Volume 1 Number 8 Dennis R. Dinger 1 June 2003
An Update
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Correlations Between
Process Properties and
Particle Size Distribution Parameters
Particle size distribution is arguably the most important property one can monitor and control in ceramic processing. We have been encouraging tight control of PSD for years. In this article, we will consider several of the parameters that can be calculated from measured PSDs. The goals of this article are to define the parameters, to show why they're important, and to show how they correlate with process properties.
PCI -- The Particle Crowding Index
One of the parameters that can be calculated from a particle size distribution is the PCI, which is the number of particles in a true cubic centimeter of powder. Jim Funk gave the PCI its name many years ago.
It is interesting how the PCI actually came to be used. We were developing calculations for a variety of parameters based on the measured particle size distributions of powders. We were calculating CPFT distributions from their corresponding histograms (and vice versa), surface areas (which required particle shape assumptions) and packing factors (which didn't.) We were plotting all of the results in a variety of ways to try to understand their interrelationships. Our blackboard was filled to capacity with ideas, calculations, and scribbles, and we were constantly erasing and squeezing more new thoughts and ideas onto the board on a daily basis.
One day, Jim asked me if it was possible to calculate the numbers of particles in a PSD. In fact, I had already been calculating the number of particles in each size class as a step in the procedure to calculate surface areas. He was pleasantly surprised to learn that I'd already been doing it. He didn't know I was calculating numbers of particles because it was an intermediate step in the calculation, and I was simply not printing the intermediate results anywhere that it could be seen.
From there, we began discussing how to calculate the total number of particles in the whole distribution, considering that particle size analyzer limitations don't allow the measurement of all particle sizes. Colloids, for example, were not measurable. Eventually, we put together a process to complete the particle size distribution measurement down into the colloidal size range by extrapolating the distribution so the calculated surface area equaled the measured BET specific surface area. (Unlike the limitations of particle size analysis, specific surface area measurements include all exposed surfaces of all particles.) The PCI was then born.
I have never really liked the extrapolation procedures because they were always neat, linear extrapolations of the measured particle size distributions, and the measured particle size distributions were everything but neat and linear. Even with its perceived flaws, however, we used the PCI successfully for years, and we still use it.
Following the PCI calculation, we developed a simple method to calculate the InterParticle Spacings (IPS) between particles in suspension. It becomes relatively easy to understand the practical meaning of the 'particle crowding index' when one thinks about it from the point of view of IPS. As the number of particles in a cubic centimeter of powder increases, IPS decreases and particles become crowded -- hence, particle crowding index, PCI.
1000 people squeezing into a banquet hall for a social hour will produce a much more crowded room than will 100 people in the same room. Similarly, the numbers of particles in a fixed volume can be enormous and they do become crowded. Alumina, sand, and other non-plastic powders usually contain relatively few particles -- only 1010 - 1012 particles per cm3, while clays and kaolins can have 1014 - 1016 particles per cm3. Think about it. 1010 particles represents "only a few particles." That's pretty crazy, right?! But those are the numbers. The particles do become crowded, and the label is an accurate, descriptive one.
What effect does the number of particles, or the PCI, have on a suspension or forming body? The number of particles in a cm3 of powder indicates how many particles are available to build the gel structure, and it affects rheological properties.
Not only do clays have many, many fine particles, but they are many, many colloidal particles. The fact that there are so many colloidally sized particles indicates that there are many particles that can easily enter into and form the gel structures in suspensions. Such small particles, with their tiny masses, can move quickly and can easily be affected by shear. Gel structures can break down easily, and severe dilatant behaviors between the coarser particles are minimized by the many suspended colloidal particles. If you're reading this on your computer screen, there are many, many such colloidal particles floating around in the air between your eyes and your computer screen. They move easily, and they don't collide with impacts like we'd associate with colliding 18-wheelers.
Since colloids are so small, they are strongly affected by electrostatic forces. They build gel structures quickly; they move easily; and those structures can be easily disrupted by shear. The more particles there are per cm3, the finer the powders, and the greater the numbers of such gel disruption/gel buildup phenomena that occur.
On the other hand, the fewer particles there are, the coarser they are, the more they act like boulders, the more difficult it is to change their courses of motion, and the more violent their impacts. Imagine standing in front of the bowling pins at a bowling alley and trying to stop the balls from hitting the pins. Then imagine that the balls are being launched by a 500 lb gorilla and they're coming down the alley like bullets. I don't want to try to stop such bowling balls. If they're not stopped, however, their collisions with the pins will be violent. You've all seen guys throwing bullets like this down the alleys, and the pins just explode. Even though the size scales are very different, behaviors of flowing powders are similar. No powder particles are as large (obviously) or their collisions as violent as bowling balls with the pins, but relatively speaking, coarse powder particles behave similarly relative to the fines.
So the numbers of particles crowded into suspensions, their fundamental properties (plastic or non-plastic), and their sizes all affect the rheological properties of flowing suspensions, the packing of compacts, surface areas, IPS, etc.
The numbers of each size particle of each ingredient in a multi-component body not only affects rheologies and packing behaviors, but drying behaviors, dry strengths, kinetics during firing, and fired properties of ceramics. The more particles of each type are present in a body, the better and more uniform will be their distribution throughout the body, and the better the body's firing behavior will be.
