

f ∇Sg dS +

∫

S

f gHn dS +

∫

Γ

f g cΓdΓ.

In a simple case, if S is a flat surface (i.e., H = 0) with a plane boundary curve Γ,then Eq. (2.10) can be reduced to

(2.12)

∫

S

∇Sf dS =

∫

Γ

f cΓdΓ,

which is the Gauss-Green theorem in the multivariable calculus, because∇Sf collapses

to the gradient of f in 2D, and cΓcollapses to the unit outer normal vector of Γ.

2.2. The speed method and shape derivative. In this section, the objectiveis to calculate the first variation of the energy (or shape) functional defined in (2.3).To this end, we first introduce an independent parameter ǫ ∈ [0, ǫ0) to parameterizea family of perturbations Dǫ of a given domain D ⊂ R

3, where the parameterǫ controls the amplitude of the perturbation and ǫ0 is the maximum perturbationamplitude. Furthermore, we assume that the domains D ≡ D0 and Dǫ for ǫ ∈ [0, ǫ0)have the same topological properties and regularity, e.g., D and Dǫ for ǫ ∈ [0, ǫ0) areof class Ck with k ≥ 1.

More precisely, we consider a bounded domain D ⊂ R3 with a piecewise smooth

boundary ∂D, then we can construct a family of transformations Tǫ, which are one-to-one, and Tǫ maps D onto Dǫ, i.e.,

(2.13) Tǫ : D → Dǫ, ǫ ∈ [0, ǫ0),

where ǫ is the small perturbation parameter. Generally, we assume that: (i) Tǫ andT−1ǫ belong to Ck(R3,R3) for all ǫ ∈ [0, ǫ0) with k ∈ N ∪ ∞; and (ii) the mappingsǫ→ Tǫ(x) and ǫ→ T−1

ǫ (x) belongs to C1[0, ǫ0) for all x ∈ R3 (with x = (x1, x2, x3)

T ).


Given any point X ∈ D (with X = (X1, X2, X3)T ) and ǫ ∈ [0, ǫ0), we can define

the point x = Tǫ(X) which moves along the trajectory. Here, the point X representsthe Lagrangian (or material) coordinate, while x is the Eulerian (or actual) coordinate.Therefore, the speed vector field V(x, ǫ) at point x is defined as

(2.14) V(x, ǫ) =∂x

∂ǫ(T−1

ǫ (x), ǫ).

On the other hand, the transformation Tǫ can be uniquely determined by the speedvector field V via the following ordinary differential equation (ODE)

(2.15)

ddǫx(X, ǫ) = V(x(X, ǫ), ǫ),

x(X, 0) = X.

Therefore, the transformation Tǫ and the smooth vector field V are uniquely deter-mined by each other. For a smooth vector field V, e.g., V ∈ C(Ck(D,R3); [0, ǫ0)),the equivalence between the transformation Tǫ and the speed vector field V has beenstrictly established by Theorem 2.16 in [47]. In the following, we use Tǫ(V) to de-note the transformation associated with vector field V. For simplicity, we also denoteV0 = V(X, 0).

Let J(G) be a (shape) functional defined on a shape G ⊂ R3, where G could

be a two-dimensional surface (e.g., S) or a three-dimensional domain (e.g., Ω). Thefirst variation of the functional J(G) at G in the direction of a speed vector fieldV ∈ C(Ck(D,R3); [0, ǫ0)) is given as the Eulerian derivative:

(2.16) δJ(G;V) = limǫ→0

J(Gǫ)− J(G)

ǫ,

where Gǫ = Tǫ(V)(G). Based on the transformation, we can define the materialderivative of a function over a shape.

Definition 2.2. (Material derivatives, and refer to Def. 2.71 in [47]) The ma-terial derivative ϕ(G;V) of ϕ on the shape G in the direction of a speed vector fieldV is defined as

(2.17) ϕ(G;V) = limǫ→0

ϕ(Gǫ) Tǫ(V)− ϕ(G)

ǫ,

where for X ∈ G, ϕ(Gǫ) Tǫ(V) = ϕ(Tǫ(X)).Let Ω ⊂ R

3 be a bounded domain, and J(Ω) =∫

Ωψ(Ω) dΩ with ψ ∈ W 1,1(Ω).

Using the change of variables x = Tǫ(V)(X) for J(Ωǫ), we have

(2.18) J(Ωǫ) =

∫

Ω

ψ(Ωǫ) Tǫ(V)ζ(X, ǫ) dΩ,

where ζ(X, ǫ) = det(DTǫ) with DTǫ =(

∂Tǫ,i(V)∂Xj

(X))

i,j=1,2,3. By noting the fact that


ζ(X, 0) = 1, ∂ζ(X,ǫ)∂ǫ

∣∣∣ǫ=0

= ∇ ·V0 (see Proposition 2.44 in [47]), we have

δJ(Ω;V) = limǫ→0

J(Ωǫ)− J(Ω)

ǫ

=

∫

Ω

limǫ→0

[ψ(Ωǫ) Tǫ(V)ζ(X, ǫ) − ψ(Ω) T0(V)ζ(X, 0)

ǫ] dΩ

=

∫

Ω

[

ψ(Ω;V)ζ(X, 0) + ψ(Ω)∂ζ(X, ǫ)

∂ǫ

∣∣∣ǫ=0

]

dΩ

=

∫

Ω

ψ(Ω;V) dΩ +

∫

Ω

ψ(Ω)∇ ·V0 dΩ.(2.19)

We can define the shape derivative ψ′(Ω;V) of ψ(Ω) on Ω in the direction V as

(2.20) ψ′(Ω;V) = ψ(Ω;V) −∇ψ(Ω) ·V0.

Integration by parts for the second term of Eq. (2.19), we then obtain that

(2.21) δJ(Ω;V) =

∫

Ω

ψ′(Ω;V) dΩ +

∫

∂Ω

ψ(Ω)V0 · n dΩ.

If V0 ·n = 0 on the boundary ∂Ω, we obtain that the first variation of the functionalreduces to

(2.22) δJ(Ω;V) =

∫

Ω

ψ′(Ω;V) dΩ.

The material derivative can be regarded as the derivative with respect to the geometryin the moving coordinate systems. On the other hand, the shape derivative of ψrepresents the derivative with respect to the geometry in the stationary coordinatesby subtracting the term ∇ψ ·V0. If ψ ∈ W 1,1(R3) is independent on the geometricobject Ω, then we have ψ′(Ω;V) = 0. Furthermore, for a function ϕ(S) defined overa two-dimensional manifold S, we can define the shape derivative as follows.

Definition 2.3. (Shape derivatives, and see Def. 2.88 in [47]) The shape deriva-tive ϕ′(S;V) of ϕ defined on S in the direction of a vector field V is defined as

(2.23) ϕ′(S;V) = ϕ(S;V)−∇Sϕ(S) ·V0.

This definition ensures that the shape derivative shows no dependence on the exten-sion of ϕ in the near neighbourhood. We proposed the following proposition to showthat the first variation of a functional on S is closely related to the shape derivative.

