Computational Materials Science 50 (2011) 1030–1036

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Computational Materials Science

journal homepage: www.elsevier .com/locate /commatsci

Morphological instabilities in thin films: Evolution maps

Mohsen Asle Zaeem a,⇑, Sinisa Dj. Mesarovic b

a Center for Advanced Vehicular Systems, Mississippi State University, Starkville, MS 39759, USAb School of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99164, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 30 September 2010Received in revised form 27 October 2010Accepted 29 October 2010Available online 20 November 2010

Keywords:Phase-field modelElasticityMultilayer thin filmsInstability

0927-0256/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.commatsci.2010.10.043

⇑ Corresponding author. Tel.: +1 662 325 0126.E-mail address: mohsen@cavs.msstate.edu (M.A. Z

We consider morphological instabilities in binary multilayers and the post-instability evolution of thesystem. The alloys with and without intermediate phase are considered, as well as the cases with stableand meta-stable intermediate phase.

Using the Galerkin finite element formulation for coupled Cahn–Hilliard – elasticity problem, maps ofdifferent evolution paths are developed in the parameter space of relative thicknesses of initial phases.We consider the relative importance of elastic and chemical energy of the system and develop mapsfor different cases.

The systems exhibit rich evolution behavior. Depending on the initial configuration (which determinesthe mass conservation condition), the final equilibrium varies, but even greater variety is observed inevolution paths. The paths may consist of multiple evolution steps, which may proceed at different rates.

Except for few special circumstances, the instabilities are to perturbations non-homogeneous in thefilm plane. Post-instability evolution is essentially two-dimensional, and cannot be reduced to the one-dimensional model.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Multilayer thin films have been a very active research area inthe past decade. When the thickness of an individual layer is re-duced to micro- or nano-scale, novel mechanical [1–5], optical[6], electronic [7,8], and magnetic [6–8] properties emerge, whichmake such materials attractive for a number of potential applica-tions. However, to maintain the desirable properties, the stabilityof phases is required, and this is affected by temperature andstress. In addition to externally applied load, internal stresses ap-pear as a result of enforced lattice continuity and compositionalstrain [9–14]. For example – a significant drop of hardness occursafter annealing of Ni/Ru multilayers at 600 �C [15], Ni layers break-down in Ni/Ag multilayer is annealed at 600 �C [16], and pinch-offof Co layers in Co/Cu multilayers is observed after creep test at830 �C [17].

Greer [18,19] classified the changes that can occur in thin filmmultilayers as their microstructures evolve. He divided these pro-cesses to the following categories: inter-diffusion (same phasescoexist with changed compositions) (Ag/Au [20], Ni/Al, Ag/Zn[21], Al/Ni [22]), interfacial reaction (nucleation and growth of anew phase) (Si/Ni [23], Ni/Zr [24], Al/Mn [25], Ni/Al [26]), transfor-mation in one phase of the multilayer without any changes in otherphase(s) (Si/Al [27]), and, coarsening or spinodal decomposition of

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aeem).

the layers (Ni/Ag [15], Ni/C [28]). All of these can be modeled usingthe diffuse interface (phase-field) model.

Following the initial Cahn–Hilliard [29] formulation of the dif-fuse interface model, many numerical phase-field studies [30–43]have been reported. Very few considered thin films, and theseare either based one-dimensional models [39,40] with severerestrictions on possible instabilities, or, are focused on film-fluidinteractions [41–43]. The exception is the recent work by Chir-ranjeevi et al. [14]. They consider a special two-phase multilayerwith one phase much stiffer than the other.

Here, we present a comprehensive study of binary multilayerthin films, including two-phase systems and systems with an inter-mediate phase (meta-stable or stable). Following the dimensionalanalysis, we develop the maps of the evolution behavior of multi-layers, in the parameter space describing the initial geometry, andconsidering the cases of different relative importance of elastic andchemical energy density. In addition to having different (final) sta-ble configurations, multilayered thin films may reach those stableconfigurations following different paths in configurational space.We classify the paths and map the initial configuration space cor-responding to each path.

We use the recently developed Galerkin finite element formula-tion [44,45] for the Cahn–Hilliard diffuse interface (phase-field)model [29], coupled with elasticity [40].

