Characterization of Uniaxial Compaction in Spray Dried

i

The Pennsylvania State University

The Graduate School

College of Earth and Mineral Sciences

CHARACTERIZATION OF

UNIAXIAL COMPACTION IN

SPRAY DRIED CERAMIC POWDERS

A Thesis in

Materials Science and Engineering

by

Robert David Carneim

©2000 Robert David Carneim

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

December 2000

ii

We approve the thesis of Robert David Carneim.Date of Signature

____________________________________ ___________________Gary L. MessingProfessor of Ceramic Science and EngineeringThesis AdvisorChair of Committee

____________________________________ ___________________James H. AdairAssociate Professor of Materials Science and Engineering

____________________________________ ___________________Ian R. HarrisonProfessor of Polymer Science

____________________________________ ___________________John R. HellmannAssociate Professor of Ceramic Science and Engineering

____________________________________ ___________________Virendra M. PuriProfessor of Agricultural Engineering

____________________________________ ___________________Richard E. TresslerProfessor of Materials Science and EngineeringHead of the Department of Materials Science and Engineering

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Abstract

A model granulated ceramic powder system was studied with systematically varied

binder content and binder plasticity (binder glass transition temperature, Tg). A submicron

α-alumina was used as the inorganic component and poly(vinyl alcohol)–4 wt% glycerol

was the base binder system. Four compositions were spray dried containing ~2, 3, 4 and

5 wt% binder (dry weight basis alumina). The resulting powders were classified and the 75–

150 µm granule size range powders prepared for testing. These powders were conditioned

at five different relative humidities to adjust binder Tg to five different values between –

32°C and +35°C.

All twenty compositions were tested in uniaxial compaction in a 6.34 mm diameter

cylindrical steel die to stresses from ~6 MPa to ~175 MPa. Green strengths of the samples

produced were measured using the diametral compression test. Compaction curves were

constructed and springback on ejection was measured. A selection of similar samples was

prepared and dimensional changes after compaction were measured as a function of time. It

was found that the compositions with low Tg during compaction resulted in the highest

achievable densities and green strengths. However, green strength increased and achievable

green density decreased with increasing binder content. Dimensional changes on ejection

were found to be dominated by the instantaneous springback in the axial direction (~5–8%).

Radial springback was generally less than 1% and the total dimensional change due to time-

dependent relaxation was generally less than 0.5%.

It was observed that compaction behavior was affected by sample size. A technique

was developed which determines this effect. By measuring two compaction curves of a

powder of different sample size, it was possible to calculate the force opposing compaction

due to friction at the die wall. This allowed the calculation of the intrinsic compaction curve

of the material, i.e., its compaction behavior in a frictionless die. With these two parameters

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known it became possible to predict compaction curves of the powder for different sample

sizes. Compaction curves calculated in this manner predicted experimentally determined

compaction curves with correlation coefficients greater than 0.99.

To aid in the characterization of individual granules and their interaction during com-

paction, an analysis was developed that calculates granule strength and intergranular bond

strength during compaction and the free granule strength. In a series of pellets pressed to a

wide range of pressures, granule deformation and adhesion varied greatly between samples.

The strengths of these samples were measured by diametral compression and the fracture

surfaces were analyzed to determine the relative amounts of intergranular and intragranular

fracture. A quadratic curve was found to describe the relationship between the overall green

strength of the sample and the area fraction of intergranular fracture. By applying knowl-

edge of the physical process at the 0, 50 and 100% intergranular fracture points along this

curve, this curve was deconvoluted to the unique pair of linear functions that track the

intergranular bond strength and the intragranular strength throughout the compaction cycle.

The 100% intergranular fracture point corresponds to little or no consolidation of the pow-

der; therefore, the value of the determined intragranular strength line is a measure of the

free granule strength. The free granule strength measured by this technique results in values

much lower than the often-reported granule yield point measurement due to the difference

in loading of the granules.

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Table Contents

Page

List of Tables ................................................................................ viiList of Figures .............................................................................. viiiAcknowledgments ........................................................................ xiiChapter 1: Introduction ................................................................... 1

Background................................................................................................1Previous Work............................................................................................4

Measurement of Compaction Processes ................................................................. 4Effect of Process Parameters .................................................................................. 6Compaction Modeling ............................................................................................ 9Post-Compaction Behavior ................................................................................... 10Strength Characterization ..................................................................................... 12

Research Objective ..................................................................................14Chapter 2: Powder Preparation and Physical Characterization .... 16

Introduction .............................................................................................16Experimental Procedure ..........................................................................16

Powder Production ............................................................................................... 16Powder Preparation ............................................................................................... 17Powder and Binder Characterization .................................................................... 17

Results and Discussion ............................................................................17Summary..................................................................................................21

Chapter 3: Loading and Unloading Response .............................. 22Introduction .............................................................................................22

Compaction Curves .............................................................................................. 22Post-Compaction Dimensional Changes .............................................................. 24

Experimental Procedure ..........................................................................25Compaction and Density ...................................................................................... 25Viscoelastic Relaxation......................................................................................... 25

Results and Discussion ............................................................................26Compaction and Density ...................................................................................... 26Instantaneous Springback ..................................................................................... 39Time-Dependent Relaxation ................................................................................. 45

Summary and Conclusion........................................................................48

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Table Contents (cont.)

Page

Chapter 4: A Model for Compaction Curve Prediction ................ 50Introduction .............................................................................................50

Background ........................................................................................................... 50Overview of Principle ........................................................................................... 51

Experimental Procedure ..........................................................................53Analytical Procedure ............................................................................................ 53Experimental Verification ..................................................................................... 58Computer Utilization ............................................................................................ 61

Results and Discussion ............................................................................62Evaluation of k·µ................................................................................................... 62Compaction Curve Prediction .............................................................................. 63

Summary and Conclusion........................................................................64Chapter 5: Green Strength and Granule Strength Modeling ........ 67

Introduction .............................................................................................67Green Strength ...................................................................................................... 67Granule Strength Modeling .................................................................................. 68

Experimental Procedure ..........................................................................69Green Strength ...................................................................................................... 69Granule Strength Modeling .................................................................................. 69

Results and Discussion ............................................................................71Green Strength ...................................................................................................... 71Granule Strength Modeling .................................................................................. 72

Summary and Conclusion........................................................................77Chapter 6: Conclusion .................................................................. 79

Summary and Conclusions ......................................................................79Recommendations for Future Work.........................................................81

Appendix: Mathematica® Code ................................................... 83References .................................................................................. 100

Referenced Works ..................................................................................100General References ................................................................................104

Vita Robert D. Carneim .............................................................. 111

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List of Tables

Page

Table 2.1 ........................................................................................................................... 21Total Binder Contents, Glass Transition Temperatures (Tg) and Granule Densi-ties of the Spray Dried Powders

Table 3.1 ........................................................................................................................... 42Apparent Yield Points

Table 3.2 ........................................................................................................................... 42Stage IV Compaction Onset

Table 5.1 ........................................................................................................................... 77Granule Yield Points and Granule Strengths of Tg = 6°C (43 %RH) Samples

viii

List of Figures

Page

Figure 1.1 ............................................................................................................................ 2Schematic representation of uniaxial compaction. Region I: granule rearrange-ment, II: granule elastic deformation, III: extensive plastic deformation and/orfracture of granules, IV: particle rearrangement within compact, V: instantaneousspringback and time-dependent relaxation on ejection, VI: ultimate green partdensity.

Figure 2.1 .......................................................................................................................... 18Micrograph of 75–150 µm diameter spray dried powder containing ~4 wt%binder; other compositions used have similar appearance.

Figure 2.2 .......................................................................................................................... 18Equilibrium moisture content (EMC) of the binder as measured bythermogravimetric analysis (TGA).

Figure 2.3 .......................................................................................................................... 19Density of the binder in the relative humidity range used. Error bars indicate95% confidence intervals based on 3 measurements.

Figure 2.4 .......................................................................................................................... 20Glass transition temperature (Tg) as measured by differential scanning calorim-etry (DSC, 20°C·min–1). The inflection of the step transition is reported.

Figure 3.1a, b .................................................................................................................... 27Measured compaction curve and ejected densities of Tg = 35°C, 2 (a) and3 (b) wt% NBC system in a 6.34 mm diameter die.

Figure 3.1c, d .................................................................................................................... 28Measured compaction curve and ejected densities of Tg = 35°C, 4 (c) and5 (d) wt% NBC system in a 6.34 mm diameter die.

Figure 3.2a, b .................................................................................................................... 29Measured compaction curve and ejected densities of Tg = 18°C, 2 (a) and3 (b) wt% NBC system in a 6.34 mm diameter die.

Figure 3.2c, d .................................................................................................................... 30Measured compaction curve and ejected densities of Tg = 18°C, 4 (c) and5 (d) wt% NBC system in a 6.34 mm diameter die.

ix

Figure 3.3a, b .................................................................................................................... 31Measured compaction curve and ejected densities of Tg = 6°C, 2 (a) and 3 (b) wt%NBC system in a 6.34 mm diameter die.

Figure 3.3c, d .................................................................................................................... 32Measured compaction curve and ejected densities of Tg = 6°C, 4 (c) and 5 (d) wt%NBC system in a 6.34 mm diameter die.

Figure 3.4a, b .................................................................................................................... 33Measured compaction curve and ejected densities of Tg = –25°C, 2 (a) and3 (b) wt% NBC system in a 6.34 mm diameter die.

Figure 3.4c, d .................................................................................................................... 34Measured compaction curve and ejected densities of Tg = –25°C, 4 (c) and5 (d) wt% NBC system in a 6.34 mm diameter die.

Figure 3.5a, b .................................................................................................................... 35Measured compaction curve and ejected densities of Tg = –32°C, 2 (a) and3 (b) wt% NBC system in a 6.34 mm diameter die.

Figure 3.5c, d .................................................................................................................... 36Measured compaction curve and ejected densities of Tg = –32°C, 4 (c) and5 (d) wt% NBC system in a 6.34 mm diameter die.

Figure 3.6a, b .................................................................................................................... 37Measured compaction curve and ejected densities of Tg = 35 (a) and 6 (b)°C,3 wt% NBC system in a 12.6 mm diameter die.

Figure 3.6c ........................................................................................................................ 38Measured compaction curve and ejected densities of Tg = –32°C, 3 wt% NBCsystem in a 12.6 mm diameter die.

Figure 3.7 .......................................................................................................................... 39Achievable pellet density: 6.34 mm diameter samples pressed to ~175 MPa.

Figure 3.8 .......................................................................................................................... 39Stress required to achieve 52.9% relative alumina density in a 6.34 mm diam-eter die as a function of binder content and binder Tg. (Tg = 35°C, 5% NBC datapoint was extrapolated from available compaction curve data up to ~175 MPa.)

Figure 3.9a ........................................................................................................................ 41Degree of polymer saturation as a function of compact density, binder Tg duringcompaction and compaction stress.

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Figure 3.9b ........................................................................................................................ 41Total solids content in granules and compacts as a function of compact density,binder Tg during compaction and compaction stress.

Figure 3.10a ...................................................................................................................... 43Instantaneous volumetric springback on ejection vs. binder Tg and compactionstress for samples pressed in a 6.34 mm diameter die to an aspect ratio of ~0.5.

Figure 3.10b ...................................................................................................................... 43Instantaneous volumetric springback on ejection vs. binder Tg and compactionstress for samples pressed in a 12.6 mm diameter die to an aspect ratio of ~0.4.

Figure 3.11a ...................................................................................................................... 44Example illustrating the anisotropy of instantaneous springback on ejection vs.compaction stress for 3 % NBC, Tg = 6°C samples pressed in a 6.34 mm diam-eter die to an aspect ratio of ~0.5.

Figure 3.11b ...................................................................................................................... 44Example illustrating the anisotropy of instantaneous springback on ejection vs.compaction stress for 3 % NBC, Tg = 6°C samples pressed in a 12.6 mm diam-eter die to an aspect ratio of ~0.5.

Figure 3.12 ........................................................................................................................ 45Example raw data and curve fit of the time-dependent strain recovery after ejec-tion. Example is of Tg = 6°C, 3% NBC system, 6.34 mm diameter pelletcompacted at ~175 MPa, relaxation in the axial direction.

Figure 3.13a ...................................................................................................................... 46The total, fractional, recoverable time-dependent strain as a function of binderTg for various samples as indicated in the legend. (The “A” parameter in thefunction: h = 1-A·exp(-t⁄τ)).

Figure 3.13b ...................................................................................................................... 46The characteristic relaxation time as a function of binder Tg for various samplesas indicated in the legend. (The “τ” parameter in the function: h = 1-A·exp(-t⁄τ)).

Figure 4.1 .......................................................................................................................... 59Measured compaction curves for Tg = –25°C, 5 wt% NBC samples. The aspectratio reported is the final aspect ratio of the pressed pellet. Note that this and allother compaction curve figures in this chapter plot the compaction stress as afunction of sample density.

Figure 4.2 .......................................................................................................................... 60The k·µ solution for Tg = –25°C, 5 wt% NBC samples.

xi

Figure 4.3 .......................................................................................................................... 60The k·µ solution for Tg = 6°C, 3 wt% NBC samples.

Figure 4.4 .......................................................................................................................... 61The two original, measured, compaction curves and the calculated intrinsic com-paction curve for the Tg = –25°C, 5 wt% NBC system.

Figure 4.5 .......................................................................................................................... 63The black curves illustrate the predictive ability of this analysis for the Tg = -25°C,5 wt% NBC system. The solid line is the calculated compaction curve; the dashedline is the experimentally measured compaction curve for the same conditions.

Figure 4.6 .......................................................................................................................... 64The normalized compaction curve (single curve in magenta) and predicted (un-broken curves) and corresponding measured (dashed curves) compaction curvesfor the Tg = 6°C, 3 wt% NBC system.

Figure 5.1 .......................................................................................................................... 71High magnification micrograph of a pellet fracture surface clearly showing thedifference between intergranular fracture and intragranular fracture.

Figure 5.2 .......................................................................................................................... 72a) Representative pellet fracture surface micrograph showing the two differentfracture modes. b) Analysis of the fracture surface showing intergranular frac-ture (white) and intragranular fracture (black).

Figure 5.3 .......................................................................................................................... 73Pellet green strengths at 43% RH (Tg = 6°C): a) compacted to ~175 MPa andb) compacted to 52.9 % relative alumina density.

Figure 5.4 .......................................................................................................................... 74Intergranular fracture vs. pellet green strength (R2 ≥ 0.95).

Figure 5.5a ........................................................................................................................ 75Green strength deconvolution for the 2 wt% NBC, Tg = 6°C samples.

Figure 5.5b ........................................................................................................................ 75Green strength deconvolution for the 3 wt% NBC, Tg = 6°C samples.

Figure 5.5c ........................................................................................................................ 76Green strength deconvolution for the 4 wt% NBC, Tg = 6°C samples.

Figure 5.5d ........................................................................................................................ 76Green strength deconvolution for the 5 wt% NBC, Tg = 6°C samples.

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Acknowledgments

I can not possibly express in words how I feel about all the people who have affected

my life and, consequently, helped me reach this point. I would, however, like to single out a

few and simply thank them. My amazing wife, Tammy, who, for me, has transformed exist-

ence into Life. My parents, Sergei and Erika, for twenty-nine years (and counting) of love,

support and encouragement. My advisor, Gary Messing, for having given me this opportu-

nity and helping me develop the skills required to earn this degree and succeed after leaving

Dear Old State. And the universe for going to all the bother of existing.

Of course, there are many others whose presence and friendship helped me retain some

semblance of a sane, balanced life. To name a few: Leah Witzig, Paul & Erin, Eric & Chris,

Big Ed, Mike, 10100101101111110011111001011110010 & Maureen, Matt,

C. Scott & Jana, Ender & Bahar, Scott, Mitch, R.P. & Jen, Lisa, and everyone else in our

research group, past and present.

I would also like to thank The Particulate Materials Center (an NSF I/UCRC) for fund-

ing this research and my industrial mentor, Walt Shaffer of Spang. I am grateful to Spang

for the use of their spray drying facilities to prepare the powders used in this work. Finally,

thanks go out to the faculty, staff and students that I had the pleasure of working with at

Penn State; I was fortunate in that it seemed they all conspired to make my time here,

overall, a positive experience.

xiii

“Even if there is only one possible unified theory, it is just a set of rules and equations.

What is it that breathes fire into the equations and makes a universe for them to describe?

The usual approach of science of constructing a mathematical model cannot answer the

questions of why there should be a universe for the model to describe. Why does the

universe go to all the bother of existing? Is the unified theory so compelling that it brings

about its own existence? Or does it need a creator, and, if so, does he have any other

effect on the universe? And who created him?”

