The influence of a dispersion of particles on the sintering of metal powders and wires

progress in Materials ScienceVol. 25. pp. 1 to 34

004g-5500/80/1001-0001$02.00/0

Pergamon PressLid 1980. Printed in Great Britain

THE INFLUENCE OF A DISPERSION OF PARTICLES ON THE SINTERING OF METAL POWDERS A N D WIRESt M. F. Ashby Department of Engineering, University of Cambridge, Cambridge, UK and

S. Bahk, J. Bevk and D. Turnbull Division of Applied Sciences, Harvard University, Cambridge, MA, USA CONTENTS ABSTRACT

2

1. INTRODUCTION: A REVIEW OF OBSERVATIONSOF INHIBITED SINTERING

i.I. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7.

Overview Observations of Seigle and Ballul~ (19.54) Seiole and Pranatis (1955) and Pranatis and Seiole (1956) Observations of Brett and Seiole (1966) and Sei#le and Brett (1965) Observations of Tikkanen ez al. (1962) and Tikkanen and Ylasaari (1969) Observations of Early et al. 0964) and Lenel et aL (1970) Observations of Singh and Houseman (1971) Other Miscellaneous Observations: Elliot [Reported by Kuzynski and Lavendel (1969) and Schultz

(1972)] 1.8. Previous Attempts to Rationalize These Observations 1.8.1. The Slowing of lattice Diffusion by a Dispersion of Particles 1.8.2. The Suppression of Plasticity 1.8.3. The Suppression of the Free Surface or the Grain Boundary as a Sink or Source for Matter 2. THE THRESHOLD INTERFACE POTENTIAL-BARRIER FOR SINTERING

2.1. Diffusion Control and Interface-Reaction Control 2.2. The Threshold Potential Introduced by a Dispersion of Particles 2.3. Surface Threshold Potential Ala~ 2.4. The Boundary Threshold Potential A go 2.5. The Dislocation Threshold Potential A/~ 3. THE RATE EQUATIONSFOR SINTERING WITH A THRESHOLD POTENTIAL INCLUDED 3.1. Origin of the Rate Equations 3.2. The Surface Curvatures Which Drive Sintering 3.2.1. Sinterln# of a close-pocked array of wires 3.2.2. Sinterino of an aggregate of spheres 3.2.3. Rate equations for the sinterino of an aggregate of wires 3.2.4. Rate equations for the sintering of an aogregate of spheres 4. RESULTS OF THE MODEL 4.1. Overview 4.2. Neck-Growth vs Time 4.3. Sintering Diagrams 4.4. The Position of the Cut-off

2

2 3 3 3 4 4 4

4 5 5 5 6

6 6 9 10 12 12 12 13 13 14

15 17 19 19 19 21 23

+Acknowledgements. This work has been supported in part by the National Science Foundation, under Contract DMR-72-03020, by the Division of Applied Physics, Harvard University, and by the Science Research Council under Contract GR/A 1233.5 J p.M.s 25 I--A

1

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PROGRESS IN MATERIALS SCIENCE

5. COMPARISON OF THEORY WITH EXPERIMENT

5.1. The Experiments of Brett and Seigle (1966) on Copper Containing A1203 5.2. The Experiments of Lenel et al. (1970a) on Silver Containing A1203 5.3. The Experiments of Lenel et al. (1970b) on Copper Containing SiO2 5.4. The Experiments of Singh and Houseman (1971) on Iron Containing Oxide Dispersions 5.5. The Measurement of Bahk et al. (1975) on Copper Containing BeD 6. SUMMARYA N D CONCLUSIONS REFERENCES

24 25 27 28 30 31 33 33

Abstract A dispersion of fine, stable particles is known to inhibit the sintering of metal powders and wires. This can be understood as an effect of the dispersed particles on the efficiency of the sources and sinks between which matter flows during sintering. Well-wetted particles cause the free surface of the metal containing them to become less effective as a source of matter because, as the surface recedes, particles are exposed and a new, high energy interface is created. Particles on grain boundaries, too, make the removal of matter from the boundary more difficult because they pin the boundary dislocations which (at a microscopic level) are the sites from which atoms are detached. The result is that sintering can proceed only if a threshold driving force is exceede(il. This threshold depends on the size and volume fraction of particles; its magnitude is different for free surfaces and grain boundaries; and it may change as sintering proceeds. Because of this, some mechanisms of sintering--particularly those leading to densification--are inhibited more strongly than others. These ideas are developed quantitatively, and the results presented as sintering maps which show the degree to which each mechanism of sintering is inhibited. The predictions of the model are compared with the results of published experimental observations of inhibited sintering. There is broad agreement, though certain interesting discrepancies remain.

1. INTRODUCTION: A REVIEW OF OBSERVATIONS OF INHIBITED SINTERING 1.1. Overview There have been n u m e r o u s observations of the inhibiting effect of a fine dispersion of inert stable particles on the sintering of metals. In the following sections, we review reports of the slowed, or totally inhibited, sintering of Cu, Ni, C o and Fe by dispersed particles of A1203, M g O , SiO2, TiO2, ZrO2, C a D , ThO2, W, graphite and possibly FeAI20,~. In general, the inhibiting effect increases with volume fraction up to volume fractions of a few %; for larger quantities of dispcrsoid the results are confused. F o r the same v o l u m e fraction, s o m e dispersoids are m o r e effective than others. This m a y simply be an effect of dispcrsoid particle size, a variable often b e y o n d the c o n t r o l of the investigator, or it m a y reflect the nature and properties of the particle-matrix interface. But the wide range of dispersoids which all, to a greater or lesser extent, inhibit sintering, suggests that it is the physical presence of the dispersoid, rather than its chemical nature which inhibits sintering. Particles inhibit both the m e c h a n i s m s of sintering which cause neck g r o w t h without densification and those m e c h a n i s m s which, by filling in the voids, cause the c o m p a c t to density. But it is the densifying m e c h a n i s m s which are most p r o f o u n d l y affected; in extreme cases, a l t h o u g h some neck g r o w t h occurs, densification is suppressed completely.

DISPERSION

OF P A R T I C L E S ON M E T A L P O W D E R S

AND WIRES

3

1.2. Observations of Seigle and Ballufi (1954), Seiole and Pranatis

(1955) and Pranatis and Sei.qle (I 956) The first indications of an inhibition of sintering by particles appear in marker experiments designed to study the mechanisms of sintering. Markers of colloidal graphite, WO3, and AlzOa, incorporated into the surface of copper wires by electrodeposition, were followed during the course of sintering: their movement indicates plastic flow; their failure to move indicates diffusional mechanisms. Seigle and Pranatis found that the markers did not move, but they altered the progress of the sintering: the pores spheroidized much as they did in clean copper, but their volume remained almost constant. Their micrographs show that during spherodization, matter was drawn from convex parts of the pore surface, meaning that diffusion from surface sources was not seriously inhibited. But the boundary between two powder particles--the normal source of matter during densification--had been suppressed as a source by the particles. 1.3. Observations of Brett and Seigle (I 966) and Seigle and Brett (1965) As part of general study of sintering mechanisms (in materials as diverse as cellulose acetate, borosilicate glass, nickel and copper) Brett and Seigle used a powder-metallurgical technique to prepare 150 gm copper wires containing 1 vol. O//oof 0.02/am Al2Os. These they twisted together in strands of three, and sintered at 1050°C for up to 600 hr. Compared with pure copper, densification was strongly inhibited: the central pore persisted after 600 hr, though in pure copper it was gone after 400. When the experiment was repeated with copper containing 6 vol. °~o ZrO2, an even greater inhibition was found. One feature of Brett and Seigle's experiments confuses the picture. When they remelted the copper-Al20 s to allow the Al2Os to float to the surface, and then made wires of the residual copper, they found sintering was still inhibited. It is unclear how much Al:O s remained in the metal. 1.4. Observations of Tikkanen et al. (1962) and Tikkanen and Ylasaari (1969) Motivated by the same aim as Seigle, Pranatis and Brett--to distinguish the contribution to sintering of plastic flow or creep from that of pure diffusion--Tikkanen and his coworkers studied the densification of dispersion strengthened powders. Dispersion strengthening is known to raise the yield strength and to slow the rate of creep; it was presumed to have no effect on diffusionai densification. They pressed and sintered -325 mesh Ni and Co powders containing fine, stable, dispersions of MgO and CaO, prepared by reducing mixed oxides. Compared with that of pure Ni and Co powder, the rate of densification (at 1250°C) was retarded. This retardation increased as the volume fraction of oxide increased, up to 1 wt ° o. At higher volume fractions (3-5 wt ~o) the densification rate reached a minimum, and at still higher (5-10 wt °~o)increased slightly, perhaps because of agglomeration of the oxide. The two oxides were not equally effective. CaO was the more effective inhibiter of sintering in Ni; MgO the more effective in Co. This, too, could be an effect of oxide particle size, which was not determined in these studies. Tikkanen and his co-workers observed two types of densification curve. The first, characteristic of low oxide contents, showed a rapid initial densification rate (though slower than that of pure Ni or Co), followed by a very slow final rate. The second type, characteristic of volume fractions of 8 wt °~o MgO and above showed a slow, steady densification, which was still appreciable at the end of their test period (2 hr).

