diffusion driven grain boundary migration

A<,<, ZI~rollery,~<, Vol. 29. pp. 1567 to 1572. 1981 @Xl-616O’XI 091567.06$02.00:0

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DIFFUSION DRIVEN GRAIN BOUNDARY MIGRATION

PAUL G. SHEWMON

Department of Metallurgical Engineering, The Ohio State University, Columbus, OH 43210. U.S.A.

(Rrceiced 9 December 1980; in rerisedform. 27 February 1981)

Abstract-If the grain boundary diffusion coefficient of the two species in a binary alloy differ. diffusion driven grain boundary motion should lead to predictable surface relief and shape changes. This leads to predictions and equations which would allow the determination of the separate diffusion coefficient of each species as well as the chemical diffusion coefficient for the boundary, D,. For a thick sample the stress gradients built up by differences in the fluxes of the two species can lead to a reduced apparent D,. especially if the solute boundary diffusion coefficient, &, is appreciably greater than that of the solvent. 0;. For Zn diffusion into Fe it appears that Ok, >> Oz,. The stress built up will also slow the boundary motion.

RCsumGSi les coefficients de diffusion intergranulaire des deux constituants d’un alliage binaire sont difftrents, un d&placement des joints de grains sous l’effet de la diffusion devrait conduire a des change- ments du relief et de la forme de la surface. On peut done tirer des pr&visions et des kquations qui devraient permettre de dkterminer le coefficient de diffusion de chacun des constituants. ainsi que le coefficient de diffusion chimique pour le joint, &,. Dans le cas d’un 6chantillon t-pais, les gradients de contrainte provoqu6s par les diffkrences de flux des deux constituants peuvent conduire g une valeur apparente rCduite de Db. surtout lorsque le coefficient de diffusion intergranulaire du sol& 0; est nettement plus grand que celui du solvant 0;. Dans le cas de la diffusion du zinc dans le fer on a I&, x LY’,. Les contraintes qui se produisent vont ralentir le diplacement du joint.

Zusammenfassung-Wenn die Koeffizienten fiir die Korngrenzdiffusion der beiden Komponenten einer binTren Legierung sich unterscheiden, dann sollte die diffusionsinduzierte Korngrenzbewegung zu vor- hersehbaren Anderungen in Oberfllchenstruktur und -form fiihren. Daraus ergeben sich Voraussagen und Gleichungen, mit denen sowohl die Diffusionskoeffizienten der einzelnen Komponenten als such die chemischen Diffusionskoeffizienten Db fiir die Korngrenze bestimmen lassen. Bei einer dicken Probe entstehen die Spannungsgradienten durch Unterschiede in den Fliissen der beiden Komponenten und kiinnen zu einem effektiven D, fiihren, insbesondere wenn der Koeffizient fir Diffusion der gel&ten Atome entlang von Korngrenzen 0; betrichtlich hi5her ist als der der Matrixatome 0;. Fiir Zn- Diffusion in Fe scheint O;, >> I&,. Die aufgebauten Spannungen verringern die Korngrenzbewegung.

INTRODUCTION

It has now been shown by several authors that at temperatures such that grain boundary diffusion is

the dominant transport mechanism in a diffusion couple solute diffusion along the boundary will some-

how cause the boundary to move, leaving behind a higher solute solid solution. Consider for example the experiment reported by Hillert and Purdy [l]. They diffused zinc into polycrystalline pure iron sheets at 600 C using the chips of an Fe-11 wt’;,h Zn alloy as a Zn vapor source, and observed that:

1. The bulk composition of the sheet was

unchanged except in regions swept out by grain boundaries where the composition was raised to f&8”/, Zn. Each boundary passed through the sheet and essentially remained straight during its motion.

2. The grain boundaries migrated only where a composition change occurred.

3. In thick samples the boundaries moved only to a depth appreciably less than half the thickness of the sheets studied.

4. The grain boundary motion could be driven by the dezincing of a Fe-Zn alloy in a vacuum as well as by zincification.

5. Surface relief is developed between the regions that have and have not been swept by the boundary though its magnitude was not measured.

Examples of such diffusion driven boundary motion have also been reported by Cahn, Pan, and Balluffi (Cu-Zn, Au-Ag, Au-Cu) [2], den Broeder (W-Cr), [3] and Tu (Ag-Pd) [4].

