Aula Balancim Difusao

8/2/2019 Aula Balancim Difusao

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C H A P T E R 6D I F F U S I O N

Consider the simple experiment shown in Fig. 6 .1. A vertical glass tubecontains a column of clear water above separated from a column of inkedwater be low by a th in d iaphragm. At t ime zero we gen t ly remove thed iaphragm in a ma nner such tha t the water remains s tagnant . We now ask :As t ime passes what happens to the ink? One f inds tha t the dark co lor o fthe ink s lowly migra tes upward so tha t a f ter 5 hours the co lumn mightappear as shown in Fig . 6 .1 . Somehow the molecu les p roducing the darkco lor of the ink have migra ted upwa rd . The water has rema ined s tagnant sothat this motion must have been accomplished by an actual preferredmigration of the ink molecules; that is, the movement results frommolecular motion. This is a form of mass transport that is called diffusion.Mass transport in gases and l iquids generally occurs by a combination ofconvection (f luid motion) a nd diffusion . In solids, convection d oes no toccur and consequently diffusion is generally the only available masst ranspor t mechanism; therefore , i t i s an impor tan t mechanism contro l l ingthe rate of many physical processes of interest to us.We can s tudy d i f fusion f rom two genera l approaches .

Clearwater

Ink inwater"

Figure

> Diaphragm

'ime = 0 Time = 5 hr6 .1 D i f f us i on o f i nk i n wa t e r .

137


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138 Diffusion1. Pheno meno logical approach . He re we ask : Ho w can we descr ibe thera te and , hence, the amount o f mass t ranspor t that occurs in t erms ofparameters we can measure? This approach i s impor tan t to our ab i l i ty tocont ro l such processes as carbur iz ing , n i t r id ing , t emper ing , homogeniza-t ion of cast ings, and the l ike.2 . Atomic appro ach . H er e we ask : W hat is the a tomic mech anism bywhich a toms move? This approach i s impor tan t to our unders tanding ofhow d i f fus ion mechan isms a f fect such processes as p recip i ta t ion harde ning .We wi l l d i scuss d i f fus ion under these two above categor ies ,

6 .1 P H E N O M E N O L O G I C A L A P P R O A C HA simple unidirect ional diffusion experiment involving sol ids is shown inFig. 6.2. A rod of pure i ron is but t welded to a s teel rod containing 1 wt%carbon and th i s "d i f fus ion cou ple" i s hea ted to 700C to a l low d i f fus ion tooccur a t a s ign i f i can t ra te . Af ter some t ime a t t emperature the d i f fus ioncouple i s quenched to room temperature and the compos i t ion of carbonalong the rod i s deter min ed b y chemical analys i s. Th e compos i t ion prof i l emight look as shown by the dashed curve l abeled t= t in Fig . 6 .2 . Af ter a ninf in i t e t ime the compos i t ion wi l l become cons tan t a t the average value .Our problem i s to descr ibe the ra te a t which the carbon a toms move to theright . This type of a diffusion problem was s tudied by Adolf Fick in a paperpublished in 1855.1 H e fou nd that the f lux of a toms w as propo r t ional to the .

Fe + 1% C Pure Fe

0.01 t 0. t = tV \\\t = \

\ \0 Position, Z

Figure 6.2 A d iffusion couple showing variation of com position with position andt ime.


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6*1 Phenom enological Approach 139volume concent ra t ion grad ien t , so that one has

(6.1)This eq uat i on is cal led Fic k 's f irst law of diffusion . T he fo l lowin g poin tsshould be no ted .

1. J , is defin ed as th e flux of ato ms 1 (in this case car bo n atoms ), an d i thas uni ts of ei ther g/cm 2 -sec or atoms/cm 2 -sec . The f lux may b e thoug ht o fas the rate at which atoms cross a uni t area, that is , toins pr sec/cm 2 .2 . D is simply the proport ional i ty constant and i t is cal led the diffusioncoefficient . I ts uni ts are cm 2 / sec .3 . C i i s a vo lum e concent ra t ion of com pon ent 1 , e i ther g /cm3 o ratoms/cm 3 . Not ice that a chemis t does no t d i rec t ly measure volumeconcent ra t ion bu t ra ther he measures weight percen t carbon , which i s a

fractional concent ra t ion tht we wi l l t e rm Xi . The re la t ion Between thesetwo concentrat ions is s imply Ci = Xi (densi ty), where the densi ty is ei thera mass dens i ty or an a tom dens i ty depending on the un i t s being used forvolum e conc ent ra t ion . ( In th i s t ex t we wi ll use the t erms concen t ra t ion andcompos i t ion in terchangeably . )4 . Th e minus s ign i s requi red beca use the a tom s f low towa rd the lowercon cen trat ion s. No tice in Fig. 6. 2 tha t the atom s mov e to th e right , which isthe pos i t ive d i rect ion . However , the concent ra t ion grad ien t that causesthem to migrate i s negat ive; see the curve for t= t . Hence, to make the f luxpos i t ive in Eq . 6 .1 we mus t use a minus s ign to compe nsate fo r the negat iveg rad i en t .

From the above we conclude that whenever a concent ra t ion grad ien t i spresent in a metal , a diffusion flux wil l occur. Later we wil l see that this isgeneral ly, but not always, t rue. Our problem now is : Hoyv do we actual ly-determine D? Ry cons ider ing the exper iment o f Fig . 6 .2 you wi l l see thato n e can n o t m eas u re e i t h e r h o r D t di rect ly . What we can measure i s thecompo s i t ion prof i l e af t er var ious t imes so that we me asure co mpos i t ion asa funct ion of Z and t. The concent ra t ion grad ien t o f Eq . 6 .1 wi l l vary wi thboth pos i t ion and t ime, and consequent ly so wi l l the f lux . Therefore , wemus t de term ine a d i f feren t ia l equat ion for th i s d if fus ion process* To d o th i swe s imply perf orm a mass balance upon a d i f feren t ia l vo lum e e le me ntperpendicular to the mass flow direct ion as in Figs 6.3. We may wri te forcarbon t ranspor t on th i s e lement

M a s s i n - m a s s o u t = A c c u m u l a t i o n ( 6 .2 )By cons ider ing some t ime in terval we have

R a t e i n - r a t e o u t = R a t e a c C u r i a i


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140 Diffusion

1 2Figure 6.3 A differential volume element for unidirectional diffusion.

