Table Of ContentParticle
Packing
Characteristics
by Randall M. German
Materials Engineering Department
Rensselaer Polytechnic Institute
Troy, New York
METAL POWDER INDUSTRIES FEDERATION
Princeton, New Jersey
No part of this publication may be reproduced, stored in a retrieval
system, or transmitted, in any form or by any means, electronic,
mechanical, photocopying, recording, or otherwise, without the prior
written permission of the publisher.
Library of Congress Cataloging-in-Publication Data
German, R. M.
Particle packing characteristics f by Randall M. German.
p. em.
Includes bibliographies and index.
ISBN 0-918404-83-5: $65.00 (est.)
I. Powder metallurgy. 2. Compacting. I. Title.
TN695.G4695 1989
671.37-dc20 89-50340
CIP
Copyright 1989
Metal Powder Industries Federation
ISBN 0-918404-83-5
Library of Congress Catalog Card No. 89-50340
Printed in the United State of America
vii
Contents
Preface xiii
oooooooooooooooooooooooooo•ooooooooooo••o••o•······· .... •oo ..................... ooo ....
The Author .............. xvii
0 00 00 0000 0 0. o•• 0 0000 0 00 ..... o. o o•o. o••···· ..... 0 •••• 0 .0 ... 00 ......
Chapter One
Introduction to Particle Packing
Overview ............................................................................. 1
Definitions ........................................................................... 6
Key Concerns . 9
0 ....................................................................
Applications for Particle Packing Concepts ......................... 11
References .......................................................................... 18
Chapter Two
Structures in One and Two Dimensions
Introductory Concepts ......................................................... 23
Packing of Monosized Segments in One Dimension ............. 27
Mixed Segment Sizes in One Dimension .. 29
0 ..........................
Packing of Monosized Disks in Two Dimensions ................. 32
Ordered Packing ........................................................ 32
Random Packing. ....................................................... 34
Mixed Disk Sizes in Two Dimensions .................................. 38
Ordered Packing of Other Two-Dimensional Shapes .......... .4 7
Summary and Generalizations ............................................ 4 7
References 50
0 ....... 0 .................................................................
viii Contents
Chapter Three
Factors Affecting Particle Packing
Introduction ........................................................................ 53
Stable Positions ......................... , ........................................ 54
Particle Size ................................... , ................................... 56
Particle Shape and Surface Texture ................................... 56
Agglomeration ....... , ............................................................ 59
Surface Active Agents ........................................................ 65
Internal Powder Porosity .................................................... 66
Container Wall ................................................................... 68
Segregation ........................................................................ 13
Bridging and Vibration ....................................................... 77
Summary ........................................................................... 80
References .......................................................................... 81
Chapter Four
Packing of Monosized Spheres
Introduction ........................................................................ 89
Coordination Number. ......................................................... 90
Ordered Packing ................................................................. 92
Random Packing ................................................................. 97
Random Loose Packing ............................................ 103
Random Dense Packing ........................................... 106
Summary ......................................................................... 110
References ........................................................................ 111
Chapter Five
Packing of Monosized Nonspherical Particles
Introduction ...................................................................... 121
Particle Shape Effects ...................................................... 122
Summary ......................................................................... 130
References ........................................................................ 131
Chapter Six
Bimodal Mixtures of Spherical Particles
Introduction ...................................................................... 135
Qualitative Description ..................................................... 138
Quantitative Model for Optimal Packing ........................... 141
Porosity and Particle Size Ratio Effects ............................ 144
Contents lX
Coordination Numbers ...................................................... 151
Homogeneity Effects ......................................................... 155
Summary ......................................................................... 158
References ........................................................................ 159
Chapter Seven
Bimodal Mixtures Involving Nonspherical Particles
Introduction ...................................................................... 165
Mixtures of Spheres and Fibers ........................................ 166
Nonspherical Particle Mixtures ......................................... 172
Summary ......................................................................... 178
References ........................................................................ 179
Chapter Eight
Multiple Mode and Continuous Particle Distributions
Introduction ...................................................................... 181
Trimodal Mixtures ............................................................ l83
Multimodal Mixtures ........................................................ 192
Continuous Particle Size Distributions .............................. 196
Summary ......................................................................... 210
References ........................................................................ 213
Chapter Nine
Particle Deformation and Compaction
Introduction ...................................................................... 219
Stages of Compaction ....................................................... 220
Density-Pressure Relations ............................................... 228
Changes in Particle Shape and Size .................................. 231
Effects of Material and Particle Characteristics ................ 238
Load Transmission in Packed Particles ............................. 241
Summary and Technological Implications ......................... 245
References ........................................................................ 246
Chapter Ten
Connectivity in Mixtures of Two Phases
Introduction and Measurement of Connectivity ................. 253
Monosized Spheres in a Matrix Phase .............................. 258
Mixtures of Equal Sized Spheres ...................................... 260
X Contents
Mixtures of Unequal Sized Spheres .................................. 263
Particle Shape Effects ...................................................... 269
Summary ......................................................................... 271
References ........................................................................ 2 71
Chapter Eleven
Pore Size and Shape
Introduction ...................................................................... 27 5
Pore Characterization Techniques ..................................... 278
