Long-Term Life of Ni/Cu Bellows: Effect of Diffusion on Thermomechanical Properties

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Long-Term Life of Ni/Cu Bellows: Effect of Diffusion on

Thermomechanical Properties

Olivier Arnould, François Hild

To cite this version:

Eect of Diusion on Thermomechanical Properties

OlivierArnould ?

and Francois Hild

LMT-Cachan, ENSde Cachan / CNRS-UMR8535 /UniversityParis6 61,av. duPresident Wilson,F-94235 Cachan,France.

Ph. +33(0)1 47 4022 25 Fax +33(0)1 4740 22 40

Keywords: ageing, diusion, grain growth, mechanical behaviour, mechanical loading, couplings,homogenisation.

Abstract. The aim of this paper is to discussdierent couplingsbetween diusion, re-crystallisation/graingrowth,strain/stress,heattransferandthermo-mechanicalbehaviour of metallicmaterials.The understanding ofthese dierent physicalphenomenais needed topredictthelong-termlifeofmechanicalpartsincomponents.Someresultsarepresented and furtherstudiesaresuggestedto improvetheevaluationof the(inter)diusionand re-crystallisationparameters inne-grainedelectroplatedNi/Cu materials.To illustratethe coupling between elasticity and diusion, classical results in homogenisation consistent withthemaindiusionregime areusedto predictthechangeinelasticmoduliwithtime.

1 Introduction

This paper is a more detailed presentation of a study on ageing that has been introduced in Ref.[1]. It deals with a high precisionpressure sensor that uses bellows to convert a pressure dierenceinto a displacement (Fig.1a). The high precision and airtightness of bellows require to make them as a Ni/Cu/Ni layered material (overall thickness  100m) obtained by an electrodepositionprocess onanaluminiummandrelthat isdissolved in apotassiumhydroxide bath.Nickeldeposits (Fig.1b) have ane columnarstructure withsubmicrometreto microme-tre grains joined in colonies delimited by crevices surrounding them and inducing porosities [2]. Alone, nickel cannot provide the required airtightness but possesses the needed stiness. Conversely,copperdeposits asacontinuousmultilayerthatisabletoprevent(helium)diusion throughthe thickness.Consequently,acopperlayer(thickness 3m)isinserted between two layers ofnickel.Thiscongurationallowsfortheminimisationofthe`softening'eectofcopper onthe mechanicalbehaviour of bellows since bendingis the predominantloading pattern.

Suchcomponentsareexpectedtostayinserviceformanydecades,thus, theageingmode(s) ofthematerialhavetobeknown. Thestudyofthe mainenergies availableinthisconguration (fromwhichthedrivingforcesarederived)allowsonetolisttheageingmodesthatmayhappen. Numerousageingmodesare possibleandmustbeinvestigatedbut themostimportantones for this conguration seem to be caused, rst, by the chemical gradient that gives rise to Ni/Cu diusion and, second, by the unstable microstructure that leads to recrystallisation (or more likely grain growth).

Consequently, the thermo-mechanical behaviour of bellows will change with time and the determination of the diusion coeÆcients in this case enables one to predict stiness (i.e., elasticity) changes induced by diusion whereas grain growth can cause some change in the plastic behaviour for example. The paper rst focuses on a presentation of the couplings that

?

Nickel

Nickel

Nickel

Nickel

CopperCopper

Position of the aluminium mandrel

_{during the electrodeposition process}

20 m

(a)

(b)

5mm

Fig.1.a)Generalview ofelectroplatedbellows.b) Electronmicrographofacross-sectionobtainedin anSEMafterchemicaletching[2 ].Notethemultilayeredmaterialandthenecolumnarmicrostructure ofthenickellayers.

canbeexpectedbetweenmechanicalpropertiesandloadings,heattransfer,(inter)diusionand graingrowth, and some ways of modellingthe dierentinteractions.Then, practicalstudiesof diusionand arst estimateof the associatedcoeÆcients are performed formoderate temper-atures. This leads us to suggest some further improvements in the diusion curves treatment sincegraingrowth andthe diusionkinetics canchange.Finally,homogenisationtechniques in elasticity allowus to evaluate the change of the thermo-mechanical properties of the eective materialwith time.

