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Acta Materialia 61 (2013) 32–39

Formation of multiple stoichiometric phases in binary systemsby combined bulk and grain boundary diffusion: Experiments

and model

J. Svoboda a, F.D. Fischer b,⇑, W. Schillinger c

a Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zizkova 22, CZ-616 62 Brno, Czech Republicb Institute of Mechanics, Montanuniversitat Leoben, Franz-Josef-Str. 18, A-8700 Leoben, Austria

c Wieland-Werke AG, D-89070 Ulm, Germany

Received 20 June 2012; received in revised form 3 September 2012; accepted 4 September 2012Available online 19 October 2012

Abstract

The thermodynamic extremal principle has been used by the authors to treat the evolution of binary and multicomponent systemsunder the assumption that all phases are nearly stoichiometric. Up to now only bulk diffusion has been taken into account. The conceptis now extended to combined bulk and grain boundary diffusion possible in each newly formed phase. The grains are approximated bycylinders allowing interface diffusion along the top and bottom of the grains and grain boundary diffusion along the mantle with differentinterface/grain boundary diffusion coefficients. A consistent analysis yields an effective diffusion coefficient taking into account thecombined interface/grain boundary and bulk diffusion of each individual component. The current concept is applied to the Cu–Sn couplewhich has been studied by a number of researchers. The results of simulations are compared with experiments at 200 �C on solid systemsreported in the literature as well as with our experiments at 250 �C with liquid Sn.� 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Bulk diffusion; Grain-boundary diffusion; Intermetallic phases; Thermodynamic modeling; Kirkendall effect

1. Introduction

If two different alloys come into contact, e.g. by solder-ing or welding, several intermetallic layers usually form andgrow at elevated temperatures in the region of the originalinterface. Interdiffusion is responsible for this phenome-non, and a significantly pronounced Kirkendall effect canresult in the formation of pores and cracking. In addition,splitting of the Kirkendall plane has been observed andstudied theoretically in such systems—see e.g. the extensivework by van Loo and co-workers [1–3], who introduced anaverage interdiffusion coefficient for each phase and withthe ratio of the intrinsic diffusion coefficients of the speciesin each phase being a fitting constant. This approach can

1359-6454/$36.00 � 2012 Acta Materialia Inc. Published by Elsevier Ltd. All

http://dx.doi.org/10.1016/j.actamat.2012.09.008

⇑ Corresponding author. Tel.: +43 38424024001; fax: +43 384246048.E-mail address: mechanik@unileoben.ac.at (F.D. Fischer).

thus treat the parabolic growth of layers controlled by dif-fusion as well as the splitting and motion of the Kirkendallplanes. In addition, other theoretical concepts have beenused, such as the quasi-steady-state concept proposed byMei et al. [4] for sharp interfaces and by Danielewskiet al. [5] for thick interfaces. A strong impact on modellingthe Kirkendall effect has come also from the multiphase-field approach—see e.g. Steinbach and coworkers [6] andthe adaption to the current soldering problem proposedby Park and Arroyave [7,8].

Recently Svoboda et al. (e.g. [9,10]) introduced athermodynamic model based on the application of thethermodynamic extremal principle (TEP), using the fixedchemical composition of individual stoichiometric phases,their molar volumes, their molar Gibbs energies and tracerdiffusion coefficients of all components in individual phasesas input parameters. The model simulates the growth of an

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J. Svoboda et al. / Acta Materialia 61 (2013) 32–39 33

optional number of layers in the system as well as themotion of the split Kirkendall plane. The model also pro-poses a realistic image of how the Kirkendall plane splits.This approach works with standard thermodynamicquantities such as Gibbs energies, interface energies anddiffusivities, and requires no fitting parameters. It evenallows the tracer diffusion coefficients to be determinedfrom measurements if these coefficients are unknown. Inco-herent interfaces are considered to have infinite mobilityand to act as ideal sources and sinks for vacancies (nodissipation occurs due to interface migration and due togeneration/annihilation of vacancies at the interface). Thebulk of the phases is taken as free of sources and sinksfor vacancies. With respect to bridging of the TEP to thephase field method, the reader is referred to Ref. [11].

In nearly all the modelling studies, only bulk diffusion isassumed. A typical example is a solid Cu–solid Sn couple, aswas studied in detail by Paul in his thesis [12] and by Paulet al. [13,14] and most recently by Kumar et al. [15]. How-ever, soldering systems exist in which the nucleation andgrowth of the intermetallic phases is extremely fast. Thephases have rather the microstructure of fine polycrystalsthan a single crystal, and grain boundary diffusion interactsstrongly with bulk diffusion. The effect of interacting bulkand grain boundary diffusion occurs predominantly in theg-phase Cu6Sn5. The g-phase shows a “scalloped” morphol-ogy (e.g. [7,8,12,16,17]), which can only be explained byintensive nucleation of the g-phase and cooperativediffusion through the bulk and along the grain boundaries.