Proposition 2.1. Let S be a two-dimensional manifold in D with smooth bound-ary Γ, and V be a speed vector field such that V ∈ C(Ck(D,R3); [0, ǫ0)). ϕ(S) ∈W 1,1(S) is a function such that the material derivative ϕ(S;V) exists or exists in theweak sense and ϕ(S;V) ∈ L1(S). Given the shape functional J(S) =

∫

Sϕ(S) dS, we

then have

(2.24) δJ(S;V) =

∫

S

ϕ(S;V) dS +

∫

S

ϕ(S)∇S·V0 dS.

Furthermore, if the shape derivative ϕ′(S;V) exists and (∇Sϕ ·V0) ∈ L1(S), we have

(2.25) δJ(S;V) =

∫

S

ϕ′(S;V) dS +

∫

S

ϕ(S)HV0 · n dS +

∫

Γ

ϕ(S)V0 · cΓdΓ.


Proof. According to the change of variables x = Tǫ(V)(X), and by using thetransformation Tǫ(V), the shape functional J(Sǫ) over the perturbed surfaces S canbe expressed as follows:

(2.26) J(Sǫ) =

∫

Sǫ

ϕ(Sǫ) dSǫ =

∫

S

(ϕ Tǫ(V ))(X)ω(X, ǫ) dS,

where ω(X, ǫ) is defined as

(2.27) ω(X, ǫ) = det(DTǫ) |((DTǫ)−1)Tn|, with DTǫ =

(∂Tǫ,i(V)

∂Xj(X)

)

i,j=1,2,3.

Note that the following expressions hold according to the Lemma 2.49 in Page 80 of[47],

(2.28) ω(X, 0) = 1,∂ω(X, ǫ)

∂ǫ

∣∣∣ǫ=0

= ∇S·V0.

Therefore, based on the definition of the first variation, we have

δJ(S;V) = limǫ→0

J(Sǫ)− J(S)

ǫ

=

∫

S

limǫ→0

[ϕ(Sǫ) Tǫ(V )ω(X, ǫ)− ϕ T0(V )ω(X, 0)

ǫ

]

dS

=

∫

S

ϕ(S;V)ω(X, 0) dS +

∫

S

ϕ(S)∂ω(X, ǫ)

∂ǫ

∣∣∣ǫ=0

dS

=

∫

S

ϕ(S;V) dS +

∫

S

ϕ(S)∇S·V0 dS.(2.29)

Furthermore, by using the integration by parts and also making use of Eq. (2.23), wealso obtain

δJ(S;V) =

∫

S

ϕ(S;V0) dS −

∫

S

∇Sϕ(S) ·V0 dS +

∫

S

ϕ(S)HV0 · n dS

+

∫

Γ

ϕ(S)V0 · cΓdΓ

=

∫

S

ϕ′(S;V) dS +

∫

S

ϕ(S)HV0 · n dS +

∫

Γ

ϕ(S)V0 · cΓdΓ,

which completes the proof.From Proposition 2.1, we have shown that the shape derivative ϕ′(S;V) is an

important tool to derive the first variation of a shape functional over a surface. Wecan assume that ψ is a function defined on the domain Ω, such that its restriction onS is equal to function ϕ(S), namely

(2.30) ψ(Ω)∣∣∣S= ϕ(S).

By using Eq. (2.20) and Eq. (2.24), we can reformulate Eq. (2.25) in terms of theextension function ψ as

(2.31) δJ(S;V) =

∫

S

ψ′(Ω;V)∣∣∣SdS +

∫

S

(∂nψ + ψH

)V0 · n dS +

∫

Γ

ψV0 · cΓdΓ.


In the following, we will apply Eq. (2.31) for calculating the first variation of theenergy (or shape) functional defined in (2.3), where the integrand is the surface energydensity γ(n). To calculate the shape derivatives and obtain the first variation, we shallmake use of the signed distance function, which is a powerful tool in shape sensitivityanalysis. Consider a closed domain Ω ⊂ R

3 with a smooth boundary surface ∂Ω, andthen the signed distance function is defined as

(2.32) b(x) =

d(x, ∂Ω), ∀x ∈ R3\Ω,

0, ∀x ∈ ∂Ω,

−d(x, ∂Ω), ∀x ∈ Ω.

Here, d(x, ∂Ω) = infy∈∂Ω ||x− y||. The signed distance function b(x) can be used todetermine the unit outer normal vector n and the mean curvature H on the boundarysurface ∂Ω. More precisely, we can extend the functions n and H which are definedon ∂Ω in terms of b(x) in a tubular neighbourhood such that

(2.33) n(x) = ∇b(x)∣∣∣∂Ω, H(x) = ∆b(x)

∣∣∣∂Ω, ∀x ∈ ∂Ω.

The shape derivative of the signed distance function in the direction of a vector fieldV is calculated as b′(Ω;V) = −V0 · n (see Section 3 in [21] for details). Moreover,based on the extension, the shape derivatives of the two extension functions restrictedon ∂Ω are also obtained in [21], i.e.,

(2.34) n′(Ω;V)∣∣∣∂Ω

= −∇S(V0 · n), H′(Ω;V)

∣∣∣∂Ω

= −∆S(V0 · n).

2.3. First variation. By applying Eq. (2.31) and making use of the shapederivative of the unit outer normal vector, we obtain the following lemma.

Lemma 2.1. Assume that S ⊂ D is a two dimensional manifold of class C2 withsmooth boundary Γ. Let n be the unit outer normal vector of S, and V be a speed vectorfield such that V ∈ C(Ck(D, D); [0, ǫ0)). If the shape functional J(S) =

∫

Sγ(n) dS

with a surface energy (density) γ(n), then the first variation of J(S) is given as

(2.35) δJ(S;V) =

∫

S

(∇S· ξ) (V0 · n) dS +

∫

Γ

V0 · cγΓdΓ,

where ξ := ξ(n) is the Cahn-Hoffman vector, which is defined previously in Eq. (1.2),and V0 ·n represents the deformation velocity along the outer normal direction of theinterface S, and the vector cγ

Γ:= (ξ · n) c

Γ− (ξ · c

Γ)n with c

Γrepresenting the unit

co-normal vector (shown in Fig. 2.1).Proof. We firstly assume γ is a homogeneous extension of γ,

(2.36) γ(p) = |p|γ

(p

|p|

)

, ∀p ∈ R3\0,

where the definition domain of the function γ(n) changes from unit vectors n toarbitrary non-zero vectors p ∈ R

3.We next consider a bounded domain Ω ⊂ R

3 such that S ⊂ ∂Ω. Then, basedon the signed distance function defined in (2.32), we can define ∇b(x) ∈ R

3 as anextension of the normal vector n in the neighbourhood of S. Thus we can reformulate

(2.37) J(S) =

∫

S

γ(∇b(x)

)∣∣∣SdS =

∫

S

ψ(Ω)∣∣∣SdS,


with ψ(Ω) := γ(∇b(x)

). Using the chain rule for shape derivatives and the definition

of Cahn-Hoffman ξ-vector in Eq. (1.2), we conclude that the following expressionholds,

(2.38) ψ′(Ω;V)∣∣∣S= ∇γ

(∇b(x)

)∣∣∣S· n′(Ω;V)

∣∣∣S= −ξ · ∇

S(V0 · n).