All of the mass com es in to the vo lume e lem ent th ro ugh p lane 1 so that th erate of mass in is s imply the flux at 1 t imes the area at 1,Ra te mass in = (JA)i (6 .4)Since dZ is a d i f feren t ia l l eng th we ob ta in the ra te ou t o f the vo lumeelement a t p lane 2 by adding the change in ra te go ing across the vo lumee l em en t a s

R a te m ass o u t = ( J A ) , + ^ ^ d Z (6.5)Th e ra te of accumulat ion i s now wri t t en in t erms of the vo lume conce nt ra-t ion as

R a t e a c c u m u l a t i o n A dZ ^ (6.6)at atWe now subs t i tu te Eqs . 6 .4 , 6 .5 , and 6 .6 in to Eq . 6 .3 , cancel t erms , andobta in

M J S (6.7)dZ dt Vwhe re we have assum ed that the are a A i s cons tan t in the Z d i rect ion . Thisi s a very impor tan t equat ion , ca l l ed the cont inu i ty equat ion , which i sl imi ted by our der ivat ion to un id i rect ional t ranspor t . Not ice that ourt rea tm ent here assum ed that mass t ranspo r t occurred in on ly on e d i rect ion .For the general th ree-d imens ional case the der ivat ion i s s imi lar and weobta in Eq . 6 .7 wi th d/dZ rep laced by the del opera tor . The equat ion ho ldsfor a l l mater ia l f low processes when no mater ia l i s generated wi th in thevolume e lement , fo r example , f low of heat , neu t rons , e lec t rons , and so on .Our flow process is mass diffusion and by subst i tut ion of the diffusion fluxequat ion , Eq . 6 .1 , we ob ta in for one-d imens ional d i f fus ion

d[ D idC i/dZ]_dC1 f68ydZ dt v 'This equat ion is sometimes cal led Fick 's second law. It is a part ial


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6.1 Phenomenological Approach 141

- -< o +vi. Distance, T. Figure 6.4 Composition profiles in an iron-steel diffusion couple.

d i f feren t ia l equat ion wi th C i as the dependent var iab le and Z and t as thetwo indep end ent va r iab les . Henc e, so lu t ions to th i s equ at ion w i ll g ive C t asa funct ion of Z , t, an d D i . S ince our exp er ime nta l data g ive us C i = funct ion(Z, i ) as shown on Fig . 6 .2 , we may f i t these data to the mathemat icalso lu t ion of Eq . 6 .8 for var ious values of Di and then determine D t as thatvalue g iv ing the bes t f i t* An example so lu t ion wi l l now be presen ted toi l lus t ra te th i s t echnique.We co n s i d e r t h e Fe -C ex am p l e m en t i o n ed ea r l i e r an d r ed raw t h ecompos i t ion prof i l es as shown in Fig . 6 .4 . The vo lume concent ra t ion C i sp lo t t ed here versus d i s tance for var ious t imes . We assume that D i s acons tan t , so that Eq . 6 .8 becomes a l inear par t i a l d i f feren t ia l equat ion ,

az2 at (6.9)W e may so lve th i s p rob lem fa i r ly eas ily us ing the Laplac e t rans fo rma t ion ifwe make two addi t ional assumpt ions , which tu rn ou t to be very real i s t i c .As sum e: ' . ,

1 . A t a ll t i m es f> 0 , t h e co n cen t r a t io n a t t h e i n t e r f ace , Z = 0 , r em a i n sCo/2 . This assumpt ion requi res that the a tom veloci t i es do no t dependon concent ra t ion so that the decay on the l ef t i s symmet r ic to the bu i ldupon the r igh t .: 2. Th e ba r is sufficient ly long tha t th e con cen trat io ns at ei ther- end ar e


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142 Diffusionunaffe cted by the d i f fus ion process . To so lve the proble m w e cons ider on lythe por t ion of the rod whe re Z > 0 and we wri t e CBoundary condi t ions : C(Z~0,t) = ^

C ( Z = , i ) = 0 (6 . 10 )Ini t ial con di t ion : C(Z , 0) = 0

The Laplace t rans format ion us ing t as the independent var iab le i sappl ied to Eq . 6 .9 and an ord inary d i f feren t ia l equat ion i s ob ta ined thatmay be eas i ly so lved for the boundary condi t ions . The t rans formedsolut ion is found to beC (Z , s ) = y [ V ^ ] (6.11)

Taking the inverse t rans form we f ind as the so lu t ion of Eq . 6 .9 undercondi t ions 6 .10 r ^ rxn-jot "I

This equat ion is not nearly as messy as i t looks at f i rs t encounter becausethe funct ion e~y* decays rap id ly f rom one to zero . The in tegra l funct ion i scal led the error function and i t is defined as ,E r ro r fu n c t i o n o f 0 = E r f [0 ] = -j= f V y 2 dy (6.13)V 7T Jo

Consequent ly we may wri t e the so lu t ion asf^^fc^O


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OSt--


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144 DiffusionA . C A R B U R I Z I N GPerhaps the most important appl icat ion of the principles of diffusion inmetal lurgy involves the carburizat ion of s teel . Suppose a rod of pure i ronhas one end packed against graphi te as shown in Fig. 6.5 and i t is heated to700C. Wi th in a few minutes o f ach iev ing t emperature a local equilibriumwil l be es tab l i shed a t the graphi te- i ron in ter face . This means that thecompos i t ions of the two phases touching each o ther a t the in ter face areg iven by the equi l ib r ium ph ase d iagram at 700C. From F ig . 6 .5 (b) we seethat at 700C a i ron is in equi l ibrium with the carbide phase, Fe 3 C, cal ledcement i t e . (Actual ly , Fe 3 C i s a metas tab le phase , bu t i t never thelessusual ly forms). In order to establ ish the local equi l ibrium i t is necessary tomix the graphi te wi th a ca ta lys t o r to use cer ta in gas a tmospheres , bu t wewil l not discuss this difficul ty here. A carbide layer forms on the surface ofthe i ron as a resul t of the local equilibrium and the carbon compos i t ion inthe i ron r igh t a t th is in ter fa ce is deter min ed f rom th e phase d iagra m as C, .Phys ical ly , th i s me ans tha t a t th e l ef t sur face of the i ron bar t he comp os i -t ion jumps to a value of C s at t ime zero and remains there . This causes avery l arge carbon con cent ra t ion grad ien t to be gene rated a t the l ef t end ofthe i ron rod and so carbo n d i f fuses in to the rod a t a h igh ra te p roducingcompos i t ion prof i l es in the rod th at vary wi th t ime as shown, for ex ample ,on Fig . 6 .5 (a) . To determine these concent ra t ion prof i l es we mus t so lveFick 's second law for the boundary condi t ions of th i s example . We maywri te these condi t ions as

B o u n d ary co n d i ti o n s:In i t ia l co ndi t ion :

C(Z=0,t) = aC ( Z = o , t ) = 0C ( Z , 0 ) = 0

(6.16)

Graphite Iron

( a)

700" C

(b )Figure 6.5 (a) Com position profiles for carburizing iron, (b ) Pertinent portion ofi ron-carbon phase diagram.


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6.1 Phenomenological Approach 145whe re we ha ve assu med the ba r suffi c ien t ly long so that n o carbo n d i ffusesto the right-hand end during the 700C anneal . Notice that these condi-t ions are iden t ica l to condi t ions 6 .10 wi th Co/2 rep laced by C,. H e n c e t h eso lu t ion to our carbur iz ing problem i s

C ( Z , t) = G [ l - e r f Z / 2 V D i ] ( 6.1 7)This equat ion i s res t r i c ted to carbur izat ion of a rod that conta ins nocarbon . If the rod to be carbur ized a l ready conta ins a un i form compos i t ionof carbon less than C cal l i t C, , then you may show yourself from Eq. 6.15that the solut ion is

C (Z , t) = G [ \ - ( l ~ ) erf Z /2 V D t] (6.18)Su p p o s e we d e f i n e t h e case depth as the dep th in to the bar that conta insa carbon concent ra t ion above some arb i t rary value C c. The case dep ths a tthe th ree t imes ( i , t2, and t3 of Fig. 6.5(a) are shown on the figure for thecondit ion of Cc = CJ2. Suppose we ask : How far does the case dep thextend in to the bar as a funct ion of t ime for the condi t ion C c = CJ2.Assuming the bar o r ig inal ly conta ined no carbon we wri t e Eq . 6 .17 as

w h e r e Z0.s i s the case de p th fo r G = 0 .5C, . Rearran ging w e hav e, - ( i s )

