484 Diffusion in Solids

and xj is that value of x where C = Co. Then

(13.57)

Example 13.7 After the predeposit step, the wafer of Example 13.6 is subjeeted to drive-inat 1423 K for 2 h. Calculate: a) the junetion depth; b) the eoneentration at that depth; ande) the surfaee eoneentration.

Solution. a) Equation (13.57) applies with e. C, and Cogiven in Example 13.6. From Table13.3 D = 1.6 X 10-1 J.lm2 h-I = 4.44 X 10-13 em2 S-I; also t = 7200 s.

x, ~ /4 x 4.44 x 10-"

[ [

~) [

4.5 X 1020

) [

2.832 X 10-12

)

112

] )

1/2

X 7200 ln 7r 5 X 1015 4.44 X 10-13 x 7200

xj = 3.19 X 10-4 em = 3.19 J.lm.

b) C "" Co = 5 X 1015atoms em-3. We obtain an exaet result by using Eq. (13.55).

2

[

2.832 X 10-12

)

1/2

[

3.192 X 10-8

]-:;r 4.44 X 10-13x 7200 exp - 4 x 4.44 X 10-13x 7200

1.115 X 10-5.

Therefore,

C = (1.115 x 10-5)(4.5x 1020- 5 x 1015)+ 5 X 1015

= 1.00 X 1016atoms em-3.

e) We use Eq. (13.55) with x = O.

C - 5 X 1015 2

[2.832 X 10-12

]

1/2

4.5 X 1020 - 5 X 1015 = -:;r 4.44 X 10-13 x 7200 = 0.0319.

C = 1.43 X 1019atoms em -3.

T

j

Dlffuslon In SoIlds 485

13.5 HOMOGENIZA TION OF ALLOYS

During solidificationof alloys, coring oecurs. because the rate of diffusion for most alloyingelements in the solid state is toe slow to maintain a solid of unifonn coneentration inequilibrium with the liquid. A cast structure is exemplified in Fig. 13.9 showing therepetitive pattem of microsegregation. In the case at hand we presume the alloy to be asingle phase. Dendrilearms

~

Jo ""I11 11" "

Fig. 13.9 A dendritie strueture showing the coring or microsegregationof thealloying element. L is one-half of the dendrite arm spacing.

The dendrite anns develop during solidificationand their spacing depends upon the alloyand the rate of cooling from the liquidus temperature to the nonequilibrium solidustemperature. Depending on the cooling rate. the dendritic arm spacing can be between 5 J.lmand 400 J.lm.with typical values ranging from 50 J.lmto 200 J.lm.

Dendritic structures occur in cast products that encompass ingots. shaped castings.continuous castings. and weldments. The coring. or microsegregation. is coincidental withthe growth of the dendrites. Imagine a local region in a solidifying casting that comprisesseveral to many dendrite arms. As solidificationproceeds within the local region. solute isrejected to the interdendritic liquid, when the concentration of solute in the solid is less thanthat of the interdendritic liquido The microsegregation can be closely approximated by asimple solute balance.3 That is. the solute rejected from the dendritic solid equals theincrease of sollite in the interdendritic liquido Therefore. within a local region undergoingsolidification we have

:1

j'.

(13.58)

whereCL = weight percent of solute in the interdendritic liquid;

C,* = weight percent of solute in the dendritic solid at the solid-liquid interface;and

1, = local weight fraction of solid.

3M. C. Flemings. SolidificationProcessing, McGraw-HiII. New York, NY, 1974, pages 34-36,142.

486 Diffusion in Solids

When we use Eq. (13.58), it is assumed that there is sufficient diffusion of solute in the liquidso that it is uniform. This is usually valid because the diffusion length in the liquid is manytimes greater than the dendrite arm spacing. On the other hand, the diffusion coefficient ofthe solute in the solid is typically 5 or 6 orders of magnitude less than in the liquid, so weassume no diffusion in the solid. Consequently, as a layer of solid with a concentration C,*forms, its concentration never changes as subsequent solid of different C/ grows from theinterdendritic liquid.

