On the plastic deformation associated with the Kirkendall effect

OS

THE

PLASTIC

DEFORMATIOS

ASSOCIATED J.

WITH

THE

KIRKENDALL

EFFECT*

SCHLIPF:

Plastic deformation accompanying Kirkendall experiments has long been known. The present theory shows that plastic deformation is a necessary and unavoidable consequence of the fact that marker displacement (i.e. a Kirkendall effect) occurs only in the direction of diffusion. The interrelation of plastic strain, vacancy concentration, and marker movement is investigated, and it is found that the strain rate The position of the porous zone n-irh is proportional to the velocity gradient of the marker displacements. respect to the vacancy supersaturation is determined. It isshownthat themaximum of the marker velocit> coincides with the center of the porous zone. Assuming edge dislocations as sources and sinks for vacancies the stress produced by climbing dislocations is calculated. Its influence on plastic deformation is discussed. It turns out that plastic deformation starts long before marker displacements can be detected, i.e. far outside the macroscopically defined diffusion zone. DEFOR3L~TIOS

PLASTIQUE

ASSOCIEE

A L’EFFET

KIRKESDALL

11 est connu depuis longtemps que la deformation plastique est un effet secondaire dens les experiences de Kirkendall. La presente theorie montre que la deformation plastique eat une consequence necessaire et inevitable du fait que le deplacement du marqueur (c.a.d. I’effet Kirkendall) se produit seulement clans la direction de diffusion. L’interrelation de la deformation plastique, de la concentration de5 lacunes et du mouvement du marqueur est etudiee, et l’auteur trouve que la vitesse de deformation eat proportionnelle au gradient de vitesse des d&placements du marqueur. La position de la zone poreuse est determinCe par rapport a la sursaturation des lacunes. L’auteur montre que le maximum de vitesse du marqueur coincide avec le centre de la zone poreuse. Si on suppose que les dislocations coin sent des sources et des puits de lacunes, on peut calculer la contrainte produite par la mont&e des dislocations, On en deduit que la deformation plastique et discuter de son influence sur la deformation plastique. commence longtemps avant que les d&placements du marqueur puissent 6tre detect&, c.a.d. loin de la zone de diffusion definie macroscopiquement. MIT

DELI KIRKESDALL-EFFEKT

VERBUSDESE

PLASTISCHE

VERFORMUSG

Es ist seit langem bekannt, da13 als Sebeneffekt des Kirkendall-Effekts eine plastische Verformung auftritt. Die vorliegende Theorie zeigt, da13 die plastische Verformung eine notwendige und unvermeidliche Konsequenz der Tatsache ist, da13 die Verschiebung der Markierung (d.h. der KirkendallEffekt) nur in Richtung der Diffusion erfolgt. Die Zusammenhange zmischen plastischer Dehnung. Leerstellenkonzentration und Bewegung der Markierung werden untersucht und es zeigt sich, daR cl:e Dehngeschnindigkeit proportional zum Geschwindigkeitsgradienten der Verschiebungen der ZIarkierung ist. Die Lage cles porcren Bereichs in Bezug auf die Leerstellentibersattigung wird bestimmt. Es wircl gezeigt, da13 das Maximum der Geschwindigkeit der Markierung mit dem Zentrum des poriisen Bereichj tibereinstimmt. Unter der Annahme, dal3 Stufenversetzungen als Leerstellenquellen und -senken wirken, wird die van kletternden Versetzungen verursachte Spannung berechnet. Ihr Eintlul3 auf die plastische Verformung wird diakutiert. Dabei zeigt es sich, daO die plastische Verformung einsetzt lange bevor eine Verschiebung der Markierung nachgewiesen nerden kann, d.h. weit aul3erhalb cler makroskopisch definierten Diffusionszone.

1.

Diffusion usually during

INTRODUCTION

in 1958, Doo and Balluffi(j)

couples of two different, metals

show a displacement a diffusion

experiment.

attributed

by Darken t2) to the unequal

initially

and Kirkenclall(r)

the slower atoms

to large displacements in the field already solely

diffusirities

However,

in the

direction

arises

accidental

or whether

feature

of the Kirkendall to investigate

the marker movements

early workers

will be demonstrated

of diffusion.

this by assuming

whether

merely

be given.