2 Thermo-mechanical, diusion and recrystallisation couplings

The dierent couplings of interestare presented in Fig.2. The linear behaviourof materials is nowwellknown, thus the thermo-elasticcouplings can bestudiedwith aclassical formulation, which is based on the expression of thermodynamic potentialsand the derived state laws [3]. For example, the second order thermal expansion tensor,  , has been evaluated at a submil-limetre scale by digital image correlation [4, 5]. A CCD camera is used with a long-distance microscope. Images are taken at each temperature step and the intercorrelation between the considered image and the initial one (at room temperature) allows one to determine the in-planedisplacementeld.TheisotropyofthecoeÆcientofthermalexpansionandtheassociated valuesare determined.In the presentcase, thesecoeÆcients have been obtainedonthe surface of the bellows. The behaviouris essentially isotropicand corresponds mainlytothe electrode-positednickelcontribution(Fig.3).The average value ofthe coeÆcient ishigher thanfor bulk nickel and seems to be closer to bulk copper when the temperature T increases. This could be explained by the copper layer that is inside the nickel layers. However, the copper layer shouldnot`impose' itsdeformationasitsthickness istoosmallcompared tothatofnickel(see Fig.1b). Furthermore, the Young's modulus of copper is less than that of nickel. Therefore, this change in the thermal expansion coeÆcient may be due to the particular microstructure of the nickelelectrodeposits.

"Classical" coupling

Heat Transfer

Kirkendall

effect

E

_ij

(C,

f

)

α

(

C, f,

T

)

h(

C

,

f

,

d

)

Diffusion

Change in composition

C

and in porosity f

Recrystallisation/Grain Growth

Change in grain size

d,

texture and dislocation density

Mechanical

behaviour and loading

Strain

plasticity

Thermal activation

(Arrhenius law)

DIGM

DIR

(Fig. 3)

Moving grain

boundary

Change in diffusion

regime

Thermal activation

(Arrhenius law)

Temperature gradient

Gorsky effect

(Stress gradient)

Stress

σ

y

(

C

,

f

,

d

)

~

Fig.2. Overall view of the assumed thermo-mechanical, diusion and recrystallisation couplings. DIGM standsfor`Diusion-InducedGrain Migration',DIR for`Diusion-InducedRecrystallisation',  is the thermalexpansion second order tensor, h theheat conductivity tensor, E

ij

the components of thestiness tensorand ~

y

theyieldstress and its possibleanisotropy.The presentstudyis mainly focused onthe determinationof thechangeof E

ij

withthe concentration C.

tensor h, depend onthe local (i.e.,microscopic)concentrationin nickel andcopperC(x). This canbeobtained byhomogenisationtechniques withthesame approachfor allthesecoeÆcients asthosepresented inSection4 forthestiness. Therefore,the change ofthesematerial param-eters with time depends on the concentration prole that will be determined by the diusion coeÆcients.Itisimportanttonote herethatthe (average)grain size,d,andtexturecan havea non negligiblein uenceonthe previousparameters.First, thehomogenisationtechniques used hereinassumethattherepresentativepatternissmallcomparedwiththestructure lengthscale (i.e., layers thicknesses) and loading gradients. If these conditions are not fullled, boundary eects have to be accounted for and other models must be used. Lastly, the global pattern (i.e., statisticaldistribution) can aect the macroscopic behaviour, especiallyfor the thermal conductivity (see Section4).

Theeect of temperatureondiusionandgrain growthismodelledby anArrheniuslawas athermally-activatedprocess.Moreover,atemperaturegradientcanaect theatomicdiusion uxasit willbedescribed below. The eect of diusion onthe mechanicalloadingarises from the dierence between the dierent diusing species in the product V

i M i , where V i stands for the molar (or atomic) volume of the species i, and M

i

its mobility (linked tothe intrinsic diusion coeÆcient D

i

Isotropic thermal expansion

coefficient

α

(

m/m.

C)

Temperature

θ

( C)

7

12

17

22

40

60

80

100

120

Image correlation measurements

Linear interpolation:

α

(

θ

) = 0.038

θ

+ 13

Bulk nickel [6]:

α

(

θ

) = 0.0092

θ

+ 13.2

Bulk copper [6]:

α

(

θ

) = 0.0087

θ

+ 16.3

Fig.3. Image correlation measurements of the (nearly) isotropic thermal expansion coeÆcient of electroplated Ni/Cu multilayers (reference temperature: 20

Æ

C). Comparisons with bulk nickel and copper coeÆcients. The scatter (i.e., the error bars) is mainly due to the accuracy in displacement measurement combined with the fact that, the lower the temperature, the lower the dilatational displacements.

