Persistence of Granular Structure during Compaction Processes

Compaction of granulated powder is a common forming process used in the ceramics industry. Glass spheres were used as a model system to investigate granule failure during die compaction. Stresses within an assembly of spheres follow a network of pathways. Results demonstrate the statistical nature of granule failure during compaction, with some granules failing at very low applied pressures while a large fraction persist at even the highest applied loads. At high compaction pressures, size distributions of compacted spheres were seen to approach the Dinger-Funk distribution for maximum packing. In the limiting case of maximum density, the Dinger-Funk equation predicts 33% of the volume of granules will have sizes in the range of the initial size distribution. * Alfred, NY 14802 † Accepted: September 9, 2003 sistent interfaces can be clearly observed. The results clearly reveal a granule structure that persists after sintering. The coarse fraction of the granule size distribution has been shown to be responsible for the largest defects that persist after compaction. Strength testing is a very effective way to locate large defects. With 109 granules per cm3, statistically there will be cases where the largest granules are positioned in such a way to create large triangular or tetrahedral pores. In unfired compacts of spray-dried alumina, Mosser [3] demonstrated that strength could be reduced through the introduction of oversized granules. Walker and Reed [4] found that, with a constant maximum granule size, the number of granules in the coarse end of the distribution was the dominating factor responsible for the formation of strength-limiting defects in sintered alumina, even when the total number of defects could be changed through changes in the strength and density of the granules. Fractography confirmed that the strength-limiting defects were granule related. Stress transmission during compaction has been modeled by a number of researchers using a continuum approach [1]. This method provides a good macroscopic description of the pressure gradients that occur within compacts and the resulting density gradients that are observed. However, a discrete granule approach provides better insight into granule deformation and the origin of granule-related defects. Computer simulation [5] and a photoelastic disk method [6] have been used to model stress transmission through a packed bed of discrete granules. Stress distributions in photoelastic materials are easily observed as color changes when viewed under polarized light. Thus, stress distributions through a two-dimensional cell packed with photoelastic disks can be directly observed. Results from both computer simulations and photoelastic disk studies show that stress is transmitted through a granular material along a branched network of pathways which concentrate stress in some disks and bypass others. Flow occurs from sliding of block-like regions along slip planes. These methods are effective for modeling stress distributions below the elastic limit of the granules (Stage I). Yielding of granules due to brittle failure or plastic deformation, which occurs at approximately Py, results in rearrangement of localized stress distributions. In this work, glass beads were used to model granule breakdown during compaction. Previous studies using glass beads have been conducted by the author [7-9] and by others [10]. The glass beads are spherical, as are spray-dried granules, and were selected to be of similar size distribution to granules commonly used for processing of ceramics. Glass beads are elastic, brittle, have low bulk compression and fail by brittle fracture. Granulated powder consists of porous agglomerates, which exhibit moderate bulk compression, and may only be elastic at low loads. While granules may be brittle they are often engineered to be plastic and thus deform rather than fracture under load.


