Computer simulation of sintering in powder compacts

Computer simulation of sintering in powder compacts

0M)l-61~/82/010203-10~3.00M Copyright 0 1982 Per~amon Press Ltd COMPUTER SIMULATION OF SINTERING POWDER COMPACTS IN J. W. ROSSt, W. A. MILLER and ...

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0M)l-61~/82/010203-10~3.00M Copyright 0 1982 Per~amon Press Ltd




J. W. ROSSt, W. A. MILLER and G. C. WEATHERLY Department of Metallurgy and Materials Science, University of Toronto, Toronto, Canada MSS lA4 (Rereiwd 21 April 1981) Ahtract-A computer simulation procedure has been applied to study the dlccts of packing geometry on the sintering behaviour of two- and three-dimensional powder compacts comprised of randomly packed equivalent spheres. The shrinkage rate of the compacts was found to be very sensitive to the average particle coordination number; both two- and three-dimensional ‘dense’ compacts shrank in a homogeneous fashion, white their more loosely-packed counterparts shrank i~omo~~~sly and at a much slower rate for a given ‘two-sphere* behaviour. This was shown to be associated with the phenomenon of chain straightening in the toose packings. The grain structure that evolved during sintering was characterized by the Voronoi cell network associated with the particles, from which the geometric parameters describing the microstructure were determined. Rhwn&-Nous

avons Ctudii par simulation sur ordinateur fes dfets de la giomttrie du compactage sur It f&age de poudres b~im~si~~~ et t~djrn~onneii~ constituees de spheres tquivalentes tas&es de maniere aiiatoire. La vitesse de f&age est t&s sensible au nombrc de coordination moyen des particules; les ensembles “denses” bidimensionnels et tridimensionnels frittaient de man&e homogene, alors que ceux qui itaient moins denses frittaient de man&e h&Crogtne et plus lente, pour un comportement “a deux spheres” donnb Nous avons montre que ce fait est lit au phinomtne de rectification des chaines dam les ensembles peu denses. Au tours du f&age, l’evolution de la structure des grams Ctait cam&&e par un r&au de celluies de Voronoi, a partir duquel nous avons d&ermine les parametres g~rn~t~u~ qui permettent de d&ire la ~crostr~ture. Zutmmsz&aemtog-Der EinfluB der Packungsgeometrie auf das Sinterverhalten von zwei- und dreidimensionalen Pulvcrkompakten, die aus zurallig zusammengeschtitteten gleichen Kugeln bestehen, wurde mit einer Rechaersimulation untersucht. Die Schrumpfrate da Kompaktes hing empfindhcfi von der mittieren Teifchen-Koordinationszahl ab. Sowohl zweials such dreidimensionale Kompaktc schrumpften in homogener Weise, wohingegen deren eher bcker gepackte Gegenstticke inhomogen ~hr~~t~ und dazuhin mit weit geringmr G~h~~ig~t ah fiir ein gegebenes ‘Z~-Kuge~-V~h~t~. Dieses Verhahen hiingt zusammen mit dem Phanomen der Kettenstreckung in der lockeren Packung. Die wahrend des Sintems entstehende Komstruktur entsprach dem mit den Teilchen zusammenhangenden Voronoi-Zellnetzwerk, aus welchcm die geometrischen Parameter zur Beschreibung der Mikrostruktur ermittelt wurden.

1. INTRODUCl-TON The ultimate goal of any sintering theory is to predict the sintering behaviour of a powder compact, starting with only a knowledge of the initiai geometry of the compact and a ‘two-sphere’ model for neck growth and shrinkage. Such theories as exist are generally couched in terms of unrealistically simple systems, e.g. lines of wires or spheres or regular threedimensional packings. The irregular or random nature of the initial packing of a typical powder (which exists even in the case of equal-sized spheres) has been ignored in these models, with the result that the inhomogeneity in shrinkage that is associated with an irregular packing is not accounted for. That such effects are indeed important has been t Presently with Into Ontario.