Bardeen

by

i.e. that

the

plastic

occurs.

effect.

In Section

these they The

this problem

In the following two sections

movements

confirmed

is accompanied

and glide of dislocations,

question

undertakes

experimentally

effect

perpendicular

noticedf3) that marker

and Herringc4) explained

the region

should give rise

also in directions

to the direct,ion of diffusion. occur

The

species.

containing

Kirkendall

deformation

has been

net flow of atoms towards

the

production

This effect which was

by Smigelskas

The resulting

that

of the melding interface

discovered

of the two atomic

or alloys

findings are

are

a general

present, paper theoretically.

the relationship

between

and the plastic flow of material and quantitat,ive

relations

will

4 it will be shown that with the

usual diffusion geometry plastic deformation via glide processes is a necessary corrolary of the Kirken-

that climb

of edge dislocations with a Burgers vector parallel to the direction of diffusion is more effective than climb of dislocations of other orientations. Finally,

da11 effect. In Section 5 the results are discussed some numerical estimates are given.

* Received July li, 1953. i Institut fur allgemeine Metallkunde und Metallphysik, Technische Hochschule Aachen, West Germany.

Let us consider the usual unidirectional diffusion geometry with the z-axis of the coordinate svstem

_1CTA METALLURGICA,

VOL.

91, APRIL

1953

2.

435

MARKER

MOVEMENT DEFORMATION

AND

and

PLASTIC

436

_iCT_i

JIETALLTRGIC.1.

parallel to the direction of diffusion. Let cl, c, denote the mole fraction of the two interdiffusing components d and B respectively. With D, the intrinsic diffusion coefficient of component, -4, we write the flux density of this component with respect to an inert marker at 2:

(1) -4 similar expression is obtained for component B. In the following we adopt the convention DA < D,. Let us define a volume element (&Z)~ in a slice of thickness Aa which is cut out perpendicular to the direction of diffusion. We may envisage both faces of the slice being marked by inert markers. The net flux of atoms through the volume element is given by J = j,

+ j,

=

-(DA

-

DB)acA/&

where c, + cB = 1 has been used. In traversing volume element the flux changes by

(2) the

(3) where n-e have set D, = DB - DA for abbreviation. AJlAu represents the mole fraction of atoms which have entered the volume element in excess over those having moved out. In order to accommodate the excess atoms on regular lattice sites, an equal number of new lattice sites has to be generated. Naively we would expect that the volume element expands equally along all three dimensions in space, increasing the 1engt.h of a cube edge by dAa. The volume change produced within a time interral dt is given by d(Aa)3 = S(Aa)*dAa

= -AJ(Aa)*dt

(4)

VOL.

21,

1973

originally containing the slower moving species, because more atoms enter this region than leave it. The reverse would be true for the other part originally containing the faster moving species only: here the slices would shrink appreciably due to the fact that more atoms move out than come in. It is interesting to note that both swelling and shrinkage can be observed experimentally.(6) However, this phenomenon is restricted to a very thin surface layer (~10-4 cm). The reason is simple enough. The slices cannot slide over each other, but are rigidly connected to each other. Therefore radial tensile or compressive stresses are built up which increase monotonically until the flow stress of the material is reached. From then on the radial displacements are converted into axial displacements by means of plastic flow. Then the marker displacements in the direction of diffusion represent the total three dimensional volume change. Thus the local velocity increment of an inert marker dAa/dt is three times that given by equation (5): dha~A~=-_hJ dt

. .

This may immediately be integrated to give Darken’s equation for the velocity of a marker v = -J

= -DDac,ial.

de = d(Aa)?/(Aa)’ Inserting

lAJat 3 Aa

(7)

The preceding analysis provides us also with the key to handle the plastic deformation quantitatively. Again, the increment of plastic deformation due to the conversion of the radial volume change is given by = -“dAnlila.

which gives us dha -_=--Aa

(6)

equations

(8)

(5) and (6) gives us

.