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P R O G R E S S IN M A T E R I A L S S C I E N C E

1.5. Observations of Early et al. (1964) and Lenel et al. (1970) Seeking to distinguish the plastic from the diffusional contribution to sintering, Early et al. (1964) compared the sintering of copper containing A1203 (made by internal oxidation) with that of pure copper. They recorded the dimensional change I of compacts as a function of temperature and stress amthis last variable was included to distinguish diffusional mechanisms (for which, if friction and other effects are ignored, i ~c a) from those involving creep (for which l oc on). Their measurements show that A1203 reduces the rate of shrinkage between 800 and 1025°C; and optical microscopy showed that the neck size was smaller in the oxide-containing powder--so both densifying and non-densifying mechanisms were retarded by the oxide. In a separate comparison of the sintering of Cu-SiO2 with pure Cu, Lenel et al. (1970) measured the neck growth-rate as spherical particles sintered to plates, and to each other: the oxide significantly retarded neck growth.

1.6. Observations of Sin#h and Houseman (I 971) These workers studied the densification of 7/am carbonyl-iron powder, between 1300 and 1490°C, with and without additions of TiO2, A1203 and ZrO2. The oxides were finely dispersed in the iron as particles of size 50-500 A, and in amounts between 0.5 and 2 wt °,o. All three inhibit densification to an extent which generally increases with the amount of oxide. Of the three, ZrO2 is the most effective: as little as 0.5 wt O/~ovirtually halts densification up to 1490°C. 1.7. Other Miscellaneous Observations: Elliot [reported by Kuzynski and Lavendei (1969) and Schultz (1972)] Other observations of inhibited sintering in Fe powders exist. Elliot noted the retardation of sintering in both Fe and Ni by ThO2. Such effects are important in catalysis, where sintering reduces the surface area and thus the effectiveness of metallic catalysts in chemical processes. The commercial importance of this has stimulated several studies of inhibited sintering: they are listed in the paper by Schultz (1972). Typically, it is found that fine iron powder (500 A) containing what is described as "a molecular dispersion of FeAI204" agglomerates more slowly than pure iron, and has a longer useful life as a catalyst. 1.8. Previous Attempts to Rationalize these Observations Attempts to explain the slowing or suppression of sintering by a dispersion of particles fall into three broad classes. First, since much sintering is diffusion-controlled, it is natural to ask whether particles slow lattice diffusion. One proposal to this effect is reviewed in Section 1.8.1 below, where we conclude that a small volume fraction of particles is more likely to accelerate than to retard diffusion, a conclusion born out by direct experimental measurement. Second, since a particle dispersion retards plastic flow, one might ask whether it is throuffh their influence on plasticity that particles slow sinterinff (Section 1.8.2, below). The experimental evidence strongly supports the view that, in the absence of an applied stress or pressure, plasticity (including power-law creep) contributes only to the earliest stages of sintering, so the idea, though valid, and incorporated in our treatment cannot explain the broad influence of particles on all stages of sintering.

DISPERSION

OF PARTICLES

ON METAL POWDERS

AND WIRES

5

We are left with the class of model which is developed in later sections of this report: that particles interfere with the sources from which the d!ffusive flow of matter is drawn, or the sinks to which it.flows, discussed in 1.8.3 below. 1.8.1. The slowing of lattice diffusion by a dispersion of particles Easterling and Gessinger (1972) develop a qualitative model which suggested that immobile particles might suppress lattice diffusion. Their model neglects the tractions exerted by the particle on the surrounding matrix, and for this and other reasons, it appears to be physically dubious. An upper limit on the effect they seek to model would appear to be that lattice diffusion be slowed by the factor (l-f), w h e r e f is the volume fraction of particles: a trivial effect for the volume fractions with which we are normally concerned (typically 1'~/o). But if the particles are incoherent, or have a strain field associated with them, the additional internal surface, or the elastic distortion of the matrix, may well accelerate diffusion. In fact, the only relevant measurements of lattice diffusion, carried out on silver contained 2% A1203, showed a slight acceleration (Imai and Miyazaki, 1967). 1.8.2. The suppression of plasticity Early et al. (1964), Tikkanen (1969) and Lenel et al. (1970a, b) among others, use the suppression of sintering by particles as evidence that sintering requires plasticity (slip or power-law creep). All theoretical treatments, and almost all experimental data using markers, indicate that the stresses are too low to cause plasticity when x/a > 0.1, that is, in all but the earliest stages of sintering. We have included the pinning effect of particles on dislocations in our treatment of the transient contribution of plasticity to neck growth, but we find that it is seldom the dominant mechanism of sintering, even at small neck sizes. Yet the experiments reviewed above show that particles influence all stages of sintering~ It must be concluded that the suppression of plasticity is one way in which particles may suppress sintering, but that it is not their most important effect. It should be clearly recognized that our discussion refers to sintering in the absence of applied pressure or stress. Power-law creep becomes a dominant mechanism in pressure sintering (Wilkinson and Ashby, 1975) and particles can profoundly affect its rate. 1.8.3. The suppression of the free surface or the grain boundary as a sink or source for matter Kuczynski and Lavendal 0969) analyse the action of a free surface as a sink, and conclude that poorly wetted particles should suppress it. But as we point out in Section 2, as soon as some diffusional sintering has occurred, the sink becomes particle-free, and is therefore uninteresting. It is the source of matter that has particles in its plane. We then find (Section 2, below) that poorly wetted particles do not pin the free surface; it is the wellwetted particles which do so, thereby introducing an interface potential barrier. [Johnson (1975) has developed one aspect of this idea by considering the possibility of the dragging of particles by a surface. At the present level of our understanding, it is not possible to say whether this refinement is important. It does not appear to be consistent with the observation that small particles inhibit sintering more strongly than large ones.] The complementary idea of the inhibition of the grain boundary (which is the source of matter during densification) as a source of matter has been treated by Ashby (1969, 1972) and in a somewhat different way by Johnson (1973). The ideas are fully discussed in the next section.

6

PROGRESS IN MATERIALS SCIENCE

2. THE THRESHOLD INTERFACE POTENTIAL-BARRIER FOR SINTERING 2.1. Diffusion Control and Interface-Reaction Control In modelling the kinetics of sintering in the absence of applied stress, it has been customary to assume that equilibrium exists across the interfaces that act as the sinks and sources for matter. Diffusion is then rate-controlling; the rate of any one mechanism of sintering is determined by the rate of diffusion of matter from the sources to the sinks. The shrinkage of a pore during sintering has certain features in common with the shrinkage or growth of a precipitate particle in a solid or a liquid with which it is not in equilibrium. Studies of precipitate growth and shrinkage show that diffusion does not always control the kinetics; under some conditions the rate of the process is determined instead by an interface reaction, that is, by the rate of transfer of atoms across the interface to or from the particle I'see for example Turnbull (1953) for a study of the interface-reaction controlled growth of particles from a liquid, and Nolfi et al. (1969) for a study of both growth and shrinkage of particles in the solid state]. The basis of the model developed here is that pore shrinkage, like precipitate dissolution. can sometimes be controlled by an interface reaction. Pores differ from precipitates in that the components leaving a shrinking precipitate in general go into solution in the matrix, whereas the vacancies leaving a shrinking pore diffuse to a sink--usually a free surface, or the grain boundary in the neck between the two particles. This means that two interfaces are involved in the growth or shrinkage of a pore: the pore surface itself, at which the vacancies are created, and the surface to which they flow and are annihilated. The kinetics of either process may, under the right circumstances, be rate-controlling. When equilibrium does not obtain at an interface, the chemical potential,/a, of the atoms is discontinuous there. The flux of atoms (or vacancies) across the interface depends on the magnitude of this discontinuity. For precipitate it is often found that the flux is a linear function of the interface potential discontinuity A#~, so that the faster the precipitate grows, the bigger Ap~ becomes, and the more likely are the kinetics of growth to be interfacereaction controlled [e.g. Turnbull (1953), Nolfi et al. (1969)]. In the following section of this chapter we consider simple models for the potential discontinuity introduced by a dispersion of fine particles at an interface, and find that at the lowest level of approximation, A#~ is constant; this means that no flux crosses the interface until the constant A/a~ is exceeded there, and for this reason we refer to the A#~'s we calculate as threshold potentials. At this point it is helpful to list the symbols used in subsequent sections. They are listed in Table 2. I. 2.2. The Threshold Potential Introduced by a Dispersion of Particles Figure 1 shows a neck between two powder granules during sintering. Six independent diffusion paths contribute to sintering, three originating on the pore surface, two on the grain boundary separating the granules, and one at the dislocations within the granules themselves. Arrows show the direction in which matter flow. In the absence of applied loads, these fluxes are driven by the difference in surface curvature between the source of the matter and the sink. Take the surface flux as an example. Figure 2 shows the potential of atoms at the neck (A) as/~,4 and that at the surface well away from the neck, at B, as ~L8. Experiments suggest that, when no particles are present, sintering is diffusion controlled--that is, that the entire potential difference/~,_#A is available to drive a diffusive flux from A to B. Then, at steady