The goal of this paper is to present a model in which themacroscopic diffusive fluxes of the components A, B insome of the newly formed phases between the parentphases consist of bulk and grain boundary contributions.The TEP, as used in Refs. [9,10], is used to analyze the indi-vidual contributions to the fluxes as well as the kinetics ofthe system.

2. The model

2.1. General mass balance and interface velocities

We briefly outline here the background to the problemas this has already been presented in Refs. [9,10]. There-fore, we avoid repeating corresponding figures and referthe reader to Refs. [9,10]. Details with respect to theapplied equations (kinematics, mass balance) can also betaken from Refs. [9,10].

Let us assume a binary system of components A and B

forming n stable stoichiometric phases Ari B1�ri with ri beingthe mole fraction of the component A in the phase i,0 6 ri 6 1, i = 1, . . . ,n, and ri increasing with i.

As a starting configuration we have a diffusion couple ofunit cross-section consisting of the phases Ar1

B1�r1at the

left side and Arn B1�rn at the right side of the interface withinert markers on it. We assume that during a negligibletime interval very small nuclei of all other phases areformed at the original interface and these grow into a

sequence of phases. All newly nucleated phases containsome markers originally deposited at the interface of thediffusion couple. The assumption of stoichiometry guaran-tees a constant chemical composition and a constant molarvolume Xi in each phase. As an approximation

Xi ¼ riXA þ ð1� riÞXB ð1Þcan be taken, where XA, XB are the molar volume of A, B,respectively.

We assume that all interfaces act as ideal sources andsinks for vacancies and are ideally mobile. In this casethe sources and sinks for vacancies in the bulk are not acti-vated [18]. This implies that both macroscopic diffusivefluxes jAi and jBi as well as the material velocity must beconstant in each region occupied by an individual phase.The reason for the existence of fluxes is due to the fact thata very small alteration of the mole fraction in each phasecauses large changes in the chemical potentials of bothcomponents. The treatment of this complicated task can,however, be avoided by application of the TEP. The mac-roscopic diffusive fluxes may consist of two contributions:flux through the bulk of grains and flux along the com-bined path of the grain boundaries and interfaces.

Let us analyze the behavior of an interface between thephases i and i + 1 where the macroscopic diffusive fluxesjAi, jBi and jAi+1, jBi+1 occur. The splitting of the diffusivefluxes into a bulk contribution and a grain boundary/inter-face contribution is treated in Section 2.2. The balancebetween the fluxes at each interface is coupled with boththe interface migration and the thickening or thinning(deposition or removal of new atomic layers) at the inter-face. The second effect causes a motion of the lattices ofthe individual phases relative to each other.

Let ui be the velocity of the interface between the phases i

and i + 1 relative to the lattice of the phase i, and vi+1 be thevelocity of the interface between the phases i and i + 1 rela-tive to the lattice of the phase i + 1. The mass conservationfor both the A and B components leads to the equations:

uiri

Xi� viþ1riþ1

Xiþ1

¼ jAi � jAiþ1; i ¼ 1; . . . ; n� 1; ð2Þ

uið1� riÞXi

� viþ1ð1� riþ1ÞXiþ1

¼ jBi � jBiþ1; i ¼ 1; . . . ; n� 1:

ð3ÞThe velocities ui and vi can be calculated from Eqs. (2)

and (3) as:

ui ¼Xi

ri � riþ1

½ðjAi � jAiþ1Þð1� riþ1Þ � ðjBi � jBiþ1Þriþ1�;

i ¼ 1; . . . ; n� 1; ð4Þ

viþ1 ¼Xiþ1

ri � riþ1

½ðjAi � jAiþ1Þð1� riÞ � ðjBi � jBiþ1Þri�;

i ¼ 1; . . . ; n� 1: ð5Þ

We consider a closed system with no deposition of mat-ter at the surface of the system, yielding the boundaryconditions:

Fig. 1. Sketch of a scalloped grain with diffusive fluxes (k = A, B) in theinterior (bulk) of the grain (phase i) and along the grain surface. The layerswith thickness di, di�1, dmi/2 are assigned to one grain “unit cell”. Thethickness of phase i is di.

