lecture 15

NPTELPhaseII:IITKharagpur:Prof.R.N.Ghosh,DeptofMetallurgicalandMaterialsEngineering|||||

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Module 15

Diffusion in solids I

Lecture 15

Diffusion in solids I


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Keywords:Lawsofdiffusion,natureofconcentrationprofileininfiniteandsemiinfinitediffusioncouples,diffusivity,diffusionpaths,effectoftemperatureandcrystalstructureondiffusivity,

homogenization,hotforging,uphilldiffusion,chemicalpotential,activityIntroductionThreemodulesofthiscoursehavebeendevotedtointroducetheconceptofdiffusioninsolids.Atoms&moleculesinsolid,liquidandgasareneverstationary.Theykeepmoving.Inaliquidoragasonecouldseeorfeelthatthishappens.Ifyouleavealightedcandleinthecornerofaroomyoudoseethesmokecomingoutandspreadallovertheroom.Similarlyifyouleavesomearomaticsubstance(naphthaleneball) inadishyoucould feel thearomaeven fromadistance.Thishappensbecause themolecules /particles responsible for the smokeor thearomaareable tomovearound.Suchamovement in theabsenceofanyexternalforce iscalleddiffusion.Thistakesplace insolidsaswell.Weshall lookatthephenomenologicalaspectandlearnaboutthelawsgoverningdiffusioninthesemodules.Weshallalsolearnaboutexperimentaltechniquesbasedondiffusioncouplesthathelpsunderstandthemechanismsofdiffusion.Whatisdiffusion?Diffusion is themovementofany species inamediumalongagivendirection. It takesplace in solidliquidorgas. It isassociatedwiththemovementofatoms/moleculesofthemediumandthespeciespresentinit.Itoccursevenintheabsenceofanyexternalforce.Inliquidorgassuchmovementscanbeseenor felteasily.Slide1 suggestsa simpleexperiment todemonstrate theprocessofdiffusion inaliquid.

This illustratesa caseofmixing inabsenceofanyexternal force. Ifwe stir the liquid (orheat it) themixingwilltakeplacemuchfaster.Mixingintheabsenceofanyexternalforcedoesoccureveninsolid.Lawsofdiffusion:Diffusionistheprocessbywhichaspeciesmovesinagivendirection.Therateatwhichitmovescanbemeasuredintermsofthenumberofspecies/unitarea/unittime.Thisistermedasflux(J).Higherthedifferenceintheconcentrationofthespeciesinneighboringregionhigheristheflux.Theconcentration(c) isdefinedas thenumberof species inaunitvolumeof themedium (ormatrix).Thediffusionof

Slide 1: Take a glass of water with some soluble ink orpigment at the bottom as shown in the sketch on theextremeleft.Thegraphbesideitshowstheconcentrationofinkalongthexaxis&theheightalongtheyaxis.Noteinthebeginningonly theextremebottomappears red.With timethecolorkeepschanging.Theglass in themiddleshows itsappearance after a short time. The graph beside it showshow the concentration changeswithheight.The top is stillcolorless.Thecolorbecomesuniformafteralongtime.


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species (ormaterial) is governed by Ficks first law. This states that the rate of flow ofmaterial isproportional to its concentration gradient and it occurs down the gradient. In the case of aunidirectionalflowthisisexpressedas:

(1)

D isknownasdiffusivity.OftenJ isexpressedasnumber/(cm2sec)andconcentration ismeasuredasnumber/cm3,thereforethedimensionofDiscm2/sec.ThecorrespondingSIunitism2/sec.DerivationofFicksfirstlaw:Atoms or molecules in a medium are not stationary. They keep moving randomly in all possibledirections. Using this it is possible to derive the first law of diffusion. Let the concentration of aparticularspecies inamediumbec(x)atadistancexalongxaxis. Itsconcentrationatx+x isc(x+x)anditsconcentrationatxxisc(xx).Letusconsidera2dimensionalcasewhereatomscanmoveinallthefourdirections.Thereforethechancethatthespeciesmovesalongthedirectionxis.Fromtheexpressiongiven infig1theexpressionforthenetflux(J)canbeobtained.Notethat inthederivationtheconcentrationtermhasbeenwrittenintheformaTaylorseries.Thehigherordertermshavebeenneglected.