D I S P E R S I O N OF P A R T I C L E S ON METAL P O W D E R S AND W I R E S Table

2.1

powder granule radius radius of disc of contact of two powder granules the final value of x when I00'~, density has been reached Xf P,Pl,P2 = radius of curvature of the neck K I , K 2 , K 3 = curvature differences which drive diffusive fluxes ,u = chemical potential of diffusing atoms or ions Ala i = discontinuity of chemical potential required to drive an interface as a sink or source for matter threshold potentials for a free surface, a grain boundary and a dislocation source re= rate of flow of matter into the neck O v = lattice diffusion coefficient D n = grain boundary diffusion coefficient O s = surface diffusion coefficient effective surface thickness 3n = effective grain boundary thickness P r =" vapor pressure [P,. = P o exp - ( Q , ~ p / k T ) ] surface free energy of powder granule grain boundary free energy "~p = surface free energy of a dispersed particle T ps - ~ interracial free energy of the particle/matrix interface d = diameter of dispersed particles f = volume fraction of dispersed particles f 2 = atom or molecular volume k = Boltzmann's constant (1.38 × 10- 23 J/K) T = absolute temperature (KI Z~, = melting temperature F = ">,,t2/kT (typically of magnitude 10 -8 ml rate of approach of particle centers A o = theoretical density A i =. initial density of powder compact G - - - shear modulus b = Burger's vector of dislocation, or the atomic or molecular diameter N= dislocation density N a = area density of particles X=

PATI SURFJ

IVE

FIG. 1. Two wires or spheres in contact, showing, on the left, the various diffusion paths, and on the right, three threshold potentials.

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P R O G R E S S IN M A T E R I A L S S C I E N C E

CHEMICAL POTENTIAL, p. /u. B

t

-....x...

SOURCE, B

DISTANCE

I SINK, A

FIG. 2. The variation of chemical potential with distance, illustrating how an interface threshold, A#l, lowers the gradient and slows the rate of sintering.

state, the chemical potential of an atom varies in a smooth way from #A at A to #B at B, as shown by the full line on Fig. 2. Suppose, now, that at the surface B an interracial barrier of magnitude A#~ exists. The chemical potential gradient within the material is reduced, and the rate of sintering slowed down. More importantly, if A#T is fixed, then when #B _ gA < A#~

no sintering will occur at all. In what follows, we assume that each source of matter has a characteristic interface potential: Source

Threshold potential

Free surface Grain boundary Dislocations

A#[ A#~ A#p

At the sink (A) we assume that there is no such barrier. This is because, in the early stages of sintering, the driving potential gs /~,~ is large--large enough to overcome any threshold and therefore able to deposit new, particle-free material in the neck (Fig. 1); and particlefree material, we assume, can act as a perfect sink. In the next three sections we examine simple models for the threshold potentials A#~, A#~ and A#~a. They are based on the following ideas: (a) that the particles interact with free surfaces, and, if well wetted by the matrix in which they lie, they may restrain the motion of the surface and so introduce a threshold potential which is constant with time (Section 2.3); (b) that dislocations move on a grain boundary when it acts on a sink or source for point defects. The dislocation cores are the sinks or sources. A dispersion of particles interacts

DISPERSION

OF PARTICLES

ON METAL POWDERS

AND W1RES

9

with these dislocations, and introduces a finite driving force below which they cannot move. If the boundary separates two grains which contain a dispersion of fine particles, then the number-density of particles on the boundary increases as it acts as a source of matter, and the threshold potential increases with time (Section 2.4): (c) that a dispersion of particles interacts with lattice dislocations, reducing the segment length that is free to climb and so contribute to sintering, and by introducing a threshold potential opposing long-distance climb (Section 2.5). These ideas are not new. Kuczynski and Lavendal (1969), and Johnson (1975) have suggested that free surfaces might be pinned by particles: Ashby (1969) and Bahk et al. (1975) that grain boundaries might be inhibited as sources by a dispersion of particles; and the idea that dislocations might be pinned by particles dates back to the work of Orowan (1948). But ours is the first treatment to evaluate all three in the ways outlined below. We should make it immediately clear that these ideas may be too simple, and may require modification later. Both the thresholds described under (a), (b) and (c) above turn out to be essentially independent of temperature--yet measurements of diffusional creep (which has much in common with sintering), in dispersion-hardened alloys, show that the threshold may increase with decreasing temperature. This suggests an additional threshold caused by segregated solute, or due to a pinning of dislocations and steps that is caused by impurities on an atomic scale. Such effects are discussed elsewhere (Verrall and Ashby, 1976). 2.3. Surface Threshold Potential A#~ Consider a particle which lies in a free surface, as shown in Fig. 3. Since the surface interacts with many particles simultaneously, we are justified in taking an averaoe interaction energy, per atom removed from the surface, and equating this with the potential barrier A/z~. Suppose the surface moves from position 1 to position 2, by the removal of 12d/2G atoms, where f2 is the atomic volume, d the particle diameter, and I the particle spacing. If the surface free energy of the matrix is 7s, that of the particle is 7p and that of the particle/

4 [

i '

I

iYs

;i ;il

i ,

j_ I/

=r

:l P O S I T I O N

1

ii!i!i;i;i;i!ii!;i!iii;!;iliiiiiiiiiiii;ii;i;i;ii;ii!;i;!i!iiii!!iii;i

liii!iiii?i:.!i!ii!ii!i!ii!i!iiii;iii!ii!iiii

FIG. 3. A particle in a free surface. Depending on whether it is wetted or non-wetted, its presence m a y enhance or detract from the action of the surface as a source.

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P R O G R E S S IN M A T E R I A L S S C I E N C E

matrix interface is 7p, then the free energy change per atom is nd But the volume fraction of particles, f, is given (from Fig. 3) by d2

f =-~ l--T and thus A#~-

3ffl d

{2"i'p-2~%+~sl.

Two limiting cases are important. First, if the particles wet the matrix, then 7ps is negligible and A/~ is positive (it opposes sintering)

But if the particles are completely non-wetting, then ~'ps = "~p @ 7s and the interface potential A#f is negative (it aids sintering) A#~ = -

3ff~ d y'"

In our calculations we have assumed that the particles are partly wetted, and that ~% = ~,p when Agt - 3ffly~ d

2.4. The Boundary Threshold Potential A# .a, The grain boundary between two dispersion-strengthened powder granules will contain particles. Let the area density of the particles by N,~, and assume that dislocations move in the boundary plane when the boundary acts as a source of matter (Ashby, 1969, 1972). If these dislocations interact with the particles, then a threshold chemical potential per atom A#~ will be necessary to make the dislocations move, and hence to make the boundary act as a source. If there is a difference in chemical potential between the dislocation source and the sink, (the pore) the dislocation will bow out, in climb, between the points where particles obstruct its motion, releasing matter as it does so (Fig. 4). A small potential difference between this dislocation-source and the sink (the pore) merely causes the dislocation to bow until it is in equilibrium with the potential of matter in the surrounding matrix. For extensive, steady motion, the potential difference must be at least sufficient to detach the dislocation from the particles which pin it. It is readily shown that this requires a potential difference A#~ =

ctG5~ ( I / ~ / V ~ - d)

DISPERSION

OF P A R T I C L E S

BOUNDARY .

1I t

I

ON METAL POWDERS

I L

.

.

.

.

II I

--

.

.

.

.

/

;n

f ..¢"

/

| •

._

,__ L _

.

.

.

.

I I I

=

L ~

..:'iii::::::i~:i-"i'=~'"!:"::~::-i!i :': ":i~;:!'~iiiii::'" .

11

DISLOCATIONS

......... | I

AND WIRES

GRAIN BOUNDARY

PLANE

.