—Stephen W. Hawking, (A Brief History of Time, 1988)

1

Chapter 1:

Introduction

Background

Die compaction of ceramic powders is a widely used forming process which is auto-

mated to rapidly produce parts of low to moderate geometric complexity. Dry pressing

involves the uniaxial compaction of spray dried granules consisting of ceramic particles

bound by an organic binder. A schematic compaction curve for this process is shown in

figure 1.1. During compaction, part density increases due to granule rearrangement and

deformation, and particle rearrangement. Upon releasing the compaction pressure the part

expands, relieving internal stresses. The expansion occurs as both an elastic (instantaneous)

springback, due to the energy stored in the structure, and a viscoelastic (time-dependent)

relaxation, due to the viscoelastic character of the organic binder.

For the application of dry pressing, submicron powders are often intentionally granu-

lated to improve flow properties. This is necessary since submicron particles tend to

spontaneously form large, irregular, low-density aggregates as a result of the dominance of

surface forces (e.g., van der Waals) over inertial forces (i.e., particle weight). The presence

of such aggregates results in poor powder flow and inhomogeneous and inconsistent die

filling. Organic binders are added to impart strength to the granules and green part, and to

enhance compaction. A popular and effective granulation method, which also facilitates the

addition of a binder phase, is spray drying. In this process, the powder is dispersed in a

liquid medium with a chemical dispersant and a binder. This slurry is atomized in the spray

drier and exposed to a flow of heated air. The liquid evaporates and is carried away leaving

relatively dense, spherical granules on the order of tens to hundreds of micrometers in di-

ameter. As a result of the increase in scale of the “particles” comprising the powder, inertial

2

forces dominate the interaction between the now-granules and the powder flows easily and

consistently.

A raw submicron ceramic powder, if placed in uniaxial compression, is difficult to

efficiently compact. Aside from the problems associated with poor flow and die-filling (which

are largely alleviated by the scale increasing effect of granulation), difficulties arise from

friction between the rigid and potentially rough and irregular particles.1 During compac-

tion, particles can lock together forming rigid spans which dramatically impede

consolidation.1–3 The incorporation of an organic binder phase between particles minimizes

this problem. The binder is, effectively, an interparticle lubricant.1,4 With an appropriate

binder selection, the force required to rearrange particles by shear deformation of the

interparticular binder is much less than the force required to overcome the friction between

particles in direct contact.

Given the preceding discussion, it becomes apparent that the compaction behavior of a

binder-containing granulated powder will be dominated by the type and amount of binder

I II

III

IV V VI

Time

Com

pact

Rel

ativ

e D

ensi

ty

Applied U

niaxialC

ompressive Stress

0

Figure 1.1: Schematic representation of uniaxial compaction. Region I: granule rearrangement,

II: granule elastic deformation, III: extensive plastic deformation and/or fracture

of granules, IV: particle rearrangement within compact, V: instantaneous springback

and time-dependent relaxation on ejection, VI: ultimate green part density.

3

used. Easily measured effects of the binder on compaction include the onset stress of stage III

compaction (as defined in figure 1.1) and the stress required to attain a given part density (or

the density achieved when pressed to a given compaction stress).

The polymer binder used to aid compaction and provide green strength also affects the

type and amount of post-compaction dilation. Viscoelastic polymers can react to an applied

stress by short-range bond bending or stretching as well as by long-range rearrangement of

the molecular strands. The former process occurs instantaneously, propagating through the

polymer at the speed of sound. The latter process, however, requires a finite amount of time

and depends on the difference between the polymer’s glass transition temperature (Tg) and

the working temperature. When working below the Tg of the polymer, the molecular strands

can be viewed as frozen in place and stress induced molecular rearrangement is slow; the

mechanical behavior of the polymer should be mostly elastic. When working above the Tg

of the polymer, there is sufficient random thermally induced motion of the molecular strands

to allow molecular rearrangement under stress and the mechanical response of the polymer

will be time-dependent in nature. There may also be an intermediate, rubbery stage in which

the binder can undergo a large recoverable strain.5

The dimensional change after compaction can pose at least two problems: (1) the ex-

pansion can cause the part to fail, producing defects such as end caps or delamination, and

(2) dimensional variability. While it can be argued that some amount of springback is nec-

essary to facilitate release of the part from the punch surfaces6, the amount of expansion

should be as small as possible and/or well controlled.

In order to modify the Tg of a binder, plasticizers such as glycerol, ethylene glycol, low

molecular weight poly(ethylene glycols) and water are added. Such additives lower the Tg

by interrupting the regular hydrogen bonds between the binder molecules. Furthermore, by

seprating the polymer molecules the volume increases resulting in more space for molecu-

lar motion. A water-soluble binder will, generally, adsorb water from the atmosphere to an

4

equilibrium moisture content (EMC) dependent on the ambient relative humidity (RH) and

temperature.7 Furthermore, some plasticizers (such as glycerol) are humectants, which act

to increase the moisture capacity of the binder.

Previous Work

Measurement of Compaction Processes

Many researchers have studied various aspects of the compaction of granulated ce-

ramic powders and the effect of the binder phase. Cooper and Eaton8 studied and compared

the compaction of calcite, magnesia, silica, and alumina powders. These materials range in

hardness from 3 to 9 Mohs and it was clearly shown that softer powders compacted more

efficiently. For example, pressing calcite to ~610 MPa resulted in a density of nearly 90%

while the much harder alumina powder achieved only ~69% density. They also concluded

that the compaction behavior could not be described by assuming a single compaction mecha-

nism—either the filling of large voids by particle rearrangement or the filling of small voids

by particle fracture or plastic flow, but rather a combination of these two mechanisms occur-

ring at different stages during compaction. Murthy et. al.4 reached a similar conclusion

based on compaction studies of powders composed of fairly dense, sintered fire clay gran-

ules. They decided, however, that the two mechanisms are occurring simultaneously.

The work of Lukasiewicz and Reed9 shows the basic effect of granulation on powder

compaction. For various systems, the compaction of the bulk powder was compared with

the compaction of a binder-containing spray dried version of the same powder; in some

cases, debound and partially sintered granules were also examined. The effect of granula-

tion can be clearly observed on compaction diagrams in which the relative density is plotted

versus logarithmic stress. At low compaction stresses, less than ~1 MPa, the spray dried

powders experience much slower consolidation than the corresponding bulk powder—nearly

5

zero. At a certain stress, the rate of compaction of the spray dried powder becomes greater

than that of the bulk powder. The stress at which this sudden change in slope occurs on the

compaction diagram was termed the “yield pressure” and, based on SEM examination, was

associated with the onset of plastic deformation of the granules to fill intergranular voids. At

higher compaction stresses, the compact densities of the spray dried and bulk powders even-

tually converge and further compaction of the spray dried powder is indistinguishable from

that of the bulk powder. The stress at which this occurs was termed the “joining pressure.”

The shapes of the compaction diagrams for the debound and partially sintered powders

were the same as that for the standard spray dried powder. The yield and joining pressures

shifted to higher values with increasing heat treatment, however, as the strength of the gran-

ules increased. Dynys and Halloran10 studied mixtures of alumina powders consisting of

~1 µm diameter particles and ~10 µm diameter hard aggregates in ratios from 6 wt% to

92 wt% aggregated powder. This work showed a smooth progression from bulk-type be-

havior in the nearly aggregate-free sample to granular powder behavior in the aggregate-rich

sample.

Compaction diagrams, such as those described above, have been widely used to char-

acterize powder compaction. An early example is Bruch1, who constructed these diagrams

by pressing samples to various densities, ejecting them from the die and measuring and

weighing them. This method of discreet measurements was used by others as well, includ-

ing Lukasiewicz and Reed9 and Niesz et. al.11. Youshaw and Halloran12 and Messing et. al.13

were among the first to construct such diagrams by collecting continuous data during the

compaction cycle with an instrumented testing apparatus on a strip chart recorder. This

method allows for greater accuracy and reproducibility of compaction data by eliminating

operator-induced variability. In these studies, the joining pressure at high compaction stress

observed by Lukasiewicz and Reed9 was not observed. In its place, a “high pressure

breakpoint” was observed: an increase in compaction rate at high pressure. It was postu-

6

lated that this breakpoint was an artifact arising from the elastic compliance of the testing

apparatus and load cell. Matsumoto14 pointed out that later work by DiMilia and Reed15

confirmed this hypothesis and recommended that such data be corrected for the compliance

of the load cell. More recently, Glass and Newton16 also illustrated this effect, though they

attribute the error to die plunger deformation.

Effect of Process Parameters

The effects of many powder parameters can be evaluated using compaction diagrams.

Huffine and Bonilla17 studied the compaction of powders of sodium chloride, sucrose and

quartz composed of particle sizes ranging from ~80 µm to ~650 µm. It was readily apparent

that larger particle size powders achieved a given density at lower stresses than smaller

particle sizes. They concluded that this result was due to the retardation of the compaction

mechanisms of particle elastic deformation, plastic deformation and fracture for smaller

particle sizes because of lower contact forces at a given applied pressure. Similar results

were reported by Hersey and Rees18 studying comparable powders which included smaller

particle size samples (as low as 4.2 µm for sodium chloride and as low as 2 µm for lactose).

Compaction diagrams are also used in other industries which utilize pressure forming tech-

niques such as powder metallurgy19–21, and pharmaceuticals22–24.

Compaction rate has also been shown to have an effect on compaction efficiency.

Youshaw and Halloran12 studied the effects of various parameters on the compaction of two

binder containing (poly(vinyl alcohol)–gum arabic) Mn–Zn ferrite powders, including com-

paction rate. In both powders, it was observed that consolidation became more

difficult—significantly greater stress was required to reach a given density—with increas-

ing compaction rate in the range tested (0.508 mm·min–1 to 5.08 mm·min–1). It was assumed

that this rate dependence was due to viscous flow of the binder phase.

7

Also examined were the effects of temperature and humidity. When different tempera-

ture during compaction were used (0, 25 and 50°C), higher densities were achieved at all

compaction stresses at the elevated temperatures. This was attributed to the softening of the

binder phase, and therefore, the granules, with increasing temperature. A similar trend was

observed when the water content of the binder was increased. As the binder moisture con-

tent is increased, the granules become softer, allowing higher densities to be reached at a

given pressureand temperature. Both of these observations can be attributed to the same

mechanism: the effective softening of the granules. In both cases the difference between the

pressing temperature and the characteristic glass transition temperature (Tg) of the binder is

increased. With temperature increase, the Tg remains constant while the working tempera-

ture is increased, and with moisture addition, the working temperature remains constant

while the binder Tg is decreased.

Numerous researchers have studied the influence of Tg using a variety of methods.

Nies and Messing25 tested a series of spray dried alumina powders containing mixtures of a

poly(vinyl alcohol) binder and poly(ethylene glycol) (PEG) plasticizer. The total binder

content (PVA + PEG) was kept constant at 3 wt% and moisture content was considered

constant in that all powders were conditioned at 50% relative humidity. Binder Tg was var-

ied from 47 to 17°C by varying the ratio of PVA to PEG from 100% PVA to 20% PVA and

compaction experiments were carried out at temperatures between 25 and 90°C. It was

shown that the granule yield point (AYP; equivalent to the previously discussed “yield pres-

sure”) decreased with decreasing Tg at constant temperature by a factor of ~3 over the range

studied. Additionally, it was shown that when pressing to a specified pressure, higher densi-

ties are achieved with increasing pressing temperature, and this trend becomes more

pronounced when pressing at greater than ~10°C above the Tg of the binder.

DiMilia and Reed15 examined ten spray dried alumina powder compositions contain-

ing 1.0 to 2.7 wt% binder (PVA and/or PEG) conditioned at relative humidities between 15

8

and 92%. Though the glass transition temperatures of the compositions are not reported, the

observed trends in the compaction behavior are consistent with other work: decreasing AYP

and increasing density with increasing relative humidities. They also measured other funda-

mental characteristics of these systems. The moisture content of the powders was shown to

increase only slowly at relative humidities below ~70%. By 92% RH, however, moisture

content increased to values two to four times the 70% RH value. The pore size distribution

was also studied by Hg-porosimetry. At low compaction stresses, a bimodal pore size distri-

bution was evident—large, intergranular pores and smaller, interparticle pores within

granules. As compaction stress increased through “Region II” compaction (corresponding

to region III in figure 1.1), the volume of intergranular porosity decreased rapidly. It was

also concluded that during “Region III” compaction (corresponding to region IV in

figure 1.1), above the joining pressure, the compaction mechanism is sliding and rearrange-

ment of individual particles as could be inferred from previous work9.

Perry and Carty26 studied the effect of the type and amount of plasticizer in PVA on

compaction. A constant 1 wt% PVA was used and 0.5, 1.0 and 2.0 wt% of each of the three

plasticizers (glycerin, PEG 400 and PEG 8000) were added. Relative humidity was moni-

tored and reported as being between 44 and 51% during pressing. The study showed that the

apparent yield points and joining pressures were significantly reduced with the addition of

glycerin; PEG 400 had little effect and PEG 8000 actually increased the stress at which

these occur. They also examined the breadth of the pressure range the AYP and joining

pressure occurred by measuring the full width at half maximum of the peaks in the second

derivative of the compaction curve. It was found that the PEG compositions had no effect,

but the addition of glycerin caused a broadening of the transition region. It is difficult to

fully interpret these results as the total amount of binder material was not constant.

Other examples of similar work include: Brewer et. al.27, studying spray dried barium

titanate and Mn–Zn ferrite powders with PVA/PEG binders, Whitman et. al.28, studying

9

alumina spray dried with three similar acrylic emulsion binders with Tgs of –16, 29 and

75°C, and Novak and Spino3, again examining PVA/PEG binders as well as the effect of

adding a wax-type lubricant. In this last study, it was concluded that the main effect of the

wax was to improve stress transmission during compaction and, dubiously, that PEG did

not plasticize the PVA binder.

Compaction Modeling

Many attempts have been made to model compaction curves (compaction diagrams)

with a general equation. Though rarely reported, the first of these was proposed by Walker29,

V C K P= − ( )log (1.1)

where V is the relative volume, P is the applied pressure and K and C are empirical constants

obtained by evaluating the slope and intercept of the linear region in a V vs. log(P) plot,

respectively. A mathematically identical equation was proposed by Balshin30,

ln P LV C( ) = − + (1.2)

where L is the empirically determined “pressing modulus,” which may still be found in lists

compiling compaction equations2,31–34.

Other popular compaction equations include the Heckel31 equation,

ln1

1 −

= +

DKP A (1.3)

where D is the relative sample density, and K and A are empirically determined constants,

and the Kawakita32 equation,

CabP

bP=

+1(1.4)

where C is the volumetric shrinkage, and a and b are empirically determined constants. The

constants for the Heckel equation are the slope (K) and intercept (A) of the linear region of

the compaction curve of a powder plotted as ln(1/(1–D)) vs. P. The constants for the Kawakita

10

equation are determined in a similar manner where (ab)–1 is the intercept and a–1 is the slope

of the linear region of a compaction curve plotted as P/C vs. P.

A slightly different approach was proposed by Cooper and Eaton8 in which the com-

paction equation is composed of individual terms describing the portion of compaction due

to various mechanisms:

V ak

Pa

k

P

*

exp exp= −

+ −

1

12

2 (1.5)

where V*

is the fraction volumetric compaction, a1 and a2 are the fractions of the total

compaction each mechanism (particle rearrangement and particle deformation) is respon-

sible for (a1 + a2 = 1), and k1 and k2 are the compaction stresses at which the corresponding

mechanism produces its maximum effect. The a and k values are determined for a particular

powder by a curve fitting technique of V*

vs. P. Kenkre et. al.2 derived a similar equation;

the fundamental difference arising from the assertion that particle rearrangement is not a

pressure activated process, resulting in:

V aP

ka

k

P

*

exp exp= − −

+ −

1

12

21 (1.6)

where the original form of this equation was rearranged to allow easy comparison to equa-

tion 1.5 using similar symbols. Though perhaps more physically accurate, the a and k values

must still be determined empirically.

Post-Compaction Behavior

The dimensional changes that occur after ejection have rarely been addressed in great

detail. Novak and Spino3 reported some stress relaxation and springback data for the sys-

tems they studied. Stress relaxation was reported throughout the compaction cycle as the

ratio of the load after three minutes at constant strain to the initial load. It was found that at

low stresses (<1 MPa) 30–40% of the stress was dissipated within three minutes. The amount

of stress relaxation decreased asymptotically to ~5% as the compaction pressure was in-

11

creased to ~30–100 MPa, depending on the composition. Furthermore stress relaxation in-

creased with increasing binder content; this corresponded roughly to increasing granule

hardness inferred from the AYP.

