pinch-off maps for the design of morphologically stable multilayer thin films with immiscible phases

Acta Materialia 53 (2005) 5089–5099

www.actamat-journals.com

Pinch-off maps for the design of morphologically stablemultilayer thin films with immiscible phases

Jue Wang, Peter M. Anderson *

Department of Materials Science and Engineering, Ohio State University, 2041 College Road, Columbus, OH 43210-1179, United States

Received 8 March 2005; received in revised form 22 June 2005; accepted 20 July 2005Available online 15 September 2005

Abstract

Design maps are developed for the time to pinch-off via grain boundary grooving in multilayer thin films consisting of two immis-cible phases. Variables include the (isotropic) interfacial/grain boundary energies, interfacial diffusivities, initial grain aspect ratio,and imposed in-plane strain rate. The maps are based on a new 2D analysis of grooving in multilayer thin films that extends earlierwork by Thouless [Thouless MD. Acta Metall Mater 1993;41:1057] for single phase films. A modified Mullins [Mullins WW, J ApplPhys 1957;28:333] parameter eBmulti and time scale smulti are found to control grooving kinetics, with initial groove depth scaling as t1/4,t3/4, or t, depending on the magnitude of in-plane strain rate and geometry. The maps predict that nano-scale films may or may notbe more unstable to pinch-off at elevated temperature compared to micro-scale counterparts, depending on morphology, interfacial/grain boundary energies, and in-plane strain rate.� 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Multilayer thin films; Creep; Interface migration

1. Introduction

Several experimental investigations have documentedpinch-off of individual layers during heating of multi-layer thin films. This process involves thermal groovingalong grain boundaries in polycrystalline layers, so as toreduce system grain boundary energy. Pinch-off prohib-its measurement of steady-state zero creep stress in free-standing Co/Cu and Fe/Cu multilayer thin films withbilayer period K � 1 lm [1,2]. For Ni(Al)/Ni3Al multi-layer thin films with K = 40–240 nm, the process is quiteinhomogeneous, with pinch-off occurring among iso-lated groups of grain boundaries within a columnarmorphology [3,4]. Here, pinch-off occurs more rapidlyfor Æ111æ versus Æ001æ textured films, and at smaller Kwhere interfaces are expected to be more coherent. A

1359-6454/$30.00 � 2005 Acta Materialia Inc. Published by Elsevier Ltd. A

doi:10.1016/j.actamat.2005.07.020

* Corresponding author. Tel.: +101 614 292 1537.E-mail address: [email protected] (P.M. Anderson).

survey of pinch-off in various metal/metal systems re-veals that the larger melting temperature (Tm) phase isusually less stable, since grain boundary energy scaleswith Tm in those cases [5]. Stable multilayer thin filmssuch as Ag/Fe and Ag/Ni with K � 1 lm [6] and Cu/Nb with K � 75 nm [7,8] have been reported. In the lat-ter example, Cu and Nb grain boundaries migrate toform columnar, zigzag arrangements of equi-angled tri-ple junctions that are stable, despite a large (400 nm)grain size. Although thermal grooving clearly has detri-mental effects, it aids stagnation of coarsening in thinfilms, via grain boundary trapping within grooves[9,1,10]. For thin films on substrates, thermal groovingis predicted to generate large, crack-like stress fields thatmay serve to inject dislocations into the film [11].

The theory of thermal grooving has been developedextensively, beginning with the classic 2D treatment byMullins [12,13] of an isolated grain boundary that termi-nates at a free surfacewith isotropic (orientation-indepen-dent) energy. The groove assumes a time-independent

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β

α

β

∞ ∞. .

x

y

α

β midplane

α midplane

h (x,t)β

s

L(t)/2

Jgb(y,t)b

Jgb(y,t)a

Jint(y,t)a

Jint(y,t)b Λ(t)/2

hB(t)

B

B′

B′′

hA(t)C

θB

h (x,t)α

εε

a

b

Fig. 1. (a) 2D model of a multilayer thin film with aligned grains and(b) a unit cell thereof.

5090 J. Wang, P.M. Anderson / Acta Materialia 53 (2005) 5089–5099

shape with linear dimensions � t1/2 for evaporation/con-densation, t1/3 for bulk diffusion transport, and t1/4 forsurface diffusion transport. The latter is expected to dom-inate when Dsa � Dvk and I0k

2, where k is the transportdistance, a is an atomic dimension, I0 is the condensingflux in volume/area Æ t, and Ds, Dv, are surface, volumediffusion coefficients [14]. Indeed, a t1/4 dependence hasbeen observed, originally with interference microscopyin Cu at 930 �C in H2 [15,14] and recently with atomicforce microscopy in lm-scale Cu/Agmultilayer thin filmsannealed in Ar at 700–900 K [16] and in polycrystallinealumina at 1798 K [17].

Significant extensions to the original Mullins analysisprovide insight for internal grooving in multilayer thinfilms. First, the small slope approximation in theoriginal Mullins work and several subsequent analyses[14] is in excellent agreement with numerical analyses,provided the equilibrium grooving slope m 6 0.2[18,19]. In binary alloys, a t1/4 dependence is predicted,but the grooving shape can be asymmetric and faceteddue to variation in composition and thus energy alongthe surface [20,21]. An applied tensile (compressive)stress normal to a boundary aids (suppresses) groovingvia flow of matter into (out of) the boundary. This pro-cess superimposes a t3/4 dependence of groove depthonto the Mullins t1/4 dependence mentioned earlier, pro-vided the flux into the grain boundary is constant with t

[22–24]. For finite grain size L, the interaction betweenadjacent boundaries introduces a time scale s � L4 overwhich the Mullins t1/4 dependence decays to a steady-state value [22]. Finally, analytic [25,26] and detailednumerical analyses [27] suggest that films will firstperforate at pits along triple points rather than alonggrain boundary grooves. The latter analysis predictsthe depth of isolated pits �t1/4, with pit depth to groovedepth ratios varying from 1.5 to 2.5.