I

111 FIG. 4. Particles in a grain boundary. In general they will tend to suppress the action of the b o u n d a r y as a source of matter.

where G is the shear modulus of the matrix metal, and b the Burger's Vector of the boundary dislocations, and ~t measures the strength of the interaction. (Its maximum value, corresponding to the bowing of dislocations between particles, is about 0.5. We have taken 0.1 as a reasonable value). If, in spite of the threshold, the boundary acts as a source of matter, then particles accumulate on it and A#~a rises. This we have allowed for by calculating (incrementally) the amount of matter I;"drawn by diffusion out of the boundary; we have then calculated N~ from the number of particles contained in this volume, plus those within a distance d/2 on either side of the boundary. N,~, strictly, is not uniform; we have taken an average, per unit area. in the boundary plane. The calculation is as follows. Wilson and Shewman (1966) give the rate of approach of two powder granules as 2 y ~ D~ 1 + = kTp----~ 2--~x) where x is the neck radius and p its curvature, and D~, is the bulk diffusion coefficient and fiDB that for boundary diffusion, multiplied by the boundary width. This deposits particles on the boundary at a rate dNA 6f . dt = ~td3 y

since 6fled a is the number of particles per unit volume. Using the substitution p = x2/2a, we find dNA 4?st3a 6f ( DB6"] d---~ = k T x a rcda Dv 1 + 2Dt, x / "

In practice the densification is retarded by the existence of the threshold potential A/~, by a factor which we calculate in the next section: it is (1 -(A/a~/K~,Q)), where K is the

12

PROGRESS

IN MATERIALS

SCIENCE

curvature of the neck. The rate of accumulation of particles then becomes

dNa _ 4ysf~a 6 f Do 1 + dt kTx 3 red3 2Dox/

1 K~A~J"

The result for wires has the same form, but the constant of 4 is replaced by 1. We have integrated this numerically when computing the sintering diagrams, and used the result to calculate a (continuously increasing) threshold potential, A/a~. One result of the accumulation of particles on the grain boundary is that it ceases to act as a source earlier in the sintering than free surfaces do. 2.5. The Dislocation Threshold Potential A#~° If the lattice dislocations interact with dispersed particles, a potential difference A#~ = ~Gbf~x/ N a is required to make them climb and thus act as sources of matter over distances large compared with the particle spacing. When long-distance climb occurs, we may take the interaction constant, at = 0.2, and the area density of particles, NA, as 6f Na

-

~d 2

giving A# ° = 0.2 Gbfl

x/6f

n--dr"

This is constant because a climbing dislocation escapes from particles--they do not accumulate on it. More usually, the driving force is too low to cause any long-distance climb, though dislocations may bow out by climbing between points where they are pinned until the curvature they thereby acquire limits further climb. 3. THE RATE EQUATIONS FOR SINTERING WITH A THRESHOLD POTENTIAL INCLUDED 3.1. Orioin of the Rate Equations It is conventional to think of sintering as occurring in three sequential stages. Stage 1 is the early stage of neck growth: the individual powder granules or wires are still distinguishable. Stage 2 is the intermediate stage: the necks are now quite large and the pores are roughly cylindrical. For powder granules (but not for wiresl a final stage, Stage 3, follows: the pores are now isolated and spherical. Several mechanisms contribute to each stage (Fig. 1). Those describing the sintering of spherical powder granules have been reviewed recently by Ashby (1974); he considers one set of rate equations for Stage 1 a~d a second set for Stages 2 and 3, which, to an adequate approximation, can be treated as a single stage. In this section, we consider how these rate equations and those describing the sintering of wires, are changed by the existence of a threshold potential, A/~,.

DISPERSION

OF P A R T I C L E S

ON M E T A L P O W D E R S

AND WIRES

13

3.2. The Surface Curvatures which Drive Sinterino In the absence of pressure (or stress), sintering is driven by the difference in surface curvature between sources and sinks. This driving force differs for different diffusion paths; and, for a given path, depends on the geometry of the sintering granules or wires. The formulation used here is the same as that of an earlier paper {Ashby, 1974). 3.2.1. Sintering of a close-packed array of wires Consider first an array of wires wound on a drum, or twisted together, so that they form a close-packed array, Figure 5 shows the geometry. The non-densifying mechanisms transport matter from the convex surface of the pore [curvature l/a) to the concave [curvature 1/p~) so'that the curvature difference is

1

1

p~

a

K=--+where X2

P' = 2 ( a - x ) ' This driving force must go to zero when the pore becomes cylindrical; to achieve this, we multiply it by

1-

x ) Xcrit

where xcr,=xr

0.74(A° ~o A~)a/:

a.

Here x s is the final neck size when full density is reached x I = 0.55a and Ao is the pore-free density and A~ is the initial density of the compact Ai = 0.9Ao. The final result is K1 =

t'

~

1

+

.x'~ - 0.74(A° - Ai~

~, ao

}

1/2

a

1

"

For the densifying mechanisms in Stage I, matter flows from the grain boundary, which is fiat.to the pore. The curvature difference is K2

=

1

2(a - x)

px

xz

Finally, when pores become cylindrical [Fig. 5(b)] the curvature difference is

1 K 3 = xf

-- x "

14

PROGRESS IN MATERIALS SCIENCE

= STAGE I

I

G

a

t~-Xf JX STAGE n & ~[

B

\ FIG. 5. The geometry of the neck during Stage I (above) and Stages 2 and 3 (below) of sintering.

3.2.2. Sintering ~?fan aggregate of spheres The driving force for sintering, in this geometry, has been described elsewhere. For the non-densifying mechanisms we take the curvature difference of the pore surface

DISPERSION

OF P A R T I C L E S

ON M E T A L P O W D E R S

AND WIRES

15

and multiply it by a factor which causes it to go to zero as the pore becomes cylindrical. The result is gl

=

Pl

.~f +

1 --

_ (Ao~

mi~l,3

\

/

xl

. "

For the densifying mechanisms of Stage 1, we have

and for the densifying mechanisms of Stages 2 and 3 2 K3----. P2 3.2.3. Rate equations for the sintering of an aggregate of wires Figure l shows the six alternative diffusion paths which contribute to the sintering of an array of wires or spheres. Consider the following calculation of a rate equation--that for path l, surface diffusion-which includes the effect of the interface threshold potential, A#~. Diffusion along the surface (Fig. l) is driven by the difference in chemical potential between atoms on the surface at A and those at B. If no interface potential discontinuity existed, then the atoms on the surface would be in equilibrium with, and at the same potential as, those in the crystal immediately below the surface; but it is our postulate that, at B this is not the case: there is a potential discontinuity of A#~ there. Let the difference in surface curvature between B and A be K~. Then the potential gradient driving diffusion is 1 V/~ ~ ~1 l(,u~ - A~tr) - ~t A ', = ~(K~y,fl - A/~)

(3.1)

where S is the distance from B to A. The flux of matter from B to A along the surface is given by D, 7,taK1 (1 Js = f~k r S

A#] , K-~7,/t

(3.2)

and the volume of matter flowing into one side of the neck, per unit length of wire is I?= 2J~6s~) where 62 is the effective width of the surface as a diffusion path, and the factor 2 enters because matter flow in from below (Bit as well as above (B). The neck growth rate is related to this by I? -,- 2p.~. If we approximate the diffusion distance. S, by p, we obtain

-~ =

p2

k T K1

1

Klfl'AJ"

16

PROGRESS IN MATERIAL S S CIE NCE

When no interface potential barrier exists, the term in brackets becomes unity. And since. in Stage 1 of sintering (the only stage to which surface diffusion contributes) the curvature difference Kt is almost identical with the neck curvature, 1/p, the diffusion controlled limit becomes 7~2 =

K,

It is possible to rederive rate equations for all six stages of sintering, in a more or less approximate way depending on the geometry of the diffusion path, and including a threshold potential barrier A/~. But, at the level of approximation of interest here, the result parallels that just derived. A fixed threshold simply reduces the potential gradient driving the sintering process; and if this gradient is linear (or approximately so), then the rate of sintering is lowered by the factor (1

A . , '~

K ,n/

where A#i is the appropriate threshold potential, and K the appropriate curvature difference, for a given diffusion path and stage of sintering. In what follows we shall comment briefly on any special features off or approximations in, the rate equations. Sintering by lattice diffusion from the surface is assumed to follow the same rate equations as that for surface diffusion, with the area across which the diffusive flux flows (6~ per unit length) replaced by a dimension which we take to be p. The case of sintering by evaporation and condensation (path 3) is slightly more complicated, although the result is the same. It is usual to assume that the rate of this mechanism is not controlled by diffusion, but by the kinetics of evaporation or condensation at the sink or source. (This will be true for sintering in a vacuum, or under low pressure of gas, when the mean free path of evaporated atoms or molecules is large; but under a sufficiently high gas pressure the mean free path may be reduced sufficiently that diffusion through the gas phase becomes rate-controlling. We ignore this possibility and thus calculate an upper limit for the rate of neck growth by vapour transport.) When this assumption is made, the upper bound for the neck growth rate is given, by the arguments of Kingery and Berg (1955), by ~:=AP

H

where AP is the difference between the equilibrium vapour pressure over the sink and that at the source. We shall assume this to be given by kTIn

1+

= (?~Ktf~ - A ~ ) .

