Surface Tension Gradient as Additional Driving Force for Grain Boundary Diffusion. Equilibrium and Non-Equilibrium Cases

1223

PACS numbers: 61.72.Mm, 61.72.S-, 68.35.Fx

Surface

Tension

Gradient

as

Additional

Driving

Force

for

Grain

Boundary

Diffusion.

Equilibrium

and

Non-Equilibrium

Cases

B.BoksteinandA.Rodin

NationalUniversityofScienceandTechnology«MISiS»,

4LeninskiyProspekt,

119049Moscow,Russia

Classic Fishermodel of grain-boundarydiffusion withleakage into bulkis generalized byconsideration of thesurface-tension changealong the grain boundary. КласичнаФішеровамодельдлязерномежовоїдифузіїзвідсмоктуванням в об’єм зерен узагальнено з урахуванням зміни поверхневого натягу вздовжмежізерна. КлассическаямодельФишерадлязернограничнойдиффузиисотсосомв объём зёрен обобщена с учётом изменения поверхностного натяжения вдольграницызерна.

Keywords: diffusion,grainboundary,surfacetension,chemicalpotential.

(ReceivedJune21,2013)

1.INTRODUCTION

It is well known that diffusion flux of i-th component is governed mainlybythegradientofitschemicalpotential.Consequently,at con-stanttemperature,thedrivingforcesusuallydiscussedarethe concen-trationgradientinanidealsolutionorthermodynamicactivity gradi-entinnon-idealsolution[1,2].Stressgradientisanexampleof addi-tionaldrivingforce,whichcanchangethediffusionprocess[3].These forcesfitforbulkandinterfacediffusion.

DuringroundtablediscussionattheInternationalConference DSS-2010[4],Prof.L.Klingerstatednewideathatifgrainboundaries(GB) energy is notconstant forall points of the GB,its gradient provides

Фотокопирование разрешено только

the additionaland independentdrivingforceforatomsmotion inGB (byanalogywithMarangonieffect).

Some later, L. Klinger and Eu. Rabkin [5] applied this ideato the penetrativewettingofGB’s.

ThemaingoalofthispaperistodevelopthemodelofGBdiffusion takingintoaccountthesurfacetensiongradientasanadditional driv-ingforce.

2.DESCRIPTIONOFTHEMODEL

Grain boundariesare thepaths of fastmass transport in polycrystal-line solids. Mainly, the macroscopic description of GB diffusion is basedontheFisher’smodel[6].Accordingtothismodel(Fig.1),GBis a thin plate with the thickness  (5109_{m) and diffusion}

coeffi-cient, which is much higherthan thebulk diffusivity (Db Dv). The

model takes into account the fast diffusion flux from the surface (y0)alongGBanddiffusioncleavagefromGBtothebulk.

LaterGibbs[7]developedthismodelforthecaseofheterodiffusion and took into accountenrichment coefficients, which gives the rela-tionbetweentheGBandbulkconcentrations

b v /2 . x C s C _  (1)

Fig.1.SchematicdescriptionofGBdiffusionmodelaccordingtoFisherwith conditionCb(y,t)Cv(x/2, y, t).

The simplest solution was obtained with following approximations: concentrationonthesurfaceisconstant;GBconcentrationweakly de-pendsontime(C_b t0,quasi-stationaryapproximation);diffusion fluxalongy-axisinthegrainbulkisneglected;segregationisothermis linear(sdoesnotdependonconcentration).

Thissolutioncanberepresentedas:

b b v b ( , ) (0) exp( / ), ( , , ) ( , ) erfc /(2 ) , C y t C y L C x y t C y t x Dt      _{ } (2) where 1/2 1/2 b v ( [ /(4 )] )

L s D t D , and Fisher’s solutioncorresponds to

s1(self-diffusion).

ItshouldbenotedthatlatertheexactsolutionsfortheGBdiffusion problem were obtained by Whipple [8]. Different adsorption models were used todescribe the effect of segregation onGB diffusion(see,

e.g.,[9—13]).

TheimportantpointisthatifGBadsorptiontakesplace,thesurface tensionbecomesdependentonconcentration.Thegradientofsurface tensionisanadditionaldrivingforceforthemasstransportalongthe interface.Totakeitintoaccount,theFisher’smodelcanbecombined withZhukhovitskiitheoryofsurfacephenomena[14].

Accordingtothistheory,GBchemicalpotentialofasoluteinan ide-alsolutioncanbewrittenas

, ln _b

st

b  RT C f

 (3)

wherefistheareacorrespondingto1moleofatoms,f/(isa mo-larvolumeforthesolvent).

ThediffusionfluxalongGBcanbewrittenas: , ) , ( grad ) , ( b b b b b y M t y M t y j         (4)

whereMb isadiffusantmobilityatGB.

