Volume 6  Number 1                            Dennis R. Dinger                                1 November 2007

Updates

The E-zine

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"... 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.)          

Frequently, I am asked to address concerns about the effects of non-spherical particles on packing and rheological properties.  In this article, we will address this subject.

Non-spherical Particles, Rheology, & Packing

Most particles in production powders are non-spherical.  It is the rare, exceptional powder in which the natural shapes of the particles are spheres.  Some small polymers can be purchased in spherical form.  Some high temperature products, such as fly ash (if they are solidified molten droplets) are spherical.  Spray dried granules are usually spherical.  Most other powders are not spherical.  Many composite materials use fibrous particles which are extremely non-spherical.

Many production operations must therefore utilize non-spherical particles.  How do these non-spherical particles affect packing and rheology?  Those operations that actually use spherical particles benefit from the spheres.  How?

Modelling

Computer models are easy to write for spherical particles.  It is not necessary to randomly rotate each sphere to achieve random orientation within the model.  Cubes and other non-spherical particles, however, must be randomly rotated to achieve truly random orientations.  All particles used in the computer model must them also be randomly positioned.  Or, if the particles are in simulated motion, all particles must be repositioned for each time increment and all particles must be checked to prevent interference of particles.  That is, particles interfere with one another when they try to occupy the same space.  This is a difficult task to calculate for non-spherical particles but it is not difficult at all for spheres.  For these reasons, it is much easier and quicker, and it utilizes far less computer storage for spheres to be modelled than for cubes and other non-spherical shapes.

          Memory Requirements

When a computer model is written, one only needs to store the X, Y, and Z coordinates of the center and the radius, R, of each sphere.  If one wants to model ellipsoids, one must store the X, Y, and Z coordinates of the center, the major and minor axes lengths, R, S, and T, and two angles of rotation, A and B, to define the orientation of the axes of the ellipsoid.   (Just so we're clear -- the GoodYear blimp and M&Ms are ellipsoids.)  For cubes, one must store the X, Y, and Z coordinates of the center, the edge length, L, and two orientation angles, A and B.  For other rectangular solids, one must store the X, Y, and Z coordinates of the center, the edge lengths, L, M, and N, and two orientation angles, A and B.  For more complex shapes, more values must be stored for each particle used in the model.  Four values must be stored for each sphere, while six to eight (or more) must be stored for each non-sphere.  

The memory requirements for models of spherical particles is considerably smaller than the memory requirements for any type of non-spherical particle.  The larger the computer memory, the more particles can be used in the model -- but for a fixed amount of memory, more spherical particles can be used in a simulation than non-spherical particles.

          Calculation Requirements -- Processing Times

Regarding the number of calculations and the processing times of computer models:  It is very easy to determine when two spheres touch.  The sum of the radii of the two particles must be less than the distance between their centers.  When the distance between centers is smaller than the sum of the radii, both particles are trying to occupy the same space and one of the two must be moved.  On the other hand, it is very difficult to determine when two non-spheres touch.  One must calculate which part of the surface of each particle lies on the line between particle centers.  Then, one must calculate whether other points on the surfaces of both particle (that are not on the line joining centers) interfere.  This second part of the calculation is the most difficult for non-spherical particles.  Calculating non-interference of non-spheres therefore can take several large calculation routines, lots of trials of lots of particles, and lots of calculation time for each test for interference between each two particles (unlike the single, simple calculation for spheres.)

Calculating interferences between spheres is fast and easy, while calculating interferences between non-spherical particles is very difficult and very time-consuming.

          Influences of the Numbers of Particles in a Model

Within computer models, it is not a simple task to determine which particles are near one another.  This determination requires more calculations of the particles in the model to determine which ones could interfere with each other.  Unless one considers how to store particles in memory in such a way as to make this determination fast and easy, or unless one comes up with an easy way to identify particles that are in close proximity to the one of interest, it may be necessary to calculate possible interferences between many, many particles for each one particle of interest.  The longer it takes to perform this task, the longer the model takes to run to completion.  The combination of all such routines can increase run times to many, many hours.    