From Table 6 .1 we f ind er f 0 .477=5 so that we ob ta inZo. 5 = 0 . 9 5 W D i ( 6 .2 1 )

I f we had t aken C c = 0 . 2 5 G we wo u l d h av e fo u n d t h e s am e r e s u lt w i th t h econs tan t 0 .954 rep laced by 1 .6 . The general resu l t may be wri t t en 9sZCc = const yfDt (6.22)This is a very s ignificant resul t because i t shows that the thickness of a"case dep th" i s p ropor t ional to > /Dt . Of ten in cons ider ing anneal ingprocesses i t i s des i rab le to b e ab le to es t imate ho w far a tom s wi ll mo ve bydiffusion in a given t ime. In the absence of a solut ion to Fick 's second lawfor the problem at hand one may es t imate the d i f fus ion d i s tance s imply asVDf for a reasonable f i r s t -o rder approximat ion .Suppose one had heated the i ron rod of Fig . 6 .5 to a t emperature T t


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14 6 Diffusion

Figure 6.6 (a ) Pert inent port ion of i ron-carbon phase diagram. (b) Composi t ionprofiles for carburizing iron above the eutectoid temperature.above the eu tecto id t emperature as shown on Fig . 6 .6 (a) . In th i s case avery in teres t ing fact i s observed when the rod i s examined . A sharp phaseboundary i s found to move down the bar as d i f fus ion proceeds . The localequi l ib r ium at the graphi te- i ron in ter face f ixed the compos i t ion a t the l ef tend of the bar at C 3 . A t t esmperature T\ i ron wi th a carbo n conten t betw eenC 3 a n d C 2 must be y i ron (fee) and i ron wi th C < C i mus t be a i ron (bcc).Since the carbon content at the surface is f ixed at C 3 we m u s t h av e y i ron a tthe i ron-graphi te in ter face and we mus t have a i ron far f rom the in ter facewh ere C < C i . N o t i ce o n t h e p h as e d iag ram t h a t a t t em p era t u re T i i t i s n o tposs ib le to have i ron conta in ing a compos i t ion be twee n C i and C 2 . Ci is themax imum a mo unt o f carbon th e a i ron wi ll d i s so lve and C 2 i s the min imuma m o u n t y i ron wi l l ho ld . However , the i ron bar could have an averagecompos i t ion between Ci and C2 if it consisted of a mixture of a i ron( C = C i ) a n d y i r o n (C = C 2 ) . Hence, on a p lo t o f compos i t ion versusd i s tance as shown in Fig . 6 .6 (b) one could on ly have compo s i t ions be twe enCi a n d C 2 i f a two-phase mix ture were to occur . However , i t i s found that :Two phase regions never form in diffusion couples. C o n s eq u en t l y , o n ealways observes sharp phase boundar ies in d i f fus ion couples . This i si l lus t ra ted for you in the photograph of Problem 6 .7 . These phaseboundar ies are po in t s o f local equilibrium, s ince the two phases contact ingeach o ther are essen t ia l ly in equi l ib r ium wi th each o ther a t the t e mp era tureof the exper iment . The phase boundary moves down the rod , and one maycalcu la te the ra te o f th i s mot ion f rom a so lu t ion of Fick 's second law forth i s p rob lem . 3 This solut ion is qui te complex and wil l not be discussed here.Later we wi l l d i scuss why two phase reg ions do no t fo rm in d i f fus ioncouples . The s tudent i s u rged to work Problems 6 .1 and 6 .2 to apprecia tebet ter the appl ica t ion of the pr incip les o f d i f fus ion to th i s impor tan tpract i ca l p rob lem of carbur iz ing .


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6.1 Phenom enological Approach 147B . S U B S T I T U T I O N A L D I F F U S I O NIn the example of Figs. 6.2 and 6.4 we considered the diffusion of C in Fe,which is an example involving the diffusion of an inters t i t ial solute. In thiscase we ma de no m ent ion of a d i f fus ive mot ion of the Fe a tom s be causeany such mot ion i s neg l ig ib le comp ared to the d i f fus ive mo t ion of thesmal ler and more mobi le C a toms . Suppose , however , the d i f fus ion couplewere made of Cu and Ni as shown in Fig . 6 .7 (a) . These a toms are near lythe same size and so they dissolve in each other as subst i tut ional solutesand one expect s thei r mobi l i ty to be about the same order o f magni tude.Conseq uent ly , we mus t cons ider bo th the d i f fus ion of Cu to the r igh t a ndthe diffusion of Ni to the left . In general , subst i tut ional solutes do no tdiffus e into each other-, at equa l and o ppo si te rates . Su ppo se tha t the N iatoms d i f fuse to the l ef t fas ter than the Cu a tom s d i f fuse to the r igh t . Toass i s t us in determin ing what ef fect th i s re la t ive mot ion has upon thed i f fus ion couple we p lace iner t markers on the weld in ter face . Thesemarkers could be an iner t mater ia l such as Mo or Ta wi res , o r ox idepar t i c les , o r even the smal l vo ids generated by the weld ing process a t thein ter face . Af ter d i f fus ion has occurred for a number of hours we wi l l haveprodu ced a net t ranspor t o f a tom s f rom the r igh t o f the m ark ers to thei r l ef tbecause the Ni a toms are moving fas ter . The ex t ra a toms that ar r ive a t thelef t -hand s ide of the markers wi l l cause the l a t t i ce to expand on the l ef t ,whe reas the loss o f a toms f rom th e r igh t -han d s ide wil l cause th e l a t t i ce onthe r igh t to shr ink . Consequent ly , the en t i re cen ter sect ion of the bar wi l lshift to the right as shown in Fig. 6.7(b) as d i f fus ion causes a toms to bedepos i t ed on the l ef t and removed f rom the r igh t . Hence, i f the a tomsmo ve a t d i f feren t ra tes on e expect s to see a sh i ft o f the ma rke rs re la t ive tothe ends of the bar as shown in Fig. 6.7(b). Such a shift does occur. I t wasf i rs t repor ted in meta l s by Ki rkendal l and i t has come to be cal l ed the.Ki rkendal l ef fect . Th e occu rrence of th i s sh if t me ans that the e n t i re crys ta ll a t t ice i s ac tual ly m oving wi th respec t to the observer dur ing th e d i f fus ionprocess . This is a type of bulk motion s imilar to convect ive motion inl iqu ids and we mu s t t ake th i s in to account in analyzing the d i f fus ion processoccurring here. Such an analysis was fi rs t done for al loys in 1948 by L.D a r k e n . 4

(a)


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148 DiffusionL , u ,

Vb = bulk veloci ty of the lat t ice= veloci ty of the markers=Vm

vD = veloci ty due to diffusion alone=veloci ty of a toms re la t ive to markersThen we may wri t e u,a,=t>B + OD = u m + u D . Earl ier we defined a flux ofa t o m s i as the rate of t ransport of i per un i t are a in a toms per sec/cm 2 . On emay show that if the atoms i have a volume concentration, Q, and m ove atvelocity v their flux may be written as Cv. This is a very usef ul resul t that wewil l have occasion to use many t imes. Hence, the total f lux of atoms 1 withrespect to an observer is s imply Ci[r m +(t> D )i ] . But C,(v D)i is simply thediffus ive flux of atom s 1, which w e may wri te as -Di dCJdZ. We h av e fo rb o t h co m p o n en t s