34 J. Svoboda et al. / Acta Materialia 61 (2013) 32–39

jA1 ¼ 0; jB1 ¼ 0; jAn ¼ 0; jBn ¼ 0; v1 ¼ 0

and un ¼ 0: ð6Þ

Then the rate of change of the thickness _di of the regionof phase i is given by:

_di ¼ ui � vi; i ¼ 1; . . . ; n: ð7ÞWith Eqs. (4)–(6), _di can be expressed by means of the mac-roscopic diffusive fluxes jAi and jBi, i = 2, . . . ,n � 1.

If we assume that the left end of the specimen is situatedin the origin of a spatially fixed coordinate system, one candetermine the velocity wi of the lattice of the phase i from:

wiþ1 ¼ wi þ ui � viþ1; i ¼ 1; . . . ; n� 1; ð8Þand with w1 = 0 as:

wiþ1 ¼Xi

k¼1

ðuk � vkþ1Þ; i ¼ 1; . . . ; n� 1: ð9Þ

The velocity Vi of the interface i between the phases i

and i + 1 can be calculated from:

V i ¼Xi

k¼1

_dk ¼Xi

k¼1

ðuk � vkÞ; i ¼ 1; . . . ; n� 1: ð10Þ

The position Xi of the interface i at time t is given by theintegration of Eq. (11) with respect to time as:

X i ¼ X 0i þZ t

0

V idt; i ¼ 1; . . . ; n� 1; ð11Þ

where X0i is the position of the interface at time t = 0.Let us assume that, at time t = 0, the position x0k of the

Kirkendall plane k and its actual correspondence to aphase l are known. The corresponding phase index l mustbe determined at any time from the actual positions ofthe interfaces Xi and the actual position of the Kirkendallplane xk. Then, the actual velocity of the Kirkendall planek is wl, and the position of the Kirkendall plane k at time t

is given by:

xk ¼ x0k þZ t

0

wldt: ð12Þ

2.2. Splitting of diffusive fluxes

The relevance of grain boundary diffusion in the contextof the Kirkendall effect was already shown by Thorntonand coworkers [19,20] through application of the phase-field method.

We now introduce the splitting of the macroscopic diffu-sive fluxes jAi, jBi into bulk and grain boundary/interfacecontributions specially adapted for an intermetallic layer(with the label “i”) consisting of columnar grain assketched in Fig. 1 The scallops are approximated by cylin-ders of a height identical to the layer thickness di and radiusqi; one cylinder represents a unit cell. So far, the model issimilar to that used in surface science (e.g. [21]). However,in our case we consider different flow paths and split the

macroscopic total fluxes jAi, jBi into macroscopic fluxesJ b

Ai; JbBi through the bulk of the grain in the axial direction

and in macroscopic diffusive fluxes J gbAi and J gb

Bi along thegrain boundary layer of the unit cell represented by layersof finite thicknesses di on the top, dmi/2 on the mantle anddi�1 on the foot of the cylinder (di is the thickness of theinterface i) (see Fig. 1). Consequently, the macroscopic dif-fusive fluxes jki are given by the sum:

jki ¼ J bki þ J gb

ki ; k ¼ A;B: ð13ÞThe fluxes along the boundary layer are correspondingly

denoted as jkti at the top, jkmi at the mantle and jkfi at thefoot for k = A, B. Mass conservation yields for jkti, jkmi, jkfi:

jktið~qÞ ¼ J gbki ~q=ð2diÞ for 0 6 ~q 6 qi; ð14Þ

jkmi ¼ J gbki ~q=ðdmiÞ at ~q ¼ qi; ð15Þ

jkfið~qÞ ¼ J gbki ~q=ð2di�1Þ for 0 6 ~q 6 qi: ð16Þ

Different concepts exist in the literature to describe theinteraction of bulk diffusion, grain boundary diffusionand finally surface segregation, e.g. a modified Darkenapproach, the so-called “Darken–du Plessis” concept,developed by du Plessis et al. (e.g. [22,23]) and a recentapplication [21] with the concentration rate directly relatedto the gradient of the chemical potential. Contrarily, weprefer to avoid some rather artificial assumptions and willuse the TEP to find the evolution equations for combined

J. Svoboda et al. / Acta Materialia 61 (2013) 32–39 35

bulk and grain boundary/interface diffusion. A first steptowards this goal has already been done by Kozeschniket al. [24], who have described the evolution of precipitatesat grain boundaries. The goal of the current paper is todescribe the kinetics and the Kirkendall effect in a diffusioncouple, where the individual stoichiometric phases maygrow via combined bulk and grain boundary/interfacediffusion. The kinetics of the system is unambigously deter-mined by the diffusive fluxes J b

Ai; JgbAi and J b

Bi; JgbBi ; i ¼ 2; . . . ;

n� 1, which represent independent kinetic variables. Theactual values of these fluxes are determined by applicationof the TEP similar to that described in Refs. [9,10].

