Volume 1 Number 9 Dennis R. Dinger 1 July 2003
An Update
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Several people, within a few days of each other, all asked about the capability to calculate pore size distributions when performing the expected minimum porosity calculation. I have given it some thought over the years, but I have never actually tried to do any pore size distribution calculations. As Jim Funk said many times, "each individual pore is a whole pore size distribution in itself." So that complicates both the question and the possible answers. I decided, however, that I would begin to answer this question in this issue. There are a few fundamental points that derive from our packing models that apply to this topic. I will try to go over them in the following article.
Particle Packing and Pore Size Distributions
To introduce this subject, I will give some history behind our particle packing research, introduce the two approaches to particle packing (the discrete and continuous approaches), and then describe briefly our packing models and results. Our packing models lead to the point I want you to think about concerning pore size distributions. It's a long, round-about route, but I hope it will be interesting and informative.
Some History
Approximately twenty-five years ago, we started the project in which the computer packing models were developed. At that time, I wrote several Fortran routines to pack spherical particles into a box. The initial research direction I had been given was to consider the particle packing work of Andreasen. I was to figure out what he did, determine whether it applied to our problem, and if so, apply it to our problem and extend it as necessary.
Andreasen's packing work was performed in the 1920-1930s. Since they didn't have computers back then (they didn't have calculators either), he had to do all of his calculations by hand. He based his theory on continuous distributions of particles. Then he went into the lab and tested the packing of continuous distributions of particles. Because he was working with continuous distributions in the lab, he felt it necessary that his theory applied to continuous distributions. So we describe his approach as the continuous approach to particle packing.
A continuous distribution of particles is one in which all possible particle sizes within a range are represented. If a size class of particles is defined as 100-90 micrometer particles, all possible sizes within that range are expected to be and will be present. If a ceramic body has a particle size distribution described as 100% passing 150 micrometers, all particle sizes below 150 micrometers will be present.
The Discrete Approach to Particle Packing
The alternative approach to continuous is known as the discrete approach. All particle sizes within the range of a discrete distribution are not represented. A discrete distribution contains only a small number of tightly defined, discrete particle sizes. For example, a discrete distribution could contain three monodispersions. A monodispersion is a system in which all particles are exactly the same size. If I went out and purchased 1000 of my favorite brand and style of golf balls, that carton of balls would constitute a monodispersion of golf balls (all of exactly the same size).
If I wanted to make a ceramic body consistent with the discrete approach, I might use a monodispersion of 1500 micrometer particles, a monodispersion of 15 micrometer particles, and a monodispersion of 0.15 micrometer particles. The three sizes would constitute a polydispersion of particles. No sizes other than 1500, 15, and 0.15 micrometers would be present.
The discrete approach to particle packing says that you pack the coarse particles as densely as possible. Then, without disturbing that pack (by holding those particles in place as necessary), you pack the pores with the medium size particles. Then, without disturbing the pack of the coarse and medium particles, you pack the fines into the pores from the medium and coarse particles. This procedure will produce a dense pack consistent with the discrete approach.
But most powders aren't monodispersions. Most powders are continuous distributions over wide ranges of sizes. Chemically prepared materials are not truly monodisperse because even when they are supposed to all be a single size, they will have some size variations about their mean size. Sieved size classes are also not truly monodisperse because there is a finite width of sizes possible within each size class.
The discrete theory, however, is based on true monodispersions and the fact that the diameter ratio between adjacent size classes is at least 100:1 (greater is better). Taking short cuts with the definitions and using artistic license with the discrete packing rules negates the validity of the predicted packing behaviors. But short cuts, artistic license, and twisting of the rules, define where most of the discrete packing research over the last 70 years has been spent. During that time, many have tried to modify the discrete rules to apply the results to continuous distributions.
One last point about the discrete approach. It does not allow you to move or stir a system once it has been packed. Stirring or randomly distributing discrete particle distributions is not possible according to the theory. For example, if we densely pack a room with basketballs, then densely pack the pores with golf balls, and then densely pack those pores with BBs, we have a discrete dense pack of particles that is reasonably consistent with the rules. But when we have finished packing the BBs, rearrangement of the particles is not allowed. The basketballs, golf balls, and BBs cannot then be dumped into a mixer or into an extruder or into any other type of processing device. The process is complete after the fines have been packed! So the original volume filled with the coarsest particles had better be the exact shape of the ceramic object to be formed.