(2)

Thisshowsthatthefluxisdirectlyproportionaltotheconcentrationgradientandthediffusivityisgivenby

Numberofspeciesmovingxalongxintimet:

Fromlefttoright

Fromrighttoleft

xc(x)

xxc(xx) c(x+x)

x+x

xy

Fig.2Concentrationsofspeciesatthreelocationsnormaltothexaxisinamediumaregiveninthesketch.Theexpressionsontherightgivestherateatwhichspecieskeepsmovingfromlefttoright&viceversa

C

x

Figure 1 illustrateshow concentration varieswithdistance.Notethattheconcentrationofthespeciesatanypointdoesnotchangewithtime.Thisiswhytheconcentrationprofileofa species in a medium is linear. This means that theconcentrationgradientandthefluxJremainconstant.


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(3)

Thisdefinesthemobilityofthespeciesinthemedium.Itisdeterminedbytheshortestdistance(jumpstep)itcanmoveandthetimeittakesforthistohappen.Fickssecondlaw:Whatwouldhappen if the concentrationprofile at a given time t isnot linear as in fig 1.Here is asituationwheretheconcentrationgradientkeepschangingwithdistance.Accordingtothefirstlawtheflux isdirectlyproportional to the localconcentrationgradient.Therefore the flux Jatagiven time twouldbedifferentatdifferentlocations.Thisisillustratedinfig3. Let J denote the flux at a point x. The flux at the two points at xx and x+x are given by the and

respectively.Thedifferencebetween the twogives theamountof

thespeciesthataccumulateinthisspaceoverasmalltimet.Sincethevolumeofthespacebetweentheplanesshowninfig3is2xtherateofchangeoftheconcentrationduringthetime,t,isgivenby

(4)

On substitution of equation 1 andmaking the time interval extremely small the expression for thechangeinconcentrationcouldbewrittenas

(5)

Thisshowsthattherateofchangeofconcentrationatagivenpointwithtimeisproportionaltotherateofchangeofconcentrationgradientandthediffusivitywiththedistance.ItisknownastheFickssecondlawofdiffusion.Ifthediffusivityisindependentofconcentrationthelawcanbedescribedasfollows:

(6)

Theequation6describestherateofchangeofconcentrationduetodiffusionalongxaxisonly.Howeverinasolidsuchaprocesscantakeplacealonganydirection.Thereforeweneedamoregeneralformofthelawtodescribediffusionofspeciesin3D.Letusfirstlookatthenatureofdiffusivity.Isthisascalar,vectoror somethingelse? If you again goback to Ficks first law you find that itprovides a relation

x

CJ2

J1

xx x+x

Fig3:Thesketchshows theconcentrationofanelement inasolid as a function of distance x at given time t. Flux at anypoint is directly proportional to concentration gradient. Thefluxatxx isJ1andthatatx+xisJ2.J1>J2.Thisshowsthatmorenumberofatomsmove intothespacebetweenthetwodotted lines than that leaving this space. Therefore theconcentrationofatominthiszonewouldincrease.


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betweentwovectors.BothfluxJandconcentrationgradientarevectors.Youmayrecallthatstresstoorelatestwovectors;forceandarea.Weknowthatstress()isatensorofrank2.Therelationbetweentheforce(Fi)andthearea(Aj)intensornotionisgivenby wheretheuseofrepeatedsuffixdenotessummation.Thereforediffusivitytooisatensorofrank2.Followingtheaboveanalogyfluxduetodiffusionalongagivendirectionmaybewrittenas .Notethatconcentrationisascalar.Itsnature is something similar to thatof temperatureorpotential.Youneed two suffixes todenotediffusivity.Dependingonthematerialitmayhavedifferentvaluesalongdifferentdirections.SolutionofFickssecondlawina1Dcase:Letuslookatasimplecaseofmixinginsolidduetothemovementofaparticularspeciesthroughthelattice.Takethecaseofdiffusionofcarboniniron.Carboncandissolveiniron.Itsconcentrationcanbeincreasedthroughsolidstatediffusion.Keepalongpieceofironinafurnaceatagiventemperature.Infig4this isrepresentedbyredcolorbar indicatingthat it isuniformlyheatedtoatemperatureT.Forsimplicitywewouldconsidera1Dcaseonly.Thereforeallthesidesexceptthefrontfacelocatedatx=0arecovered.Thisensuresthatcarboncanonlydiffusealongthexaxis.Theconcentrationofcarbonatx=0 ismaintainedatcs for t>0.Thiscanbeachievedbymaintaininga suitablegaseousatmospherehavingtheappropriatecarbonpotentialtomaintaintheconcentrationofcarbonatadesiredlevel.Theboundaryconditionandthestatementoftheproblemareillustratedinfig4. There are standardmethods to solve equation 6 for specifiedboundary conditions. Ifdiffusivity is afunctionofcomposition suchequationsare solvednumerically.For thecasedescribed in fig4 ifD isassumedtobeindependentofconcentrationthesolutionisgivenbythefollowingequation. 1