Surface Area Measurement .............................................. 289
Pores in Ordered Sphere Packings .................................... 294
Pores in Random Packings ................................................ 298
Summary ......................................................................... 300
References ........................................................................ 302
Chapter Twelve
Capillarity and Liquid Interactions with Packed Particles
Overview and Defmitions .................................................. 309
Contact Angle ......................................................... 312
Spreading ................................................................ 314
Hysteresis ............................................................... 315
Capillary Force ................................................................. 316
Condensation .................................................................... 327
Imbibition and Wicking ..................................................... 329
Drainage .......................................................................... 332
Evaporation and Drying ................................................... 337
Fingering in Liquid Displacement ..................................... 343
Summary ......................................................................... 346
References ........................................................................ 34 7
Chapter Thirteen
Flow of Fluids Through Packed Particles
Introduction ...................................................................... 353
Diffusion ........................................................................... 35 7
Slip Flow .......................................................................... 359
Viscous Flow .................................................................... 361
Incompressible Fluids .............................................. 361
Compressible Fluids ................................................ 363
Mixed Phases .......................................................... 366
Contents xi
Permeability Coefficient.. ......................................... 369
Inertial and Turbulent Flow ............................................. 3 77
Summary ......................................................................... 382
References ........................................................................ 384
Chapter Fourteen
Interparticle Strength
Overview .......................................................................... 3 91
Loose Particles ................................................................. 393
Adhesion and Cohesion ............................................ 393
Shear Induced Flow ................................................. 397
Compacted Particles ......................................................... 400
Role of Particle Characteristics ......................................... 404
Summary ......................................................................... 407
References ........................................................................ 409
Chapter Fifteen
Summary on Particle Packing Concepts
Review of Important Concepts ......................................... .413
Comments on Computer Simulations ................................ 423
Practical Approaches to High Packing Densities ............... 424
Opportunities for the Future ............................................ .427
References ........................................................................ 4 2 9
Index ......................................................................................... 435
Xlll
Preface
Many research and development projects have been conduc
ted that use powders. However, insufficient attention is often given
to the particle packing characteristics. I have studied this topic in
great detail and am convinced that more advantage can be made
from a better understanding of particle packing. My interest has
been heightened by the advent of powder injection molding. In this
manufacturing process, a powder is mixed with a polymeric binder
and the mixture is molded into a complex shape using conventional
plastic injection molding technologies. After molding, the binder is
removed and the porous structure is densified by sintering. Unlike
die compaction, the particle packing density after molding is close
to that for the loose powder. Consequently, the overall success in
fabricating high performance structures depend on the ability to
work with high packing density particulate systems. A high
compaction force could be applied to the powder to overcome poor
packing. However, with powder injection molding the inherent
packing behavior must be attended to directly. Accordingly, my
attention to this subject resulted in my eventual decision to produce
this book.
One reason for orgamzmg the information on particle
packing characteristics was the discovery that researchers and
practitioners in many fields of technology are immersed in very
similar problems. Interest in particle packing exists in studies
dealing with particulates, whether they are oxide ceramics,
xiv Preface
pharmaceuticals, filter media, catalysts, minerals, nuclear fuels,
metal powders, soils, cereals, casting sands, or seeds. Unfortu
nately, insufficient communication takes place between the various
groups working in these diverse fields, resulting in replicated effort
in learning the same basic principles over and over. To assess the
contributions of these diverse fields, I conducted a broad survey of
thP. literature. I was immediately overwhelmed by the quantity of
information available. By no means can any book do justice to the
many prior investigations. As just an example of the problem,
there are well over one thousand prior citations of work on pore
characterization using mercury porosimetry. Likewise, ma,jor liter
ature compilations deal specifically with permeability, packing den
sity, computer simulations, surface area, capillarity, compaction,
strength, percolation theory, and several other topics treated here.
Thus, my focus has been on distilling this diverse and large body
of knowledge into a unified treatment that will serve as a basis for
teaching, research, and improved industrial practices. The book
contains many references (approximately 650) for the reader
interested in the details. For those with the time to read the book
from cover to cover, they will learn the essential elements of parti
cle packing. This is appropriate for students newly discovering this
field. Alternatively, since there is often need for quick access to
the knowledge base on selected topics, the presentation is organized
so that specific segments can be consulted as needed. This will be
most useful to the practicing engineers and scientists that face the
day-to-day problems associated with particulate systems. Bach of
the book's chapters contains its own set of references. The table of
contents provides a quick overview of the scope and organization of
the book.
Since my background is in materials science and engineer
ing, I have a natural bias to select examples and literature that
relate to this field. However, I have considered the many areas
where particle packing characteristics are important and h~ve used
illustrations from several fields. The text is useful to those with
practical or research interests in almost all forms of technology,
including aeronautics, agriculture, biology, ceramics, chemical
engineering, chemistry, civil engineering, composites, electrical
engineering, foods, geology, mechanical engineering, medicine,
metallurgy, nuclear engineering, paints, phar.maceuticals, physics,
and polymers. Obviously, the packing of particles is fundamental
to a wide variety of disciplines.