The aim of this paper is to produce design maps forthe pinch-off of individual polycrystalline layers in a/bmultilayer thin films with immiscible, single componentphases. Motivated by observed t1/4 groove depth depen-dence in Cu/Ag films mentioned earlier [16], the analysisadopts a 2D geometry in which interfacial transportdominates over bulk transport, akin to surface transportdominating over bulk transport in earlier free-surfaceanalyses. Detailed solutions for the interface profileand normal stress across grain boundaries and interfacesare obtained as a function of the diffusivities, individuallayer thickness, in-plane grain size, equilibrium groovingangles, and remote in-plane creep rate of the film. Fromthis, maps are predicted for time to pinch-off as a func-tion of dimensionless collections of system parameters,contrasting films with micron-scale versus nm-scale bi-layer period. The results combine t1/4, t3/4, t, and expo-nential groove depth dependences cited in earlier work,and introduce an effective diffusivity for the multilayer.Similar to equilibrium 3D calculations by Josell [1,28],

needle-like grain morphologies are predicted to be morestable to pinch-off than one with pancake-like ones.

The organization of the manuscript is to define thegoverning equations and geometries, initial and bound-ary conditions, and solution method in the followingsection. The results section to follow presents evolutionof groove position, interface profiles and interface stressassuming diffusivity parameters representative of Cu inthe range 650–680 K. The section concludes with apinch-off map for multilayer systems with immisciblea/b phases and discussion of 2D versus 3D geometries.The final section describes how the results can be usedto interpret experimental grooving data and contrastspredictions for micron-scale versus nm-scale multilayerthin films.

2. Model development

This section defines the geometries and associatedmathematical models for two 2D multilayer thin filmmorphologies shown in Figs. 1 and 2. Here, phases aand b alternate with bilayer period K. Each phase is as-sumed to be single component and immiscible, with aconsisting of pure a atoms and b consisting of pure batoms. Grains are assumed to be rectangular initiallywith aspect ratio (ha/L)0 and (hb/L)0 for the respectivephases. Fig. 1 shows a columnar structure with grainsthat are aligned vertically, compared to a staggered grainmorphology in Fig. 2. An in-plane strain rate _e1 at ele-vated temperature extends the film along the x-direction.During the process, initially flat interfaces may becomenonplanar via grooving, thereby reducing system grainboundary energy. For the aligned morphology, the en-ergy reduction occurs via a decrease in area of b grainboundaries, which are assumed to have a larger energy

s

Jgb(y,t)b

Jgb(y,t)a

Jint(y,t)a

Jint(y,t)b

αα midplane

β

α

β

∞.

∞.

hB(t)

B

A

y

B′

A′

β midplaneL(t)/2

h (x,t)β

h (x,t)

β

hA(t)

x

θB

θA

ε ε

a

b

Fig. 2. (a) 2D model of a multilayer thin film with staggered grains and(b) a unit cell thereof.

J. Wang, P.M. Anderson / Acta Materialia 53 (2005) 5089–5099 5091

per area than a grain boundaries. For the staggeredmorphology, the energy reduction occurs via a decreasein both a and b grain boundary area.

The proposed mathematical models assume that evo-lution may be studied via periodic cells, shown for thealigned and staggered cases in Figs. 1(b) and 2(b),respectively. Symmetry boundary conditions on eachcell face permit a/b interfaces to become non-planar,but a and b grain boundaries remain planar and vertical.Voids are assumed to not form along grain boundaries,interfaces, or in grain interiors. Thus, each cell is re-quired to change shape in a self-similar manner to themacroscopic film.

Deformation at elevated temperature is assumed tobe diffusional transport along a/b interfaces and a andb grain boundaries. Thus, elastic deformation and bulkdiffusion in grain interiors are neglected. As a conse-quence, grains elongate along the x-direction via uni-form deposition of matter along grain boundaries.Further, the immiscible nature of the species imposesthe restriction that only a atoms deposit on a grainboundaries and b atoms deposit along b grain bound-aries. The section to follow develops a mathematicalframework for creep of multilayer thin films, based onthe assumptions outlined here.

2.1. Development of governing equations

The applied in-plane strain rate imposes relations forthe flux of atoms along the interface and grain bound-aries. Conservation of matter requires that the rates ofchange in thicknesses ha and hb are related to the localflux J a

int [atoms/unit length Æ t] of a atoms and corre-sponding flux Jb

int of b atoms along the interface,

_haðx; tÞ � �Xa oJaint

ox; _hbðx; tÞ � �Xb oJ

bint

ox; ð1a; bÞ

where Xa and Xb are the atomic volumes of a and batoms, respectively. Eqs. (1a,b) are valid only in the limitof small interfacial slope; otherwise, x must be replacedby a local coordinate s along the interface and _h must bereplaced by velocity normal to the interface (e.g. [14]). Inthe absence of any internal void growth, macroscopicconservation of volume requires that the rate of changeof bilayer period is

_KðtÞ=2 � _haðx; tÞ þ _hbðx; tÞ ¼ �_e1KðtÞ=2� �_e1ðhaðx; tÞ þ hbðx; tÞÞ; ð2Þ

where K is independent of x due to the periodic cellstructure. Eqs. (1a,b) and (2) furnish the kinematiccondition

Xa oJaint

oxþ Xb oJ

bint

ox¼ _e1

K2. ð3Þ

The absence of internal void growth and deformationmechanisms other than diffusional flow along interfacesand grain boundaries imposes that the rate of grainboundary thickening must be uniform along the grainboundary and equal to _L. For the immiscible systemconsidered here, the thickening occurs via depositionof a atoms onto a grain boundaries and b atoms ontob grain boundaries, so that

_L ¼ _e1L ¼ �XaoJ a

gb

oy¼ �Xb

oJbgb

oy. ð4Þ

The Nernst–Einstein equation specifies that the atomflux Ji of species i along a grain boundary or interface isdriven by the gradient in the chemical potential li peratom

J i ¼ � cixi

Xi

dli

ds¼ � Didi

XikT

dli

ds. ð5Þ

Here, the product cixi of atomic fraction and mobility ofspecies i is expressed as Didi/kT, where Di is a tempera-ture-dependent diffusivity, di is the diffusion thickness ofthe grain boundary or interface, k is Boltzmann�s con-stant, and T is absolute temperature. As discussed byMullins for surfaces in local equilibrium [12,14],di ¼ vi0X

i, where vi0 is the nominal density of interfacialatoms of species i. Alternately, Rabkin et al. [29] inter-pret di as the thickness of a sublayer of species i residingin an interface of total thickness d, so that d = da + db inthis analysis.