3. Total Gibbs energy and its rate

The total Gibbs energy per unit area

G ¼Xn

i¼1

gi

Xiþ cmi

qi

� �di þ

Xn�1

i¼1

ctfi ð17Þ

is given by the sum of the chemical Gibbs energy (the molarGibbs energies gi and the molar volumes Xi are supposed tobe known and are fixed quantities at a given temperature),the grain boundary energy cmi and the interface energy ctfi

representing a fixed quantity; we now introduce~gi ¼ gi=Xi þ cmi=qi with dimensions of J/m2. The rate ofthe total Gibbs energy can then be calculated by usingEqs. (4)–(7), (and) (14) as:

_G ¼Xn

i¼1

~gi

J bAi þ J gb

Ai � J bAiþ1 � J gb

Aiþ1

� �1�riþ1

ri�riþ1

� J bBi þ J gb

Bi � J bBiþ1 � J gb

Biþ1

� � riþ1

ri�riþ1

� J bAi�1 þ J gb

Ai�1 � J bAi � J gb

Ai

� �1�ri�1

ri�1�ri

þ J bBi�1 þ J gb

Bi�1 � J bBi � J gb

Bi

� �ri�1

ri�1�ri

8>>>>><>>>>>:

9>>>>>=>>>>>;: ð18Þ

The partial derivatives with respect to diffusive fluxes aregiven by

@ _G

@J bAi

¼ @ _G

@J gbAi

¼ ~gi1� riþ1

ri � riþ1

þ 1� ri�1

ri�1 � ri

� �

� ~gi�1

Xi�1

Xi

1� ri

ri�1 � ri� ~giþ1

Xiþ1

Xi

1� ri

ri � riþ1

¼ �F Ai;

i ¼ 2; . . . ; n� 1; ð19Þ

@ _G

@J bBi

¼ @ _G

@J gbBi

¼ �~giriþ1

ri � riþ1

þ ri�1

ri�1 � ri

� �

þ ~gi�1

Xi�1

Xi

ri

ri�1 � riþ ~giþ1

Xiþ1

Xi

ri

ri � riþ1

¼ �F Bi;

i ¼ 2; . . . ; n� 1: ð20Þ

The quantities FAi (FBi) can be understood as the drivingforces conjugated to the fluxes J b

Ai and J gbAi (J b

Bi and J gbBi ).

4. Total Gibbs energy dissipation

In this section we study the dissipation due to diffusionin the bulk and along the grain boundaries/interfaces andcalculate all quantities per unit area.

In phase i the dissipation in the top layer due to interfacediffusion follows as:

Qgbit ¼

2pdiXiRgTpq2

i

Z qi

0

ðjAtiÞ2

riDintAi

þ ðjBtiÞ2

ð1� riÞDintBi

!~qd~q; ð21Þ

where Rg is the gas constant, T is the absolute temperatureand Dint

ki is the tracer diffusion coefficient of component k inthe interface i. Inserting Eq. (15) into Eq. (22) yields thedissipation Qgb

it as:

Qgbit ¼ RgT Xi

q2i

8di

ðJ gbAi Þ

2

riDintAi

þ ðJ gbBi Þ

2

ð1� riÞDintBi

!: ð22Þ

In an analogous way one calculates Qgbif for the base

layer as:

Qgbif ¼ RgT Xi

q2i

8di�1

ðJ gbAi Þ

2

riDintAi�1

þ ðJ gbBi Þ

2

ð1� riÞDintBi�1

!: ð23Þ

For the dissipation Qgbim due to diffusion along the mantle

related to unit area, one finds:

Qgbim ¼ RgT Xi

qidi

dmi

ðJ gbAi Þ

2

riDgbAi

þ ðJ gbBi Þ

2

ð1� riÞDgbBi

!: ð24Þ

Note that Dgbki is the tracer diffusion coefficient of compo-

nent k in the mantle in phase i.The dissipation Qb

i due to bulk diffusion in phase i fol-lows with the tracer diffusion coefficient Db

ki in the bulkfor component k as:

Qbi ¼ RgTXidi

ðJ bAiÞ

2

riDbAi

þ ðJ bBiÞ

2

ð1� riÞDbBi

!: ð25Þ

The total dissipation Q in the system is then given by:

Q ¼Xn�1

i¼2

Qgbit þ Qgb

im þ Qgbif þ Qb

i

� �: ð26Þ

The partial derivatives of the total dissipation Q withrespect to the independent kinetic variables read for2 6 i 6 n � 1 as:

1

2

@Q

@J bAi

¼ RgTXidi

riDbAi

J bAi;

