Effect of particle size in aggregated and agglomerated ceramic powders

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Acta Materialia 58 (2010) 802–812 www.elsevier.com/locate/actamat

Effect of particle size in aggregated and agglomerated ceramic powders A. Balakrishnan a, P. Pizette a, C.L. Martin a,*, S.V. Joshi b, B.P. Saha b b

a Laboratoire SIMAP, GPM2 Grenoble-INP, UJF, CNRS 101 Rue de la Physique, BP46 38402 Saint Martin d’He`res Cedex, France International Advanced Research Centre for Powder Metallurgy and New Materials (ARCI), Balapur (PO), Hyderabad 500 005, Andhra Pradesh, India

Received 23 June 2009; received in revised form 21 September 2009; accepted 27 September 2009 Available online 21 October 2009

Abstract This work describes the compaction of agglomerated and aggregated ceramic powders with special emphasis on the role of primary particle size. Discrete element simulations are used to model weakly bonded agglomerates as well as strongly bonded aggregates. Crushing tests are carried out to obtain the characteristic strength of single agglomerate and aggregate. Microstructure evolution and stress– strain curves indicate that aggregates undergo a brittle to plastic-like transition as particle size decreases below 50 nm. It is shown that agglomerates made of nanoparticles exhibit much greater strength than those made of micron-sized particles, with an approximately inverse linear relationship with primary particle size. Simulation of the uniaxial compaction of a representative volume element of powder demonstrates that adhesive effects are responsible for the difficulty to compact nanopowders and for the heterogeneity of microstructure prior to sintering. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Ceramics; Powder consolidation; Molecular dynamics simulations; Aggregates; Agglomerates

1. Introduction In the last 30 years, the smallest achievable particle size in powders has steadily shifted from the micron to the nanometer scale. For such fine particles, adhesive forces play a dominant role in the handling, storage and forming of the powder. Depending on the synthesis temperature and on the time spent in the reactor, fine particles in soft agglomerates undergo sintering and form harder aggregates. In the context of the present study, agglomerates are defined as weakly bonded particles sticking together under van der Waals forces, whereas aggregates are considered to be constituted of particles which are bonded together by solid bridges (typically formed during the calcination process). Many examples of aggregated or agglomerated ceramic with submicron-sized primary particles can be found in the literature pertaining to very diverse synthesis routes [1–6]. *

Corresponding author. E-mail address: [email protected] (C.L. Martin).

Depending on the processing stage considered, agglomerates and aggregates may or may not be desirable. A certain amount of aggregation is useful to enhance the flowability of the powder, which is inherently poor for fine particles. However, the agglomerate and aggregate length scales should be eliminated during further processing to ensure green strength and acceptable sintering conditions. Indeed, if these agglomerates/aggregates are not disintegrated into smaller units during compaction, their size may determine the sintering and coarsening kinetics and the scale of the microstructural heterogeneities. Notwithstanding the above, elimination of agglomerates/ aggregates is not a simple task for submicron-sized powders [7]. For example, consolidation of ceramic powders is difficult in case of aggregates made of submicron crystallites, resulting in low green densities. The difficulty in compacting submicron powders is generally attributed both to the low tap density due to agglomeration phenomena and to the large interfacial area per unit volume for such powders [6,8]. In particular, it has been stated that “the total frictional resistance to compaction can be much greater for a nano-