The chemical potential li is dictated by a referencevalue l0 for a flat, unstressed grain boundary or inter-face plus contributions due to a curvature j and tensilestress r normal to it. Accordingly, the chemical potentialof species a along a stressed, planar a grain boundaryand stressed, curved a/b interface are, respectively,

lagbðyÞ ¼ la

gbð0Þ � ragbðyÞXa; ð6aÞ

laintðxÞ ¼ la

intð0Þ � rintðxÞXa þ /ajintðxÞcintXa; ð6bÞ


and the corresponding expressions for species b are

lbgbðyÞ ¼ lb

gbð0Þ � rbgbðyÞX

b; ð7aÞlbintðxÞ ¼ lb

intð0Þ � rintðxÞXb � /bjintðxÞcintXb. ð7bÞ

Here, jgb = 0, consistent with the periodic cell geome-tries in Figs. 1 and 2. jint is defined for an outward nor-mal to the b phase, so that jint < 0 in Fig. 1(b). Thefractions /i partition the contribution of the interfacestress, jintcint, to the chemical potential of each species.Rabkin et al. [29] show that for migrating interfaces be-tween dissimilar phases, the values of /i depend on thediffusivities, atomic volumes, and fraction, di/d, of inter-phase associated with each species. For ‘‘rigid’’ inter-faces, a variation in position requires the concertedremoval/addition of species a and b in a volume con-serving fashion, so that

/a þ /b ¼ 1. ð8ÞInterfaces for which (da/d)DaXa = (db/d)DbXb have thefeature that /a = /b = 1/2.

The governing differential equation for rint is ob-tained by using Eq. (5) to replace ðJ a

intÞ0 and ðJb

intÞ0 in

Eq. (3), where ( ) 0 denotes o( )/ox. Eqs. (6b) and (7b)are then used to express ðla

intÞ00 and ðlb

intÞ00 in terms of

r00int and j00

int (� h0000

b for small interfacial slope),

r00int ¼

_e1K=2þ ð/aDaint � /bDb

intÞcinth0000

b

Daint þDb

int

; ð9Þ

where the temperature-dependent interfacial diffusionparameter for species i is

Diint ¼

Diintd

iintX

i

kTði ¼ a; bÞ. ð10Þ

Governing equations for the normal stress on a and bgrain boundaries are obtained by using Eq. (5) to re-place oJ a

gb=oy and oJbgb=oy in Eq. (4). Eqs. (6a) and

(7a) are then used to express o2lagb=oy

2 and o2lbgb=oy

2

in terms of o2ragb=oy

2 and o2rb

gb=oy2

o2ragb

oy2¼ � _e1L

Dagb

ando2rb

gb

oy2¼ � _e1L

Dbgb

ð11Þ

with

Digb ¼

Digbd

igbX

i

kTði ¼ a; bÞ. ð12Þ

The governing equation for interface evolution is ob-tained from Eqs. (1b), (5), and (7b),

_hb ¼ �Dbintðr00

int þ /bcinth0000

b Þ. ð13Þ

Substitution of r00int from Eq. (9) furnishes

_hb ¼ �eBmultih0000

b � Dbint

Daint þDb

int

_e1K2;

eBmulti ¼ cintDa

intDbint

ðDaint þDb

intÞ. ð14Þ

Similar to the original analysis by Mullins [12] of ther-mal grooving of an isolated grain boundary intersectinga free surface, the governing differential equation ex-presses _hb in terms of h

0000

b . However, the multilayer anal-ysis introduces a new term due to the remote in-planestrain rate and a modified Mullins coefficient eBmulti

dependent on Daint, D

bint. The strain-rate-dependent term

in Eq. (14) arises from the mechanical coupling betweenphases. Thus, it is not present in the Mullins analysis norin the study by Thouless [22] of thermal grooving in asingle phase polycrystalline thin film under a remotestrain. In the limit Da

int=Dbint ! 1, Eq. (14) reduces

to that obtained by Mullins and Thouless, with cintreplacing cs.

Expressions for eBmulti have been obtained by Klingeret al. [30] for interface stability in the presence of an elec-tric field and by Rabkin et al. [29] for interfaces that arepartitioned into sublayers of thickness da and db and areperturbed with a sinusoidal shape. In the former case,the phases adjoining the interface are permitted to de-form elastically via shear moduli Ga, Gb so thateBmulti ¼ ðGaDa

int þ GbDbintÞ=ðGa þ GbÞ. In the latter case,

the same expression for eBmulti in Eq. (14) is obtained,provided the principal components of the interface stresstensor equal cint.

2.2. Boundary and initial conditions

The boundary conditions are derived from symmetryand equilibrium of the unit cells. For the alignedgeometry,

initial condition :

hbðx; 0Þ ¼ hbð0Þ; ð15aÞat point B :

h0bð0; tÞ ¼ tan hB � mB; ð15bÞ2J a

intð0; tÞ ¼ �J agbðhB; tÞ; 2Jb

intð0; tÞ ¼ JbgbðhB; tÞ; ð15c; dÞ

laintð0; tÞ ¼ la

gbðhB; tÞ; lbintð0; tÞ ¼ lb

gbðhB; tÞ; ð15e; fÞat point B0 :

Jbgbð0; tÞ ¼ 0; ð15gÞ

at point B00 :

J agbðK=2; tÞ ¼ 0; ð15hÞ

at point C :

h0bðL=2; tÞ ¼ 0; ð15iÞJ aintðL=2; tÞ ¼ 0; Jb

intðL=2; tÞ ¼ 0; ð15jÞavg normal stress :Z L

0

rintðx; tÞdx ¼ 0. ð15lÞ


The initial condition imposes that the interface is flat ini-tially at y = hb(0). The conditions at node B in Fig. 1 en-sure an equilibrium grooving angle hB, conservation ofspecies a and b, and chemical equilibrium of species aand b. hB is determined by ensuring that at node B,the vertical force generated by an imbalance in the a,b grain boundary energies cagb, c

bgb is balanced by the pro-

jected vertical force from interfacial energy cint [31]

hB ¼ sin�1cbgb � cagb2cint

!; mB ¼ tan hB ðalignedÞ.