The paper is organized as follows. In Section 2, formulation,dimensional analysis, and numerical solver for the coupled phasefiled-elasticity equations are presented. In Section 3, the maps of

Fig. 1. The non-dimensional triple-well free energy density, �f , as function of scaledcomposition, �c.

M.A. Zaeem, S.Dj. Mesarovic / Computational Materials Science 50 (2011) 1030–1036 1031

transformations for two-phase systems are developed. In Sections4 and 5, similar maps are developed for systems with stable andmeta-stable intermediate phase. Discussion and conclusions are gi-ven in Section 6.

2. Formulation and numerical method

Concentration of a component in a binary system is subjected toa conservation law. When used a phase-field variable, it results in anonlinear 4th order diffusion partial differential equation (PDE),which is then coupled to the 2nd order elasticity PDEs. Considera binary system, with components A and B, which forms up tothree phases (a, b, and c), and characterized by the molar fractionof the component B, c. The total free energy, F, of a non-homoge-neous system with volume V, bounded by surface S, is constructedas:

Fðc;uiÞ ¼Z

Vf ðcÞ þ Uðeij; cÞ þ

12jðrcÞ2

� �dV �

ZS

tiuidS; ð1Þ

where f(c) is the Helmholtz free energy density of a uniform, stress-free system, j is the isotropic gradient energy coefficient, ti and ui

are the components of surface traction displacement. The last termis the loading potential. The surface energy of the outer boundary Sis neglected. U(eij,c) is the strain energy density and eij are the com-ponents of the total strain tensor, consisting of elastic (stress-in-duced), and compositional (stress-free) strain [46]. The latter ismodeled as purely volumetric and linear in composition (Vegard’slaw) [47]:

eij ¼12ðui;j þ uj;iÞ ¼ ee

ij þ edij; e ¼ gðc � c0Þ: ð2Þ

The reference composition c0 is taken to be the average ofcomposition over the domain, and g is the compositional strainparameter. g varies widely in binary metallic alloys, e.g.,g = 0.094 for Al-2 at.% Cu and g = 0.264 for Fe-5 at.% C [48].

The 2nd order PDE governing mechanical equilibrium, whichcan be obtained by dF=d ui ¼ 0, has this form:

ðCijkluk;lÞ;j ¼ ~Cije;j; ~Cij ¼ Cijkldkl: ð3Þ

The conserved phase field formulations results in a 4th orderevolution PDE [44]:

_c ¼ Br2M; Mðc;r2cÞ ¼ @ðf � UÞ@c

� jr2c: ð4Þ

The chemical potential, M, is a variational derivative of the freeenergy (1), and B is the isotropic mobility coefficient.

The Helmholtz free energy density of a uniform unstressed sys-tem, is assumed to be a triple well potential that can produce two-phase and three-phase binary systems:

f ðcÞ ¼W1ðc � caÞ2ðc � cbÞ2ðc � ccÞ2 þW2ðc � caÞ2ðc � ccÞ2; ð5Þ

W1 and W2 are the energy coefficients, ca, cb and cc are the molefractions at which f(c) has extrema. The non-dimensional free en-ergy density is shown in Fig. 1, as function of the scaledcomposition.

In this research, we use particular non-dimensional parameterssuch that the results can be used to predict the transformation andevolution of different binary multilayer thin films. To analyze thespace of non-dimensional parameters of the problem, we use thefollowing dimensional reduction [40,44]. The scaled composition�c varies between �ca ¼ �1 and �cc ¼ 1, with intermediate phase at�cb ¼ 0. With Dc = (cc � ca)/2:

�c ¼ ð2c � cc � caÞ=2Dc: ð6Þ

Let q0 be the density of lattice sites, k � the Boltzmann constantand T � the temperature. Denote x = q0kT. The characteristiclength X, the characteristic time X, and the characteristic energydensity W, are:

X ¼ffiffiffiffiffiffiffiffiffiffij=x

p=Dc2; X ¼ j=ð2Bx2Dc8Þ; and W ¼ xDc6: ð7Þ

We note that j affects only the length and time scale, and Baffects only the time scale. Non-dimensional gradient energycoefficient and mobility are �j ¼ �B ¼ 1. The isotropic elastic moduliare scaled by E� ¼ E=ð1� mÞ, where E is the Young’s modulus and mis the Poisson ratio, so that their non-dimensional versions dependon the Poisson ratio only: �C11 ¼ �C44ð1� mÞ=ð1� 2mÞ, �C12 ¼�C44m=ð1� 2mÞ, and �C44 ¼ ð1� mÞ=ð1þ mÞ. The scaled compositionalstrain parameter is:

�g ¼ gffiffiffiffiffiE�p

=x=Dc2; ð8Þ

Assuming a fixed Poisson ratio (v = 0.3 in this study), the systemis governed by only three independent non-dimensional parame-ters, �g, �W1 ¼W1=W and �W2 ¼W2=W . We will consider three com-binations of �W1 and �W2, as shown in Fig. 1:

� �W1 ¼ 0; �W2 ¼ 0:02 (no intermediate phase),� �W1 ¼ 0:4; �W2 ¼ �0:04 (stable intermediate phase), and,� �W1 ¼ 0:4; �W2 ¼ 0:02 (meta-stable intermediate phase).

The details of coupled phase-field – elasticity formulation andthe corresponding Galerkin finite element implementation can befound in [44]. The optimal element size was determined to beabout 1/5 of the interface width, which guarantees the uniformquadratic convergence, expected for conforming elements. Thenominal (equilibrium) interface thickness is [29]:

l0 ¼ Dcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

j2Dfmax

r; ð9Þ

where Dfmax is the difference between the neighboring maximumand minimum of the free energy density, as illustrated in Fig. 1.

In the following three sections, we summarize the results of acomprehensive set of computations in form of evolution behaviormaps, for the three cases of multilayers (two phases, stable andmeta-stable intermediate phase). Total of 580 finite element runswere performed, with 215,000–360,000 degrees of freedom, and108–1011 time steps. The cpu time per run varied between 9 and35 h. Both, home-grown code and the commercial COMSOL Multi-physics [49] software were used. The boundary conditions for thethin film multilayers are illustrated in Fig. 2: the left and right

4103×=t

510=t

0=t

Fig. 2. Evolution of a 3-layers configuration (type I). Initially �dc ¼ 15, �da ¼ 30, and �g ¼ 0:07. Color bar shows the composition. The FE mesh consists of 29,243 triangularelements. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

1032 M.A. Zaeem, S.Dj. Mesarovic / Computational Materials Science 50 (2011) 1030–1036

boundaries are subjected to periodic boundary conditions, whilethe top and bottom boundaries are traction-free and flux-free. Byapplying these boundary conditions and imposing plane strainconditions, the results of two-dimensional simulations can be usedto predict the behaviors of three-dimensional multilayer thin films.

3. Two-phase systems

We first consider binary multilayers without an intermediatephase. Denote the initial thicknesses of a and c layers by da anddc (Fig. 2). In the 3-layers configuration shown in Fig. 2, the centrallayer breaks up into particles for �g P 0:07. For the purpose of map-ping, we will denote this type of behavior as type I. For �g P 0:15the breakup is followed by full homogenization at concentrationequal to the average composition of the system. This is of coursepossible only if the c-layer is thin enough.

In Figs. 3 and 4, the cases where the average compositions arenot close to the a-composition are shown. The final equilibriumis (qualitatively) the same in both cases – a thicker c-layer, butthe evolution paths are different. In Fig. 3 (type II behavior), slow

0=t

42.8 10t = ×

43.7 10t = ×

Fig. 3. Evolution of a 5-layers system (type II). Initially �dc ¼ 10, �da ¼ 1

fragmentation of both c-layers is followed by fast joining of theresulting patches to form a single, thicker c-layer. In Fig. 4 (typeIII), slow necking of the central a-layer is followed by fast fragmen-tation and disappearance of this layer.

The maps of transformations for the 5-layers initial configura-tion are shown in Figs. 5 and 6, for �g ¼ 0 and �g ¼ 0:1. There are fivedifferent regions in this parameter space. In addition to the abovedescribed types I, II, and III, homogenization occurs for initiallyvery thin c-layers, while for very thick layers, the initial configura-tion is stable.