(7)

Whereerf()denoteserrorfunction.Itisgivenby:

(8)

Therearestandardtablestoevaluatesuchfunctions.Mostelectronicspreadsheetshavethisasabuiltin function. Figure 5 gives a typical error function plot obtained using Excel. Note that the initialconcentrationofcarboninequation7hasbeenenteredasc0.Inthisspecificcaseitiszero.Thereforeondirectsubstitutionoftheinitialconditioninequation7itispossibletoshowthatatt=0c=csatx=0astapproachesinfinitycwouldapproachcsatallvaluesofx.

0, , 0 0 0, , 0

x

ccs

Fig4:TherectangledenotesapieceofironatatemperatureTinafurnace.Thegraphshowsthatthecarboncontentatx=0iscswhereasatx>0 it is0.Thisdefinesthe initialcondition.Thecarboncontentat theexposed face ismaintainedat thislevelatalltimes.Theequation6istobesolvedtogetc(x,t).


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It isthereforepossibletogenerateconcentrationprofilesatanytimetasa functionofdistance foragivenvalueofD.Figure7presentsasetofsuchprofilesfordifferentvaluesoftime.InthiscaseDhasbeentakenas104cm2/sforanarbitraryspeciesandtimesinthefig7areinhours.Notethatatshortertimestheprofile isstiffat lowervaluesofxand itbecomesasymptoticathighervaluesofx.Astimeapproachesinfinitytheconcentrationtendstobecomeuniformallthroughthesection.Thereisaclosesimilarity between the error function and standard normal distribution. This iswhy it is possible tosimulatediffusionasarandomwalkprocess.

Fig6:Usingequation7theconcentrationofthespecieshasbeenplottedasafunctionofdistancefor

differentvaluesoftime.Diffusioncoupleconsistingoftwosemiinfinitebarsofmetalsjoinedataninterface:Diffusionisaprocessofmixing.Itisassociatedwithmovementofatomsinthelattice.Thedifferenceinconcentrationofagivenspeciesinthematrixisthedrivingforce.Letusconsidertwoalloysmadeoftwometals (sayCu&Ni)havingdifferent compositions joined at an interface as shown in fig7where c

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

erf(z

)

z

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.50 1.00 1.50 2.00 2.50

C

x

1

2

5

1000

Fig5:Atypicalerrorfunctionplotobtainedfromexcelspreadsheet.


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denotesconcentrationofNioneithersidesoftheinterfaceasfunctionofdistance.Thealloyonthelefthas lower concentration of Ni than that on the right (c1 0x0x0x


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Self diffusionDoes atoms in pure metal move about?

Is there a way to find out?

No. atoms moving towards right = No. atoms moving towards left

Net flux = 0

Use radioactive tracer

Isotope has same electronic structure but different mass. Distinguishable.