ð16a; bÞThe conditions at points B 0, B00, and C are imposed bythe symmetry of the cell and the last condition ensureszero average stress normal to the interface.

The boundary conditions for the staggered configu-ration are similar, but reflect two nodes A and B (seeFig. 2) with corresponding equilibrium groovingangles

hA ¼ sin�1cagb2cint

� �; mA ¼ tan hA ðstaggeredÞ;

ð17a; bÞ

hB ¼ sin�1 cbgb2cint

!; mB ¼ tan hB ðstaggeredÞ.

ð17c; dÞ

Further, species b diffuses onto b grain boundaries via aflux through node B and species a diffuses onto a grainboundaries via a flux through node A.

initial condition :

hbðx; 0Þ ¼ hbð0Þ; ð18aÞat point B :

h0bð0; tÞ ¼ tan hB � mB; ð18bÞJ aintð0; tÞ ¼ 0; 2Jb

intð0; tÞ ¼ JbgbðhB; tÞ; ð18c; dÞ

lbintð0; tÞ ¼ lb

gbðhB; tÞ; ð18eÞat point A :

h0bðL=2; tÞ ¼ tan hA ¼ mA; ð18fÞ2J a

intðL=2; tÞ ¼ J agbðK=2� hA; tÞ; Jb

intðL=2; tÞ ¼ 0;

ð18g; hÞlaintðL=2; tÞ ¼ la

gbðK=2� hA; tÞ; ð18iÞat point B0 :

Jbgbð0; tÞ ¼ 0; ð18jÞ

at point A0 :

J agbðK=2; tÞ ¼ 0; ð18kÞ

avg normal stress :Z L

0

rintðx; tÞdx ¼ 0. ð18lÞ

2.3. Solution of the differential equations for interface

profile and normal stress

The approximate solution for the interface profilehb(x,t) is partitioned into three contributions, similarto Thouless [22],

hbðx; tÞ ¼ huniform thinningðtÞ þ hsteady stateðxÞ þ htransientðx; tÞ.ð19Þ

The first term reflects uniform thinning due to in-planestraining, so that _huniform thinningL ¼ �hB _L. For a constantin-plane strain rate _e1,

huniform thinningðtÞ ¼ hbð0Þe�_e1t. ð20Þ

The steady-state profile hsteady state signifies that for longtimes, the interface evolves to a shape that is indepen-dent of time and translates vertically by huniform thinning.The transient profile htransient is the time-dependentdeparture from the steady-state and uniform thinningdescriptions. Inserting the proposed forms (Eqs. (19)and (20)) into Eq. (14) yields

h0000

steady stateðxÞ ¼_e1cint

hBDb

int

� hADa

int

!; ð21aÞ

_htransientðx; tÞ ¼ �eBmultih0000

transient. ð21bÞ

The solution for the steady-state profile is obtained byintegrating Eq. (21a),

hsteady stateðxÞ ¼ A4

xL

� �4þ A3

xL

� �3þ A2

xL

� �2þ A1

xLþ A0.

ð22Þ

The transient profile is represented via a Fourier series

htransientðx; tÞ ¼X1n¼1

anLe�n4t=s cos2npxL

� �. ð23Þ

The corresponding distributions of normal stressacross the interface and grain boundaries are obtainedvia integration of Eqs. (9) and (11)

rintðx; tÞ ¼ B2

xL

� �2þ B1

xLþ B0 þ

X1n¼1

bne�n4t=s cos2pnxL

;

ð24aÞ

ragbðy; tÞ ¼ � _e1L

2Dagb

y2 þ C1y þ C0;

rbgbðy; tÞ ¼ � _e1L

2Dbgb

y2 þ E1y þ E0. ð24b; cÞ

The coefficients in Eqs. (22)–(24) are determined by sat-isfying the initial and boundary conditions in Eqs. (15)for the aligned case and Eqs. (18) for the staggered case.The time scale s in Eq. (23) is obtained by satisfying Eq.(21b). The solutions to follow use the dimensionlessstrain rate parameters


fA ¼ _e1L3hA24Da

intcint; fB ¼ _e1L3hB

24Dbintcint

. ð25a; bÞ

2.3.1. Aligned case

The resulting steady-state profile for the aligned caseis given by Eq. (22) with

A4 ¼ ðfB � fAÞL; A3 ¼ �2A4;

A2 ¼ A4 � A1; A1 ¼ mBL; A0 ¼ �A4

30� A1

6ð26aÞ

and the transient profile is given by Eq. (23) with

an ¼mB

n2p2þ 3

fB � fAn4p4

; s ¼ L4

16p4eBmulti

. ð26bÞ

The stress normal to the interface is provided by Eq. (24)with coefficients

B2 ¼ 6B0; B1 ¼ �6B0; B0 ¼ cintfA þ fB

L;

bn ¼ � 2cintL

Daint �Db

int

Daint þDb

int

mB þ 3fB � fAn2p2

� �. ð26cÞ

The stress normal to a grain boundaries is given by Eq.(24b) with coefficients

C1 ¼_e1LK2Da

gb

;

C0 ¼ � _e1L2Da

gb

hAhB þlagbð0Þ � la

intð0Þ

Xa þ rintð0; tÞ

� /acinth00bð0; tÞ ð26dÞ

and that normal to b grain boundaries is given by Eq.(24c) with coefficients

E1 ¼ 0;

E0 ¼_e1L

2Dbgb

h2B þlbgbð0Þ � lb

intð0Þ

Xbþ rintð0; tÞ þ /bcinth

00bð0; tÞ.

ð26eÞ

C1, E1 are determined from Eqs. (15h,g). C0, E0 aredetermined from Eqs. (15e,f).

