A diﬀuse-interface approach for solid-state dewetting with anisotropic surface energies Harald GarckePatrik KnopfRobert NürnbergQuan Zhao

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A diﬀuse-interface approach for solid-state dewetting with

anisotropic surface energies

Harald Garcke∗Patrik Knopf∗Robert Nürnberg†Quan Zhao∗

Abstract

We present a diﬀuse-interface model for the solid-state dewetting problem with anisotropic

surface energies in Rdfor d∈ {2,3}. The introduced model consists of the anisotropic

Cahn–Hilliard equation, with either a smooth or a double-obstacle potential, together with

a degenerate mobility function and appropriate boundary conditions on the wall. Upon

regularizing the introduced diﬀuse-interface model, and with the help of suitable asymptotic

expansions, we recover as the sharp-interface limit the anisotropic surface diﬀusion ﬂow for

the interface together with an anisotropic Young’s law and a zero-ﬂux condition at the contact

line of the interface with a ﬁxed external boundary. Furthermore, we show the existence of

weak solutions for the regularized model, for both smooth and obstacle potential. Numerical

results based on an appropriate ﬁnite element approximation are presented to demonstrate

the excellent agreement between the proposed diﬀuse-interface model and its sharp-interface

limit.

Key words. Solid-state dewetting, Cahn–Hilliard equation, anisotropy, sharp-interface limit,

weak solutions, ﬁnite element method.

1 Introduction

Deposited solid thin ﬁlms are unstable and could dewet to form isolated islands on the substrate

in order to minimize the total surface energy [53,70]. This phenomenon is known as solid-state

dewetting (SSD), since the thin ﬁlms remain in a solid state during the process. SSD has attracted

a lot of attention recently, and is emerging as a promising route to produce patterns of arrays of

particles used in sensor technology, optical and magnetic devices, and catalyst formations, see

e.g. [6,7,19,23,65,67].

The dominant mass transport mechanism in SSD is surface diﬀusion [68]. This evolu-

tion law was ﬁrst introduced by Mullins [57] to describe the mass diﬀusion within interfaces

in polycrystalline materials. For surface diﬀusion, the normal velocity of the interface is pro-

portional to the surface Laplacian of the mean curvature. In the case of SSD the evolution of

the interface that separates the thin ﬁlm from the surrounding vapor also involves the motion

∗Fakultät für Mathematik, Universität Regensburg, 93053 Regensburg, Germany,

(harald.garcke@ur.de,patrik.knopf@ur.de quan.zhao@ur.de)

†Dipartimento di Mathematica, Università di Trento, 38123 Trento, Italy,

(robert.nurnberg@unitn.it)

arXiv:2210.01698v2 [math.AP] 11 Dec 2022

of the contact line, i.e., the region where the ﬁlm/vapor interface meets the substrate. The

equilibrium contact angle is given by Young’s law which prescribes a force balance along the

substrate. Many eﬀorts have been devoted to SSD problems in recent years. For example, a

large body of experiments have revealed that the pattern formations could depend highly on

the crystallographic alignments, the ﬁlm sizes and shapes, as well as the substrate topology, see

e.g. [5,23,59,70,76]. In addition, mathematical studies based on diﬀerent models have been

considered in [22,24,34,36,40,46–48,59,68,73].

In this work, we aim to study the SSD problem with anisotropic surface energies in the

diﬀuse-interface framework. In the isotropic case, diﬀuse-interface models are based on the

Ginzburg–Landau energy

Eiso(ϕ) = ZΩ

2|∇ϕ|2+ε−1F(ϕ) dx, (1.1)

where Ω⊂Rdis a given domain with d∈ {2,3},ϕ: Ω →Ris the order parameter, ε > 0is a

small parameter proportional to the thickness of the interfacial layer, and F(ϕ)is the free energy

density. The following three choices for Fare mainly used in the literature:

(i) the smooth double-well potential [69]

F(ϕ) = 1

2(1 −ϕ2)2,(1.2a)

which has two global minimum points at ϕ=±1and a local maximum point at ϕ= 0;

(ii) the logarithmic potential [27]

F(ϕ) = 1

2θ[(1 + ϕ) log(1 + ϕ) + (1 −ϕ) log(1 −ϕ)] + 1

2(1 −ϕ2),(1.2b)

where θ > 0is the absolute temperature. This potential has two minima ϕ=±(1 −˜

k(θ)),

where ˜

k(θ)is a small positive real number satisfying ˜

k(θ)→0as θ→0, and its usage

enforces ϕto attain values within (−1,1);

(iii) the double-obstacle potential [21]

F(ϕ) = 





2(1 −ϕ2)if |ϕ| ≤ 1,

∞otherwise.(1.2c)

It can be characterized via the deep quench limit of the logarithmic potential, i.e., the limit

of (1.2b) as θ→0.

The (isotropic) Cahn–Hilliard equation can be interpreted as a weighted H−1-gradient ﬂow of

the free energy (1.1). It reads as

∂tϕ=∇ · (m(ϕ)∇µ), µ =−ε∆ϕ+ε−1F0(ϕ),(1.3)

where m(ϕ)is a mobility function, together with Neumann boundary conditions for µand ϕ.