1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2009.09.058


A. Balakrishnan et al. / Acta Materialia 58 (2010) 802–812

crystalline powder than for a powder composed of larger particles” [8]. To the authors’ knowledge, this point has not been unambiguously established although molecular dynamics simulations have shown that nanoparticles may exhibit specific frictional characteristics [9]. One of the most important properties of agglomerates and aggregates is their strength. However, determining the strength characteristics of agglomerates and aggregates made of fine particles is difficult. In this context, numerical simulations may provide information that is unattainable experimentally. Discrete element method (DEM) simulations are a natural tool for such a task since the deforming material is made of discrete particles. A number of DEMbased investigations have studied the breakage of agglomerates made of large primary particles, with a specific interest on the issue of impact, which is relevant for applications in pharmaceutical, detergent and food manufacturing industries [10–12]. For ceramic powders, DEM has been used for investigating force transmission during high-pressure compaction [13]. Some studies have also taken into account the fracture of agglomerates [14] or aggregates [15] for understanding the effect of microstructure on the macroscopic compaction behavior. Recently, MorenoAtanasio and co-workers [16,17] have used DEM simulations to generate realistic agglomerates and to obtain their yield strength. Their simulations included adhesive forces due to surface energy and were limited to the behavior of individual agglomerates. Gilabert et al. [18] have also introduced cohesive forces into two-dimensional DEM simulations to study the structure and the mechanical properties of a cohesive granular material. Although the above simulation studies have served to advance the understanding of agglomerate or aggregate behavior, a general relationship between the primary particle size and agglomerate or aggregate strength has not yet been established. The effect of the processing route which may lead to strongly bonded aggregates or weakly bonded agglomerates has also not been analyzed. Hence, the present study investigates the agglomerate and aggregate crushing behavior, as this is a common damage process during handling or compaction of porous objects. Building on this knowledge at the scale of one aggregate/agglomerate, a representative volume element (RVE) of the powder, which is made of several aggregates/agglomerates, has been constructed. The RVE is then compacted under uniaxial conditions to mimic closed-die compaction. The aim of these last simulations is to obtain results at the macroscopic length scale to illustrate the effect of primary particle size and to develop an improved understanding of the limitations that make submicron powders so difficult to compact. 2. Model description Aggregates and agglomerates have been modeled as three-dimensional random assemblies of spherical particles. The type of interaction that bonds particles together defines an agglomerate or an aggregate. In agglomerates,

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adhesion is the only force bonding particles together; in aggregates, particles are assumed to be bonded together by solid bridges formed during the high-temperature processing of the powder. Particles as small as 10 nm have been used in the simulations. For this size, it is important to ensure the validity of the contact law adopted in the model. First, it has been assumed that primary particles do not fracture, since such a phenomenon is only observed under nanogrinding conditions in which the mills provide for numerous collisions and very large local stresses [19]. Molecular dynamics simulations have been used to obtain the load–displacement curve for indentation of a 10 nm sphere [20,21]. It has been reported that the above curve is adequately described by a power-law with an exponent of 1.5, which is consistent with the Hertz theory [20]. Furthermore, it has been shown that a power-law exponent of 1.5 is also obtained for loads smaller than 0:3lN, and that the deformed contact profile is given by the Hertz theory [21]. Thus, it can be inferred from the above studies that, for spheres as small as 10 nm and with loads of the order of 0:01lN (typical of those obtained in our simulations), the Hertz theory is applicable. The latter theory is adjusted for adhesion by the Derjaguin–Muller–Toporov (DMT) model which works under the important assumption that deformation fields are given by the Hertz theory [22]. 2.1. Weakly bonded agglomerate The normal force, N e , between two weakly bonded spherical particles of radii R1 and R2 , characterized by their elastic properties (Young’s modulus E and Poisson’s ratio m), accounts for the adhesion effect (work of adhesion w ¼ 2c with c being the surface energy). N e is given by the DMT model [22], which is well suited for “small, hard” particles: N e ¼ N rep þ N DMT ¼

2 E a3e 2pwR ; 3 ð1 m2 Þ R

ð1Þ

R2 , while N rep and N DMT denote the repulsive where R ¼ RR11þR 2 and adhesive forces, respectively. In the DMT model, the Hertz theory leads to the contact radius ae :

a2e ¼ R he ;

ð2Þ

where he is the indentation between the two particles. Decohesion occurs in the DMT model for the pull-off force N DMT . Friction is included in the agglomerate model with a Hertz–Mindlin-type tangential force model in the sticking mode, while a Coulomb friction (friction coefficient l) limits the norm of the tangential force jT j during sliding. It should be noted that the Coulomb limit applies to the repulsive part of the normal force: jT j 6 lN rep :

ð3Þ

When N e ¼ 0 (equilibrium between elastic and adhesives terms), Eq. (3) implies that friction forces may be non-zero. This is important since it allows the frictional role of


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contacts for small adhesive particles to be taken into account. Prior work on shearing of two spherical nanoparticles has shown that there is indeed a shift in the Coulomb cone (see Fig. 5 of Ref. [9]). Experimental work on silica microspheres using friction force microscopy also points to the same result [23]. Finally, rotations are fully included in the simulations while no rotational resistance at the contact is introduced. 2.2. Strongly bonded aggregate For aggregates, a more elaborate model, involving two types of contacts, has been developed. It allows the description of: (i) a contact between two bonded particles that have formed a solid bridge and (ii) contact between two particles whose bond has broken, with the contact being resumed after fracture. The solid bridge is defined by its radius ab (see Fig. 1). The normal and tangential forces between bonded particles have been derived from the model and the finite-element calculations given by Jefferson et al. [24]. More details may be found in an earlier application of this model [15]. For two particles with an accumulated normal displacement uN (as compared to the initial stage of the bond shown in Fig. 1), the normal force may either be compressive ðuN > 0Þ or tensile ðuN < 0Þ: Nb ¼