2.3.2. Staggered case

The resulting steady-state profile for the staggeredcase is given by Eq. (22) with

A4 ¼ ðfB � fAÞL; A3 ¼ �2fBL;

A2 ¼fA þ 2fB

2þ mA � mB

� �L;

A1 ¼ mBL; A0 ¼ � 7fA240

þ fB30

þ 2mB þ mA

12

� �L

ð27aÞ

and the transient profile is given by Eq. (23) with

an ¼mB � mA cos np

n2p2þ 3

fB � fA cos npn4p4

;

s ¼ L4

16p4eBmulti

. ð27bÞ

The stress normal to the interface is provided by Eq. (24)with coefficients

B2 ¼6cintfA þ fB

L;

B1 ¼� 6cintfBL; B0 ¼ � cint

2

fA � 2fBL

;

bn ¼� 2cintL

Daint �Db

int

Daint þDb

int

� mB � mA cos npþ 3fB � fA cos np

n2p2

� �.

ð27cÞ

The stress normal to a grain boundaries is given by Eq.(24b) with coefficients

C1 ¼_e1LK2Da

gb

;

C0 ¼ � _e1L2Da

gb

K2

� �2

� h2A

" #þlagbð0Þ � la

intð0Þ

Xa

þ rint

L2; t

� �� /acinth

00b

L2; t

� �ð27dÞ

and that normal to b grain boundaries is given by Eq.(24c) with coefficients

E1 ¼ 0;

E0 ¼_e1L

2Dbgb

h2B þlbgbð0Þ � lb

intð0Þ

Xbþ rintð0; tÞ þ /bcinth

00bð0; tÞ.

ð27eÞ

2.4. Limitations to the solution

The long time and transient parts of the solution out-lined in Sections 2.3.1 and 2.3.2 satisfy Eq. (14) approx-imately. In particular, hlong time = huniform thinning(t) +hsteady state(x) is an exact solution only if A0, A1, A2,A3, A4 in Eq. (22) are independent of x and t. However,they are functions of fA, fB, and L, which vary withtime. If this time-dependence is included, then

_hlong time ¼� _e1hB þ _e1mBLx2

L2� 1

6

� �þ Oð_e21Þ ðaligned caseÞ; ð28aÞ

_hlong time ¼� _e1hB þ _e1L ðmB � mAÞx2

L2� 2mB þ mA

12

� �þ Oð_e21Þ ðstaggered caseÞ. ð28bÞ

Thus, the approximate solution is accurate only when inEqs. (28a,b), the leading term is much larger than theremaining terms. This occurs for


j_e1j � 1 and hB=mBL � 1=6 ðaligned caseÞ. ð29aÞj_e1j � 1 and hB=ð2mB þ mAÞL � 1=12 ðstaggered caseÞ.

ð29bÞ

The transient solution is approximate also, in that it sat-isfies Eq. (21b) exactly only if an, L, and s are indepen-dent of x and t. However, these quantities also dependon fA, fB, and L and thus vary with time. The contribu-tion to _hb from the time dependence of these quantities isnegligible provided

j_e1jt � 1=4 ðaligned and staggered casesÞ. ð30ÞFinally, the approximation d( )/ds � d( )/dx is validwhen the slope h0b is small. This approximation is goodprovided m < 0.2 [17].

ln(d /L )

3. Results

3.1. Parameters

The material values adopted for the a and b phasesare reported in Table 1. Thus, a multilayer thin filmwith initial layer half-thicknesses ha(0) = hb(0) = 1 lmand in-plane grain size L = 1 lm is considered (seeFig. 1). An in-plane strain rate _e1 ¼ 10�6=s, interfacialenergy cint = 2 J/m2, and T = 600 K are used, withdiffusivities representative for Cu in the rangeT = 650–680 K, based on data in Chuang et al. [32]and Thouless [22]. The diffusivity for species a is twicethat for b, so that initial values of the strain rateparameters are 2fA(0) = fB(0) = 1.73 · 10�1 and the ini-tial time scale is s(0) = 4.0· 103 s. Grooving slopesmA, mB are consistent with cbgb � cagb ¼ 0.4 J=m2 for

an aligned configuration and cbgb ¼ cagb ¼ 0.4 J=m2 fora staggered configuration. Thus, there are equal drivingforces to pinch-off phases a and b in the staggered con-figuration, but phase b is preferred to pinch-off in thealigned configuration.

3.2. Evolution of groove position

The depth of the groove at node B, relative to the ini-tial interface position, is defined as

Table 1Parameters used for simulations unless noted otherwise

L0 (m) 10�6

cint (J/m2) 2

mA = mB 0.1T (K) 600s0 ¼ ðL4=16p4eBmultiÞ0 ðsÞ 4.0 · 103

/a = /b 0.5ha(0) = hb(0) (m) 10�6

ðDdÞaint ¼ 2ðDdÞbint ðm3=sÞ 2 · 10�22

Xa = Xb (m3/atom) 1 · 10�29

_e1 ð1=sÞ 10�6

2fA(0) = fB(0) 1.73 · 10�1

dBðtÞ � hbð0; 0Þ � hbð0; tÞ

¼ hbð0Þð1� e�_e1tÞ � A0 �X1n¼1

anLe�n4t=s. ð31Þ

Fig. 3 displays ln(dB/L0) versus ln(t) for the aligned casewith parameters indicated in Table 1 and a constant ap-plied strain rate, so that L ¼ L0e

_e1t. This solution wasobtained from Eq. (19), with incremental updating offA, fB, L, and s due to straining. For t � s0, dB/L�t1/4

is observed as reported by Mullins for grain boundarygrooving in the absence of in-plane strain. For t � s0and _e1t � 1, dB / t, which is indicative of groovingdue to uniform thinning from the imposed strain rate.Thus, s0 is the critical time scale below which pinchingoff is dominated by transport in the vicinity of the nodeand above which grooving is dominated by steady-statethinning of layers.

A numerical analysis of the Fourier terms in Eq. (31)indicates that for t� s,

dBðtÞ¼ hbð0Þð1� e�_e1tÞþ0.77mBðeBmultitÞ1=4

þ0.38 _e1L

cint

hBDb

int

� hADa

int

� �ðeBmultitÞ3=4 aligned;

0.38 _e1LhBcintD

bint

eBmultit� �3=4

þ _e1hAcintD

aint

eBmultit� �

staggered.