Metals, (Canada), Sudbury,

shown experimentally for both two- and three-dimensional packings [l-5]; this topic has also been considered theoretically by Exner and Bross [6] using a 3-sphere model. The key observation in the previous work is that den&cation proceeds only in areas that are initially densely packed, while more .loosely packed regions tend to open up on sintering. These two opposing factors explain why the shrinkage of irregular compacts is often less than that predicted from a two-sphere sintering model. In two-&mensional packings, Petzow and Exner [5] have demonstrated the important role that chain straightening plays in this process, which they attributed to the development of asymmetric curvatures on opposite sides of the neck during sintering. The Exner and Bross [6] model accounts qu~titatively for the rate of chain straightening by this mechanism. Other situations leading to chain straightening (e.g. where a degree of constraint operates to prevent the free 203



motion of spheres along their line of centres) can be envisaged, and these are discussed further below. The behaviour of real compacts is further complicated by the formation during sintering of new interparticle contacts which will attempt to shrink faster than the original contacts. However, these new contacts will in general be constrained from shrinking at a faster rate by the surroun~ng necks, and a complicated set of a~omm~ation stresses will be set up at each neck. A preliminary consideration of these effects was undertaken by Eadie et af. [7J; they determined a relationship between the number of contacts per particle and the compact shrinkage for various ratios of local shrinkage rate to compact shrinkage rate. Swinkels and Ashby [S] have included the effect of a changing neck distribution in their sintering ‘maps’ by assuming that the number of contacts per particle and the local packing density increase with densification, althou~ their analysis starts with a regular rather than a random packing geometry. The present work extends the analysis of Eadie et al. [fl to both two- and three-dimensional packings, using a simulator that has three components: a twoparticle shrinkage model, a realistic means of characterizing the compact geometry, and a rule which determines the local sintering behaviour of a particle, taking account of the constraints imposed by its neighbours. These elements of the simulator are considered in the next section. 2 THR SINWRING


2.1 The two-sphere model For simplicity, we have chosen to model shrinkage and neck growth by assuming that grain boundary diffusion alone contributes to both. Since we are interested in effects which result from the topology of the packing this assumption is not crucial to any of the conclusions that can be drawn, and could be modified if necessary. For grain boundary diffusion, the shrinkage rate is given by [9] dL -=--dt




where L is the distance between particle centres, Db is the grain boundary diffusion coefficient, & is the width of the diffusion path at the grain boundary, y is the surface energy, n is the atomic volume, x is the neck radius, p is the radius of curvature at the neck root, and kT has its usual meaning. The system can be effectively normalized by setting the product of the non-geometrical constants to unity so that dL

-=&+l) dt



Shrinkage, neck radius and neck curvature can be related by applying the tangent+ircle neck geometry model That is. we assume the neck protile to be cir-

cular and that the material in the neck comes from the interpenetration volume of the two spheres, a fairly reasonable assumption for a grain boundary diffusion mechanism. This relation is best solved numerically; the solution may be conveniently expressed as an empirical equation relating shrinkage rate, L as a function of centre+entre separation L: i = L”exp(6.L

+ c)


where the constants a = -2.030, b = -0.9128, c = -0.1890 were evaluated by a Ieast squares regression analysis. 2.2 Packing geometry The starting geometries for both the two- and three-dimensional powders were based on random close packing of equal-sized incompressible spheres. Such structures are dense in the sense that they contain no holes large enough to a~omm~te additional spheres, and random in that there are only weak correlations between the sphere positions and no regions of crystal-like regularity. Remal [l&12] first suggested that these models could be used to simulate the structure of simple liquids. and more recently they have also been applied to metallic glasses [13,14-J. The geometry of random sphere packings is completely defined by a list of the coordinates of all of the sphere centres. The main ex~~mental source of such data has been from work with random packings of ball bearings. Scott [15] measured the coordinates of a random packing of about 1,000 ball bearings, while Finney [16] later repeated the experiment using about 8,000 ball bearings. In careful, independent studies, Scott and Kilgour [I 71 and Finney [16] have measured packing densities of 0.6366 f 0.0008 and 0.6366 + O.WO4, respectively. Scott [18] had previously constructed a ‘toose’ random packing having a density of 0.601, which is thought to be the lower limit for mechanical stability for a packing of equal spheres. The value 0.637 is generally accepted as the packing density of a random close packing of equal hard spheres. The average number of contacts per sphere or average coordination number is important in sintering. This must be at least 4 to ensure mechanical stability [19] and the average is thought to be about 6 for a dense random packing[ZOJ. This quantity is seldom reported in the literature, however, due to experimental un~rt~nty in defining a contact; instead the average number of geometric neighbours per sphere (usuaily about 14) is more often used. The geometric neighbours of a particular sphere are those which contribute a face to its Voronoi cell 1123. The Voronoi cell or polyhedron associated with a point in an array of discrete points is defined as that region of space (sometimes called the Dirichlet region) lying closer to the given point than to any other point in the array. The cell which surrounds a point is constructed by connecting the point with all of the other