2ER fN ða ; wÞa uN ; 1 m2

ð4Þ

where the function fN ða ; wÞ depends on the size of the bond ða ¼ ab =ð2R ÞÞ and w is a geometric factor which allows bond interaction to be taken into account. Here, w ¼ 0:37 has been taken based on the upper bound of the model of Jefferson et al. [24]. The tangential contact force, T b depends linearly on the accumulated tangential displacement at the contact, uT , as [24]: Tb ¼

4ER fT ða Þa uT ; ð2 mÞð1 þ mÞ

ð5Þ

where the function fT ða Þ depends only on the size of the bond a . For the typical value of the size of the bond used here ða ¼ 0:3Þ, it has been determined from the model of Jefferson et al., that fN ða ; wÞ ¼ 1:4 and fT ða Þ ¼ 1:1. The

strongly bonded contact transmits resisting moments, M N and M T , in the normal and tangential directions, respectively: MN ¼

8ER3 fT ða ÞhN ; ð2 mÞð1 þ mÞ

MT ¼

2ER3 fN ða ; wÞhT ; ð1 m2 Þ

ð6Þ

where hN and hT are the accumulated relative rotations along the normal and tangential axis of the contact, respectively. Approximating the solid bridge to a cylindrical beam of radius ab and using beam theory, the maximum tensile and shear stresses at the bond periphery may be evaluated as: Nb jM T j ; 2 2 2pa3 R3 4pa R jT b j jM N j þ : rT ¼ 4pa2 R2 4pa3 R3

rN ¼

ð7Þ

Thus, bond fracture may occur due to tensile, shear or bending deformation of the beam. It is assumed that fracture occurs whenever rN > rb or rT > rb , where rb is the fracture stress of the dense material forming the solid bridge. Two particles may resume contact after the original bonded contact has failed. We assume that this occurs when the interparticle distance is the same as that when the solid bridge failed (Fig. 1c). When the contact resumes, adhesive forces between the two particles have to be taken into account since a free surface has been created. Furthermore, the resumed contact behaves with the same normal stiffness in compression ðuN P 0Þ as a bonded contact with the adhesive term of Eq. (1) added to Eq. (4): 2ER fN ða ; wÞa uN þ N DMT : ð8Þ 1 m2 The two particles separate again for a pull-off force equal to N DMT . Thus, a broken solid bridge behaves asymmetrically in tension (as a weakly bonded contact) and compression (as a strongly bonded contact). Tangential force for a broken bond follow the Hertz–Mindlin law described previously for weakly bonded contacts N bb ¼

Fig. 1. The three stages of a solid bridge forming a strong bond: (a) initial status (unbroken); (b) broken bond; (c) resumed contact after fracture.



(Eq. (3)). Accordingly, the contact transmits a resisting moment in the tangential direction but none in the normal direction. This model allows the existence of a previous solid bridge to be taken into account in a very simplified manner. It is well suited for particles that resume their contact after a limited amount of relative rearrangement (Fig. 1). Clearly, this model is inaccurate if the two particles resume contact after a large relative rotation. The above equations have been incorporated into the DEM code dp3D. An explicit scheme has been used with dynamic resolution of the new position of each particle at each time step. Mass renormalization, together with very slow strain-rates, ensures quasi-static conditions. Details concerning the methodology adopted for the code dp3D can be found in a previously published work [25]. 2.3. Agglomerate and aggregate preparation The morphology of aggregates and agglomerates is determined in a very complex way by the synthesis route. Generally, the associated coagulation, sintering and coalescence phenomena lead to more or less compact structures [17,26]. For the purpose of this study, the morphology of aggregates and agglomerates has been greatly simplified by considering the shape to be spherical. This allows for a simpler analysis of the mechanical tests which are used to obtain aggregate and agglomerate strength. Agglomerates and aggregates have thus been generated by selecting a number of particles within a sphere randomly located in a sample made of 100,000 particles. This large sample itself has a randomly packed structure and was prepared using a procedure described in [27]. The packing fraction is 0.61, and particles have only point contacts in this packing. The primary particle size was varied from 10 nm to 1 lm in order to simulate a wide range of sizes. Quasi mono-sized particles were considered (with a small random deviation ð5%Þ). Aggregates and agglomerates cut from the large sample were made either of 500 or 5000 primary particles. The indentation, he , between primary particles in agglomerates was calculated to approach force equilibrium he in Eq. (1) ðN e ¼ 0Þ. Since the relative indentation 2R is sizedependent, the agglomerate internal relative density, D, is also slightly size-dependent. It varies from 0.61 for the largest particles ð1lmÞ to 0.65 for the smallest ones (10 nm). For aggregates, densification is stopped when the average normalized size of the sintered solid bridge, a ¼ ab =ðR Þ, is 0.3. This leads to an internal relative density of 0.65. Table 1 summarizes the material parameters used for the simulations. These parameters are typical of zirconia, mullite, yttria or uranium dioxide. The surface energy value is the most difficult parameter to ascertain since it depends very much on the surface properties. A value of 1 J m2 has been chosen to represent typical covalent/ionic ceramic materials for which surface energy ranges from 0.5 to 1:5 J m2 [28–30]. Concerning adhesive strength, it has been shown that care must be taken when using the