To get a quantitative picture of the microsegregation, we integrate Eq. (13.58). Then

CL

JcL -co

dCL

CL -:-C,*

I, dI,= J 1 -I,'1,=0

or

1 - I, = exp[

- fL (CL - C; tdCL]

,cL-co

(13.59)

where Cois the concentration of solute in the unsolidified melt (i.e., the alloy composition).In dendritically freezing alloys, the solid-liquid interface is very dose to equilibrium so thatC/ can be taken as the solid in equilibrium with the liquido Many binary alloys have phasediagrams in which the ratio C/fCL = k can be approximated as a constant, called theequilibrium partition ratio. With a constant k, we can substitute C,* = kCL into Eq. (13.59)and carry out the integration. The result is

CL

C = (I - !.)k-I

o "

or

C*,

C; k(1 - I,tl. (13.60)

As an example, suppose k = 0.3; then the microsegregation according to Eq. (13.60) isshown in Fig. 13.10. The first solid that forms has a concentration C,* = kCo and thensucceeding solid has progressively higher values of C,*. But with no diffusion in the solid,the microsegregation, which develops, remains.

As /, --+I, Eq. (13.60) predicts C/ --+00. In reality, a reaction, such as a eutecticreaction, takes place near the end of solidification when the interdendritic liquid becomessufficiently enriched in solute. In any event, after solidification is complete there ismicrosegregation, and in many applications it is necessary to subsequentlyheat treat the alloyto reduce the microsegregation by diffusion. This process is called homogenization.

During homogenization, the alloy naturally tends toward a uniform concentration(Fig. 13.11). We need only examine what happens within one dendritic element since theprofile is periodic, and there is no net flow of solute from any dendritic region to the next.

.."..

.....

Diffusion in Solids 487

9Co

oO 0.2 0.4 0.6 0.8 1.0

fI

Fig. 13.10 Microsegregation in a dendritic solid with k = 0.3.

Completelyhomogenized, C = Co

==:::=====_-::Short time

~

O Lx

Fig. 13.11 Dendritic element, O < x < L, used for describing homogenization kinetics.

To describe the homogenization kinetics, a solution to Fick's second law is needed thatsatisfies

I.C.: C(x,O) = f(x),

éJCB.C.: éJx(O,t) = O,

(13.6Ia)

(13.6Ib)

éJCéJx (L,t) = O, t > O. (13.6Ic)

The solution can be obtained by applying the method of separation of variables. Here wesimply present the solution as given by Crank":

C(x,t) = Co + .E A. exp.-1[

Dt

]

n~x

-n2,r [2 cos T'(13.62)

4J. Crank, The Mathematics of Diffusion, Oxford University Press, London, 1957, page 58.

488 Diffusion in Solids

with

L2

Jn7rX

An = I f(x) cos L d.x.o(13.63)

In Eq. (13.62), Co is the overall or average alloy content, and we evaluate the Ans byEq. (13.63) using f(x) as the initial solute distribution.

A useful parameter to describe the homogenization kinetics is the residual segregationindex fi, which is defined as

CM - Cmfi '5 o o'

CM - Cm(13.64)

where CM= maximum concentration, that is, CM= C(O,t); ct = initial maximumconcentration, that is, ct = C(O,O);Cm= minimumconcentration, that is, Cm= C(L,t); andc;:. = initial minimum concentration, that is, c;:. = C(L,O). For no homogenization, fi = I,and after complete homogenization, fi = O.

We can find CM- Cmby applying Eq. (13.62) to x = Oand x = L, and performingtheindicated subtraction. The residual segregation index can then be written as

fi 2 n=ltodd An exp [-n2r V]

C~ - C~

(13.65)

Equation (13.65) has been used to analyze the homogenizationof chromium in cast 52100steel (1%C-1.5%Cr). Figure 13.12 shows the results. The practical conclusions of suchstudies show that:

1. In commercial material, with relatively large dendrite arm spacing (200-400 Jlm),substitutional elementsdo not homogenizeuniess excessively high temperatures and longdiffusion times are employed. For example, in laboratory-cast 52100 ingots the dendritearm spacing could typically be 300 Jlm, which would have to be held at 1450 K forabout 20 hours to reduce fi to 0.2 for chromium. In large commercial ingots, dendritearm spacings are larger and homogenization is even more difficult to achieve.

2. Significant homogenization of substitutional elements is possible at reasonabletemperatures and times oniy if the material has fine dendrite arm spacings « 50 Jlm),which result from rapid solidification.

3. Interstitial elements (such as carbon in steel) diffuse very rapidly at austenizingtemperatures.

1.0

0.8

O.~53

oO 4 8 -ii 16 20' 36 40 44 48

/]Jx 10'L'

Fig.13.12 The residualsegregationindex for chromiumin cast 52100 steel. (FromM. C. Flemings, D. R. Poirier, R. V. Barone, and H. D. Brody, l/S/, 371, April1970.)