whereas the other one will expand. The corresponding stress-free strain can be expressed by [7,8,9] _ " SF = 1 3 div  V A J 0 A +V B J 0 B  1; (1) whereJ 0 i

isthe uxofthespeciesiinthelatticeframe(J i =J 0 i +v K (x )C i ,withJ i

the uxinthe laboratoryframeandv

K

thevelocityofthelatticeframerelativetothelaboratoryframeusually measuredbyKirkendallmarkers).ThestraininEq.(1)inducesstressesthatcanbevisualisedby thebendingofthin-sheetdiusioncouples[10,11,9]evenforaNi/Cucouple[12].Ifthestresses relaxfasterthanthediusioncharacteristictimeandcompletely,theconvectivetransport(i.e., plastic ow) that occurs leads tothe `classical' Kirkendall eect (i.e., the displacement of the Kirkendall plane is proportional to the square root of time [13, 14, 15, 16]) and the classical Darken modelis valid[17, 7].In the opposite case, the stress developpingslows down diusion (seeEq.(3))anditsrelaxationcanmainlycontroldiusion.Thiscasecorrespondstothe Nernst-Planck limit(with approximatelyno marker displacement [8]). Diusion kinetics in these two extreme casesare completelydierentand allthe possiblecases inthe transitionregimecanbe obtained by varying the layerthicknesses in compositionallymodulated multilayereddiusion couples [8, 18]. Moreover, the Kirkendalleect implies adiusion of vacancies in the direction opposite to the net ux of atoms that can lead to pore formation (in the side containing the species with the higher product V

i M

i

) by the so-called Frenkel eect [14, 19, 20] in the bulk as well as on the grain boundary [21, 22]. Since copper is the species in which this porosity could appear, it can lead to the loss of bellows airtightness. Furthermore, the development of stresses can lead to crack initiation and propagation in the nickel or copper layers. Finally, grainboundary diusioncan accelerate creep in materials[23].

Distance ( m)

W

eight fraction

in Cu (%)

0

10

20

30

0

10

20

30

-10

-20

-30

Initial copper

layer

EDS measurements

Numerical predictions

Interdiffusion affected area

20 m

}

Fig.4. Measuredand predictedweight fractionincopperof abellows afterheat treatment at 780 Æ

C for 15min. Note the (strong) grain growth and its interaction with diusion revealed by chemical etching[2 ] (SEMmicrograph).

relatedto the diusion coeÆcient by

D p =D p=0 exp  p
V R T  ; (2)

where p is the pressure (i.e., p = trace ()=3 with  the stress tensor),
V the activation volumethatcorresponds,whenselfdiusionisconcerned,tothechangeinvolumeofthecrystal associatedwithdefectformationandatomicjump(i.e.,migration).Inthecaseofinterdiusion, the activation volume contains additional terms corresponding to the presence of impurities [27]. Finally, R is the gaz constant and T the absolute temperature. The previous equation only depends on the hydrostatic part of the stress tensor but it is not excluded that other componentsof the tensor (i.e.,shear stresses) can aect diusion by a similarthermodynamic eect [28]. The eect of pressure on grain boundary diusion has also been studied [29, 30]. Equation(2) shows thatanincreasingpressure slows down diusionif
V ispositive(whichis generally the case [27]). The other aspect concerns a stress gradient that has been introduced by Larche and Cahn [31, 32,33] and Stephenson [7]. This aects the ux of species i and can be summarised by the following modication in the Fick's law for a one dimensional binary diusioncase undera pressure gradientonly [10, 8]

J 0 i = D i @C i @x D i C i V A V B R T @p @x ; (3)

signif-nomenondescribed by Eq.(3)has beenrecognised earlierfordiusionof hydrogen insteeland has been referred to as the Gorsky eect [34, 35]. The Gorsky experiment is based on a steel bar in which a uniform hydrogen concentration is initialy present. Bending this bar leads to an enrichment of hydrogen in the tensile part of the specimen and the opposite eect in the compressive part. An other example is the formation of impurity clouds around edge disloca-tions, e.g., the Cottrelatmosphere [36]. Inthis case, the impuritiesare attracted (orrepulsed) by the dislocations via the stress eld created by the dislocations. Similareects are observed for water transport in Agar gel [37, 38] where a modication in the dependence of apparent diusion coeÆcientswiththe concentrationisdescribed.Furthermore,inthesecases,the diu-sion species are mainlyinterstitialatoms but the previous remarksapply forvacancydiusion mechanisms. This study has been extended to the case of grain boundary diusion [29, 39]. Furthermore,the previous equation is expressed for apressure gradient, itsgeneral expression contains any component of the stress gradienttensor since the modication in the ux is due to agradientintheelasticenergy density r(:"