For small AP, AP =

PvT, f~Kt kT

{l KtDT, A/~:_~J

where Pv is the equilibrium vapour pressure over a flat, particle-free surface. The rate equation for neck growth then becomes /c = P~K, ~

\2FIAokTJ

_l

Ktn?J

which reduces to the equation of Kingery and Berg (1955), when AF~ is zero.

DISPERSION

OF P A R T I C L E S O N M E T A L P O W D E R S

AND WIRES

17

The densifving mechanisms of Stage 1 and 2 remove matter uniformly from the boundary plane. When grain boundary d!ffusion is the rate controlling process, the problem becomes particularly simple: we require a solution to the governing equation d2b/ dz 2 = constant

subject to the boundary conditions p = Klft),~

d/a --=0 dz

at

Z=x;

at

;~ = 0;

at

Z = 0symmetry.

The desired solution is

pxk T

1

}

where K = Ks in Stage 1 and K3 in Stage 2. Dislocations are also made less effective as sources by a dispersion of particles. If we assume that dislocation segments bow-out by climb when they contribute to sintering, then the effect of particles is to shorten the dislocation segments which can do this, reducing the amount of matter each can release before its curvature prevents further climb. A straightforward reworking of the calculation given by Ashby (1974) then gives

FI Nx2Dv~r'~K2 (1 = T pkT _

3Gx "~ 212a---~7,K2J

where l is the average length of a dislocation segment, given by

?=

N

N is the dislocation density and G the shear modulus. The equations used to compute maps for an aggregate of wires are summarized in Table 3.1. 3.2.4. Rate equations for the sintering of an aggregate of spheres As pointed out in the last section, when potential gradients are linear, or almost so, the effect of an interracial potential barrier is simply to reduce the rate of a given mechanism of sintering by a factor

where A~uiand K are the potential barrier and the curvature difference appropriate to the mechanism. For the first two mechanisms of Fig. 1, surface and lattice diffusion from the surface, the gradients are almost linear, and we are justified in using such a correction factor. Although mechanism 3, evaporation and condensation, is not diffusion controlled, the correction factor J.p.M.S 25 |--B

18

PROGRESS

IN MATERIALS

SCIENCE

e.

..s

O~

m 0 ¢.. .,s

e-

=

e,

"

m

._.q .=

ca ._=

o= t.. 2

o

N

N

<

u~

li m

O

o~

1.

•<

th)

e~

o

ca ~¢"4

m

.-.q ¢,t}

~-I, ~

~

~

II

II

II

II

cO O es

~=

e-

eQ

o=

oo

P,

o

E -"

o° oS

~

e, ~ •

~1

~"

k.

0 ¢¢

o

..O

LD

tad J

Su!,(J!suap-uo N

I aSelS

II

D I S P E R S I O N OF P A R T I C L E S ON M E T A L P O W D E R S A N D W I R E S

19

is the same. For the dens(/)'in.q mechanisms (4 and 5), the gradients are not linear, and without rederiving the results from first principles, including the effect of a threshold, it is not obvious that such a correction factor is adequate. When, however, this rederivation is performed by solving (as before) V2~ = constant with appropriate boundary conditions, we find that the result is so close to that obtained by applying the correction factor that, at the low level of approximation with which we are concerned, it is adequate. The dislocation mechanism is modified as in Section 3.3. Because the geometry is different, the constant differs from that of the earlier section. The resulting modified rate equations are listed in Table 3.2.

4. RESULTS OF THE MODEL

4.1. Overview This section describes the predicted effects of a dispersion of particles on the sintering of silver. The results are displayed in two ways: as curves of neck growth against time at a fixed temperature (Fig. 6); and as sintering maps (Figs 7-10) which identify the dominant mechanism of sintering, and show how its rate varies with both time and temperature. In general, a dispersion of particles inhibits sintering, the inhibition becoming greater as the volume fraction increases, and as the particle size decreases. Densifying mechanisms tend to be more strongly inhibited than those that do not densify, so that the dispersion can change the dominant mechanism of sintering. A crude measure of the effectiveness of the dispersion in suppressing densification is given by the quantity fl = a \ / f d where a is the powder-granule radius, f the volume fraction of particles and d their diameter. When fl < 1, inhibition is negligible; when 10 < fl < 100, it is marked, and when fl > 100 it is very strong. When fl > 100 a small neck may form between powder granules, but densification is almost completely suppressed unless a pressure is applied to the compact. It must be remembered that the particle dispersion may coarsen with time, and that the relevant d is the current value of the particle size. The figures of this section are computed on the assumption that the particles are stable and do not grow. 4.2. Neck-Growth Vs Time The neck size at a given time is calculated by summing the rate equations for a given stage, and for a given geometry (wires, spheres etc.) and integrating the sum with respect to time. The result is a figure such as Fig. 6 for silver spheres of radius 10 -4 m. It is restricted to a single temperature--in this instance, 900°C--and contains less information than the sintering maps shown later in this section, but it lends itself to quantitative checks of the theory. This figure shows the predicted influence of a volume fraction f of spherical particles of diameter d on the sintering of silver spheres of radius 10 -4 m. It was computed using the

20

PROGRESS

IN

MATERIALS

SCIENCE

m

e-

c~ e.

9

~o

m

e~

..~

._~

P

<

<

-d8 q g

il :~ I

r~

i¢'.4

..... "~

<

I

I -

i~ =-i~ ,el i ,~ ,~

I I I~

:¢='1

i +

I

II ,'-i

,,4

II

tl

il

•2

-2

.2

!l

II

]1

= 0

0

o

e.. 0

P. C) t-

.o_ Q

0

.4 J

~u!Kj!s~p-uoN

DISPERSION

OF P A R T I C L E S

1 ~

0

ON M E T A L P O W D E R S

TIME (HOURS) 100

10

i

I000

i

i

SILVER

10,000

,

FULL DENSITY REACHED~..,,.

TEMPF_RATURE 9(X)*C a • lO'*m

-.2

--~

i

21

AND WIRES

-.4-

f = O, ]O',ZlO,",alO"4

I

d • IO'~,IO"7, IO "Irn

'

PARTICLE'FREESILVER~,\ / ~ ~ , , ~

-.6--SI03 EONI#5._E/76 ~

~-.8-

~

,. . . . .

I

" IO"

IO"s_

IO

,o'" ":00]:

,o" I10::

-,o36 ;'

10"2

I0 "'z

100

10-2

10"e

1000

~

~-1.0-_J I

-1.~ f i

10 3

1

!

|

I iil~

I

1

104

10s

I0 s

I

I

I I I11

107

¢1 (m)

oqT d

t 01

TIME (SECONDS) FIG. 6. The influence of a dispcrsion of particles, of volume fraction f a n d diameter d, on the sintering of silver powder, of granule radius l0 -4 m, at 900°C.

equations of Table 3.2 and a program, the operation of which is described by Ashby (1974). It shows that particles have most effect on the later stages of sintering, and that the dispersion has little effect unless fl > 10. A plot of this sort is compared with experimental data in Section 5. 4.3. Sintering Diagrams Consider a two dimensional space with neck-size and temperature as coordinates (Fig. 7). It is convenient to use as axes the normalized neck-size x/a and homologous temperature T/TM, where Tu is the melting temperature of the material. The construction of a diagram involves two stages. We first ask: in what field of neck size~temperature space is a given mechanism dominant--that is, where does it contribute more to the neck growth-rate than any other single mechanism? The boundaries of these fields are obtained by equating pairs of rate equations and solving for neck-size as a function of temperature. At field boundaries (shown as heavy lines on Fig. 7) two mechanisms contribute equally to the sintering rate. A heavy horizontal line marks the transition from Stage 1 to Stages 2 and 3: the mechanism does not change here, though the equation used to describe it does (Section 3). Above this line the rate of sintering increases with neck size: it might not do so if the pores contained trapped gas, but we have assumed that they do not. Superimposed on the fields are contours of constant time. The neck growth-rate is the sum of the contributions due to each of the mechanisms listed in Section 3--with one restriction.

22

P R O G R E S S IN M A T E R I A L S S C I E N C E

TEMPERATURE 400

/_/

1

1

.I

11

~DIFFUSIONI FROM THE / ~ BOUNDARY .., ~ , ~ ~ , /

DIFFUSION

DIFFUSION

800

tt.arrcs O FF. / • III

BOUNDARY DIFF.