For complete understanding ofthe problem, we need todividetwo differentstatesofGB’s:equilibriumandnon-equilibrium.

EquilibriumState.ItiswellknownthatGB’sarethenon-equilibrium

defects of crystals. From thermodynamicpoint of view, there are no conditionsforequilibriumbetweenGBandthebulk.Nevertheless,due togeometricalrestrictions,itisvalidtoassumethatGB’sarein meta-stableequilibriumwiththeadjacentbulkregion[15,16].Inthiscase, wehavetousetheconditionofequalityofchemicalpotentialsforeach componentatGBandinthebulk:

v b i i 

and,consequently,theequation(4)canberewrittenforadilute solu-tionas . ) , ( v v b v b b y C C RT M y M t y j          (6)

Forthelinearsegregationisotherm(seeEq.(1))withconstants,we canobtain: . ) , ( b b b b b b y C D y C C RT M t y j         (7)

Here,therelation D_b  MRT C/ _b const wasused.

Thus, the same equations, as in Fisher—Gibbs model, will be re-tained,neglectingdependenceofonCb [17].

Non-EquilibriumState.LetusassumenowthatGBchemicalpotential

for the diffusant is not equal to its value in the bulk. On the other hand,wewillkeeptheconditionofequalityofchemicalpotentialsfor thematrixelementatGBandinadjacentbulk.

TheGBconcentrationchangewithtimecanbedescribedas

. 2 div 2 div 2 ) , ( div 2 b b b 2 b b b b 2 b b                                              x x x x x x j D y RT f C y C D j D y f M y C C RT M j D t y j t C (8)

For demonstration ofthiseffect,the approximationsindicated above arekept.

The assumption of equality of chemical potentialof solvent at GB and inthe bulk, according [14], leads to the following concentration dependenceofGBsurfacetension:

1 (RT f/ ) ln(a1b/a1),

    (9)

where and 1 are theGB surfacetensionsof thesolutionwith given

concentration of solute (Cv) and of pure solvent, a1 and a1b are the

thermodynamic activities ofthe solventinthe bulk and inGB.With theuseofidealdilutesolutionapproximation(ai Xi)andassumption

thatEq.(1)isstillvalid,wearriveto

1b 1 1b 1 b b b b

ln(a / )a lnX lnX ln(1X)ln(1X) X X (X X/s). Theexpression(9)canberewrittenas

1 X RT f sb( / )( 1)/s

or,takingintoaccountthatf/andCX/,

1 C RT sb ( 1)/ .s

      (11)

Usingthesubstitution:

y C RT s s y C C y                      b b b 1 (12) and the solution for bulk diffusion along x-axis from the constant

source,onecanobtain:

). 0 ( 2 1 ) 0 ( 2 1 v b b 2 b 2 b v b b b b b                                                              x C t D y C s s C y y C D x C t D y C RT s s RT f C y C y D t C

Forthequasi-stationaryregime,wefinallygetthefollowingequation: 0 s 2 1 b b b b 2 b 2                                t D C D y C s s C y y C (13) or , 0 ) 1 ( ₂b 2 b 2 b 2 b               L C y C A y C AC (14) where A   (s 1)/s,and L2  s D_b[t/(4 )]D 1/2,or . 0 ) 1 ( ) 1 ( _b 2 b 2 b b 2 b 2                L AC C y C AC A y C (15) 3.RESULTSOFCALCULATIONS

SolutionofthisequationwithboundaryconditionСb(y0)Cb0 const

andСb(y)0canbewrittenasanintegral: b b0 1/2 (1 ) . (1 2 /3) C C L A y d A        



(16)

It iseasytoseethat,ifA0(s1),wegetFisher’ssolution.To illus-tratethissolution,thediffusionparametersweretakensuitabletoCu self-diffusion[18]. Forbulkdiffusion, 5 196800/( )

10 RT

D   e m2_/sec.For

2, the GB-concentration profilesfordifferents values in comparison withGibbs’solutionareshown.

Theresultsofcalculationshowthat:GBsurfacetensiongradient,as an additionaldriving force for GBdiffusion, alters the GBdiffusion profile; the concentration profile in this case is not linear in semi-logarithmic coordinates;the effect of GBsurface tension gradient is the mostimportant inthecase ofnegativesegregation, whilefor the

s1,itisnegligible.

OnecanseethateffectiveFisherlengthL(yvalue,correspondingto

b b0

ln(C /C )  )1 decreases with decreasing of s. Note that the less is thevalueofs,thelessisL.Consequently,theconcentrationgradient andthegradientofsurfacetensionincrease.Thatisthereasonwhythe additionalforceconnectedwithsurfacetensiongradientisimportant fornegativeadsorptiononly(s1).

