Microstructural coarsening of the PbAg eutectic

Materials Science and Engineering, 36 (1978) 1 - 9

1

© Elsevier Sequoia S.A., Lausanne -- Printed in the Netherlands

Microstructural Coarsening of the Pb-Ag Eutectic

G. G. COLLINS, I. O. SMITH and G. A. CHADWICK

Department of Mining and Metallurgical Engineering, University of Queensland, St. Lucia, 4067, Queensland (Australia) (Received February 14, 1978)

SUMMARY

The coarsening behaviour of Pb-Ag eutectic alloys has been investigated under isothermal and temperature gradient environments. No evidence was found for fibre migration or fibre coalescence. Under both isothermal and temperature gradient conditions the rate of coarsening occurred with an activation energy equivalent to the sum of the activation energies for self-diffusion of lead and dissolution of silver in lead.

1. INTRODUCTION

The thermal stability of unidirectionally solidified eutectic alloys is a property of considerable practical importance. The rates of coarsening of fibrous alloys in isothermal environments are reasonably well documented and have been shown to follow the normal Ostwald ripening kinetics. Accelerated coarsening kinetics have been reported for coarsening in a transverse temperature gradient, although, as is indicated later, conflicting results have been obtained and several different coarsening mechanisms have been proposed. It is important to understand this phenomenon since it might be of critical importance in the utilization of eutectic alloys as turbine blade materials where transverse temperature gradients of the order of 100 K mm -1 might be expected. The background to the problem is outlined below before we present our own experimental data and analysis. Jones [1, 2] has suggested that fibres of Ag in the Pb-Ag eutectic migrate under an applied transverse temperature gradient with constant volume, in a manner analogous to

that of liquid or gaseous inclusions in metals [3 - 5]. Mass transport is deemed to be by interphase boundary diffusion and the velocity of fibre migration is given by Di~Q*G v -

RT2r

(1)

where v is the velocity of the inclusion relative to the matrix, Di the diffusivity of Pb in the interphase boundary, 5 the thickness of the high diffusivity layer, Q* the heat of transport of Pb, R the gas constant, T the absolute temperature, r the fibre radius and G the temperature gradient. The resulting variation in fibre velocity with fibre radius means that it is possible that the contact and coalescence of fibres could contribute significantly to fibre coarsening. A second, though less significant, mechanism of particle migration was suggested to result from the movement of Pb atoms up or down the concentration gradient owing to the necessity of satisfying the conditions of chemical equilibrium at the interphase boundary at all points in the temperature gradient. Several models were examined in which mass transport was both by interface diffusion and by volume diffusion either in the matrix phase or within the particle, and it was shown that the fibre velocity would, in general, be less than that predicted by the previous model of simple thermal diffusion. McLean [6] has developed a theory of temperature gradient coarsening in which he argues that the basic assumption of Jones' analysis, that the inclusion does not change in volume, does not necessarily apply to high melting point fibres in a low melting point matrix. For a fibre of Ag in a Pb matrix, the temperature difference between both sides of the fibre will induce a net flow of atoms in

d log (r - - ro) dz

the matrix, and the coarsening rate will be given by dr dt

-

•v0

{1 -- exp(--2ar)}

2

(2)

where r is the radius of the fibre, ~ = Q G / R T 2, Q is the activation energy for the diffusion of Ag in Pb, G the temperature gradient, R the gas constant, T the absolute temperature and v0 the velocity of the hotter fibre surface; p is a dimensionless constant dependent on particle geometry and approximately equal to unity for both spheres and cylinders. This model for temperature gradient coarsening based upon matrix diffusion is an alternative to that proposed by Jones [1, 2] and should predominate at large fibre radii. McLean compares the evidence for this model of temperature gradient coarsening with that for an Ostwald ripening process, where minimization of the matrix-particle interfacial energy is the driving force. The coarsening rate for this process has been derived by Wagner [7] and Lifshitz and Slyozov [8] as dr _ ~,C~ ~2~D~/ dt