The more particles one wants to include in a computer model, the more calculations are required, and the longer the duration of the run time of the model.

Although this was a very simple explanation of model considerations, hopefully, the point was made that the simplest particle shape to use in any computer model is the sphere.  The sphere is not totally applicable to all particle types, but its utilization in computer models minimizes memory size and run-time requirements.  For these reasons, most computer models simulate interactions between spherical particles.  After the simple spherical models are successful, then the more complex shapes with their more complex calculation algorithms and longer run-times can be developed.

Packing

Many have asked how non-spheres pack compared to spheres.  This answer can be approached from two points-of-view:

          Ideal Results -- Considerations Following Continuous vs Discrete Packing Theories

The considerations in this section apply to real powders in lab and plant forming processes.

                    Packing of Discrete Distributions

Those who prefer to follow the discrete packing theories must realize that packing of particles in a distribution must be performed according to the discrete packing requirements:  (1) all particles of the same size must be densely packed so that they touch one another;  (2) all packing must take place from coarse to fine particle sizes, and (3) all smaller particles must pack densely (as in #1) within the pores remaining after all larger particles have been packed.

The fact that particles of each size must touch means that major surface/surface interactions, including touching and sliding of particles (which produces friction), will take place.  These two phenomena will hinder packing capabilities.  Also, particle orientations play a big role in this.  Cubes, for example, will pack very densely when particles are oriented properly.  Bricks and concrete blocks pack very well in the walls of buildings.  But note that each brick and/or concrete block was positioned individually by hand in a very specific orientation.  Most production bodies do not have this luxury.  Particles are dumped into mixers and bodies are mixed randomly.  When cubes and other non-spheres are randomly oriented, packing can be expected to be quite poor.  Spheres have no orientations to consider.  They will pack reasonably easily, touching and sliding interactions are minimized, although surface/surface interactions and friction/lubricity still determine the ease with which the particles can move into their proper random orientations and packing positions.  

The requirements of the discrete approach are major hindrances to the packing of particles of any shape, and to the packing of non-spherical particles in particular.

                    Packing of Continuous Distributions

Those who prefer to follow the continuous packing theory will find that normal processing methods are consistent with the requirements of the theory.  There is no requirement in the continuous packing theory that prevents moving or stirring particles after they are placed into a system.  The requirement for packing continuous distributions is that all particles are randomly positioned and oriented throughout the body.

Isn't that exactly what we are doing when we mix bodies in blungers and mixers?  If we don't achieve a good mixture, we haven't achieved random distributions of the particles within the body.  If we do achieve a good mixture, we have achieved a good random distribution of all powders throughout the body.

The only remaining variable of concern with the continuous theory of packing is the actual particle size distribution of the body.  That, however, is another subject.

Comparisons of the Two Approaches

To consider this more carefully, we will consider the requirements for packing particles in a computer model consistent with both theories.  The model contains a box, within which we are going to individually place particles consistent with the two packing theories.  

Consistent with the discrete theory, if a monodispersion of a particular shape of particle can pack to 75% packing factor (PF), the discrete packing theory requires that each particle size class in the discrete distribution be packed to 75% PF.  This is true in the lab, in the plant, and in computer models.  Only after this has been achieved for the coarsest particle size can the next smaller particle size be packed.   Etc.  This is an unusual method for making ware in labs and plants, but the approach requires it.

If a monodispersion of a particular shape particle can pack to 75% packing factor, the continuous packing theory, however, requires that only a small percentage of those particles that could fit into a container must be packed into the container.  

Let's consider a particle shape that can pack to 75% PF.  The discrete packing theory requires that the coarsest size of those particles be packed in the container to achieve 75% PF.  This is difficult to do -- even with spheres.  But the discrete approach to packing requires it.  