Moreover, By noting the fact |∇b(x)| = 1, we obtain

(2.39) ∂nψ∣∣∣S= ξ ·

((∇∇b(x)) ∇b(x)

)∣∣∣S= 0,

where ∇∇b(x) ∈ R3×3. By making use of Eq. (2.31) and combining Eq. (2.38) and

(2.39), we immediately have

δJ(S;V) = −

∫

S

ξ · ∇S(V0 · n) dS +

∫

S

γ(n) (V0 · n)H dS +

∫

Γ

γ(n) (V0 · cΓ) dΓ

:= I + II + III.(2.40)

For the first term, by using integration by parts, we obtain

(2.41) I =

∫

S

(∇S· ξ) (V0 · n) dS −

∫

S

(ξ · n) (V0 · n)H dS −

∫

Γ

(ξ · cΓ) (V0 · n) dΓ.

Based on Eq. (1.3), we have γ(n) = ξ · n. Thus we can rewrite

II =

∫

S

(ξ · n) (V0 · n)H dS.(2.42)

III =

∫

Γ

(ξ · n) (V0 · cΓ) dΓ.(2.43)

Finally, by combining the above three terms together, we immediately have

δJ(S;V) =

∫

S

(∇S· ξ) (V0 · n) dS +

∫

Γ

[

(ξ · n) cΓ− (ξ · c

Γ)n

]

·V0 dΓ

=

∫

S

(∇S· ξ) (V0 · n) dS +

∫

Γ

cγΓ·V0 dΓ,(2.44)

with cγΓ= (ξ · n) c

Γ− (ξ · c

Γ)n.

By using the above Lemma, we can easily obtain the first variation of the energyfunctional for solid-state dewetting problems defined in (2.3).

Theorem 2.1. The first variation of the free energy (or shape) functional (2.3)used in solid-state dewetting problems with respect to a smooth vector field V can bewritten as:

(2.45) δW (S;V) =

∫

S

(∇S· ξ) (V0 · n) dS +

∫

Γ

(cγΓ· n

Γ+ γ

FS− γ

V S)(V0 · nΓ

) dΓ.

Proof. From (2.3), we observe that the total free energy consists of two parts:the film/vapor interface energy Wint and the substrate energy part Wsub. First, byusing Lemma. 2.1, we can directly obtain the first variation of the film/vapor interfaceenergy Wint as follows,

(2.46) δWint(S;V) =

∫

S

(∇S· ξ) (V0 · n) dS +

∫

Γ

V0 · cγΓdΓ.


Here, cγΓis a linear combination of c

Γand n, which is defined on the contact line Γ.

Therefore, as shown in Fig. 2.1, we have

(2.47) cΓ⊥ τ

Γ, n ⊥ τ

Γ⇒ cγ

Γ⊥ τ

Γ.

For solid-state dewetting problems studied in this paper, we assume that the contactline Γ must move along the substrate plane Ssub, i.e.,

(2.48) TǫΓ ⊂ Ssub, V0(x) Ssub, ∀x ∈ Γ.

Therefore, for any x ∈ Γ, V0(x) can be decomposed into two vectors along thedirections n

Γand τ

Γ, i.e., V0(x) = k1nΓ

+ k2τ Γ, where k1 and k2 represent the

corresponding components. By making use of (2.47), we can obtain

V0(x) · cγΓ= (k1nΓ

+ k2τ Γ) · cγ

Γ= k1(nΓ

· cγΓ)

= (V0(x) · nΓ) (cγ

Γ· n

Γ), ∀x ∈ Γ.

Thus we can reformulate (2.46) as

(2.49) δWint(S;V) =

∫

S

(∇S· ξ) (V0 · n) dS +

∫

Γ

(cγΓ· n

Γ)(V0 · nΓ

) dΓ.

On the other hand, we can rewrite the substrate energy Wsub as

(2.50) Wsub = (γFS

− γV S

)A(Γ) = (γFS

− γV S

)

∫

SFS

dSFS.

By using Lemma 2.1, and noting that the integrand is a constant and SFS

is a flatsurface with a plane boundary curve Γ (i.e., H = 0 and n

Γis the unit co-normal

vector of the flat surface SFS

), we directly have

(2.51) δWsub(S;V) = (γFS − γV S)

∫

Γ

V0 · nΓdΓ.

By combining Eqs. (2.49) and (2.51), we obtain the following conclusion

(2.52) δW (S;V) =

∫

S

(∇S· ξ) (V0 · n) dS +

∫

Γ

(cγΓ· n

Γ+ γFS − γV S)(V0 · nΓ

) dΓ,

which completes the proof.

3. Equilibrium shapes. The equilibrium shape of the solid-state dewettingproblem can be stated as follows [5]:

(3.1) minΩW :=W (S) =

∫

S

γ(n) dS + (γFS

− γV S

)A(Γ) s.t. |Ω| = C,

where C > 0 is a prescribed constant representing the total volume of the dewettedparticle, and Ω represents the domain (or the particle) enclosed by the interface Sand the substrate plane Ssub.

The Lagrangian for the above optimization problem can be defined as

(3.2) L(S, λ) =

∫

S

γ(n) dS + (γFS

− γV S

)A(Γ)− λ(|Ω| − C),


with λ representing the Lagrange multiplier. The first variation of the total volumeterm can be obtained by simply choosing the integrand ψ(x) ≡ 1, ∀x ∈ Ω in Eq. (2.21).Therefore, by combining with Eq. (2.45), the first variation of the Lagrangian withrespect to a smooth vector field V can be given as

(3.3) δL(S, λ;V) =

∫

S

(∇S· ξ−λ)(V0 ·n) dS+

∫

Γ

(cγΓ·n

Γ+ γ

FS− γ

V S)(V0 ·nΓ

) dΓ.

Based on the above first variation, we have the following lemma which yields thenecessary conditions for the equilibrium shape of solid-state dewetting problem.

Lemma 3.1. Assume that a two-dimensional manifold Se with smooth bound-ary Γe is the equilibrium shape of the solid-state dewetting problem (3.1), then thefollowing conditions must be satisfied

∇Se

· ξ = λ, on Se.(3.4a)

cγΓ· n

Γ+ γ

FS− γ

V S= 0, on Γe.(3.4b)

where the constant λ is determined by the prescribed total volume, i.e., the constantC.

Proof. If Se is the equilibrium shape, then Eq. (3.3) must vanish at S = Se forany smooth vector field V. Thus we immediately obtain the above two conditions.

Film/Island

θ i(x)

Substrate

nΓ(x)

n(x)c

Γ(x)

Vapor

γV S

γF S

γF V

(a) (b)

∂γ

∂θτ θ

γ n

ξ(n)

1

sin θ

∂γ

∂φτ φ

Fig. 3.1. (a) The cross-section profile in configuration of the vectors along the contact line Γ.(b) The three components of the Cahn-Hoffman ξ-vector.

Eq. (3.4b) can be regarded as the Young equation for anisotropic surface energyγ(n) in 3D. For isotropic surface energy, i.e., γ(n) ≡ 1 (scaled by a constant γ0), wehave ξ = n and cγ

Γ= c

Γ. By simple calculations, Eq. (3.4a) will be reduced to the

condition of constant mean curvature. Denote Γe as the boundary of Se, for arbitraryx ∈ Γe, let θ(x) represent the corresponding contact angle at boundary point x. ThenEq. (3.4b) will reduce to

(3.5) cos θ(x) = σ, ∀x ∈ Γe,

where the material constant σ :=γV S

−γFS

γ0, and it is the well-known isotropic Young

equation [60]. For the anisotropic case, we can write the surface energy densityin terms of the spherical coordinate, i.e., γ

FV= γ(θ, φ) (scaled by a constant γ0).