(3)

3r2RT

for a particle of/3 in an ~ matrix where ~ is a constant dependent on geometry, C ~ the concentration of 6 in ~, ~ the atomic volume of ~, D~ the diffusion constant of/3 and ~/ the energy of the ~--6 interface. The relevant mechanism of temperature gradient coarsening or Ostwald ripening will be illustrated by the final distribution of particle sizes in the sample. For Ostwald ripening the size distribution will tend to be more uniform as the smallest particles disappear and the larger particles increase in size. However, for a temperature gradient coarsening mechanism all particles will grow as well as migrate up the temperature gradient. Using this fact, data can be plotted as graphs of log (r -- ro) and log (r 3 - - r3o) v e r s u s z , the distance along the temperature gradient. For an Ostwald ripening mechanism, d log (r 3 - - r~) _ dz

QG

2.303RT 2

G + --

T

(4)

while for a temperature gradient coarsening mechanism,

QG

2.303RT 2

+

G -

T

(5)

The gradient of these plots can be used to calculate Q, the activation energy for the coarsening process. Experiments which have been performed to date on the coarsening behaviour of fibrous eutectics have not clarified which mechanism is operative. In his work on the coarsening of the Pb-Ag eutectic, Jones [1, 2] presented data which indicated an enhanced coarsening of the Ag phase in a temperature gradient in comparison with isothermal coarsening data. Microstructural evidence was also provided to support the model of fibre coalescence, in that "precipitate densities in the coarser regions of the temperature gradient specimens" were less than in the isothermally annealed specimens. Jones e t al. [9] investigated the stability of the Pb-Ag eutectic under conditions of thermal cycling, in which severe temperature gradients are generated in the specimens. However, no evidence was found for accelerated coarsening. The temperature gradient coarsening of A1-A13Ni eutectic specimens has been investigated by Jones and May [10] and by McLean [11]. In the earlier work it was shown that enhanced coarsening consistent with fibre migration occurred. However, McLean investigated the same system by thin-foil hot-stage electron microscopy and by conventional microscopy of bulk samples and found no evidence of fibre migration. In this latter work the coarsening of the microstructure could be explained solely on the basis of Ostwald ripening. The absence of fibre migration was explained by McLean, following a model proposed by Doherty and Strutt [12], in terms of fibre pinning by matrix dislocations. Stohr e t al. [13] have investigated the coarsening behaviour of COTAC-type eutectic alloys under isothermal and temperature gradient conditions. They found that there was no accelerated coarsening in a temperature gradient provided T < 0.95Tin and that the coarsening kinetics were adequately described by the Ostwald ripening mechanism. McLean [14] has also studied coarsening in the (Co, Cr)-CrTCa system and has found an enhanced coarsening effect in a transverse

temperature gradient. The results were shown to be consistent with his temperature gradient coarsening model. However, transverse fibre migration was observed at temperatures within 2 K of the eutectic melting point. In view of the contradictory nature of much of this experimental work and the further conflicts between experiment and the theoretical analyses, it was considered that a re-examination of the coarsening behaviour of the Pb-Ag systems was warranted. 2. EXPERIMENTAL Weighed amounts of Pb and Ag were melted together under nitrogen and cast into two bars. These were subsequently directionally solidified at growth rates of 5.6/~m S- 1 and 0.4 gm s-l, producing polycrystaUine samples of predominantly rod-like and broken lamellar morphologies respectively. Specimens 12 m m × 5 mm thick were cut from the centre of the bar of the fibrous eutectic and were polished on a plane normal to the growth direction. The samples were first photographed in the as-grown state in defined positions. Samples were subsequently annealed isothermally at 448 K, 495 K and 546 K for up to 1.26 × 10 s s, and in temperature gradients of 4 K mm -1 and 16 K mm -1 for up to 0.77 × 10 s s. (Only specimens annealed under the larger temperature gradient were used in the computation of activation energies.) Samples were polished back approximately 50/~m and then photographed in the same positions to compare the density of the Ag rods. This was measured directly by counting the number of Ag rods cutting the polished plane within a standard grid pattern. The fibre density was plotted against annealing time at the three temperatures for both types of anneal. The crystallography of the Pb-Ag eutectic was investigated with the aim of discovering whether a preferred orientation relationship existed between the Ag rods and the Pb matrix which could possibly account for the elongation of rods reported by Jones by preferential coarsening of the Ag along one low energy plane. To this end, pole figure and extraction replica techniques were used to try to establish the orientation relationship between the Pb and Ag phases and the facet plane of the Ag rods.