Axial springback was also reported3, though the exact measurement method is not

known. Below ~5 MPa the magnitude of the springback changes slowly and is well below

1%; for the raw alumina powder and the wax-only containing system, it appears to be zero

below ~3 and ~2 MPa, respectively. At higher compaction stresses, springback increases

rapidly with the harder granules exhibiting greater springback than the softer granules. It is

important to point out that the raw alumina powder had the greatest springback, ~50% more

than the binder containing compositions, and the wax-only containing powder had the low-

est springback, ~50% less than the binder containing powders (~3%, ~2% and ~1% for the

raw alumina, binder containing and wax-only containing compositions, respectively, at

10 MPa). This trend is due to the presence or absence of continuous particle chains as dis-

cussed earlier: particles in the raw alumina powder can not easily slide past each other at

high stresses due to friction; it becomes easier for particles to rearrange with a binder present,

however, the particles are still bonded together by the binder; with only the wax lubricant

present, it has become very easy for particles to rearrange due to very low friction and only

weak adhesion between particles.

Whitman et. al.28 reported radial springback of the acrylic binder containing systems

studied. This was evaluated by comparing the diameter of the pellet pressed to 70 MPa 1 h

after ejection to the diameter of the die. The same trend was observed that harder granules

(high Tg binder) resulted in greater springback. The magnitude of the springback, however,

was much lower at ~0.7–0.85%. This is most likely due to springback in the radial direction

being inherently smaller than in the axial direction; there may be a binder related difference,

however, that cannot be evaluated without identical testing conditions.

12

A more targeted study was done by Moeggenborg35, who introduced time after ejec-

tion as a parameter worth considering. Spray dried alumina powders “containing 4 wt% of

several different binder systems” were used (a “hard” PVA/PEG blend and a “soft” Nalco

blend). A general result of this work was the conclusion that the vast majority of springback

occurred within ten minutes of ejection and, though measurements were taken up to 130 h

after ejection, little or no dimensional change occurred after the first hour. An interesting

result shows the springback in both the axial and radial directions at pressure, in the die

after the compaction stress has been released and at various times after ejection from the

die. It was observed that the axial dimension increases by ~5.5% on releasing the compac-

tion stress. After ejection from the die, however, the axial dimension shrinks to ~2% (greater

than the dimension at pressure) as the pellet is allowed to expand in the radial direction. It is

also apparent that the radial springback is inherently smaller than the axial springback, after

ejection, by a factor of ~3–4 times for this system.

Strength Characterization

The final topic relevant to this work is the strength characterization of the granules

making up the powder and the pressed parts. The standard tensile test generally used for

metals and polymers can rarely be applied to either green or sintered ceramics. These mate-

rials have little or no ability to deform plastically during the tensile test resulting in a high

probability of additional stress components which will distort the results. Furthermore, fail-

ure is generally initiated by stress concentrations at surface flaws rather than the intrinsic

strength of the material being exceeded and it is not feasible to eliminate surface flaws in

green samples. Other, indirect, methods of tensile strength testing are used for ceramics and

glasses to avoid these problems. One such test is the 3- or 4-point bend test. The bend tests

are also sensitive to surface flaws and therefore can not provide reliable data for green

bodies.

13

The most reliable and widely used mechanical test for green ceramics25,27,36,37 and simi-

lar materials, such as pharmaceuticals38,39, is the diametral compression test. Rudnick et.

al.40, published an analysis outlining the mechanisms, advantages and limitations of this

method. The diametral compression test involves the loading of a right circular cylindrical

sample across a diameter. A uniform tensile stress is established perpendicular to the load-

ing direction. If the load is applied perfectly on lines along the cylinder the induced tensile

stress will be uniform across the full diameter. However, in this idealized case, the compres-

sive and shear stresses approach infinity at the contact points resulting in a non-tensile

failure. When the load is evenly distributed over an area along the cylinder, the compressive

and shear components near the contacts become finite, though the previously tensile stress

in these regions becomes compressive. This is, nevertheless, the desired condition. For ex-

ample, if the load is applied over an area with a width of one tenth the diameter of the

sample, a uniform tensile stress is produced over ~80–90% of the diameter with shear stresses

vanishing and the compressive stress components reaching a maximum, hydrostatic, value

at the contacts.

The diametral compression test measures the tensile strength in the radial direction. It

should be apparent, given the directional nature of the uniaxial pressing technique, that

green pellets may have anisotropic properties. Some work has been done in which the strength

of pellets in the axial direction was measured. Jarosz and Parrott38–39 compared the radial

and axial strengths of pharmaceutical tablets. The radial strength was determined by diame-

tral compression as discussed above. Axial strength was measured by what was essentially

a standard direct tensile failure test. The top and bottom surfaces of the pellet were superglued

to the testing apparatus and pulled apart. Since this method can produce unpredictable fail-

ures, measurements were only used from pellets that failed in tension within the body of the

sample. The results consistently showed significantly higher radial strengths. This result is

14

not necessarily conclusive due to the different techniques used to measure the strengths in

the different directions.

Similar concerns apply to the measurement of individual granule strengths—they are,

essentially, green ceramic bodies. Additional difficulties arise due to their small size, typi-

cally tens to a few hundred micrometers. Two techniques are commonly used to evaluate

granule strengths. It is possible to apply the diametral compression test to small spherical

samples with the availability of a sufficiently sensitive testing apparatus. Spray dried ce-

ramic granules have been studied in this way by Reetz41, Kamiya et. al.42 and Glass et. al.43,

and on green and partially sintered ceramic granules by Glass et. al.16. Another method

requires the simplifying assumption that the AYP of a compaction curve, as described above,

reflects the average strength of the granules. Work by Glass et. al.16 with partially sintered

granules showed that the strength measured by diametral compression was up to twice the

value obtained from examining the compaction curve. This is not necessarily a general

relationship between the two methods; plastically deformable granules may behave very

differently than the brittle granules studied. Walker and Reed44, working with d50 = 80 µm

glass beads, showed that at a given applied compaction stress, smaller particles in the sample

bed are under greater stress than larger particles. It was found that the smaller particles

break at lower compaction stresses and fracture of the largest particles in the distribution

occurs at approximately the stress of the apparent yield point. The implications of these

results, with respect to the compaction of plastically deformable spray dried granules, is not

yet clear.

Research Objective

The purpose of this work is to advance the fundamental understanding of the compac-

tion process as applied to ceramic processing. A systematic study of the behavior of a model

powder system will be described in which the binder content and binder Tg were varied over

15

a wide range. The mechanical response of the powder both during and after compaction, as

well as the green strength of pressed samples, will be analyzed and the influence of binder

parameters evaluated.

In order to better understand the interaction between the powder and the die during

compaction, a model will be introduced which allows the determination of the intrinsic

uniaxial compaction behavior of a powder. In the course of this analysis, a parameter which

includes the powder/die wall friction coefficient will be determined. This will ultimately

lead to the ability to predict the compaction of the powder in an arbitrarily sized die.

Finally, the behavior of individual granules and the interaction between granules will

be examined. A technique will be developed for which granule strength and intergranular

bond strength during the compaction cycle can be determined, as well as the pre-compacted

granule strength.

16

Chapter 2:

Powder Preparation and Physical Characterization

Introduction

This work deals with the testing, characterization and modeling of a model granulated

ceramic–binder powder system. This chapter details the preparation and physical character-

istics of the model system used. The ceramic component was chosen based on its physical

characteristics of a median particle size of ~0.5 µm and chemical stability. Poly(vinyl alco-

hol) (PVA) was chosen as the binder component due to extensive interest in this system and

the fact that its properties can easily be adjusted by varying the water content (this is actu-

ally a problem in industrial applications since environmental changes will affect powder

behavior). Spray drying was chosen as the granulation method for its ability to produce

spherical granules.

Experimental Procedure

Powder Production

An alumina powder (Alcoa, A16-SG) was dispersed in deionized water with an ammo-

nium polyelectrolyte at the recommended level of 0.5 mg·m–2 (Rohm & Haas, Duramax

D-3021). A stock binder solution was prepared according to manufacturer’s directions by

dissolving the PVA (Air Products, Airvol 205S; MW = 30,000–51,000, 87–98% hydro-

lyzed) in deionized water to make a ~20 wt% PVA solution (exact concentration determined

from the known mass of PVA added and the final total mass of the solution). Four weight

percent (PVA basis) glycerol was added as a plasticizer. Binder solution was added to the

alumina suspensions to produce approximately 3.5 l each of 30 vol% alumina slurries with

~2, 3, 4 and 5 wt% binder (d.w.b. alumina). The slurries were agitated with a laboratory

17

mixer for over 12 hours with octanol added drop-wise, as needed, to control foam (20–

30 gtt). The slurries were transported to the spray drying facility, agitated again with a

laboratory mixer and spray dried at a rate of ~15 l·h–1 (Bowen Engineering, Laboratory

Spray Dryer BE 665) with an inlet temperature of 340°C and an outlet temperature of 120°C.

Powder Preparation

The spray dried powders were sieved (Hosokawa, Micron AirJet Sieve; –100+200) to

obtain granules in the size range of 75–150 µm. These powders were stored at controlled

relative humidities of ~0, 31, 43, 79, and 97% RH over saturated salt solutions45 for more

than one week to adjust the moisture content of the binder to its equilibrium moisture con-

tent (EMC).

Powder and Binder Characterization

The powders were then analyzed by TGA to determine their exact binder contents.

TGA was also performed on films of the binder composition to determine their water con-

tents. The binder glass transition temperature was determined by DSC (20°C min–1) on cast

polymer films of the binder compositions using the inflection point of the step transition

(TA Instruments DCS 2920). Binder densities were measured to allow conversion between

weight percent and volume percent binder content by geometry. Granule densities were

estimated by dividing the vibrated density by 63% (representing random close packing of

spheres46).

Results and Discussion

A representative micrograph of the spray dried and sieved powder is shown in fig-

ure 2.1. The majority of the granules are smooth and appear solid. There are some granules

present which are composed of smaller granules fused together. The effect of having a small

number of these irregular granules should be minimal as they are more likely to interact

18

500 µm

Figure 2.1: Micrograph of 75–150 µm diameter spray dried powder containing ~4 wt% binder;

other compositions used have similar appearance.

Figure 2.2: Equilibrium moisture content (EMC) of the binder as measured by

thermogravimetric analysis (TGA). fit: EMC = 2.0185·exp(0.0301·RH); R2>0.99.

0 20 40 60 80 100Relative Humidity (%)

40

35

30

25

20

15

10

5

0

Equ

ilibr

ium

Moi

stur

eC

onte

nt (

wt%

)

with a regular granule than another irregular granule. This precludes the development of

detrimental effects such as the formation of long-range mechanically interlocked granule

networks.

The equilibrium moisture contents of the binders conditioned at the various relative

humidities is shown in figure 2.2. The EMC increases exponentially with relative humidity

19


1.40

1.35

1.30

1.25

1.20

1.15

1.10

1.05

1.00

Den

sity

(g·

cm–3

)

Figure 2.3: Density of the binder in the relative humidity range used. Error bars indicate 95%

confidence intervals based on 3 measurements.

from 1.8 wt% at ~0% RH to 36.1 wt% at 97% RH. The incorporation of the water in the

polymer structure causes a change in the density of the binder as shown in figure 2.3.

The measured glass transition temperatures of the binder compositions are shown in

figure 2.4. The “s” shape of the Tg vs. RH data is due to the preferential manner in which

water is incorporated in the system. Water is adsorbed slowly at first, at low relative humid-

ity, and more rapidly with increasing relative humidity as seen in figure 2.2. At the low

relative humidities (below 80% RH) water molecules have an affinity to the sites of hydro-

gen bonding between polymer chains. This disrupts bonding between polymer molecules

and causes the separation distance between the polymer molecules to increase resulting in

an increase in the volume of the system. Therefore, an accelerating decrease of the binder Tg

with increasing relative humidity is observed—as the bonding between polymer strands

becomes more disrupted by water molecules, the activation energy for polymer strand mo-

bility is reduced. The Tg begins to stabilize above 80% RH, however, despite the large quantity

of water in the system. At ~80% RH, the sites of polymer chain bonding have become

20

saturated. As the relative humidity is increased beyond 80% RH, additional water mol-

ecules—with no preferred sites available to be attracted to—increase the volume of the

material. Additional water beyond this point has little effect on the structure of the polymer

and, therefore, little effect on its glass transition temperature.

It should be noted that the glass transition temperatures reported for polymer films are

slightly lower than the actual glass transition temperatures of the binder in the spray dried

powder. The relatively large polymer samples used in the Tg measurement behave as uncon-

strained bulk solids. An identical polymer composition in the spray dried powder behaves

more as a thin polymer film on a rigid ceramic substrate. The constraint imposed by the

rigid substrate impedes molecular motion in a region near the polymer/ceramic interface

resulting in a slightly higher effective Tg. For the purpose of this study, the unconstrained

polymer Tg is reported; this point must be considered when comparing results from differ-

ent sources.


50

40

30

20

10

0

–10

–20

–30

–40

Gla

ss T

rans

itio

nTe

mpe

ratu

re (

C)

Figure 2.4: Glass transition temperature (Tg) as measured by differential scanning calorimetry

(DSC, 20°C·min–1). The inflection of the step transition is reported.

21

Summary

Four different powders were produced containing ~2–5 wt% binder. Conditioning these

powders at five different relative humidities between ~0 and 97% RH resulted in twenty

different powder compositions for study. The powders were sieved to produce a narrow

granule sized distribution. The results of the powder and binder characterization are sum-

marized in table 2.1: binder Tg, binder density, total binder content of each of the powder

compositions, and the estimated granule densities of the four different spray dried powders.

Table 2.1: Total Binder Contents†, Glass Transition Temperatures (Tg)‡ and Granule Densities of the Spray

Dried Powders

† Total Binder Content (TBC) is defined as: TBCm m m

m m m m=

+ ++ + +

PVA glycerol water

PVA glycerol water alumina

.

‡ Based on Tg measurements of unconstrained binder films.

* NBC (Nominal Binder Content) is defined as: NBCm m

m m m=

++ +

ROUND PVA glycerol

PVA glycerol alumina

.

Relative 2 wt% 3 wt% 4 wt% 5 wt%Humidity NBC* NBC* NBC* NBC*

(%) Tg (°C)‡ Powder Total Binder Content (wt%)†

0 35 2.45 3.42 4.20 5.2231 18 2.55 3.56 4.37 5.4343 6 2.61 3.63 4.46 5.5479 –25 3.03 4.21 5.16 6.4097 –32 3.72 5.16 6.32 7.80

Granule Relative Alumina Density (vol%)47 43 43 41

22

Chapter 3:

Loading and Unloading Response

Introduction

Compaction Curves

There are many possible methods to characterize the compaction behavior of a powder

system. A procedure often used in industrial processes is to simply monitor the stress re-

quired to produce the green part. This is useful in such a setting for insuring the process is in

control. A change in the measured parameter indicates that something in the process has

changed; though it is difficult to pinpoint, from this measurement alone, what has changed.

For example, if there is an increase in the stress required to produce the part, there are

several possible causes: (i) the powder has become less compliant, (ii) a change has oc-

curred in the powder which allows it to fill the die more efficiently, providing a greater

amount of powder which needs to be compacted to the specified size, requiring a higher

stress to produce the higher density green part, (iii) a change has occurred in the powder

which causes die-sticking—a section of the green part is left in the die after ejection, result-

ing in a situation similar to (ii), above.

The procedure described above can be supplemented by monitoring the ejection force

and/or weighing the green part after compaction. These additional procedures can be help-

ful in identifying what has changed in the process. This method of evaluating a powder

system and compaction process is relatively easy to implement in an industrial, on-line,

process; however, it is a qualitative measurement useful mainly for comparison of different

samples within the same process.

A more comprehensive method of characterizing a powder system for a compaction

application is to construct a compaction curve. This is essentially a stress–strain curve in

23

which the sample density is plotted as a function of the compaction stress—commonly, the

log of the compaction stress. There are several ways compaction curves can be constructed,

including using an appropriately instrumented industrial press and, in a laboratory setting,

using a universal mechanical testing machine in conjunction with a strip chart recorder or

computer interface to continuously record the stress–strain data. The former has the advan-

tage of being able to measure the compaction curve at compaction rates that are actually

seen in production. The latter, though limited to lower strain rates, is more readily available

in laboratory settings and allows a great deal of control over the applied stress or strain path.

A third procedure for measuring a compaction curve involves pressing a sample to

incrementally increasing stresses. After each compaction step, the sample/die assembly is

removed from the press and the ram position measured with calipers. After the measure-

ment, the assembly is placed into the press and compacted to the next higher stress. This

method can be used if a universal mechanical testing machine is not available; there are,

however, numerous disadvantages to the step-wise compaction approach, including: (i) ex-

treme care must be taken while handling the sample/die assembly and with the measurement

itself; the operator could introduce random error into the measurement, (ii) since the stress

must be released, the sample has undergone some amount of springback before the mea-

surement takes place, resulting in an increase in the measured sample size, and (iii) it is

especially difficult to generate reproducible measurements at low stresses where a feature

often of interest in a compaction curve generally occurs (< 5 MPa).