Therefore, the Cahn-Hoffman ξ-vector can be decomposed into the following threecomponents

(3.6) ξ(n) = ∇γ(n) = γ(θ, φ)n +∂γ(θ, φ)

∂θτ

θ+

1

sin θ

∂γ(θ, φ)

∂φτ

φ,

where in these expressions,

n = (sin θ cosφ, sin θ sinφ, cos θ)T ,(3.7a)

τθ= (cos θ cosφ, cos θ sinφ,− sin θ)T ,(3.7b)

τφ= (− sinφ, cosφ, 0)T .(3.7c)

Therefore, we obtain that the following expressions hold

ξ · n = γ(θ, φ), cΓ· n

Γ= cos θ(x).(3.8a)

ξ · cΓ=∂γ(θ, φ)

∂θ, n · n

Γ= sin θ(x).(3.8b)

Thus we can rewrite Eq. (3.4b) as

(3.9) γ(θ, φ) cos θ(x)−∂γ(θ, φ)

∂θsin θ(x) − σ = 0, ∀x ∈ Γe,

which is consistent with the anisotropic Young equation discussed for the solid-statedewetting problem in 2D [5, 52].

If X := X(θ, φ) represents the position vector of a surface (e.g., the equilibriumshape Se for a given γ(θ, φ)), then we have ∇

S· X = 2 by using Definition 2.1.

Therefore, we obtain that an equilibrium shape of the solid-state dewetting problemcould have the similar shape with the ξ-plot from (3.4a). Based on the Winterbottomconstruction [53] and recent work for the generalized Winterbottom construction [5],we can construct its analytical expression for the equilibrium shape. First, we definea domain of definition U

φfor θ under a fixed value φ as

(3.10) Uφ:=

θ∣∣∣γ(θ, φ) cos θ −

∂γ(θ, φ)

∂θsin θ − σ ≥ 0, θ ∈ [0, π]

,

where σ =γV S

−γFS

γ0

. Based on Lemma 3.1, we can explicitly construct its equilibrium

shape in the parametric formula as Se(θ, φ) := X(θ, φ) = (x(θ, φ), y(θ, φ), z(θ, φ))T ,

(3.11)

x(θ, φ) = λ[γ(θ, φ) sin θ cosφ+ ∂γ(θ,φ)

∂θcos θ cosφ− 1

sin θ

∂γ(θ,φ)∂φ

sinφ],

y(θ, φ) = λ[γ(θ, φ) sin θ sinφ+ ∂γ(θ,φ)

∂θcos θ sinφ+ 1

sin θ

∂γ(θ,φ)∂φ

cosφ],

z(θ, φ) = λ[γ(θ, φ) cos θ − ∂γ(θ,φ)

∂θsin θ − σ

],

where φ ∈ [0, 2π], θ ∈ Uφ, and λ is the scaling constant determined by the total

volume |Ω| and γ0.Based on the formula (3.11), the equilibrium shape under different types of surface

energy anisotropies, e.g., the cubic anisotropy and regularized “cusped” anisotropydefined in Eq. (1.4), can be easily constructed. Fig. 3.2(a)-(c) depicts the equilib-rium shapes for isotropic surface energy with the material constant σ chosen asσ = cos(π/3), cos(π/2), cos(3π/4), respectively. It clearly demonstrates the effect


Fig. 3.2. The equilibrium shape defined by Eq. (3.11), where (a)-(c) is for isotropic surfaceenergy, i.e., γ(n) ≡ 1, but with different material constants σ = cos(π/3), cos(π/2), cos(3π/4),respectively; (d) γ(n) = 1 + 0.2(n4

1+ n4

2+ n4

3), σ = cos(3π/4); (e) The surface energy density is

given by Eq. (1.4), where σ = cos(3π/4), ε = 0.01 ; (f) The surface energy density is given byγ(Mx(π/4)n) where γ(n) is defined by Eq. (1.4) and Mx(π/4) represents the rotation for a matrixby an angle π/4 about the x-axis in 3D, using the right-hand rule, where σ = cos(3π/4), ε = 0.01.

of the material constant σ on the equilibrium shape by influencing the equilibriumcontact angle via (3.5). Moreover, we also present equilibrium shapes for the cubicanisotropic surface energy, i.e., γ(n) = 1+a(n4

1+n42+n

43) and regularized “cusped” sur-

face energy defined in Eq. (1.4) with σ = cos(3π/4) in Fig. 3.2(d)-(e). The anisotropyfor Fig. 3.2(f) is chosen by an anti-clockwise rotation along the x-axis by 45 degreesunder the right-hand rule for the “cusped” surface energy. We can observe that thisrotation results in a corresponding rotation of the equilibrium shape.

4. A sharp-interface model and its properties. In this section, we proposea kinetic sharp-interface model for simulating solid-state dewetting of thin films withanisotropic surface energies, and then we show that the proposed model satisfies themass conservation and energy dissipation.

4.1. The model. Based on Eq. (2.35), we can define the first variation of thetotal interfacial energy functional with respect to the film/vapor interface S and itsboundary curve (i.e., the contact line Γ) as

(4.1)δW

δS= ∇

S· ξ,

δW

δΓ= cγ

Γ· n

Γ+ γ

FS− γ

V S.

From the Gibbs-Thomson relation [37, 50], the chemical potential can be defined as

(4.2) µ = Ω0δW

δS= Ω0∇S

· ξ,

with Ω0 representing the atomic volume. The normal velocity of the moving interfaceis controlled by surface diffusion [8, 37, 52, 25], and it can be defined as follows byFick’s laws of diffusion[4]

(4.3) J = −Dsν

kB Te∇

Sµ, vn = −Ω0(∇S

· J) =DsνΩ0

kB Te∇2

Sµ.

In these expressions, J is the mass flux of atoms, Ds is the surface diffusivity, kB Teis the thermal energy, ν is the number of diffusing atoms per unit area, ∇

Sis the


surface gradient. In addition to the surface diffusion which controlled the motion ofthe moving interface, we still need the boundary condition for the moving contactline. Following the idea for simulating solid-state dewetting in 2D [52, 25], we assumethat the normal velocity of the contact line Γ is simply given by the energy gradientflow, which is determined by the time-dependent Ginzburg-Landau kinetic equations,i.e.,

(4.4) vnΓ= −η

δW

δΓ= −η[cγ

Γ· n

Γ+ γ

FS− γ

V S],

with 0 < η < ∞ denoting the contact line mobility, which can be thought of as areciprocal of a constant friction coefficient. For the physical explanation behind thisapproach, please refer to the recent paper [52].

We choose the characteristic length scale and characteristic surface energy scale

as h0 and γ0, respectively, the time scale as

h40

Bγ0with B =

DsνΩ20

kB Te, and the contact

line mobility is scaled by Bh30

. Let X(·, t) = (x(·, t), y(·, t), z(·, t))T be a local parame-

terization of the moving film/vapor interface S, then we can obtain a dimensionlesskinetic sharp-interface model for solid-state dewetting of thin film via the followingCahn-Hoffman ξ-vector formulation as

∂tX = ∆Sµ n, t > 0,(4.5)

µ = ∇S· ξ, ξ(n) = ∇γ(p)

∣∣∣p=n

,(4.6)

where t is the time, n is the unit outer normal vector of S, and ξ := ξ(n) is the Cahn-Hoffman vector (scaled by γ0). Here, for simplicity, we still use the same notationsfor all the dimensionless variables.