J. W, ROSS et cd.: SINTERING IN POWDER COMPACTS points in the array, and perpendicularly bisecting each connecting line with a plane. The Voronoi cell is then the smallest polyhedron formed by these planes, including the given point. Since the points are uniquely associated with their ceils, the volume of a Voronoi cell provides a means of calculating the volume associated with a point, or in the case of sphere packings, for calculating the local packing density. A detailed study of the Voronoi cell topology for a particular packing is a means of quantitatively characterizing the packing geometry. Two planar packings and two thr~-dimensional packings were studied in this work (see Section 3). The planar packings were computer-generated, while the three-dimensional packings were derived from Scott’s [ 15J measurements using ball bearings.


Now we may define a set of new geometrically compatible shrinkage rates as /ij = lij (1 - do)


In general, the shrinkage rate of contact BC, iac determined in this way would not be the same if triangle BCD had been used as the basis. However, by repeating the calculation a number of times, first for triangle BCD using the value of is, just determined, and then again for triangle ABC using the latest i,, until1 i,, eventually converges to a constant, we have a set of compatible, compromise shrinkage rates for each of the five contacts in which the fast-shrinking necks are retarded by their slower-shrinking neighbours and vice versa. This procedure can be easily extended to large packings consisting of many triangles. With minor modifications, the same procedures were apphed to thr~-dime~ional packings consisting of As noted earlier, any particular neck in a compact tetrahedral rather than triangular units. containing a spectrum of neck sizes is not free to The determination of a set of compatible shrinkage shrink at the rate it would assume in isolation, but it rates for a packing is the first step in the simulation. must be constrained in the sense that its shrinkage Based on these shrinkage rates, a projected length of rate is accommodated with the shrinkage of neigheach contact is calculated at some short time in the bouring necks. For example, Fig. I shows a small unit future. Then each pair of spheres is considered in turn of a tw~m~sional random packing after some by moving them along the line joining their centres shrinkage, where neck BC will attempt to shrink until this projected separation is realized. Since each faster than AB, which in turn attempts to shrink faster sphere has several contacts, this process must be than AC. A compromise rate must 6e achieved by repeated many times until all of the contacts have each neck. The device of dividing the packing into the appropriate lengths. At this point, one cycle of the triangles illustrated in Fig. 1 is proposed as a means simulation is complete, and the next is initiated by of effecting these compromises locally. calculating a new set of compatible shrinkage rates, Suppose necks AB, AC, BC, BD and CD in Fig 1. etc. have lengths, lABII*,-, I,,,,-,Iso and Icu respectively, and No special provision is made for new contacts which occur during a cycle, but they are incorporated that the shrinkage rates of these contacts are iAt.,,I,,, &c, I,, and /c,. as determined by Equation (3). If we at the start of the next cycle. It was found by careful assume that triangles ABC and BCD remain geoexamination of the simulator that contacts which metrically similar, we may. for triangle ABC, define a form during a cycle do not separate, but remain relative decrease in length for each contact touching until the end of the cycle when they are ‘noticed’ by the program and incorporated into the dij = (/,j - iij)lij (ii = AB, AC or BC) (4) contact list. Time periods were kept very short which The average decrease in contact length is therefore typically restricted the shrinkage to lie between 0.0025 and 0.01 sphere diameters per cycle. This minimizes do = id,, + dAc + d,,).P CT) the number of new contacts formed in a particular cycle, and ensures that the packing changes very little from one cycle to the next. After each cycle was completed, a tabulation was made of the number of contacts per sphere. the contact lengths and other geometrical information derived from the Voronoi cell array such as bulk packing geometry, local packing density, and the number of geometric neighbours of each particle. The simulation was run until an average interparticle shrinkage of 8% was reached. Beyond this stage, the structure of the sinter body is essentially fixed and a discrete particle simulator is no longer required. 3. TWO-DIMENSIONAL Fig. 1. A small unit of four spheres in a two-dimensional

