Volume 5  Number 12                            Dennis R. Dinger                                1 October 2007

Updates

"... for Ceramists" Series Books

          Spanish Language Books

For those of you who speak Spanish as your primary language, a downloadable PDF version of Rheology for Ceramists in Spanish is currently in progress.  Reología para Ceramistas is currently being edited to be made available as soon as possible.  Best estimate at this time is that it will be available sometime in 2008 because the editing process is proceeding slowly.  The PDF file will be set up so it can be printed on your printer if you prefer a hard copy.  Depending on the reception this version receives, I will then consider translating the Particle Calculations book as well.  I will also then consider translating it into Portuguese.  Any thoughts, comments, and/or suggestions will be appreciated.

          English Language Books

The paperback version of Characterization Techniques for Ceramists is available on the Books and Downloads page at the web site!    Retail price is $29.95 plus shipping and handling. The book has 256 pages and it covers 34 different characterization techniques that are commonly used by ceramists.  Purchase a copy NOW!

The book sets on the web site have also been revised to include this new book.  A 3-book set of paperbacks, including one each of Particle Calculations for Ceramists, Rheology for Ceramists, and Characterization Techniques for Ceramists, is now available for $64.85 plus shipping and handling.  This is a $10 saving off the total retail price of the 3 paperback books.  A 3-book set of downloads is also available for $52.85.  This, too, represents a $10 saving off the total retail price of the 3 downloadable books.  

The E-Book version of Characterization Techniques for Ceramists is available for downloading at the Books and Downloads page of the website for $24.95.  The download is a 2.889 Mb self-extracting Zip® file for the Windows® environment which unzips to the 2.998 Mb book in PDF file format.  Those of you who order the downloadable book will want to know that the PDF book is formatted to print on 5.5" X 8.5" paper (i.e., 8.5" X 11" sheets cut in half.)

The other two books, Rheology for Ceramists and Particle Calculations for Ceramists, continue to be available for purchase as downloadable E-books and as paperback books at the Books and Downloads page of the web site.

          Requests for Multiple Copies

I have had several recent inquiries about the purchase of multiple copies of these books.  Here are my two suggestions:  

          (1)  If you purchase downloadable versions, purchase the required number of copies (please be honest about the number) from the Books and Downloads page of this website.  Then download a single copy and distribute it (or print it and distribute it) to the people for whom you purchased the copies. ... or ... 

          (2) Purchase the required number of paperback copies from the Books and Downloads page of this websiteand distribute them to your people.  My books are priced $19.95, $24.95, and $29.95 with this in mind.  You won't find many other good ceramics books in this price range.  Most others start at $80 to $100 each and prices rise from there.  For example, our PPC book (when it was available) was $195 per copy.  (I had no input when that price was set.  During one phone conversation, after they made sure I was sitting down, they simply told me the price.)          

The E-zine

If this is the first issue of the Ceramic Processing E-zine that you've seen, you can add your name to the mailing list by clicking HERE.  All back issues can be accessed from the Publications page at the web site.  For those of you whose e-mail programs don't properly show the figures in these E-zines, go to the Publications page of the web site using your web browser to open any and all issues.  All figures should open properly when issues are accessed from the web site.  Questions, suggestions, and/or requests for topics to be covered in future issues of this e-zine can be sent to QuestionsandComments@DingerCeramics.com   .

If you have friends, business associates, etc., who are ceramists, materials engineers, or any other type of engineer or technician, and they are interested in receiving this e-zine, please forward this issue to them and encourage them to sign up.  Or simply point them to the Dinger Ceramics web site.  Also -- whether you are a new or continuing reader -- please send suggestions for topics you'd like to see addressed in future issues of this E-zine.

For this month's article, I decided to go back to basics and give a subjective review of particle packing research.  

A Subjective Review of Particle Packing Research During the 20^th Century

Introduction

The complete review with all references is in our paper (Dinger and Funk) entitled "Particle Packing Phenomena and Its Application in Materials Processing" which was published in the Materials Research Bulletin, 22[12]19-23 (Dec, 1997).  Since many of our equation derivations are in our green PPC textbook and in several other publications, I do not want to get into equations and super details here.  I want to subjectively review this field of research.  If you want to see the equations, derivations, discussions, and all the technical details, check out the other papers.