The continuous approach to packing, however, only requires that a small percentage of the 75% maximum be packed.  Instead of requiring that the coarsest size particles be packed perfectly and densely so they all touch and so they produce an actual pack at 75% PF (like the discrete approach), the continuous approach only requires, for example, 5% of the particles to be packed.  Within the container to be packed, one calculates how many particles are necessary to be densely packed to achieve 75% packing factor in the container, but then one only needs to pack the consistently small fraction of the maximum possible.  In this case, we will continue to use the 5% value.  Then, for example, if one calculates that 300 particles could densely pack and fill the container, the continuous theory only requires that 15 particles be actually packed.  5% are packed into the volume that could hold 75%.  Lots of volume remains between all of the particles.

It is easy to randomly locate and position these few particles of the coarsest particle size uniformly throughout the container in the computer model.  Few, if any, particles will touch.  Most will be spaced reasonably far from one another.  

Using this procedure where one only needs to pack a few percent of the maximum possible that can be packed, it is very easy to position new particles of a variety of sizes into the container before particles get close to one another and  interferences begin to dominate.  In our computer packing models developed 30 years ago, particles were only beginning to get close to one another after we had packed two log-decades (e.g., particle diameters from 100 to 1 make up two log-decades) of particles into the container.  

Here is the important point regarding particle shape:  Because so few particles need to be packed following the continuous packing theory, it really doesn't matter what shape they are.  If 50 basketballs must be located randomly and uniformly in 3D throughout a 10' by 10' by 10' room, there will be lots of space between balls.  None will be close to any other.  For this reason, the basketballs could even be placed into the same positions while they are still in the cubical boxes in which they are delivered -- and this would still leave lots of volume between all boxes.  

The continuous packing approach then says that each smaller particle size will be packed similarly within the new available space.  If only 5% of the basketballs that could be packed into the room must actually be placed there, then only 5% of the next smaller ball (volleyballs) must be packed into the remaining volume within and around the basketballs.  Due to the similarity condition required by the continuous packing theory, there will be just as much available volume for the volleyballs within the basketballs as there was available volume for the basketballs within the whole room.  The percentages of each size particle that must be packed are the same and the volumes available for all smaller particles will be proportionally similar.  Such a huge volume is available for each step in the packing that the actual shape of the particle makes little matter.

The bottom line is that to follow the discrete packing theories, particles must always be packed as densely as possible.  This will always maximize particle/particle contacts and interferences when attempting to achieve the dense packs.  To follow the continuous packing approach, it is possible to easily pack particles of any shape within the available packing volume.  

The second major difference between the two approaches is this:  Particles, once packed, CANNOT be moved according to the discrete packing theory.  Moving of particles corresponds to mixing, flow, and forming processes in ceramic production environments.  None of this is allowed by the discrete packing theory.  Particles must be packed into the exact, final shape of the ware, because once packed, they cannot be moved.  This is not the way we produce ceramics, but it is the way the discrete theory requires packing to occur.  

But since the particles CAN be moved once positioned, according to the continuous packing theory, all of our routine mixing, dewatering, drying, casting, and forming procedures are 100% consistent with the requirements of the continuous packing theory.  This means that packing can be done at lower solids contents where viscosities are low and rheologies are good, and that the solids contents can then be raised after all particles well mixed and randomly placed.  

Another way to say this is that the continuous approach to packing is consistent with all current ceramic forming techniques.  The discrete approach to packing is not.  Specifically, both approaches lead to the definition of particle size distributions (PSDs) that will pack well.  But if the required packing procedures are totally inconsistent with processing methods, there is little sense using the discrete approach to calculate the PSD when the particles cannot be placed into the positions required by the theory.

And if we follow the continuous theory and actually pack a container step by step consistent with the requirements of the theory, we see that particle shape has little effect on particle packing.

          Practical Results -- Particle/Particle Interactions

Consistent with the continuous theory of packing, in actual practice -- in the lab or in the plant -- bodies can be formed and mixed at low solids contents (while all particles are far apart) to achieve random distributions and orientations of all particles within the body.  Then, solids contents can be increased to bring all of the randomly arranged particles into closer proximity to achieve the higher packing densities required in production ware.  This process is totally consistent with the requirements of the continuous packing theory.  This process is not allowed by the discrete approach to packing.  