Each map is constructed from about 150 computer runs. Toillustrate the accuracy, we show some computational points inFig. 5, namely those that determine the geometry of the triple junc-tion between regions I, II and III.

4. Systems with stable intermediate phase

We now consider binary multilayer systems with the stableintermediate phase, b. An example is shown in Fig. 7 (type IV). First,a fast growth of the intermediate phase takes place: the whole

42.2 10t = ×

43.2 10t = ×

51.8 10t = ×

5 and �g ¼ 0. The FE mesh consists of 16,158 triangular elements.

4109t 421 01×= ×=t

0=t 4107.8 ×=t

Fig. 4. Evolution of a 5-layers system (type III). Initially �dc ¼ 25, �da ¼ 15 and �g ¼ 0:1. Color bar shows the composition. The FE mesh has of 18,521 elements. (Forinterpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. Evolution map for 2-phase systems with �g ¼ 0. The non-dimensionalinterface thickness (10) is �l0 ffi 7. Some computational points are shown.

Fig. 6. Evolution map for 2-phase systems with �g ¼ 0:1. The non-dimensionalinterface thickness (10) is �l0 ffi 7.

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c-phase and a part of a-phase are consumed. Then, slow thinningand fragmentation the central a-layer follows until it disappears.A thick b-layer remains in the final equilibrium configuration.

The maps of transformations for 5-layers system (such as theone shown in Fig. 7) with stable intermediate phase, are shownin Figs. 8 and 9, for �g ¼ 0 and �g ¼ 0:2. Since the intermediate b-phase has a lower energy level than both a-phase and c-phase,growth of the intermediate phase occurs until all the smaller-vol-ume (in our case c-phase) is consumed. We define four differenttypes of evolution.

Type I: Slow fragmentation of initially thin c-layers is overtakenby fast c-to-b transformation of the resulting particles. b-particlesin a-matrix remain as the final equilibrium configuration.

Type II: Fast b-growth overtakes slow fragmentation and disap-pearance of c-layers. This is followed by fast coalescence of b-par-ticles into a new, thick b-layer.

Type III: Fast uniform growth of b-phase consumes c-layers andtwo thick b-layers will remain in the final configuration of thesystem.

Type IV: Shown in Fig. 7 and discussed above.For larger values of �g and very thin c-layers, there is a small

homogenization region (the lower left corner of Fig. 9). The homo-geneous phase is a (not exactly at the minimum). The system isconstrained by mass conservation. A homogeneous b-phase wouldbe the minimum energy configuration for an unconstrained sys-tems. In the constrained system, b-phase first consumes c-layers,then disappears into a, so that the final uniform phase has theaverage composition of the system.

The initial configuration, consisting of a and c layers is neverstable.

5. Systems with meta-stable intermediate phase

A meta-stable intermediate phase produces a more complexbehavior than the stable one. The evolution is often characterizedby partial growth of the intermediate phase, which is arrested be-fore the other phase is consumed. Moreover, the intermediatephase itself can subsequently be consumed by another phase. Con-sequently, the final configurations and paths are more complexthan in the case of a stable intermediate phase.

0=t310=t

3101.3 ×=t 3105×=t

3107.5 ×=t 3108×=t

Fig. 7. Evolution (type IV) of a system with stable intermediate phase. Initially �dc ¼ 15, �da ¼ 20 and �g ¼ 0:2. Color bar shows the composition. The FE mesh consists of 30,744triangular elements. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 8. Evolution map for the system with stable intermediate phase; �g ¼ 0. Thenon-dimensional interface thickness between b and c (10) is �l0 ffi 3:2.

Fig. 9. Evolution map for the system with stable intermediate phase; �g ¼ 0:2. Thenon-dimensional interface thickness between b and c (10) is �l0 ffi 3:2.