D*=Dt

Slide2suggeststhatthattheselfdiffusioncoefficientcanbeestimatedbyaddingradioisotopesatoneend of the solid andmonitoring how these atomsmove into the solid. Isotopes are atoms havingidenticalelectronicstructurebutdifferentmass.Oftenthemassdiffersbyoneatomicmassunit.Suchatomsareunstabletheytrytogetbacktothemorestablestructurebyemissionofradiations.Thiscanbedetectedby suitablemeasuringdevices like counters. If the concentrationof traceratoms canbeobtainedasafunctionoftime&distance;thediffusivityofthetracercanbeestimated.Sincethesizeand themassofthetraceratomandthebaseatomarenearlythesame itmaybeassumedthatthediffusivityofthetwospecieswouldbethesame.Temperaturedependenceofselfdiffusioncoefficient:We have just seen how the self diffusion coefficient can be estimated using radioactive tracer.Diffusivity (D) of a given species in a specificmedium can also be obtained from the concentrationprofile along a diffusion couple. Diffusion of species is associatedwith themobility of atoms. Thisincreases with increasing temperature (T). Therefore it is expected to be a strong function oftemperature.Thetemperaturedependenceofdiffusivityisoftenrepresentedasfollows:

(10)

WhereRistheuniversalgasconstant,QistheactivationenergyfordiffusionandD0isaconstant.IfDisknownatseveraltemperaturesQ&D0canbeobtainedfromtheslopeandinterceptsoftheplotgiveninfig8.

Slide 2: The sketch on the left illustrates howatoms are arranged in the lattice along withvacantsites.Thelinedenotesanimaginaryplane.Thearrowsrepresentthedirectionofmovementof atoms.At any instant the numbermoving tothe right issamethenumbermovingtothe left.Therefore thenet flux is zero. Is there away toestablish thatdiffusion takesplace even inpuremetals? Is itpossible to find itsdiffusivity? Thesketchontherightsuggeststhatbyaddingtraceratomsitmaybepossibletodoso.

ln(D)

1/T

D0Slope=Q/R Fig8:Illustratestemperaturedependenceofdiffusivity.

Iffollowsfromequation10that ln .Thereforeintercept=ln(D0)andslope=Q/R.


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Diffusionpathsinsolids:Diffusionofspecies insolidsalsodependsonthepaththat itfollows. Inasolidwecanthinkofthreedistinctpaths.Metalsaremadeofseveralcrystalsthatmeetalonggrainboundaries.Ifaspecieshastomovethroughthisitcouldeithermovethroughthegrain,thegrainboundaryorthetopsurface.Thisisillustrated in slide 3. Thediffusivity through the grain isdenotedbyDg. This isoften known asbulkdiffusioncoefficient.Thediffusivitythroughthegrainboundary isdenotedbyDgb.Thespacebetweenatomsbecauseofirregulararrangementismoreatthegrainboundarythanthatwithinthegrains.Thisiswhythemobilityofatomsthroughthegrainboundaryisexpectedtobehigherthanthatthroughthegrain. The same logic can be extended to the exposed top surface. There is enough space toaccommodateextraatomsatthefreesurfaceifrequired.Thereforethemobilityoftheatomsalongthefreesurface(Ds)ismuchhigher.TherelationbetweenthethreecouldbedescribedasDs>Dgb>Dg.

Diffusion paths in solids

surface

Grain boundary

grain

D

1 / T

Ds

DgbDg

Ds > Dgb > Dg

Theslide3alsoillustratesthetemperaturedependenceofthediffusivitythroughthreedifferentpaths.Theactivationenergyofsurfacediffusionislikelytobethelowestandthatforthegrainisthehighest.MathematicallythisisdenotedasQg>Qgb>Qs.Diffusionasarandomwalkprocess:Theprocessofdiffusion is governedby themovementof atoms. In solidswhere atoms are closelypackedsuchmovementswillbedifficult intheabsenceofvacantsites.Atagiventemperatureseverallatticesitesarevacant.Ifthereareseveralsitesaroundanatomhowwouldanatomdecidewheretomove?An obvious option could be a random selection. Slide 4 illustrates the difference between anormalandarandomwalk.

Slide3:Showsthreedistinctpathsinasolidthroughwhichanatomofagivenspeciescoulddiffuse.Thediffusivity of the species through these paths isdenoted as Ds (surface diffusion), Dgb (grainboundary diffusion) and Dg (diffusion through thegrain). These are strong functions of temperature.The temperature dependence is verymuch similartotheexpressiongiveninequation10.HoweverthemagnitudesofQ&D0arelikelytobedifferent.