LetXΓ(·, t) = (x

Γ(·, t), y

Γ(·, t), z

Γ(·, t))T represents a parameterization of the mov-

ing contact line Γ(t). The initial condition is given as S0 with boundary Γ0 such that

(4.7) S0 := X(·, 0) = (x0, y0, z0), Γ0 := X(·, 0)∣∣∣Γ.

The above governing equations are subject to the following boundary conditions:(i) contact line condition

(4.8) zΓ(·, t) = 0, t ≥ 0;

(ii) relaxed contact angle condition

(4.9) ∂tXΓ= −η

(

cγΓ· n

Γ− σ

)

nΓ, t ≥ 0;

(iii) zero-mass flux condition

(4.10)(

cΓ· ∇

Sµ)∣∣∣Γ= 0, t ≥ 0;

where η represents a (dimensionless) contact line mobility, cγΓis the anisotropic co-

normal vector which is defined as cγΓ

:= (ξ · n) cΓ− (ξ · c

Γ)n, c

Γrepresents the

co-normal vector, and nΓ= (n

Γ,1, n

Γ,2, 0)T is the outer unit normal vector of Γ on

the substrate (cf. Fig. 2.1), and σ =γV S

−γFS

γ0

is a (dimensionless) material constant.

Remark 4.1. The contact line condition in Eq. (4.8) ensures that the contactline must move along the substrate plane. Because the contact line Γ lies on the


substrate (i.e., Oxy plane), the third component of nΓis always zero, i.e., n

Γ,3= 0.

As long as the initial condition satisfies zΓ(·, 0) = 0, it can automatically satisfy the

boundary condition (i) zΓ(·, t) = 0, ∀ t > 0 by using the boundary condition (ii). The

last boundary condition (iii) ensures that the total volume/mass of the thin film isconserved during the evolution, i.e., no-mass flux at the moving contact line.

4.2. Mass conservation and energy dissipation. In the following, we willrigorously prove that the proposed sharp-interface model satisfies the mass conserva-tion and the total free energy dissipation during the evolution.

Proposition 4.1. Assume that X(·, t) is the solution of the sharp-interfacemodel, i.e., Eqs. (4.5)-(4.6) with boundary conditions (4.8)-(4.10), and denote S(t) :=X(·, t) as the moving film/vapor interface. Then, the total volume (or mass) of thethin film, labeled as |Ω(t)|, is conserved, i.e.,

(4.11) |Ω(t)| ≡ |Ω(0)|, t ≥ 0.

Furthermore, the (dimensionless) total interfacial free energy of the system is non-increasing during the evolution, i.e.,

(4.12) W (t) ≤W (t1) ≤W (0) =

∫

S(0)

γ(n) dS − σA(Γ(0)), t ≥ t1 ≥ 0.

Proof. By making use of the first variation (2.21) and simply choosing the inte-grand ψ(x) ≡ 1, ∀x ∈ Ω, and using the governing equation (4.5), we can calculate thetime derivative of the total volume as (noting that V0 = ∂tX)

(4.13)d

dt|Ω(t)| =

∫

S(t)

∂tX · n dS =

∫

S(t)

∆Sµ dS = 0, t ≥ 0,

where the last equality comes from the integration by parts and the zero-mass fluxcondition (4.10), and it indicates that the total volume/mass is conserved.

To obtain the time derivative of the (dimensionless) total free energy, by makinguse of Theorem 2.1 and Eq. (2.45), but replacing the perturbation variable ǫ with thetime variable t, we can immediately obtain

d

dtW (t) =

∫

S(t)

(∇S· ξ) (∂tX · n) dS +

∫

Γ(t)

(cγΓ· n

Γ− σ) (∂tXΓ · n

Γ) dΓ.

By substituting the governing equations and the relaxed contact angle boundary con-dition, i.e.,

(4.14) µ = ∇S· ξ, ∆

Sµ = ∂tX · n, ∂tXΓ · n

Γ= −η(cγ

Γ· n

Γ− σ),

into the above equation and integrating by parts, we obtain

d

dtW (t) =

∫

S(t)

µ∆Sµ dS − η

∫

Γ(t)

(cγΓ· n

Γ− σ)2 dΓ

= −

∫

S(t)

|∇Sµ|2 dS − η

∫

Γ(t)

(cγΓ· n

Γ− σ)2 dΓ ≤ 0, t ≥ 0,(4.15)

where the constant η > 0. The last inequality immediately implies the energy dissi-pation.


Remark 4.2. In the above proof, we need to calculate the time derivatives of thetotal volume and the total free energy. These two derivatives can be easily obtainedby making use of the speed method and the first variation presented in Section 2. InSection 2, we consider any type of smooth perturbations. In fact, a family of evolvinginterface surfaces S(t)t≥0 can be also thought of as a type of perturbations, onlyby replacing the perturbation variable ǫ with the time variable t. Therefore, the timederivatives can be directly obtained by using the first variation of the total volumefunctional and the total free energy functional.

5. Numerical results. In this section, we perform numerical simulations forsolid-state dewetting in 3D to investigate the morphological evolution of thin filmsin various cases. We implement the parametric finite element method (PFEM) forsolving the proposed sharp-interface model [6, 7].

5.1. Equilibrium convergence. We have presented a mathematical descrip-tion of the equilibrium shape based in Section 3. Here, we present some numericalconvergence to equilibrium shapes by solving the proposed kinetic sharp-interfacemodel.

From the relaxed contact angle boundary condition (4.9), which describes the mi-gration of the contact line, we know that the contact line mobility η precisely controlsthe relaxation rate of the contact angle towards its equilibrium state. The large ηwill accelerate the relaxation process [52]. Therefore, we numerically investigate theeffect of η on the evolution of the dynamic contact angles. The evolution surfacesS(tm)Mm=1 are discretized by polygonal surfaces such that Sm =

⋃Nj=1 D

mj , , where

Dmj Nj=1 are mutually disjoint triangle surfaces, and the polygonal surface Sm has

K vertices given as qmk Kk=1. The boundary of the polygonal surface Sm given by

polygonal Γm =⋃Nc

j=1 hmj , where hmj Nc

j=1 are line segments of the curve ordered incounter-clockwise direction when viewing from the top. We define the following meancontact angles as the indicator,

(5.1) θm =1

Nc

Nc∑

j=1

arccos(cΓm,j

· nΓm,j

),

where cΓm,j

and nΓm,j

are unit vector defined on line segment hj . Fig. 5.1 shows

the temporal evolution of θm and the normalized energy W (t)/W (0) under differentchoices of the contact line mobility η. The initial thin film is chosen as a unit cube.From the figure, we can observe that the larger mobility will accelerate the process ofrelaxation such that the contact angles evolve faster towards its equilibrium contactangle 3π/4. Similarly, as shown in Fig. 5.1(b), the energy decays faster for largermobility, but finally the equilibrium contact angle converges to the same value. Itindicates that the equilibrium contact angles as well as the equilibrium shape areindependent of the choice of the contact line mobility η. In the following numericalsimulations, the contact line mobility is chosen to be very large, e.g.,η = 100. Thischoice of η will result in a very quick convergence to the equilibrium contact angle(defined by Eq. (3.4b)). The detailed investigation of the influence of the parameter onthe solid-state dewetting evolution process and equilibrium shapes has been discussedin [52].