When we were doing this research 25+ years ago, there were a few simple points that appeared to be missing from most of the then current particle packing research.  I think those points are still missing and should be highlighted somewhere.  This e-zine is as good a place to highlight these points as one will find.

Discrete vs Continuous Packing

Right from square one, there have been two primary, fundamental approaches to particle packing:  (1) pack discretely sized systems of particles, and (2) pack continuous systems of particles.  We will consider both of these briefly. 

     Approach #1 -- Furnas -- Discrete Packing

The discrete packing approach goes back to Furnas who performed his research in the 1920s-1930s.  The 'discrete approach' means that one packs well defined particles of distinct, discrete sizes.  For example, several cases of identically sized billiard balls defines a single, discrete size class.  Several cases of identically sized marbles defines a second discrete size class.  Mixing the two together defines a multimodal discrete particle size distribution.

Very few distributions used in industry are truly discrete particle size distributions.  A log-normal particle size distribution has width to it -- that is, there are many particle sizes represented in a single log-normal distribution.  Some try to define a bimodal discrete PSD as (for example) a log-normal distribution with median particle size of 100microns mixed with a second log-normal PSD with median particle size of 1 micron.  This is actually a bimodal continuous distribution.  If the narrow, median particle size class in each distribution can be extracted from each log-normal distributions, a mixture of 100 micron particles plus 1 micron particles (for example) defines a bimodal distribution that is very close to being a true bimodal discrete PSD.  But such distributions in industry hardly ever exist.   Surely, there are exceptions which I can name, but most companies routinely use continuous distributions of powders. 

The single reason that is most damaging to the discrete approach to particle packing is that discrete distributions are few and far between in ceramic processing.  They are the exception -- certainly not the rule.

     Approach #2 -- Andreasen -- Continuous Packing

The 'father' (if we can call him that) of the continuous approach to particle packing is Andreasen who also performed his research in the 1920s-1930s.  A continuous particle size distribution between 100 microns and 1 micron includes particles of all possible sizes between those two limits.  When we buy powdered silica, or clay, or feldspar, or barium titanate, etc., we are almost always buying continuous distributions of powders.  Even the particles that make up the agglomerates in calcined alumina, the individual particles (when released from the agglomerates) form narrow, continuous distributions.  Many distributions distributions of chemically prepared materials are very narrow (that is, they only contain a small range of particle sizes) but they are almost always continuous distributions.  Even the powders contained on a single sieve size class form a continuous particle size distribution.

Andreasen was working with continuous powders which he recognized as such from the start.  For this reason, his packing considerations were all based on the packing of continuous particle size distributions.  

The primary, obvious advantage to Andreasen's approach is that he used continuous powders and the assumption that his packing theory be based on continuous distributions of particles.  For this reason, there has been no need to try to extend his theory to process distributions because his starting assumptions used continuous distributions of particles which are exactly typical of normal process particle size distributions. 

Both men and the research teams to follow each methodology appeared to be after the same goal:  they all wanted to be able to pack particles as densely as possible.  Those following Furnas had to find ways to extend his discrete approach to include and apply to continuous distributions.  Those following Andreasen had no need to extend his theories to continuous distributions because his theories are based on continuous distributions.

Rheological and Flow Considerations

Both methodologies lend themselves to mathematical definition.  The factor that is routinely missed in most of these discussions is that we are wanting to process ceramics and other materials to achieve dense packing.  Simply playing mathematical games with the distributions ignores one's ability to achieve the final distributions and follow the underlying assumptions to real bodies produced in the lab or in the plant.  Most discrete models ignore particle flow.  They ignore the capability for particles to move into the positions required by the underlying theory to achieve the dense packs calculated from the distributions.