The most reported feature of compaction curves is the apparent yield point (AYP).

This is the stress during compaction at which the compaction curve shows a sudden dra-

matic increase in slope. For a compaction system, this occurs between approximately 0.1

and 10 MPa. At this stress, the granules comprising the powder begin to permanently de-

form by plastic strain and/or fracture. This corresponds to the beginning of stage III

compaction (as defined in chapter 1).

24

An additional feature of interest in a compaction curve is the beginning of stage IV

compaction (as defined in chapter 1). This feature is seen on the compaction curve as a

subtle decrease in slope and indicates the point at which intergranular porosity has been

essentially eliminated. The onset of stage IV compaction occurs at a sample density at or

above the initial density of the granules. Exactly when this occurs can provide additional

information on the behavior of the granules during compaction: if stage IV onset occurs at

approximately the initial granule density, granules have deformed without substantial com-

paction of the granules themselves; if stage IV onset occurs at a density significantly above

the initial granule density, granules have undergone a significant volume reduction early in

the compaction cycle.

Post-Compaction Dimensional Changes

When the compaction cycle is complete, the part must be ejected from the die. During

ejection, the ultimate compaction stress is removed and the sample is forced out of the die.

When the stress is removed, the part experiences an instantaneous elastic expansion (here-

after referred to as “springback” or “instantaneous springback”). Furthermore, if a polymer

binder phase is used in the powder system, there may be a time-dependent dimensional

expansion as residual stress within the polymer is relieved (hereafter referred to as “relax-

ation” or “time-dependent relaxation”).

Instantaneous springback data for a system is sometimes reported, while time-depen-

dent relaxation is rarely measured. Springback is usually characterized by measuring ram

displacement after removal of the compaction stress. This is a straightforward measurement

in an instrumented industrial press or laboratory universal testing machine. However, since

the sample is still in the die this measurement is not a comprehensive measure of the

springback: the part is still constrained radially by the die wall and constrained axially—at

least partially—at the die wall due to friction and the mass of the upper ram. The one case in

25

which time-dependent relaxation has been reported, the measurement was done by manu-

ally measuring sample dimensions at various times after ejection.35


Compaction and Density

Pellets were uniaxially compacted by single action in a universal mechanical testing

machine (Instron, Model 4202) at 0.5 mm·min–1 at room temperature (22±2°C). The cylin-

drical steel die had a diameter of 6.34 mm with the wall lubricated with zinc stearate and the

punch surfaces polished with 1 µm diamond paste. The die was charged with 0.16–0.19 g

powder to produce a pellet aspect ratio (height ÷ diameter) of ~0.5. A limited set of experi-

ments was also performed using a 12.6 mm diameter die charged with 1.1–1.2 g powder.

Stress–strain data was collected during compaction via a PC interface. Measurements were

corrected for the compliance of the die and testing apparatus by subtracting the stress–strain

characteristics of a blank compaction run from the sample data.14 The pellets were mea-

sured and weighed immediately after ejection to allow measurement of the instantaneous

springback on ejection.

Viscoelastic Relaxation

In order to evaluate how pellet dimensions changed with time after ejection, pellets

were pressed, ejected from the die and immediately placed in a thermomechanical analyzer

(TMA; Shimadzu, TMA-50) which allows the measurement of very small strains. For the

measurement, a constant load of 0.1 g (instrument minimum) was used to keep the probe in

contact with the sample and the height was measured as a function of time. In addition to

the main parameters examined—binder content, binder Tg, sample size, and direction within

the sample—the compaction rate was varied between “slow”, in which the ultimate load

was attained in 30 s, and “fast”, in which the load was reached in approximately 1 s. This

26

range of compaction rates showed no statistically significant effect and will not be dis-

cussed further here.

The data was fit to a decaying exponential function of the form:

y

yA

t

∞

= − −

1 exp

τ (3.1)

where y = dimension at time t, y∞ = dimension at t = ∞, A = total (normalized) dimensional

change, and τ = the characteristic relaxation time. This equation is related to the creep com-

pliance of a Voigt element (a viscoelasticity model utilizing an idealized spring and dashpot

in parallel).


Compaction and Density

Compaction curves were constructed, after correction for instrument compliance and

binder content, as a function of applied load. These curves are shown in figures 3.1–3.5 for

the 6.34 mm diameter samples and figure 3.6 for the 12.6 mm diameter samples. Figures 3.1,

3.2, 3.3, 3.4, and 3.5 shows the set of compaction curves for samples containing binders

with Tgs of 35, 18, 6, –25, and –32°C, respectively. Sections a, b, c, and d within figures 3.1–

3.5 are the compaction curves for binder contents of 2, 3, 4, and 5 wt% NBC, respectively,

for each Tg. Figures 3.6a–3.6c show the compaction curves for the system containing 3 wt%

NBC at Tgs of 35, 6 and –32°C, respectively.

With less binder in the powder, it is possible to achieve slightly higher densities at a

given compaction stress. This effect seems greater at low stresses, but this is an artifact

caused by the small difference in granule and/or fill densities between the powders. Fig-

ure 3.7 summarizes the compaction results of the various powders pressed to the maximum

stress of ~175 MPa. In this plot, it is clearly seen that decreasing Tg and decreasing the

binder content produces higher density parts for a given load.

27

The stress required to achieve a specified density is shown in figure 3.8. In this case,

52.9% was chosen for the comparison because this density was included in most of the

experimental sets. In the three cases in which this density was not attained, the stress re-

quired to achieve 52.9% density was extrapolated from the available compaction curve

0.01 0.1 1 10 100 1000Compaction Stress (MPa)

65

60

55

50

45

40

35

30

25

Rel

ativ

e A

lum

ina

Den

sity

(%

)

Figure 3.1a: Measured compaction curve and ejected densities of Tg = 35°C, 2 wt% NBC system

in a 6.34 mm diameter die.


65

60

55

50

45

40

35

30

25

Rel

ativ

e A

lum

ina

Den

sity

(%

)

Figure 3.1b: Measured compaction curve and ejected densities of Tg = 35°C, 3 wt% NBC system


28

data. Two effects are immediately evident. First, increasing the Tg increases the stress re-

quired to achieve a desired density by as much as a factor of four in the range tested. This is

due to the greater stiffness of the high Tg binder.


65

60

55

50

45

40

35

30

25

Rel

ativ

e A

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ina

Den

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Figure 3.1c: Measured compaction curve and ejected densities of Tg = 35°C, 4 wt% NBC system



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Figure 3.1d: Measured compaction curve and ejected densities of Tg = 35°C, 5 wt% NBC system


29

The resistance to deformation is due to a combination of effects. At low Tgs the binder

is easily deformed; however, it requires more stress to displace the greater volume of binder

into increasingly smaller pores. When increasing the amount of binder in a high Tg system,

it becomes increasingly difficult to deform or fracture the binder phase. This effect, in addi-


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30

tion to pore volume and size considerations, leads to the increase in stress required to attain

the target density with increasing binder volume in the high Tg case. A more subtle effect is

the increase in slope of this trend (see figure 3.8) with increasing Tg. This is due to the

increasing significance of the second result above.


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31


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To justify the aforementioned pore volume/size consideration, it is useful to examine

the degree of polymer saturation (DPS) and total solids content (TSC) of the granules and

pellets. Figure 3.9a plots the degree of polymer saturation (percent of volume not occupied

by ceramic, occupied by binder) vs. relative alumina density, compaction stress and binder

32


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Tg during compaction. The contours labeled ‘8–20 vol% Binder’ show the fixed relationship

of DPS vs. density for the range of binder contents used in this study. The three shaded

regions show the measured responses of the powders compacted to the indicated stress. The

shading within each of these regions indicates the binder Tg during compaction, ranging

33


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Figure 3.4a: Measured compaction curve and ejected densities of Tg = –25°C, 2 wt% NBC

system in a 6.34 mm diameter die.


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Figure 3.4b: Measured compaction curve and ejected densities of Tg = –25°C, 3 wt% NBC


from –32°C at the dark ends to 18 or 35°C at the light ends (white lines are approximate,

intermediate iso-Tg lines). The discrete points on this plot indicate the DPS and densities of

the starting granules. Figure 3.9b is an identically formatted plot with the exception that the

dependent variable is TSC (percent of the sample volume occupied by ceramic + binder).

34


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Figure 3.4c: Measured compaction curve and ejected densities of Tg = –25°C, 4 wt% NBC



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Figure 3.4d: Measured compaction curve and ejected densities of Tg = –25°C, 5 wt% NBC


Figure 3.9a shows that with increasing green density, the DPS increases exponentially.

For the high binder content samples compacted at ~174 MPa as much as 25% of the

non-ceramic volume must be occupied by the binder phase. At lower binder contents, the

35


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Figure 3.5a: Measured compaction curve and ejected densities of Tg = –32°C, 2 wt% NBC



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Figure 3.5b: Measured compaction curve and ejected densities of Tg = –32°C, 3 wt% NBC


DPS is as low as 8% at this compaction stress and higher green densities are achieved—

54% vs. 50% at a Tg of 35°C, and 57% vs. 55% at a Tg of –32°C.

In figure 3.9b it can be seen that the material tends from a state of constant total solids

content (for a given binder Tg and compaction state) to a more constant relative alumina

36


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Figure 3.5d: Measured compaction curve and ejected densities of Tg = –32°C, 5 wt% NBC


density as compaction progresses. The granules begin with total solids contents of ~47.5%–

50%, corresponding to relative alumina densities of ~41–47%. After compaction at

~174 MPa, the range of relative alumina density is down to ~2.5 percentage points. These

trends suggest several interesting conclusions. For given spray drying conditions (and slurry

37


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dispersion state) the droplets will dry to form granules of constant total solids content re-

gardless of the binder content (within the limits examined in this study). At low to intermediate

compaction stresses, the binder behaves simply as a solid component in the material, i.e., it

does not deform sufficiently to allow a more efficient compaction of the ceramic compo-

38


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nent. As the binder Tg is lowered, the binder does deform more readily, as expected. This is

evidenced in figure 3.9b by the changing slope of the iso-Tg lines for a given compaction

stress. At high compaction stresses, the binder has deformed sufficiently to produce a sig-

nificant deviation from a constant level of total solids content, tending toward a constant

relative alumina density.

The apparent yield points and stage IV compaction onset points were determined from

the compaction curves and are listed in tables 3.1 and 3.2, respectively. The AYP data agrees

with published results. In samples with extremely low Tg binders, the AYP occurs at low

stresses: ~0.5 MPa at Tg = –32°C. As Tg increases, the AYP also increases; to 3–4 MPa at

Tg = 35°C, in this case.

The stage IV compaction onset points show no statistically significant trend with re-

spect to binder content or binder Tg. The average value is 49.3% NBC with a range of

{46.1, 51.9}. This indicates that granules in the low binder content samples (initial granule

density ~47%) have undergone almost exclusively shear strain to this point in the compac-

tion cycle. Granules in the highest binder containing powder (initial granule density ~41%),

39

however, have experienced a small amount of bulk compression in addition to shear defor-

mation.

Instantaneous Springback

The compaction curve in figures 3.1–3.6 show the in-die density (solid line) as well as

the green density after ejection for all samples. The difference between the two values at a

given compaction stress is due to the instantaneous springback. Figure 3.10a shows the

–30 –15 015 30

Binder Tg ( C) 2

34

5

Nomin

al B

inde

r

Conte

nt (w

t%)

565452

Rel

ativ

e

Alu

min

a

Den

sity

(%)

Figure 3.7: Achievable pellet density: 6.34 mm diameter samples pressed to ~175 MPa.

2 3 4 5 6 7 8Total Binder Content (wt%)

200

175

150

125

100

75

50

25

0

Com

pact

ion

Stre

ss (

MP

a)

T g = 35 C

Tg = –32 C

Tg = 6 C

Figure 3.8: Stress required to achieve 52.9% relative alumina density in a 6.34 mm diameter

die as a function of binder content and binder Tg. (Tg = 35°C, 5% NBC data point

was extrapolated from available compaction curve data up to ~175 MPa.)

40

volumetric springback of the 6.34 mm samples as a function of Tg at various compaction

stresses. For Tg values greater than about –7°C, the sample volumes increase by ~8% after

ejection at all compaction stresses tested. When the Tg is lowered to < –7°C, the amount of

springback decreases to 4–5.5% at Tg = –32°C. The corresponding data obtained from the

12.6 mm diameter samples is shown in figure 3.10b. In this case, the springback at high Tg

values is ~6.3%, dropping off to ~5.2% at Tg = –32°C. Instantaneous springback was inde-

pendent of compaction stress for the larger samples.

The trends observed in figure 3.10 can be interpreted in terms of the stress–strain be-

havior of the binders. At Tg values lower than the usual compaction temperature (22°C) in

this study the binder is easily deformed, while at Tg higher than the working temperature the

binder is quite stiff. In the case of the high Tg samples, regardless of the stress applied to the

pellet, when the stress is released, the pellet expands elastically by approximately the same

amount. In the case of the low Tg samples, elastic springback is dependent on the compac-

tion stress.

The instantaneous volumetric springback is smaller in the larger samples. This can be

explained if a large part of the springback occurs in only a shallow surface layer of the

pellet. If the thickness of this region is independent of sample size, then the springback

relative to the total dimension of the pellet will appear to decrease with increasing sample

size. In addition, the sensitivity to Tg is smaller as evidenced by the 17% decrease in

springback at low Tg compared to the up to 50% decrease in springback at low Tg for the

smaller samples. It also appears that there is no dependence of springback on compaction

stress in the larger samples. It is likely that the sensitivity to compaction stress diminished

to the point where the differences could not be seen in the limited data set obtained for the

12.6 mm diameter samples.

Figure 3.11 shows the relative springback in the axial and radial directions. The equiva-

lent isotropic linear springback, calculated from the volumetric springback, is included for

41

30 35 40 45 50 55 60 65Relative Alumina Density (vol%)

30

25

20

15

10

5

0

Deg

ree

of P

olym

er S

atur

atio

n (v

ol%

)

Compaction Stress

6 MPa

48 MPa

174 MPa

Binder Glass Transition Temperature

During Compaction18 C–32

C

35C

35C

–32C–32

C

Figure 3.9a: Degree of polymer saturation as a function of compact density, binder Tg during

compaction and compaction stress.

30 35 40 45 50 55 60 65Relative Alumina Density (vol%)

70

65

60

55

50

45

40

35

Tota

l Sol

ids

Con

tent

(vo

l%)

Compaction Stre

ss

6 MPa

48 MPa

174 MPa

Binder Glass T

ransition Tem

perature

During Compactio

n

6 C

–32 C

35C

35C

–32C

–32C

Figure 3.9b: Total solids content in granules and compacts as a function of compact density,

binder Tg during compaction and compaction stress.

42

comparison. In the 6.34 mm diameter samples (figure 3.11a), springback in the radial direc-

tion is ~9–13% of the springback in the axial direction. In the 12.6 mm diameter samples

(figure 3.11b), the radial springback is ~25% of the axial springback.

In the 6.34 mm diameter samples, ~89% of the volume change during ejection occurs

due to springback in the axial direction. This is due to the nature of the compaction and

ejection processes. First the compaction stress is removed, allowing the pellet to expand in

the axial direction while it is still constrained in the radial direction. If the pellet contains

enough strain energy to overcome the static friction with the die wall surface, the pellet will

either (i) expand uniformly or (ii) form cracks between the regions which were able to slip

and the regions which were not able to slip (delamination). If the pellet does not contain

enough energy to overcome the die wall friction, the pellet should simply bulge in the cen-

ter. Next, the pellet is ejected from the die by an external force. Since the ejection stresses

force the pellet to overcome the static friction barrier with the die wall, the pellet is now

Table 3.2: Stage IV Compaction Onset

6.34 mm ∅ samples 12.6 mmRelative Nominal Binder Content (%)

Humidity Tg 2 3 4 5 3(%) (°C) Stage IV Onset (Relative Alumina Density, vol%)

0 35 50.0 50.7 46.1 47.9 46.931 18 47.9 49.3 48.9 49.143 6 49.1 49.7 51.4 50.4 49.679 –25 49.8 48.7 49.4 51.597 –32 50.2 48.8 48.7 48.4 51.9

Table 3.1: Apparent Yield Points

6.34 mm ∅ samples 12.6 mmRelative Nominal Binder Content (%)

Humidity Tg 2 3 4 5 3(%) (°C) Apparent Yield Point (MPa)

0 35 3.5 2.8 4.0 4.0 1.531 18 1.5 1.5 2.6 2.643 6 1.3 1.5 1.3 1.6 0.9079 –25 0.70 0.85 0.73 0.6597 –32 0.60 0.38 0.40 0.50 0.28

43

freer to expand axially while still fully constrained radially. At this point, much of the stress

within the pellet has been relieved as strain in the axial direction. When the pellet is finally

expelled, it is free to relieve the remaining stress by expanding in the radial direction. Ra-

dial springback can cause the pellet faces to suddenly shear against the die punch surfaces,

–35 –21 –7 7 21 35Tg ( C)

9

8

7

6

5

4

3

Inst

anta

neou

s Sp

ring

back

(vol

%)

175 MPa

110

MPa

50M

Pa

15M

Pa

Figure 3.10a: Instantaneous volumetric springback on ejection vs. binder Tg and compaction

stress for samples pressed in a 6.34 mm diameter die to an aspect ratio of ~0.5.