(Jih = CiVm-Di-r^ c , < 6 - 2 3 )(J 2)t = ClVm D2

Darken now assumed that the molar dens i ty (a toms/cm 3 ) r em a i n ed co n -s tan t , which tu rns ou t to be a good assumpt ion here . This requi res( j 1 ) T = - ( j 2 ) T s ince the d i f fus ion process may not change the number ofa toms pe r un i t vo lume. By eq uat ing Eq s . 6 .23 , recogniz ing that , (a) vo lumeconcent ra t ion i equals molar densi ty t imes atom fract ion i , and (b) molardensi ty is constant , we obtain

^ ( D t - D j j ^ (6 .24)wh ere x i i s the m ole (or a tom) f rac t ion . Subs t i tu t ing th is equat ion back in toEqs . 6 .23 we ob ta in

(J 1)r = -(D 1x2+D 2x1)^-D^ (6.25)(J2)t=-(D iX2 +D2X i)^=-D^

This resul t shows that we may analyze the diffusion process of Fig. 6.7 withFick 's f i rs t law even though a bulk motion of the rod occurs . However, thequant i ty we measure as D is not a s imple diffusion coefficient but is relatedto the s imple diffusion coefficients as shown in Eqs. 6.25. The quant i ty D iscal led the mutual diffusion coefficient. By measuring the veloci ty of themarkers and D one may calcu la te the intrinsic diffusion coefficients D i andD2 f rom Eqs . 6 .24 and 6 .25 .


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6.1 Phenomenological Approach 14987

I 6

ox 4' a 2a

1

0 10 20 30 40 SO- 60 70 8 0 90 100At. % Au--Figure 6 . 8 Concen t ra t ion dependenc e o f diffusion coefficient in some gold alloys.(From Ref. 5, used with permission of Springer-Verlag.)

Many exper iments have been done and a great deal has been wri t t en onthe Ki rkendal l ef fect , mos t o f which i s main ly of theoret i ca l in teres t . 4 , 5From a pract i ca l po in t o f v iew one mos t o f ten wants to know D since thisg ives us the to ta l f lux re la t ive to the observer . When one does a Grubeanalysi s on subs t i tu t ional so lu te d i f fus ion , the mea sured d i f fus ion coeff i -cient is actual ly JD. It is possible to measure D by a method cal led theM a t an o i n t e r f ace t ech n i q ue ,3" 6 which a l lows one to re lax the assumpt ionthat D i s inde pen den t o f concent ra t ion . This t echniq ue i s general ly used ins tud ies on subs t i tu t ional d i f fus ion ; i t al lows on e to dete rmin e the conc en-t ra t ion dep end ence of the d i f fus ion coeff ic ien t . Sum marie s of expe r imen ta lresu l t s on the concent ra t ion dependence of D in many a l loys may be foundin Chapter 5 of Ref . 3 andChapter 6 of Ref . 5 . Figure 6 .8 shows a typ icalresu l t fo r the var ia t ion of D w i th concent ra t ion .5 The se re su l t s show that ifd i f fus ion i s occurr ing over a wide concent ra t ion range one mus t be verycarefu l about assuming D to be cons tan t in so lv ing Fick 's second law.C . D R I V I N G F O R G E F O R D I F F U S I O NFick 's fi r st law of d i f fus ion was form ulated on a n em pi r ica l basis . W e w o u l dl ike to cons ider the d i f fus ion process f rom a more fundamental approach,but in o rder to do so we mus t f i r s t de termine an expression for the fcHCe


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150 Diffusion

Figure 6.9 Varia t ion of potent ia l energy with pos i t ion.

that causes diffusion to occur. We are used to thinking of s imple mechani-cal forces or of f ield forces such as the electrical , magnet ic, or gravi tat ionalforce . Here , howeve r , we are deal ing wi th what might be ca l led a chemicalforce. I t arises as a resul t of the interatomic forces in a very complex way,and i t may no t be determined f rom a s imple force equat ion as the abovefield forces may. Consider the fol lowing analogy. The upper part of Fig. 6.9shows a bal l on a h i l l . We know that the bal l exper iences a force f rom thegrav i ta t ional f i e ld , and f rom phys ics we know the force funct ion to be

kMearth n i w co n s t an t (f xFdown = 2 = 2 s (O.zb)The poten t ia l energy i s now determined as a funct ion of heigh t , Z , bysimply integrat ing this funct ion as

P.E . = [ Z F d o w n d Z = - 5 5 S | 5 i L t ( 6 . 2 7 )From Eq. 6 .27 we may now determine the po ten t ia l energy a t any pos i t ionalong the h i ll and so we are ab le to cons t ruct a map of th e po ten t ia l energyas shown in the lower part of Fig. 6.9.No w suppose th at the lower pa r t of Fig . 6 .9 was a ll the in form at ion thatyou had , and you des i red to know the downw ard force , Fd0wn, up on the bal l .


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6.1 Phenomenological Approach 151I t should be c lear f rom above that you could wri t e

F < t o w n = ^ ) ( 6.2 8)Fro m F i g . 6 . 9 y o u co u ld d e t e rm i n e t h a t P . E . = - co n s t / Z an d f ro m E q . 6 . 2 8you could then determine the force funct ion by d i f feren t ia t io i i i We wantto emphasize two points about this example. (1) The bal l wil l seek theposi t ion of lowest potent ial energy and (2) the force on the bal l is given bythe der ivat ive of the po ten t ia l -energy funct ion , Eq . 6 .28 .

F ro m t h e rm o d y n am i cs we k n o w t h a t s y s t em s a t co n s t an t t em p era t u reand pressure seek the lowes t Gibbs f ree energy of the sys tem. To cons iderthe f ree energy per a tom of a cer ta in type in the sys tem we use the par t i a lmolal Gib bs f re e energy , which is o f te n call ed the chemical po ten t ia l and i sdef ined asChem ical po ten t ia l o f e lem ent =fM (6 .29)\OHi/T,P,n,

where G i s the f ree energy of the sys tem cons idered , n , i s the number of ia toms , and rij i s the nu mb er of o th er a tom s . The d ef in i t ion of chemicalpotent ials , fu, appears qui te formal but i t may be shown that , physical ly, (mi, s imply the Gibbs f ree energy of an i atom when i t is in solut ion in tKe.alloy.By analogy to the above example we now haveChemical fo rce per i a to m in Z d irec tio n = ( 6 . 3 0 )

Th e minus s ign d id no t app ear in Eq . 6 .2 8 becau se the in tegra l o f Eq . 6 .2 7i s t aken by convent ion f rom inf in i ty to Z . Equat ion 6 .30 i s a fundamentalresu l t showing that Whenever there i s a g rad ien t in thq chemical po ten t ia lQtafcams t in m.aUQMhesf i . J t toms wi l l exper ience a fnrce that may causet h em t o m o v e .D . M O B I L I T Y A N D D I F F U S I O N C O E F F I C I E N TIn discussing diffusion i t is qui te useful to introduce the concept ofmobi l i ty . Cons ider the force balance upon a parachute as shown in Fig .6 .10(a) . The m an of mass m i s in a g rav i ta t ional f ie ld an d so he exper ien cesa downward grav i ta t ional fo rce , F g , p ropor t ional to h i s mass , F t m g ,where g i s the grav i ta t ional accelera t ion . In i t i a l ly af t er b i s jump there i szero force ho ld ing the man up . However , as h i s downward veloci tyincreases the a i r co l l id ing wi th h i s parachute produces an upward dragforce that wil l be proport ional to his veloci ty, F d - kv , s o t h a t w e h a v e f o r