How many ceramics are made this way? This is not the way we make ceramics. We put the coarse, medium, and fine particles into a mixer of some sort and we mix them. Then after mixing, we use other processes to form the ware. Such procedures are not allowed by the discrete approach. The particles only pack consistent with the discrete theory when packed one at a time, coarse to fine, as discussed. If you mix all the particles together and pack them in one process, the result will not be consistent with the discrete theory. The resulting pack will be less dense than the theory predicts.
So the discrete approach is not a good approach to use to calculate the packing capability of a particle size distribution when the goal is to pack particles in the course of making ceramic wares by all traditional methods. The method of packing described above is certainly not a traditional method.
But the discrete approach is used all the time to calculate particle size distributions for dense packing. All of the discrete rules are routinely bent and stretched. Narrow particle size classes are not truly monodispersions, but that's overlooked. The size ratio between adjacent size classes is supposed to be greater than 100:1, but if we follow this rule, we don't have enough valid size classes for a body. So ..... well ..... 50:1 is close to 100:1, and 20:1 is close to 50:1, and 10:1 is close to 20:1, and soon, continuous distributions are being used in place of the monodispersions required by the theory. And finally, this approach calculates what will pack following the unnatural packing procedure described above. Then we throw the powders into a mixer, or extruder, or whatever, we ignore the rules, and we process them using traditional processing techniques. None of that is allowed by the discrete theory. But it is routinely done anyway.
The Continuous Approach to Particle Packing
Andreasen's continuous theory for packing was based on a similarity condition. Some have tried to say Andreasen's packing work was all empirical -- that he didn't have a theory. He did have a theory. He required that a similarity condition apply to the packing of continuous distributions. The similarity condition is a fundamental requirement in the formation of fractals. But fractal geometry and fractal systems were not formally defined until the 1960s, so Andreasen was somewhat ahead of his time in this regard. If you zoom into a fractal picture, you will see exactly similar structures at the new, higher magnification as were present at the lower magnification. If you continue to zoom in, you will continue to see the same structures. In a densely packed, perfect, continuous particle size distribution, if you zoom in, you will see similar particle size variations and packing arrangements as you saw at the lower magnification. If you continue to zoom in, particle size variations and packing arrangements will continue to be similar. This is the similarity condition Andreasen used to formulate his continuous packing theory.
To produce this similarity condition, Andreasen realized that a linear CPFT (Cumulative Percent Finer Than) versus particle size plotted on log-log axes would work. His equation is:
CPFT/100 = (D/DL)n
where CPFT
= Cumulative Percent Finer Than;
D = particle diameter;
DL = the largest
particle size; and
n = the distribution modulus.
This is a simple equation that produces straight lines on log-log axes. The similarity condition is a characteristic of any particle size distribution that follows this equation.
The error in Andreasen's approach and in this equation, however, is that he did not recognize a smallest particle size, Ds. Andreasen knew this was a problem, but he reasoned that it would be such a small error, it would make little difference. He then ignored it. But straight lines on log-log plots continue forever. They can reach huge numbers on the one extreme, and they can reach especially small values at the other extreme. For example, Andreasen's equation allows you to calculate the CPFT of a particle diameter than is 10-50m. A particle of that size is absurd. But it's possible with Andreasen's equation.
This is not a trick question either. There is no characteristic of a quartz crystal, or any other crystal, that is in that size range. There may be a characteristic of a sub-sub-sub-sub-atomic particle that is in that order of magnitude, but we are talking about powders here.
To fix this problem, we added a smallest particle size, Ds, to Andreasen's equation. We did this about 50 years after Andreasen first defined the equation. Our equation is:
CPFT/100 = (Dn - DSn) / (DLn - DSn)
Figure 1 shows an Andreasen distribution with a largest particle size, DL = 1000 micrometers and a distribution modulus, n = 0.37, and a Dinger-Funk distribution with a DL = 1000 micrometers, a Ds = 10 micrometers, and a distribution modulus of n = 0.37.
Figure 1. Andreasen and Dinger-Funk Particle Size Distributions with n = 0.37
Notice that the Andreasen distribution in Figure 1 continues off towards smaller and smaller sizes at the left edge of the diagram. Two decades off the chart to the left is 1 nanometer (10 Angstroms). Is it possible to have a particle of a particular crystal that is that small? No. Any particle of any distinct mineral will be larger than that. So Andreasen should have used a Ds in his equation if only to enter a Ds somewhat larger than 1 nanometer. He didn't, so we did.
