A Metallographic Study of Diffusion-Induced

A METALLOGRAPHIC STUDY OF DIFFUSION-INDUCED GRAIN BOUNDARY MIGRATION IN THE Fe-25 SYSTEM

LI CHONGMOt and MATS HILLERT

Division of Physical Metallurgy. Royal Institute of Technology. 10044 Stockholm, Sweden

Abstrsct-Diffusion-induced grain boundary migration has been studied by zinci~~tio~ of pure iron. The temperature dependence of the rate was measured at 460~65o’C. A change in morphology was observed between 460 and 650 C and may explain the fact that the phenomenon changed character at higher temperatures. The temperature dependence of r’D”r5 may be decisive. I‘ being the rate of migratton. D’ the gram boundary diffusivity and i, the thickness of grain boundaries. The concentration behind the migrating boundaries was studied as function of the distance from the surface by microprobe measurements and allowed the grain boundary diffusrvity to be evaluated as function of temperature.

Fine zinc-rich grains of ferrite sometimes nucleate on the surface a form and complete fine-grained surface layer. After a while it disappears by abnormal grain growth but the increased zinc content remains.

R6sumtLNous avons etudii la migratton de joints de grains provoquie par la diffusion de zinc dans le fer pur. lvous avons mesure la variation de la vitesse en fonction de la temperature entre 460 et 650 C et observe un changement de morphologie entre ces temperatures: il pourrait exphquer le changement de caractere du phenomene aux temperatures eievees. La variation de t/f?*6 en fonction de la temperature peut itre decisive. 1 itant fa viresse de migration. D” la diffusivit~ intergranulaire et d Tepaisseur des joints de grains. Nous avons etudie la concentration derriere les joints de grains en migration en fonction de la distance a la surface. par microsonde. et nous avons pu evaluer la diffusivite intergranu- lair, en fonction de la temperature.

Des petits grains de ferrite riches en zmc germent parfois a la surface et forment une couche superfi- cielle complete a petits grains. Au bout d’un certain temps. elle disparatt par croissance anormale de grains, mais I’augmentation de la teneur en zmc reste.

Zu~rn~nfa~~-Die di~usionsIndu~ierte Korngren~enwanderung wurde mittels Verzinken von Reineisen untersucht. Die Temperaturabhlngigkeit der Geschwindigkeit wurde zwischen 460 und.G.t_YC gemessen. In diesem Temperaturintervall anderte sich die Morphologie. welches die beobachtete Ander- ung im Charakter der Erscheinung bei hoherer Temperatur erkllren kann. Die Temperaturabhangigkeit von I’ D”8 kann ausschlaggebend sein: r bedeutet die Wanderungsgeschwindigkeit. Dh die Korngrenzdif- fusivitat und ci die Dicke der Korngrenzen. Die Konzentration hinter der wandernden Korngrenze wurde mittels Mikrosonde~messungen m Ab~n~igkeit von der Tiefenlage unter der Oberflache ermit- telt. Diese Messungen ermoglichen. dte Korngre~zdiffus~vit~t als Funktion der Temperatur auszuwerten.

Kleme zinkreiche Ferritkorner entstehen manchmal an der Oberflache und bilden eine vollstandige. femkornige Oberflachenschicht. Nach einer Weile verschwindet diese durch aul3ergewohniiches Korn- wachstum. wobei der erhijhte Zinkgehalt erhalten bleibt.

rNTRODU~lON differences in chemical potentials in the matrix phase

The gain boundaries in the matrix phase migrate on the two sides of the boundary. caused by the dis-

together with the advancing tips of the precipitate continuous change in alloy content. He also showed that Sulonen’s mechanism can be included in his

particles in so-called discontinuous precipitation (cellular precipitation). As a grain boundary passes.

thermodynamic treatment [4]. Even though the exact

there is a discontinuous change of the alloy content in nature of the driving force is still under debate, it now

the matrix. This migration was long befieved to be seems to be generally agreed that it is caused by the

due to a pulling effect of the precipitate particles [l]. change in alloy content and is thus dependent upon diffusion. The first observation of diffusion-induced

Another mechanism was suggested by Sulonen [2] who pointed out that the difference in atomic size

grain boundary migration in an experiment without

would give rise to strain energy in the concentration any precipitation was made by den Broeder [S] when

gradient in front of the migrating grain boundary. studying Cr-W diffusion couples. In order to test that

Later on. Hiilert [3] suggested that the driving force the grain boundary migration during discontinuous

for the grain boundary migration originates from the precipitation in Fe-Zn afloys is actually caused by the diffusion of zinc and not by the growth of the zinc- rich phase, Hillert and Purdy [6] reacted Fe--&

t On leave from Beijing Institute of Aeronautics and alloys with a vapour phase which was in intimate Astronautics. contact with an Fe-Zn alloy of a different compo-

1949

1950 LI AND HILLERT: GRAIN BOUNDARY MIGRATION IN Fe-Zn

sition. They observed how the grain boundaries, which have contact with the surface, migrate during the experiment. As they migrate through the volume of the material, the composition changes discon- tinuously. The rate of migration is thus coupled to the rate of grain boundary diffusion, the volume diffusion being negligibly small at the temperature of their experiment. Similar experiments were later carried out by Cahn et al. [7] who studied the. Cu-Zn, Cu-Au and Ag-Au systems.