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152 Diffusion

(a)Figure 6.10 Atta inm ent of a termin al veloci ty .

t h e n e t fo rceF n = F g - F , i = m g - kv= ma (6.31)

At sufficient ly high veloci ty the drag force equals the gravi tat ional force.At th i s po in t the net fo rce i s zero and , hence, the accelera t ion i s zero .Consequent ly, as shown on Fig. 6.10(b) a terminal veloci ty is reached atth i s po in t o f fo rce balance. The mobility is defined asTerminal veloci ty _ ^M o b i l i t y = U n i t a p p H e d f o r c = (6.32)

As an example cons ider an e lec t ron moving th rough a meta l under anapplied electric f ield E. One may consider the electron to move with aterminal veloci ty that is achieved when the drag force of the lat t icebalances the appl ied force of the electric f ield. The resis t ivi ty, p, gives amea sure of the drag for ce and the appl ied forc e i s s imply eE , where e is theelect ron charge . As a s imple exerc i se you may show yoursel f us ing Ohm'slaw,B = -pn e (6.33)where n i s the vo lume concent ra t ion of e lec t rons .Equat ion 6.33 relates the mobil i ty of the electron to the resis t ivi ty.Similarly, we may consider the mobil i ty of an atom under a chemical forceand relate this mobil i ty to the diffusion coefficient . As previously men-t ioned we may wri t e the f lux of component i as a p roduct o f vo lumeconcent ra t ion and veloci ty , J, = C vu From the defini t ion of mobil i ty wehave for the veloci ty u, = BiF, wh ere F t i s the forc e on th e a tom s i. Us i n g E q .


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6.30 we ob ta in6.1 Phenomenological Approach 153

J. = -GB,ff (634)In wr i t ing th i s equat ion w e have assume d that the re i s no ne t fo rce exer tedon the a toms i by the flow of addi t ional solutes in the al loy. 4 T h i sassumption is not required in binary al loys and i t is general ly a goodapproximat ion in h igher-order sys tems . From thermodynamics we have forthe chemical po ten t ia l ,

dfM = kTd In & (6 .35)w h e r e at is the act ivi ty of atoms i. Subs t i tu t ing in to Eq . 6 .34 and equat ingto Fick 's f i rs t law we have(6 .36)

By a lgebraic ampl i f i ca t ion we determ ine the re la t ion betw een the d i f fus ioncoefficient , Dh and the mobi l i ty , B, , asD > = B ' k T j t i

The act ivi ty is usual ly related to the atom fract ion concentrat ion, Xi , asai = YiXj, w h e r e y t is cal led an act ivi ty coefficient . As sum ing con stan t m olardens i ty , Eq . 6 .37 becomes

In ideal solut ions or in di lute solut ions, y t i s cons tan t , so thatj [D i = Bik Tjideal or dilute soln V (6.39)

These equat ions show that there ex i s t s a d i rec t re la t ionsh ip betweenmobil i ty and diffusion coefficient . If an atom has a high mobil i ty i t "hasahigh diffusion coefficient .Th e absence of two phase zones in a d i f fus ion couple may b e unde rs toodqui te read i ly f rom Eq. 6 .34 . At any average compos i t ion wi th in thetwo-phase boundar ies on the phase d iagram the a l loy wi l l be composed oftwo phases each of which has a cons tan t compos i t ion independent o f theaverage compos i t ion . Consequent ly , in a two-phase reg ion on a phasediagram the chemical potent ial , yu, is constant . Equat ion 6.34 shows that i fa two-pha se reg ion d id form, the flux th rough i t would be zero b ecause thechemical potent ial gradient would be zero in the region. I t is a s implemat ter then to show yoursel f that i f a two-phase reg ion d id form in a


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154 Diffusiondiffusion couple, diffusion into and out of i ts boundaries would cause i t tod i sappear because of the fac t that d i f fus ion wi th in the reg ion i s zero ; seeProblem 6.5 .E . T E M P E R A T U R E D E P E N D E N C EThe d i f fus ion coeff ic ien t i s a very s t rong funct ion of t emperature ; i tvirtual ly always may be expressed as

D = D 0 e x p [ - ^ ] (6 .40 )wh ere D 0 is a constant and Q is a constant cal led the act ivat ion energy. Thevalues of D are almost always quoted in cgs uni ts and the uni ts of D arecm 2 / s ec . Ap p ro x i m a t e v a l u es fo r t h e t em p era t u re d ep en d en ce o f D a reshown in Fig. 6.11. Note the fol lowing: (1) The diffusion coefficients ofinters t i t ial solutes are s ignificant ly higher than fo r subst i tut io nal solutes . (2)The diffusion coefficient in the sol id at the melt ing point (along sol idus) isnearly the same for different al loys, and s imilarly, the diffusion coefficientin the liqu id a t the f reez ing t em per ature (a long l iqu idus) is near ly the sam efor d i f feren t a lloys; see Fig . 6 .11 for typ ical D values . (3) The t em per atur edependence i s very s t rong , wi th h igh-mel t ing meta l s hav ing the h igherroom-temperature D values shown on Fig . 6 .11 and low-mel t ing meta l shaving the lower D values shown. Extens ive data on the t emperaturedep end enc e of D may be fou nd in Chapte r 5 of Ref . 3 , and in Chapter 4 ofRef . 5 .F . I N T E R F A C E D I F F U S I O NFigure 6 .12(a) shows that in po lycrys ta l l ine meta l s d i f fus ion may occuralong the gra in bounda r ies an d the surface as wel l as th rough the vo lum e of

Figure 6.1 1 Ran ge of diffusion coefficient values nea r liquidus, solidus, and ro omtemperatures .


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6.2 Atomistic Approach 155

10"Temperature -15001750 2000*0

Surface

Grainboundaries

10-

10-'?I(JC 1 0 - 1 2

5 10-'310-'*10-15

( a) 1 0 " ' 6

I I IS u r f a c e /

/ / ~' ' ' /Grain / -

b o u n d a r i e s /f

L a t t i c e / -

/ "1 1- f X 104Diffusion of Th in W(b )

Figure 6.12 Lattice, grain boundary, and surface diffusion. (From R ef. 5, usedwith permission of Springer-Verlag.)the gra ins . In recen t years many exper imenta l measurements have beencondu cted on surfac e d i f fus ion and gra in-bou ndary d i f fus ion . O ne expe ct sthe mobi l i ty o f an a tom along a gra in boundary or a surface to be h igherthan th rou gh the crys ta l vo lume beca use these in ter faces hav e a mo re ope ns t ructure and , henc e, should offer l ess res i s t ance to a tom mo t ion . Con se-quent ly , o ne expect s in ter fa ce d i f fus ion coeff ic ien t s to b e h igh er , thanvolume diffusion coefficients s ince the diffusion coefficient is direct lyre la ted to mobi l i ty , Eq . 6 .39 . Exper imenta l resu l t s bear th i s ou t and Fig .6 .12(b) p resen t s some typ ical exper imenta l data . Grain-boundary d i f fus ionma kes a s ign i fi can t cont r ibu t ion t o the to ta l d i f fus ion on ly when t he gra insize is qui te small .6 . 2 A T O M I ST I C A P P R O A C HAs we wil l see later the atoms of a metal are not f ixed on their lat t ice Si tebut ra ther they cont inual ly move about . Consequent ly , i f a


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156 Diffusion

a(1) (2)Figure 6.13 Lat t ice plane spacing for diffus ion mod el .

gradient exis ts in a metal the random motion of the atoms wil l eventual lycause the grad ien t to d i sappear . Therefore , we may re la te the d i f fus ioncoeff ic ien t o f an a to m to i t s jum ping prope r t i es in the crys ta l. Cons ider twoneighboring planes in a set of occupied (hkl) planes as shown in Fig. 6.13where the spacing between the p lanes i s ca l l ed a . We now def ine thefo l lowing th ree t erms ,( a) T = j u m p f r e q u e n c y= No. t imes per second a g iven a tom jumps to aneighbor ing pos i t ion(b) p= pro ba bi l i ty that any jum p of an averag e a tomon plane 1 wil l carry i t to plane 2=fract ion of jumps go ing f rom p lane 1 to p lane 2(c) Mi, n 2 = N o . a t o m s / c m 2 on p lanes 1 and 2 .