4.APPLICATIONTOTHEFeGBDIFFUSIONINCu

In study ofFe diffusionin Cu[19],verystrange resultwasobtained. With the useofthemicroprobe analysis, itwasshownthat in awide temperaturerange(550—800C)thedifference betweenconcentration profiles nearandfarfromGBisverysmall.Theauthorscouldnot ex-plain that. It should benoted that the temperaturerange550—600C corresponds toregime «B» ofGB diffusion [20], ifannealing timeis about 100h. Let us take as approximation that GB diffusion coeffi-cient of Feis not very differentin comparison with CuGB diffusion coefficient(boththeatomicsizeandthebulkdiffusioncoefficientare almost thesame) and use themodel described above. Tocompare the

a b

Fig.2. Calculated GB-concentration profile for100h of annealing at 600C (Db 1.4107andDv 1.81012cm

2_{/s)fordifferent}

svalues(dashedlines cor-respondtothe Gibbssolution).

penetrationdepth,wecancomparethedepth(y*_),atwhichthe

concen-tration in e2 _{times smaller incomparison with surface concentration}

(ln(C0_/C)_{2). In the bulk for 100h, y}*₁₇m, _{and, in GB if s}_1,

y*₃₅m_{(Fig.2).Depending ons value,thedepthswillbeasfollow:}

y*_(s_0.1)₇m,_y*_(s0.5)₂₀m.

Unfortunately,wedo notknowthesegregationfactorforthis sys-tem, butaccording tosurface tension measurements:surface_(1200K)

1.8J/m2_forpureCuandsurface_(1200K)_1.9J/m2_{forCu—0.72at.%}

Fe [21]. Using the estimation GB  surface/3, the enrichment coeffi-cientcanbecalculatedfromEq.(10)(7.13106_m3_/mole):













1 1 b 6 2 9 1 1 3 3 7.13 1.8 1.9 10 1 1 0, 67 0,33. 3 0.72 10 8.31 1220 5 10 s s X RT XRT                                

Ofcourse,itisveryroughlyestimatedvalue.However,thetendency can beclearlyseen.Ifthesegregationfactorislessthan0.5,the esti-mated inFisher’smodelpenetrationdepthalong GBisless thanbulk diffusiondepth.

Finally,wecanconcludethatGBsurfacetensiongradientactsasan additionaldrivingforceanditmustbetakenintoaccountinthecaseof negativeadsorption(s1).

TheworkwasperformedwithfinancialsupportofRussianMinistry of Education and Science, Contract No.16.513.12.3009. Special thanks to Prof. L. Klinger (Technion, Haifa) for his very useful re-marks.

REFERENCES

1. J.Philibert,AtomMovements:DiffusionandMassTransportinSolids(Les EditionsdePhysique:1991).

2. B.S.Bokstein,M.I.Mendelev,andD.J.Srolovitz,Thermodynamicsand Ki-neticsinMaterialScience(OxfordUniversityPress:2006).

3. L.M.KlingerandE.Rabkin,DefectandDiffusionForum,309—310:19(2011). 4. B.Bokstein,A.Rodin,andB.Straumal,Proc.oftheInternationalWorkshop

‘GrainBoundaryDiffusion,Stresses,Segregation—DSS-2010’(June1—4,2010, Moscow).

5. L.KlingerandEu.Rabkin,ScriptaMat.,62:918(2010). 6. J.C.Fisher,J.Appl.Phys.,22,No.1:74(1951).

7. G.B.Gibbs,physicastatussolidi,16,No.1:27(1966). 8. R.T.P.Whipple,Philos.Mag.,45,No.371:1225(1954).

9. G.MartinandB.Perraillon,GrainBoundaryStructureandKinetics(Ed.R.W. Balluffi)(MetalsPark,OH:ASM:1980),p.239.

(1992).

11. B.S.Bokstein,A.S.Ostrovsky,andA.O.Rodin,Philos.Mag.A,72,No.4:829 (1995).

12. V.Esin,B.Bokstein,andA.Rodin,DefectandDiffusionForum,309—310:29 (2011).

13. Yu.MishinandChr.Herzig,J.Appl.Phys.,73:8206(1993). 14. A.A.Zhukhovitsky,J.Phys.Chem.,15:296(1944).

15. B.S.Bokstein,Ch.V.Kopetsky,andL.S.Shvindlerman,Thermodynamicsand KineticsofGrainBoundariesinMetals(Moscow:MetallurgyPress:1986). 16. G.GottsteinandL.S.Shvindlerman,GrainBoundaryMigrationinMetals

(Bo-caRaton:CRCPress:2010). 17. L.Klinger,Privatecommunication.

18. T.SurholtandChr.Herzig,ActaMater.,45:3817(1997).

19. D.Prokoshkina,V.Esin,andA.Rodin,DefectandDiffusionForum,323—325:

171(2012).

20. L.G.Harrison,Trans.FaradaySoc.,57:119(1961). 21. S.ZhevnenkoandE.Gershman,Privatecommunication.