Equation (5) allows a simple interpretation: one third of the surplus atoms is responsible for the expansion of the volume element in each of the three dimensions in space respectively. The expansion parallel to the direction of diffusion is uniform over the nhole slice. Therefore no restrictions are imposed in this direction, the slice simply becomes thicker by an amount dAa. However, trro-thirds of the volume change take place along the directions perpendicular to the z-axis, and here the contributions of the individual volume elements are additive. Let us assume for a moment the n-hole specimen be cut into thin slices which are able to slide over each other without friction. Then considerable swelling would be observed in that part of the diffusion couple

AriAa is the increase of marker velocity in going from z to z + An. In the limit. Au - dc, Aa -+ dz ve have -” dv d& -=--* dt 3 a:

(10)

This equation relates the rate of plastic deformation to the velocity gradient of a marker at any point, z along the diffusion direction, a quantity Fvhich is easily accessible theoretically as well as experimentally.“) dt the same time it becomes clear that the plastic deformation associated with the Kirkenda11 effect is not an accidental and uncontrollable

J. SCHLIPF:

PLASTIC

DEFORMATIOS

KITH

KIRGESDALL

EFFECT

137

maximum supersaturation. Similarly, because of Equation (10) the rate of plastic deformation t- also has extreme values at de-c/d? = 0. Therefore, in problems involving marker movements 3. S-I-RAIN fihTE AND VACASCY CONCENTRATIOS and plastic deformation it should be farourable to use -1s has been shown in a previous paper’” dc1d.z a moving reference frame. Its origin is located at the is likewise important in determining the concentration center of the porous zone, or, more precisely, at the of vacancies in the diffusion zone. With a vacancy point where dcldz equals zero. Denoting by 2 the mechanism of diffusion, -J of Equation (2) represents coordinates with respect to the resting laboratory the net amount of vacancies “pumped” against the system, and by t: the coordinates with respect to the net flux of atoms, as has been shown by Seitz.@) moving system me have .&cordingly, an undersaturation of vacancies will be Z = 2’ - :L’(t) maintained in the region originally containing the slower diffusing atoms, i.e. component -4. Similarly, where * ‘(t) - 1 t designates the position of the . . a supersaturation of vacancies will build up in the . 1: .stem. orrwn o?movmQ c I region of component B. Therefore, a plane must exist I”t has been shown(7) that in this system v = in the diffusion zone where the concentration of -DnacA/& can be closely approximated by an vacancies exactly equals the equilibrium concentraexpression of the form tion. The position of this plane has long been a matter of speculation. Some authors identified it with the v = =cc& esp (--_‘/iD,t) (11) JrrD,t welding interface,(g*s) others with the point where porosity just tends to form.clO) It can easily be shown where co stands for c, at E = 0, while D, = cAD, f that both assumptions are wrong. The reason %?DLl. In the region z < 0 we replace D, and D, for their failure is, that they refer to pore formation by their mean values D,, and D, respectively. only. Experiments,@) however, show clearly that the For z ~0 we have accordingly D,, and DC,. porous zone wanders by growth of the pores facing Inserting this into (10) we get: the side of the faster diffusing component B, and by z . ” coDD dissolution of the pores facing the side of the sloner &=-exp ( --z’j4Dct). (12) 3 $+ D,t 2 Jqt diffusing component 3. This experimental fact immediately suggests that there exists a superThe extrema of this function occur at z = 2Y/G. saturation of vacancies in the side B of the porous Thus the maximum strain rate is given by zone, and an undersaturation in the side facing -4. “0 DD;. Thus the plane of zero supersaturation and under(13) 4m = 3,Jlne D,t saturation must be located in the middle of the porous zone. Since D,/D, usually is of the order of 1 and its This has some interesting consequences. (1) -Is temperature dependence is relatively weak, the has been shove in(‘) the divergence of the marker maximum strain rate is given essentially by the velocity v is proportional to AC,, the deviation of reciprocal of t. In the beginning of a diffusion run, vacancy concentration from equilibrium (if minor the strain rate is seen to be very high, but even after effects such as vacancy production by plastic de- a diffusion time of several hrs., still i: GZlOA set-l. formation and the dependence of atomic volume on 4. MICROSCOPIC THEORT chemical composition can be neglected). In other We now want to discuss the nature of the plastic words, dvldz: goes through zero on the same plane where the undersaturation of vacancies changes into a deformation. It is most convenient to do this in supersaturation of vacancies, namely in the middle of terms of the vacancies generated and annihilated in the process of diffusion. The following considerations the porous zone. Thus v has its maximum there, are based on the assumption that edge dislocations as has been verified in{” by evaluation of experimental are the only sources and sinks for vacancies, although results of Neumann and Walther.‘6) others (e.g. free surfaces and gram boundaries) (2) Because dv/dz has extremes at the points of might also be operative. imlexion of L’, dsv/dz* = 0, also the maximum superti edge dislocation can absorb or emit a vacancy saturation and undersaturation occur at these points. by climbing. In each of these elementary climb Thus that part of the diffusion zone where porosity just tends to form is centered around the plane of processes an atomic volume is annihilated or generated.