1

2

@Q

@J bBi

¼ RgTXidi

ð1� riÞDbBi

J bBi; ð27Þ

1

2

@Q

@J gbAi

¼ RgTXiqi

ri

qi

8diDintAi

þ di

dmiDgbAi

þ qi

8di�1DintAi�1

!J gb

Ai ; ð28Þ

1

2

@Q

@J gbBi

¼ RgTXiqi

1� ri

qi

8diDintBi

þ di

dmiDgbBi

þ qi

8di�1DintBi�1

!J gb

Bi : ð29Þ

5. Kinetic equations of the system

The evolution of the system is given according to theTEP (e.g. [9,10]), as:

1

2

@Q@J b

ki

¼ � @ _G@J b

ki

¼ F ki;1

2

@Q

@J gbki

¼ � @ _G

@J gbki

¼ F ki;

i ¼ 2; . . . ; n� 1; k ¼ A;B ð30Þ

36 J. Svoboda et al. / Acta Materialia 61 (2013) 32–39

Solving the equations with respect to the independentkinetic variables J b

Ai; JgbAi and J b

Bi; JgbBi ; i ¼ 2; . . . ; n� 1,

provides:

J bAi ¼

riDbAiF Ai

RgTXidi; J b

Bi ¼ð1� riÞDb

BiF Bi

RgT Xidi; ð31Þ

J gbAi ¼

riF Ai

RgTXiqi

qi

8diDintAi

þ di

dmiDgbAi

þ qi

8di�1iDintAi�1

!�1

; ð32Þ

J gbBi ¼

ð1� riÞF Bi

RgT Xiqi

qi

8diDintBi

þ di

dmiDgbBi

þ qi

8di�1iDintBi�1

!�1

: ð33Þ

The macroscopic diffusive fluxes jAi and jBi are thengiven by:

jAi ¼ J bAi þ J gb

Ai

¼ riF Ai

RgTXidiDb

Aiþdi

qi

qi

8diDintAi

þ di

dmiDgbAi

þ qi

8di�1DintAi�1

!�124

35;ð34Þ

jBi¼ J bBiþJ gb

Bi

¼ð1�riÞF Bi

RgTXidiDb

Biþdi

qi

qi

8diDintBi

þ di

dmiDgbBi

þ qi

8di�1DintBi�1

!�124

35:ð35Þ

The expressions in square brackets can be considered aseffective diffusion coefficients. As the values of di evolvewith time, the effective diffusion coefficients must be takenin an updated form. Moreover, the values of qi maydevelop according to any established grain growth law(e.g. [25]). The evolution of the system is then given by timeintegration of Eqs. (4)–(13) as presented in Refs. [9,10].

Here a short discussion on effective diffusion coefficientsseems to be useful. We can base this discussion on anelementary study by Belova and Murch [26], in which theyconsidered different arrangements of grains and grainboundaries in relation to the diffusion direction (parallel,series, grains in a square pattern). In our case we havecolumnar grains with both types of interfaces (see Fig. 1),namely a parallel arrangement of the mantle grain boundaryand a series arrangement of the top and foot interfaces (seealso Section 2.2). In addition, the fluxes in these interfacesare coupled by mass conservation, Eqs. (15)–(17). Thus farthe configuration dealt with in this paper cannot be appliedto one of the above-discussed arrangements. However, ifone assumes that the diffusion coefficients of the top andbase interfaces tend to infinity, Dint

ki ; Dintki�1 !1, k = A, B,

then the bulk and the mantle grain boundary form a parallelarrangement, which Hart [27] investigated in 1957, conclud-ing that the effective diffusion coefficient Deff follows as:

Deff ¼ fDgb þ ð1� f ÞDb;

where f is the volume fraction of the grain boundary. In theactual case f can be calculated as f = dmi/qi� 1. With theabove assumptions, we find for Deff from Eqs. (34) and (35):

Deff ;ki ¼dmi

qiDgb

ki þ Dbki;

representing a Hart-type relation.

6. Examples

Let us consider the Cu–Sn couple with the phasesequence Cu–Cu3Sn (e-phase)–Cu6Sn5 (g-phase)–Sn asstudied by Paul [12] and by Paul et al. [13,14], and mostrecently by Kumar et al. [15]. Although material data canbe found in those publications, we will collect them below.