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Diffusion as random walk

Distance cover in time t during normal walk along a specific direction= velocity x time = v t

Distance covered in time t during random walk along a specific direction


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higherthanthatofBCC.Ifwewantthesteeltodissolvesubstantialamountofcarbonwemusttakeittobeyond 910C.However carbon can also diffuse in its BCC state. The rate of carbon pick upwoulddependonitsdiffusivity.TheBCCformofironisknownasferriteandoftenrepresentedas.TheFCCformofironisknownasaustenite.Itisrepresentedas.Thediffusivitiesofcarboninthesetwoformsofironareasfollows:

2 10 /(15)

2 10 /(16)

Ifyouestimatetheseoverarangeoftemperatures(6001000C)andplotthedataagainstreciprocaloftemperatureinKyougetaplotasshownin9.

Fig9:Diffusivityofcarbonintwoformsofironferriteandausteniteasfunctionoftemperature

Austenite is a close packed structure in comparison to ferrite. Therefore diffusivity of carbon inaustenite is lowerthanthat in ferrite.Thesolubilityofcarbon inaustenite ismuchmorethanthatofferrite. Therefore in order to have uniform distribution of carbon in steel it has to be heated toausteniticstate.Thediffusivityofcarbon inaustenitebeingvery lowthetimeneededhomogenizationofcaststeelcanbeverylong.Howtoreducetimeneededforhomogenization:Solidificationofmetalsandalloysalwaysresultsinsegregation.Thisisbecausetheonlywayaspeciesinasolidcanmoveisbydiffusionwhichisaslowprocess.Itmeanscaststructureisrarelyhomogeneous.In order tomake the solid homogeneous it is oftenheld at a high temperature for long hours. Theeffectivediffusiondistance isapproximatelyequal to.A simplecalculationwould show that thetimeneededforhomogenizationevenifthedistanceoverwhichheterogeneityexistsisoftheorderofa

1.00E12

1.00E11

1.00E10

1.00E09

6.00 7.00 8.00 9.00 10.00 11.00 12.00

Dm

2 /s

10000/T,K1

Ferrite

Austenite


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fewmmcouldbevery long.Hotforging isaprocessthatcombinesdiffusionwithmechanicalworking.Thecombinedeffectsofdiffusionandappliedstresshavingachurningeffectonthesolidcanhelpcutdownthetimeneededforhomogenizationsignificantly.Solidificationofmetalsandalloysalwaysresultsinsegregation.Thisisbecausetheonlywayaspeciesinasolidcanmoveisbydiffusionwhichisaslowprocess. Itmeanscaststructure israrelyhomogeneous. Inordertomakethesolidhomogeneous it isoftenheldatahightemperatureforlonghours.Theeffectivediffusiondistanceisapproximatelyequalto.Asimplecalculationwouldshowthatthetimeneededforhomogenizationevenifthedistanceoverwhichheterogeneityexistsisoftheorderofafewmmcouldbeverylong.Hotforgingisaprocessthatcombinesdiffusionwithmechanicalworking.Thecombinedeffectsofdiffusionandappliedstresshaving a churning effect on the solid can help cut down the time needed for homogenizationsignificantly.Applicationsthatdemandhomogeneousalloysprefertousecomponentsthathavebeenproducedbyhotupsetforging.Uphilldiffusion:Diffusion in solid as described above has been visualized as a process ofmixing. The concentrationgradienthasbeenassumedtobe itsdrivingforce.Themassflowtakesplacedowntheconcentrationgradient.Thismaytrueforsolidsolutionshavingasingleminimuminitsfreeenergycompositionplotasshown in fig11.However thereareseveralalloysystemswherediffusionmay takeplaceagainst theconcentration gradient. Such a system has free energy composition plots havingmultipleminima asshowninfig12.Thisisknownasuphilldiffusion.Thisisbecausethetruedrivingforcefordiffusionisthechemicalpotentialorthepartialmolarfreeenergyandnottheconcentration.Themolarfreeenergy(G)ofabinarysolidsolutionconsistingoftwometalsA&Bisgivenby: NA&NBdenoteatomfractionA&B,&denotethefreeenergiesofpureA&B,Ristheuniversalgasconstant,TisthetemperatureinKelvin,A&BdenotethechemicalpotentialsofA&BandaA&aBdenotetheactivities(effectiveconcentrations)ofA&B.Thetwoactivitiesaregivenby :Activitycoefficient 1inanidealsolidsolution :Activitycoefficient 1inanidealsolidsolutionFigure10showsatypicalfreeenergycompositiondiagramofabinaryalloyatagiventemperature.TheinterceptofthetangentatapointG1withtheverticalaxisatAdenotesdenotesthepartialmolarfreeenergyofAinalloy1.Itisalsoknownasitschemicalpotential.TheinterceptofthesametangentwiththeaxisBdenotesthechemicalpotentialofBinalloy1denotedas .NotethemagnitudesofthechemicalpotentialsofA in2andB in2 ( ).Figure10 includesthesketchofadiffusioncouplemadeoftwoalloys1&2havingdifferentcompositions.Alloy1isrichinAwhereasalloy2isrichinB.Inthiscase thereforeAdiffuses from1 to2.Here aswell. It representsacasewherediffusiontakesplacedowntheconcentrationgradient.Figure11showsatypicalfreeenergycompositiondiagramofabinaryalloyatagiventemperature. Ithastwominima.TheinterceptofthetangentatapointG1withtheverticalaxisatAdenotesdenotesthepartialmolarfreeenergyofAinalloy1.TheinterceptofthesametangentwiththeaxisBdenotes