INTRODUCTION
Die compaction of granulated powder is a common forming process used in ceramics and other industries.Granulation of fine powder results in free f lowing feed material with controlled composition and properties that can be easily used in high speed presses.In ceramic processes, application of pressure consolidates the granulated material into a green body, which is subsequently sintered.Artifacts of the granule structure may persist as pores and laminations after compaction, and remain as defects in the sintered microstructure.Such defects can be detrimental to the properties of the final part.Thus it is desirable to eliminate the granule structure during compaction.In this work, statistical analysis of the fragmentation of glass beads during die compaction was used as a model granular system to achieve a better understanding of persistent granular structures in compacts.
Compaction of granulated powder occurs in three pressure-dependent stages [1].In Stage I, granule rearrangement at low pressures results in a small increase in density of the granular assembly.Above an apparent yield pressure P y , which marks the onset of Stage II, the interstitial pores between granules (intergranular pores) are reduced in size as the granules break down or deform, resulting in a linear increase in density with log (compaction pressure).In Stage III, the intergranular pores are mostly eliminated, and particle rearrangement within the granules causes increased densification at high pressures.The types of granule-related defects that may persist after compaction include persistent intergranular pores and poorly joined interfaces between granules.
An assembly of granulated powder has a hierarchical structure consisting of packed granules, which are comprised of packed particles.The pore size distribution is bimodal; large intergranular pores are formed from the packing of the granules and small intragranular pores are formed from the packing of the primary particles.During compaction, the intergranular pores are reduced in size, and largely eliminated, as granules deform in response to applied pressure in Stage II and Stage III.The intragranular pores are eliminated during sintering.However, the size of persistent intergranular pores is sufficiently large that these pores cannot be removed by the sintering process.Consequently these pores can persist into the sintered part.Interfaces between granules may persist as well when the granule surfaces do not meld together.Uematsu [2] has developed an optical microscopy method that is useful for observing persistent granule structures.By immersing a thin section of sintered material in a liquid with refractive index similar to that of the ceramic powder, the per-sistent interfaces can be clearly observed.The results clearly reveal a granule structure that persists after sintering.
The coarse fraction of the granule size distribution has been shown to be responsible for the largest defects that persist after compaction.Strength testing is a very effective way to locate large defects.With 10 9 granules per cm 3 , statistically there will be cases where the largest granules are positioned in such a way to create large triangular or tetrahedral pores.In unfired compacts of spray-dried alumina, Mosser [3] demonstrated that strength could be reduced through the introduction of oversized granules.Walker and Reed [4] found that, with a constant maximum granule size, the number of granules in the coarse end of the distribution was the dominating factor responsible for the formation of strength-limiting defects in sintered alumina, even when the total number of defects could be changed through changes in the strength and density of the granules.Fractography confirmed that the strength-limiting defects were granule related.
Stress transmission during compaction has been modeled by a number of researchers using a continuum approach [1].This method provides a good macroscopic description of the pressure gradients that occur within compacts and the resulting density gradients that are observed.However, a discrete granule approach provides better insight into granule deformation and the origin of granule-related defects.
Computer simulation [5] and a photoelastic disk method [6] have been used to model stress transmission through a packed bed of discrete granules.Stress distributions in photoelastic materials are easily observed as color changes when viewed under polarized light.Thus, stress distributions through a two-dimensional cell packed with photoelastic disks can be directly observed.Results from both computer simulations and photoelastic disk studies show that stress is transmitted through a granular material along a branched network of pathways which concentrate stress in some disks and bypass others.Flow occurs from sliding of block-like regions along slip planes.These methods are effective for modeling stress distributions below the elastic limit of the granules (Stage I).Yielding of granules due to brittle failure or plastic deformation, which occurs at approximately P y , results in rearrangement of localized stress distributions.
In this work, glass beads were used to model granule breakdown during compaction.Previous studies using glass beads have been conducted by the author [7][8][9] and by others [10].The glass beads are spherical, as are spray-dried granules, and were selected to be of similar size distribution to granules commonly used for processing of ceramics.Glass beads are elastic, brittle, have low bulk compression and fail by brittle fracture.Granulated powder consists of porous agglomerates, which exhibit moderate bulk compression, and may only be elastic at low loads.While granules may be brittle they are often engineered to be plastic and thus deform rather than fracture under load.

EXPERIMENT
Glass beads (3M Company, St. Paul, MN) with a log normal size distribution (mean diameter 85.5 µm and geometric standard deviation 1.25) were compacted in a steel die using a laboratory press (Laboratory Press Model M, Fred S. Carver, Inc., Menomonee Falls, WI.) at pressures ranging from 17.5 MPa to 1.05 GPa.The die had a cylindrical cavity with diameter 12.7 mm, and was fabricated in two halves that could be separated to facilitate removal of the specimens without an ejection step.Size distributions of the compacted beads were measured using laser scattering (Microtrac 9200, Leeds and Northrop, North Wales, PA) after dispersing the compacted material in water using ultrasound.
Compacts were vacuum infiltrated with epoxy and polished cross-sections were prepared and evaluated with a scanning electron microscope.For specimens pressed to low pressure, it was necessary to modify the die to allow a portion of one punch to be removed so the epoxy could be introduced while the compact was still constrained in the die.

RESULTS
Figure 1 shows that the glass beads followed similar compaction behavior to that of granulated ceramic powders.P y was 170 MPa, which is 10 2 to 10 3 that of typical spray-dried granules.This large difference is because of the much higher strength of the glass beads compared to that of a granulated powder.Just as in the case of granulated powder, a small increase in density was observed below the P y from rearrangement of the spheres; and above P y , density increased rapidly as spheres were fractured.Stage III, attributed to bulk compression of granules, was not observed in the case of the glass beads.Some beads were observed to fracture at pressures as low as 53 MPa, as is shown in Figure 2(a).As pressure was increased, larger numbers of fragments were observed, but some large spheres remained intact at the highest compaction pressures tested, as is shown in Figure 2(b).The internal structure of the compacts is seen by examining polished cross-sections of compacts shown in Figures 3(a) and 3(b).Evidence of stress pathways through the assembly of spheres is evident even at low pressure.Large spheres persist intact at high pressures.Densification occurs as some spheres fracture and their fragments rearrange to fill packing voids between the spheres that persist intact.
With the large number of spheres (ȁ10 9 /cm 3 ) that are present in a single compact, simple observation does not give an accurate indication of trends involving so many particles.Particle size analysis provides a statistical method.the 70 to 110 µm size range decreased and a tail of fine particles in the 10 to 60 µm range increased as more beads became fractured.