TDiffusion in Solids 489

Homogenization studies have also been carried out for 4340 low-alloy stee1S and 7075aluminum alloy.6 In Reference 6, the analysis and discussion of homogenizationemphasizesthe dissolution of a nonequilibrium second phase during solutionizing.

Example 13.7 An ingot of 52100 steel goes through the processing schedule indicatedbelow.Estimate the residual segregation index of chromium for the material in the center and theoutside surface of the finished bar. Neglect the small amount of diffusion that occurs duringblooming, rolling, and final cooling. Near the surface of the original cast ingot, the dendritearm spacing is 40 r.m, and in the center, it is 800 Jlm. The diffusioncoefficient of chromiumin this steel at these temperatures is given by

D = 2.35 x 10-5exp [-17 300/T(K»), cm2S-I.

LD ~ <::])J

Illgor24 in. X 24 in.

Biller6 in. X 6 in.

BarI-in. diam.

°F

.

Cooling I

h_ Surface___Center

Reduction schedule for 51100 alloy steel bano

Solution. Before proceeding directly to the solution of this problem, we should recognizethat, although the basis for Eq. (13.65) assumes a constant diffusion coefficient (henceisothermal treatrnent), we can apply it to non-isothermal conditions.

For example, if a part is subjected to n heat treatrnent steps, each at a differenttemperature and for different times, we can compute the total magnitude of Dt/L2 as

Dt 1 n

L 2 = L 2 L DJj',-I

ST. F. Kattamis and M. C. Flemings, Trans. TMS-A/ME 233,992 (1965).

6S. N. Singh and M. C. Flemings, Trans. TMS-A/ME 245, 1803 (1969).

00

><...."

.:

,2

;; 0.6..

0.4..

]

..'"

'\

/1

/1 /1 /1 1:.-10 5

490 DiffusloD In SoUds

We can then use this value of DI/L2 in Eq. (13.65). For a continuous nonisothermalsituation, we compute the total magnitude of DI/L2 to be used as

DI 1 I

L 2 = u J D(I) dI .o

For hot-working in which D and L are both time dependent, DI/L2should be evaluated asI

DI

fD(I)

U = [L(t)F dt.o

Now we determine D(t), given the thermal process schedule and D(I).

~8

~4

20

oO 20 3010

Time.h

During the processing, assume that the dendrite spacing decreases in proportion to thechanges of linear dimensions. Based on this, L(t) is given in the table below.

..Diffuslon In SoUds 491

Now DI/L2 for the "surface" material can be evaluated by determining the area under thecurve in the figure below.

10 X 10"

D/L1. 8S-I

6

4

O 10 ~O 26 30

Time. h

Under area IUnder area 11

Total

112 X 10-5 h S-I

141 X 10-5 h S-1

253 X 10-5 h S-1

Dt

U (surface) = 253 X 10-5 X 3600 = 9.10.

~

According to Fig. 13.12, this value of Dt/L2 indicates that the surface material would behomogeneous since ó ==O.

In the center of the ingot, since the dendrite spacing is 20 times the spacing of thesurface, then (neglecting small differences in the thermal history)

Dt Dt 1 9.10U (center) == U (surface) X 202 = 400 = 0.0228.

II

From Fig. 13.12, we see that ó ==0.27, and a significant amount of microsegregationremains in the material.

13.6 FORMATION OF SURFACE LAYERS ~~

The rate of formation of oxide (sulfide) layers on metais and alloys exposed to oxidizing(sulfidizing) conditions is a matter of considerable technological importance. In general, itis not possible to say a priori that the rate of formation of a nonmetallic layer will becontrolled by diffusion. For example, the initial rate of formation of the layer is oftendetermined by the rate of an interface reaction between the gas and solid. As growthproceeds, if the specific volume of the oxide is much larger than that of the metal substrate,separation of the two phases may occur, causing an interruption in the growth of the oxidelayer. However, in many cases this separation does not occur, and growth continues bydiffusion of either the metal out through the oxide layer, or diffusion of oxygen into themetal-oxide interface, or a combination of both. In the case of adherent oxides, eventually

II

I

\

I "\ I

\ :'--Cenler II\ I I

-Surface I\ !

i

'"-16-

x12

8

4

Center of ingot, Surface of ingotTime L, cm L, cm

0-26 h 0.040 0.002026-29 h 0.010 0.0005