The experimental technique, introduced by Hillert and Purdy, is very simple and may thus be a valuable tool for a further study of diffusion-induct grain boundary migration. It was employed in the present work in order to study how this phenomenon is affected by temperature.

EXPERIMENTAL

Three different Fe-Zn alloys. called A, B and C (33.00/i, Zn, 18.80,; Zn and 9.y/; Zn) were prepared by powder metallurgy and turnings were produced by machining. The specimens were cut from an iron foil of purity 99.9989; Fe and thickness 0.127 mm. Each specimen weighed 0.0~~0.015 g and was sealed in an evacuated silica capsule and recrystallized at 8SO’C for 2.5 h. Each specimen was then sealed in a new capsule with 0.1-0.3 g of turnings or powder of an Fe-Zn alloy.

The capsules were U-shaped. The pure iron specimen was put in one leg and the Fe-Zn alloy in the other. The connecting part was made very thin in order to prevent direct contact. The U-shape ensured that the specimen and Fe-Zn alloy were in the same temperature zone of the heat-treatment furnace during zinc&cation.

Experiments were carried out for different tempera- tues from 350-8OO’C. The specimens were mainly examined in a light microscope and an electron microprobe.

Most of the specimens were mounted in bakelite and prepared metaIiographically by a very light polishing of the original surface of the foil. They were etched for the microscopic examination. Interesting features were marked by microhardness indentations for identification and then repolished before microprobe measurements. The distance of the grain boundary migration as shown on these specimens varied from one grain boundary to another due to a number of reasons. One important factor is the angle between the grain boundary and the surface. How- ever. it should usually be close to a right angle because of the initial annealing at 850 C. In order to obtain a representative value, the very largest values were excluded and an average -was taken between the next Iargest values.

MICROSCOPIC OBSERVATIONS

The three Fe-Zn alloys, which were used as a zinc

source. were in a two-phase state, z + f, in most of the experiments, for alloy A up to 72o’C, for alloy B up to 660°C and for alloy C up to 55OC. Even a very small temperature difference between the iron specimen and the alloy powder may thus cause a precipitation of f on the specimen surface. In addition, in some of the experiments there seemed to be an impur- ity of carbon which caused the precipitation of an Fe-Zn carbide on the specimen surface. Such precipitation did not seem to affect the processes inside the iron specimens. Xt was removed by light polishing and the migration of the grain boundaries in the ferritic material was then studied as close as possible to the initial specimen surface.

Grain boundary migration was observed down to 46OC. The temperature did not seem to have any effect on the morphology of the grain boundary migration ciose to the surface. As a demonstration, Fig. 1 shows that a very long segment of a grain boundary can move in the same direction at 500°C and this even happens at the low temperature of 460°C. Other boundaries split up in several segments which move in opposite directions and some boundaries seem to have a special tendency For this. Figure 2 shows an example where the boundary ap~rently can form facets of low mobility. It is interesting to note that the same facets form on both sides.

At 500 and 550-C. the grain boundaries were observed to continue migrating across the surface of the specimen until the whole surface has been swept and has thus increased its zinc content. No incubation time was found and the rate of migration was constant within the experimental uncertainty.

At 600°C. a new phenomenon was observed. The migrating boundaries seem to stop before complete reaction and may then even start to move back. An example is shown in Fig. 3. This boundary has initially moved upwards in the picture. it has then stopped its uniform motion and instead made small

Fig. 1. Grain boundary migration close to the surface in an experiment with alloy B for 2900 min at 500 C. Magnifica-

tion 500 x

Fig. 2. Grain boundary migration close to the surface in an expertment with alloy B for 5910min at $20 C. Magnifica- tion 1000 x , The boundary has moved m both directions

and formed the same type of facets.

adjustments in both directions. In view of this phenomenon, most of the values for the rate of migration were taken from short-time specimens, In a few long-time specimens a related phenomenon was observed. Fig. 4. The sharp lines show the present positions of the grain boundaries and more diffuse lines, some bright and some dark because of the action of oblique illumination, show d&continuities in composition The geometric arrangement of the diffuse lines does not represent a natural grain boundary net-work in an annealed specimen whereas the sharp lines do. It may thus seem that the grain boundaries have moved by a rather uniform motion from their initial positions, to the positions of the diffuse lines and then back to their original positions. This process will be further discussed later on. Of course, it is not known what part of the experimental time was spent on the first motion but it is interesting to note that a rate which was evaluated by dividing the distance between the two kinds of lines and the totaLexperimental time agreed fairly well with the rate evaluated from short- time experiments.