6.1. Material data

The molar volumes of solid Cu and solid Sn are calcu-lated as the ratio of standard atomic weight and the density(http://en.wikipedia.org/Copper and Tin) as XCu =7.11 � 10�6 m3/mol, XSn = 16.31 � 10�6 m3/mol and forliquid Sn as XL

Sn ¼ 16:98 � 10�6 m3=mol. The molar volumesof the e- and g-phase are calculated as the ratio of weightedstandard atomic weights of Cu and Sn with the densities ofthe e- and g-phase. These quantities are taken from aresearch publication of the Materials Performance Groupas 8.90 g/cm3 and 8.28 g/cm3, respectively. The molar vol-umes then follow as Xe = 8.69 � 10�6 m3/mol, Xg =10.70 � 10�6 m3/mol (see also http://www.metallurgy.nist.gov/mechanical_properties/solder_paper.html). The Gibbsenergies gi for the pure phases as well as the e- and g-phasecan be taken from Shim et al. [28] and easily programmed.Here we would like to mention that a contribution due thehydrostatic stress state rH, i.e. �XirH, is not added to thechemical potentials. As Klinger and Rabkin have shownin several papers (e.g. [29,30]), diffusion of componentsfrom a liquid layer into the grain boundaries of a solidphase beneath the liquid layer may cause formation of awedge-like second phase, whose shape is controlled bythe local stress state. However, as one can see fromFig. 3, such wedges do not appear in the grain boundariesof the solid phase beneath the liquid (Sn) layer. Therefore,it is justified to omit the hydrostatic stress term in theGibbs energy for the system under investigation.

The following relations for the tracer diffusion coeffi-cients in the bulk are used

e-phase DbCue ¼ 1:8 � 10�8 expð�78:8 kJ mol�1=RgT Þ m2=s;

DbSne ¼ 7:9 � 10�10 expð�79:7 kJ mol�1=RgT Þ m2=s;

g-phase DbCug ¼ 6:2 � 10�8 expð�80:5 kJ mol�1=RgT Þ m2=s;

DbSng ¼ 5:9 � 10�7 expð�85:2 kJ mol�1=RgT Þ m2=s:

The above data are taken from Fig. 6 in Paul et al. [14].Here we would like to refer also to the possibility to calcu-late the a/m diffusion coefficients in the intermetallics bydensity functional theory [31].

Fig. 2. Sketch of the phase arrangement in a Cu–Sn diffusion couple asinvestigated by Paul et al. [8] and Kumar et al. [9]: (a) startingconfiguration; (b) calculated positions of the interfaces and the Kirkendallplanes after 225 h, marked by vertical lines along the length coordinateaxis (0 6 x 6 45 lm) in the reference configuration. Relations to theexperiments are listed in the text, Section 6.2.

J. Svoboda et al. / Acta Materialia 61 (2013) 32–39 37

The tracer (i.e. self) diffusion coefficients of Cu and Sn inliquid Sn are taken from Smithells et al. [32] for Sn andfrom [33] for Cu as:

DLCu ¼ 3:66 � 10�8 expð�11:6 kJ mol�1=RgT Þ m2=s;

DLSn ¼ 3:18 � 10�8 expð�11:6 kJ mol�1=RgT Þ m2=s:

Finally we report the grain boundary tracer diffusion coef-ficient for Cu from Ref. [34] as:

DgbCud ¼ 8:8 � 10�14 expð�120 kJ mol�1=RgT Þ m3=s

with d being the grain boundary thickness. Unfortunatelygrain boundary and interface diffusion coefficients for Cuand Sn in the e- and g-phase are not available.

6.2. Comparison with existing results

First let us compare two very recent investigations[14,15] on Cu–Sn solid diffusion couples performed at200 �C over �10 days. Paul and co-workers [14]report a thickness of (9 ± 3.6) lm for the e-phase and(16.4 ± 2.1) lm for the g-phase and Kumar et al. [15],(4.1 ± 1.8) lm for the e-phase and (7.9 ± 3.4) lm for theg-phase. Although the results of both groups differ by afactor of �5 in their kinetics (if a parabolic growth law isassumed), both groups confirmed their experimental resultswith their own simulations.

Motivated by this discrepancy we used our model withthe tracer diffusion coefficients from Section 6.1 and assum-ing no grain boundary/interface diffusion (qi!1 in theeffective diffusion coefficients in Eqs. (33), (34)). We foundnearly perfect agreement for the thickness of the e- andg-phase and the positions of the split Kirkendall planesrelative to the position of the interfaces with the resultsof Paul and coworkers (see Fig. 10 in Ref. [14]), and ourFig. 2, where the positions of the interfaces as well of theKirkendall planes are marked by vertical lines along thelength coordinate axis (0 6 x 6 45 lm) in the referenceconfiguration.

Any information about the absolute position of theinterfaces as well the shrinkage of the specimen cannot bededuced from Fig. 10 in Ref. [14]. Our results are referredto the spatially fixed left end of the specimen and show ashrinkage in the transformed region of nearly 8%. Thiseffect is discussed by Kumar et al. [15], whose results wecan reproduce only if the tracer diffusion coefficients inour simulations are reduced by a factor of 5. Moreover,Kumar et al. [15] report no splitting of the Kirkendallplane, which is obviously situated in their results in theg-phase.