side effect but is intimately related to the marker movements.

9

ACT-1

438

JIET,\LLCRGICA,

However, since climb results in growth or shrinkage of the extra plane, the associated volume change takes place perpendicular to this plane only, i.e. in the direction of the Burgers vector. In Fig. 1 the situation is displayed schematically. -4 cube of side length .L contains an edge dislocation running parallel to the +-axis and having a Burgers vector b. Climbing a distance PC from x, = w/2 to x2 = --w/2 the dislocation produces a dilation of magnitude b parallel to the zraxis which gives rise to a (plastic) strain &II = b/L. If the dilation is prevented an (elastic) compressive stress -ull is set up acting over a cross sectional area wL. Since this is a problem of plane strain, i.e. Ed = 0, isotropic elasticity yields : El1 =

Gil -

E

;

-

Ez2 = 2% E

Y E(

(ozL -i- crs3) = -b/L

t

011

G33)

with the boundary condition that planes x2 = &w/2 of Fig. 1 must be stress free, i.e. crP2= 0. We then obtain : c33

=

vu11

(711 ----* -

E

ll=l--y

(15)

21,

1973

~~~ is the stress produced by p dislocations per cm? after they have climbed a distance K. The same result may be obtained bv superimposing the stress fields of the climbing dislocations, as shown in appendix A. Since bwp s A V/ V represents the fractional volume change, we may say that the normal stress in a volume plane is -_“G/(l - v) times the fractional change taking place perpendicular to that plane. Four simple types of dislocation arrangement with respect to 2 the diffusion direction may be visualized for which the normal stress distribution is easily obtained. (1) All Burgers vectors are parallel to the z-axis. This would imply cZi = (Tag. However, since no constraints esist in the direction of diffusion, the volume change takes place without any restriction, rendering cLL zero, as well as all other stresses. (2) 911 Burgers vectors are parallel to the Zaxis. This gives o,, = cll, since no volume change can take place in the x direction, and G,, = 0. Thus d zz will increase until the 5.0~ stress is reached and t.he material will be extruded in the I direction. (3) All Burgers vectors are perpendicular to the z axis and randomly distributed in the z--y plane. NOK all planes parallel to the z axis have equal normal by averaging over all oristress grr. It is obtained entations of the Burgers vector:

l-vy?L

Substituting G = E/2(1 f densit>- p = l/wL we have a

b

VOL.

v)

and

the

dislocation

--2G

bwp.

I

b

Fx. 1. An edge dislocation with its glide plane at x2 = w/f climbs a distance 2~’ to a new position at I? = -w/2 producing a compressive stress -gII, if L is kept fixed between z2 = &w/2.

?3 1 fJ11co?? v dg = tall crTT= 2ir I 0 2G 1ilV =---. l-Y’7V

(17)

Since for a fractional volume change IV/V = 1 the stress produced is -2G/(l - v). this may be interpreted as half of the volume change being active in each of the two dimensions of the x-y plane. (4) Random spatial distribution of Burgers vectors. This idealized arrangement of dislocations n-ill depict most closely the situation in a real diffusion experiment. Sgain all planes parallel to the z axis have equal stress CI~,, which is obtained now by averaging over all orientations in space. The contribution of a Burgers vector making an angle 0 with the z axis and an angle v with the normal of the plane under consideration is (ill sin? 0 cos2 y. Thus 27 7 1 oll sin’ 0 ~0s’) 7 sin 0 d0 da, (1s) aPr = 4x i‘s0 0 which gives us u IT = *a,,=

2G ----. l--v3

1A.T’ J’

(19)

J. SCHLIPF:

PLASTIC

DEFORX_kTIOS

_&gain this can be interpreted as one third of the fractional volume change being active in each of the three dimensions in space, and this is esactly what we expect according to the phenomenological approach in Section 9. Wth 11*/J* increasing from zero the stresses IS,, increase until the flow stress T,,~~of the material is reached. Ke thus may define a critical fractional volume change (AVV/fl),tit in order to begin plastic deformation : (--\FI ~)cril = [(l -