These authors have speculated on the driving force for boundary motion and some noted that the boundary diffusion coefficient was orders of magnitude greater than observed using tracers and pure metal bicrystals. However, no discussion of shape changes was given. We will show that these shape changes in sheets can be used to determine the boundary counterparts of the three diffusion coefficients Darken used in his analysis of the Kirkendall shift, that is, a boundary diffusion coefficient of the solvent, O;, the solute 0;. and the chemical boundary diffusion coefficient, D,,. The constraints inherent in a thick sample

1567

1568 SHEWMON: DIFFUSION DRIVEN GRAIN BOUNDARY MIGRATION

(WPor) f

mO”,“g

boundary

’ W !c~m+,al boundary

I

I posItIon

I I I ,

I

cp = AC c2 =o

Fig. 1

suppress some of these shape changes. This leads to the suggestion that the values of the boundary diffusion coefficient found in discontinuous precipitation (a constrained system) could be lower than that found in a thin sheet with boundaries passing through the sheet (unconstrained).

ArUZlYSiS

Due to the differing constraints on shape change we consider first diffusion driven boundary migration (DDBM) in a thin sheet with cylindrical grains passing entirely through the sheet. Then the additional constraint imposed by a thick or bulk equiaxed

sample is treated. Thin sheet. Consider a thin alloy sheet of thickness

2w, with a grain boundary running through it, and a source, or sink, for solute on both surfaces (Fig. 1). Assume the boundary is moving at a steady state velocity, u, in the x-direction, and that a composition change, AC, is observed between the two sides of the moving boundary. The sheet is taken to be so thin that AC is nearly constant through the sheet. At steady-state the divergence of the flux at a given depth along the grain boundary will be a constant indepen- dent of time.

Conservation of solute for the moving boundary

gives the equation

ir2Cb u(Ac) = DJ 2.

dY

Here Db is the diffusion coefficient in the boundary and 6 the boundary thickness. For the limiting case of (C,(O) - C,) < AC one obtains

C,(y) = c, + & (w2 - y2). (2) b

Here C, is the solute concentration in the boundary at the surface y = w. Or, if one observes the lattice concentration c, in equilibrium with these boundary concentrations, the equation becomes

vAc c(y) = c, f ~ 2D,Sk (w2 - 9)

where k = C,/c is the distribution coefficient between boundary and lattice. For the case where AC varies appreciably through the foil the solution involves

tanh functions and has been discussed by Cahn in his analysis of discontinuous precipitation [S].

Consider now the shape change accompanying dif-

fusion into or out of this thin sheet of binary single phase alloy in which the sheet is surrounded by, and exchanges solute with, only a vapor phase. Thus there are no external restraints on its shape change. A unit

of volume inside the sample with dimensions Ay, AZ, and extending in the x-direction far on either side of the moving boundary is indicated in Fig. 2. At time to this volume of V, contains n atoms with an average atomic volume R. A brief time later the region bounded by the same atomic planes fixed in the lattice away from the boundary will have changed its volume due to (1) the net flux of atoms into and out of the volume along the boundary, and (2) the change in average atomic volume due to lattice parameter change from the composition change AC across the moving boundary. If the strain energy in the grains on either side of the boundary is to be minimized any divergence of the fluxes will be accommodated by a change in length normal to the boundary. This leads

to the equation

d& _ ma dt

- $Jl + J2) + 3d$~Ac). (4)

Let the boundary diffusion coefficient of the ith species

be defined by the equation

Ji E _Df aci, ay

If the molar volume is constant along the boundary in a binary system

Letting /I = d In a,,/dc, equation (4) becomes

= + 3jl >

vAc.

The last equality comes from using equation (1). Thus the sheet may lengthen or contract with the addition

of solute to the sheet depending on the relative magnitude of D\ and 0;. Note we’ve assumed that all boundaries in the sheet must move at the same velocity to eliminate constraint in the plane of the sheet.

This leads us to the question of how much solvent diffuses to the surface as solute vapor enters the boundary. Consider Fig. 3. The number of solvent atoms emerging onto the surface along unit length of the boundary in the time At is Ji6At. If this material

deposited on the surface as a slab is Ay high, Ax wide, and of solvent concentration ci, then

.I16 = Ay $ ci = vAy(c,) = -D;S s dY

(7)

where Ay is an observable step height.