6.3. Reaction of Cu–4 wt.% Sn with liquid Sn–1 wt.% Cu:

experiment and simulation

In the experiment a strip of Cu–4 wt.% Sn alloy 0.8 mmthick comes into contact at both surfaces with the liquidSn–1 wt.% Cu at 250 �C for 5 s. The liquid with such

chemical composition is saturated with respect to theneighboring g-phase. Thus, no flux of Cu into the liquidexists, which means that the g-phase does not dissolve inthe liquid phase. In this case the liquid behaves as a stoichi-ometric phase with a fixed chemical composition, which isin agreement with the preposition of the model. This situ-ation differs from the experimental settings in various otherworks (e.g. [35–37]), where a Cu-free solder was used,giving rise to an interaction between the g-phase and thesolder. The temperature in the strip is equilibrated over afraction of a second (an unpublished study estimates4 � 10�3 s). Thus, the experiment can be treated as anisothermal one. Within 5 s an 80 nm thick e-phase layerand a 1000 nm thick g-phase layer grow at the Cu–4 wt.%Sn strip (see Fig. 3). The e-phase exhibits an extremely finegrain structure of typical size 2qe = 50 nm and the g-phaseexhibits a scallop morphology characterized by the grainsize 2qg = 600 nm. The scallops are formed according tothe dihedral angle given by the local equilibrium of specificenergies of g-phase/Sn–1 wt.% Cu liquid interface and theg-phase grain boundary.

If the growth of the layers is simulated by assuming nograin boundary/interface diffusion, i.e. by using the bulkdiffusion (qi!1 in the effective diffusion coefficients inEqs. (33), (34)), the calculated kinetics of the system isnearly two orders of magnitude slower than the observed

Fig. 3. Cross-section through a strip of Cu–4 wt.% Sn in contact with Sn–1 wt.% Cu at 250 �C for 5 s. The e-layer is marked by its thickness of 63–83 nm and the g-layer is marked by its thickness of 820–1090 nm. Belowthe e-layer the Cu–4 wt.% Sn solid layer is situated. Above the g-layer theSn–1 wt.% Cu originally liquid layer (but now a solid layer with tiny g-precipitates (grey)) is situated. Focused ion beam image courtesy of Chairof Functional Materials, Department Materials Science and Engineering,Saarland University, 66123 Saarbrucken, Germany.

0,0 0,5 1,0 1,5 2,03,996x10-4

3,998x10-4

4,000x10-4

4,002x10-4

4,004x10-4

4,006x10-4

Sn-1wt%Cu

η

η

K2

K1

εCu-4wt%Sn

Posi

tion/

m

(t/s)1/2

Fig. 4. Simulated positions of the interfaces measured as the distances fromthe midplane of the strip with initial thickness of 8.000 � 10�4 m. Only theinterface between the Cu–4 wt.% Sn phase and the Sn–1 wt.% Cu phaseexists at t = 0 coinciding with Kirkendall plane K1 � K2. The Kirkendallplanes K1 and K2 remain coincident throughout the system evolution.

38 J. Svoboda et al. / Acta Materialia 61 (2013) 32–39

one (after 5 s the e-phase is 114 nm thick and the g-phase is55 nm thick). This indicates that it is necessary to invokegrain boundary/interface diffusion to account for theexperimental observation. Thus, a radius qe of 25 nm isused for the e-phase, and qg = 300 nm is used for theg-phase. The grain boundary/interface thicknesses de, dg

are selected as 0.3 nm. In the first attempt to account forthe experimental observation all grain boundary/interfacediffusion coefficients are approximated by the diffusioncoefficients DL

Cu;DLSn for the liquid except the diffusion coef-

ficients of Cu and Sn at the g-phase/liquid Sn–1 wt.% Cuinterface, which are taken as infinite. The kinetics of the sys-tem in this case is much faster than the observed one (after5 s the e-phase is 623 nm thick and the g-phase is 36550 nmthick). To account for the experimental observation, it isobviously necessary to decrease the diffusion coefficients.Compared to the former case the diffusion coefficients (forSn and Cu, abbreviated by X) are set as DL

X=1:2 � 106 atthe Cu –4 wt.% Sn/e-phase interface, at the mantle and atthe e-phase/ g-phase interface as DL

X=15 and as infiniteat the g-phase/liquid Sn–1 wt.% Cu interface. The men-tioned diffusion coefficient for the Cu–4wt.% Sn/e-phaseinterface Dint

Cui ¼ 2:17 � 10�15 m2 s�1 ranges betweenDgb

Cu ¼ 3:04 � 10�16 m2 s�1 and DLCu ¼ 2:54 � 10�9 m2 s�1 for

T = 250 �C, so the value of DintCui can be considered as an

acceptable assumption. The simulation then providesresults (depicted in Fig. 4) that are in agreement with theexperiment, predicting that after 5 s the e-phase is 81 nmthick and g-phase is 965 nm thick compared to the experi-mental observations of 80 and 1000 nm, respectively.