–35 –21 –7 7 21 35Tg ( C)

9

8

7

6

5

4

3

Inst

anta

neou

s Sp

ring

back

(vol

%)

Figure 3.10b: Instantaneous volumetric springback on ejection vs. binder Tg and compaction

stress for samples pressed in a 12.6 mm diameter die to an aspect ratio of ~0.4.

44

0 40 80 120 160 200Compaction Stress (MPa)

8

6

4

2

0

Inst

anta

neou

s Sp

ring

back

(%

)

Axial

Radial

EquivalentIsotropic

Figure 3.11a: Example illustrating the anisotropy of instantaneous springback on ejection vs.

compaction stress for 3 % NBC, Tg = 6°C samples pressed in a 6.34 mm diameter

die to an aspect ratio of ~0.5.

0 40 80 120 160 200Compaction Stress (MPa)

8

6

4

2

0

Inst

anta

neou

s Sp

ring

back

(%

)

Axial

Radial

EquivalentIsotropic

Figure 3.11b: Example illustrating the anisotropy of instantaneous springback on ejection vs.

compaction stress for 3 % NBC, Tg = 6°C samples pressed in a 12.6 mm diameter

die to an aspect ratio of ~0.5.

which is necessary for a clean release of the pellet from the punches. If springback is exces-

sive, however, the pellet is more likely to form radial cracks or end-capping defects.

45

0 100 200 300 400 500 600 700Time After Ejection (s)

2.297

2.296

2.295

2.294

Sam

ple

Hei

ght

(mm

)

Figure 3.12: Example raw data and curve fit of the time-dependent strain recovery after ejection.

Example is of Tg = 6°C, 3% NBC system, 6.34 mm diameter pellet compacted at

~175 MPa, relaxation in the axial direction. Fit equation is of the form:

h = h∞-A´·exp(-t⁄τ), where t is the time after ejection, h is the sample height at time = t,

h∞ is the sample height at time = ∞, A´ is the total recoverable time-dependent

strain, and τ is the characteristic relaxation time.

The springback anisotropy is still apparent in the 12.6 mm diameter samples, but the

degree of the anisotropy is reduced relative to the smaller samples. This can also be ex-

plained by the limited region of expansion concept. If the bulk of the pellet changes only

slightly in size, then differences in expansion between the faces of the pellet and the wall of

the pellet will appear less extreme in larger pellets.

Time-Dependent Relaxation

Figure 3.12 shows an example of the data collected showing the time-dependent strain

relaxation that pellets undergo after ejection. Figure 3.13 summarizes the time-dependent

relaxation parameters for pellets pressed at 0, 31, 43, 79 and 97% RH (Tg=35, 18, 6, –25

and –32°C). The data compares the following parameters: 5 wt% vs. 3 wt% nominal binder

content, axial relaxation vs. radial relaxation, and 6.34 mm vs. 12.6 mm diameter pellets.

Figure 3.13a shows the total (normalized) dimensional change (A) of the part after ejection

46

–35 –21 –7 7 21 35Tg ( C)

1600

1280

960

640

320

0

“τ”

Par

amet

er (

s)

3% NBC/6.34 mm/axial5% NBC/6.34 mm/axial5% NBC/6.34 mm/radial3% NBC/12.6 mm/axial

Figure 3.13b: The characteristic relaxation time as a function of binder Tg for various samples as

indicated in the legend. (The “τ” parameter in the function: h = 1-A·exp(-t⁄τ), where

t is the time after ejection, h is the sample height (or diameter) at time = t, A is the

normalized recoverable time-dependent strain, and τ is the characteristic relaxation

time.) Error bars indicate 95% confidence intervals based on 4–7 measurements.

Series are offset for clarity.

–35 –21 –7 7 21 35Tg ( C)

0.020

0.015

0.010

0.005

0.000

–0.005

“A”

Par

amet

er (

dim

ensi

onle

ss)

3% NBC/6.34 mm/axial5% NBC/6.34 mm/axial5% NBC/6.34 mm/radial3% NBC/12.6 mm/axial

Figure 3.13a: The total, fractional, recoverable time-dependent strain as a function of binder Tg

for various samples as indicated in the legend. (The “A” parameter in the function:

h = 1-A·exp(-t⁄τ), where t is the time after ejection, h is the sample height (or diameter)

at time = t, A is the normalized recoverable time-dependent strain, and τ is the

characteristic relaxation time.) Error bars indicate 95% confidence intervals based

on 4–7 measurements. Series are offset for clarity.

47

from the die. It can be seen that the amount of relaxation decreases with decreasing Tg. For

example, the axial relaxation of the 5% NBC pellets decreases from ~1.6% at Tg = 35°C to

~–0.2% at Tg = –32°C (negative relaxation indicates that the observed dimension decreases

with time). The following statements can be made with 95% confidence: there is a differ-

ence in the relaxation behavior between the axial and radial directions, and binder content

and sample size also have an effect. Figure 3.13b shows the corresponding characteristic

relaxation times (τ) for the pellets. In this case there is no discernible trend in the behavior

as a function of any of the parameters considered.

Decreasing the Tg relative to the working temperature reduces the total time-dependent

dimensional change (figure 3.13a). The variability of the relaxation time (figure 3.13b) in-

creases with decreasing Tg but this is most likely an artifact caused by the decrease in the

magnitude of dimensional change. Relaxation in the radial direction is significantly less

than in the axial direction. The arguments used for the similar result for instantaneous

springback can also be applied here.

The amount of binder also has a significant effect on the viscoelastic parameters with

lower binder content samples exhibiting significantly less time-dependent relaxation. How-

ever, the amount of time-dependent relaxation was significantly less for the 12.6 mm diameter

samples as compared to the smaller samples. The limited–expansion–region concept can

also be applied here. One result that is not fully understood is the negative relaxation at the

two lowest Tg values. Though shrinkage of the part in the axial direction was observed in the

previous work, it was complemented an expansion in the radial direction. Further investiga-

tion of this may be necessary while remaining aware that other factors may be responsible

for this result.

48

Summary and Conclusion

This study clearly shows the effects of binder content and binder plasticity on the

compaction behavior of a PVA–containing spray dried ceramic powder and the bulk proper-

ties of the pressed part. When the binder content is increased, the compaction stress required

to attain a specified density increases due to the greater amount of binder which needs to be

deformed and displaced.

Modifying the binder plasticity also has a great effect on the compaction of the pow-

der. When pressing a powder with a binder Tg much lower than the working temperature,

the binder is easily deformed and high green densities can be achieved at moderate stresses.

As binder Tg is increased, it becomes stiffer and more brittle, resulting in the requirement of

a higher compaction stresse to achieve a given density.

Sample size was shown to have a significant effect on compaction of the powder. In the

larger samples, the surface area to volume ratio is smaller than in the smaller samples reduc-

ing the influence of die wall friction on the bulk compaction of the powder.

The dimensional change of the pellets after ejection from the die occurs mainly in the

axial direction and the total dimensional change is dominated by the instantaneous springback.

Factors influencing the amount of time-dependent relaxation include the amount of binder,

the binder plasticity, and the direction in the pellet. The rate at which the pellets were com-

pacted did not have an effect on the relaxation behavior in the range of rates examined.

However, it is likely that a more substantial change in compaction rate would produce a

detectable change in the relaxation behavior.

In order to achieve optimal control over the pellet dimensions for this binder system, it

is desirable to carry out the process between ~20 and 60% RH (26°C ≥ Tg ≥ –10°C). Below

~60% RH, the instantaneous springback is rather large at ~8% but this value is not sensitive

to the compaction pressure or changes in relative humidity (below ~60% RH). For example,

49

if, in a process, a part is mechanically pressed with the powder normally stored and used at

45% RH; if environmental conditions change over time to 35% RH, a higher stress will be

required to achieve the desired density, yet the amount of springback will not have changed

and a similar part will be produced despite the change in RH and compaction stress. At

relative humidities below 20%, however, the time–dependent relaxation can become no-

ticeably large. Depending on the specific application of the product, other factors also need

to be considered. For example, processing at the low end of this relative humidity range

may require more compaction stress to achieve the desired density and will result in less

density gradients, while working at the high end of this range will allow lower compaction

stresses and higher aspect ratio parts, but will also cause larger density gradients.

50

Chapter 4:

A Model for Compaction Curve Prediction

Introduction

Background

Compaction curves such as those shown in chapter 3 are useful tools for qualitative

evaluation of compaction processes. It is possible to compare the compaction efficiencies of

different systems, or of different samples, of the same system under different conditions

(such as temperature and relative humidity). Examining merely one compaction curve, how-

ever, does not reveal the underlying factors responsible for the behavior of the system. Nor

does a compaction curve accurately represent how the interaction between powder and die

wall and the sample size affect the measured stress–strain relationship. These consider-

ations also make it difficult to predict the compaction behavior of a powder under different

conditions.

There are numerous equations available, as discussed in chapter 1, that attempt to de-

scribe the stress–stain relationship during compaction. Many such equations provide a good

fit to experimentally measured data; however, all of these models rely on adjustable param-

eters that have no fundamental physical basis.

At the other end of the spectrum of available compaction models are discrete element

and finite element models (DEM and FEM, respectively). In DEM, the stress state of each

particle in the system and all of the contact forces between them are calculated for each time

or stress increment. Though this approach is mathematically fairly basic, simply the num-

ber of calculations required to simulate a useful sample size often makes the application of

this method impractical. For example: an initially 25% dense, 1 cm3 sample composed of

0.5 µm diameter particles (assuming spherical, monodisperse) contains 3.82×1012 particles.

51

The stress state of each of these particles, and the interaction between them, and between

them and the die wall must be calculated for up to several hundred steps (time or stress

increments). If it is (optimistically) estimated that ten calculations are required per particle

and it would take an efficient algorithm ten computational cycles per calculation, a modern

desktop computer with an average sustained speed of 2 GFLOPS would take over 2 days

per increment—a rate of approximately 165 increments per year. This is clearly a drawback

of this approach.

FEM, on the other hand, treats the sample as a continuum which is divided into a small

number (compared to DEM) of simple geometric elements. For this reason, there is a trac-

table number of elements which need to have their stress state calculated for each time or

stress increment. The disadvantage of this approach is that the mechanical response of the

hypothetical continuum material must be known and must be a sufficiently accurate repre-

sentation of the response of the actual material. The process of simulating the material

requires extensive testing of the actual system, generally requiring large amounts of the

powder, to determine the parameters required for the particular material model used.

The methods mentioned above all have their place and can be extremely informative.

DEM can be applied to reasonably large two-dimensional systems, as well as small three-

dimensional systems, which is useful for some aspects of fundamental research in this area.

Useful results of FEM include the determination of various system characteristics such as

the failure surface, cohesion and the angle of internal friction, and the prediction of density

gradients within the sample.

Overview of Principle

The procedure for modeling compaction described below complements the other avail-

able options. This method takes into account the stress distribution within the sample based

on a theoretical model47 and the frictional force between the powder and die wall which

52

opposes compaction. The basis for the development of this analysis is the assumption that a

given powder system has a certain compaction behavior (represented by a compaction curve)

that is independent of sample size or shape. This intrinsic compaction curve would also be

independent of compaction rate were it not for the presence of organic binder phases, which

have time/rate dependent properties. In the case of a measured compaction curve, the intrin-

sic stress required to compress the sample to a given average density would be the difference

between the actual (applied) stress and the opposing stress developed by friction at the

powder to die wall interface.

In order to calculate the opposing stress, the following parameters must be known: the

friction surface area, the effective powder to die wall friction coefficient (µ) and the magni-

tude of the normal force the powder exerts on the die wall (radial stress). The magnitude of

the normal force at the die wall can be represented by the stress parallel to the die wall (axial

stress) multiplied by the powder fluidity factor (k), which is the ratio of the induced radial

stress to the applied axial stress. The friction surface area is known from the perimeter of

the sample and its height at any given time. The values of k and µ, however, are unknown

and are likely to vary with sample density and applied stress. It will be shown that these two

quantities always appear together in the governing equation as the product, k·µ.

There are, therefore, only two unknown quantities: the intrinsic stress required to achieve

a given density (σz,eff) and the product k·µ. To solve for these quantities, two independent

equations are required. Each equation is based on an experimentally measured compaction

curve; the two curves are for different sized samples of the powder system. By changing the

sample size (e.g., by changing the aspect ratio), the effect of friction at the die wall on bulk

compaction (measured compaction curve) will change; this allows the construction of inde-

pendent equations.

In the course of this analysis the product k·µ—as a function of sample density during

compaction—is determined. Once this relationship is known, an expression for the intrinsic

53

compaction curve can be written. The k·µ relationship is a useful tool in and of itself. It can

be used to evaluate the effectiveness of processing additives such as binders and lubricants.

It is desirable to have this quantity small throughout the compaction cycle; this would indi-

cate that friction between the die wall and powder and/or the normal force on the die wall is

low, resulting in minimal stress gradients within the sample and a very uniform density

green part.

An additional powerful feature of this analysis is the potential to predict the compac-

tion behavior of an arbitrary sized (or shaped) sample of the system. This statement must be

qualified due to limitations of the model describing the stress distribution within the sample.

At extreme aspect ratios or very large sizes, the stress distribution model may fail. For

example, a very long—relative to its diameter—cylindrical sample may form a plug in the

region near the moving ram. This would act more as an extension of the ram rather than a

part of the sample being compacted. Such effects would, of course, disrupt the ability of the

model to determine k·µ and, consequently, σz,eff in the first place. It may be possible, how-

ever, to account for these deviations in the stress distribution function.


Analytical Procedure

This analysis originates with the straightforward assumption that a relationship exists

between an effective stress function (σz,eff) and sample density (ρ) which is the sum of the

applied compaction stress (σz0) and an opposing stress due to friction between the powder

and the die wall (σz,f):

σ σ σz eff z z f, ,= −0 (4.1)

The function σz,eff (shorthand for σz,eff(ρ)) describes the intrinsic compaction behavior of the

powder, i.e., the compaction behavior of the powder in a frictionless die or a die of infinite

54

cross-section. It is clear, since σz,eff is the function to be determined and σz0 is the known

applied stress, that the bulk of this analysis deals with the proper determination of σz,f.

If the shear stress at the powder to die wall interface (σ f ) is known, the stress opposing

compaction (σz,f) can be calculated by adjusting for the difference between the sample wall

area and sample cross-sectional area, which, for the cylindrical geometry used in this analy-

sis is:

σ σz f f

t

R, = 2(4.2)

where t is the sample thickness and R is the sample radius. The shear stress at the powder to

die wall interface is simply:

σ σ µf z R k= ( ) × (4.3)

where σ z R( ) is the average axial stress at r (radial distance from the centerline) = R, k is the

ratio of the induced radial stress to the axial stress and µ is the effective coefficient of

friction between the powder and the die wall. σ z R( ) will be determined using:

σ z z r f z r C z,( ) = ( ) × + ( )2 (4.4)

which describes the axial stress (σz) at any point (z = height from the bottom, fixed, end of

the sample, and r) within a cylindrical sample.47 This model of the stress distribution within

a sample in uniaxial compaction assumes a stress distribution along the centerline of C(z)

with the stress changing radially as the distance from the centerline squared. The magnitude

of the radial change in stress is given by f(z). The two unknown functions (C(z) and f(z))

must be determined.

The function f(z) is determined first by averaging equation 4.4 over the cross-sectional

area:

σπ

πz

R

zR

f z r C z r dr( ) = ( ) × + ( ) ×( )∫122

3

0

(4.5)

55

σ z

R

zR

f z r C z r dr( ) = ( ) × + ( ) ×( )∫22

3

0

(4.6)

σ zr

R

zR

f z r C z r( ) = ( ) × + ( ) ×

=

2 14

122

4 2

0

(4.7)

σ z zR

f z C z( ) = ( ) + ( )2

2(4.8)

where σ z z( ) is the average transmitted axial stress at height z in the sample. Equation 4.8 is

then equated to the following relationship48:

σ σ µz zz

k t z

R( ) = − −( )

0 2exp (4.9)

σ σ µ µz zz

k z

R

k t

R( ) = −

0 2 2exp (4.10)

σ σ µ µz zz

k z

R

k t

R( ) =

−

0 2 2exp exp (4.11)

which is a generally accepted and reasonably accurate approximation of the transmitted

axial stress in a cylindrical sample during uniaxial compaction:

σ σ µ µz zz

Rf z C z

k t

R

k z

R( ) = ( ) + ( ) = −

20

22 2

exp exp (4.12)

By rearrangement and the introduction of the simplification:

C z C( ) ⇒ (4.13)

the function f(z) can be written:

f zR

k t

R

k z

RCz( ) = −

−

2 2 22

0σ µ µexp exp (4.14)

Replacing the function C(z) with a simple constant, C, is in accordance with the theory

behind equation 4.4; there is no shear along the centerline, therefore C(z) must be constant.