BOUND/~Y'

/ io seooN.

I¢1 (J

I

o

~/

600

4O0

BOUNIDARY

.IOI / ' 10~' / "

i

(*C)

8O0

6OO

~7 ,¢"~

/

/

,o.2/L/r /

RFACE

pFFUSION AO.Es

-1.5

s /

o

SILVER ,n = | O ' l m

~/~0

"2.0 0.4

0.8

f

/

¢"

/

t

o ~ lO'4m

l O,8

CI:MPACT $103 E0N#'1 5/7E 0.8 LO 0.4 0.6 HOMOLOGOUSTEMPERATURE T/TM

COMPACT

~03 EON•1

5/76

0.8

FIG. 7. Sintering diagrams for pure silver powders of two sizes. The left-hand diagram describes powder granules of radius 10 -6 m; the righthand diagram describes granules of radius 10-" m.

Two rate-equations are listed for each of the two densifying mechanisms: one pair is appropriate in Stage 1, the other in Stages 2/3. These pairs are treated as alternative, not additive, processes (for obvious reasons): the faster pair is added into the total rate; the slower pair is omitted from the sum. Contours of constant time are computed from an integral of the sum of the rateequations with respect to time. The diagrams are constructed by numerical computation. The data used to do this is listed in Table 4.1. Figure 7 shows maps for pure silver with two granule-sizes: 10-6m and 10-4m. The fields and their characteristics have been described elsewhere (Ashby, 1974). Figure 8 shows the influence on Fig. 7 of 1% of inert particles of diameter d = 10 -7 m. The sintering of the smaller granules for which / / = 1, is scarcely affected because the driving force for sintering is high. But that of the larger granules, for which fl = 100, has been suppressed. Densification ceases at a neck size x/a of 0.14, and although neck growth without densification continues beyond this point, it too ceases when x/a = 0.16. Figure 9 shows the influence of particle size. Both diagrams describe the effect of l%of dispersoid on the sintering of 1 0 - 4 m silver spheres, but in the left-hand diagram the particle diameter d is 10 -s m (fl = 1000) while in the right hand diagram it is 10 -6 m

1.0

DISPERSION

OF P A R T I C L E S

ON M E T A L P O W D E R S

23

AND WIRES

TEMPERATURE ("C) 400

600

800

400

,

~

800

m|

1,,'1 Y ;";

600

0

-.5

1oEOONOS

{1: w _J

I

LATTICE DIFFUSION

FROMTHE SL~ACE

L A T T I C E DIFFUSION

FROM THE ~

u_ P" iv,

BOUNQ ) A R Y

DIFFUSION -, .o

:3 r", n,"

,

r DIFFUSION

/_DIFFUSION/

,

Z

[

ADHESION,

-I.5

q * • I0"* ,.

•

, ..o, a

W03 0.6

/

• 10"I' m B~S1 0,8

5/71

I SlOS

-2 1.0

• |O'~m

f • .01 a " lO'?m 0.6

F.~ • ! 0.8

5/'/6

1.o

HOMOLOGOUSTEMPERATURET/TM FIG. 8. The influence of inert spherical particles (volume fraction f = 1%; diameter d = 10- ~m) on the sintering shown in Fig. 7. The dispersion has almost no effect on the fine powder, but suppresses sintering of the coarser one.

(fl = 10). The small particles have a profound effect on sintering, and (provided they do not coarsen) they largely suppress both densification and neck growth. Finally, Fig. 10 shows the effect of changing the volume fraction, while holding the particle and granule size constant. In b o t h diagrams (for which/~ = 36 and 10, respectively) there is some inhibition, though it is slight for the smaller volume fraction (10-2%).

4.4. The Position of the Cut-Off The effect of a dispersion is to impose a cut-off on each mechanism. Those which use the free surface as a source of matter are least affected. For these, sintering stops when K1 < - -

7~"

With A/I~ = 3ff~7~/d and KI ~ 2a/x 2, this condition becomes x

-_> 12

24

P R O G R E S S IN M A T E R I A L S S C I E N C E

TEMPERATURE ('C) 4OO I

0 SILVER

SILVER G = JO" 4 m

0 I ~0"4 R'I

f

f • .01

-.Of

Q=

¢1 • IO'lr m

5

SI03

EQNdH

]0"8 fit

5/i~

10 5 ,

/

"°" ,

0.6

/

I

0.8

1.0

0.6

I DIFFUSION

0.8

HOMOLOGOUSTEMPERATURE T/TM FIG. 9. The effect of dispersed particle size. For a fixed volume fraction f = 1%, small particles (left-hand diagram, d = 10-s m) have a much larger effect than do larger particles (right-hand diagram, d -- 10-* m). or, for particles to have any influence,

fa

t.

Those mechanisms which use the boundary as a source (and so cause densification) are more strongly influenced. This is because, as matter is drawn from the boundary, particles accumulate on it and the threshold potential rises. Because densifying and non-densifying mechanisms proceed simultaneously, and only those which densify cause this accumulation, no simple equation for the cut-off can be written; the effect is, however, properly included in the figures of this report. In general, the result is that mechanisms which densify are cut off sooner than those which do not, and thus that densification is suppressed more strongly than neck growth. As a rule of thumb, we find that, for particles to have a detectable influence a~/-~Ff >> 1. d 5. COMPARISON OF THEORY WITH EXPERIMENT In this section we compare the results of experimental studies on copper, silver and iron, each containing a dispersion of an oxide, with the theory of Sections 2 and 3.

1.0

DISPERSION

400 I

o

OF P A R T I C L E S

ON M E T A L P O W D E R S

TEMPERATURE 800

600

(*C) 400

s

~O'4m

o

•

f

•

.001

¢1

z

f a

• .0001 = 10 "I m

1 0 "? m

sxoa

5/76

10 -4m

~.1

~

NEX~K GROWTH "

BOUNDARY ~ / _ DI F F U S I O N ~

{

8( 0

SILVER

0

EONII

600

I

0

SILVER

SI03

25

AND WIRES

'

Ju

III II I

~

t s v

I0"

I

li ~ / v

I

j~ /

eOU.0ARV'I/ /DIFFUSION

j"

K)3T,

i/b///;/<,;/

'

-I.0

/

~~

v

~,loEcs

W Z

-! .5

,jr

DIFFUSION

Z

I

SURFACE DIFFUSION

/ 0.6

0.8

1~0

~A

0.6

i

O.8

1.O

HOMOLOGOUS TEMR RATURE T/TM

FIG. 10. The effectof volume fraction of dispersed particles. A volumefraction of 1~/o(Fig. 8, icft-hand side) has a large effect; 0.1% (this figure, left-hand side) has less, and 0.01~ (this figure right-hand side) has less still. But if the particles are stable, inert, and fine enough, even 0.01~ocan have a small effect on sintering.

5.1. The Experiments of Brett and Seigle (1966) on Copper Containing A1203 Brett and Seigle (1966) sintered wires of copper containing A12Oa, and observed that densification was strongly inhibited: pure copper wires showed a cylindrical void after 50 hr, and reached full density after about 300 hr at 1050°C; wires v~ith Al2Oa in them still showed a central void after 584 hr. In all cases the material in the neck was particle-free, indicating diffusive sintering. The data is analysed in Fig. 11 which was constructed from the data of Table 5.1 and the material properties of Table 4.1. The left hand side of the figure describes pure copper. It is in tolerable agreement with Brett and Seigle's observations at 1050°C in predicting full density after about 10 6 s e c (280 hr; they observed 250--400) and a transition from Stage 1 to Stage 2 sintering after about 4 x 105 sec ( l l 0 h r ; they observed 50 but the definition of this transition is not precise). The right hand side of the figure describes copper containing 1~ A1203. Although the alumina used to make the alloy had a particle size 2 x 10- 8 m, it is clear from Brett and Seigle's micrographs that coarsening or aggregation has occurred. Our measurements of their micrographs suggest a particle diameter of 6 x 10 -7 m, which we have used in

26

PROGRESS

IN. M A T E R I A L S

SCIENCE

L..

×

.

.

.

.

~

×

×

.

A A A A A A A A A A A

X X~

.

.

X

.

X

.

X

.

X

X.

.

X

~N~

A

A I/3

,T

T

X

X~

X

X

X

T

" X

X

x .

o

°_

"~

v ~ v v v m I

I

X

I

X~

X

X

I~

X~

X~

X~

-

~

X~

A A A A A

?,

,

,

x

X

~rkl

x

x~

--~

X

X

X

~ ~

X

~

~

~

.