random packing after some shrinkage; note the ‘accommodation’ triangles ABC and BCD as discussed in the text.


Two planar arrays of spheres were studied corresponding to random loose and random close packing


Fig. 2. Sintering behaviour of a two-dimensional random close packing of qual-sized spheres after (a) Oy6 (b) 4%

and (c) 8% average interparticle shrinkage. The relative motion of each sphere is indicated in (d). respectively. The sphere coordinates for each were generated numerically. The dense packing was generated using an algorithm suggested by Ross [21]. This involves removing spheres at random from a regular planar array and then relaxing the remaining spheres into the voids. The loose packing was generated by a sequential addition algorithm which operated by choosing a random angle in turn from each of the four quadrants in a plane. A sphere was than translated along the vector defined by this angle from infinity toward the origin, until it contacted the origin or a previously located sphere.



The two packings depicted in Figs 2 and 3 each consist of 100 spheres. The packing densities, based on the Voronoi cell areas of the 60 spheres closest to the packing centre are 0.824 and 0.598 (cf. 0.785 for square.packing and 0.907 for planar closest packing). The positions of ten pairs of diametrically opposed spheres located near the packing perimeter were accurately located to monitor the dimensions of the array. Starting with the geometries illustrated in Figs 2 and 3, the arrays were allowed to sinter until the average contact had shrunk by 8Y, (progressive changes in the two configurations are shown for 2, 4 and 8% average interparticle shrinkages). Figure 4, which is a plot of the compact shrinkage vs average particle shrinkage, illustrates the essential difference in sintering behaviour between the two packings. It is evident that the dense packing shrinks in a nearly homogeneous or uniform manner, as represented by the 45” straight line, while the loose packing experiences considerably less overall shrinkage for the same degree of interparticle shrinkage. The reason why the loose packing undergoes relatively little overall shrinkage may be seen in Fig. 3. As sintering proceeds, irregular chains of particles tend to straighten. In this manner, interparticle shrinkage is accommodated, but compact shrinkage is inhibited. The behaviour of the two planar arrays is consistent with the experiments of Exner and his coworkers [4,5]. The variation of compact shrinkage with time is illustrated in Fig. 5. The solid line represents the shrinkage behaviour predicted by the two-sphere model [Equation (2)]. Again the strong dependence of shrinkage on packing density is observed. En the final stages, the dense array has undergo& more shrinkage

cl Dsnro army

* Laawarmy


. . .

, . . . . *. . . . .. . \ . . .. I



Fig. 3. Sintering behaviour of a two-dimensional random loose packing of equal-sized spheres after (a) O”/, (b) 4% and (c) 8’5; average interparticle shrinkage. Note the behaviour of irregular chains of particles which tend to straighten during sintering, leading to (d) little overall shrinkage. [Compare Figs 2(d) and 3(d)]


Average interperticlo

shrinkage (%I

Fig. 4. Compact shrinkage vs the average interparticle shrinkage for two-dimensional close and loose random packings.