An ind iv idual a tom wi ll jum p f rom p lane 1 to p lane 2 p r t imes per second .Consequent ly we may wri t eNo. a toms/cm 2 jump ing 1 - 2 in t ime 8t=Mi (p O 8tNo. a toms/cm 2 jump ing 2 1 in t ime 8t= n2 ( p O 8t

Net No. a toms/cm 2 jump ing 1 - 2 in t ime 8t=(ni-n2) ( p O 8tIf J is the flux of atoms from 1 to 2, in atom/sec-cm 2 , we have

J8t=(th~n2) p T 8t (6.41)The vo lume concent ra t ion a t p lane 2 , C 2 , may be wri t ten as

G - G + g - a (6 .42)where Z i s the d i s tance normal to the p lanes shown in Fig . 6 .13 . Thevolume concent ra t ion , C , may be re la ted to the area concent ra t ion , n , as ,


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6.2 Atomistic Approach 157C=n/a . Subs t i tu t ing th i s re la t ion we ob ta in

scn2-n l = a 2 (6.43)Combin ing Eqs . 6 .41 and 6 .43 we ob ta in the fo l lowing express ion for theflux:

J = - a 2 - p - r - | | (6.44)Comparing this equat ion with Fick 's f i rs t law the final resul t is obtained,

(6 .45)To determine the phys ical s ign i f i cance of Eq . 6 .45 we wi l l cons ider thediffusion of carbon in i ron. I t is given as Problem 6.4 to show that for aninterst i t ial atom diffusing in an fee lat t ice the diffusion coefficient is

(6.46)w h e r e a i s the l a t t i ce param eter . Ta king D for carbon in aus ten i te f rom thedata in Problem 6 .1 we ob ta in a t 925C

r925= 1.7 x 109 j u m p s / s ecAt room temperature we f ind the fo l lowing value of T for carbon inre ta ined aus ten i te :

r 2 oc= 2.1 x 10~9 j u m p s / s ecThis resu l t po in t s up two very in teres t ing th ings : (1 ) A t h igh t em pera ture sthe in ters t i t i a l carbon a toms are changing thei r l a t t i ce pos i t ions a t afantast ic rate, on the order of a bi l l ion t imes a second, and (2) t j iS-j iUPPfrequency i s ex t remely sens i t ive to t emperature . I t i s apparen t f rom theabove analysis that the basic mechanism of the diffusion process is closelyre la ted to the jumping character i s t ics of the a toms involved . C onsequ ent ly ,i f one carries out a s tat is t ical analysis relat ing the net motion of an atoip toi ts individual jumping characteris t ics i t is possible to gain further insightin to the d i f fus ion problem . Cons ide r an a tom located a t som e pos i t ion on acrystal lat t ice cal led posi t ion zero. Now,

1 . Al low the a tom to ma ke jum ps of l eng th r on ly .2 . Assum e jumps in any d i rect ion are equal ly proba ble , that is , eachj u m p i n d ep en d en t o f p reced i n g j u m p .3 . Let R be the net d i sp lacement of the a tom from pos i t ion zero af tern j u m p s .


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158 DiffusionWe n o w as k : Ho w d o es Rn vary with n_? Since the jum ps ar e rand om , o nemight expect R to approach zero as n becomes suff i c ien t ly l arge . How-ever, s tat is t ical analysis shows that this is not t rue and one finds 4

R ^ n r 2 (6.47)whe re i s the roo t -m ean- squa re d i sp lacem ent . Hen ce, we see thatR==\ /n- r , which i s cons iderab ly d i f feren t f ro m zero . This p rob lem i scal led the random-walk problem and i t is discussed in detai l in Ref. 4, p.4 7 .

Consider the diffusion of carbon atoms in a rod of y i ron. In this case thejum p dis tance, r , is relate d to a as r 2 = 2 a 2 , and p is | so that we obtain fromEq. 6 .45 and Eq . 6 .47(6.48)r n

Th e ra t io n /T is the num ber of jumps d iv ided by the jump s per sec which i ss imply the t ime, i , so that we may wri te Eq. 6.48 asR l = 6 Dt, (Rn)n.M. = 2.45-v/Df (6.4 9)

This resu l t shows that the roo t mean d i sp lacement i s p ropor t ional to VDt .Note the s imilari ty of this resul t with the resul t obtained earl ier for theprob lem of the case dep th u pon carbur iz ing : bo th go as the VDt . It is qu i tein teres t ing to examine the phys ical s ign i f i cance of Eq . 6 .49 . Suppose weask : What i s the roo t me an d i sp lacem ent o f a carbo n a tom in y i ron af ter 4hours . The resu l t i s g iven a t 925C and a t room temperature in Table 6 .2 .The to ta l d i s tance that the carbon a tom moved may be calcu la ted bymul t ip ly ing the jump d i s tance by the to ta l number of jumps , and theseresu l t s are a l so included in Table 6 .2 . Two resu l t s are apparen t f rom thedata of this table: (1) The root mean displacement is qui te sensi t ive totemperature , and (2) the a toms mus t t ravel a t remendous to ta l d i s tance inorder to ob ta in a s ign i f i can t roo t -mean-square d i sp lacement .

Table 6.2 Migrat ion Dis tances of Carbon Atoms in yI ro n fo r 4 H o u r sT e m p . To ta l D is tan ce(C) (Rh)R.M. M o v ed