In actual practice, one wants to achieve random distribution of particles without major particle/particle interferences.  This can be done as described above.  Particle/particle interferences only come into play when solids contents rise.  With all particles should be in their random positions, the final compaction is only a matter of particles sliding into their densest orientations without major changes to particle locations.  Mixing in production equipment achieves the random distribution of all particles.  Later forming processes should not alter the random distributions of the particles throughout the body. 

Is particle packing a function of particle shape?  No.  From the point-of-view of pure packing potential, the shape of the particles does not appear to be an issue.  If a polydisperse system of spheres can pack to 85% packing factor, it should be possible to replace those spheres one-for-one with cubes and the polydisperse system of cubes should be able to pack to the same 85% packing factor.  One's ability to achieve this in the plant or lab is totally dependent on the effects of particle shape on rheology -- which will be covered next.

Rheology

This is where non-spherical particles have the greatest influence on forming operations.  Non-spherical particles simply do not flow well.  When cubical particles flow, they will spin and roll and their pointy corners will interfere with other pointy corners of other cubical particles.  The result will be poor (high) viscosities and poor (dilatant) rheologies.  

Spheres have no such corners or protrusions to interfere with other particles.  For this reason, spheres flow well.  As the aspect ratios of non-spherical particles increases, particle/particle interferences during flow increase and flow becomes more and more difficult, viscosities rise, and rheologies become dilatant.

Consider fiber reinforced composites.  Flow in such bodies will be slow, viscosities will be high, and efficient mixing will be difficult to achieve.  The fibers can tangle with one another which can prevent their uniform distribution throughout the body.  The higher their aspect ratios, the more problematic their mixing will be.

Rheology is a measure of the behavior of viscosity as a function of shear rate.  Viscosity is a measure of the ease of flow of fluids, suspensions, and bodies.  Neither of these will be very good when the body consists of non-spherical particles.  This will require extreme care during mixing operations.  Dilatancy will be a greater problem as aspect ratios increase.  Mixing speeds must therefore be reduced as aspect ratios of the particles increase, as solids contents increase, and as particles become more and more crowded.

Summary -- Packing and Rheology

To summarize, the shapes of particles do not appear to control packing potentials.  If spheres can pack well, so should other particle shapes.  But the ability of the process engineer and the production staff to achieve uniform mixing, excellent rheologies and viscosities, and desirable forming processes are EXTREMELY affected by the shapes of the particles.  PACKING is not the problem.  FLOW is the problem.  Particle shapes have little effect on packing behaviors, but major effects on flow properties required to achieve the packing behaviors.

If it is not possible to achieve good packs of non-spherical particles, it should not be a function of packing capabilities, but a function of the poor rheological and viscous properties of the non-spherical particles.

Spray-Dried Granules

Spray-dried granules present an interesting example of the combination of these points.  Spray drier feed suspensions are low solids content suspensions -- which means most particles are far apart, particle/particle interactions are few and far between, viscosities are fairly low, and rheological properties can be moderately good.  As the droplets in the spray drier are formed, particles are pulled into close proximity to one another without having to make major shifts of position.  The random distribution of all powders during mixing of the spray dryer feed suspensions should then translate to uniform distribution of all powders within the spray dried granules.  Each granule should contain a representative distribution of all powders used in the spray drier feed body.  The granules, however, will be generally spherical and spherical granules will flow easily.

Where a high solids forming body of the non-spherical feed materials will probably not flow easy during forming operations, the non-spherical particles can be contained in relatively dense spherical granules (which do flow well).  The non-spherical particles should be well distributed and randomly oriented within the granules, and the granules (which flow well) can be used to fill production dies.  

During dry pressing operations, relatively close-packed non-spherical particles must then move relative to one another to densify within the high pressure pressing operations.  If the granules flow well and all particles are well-distributed within each granule, minimal rearrangement of the non-spherical particles should occur during dry pressing.  The resulting wares should contain uniform distributions of all feed particles.  If there are forming problems, it will be that the particles within the granules do not flow easily during pressing operations.  This will not be a function of the packing capabilities of the non-spherical particles, but a function of the resistance of non-spherical particles to flow at high solids contents.