     Flow of Particles to Achieve Dense Packing

Many wares are made using suspensions or high solids extrusion bodies.  Even many so-called dry processing methods have all particles coated with additives and/or fluids to provide some particle flow capacity.  The discrete packing approach, if you follow it exactly, requires not only that each particle size be packed one at a time, coarse to fine, but that each particle system packs densely within the pores of the previous (larger) particles' pores.  The goal is dense, particle to particle contact of each particle size as it is packed.  This may work reasonably well for really coarse particles, but as particle sizes approach colloidal sizes, surface texture begins to affect and destroy the capability for dense packing.

Coat bowling balls with epoxy and then dip them in coarse sand so their surfaces are perfectly rough.  Now ask yourselves -- will bowling balls coated with coarse sand pack as densely in a 55 gallon drum as normal smooth bowling balls will?  No -- the sand-covered balls will hit and stick and define larger porosities than expected.  When trying to put 1 micron powders into the pores of a pack of 20 micron powders, will they flow into the holes and fill all pores easily and well?  No they won't, either -- for similar reasons.  But this is required -- that is, perfect packing of each size class is required -- by the discrete packing method.

The discrete packing theory requires dry to dry particle contacts and flow -- in order of particle size from coarse to fine.  Continuous packing requires no such flow.

What happens with continuous distributions?  When the proper amounts of each continuous powder are dumped into the mixer, the only question concerns how long it will be necessary to mix the powders to achieve a homogeneous distribution of particle sizes.  It isn't a question of forcing fine particles into pores into which they may not fit, or into which they won't easily flow.  With continuous distributions, it is only a matter of achieving a good, homogeneous mixture of particles.  Then, wares are formed, structures dry and shrink, and homogeneously distributed particles are locked into place.

     Flow of Particles after Achieving Dense Packing

The fundamental requirement of the discrete packing theory is that the coarsest particles must be packed densely first.  Then the next smaller discrete size class must be packed densely within the pores defined by the first coarse discrete size class.  This process repeats similarly for all n discrete size classes to be packed -- in order -- from coarse to fine. 

When this distribution has been completely formed and is densely packed, IT CANNOT BE MOVED!!!  To mix such a distribution destroys all the packing density and particle arrangements achieved by this one-at-a-time packing procedure.  Imagine your process where all particles must be packed one size at a time into their final locations in the ware.  Then, once packed, they cannot be moved, stirred, extruded, or whatever!  I only ever saw one publication in which this technique was actually employed and it was in the making of fuel pellets for nuclear reactors.  Each pellet was to be a small cylinder and price did not appear to be an object.  

Try to make a toilet bowl, or an insulator, or a brick, or a dinner plate, or etc., and not be able to dump the particles into a mixer to mix them to achieve a homogeneous distribution, or not be able to extrude them, or to slip cast them, or to dry press them, or etc.  It is one unique ware that allows forming to accurately and perfectly follow the requirements of the discrete packing theory.

There was a method of processing of electronic ceramics touted ~30 years ago at MIT which insisted that all powder grains be an identical single ultrafine size.  They needed fine grains for their electronic ceramic properties, so starting with all extremely fine, monodisperse powders could provide them with that structure.  But guess what?  Such systems were extremely dilatant and did not flow easily.   Monodisperse discrete particles simply do not flow well.  So trying to achieve reasonable flow while densely packing with a single discrete size class of particles is nigh unto impossible.  

From crystallography, we know that atoms in perfect close packing arrangements can pack to ~74 volume percent.  But dump spheres into a box, and the packing factor achieved will be about 60%.  Particles simply don't pack as easily and as well as the theories say they will.  But this is not a problem with continuous distributions because one is relying on homogeneity (achieved at lower solids contents) to move particles into their proper locations, followed by dewatering and shrinkage when all particles are in their proper locations.  There is no shrinkage in the discrete packing theory because all particles touch as soon as they are densely packed.

Simple Mathematics -- But Impossible Real Conditions

One paper that was published started with a broad continuous distribution with about 40 different size classes.  They showed that every tenth class, 1st, 11th, 21st, and 31st class, for example, defined a 4-mode discrete size distribution.  Then they showed that they could achieve similarly dense packing using the 2nd, 12th, 22nd, and 32nd size classes.  They repeated this with the 3rd, 13th, 23rd, and 33rd size classes.  And they repeated this with the 4s group, the 5s group, .... and so on to the 10s group  (10th, 20th, 30th, and 40th size classes).  Since they could pick apart this broad continuous distribution and break it into ten 4-mode discrete size distributions, they had achieved their goal -- which was to explain how dense packing of broad continuous distributions can be explained by the discrete packing theory.