925 1.3 mm 3.9 miles (6.3 km)20 1 .4 x 1 0 mm zero


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6.2 Atomistic Approach 159A . D I F F U S I O N M E C H A N I S M SThere are a number of d i f feren t d i f fus ion mechanisms that have beenpos tu la ted . 4 The two mos t impor tan t mechanisms , in ters t i t i a l and vacancydiffusion wil l be discussed here. In inters t i t ial diffnsipp thp. atomss imply jump through the in ters t i t i a l vo ids in the l a t t i ce . This mecha-nism general ly occurs for small atoms in metals such as C, O,N, and H. In vacancy d i f fus ion the a toms migrate by jumping in tonear-neighbor vacancies . This mechanism occurs p redominate lyfor subst i tut ional so lutes an d also for self-diffu sion. Th e self-diffusion of metal A is measured by observing the migrat ion ofan isotope of element A in a crystal of metal A. Self-diffusion lendsi t se l f more eas i ly to theoret i ca l in terpre ta t ion s ince one does no thave to worry about chemical in teract ions between so lu te and so lventa toms . Note that in vacancy d i f fus ion one has a vacancy f lux equal andoppos i t e to the a to m f lux . I t was po in ted ou t ear l i er that a Ki rke ndal l sh i f toccurs when the fluxes of the two subst i tut ional species are not equal . Sincethese fluxes occur by a vacancy mechanism, there must exis t a net vacancyflux in a direct ion opposi te to the net atom flux. In Fig. 6.7(a), for example,a net vacancy f lux would occur to the r igh t , toward Ni . This requi res thatvacancies be cont inual ly gene rated on the l ef t in the Cu an d annih i l a ted onthe right in the Ni . This is a very interest ing conclusion, and one may find amo re ex tens ive d iscuss ion of i t in Ref . 4 . On e of the mec hanism s prop osedfor the cont inual generat ion of vacancies i s a Frank-Read generat ion ofedge d i s locat ions whe re the edge d i s locations are a lways moving by c l imb.4B . R A T E D A T A A N D T H E A R R H E N I U S E Q U A T I O NQu i te of te n in phys ical me ta l lu rgy i t i s necessary to be ab le to desc r ibe thera te o f some proce ss in an a l loy as a funct ion of t emper atur e . Fo r e xam ple ,in heat - t rea t ing operat ions i t i s necessary to have a knowledge of the ra teof g ra in growth as a funct ion of t emperature in order to avoid excess ivegrain growth ; the ra te o f creep a t h igh t emperatures in a l loys used inturb in e b lades i s a cr i ti ca l p rop er ty in the de velopm ent o f usef u l a l loys forth i s purp ose; th e ra te o f d i f fus ion as a funct ion of t em per atu re i s a cr i ti ca lfactor in the control of the carburizing process , and so on. I t is almosta lways found that i f one p lo t s the log of the ra te concerned as an inversefunct ion of t emperature a s t ra igh t l ine i s observed as shown in Fig . 6 .14 .Th e equat ion f or th i s s t ra igh t - l ine funct ion is ca l l ed the Arr he niu s eq uat ionand i t is wri t ten as

R = Ae~Q,RT (6 .50)where A i s the in tercep t and Q i s ca l l ed the ac t ivat ion energy . This


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160 Diffusion

\/TFigure 6.14 The Arrhenius plot .

empi r ica l equat ion wi l l f requent ly be encountered when cons ider ing thetemperature dependence of so l id -s ta te p rocesses .Fr om a compar i son of Eqs . 6 .40 and 6 .50 it i s app are n t that the d i f fus ioncoeff ic ien t fo l lows an Arrhenius equat ion . Us ing the a tomis t i c model todescr ibe the d i f fus ion process , one can ar r ive a t a phys ical in terpre ta t ionfor the ac t ivat ion energy , which is determ ined f r om the Arrh enius p lo t . W ewi ll cons ider two cases dep ending o n wh ether t he d i f fus ing a tom s a reinters t i t ial or subst i tut ional .C . I N T E R S T I T I A L D I F F U S I O NIn this case the solute is an inters t i t ial atom that migrates by jumpingbetween the inters t i t ial voids, for example, carbon in i ron. The fi rs tques t ion we ask ourselves i s how may we determine an express ion for T ,the rate at which an atom changes i ts posi t ion in the lat t ice. Figure 6.15shows a plot of the free energy of a solute atom as a funct ion of i ts posi t ionin the lat t ice. A t the tw o lat t ice posi t ions shown, 1 an d 2, i ts fre e ener gy is amin imum. We now def ine the fo l lowing terms :

1. f= fract ion of atoms at any t ime having sufficient energy to changeposi t ion, that is , having G>G 2.2. Z = n u m b e r of neare s t -neigh bor in ters ti t i a l vo ids arou nd each so lu teatom. We assume they are a l l unoccupied .3 . v= frequency of vibrat ion tow ard each of these Z voids.Our problem i s f i r s t t o ca lcu la t e I \ t he ra t e a t which .any g iven so lu te a tomc h a n g e s p o s i t i o n . I f a s o l u t e a t o m w e r e t o c h a n g e p o s i t i o n s o n e v e r y


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6.2 Atomistic Approach 161vibra t ion we would have Y = v- Z. However , an a tom wi l l change pos i t iononly i f i t has sufficient energy to make the jump. We wil l assume that theprobabi l i ty that an a tom has suffi c ien t energy to jump, G > G 2 , is given by f .Hen ce we h av e

r = v Z - f (6.51)Again, in order to proceed i t is necessary to cal l upon studies of s tat is t icalmechanics , which show that the d i s t r ibu t ion of f ree energ ies over the a tom sfo l lows a Maxwel l -Bol tzman law. Accord ing ly we have for the f rac t ion 6fa toms wi th f ree energy G>G t

n ( G > G Q _G|/kTNw h e r e N refers to the to ta l number of a toms . Therefore , we may wri t e

n(G>G2) (Cj2-Gi)/kT -Ao/fcT ,( rn ( G > G i ) e cwh ere G 2 and Gi a re def ined in Fig . 6 .15 . Gi i s the f ree energy of the a tomwhen i t l ies direct ly on the lat t ice s i te, and hence i t is the minimum freeen e rg y . T h e re fo re a l l a t o m s h av e a f r ee en e rg y G> Gi . C o n s eq u en t l y E q .6.52 gives us the value of / ,

^ ( G g G ^ ^ ( 6 5 3 )

1Position in latticeFigure 6.15 Varia t ion of fre e energy of an a tom with pos i t ion in the la t t ice .


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162 DiffusionNow by combin ing Eqs . 6 .45 , 6 .51 , and 6 .53 we ob ta in for D

D = a 2 p ( Z v e 4 S / V 4 E / k T (6-54)w h e re w e h a ve t ak e n A G = A E - T AS. In general , AS i s no t very t empera-ture dependent so that the on ly t emperature-sens i t ive t erm in Eq . 6 .54 i sthe f inal t erm. Hen ce, by compar ing Eq . 6 .54 to Eq . 6 .50 one sees that theact ivat ion energ y Q i s iden t ica l to AE. Th e abo ve analys i s shows that AE i sthe energy d i f ference for an a tom p laced a t pos i t ion l a an d pos i tion l ofF i g . 6 . 1 5 . In o rd e r t o m o v e f ro m p o s i t i o n 1 t o 2 t f i e^ o m T n u s f p as sthrough the maximum energy pos i t ion a t l a . The increase in energyrequi re d to mo ve the a tom to th i s pos i t ion i s ca l l ed the qr t ivat inn energy ,for obvious reasons . Hen ce in th i s case Q i s the t ru e act ivat ion energy , AE,for the process that we are cons ider ing , a tom migrat ion . In general , the Qof an Arrhen ius equa t ion wi l l no t correspo nd d i rect ly to the t rue act ivat ionenergy for the ra te p rocess being cons idered . This may be demons t ra tedfor our second case .D . S U B S T I T U T I O N A L D IF F U S I O NIn this case the solute atom is a subst i tut ional solute and, hence, iscons t ra ined to move on the l a t t i ce s i t es . We now le t Z be the number ofnearest-neighbor lat t ice s i tes . A significant difference occurs in this casebecause in order for a jump to occur to a near-neighbor s i t e the s i t e mus tbe vacant . We take the number of near-neighbor s i t es that are vacant as Zt imes the fract ion of al l s i tes that are vacant . This lat ter expression is givenby Eq. 5 .9 . Hence for th i s case Eq . 6 .51 becomes