3. RESULTS The as-grown eutectic of predominantly rod-like morphology exhibited a wide range of rod densities, ranging from very fine rods to branched rods and broken lamellar structures as shown in Fig. 1. Examination of the scanning micrographs shows that the fibres were faceted and ribbon like, not cylindrical. Using pole figures no constant relationship was found to exist between the growth direction of the Ag fibres and the Pb matrix (Table 1). Attempts to define the Ag facet plane by extraction replica techniques were unsuccessful because of the difficulty of penetrating the Ag rods by the electron beam and of controlling the position of the extracted rods on the grid. In the as-grown eutectic of predominantly lamellar morphology the relationships were more regular. The growth direction of both phases was (111) and the lamellar interface plane, measured by trace analysis, was {100) for both phases. In Figs. 2 and 3, optical and scanning micrographs are compared of typical structures obtained after annealing isothermally and in a temperature gradient of 4 K ram- 1, respectively. In the temperature gradient coarsened samples the microstructure ~;aried from very coarse to very fine fibres along the length of the temperature gradient. At the same annealing temperature there was no observable difference in the microstructure of the temperature gradient and isothermally annealed samples. The degree of faceting of the rods was the same in both cases and there was no evidence of rods being pinched off and spheroidizing during coarsening. In no case was it observed that rods migrated along the temperature gradient and coalesced to form plates. The alignment of the Ag rods after the temperature gradient anneal was clearly independent of the direction of the temperature gradient, and seemed to depend on the original fibre orientation (Figs. 3(c) and (e)). No sign of Ag particle build up or deficiency was observed at grain boundaries normal to the temperature gradient as might have been expected if Jones' analysis [2] were correct (Fig. 3(e)) bearing in mind that the temperature gradient here was four times that used by Jones and that the initial fibre diameter was approximately half that in Jones' work.

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.

.

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(a)

(b)

(c)

(d)

Fig. 1. The microstructure of the as-grown Pb-Ag eutectic. The fibre radius varied considerably along the bar and also locally: (a) and (b) (270×); (c) a region of lamellae adjoining rods (900×); (d) fibres are sometimes branched and have an aspect ratio ¢ 1 (2025×).

The graphs of fibre density v e r s u s annealing time {Fig. 4) show that there was no appreciable difference in coarsening rate between isothermal and temperature gradient anneals. The effective radii of the fibres before and after isothermal and temperature gradient heat treatments were calculated from the compositions given by the phase diagram and the observed fibre density. For the temperature gradient anneals, the results were plotted as graphs of log (r -- ro) and log (r 3 -- r~)

z (eqns. (4) and (5)) following the method of McLean [ 1 2 ] . The plot for an annealing time of 0.60 × 106 s is shown in Fig. 5; similar plots with similar gradients were found for all the annealing times. The gradients of these lines were used to calculate the activation energy for coarsening according to eqns. (4) and (5), and the computed activation energies, calculated at three separate time intervals, are given in Table 2. As these data are collected over a distance of versus

TABLE 1 Specimen

R (ms -I)