8><>:ð32Þ

Thus, the familiar t1/4 dependence observed by Mullins[12,13] is accompanied by several strain-rate dependentterms. The t3/4 dependence discussed by Thouless [22]and Genin [23,24] stems from the flux of matter intograin boundaries at node B, due to straining. The lineardependence in the staggered configuration stems fromflux into node A. For t� s,

dBðtÞ ¼ hbð0Þð1� e�_e1tÞ

þmBL6þ _e1L4

24cint130

hBDb

int

� hADa

int

� �aligned;

2mBþmA

12Lþ _e1L4

24cint130

hBDb

int

þ 78

hADa

int

� �staggered.

8><>:ð33Þ

-8

-6

-4

-2

0

-6 -2 2 6 10 14

0

ln( t [s])

d ~ t

τ0

Mullins regime regimeε.

B

d ~ t1/4B

B

Fig. 3. Groove depth dB at node B for an aligned geometry with_e1 ¼ 10�6=s and parameters given in Table 1.


Thus, the staggered case has a larger grooving depth, atleast for _e1 P 0. For the aligned case, a positive strainrate suppresses grooving into the layer with smallerh=Dint.

3.3. Evolution of the interface profile

Fig. 4 presents the interface profiles, hb(x,t) � hb(0), asa continuous function of x/L for discrete values of nor-malized time, t/s0. The results in Fig. 4(a) for the alignedcase show grooving into the b phase at node B while re-sults for the staggered case in Fig. 4(b) show groovinginto the a phase at node A also. For t < 0.5 s0, thealigned profile displays an upward ‘‘bump’’ that propa-gates over time from the grooving site (x = 0) to graincenters (x = ± L/2). The scenario for the staggeredgeometry is similar, except that opposing bumps fromB and A grooving sites propagate toward one another.For t > s0, the interface profile appears approximatelylinear for the parameters assumed in Table 1; such aprofile is possible only for mA = mB andfA,fB � mA,mB. The gradual displacement of the profilebelow the x-axis is due to thinning from a positive _e1.

3.4. Evolution of the normal stress across the interface

Fig. 5 presents the profiles, rint(x, t), of normal stressacross the interface for the aligned and staggered geom-etries, with parameters reported in Table 1. The resultsfor the aligned geometry (Fig. 5(a)) also reveal s0 as a

-0.035

-0.025

-0.015

-0.005

0.005

-0.5 -0.3 -0.1 0.1 0.3 0.5

t/τ0 = 0.001 , 0.005 , 0.01, 0.05 ,5,1,10

x/L

h /Lβ node Cnode B

x/L

h /Lβ

-0.04

-0.02

0

0.02

-0.5 -0.3 -0.1 0.1 0.3 0.5

t /τ0 = 0.0005, 0.05 ,0.3 ,0.5 ,1,3

node A

node B

a b

Fig. 4. Evolution of the interface profile with position and time for the(a) aligned and (b) staggered configurations. The vertical position ismeasured relative to the initial (flat) interface. Material and geometricparameters are provided in Table 1.

x/L

-0.25

-0.05

0.15

0.35

0.55

-0.5 -0.3 -0.1 0.1 0.3 0.5

t /τ0 = 0.001 , 0.005 ,0.5 ,2

σ int (MPa)

-0.4

0.

0.4

0.8

-0.5 -0.3 -0.1 0.1 0.3 0.5

t/τ0 = 0.001 , 0.005 ,1,2

σ int (MPa)

x/L

node C

node B node A

node BS

a b

Fig. 5. Normal stress across the interface as a function of position andtime for the (a) aligned and (b) staggered configurations. Material andgeometric parameters are provided in Table 1.

critical time scale. For t = 0.001s0, large tensile peaksoccur at S. This tension is generated ahead of the‘‘bump’’ or diffusional wedge of matter discussed in Sec-tion 3.3. At t = 0.005s0, the tensile peak propagates to-ward larger x and a secondary tensile peak forms atthe node. This latter peak is due to diffusional flow intothe grain boundaries to accommodate the positive _e1.For t > 0.01s0, the largest tensile stress is at node Band for t > 0.5s0, the entire stress profile is consistentwith diffusion of species a and b toward node B.

The results for the staggered case in Fig. 5(b) reflectthe same trend that initially, species b is transportedaway from node B to create a groove there. At longertime, species b is transported toward node B to accom-modate the positive _e1. In contrast, node A has a tensilerather than compressive stress at short time. This is dueto the smaller interfacial diffusivity of species b (Table1). The slower species (b) requires a compressive stressto drive transport away from node B and a tensile stressto drive transport toward node A. At larger time, thestress profile is consistent with transport of species b to-ward node B and species a toward node A. Node B has alarger tension in this regime due to the smaller diffusivityof species b.

3.5. Steady-state interface profiles

In the limit t� s0, steady-state interface profiles,Yss(x) = hb(x,t) � hb(0,t), develop as shown in Fig. 6.They are specified by Eq. (22), using parameters re-ported in Table 1. For the aligned case (Fig. 5(a)), theinterface roughness, R = Yss(L/2), increases withincreasing grooving slope mB (compare profiles 3 and5) and increasing difference, fB � fA, in strain rateparameters. For the staggered case (Fig. 5(b)), R in-creases with larger grooving slope (mA = mB) and strainrate parameters (fA,fB). The analytic results (Eq. (22)with coefficients from Eqs. (26a) and (27a)) give

Rss ¼ fB � fA þ mB

2

� �L aligned; ð32aÞ

x/L

0

0.02

0.04

0.06

-0.5 -0.3 -0.1 0.1 0.3 0.5

Case:1,2,3,4,5

0.1

0

0.3

-0.5 -0.3 -0.1 0.1 0.3 0.5

x/L

0.2

Case:1,2,3,4,5

Y /Lss Y /Lss

node C

node B

node A

node B

R

R

ss

ss

Case 1 2 3 4 5 ζ B-ζA -0.1 0 0.1 0.2 0.1 mB 0.1 0.1 0.1 0.1 0.2

Case 1 2 3 4 5 ζ A,ζ B 2.0,1.0 1.5,1.5 1.0,2.0 1.5,1.5 2.0,2.0

mA,mB 0.1,0.1 0.1,0.1 0.1,0.1 0.2,0.2 0.1,0.1

a b

Fig. 6. Steady state interface profiles Yss(x) = hb(x,t) � hb(0,t) for the(a) aligned and (b) staggered configurations. Material and geometricparameters are provided in Table 1 except for m and f values, whichare provided below each figure.