Y)/2]rc~~r/G.

(20)

Kith AJ’/V = b wp this defines a critical climbing _&cording to the theory of work distance ~~~~~~~~~ hardening

rcrit = xGbt /-p, where ‘A is a numerical

factor
‘w&t w %/lO.

&other useful quantity is the mean climbing distance iE of dislocations after a certain diffusion time t has elapsed. It is obtained by relating the number of vacancies to the total volume change they bring about. Using the moving reference system of Section 3, x1-ehave for the number N, of vacancies produced in the region z 5 0 after time t 3,

j”2)

dr cir

where (23) represents the net production rate of vacancies(‘), L2 is the cross-sectional area of the specimen and V, denotes atomic volume. Introdllcing equation (23) into equation (22) and following the same line of arguments that leads to equation (11) 3-e get:

WITH

D III

KIRKESD.kLL

andDC, respectively

average fractional the diffusion time. dislocation density sideration we may climbing distance

a!+

EFFECT

holds for z 2 0, i.e. the

rolume change is independent of Wth p(t) designating the acerage in the diffusion zone under conrewrite (25) in terms of the mean

Since p(t) will initially increase and erentuaily approach a steady state value the mean climbing distance of dislocations will also become independent. of time for long diffusion times. ObviousI)* the shorter climbing distances near the margin of the diffusion zone w-ill then be exactly compensated by the larger climbing distance in the interior. 5. DISCUSSION

AND

CONCLUSIONS

(1) From equation (21) we learn that plastic deformation starts after very short climbing distances. This can be put into an even more informative version. With the volume element f’ = (Au)* we have A V/V = 3dAu/Aa and Equation (20) yields the critical marker displacement : _ 1 6

y T&f

.

(“7)

t7

Introduei~g rc,.&2 .u lo4 and Y N f, we obtain (dA~lAa)~~i~m 10-j. In order to become detectable the reIative marker displacements dAila/Anhave to be Thus plastic deformation starts larger than *lo-“. long before marker displacements are detectable, or in other words, plastic deformation takes place in regions far outside the macroscopically defined diffusion zone. This forms the basis of a theory of these processes published previously.(ll) (2) It is interesting to relate the mean climbing distance 9 to the mean dislocation distance 2. = I/\,‘;. From equation (26) we have

(24) _Issuming 2-t’ to be the width of the diffusion zone in the side z < 0 w-e have for the total volume considered 1‘ = 22./D& L”, and from equation (21)

Thence

-4. similar relation

nith

D,,

and DC,

instead

of

Inserting

typical numbers, for instance for a silver

gold couple,“) we find c,/%‘& YY10-l; D,,jDcA Q 1, Since always i. > lob we obtain from equation (28) G > 1, i.e. the mean climbing distance is large compared to the mean dislocation spacing at any time, and the more so the lower the dislocation density is. As a consequence we must regard dislocation climb as a possible source of dislocation multiplication in addition to the dislocation proThe multiplication duction by glide processes. process is very similar in both cases: a Frank-Read

ACT-1

440

METALLURGIC_\,