Area

Growth direction Pb

Growth direction Ag

(1) As grown Isothermal anneal

5.6

R o d region

<113> <100>

<111>

(2) As grown

5.6

Rod region

<113> <310> <100> <113>

<133> <111> <110> <110> <133>

<111> <111>

<111> <111>

Rod region with scattered plates Lamellar

(3) As grown

0.4

Lamellar

6 mm the applied temperature varies linearly over a 100 K range; an average temperature of 473 K has been used to evaluate the activation energies. For the isothermal coarsening experiments, data were plotted as (r 3 - - r 3) v e r s u s annealing time for three separate annealing temperatures (Fig. 6(a)) assuming an Ostwald ripening process. From the gradients of these lines an activation energy was calculated (Fig. 6(b)) and this is given in Table 2. TABLE2 Experimental condition

Q (kJ tool - 1 ) Temperature gradient coatsening model

Q (kJ tool -1) Ostwald ripening process

Isothermal anneal

-

119.5

Temperature gradient anneal 0.18 × 106 s 0.60 × 106 s 0.77 × 106 s

94.1 80.7 73.3

-121.4 111.8 104.6

4. DISCUSSION

No microstructural evidence was found for phase migration in the Pb-Ag eutectic in contrast to the results reported b y Jones [1], nor was there observed any enhanced coarsening of the microstructure due to the application of a transverse temperature gradient. The microstructure of the directionaUy solidified eutectic -- fine rods, branched rods, and

broken lamellae -- was consistent with that reported by Moore and Elliott [15]. However, the conditions necessary for the growth of fibres, as distinct from other growth forms, are quite stringent and areas of broken lamellae quite often appeared in areas of coarser fibres. To avoid any anomalous results from this effect the coarsening data were obtained from areas of fine fibrous eutectic (an average value being 70 000 fibres mm -2) well away from lameUar regions. By comparison, the eutectic used b y Jones [1] had at the most 22 000 fibres mm -2 so that anomalous transition effects become more likely. This could possibly explain the areas of "highly anisometric" shapes he reported on temperature gradient annealing. The calculated values for the activation energy for the coarsening processes are shown in Table 2, and accepted values for the activation energy for diffusion of Ag in Pb and the self-diffusion of Pb are given in Table 3. It has been pointed o u t by Courtney e t al. [22, 23] that the Ostwald ripening equation contains a solute concentration term which itself is temperature dependent. From the position of the solvus line in the published P b - A g equilibrium diagram the activation energy for the dissolution of Ag in Pb was calculated to be 16 kJ mo1-1. When analysed in terms of the standard Ostwald ripening mechanism the data from both the isothermal and the temperature gradient coarsening experiments yield activation energies which are comparable with the sum of the activation energies for dissolution and for the self-diffusion of Pb. McLean's temperature gradient coarsening

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analysis leads to an activation energy that corresponds neither to that for the selfdiffusion of Pb nor to that for the (interstitial) diffusion of Ag in Pb. The ratio R of the coarsening rate expressions for temperature gradient coarsening and Ostwald ripening may be used to calculate which of the two mechanisms will be operative in any one system under given experimental conditions. From McLean [ 6 ] ,

R =

3pT?QG 2r3 2XTT2~(C~

--

C~)

(6)

where 3~/2X is a constant depending on particle geometry, O the activation energy for the self-diffusion of Pb, C ~ the concentration of Ag atoms in fl, C ~ the concentration of Ag atoms in o~, rl = Q*/kT + T / C ~ m ~ , Q* is the heat of transport of fl atoms and ms the gradient of the ~ solvus. The value of R was

(a)

(b)

(c)

(e)

~

,~ . . . . . . . . . .

