Introduction

The two major approaches to particle packing are the (1) Discrete and (2) Continuous approaches.  A few brief comments will follow regarding the practical applications of these theories.  The natures of the particles size distributions that will be mixed and similarities and differences between the two approaches will be discussed.  The desire is that these few brief comments will enable you to approach the subject of packing more knowledgeably and to achieve success in your production goals.

Background to the Two Approaches

A lot has been written on these two subjects, so everything in these comments will be brief.  These comments will either answer some of your questions, or whet your appetites for more.  If the second is the case, please see the various reference lists that are available in our textbook and in my website.

          Discrete Distributions

The name for the 'discrete' approach to packing is derived from the fact that the particles to be mixed are defined as narrow 'discrete' size classes of particles.  For example, one class of particles may include all particles in the range 1.00μm to 1.05μm.  If that is the smallest size class, the ideal next size class should be all particles between ~200μm and ~210μm.  The ideal next size class should then include particles between ~4000μm and ~4200μm.  The ideal ratio between size classes is 200:1.

This describes a tri-modal discrete particle size distribution.  Note that there are no particles represented in the size range from ~1.05μm to ~200μm.and between ~210μm and ~4000μm.  Theoretically, this type of distribution will work well.  The middle size particles will pack densely within the coarse particles' pores, and the fines will pack densely within the middle particles' pores.  A perfect pack of these three distributions will achieve a packing factor of 0.973.

If ~4000μm particles are too large, the next smaller particle size class can be used, which is ~0.005μm to ~0.00525μm (which is ~5nm to ~5.25nm).  This class plus the ~1.00μm and ~200μm classes will make a similar tri-modal discrete particle size distribution.

          Continuous Distributions

The name for the 'continuous' distributions comes from the use of continuous size distributions.  Continuous distributions contain all possible particle sizes from the largest, DL, to the smallest, Ds.  All possible sizes are represented.  For example, the fines may be all particles smaller than ~1μm;  the middles may include all particles smaller than ~200μm;  and the coarse particles may include all particles smaller than ~4000μm.   There are no gaps in continuous distributions -- that is, all particle sizes will be represented between the largest ~4000μm particles and the smallest particle size Ds (which is usually well into the colloidal size range -- for example, down to ~0.01μm to ~0.001μm.)

This description describes a tri-modal continuous particle size distribution.   A perfect pack of these three distributions will achieve packing factors higher than the densest discrete tri-modal pack of 0.973.  In addition to not having gaps (size classes in which no particles reside) in continuous distributions, the three distributions used usually overlap.  In this example, all three distributions will contain colloidal particles in the colloidal size range down to Ds for this material.  When some coarse is added to the recipe, it will bring colloids along with it.  For excellent packing, however, the goal is to bring insufficient colloids along with the coarse and the middles, so some fines are needed to improve packing.

Practical Limitations

          Discrete Mixtures

The first, and obvious, question is:  Can you use a sufficiently broad range of particle sizes to have 3 distributions to pack perfectly?  That is, can you use 4000μm particles in your body?  If not, and you still want 3 distributions to pack, then you will need some of the 5nm particles to give you the third discrete size class.  If this, too, is not possible, then you have to start making approximations and changes.

Next, (assuming you have 3 or more discrete size classes available) the theory says that the coarsest discrete size class must be packed densely first.  Then, it cannot be moved or changed.  Once packed, all coarse particles must be locked in place.  NO FURTHER MOVEMENT IS ALLOWED!!  In practice, this means that once packed, the powders cannot be mixed, stirred, poured from one container to another, etc.  They need to be packed into the desired shape, and then their part in the processing is finished!

Within the dense coarse pack, the middles must be added and allowed to sift down through the pores until they are densely packed as well.  Then, they too must be held rigidly in place while the fines are added to the top of the mixture and allowed to sift down through the pores between the middles.  When they are densely packed, the process is complete.  Voilá!  You have a dense discrete pack!

This is a somewhat unusual processing technique, but this is REQUIRED for the theory to work.

There are a few points that usually are forgotten to be mentioned regarding dense discrete packs and the discrete packing theory:

     1 -- If the packing procedure is not faithfully followed (to the letter), the high calculated density will never be achieved in the pack.