The Cahn–Hilliard equation was ﬁrst introduced to study the spinodal decomposition in binary

alloys [25,27] and has since then been used to model many other phenomenon, e.g., [1,20,41,49].

We note that the double-obstacle potential is not diﬀerentiable at ϕ=±1, and the deﬁnition of

the generalized chemical potential in this case becomes

µ∈ −ε∆ϕ+ε−1∂F (ϕ),(1.4)

where ∂F (ϕ)is the Fréchet sub-diﬀerential of Fat ϕand ∆ϕhas to be understood in a weak

sense, see [14,21]. In the case of a constant mobility m(ϕ)≡1, (1.3) converges to the Mullins–

Sekerka problem [58] as ε→0[3,61]. In order to obtain the surface diﬀusion equation in the

sharp-interface limit, a degenerate mobility needs to be chosen. For example, it was shown in [26]

by a formal asymptotic analysis that the surface diﬀusion ﬂow is recovered by considering a slow

time scale τ=O(ε−1t)of (1.3) with m(ϕ) = (1−ϕ2)+and with the potential F(ϕ)either chosen

as in (1.2c), or as in (1.2b) with θ=O(εξ),ξ > 0. When using the smooth double-well potential

(1.2a) the situation is less clear. While the limiting motion of surface diﬀusion is obtained with

the choice m(ϕ) = (1 −ϕ2)2[30,46,63,72], using the less degenerate mobility m(ϕ) = (1 −ϕ2)+

may not lead to pure surface diﬀusion in the limit ε→0, since an additional bulk diﬀusion

term is conjectured to be present due to the non-zero ﬂux contributions [30,51,52]. However,

in all these cases, no rigorous proof for the sharp-interface limit or the presence of non-zero ﬂux

contributions are available so far.

A natural generalization of the free energy (1.1) to the case of anisotropic surface energies

is given by

Eγ(ϕ) = ZΩ

2|γ(∇ϕ)|2+ε−1F(ϕ) dx=ZΩ

εA(∇ϕ) + ε−1F(ϕ) dx, (1.5)

see e.g. [37,50]. Here, γ:Rd→[0,∞)is the anisotropic density function, which is positively

homogeneous of degree one, and A:= 1

2γ2. This then gives rise to the anisotropic Cahn–Hilliard

equation

∂tϕ=∇ · (m(ϕ)∇µ), µ =−ε∇ · A0(∇ϕ) + ε−1F0(ϕ),(1.6)

where A0represents the gradient of the map A:Rd→[0,∞). In contrast to the isotropic case,

diﬀuse-interface models based on (1.5) result in a nonuniform asymptotic interface thickness,

which in fact depends on the anisotropic density function γ(∇ϕ), see [2,18,39,74,75]. To remedy

this issue, an alternative energy of the form

Eγ(ϕ) = ZΩ|∇ϕ|−1γ(∇ϕ)ε

2|∇ϕ|2+ε−1F(ϕ)dx(1.7)

can be considered, see [64,71], so that a constant thickness of the asymptotic interface is achieved.

However, the resulting diﬀuse-interface models based on (1.7) become more nonlinear and are

singular at ∇ϕ= 0, which poses great challenges in the mathematical analysis and the stable

numerical approximation. Therefore, in this work, we will restrict ourselves to the classical

energy in (1.5). We also note that to guarantee that (1.6) converges to the anisotropic surface

diﬀusion ﬂow as ε→0, a rescaled anisotropic coeﬃcient needs to be introduced to the degenerate

mobility [54,63]. We refer to Section 2below for the precise details.

Figure 1: Sketch of the structure for SSD near the contact line (green point), where the vapor, ﬁlm and

substrate phases meet.

When it comes to SSD, as shown in Fig. 1, the total surface energy of the system consists

of the ﬁlm/vapor interface energy Einf and the substrate energy Esub,

Einf =ZΓ(t)

γ(ν) dS, Esub =γF S ZΓF S

dS+γV S ZΓV S

dS, (1.8)

where Γ(t)is the dynamic ﬁlm/vapor interface with νbeing the interface normal pointing into

the vapor phase, ΓF S and ΓV S are the interfaces between ﬁlm/substrate and vapor/substrate,

respectively, and γF S and γV S are the corresponding surface energy densities. In order to model

SSD by the diﬀuse-interface approach, we associate the vapor phase with ϕ≈1and the ﬁlm

phase with ϕ≈ −1. Then the Ginzburg–Landau type energy (1.5), up to a multiplicative

constant, will approximate the sharp interface energy Einf . Moreover, the contribution to the

wall energy Esub can be approximated by

Ew(ϕ) = ZΓF S ∪ΓV S

γV S +γF S

2+ (γV S −γF S )G(ϕ) dS, (1.9)

where G(ϕ)is a smooth function satisfying G(±1) = ±1

2, see [7,36,44,46] for SSD and [45,62]

for moving contact lines in ﬂuid mechanics.