The kinetics in Fig. 4 shows for the interfaces a slightdeviation from the

ffiffitp� behavior. This is due to the fact

that the effective diffusion coefficients given by Eqs. (34)

and (35) are dependent on the phase thickness di, i.e. theheight of the grains, for fixed values of grain radii qi. InFig. 4 the positions of the Kirkendall planes K1 and K2, cal-culated according to Eq. (13), are also depicted.

Finally, let us briefly discuss the motion and location ofthe Kirkendall planes. If the Kirkendall plane is located inthe bulk, it moves with the lattice at a velocity given byEq. (9). If the interface i between the phases i and i + 1crosses the Kirkendall plane, four possible situations canoccur after the crossing. If ui 6 0 and vi+1 P 0, the Kirken-dall plane remains located at the interface. In the oppositecase the Kirkendall plane enters the phase i if ui > �vi+1,enters the phase i + 1 if ui < �vi+1 and gets split and entersboth i and i + 1 phases if ui = �vi+1 (a rather rare case). Sincewe work with effective bulk diffusion coefficients, we do notdistinguish between the bulk and the grain boundary in thephase. Consequently, we assume that the marker definingthe Kirkendall plane located in the grain boundary moveswith the same velocity as the marker located in the bulk.This is also in accord with the assumption that the adjacentgrains do not move relatively and the atoms in the grainboundary have no reason to move relative to the grains.

7. Summary and conclusion

The analysis of diffusion couples with several stoichiom-etric phases between them is extended to the combinedbulk and grain boundary/interface diffusion. This new stepin modelling diffusive processes enables us to predict thekinetics of real processes, where both diffusive mechanismsare active, in a significantly improved way. On the basis ofthe TEP, effective diffusion coefficients for each intermetal-lic layer are developed, which reflect the fluxes through thebulk and along the interfaces and/or grain boundaries. Theconcept is applied to the Cu–Sn couple with solid Cu andsolid as well as liquid Sn. The experimental results arediscussed in detail and compared with the predictions.The vacancy activity is assigned to the grain boundaries/

J. Svoboda et al. / Acta Materialia 61 (2013) 32–39 39

interfaces, an assumption which is already justified by pre-vious work by the authors [18]. The model, however, is alsosuitable for extension to vacancy activity in the bulk.

8. Outlook

Here we would like to highlight the role of vacancies. Infuture work, non-ideal sources and sinks for vacancies inthe bulk should be studied as already outlined in the recentstudy by Svoboda et al. [18]. However, we consider thecurrent study, which assumes no vacancy activity in thebulk, as relevant, since Ref. [18] shows that the assumptionof ideal sources and sinks in the interfaces yields resultsindependent of activity of sources and sinks for vacanciesin the bulk. A further improvement can be expected if afinite mobility and/or non-ideal sources and sinks forvacancies are assigned to the interfaces, so further dissipa-tive terms due to the rearrangement of the lattice and thevacancy activity in the interfaces come into play.

A detailed picture on the vacancy activity will allow alsostudy of the formation of voids as they are observed, forexample, by Kumar et al. [15] in the e-phase near itsinterfaces or by Shang et al. [38] in the intermetallic layerfollowing the Cu phase. Voids are generated as a conse-quence of vacancy condensation; however, this is a ratherdifficult topic.

Further developments can also be performed for caseswhen the diffusion couple is composed of phases of certainmixability, so that concentration profiles can develop in theparent phases.

Acknowledgements

J.S. and F.D.F. acknowledge gratefully the financialsupport by the Austrian Federal Government (in particularfrom the Bundesministerium fur Verkehr, Innovation undTechnologie and the Bundesministerium fur Wirtschaftund Arbeit) and the Styrian Provincial Government,represented by Osterreichische Forschungsforderungsge-sellschaft mbH and by Steirische Wirtschaftsforderungsge-sellschaft mbH, within the research activities of the K2Competence Centre on “Integrated Research in Materials,Processing and Product Engineering”, operated by theMaterials Center Leoben Forschung GmbH in theframework of the Austrian COMET Competence CentreProgramme. J.S. acknowledges gratefully the financialsupport by the Czech Science Foundation in the frame ofproject P204/10/1784 and by the project RVO:68081723.

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