At 65o’C and higher, the grain boundaries showed signs of having stopped in all the experiments and no reliable rate values were obtained. However. a value was taken from a short-time experiment at 650-C.

The experimental growth rate data are presented in Table 1. Alloy B was used for this purpose but experiments with the other alloys indicated that the results

Fig. 3. Grain boundary migration close to the surface in an experiment with alloy B for 120min at 600-C. Mapnitica- bon 1000x. The migrating boundary seems to have

stopped and has started oscillating.

are independent of the alloy composition as long as the alloy is inside the 2 + r two-phase field.

The mobility can be defined by

where AC, is the driving force per mole and i& is the molar volume. The quantity AC, can be evaluated from the change in composition at the grain boundaries. assuming that all the total available chemical driving force can act on the boundaries. This assumption will now be used and the resulting values for the mobility ma! thus be regarded as a lower limit. This assumption will be further discussed later on and an alternative will be considered. An ambitious evaluation of the total chemical driving force was presented in Ref. [6], For low zinc contents one may use the approximation

AG, = RT.Y~,, (21

when the specimens are initially pure iron. .Q,, rep- resents the final composition. Microprobe measurements. to be discussed later. indicated that. close to the original surface of the specimens. sz, is very close to the solubility of zinc in r-iron. which is also the composition of the 2 phase in the alloy powder used. The solubility varies with temperature [83. as shown in Table I. These values were used in the evaluation of the mobility. .An Arrhenius plot of the velocity and the mobility is presented in Fig. 5 and the straight Iines obey the following relations

I. = r,exp - Q,.:RT (Jai

AI = .\I,exp - Q+RT t?b)

with r0 = 3.4. IO” m’s. QI = 287 kJ:mole and Me = 0.001 m4 J s. Qw = 130kJ:mole. An extrapolation to temperatures below 460 C demonstrates that the experimental times applied in the present study should be too short to produce a visible grain boundary migration at such low temperatures.

Fig. 4. Grain boundary migration close to the surface in an experiment with alloy B for 990min at 600 C. Magnifica- tion 200 x The grain boundaries seem to have migrated a long distance and then returned to their original positions.

Table 1. Growth rate and mobility of migrating boundaries in experiments with alloy B

460 t.3*10-‘z 0.03037 4.t * to-= 480 2.8*10_” 0.047 6.8- 10-‘0 500 1.4.10-” 0.058 2.7*10-‘3 520 5.1*10-” 0.07 i 7.7.10-19 550 1.7*10-‘” 0.092 1.9~10-‘@ 600 2.8. 1o-9 .0.127 2.2*10-1’ 620 5.6~10-9 0.145 3.7, lo- ” 650 1.7.10-B 0.185 8.5. to- I7

Etching suggests that after lang times at 650°C the grain boundaries have been moving back and forth several times in a zinc-rich region surrounding the original position of the boundary and it has a very irregular shape at the end of such an experiment. Fig. 6.

At even higher temperatures the boundaries are less irregular and the etching suggests that they have moved by a jerky motion, leaving ghost fines in previous positions, Fig. 7.

Under the present ex~r~mentaj conditions new grains sometimes nucleated on the surface. A fine- grained surface layer could thus form and it gradually grew into the material. This is demonstrated in Fig. 8

1.2 1.3 Inverse tempemture 1/T IK’I

Fig. 5. Kinetic data for d~~u~jon-induced grain boundary migration as function of temperature. r is tire rate of migration observed close to the surface, M is the mobility evaluated from r using equation (t), t>/Dh& is a quantity evaluated from concentration profiles in sectioned specimens and D*a was evaluated by combination with the rate values _

close to the surface.

Fig. 6. Usciliation of a grain boundary around its original position. observed cfose to the surface in an experiment with alloy A far 1000 min at 6.50X. Magni~cation IO00 x .

which was taken on sectioned specimens. This nucleation seems to be promoted by a surface rough- ness. It occurred less frequently on electropolished specimens. Rather c&mpfete fine-grained surface layers were observed in some specimens at 520°C and more ~requentIy at- higher t~m~rature. Their thickness was studied as function of time. Some data for alloy 3 are presented in Table 2. Data for alloy A were very similar.

The grain size within the surface layer seems to vary rather gradually with its thickness I at WC and probably at lower temperatures, as well. The grain diameter is approximately half the thickness of the surface layer. Figure 9 shows an example where the grain size is considerabiy liarger than in the shorter experiments from the same temperature, 600°C. The etching effects suggest that the grain boundaries have moved in a complicated fashion.

In many cases there were only a Few new grainoand they grew to a considerabIe size. Figure f0 shows that the growth of such grains may stop and the grain boundary may start to osStillate, some segments con-

for 15 min at 780-C. Magnification 500 x .