e

)=2,where " e

istheelastic straintensor(see e.g., Eqs. (40) and (48) in Ref.[7]). Moreover, stresses can modify or completely prevent the Frenkeleect ofporeformation[19, 12].Notethatthe`external'stresscanarisefromathermal expansioncoeÆcientmismatch(e.g.,

Cu >

Ni

).Itis importanttonotethat, accordingtothe Curie'sprinciple, uxes originate ingradientsof the intensivethermodynamic variables [8, 28] (e.g.,auniformstress cannotgiverisetoadiusion uxbut itcanslowdown oracceleratethe process) and similar additionalNernst-term in the Fick's law can be used to modelthe eect of atemperaturegradient ondiusion [28]. In this case, Eq.(3)can berewritten as

J 0 i = D i @C i @x D i C i Q  R T 2 @T @x ; (4) whereQ 

isathermodynamicfactor. Butthethermalconditionsduring experimentsorforthe realcomponentare generally suchthat thecharacteristic timefortemperaturehomogenisation is much smaller than diusion characteristic time. The eect of a temperature gradient is thereforeneglected here.

In the present case, since the material grain size is bound to evolve, grain growth or re-crystallisationmay occur.This willbeparticularlytrue during anannealing treatmentfor the study of diusion (see Figs.1 and 4). The eect of a moving grain boundary on the diusion analysis has been treated fordiusion ona single grain boundary [40, 41] and ina polycrystal containing a fraction of moving grain boundaries [42]. It can lead one to underestimate the grain boundary diusion coeÆcient since the penetration depth in a moving grain boundary is smaller by a factor (V t=

p Dt)

1=2

, where V is the (constant) grain boundary velocity, t the diusiontimeandDthe averagelatticediusion.Inthiscase,newdiusionregimescanappear dependingonthe velocity,V,thegrainboundarydiusioncoeÆcientD

b

,thelatticeoneDand the grain size d[40, 43]. Lastly,the dislocationdensityin uences the apparentlatticediusion since dislocations(andsubgrainboundaries)representdiusionshort-circuitpathsand,iftheir density is signicant,a new grain boundary diusion regime, say D, appears [44]. Conversely, diusion can modify or in uencethe grainsize and distribution by the diusion-inducedgrain boundarymigration(DIGM)[24,42,22]anddiusion-inducedrecrystallisation(DIR)[45].Itis interesting tonoticeinFig.4that graingrowth inthe interdiusionaectedarea seems slower than inpure nickel.

y size d by, say,the empiricalHall-Petchrelationship [47]

 y = i + k y p d ; (5)

that applies for nickel used in the present case [2] where  i

is an `internal' stress which is relatedtothe shear stress threshold fordislocation glide (approximatelyequalto 40MPa)and k

y

 0:48MPa

p

m. For very ne-grained materials, the yield stress can be governed by other mechanismsthandislocationsglideandleadtoareverseHall-Petcheect[48].This`mesoscopic' yield stress associated with the material `texture', the porosity and the concentration curve are needed to determine the macroscopic yield stress of a cross section of the multilayered material.Porosity reduces the macroscopicyield stress[49]. The diusion of copper couldlead toasignicantmodicationinthemacroscopicplasticbehavioursinceitsdiusiononthegrain boundaries couldlead to abrittle (intergranular) fracturemechanisminstead of ductilefailure asit has been observed experimentally [50].

This overview of the main couplings of interest in the present study of ageing shows the complexityoftheproblemanditisnot possibletotreat(numericallyorexperimentally)allthe eectssimultaneouslyandonlytherelevanteectsmustbeconsidered.However,itisimportant to keep all of them in mind when interpreting experimental results. The next section focuses on the determination of diusion parameters for the electroplated multilayered material that willallowfor the evaluation of the change of the elastic moduliwith time.