~

(d) calculated from the known experimental conditions of the temperature gradient anneals and from the following values of t h e r m o d y n a m i c parameters: Q* = 84 kJ tool -1 [6] ; Q ~ 100 kJ mo1-1 [ 1 8 - 21] ; a n d the interracial energy 7 = 0.4 J m -2 [24]. The resulting value of R (~ 10 -4) indicates that coarsening of the alloy is controlled by Ostwald ripening as, in fact, the experimental data indicate. The activation parameters therefore suggest that coarsening proceeds by an Ostwald ripening process similar to that proposed earlier for the coarsening of the Fe-Fe2B eutectic in the 7-Fe phase field [16]. In the present case, silver atoms diffuse interstitially

Fig. 3. The microstructure of the directionally grown Pb-Ag eutectic after a temperature gradient anneal of 4 K ram-1 imposed normal to the growth direction for 0.34 X 10 6 s. Direction of the temperature gradient is from bottom to top: (a) temperature 546 K (270X); (b) temperature 448 K (270×); (c) lamellar and fibre regions (720×); (d) a typical area of rod temperature gradient annealed at 546 K (1800×); (e) boundary at right angles to the temperature gradient (720×).

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Isothermot

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495 K

Isothermat Temperature Grodient

{

448 K E E

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x 60

c~ 40

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kZ 20

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tZ 20

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Time xl0 6

(sec.)

'3'6

(a)

(b)

i

o

Time xl0 6

(sec.)

lo8

(c)

Time xl0 "6

(sec.)

Fig. 4. Fibre density against annealing time for 16 K mm -1 temperature gradient and isothermal anneals at various temperatures. Error bars show the range of experimental values.

-18

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= 546K;

-19

~ o

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2

(r3-ro3)xl018

o 495K;(r3-ro3)X1019 -8

v 448K; ( r3-ro3 ) x 1020

o~

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4 6 z xl0 ~ (m)

3 z xl03 (m)

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=

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Fig. 5. Plots of log (r -- r0) and log (r 3 -- r03) against z, the distance along the temperature gradient 16 K mm -1, for an annealing time of 6.0 × 105 s. The gradients were used to calculate a value of Q, the activation energy for the process according to eqns. (4) and (5) taking T as 473 K, the average value.

1

o

i

0.36 (a)

0.72 1.08 1,44 Time x 106 (seconds)

TABLE 3 -52 Source

Ascoli e t al. [16] Seith and Keil [17] yon Hevesy e t al. [18] Nachtrieb and Handler [19] Hudson and Hoffman [20] Miller [21]

Activation step

Activation energy (kJ mo1-1)

QAg in Pb QAg in Pb QFo in Pb

60.7 63.7 116.8

QPb in Pb

101.4

QPb in Pb QPb in Pb

104.1 107.0

"5 -54

,'I,,"

i

-56

~ -58 -60

4

through the lead matrix with a low activation energy. However, the rate-controlling process is the diffusion of lead atoms in the opposite direction to fill the vacant lattice sites left by the migrating silver atoms. Hence the activation energy for the coarsening reaction is the sum of that of the self-diffusion of lead and the dissolution of Ag in Pb. However, substi-

(b)

2 I / T x103 (K4)

Fig. 6. (a) Values of (r 3 -- r 3) for isothermally annealed specimens plotted against time for each annealing temperature. (b) Logarithms of gradients obtained from (a) plotted against 1 / T to evaluate Q, the activation energy for the process.

tution of appropriate values in the Ostwald ripening equation leads to a value of the diffusion coefficient some three orders of magnitude greater than that for the self diffusion of lead. The cause of this discrepancy is

n o t y e t u n d e r s t o o d , a l t h o u g h a statistical analysis [ 2 6 ] o f the e x p e r i m e n t a l d a t a indicates t h a t t h e p r o b a b l e a c c u r a c y o f determ i n i n g a d i f f u s i o n c o e f f i c i e n t in this m a n n e r is very low. This s t a t e m e n t h o l d s t r u e f o r o t h e r published d a t a o n p r e c i p i t a t e c o a r s e n i n g as well as f o r t h e present results.