J. W. ROSS cr ul.:





algorithm, an operation which resulted in an unusually high density of 0.664. This will be referred to as the dense packing. Scott’s original coordinates [15] contained a number of transcription and measure-

8 -I 5 -12


I -10

I -9

I -6

1 -7

I -6

LN (fimo)

Fig. 5. The variation of compact shrinkage with sintering time for the two-dimensional close and loose packings.

than that predicted by the two-sphere model. This efrect is due to new contact formation, which induces compressive stresses in neighbouring necks, accelerating the rate of shrinkage there. The chain straightening effect depicted in Fig. 3 is caused by allowing the centres of both spheres to move during shrinkage. One might argue that for loose chains [such as are seen in Fig. 3(a)] where only one end of the chain is coordinated to denser clumps of particles, there is no need to invoke chain straightening. Shrinkage could be acomplished more effectively in this case by moving the centres of spheres sequentially along their line of centres by keeping the centre of one sphere fixed at the highly coordinated end of the chain. However, if both ends of a chain are coordinated to denser clumps of particles (a more likely situation in practice), the particles at either end of the chain will be constrained; in this case, chain straightening must occur as densification proceeds. Figure 6 illustrates the essential difference in-behaviour between these two situations.


4. THREE-DIMENSIONAL PACKINGS Two different three-dimensional packings were available for t.his study; the coordinates of both were derived from a larger ball bearing packing constructed by Scott [15]. Both packings consisted of about 250 spheres, with the 100 centralmost spheres being used to generate statistics. As with the planar arrays, the positions of pairs of spheres on opposite sides of the packings were noted in order to monitor the dimensions of the array. Although the packings originally had the same density of 0.637, they were subjected to different treatments. The first had been used to construct an atomic

model of a liquid [21,22]. In its final form, it had been subjected to a hard-sphere-gas compression

0)) Fig. 6. Sintering behaviour of irregular chains of particles coordinated to denser clumps of particles (a) at one end of the chain, and (h) at both ends of the chain. In (a), particles in the chain shrink sequentially along their lines of centres such that no chain straightening occurs while in (h) the particles are constrained at each end so that chain straightening must occur as densification proceeds. The effect of approximately 10% interparticle shrinkage is shown in each case.

J. W. ROSS er 4:




t_ @ILow


C.N. orroy

t High C.N. array





Awag 2









Numtw of contacts

Fig. 7. The distribution of the number of Contacts per particle for the three dimensional loose and close random packings prior to sintering. ment errors, which meant that many of the spheres overlapped by varying degrees. This was not suitable for a sintering simulation, and the second packing resulted from a series of operations which eliminate or minimized these overlaps [23]. This process actually increased the density of this packing slightly to 0.641. This packing will be identified as the ‘loose’ packing, ~though it is only 3% less dense than the first packing. Significant differences in the geometry of the two packings are revealed in Fig. 7 which shows the distribution of the average number of contacts per sphere in each case. It is evident that the spheres of the ‘loose’ packing have a much lower average coordination number and, in fact, many lack the necessary minimum of four contacts required to ensure the mechanical stability of a hard sphere packing. This is surprising in view of the fact that it is actually denser than the original packing of ball bearings. The low coordination number is a consequence of the overlap removal process which caused many spheres which were touching in the original packing to become slightly separated. The sintering simulator was run for both packings. The loose packing with its lower coordination was sintered to 87, interparticle shrinkage, but the dense packing could only be sintered to 4% interparticle shrinkage before the use of computer time became excessive. Although both packings have densities which are higher than the value for a random packing of equal spheres produced solely by mechanical mixing, they sinter in fundamentally different ways, with the loose three-dimensional packing behaving like the loose t#o~imensionai packing.

Fig. 8. Compact shrinkage vs average interparticle shrinkage for the three-dimensional random packings.

Figure 8 is a plot of compact shrinkage versus average interparticle shrinkage for the two packings, where the. 45” line again represents homogen~us shrinkage. The results for the dense packing lie above the homogeneous shrinkage rates because, as sintering proceeds, new contacts form. New contacts must be forming more rapidly in the dense packing than in the loose packing, and indeed this is shown to be the case. in Fig. 9 which is a plot of the average number of contacts vs the shrinkage. For the dense packing, the initial rate of new contact formation is rapid, but this subsequently decreases to a lower rate which may be described approximately by the equation dn/dp = 15, where n is the number of contacts per particle, and p is the packing density of the compact. Swinkeis and Ashby [8], as part of their recent model of the sintering of aggregates of spheres. suggested an empirical IO





ho / :


Low C.N. ormy

+ HiQh ,*-*

4-t-* 4--4-






Fig. 9. The change in the average number of contacts per particle as a function of sintering time for the three-dimensional random packings.