Fig. 7. Oscillation of grain boundaries by a jerky motion observed close to the surface in an experiment with alloy A

LI AND HILLERT: GRAIN BOUNDARY MIGRATION IN Fe-& 1953

,‘. -.; ‘. --L

1w

Fig. 8. Formation and growth of a fine-grained zinc-rich surface layer observed in sectioned specimens from experi- men& with alloy B at 600-C. fai @tImin, @) f50min. Ic)

990 min. Magnification 500 x .

tinuing and other segments moving back, This is very similar to the phenomenon shown in Fig. 3 and to the phenomenon which has probably given rise to the structure shown in Fig. 6.

The one-grainy surface layer usua~iy advances into the interior of the material with a slightly higher rate along the initial grain boundaries. This is demonstrated by Fig. I1 where the specimen was poiished down not quite enough to remove the surface layer completely. At the higher temperatures this reaction competes with the ordinary grain boundary migration and it may even dominate completely.

At 650 f a new phenomenon occurs in the’ fine- grained surface layer. A kind of abnormai grain growth takes place by which the initial, large grains grow back to the surface and the fine grains disappear completely. Fig. 12. Evidentiy, at temperatures above 720-C this reaction was so rapid that the fine grains were never observed. However, the presence of a zinc- rich surface layer with a non-uniform distribution of zinc is an indication that there has been a tine-grained iayer. At even higher temperatures. the zinc-rich layers at the surface and at the grain boundaries

fable 2. Growth of fine-pined surface layer in experiments with alioy B

7- I I CC) fmin) @ml

500 14640 5 520 5190 18 550 { 460 5

IiS0 580 I330 3&s

i

80 12-15 120 20-22

600 150 22-27 470 22-24 990 50

1200 50

i

15 _s7 30 7-9 60 lo-11

120 17-20 650 I50 7-20

470 lo-15 loo0 10 1170 15-f-I 8430 10

I5 7 700 30 6

60 6 720 15 2

obtain a more uniform composition. presumably due to volume diffusion which starts to become significant, Fig. 13.

Some specimens were sectioned perpendicular to the surface of the original foil. At the lower temperatures they showed that the grain finery migration occurred very uniformly through the whole thickness of the specimen. Figure 14. On the other hand. at 600°C it was observed that the migration was much more rapid along the surface and a surface layer of rather unifo~ thickness may thus form. Fig. 15. This micrograph shows that the grain boundary has started to move back on the lower side. possibfy while the boundary is still advancing along the surface. This is very similar to the phenomenon shown in Fig. 3.

The thickness of the surface layer formed by the process illustrated in Fig. 15. varies with temperature.

Fig. 9. Grain growth in tRe fine-grained surface Iayer observed in a sectioned specimen from an experiment with

aIIoy A for 1200 min at 600°C. Magnification 500 x

1954 LI AND HILLERT: GRAfN BOUNDARY MIGRATION IN Fe-Zn

Fig. 10. The cessation of growth of a few new grains formed on the surface in an experiment with alloy B for 480min at 600°C. Segments of the grain boundaries then seem to have started to move in different directions. Mag-

ni~cation 1000 x . Fig 13. The zinc-rich layer remaining after the disappearance of the fine grains and some volume diffusion observed at some distance below the surface in an experiment with

alloy C for 8430 min at 650°C. Magnification 200 x

Fig. 11. Favoured growth of fine-grained surface layer along grain boundaries observed at some distance from the surface in an experiment with alloy B for 280min at 600%

Magnification 200 x .

Fig. 14. Uniform migration of a grain boundary observed in a sectioned specimen from an experiment with ahoy 3

for I19,OOQ min at 460-C. Magnification 200 x

Fig. 15. Favoured migration of a grain boundary aiong the Fig. 12. The disappearance of the fine grains by abnormal surface observed in a sectioned specimen from an experi- grain growth of the original large grains observed at some ment with alloy C for 12OOmin at 600 C. Magnification distance below the surface in an experiment with alloy B 1000x. On the lower side. the migration seems to have

for 470 min at 65O’C. Magnification 200 x stopped and the boundary has started to oscillate.

L1 4x1) HILLERT: GRAIN BOCNDARI’ MIGRATION IN Fe-Zn

’ I ; migration would not continue to increase at temperatures above the experimental range. This is contrary to previous belief [7] which was based upon the increasing role of volume diffusion at higher temperatures. It may be interesting to examine this argument in detail. As already mentioned. it has been suggested [3] that the driving force is related to the discontinuity in composition at the grain boundary. Normally. one would expect a very steep concentration gradlent just in front of the migration boundary and the width of the zone with a variable composition. the so-called spike, can be estimated as d = D’:r where D’ is the ordinary, lattice diffusion coefficient in the z phase and I‘ is the rate of migration. There should be no real discontinuity in composition at the boundary if d is much larger than the atomic dimensions. On the other hand, if d is much smaller than the atomic dimensions, then the spike does not exist and instead there is a discontinuous change of the composition from the initial value of the specimen to the value of growing grain. At d values close to the atomic dimensions the discontinuity will be smaller and one would need some model in order to evaluate the size of the chemical driving force. In view of these considerations it should be interesting to evaluate D’:r from the experimental rates. This evaluation gave a value of 2A at 600°C and an extrapolation to higher temperatures would give approximately the same value since the activation energy for the mobility, 230 kJ/mole. is roughly equal to the value for lattice diffusion in s-iron. 240 kJimoIe [9]. In view of this result we should not expect the driving force for the migration to decrease towards higher temperatures. Instead, we should look for another explanation why the diffusion-induced boundary migration has not been observed at higher temperatures.