In a real system, however, there are local perturbations which invalidate equation 4.13. As a

practical matter for this analysis, though, it remains a reasonable conversion to make. The

value of C must still be determined, or, more practically, written as a universally applicable

56

variable. Substituting equation 4.14 back into equation 4.4 (and further applying the con-

version of equation 4.13) the axial stress distribution within the sample is:

σ σ µ µz zz r

r

R

k t

R

k z

RC C, exp exp( ) = −

−

+2 2 22

20

(4.15)

σ σ µ µz zz r

r

R

k t

R

k z

R

r

RC C, exp exp( ) = −

− +2 2 2 22

20

2

2 (4.16)

σ σ µ µz zz r

r

R

k t

R

k z

RC

r

R, exp exp( ) = −

+ −

2 2 21

22

20

2

2 (4.17)

At the height within the sample at which f(z) becomes zero (z´), equation 4.4 becomes:

σ z z r C′( ) =, (4.18)

Defining z´ as the product of the thickness of the sample and the fractional height within the

sample at which f(z) becomes zero (q):

′ =z qt (4.19)

C can be determined:

σ σ µ µz zz r C

r

R

k t

R

k qt

RC

r

R′( ) = = −

+ −

, exp exp2 2 2

122

20

2

2 (4.20)

Cr

R

r

R

k t

R

k qt

Rz

2 2 2 22

2

2

20= −

σ µ µ

exp exp (4.21)

Ck t

R

k qt

Rz= −

σ µ µ0 2 2

exp exp (4.22)

Substituting equation 4.22 into equation 4.17, the complete form of σz(z,r) can be written:

σ σ µ µ σ µ µz z zz r

r

R

k t

R

k z

R

r

R

k t

R

k qt

R, exp exp exp exp( ) = −

+ −

−

2 2 21

2 2 22

20

2

20

(4.23)

σ σ µ µ µz zz r

r

R

k t

R

k z

R

R

r

k qt

R, exp exp exp( ) = −

+ −

02

2

2

2

2 2 22

12

(4.24)

Thus C has been replaced by the normalized parameter, q. Though q is still technically a

third unknown quantity, it is not necessarily required to determine its value experimentally

for every system. A reasonable value (or function of sample density during compaction)

may be chosen for q based on experience, published values or relevant FEM simulations.

57

Solving equation 4.24 at r = R results in:

σ σ µ µ µz zz R

k t

R

k z

R

k qt


−

22 2 1

220

(4.25)

σ σ µ µ µz zz R

k t

R

k z

R

k qt


−

0 22

2 2(4.26)

σ σ µ µ σ µ µz z zz R

k t

R

k z

R

k t

R

k qt

R, exp exp exp exp( ) = −

− −

2

2 2 2 20 0(4.27)

which can be averaged over the sample thickness:

σ σ µ µ σ µ µz z z

tR t

k t

R

k z

R

k t

R

k qt

Rdz( ) × = −

− −

∫ 2

2 2 2 20 0

0exp exp exp exp (4.28)

σ σ µ µ σ µ µz z

t

z

tR t

k t

R

k z

Rdz

k t

R

k qt

Rdz( ) × = −

− −

∫ ∫2

2 2 2 20

0

0


σ σ µµ

µ σ µ µz z

z

t

zR tk t

R

R

k

k z

Rt

k t

R

k qt

R( ) × = −

− −

=

22

22 2 20

0


σ σ µµ

µµ

σ µ µz z zR t

k t

R

R

k

k t

R

R

kt

k t

R

k qt

R( ) × = −

−

− −

2

22

22

2 20 0exp exp exp exp (4.31)

σ σµ

µ µ σµ

µ σ µ µz z z zR t

R

k

k t

R

k t

R

R

k

k t

Rt

k t

R

k qt

R( ) × = −

− −

− −

2

22 2

22

2 2 20 0 0exp exp exp exp exp (4.32)

σ σµ

µ σ µ µz z zR t

R

k

k t

Rt

k t

R

k qt

R( ) × = − −

− −

0 012 2 2

exp exp exp (4.33)

σ σµ

σµ

µ σ µ µz z z zR

R

k t

R

k t

k t

R

k t

R

k qt

R( ) = − −

− −

0 0 02 2 2exp exp exp (4.34)

σ σµ

σ µµ

µz z zR

R

k t

k t

R

R

k t

k qt

R( ) = − −

+

0 0 2 2exp exp (4.35)

σ σµ

µµ

µz zR

R

k t

k t

R

R

k t

k qt

R( ) = − −

+

0 2 2exp exp (4.36)

σ σµ µ

µ µ µz zR

R

k t

R

k t

k t

R

k t

R

k qt

R( ) = − −

− −

0 2 2 2exp exp exp (4.37)

Equation 4.37 can now be substituted into equation 4.3:

σ σ µ µ µ µf z

R

t

R

t

k t

Rk

k t

R

k qt

R= − −

− −

0 2 2 2exp exp exp (4.38)

58

Finally, equation 4.38 can be inserted into equation 4.2 which can than be substituted into

equation 4.1:

σ σ σ µ µ µ µz eff z z

k t

R

k t

R

k t

R

k qt

R, exp exp exp= − − −

− −

0 02 12 2 2

(4.39)

σ σ µ µ µ µz eff z

k t

R

k t

R

k t

R

k qt

R, exp exp exp= −

+ −

−

0 22 2 2 2

1 (4.40)

Equation 4.40 is a final, usable form; however, it simplifies matters slightly in this

analysis to make a few substitutions: diameter (d) for radius (R) and sample density (ρ) for

sample thickness (t):

ρπ

= 42

m

d t(4.41)

σ σ µπρ

µπρ

µπρ

µπρz eff z

k m

d

k m

d

k m

d

k qm

d, exp exp exp= −

+ −

−

03 3 3 32

16 16 16 161 (4.42)

Experimental Verification

This analysis was tested by experimentally measuring two compaction curves each of

two compositions, Tg = –25°C, 5 wt% NBC and Tg = 6°C, 3 wt% NBC, and calculating the

k·µ function and the intrinsic compaction curve. All experiments were done in the same

6.34 mm diameter, zinc-stearate lubricated steel die to avoid complications which may have

arisen due to differences in die wall surface finish between two different dies. Figure 4.1

shows the two measured compaction curves for the Tg = –25°C, 5 wt% NBC samples. One

sample used a charge of 0.0705 g powder, producing a pellet with a final aspect ratio (t/d)

of 0.14. The other sample used a 0.3600 g charge, resulting in a pellet with an aspect ratio

of 0.77. For the Tg = 6°C, 3 wt% NBC powder, the two compaction curves were produced

from sample charges of 0.1069 g (AR = 0.23) and 0.3631 g (AR = 0.80).

For each sample, the mass, diameter, q, and the measured compaction curve (σz0(ρ))

were entered into the right-hand side of equation 4.42. In various published articles depict-

59

ing measured and calculated stress or density variations within a cylindrical sample49–52, the

height at which the stress across the entire cross-section appeared most constant was at

approximately two-thirds of the way up the height of the sample from the stationary ram.

Therefore, a value of 0.65 was chosen for q for all calculations. The two resulting equations

(for each powder composition) were set equal to each other and solved for k·µ (as a function

of sample density during compaction). This solution is shown graphically in figure 4.2. The

same procedure applied to the Tg = 6°C, 3 wt% NBC powder resulted in the k·µ solution

shown in figure 4.3.

Substituting the solution for k·µ and one of the original compaction curves (σz0(ρ)) into

equation 4.42 results in an expression for the intrinsic compaction behavior. This compac-

tion curve is shown for the Tg = –25°C, 5 wt% NBC powder in figure 4.4.

1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5Alumina Density (g·cm–3)

1000

100

10

1

0.1

Com

pact

ion

Stre

ss (

MP

a)

Figure 4.1: Measured compaction curves for Tg = –25°C, 5 wt% NBC samples. The aspect

ratio reported is the final aspect ratio of the pressed pellet. Note that this and all

other compaction curve figures in this chapter plot the compaction stress as a

function of sample density.

60


1.00

0.75

0.50

0.25

0

k·µ

(dim

ensi

onle

ss)

Figure 4.3: The k·µ solution for Tg = 6°C, 3 wt% NBC samples.

By substituting the expression for the intrinsic compaction curve into the left-hand

side of equation 4.42 and solving for σz0, the following equation results:

σσ ρ

µπρ

µπρ

µπρ

µπρ

zz eff

k m

d

k m

d

k m

d

k qm

d

0

3 3 3 3216 16 16 16

1

=( )

−

+ −

−

,

exp exp exp

(4.43)


1.00

0.75

0.50

0.25

0

k·µ

(dim

ensi

onle

ss)

Figure 4.2: The k·µ solution for Tg = –25°C, 5 wt% NBC samples.

61

Since σz0(ρ) and k·µ have been determined, all that is required to predict the compaction

curve of the powder for a particular sample size is to enter the sample diameter and mass

into equation 4.43.

Computer Utilization

In addition to the more routine roles of computers, such as compaction curve data

collection with LabVIEW (National Instruments) and data manipulation and preparation in

Excel 98 (Microsoft®), they were indispensable in performing the calculations required for

this analysis. Mathematica® 3.0.1 (Wolfram Research) was used on a Power Macintosh

(Apple Computer) for all calculations.

The algorithm used exactly follows the procedure outlined above. Second, third or

fourth degree polynomial curve fits were used (the lowest required for a particular instance)

in the following three situations: (1) when there was discrete data which needed to be handled

analytically (e.g., the measured compaction curves), (2) when an analytical solution could

not be expressed (e.g., the solution of k·µ as a function of ρ) and (3) to simplify expressions


1000

100

10

1

0.1

Com

pact

ion

Stre

ss (

MP

a)

Figure 4.4: The two original, measured, compaction curves and the calculated intrinsic

compaction curve for the Tg = –25°C, 5 wt% NBC system.

62

(e.g., the solution for the intrinsic compaction curve). For this reason, results may not be

extrapolated beyond the density limits imposed by the range of the original measured com-

paction curves. The Mathematica® code written for this analysis is listed in the appendix.


Evaluation of k·µ

The calculated k·µ curves for the two compositions (figures 4.2 and 4.3) have similar

characteristics. Below sample densities of ~1.8 g·cm–3 (~45% relative alumina density) k·µ

is small and reasonably constant. Above this density, the value of k·µ increases rapidly. In

the low density region, the effective friction coefficient is the dominant factor. At low den-

sities, granules still have a highly curved surface and contact the die wall at small contact

patches. The actual friction surface area is much smaller than the friction surface area calcu-

lated from the sample geometry. Therefore, the effect of friction is calculated for a much

greater area than it actually acts on. This results in a small effective friction coefficient.

Furthermore, the system is sufficiently compliant that the applied stress efficiently com-

presses the sample in the axial direction; the induced radial stress component is therefore

also small.

When the sample density exceeds the initial density of the granules, at approximately

45% density, little intergranular porosity remains. The granule contact patches on the die

wall completely cover the friction surface area calculated from sample geometry. The effec-

tive friction coefficient becomes the actual friction coefficient of the system, which is constant.

Above this density, k·µ is controlled by the powder fluidity factor. As sample density in-

creases at this late stage of compaction, the applied stress becomes decreasingly efficient in

its ability to further compress the sample (as the sample becomes less compliant). This

stress must be redirected and manifests itself as an increasing radial stress component.

63

The difference in the k·µ functions between the two powder compositions is due to the

differences in the amount and the characteristics of the binder system. The Tg = –25°C,

5 wt% NBC system is more adhesive and more easily deformed than the Tg = 6°C, 3 wt%

NBC system. These characteristics conspire to increase both the friction coefficient and the

induced radial stress. The results of this analysis do not contradict the results discussed in

chapter 3 (lower Tg binder compositions allow higher density pressed parts at a given com-

paction stress). Analysis of the k·µ data expands the understanding of the process and indicates

that low Tg systems will contain more extreme density variations than their high Tg counter-

parts.

Compaction Curve Prediction

The predictive ability of this analysis is illustrated in figures 4.5 and 4.6 (Tg = 6°C,

3 wt% NBC powder and Tg = –25°C, 5 wt% NBC powder, respectively). Figure 4.5 shows

the predicted (unbroken curve) and measured (dashed curve) compaction curves superim-

posed on the two initial compaction curves and the calculated normalized compaction curve.


1000

100

10

1

0.1

Com

pact

ion

Stre

ss (

MP

a)

CalculatedMeasured

Figure 4.5: The black curves illustrate the predictive ability of this analysis for the Tg = –25°C,

5 wt% NBC system. The solid line is the calculated compaction curve; the dashed

line is the experimentally measured compaction curve for the same conditions.

64


1000

100

10

1

0.1

Com

pact

ion

Stre

ss (

MP

a)

CalculatedMeasured

Figure 4.6: The normalized compaction curve (single curve in magenta) and predicted

(unbroken curves) and corresponding measured (dashed curves) compaction curves

for the Tg = 6°C, 3 wt% NBC system.

The test sample used a charge of 0.1417 g powder, which produced a pressed pellet with an

aspect ratio of 0.30.

Figure 4.6 shows the calculated and measured compaction curves for four test samples

of the Tg = 6°C, 3 wt% NBC powder, as well as the intrinsic compaction curve calculated

for this system. The four test samples had masses and resulting pellet aspect ratios of 0.1509 g

(AR = 0.32), 0.2113 g (AR = 0.45), 0.2671 g (AR = 0.57), and 0.3126 g (AR = 0.67).

For all five test samples (one for the Tg = –25°C, 5 wt% NBC composition and four for

the Tg = 6°C, 3 wt% NBC composition) the correlation coefficients of the predicted com-

paction curves are greater than 0.999 over the entire modeled range and greater than 0.99 at

compaction stresses above 100 MPa.


A method has been developed to analyze the compaction of a powder system which

complements other available methods such as DEM, FEM and simple curve fitting. The

current method uses the curve fitting of measured compaction curves only to allow analyti-

65

cal manipulation of the initially discrete data. This analysis is based on an equation describ-

ing the stress–density relationship of a system in uniaxial compression. The parameters of

this equation are the sample mass, size, q (the fractional height within the sample at which

the transmitted axial stress is not a function of radial position), the product of the effective

friction coefficient and the powder fluidity factor (k·µ), and the intrinsic compaction curve

of the powder (σz,eff).

Since the sample mass and size are known and an appropriate value is chosen for q,

there are two unknown quantities: (k·µ) and σz,eff. Two independent equations can be created

to solve for these quantities by experimentally determining compaction curves of two dif-

ferent sized samples of the system. The product k·µ, as a function of sample density during

compaction, is calculated first. This quantity can be used to evaluate and compare the ef-

fects of system parameters—such as binder content and properties—on compaction. In

particular, this parameter predicts the propensity of the powder to produce density gradients

within the pressed part.

With k·µ known, it is possible to calculate the intrinsic compaction curve of the pow-

der. This function describes the stress–density relationship of the powder as would be observed

in a frictionless, or infinite cross-sectional area, die. Combining the intrinsic compaction

curve function and the k·µ function in the governing equation allows the prediction of com-

paction curves of the system for arbitrary sized samples (within limits, as discussed above).

There are other factors which may influence the results of this analysis; factors which

are generally always concerns in compaction. Examples include the compaction rate, gran-

ule size distribution and die fill density and homogeneity. Compaction rate is always a

factor when a system component has time/rate dependent mechanical properties, as is the

case with viscoelastic polymer binders. Furthermore, at the high compaction rates generally

seen in industrial production, the heat generated can dramatically change the deformation

response of the binder phase. This type of effect could, in theory, be accounted for within

66

the expression for the intrinsic compaction curve (simply by performing the measurements

at the desired compaction rate). The k·µ data, however, would become difficult to interpret

since both k and µ would be changing internally (i.e., due to the heat as well as the compac-

tion process).