J. W. ROSS er ul.: SINTERING relationship between the initial density, pO, of various crystalline and random sphere packings and the final coordination number, nl, in the fully sintered material, viz. n,=





* lnital


+ 4%


30 -

16p, - 2

They further assumed that this equation could be applied to estimate the rate of new contact formation during sintering as follows: n=



where n and p are the instantaneous values of coordination number and density which develop as sintering proceeds. This approach leads to a value of dn,ldp = 16, which is only slightly greater than the value found for the dense packing in the present work. A further comparison with the Swinkel and Ashby equation may be obtained by noting values of n at equivalent densities: at the end of our simulation (after 47; of interparticle shrinkage in the dense packing), n = 9 and p = 0.755, while equation (7) yields a value of 10 at this packing density. As noted earlier, the Voronoi cell of any sphere in the packing can be used to determine the packing density in the neighbourhood of that sphere. This procedure has been used to follow the packing density distribution in the loose and dense packings, both before and after sintering (Fig. 10). The density distribution for the loose packing becomes quite spread out, i.e. some regions densify and others do not, while shrinkage occurs much more uniformly in the dense packing. These data illustrate the basic difference between the two packings. The dense packing is the threedimensional analogue of the planar array of Fig. 2, i.e. there are no regions of low density in it, and the overall shrinkage process is essentially homogeneous. The behaviour of the loose three-dimensional packing more closely resembles that of the loose planar array (Fig. 3), because the particles in both packings have low coordination numbers. It seems puzzling at first sight that two packings as similar as those used in this work should sinter so differently. The results may be explained however if the packing with the lower average coordination number actually consists of ‘chains’ of particles woven together, more tortuous and intertwined than those in a loose planar array, but having a similar effect on sintering. That is, sintering may have occurred by shrinkage and straightening of these particle chains, without substantial cross-linking occurring between them such as might have been expected due to their close proximity. The net result is that the individual particles approach one another, but the compact as a whole shrinks very little. The validity of this explanation depends upon the chains remaining distinct during sin&ring. To test this hypothesis, portions of the simulation were run with full printouts of intermediate particle positions and separations in order to determine if cross-linkage actually occurred during a A.MM,‘I--r;


0 055



I 070

I 0 75

I 000

I 0.85

Local pocking dwtrity 40 l




6 X ShitkOQS







0 60


Loco1 pocking d4nrity

Fig. 10. The change in the packing density distribution during sintering for the three-dimensional (a) random close packing, and (b) random loose packing. cycle, only to be followed by separation and therefore failure to detect the new contacts. When the intermediate printouts were inspected it was confirmed that all new contacts that formed were stable and that the low rate OFnew contact formation is a real effect. As a further test of this hypothesis, groups of particles were drawn from both packings and their positions plotted stereographically, both before and after the sintering simulation. Examination of such figures revealed that substantial changes occurred in the relative positions of the more loosely packed spheres without new contact formation. It was seen that the particles did indeed have a sufficient freedom of movement such that ‘chain’ straightening could occur without generating many new contacts. By contrast, little relative motion was observed between spheres during sintering of the dense packing. Shrinkage for this packing was accomplished by a nearly uniform contraction about the packing centre, a process which generates many new contacts. Whether the marked effect of particle coordination number on shrinkage rate predicted by the computer simulation would actually be observed experimentally is a matter of conjecture. Only careful measurements of shrinkage rate and average coordination number as