There are several results on the existence of weak solutions for the degenerate Cahn–Hilliard

equation (1.3) with homogeneous boundary conditions or its variants with inhomogeneous bound-

ary conditions, see [31,38,77]. However, little is known about the anisotropic case except the

work in [35] which focuses on a particular n-fold anisotropy in two space dimensions.

The main aim of this work is to develop a diﬀuse-interface approach to SSD in the case

of anisotropic surface energies based on the energy contributions (1.5) and (1.9). The obtained

diﬀuse-interface model consists of a degenerate anisotropic Cahn–Hilliard equation with appro-

priate boundary conditions. We study the sharp-interface limit and show the existence of weak

solutions to the diﬀuse-interface model.

The rest of the paper is organized as follows. In Section 2, we review a sharp-interface model

for SSD and then introduce a diﬀuse-interface model based on a gradient ﬂow approach. We then

derive the sharp-interface limit from a regularized model with the help of asymptotic expansions

in Section 3. In Section 4, we prove the existence of weak solutions to the diﬀuse-interface

model. Numerical tests are presented in Section 5, where a comparison between sharp-interface

approximations and diﬀuse-interface approximations is made.

2 Modeling aspects

In this section, we ﬁrst review a sharp-interface model for SSD with anisotropic surface energies

in two or three space dimensions. Then, we propose a suitable diﬀuse-interface model to approx-

imate this sharp-interface model. Here we note that there exist several works on the modelling

of SSD using a diﬀuse-interface approach in the literature. However, these works consider either

the isotropic case, e.g., [7,47], or the anisotropic case in 2d, e.g., [36].

2.1 The sharp-interface model

We consider the dewetting of a solid thin ﬁlm on a ﬂat substrate in Rdwith d∈ {2,3}, as shown

in Fig. 1. We parameterize the interface of Γ(t)over the initial interface as follows

x(·, t) : Γ(0) ×[0, T ]→Rd,

where T > 0is a prescribed ﬁnal time. The induced velocity is then given by

V(x(q, t), t) = ∂tx(q, t)for all q∈Γ(0), t ∈[0, T ],

where Γ(0) is a smooth hypersurface with boundary. The sharp-interface model for SSD (cf.

[12,28,48,69]) reads as:

V=−∇s·(D(ν)∇sκγ),(2.1a)

κγ=−∇s·γ0(ν),(2.1b)

which has to hold for all t∈[0, T ]and all points on Γ(t). Here, V=V·νis the normal velocity,

νis the unit normal to Γ(t)pointing into the vapor, and ∇sis the surface gradient operator

on Γ(t). Besides, D(ν)is an orientation dependent mobility (cf. [69]). The function Dneeds

to be deﬁned for unit vectors, but here we extend its domain to Rdsuch that it is positively

homogeneous of degree one. The term κγrepresents the anisotropic mean curvature, and γ0(ν)is

the Cahn–Hoﬀman vector, where γ0denotes the gradient of γ(cf. [43]). The above equations are

subject to the following boundary conditions at the contact line, where the ﬁlm/vapor interface

Γ(t)meets the substrate:

•attachment condition

V·nw= 0,(2.2a)

•contact angle condition

γ0(ν)·nw+σ= 0,(2.2b)

•zero-ﬂux condition

D(ν)∇sκγ·nc= 0,(2.2c)

where

σ=γV S −γF S (2.3)

denotes the diﬀerence of the substrate energy densities across the contact line. Here, nwis the

unit normal to the substrate and points in the direction of the substrate, and ncis the conormal

vector of Γ(t), i.e., it is the outward unit normal to ∂Γ(t)and it lies within the tangent plane

of Γ(t). We observe that (2.2b) enforces an angle condition between the Cahn–Hoﬀman vector

γ0(ν)and the substrate unit normal nw. For example, in the isotropic case, γ(p) = |p|, the

Cahn–Hoﬀman vector reduces to the normal ν, and so if σ= 0 the condition (2.2b) encodes a

90◦contact angle between the ﬁlm/vapor interface and the substrate.

We assume that the anisotropy function γbelongs to C2Rd\{0}∩C(Rd,R≥0), is convex

and satisﬁes γ > 0on Rd\ {0}. We further assume that γis positively homogeneous of degree

one, meaning that

γ(λp) = λγ(p)for all λ > 0,p∈Rd.

This immediately implies γ(0)=0and the gradient of γ(p)satisﬁes

γ0(p)·p=γ(p)for all p∈Rd\ {0}.(2.4)

Similarly, the orientation dependent mobility function D∈C2Rd\{0}∩C(Rd,R≥0)is assumed

to satisfy D > 0on Rd\ {0}and

D(λp) = λD(p)for all λ > 0,p∈Rd.

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Adiuse-interfaceapproachforsolid-statedewettingwithanisotropicsurfaceenergiesHaraldGarcke*PatrikKnopf*RobertNürnbergQuanZhao*AbstractWepresentadiuse-interfacemodelforthesolid-statedewettingproblemwithanisotropicsurfaceenergiesinRdford2f2;3g.TheintroducedmodelconsistsoftheanisotropicCahnHilliarde...

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A diﬀuse-interface approach for solid-state dewetting with anisotropic surface energies Harald GarckePatrik KnopfRobert NürnbergQuan Zhao

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