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thechemicalpotentialofBinalloy1denotedas .NotethemagnitudesofthechemicalpotentialsofAin2andBin2( ).Figure11includesthesketchofadiffusioncouplemadeoftwoalloys1&2having different compositions.Alloy 1 is rich inAwhereas alloy 2 is rich in B. In this case thereforeAdiffuses from alloy 2 to 1 although . It represents a casewherediffusion takesplaceagainsttheconcentrationgradient.

BA 1 2

G

Ga

Gm

1 2Arich BrichA

B

G2G1

Fig 11: The sketch at the top represents adiffusioncoupleconsistingoftwoalloys1&2.Alloy1isrichinAandalloy2isrichinB.Thesketch at the bottom gives free energycompositionplotG.G1 isthefreeenergyof1andG2isthefreeenergyof2.Gadenotesthecombinedfreeenergyof1&2.Gmisthefreeenergy of themost stable composition. Thevertical arrow head ending at this point is ameasure of the driving force for diffusion.NotethepointsofintersectionofthetangentatG1&G2with the vertical axis at A. Since A diffuses from 2 to 1. Diffusiondirection isshownwiththehelpofanarrow. : Hence B diffuses as shown fromalloy1to2.Thisisacaseofuphilldiffusion.

A B

G

1 2

G1 G2

Gm

1 2 Fig 10: The sketch at the top represents adiffusioncoupleconsistingof twoalloys1&2.Alloy 1 is rich in A and alloy 2 is rich in B. Grepresentsthefreeenergycompositionplot.G1isthefreeenergyof1andG2isthefreeenergyof 2.Gm is the free energyof themost stablecomposition.Theverticalarrowheadendingatthispoint isameasureof thedriving force fordiffusion.Notethepointsofintersectionofthetangent atG1&G2with the vertical axis atA.Since Adiffusesfrom1to2.Diffusiondirection is shownwith the help of an arrow. :HenceBdiffusesasshownfromalloy2to1.Thisisacaseofdownhilldiffusion

AB


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Thissuggeststhatitisthechemicalpotentialgradientthatdeterminesthedirectionofflowofspecies.Thereforethemoregeneralexpression forFicks lawshouldbeexpressed intermsofeitherchemicalpotentialoractivitygradient.Thebest knownexampleofuphilldiffusionwas first reportedby L SDarken inTransAIMEVol.180(1949)p430438.Adiffusioncouplewasmadebyweldingtwopiecesofsteelonehaving3.5wt%Si&0.49wt%Candtheotherhaving0.05wt%Si&0.45wt%C.Thiswaskeptto1050Cfor13days.Atsucha high temperature themost stable form of iron is austenite having FCC structure. The solubility ofcarboninausteniteismuchhigherthantheamountofcarbonpresentinthetwosteel.Figure12showswiththehelpofaschematicdiagramcarbonconcentrationprofilebeforeandafterhighertemperaturethermalexposure.Nodoubtcarbon isexpectedtodiffusefromthesteelonthe lefthaving0.49%Ctothaton the righthaving0.45%C.However%carbonnear the interfaceonthe leftwas foundtohavegonedownto0.33%whereasthatontherightofthe interfacehadgoneupto0.59%.Clearlythis isacasethatillustratesdiffusionofaspeciesinadirectionagainsttheconcentrationgradient.ThisunusualphenomenonwasattributedtothepresenceofSithatalterstheactivity(orthechemicalpotential)ofCiniron.