50 100 Dlstonce. m

Fig. 16. Measurements of the concentration behind the migrating boundaries as function of the distance from the surface. The curves represent an attempt to fit the data to

an approximate solution of the equation of diffusion.

It was about IO pm at 600-C and 2-6 vrn at 650‘C. At 70&C it was difficult to observe but there was an indication of a layer of less than 1 pm. At lower temperatures, the experimental time was not sufficient for the surface layer to develop or the thickness of the specimens was not enough (127 pm).

MICROPROBE MEASUREMENTS

By microprobe measurements on specimens, which had only been polished very slightly. it was confirmed that the composition behind the migrating grain boundaries was close to the equilibrium solubility in the experiments with alloys A and B up to 650°C. where they are both in the 1 + f two-phase state. In this kind of section, the composition was uniform in the regions swept by a grain boundary once. In regions swept several times. like the one shown in Fig. 6. there were large variations.

The concentration in regions passed by a grain boundary was also measured as a function of the distance from the surface on sectioned specimens. heat treated from 480 to 580-C. Fig. 16. Some concentration profiles were measured through the fine-

Ordinar_v grain houndarj~ miyrutim

Figure 5 shows the mobility. evaluated from the experimental rate of migration. using equations 1 and 2. It shows a rapid increase with temperature and there is no indication that the experimental rate of

grained surface layer formed at 600-C. Fig. 17.

DISCUSSlON

Fig. 17. The concentration profile through a fine-grained

1955

It may be significant that the upper temperature limit of the experimental range coincides with some new phenomena. which may be illustrated by Figs 3. 4. 10 and 15. The primary one may be the local growth close to the surface. illustrated by Fig. 15. as

i Cl

0 10 20 30 LO 50 60 Depth (pm1

surface layer. The curve is tentatively drawn.

E5 i;r \


v -

\

Fig. 18. Development of a uniform zinc-rich surface layer by favoured migration close to the surface.

different from the uniform growth in the whole material, illustrated by Fig. 14. In order to understand Fig. 15 it is important to realize that zinc must diffuse into the material in order to allow a zinc-rich grain to grow. As a consequence, the zinc concentration must decrease along the grain boundary. This leads to a decrease of the driving force for the grain boundary migration and of the rate of migration. We must con- clude that the motion, illustrated by Fig. 14, is only approximately uniform and in a very thick specimen one should see that the migration is slower in the center. The concentration profile should thus be calculated together with the migration of the boundary. This type of calculation was carried out by Hillert and Purdy. Their calculated concentration profiles are shown in Fig. 12 of their report [6] and the rate of migration was predicted to be approximately proportional to the zinc concentration. It is thus evident that, in a thick specimen, the grain boundary will lag behind considerably below a certain depth and the shape of the migrating grain boundary will develop as illustrated in Fig. 18. The final thickness of this layer, 1, may be estimated as the distance below the surface, measured along the boundary, where the concentration is hali the value at the surface. According to the calculation by Hillert and Purdy, this criterion yields

1 = y’l.4DbWJMRTx0 = \‘1.4DbSiro (4)

where c’~ is the rate close to the surface. It is thus impossible to predict the temperature dependence of I without knowing how DbS and c,, vary with temperature. As already mentioned, it was observed in this study that I decreases with increasing temperature. It is thus evident that Db6 has a smaller activation energy than the rate of migration c,,. Some quanti- tative information can actually be extracted from the experimental concentration profiles presented in Fig. 16. By fitting the data to the simple solution given by Cahn [lo]

x cosh(z, cO/Db6) -= w (5) % cosh(+ \ c,,‘Db6)

it was possible to estimate a,,/@‘& In this equation .K,, is the composition at the surface, z,, is half the thickness of the specimen and z is the distance for any point from the center line. The data are listed in Table 3 and plotted in Fig. 5. The data can be rep- resented by an equation

cO/DbG = 10” exp - Q/RT me2 (6)

with Q = 100 kJ/mole. The activation energy for r0 is thus 100 kJ/mole larger than for Db6 and the decreasing value of I with increasing temperature is explained.

The grain boundary diffusioity

By combining the data for ~+,/@b in Table 3 with information on the rate of grain boundary migration close to the surface, L’,,, which is given as c in Table 1, we can evaluate #‘a. The result is shown in Table 3 and can be described by

Db6 = 3.4.10-l exp - Q/RT m’/s (7)

where Q = 187 kJ/mole. These values of the grain boundary diffusivity confirm the result of Hillert and Purdy, who obtained values several orders of magnitude higher than for stationary boundaries. Their results were confirmed by Smidoda et al. [ll], who also observed an abnormally large activation energy which fell between the normal values for boundary diffusion and lattice diffusion. The present result indi- cates some tendency for this but may not be reliable enough for a safe conclusion. The activation energy for normal grain boundary diffusion in iron is 155 kJ/ mole and for lattice diffusion 240 kJ/mole.