Many in the mineral processing industry use Andreasen's equation (actually, they use the Gaudin-Schuhmann equation for milling -- but they are one and the same equation) for their particle size distributions. As their powders become finer and finer, their measured distributions deviate from the linear to curve down towards the Ds. Their theory (the G-S equation) says they should always have straight lines, but their measured distributions begin to show distinct curvatures when they enter the size ranges below about 200 mesh. All of our measured particle size distributions curve as well, but the D-F equation with the Ds included predicts, explains, and properly shows the curved distribution line.
Particle Packing in Continuous Systems
When I began to write particle packing models, I packed particle distributions defined by Andreasen's equation, packing particles from coarse to fine sizes. The volume of the box was 100%, and the volume of spheres to be packed was also 100%. We simply calculated the number of spheres of each size required to produce the Andreasen distribution with the particular distribution modulus we wanted. Then we packed the coarsest particles first, and continued packing as the particle sizes moved down through the size classes towards the Ds. When packing stopped at some finest size, Ds, (usually because the computer runs were taking too long, or the number of particles to pack became too large) which meant we had no finer particles to pack, we had defined the Ds for a D-F distribution. Our packed distributions therefore had the same DL and n as Andreasen's equation, but they each had a Ds. The smallest size packed defined the Ds.
As you may have perceived if you followed this explanation, D-F histograms are effectively identical to Andreasen histograms. They are all linear. D-F histograms have slightly higher percentages of particles in each size class (due to normalization to 100% total volume after a Ds is defined), but if an Andreasen distribution and a D-F distribution have the same modulus, n, the histograms will be parallel when plotted on log-log axes.
We know that perfect monodispersions pack to about 0.6 packing factor. The distribution modulus for a monodispersion can be approximated by a large positive number. At the other extreme, distributions with moduli near zero should pack perfectly using the technique in these models. There will always be space available for a few smaller and smaller particles (consistent with distributions with moduli near zero). Such distributions wouldn't pack very densely very quickly, but if the finer and finer particles are available, they would easily fit into the box. (Remember, a straight line on a log-log plot will eventually get as close to zero CPFT as you desire. Even a distribution modulus near zero will eventually approach zero CPFT if the fine particles are available.) The distribution modulus we needed to find, therefore, was the one between these two extremes which packed perfectly and filled the box most quickly as we packed particles from coarse to fine.
When packing particles into a box, since the total volume of particles was always equal to the total volume of the box, there was always sufficient volume in the box to hold all of the particles. The question was, "Is there enough contiguous volume to accept all of the particles?" The volume in which a sphere would fit had to be large enough to hold the sphere. It couldn't be divided and spread about -- a little volume here, a little volume there. It all had to be in one location and it had to be the right size to hold the sphere. So there was a distribution modulus less than about 10 (approximating a monodispersion) and greater than zero that densified most quickly, but always maintained enough contiguous space for the next particles to be packed. We went searching for that distribution modulus.
Andreasen had suggested, based on his experiments, that optimum packing occurred for distribution moduli between 1/2 and 1/3. So where Andreasen used hand calculations and distributions of powders to determine this, we used a digital computer.
After setting the computer to run 24-7 over the Christmas vacation that year, it packed spheres into a series of cubes following distribution moduli 1.0, 0.9, 0.8, 0.7, etc. We changed the distribution modulus increment to 0.01 in the 1/2 to 1/3 range, which is where Andreasen suggested we would find the optimum distribution modulus, and the computer packed spheres and spheres and more spheres.
The results showed that the optimum value for densest packing occurred when the distribution modulus, n, was equal to 0.37. This number is in full agreement with Andreasen's 1/2 to 1/3 range for optimum packing.
Pore Size Distribution
If you thought I forgot about pore size distributions -- I haven't. I wanted to set up a little of the history and then to show you Figure 1 to explain my point. Note that in Figure 1, I showed a D-F distribution with the Ds value at 10 micrometers. The Ds value is shown by a dark vertical line. I also showed that the CPFT value for the Andreasen distribution at that value is 18.2%.