SummaryIn thismodulewehave learntaboutasolidstatemixingprocesscalleddiffusion.This isgovernedbyFickslawsofdiffusion.Thefirstlawdescribessteadyastateprocesswhentheconcentrationofagivenspeciesatapointdoesnotchangewithtime.Inthiscasetheconcentrationgradientremainsconstantatallpoints. If theconcentrationgradient isa functionofdistance then theconcentrationatapointwouldkeepchangingwith time.This isgivenbyFickssecond law.This isexpressed in the formofadifferentialequation.Thenatureofsolutionofsuchanequationin1Dhasbeendiscussedwithoutgoingintoitsderivation.Attemptshavebeenmadetogiveaninsightintoitsmechanism.Itcanbesimulated

3.8Si0.49C 0.05Si0.45C

0.49%C

0.45%CTime=0

Time=0

Time=13days

Time=13days0.55%C

0.33%C

Fig12:ThesketchatthetopshowsaschematicrepresentationofadiffusioncouplemadeoftwosteelhavingdifferentamountsofCandSi.Thesketchatthebottomshowsthe initial(t=0)andthefinal(t=13daysat1050C)concentrationofCinthetwosteel.


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asa randomwalkprocess.Theaverage randomwalkdistancehasadirect correlationwithdiffusiondistance.Wealso lookedat theeffectof temperatureondiffusivity.Usingan illustration ithasbeenshownthatdiffusiondependsontemperatureaswellascrystalstructure.AlthoughFicks law isoftenwrittenintermsofconcentrationgradienttherearecaseswherediffusionmaytakesplaceagainsttheconcentrationgradient.This isknownasuphilldiffusion.This suggests that it ismoreappropriate todefinethisintermsofchemicalpotential.Exercise:

1. If iron is kept at 1200K in a carburizing atmosphere for 8hrs to obtain a carbonconcentration of 0.75 at a depth of 0.5mm. Find the time itwould take to reach samecarbon concentration at depth of 7.5mm at 1250K. (Given D0 = 0.2x104 m2/s & Q =143kJ/mole/K)

2. Asteelcontaining0.2%carbonwasheatedto950Kfor15hours.Findthedepthoflayerin

whichthereisnocarbide.Assumethatsteelconsistsofferriteandcarbide.Thesolubilityofcarboninferriteatthistemperatureis0.015%and%Catthesurfaceisnegligible.(GivenD0=2x106m2/s&Q=84.4kJ/mole/K)

3. The concentrationof carbonon the surfaceof iron ismaintainedat1.00%at1175K for

2hours.Estimatethedepthatwhich%Cwouldbe0.5%.Usethediffusivityvaluesgiveninquestion1.Assumeinitialcarboncontentofirontobenegligible.

Answer:

1. If iron iskeptathigh temperature inanenvironmenthavinghighcarbonpotential itdiffusesinto iron. The depth of carburization (x) is proportional to. Therefore

Since

Therefore ..

.

11200=10.14hours

2. If initialcarbon insteel isCi&solublecarbon inferrite isCamountcarbontoremoved/unitcross section through a distance dx = (CiC)dx = flux of carbon atom in time dt = Jdt = Assuming carbon concentrationat surfaceas0

Thus

or; 2 10

. 4.55 10m2/sNowCi=

0.2&Ca=0.015Therefore .... 0.00063m=0.63mm

3. Carboncontentat surface Cs=1.00, InitialcarbonC0=0.0&carboncontentatadistancexC=0.5


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Or,

...

Or,

0.477

Or, 0.477 2 0.2 10 . 2 3600=0.000425m=0.445mm

lecture 15

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