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Table 1 Material parameters Young’s Poisson’s Surface Friction Solid modulus coefficient energy coefficient bridge E m c fracture stress rb

Primary particle average size 2R

200 GPa 0.3

1 J m2 0–0.5

2 GPa

10–1000 nm

DMT model to ensure that the pull-off force jN DMT j does not exceed pR2 rth , where rth is the theoretical strength of the solid [31]. It was ensured that the above limit was avoided for the material and the particle sizes used in the present study. For a brittle ceramic material, the value of the solid bridge strength rb is strongly size-dependent. Both the mean fracture strength and the Weibull modulus increase when the size of the stressed material becomes submicronic. For such a size, the strength approaches the theoretical strength of the solid ðrb ! rth E=30Þ [32–34]. For instance, experiments by Namazu et al. [32] have revealed that when the diameter of a single-crystal silicon rod is reduced to the nanometer regime, not only is the measured strength above 10 GPa but also the value of the Weibull modulus exceeds 40. Hence, owing to the high Weibull modulus for the material size of interest ð2R 6 1lmÞ, the value of rb is considered to be the same for all bonds. Furthermore, the material that forms the solid bridge has been subjected to calcination or sintering. The strength of the submicron solid bridge should be smaller than the theoretical strength of the solid. Thus, rb ¼ E=100 was chosen, keeping in mind that this is a rough estimate of the strength of the solid bridge. In this context, it may be noted that the grain boundary tensile strength was chosen as 4.2 GPa ’ E=90 in a previous study by Warner and Molinari [35] to fit the macroscopic tensile strength of sintered alumina with a 500 nm grain size, thus supporting the above choice. 3. Stress and microstructure evolution during crushing In order to gain an insight into their fracture and deformation behavior, the agglomerates and aggregates were subjected to a crushing test by applying compression to the numerical sample with two platens. Stress was calculated from the total contact forces applied to the platens and the surface area of the spherical sample. A simplifying assumption that this area remains approximately constant all through the crushing test was made. The axial strain was calculated from the initial and current location of the platens during the simulation. For the sake of brevity, the discussion on microstructure evolution is restricted to aggregates/ agglomerates comprised of 5000 primary particles. 3.1. Agglomerate crushing Fig. 2 shows the typical evolution of an agglomerate made of 5000 particles (10 nm), while Fig. 3a depicts the


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Fig. 2. Section showing the microstructure evolution during crushing of a spherical agglomerate made of 5000 primary particles. Primary particle size is 10 nm. The arrow indicates the wedge opening (also shown in Fig.3b).