Care must be taken to provide identical initial conditions for the compaction curve

measurements to be used in this analysis. This means the initial die fill should be homoge-

neous and consistent for the different measurements. Since this is often achieved by vibrating

or tapping the filled mold, having a wide granule size distribution can easily lead to die fill

inhomogeneities. If significant density gradients are present from the outset, the stress dis-

tribution model would no longer be valid, disrupting the analysis. If the two samples have

grossly different fill densities, the intrinsic compaction curve would not be constant be-

tween the two samples and physically unrealistic values of k·µ (i.e., < 0 or > 1) would result.

When properly implemented, this is an excellent analytical method for evaluating com-

paction systems. It has the advantages of being relatively easy and rapid. Only two simple

experiments are necessary and the computational demands are low. Additionally, the mea-

surements done on the powder are representative of the final application—i.e., the

measurements required to characterize uniaxial compaction are done by uniaxial compac-

tion. It is also possible to expand the analysis to predict the compaction response for other

sample geometries and to incorporate additional information from other methods or mea-

surements to refine the analysis (e.g., to allow the separation of k and µ).

67

Chapter 5:

Green Strength and Granule Strength Modeling

Introduction

Green Strength

An important characteristic of green ceramic parts is their strength. A green part must

be strong enough to support itself, survive the strain on ejection and remain intact during

handling for further processing steps—which may include machining in some instances. A

compact made of a raw ceramic powder would be held together by moisture bridges and/or

weak surface forces (i.e., van der Waals) in a manner identical to that responsible for the

formation of loose aggregates in submicron powders. The strength of a compact formed of

a binder-containing powder, however, emanates from the adhesive strength between the

binder phase to the ceramic particles, the strength of the binder phase and the strength of

binder to binder interfaces.

Since it is not possible—with few exceptions—to use a standard tensile test specimen

for ceramics in any state, the diametral compression test was used to determine the strengths

of the cylindrical samples.40 A further complication to be considered when attempting to

compare the green strengths of all powder compositions in this study involves the fact that

the variation in binder properties in the range of Tg values used will have a massive influence

on the measured strengths. It was decided that after the production of the samples—at the

various binder Tg values—for strength testing, to condition the samples at a common rela-

tive humidity prior to strength testing. This was done, and a relative humidity of 43%

(Tg = 6°C) was chosen for this purpose, for several reasons. First: it was desired to observe

the effect of varying binder properties during compaction—measuring the green strengths

with the binder properties still varied would result in observing the intrinsic effect of the

68

binder properties on the strength measurement as well as the effect of the binder character-

istics on compaction. Second: the binder at this Tg is stiff enough to prevent the sample from

experiencing excessive shear strain prior to tensile failure and pliant enough to eliminate

stress concentrations due to minor irregularities in the contacts between the sample and the

test apparatus (platens). Finally: by choosing a moderate relative humidity minimized the

difference between the relative humidity at which the samples were conditioned and the

ambient conditions during testing; there was less concern for doing the testing rapidly to

avoid changes in the binder properties during testing.

Granule Strength Modeling

There are several user adjustable granule characteristics, for a given raw ceramic pow-

der, which will affect compaction behavior. Among these parameters are granule size and

density, binder content, binder type and binder properties. Granule density can be adjusted

by the selection and amount of dispersant used. Binder properties are generally fixed by the

user’s selection of binder type and amount, but they can be unintentionally affected by

factors such as relative humidity and working temperature.

Since the bulk of compaction occurs by deformation and/or fracture of the granules to

fill the intergranular voids, it would be beneficial to use granules with mechanical proper-

ties suited to this purpose. To this end, it would be invaluable to be able to characterize the

granules during the compaction process.

Mechanical testing of individual granules is a possibility; however, there are difficul-

ties associated with this approach. Since the granules are small and fragile, specialized

equipment may be necessary for handling and testing. Another factor to be considered is the

fact that the average spray dried powder will contain a distribution of granule sizes and,

therefore, a distribution of granule properties. In order to obtain useful data, a large number

of individual granules would need to be tested with corrections made for granule size.

69

The method discussed below allows the testing of thousands of granules with rela-

tively few experimental samples. The average strengths of free granules are determined,

and the development of granule strength and the evolution of intergranular bond strength

during the compaction cycle are shown.


Green Strength

Before measuring the green strengths of the pellets, the pellets were equilibrated at

43% RH (Tg = 6°C) to eliminate the effect of the binder characteristics on strength, and thus

learn how the binder properties affected the compaction process. Strength was determined

by the diametral compression method in a universal testing machine (Instron 4202). The

pellets were placed on edge between the platens and compressed at 0.5 mm·min–1. The

tensile strength (σ) of the pellets is related to the load at failure (F) by

σπ

= 2F

dt(5.1)

where d = pellet diameter and t = pellet thickness.40


The basis of this analysis is the assumption that the green strength of a pellet is an

appropriate combination of the strength of the binder bonding ceramic particles together

and the adhesive strength between two binder surfaces. These two quantities are directly

related to intragranular (i.e., granule, or particle–particle bond) strength and intergranular

(i.e., granule–granule) bond strength, respectively. A mathematical expression of this is:

σ σ σp b b g gx f x x f x x( ) = ( ) × ( ) + ( ) × ( ) (5.2)

70

where σp is the measured pellet green strength, σb is the intergranular bond strength compo-

nent, σg is the intragranular (granule) strength component, fb(x) and fg(x) are coefficients of

the component strengths, and x is an appropriate independent variable.

The appropriate quantity for the independent variable is the fractional amount of the

mode of fracture (inter– vs. intragranular) seen in a sample broken in tension. This leads to

the definition of the coefficient functions for the two strength components as the fractional

amount of the corresponding fracture mode:

σ σ σp b gx x x x x( ) = × ( ) + −( ) × ( )1 (5.3)

where x is now the fractional area of intergranular fracture (fracture between granules) of

the fracture surface. To complete the analysis, the function σp(x) must be experimentally

determined and deconvoluted to produce the component strength functions, σb(x) and σg(x).

Four series of pellets were produced as described in chapter 3 and their green strengths

measured as described above. The tested series consisted of binder Tg = 6°C with 2, 3, 4,

and 5% NBC. Samples were pressed to a maximum stress of ~175 MPa and a minimum

stress as low as ~6 MPa. This provided the wide range of sample strengths necessary for

this analysis.

The next step was to determine the relative amounts of inter– and intragranular frac-

ture modes of the pellets (figure 5.1). Low magnification electron micrographs were taken

of the pellet fracture surfaces (figure 5.2a). These images were analyzed in a commercial

image editing application (Adobe® Photoshop®) on a personal computer by manually chang-

ing the areas determined to be intragranular fracture to pure black. The software could then

transform the remaining area to pure white. The area fraction of intra– and intergranular

fracture was then automatically determined from the relative number of black and white

pixels representing intra– and intergranular fracture (figure 5.2b).

71


Green Strength

Figure 5.3a shows the strength of the pellets pressed to the highest load (~175 MPa),

and figure 5.3b shows the strength of pellets of the same density (52.9%). Both plots show

approximately the same trend of higher binder content, lower Tg systems producing stron-

ger parts. This means a binder which is compliant during compaction will produce more

intimate contact between granules and between particles, i.e., the more binder present and

the softer it is during pressing, the stronger the part.

The lower Tg binder is more easily deformed than the higher Tg compositions resulting

in more efficient deformation and compression of the granules. This, in turn, results in

larger and more confluent contact patches between granules. Add to this the fact that the low

Tg binder, under these conditions, is sufficiently self-adhesive to produce strong bonding

between granules. In the opposite case of the high Tg compositions, the binder does not

deform plastically easily and it is not as adhesive. This results in less-intimate contact patches,

possibly less overall contact area between granules due to fracture and intrinsically weaker

bonding between the granules.

Figure 5.1: High magnification micrograph of a pellet fracture surface clearly showing the

difference between intergranular fracture and intragranular fracture.

IntragranularIntragranularFractureFracture

IntergranularIntergranularFractureFracture

50 µm

72


Plotting percent intergranular fracture vs. pellet green strength reveals a trend that can

be fit to a quadratic equation (figure 5.4). In the absence of evidence to the contrary, it was

most reasonable to assume that the quadratic relationships were composed of the sum of

simpler functions (lower order polynomials). Having postulated this, it remains to deter-

Figure 5.2: a) Representative pellet fracture surface micrograph showing the two different

fracture modes. b) Analysis of the fracture surface showing intergranular fracture

(white) and intragranular fracture (black).

a)

200 µm

b)

200 µm

73

–30 –15 015 30

Binder Tg ( C) 2

34

5

Nomin

al B

inde

r

Conte

nt (w

t%)

300020001000

0St

reng

th(k

Pa)

20001000

0–30 –15 015 30

Binder Tg ( C) 2

34

5

Nomin

al B

inde

r

Conte

nt (w

t%)

Stre

ngth

(kPa

)

Figure 5.3: Pellet green strengths at 43% RH (Tg = 6°C): a) compacted to ~175 MPa and b)

compacted to 52.9 % relative alumina density.

mine the two component linear functions for each composition. This transformation can be

done analytically and exactly:

a a x a x x m x b x m x bb b g g0 1 22 1+ + = +( ) + −( ) +( ) (5.4)

where the a coefficients are known and mb, bb, mg, and mb are to be determined.

There are an infinite number of solutions to equation 5.4. By applying knowledge of

the behavior of the system at certain points, it becomes possible to find the proper unique

solution. The following points on the quadratic curve are known: (1) at 0% intergranular

fracture (the pellet fractures entirely through granules) σp = σg, (2) at 100% intergranular

74

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Pellet Green Strength (MPa)

100

80

60

40

20

0

Inte

rgra

nula

r F

ract

ure

Are

a (%

)

Figure 5.4: Intergranular fracture vs. pellet green strength (R2 ≥ 0.95)

fracture, σp = 0 because granules have not fused, therefore σb = 0, and (3) at 50% inter-

granular fracture the pellet is as likely to fracture between granules as it is to fracture through

granules, therefore σp = σg = σb.

The overall strength vs. percent intergranular fracture curve is shown with the two

component lines in each of the plots in figure 5.5. It can be seen that both intergranular bond

strength and intragranular (granule) strength increase as compaction progresses. Intragranular

strength increases due to compaction of the granules and is equivalent to the sample strength

at full compaction. Intergranular bond strength increases from 0 (loose powder) to a value

significantly higher than the pellet strength at full compaction. This value is necessarily

greater than the fully compacted pellet strength as evidenced by the absence of intergranu-

lar fracture at this extreme. The increase in intra– and intergranular strength is a result of the

granules being fused together and the increasing contact area between them as they deform

to fill intergranular voids.

75

At the 100% intergranular fracture point, no compaction or granule fusion has taken

place, and the intragranular strength function takes on the value of the average free granule

strength. The free granule strengths thus determined are listed in table 5.1; the correspond-

ing apparent yield points are repeated in table 5.1 for comparison. It is interesting to note

that the calculated free granule strength is significantly less than the apparent yield point

0 20 40 60 80 100Intergranular Fracture (%)

Ele

men

t St

reng

th (

MP

a)

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Figure 5.5b: Green strength deconvolution for the 3 wt% NBC, Tg = 6°C samples.


Ele

men

t St

reng

th (

MP

a)

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Figure 5.5a: Green strength deconvolution for the 2 wt% NBC, Tg = 6°C samples.

76


4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Ele

men

t St

reng

th (

MP

a)

Figure 5.5c: Green strength deconvolution for the 4 wt% NBC, Tg = 6°C samples.


4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Ele

men

t St

reng

th (

MP

a)

Figure 5.5d: Green strength deconvolution for the 5 wt% NBC, Tg = 6°C samples.

measure. For the examples shown in figure 5.5, the calculated free granule strengths are

between 0.27 and 0.90 times the observed apparent yield point. This difference is largely

due to the difference in the stress state during the measurement. In the current analysis,

granules experience a uniform tensile stress at the failure plane. When determining the

apparent yield point, granules are subjected to multiple contact forces resulting in a com-

77

Table 5.1: Granule Yield Points and Granule Strengths of Tg = 6°C (43 %RH) Samples

GranuleNBC TBC Apparent Yield Point Strength

(wt%) (wt%) (MPa) (MPa)2 2.61 1.3 0.353 3.63 1.5 0.454 4.46 1.3 1.175 5.54 1.6 1.21

plex stress state with significant shear stresses and little true tensile stress. Failure under

such different conditions would not be expected to occur at the same stress.


Modifying the binder plasticity has a large effect on green strength. When pressing a

powder with a binder Tg much lower than the working temperature, the binder is easily

deformed and adhesion between binder surfaces is sufficient to provide good bonding be-

tween granules. As binder Tg is increased, its surface becomes less active resulting in poor

bonding between granules during compaction. Binder content plays an equally significant a

role for green strength. While increasing binder content does not intrinsically increase the

binder strength, the true bonded surface area is increased resulting in a higher observed

strength. These two parameters also reinforce each other: the effect of changing Tg is greater

at higher binder content, and the effect of binder content is greater when working high

above the Tg.

A method has been developed to calculate granule strength and intergranular bond

strength throughout the compaction cycle, including the pre-compaction (“free”) granule

strength. Relatively few samples are required to measure the average properties of thou-

sands of individual granules. Granule strength measurements are for failure in tension; this

complements the often-reported apparent yield point measurement, in which the granule

yields plastically under a very complex stress field. From the results available here, it is not

78

possible to state a relationship between these two measures of granule strength with cer-

tainty.

79

Chapter 6:

Conclusion

Summary and Conclusions

Several aspects of the uniaxial compaction of granulated ceramic powders were ad-

dressed. A series of spray dried alumina powders were produced with various binder (PVA)

contents; these powders were further conditioned to provide a wide range of binder Tg. A

systematic study of the behavior of this model powder system was done in which the com-

paction efficiency, springback, time-dependent relaxation, and resultant green part strength

were measured. As binder content was increased, the compaction stress required to attain a

specified density increased due to the greater amount of binder which needed to be de-

formed and displaced. A higher binder content powder, however, produced a stronger part

due to increased binder volume connecting the ceramic particles.

Adjusting binder plasticity also was shown to have a great effect on the compaction of

the powder and green part strength. When pressing a powder with a low Tg binder, the

binder was easily deformed and high green densities were achieved at moderate stresses.

Such a binder would tend to conform to particle surfaces, leading to a high strength green

part. As binder Tg was increased, it became stiffer and brittle, requiring higher compaction

stresses to achieve a given density.

Binder content also significantly affected green strength. While increasing binder con-

tent does not in itself increase the binder strength, the true bonded surface area was increased

resulting in a higher effective strength. The two parameters of binder content and binder Tg

were also found to reinforce each other: the effect of changing Tg was greater at higher

binder content, and the effect of binder content was greater when working high above the

Tg.

80

The dimensional changes pellets experienced after ejection from the die was domi-

nated by instantaneous springback in the axial direction. Factors influencing the amount of

time-dependent relaxation were: the binder content, the binder plasticity, and the direction

in the pellet.

Sample size was shown to have a significant effect on powder compaction. In larger

samples, the surface area (in contact with the die wall) to volume ratio was smaller than in

smaller samples, reducing the influence of die wall friction on the bulk compaction of the

powder. To address this effect, a technique was developed which allowed the determination

of the intrinsic compaction behavior of a powder. In this analysis compaction curves were

measured of a powder pressed to two different sample sizes. The effect of die wall friction

was calculated, which could be subtracted from the measured compaction data. With the

intrinsic compaction behavior of the powder and the effect of die wall friction both quanti-

fied, it became possible to predict the compaction behavior of the powder for different

sample sizes. Compaction curves calculated in this manner for various sample sizes were

shown to match experimentally measured compaction curves with correlation coefficients

greater than 0.99. This is an excellent analytical method for evaluating compaction systems.

It has the advantages of being relatively easy and rapid. A minimum of only two straightfor-

ward experiments are necessary and the computational demands are low. Additionally, the

measurements done on the powder are representative of the final application—i.e., the mea-

surements required to characterize uniaxial compaction are done by uniaxial compaction.

A method was introduced which allows the measurement of average granule strength

as well as granule strength and intergranular bond strength during the compaction cycle.

Relatively few samples are required to measure the average properties of thousands of indi-

vidual granules. Granule strength measurements are for failure in uniform tension which

complements the often-reported apparent yield point measurement, in which granules yield

plastically under a complex stress field. Not unexpectedly, these two methods produced

81

very different values with the granule strength being significantly lower than the apparent

yield point.

Recommendations for Future Work

There are other parameters which may affect compaction which were kept constant in

this analysis. For example, only a narrow size distribution of granules was used in this work

in order to minimize granule size and granule size distribution effects. These parameters

were expected to have an effect and it would be informative to repeat this entire study with

different granule size ranges of this system.