J: W. ROSS er ul.:




a function of the sintering time in a system where there is a good knowledge of the ‘two-sphere’ behaviour could resolve this question. Intuitively, one would think that the ‘dense’ packing more closely models the behaviour of a real powder compact during sintering (particularly so in light of the suuxss of the sintering map approach [8] in predicting neck size and density for compacts of spheres), although irregular shrinkage has been reported for three- as well as two-dimensional packings [l-5]. The low coordination numbers typical of the ‘loose’ packing studied in this work would certainly not be achievable in real packings of equal spheres, since the average coordination number was below the limit for a mechanically stable loose packing. However, one might expect local regions of loose packing to exist in commercial powders where the particles have a distribution of sizes and shapes. 5. APPLICATIONS INTERMEDIATE


During the intermediate stage of sintering, originally spherical particles become grains, and the interparticle porosity retreats to the grain edges forming interconnected channels. At the beginning of this stage, the particles may still be nearly spherical, and new contacts may still occur. but generally we may regard the grains as being fixed and shrinkage as occurring homogeneously. It is therefore unnecessary to use a discrete particle simulator and simpler models will suffice to describe this stage of sintering. Information from the discrete particle simulator may nevertheless be used to provide additional insight into the sintering process. The positions of the grains and their geometry will be closely related to the original positions of the spherical sinter particles and to their local packing geometry. To a good approximation, the microstructure of the later stages of sintering can be modelled by the Voronoi cell network associated with the sphere packing at the end of the first sintering stage simulation. since there is a one-to-one correspondence between the polyhedra in this network and the sphere centres. Stereo pairs for three typical Voronoi cells are shown in Fig. 11. The network of Voronoi cells may be intersected by a sectioning plane in various orientations and the intersections of this plane and the cell edges plotted. Four such traces are illustrated in Fig. 12, and the similarity to a typical post-sinter, equiaxed microstructure is apparent. To better understand the role of the Voronoi cell concept in describing sintered microstructures, consider some simple geometrical characteristics of a few different sphere packings, as shown in Table 1. F.c.c. packings exemplify the case of homogeneous shrinkage in which no new contacts form during sintering. Thus, the packing density, p, increases during sintering, but the coordination number, CNo, is constant, and furthermore is eoual to the number of eeometric

Fig. 11. Stereographic pairs of typical Voronoi cell structures developed after 4% interparticle shrinkage in the three-dimensional random close-packed compact.

Fig. 12. Sections through the three-dimensional Voronoi cell network in the random close-packed compact after 4%

J. W. ROSS et ul.:





Table 1. Geometrical characteristics of various sphere packings Packing




1. Homogeneous shrinkage (no new contacts form during sintering) 12 0.741 f.c.c. 2. Inhomogeneous shrinkage (new contacts form during sintering) 8 0.68 1 b.c.c. -6 0.637 r.c.p.

neighbours, GN,, (tbe number of faces on the Voronoi cell). The rhombic dodecahedral shape of the Voronoi cell does not change during sintering in spite of the fact that its volume decreases steadily as porosity is eliminated. Neglecting grain growth. the final grain structure is identical to the Voronoi cell network which in turn is completely determined by the starting geometry of the sphere packing. Body centered cubic packings provide a simple, although somewhat special, example of inhomogeneous shrinkage. As Table 1 indicates, new contacts form during sintering and shrinkage may be regarded as an operation in which CN increases from CN, to GNo. However, even in this example in which the number of contacts changes during sintering, the number of geometric neighbours and the shape of the Voronoi cell (in this case a tetrakaidecahedron) remain unchanged as in the previous example of the f.c.c. packing. The sintering of random close packings provides a better example of inhomogeneous behaviour, because in this case there exists a distribution of both initial Voronoi cell shapes and local packing densities, and because the increase in the average number of contacts during sintering occurs gradually instead of abruptly at a particular value of bulk shrinkage as in the b.c.c. packing. Even for random packings, however, the distribution of Voronoi cell shapes does not change markedly during sintering. For the dense packing which was sintered to 40/b interparticle shrinkage in this work, it was found that the average Voronoi polyhedron had 13.90 faces and 5.13 edges per face, not substantially different from the average Voronoi cell shape in the original packing. Local changes in individual cell shape probably did occur, although this effect was not monitored in the present work. The fact that the Voronoi cell structure is not basically changed by sintering leads to the following interpretation of sintering processes along the lines indicated in Table 1: (1) homogeneous shrinkage when CNO = GN, in which densification involves only the reduction of Voronoi cell volume, (2) inhomogeneous shrinkage when CNo < GN, in which densification involves both the accumulation of contacts (until CN = GN) and the reduction of Voronoi cell volume. The average characteristics