. ~1"~

~

~ r ~ ~ .~"~

. .~. ~l~*dO~

_

.

~,~

-

,~,

~

~'-~ .

~.~

. ~

~

.~

~

~ . ~ . , ~

Z t~

~'~

- - ~

~

-

'

~

~'~

p ~

,,~'~,

.

. ~,, . ~/~

~-,.~

~

~

~

. ~ ,

~

',.~

~.~"

~-

.

.

~

'

~

m~'~---O ~

".:~

_ ~

~

.

~,.~

I=

~

~

~m

,.~ ~ ' ~ , .

_

8 g'.-

"~ ~ . I

~,-'~, ~ ~ . ~.~ ~.~.,'~-~ •

__

~. ~

ua,~-.

~

,..~

~ -~'~.

.-~ :! ~

~

"~

~

. . . . ~":~ ""~

~

~ .-=~ "

~

II

0

~

~ ~

~.--,,

r~ .

~.'.-~

DISPERSION

OF PARTICLES

ON METAL

TEMPERATURE 400

600

0

800

AND

600

400

I

i LATrcE ~FFU~ON I

.10

/ Lu

"7

RFACE

/

DIFFUSION

'

/'~,~

./t

.f

/ ' # "SURFACE ,,r /.,

/ /

-I .~ 0= 7 . 6 x 10"~ m

SI03

0.4

!000

.

-I .5

"2.0'

800

o

DIFFUSION

/

27

WIRES

(*C)

lO00

I

A

POWDERS

0.6

EON~'5

I

o • 7.6x

5/76

0.8

/ DIFFUSION

SI03 1.0

0.4

0.6

10"sin

EON#5

5/71

0.8

HOMOLOGOUS TEMPERATURET/TM FIG. 11. A

diagram for pure copper and for copper containing I% of AI:O3, corresponding to the experiments of Brett and Seigle (1966).

constructing the figure. The figure illustrates how the particles inhibit sintering, particularly those mechanisms which, by using the grain-boundary as a source of matter, cause densification. With the data we have used, the final neck size is predicted to be 0.32 in fair agreement with Brett and Seigle's observation that sintering stopped when x/a reached 0.36. Coarsening of the particles would, of course, permit further slow neck growth; but its rate would now be determined by the kinetics of particle coarsening, not by the mechanisms we have described here. 5.2. The Experiments of Lenel et al. (1970a) on Silver Containing AI203 Lenel et al (1970a) sintered silver wires containing 1% of A1203. Their data are analysed in Fig. 12 which is constructed from the data in Table 5.2 and from the material properties of Table 4.1. Table 5.1. Characteristics of the Copper and Copper-Al203 Wires

Material Wire radius, a ( m ) Volume fraction, f Particle diameter,

d(m)

Copper

Copper-Al203

7.62 x 10 - s ---

7.62 x 10 - s 0.01 6 x 10 -~

1.0

28

PROGRESS IN MATERIALS SCIENCE

TEMPERATURE (*C) 400

6O0

80O

0

[1

0

FULl I:~.N61TY R E ~ D I

II

,

j I

.

RE610NSTUDIED --LIMITING MB:IKSIZE

),"

OlFrUSlON . ( I

jor

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BOUNDARY OIFFUSION /

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10 106" •

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SZ03 EON#5 5/?6 SlOS EONB5 5/76 O~ 1JO O.6 0.8 1.(3 HOMOLOGOUS TEMPERATURE T/T M FIG. 12. A diagram for pure silver and for silver containing 1,% A1203, corresponding to the O.6

experiments of Lend et al. (1970a).

The figure shows how the dispersion suppresses neck growth when the neck radius, x, has reached about 0.3a, where a is the wire radius. The data of Lenei et al., shown as a shaded box on the figure, do not provide a critical test of the model, but they are consistent with it. The black squares show the observed neck size after 1 hr and 240 hr. The model predicts that, after 240 hr, the neck size is approaching its limiting value--provided, of course, that the Ai203 dispersion does not coarsen.

5.3. The Experiments of Lenel et al. (1970b) on Copper Containing SiO2 Lenel et al. (1970b) r e p o r t t h a t a d i s p e r s i o n o f S i O 2 i n h i b i t e d the s i n t e r i n g o f c o p p e r s p h e r e s o n t o plates of the s a m e material. T h e i r results a r e a n a l y s e d in Fig. 13 which was c o n s t r u c t e d f r o m the d a t a of T a b l e 5.3 a n d the m a t e r i a l p r o p e r t i e s o f T a b l e 4.1.

Table 5.2. Characteristics of the Silver and Siiver-Al20~ Wires Material Wire radius, a(m~ Volume fraction, f Particle diameter, d(m)

Silver

Siiver-Al_,O3

9 x 10 -5 ---

9 x I0 -~ 0.0 I 5 × 10-7

DISPERSION

OF P A R T I C L E S

ON M E T A L P O W D E R S

('c)

TEMPERATURE

__r-#90_

40O E~TAAI

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BOUNDARY A

to n.W ._1

DIFFUSIC)N

i

~ I/

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800

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600

800

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AND WIRES

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HOMOLOGOUS TEMPERATURE T / T M FIG. 13. A diagram for pure copper and for copper containing 1.2°o SiO2' corresponding to the experiments of L e n d er al. (197Obj.

The figure shows how the silica particles inhibit sintering, suppressing the densifying mechanisms (boundary diffusion, and lattice diffusion from the boundary) more strongly than the non-densifying one (surface diffusion) because of the way in which silica accumulates on the grain boundaries as sintering progresses. The temperature and neck sizes reported by Lenel et al., for a sintering time of 4 hr. are shown on the diagram. The time predicted by the figure is longer, by a factor of about five, than that reported by Lenel--probably because the surface diffusion coefficient we have used is too slow. But the figure clearly shows the reason for the smaller neck in the dispersion-containing material: it is because the lattice-diffusive flux has been slowed by the silica particles. The diagram predicts that the neck size observed by Lenel et al. in the Cu-SiO: was close to the final value at which--provided the silica particles did not coarsen or aggregate--sintering should stop completely. Table

5.3.

Characteristics of the Copper and Powders

Powder radius, a(m) Volume fraction, .f Particle diameter, d(m)

2.5 × 1 0 - S m ---

Copper-SiO2

2.5 × 1 0 - ~ m 0.012 1.37 x 10- ~

30

PROGRESS

-20O

0

200

IN M A T E R I A L S

SCIENCE

TEMPERATURE ('~;) 400 600 800

I000

12(:]0

1400

,¢

9 HOMOLOGOUS TEMPERATURE

T/T•

FIG. 14. A diagram for pure iron with a granule radius of 6 x 10-° m. The rates of sintering change sharply at the phase boundaries. The shaded box shows the area investigated by Singh and Houseman (1971}.

5.4. The Experiments of Singh and Houseman (1971)on Iron

Containing Oxide Dispersions Singh and Houseman (1971) sintered compacts of carbonyl iron, with and without additions of TiO2, AI2Oa and ZrO2, between 80OC and 1490 C, for times of between I and 300hr. These regions are shown as shaded blocks on the sintering diagrams shown in Figs. 14 and 15, which were constructed from the data of Table 5.4, and the material properties of Table 4.1. The diagram for pure iron shows three phase fields. Diffusive transport is faster in :t and 6 iron than in the close packed ),-iron, so there is a sudden change in the sintering rate at the phase boundary. The result is a sharp drop in sintering rate as the temperature is increased from 900 to 100OC, and a sharp increase clearly apparent in the data of Singh and Houseman, between 1350 and 1400'C. Interpreting the data from the iron/oxide compacts is difficult because the oxide was physically mixed with the iron powder (and thus is not uniformly distributed) and because the oxide tends to coarsen during sintering. Singh and Houseman found that all three oxides inhibited sintering, but that ZrO2 was the most effective. Figure 15 is constructed for

DISPERSION

OF P A R T I C L E S ON M E T A L P O W D E R S

31

AND WIRES

TEMPERATURE (PC) -2{30

0

200

400

600

800

1 0 0 ( 3 1200

1400

0

~-O.5 lad ,..I b.o-

~to

-1.0

ne

Z v

-I.5 O ._1

-2.0

0

O.2

O.4

0.6

0.8

HOMOLOGOUS TEMPERATURE

1.0

T/TM

FIG. 15. The effect of a volume fraction of 1.35% of ZrO2 on the sintering of iron. The shaded box shows the area studied by Singh and H o u s e m a n (1971).