     2 -- It is often impossible to achieve more than two size classes that are 200:1 apart to use with this theory.

     3 -- Even when it is possible to achieve three or more 200:1 size classes, the actual discrete classes are difficult (impossible?) to achieve.  How does one produce a size class that contains only particles between ~1.00μm and ~1.05μm?  If this size class were 10 times larger, this problem would still occur.  When the size classes are in the sieve size ranges, then, a sieve size class can be used.  This may still not be sufficiently narrow, but sieves usually can be used to produce relatively narrow size classes.  But when the size classes are below traditional smallest sieve sizes, lots of problems exist which will prevent ideally narrow size classes.  Even chemically prepared powders will be broader than required for discrete size classes.

     4 -- Monodispersions like each of these discrete size classes will have terrible rheology.   Think, "Dilatancy!"  Think BBs in a box. 

     5 -- The procedure required to pack and achieve dense packs by the discrete method is totally incompatible with traditional industrial practices.  If the target is uranium oxide fuel pellets, time and money may be no object.  But for almost all other products, the luxury of packing coarsest first, then locking them in position while the middles are sifted down through the pores, and then locking the coarse and middles in position while the fines are sifted down through the pores simply does not exist.

     6 -- And finally, a reminder ... once any class is packed, it CANNOT be moved again.  Those particles are to be locked in position never to be moved again.  Their packing process is done.  Complete.  Finished!  Any scrap materials CANNOT be re-used -- they must be discarded (or reprocessed to separate the three size classes again.)

          Continuous Mixtures

The packing of continuous mixtures was originally studied to be used on real systems.  That is, take any of your raw materials that cover a broad range of particle sizes, and pack then with other similar materials and distributions.  The packing process in this case is simple:  throw all ingredients into a container;  mix well;  and you are finished.  If the system changes solids contents (Did I mention?  ....continuous mixtures can be made from suspensions as well as dry powders) and it must be mixed again -- add water and mix it again.

     1 -- The ratio between particles in continuous particle size distributions approaches 1:1.  This is totally consistent with the continuous packing theory and with the actual particle size distributions of most particles.

     2 -- Traditional mixing methods are totally compatible with continuous mixtures and the continuous packing theory.  The only requirement is that the production bodies be well mixed to achieve homogeneity throughout the whole batch.

     3 -- Continuous distributions are easy to find and/or produce.  Milling powders produces continuous distributions.  Digging powders out of a pit provides continuous distributions.  They are almost everywhere to be found.

     4 -- The rheologies of continuous distributions are frequently shear-thinning (not rigidly dilatant), and the rheologies of final, dense continuous mixtures are usually excellent.  If done properly, viscosities will be the lowest possible when particles are capable of extremely dense packing. 

     5 -- The procedure for mixing continuous distributions includes any and all industrial mixing methods and equipment.

     6 -- And finally, a reminder .... wares that have been packed, and dry or wet scraps can be re-used.  Just add fluid, mix, and use again. 

Which To Use?  Discrete?   Continuous?

The more difficult of the two theories to use and to predict final packing densities is the continuous method.  Packing densities for discrete packs are relatively easy to calculate.

The easiest to actually produce in the lab or plant is totally consistent with the requirements of the continuous method.  To follow the discrete method precisely, which is required to achieve predicted results, one must work on a relatively small scale in a laboratory environment.  The discrete approach simply doesn't work well in large-scale production settings.

Errors That Scream, "BEWARE!!!" When Using The Discrete Theory

It is simple to choose the discrete approach, and then to calculate densest packing values that can be achieved.  But any deviations from discrete packing theory requirements does not allow predicted values to be achieved.

          Ignore the DO NOT MIX Requirement

When coarse particles are packed first, they will form a network throughout the pack with coarse particles touching coarse particles.  Then when the middles are added, coarse will still touch coarse, and middles will touch middles and coarse.  Etc.  To ignore the "Do not mix!" requirement, however, is to completely disrupt the dense pack.  Once mixed, the dense packing will never again be achieved.  WHY?  It won't happen because the theory requires coarse to touch coarse, middles to touch middles, fines to touch fines, etc.  Mixing disrupts this arrangement, and only the artificial method (pack the coarse first, then pack the middles, then pack the fines) can produce such packs.