The observation, that the diffusivity in a migrating

boundary is several orders of magnitude larger than the values reported from experiments with stationary boundaries, is particularly interesting in connection with the start of the grain boundary migration. It is still not understood how the migration starts but some kind of a nucleation process must be involved because a dislocation wall seems to form and remain at the position of the initial boundary [6]. Once there is a successful nucleation at some point, the reaction may spread rapidly along the boundary in view of the increased diffusivity. It may thus swallow other nuclei which have not yet developed fully and have not yet

been able to make use of the high difl’usivity. This may explain how it is possible that a very large segment of a grain boundary starts to migrate in one direction even at the low temperature of MC, where

Table 3. Evaluation of the grain boundary diffusion constant

(‘0 rO; D”6

(I&

Db6 (rn-‘) (m’is)

480 1.0. lo8 2.8.10-” 2.8. IO-“’ 520 2.7. lo8 5.1.10-” 1.9. lo-l9 550 4.7. IO” 1.7. lo- ‘O 3.6. 1O-‘9 580 8.0. 10s 1.0. 1o-9 1.3.10-‘8

LI AND HILLERT: GRAIN BOLJNDARY MIGRATION IN Fe-Zn 1957

I” I ,

o 750 *C Alloy A

Y? f 2o 5

10

0 0 500 1000 1500

lime, minutes

Fig. 19. The thrckening of the fine-pained surface layer formed during zincification of iron specimens with a rough

surface.

one might have expected that the information. which must spread along the boundary. should be too slow to prevent many independent nuclei from forming.

Oscillarions qf grain boundaries

Figure 15 shows another interesting feature. The grain boundary on the bottom side of the zinc-rich grain has started to move back. It is easy to understand that its growth has more or less stopped because the diffusional distance from the surface, as measured along the boundary, is too long to allow the zinc supply to be maintained. Furthermore. it is easy to see that the boundary will not be stable if it simply stops when the supply is interrupted. The Gibbs energy of the system can be decreased further if one segment moves back into the zinc-rich region and picks up some of the zinc, thus allowing another segment to continue into the pure iron region and increase its zinc content. In this way, the zinc content of the surface layer wilI spread over a larger volume. One may thus expect an oscillation to start once the supply of zinc is interrupted. In fact. signs of a very similar oscillation was observed at the end of a discontinuous precipitation in a ternary system after a long time when the supersaturation had probably vanished [ 121. Furthermore. it seems probable that Fig. IO show5 another exampIe of the same phenomenon.

During the oscillation of the grain boundary in the zinc-rich region. it should eventually come into contact with the surface and there establish some contact angle. e.g. a right angle. It should then be stuck to the surface and, because of the surface tension effect. it should drag the rest of the grain boundary out to the surface. The grain boundary would thus disappear from the zinc-rich region and only be found in the neighbourhood of the initial position of the boundary where it is stabilized by the presence of the boundary inside the material. This mechanism may provide part of the explanation of the diffuse lines in Fig. 4 where

the grain boundaries seem to have moved back to their initial positions. However, we cannot explain why the grain boundaries did not continue to migrate across the whole surface of the specimen. It seems that something happened to the specimen at a certain moment and that the boundaries coufd not move further.

If a specimen of this kind is polished down to just below the zinc-rich surface layer, one should still see a zinc-rich region close to the initial position of the grain boundaries. This is evident from Fig. 18. It seems probable that Fig. 3 was taken in such a way.

Fine-grui~ed swface layer

The formation of a fme-grained surface layer resem- bles a kind of discontinuous precipitation observed by Tu [13]. The data presented in Table 2 indicate that the rate of growth of the layer into the material decreases with time. The information obtained for alloy B is shown graphically in Fig. 19. The initial linear part seems to be in essential agreement with the rate of ordinary migration of grain boundaries. The data for 520 and 550°C onfy concern the initial linear part of the curve. At and above 650°C the growth seems to come to a standstill and the final thickness seems to decrease at increasing temperature. The reason for the standstill at high temperatures and the shape of the curve for 6OO’C should now be discussed.

The decreasing rate of 600°C should be intimately related to the decreasing zinc content with the distance from the surface according to the microprobe data in Fig. 17. In order to test this relation, the growth rate was evaluated at four times by reading the slope of the curve in Fig. 19. For the correspond- ing depth it was then possible to read the zinc content from Fig. I7 and the mobility could then be evaluated from equations (1) and (2) assuming that all the chemicaf driving force can be applied on the boundary. The results are shown in Table 4. Except for the result obtained close to the surface, the values are almost constant. It thus seemed natural to develop a mathematical treatment from the basic relations of the rate of grain boundary migration. equations (1) and (2)

df & = MRTx$fV, (81

where x;, is the zinc content just behind the migrating boundary. In order to estimate the rate of supply of zinc to the migrating boundary a very simple geo- metry was adopted, Fig. 20, and it was assumed that

Fig. 20. Model for the thickening of a fine-grained surface layer.