this case, and shows that a limiting neck radius x/a = 0.33 is quickly reached, after which sintering stops. The limiting density can be calculated from this: it is 0.75. The green density of these compacts is about 0.75 so that the dispersion has largely prevented sintering. Singh and Houseman report that sintering for 2 hr between 1300 and 1490°C produced no densification in this alloy. 5.5. The Measurements of Bahk et al. (I 975) on Copper Containin# BeO Bahk et al. (1975) and Bahk (1976) sintered wires of copper with and without a fine dispersion of BeO (produced by internally oxidizing a Cu-Be alloy). Their data is shown in Fig. 16 together with computed lines showing the neck size that all the mechanisms deTable 5.4. Characteristics of the Iron and I r o n - Z r O 2 Powders

Material

Carbonyl Iron

Iron-ZrO2

Powder radius, arm) Volume fraction, f Particle diameter, d(m)

6 × 10 -6 ---

6 × 10 -6 0.0135 4 x 10 - s

32

PROGRESS IN MATERIALS SCIENCE TIME (HOURS) 10 i

1 I

I ,I Q:

I00 A

1003 j

,,No , WIRES. o - 6 , 4 x ! 0 "

co,,,,-,,

m

•

-I.C --

-

i

-I.;

i

I

IOs

I

s,=

I I I111 10 4

I

J

I lillJl

i

I

f =9.3,, lo s d = S . Z ~

I I jlllj

I

105 TIME (SECONDS)

I

I il

ij

10s

i0 r

FIG. 16. The data of Bahk et al. (1975) compared with the rates predicted by the rate equations, for pure copper wires (radius 6.4 x 10-s m) and for similar wires containing 0.09~o and 0.93% BeO. scribed in the last section, operating together, will produce. The lines were c o m p u t e d f r o m the equations of T a b l e 3.1, the d a t a given in T a b l e 5.5 and the material properties of Table 4.1. T h e u p p e r m o s t plot shows d a t a for pure c o p p e r wires, packed in a close-packed way, and sintered at 1050°C. T h e full line t h r o u g h the d a t a is the result of evaluating the .equations for the sintering rate for wires, s u m m i n g them, and integrating over time. U p to a neck size ( x / a ) of 0.27 surface diffusion is the d o m i n a n t m e c h a n i s m ; beyond, lattice diffusion from the b o u n d a r y b e c o m e s m o r e important. But t h r o u g h o u t , there are i m p o r t a n t c o n t r i b u t i o n s from m o r e than one mechanism. T h e u p w a r d curvature of the curve b e c o m e s p r o n o u n c e d when Stage 2 sintering starts because as the pore shrinks, its c u r v a t u r e (and thus the rate of sintering) rises until the final full density is reached. The theory fits the d a t a (circles and crosses) well. Surface diffusion coefficients are only p o o r l y known, and the theory itself contains a p p r o x i m a t i o n s . This kind of agreement is p r o b a b l y as g o o d as one can ever expect for a sintering model. Table 5.5, Characteristics of the Copper-BoO Wires

Material Wire radius (m) Volume fraction, f Mean particle diameter before start of sintering Mean particle diameter after sintering

Copper

Copper-0.09°, BeO

Copper-0.93°0 BeO

6.4 x 10 -s --

6.4 x 10-s 9 x 10-"

6.4 x 10-s 9.3 x 10-3

---

8.2 x 10 -9

8.2

x

10 - 9

--

14 x 10-9

36

x

10 -9

DISPERSION OF PARTICLES ON METAL POWDERS AND WIRES

33

The triangles are data for the dispersion-hardened copper. The theory, evaluated using the BeO particle size at the start of sintering, predicts more inhibition than is in fact observed (thin broken lines). But when the coarsened particle size is used instead, the c o m p u t e d neck size (full heavy line) is much closer to that observed. 6. SUMMARY AND CONCLUSIONS Experimental observations of the inhibition or slowing of sintering by a dispersion of a second phase are reviewed. The wide range of dispersoids which will inhibit sintering suggest that it is its physical presence, rather than its chemical nature, which is important. The experiments suggest that particles inhibit both the mechanisms of sintering which cause densification and those which simply cause neck growth, but that the densifying mechanisms are m o r e profoundly affected. The effect of the dispersoid can be understood as one of suppressing of the efficiency of the sources and sinks between which matter flows during sintering. Well-wetted particles cause the free surface of the metal containing them to be less effective as a source of matter, because, as the surface recedes, particles are exposed and energy must be supplied to create their free surface. Particles on grain boundaries suppress its action as a source because they pin the b o u n d a r y dislocations, the cores of which are the sites from which atoms are detached. Lattice dislocations are suppressed as sources for a similar reason. The result is that particles introduce a threshold during force, below which sintering stops. Quantitative models for this threshold, and for its effect on the rate equations which describe neck growth, are developed and their predictions are illustrated and described. A detailed c o m p a r i s o n of theory and experiment is difficult: few experimenters report their work in sufficient detail to allow it. We have c o m p a r e d the observations of five separate investigations with the predictions of the model, and conclude that these predictions are, in all cases, broadly consistent with the observations. In one instance a quantitative check is possible: provided allowance is made for the observed coarsening of the dispersoid during sintering, the agreement appears to be good. REFERENCES ASHBY,M. F. (1969) Scripta MeL 3, 837. ASHBY,M. F. (1972) Surface Science, 31,498. ASHBY,M. F. 0974) Acta Met. 22, 275. BAHK, S., ASHBY,M. F., BEVK,J. and TURNBULL, D. (1975) Notre Dame Conf. on Sintering and Related Phenomena, Ed. G. C. Kuczynski. BANK,S. (1976) Ph.D. Thesis, Harvard University. BALLUFFI,R., ROSE,F, D. and SEIGLE,L. L. (1954) "Self-Diffusionof Metals and Associated Phenomena," Progress Report XL AEC Contract AAT {30-1)GEN 367, April 5. BRETT.J. and SEIGLE,L. L. (1966) Acta Met. 14, 575. EARLY,J., LENELF. V. and ANSELL,G. S, (1964) Trans. AIME 230, 1614. EASTERLING,K. A. and GFASINGER,G. H. (1972) Z. Metallk. 63, 237. ICH1NOSE,H. and KUCYZYNSKI,G. C. (1962) Acta Met. 10, 209. IMAI.Y. and MIYAZAKLT. (1967) Research Institute for Iron, Steel and Other Metals, 1285th Report, p. 61. JOHNSON,D. L. (1969) J. appl. Phys. 40, 192. JOHNSON,D. L. (1973) Scripta Met. 7, 1075. JOHNSON,D. L. (1975) Proceedings of 1975 Notre Dame Conference on Sintering and Related Phenomena, Ed. G. C. Kuczynski. KINGERY,W. D. and BERG,M. (1955) J. appL Phys. 26, 1205. KUCZYNSKLG. C. (1949) Trans. AIME 185, 169. KUCZYNSK1,G. C. and LAVENDAL,H. W. (1969) Internat. J. Powder Metallurgy 5, 19. LENEL,F. V., ANSELt.,G. S. and BARROW,D. D. (1970a) Met. Trans. 1, 1772. LENEL.F. V., ANSELL,G. S. and MORRIS,R. C. (1970b) Met. Trans. 1, 2351. ,

J.PM.S: 2 5 , 1 - - c

34

P R O G R E S S IN M A T E R I A L S S C I E N C E

NOLFI, F. V., JR., SHEWMON,P. G. and FOSTER,J. S. (1969) Trans. Met. Soc. AIME 245° 1427. OROWAN, E. (1948) Discussion, Symposium on Internal Stresses, London Inst. Met. 451. PR^NATIS, A. L. and SEIGLE,L. L. (1956)"Fundamentals of Sintering-I, "Progress Report, AEC Contract AT (30-1) GEN 366, Oct. 5. SCHULTZ,J. M. (1972) J. Catalysis 27, 64. SEIGLE, L. L. and BRL~r, J. J. (1965) Sintering and Related Phenomena, Kuczynski, G. C.. Hooton, N. A. and Gibbon, C. F., editors; Gordon and Breach, New York, p. 273. SEIGLE, L. L. and PRANATlS,A. L. (1955) Metal Progress, Dec., p. 87. SINGH, B. N. and HOUIM^N, D. H. (1971) Powder Metallurgy International 3, 26. TIKKANEN, M. H. (1969) Phys. Sinrering 1, Jl. TIKKANEN, M. H., RO~LL, B. O. and WmERG, O. (1962) Powder Met. 10, 49. TmIC~NEN, M. H. and YLASAARI,S. (1969) Phys. Sint. I, Jl. TURNBULL D. (1953) Acta Met. 1, 684. WILKINSON, D. S. and Asl~Y, M. F. (1975) Pressure Sintering Maps, Cambridge University Engineering Department Report, June. WILSON, T. L. and StIEWMON, P. G. (1966) Trans. A I M E 236, 48. VERR^LL, R. A. and ASHBY, M. F. (1976) To be published.