stress evolution under crushing conditions. Initially (Fig. 2a, and point I in Fig. 3a), only two particles are in contact with the platens. The agglomerate is then deformed plastically without significant indication of fracture (Fig. 2b). This corresponds to a continuous stress increase up to a maximum (point II in Fig. 3a). Subsequently, for all particle sizes, the stress decreases to a non-zero plateau-like value at a certain axial strain, with spikes that correspond to particle rearrangement. The stress decrease is characterized by a wedge-like opening in the microstructure as demonstrated in Figs. 2c and 3b. However, there is no evidence of complete brittle fracture with several distinct chunks for agglomerates. Fig. 3b shows the coordination number, Z, of each particle in the agglomerate, which allows for an easier detection of crack patterns (the crack surface is characterized by particles with smaller Z). We have observed that the wedge shown in Fig. 3b is unique for the strain interval tested here ðe < 0:2Þ. For agglomerates, this wedge pattern was detected for all particle sizes. While the maximum stress data will be statistically analyzed in Section 4, it is already clear from Fig. 3a that particle size plays an important role both on the magnitude of maximum stress and on the strain at which this maximum stress is attained. It is also to be noted that no significant qualitative differences were observed in the evolution of the microstructure and on the stress evolution during the crushing test for agglomerates made of 500 particles as compared to those made of 5000 particles.

3.2. Aggregate crushing Crushing tests of aggregates fall into two categories: brittle behavior and plastic-like behavior. While brittle behavior is observed for aggregates made of 100–1000 nm particles, plastic behavior is mostly seen in aggregates of 10–50 nm particles. These two behaviors may be distinguished based on the observed microstructural features, with a clear localized fracture being evident in the case of brittle behavior, while a much more diffused fracture pattern is seen for plastic behavior. The stress response during the crushing test also helps in distinguishing the brittle and plastic behaviors. Fig. 4a shows typical stress–strain curves for brittle aggregates. Starting from point (I), which depicts the initial condition, the stress increases continuously to reach a maximum at point (II). Fracture of bonds first occurs near the platen. The number of broken bonds progressively increases and a crack with a cone-like geometry develops inside the aggregate. At the end of the fracture test (point III in Fig. 4a), the brittle aggregate typically breaks into two or three chunks, as shown in Fig. 4b. Fig. 4a also shows the typical evolution of bond fracture events during the crushing test for a brittle aggregate made of 100 nm particles. It indicates that bonds break both in tensile and shear mode at the beginning of the crushing test, while the subsequent breaking of the aggregate in two or three chunks corresponds to the fracture in tension of a large number of bonds.

Fig. 3. (a) Typical stress–strain curves for agglomerates (made of 5000 particles) for two different sizes of particles. (b) Section of an agglomerate with primary particle size of 10 nm at 0.2 axial strain (loading axis normal to the section). Colors indicate the coordination number (Z) of particles.



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Fig. 4. (a) Typical stress–strain curve of a brittle aggregate made of 5000 particles of 100 nm size with corresponding profile showing number of bonds breaking in tension and shear (Eq. (7)). (b) Top view of the section of the corresponding aggregate at 0.05 axial strain (crushing axis normal to the section), with arrows pointing to cracks.

Like brittle aggregates, aggregates that behave plastically (i.e. aggregates made of 10 and 50 nm particle size) also show continuous increase in stress (Fig. 5a) from points (I) to (II) accompanied by breaking of bonds. After point (II), the stress curves consistently exhibit a small drop (Fig. 5a, point (III)) that occurs at approximately 4–5% axial strain. The microstructure in Fig. 5b shows no clear cracks at this axial strain as in brittle aggregates. However, fracture may be detected at this point from the large increase in bond-breaking events depicted in Fig. 5a. As the stress increases from point (III) to (IV), it reaches a plateau-like value. Following our model (Eq. (8)), these broken bonds are replaced by adhesive contacts that are stronger as particle size decreases, thus explaining the noted stress plateau. The number of particles per aggregates (500 vs. 5000) does not affect the qualitative response of aggregates. 4. Weibull analysis of crushing tests In the preceding section, typical stress–strain curves and typical microstructural evolution were used to draw qualitative conclusions on the effect of particle size and processing conditions on the strength of aggregates and agglomerates. However, from a quantitative viewpoint, aggregates and agglomerates exhibit endemic dispersion which must also be taken into account. Thus, for all particle sizes, a series of tests were run by randomly generating 100 different numerical samples with the same macroscopic features. Strength is defined as the maximum crushing stress except for aggregates exhibiting plastic-like behavior, for which strength is defined as the first drop in stress (point II in Fig. 5a). In the latter case, the drop in stress, although small in magnitude, is clearly observable and linked to numerous bond fractures. For brittle behavior, Weibull analysis is appropriate since the breaking of bonds in tension mode is responsible for the fracture (Fig. 4a). On the other hand, for plastically behaving agglomerates, Weibull analysis is much less appropriate since a significant number of bonds break in shear (Fig. 5a). However, in order to