1958 LI AND HILLERT: GRAIN BOUNDARY MIGRA~IGN iN Fe-Zn

the di&sivity in the migrating boundary is so rapid that one only needs to evaluate the diffusion along the stationary vertical boundaries. This assumption is supported by the observation that the horizontal boundaries are more or less flat. It is also supported by the microprobe m~surements which show that the zinc content is approximately constant at each depth.

The flow of diffusion down the grain boundary of a circular, columnar grain is

(9)

where s is the grain diameter and I&, is the composition at equilibrium with the vapour phase. The factoi) was introduced because the flow is shared with the nei~bouring grains. This flow supplies zinc for the growth of .the zinc-rich grain. By ~mbination with equation (8) we thus find

t&If2 = c&l - .Y$. ~.2~b~V~~~~RTls. (11)

For small t we see that .$“, = xi, which is self-evident. In order to solve the equation for larger 1 we must know how the grain diameter, s, varies during the growth of the surface layer. As already mentioned we may use s = i/2 as a rough estimate after the fine grains have impinged upon each other and formed a complete surface layer, although there are strong variations even within a single specimen. With this value we obtain

By inserting this expression in equation (8) and solv- ing the resulting differential equation we find

5 = 2;. + i,,’ f + i.’ + In(i, + t 1 + 2’) (13)

where r and i, are normalized coordinates defined as

T = 4~.(~~R~~~“~~~) 3.2. .(L)b(j)- I,2

i. = rqMJm~,/DbiSvm)’ 2

For smali 1 we obtain r = Ji. and thus

(14)

(1%

t/t = MR T.YgJ Y,

For large i we obtain t = i2 and thus

The thickness E. as a function of r and , t, respect- iveiy, is presented in Fig. 2 1. The shape of the curve of i, vs r has a reasonable similarity with the curve for 600°C in Fig. 19 and it is possible to evaluate the diffusivity in the vertical boundaries by fitting the two curves. In fact, equations (16) and f 17) show that the

difhtsivity can be evafuated from

In order to evaluate (Iqit’h. .,rlCci from the data in Fig. 19, which do not extend to very large I, one may use the fact that the curve of i. vs ,,‘r approaches the straight line very early. One may thus use a point at the Iongest time available and another point at a fairly short time,

The curve for 6GO’C in Fig. 19 thus yielded a value of Db6 = 1.0. 10e20 m’is. It is surprising to fmd that this value is three orders of magnitude larger than the value expected for stationary boundari~ f9J and only one order of magnitude smaller than the value obtained for migrating boundaries. The explanation may be that the vertical boundaries between the fine grains have not been quite stationary but have been moving around because of the grain growth that has been observed to occur in the surface layer. See for instance Fig. 9.

The cessation of growth of the surface layer after some time at and above 650°C coincides rather well with the occurrence of abnormal grain growth, the result of which was illustrated in Fig. 12. As an example, abnormal grain growth was observed to occur after about t5Omin at 650°C. This abnormal grain growth may be regarded as the reverse of the growth of the fine-grained surface layer. It is possible that the cessation of growth is caused by the start of abno~ai grain growth. On the other hand, it is also possible that the growth stops by a different reason and abnormal grain growth can then start. In this connection it is of interest to compare the magnitude of the driving forces for the two reactions.

in principle there is always a driving force for abnormal grain growth if a large grain is in contact

Sqwre mot of time. E

Anneolmg time, T

Fig. Ii. The thickening of the fine-grained surface layer according to a simple mathematical analysis. /. and T are

defined by equations (14) and 115).

Table 4. The grain boundary mobility evaluated from the (20). It is approximately lO.OtXl J/mole. At 6WC, the rate of thickening of the fine-grained surface layer at 600 C

in experiments with alloy B maximum value of rzn (0.127) would thus yield a driving force of 160 J zmole which is about l/6 of the total driving force according to equation (8). This is still very much larger than the value of 1 J/mole. However.

3.10-‘” 25. IO-‘& equation (21 k shows a parabolic relationship and the

0 0 0.110 ‘50 8, lo- ‘O 29 0.086 9.10-‘”

driving force is thus predicted to decrease rapidly as

500 3.10-‘0 38 0.070 6, IO‘ ” the zinc concentration is decreased. It would reach