Rss ¼fB þ fA

16þ mA þ mB

2

� �L staggered. ð32bÞ

Thus, interfacial roughness in aligned structures isdependent on property differences, fB � fA andcbgb � cagb, while staggered structures are dependent onsums, fB, fA and mA + mB. In principle, the roughnessin aligned morphologies could be minimized by selectingphases and morphologies so that fB � fA and mB areopposite in sign. This could occur, for example, if thephase with larger grain boundary energy also has asufficiently large interfacial diffusivity. However, experi-mental observations indicate that aligned morphologiesmay be prone to break up into staggered geometries[7,8]. Thus, the staggered geometry may be morerelevant.

3.6. Pinch-off maps

Fig. 7 presents the normalized time, tpinch-off/s0, topinch-off phase b as a continuous function of hb(0)/L0mB. The latter parameter combines the initial grainshape hb(0)/L0 and equilibrium grooving slope mB atnode B. Baseline predictions corresponding to _e1 ¼ 0are shown for each geometry and results for _e1 > 0are displayed only for the staggered geometry, since thatcase offers the lower bound to pinch-off time. In the limit_e1 ¼ 0 and t � s0, the solution given by Eqs. (19), (20)and (22) indicates that the thickness of the b phase atnode B is hB � hbð0Þ þ A0 ð_e1 ¼ 0Þ. Thus, pinch-off inthis limit is avoided if hB > 0 or equivalently, if

( )tpinch-off

τ0

0

-1

1

2

3

0 0.2 0.4 0.6 0.8 1

= 0 .

aligned

log

= 0 .

10-2

h0L0 mB

x =

y =

yMullins

pancakelarge θ needle

Mullins pinch-off

strain pinch-off

x Mullins

increasing staggered

.

B small θB

staggered

10-1

ε

ε

εε0 = 10-3.

Fig. 7. Time to pinch-off b phase at node B, as a function of initialgrain aspect ratio h0/L0 and grooving slope mB, for the alignedgeometry with _e1 ¼ 0, and for the staggered geometry with mA = mB

and _e1 P 0. Thinner black lines for _e1 > 0 show numerical resultsbased on Eq. (19) and dashed curves show approximate results usingEq. (36). The small slope approximation (see discussion following Eq.(1)) adopted in this analysis requires that mA, mB < 0.2, based on [17].

hbð0ÞL0mB

>hbð0ÞL0mB

� �crit

to avoid pinch-off of b layer;

ð33Þ

where

hbð0ÞL0mB

� �crit

¼1=6 aligned

ðmA þ 2mBÞ=12mB staggered

�ð_e1 ¼ 0Þ.

ð34Þ

The curves for _e1 ¼ 0 in Fig. 7 show Eq. (33) graphi-cally for both the aligned and staggered configurationswhere for the latter, (mA + 2mB)/12 = 1/4 sincemA = mB. Physically, the aspect ratio of grain heightto grain width must exceed a critical value dependenton interfacial and grain boundary energies, in order tobe stable to pinch-off at zero imposed strain rate. IfEq. (33) is not met, thermal grooving alone withoutstraining will be sufficient for pinch-off.

Numerical results for _e1 > 0 show that when Eq. (33)is satisfied, positive values of _e1 will ultimately causepinch-off due to thinning of individual layers, with larger_e1 inducing more rapid pinch-off. The numerical resultswere obtained by setting hb(0,t) defined in Eq. (19) equalto 0.05hb(0), so that tpinch-off is the time for grooving to oc-cur through 95% of the original b layer thickness. Thiscondition furnishes an iterative equation for tpinch-off,since hb(0,t) depends on L(t), fB(t), fA(t) and s(t).

An analytic expression for time to pinch-off can bedeveloped for films that satisfyEq. (33) but have apositivevalue of _e1. Fig. 7 indicates that if hb(0)/L0mB equals thecritical value defined in Eq. (34), the time to pinch-off is

tcrit � 100.5s0 � 3.2s0. ð35ÞWhen hb(0)/L0mB exceeds the critical value, tpinch-off= tcrit + te, where te is the time for hb/LmB to decreasefrom the initial to critical value via thinning. te can beapproximated by 2_e1te ¼ ln½ðhb=LmBÞ0=ðhb=LmBÞcrit�,assuming layers thin uniformly. Thus,

tpinch�off � 3.2s0 þ1

2_e1ln

ðhb=LmBÞ0ðhb=LmBÞcrit

forðhb=LmBÞ0ðhb=LmBÞcrit

> 1. ð36Þ

The dashed curves in Fig. 7 show predictions of Eq. (36)with (hb/LmB)crit = 1/4 given by Eq. (34) with mA = mB,and s0 = 4.0 · 103 s given by Eq. (27b) using parametersin Table 1. The predictions of Eq. (36) capture thenumerical results.

3.7. Assessment for micro- versus nano-scale multilayer

thin films

This analysis indicates that decreasing bilayer periodfrom the micro- to nano-scale has various outcomes,depending on hb(0)/L0mB. For hb(0)/L0mB < (hb/LmB)crit,


films are inherently unstable with tpinch�off / s0 / L40,

where s0 is defined in Table 1. This regime is labeled‘‘Mullins pinch-off’’ in Fig. 7, since the time to pinch-off is dictated primarily by thermal grooving of isolatedgrain boundaries. In this regime, a decrease in scale fromL � 1 lm to 1 nm will induce a relatively instantaneouspinch-off, regardless of whether an in-plane strain rate isimposed, since s0 for a nano-scale film is �10�12 timesthat for a micron-scale film. For hb(0)/L0mB > (hb/LmB)crit, thinning via in-plane straining is required topinch-off layers. Accordingly, this regime is labeled‘‘strain pinch-off’’. Here, films are predicted to be stableto pinch-off if _e1 ¼ 0, regardless of micro- or nano- scal-ing, since the second term in Eq. (36) approaches infin-ity. If _e1 > 0, Eq. (36) applies with s0 becomingnegligible as bilayer period decreases to the nano-scale.From a practical standpoint, it is imperative thatnano-scale thin films be designed in the ‘‘strain pinch-off’’ regime, in order to prevent instantaneous pinch-off at elevated temperature.