We next show a convergence result between the numerical equilibrium shapeby solving the proposed sharp-interface model and its theoretical equilibrium shape.Fig. 5.2 shows the convergence results of the equilibrium shapes under different mesh


0 0.05 0.1 0.15 0.2

t

0

0.2

0.4

0.6

0.8

θm/π

η=10 η=20 η=50 η=100 η=200

0 2 4

×10-3

0.5

0.6

0.7

0 0.02 0.04 0.06 0.08 0.1

t

0.8

0.85

0.9

0.95

1

W(t

)

0 2 4

×10-3

0.85

0.9(a) (b)

Fig. 5.1. (a) The temporal evolution of contact angle θm defined in Eq. (5.1); (b) the temporalevolution of the normalized energy W (t)/W (0) for different choices of mobility, where the initialshape of the thin film with isotropic surface energy is chosen as a unit cube, and the computationalparameters are chosen as σ = cos(3π/4).

-1 -0.5 0 0.5 1 1.5x

0.2

0.4

0.6

0.8

1

1.2

1.4

z

EquilibriumMesh1Mesh2Mesh3

-0.1 0 0.10.95

1

Fig. 5.2. Comparison of the cross-section profiles in x-direction of the numerical equilibriumshapes under different meshes with its theoretical equilibrium shape, where the initial shape is chosenas a (1, 2, 1) cuboid, the surface energy γ(n) = 1 + 0.25(n4

1+ n4

2+ n4

3), and σ = cos(15π/36)

sizes, where σ = cos 15π36 , γ(n) = 1+0.25(n4

1+n42+n

43). The initial shape is chosen as

a (1, 2, 1) cuboid with different meshes, which are given by a set of small isosceles righttriangles. If we define the mesh size indicator h as the length of the hypotenuse of theisosceles right triangle, then Mesh 1 represents the initial mesh with h = h0 = 0.125,and the time step is chosen as τ = τ0 = 0.00125 for numerical computation. Mean-while, the time step for Mesh 2 (h = h0

2 ) and Mesh 3 (h = h0

4 ) are chosen as τ = τ04

and τ = τ016 , respectively. For a better comparison, we plot the cross-section profiles

along the x-direction for the numerical equilibrium shapes and the theoretical equi-librium shape. As shown in Fig. 5.2, we can clearly observe that as the computationalmesh size gradually decreases, the numerical equilibrium shapes uniformly convergeto the theoretical equilibrium shape (constructed by Eq. (3.11)).

5.2. Morphological evolution. We firstly focus on the case for isotropic sur-face energy, i.e., γ(n) ≡ 1. We start with numerical examples for an initially, shortcuboid island with (2, 2, 1) representing its length, width and height, respectively (as


shown in Fig. 5.3(a)). The computational parameter is chosen as σ = cos 5π6 . As can

be seen in Fig. 5.3, we show several snapshots of the morphology evolution for theshort cuboid towards its equilibrium shape. As time evolves, the initial sharp cornersand edges on the island become smooth in a very short time (Fig. 5.3(b)), and finallythe thin film form a spherical shape as its equilibrium shape (Fig. 5.3(f)).

Fig. 5.3. Several snapshots during the evolution of an initially, cuboid island film with isotropicsurface energy towards its equilibrium shape: (a) t = 0.0; (b) t = 0.1; (c) t = 0.2; (d) t = 0.5; (e)t = 0.7; (f) t = 1.40, where the initial shape of the thin film is chosen as a (2, 2, 1) cuboid, and thematerial constant is chosen as σ = cos(5π/6).

Fig. 5.4. Several snapshots during the evolution of an initial, cuboid island film with isotropicsurface energy until its pinch-off time: (a) t = 0; (b) t = 0.01; (c) t = 0.30; (d) t = 0.50; (e)t = 0.80; (f) t = 1.03, where the initial shape is chosen as a (1, 12, 1) cuboid, and the materialconstant σ = cos(3π/4).

Short cuboid island films tend to form a single island as its equilibrium shapewith the spherical shape minimizing the total free energy (i.e., the minimal surface


area). However, the morphological evolution for long cuboid islands could be quitedifferent during the evolution. Due to the Plateau-Rayleigh instability [44, 35], longcuboid islands will break up into a number of small isolated particles on the substratebefore they are able to form the single spherical shape as its equilibrium. In orderto investigate this phenomenon, we start the simulation by fixing the same materialconstant as σ = cos(3π/4), and choosing the initial thin film as a long cuboid with(1, 12, 1). For isotropic case, as can be seen in Fig. 5.4, the island quickly evolves intoa cylinder-like shape during the evolution, and then exhibits variations of the radius,and finally breaks up into two small isolated islands on the substrate. Under cubicanisotropic surface energies, long cuboid islands exhibit similar pinch-off phenomenonto the isotropic surface energy case. We test the numerical example for an initiallycuboid island with the same material constant and initial island, as shown Fig. 5.5.From the figure, we observe that it finally forms three small isolated islands, while theisland initially with the same shapes only form two isolated islands in the isotropicsurface energy case. This indicates that in the cubic anisotropic surface energy, thesolid island tends to dewet more easily than in the isotropic surface energy.

Fig. 5.5. Several snapshots during the evolution of an initial, cuboid island film with anisotropicsurface energy until its pinch-off time: (a) t = 0; (b) t = 0.020; (c) t = 0.10; (d) t = 0.24;(e) t = 0.54; (f) t = 0.695, where the initial shape is chosen as a (1, 12, 1) cuboid, the materialconstant σ = cos(3π/4), and the anisotropic surface energy is chosen as the cubic type, i.e., γ(n) =1 + a(n4

1+ n4

2+ n4

3) with a = 0.25.

We next examine the morphological evolutions of square island films with (m,m, h).We start by simulating the evolution of an initial, short square island with (3.2, 3.2, 0.1),and the material constant is chosen as σ = cos 5π

6 . As can be seen in Fig. 5.6, the fourcorners of the square island retract much more slowly than the middle points of thefour edges at the beginning, thus resulting in a near cross shape for the island film (seeFig. 5.6(d)). This phenomenon of corner accumulation has also been observed in theexperiments [51, 57, 59] or numerical simulations by the phase-field approach [23, 38].These corners at last catch up with the edges and the contact line moves towards acircular shape in order to form a spherical shape as its equilibrium. Meanwhile, wecan also observe that a valley forms at the center during the evolution, but finallydisappear. However, if we enlarge the square size but fix the thickness of the initialthin film, the formed valley will become deep and finally touch the substrate, and


Fig. 5.6. Several snapshots during the evolution of an initial, cuboid island film with isotropicsurface energy towards its equilibrium shape: (a) t = 0; (b) t = 0.004; (c) t = 0.008; (d) t = 0.0120;(e) t = 0.020; (f) t = 0.080, where the initial shape is chosen as a (3.2, 3.2, 0.1) cuboid, and thematerial constant σ = cos(5π/6).