Fig. 5. (a) Typical stress–strain curve of a plastic aggregate where primary particle size is 10 nm. Section of an aggregate (loading axis normal to the section) at (b) 0.05 and (c) at 0.2 axial strain. Colors indicate the coordination number (Z) of particles.

describe and compare the variability of strength from one specimen to another, we use Weibull statistics for all samples. For a test ranked i for fracture stress, the median ranking method gives the probability of fracture for N ¼ 100 tests as: Pf ¼

i 0:3 : N þ 0:4

ð9Þ

For a given sample volume, the cumulative probability of fracture, Pb f , for an applied stress r may then be fitted to the points calculated with Eq. (9) using the Weibull distribution [36]: Pb f ¼ 1 exp½ðr=r0 Þm ;

ð10Þ

where r0 is the characteristic strength and m is the Weibull modulus. The logarithmic plots of the Weibull relationship


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for agglomerates and aggregates are shown in Fig. 6, and indicate a reasonable fit in both cases. The relation between particle size ð2RÞ and strength of an agglomerate may also be evaluated by Rumpf’s equation [37]: rf ¼

DZ N c ; p ð2RÞ2

ð11Þ

where D is the relative density of the porous object and N c represents the magnitude of the cohesive force between particles. Examination of Eqs. (1) and (11) shows that strength should approximately scale with wR for agglomerates for which the cohesive force is solely given by adhesion ðN c ¼ jN DMT j pwRÞ. This simple correlation is partially supported by Fig. 7. We have verified that r0 / Rw1:2 for agglomerates made either of 500 or 5000 particles. The deviation from the inverse linear relationship with particle size may be explained by the slight density increase of agglomerates and also by the shift in the Coulomb cone with decreasing particle size. Although these results are based on a large assumed friction coefficient ðl ¼ 0:5Þ, additional simulations for l ¼ 0 have also yielded a power-law coefficient of about 1.1 (see Section 6). For aggregates, the particle size effect becomes discernible only when size reduces from 100 to 10 nm. For such sizes, strongly adhesive bonds replace solid bridges that have broken. Aggregates become size-dependent when jN DMT j rb pa2b . The particle size at which this transition occurs is 4cs =ðrb a2 Þ ’ 20 nm which is consistent with the results depicted in Fig. 7. For size-independent aggregates, we have verified that r0 / rb a2 . This shows that aggregate strength is proportional to the bond strength rb and that it depends on the extent of the calcination process since a must increase with calcination. Fig. 6 indicates that the particle size also plays an important role in strength dispersion. Agglomerates/aggregates made of smaller particles are seen to exhibit a larger m value. This trend is particularly pronounced in case of agglomerates for which very large m values are noted.

Fig. 7. Characteristic strength r0 for aggregates and agglomerates calculated from 100 crushing tests. The Weibull modulus characterizing the dispersion is given in Fig. 6.

More generally, comparing agglomerates and aggregates, Fig. 7 shows that the processing route has a strong influence on the mechanical behavior of the powder. 5. Closed-die compaction of aggregated and agglomerated powders Typically, aggregated or agglomerated ceramic powders are formed before sintering by cold or hot isostatic compaction, or closed-die compaction. Since closed-die compaction is a popular route for ceramic powders, the uniaxial compaction of a small volume element made of 100 spherical aggregates/agglomerates (for 500 particles per aggregate/agglomerate) or 10 spherical aggregates/ agglomerates (for 5000 particles per aggregate/agglomerate) has been simulated. These numbers are dictated by CPU time considerations, which do not allow for a larger number of particles. The aggregates/agglomerates used in this section are similar to those which have been studied in the previous section. Agglomerates or aggregates were initially randomly located in a simulation box and slowly packed together.

Fig. 6. Weibull plots of (a) agglomerates and (b) aggregates made of 5000 particles. r0 and m are obtained from the straight lines fitted as per Eq. (10).



The relative density at the end of this stage is of the order of 0.30–0.35, which is typical of the tap density of ceramic powders. The procedure for packing aggregates has been explained in detail elsewhere [15,27]. Periodic boundary conditions are used on all three axes such that, when a particle protrudes outside the periodic cell through a given face, it interacts with the particles on the opposite face. It has been shown that a relatively small number of particles ð 50; 000Þ is sufficient to represent a volume element when periodic conditions are used [25]. During the preparation stage, friction and adhesion were not included and aggregates/agglomerates were not allowed to break (the macroscopic pressure during this stage is very small,