An important additional feature is that interfacialstructure and thus interfacial energy may decrease as bi-layer period decreases to the nano-scale. For example,the interfacial energy in nano-scale c-Ni(Al)/c 0-Ni3Almultilayer thin films is estimated to be approximately10% of that for a micron-scale counterpart, due to acoherent versus semi-coherent interfacial structure fornano-scale versus micro-scale multilayer thin films[3,4]. Thus, m and therefore (h/L)crit may increase withdecreasing scale. Consequently, it may not be sufficientto synthesize nano-scale films with the same grain aspectratio as micron-scale films.

The magnitudes of interfacial and grain boundarystress also depend on multilayer scale. According toEq. (24) and the accompanying coefficients in Eqs.(26c) and (27c), the normal stress across the interfacescales as 1/L for short time (t� s0) when the ‘‘Mullinspinch-off’’ process dominates, to L2h_e1 at long time(t � s0) when the ‘‘strain pinch-off’’ process dominates.Thus, decreasing from the micro- to nano-scale increasesthe magnitude of normal interface stress in the Mullinspinch-off regime but dramatically decreases it in thestrain pinch-off regime. Eqs. (24a) and the accompany-ing coefficients in Eqs. (26d,e) and (27d,e) indicate thatthe normal stress across grain boundaries depends oncint/L and L2h_e1. Therefore, a transition from micronto nano-scale films is expected to increase the thresholdapplied stress needed for zero creep conditions and thesensitivity of _e1 to applied stress. Indeed, zero creepdata for multilayer thin films supports this [1,6].

3.8. 2D versus 3D grooving and pitting geometries

The results for equilibrium groove depth and criticalgrain aspect ratio in the present 2D analysis can be as-sessed in terms of existing equilibrium models for 3D

and pitting geometries. Srolovitz and Safran [26] ob-serve that a 2D single-phase film of cylindrically-cappedgrains will be stable to pinch-off if the original aspectratio

hbð0ÞL0

>2� 3 cos hþ cos3h

6sin3h� h

8for small h

� �. ð37Þ

The critical ratios cited in Eq. (34) range from h/6 forthe aligned case to h/4 for the staggered case withmA = mB � h and thus compare reasonably to the smallh limit in Eq. (37). However, results from 3D geometriessuggest that 2D approaches underestimate the criticalaspect ratio. In particular, Josell et al. [2] furnish an inte-gral expression (see Eq. 7 in [16] for a corrected version)for the critical aspect ratio in a multilayer film withgrains of square in-plane shape that are staggered rela-tive to grains in adjoining layers. For Co/Cu multilayerfilms with hNi = 31� and hAg = 24�, Josell et al. predict acritical aspect ratio of 0.72 [2], compared to 0.06–0.09obtained from Eq. (34) (staggered case). Genin et al.[27] note that pits, which are captured in 3D geometries,are approximately 1.5–2.5 deeper than correspondinggrain boundary grooves. Thus, the 2D results for(h/L)crit in Eq. (34) are smaller than those from 3Danalyses.

4. Conclusions

A 2D analysis of grain boundary grooving in poly-crystalline a/b multilayer thin films indicates thatnano-scale films may or may not be more unstable topinch-off at elevated temperature compared to micron-scale counterparts, depending on morphology, interfa-cial/grain boundary energies, and applied in-plane strainrate _e1. The analysis assumes that grooving occurs pri-marily via diffusional transport along grain boundariesand interfaces, compared to bulk diffusion or disloca-tion-based plasticity. A critical initial aspect ratio,hb(0)/L0, of columnar grain half-height to in-plane widthis identified in terms of _e1 and equilibrium groovingangles hA and hB at junctions where grain boundariesextend into the a and b phases, respectively. If _e1 ¼ 0,these critical values are (tanhB)/6 for a geometry withaligned columnar grains versus [(tanhA + 2tanhB)/12]for staggered columnar grains. If the grain aspect ratiois less than the appropriate critical value, pinch-off timeis predicted to scale as L4

0, so that nano-scale films pinch-off instantaneously compared to micro-scale counter-parts. If the grain aspect ratio is greater than the criticalvalue, both nano- and micro-scale films requireadditional, comparable times to pinch-off via thinningdue to a positive _e1. Overall, these conclusions areconsistent with the concept by Srolovitz and Saffran[23], Josell et al. [1,2,6], and Knoedler et al. [14] of


critical grain aspect ratios for grooving stability,although the 2D analysis here provides lower criticalvalues than 3D analyses.

The analysis also identifies a modified Mullinsparameter eBmulti that couples with _e1 to govern theinterface transport kinetics for thermal grooving. eBmulti

is the product divided by the sum of eBMullins for eachphase (see Eq. 14 and also Eq. 22 of [29]), with surfaceenergy and diffusivity replaced by interfacial values. Atshort times, groove depth scales as t1/4, t3/4, and t, withthe dominant term dictated by grain geometry and _e1.At long times, the quasi-static groove depth is dictatedby hA, hB, _e1, and columnar grain dimensions.

Acknowledgements

PMA and JW gratefully acknowledge the support ofthe Air Force Office of Scientific Metallic Materials Pro-gram (F49620-01-1-0092) and thank Sridhar Naray-anaswamy, Carl V. Thompson, and Suliman A. Dregiafor helpful discussions.

References

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[2] Josell D, Carter WC, Bonevich JE. Nanostruct Mater1999;12:387.

[3] Fain JP, Banerjee R, Josell D, Anderson PM, Fraser H, TymiakN, Gerberich WW. MRS symposium proceedings, vol. 581. War-rendale, PA: MRS; 2000. p. 603.

[4] Sperling EA, Anderson PM, Hay JL. J Mater Res 2004;19:3374.[5] Lewis AC, Josell D, Weihs TP. Scr Mater 2003;48:1079.[6] Josell D, Spaepen F. MRS symposium proceedings, vol.

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pinch-off maps for the design of morphologically stable multilayer thin films with immiscible phases

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