The mathematics of discrete packing theories is simple.  Packing factors can be calculated on a simple, 4-function calculator.  There's no rocket science here.  The first class fills 60% of the space available.  The second size class fills 60% of remaining pores.  The third size class fills 60% of the next remaining pores.  Finally, the fourth size class fills 60% of all remaining pores.  Then, voilá!  2.56% porosity!  (After the first size class, 40% pores remain.  After the second size class, 60% filling of the 40% pores leaves 16% porosity.  After the third class, 60% of those pores are filled, leaving 6.4%.  After the 4th size class, 2.56% pores remain.)  If you don't like the 60% assumption, change it, and recalculate.  It is easy.     

Back to the example with 40 size classes.  If each set of 4 size classes packs to 97.44% (2.56% porosity), then the whole distribution can pack to 2.56% porosity.  That was easy.  Problem solved!

But what requirement was ignored?  You cannot throw these particles into a mixer and mix them when following the discrete approach.  You must separate each size and pack each set of four classes one at a time to their densest positions to follow the discrete packing approach.  Then you must do the same with the 2nd through 10th sets of four classes.  This is a completely artificial way of making a ceramic product which, if you followed it perfectly -- it would produce a layered structure within the container.  

Most researchers who follow the discrete approach do so because the mathematics are simple.  Then, they simply ignore all requirements regarding the actual fabrication (packing) of the ware.

Do Fine Particles Fit into the Entrance Pores?

Another thing that is ignored with abandon is the 200:1 size ratio requirement by the discrete packing theory to pack fine particles within coarse particle pores.  The reasoning is something like this:  If a 200:1 size ratio is required, well then ---------- 100:1 is very similar.  And if 100:1 works is a reasonable substitute, --------------- then 50:1 will work similarly, too.  This goes on and on until the ratios are 10:1, 5:1, and approaching 1:1.  

A recent paper used the discrete approach to determine optimum packing for broad continuous distributions where size ratios approached 1:1.  It is a mathematical paper.  They did some experimental tests, even though the theories they were supposedly following forbid ratios like those approaching 1:1.

Entrance pores require a minimum of 7:1 to 10:1 size ratios of coarse to fine particles  in order that the fine particles can pass through the entrance pores into the coarse pores of the larger particles.  Take 4 basketballs.  Put three into a triangular arrangement and set the 4th ball on the top.  Will a soccer ball fit through the entrance pore and fall into the central pore?  (That approaches a 1:1 size ratio.)  Will a softball?  Will a baseball?  Will a golfball?  Must you use a smaller ball?

All such considerations were included in the original discrete packing theory.  In everyone's attempts to extend the discrete approach to cover continuous distributions, all such considerations are routinely ignored and seemingly lost.

Summary

Many recent research projects that follow the discrete packing methodology are attempting to extend the discrete approach to explain the packing of continuous distributions.  Many of these projects take great advantage of the simplicity of the mathematics of discrete calculations.  Many of these projects ignore the practical requirements of particle packing.  A successful packing theory (and equation) must define the resulting distributions and be consistent with the underlying theory's requirements.  If it does not obey the packing requirements of the fundamental approach -- it should be considered to be useless.

If the mathematics say you can achieve 96% packing with a system of particles, but all of the fundamental requirements of the discrete packing theory were violated or ignored, can one really make the claim that this result can be applied to real, continuous distributions?

On the other hand, starting with a theory that is based on real, continuous distributions -- one that requires only that powders be placed in a mixer and mixed well to achieve homogeneity and dense packing arrangements -- may require more difficult calculations, but provide actual, useful results.

Beware of the results of the packing studies that extend the discrete approach to real, continuous distributions.  If during the research, the fundamentals of the discrete packing approach were violated or ignored, but the mathematical results appear to work well -- there is an excellent chance the good results are simply serendipity!