Journal

J. Am. Ceram. Soc., 89 [7] 2311–2313 (2006)

DOI: 10.1111/j.1551-2916.2006.00983.x

r 2006 The American Ceramic Society

A Representative Volume Element for the Densification of Powders

Kumar Krishnan and William S. Slaughter

Department of Mechanical Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261

Richard J. McAfee and Ian Nettleshipw,*

Department of Materials Science and Engineering and University of Pittsburgh, Pittsburgh, Pennsylvania 15261

Previous studies have shown that the traditional phenomenolog-ical models of sintering contain pores that are generally toosmall and too numerous to represent the microstructural evolu-tion of alumina ceramics at temperatures below 14001C. In thisstudy a representative volume element was simulated by therandom placement of 1000 non-overlapping spheres, which wassubsequently densified by contact flattening. The resulting di-mensionless microstructural pathway based on the fraction ofsolid-solid surface showed very good agreement with experimen-tal results for alumina sintered at 13501C. Increasing the widthof the sphere size distribution for this simulation had little effecton the chosen dimensionless microstructural pathway.

I. Introduction

OVER the past 40 years, many experimental studies have beenconducted on the dependence of densification on particle

packing.1–10 Although few of these studies have directly quan-tified the effect at the microstructural scale, there are plenty ofinvestigations that have correlated the effect of green densityand powder processing variables with sintering behavior. Themodeling efforts have commonly focused on the traditionalphenomenological models of sintering that define densificationas contact flattening resulting in isotropic volume shrinkage.The initial stage models are usually based on ordered arrays ofspheres with specific green density11 or dense random packingfor which the coordination number changes in a prescribed wayas the solid volume fraction increases.12 Intermediate and finalstage models are based on ordered microstructures made up oftetrakiadecahedra with either cylindrical or spherical pores.13

Implicit in all of these models is the representation of averagebehavior by a volume element that contains only one grain orparticle. Under such a constraint, it is not possible to directlymodel the effect of particle packing behavior on densification.Furthermore, recent comparisons of the evolution of these mod-els with experimental microstructures have shown substantialdisagreement which has been interpreted as the effect of the ge-ometric assumptions in the models that result in pores that aretoo small relative to the grain size.14

In situ experiments on the sintering of particle rafts15 and two-dimensional simulations of contact flattening for randomly gen-erated arrays of circles16,17 have clearly showed the dependence ofmicrostructural evolution on particle packing. The main compli-cating issue for these models is their extension from rafts of par-ticles to three-dimensional arrangements of particles. Obviously,

the topological properties of two-dimensional and three-dimen-sional models will be different, thereby suggesting that effectsassociated with particle packing should be simulated in three di-mensions to arrive at meaningful phenomenological descriptionsof sintering. A three-dimensional simulation has been conductedwith a relatively large number of spheres.18 However, the outputswere designed to study the topological properties of densificationwhich are not directly measurable from sections. The objective ofthis study was to develop a three-dimensional model with dimen-sionless microstructural output parameters that are experimen-tally measurable for real ceramic microstructures.

II. Simulation Procedures

The simulation begins with the creation of a population ofspheres, with either uniform radii or a prescribed lognormaldistribution of radii. In this case 1000 spheres were used; thischoice was made to limit the computational time required foreach simulation. The spheres were then randomly placed one-at-a-time into a predefined virtual box. Sphere placements that re-sult in overlap were discarded. The spheres were then moved in amanner meant to simulate their movement under gravity, so asto minimize their potential energy. At this stage, spheres may bein contact, but they do not overlap. A representative volumeelement (RVE), smaller than the original virtual box, was chosenso as to avoid boundary effects. Some spheres were only partlywithin the RVE. The solid volume fraction is the sphere volumewithin the RVE divided by the RVE volume. The configurationwas not one of dense random packing, therefore the solid vol-ume fraction was on the order of 0.4–0.44.