LOW PRESSURE BEHAVIOR
A more accurate interpretation of the change in size distribution can be obtained by plotting 10 th , 50 th and 90 th percentile sizes (a 10 , a 50 and a 90 , respectively) as a function of compaction pressure (see Figure 5).The size distribution remained constant at low pressures.At a compaction pressure of 60 MPa, which is about P y /3, the size distribution began to change as glass beads fractured in response to the applied pressure.The median size and the fine end of size distribution both began to decrease at about 60 MPa (ȁP y /3).The coarse end of size distribution remained constant until a point near P y (ȁ150 MPa).As pressure was increased above P y , the entire distribution shifted toward smaller sizes as more spheres became fractured.The increase in fines in the size distribution at compaction pressures below the point where the coarse end of the distribution begins to change indicates that smaller granules begin to fail at lower applied pressure, and that the larger spheres are more likely to remain intact at low compaction pressures.Similar behavior has been reported by Deis and Lannutti [11], who observed a tendency for small spray-dried alumina granules to yield at lower pressures than large granules.This behavior can be explained by considering Hertzian contact stresses between spheres.
Granules will act as elastic spheres at low loads.Above some critical level of stress, yielding will occur as granules fracture or deform.Within an assembly of granules, yielding of any individual granule depends on the local loading conditions resulting from stresses transmitted from neighboring granules.Below the elastic limit, the level of stress in any individual granule will follow Hertzian behavior [12], and be dependent on the locally transmitted load P, the diameters of the two contacting spheres a 1 and a 2 , and the elastic modulus of the granules E. The maximum compressive stress σ c occurs along the loading axis through the spheres and is given by where The maximum tensile stress occurs radially at the edge of the contact area, and is given by and the maximum shear stress is given by KONA No.21 (2003) and occurs along the loading axis at a distance of r/2 below the contact surface, where r is the radius of the contact area, given by r҃0.881 3 ͱහ. ( Since the maximum tensile and shear stresses are equal to the maximum compressive stress multiplied by some constant, a generalized equation can be written where i is compression, tension or shear.
Since the force through an assembly of spheres is not uniformly transmitted, but instead follows a network of pathways, some spheres will experience much higher stresses than others.For spheres of non-uniform size, the maximum stress in each sphere will depend on the sizes of both spheres in a pair and the locally transmitted load.
Figure 6 illustrates the range of stresses that might be observed in an idealized single chain of granules within an assembly.In this idealized example, the spheres are arranged in such a way that the load is transmitted along a single straight line which passes through the center of several granules, with none of the load being transmitted to neighboring granules.Since the magnitude of the stress resulting from a load being transmitted by the contact of two spheres is dependent on the diameters of both spheres, the highest stresses result from the contact of two small spheres and the lowest stresses result from the contact of two large spheres.Thus it would be expected that large spheres are less likely to experience the level of stress required for deformation or fracture to occur before smaller granules yield, rearranging the local distribution of stress pathways.The figure also shows that the level of stress at the contact between a granule and the die surface is very low.In the case of a sphere and a f lat surface, a 2 is infinite and K a ҃a 1 .Elimination of surface roughness caused by granule artifacts at the surfaces of pressed ceramic components is a common concern.Several researchers have achieved limited success in efforts to reduce surface roughness [3,13].
The spray-dried granules used in the processing of ceramic material are themselves assemblies of fine particles and will thus fail in tension or shear.If the yield strength of the matrix comprising the granules is the same, regardless of the size of the spheres, failure would occur in each of the spheres.However, PK a E since the volume where maxima occur in both shear and tensile stresses is related to the contact area (given in Eqn.5), a larger relative volume of small granules is affected by the stress field, and small granules would be expected to experience more severe damage as a result.
The assumption that the strength of the matrix material is independent of granule size may not always be valid.Spray-dried ceramic powders are commonly processed using organic additives such as binders in order to impart lubricity during compaction and green strength after compaction.It has long been suspected, and recently been demonstrated [14] that some binders migrate to the granule surfaces during drying giving the granules hard shells.Larger granules, with more binder to migrate, might have thicker and thus harder shells than smaller granules, making the larger granules relatively stronger, and less prone to deformation.It has also been suggested that smaller granules might be subject to a different heating profile than large granules during drying [15].If the trajectory and residence time of the dr ying droplets is constant with respect to droplet size, the smaller droplets will dry more rapidly, and the protection from temperature of evaporative cooling on the organic binder will be shortened.Consequently, smaller granules may become harder due to temperature effects on the binder.Neither the temperature effect nor the binder-shell thickness effect has been experimentally verified as mechanisms that produce a size-dependence on granule strength.