source emits dislocation rings. In the case of multiplication by glide the rings lie in the glide plane. In the case of multiplication bv climb they lie in the climb plane, i.e. in the plane perpendicular to the glide plane. In climb processes of the estent described above, the vacancies eat up substantial parts of the extra plane. Rings emitted from Frank-Read sources near a surface will therefore emerge at the surface and produce steps similar to the slip lines However, while produced by glide dislocations. the latter are formed at intersections of the surface with (111) planes, the steps due to climb are formed at intersections with (110) planes, in f.c.c. crystals. This is exactly what has been observed in experiments by Barnes -(l”) in the &-Xi system and by Ruthc’3) in the Au-Ag system. They found “ripples” along the intersection of (110) planes and correctly interpreted this effect to be due to dislocation climb. (3) In an effort to explain the fact t,hat marker movements occur only in the direction of diffusion, Bardeen and Herring“‘) proposed a model resembling case 1 discussed in Section 4. They suggested that dislocation rings with a Burgers vector perpendicular to the direction of diffusion would rapidly outgrow the diffusion zone, while rings with parallel Burgers vectors could continue to grow almost indefinitely. However, this argument is greatly invalidated by the fact that climb is not the only production process for dislocations. As has been shown in Section 4, slight amounts of climb n-ill lead to dislocation multiplication by glide. Moreover, assuming a dislocation density of lOlo cm-? in the interior of the diffusion zone, E % 10-4 cm which is much less than the width of the diffusion zone in an ordinary Kirkendall experiment. We thus conclude that dislocations of all orientations participate in climbing and therefore case 4 of the preceding section represents an acceptable model of the Kirkendall effect. ACKXOWLEDGEMENT

I wish to thank Prof. Dr. K. L&&e for his continued encouragement and advice, and for many helpful discussions. REFERENCES and E. 0. KIRKESDALL, Trans. rim. Engrs. 171, 130 (194;). 2. L. S. DARKEF, Trans. Am. Iwt. Xin. Engre. 175, 181 (1945).

1. -1. D. SJIIGELSKAS

Inst. Xin.

\-OL.

2 I,

1973

3. L. C. C. DI s1~v.1 and R. F. MEHL, Trans. dm. In&. Mire. E,bgra. 191, 15.5 (195 I). 4. .T. B.IRDEEX and C. HERR~SG, Symposium on Imperfectiorra in A-early Perfect Cryat&, p. 2SO. John Wiley (1950). 5. V. T. Doo and R. W. B.~LLUFFI. dcta_Ifel. 6, 428 (19%). 6. T. HEL>L~S and G. W.*LTHER. 2. Metallkde. 48, 151

(19.5;). 7. J. SCHLIPF. &to Met. 14, ST7 (1966). S. F. SEITZ, Phys. Rev. 74, 1513 (194s). 9. I’. -ADDI. G. BREBEC. S. V. Doas. >I. GERL and J. PHILIBERT. Thermoduttamics II. p. 5~5. International .Itomic Energy .Qen& (1966). 10. R. W, BALLCFFI and L. L. SEIGLE, J. appl. Phys. 25, 13so (1934). 11. J. SCHLIPF. 2. Netallkde. 59, TO8 (196S). 12. R. S. BUZSEJ, Report of a conference on defects in cryJtslline solids. p. 359, Physics Society (19.55). 13. 1.. RLTH. Trans. Am. Inet. Min. Engrs. 227, 575 (1963). 14. J. P. HIRTH and J. LOTHE, Theory of Dislocalions, p. 74.

JIcGraw-Hill

(1968).

APPENDIX

A

Consider the cube of Fig. 1 containing now a large number S of parallel edge dislocations per cm situated along the line x2 = w/2, and running parallel to the x,-axis. The Burgers vector of the dislocations is in the ~,-direction. A dislocation at x1 = E, x2 = to/-l produces a stress field a?((, w/Z). After having climbed a distance tc all dislocations are lined up In the new position the stress along x, = -w/2. field is given by G,j(=‘,[r) = G;,!‘(i, - W/2) -

G;i’“(+/‘).

(Al)

The number of dislocations beWeen [ and 5 f df is Sdz. Thus the e_stra stress field due to climb of the SL dislocations is .-L/B

Gij

=

J

-Ll2

Gij(t,W)a:.

(*AZ)

Substituting in (Al) the stress components ~~ of the equation (82) is easily integrated. edge dislocation,“4’ In the limit L - co we obtain G

11=

G33 =

-%26x/( YG,,;

1-

V);

Gt2 =

0

G1? =

0.

Introducing the dislocation density p = N/w, equations (15) and (16) of the text are reproduced. The special arrangement of the dislocations used here for the calculations is equivalent to a random diskibution provided all the dislocations are climbing an equal distance tc.