Densification was modeled as contact flattening withoutrearrangement. At each step of the simulation, the sphere radiiwere increased by a common percentage and the resultant over-lapping volume discarded. For the purposes of modeling theevolution of dimensionless quantities, this is equivalent to a uni-form movement of the sphere centers towards the center of theRVE, with overlapping solid volume being uniformly redistrib-uted to the solid-vapor interfaces. At each step, the solid volumefraction, the solid–solid surface area per unit volume, and thesolid–vapor surface area per unit volume were calculated explic-itly. Each of these quantities is determined analytically, directlyfrom the three-dimensional RVE, whereas the experimentalquantities against which they are compared are determined viastereological techniques applied to two-dimensional sections.

The solid volume fraction was increased incrementally in thisfashion and the evolution of solid-solid surface area per unitvolume and solid–vapor surface area per unit volume was re-corded as a function of the solid volume fraction. Figure 1shows a partially densified RVE and a section through it thatillustrates the effect of random placement of the spheres on thepore structure. Many pore sections are larger than one wouldget in an ordered array of spheres. Figure 2 shows a typical‘‘grain,’’ with faceting, similar to randomly generated polyhedra,and grain edges formed by neck impingement.

2311

L. Klein—contributing editor

This work was financially supported of the National Science Foundation under awardCMS-9700062.

wAuthor to whom correspondence should be addressed. e-mail: nettles@pitt.edu*Member, American Ceramic Society.

Manuscript No. 10347. Approved January 25, 2006.

III. Experimental Procedure

The processing of the alumina samples and the microstructuralanalysis procedures are fully described elsewhere.19 In brief, alu-mina powder (Premalox, Alcoa, Pittsburgh, PA) with a medianparticle size of 0.3 mm was slip cast to give a green density ofapproximately 58%. The sample was pre-fired at 13501C for 0.1h and cut into sections before being heat treated for varioustimes at 13501C ranging from 0.1 to 60 h. Polished sections werethermal etched at 13251C for 18 min before SEM observation.

The volume fraction of solid was evaluated using the areafraction of solid measured from at least six processed SEM im-ages. The surface area to volume ratio of solid–solid (i.e., grainboundary) interface, SSS

V , and solid–vapor (i.e., pore boundary)interface, SSV

V , were measured using intercepts and the surfacearea fraction was calculated for the experiments as follows

SSSS ¼

SSSV

SSSV þ SSV

V

(1)

Finally the cross-sectional areas of at least 1300 grains weremeasured for each sintering condition. The results were unfold-ed into three dimensions using the Schwartz–Saltikov techniquecorrected for non-spherical grains.20 The lognormal distributionfunction was the best fit in the majority of cases and so this waschosen to characterize the data. The standard deviation was ap-proximately 1.25 for solid volume fractions below 0.9. There-fore, this value was use to model the lognormal grain sizedistribution in the simulations described above.

IV. Results

Predictions of the surface area fraction were made for the phen-omenological models of sintering in the range of solid volumefractions for which the geometries were valid.14 These modelsincluded face centered cubic (fcc) packing of spheres, the inter-mediate stage model with cylindrical pore channels on allthe edges of tetrakaidecahedra, and the final stage modelwith spherical pores on all the apexes of tetrakaidecahedra.Figure 3 compares their predictions of SSS

S with those from thepreviously described simulation and experimental results foralumina fired at 13501C. All of the traditional phenomenolog-ical models gave lower values of SSS

S than the experiments at thesame solid volume fraction. For example, at a solid volumefraction of 0.85, the fcc model, the intermediate stage model,and the final stage model predicted solid surface area fractionsof 0.35, 0.33, and 0.38, respectively. The experimental result forthe same solid volume fraction was 0.53, which is in very goodagreement with the simulation using equisized spheres, whichpredicted a value of 0.56. The systematic under prediction of SSS

Sby the traditional phenomenological models of sintering wasobserved throughout the range of solid volume fraction studied.