Now we must do a little mental exercise. The packing model packed particles from coarsest to finest along the Andreasen line in Figure 1 -- until we had no finer particles to pack. Actually, the number of particles that needed to be packed became enormous when particle sizes became small. Computer run times became increasingly long as well. (Computers in 1980 weren't anywhere near as fast or as sophisticated as they are today. In fact, PCs today are probably comparable to, or faster than, some of the big mainframes used in those days.) So it took lots of time to pack the finest particles, and it took huge computer memories to hold the data. For each particle packed, we had to store its diameter, and the X, Y, and Z coordinates of its center point. Four data points therefore defined each particle that had been packed. Working in a computer with 64kb of memory, one was somewhat limited in terms of how many particles could easily be stored. So we didn't pack distributions that were particularly broad.
The results showed (to the best of our abilities) that when the distribution modulus was n = 0.37, the required number of particles in each size class could always be packed into the box with a constant percentage of space to spare.
Now, consider Figure 1 again. We calculated the number of particles from each size class to pack according to Andreasen's equation. When we stopped packing in each case, we calculated the corresponding D-F distribution which accurately described the distribution that had been packed. In Figure 1, we packed from coarse to fine, moving down the Andreasen line. If we stopped packing at 10 micrometers (as we did in Figure 1), the D-F distribution shows the particle size distribution that actually had been packed.
What is the significance of the 18.2% at 10 micrometers on the Andreasen line? This represents the porosity in the box within which we could have continued to pack particles. Had we had smaller particles to pack, we could have followed the Andreasen line below 10 micrometers and continued our optimum pack.
What is the significance of the continuation of the Andreasen line below 10 micrometers? It represents the size distribution of particles that we could have packed into the box if we continued packing. If we didn't have any finer particles to pack, the continuation of the Andreasen line represents the size distribution of pores available to hold all those particles we didn't pack.
This is the answer for which we've been searching. The pore size distribution in the pack represented by particles in Figure 1 is represented by the continuation of the Andreasen line below the Ds.
It's Not Quite This Simple!
This is only one part of the story of pore size distributions from particle packing results (but it is a major part of it). If you don't have particles to pack, but you know the distribution of particles that could fit into the pack if you did, you have defined the sizes of the available pores.
It didn't matter what distribution modulus we tried to pack. When the required number of particles (to fit the Andreasen distribution) no longer fit into the box (this occurred for distribution moduli > 0.37), if we continued to pack as many particles into the box as we could fit, we always saw a 0.37 distribution modulus. This leads me to believe that the distribution modulus of pores will always be 0.37.
We must also remember that each pore is formed by curved edges and surfaces. At their simplest, pores can be curvy pyramids. At their most complex, they can have many more than 4 sides, but they will always have curved edges, surfaces (for spherical particles), and at least 3-sided (4-, 5-, 6-, etc., are also possible) curved entranceways. Permeability within a compact is not a function of total porosity, or even the size of the pores, but a function of the size and number of entranceways.
Stack four golf balls in a tetrahedral arrangement. The pore between the balls is a larger diameter than any of the entranceways from the outside into that pore. Permeability will be a function of entranceway size and number of such entranceways the fluid must negotiate as it travels through a compact. Total porosity only determines the amount of fluid that will reside within a compact. Pore size distribution and in particular, entranceway size distribution, determine permeabilities.
Entranceway diameters into pores can be as small as 1/7 of a particle's diameter, but they can be much larger. In a suspension of powders, the level of additive chemistry and the state of flocculation/deflocculation will have a major effect on the porosity. Flocculation will generally open up the pore system to make it easier for fluid flow. Deflocculation can keep the pores and entranceways small.
With all of this information I have just presented, you now can ponder this subject when you're resting. If you're interested in pore size distributions, add these details to your thought processes and see how far you can take them.
References Cited
Andreasen, A.H.M., and Andersen, J., "Ueber die Beziehung zwischen Kornabstufung und Zwischenraum in Produkten aus losen Koernern (mit einigen Experimenten)," Kolloid-Z., 50, 217-228 (1930).
Miscellany
Paperback versions of Rheology for Ceramists and Particle Calculations for Ceramists are available in quantity at discounted prices. My goal for writing each of them was that all ceramists (regardless of level of formal education) would find them easy to read and understand. Feedback suggests I have successfully accomplished this goal. So if you are interested in purchasing multiple copies of either of the books to distribute to employees, please contact me.
I look forward to hearing from you. Please send me suggested topics that you would like to see in upcoming issues. See you next time. Thanks.
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