Fig. 5.7. Several snapshots during the evolution of an initial, cuboid island film with isotropicsurface energy until its pinch-off time: (a) t = 0; (b) t = 0.005; (c) t = 0.010; (d) t = 0.031, wherethe initial shape is chosen as a (6.4, 6.4, 0.1) cuboid, and the material constant σ = cos(5π/6).

produce a hole in the centre of the island, as shown in Fig. 5.7. We stop the numeri-cal simulations at the time when there exists at least one mesh point which touchesthe substrate. For a better illustration, we also show the corresponding cross-sectionprofiles of the thin film during the evolution in Fig. 5.8.

6. Conclusions. We proposed a sharp-interface approach for simulating solid-state dewetting of thin films in three dimensions (3D), and this approach can handlewith the effect of the surface energy anisotropy. Based on the Cahn-Hoffman ξ-vectorformulation and the speed method, we derived rigorously the first variation of the totalfree energy functional of the solid-state dewetting problem. From the first variation,necessary conditions for the equilibrium shape of solid-state dewetting were rigorouslygiven in mathematics. Furthermore, a kinetic sharp-interface model was also proposed


-4 -2 0 2 4

z

0

0.2

0.4

0.6

(a)y-directiondiagonal-direction

-4 -2 0 2 4

z

0

0.2

0.4

0.6

(b)

-4 -2 0 2 4

z

0

0.2

0.4

0.6

(c)

-4 -2 0 2 4

z

0

0.2

0.4

0.6

(d)

Fig. 5.8. The cross-section profile of the thin film along its y-direction and diagonal directionfor the example shown in Fig. 5.7: (a) t = 0; (b) t = 0.005; (c) t = 0.010; (d) t = 0.031.

for simulating the solid-state dewetting of thin films in 3D. The governing equationsdescribed the interface evolution which is controlled by surface diffusion and contactline migration. A lot of numerical examples were performed for solving the model,and numerical results reproduced the complex features in the solid thin film dewettingobserved in experiments, such as hole formation, corner accumulation, pinch-off andRayleigh instability.

REFERENCES

[1] D. Amram, L. Klinger, and E. Rabkin, Anisotropic hole growth during solid-state dewettingof single-crystal Au–Fe thin films, Acta Mater., 60 (2012), pp. 3047–3056.

[2] L. Armelao, D. Barreca, G. Bottaro, A. Gasparotto, S. Gross, C. Maragno, and

E. Tondello, Recent trends on nanocomposites based on Cu, Ag and Au clusters: Acloser look, Coord. Chem. Rev., 250 (2006), pp. 1294–1314.

[3] R. Backofen, S. M. Wise, M. Salvalaglio, and A. Voigt, Convexity splitting in a phasefield model for surface diffusion, arXiv:1710.09675, (2017).

[4] R. W. Balluffi, S. Allen, and W. C. Carter, Kinetics of materials, John Wiley & Sons,2005.

[5] W. Bao, W. Jiang, D. J. Srolovitz, and Y. Wang, Stable equilibria of anisotropic particleson substrates: a generalized winterbottom construction, SIAM J. Appl. Math, 77 (2017),pp. 2093–2118.

[6] W. Bao, W. Jiang, Y. Wang, and Q. Zhao, A parametric finite element method for solid-state dewetting problems with anisotropic surface energies, J. Comput. Phys., 330 (2017),pp. 380–400.

[7] W. Bao, W. Jiang, and Q. Zhao, A parametric finite element method for solid-state dewettingproblems in three dimensions, in preparation, (2018).

[8] J. Cahn and D. Hoffman, A vector thermodynamics for anisotropic surfaces: I. curved andfaceted surfaces, Acta Metall., 22 (1974), pp. 1205–1214.

[9] J. W. Cahn and C. A. Handwerker, Equilibrium geometries of anisotropic surfaces andinterfaces, Mater. Sci. Eng: A, 162 (1993), pp. 83–95.

[10] W. C. Carter, A. R. Roosen, J. W. Cahn, and J. E. Taylor, Shape evolution by surfacediffusion and surface attachment limited kinetics on completely faceted surfaces, ActaMetall. Mater., 43 (1995), pp. 4309–4323.

[11] D. T. Danielson, D. K. Sparacin, J. Michel, and L. C. Kimerling, Surface-energy-driven dewetting theory of silicon-on-insulator agglomeration, J. Appl. Phys., 100 (2006),p. 083507.

[12] K. Deckelnick, G. Dziuk, and C. M. Elliott, Computation of geometric partial differentialequations and mean curvature flow, Acta Numer., 14 (2005), pp. 139–232.

[13] G. Dogan and R. H. Nochetto, First variation of the general curvature-dependent surfaceenergy, ESAIM: M2AN, 46 (2012), pp. 59–79.


[14] E. Dornel, J. Barbe, F. De Crecy, G. Lacolle, and J. Eymery, Surface diffusion dewettingof thin solid films: Numerical method and application to Si/SiO2, Phys. Rev. B, 73 (2006),p. 115427.

[15] M. Dufay and O. Pierre-Louis, Anisotropy and coarsening in the instability of solid dewettingfronts, Phys. Rev. Lett., 106 (2011), p. 105506.

[16] G. Dziuk and C. M. Elliott, Finite element methods for surface PDEs, Acta Numer., 22(2013), pp. 289–396.

[17] M. Dziwnik, A. Munch, and B. Wagner, Sharp interface limits of an anisotropic phase fieldmodel for solid-state dewetting, IFAC-PapersOnLine, 48 (2015), pp. 394–395.

[18] Y. Fan, R. Nuryadi, Z. A. Burhanudin, and M. Tabe, Thermal agglomeration of ultrathinsilicon-on-insulator layers: Crystalline orientation dependence, Jpn. J. Appl. Phys, 47(2008), p. 1461.

[19] C. Herring, The Physics of Powder Metallurgy, edited by W.E. Kingston, McGraw-Hill, NewYork, 1951.

[20] A. Herz, A. Franz, F. Theska, M. Hentschel, T. Kups, D. Wang, and P. Schaaf, Solid-state dewetting of single-and bilayer Au-W thin films: Unraveling the role of individuallayer thickness, stacking sequence and oxidation on morphology evolution, AIP Adv., 6(2016), p. 035109.

[21] M. Hintermuller and W. Ring, A second order shape optimization approach for image seg-mentation, SIAM J. Appl. Math., 64 (2004), pp. 442–467.

[22] D. W. Hoffman and J. W. Cahn, A vector thermodynamics for anisotropic surfaces: I. funda-mentals and application to plane surface junctions, Surface Science, 31 (1972), pp. 368–388.

[23] W. Jiang, W. Bao, C. V. Thompson, and D. J. Srolovitz, Phase field approach for simu-lating solid-state dewetting problems, Acta Mater., 60 (2012), pp. 5578–5592.

[24] W. Jiang, Y. Wang, D. J. Srolovitz, and W. Bao, Solid-state dewetting on curved substrates,Phys. Rev. Mater., 2 (2018), p. 113401.

[25] W. Jiang, Y. Wang, Q. Zhao, D. J. Srolovitz, and W. Bao, Solid-state dewetting and islandmorphologies in strongly anisotropic materials, Scripta Mater., 115 (2016), pp. 123–127.

[26] W. Jiang and Q. Zhao, Sharp-interface approach for simulating solid-state dewetting in twodimensions: a Cahn-Hoffman ξ-vector formulation, Physica D, (2019).