HIGH PRESSURE BEHAVIOR
By recognizing that size distributions shown in Figure 5 are bimodal with the coarse fraction consisting of largely unfractured beads and the fine fraction consisting of fractured fragments, it is possible to deconvolute the distribution curves (see Figure 7) into a log normal mode consisting of unfractured spheres, and an asymmetrical mode consisting of fragments.In Figure 8, the amount of material in each mode is plotted with respect to compaction pressure.The amount of material in each fraction changes rapidly between pressures of about 75 and 400 MPa.At higher pressures, the rate of change decreases.Above about 500 MPa the amount of material in each fraction becomes constant with respect to compaction pressure, with the size distribution consisting of about 40% coarse and 60% fine.Thus, the increase in density during compaction is a result of fragmentation of 60% of the material while about 40% of the volume of the compact consists of granules which are relatively unchanged from the precursor material.These results are consistent with the behavior observed using granulated alumina and other ceramic powders [2,4,8].However, in the case of alumina, it was not possible to quantitatively describe the degree to which the precursor granular structure persisted after compaction.
As compact density increases with increasing applied pressure, the particle size distribution changes by continued fragmentation of the glass beads.The particle size distribution must necessarily change in the direction toward higher packing density.Therefore, predictions regarding the breakdown of granules during compaction can be made based on models for continuous size distributions that result in maximum packing.
Andreasen [16] postulated that maximum packing density would follow the distribution F(a)҃ m (7) where F(a) is the size cumulative distribution function, a is particle size, a max is the maximum particle size and the exponent m is in the range of 0.33 to 0.50.The basis for Andreasen's model was a fractal-like concept that in order to achieve maximum packing with a continuous size distribution, the relative sizes of neighboring particles would be the same for particles of any size.The range for the exponent was determined experimentally.
Dinger and Funk [17] recognized that any real size distribution must have a finite lower size limit a min and modified Andreasen's equation to ing of spheres would occur when the exponent m҃0.37.Thus, it is expected that during compaction of brittle granules, the size distribution will approach Equation ( 8) with exponent m҃0.37 as the applied pressure increases.
Figure 9 shows the size distribution of the glass beads along with calculated distributions from Eqn (8) using a max and a min from the experimental data, and various exponents.It can be seen that as pressure is increased, the slope of the distribution curve decreases.This change in slope corresponds to a decrease in the exponent m from Eqn. 8.At the highest compaction pressure that was tested, the data closely follows Eqn. 8 with exponent m҃1.With further increase in compaction pressure, it is expected that the exponent m would continue to decrease until m҃0.37, at which point maximum packing density would be achieved.
As previously discussed, the amount of material that remained largely unchanged from the initial size distribution was about 40% by weight over a range of the highest compaction pressures tested.In the limiting case of maximum packing density at m҃0.37, the amount of material that would remain in this size range would be 33% by volume.Thus it can be concluded that in the case of brittle granules, 33% or more by volume of the granules will persist largely intact, even at the highest compaction loads.The increase in packing density of the compact is a result of disruption and rearrangement of 67% or less of the mass of the granules.
Algebraic manipulation of an equation developed earlier by Furnas [18] results in an equation of the same form as Equation ( 8) with the exponent m҃logV (b/c)  where V is the pore volume of a single sieve fraction (with M2 sieve size ratio) of particles, b and c are constants based on the width of the distribution.The values of b and c are based on experimental data, and can be determined from graphs presented in Ref 18.Based on a min and a max of the distribution obtained in the present work, the Furnas model predicts that maximum packing will occur with m҃0.037 (also shown in Figure 9).This would result in 18% of the volume of the particles remaining in the size range of the original distribution Ҁ a significant improvement over the prediction based on Dinger and Funk's m҃0.37.However, since Furnas's work is based on experimental packing of fine particles, his results are subject to error due to factors that hinder packing, such as friction between particles and agglomeration of fines due to surface forces.
The analysis by Dinger and Funk was based on a continuous size distribution of spheres.In this experiment, the size distribution consists of spherical particles only at the coarse end of the distribution, and fragments of irregular shape throughout the medium to fine range.Dinger and Funk did not address the question of particle shape.Zheng [19] expanded on the work of Dinger and Funk and addressed the question of a size-dependence on particle shape.Their analysis considered the Dinger-Funk distribution as a mixture of an infinite number of discrete size fractions.The basic model for packing discrete size classes was also developed by Furnas [20].Maximum packing occurs when the interstitial pores of a coarse fraction are filled by a finer fraction of size that can fill the pores without disrupting the packing of the coarser particles.The resulting pores are filled by a third even finer fraction, and so on.Zheng considered that a continuous size distribution following Equation 8 could be separated into an infinite number of Furnas packing sequences.He related the exponent m to the pore fraction φ of packed monosized particles taken from the distribution: where R is a constant related to the size ratio needed for unhindered packing of two discrete size classes.Particle Size (µm) 100 Fig. 9 As pressure is increased, the size distribution of compacted spheres approaches the Dinger-Funk distribution (Eqn.8) with exponent n҃1.Maximum packing is predicted by Dinger and Funk when n҃0.37.Maximum packing is predicted by Furnas when n҃0.037.
When particle shape is constant across the entire size distribution, φ is the same for each size class.Spherical particles have the highest packing factor, and thus the lowest φ, and consequently, the highest value for m in the Dinger-Funk equation.As particle shape deviates from spherical, packing factor decreases and φ increases, thus the exponent m would be expected to decrease following Equation 9.In the case where φ is not constant across the distribution, They suggest that an average exponent for the different size classes will describe the conditions for maximum packing of the system.In the present work, the coarse end of the distribution consists of spheres, and thus has the highest packing factor.The coarse spheres will be surrounded by irregular medium and fine particles that pack to somewhat lower density, thus m would less than 0.37 according to Zheng.
The work on particle packing by Andreasen and by Furnas is based in part on experimental data, and thus interparticle forces may have hindered rearrangement of particles and prevented maximum packing.Dinger and Funk utilized a computer simulation and thus were able to eliminate interparticle forces.They found that equivalent maximum packing could be obtained using a variety of packing algorithms.In the present work, changes in particle size distribution occur as a means of reducing bulk volume in response to an applied load.It is not known whether the same packing density would be obtained if the compacts were disrupted and the particles repacked, even if interparticle forces could be eliminated.However, the application of a load provides a driving force to achieve maximum packing density, by both particle rearrangement and by the fracture of particles into finer particles.By this path, a particle size distribution is obtained that will result in maximum packing for a given load.Such a size distribution will contain a significant fraction of coarse particles.