T= v Z e~ AEJkT e i S u / k / (6.55)wh ere AE i s the energy p er vacancy an d A S i s the v ibra t ional en t rop y pervacan cy. In this case then the expre ssion fo r th e diffusio n coefficientb e c o m e s D = a 2p(Zve (iS+lis")"')e-E+AE ,/kT (6.56)C o n s eq u en t l y we o b t a i n Q= AE + AE . T h e re fo re Q i s n o t s i m p l y anact ivat ion energy but in this case i t is the sum of the t rue act ivat ion energyand the energy per vacancy. Q is a t rue act ivat ion energy only for verysimple processes, and in general i t is best thought of as an empiricalconstant . The above discussion shows that a s tat is t ical interpretat ion of thediffusion process provides one with a physical insight into the actual atomicprocesses g iv ing r i se to the mass t ranspor t p roduced by d i f fus ion .R E F E R E N C E S

1. A. Fick , Pogg endorfP s Ann alen 94 , 59 (1855) .2. G. Gru b e an d A . J ed e l e , Z. Elektrochem. 38, 799 (1932) .


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JfeFdh|eni8l63195*2 W J O S t ' D i ^ u s i o n S o l i d s ' L i e * u i d s ' G a s e s > Acad emic , New Yo r k ,

4 . P . Shewmon, Diffusion in Solids, McG raw-Hil l , NW York , 1963 .5 . W. Sei th and T. He um ann, Diffusion in Metals: ixeifc aMM ,S p n n ge r -V e r la g , N e w Y o r k , 19 5 5 . T r a n sl a ti o n fro m t h e . e ^ a n . O f f i ^ 0 fTech n ica l Se rv ice s, Dep t . o f C o mm er ce , W ash in g to n 2 5 1 x 4 , 4 9 6 2 , A g C -T R - 4 5 0 6 . , , t t i ,6 . R. E. Re ed Hill , Principles of Physical Metallurgy, Vari NositrMd,New York , 1964 . ,

PROB L E M S m6.1 You have a p late of 1010 s teel (0 .1 Wt % carbon) that mttSt act s b k r i gsurface. To achieve the necssary hardne ss you decide to carbur ize th s i ir face andthen heat t reat . You would l ike to achieve an as-quenehed hard iissof $ t let 60Rockwell C in the outer 1 mm layer . The as-quen ched ha rdness var is with ca&oncontent as shown below. T he hardn ess fal ls at h igh carbon conter t ts d i ie to r etaine d

0 .2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0Weight percent carbonRockwell "C" hardness as a function of carbon content for as-queie|ied i^OHcarbon al lpys . (From A . Litwinchuk, "De nsit ies , Microstructures , andAs-Q uench ed I ron C arbon A lloys ," M.S. thes is , Iowa State Univ . Lib*


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164 Diffusionaustenite. T he diffusion coefficient of C in 7 iron goes as D =0 12 exp (32000/RT) where R = 1.987 cal/mole-K an d D is in un its of cm Vsec.(a) You pack carburize (as discussed on p. 144) for 2.4 hours at 1050C. Usingthe appropriate equation and the phase diagram of Fig. 9.64 (p. 306), calculate thecomposition as a function of position in the steel surface region. Now plot thecomposition at 0.02-cm intervals from the surfac e to a dep th of 1 mm. B elow this,plot the as-quenched hardness that would be obtained on heat t reatment .(b) You will notice that the hardness is lower than desired at the surface due toretained austenite. To overcome this problem it is necessary to reduce the carboncontent at the surface during carburization. As an alternative to pack carburizingyou may gas carburize in an atmosphere of methane or carbon monoxide. Bycontrolling the CH/H 2 o r C 0 / C 0 2 ratio you can achieve various surface concentra -tions in gas carburizing below the saturation value shown on the phase diagram.*Suppose you pick a ratio to give a surface concentration of 0.8% C. Now design acarburizing treatment to achieve your objective of a minimum 60 Rockwell Cas-quenched hardness in the outer 1 mm layer (that is, specify tem peratu re andtime). Note: Do not carburize much above 912C as excessive grain growth willoccur.(c) If you were to choose a temperature between 727 and 912C you could notanswer part (b) from equations developed in this chapter. Explain why not. If youwere to choose a temperature below 727C you could not achieve your objective.Explain why not .6 .2 Phase boundaries revealed bymetallographic etch

Phase 1Phase 2Phase 1

No carburizing-on ends ofplate

Section A AA11 iron plate is packed in graphite as shown abo ve and heat ed to 740 C for 4hours. A metallographic examination revealed the above two-phase picture.(a) Ma ke a plot of how the comp osition of carbon (in wt % ) might vary across thesection A-A. Label the phase regions and those compositions you can determine

from the phase diagram. (Fig. 9.64, p. 306).(b) The graphite is now removed from the plate and it is again heated to 740C.Assume that no decarburization occurs at the surface. On your diagram for (a)above, show how the composition profile would look after many weeks at 740C(equilibrium). Explain.(c) If the temperature were now increased to 800C show how the compositionprofile would look after equilibrium was again reached.* It is also possible to reduce th e surface concentration below the equilibrium saturationvalue in pack carburizing, apparently by control of the catalyst. .


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Problems 1656.3 Equ ation 6.8 was derived for the case of unidirectional diffusion along a bar.Suppose a round bar were packed in graphite and diffusion occurred in the radialdirection of the bar. As o ne m oves in the radial direction the cross-sectional area ofa volume elem ent in cylindrical coordinate s is not a con stant. Hen ce, the derivationof Fick's second law (Eq. 6.8) do es not apply directly to th e case of radial diffusion.Yo ur pr oblem is to derive Fick's second law fo r the case of radial diffusion; that is,your independent variable will now be the radius, r, rather than distance Z.6.4 A derivat ion is presented on pp. 15 6-15 7 for the equat ion D=a2 p-T,which relates the diffusion coefficient to a spacing between planes, a, a jumpfrequency, I\ and the probability, p, that a jump of an average atom will carry itfrom plane 1 to plane 2.(a) Calculate p for a carbon atom moving betw een o ctahed ral sites in (1) fee iron,and (2) bcc iron. No te that in bcc crystals the octahed ral sites are located at bot h theface centers and the edge centers of the unit cell .(b) From y our answer show tha t in term s of their respective lattice par ame ters,

D b c c =

D, fcc=aL T24aL r12

6.5 It was stated on p. 146 that a two-p hase region never appea rs on a diffusioncouple. If a two phase region did appear it might look like the sketch shown here.Particles of j Fe

7 Fe a Fe

TwophaseregionYou r problem is to show that th e two-ph ase region will disapp ear. Ma ke a sketch ofthe above. Directly underneath this sketch make a plot of how the chemicalpotential of carbon varies from the left-hand end to the right-hand end of the bar.Rem ember, the chemical potent ia l i s proport ional t o carbon content and is constantin a two-phase region. Now from an analysis of this plot determine how the carbonwill diffuse with respect to the tw o-pha se region an d explain why the tw o-ph aseregion will slowly disappear as diffusion continues.6.6

+ Anode CathodeWhen a dc electric current passes through a rod of steel i t causes the carbonatoms to move relative to the iron atoms. This phenomenon is called electtotWJM-port or electromigration. The mobility of carbon atoms due to the electric current

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