SHEWMON: DIFFUSION DRIVEN GRAIN BOUNDARY MIGRATION IS69

Fig. 2

A correction stems from any change in lattice parameter resulting from the composition change AC across the boundary. If h is half the thickness of sheet, the contribution to Ay from this source is h/IAc. Equation (7) thus becomes

= u(Ay - /@AC) = - @)

Equations (6) and (8) along with equation (3) allow the measurement of the three boundary diffusion coefficients Db, 0; and 0;. Presumably Darken’s equation relating the three should also be valid.

Db = D;N2 + l&N,

for this unrestrained case. To date there are no pub- lished measurements of

dL Ay and z.

However, the different results found by Hillert and Purdy between sheet and massive specimens provides strong evidence that the diffusitivites of Fe and Zn in iron boundaries differ substantially; this is discussed below.

This analysis leads to some interesting predictions on shape change for thin sheets. Consider the following limiting cases for the evaporative surface loss of one component, e.g. Zn from brass. For simplicity assume no variation of a0 with composition, i.e. /I = 0. As a reference case note that if the solute was lost by lattice diffusion only the contraction would be isotropic, and the volume change would stem from the loss of material. Then AVIV= AN2, the atom fraction change due to evaporation, and

Ax By AZ AN, -=--_--_-, X Y = 3

(IO)

Case I-Loss by DDBM only and D; = D;. Here J1 = -J2 and VJ, = -VJz so the sweeping out of a region by a moving boundary will change its composition by AN2 but lead to no change in length in the

plane of the sheet, i.e. AX/X = 0. Thus all volume change is due to loss from the surfaces. This surface shift results from the flux of solvent into the specimen to compensate for the solute flux. If the grains are equiaxed ‘cylinders’ through the sheet with their axes perpendicular to the sheet then

Ax AZ _=_= 0 and 9=AN,. (11) X 3 J

Case II-Loss by DDBM on/y and D; >> D’i. Here there is a substantial flux of atoms out of the sheet to the surface giving rise to a contraction in the plane of the sheet. However, these atoms evaporate and the low value of D1 gives a negligible flux of solvent from the surface into the sheet. Given the constraint that AVIV = AN1 for the regions swept by a boundary, the following is true for these regions

AZ Ax ANz and 2 1 o, _=_=-

2 (12)

Z X J

Case III-Loss by DDBM only, D’, >> D;. Equa- tion (9) indicates the boundary diffusion coefficient would be low in this case since the slowly diffusing solute must still make a major contribution to D,. Thus, J1 >> J2 and there will be a substantial expansion in the plane of the sheet even though solute is being lost from the surface. This will exaggerate the drop in the surface. Thus for a boundary in the Y-z plane moving in the x direction

Ax - > 0,

A\ := -k+AN,. X J X

(13)

Thus it is seen that the relative sign and magnitude of Ax/x and Ay/y allow one to draw firm conclusions about the relative magnitudes of 0; and 0;.

For the case of solute gain, e.g. Zn diffusing into Cu sheet, the sign of AN2 reverses and the expressions for Cases I and II are still valid. For Case III Ax/x < 0 and the expression for AYjy is unchanged.

The surface relief measurement. A~/Y, is relatively simple to measure since the displacement would be a sharp step across the grain boundary. The displace- ments in the sheet will be harder to measure accu- rately since the Ax/x in equations (10) to (13) must

flux Of Solute /

/’

Fig. 3

A.M. 29,9--s


vopor

gr. A gr. B

I Gyc -=‘I’

(0)

gr. A

\ gr. B

Gyc > Y

( b) Fig. 4. Differing grain shapes expected when the net driving

force G is (a) less than y/y, and (b) greater than y/y,

be calculated by multiplying marker shifts by the fraction of the volume between the markers that has been swept by boundaries.

Thick samples. In thick samples two new effects enter. The thickness of the sample is greater than the depth that boundary diffusion can supply, or remove, solute from. Also, the bulk of the sample restrains the shape change of the surface layer. The surface relief effects discussed above should still occur, especially in the initial stages of boundary motion.