of the Voronoi


12 14 - 14

may be used in a statistical model of sintering or on the other hand. the evolution of the full Voronoi network may be followed in terms of a discrete particle model of sintering as suggested in the present study. To date. there have been no other discrete particle sintering models to compare with ours, but Coble [24] and Swinkels and Ashby [8] have proposed statistical models of the latter stages of sintering which assume that the grains are regular tetrakaidecahedra having 14 faces and an average of 5.14 edges per face, cf. the average Voronoi cell with 13.90 faces and 5.13 edges per face for the dense packing modelled in this study. The packing density calculated from the Voronoi cell volumes showed that porosity decreased from 33.63; in the unsintered, dense packing to 24.50/ after 4:/, mterparticle shrinkage. If it is assumed that the porosity is distributed as a cylinder of constant radius along the cell edges, the cylinder radius r may be calculated by measuring the total Voronoi cell edge length. For the case of the dense packing at 4”/; shrinkage, r = 0.208~. where a is the original sphere radius. Similarly, any microstrudural parameter of the sintered compact may be determined to a quite high degree of accuracy. The Voronoi cell network is also potentially useful as a base in modelling the final stage of sintering. At this point, the porosity is essentially located at the vertices of the Voronoi cells. By allowing the angles between the grain faces (initially Voronoi cell faces) to adjust to equilibrium 120’ angles under the influence of the grain boundary tensions, the grain growth process could be simulated. The ultimate sintered microstructure will be markedly different from the original Voronoi network from which it evolved. but the existence of a quantitative relationship between the two means that the former may be derived from the latter. 6. SUMMARY The simulator was first applied to two planar packings. The more densely packed array was found to shrink nearly homogeneously, but the loosely packed array exhibited much less overall shrinkage for the same amount of interparticle shrinkage. The inhibition of overall shrinkage in the looser packing was observed to be the result of a chain straightening mechanism. Two randomly packed threh-dimensional arrays were studied. It was found that the packing with the lower coordination number sintered in much the



same way as the loose planar array. The individual particles are not fully constrained in a loose packing,

and thus have considerable freedom of motion in relation to their neighbours. The result is that substantial interparticle shrinkage may occur without much overall compact shrinkage, The particles of the highly coordinated array on the other hand were constrained to such an extent by their neighbours that they could not move so freely, and shrinkage in this case was found to be very uniform. It was also found that this packing shrank at a slightly greater rate than that predicted by a two-sphere model, a consequence of new contact formation which induces compressive stresses in the packing, thereby accelerating the shrinkage rate. The shrinkage behaviour of a powder compact during sintering appears to be a very sensitive function of the local par.ticle coordination number. The Voronoi cell network associated with the particies in the packing was shown to be useful in modelling the grain structure of compacts in the fatter stages of sintering. This was also shown to be an alternate source of data for the geometric parameters which characterize such a structure. To a certain extent, this simulator must be regarded as an approximation to the actual sintering behaviour of a compact. Nevertheless, the basic premise of applying a discrete particle simulator to a random packing of spheres is sound. In particular, the technique is useful and should be applicable to any type of two-particle sintering model. The simulator itself is modular in nature and may be improved as new information becomes available. However, it seems evident that it is the powder geometry which is most important in determining compact sintering behaviour, rather than the nature of the sintering mechanism involved.



Acknowledgements-The authors gratefully acknowledge the provision of financial support for this work by the Natural Sciences and Engineering Research Council, Canada. J.W.R. also acknowledges the Steel Company of Canada and the National Research Council of Canada for support in the form of graduate fellowships.

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