3 Determination of diusion coeÆcients

Bellows are usually utilised at an average temperature close to room temperature. For this rangeof temperature, grain boundarydiusion is predominant overvolume diusion asit can beseen inFig.5.However, the microstructureissmall(Fig.1b) comparedtospatialresolution of concentration measurements (electron microprobe is used for measuring diusion curves, i.e., the characteristic length varies between many hundredths of one micrometer and one micrometer) so that the diusion curves seem homogeneous in the direction parallel to the interface.Theinvestigationsare,forthemoment,limitedtothedeterminationofglobaldiusion coeÆcients including grain boundary diusion, (weak) lattice diusion and grain boundary motiondue to grain growth (see Fig.4).

0.5

1

1.5

2

2.5

3

3.5

12.7

60.3

127

227

393.7

727

Temperature (10

3

/T(K))

Temperature (

θ

( C))

Diffusion coefficient

D

(

m

2

.s

-1

)

10

-10

10

-8

10

-6

10

-4

10

-2

1

Grain boundary diffusion Ni in Cu

1727

10

2

(Grain boundary velocity: 0.07nm.s

-1

₎

(C regime, no migration)

Bulk diffusion Ni in Cu

Bulk diffusion Cu in Ni

Grain boundary diffusion Cu in Ni

~Temperature threshold for grain

boundary migration [23, 60]?

~Transition

temperature between

grain boundary and

bulk predominance [14]

*

?

GB

GB

GB

*

θ

m

(Cu)

θ

m

(Ni)

Lattice diffusion

Yukawa et al. [51], Mackliet et al.

[52], Ikushima [53] and Almazouzi

et al. [54] (and inner references): Ni

in Cu (single crystal)

Ref. [22] in Kolobov et al. [23]:

Cu in Ni (bulk)

Cu -

>100%

Ni -

>100%

Masson [14]: Interdiffusion in

coarse-grained couples

Cu -

>100%

Ni -

>100%

Thomas et al. [55]: Interdiffusion in

coarse-grained couples

Austin et al. [21]: Interdiffusion (bulk)

Cu -

>100%

Ni -

>100%

Hasaka et al. [56]: Cu in

poly-crystalline Ni

Melt-spun ribbons (

<

d

Ni

>

=10 m,

columnar grains)

Pre-annealed blocks (15 m<

d

Ni

< 100 m)

Benhenda [57]: electroplated Ni on

bulk Cu, studies in Ni grain boundaries,

columnar grains (

<

d

Ni

>

= 30 nm) joined

in submicrometre colonies

Tronsdal et al. [58]: Inter-diffusion

between electroplated foils, Cu ~ 60.6%,

observed B

2

’ regime, 30 m< d

Ni

<50 m

after annealing

*

Present study: Boltzmann-Matano/Hall

(lattice) estimate

Cu -

>100%

Ni ~50%

Average value (bellows)

Ni -

>100%

Grain-boundary estimate

Cu -

>100%

Ni -

>100%

GB

GB

Grain boundary

diffusion

Yukawa et al. [51]: Ni in Cu (bicrystal),

Misorientation angle: 45

Misorientation angle: 10

Kolobov et al. [23] with grain

boundary migration correction

Cu in sub-microcrystalline Ni

(polycrystals,

<

d

Ni

>

=300nm)

Cu in coarsed-grained Ni

(polycrystals,

<

d

Ni

>

=20 m, B

1

regime)

Grabovetskaya et al. [60], Cu in

ultrafine grained Ni,

<

d

Ni

>

~0.2 m,

grain boundary migration correction

(average velocity - 300 C: 0.07nm.s

-1

₎

Johnson et al. [59]: Cu/Ni thin-film

couples obtained by vapor deposition,

<

d

Ni

>

=24nm,

<

d

Cu

>

=200nm,

Ni in Cu

Cu in Ni

Austin et al. [21]: Ni in Cu

(bicrystal), misorientation angle: 45 ,

average value of concentration

dep-endent coefficients at 0.93% in Ni.

Rabkin [61]: Ni in Cu (bicrystal),

misorientation angle: 45 , average

value of concentration dependent

coefficients at 0.93% in Ni.