REFERENCES 1 2 3 4 5 6 7 8

CONCLUSIONS

9 10

(1) N o e x p e r i m e n t a l evidence was f o u n d f o r m i g r a t i o n o f Ag fibres in Pb w h e n e x p o s e d t o t e m p e r a t u r e gradients o f 4 K m m -1 and 16 K m m -1 with a m e a n t e m p e r a t u r e o f 4 7 3 K. (2) T h e spatial d i s t r i b u t i o n s o f fibre radii yield a realistic a c t i v a t i o n e n e r g y f o r coarsening b y O s t w a l d ripening o f b e t w e e n 1 0 4 . 4 and 1 2 1 . 4 kJ mo1-1 in the t e m p e r a t u r e gradient. These values o f a c t i v a t i o n e n e r g y are similar t o t h a t o f 1 1 9 kJ mo1-1 o b t a i n e d f o r O s t w a l d ripening during i s o t h e r m a l annealing. T h e a c t i v a t i o n energies derived f r o m a t e m p e r a ture gradient c o a r s e n i n g m o d e l are unrealistically low. (3) T h e m e a s u r e d a c t i v a t i o n e n e r g y o f t h e c o a r s e n i n g process c o r r e s p o n d s t o t h e s u m o f t h e a c t i v a t i o n energies f o r dissolution o f A g in Pb a n d the self-diffusion o f Pb.

11 12 13 14

15 16 17 18 19 20 21 22 23 24

ACKNOWLEDGMENTS T h e a u t h o r s wish t o t h a n k the A.R.G.C. f o r financial s u p p o r t and Dr. M. M c L e a n f o r detailed c o m m e n t s o n the m a n u s c r i p t .

25 26

D. R. H. Jones, Mater. Sci. Eng., 15 (1974) 203. D. R. H. Jones, Metal Sci. 8 (1974) 37. P. S. Ho, J. Appl. Phys., 41 (1970) 64. P. G. Shewmon, Trans. Metall. Soc. AIME, 230 (1964) 1134. R. S. Barnes and D. J. Mazey, Proc. R. Soc. London, Set. A, 275 (1963) 47. M. McLean, Scripta Metall., 9 (1975) 439. C. Wagner, Z. Electrochem., 65 (1961) 581. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids, 19 (1965) 35. D. R. H. Jones, J. P. Benson and K. T. Ison, J. Mater. Sci., 11 (1976) 1172. D. R. H. Jones and G. J. May, Acta Metall., 23 (1975) 29. M. McLean, Acta Metall., in the press. R. D. Doherty and T. R. Strutt, J. Mater. Sci., 11 (1976) 2169. J.F. Stohr, J. M. Hauser, T. Khan, M. Rabinovitch and W. Bibring, Scripta Metall., 10 (1976) 729. M. McLean, in M. R. Jackson, J. L. Walter, F. D. Lemkey and R. W. Hertzberg (eds.), In Situ Composites -- II, Xerox Individualized Publishing Program, Lexington, Massachusetts, U.S.A., 1976. A. Moore and R. Elliott, J. Inst. Met., 96 (1968) 62. A. Ascoli, L. Filoni, G. Poletti and S. L. Rossi, Phys. Rev. B, 10 (1974) 5003. W. Seith and A. Keil, Phys. Chem., 22 (1933) 350. G. yon Heresy, W. Seith and A. Keil, Z. Phys., 79 (1932) 197. N. H. Nachtrieb and G. S. Handler, J. Chem. Phys., 23 (1955) 1569. J. B. Hudson and R. E. Hoffman, Trans. Metall. Soc. AIME, 221 (1961) 761. J. W. Miller, Phys. Rev., 181 (1 - 3) (1969) 1095. H. B. Smartt and T. H. Courtney, Metall. Trans. A, 7 (1976) 123. L. Y. Lin, T. H. Courtney, J. P. Stark and K. M. Ralls, Met. Trans. A, 7 (1976) 1435. E. D. Hondros, in R. C. Gifkins (ed.), Interfaces -Proceedings of the International Conference, Melbourne, 1969, Butterworths, Sydney, 1969, p. 84. A. R. T. de Silva and G. A. Chadwick, Metal Sci. J., 6 (1972) 157. W. R. Thorpe, University of Queensland, personal communication, 1977.