The calculations of SSSS were also carried out for a simulation

with a lognormal distribution of sphere sizes. The standard de-viation of 1.25 was chosen to simulate the distribution in size ofthe alumina grains used in the experiments. Figure 4 shows acomparison of SSS

S for the simulations with equisized spheres

Fig 1. (a) A representative volume element made of 1000 spheres and(b) a section through the volume element showing the heterogeneouspacking.

Fig. 2. A grain taken from the simulation showing the random facetingand the grain edges developed by neck impingement.

0.00

0.25

0.50

0.75

1.00

0.7 0.8 0.9 1.0

So

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olid

Su

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n Alumina 1350°CFinal Stage ModelIntermediate Stage ModelFCC ModelSimulation: Equi-sized

Solid Volume Fraction

Fig. 3. Comparison of the various model predictions with those of thesimulation and the experiment results for alumina fired at 13501C.

2312 Communications of the American Ceramic Society Vol. 89, No. 7

and lognormally distributed sphere sizes over the full range ofsolid volume fraction of the simulations. In both cases, the sur-face area fraction increased smoothly. There appears to be com-paratively little difference in the results of the two simulations atthe same solid volume fraction when a standard deviation of1.25 was used for the lognormal distribution. The other notablecharacteristic of the simulations is the initial packing fractionproduced by packing the spheres in the box before the densifi-cation algorithm was applied. The log-normally distributedspheres gave 0.4 while equi-sized spheres resulted in 0.44.

V. Discussion

In the simulations the spheres are randomly placed in the vol-ume so they do not overlap before they are rearranged to im-prove the coordination number. This naturally results in looserandom packing that increases the pore channel size relative toan ordered arrangement. Consequently, the densification bycontact flattening results in a comparatively high solid-solid sur-face area fraction at any particular value of solid volume frac-tion. Despite the low starting solid volume fraction in thesimulation, there is very good agreement with the experimentalresults over the range of solid volume fraction for which thecomparison was made, certainly an improvement over the tra-ditional models.

The results of the simulations also showed that there is verylittle effect of the size distribution on the solid–solid surfacepathway. This agrees with an experimental study in which themixing of alumina powders designed to widen the particle sizedistribution caused an increase in green density, profoundlyaffected the sintering kinetics but had little effect on the dimen-sionless microstructural pathway at 13751C.21 The increase inthe width of the particle size distribution in the simulationreduced the packing fraction from 0.44 to 0.4 which contrastswith the expectation that widening the distribution wouldincrease the packing fraction if the smaller particles were ableto fit into the interstices between the larger particles. It may bethat a larger standard deviation is required to see the expectedvariation in the packing simulations. The packing fractions forthe simulations are low compared with the experimental greendensity of 58% for the slip cast alumina. This is thought to bedue to the way in which the spheres were placed in the volume

and may explain why the chosen particle size distribution hadlittle effect. More efficient packing can be achieved in simula-tions by randomly placing the spheres in the volume followed byrearrangements, which remove the overlaps. Although this ismore computationally intensive, packing fractions of 0.64 havebeen achieved by this technique.22,23 The effect of the initialpacking algorithms on the microstructural pathway in densifi-cation remains to be studied. Nonetheless, the use of three-di-mensional RVEs based on the packing of a large number ofspheres represents a promising method for directly exploring theeffect of particle packing on the evolution of sintered micro-structures, especially those aspects that are difficult to probe ex-perimentally, such as the direct observation of particle packingin green microstructures. Representative volume simulationsalso avoid the traditional approach that requires the divisionof microstructural evolution into discrete stages and consequentgeometrical discontinuities at arbitrarily chosen values of solidvolume fraction.

References

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0.4 0.6 0.8 1.0

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n Simulation: LognormalSimulation: Equi-sized

Solid Volume Fraction

Fig. 4. Comparison of the predictions of the simulation with equisizedspheres and a lognormal distribution in sphere size.

July 2006 Communications of the American Ceramic Society 2313