Consider first the case in which 0; = 0; and solute exchange is with the vapor. Near the free surface the grain boundaries will initially move at a velocity com- parable to that in thin foils. Below some critical depth, y,, the boundary diffusion distance will be too great and the lattice composition in equilibrium with the boundary C~ will not differ from the bulk composition, co, by enough to drive the boundary. Below this depth the curvature will change since any boundary motion will come primarily from curvature. For the case of solute loss, the depth of this transition will be given by

cdy) = co - AC > 0 (14)

This would suggest that the transition depth would increase with the bulk concentration of the alloy co.

Hillert and Purdy (hereafter H-P) found that in sheets 45 pm thick boundaries passing through the sheet moved essentially as planes with only a few per- cent drop in AC between edge and center of the sheet [l]. However, in thick samples the boundary moved only to a depth of 6-7 pm. This suggests that Db is appreciably larger in the case of the unrestrained sheet, and that unequal boundary fluxes of solute and solvent set up substantial stresses in the thick sample which stop the boundary motion.

The morphology of the migrating boundary will give an indication of the magnitude of the driving force that is moving the boundary. For example, if the product of the critical depth, y, and the free energy change driving the boundary, G, is small, the boundary will sweep only to a position such as that shown in Fig. 4(a). The condition for continuous sweeping of the boundary as shown in Fig. 4(b) is

GY, ’ Y (15)

where y is the grain boundary tension. H-P found the Fig. 4a configuration with y, ‘v 7 pm. If y is 0.8 J/m2 the net driving force on the boundary when it stopped was less than y/y, or 0.13 MJ/m3. This is two orders of magnitude less than H-P calculate for the driving force by equating it to the free energy of mixing zinc into iron. Strain energy probably played a dominant role in stopping the boundary motion in their thick sample.

If 0; and 0; differ the initial flux of the faster diffusing species will build up a stress which inhibits its flow and increases the chemical potential drop of the slower diffusing species, by a.$ where cX is the stress normal to the boundary. If the driving force for grain boundary motion and relative diffusivities per- mit, this stress will build up until J1 = -J2 in the boundary. In this case the stress builds no farther with grain boundary motion and the configuration of Fig. 4(b) will grow as a steady state. If the driving force is too low and/or the difference between 0; and 0; too large, the boundary will only sweep a limited distance and stop due to the high elastic stresses that develop.

The calculation of the solute flux as the stress builds up is quite difficult since it requires a calculation of the stress gradients. However, some insight into the problem can be obtained from a consider- ation of the situation when the fluxes of the two species are equal and opposite along the boundary. If the boundary is planar and 1 J1 1 # 1 .I2 1, a stress c,(y) will build up normal to the boundary. The chemical potential of the i-th species will be

pi = pco + kT In N1 + a,R

and the gradient in pi along the boundary as

Vpi=kTVlnN, +RVa,.

SHEWMON: DIFFUSION DRIVEN GRAIN BOUNDARY MIGRATION 1571

The flux equation will then be of the form

JI= -D1 (16)

Assume D, = n D, (n > 1). VNi should still equal -VN,, so for J1 + J2 = 0 one obtains

-nD, +VN, + N$V,, >

where the stress gradient works to increase the flux of the slower diffusing species and decrease that of the faster species. Thus

VN,=(n---~l)N+(+l). (17)

If n = 10, initially J1 = - 10 J2, but as stresses build up J1 would decrease by a factor of 5 while J2 would only double to give J, = - J2. Since the stress gradients would not be expected to change the concentration gradients, at least to first order, the apparent values of 0; and 0; in the equation for Db [equation (9)] would be by roughly l/S and 2 times the value in the stress free state, respectively. This would in general decrease Db though in a sufficiently dilute solution the apparent doubling of 0; could increase

D,. If the relative values of 0; and 0; are inverted so

that, 0; = 10 0; then the relative change in the fluxes as stress builds up would be the same, i.e. initially J2 = - 10 J1 but with J2 dropping by a factor of 5 as JI approaches J2. However, in a dilute solution 0; now dominates Db so Db would also drop by almost a factor of 5.

The surface relief effects should provide the most unambiguous indication of the degree to which re- sidual stresses influence the atom fluxes. For example if zinc is being added to iron and, Oz. >> L&,, then one would expect Ay 1: 0 for the step left at the initial boundary position in Fig. 5a and a finite By where the boundary came to rest, since by this time the fluxes of the iron would be greater than it was initially even though not yet equal in magnitude to that of the zinc. If Oz, << L&, there would be a substantial A?’ at the initial boundary position due to the high outward flux of iron but A? would decrease as the boundary approached its final position [Fig. 5(b)].