SUMMARY AND CONCLUSIONS
During die compaction of granulated ceramic powders, the load is transmitted along a branched network of contacting granules, concentrating the stress in some granules while others are bypassed.As the applied load is increased, granules fail by deformation or fracture, the network of stress pathways rearranges to some extent, and the packing density of the granular assembly increases.However, evidence of the granular precursor material can be seen to persist in the microstructure of unfired and fired ceramics prepared by compaction.Glass spheres were used as a model granular system to investigate the degree to which the granular structure persists during die compaction.
The state of stress within any individual granule depends on contact stresses from the locally transmitted load.The magnitude of the load, the diameter of the sphere, and the diameter of the neighboring spheres which transmit the load are all factors determining the state of stress within a sphere.Within a single chain of contacting spheres, those with smaller diameter experience higher levels of stress and will thus fail at lower loads.Within an assembly of a large number of spheres, a general trend observed is that smaller granules fail at lower compaction loads.
The size distribution of the glass spheres changed with applied compaction load as the spheres failed due to fracture.As the compaction load approached the apparent yield pressure of the granular assembly, a tail of fines was seen in the particle size distribution before any change is observed in the coarse fraction, indicating that fines are formed due to fragmentation of smaller spheres in the initial sample.As pressure was increased, the size distribution became bimodal, consisting of unfractured spheres in the coarse fraction, and fractured fragments in the fine fraction.The amount of material in each fraction changed rapidly during the early stages of compaction, but became constant at higher loads, with about 40% of the volume consisting of unfractured spheres.The size distribution approached that of a Dinger-Funk distribution with an exponent of m҃1.In the limiting case of m҃0.37, which Dinger and Funk determined to be the distribution with maximum packing density based on computer simulation, 33% of the granules would persist nearly intact.Advanced Ceramic Technology at Alfred University.

Fig. 4 Fig. 5
Fig.4 Change in particle size distribution of glass beads as a result of compaction.

Fig. 6
Fig. 6 Schematic of a stress pathway in an assembly of granules.The load follows a chain of spheres of different sizes.The relative stress (in arbitrary units) in the contacting spheres is indicated.