1034 M.A. Zaeem, S.Dj. Mesarovic / Computational Materials Science 50 (2011) 1030–1036

An example of evolution of systems with meta-stable interme-diate phase is shown in Fig. 10 (type IV). Initially, b-phase growsfast to produce very thin b-films between a- and c-layers. Sinceb-phase is meta-stable, the growth will stop. This appears to beeither a meta-stable or neutrally stable equilibrium. Perturbed bynumerical errors, the system slowly evolves; first by necking, then

pinching of the central a-layer. b-phase then quickly consumes thea-layer, only to be (as quickly) consumed by the growing c-phase.

The evolution maps for 5-layer systems with meta-stable inter-mediate phase are shown in Figs. 11 and 12, for �g ¼ 0 and �g ¼ 0:2.Owing to its high energy, b-phase growth is limited and typicallyconsumes the thinner layer (in our case, c-phase). In addition to

0=t 3105.0 ×=t

4568 01.5 ×=t 450 01.6 ×=t

452 01.6 ×=t 454 01.6 ×=t

Fig. 10. Evolution (type IV) of a system with meta-stable intermediate phase. Initially �dc ¼ 15, �da ¼ 10 and �g ¼ 0:2. Color bar shows the composition. The FE mesh consists of24,974 triangular elements. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 11. Evolution map for the system with meta-stable intermediate phase; �g ¼ 0.The non-dimensional interface thickness between b and c (10) is �l0 ffi 2:8.

Fig. 12. Evolution map for the system with meta-stable intermediate phase;�g ¼ 0:2. The non-dimensional interface thickness between b and c (10) is �l0 ffi 2:8.

M.A. Zaeem, S.Dj. Mesarovic / Computational Materials Science 50 (2011) 1030–1036 1035

homogenization to the composition (close to) a-phase (Fig. 12,lower left corner), explained in the previous section, we observefive different types of evolution.

Type I: Slow fragmentation of initially thin c-layers is overtakenby fast b-growth resulting in b-particles in a-matrix.

Type II: Starts as type I but this is followed by fast coalescence ofb-particles to form new, thick b-layer.

Type III: Stars as type I but b-growth is limited, so that the inter-mediate results are c-particles surrounded by a thin b-films in a-

matrix. This is followed by fast coalescence of particles to make athick c-layer surrounded by thin b-films.

Type IV: Shown in Fig. 10 and discussed above.Type V: Fast partial uniform b-growth, with final configuration

similar to picture in the top right corner of Fig. 10.As in the case of stable intermediate phase, the initial configu-

ration, consisting of a and c layers is never stable. However, thetype V behavior may include very little uniform b-growth, so thatthe equilibrium configuration may be acceptable in practicalapplications.

1036 M.A. Zaeem, S.Dj. Mesarovic / Computational Materials Science 50 (2011) 1030–1036

6. Conclusions and discussion

The morphological instabilities and post-instability evolution ofbinary multilayers, with and without intermediate phase (stableand meta-stable), is considered. Using the Galerkin finite elementformulation [44,45] for coupled Cahn–Hilliard – elasticity problem,maps of different evolution paths are developed in the parameterspace of relative thicknesses of initial phases.

Dimensional analysis reveals that, for isotropic materials, gov-erning equations depend on four non-dimensional parameters:Poisson ratio, scaled compositional strain parameter (8), and twoscaled energy density coefficients (5). The scaled compositionalstrain parameter determines the relative importance of elasticand chemical energy densities. We consider two values of thisparameter. The two energy density coefficients determine the nat-ure of the system, i.e., whether it has an intermediate phase or not,and the stability of the intermediate phase. We consider threecharacteristic combinations of these parameters, correspondingto: two-phase system, stable and meta-stable intermediate phase.

The systems exhibit rich evolution behavior. Depending on theinitial configuration (which determines the mass conservationcondition), the final equilibria vary, but even greater variety is ob-served in evolution paths. The paths may consist of multiple evolu-tion steps, which may proceed at different rates. In this report, wehave characterized the rates only qualitatively, as fast or slow, rel-ative to the other steps in the evolution. The rates vary in each re-gion of the evolution map.

Except for few special circumstances, the instabilities are toperturbations non-homogeneous in the film plane. Post-instabilityevolution is essentially two-dimensional and cannot be reduced tothe one-dimensional model such as the one considered in[40,45,50].

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