[27] W. Jiang, Q. Zhao, T. Qian, D. J. Srolovitz, and W. Bao, Application of Onsager’svariational principle to the dynamics of a solid toroidal island on a substrate, Acta Mater.,163 (2019), pp. 154–160.

[28] E. Jiran and C. Thompson, Capillary instabilities in thin films, J. Electron. Mater., 19 (1990),pp. 1153–1160.

[29] E. Jiran and C. Thompson, Capillary instabilities in thin, continuous films, Thin Solid Films,208 (1992), pp. 23–28.

[30] W. Kan and H. Wong, Fingering instability of a retracting solid film edge, J. Appl. Phys., 97(2005), p. 043515.

[31] G. H. Kim and C. V. Thompson, Effect of surface energy anisotropy on Rayleigh-like solid-state dewetting and nanowire stability, Acta Mater., 84 (2015), pp. 190–201.

[32] G. H. Kim, R. V. Zucker, J. Ye, W. C. Carter, and C. V. Thompson, Quantitative analysisof anisotropic edge retraction by solid-state dewetting of thin single crystal films, J. Appl.Phys., 113 (2013), p. 043512.

[33] O. Kovalenko, S. Szabo, L. Klinger, and E. Rabkin, Solid state dewetting of polycrystallineMo film on sapphire, Acta Mater., 139 (2017), pp. 51–61.

[34] F. Leroy, F. Cheynis, Y. Almadori, S. Curiotto, M. Trautmann, J. Barbe, P. Muller,

et al., How to control solid state dewetting: A short review, Surface Science Reports, 71(2016), pp. 391–409.

[35] M. S. McCallum, P. W. Voorhees, M. J. Miksis, S. H. Davis, and H. Wong, Capillaryinstabilities in solid thin films: Lines, J. appl. phys, 79 (1996), pp. 7604–7611.

[36] J. Mizsei, Activating technology of SnO2 layers by metal particles from ultrathin metal films,Sensors and Actuators B: Chemical, 16 (1993), pp. 328–333.

[37] W. W. Mullins, Theory of thermal grooving, J. Appl. Phys., 28 (1957), pp. 333–339.[38] M. Naffouti, R. Backofen, M. Salvalaglio, T. Bottein, M. Lodari, A. Voigt, T. David,

A. Benkouider, I. Fraj, L. Favre, et al., Complex dewetting scenarios of ultrathinsilicon films for large-scale nanoarchitectures, Sci. Adv., 3 (2017), p. eaao1472.

[39] M. Naffouti, T. David, A. Benkouider, L. Favre, A. Delobbe, A. Ronda, I. Berbezier,

and M. Abbarchi, Templated solid-state dewetting of thin silicon films, Small, 12 (2016),pp. 6115–6123.

[40] D. Peng, S. Osher, B. Merriman, and H.-K. Zhao, The geometry of Wulff crystal shapes andits relations with Riemann problems, Nonlinear Partial Differential Equations: Evanston,


IL, (1998), pp. 251–303.[41] O. Pierre-Louis, A. Chame, and Y. Saito, Dewetting of ultrathin solid films, Phys. Rev.

Lett., 103 (2009), p. 195501.[42] E. Rabkin, D. Amram, and E. Alster, Solid state dewetting and stress relaxation in a thin

single crystalline Ni film on sapphire, Acta Mater., 74 (2014), pp. 30–38.[43] S. Randolph, J. Fowlkes, A. Melechko, K. Klein, H. Meyer III, M. Simpson, and

P. Rack, Controlling thin film structure for the dewetting of catalyst nanoparticle arraysfor subsequent carbon nanofiber growth, Nanotechnology, 18 (2007), p. 465304.

[44] L. Rayleigh, On the instability of jets, Proc. Lond. Math. Soc, 1 (1878), pp. 4–13.[45] V. Schmidt, J. V. Wittemann, S. Senz, and U. Gosele, Silicon nanowires: a review on

aspects of their growth and their electrical properties, Adv. Mater, 21 (2009), pp. 2681–2702.

[46] A. Shklyaev and A. Budazhapova, Submicron-and micron-sized sige island formation on si(100) by dewetting, Thin Solid Films, 642 (2017), pp. 345–351.

[47] J. Soko lowski and J. Zolesio, Introduction to shape optimization: Shape sensitivity analysis.1992.

[48] D. J. Srolovitz and S. A. Safran, Capillary instabilities in thin films: I. Energetics, J. Appl.Phys., 60 (1986), pp. 247–254.

[49] D. J. Srolovitz and S. A. Safran, Capillary instabilities in thin films: II. Kinetics, J. Appl.Phys., 60 (1986), pp. 255–260.

[50] A. P. Sutton and R. W. Balluffi, Interfaces in crystalline materials, Clarendon Press, 1995.[51] C. V. Thompson, Solid-state dewetting of thin films, Annu. Rev. Mater. Res., 42 (2012),

pp. 399–434.[52] Y. Wang, W. Jiang, W. Bao, and D. J. Srolovitz, Sharp interface model for solid-state

dewetting problems with weakly anisotropic surface energies, Phys. Rev. B, 91 (2015),p. 045303.

[53] W. Winterbottom, Equilibrium shape of a small particle in contact with a foreign substrate,Acta Metall., 15 (1967), pp. 303–310.

[54] H. Wong, P. Voorhees, M. Miksis, and S. Davis, Periodic mass shedding of a retractingsolid film step, Acta Mater., 48 (2000), pp. 1719–1728.

[55] G. Wulff, Zur frage der geschwindigkeit des wachstums und der auflosung der krystallflachen,Z. Kristallogr, 34 (1901), pp. 449–530.

[56] J. Ye and C. V. Thompson, Mechanisms of complex morphological evolution during solid-statedewetting of single-crystal nickel thin films, Appl. Phys. Lett., 97 (2010), p. 071904.

[57] J. Ye and C. V. Thompson, Regular pattern formation through the retraction and pinch-offof edges during solid-state dewetting of patterned single crystal films, Phys. Rev. B, 82(2010), p. 193408.

[58] J. Ye and C. V. Thompson, Anisotropic edge retraction and hole growth during solid-statedewetting of single crystal nickel thin films, Acta Mater., 59 (2011), pp. 582–589.

[59] J. Ye and C. V. Thompson, Templated solid-state dewetting to controllably produce complexpatterns, Adv. Mater., 23 (2011), pp. 1567–1571.

[60] T. Young, An essay on the cohesion of fluids, Philos. Trans. R. Soc. London, 95 (1805),pp. 65–87.

[61] R. V. Zucker, G. H. Kim, W. C. Carter, and C. V. Thompson, A model for solid-statedewetting of a fully-faceted thin film, Comptes Rendus Physique, 14 (2013), pp. 564–577.

[62] R. V. Zucker, G. H. Kim, J. Ye, W. C. Carter, and C. V. Thompson, The mechanism ofcorner instabilities in single-crystal thin films during dewetting, J. Appl. Phys., 119 (2016),p. 125306.

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SHARP-INTERFACE APPROACH FOR SIMULATING SOLID-STATE ... · Sharp-interface approach for simulating solid-state dewetting in three dimensions 3 Fig. 1.1. γ-plot, 1/γ-plot and ξ-plot