Fig.5.Arrheniusplot forthechangeofglobaldiusioncoeÆcientswithtemperatureforNi/Cu cou-ples. All compositions are given in weight fraction, 

m

studies are performed by an EDS (i.e., Energy Dispersive Spectroscopy) method in an SEM. It enables one to identify and quantify the elemental chemical composition within a depth of aboutonemicrometer.Amassconcentrationresolutionofaboutonepercentcanbeobtainedin suitableconditions, i.e.,measurement artefacts,or eects, must be considered. Softwares used toanalyse the resultsof anEDS microprobeusually correctstandard measurement eects but they assume a homogeneousconcentration in the analysed area. However, in the present case, diusion occurs only over a few micrometers (see Fig.4) and the concentration is no longer constant in the analysed area. One measures an average of the real concentration and that can lead to overestimate the actual diusion coeÆcient [62]. It is called average eect and is caused by the X-ray emissionvolume whosecharacteristiclength,l

X

,can be of the sameorder of magnitude as the penetration length of diusion, 2

p

Dt, where D is the (true) diusion coeÆcient. A simplied quantitative study of this phenomenon in the case of close atomic numberelements (such asnickel and copper), if nodependence ofthe diusion coeÆcientwith the composition isassumed, has shown that the apparent diusioncoeÆcient D



measured by anEDS probefollows the simple law

D  =D+ l 2 X 2t ; (6)

that shows the need of taking this eect into account in the determination of the diusion coeÆcient [62]. A more detailed study of the specic physical eects occurring in this kind of measurements has been developped in Ref.[63] and a complete deconvolution procedure of concentration measurements withconcentration-dependent diusioncoeÆcients isproposed in Ref.[64].

Usually, theBoltzmann-Matano method[24] isappliedto evaluatethe diusioncoeÆcient, which isdependent onthe chemicalcomposition,fromthe deconvoluted concentration proles for intermediate concentrations. An additionalmethod is necessary for the low and high con-centration parts of diusion curves since the previous one is less accurate in these extreme domains.The Hallmethodiswell-adaptedtothis kindof study [24].The diusion curves have beenobtainedonspecicspecimen(i.e.,`semi-innite'bilayereddiusioncouplesforwhichthe layer thicknesses are suÆcient toprevent the diusing species to reach a free surface) and not on bellows. But three importantremarks (orimprovements)have tobe made:

{ this method applies for lattice and only for grain boundary diusion in the `A' regime [42] when dealing with the average diusion curve in the direction perpendicular to the diusion one.Inthis kindof grainboundarydiusion,the globalinterdiusioncoeÆcient followsamixingruleofthelatticeandgrainboundarydiusioncoeÆcientsviathevolume fractionof grainboundaries(i.e.,the so-calledHartequation[42]whosevalidityhas been recently studied for ne-grained materials [65]). The grain boundary diusion regime(s) has to be clearly specied in the present study and if other regimes of grain boundary diusionhappen,aspecictreatmentmaybedevelopedtotreatthediusioncurve[27].It can bebasedonthedeterminationofinterdiusioncoeÆcientsongrainboundary[21,61] that is, for the moment, `limited' to the case of constant lattice diusion and for dilute solidsolutions.Grainboundarydiusionhasbeenconrmedby plottingaveragediusion curves, onmany positionsalong the initialinterface, inthe (x

6=5

1

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

1

0

0.2

0.4

0.6

0.8

Weight fraction in Ni

Interdiffusion coefficient

D

(

m

2

.s

-1

)

~

Ref. [6] in Tronsdal et al. - 750 C - electroplated foils

Thomas et al. [55] - 1022 C - lattice diffusion

Masson [14] - 1030 C - lattice diffusion

Present study - 700 C - electroplated materials

Austin et al. [21] - 750 C - lattice diffusion

Tronsdal et al. [58] - 800 C - electroplated foils

Present study - 550 C - electroplated materials

Nearly constant coefficient [55,

14]

Fig.6. Change of the global interdiusion coeÆcient with the weight fraction in nickel for Ni/Cu couple. Scatter in the coeÆcient (depicted by the error bars) is due to the precision of the EDS measurements and (weak) variations of the diusion curves in the direction parallel to the initial interface.The straight linesdepictthemaintendencyof thedierentresultsand thevaluesobtained inthepresentcaseforaverage diusioncurvesformanymeasurementsatdierentpositionsalongthe initialinterface.

even if the Le Claire's analysis can be applied to polycrystals when the average grain boundarymisorientationangleis known [42, 27]). Furthermore,the initialand boundary conditions in concentration for the grain boundary diusion analysis is not well-dened here even if itseemstobeclosetoaconstantconcentration.Inspite ofthese restrictions, the extracted values are surprisingly well correlated with other grain boundary results (since the next item has to be considered too),

{ strong grain growth occurs for ne-grained materials and this eect must be accounted for sincethe apparentdiusion coeÆcientisless thanthe real onedue tograinboundary migration(seetheprevioussection).ThishasbeenperformedinRefs.[23,60]forthiskind of materialbut it appears that it exists nospecic techniques when the grain boundary interactsduring the grain growth process,

{ theKirkendalleectimpliesthatalocalvolumechangeoccursandtheBoltzmann-Matano methodcould lead towrong interdiusioncoeÆcients. This can beassessed by using the Sauer-Freiser method[66].