Another important parameter is compaction rate. Youshaw and Halloran12 clearly ob-

served a significant difference in compaction response when changing the compaction rate

by one order of magnitude. No such difference was seen in this study when the compaction

rate was changed over the same range. It is the nature of the materials used to have time- or

rate-dependent properties and additional study in this area would be beneficial, especially at

very high compaction rates. An additional concern at extremely high compaction rates is

that the heat generated can dramatically change the deformation response of the binder

phase. In the compaction curve normalization analysis, this type of effect could be accounted

for within the expression for the intrinsic compaction curve (simply by performing the

measurements at the desired compaction rate). The k·µ data, however, would become diffi-

cult to interpret since both k and µ would be changing internally (i.e., due to heating as well

as the compaction process).

It should also be possible to expand the predictive ability of the compaction curve

normalization analysis to include other sample geometries. This technique may also be

used in conjunction with other methods to refine the analysis. For example, experimental

techniques are available that measure sample ejection force. Such measurements could be

made at a series of confining pressures from which the friction coefficient may be calcu-

82

lated and correlated to corresponding points during a compaction cycle. With the powder/

die wall friction coefficient known, it would become possible to separate k and µ.

Finally, it is necessary to further develop the mathematical aspect of the method de-

scribed to measure granule and intergranular bond strengths. Currently, it is assumed that

the fracture surface of the pellet is a plane. This is obviously not the case in reality. The two

different fracture modes each produce different types of fracture surfaces. Intragranular

fracture sites may have a roughness on the order of a few times the primary particle size

(~0.5 µm, in the system studied) and, potentially, some larger scale curvature due to the

influence of surrounding intergranular fracture sites. Depending on the level of accuracy

desired, it may still be acceptable to treat intragranular fracture sites as planes. Intergranular

fracture sites, however, almost necessarily deviate significantly from the hypothetical frac-

ture plane by an amount on the order of the granule size (~100 µm, in the system studied).

Furthermore, the contact patches between granules that separate during fracture are most

often not even parallel to the hypothetical fracture plane. The chaotic nature of the fracture

surface needs to be taken into account in the analysis. This adjustment is expected to have a

significant, but not gigantic, effect on some of the results obtained from this analysis.

83

Appendix:

Mathematica® Code

In[2]:=

First Step: Set the directory

this file and the two compaction curve files are in

SetDirectory "MacInbob:Documents:Robert Carneim:Work:Current

Work:Data Analysis:Instron:1 4˝ 0.5mm min FCM" ;

84

In[3]:=

Select low density cutoff

ldco 1.1

Out[3]= 1.1

In[4]:=

Select definition for q : constant or f

qtemp .65;

In[5]:=

Compaction curve data file format:

tab delimited

file names: CC1.td and CC2.td

data format:

Col1 Col2

R1 mass diameter

R2 stress density

R3 stress density

Rn stress density

be consistent with units: i.e.,mass in grams,

diameter in centimeters, density in g cm^3, Stress is seperate

and will work for whatever units you want; MPa is prefered.

Make CC1.td the one which goes to

the higher density smaller mass or larger diameter

Definitions of symbols used, in order of appearance:

ldco: low density cutoff selected by user, based on experience

qtemp: future definition of q,

the fractional height in the pellet of the constant stress plane

selected by user as either a constant or a function of density,

based on experience

cc1r and cc2r: arrays containing the

raw data files read in from the tab delimited files,

these include the masses and diameters in the first rows

m1, m2,

d1 and d2: masses and diameters of the two samples consistent units

cc1t1 and cc2t1:

cc1r and cc2r without the masses and diameters temporary

cc1t2 and cc2t2:

placeholdes while cropping low and high ends temporary

cc1 and cc2: arrays containing

the useable portions of the compaction curve data

hdco: high density cutoff highest density in cc1 and cc2

cc1l and cc2l: lengths of cc1 and cc2

cc1i and cc2i: inverses of cc1 and cc2 switch columns

85

cc1ilog and cc2ilog: cc1i and cc2i with Log 10,stress relacing stress

cc1log and cc2log: cc1 and cc2 with Log 10,stress relacing stress

1log and 2log : curve fits of cc1ilog and cc2ilog

1 and 2: 10^ 1log and 10^ 2log

r: die radius d 2 consistent units

t: pellet thickness,

defined in terms of mass, diameter and density consistent units

m: ceramic mass of pellet consistent units

d : generic die diameter consistent units

: pellet density consistent units

zeff: the main equation for the effective stress,

generic at this point

: product of k

fluidity index and powder wall friction coefficient

zeff1 and

zeff2: specific effective stress equations for the two samples

q : gets defined here qtemp

zeffd : difference of the two equations zeff1 zeff2

fit: array of vs density

fun : curve fit of fit

zeff1s and zeff2s: solved effective stress equations,

should be equivalent

fitlog: array of Log 10, zeff1s vs. density

fit: 10^ fitlog

funlog: curve fit of fitlog

In[6]:=

This reads the files

cc1r ReadList "CC1.td", Number, Number ;

cc2r ReadList "CC2.td", Number, Number ;

86

In[7]:=

This defines the masses and diameters for the two cases

Check to make sure they are correct

m1 cc1r 1, 1

d1 cc1r 1, 2

m2 cc2r 1, 1

d2 cc2r 1, 2

Out[7]= 0.107

Out[8]= 0.634

Out[9]= 0.363

Out[10]= 0.634

In[11]:=

This drops the mass and diameter from the raw data files

cc1t1 Delete cc1r, 1 ;

cc2t1 Delete cc2r, 1 ;

87

In[12]:=

Crops high end to highest

common stress and low end to specified density ldco

If First Last cc1t1 First Last cc2t1 ,

While First Last cc1t1 First Part cc2t1, 2 ,

cc2t2 Delete cc2t1, 1 ;

cc2t1 cc2t2 ,

While First Part cc1t1, 2 First Last cc2t1 ,


cc1t1 cc1t2

cc1t2 .;

cc2t2 .;

While Last First cc1t1 ldco,


cc1t1 cc1t2

While Last First cc2t1 ldco,


cc2t1 cc2t2

cc1t2 .;

cc2t2 .;

cc1 cc1t1;

cc2 cc2t1;

cc1t1 .;

cc2t1 .;

hdco Max Last Last cc1 , Last Last cc2

General::spell1 : Possible spelling error: new symbol name "hdco" is

similar to existing symbol "ldco".

Out[15]= 2.431

88

In[18]:=

Create

inverse and inverse log stress arrays of the compaction curve data

cc1l First Dimensions cc1 ;

cc2l First Dimensions cc2 ;

cc1i Array dummy1, cc1l, 2 ;

cc1ilog Array dummy2, cc1l, 2 ;

cc1log cc1;

dummy3 1;

While dummy3 cc1l 1,

cc1i dummy3, 1 cc1 dummy3, 2 ;


cc1ilog dummy3, 2 Log 10, cc1i dummy3, 2 ;

cc1ilog dummy3, 1 cc1i dummy3, 1 ;

cc1log dummy3, 1 cc1ilog dummy3, 2 ;

dummy3

cc2i Array dummy4, cc2l, 2 ;

cc2ilog Array dummy5, cc2l, 2 ;

cc2log cc2;

dummy3 1;

While dummy3 cc2l 1,



cc2ilog dummy3, 2 Log 10, cc2i dummy3, 2 ;

cc2ilog dummy3, 1 cc2i dummy3, 1 ;

cc2log dummy3, 1 cc2ilog dummy3, 2 ;

dummy3

In[20]:=

Load curve fitting package

Statistics`NonlinearFit`

89

In[21]:= 1logtemp NonlinearRegress cc1ilog, a0 a1 a2 ^2 a3 ^3,

, a0, 1 , a1, 1 , a2, 1 , a3, 1 , ShowProgress False,

MaxIterations 1000,

PrecisionGoal 15,

AccuracyGoal 15,

RegressionReport BestFit, BestFitParameters, EstimatedVariance ;

2logtemp NonlinearRegress cc2ilog, a0 a1 a2 ^2 a3 ^3,


MaxIterations 1000,

PrecisionGoal 15,

AccuracyGoal 15,

RegressionReport BestFit, BestFitParameters, EstimatedVariance ;

1log Last First 1logtemp

2log Last First 2logtemp

1 10^ 1log;

2 10 ^ 2log ;

Out[21]= 3.16815 4.70777 2.08147 2 0.439008 3

Out[22]= 2.71664 4.09004 1.87138 2 0.4471 3

In[24]:= ListPlot cc1ilog

ListPlot cc2ilog

Plot 1log, 2log , , ldco, hdco , PlotRange All

Show %, %%, %%%

1.2 1.4 1.6 1.8 2.2 2.4

0.5

1

1.5

2

Out[24]= Graphics

90

1.2 1.4 1.6 1.8 2.2

0.5

1

1.5

2

Out[25]= Graphics

1.2 1.4 1.6 1.8 2.2 2.4

0.5

1

1.5

2

2.5

Out[26]= Graphics

1.2 1.4 1.6 1.8 2.2 2.4

0.5

1

1.5

2

2.5

Out[27]= Graphics

91

In[28]:=

Main Equation

r d 2;

t 4 m d 2 ;

zeff Simplify

z z 2 t r r t r t Exp 2 t r Exp 2 t r Exp 2 q t r

Out[28]= z 1 2 E16 md3

16 E16 m 1 q

d3 md3

In[29]:=

The two effective stress equations are defined

z 1;

m m1;

d d1;

zeff1 Simplify zeff

z .;

m .;

d .;

z 2;

m m2;

d d2;

zeff2 Simplify zeff

z .;

m .;

d .;

Out[29]= 100.439008 1.05651 6.83065 3.6848 2

1 2 E2.13839 2.13839 E

2.13839 1 q

Out[30]= 100.4471 1.03041 5.89682 3.15518 21 2 E

7.25452 7.25452 E7.25452 1 q

In[33]:=

define difference equation

zeffd zeff2 zeff1

Out[33]= 100.439008 1.05651 6.83065 3.6848 21 2 E

2.13839 2.13839 E2.13839 1 q

100.4471 1.03041 5.89682 3.15518 21 2 E

7.25452 7.25452 E7.25452 1 q

92

In[34]:=

Plot for visualization

q qtemp;

Plot3D zeffd, ,ldco,hdco , ,0,1

ContourPlot

zeffd , , ldco, hdco , , 0, 1 , Contours 0 , PlotPoints 31

1.2 1.4 1.6 1.8 2 2.2 2.40

0.2

0.4

0.6

0.8

1

Out[34]= ContourGraphics

In[35]:=

make array of vs.

dummy7 1;

.;

fit Array dummy8, 20, 2 ;

While dummy7 21,

fit dummy7, 1 ldco dummy7 1 hdco ldco 19 ;

fit dummy7, 1 ;

fit dummy7, 2 Last Last FindRoot zeffd 0, , 0, 0, 1 ;

dummy7

93

In[36]:= ListPlot fit, PlotRange All

1.2 1.4 1.6 1.8 2.2 2.4

0.1

0.2

0.3

0.4

0.5

Out[36]= Graphics

In[37]:=

curve fit previous array

.;

fun .;

fun NonlinearFit fit,

a0 a1 a2 ^2 a3 ^3, , a0, a1, a2, a3 , ShowProgress False,

MaxIterations 1000,

PrecisionGoal 15,

AccuracyGoal 15 ;

94

In[38]:= ListPlot fit, PlotRange All

Plot fun, , ldco, hdco , PlotRange All

Show %, %%

1.2 1.4 1.6 1.8 2.2 2.4

0.1

0.2

0.3

0.4

0.5

Out[38]= Graphics

1.2 1.4 1.6 1.8 2.2 2.4

0.1

0.2

0.3

0.4

0.5

Out[39]= Graphics

1.2 1.4 1.6 1.8 2.2 2.4

0.1

0.2

0.3

0.4

0.5

Out[40]= Graphics

95

In[42]:=

define solved equations

fun;

zeff1s Simplify zeff1 ;

zeff2s Simplify zeff2 ;

In[43]:= Plot Log 10, zeff1s , Log 10, zeff2s , , ldco, hdco

ListPlot cc1ilog, PlotJoined True

ListPlot cc2ilog, PlotJoined True

Show %, %%, %%%

1.2 1.4 1.6 1.8 2.2 2.4

0.5

1

1.5

2

Out[43]= Graphics

1.2 1.4 1.6 1.8 2.2 2.4

0.5

1

1.5

2

Out[44]= Graphics

96

1.2 1.4 1.6 1.8 2.2

0.5

1

1.5

2

Out[45]= Graphics

1.2 1.4 1.6 1.8 2.2 2.4

0.5

1

1.5

2

97

Out[46]= Graphics

In[47]:=

make array of effective stress vs. density solved equation

dummy9 1;

.;

fitlog Array dummy10, 20, 2 ;

fit Array dummy11, 20, 2 ;

While dummy9 21,

fitlog dummy9, 1 ldco dummy9 1 hdco ldco 19 ;

fit dummy9, 1 fitlog dummy9, 1 ;

fitlog dummy9, 1 ;

fitlog dummy9, 2 Log 10, zeff1s ;

fit dummy9, 2 10^ fitlog dummy9, 2 ;

dummy9

In[48]:=

curve fit previous array

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funlog NonlinearFit fitlog , a0 a1 a2 ^2 a3 ^3,


MaxIterations 1000,

PrecisionGoal 15,

AccuracyGoal 15 ;

In[49]:= ListPlot fit

Plot 10^ funlog , , ldco, hdco

Show %, %%

ListPlot fitlog

Plot funlog, , ldco, hdco

Show %, %%

1.2 1.4 1.6 1.8 2.2 2.4

20

40

60

80

100

120

Out[49]= Graphics

98

1.2 1.4 1.6 1.8 2.2 2.4

20

40

60

80

100

120

Out[50]= Graphics

1.2 1.4 1.6 1.8 2.2 2.4

20

40

60

80

100

120

Out[51]= Graphics

1.2 1.4 1.6 1.8 2.2 2.4

0.5

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99

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2

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1.5

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100

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Vita

Robert D. Carneim

Education

Experience

Publications

Presentations

Bachelor of Science, Ceramic Engineering – January 1995New York State College of Ceramics at Alfred University, Alfred, NY

Doctor of Philosophy, Material Science (Ceramics) – December 2000The Pennsylvania State University, University Park, PA

March 1999–July 2000Computer Lab Administrator – Particulate Materials Center, Materials ResearchLaboratory, The Pennsylvania State University, University Park, PA

August 1995–September 2000Graduate Research Assistant – Particulate Materials Center, Materials ResearchLaboratory, The Pennsylvania State University, University Park, PA

January 1995–August 1995Research Assistant – Associated Western Universities Fellowship at Pacific NorthwestNational Laboratory, Battelle Memorial Institute, Richland, WA

1990–1994Computer Lab Assistant – New York State College of Ceramics at Alfred University,Alfred, NY

R.D. Carneim, G.L. Messing, “Response of Granular Powders to Uniaxial Loading andUnloading,” in press, Powder Technology (2000).

R.D. Carneim, G.L. Messing, “Characterization of Agglomerate Mechanical Properties,”pp. 377–79 in 1999 Fine Powder Processing International Conference Proceedings(Proceedings of Fine Powder Processing ’99, University Park, PA, September 20th–22nd,1999), Edited by V.M. Puri, J.H. Adair, C.L. Knobloch, C.C. Huang, The PennsylvaniaState University, University Park, PA, 2000.

J.W. Stevenson, T.R. Armstrong, R.D. Carneim, L.R. Pederson, W.J. Weber,“Electrochemical Properties of Mixed Conducting Perovskites La1–xMxCo1–yFeyO3–d

(M = Sr, Ba, Ca),” J. Electrochem. Soc., 143[9] 2722–29 (1996).

R.D. Carneim, G.L. Messing, “Compaction Curve Normallization With Respect toSample Size,” Presented at the 102nd Annual Meeting of the American Ceramic Society,St. Louis, MO, May 2st, 2000.

R.D. Carneim, G.L. Messing, “Granule Characterization During Uniaxial Compaction,”Presented at the 102nd Annual Meeting of the American Ceramic Society, St. Louis, MO,May 1st, 2000.

R.D. Carneim, G.L. Messing, “Processing Effects on the Sintered Properties of AluminaPellets,” Poster Presentation at Sintering ’99, University Park, PA, November 1st–3rd, 1999.

R.D. Carneim, G.L. Messing, “Characterization of Agglomerate Properties,” Presented atFine Powder Processing ’99, University Park, PA, September 21st, 1999.

R.D. Carneim, G.L. Messing, “Dimensional Stability of Dry–Pressed, Organic Binder–Containing Ceramic Powders,” Presented at the 101st Annual Meeting of the AmericanCeramic Society, Indianapolis, IN, April 26th, 1999.

R.D. Carneim, G.L. Messing, V.M. Puri, “Organic Additive Effects on the Compaction ofGranulated Ceramic Powders,” Presented at the 100th Annual Meeting of The AmericanCeramic Society, Cincinnati, OH, May 6th, 1998.