          Put Other Size Classes Into The Gaps

Some have suggested that within the gaps in discrete distributions, other size classes can be defined that will improve the packing.  But this necessarily requires even more separation of the distribution into individual size classes that cover the range of sizes, and it requires an even more absurd method of packing.  The one-at-a-time packing procedures must be applied to each group of packing size classes.  Without going into more detail about the procedure required, the resulting packing process will be even more complicated and a layered structure which still cannot be moved will be produced.  Even though this may use 1000 different size classes, the results will be no better than the packing factors achieved with three size classes.  This is a wasted effort. 

          Ignore the 200:1 Ratio Requirement

Much of the work done over the past 80 years regarding the discrete theory has been to make slight variations to the rules.  The suggested ratio between size classes for optimum packing is 200:1.  Well, 100:1 is really quite similar to 200:1.  Then, 50:1 is similar to 100:1.  And 20:1 is similar to 50:1.  So you see, eventually the ratios begin to approach 1:1 (which is consistent with the continuous theory.)  It is simply not possible to pack 1:1 particles consistent with the discrete, 200:1 theory.  The reason for the 200:1 requirement is that the fines can sift through the pores and pore openings in coarse packs.  As size ratios approach 1:1, the smaller particles cannot fit through the pore openings of the coarse particles, so it is impossible to approach 1:1 and still follow the recommended packing procedures.  It cannot be done.

The attractiveness of the discrete approach is its ease of calculation, and its ease of visualization.  It's easy to understand the rules.  Pack the room full of basketballs.  When that's done, pack the pores full of golf balls.  Then, pack the pores with BBs.  That's easy to visualize and understand.  But to take a theory like that, and then to begin to ignore requirements, or to start bending rules, or to make approximations, etc., won't fly.  Finally, to say, "It won't really matter if we mix such distributions, so we'll ignore the funny packing requirement and throw our powders into the mixer and use our normal processing techniques.", is to totally throw out all predictions from the theory.  It does matter.  The screwball packing procedures must be followed to achieve the theoretical results.  Otherwise, it will be sheer serendipity if the final powders pack densely. 

Simplicity of theory, simplicity of calculation, and prediction of high packing densities does not mean you can actually achieve them in the plant with normal plant processes.  Many short cuts, approximations, and bending of the rules have been taken over the years which lead the practical results further and further away from predicted values.

Using The Continuous Approach

The continuous approach was designed to work well with continuous particle size distributions.  Why?  Because continuous particle size distributions were the types of distributions routinely found in the plant and lab. 

The continuous approach was less easy to understand, but to follow its rules was easy.  It simply had particle size distribution requirements -- but once the distributions were set, further processing, mixing, forming, etc., were totally consistent with traditional processing methods and equipment.  And over the last 80 years, nothing about these requirements has changed.

Recommendations

I have studied particle packing for many years, and I like the subject.  I am partial to the continuous approach simply because it uses real distributions and it is consistent with traditional processing methods.

I do not like the discrete approach (Come on, tell us what you really think!) because it is a very easy theory to see, understand, and calculate. but it is almost impossible to follow in practice.  As the rules over the years have been twisted, ignored, approximated, etc., the implications have always been that it these twists, approximations, and ignored requirements were making it easier and easier to overcome the original theory's processing requirements.  Not true.  If it requires a cumbersome packing method to pack 200:1 size classes perfectly, no amount of explaining will ever allow 20:1 or 10:1 size classes to pack the way 200:1 size classes will pack, and no amount of explaining will ever allow the substitution of normal mixing techniques to replace the cumbersome packing methods required.. 

My recommendation is to learn and understand the continuous packing theory and apply it to your particle size distributions.  Any and all processing and forming methods are compatible with the continuous approach.  Any and all mixing methods are compatible with the continuous approach.  Mixing itself is compatible with the continuous approach.

Miscellany

If these brief comments have whet your appetite for more on this subject, much is available in the literature.  Start with Jim Funk's and my "Predictive Process Control" book, and the many papers we've written on this subject.  Check the publications page at my web site for more info.  Then, if you still want more, write to let me know your questions, and I'll either write more, or point you to specific articles.

Suggested topics for future issues of this E-zine .... Please continue to send your ideas or questions for future topics.  Thanks.  Until next time ...

Merry Christmas and Happy New Year to you all.

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