Experimental results. To check on the surface relief in a system that exhibits DDBM the Fe-Zn system was studied. A sheet of 0.05 mm thick steel shim stock was decarburized in wet hydrogen and furnace cooled. After heat treatment the sheet was electro- polished and annealed in an evacuated Vycor tube with a piece of a 75 wt%Zn-257; Cu alloy at one end to serve as a zinc source. The zinc partial pressure was sufficient to form a zinc rich phase near the Zn-Cu alloy but farther away no surface film formed and

gr. bdry. I

Initial 4

D; > D;

(a)

D2’ < D;

(bl

Fig. 5. Surface relief expected in DDBM when stress gradients develop with boundary motion for (a) 0; > 0;

and (b) 0; < 0;.

DDBM was noted. It often seemed to nucleate at discrete spots and spread out from there. Using an interference microscope it could be shown that there was a step 3 pm high at the initial boundary position. The surface sloped away from this ledge, meeting the surface often after the boundary had migrated about

20pm. Where the boundary stopped the step across the grain boundary was about half a fringe or 0.1 pm.

Since the maximum surface elevation forms on the initial boundary position, and decays by roughly an order of magnitude as the boundary moves, it is clear from the analysis given above that D;, is much greater than Oz.. The stresses tending to make the fluxes of iron and zinc equal would be tensile in nature even though zinc is being added to the sample as the boundary moves. The hillocks seemed to nucleate at points on the boundary and spread from there. At other points the boundary moved without leaving significant surface relief.

DISCUSSION

This analysis indicates clearly that the shape change resulting from DDBM in thin foils provides information on the relative diffusivities of the two


species in the grain boundary. However, any experiment to measure D’i and D; must cope with the problems of stress buildup in the foil due to non-uniform boundary motion. Equation 6 will be difficult to use because the fraction of a foil swept by DDBM has been so small in the systems studied to date. The reason for this is not clear but probably results from the buildup of stresses due to the volume change occurring.

The buildup of stress influences the DDBM process in two inde~ndent ways, through the effect of stress gradients on boundary fluxes, and through the buildup of an average sfress that the volume change must do work against. From equation (16) and the discussion that follows one can estimate the stress gradient term from the inequality

ANi

i-l

R do

N,AY %i-&

H-P found in their foils that AN, = 0.02, Ni = 0.93. If T = 850 K, Aa is less than 23 MPa.

To estimate the degree to which PdV work reduces AG, assume that the average stress builds to a flow stress of 100 MPa and that dV/V = 2% for the trans- formation of a given unit volume. This leads to 2 MJ/m3 which is much less than the free energy of mixing which H-P give as 40 MJ/m3.

Thus the stress difference needed to make J, = -Jz is much less than the Aow stress and the flow stress cannot exert enough force to stop the expansion even if the volume change is not relieved

by the two fluxes becoming equal and opposite. This casts doubt on H-P’s suggestion that all of AG,,, can go into driving the boundaries since it is unclear why they move such short distance and are stopped so easily. This supports Cahn and Ballulli’s argument that only a fraction of AG, (they suggested the non- ideal part) could go into driving the grain bound-

ary PI. The evidence that O;, >> 4, stems primarily from

our observation that DDBM generates large initial step heights which then decay as the boundary moves. Figure 1 of H-P’s paper clearly shows appreciable relief which often appears to resemble this sort of ‘hip roof with a ridge along the original grain boundary. Private communication with G. Purdy confirmed that a slope of this sort was seen in their work. The accu- rate determination of 0; and D; must await the de- velopment of an experiment in which stress gradients can be shown not to influence the fluxes. A test of the postulate that DDBM is caused by the difference between f, and Jt [2,3] would be to see if boundary velocity decreases monotonically with this observed decrease in surface step height.

REFERENCES

1. M. Hillert and G. R. Purdy, Acra metall. 26, 333 (1978). 2. J. W. Cahn, J. D. Pan and R. W. Balluffi, Scripra

metal/. 13, 503 (1979). 3. F. J. A. den Broeder, Acta metall. 20, 319 (1972). 4. K. N. Tu, J. appf. Ph_w 48, 3400 (1977). 5. J. W. Cahn, Acta mefa/Z. 7, 18 (1959).

diffusion driven grain boundary migration

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