As a rst approximation, with the Boltzmann-Matano/Hall method, the change of the in-terdiusioncoeÆcientwiththeconcentrationhasbeenobtainedat700

Æ

Cand550 Æ

shape of the curves is consistent with classical results, i.e., the dependence of the diusion coeÆcientwiththeelementalcompositionincreaseswiththecopperconcentration.Thisis gen-erally thecase,i.e.,adiusioncoeÆcientincreaseswiththe concentrationofthelowest melting temperature compound [14, 67]. But some results ongrain boundary concentration-dependent diusion coeÆcients should be mentionned where the opposite dependence is observed [61]. However, thepresentdiusion coeÆcientsaregreaterthanthose obtainedforlattice interdiu-sion (and less than those forgrain boundary interdiusion). This follows the previousremarks onthe possible improvementsin the determination of the diusion coeÆcient from concentra-tion proles. In fact, the prevalent mechanism is grain boundary diusion and the eect of a movinggrain boundary has to be included.

Since the (inter)diusion coeÆcients have been identied at two testing temperatures, the coeÆcientof thermalactivationof the Arrheniuslawis known, with anaveragevalueof about 110kJ:mol

1

fortemperaturesrangingfrom550 Æ

Cto700 Æ

C.Theresultsare showninFig.5 to-getherwithaverageestimates ofdiusioncoeÆcientsof bellowsobtainedby ttingthesolution of the Fick's law for a constant lattice diusion coeÆcient and bilayered material congura-tions. Itisimportanttonotethat theresultsobtained by theBoltzmann-Matano/Hallmethod seem to be consistent with other studies on electroplated or ne-grained copper/nickel mate-rials [58, 57] by using similar approaches (i.e., no correction for grain boundary motion and determinationofanapparent`lattice'diusioncoeÆcientwiththe Boltzmann-Matanomethod or equivalent) even if the coeÆcient of thermal activation seems to be lower in other results than inthe present study.

Discretisation

Transverse

direction

x

Longitudinal

direction

y

Initial bilayered material

Nickel

Nickel

Copper

100

0

Weight fraction

in Ni (%)

x

Multilayered material

at time

t

Multilayered material with piecewise

constant chemical concentration

Ageing

(Diffusion)

Composite

Homogenisation

Effective elastic properties

Longitudinal

modulus

E

_y

(t)

Transverse

modulus

E

_x

(t)

Shear

modulus G

_xy

(t)

Elastic modulus

x

Laminate composite

with isotropic layers

Three-phase self-consistent

homogenisation pattern

Core

Diffusing species

on grain boundaries

Equivalent homogeneous

medium

e

k

e

Fig.7.Homogenisationstepsto predicttheoverall(i.e.,macroscopic)elasticpropertiesduring chem-icaldiusionofNi/Cu bellows.

4 Changes of elastic moduli induced by diusion

The change of the eective elastic properties (i.e., transverse E x

and longitudinal E y

Young's modulus, and in-planeshear modulus G

xy

where longitudinal means parallel to the interfaces between nickel and copper, see Fig.7) of the bilayered material with the diusion process is evaluated.Since the Poisson's ratios, , of the two materialsare very close, they are assumed to be equal. The determinationof the eective coeÆcients, when the chemical composition is known, is illustrated in Fig.7. The diusion proles are discretised into N layers of thickness e

k

toobtainamultilayeredmaterialinwhicheachlayerisassumed tohaveaconstantchemical compositionandtobeisotropic.Theeectiveelasticmoduliofeachlayerisobtainedbyusinga homogenisationscheme consistent with the assumed principaldiusion mode: grain boundary diusionin ane-grainedmaterial (see microradiogramsin Ref.[58]), i.e.,between the C