1000 1.10”“0 48 0.051 6. IO- ” the value of 1 J/mole at .xzn = 0.010. This is a quarter of the value at the front after 90 min at 600-C according to Fig. 16. In that case. the reaction has not yet

with a region of fine grains. The driving force supphed stopped and it thus seems quite possible that there by the surface energy may be estimated as 201’*, s will be a further decrease. Of course. further measure- where (T is the specific surface energy and s is the ments are required in order to test whether the driv- average grain diameter in the fine-grained region [ 143. ing force is only due to the coherency energy and A typical value of s at 6oO.C is 1Ob.m and it yields a whether the reaction stops when the driving forces for drivmg force of about 1 J:mole. At the start of the the two reactions balance each other. Primarily one reaction one has .Y?” = 0.12 at 600 C and the total should test whether the rate of migratjon is propor- chemicai driving force for the growth of the fine zinc- tional to .x;, or ~z~. If it is proportional to .&. then rich grains is 900 J/mole according to equation (2). it equation (111 t should replace equation (21 and result in is evident that there must be a drastic decrease of this many modifications of details in the present dis- driving force before the driving force for the reverse cussion. One will for instance find that the mobility. reaction can stop the further growth. However. it has deduced from Table 1. should be described by been questioned that all of the chemical driving force &lo = O.oooOl3 m”,J s and Q = 185 kJlmole and the can act as a force for the grain boundary migration mobihties. deduced from Table 4. will only vary bt a and one cannot rule out the possibility that the two factor of 2 instead of 4. Finally. it should be empha- forces may balance each other when the surface layer sized that AG, in equation (I) should be identified has reached a critical thickness. with the difference in driving force for the two reac-

A lower limit jior the driciny jiorce tions. The theoretical treatment would thus be even more complicated. As the growth rate decreases, the

It has been suggested by Cahn er al. [7] that there effect of the diffusion will be stronger and the concen- is no mechanism by which the decrease of the Gibbs tration at the front may not decrease-as rapidly as energy due to the increase of positional entropy can Indicated by an extrapolation of the curve in Fig. 16. act as a driving force for the boundary. It has also It seems most probable that the growth reverses due been pointed out that a considerable part of the driv- to some fluctuation which may occur as the growth ing force may be lost due to diffusion of zinc into a rate has decreased to a IOU. value. thin layer of the iron grain in front of the migrating boundary [4]. At the same time it was pointed out that the driving force should never be smaller than

CONCLLSlONS

the contribution due to coherency strains in the thin Many nevv phenomena were observed in the

zinc-rich layer which may form in front of the migra- present work and some attempts have been made to

ting boundary. The strain energy of a thin layer of explain them. These explanations should be regarded

composition s. which is coherent with a thick grain of as first attempts, only. and they are presented mainly

composition .x’. was estimated to in order to stimulate further work.

It does not seem probable that the increasing role of lattice diffusion is responsible for the lack of obser-

&c”h = m (2 - x)‘. (20) vations of diffusion-induced grain boundary migration at htgh temperatures. It seems more probable

This expression can also be used as an estimate of the that an increase of the rate of migration. relative to

driving force as long as the composition of the coher- the grain boundary diffusivity. is responsible.

ent layer at the contact with the migrating boundary In future work it should be important to examine if

is approximately equal to the composition of the the rate of migration is proportional to rr,, or xg,,.

growing grain. For the present experimental con- ~~lirtc~l~I~,d!,c~?~~,~~/.~-~ Stimulating discussions with Drs Paul ditions. where the starting material is pure iron. we Shewmon and John W. Cahn are gratefully acknowledged.

would thus have

REFERENCES AG$’ = Xx;, (21)

I. R. A. Fournelle and J. B. Clark. Mrrcrlf. 7ran.t. 3, 2757

where x7” is the composition behind the migrating (1971).

2. M. S. Sulonen. .4crtr rnrrall. 12, 748 (1964). boundary. The parameter K is defined by equation

_~ 3. M. Hillert. 7%r Mrc-ltorlis~t 01 Pku.~ Trutlsfornlarrons irt

LI WD HILLERT: GRAIN BOUNDARY MIGRATION IN Fe-& 1959


Crysralline Solids, Monograph and Report Series No. 33, The Institute of Metals, London (1969).

4. M. Hillert. Metall. Trans. 3, 2729 (1972). 5. F. J. A. Den Broeder. Acta memll. 20, 319 (1972). 6. M. Hillert and G. R. Purdy, Acta metal/. 26, 333 (1978). 7. J. W. Cahn, J. D. Pan and R. W. Balluffi, Scripm

metal/. 13, 503 (1979). 8. G. Kirchner, H. Harvig, K.-R. Moqvist and M. Hillert,

Arch. Eisenhiitt Wes. 44, 227 (1973).

9. J. Fridberg, L.-E. Tijrndahl and M. Hillert, Jernkont.

A&r. 153, 263 (1969). 10. J. W. Cahn, Acra metall. 7. 18 (1959). 11. K. Smidoda, W. Gottschalk and H. Gleiter. Am

metall. 26, 1833 (1978). 12. M. Hillert and R. Lagneborg. J. Mater. Sci. 6, 208

(1971). 13. K. N. Tu, J. appl. Phys. 48, 3400 (1977). 14. M. Hillert, Acto merall. 13, 227 (1965).

Documents

A Metallographic Study of Diffusion-Induced