0 and B

0 2

regimesdenedinRef.[43].Therefore,intheinitialnickellayer,nickelgrainsaresurrounded by copper and the opposite situation is found in the copper layer (Fig.8a). The generalised three-phaseself-consistentscheme[68,69]isthemostsuitableone.Itconsistsindeterminingthe eectiveequivalentmodulus ofasphericalisotropicinclusionsurroundedbya secondisotropic phase, this whole pattern being embedded in an innite medium whose elastic moduliis the expected mesoscopic one, i.e., the equivalent homogeneous medium (see Fig.7). The eective isotropicelasticmodulus forthek

th

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

2

4

6

8

10

12

14

Distance ( m)

Weight fraction or relative modulus (%)

Effective modulus

Nickel Modulus

Weight fraction in Ni

Initial nickel

layer

Initial interface

Initial copper

heart

Nickel grain

Copper on the

grain boundary

Nickel on the

grain boundary

Copper grain

Composite

Homogenisation

(a)

183

185

187

189

191

193

195

0

1.3

2.6

3.9

4.2

5.5

6.8

7.1

Time (h)

Common limit of the modulus ~ 189 GPa

Initial longitudinal value ~ 194 GPa

Initial transverse value ~ 183 GPa

Longitudinal modulus E

y

Transverse modulus E

x

Effective modulus (GPa)

θ

= 700 C

0

250

20

(Estimated) Time (years)

θ

= 20 C

(Measured diffusion)

(b)

Fig.8.Firsthomogenisationstep(Self-consistent scheme ofapiecewiseconstant chemical concentra-tion)for one half of thebilayered material, with a thickercopperlayer than inFig.1b, for 37minof diusion at 700

Æ

C. Eective elastic modulusvs. time for diusion at 700 Æ

C and an estimate of the equivalenttime for20

Æ C. modulusK k K k =K gb + C 0 g k (K g K gb ) 1+(1 C 0 g k )(K g K gb )=(K gb +4G gb =3) ; (7) whereC 0

stands forthe volume fraction determinedby the weightfraction C and the densities ofthedierentmaterial,

g

isthequantity
ofthematerialoftheconsideredgrain(innercoreof the homogenisationpattern, Fig.7)and

gb

for the materialonthe grainboundary(concentric shellof the homogenisationpattern, Fig.7).The shear modulusG

k

isthe positivesecond root of asecond order equation

 G k G gb ! 2 + G k G gb ! + =0; (8)

where  , and dependonG g ,G gb ,C 0 g k

andPoisson'sratio [69]. TheYoung'smodulusand the Poisson's ratio are then deduced

E k = 9K k G k 3K k +G k ; (9)  k = 3K k 2G k 6K k +2G k : (10)

The two results are needed (see Fig.8a) since it is assumed that copper diuses on the grain boundaries of nickel grains inthe layers containing nickelinitiallyand conversely inthe initial copper layer. When this rst step is completed, a composite-like material is obtained with N isotropic elastic layers of eective Young's modulus E

k

and Poisson's ratio  k

(Fig.8a). A homogenisation scheme applied to isotropic laminated composite materials [70] is used to evaluatethe overall transversely isotropicelastic properties (Fig.7)

E x = X k e k e 1 E k ! 1 ; (11) E y = X k e k e E k ; (12) G xy = X k e k e 2(1+ k ) E k ! 1 : (13)

The nal result is shown in Fig.8b for E y

and E x

. The magnitude of variation of the moduli depends on the relative thickness of the copper and nickel layers. The maximum magnitude of variation is obtained in a relatively short time duration, i.e., an ageing mode induced by diusion could be signicant even for low time durations or temperatures. For example, a rough approximation based on an equivalence on the average rate of diusion for a 20 year agedmaterialgivesanideaof the correspondingtimedurationatroomtemperature. Sincethe stiness of bellows is proportional to the longitudinal modulus (for relatively low de ections), diusion could have a non-negligible eect on the overall behaviour of the component if the copper layeris too thick. The same treatment can beapplied tothe thermalconductivity and coeÆcient of thermalexpansionwith the same self-consistent schemes [71, 72]. It isimportant tonote that the previous results are all based onthe assumption that each materialgrain can be modelled by a homogeneous spherical inclusion surrounded by a homogeneous shell. This assumption is only valid for small grain sizes compared to the layer thickness. This will no longerbethe caseformaterialswithstrongrecrystallisationrates.Moreover, therealgrainand grain boundary morphologiescan play an importantrole inthe eective properties [73, 74].

5 Conclusions

AER company and Dr. N. Schmitt. They alsowish to thank Dr D.J. Fisher for invitingthem towrite this paper.

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