Porosity and Pore Size Distribution
The interstitial porosity in a powder compact is a measure of the packing efficiency of the powders: the lower the porosity, the higher the packing efficiency. We can calculate the ideal, expected porosity within a compact from the particle size distribution of the powders. This does not tell us the pore size distribution within the compact, but simply the total porosity.
Porosity and pore size distributions are two different parameters. The porosity in a compact tells how much fluid can be contained in that compact. The pore size distribution of the porosity in a compact, however, indicates how easily fluid can move through the compact. This difference causes each of the two parameters to correlate with different process parameters.
The interparticle porosity determines the packing ability of a powder sample, and the packing ability determines the maximum solids loading for a suspension of the sample. If a powder sample can pack to an 80% packing factor, 20% by volume of fluid will be required to fill the pores (without separating particles). Fluid in excess of that amount is required to separate particles and allow fluidity. Similary, a 60% packing factor requires 40% fluid just to fill pores, and more than 40% fluid to separate particles and allow fluidity.
Interparticle porosity therefore correlates with total drying energy (How much water can the compact hold? How much energy is required to remove that amount of water?), and total shrinkage (How much porosity must be removed during firing?)
The pore size distribution within the compact, on the other hand, determines how easily fluids can move through the compact. The larger the pores, the more easily the fluid can flow. Casting rate, filter pressing rates, and drying rates are all functions of pore size distribution. Firing rates, where oxygen needs to flow into a body so organics can be oxidized, and decomposition and reaction products need to flow through the pores and out of a body, depend upon pore size distributions.
Remember -- porosity alone is no indication of pore size distribution. Distributions that pack to 60% packing factors can have large pores, or they can have small pores. The same can be said of distributions that pack to any other packing factors. All by itself, the maximum packing factor of a distribution of particles gives no indication of the size distribution of the pores. And the same is true in reverse: The size distribution of pores says nothing about the total volume of pores present in a compact.
InterParticle Spacing -- IPS
The InterParticle Spacing, which is the average separation distance between all particles in a suspension, is fundamental to predicting rheological behaviors. Generally speaking, the greater the IPS, the fewer collisions will occur during shear, and the lower will be the measured viscosity. As IPS decreases, however, viscosities will increase because there is less fluid, the particles are closer together, and the particles will collide more frequently.
IPS also affects the effectiveness of chemical additives which cause flocculation or deflocculation. Since some portion of any additive remains out in the liquid while other parts adsorb onto particle surfaces, the IPS affects how such coated particles will behave. Even when particles are highly electrostatically charged by additives, the effects of those coatings may be neutralized when suspensions are overcrowded and insufficient distance exists between particles for the additives to function properly.
Since IPS is a function of PSD, one should consider whether it is more valid to control solids content, or to control IPS, in process suspensions and bodies. Additives will behave more uniformly from day to day when the IPS in which they are working is relatively constant. IPS may be more difficult to control, but it may also be more valuable to the process to have it under control.
Distribution Modulus
The distribution modulus, n, of a particle size distribution indicates the steepness of the slope of the histogram. The distribution modulus is part of the Dinger-Funk particle size distribution equation:
CPFT/100 = (Dn - Dsn)/(DLn - Dsn)
where n = distribution modulus,
D = particle diameter,
Ds = smallest particle diameter,
DL = largest particle diameter, and
CPFT = cumulative percent finer than.
This equation fits many particle size distributions. It is characterized by a linear histogram on a log-log chart (see Figure 1 of Issue 6). Distribution moduli of milled particles are almost always positive, which means the histograms will contain greater masses of coarse particles and less of fines. Distribution moduli of mixtures of particles, and especially of clays and kaolins (which are natural mixtures) will frequently be negative.
Distribution moduli greater than about 0.4 usually occur in dry milling operations. Distribution moduli less than 0.4 can be produced by wet milling systems under proper milling conditions. Low solids content wet milling procedures, however, also produce distribution moduli greater than about 0.4.
Computer modelling results indicate that the distribution modulus for 'perfect' packing and minimum porosity is n = 0.37. Experience suggests that distribution moduli from ~0.20 to ~0.37 will all produce excellent packing. Distribution moduli below 0.37 can reduce dilatant effects, while distribution moduli greater than 0.37 enhance dilatant properties. Remember: dilatancy is produced by particle-particle collisions. Therefore, distribution moduli in the right range to minimize dilatant effects (<0.37) can still produce extremely dilatant suspensions when suspensions are crowded and shear rates are high.
Largest Particle Size, DL
The largest particle size in the distribution, DL, has historically been a major 'control' point in manufacturing. In fact, it is far less important than most of the others mentioned (although it is still important). In particular, it makes a major contribution to particle packing efficiency.
We have frequently encountered particle size distributions in which the specifications tightly defined the largest particles, while totally ignoring the rest of the distribution. For example, coal powders to be burned in boilers were to be 98.5% less than 300 micrometers. Some ceramic powders, such as quartzes or feldspars have to be 100% passing 200 mesh, or 100% passing 325 mesh, with few other requirements. For example, a -325 mesh powder also happens to be -200 mesh. What happens when someone substitutes a -325 mesh powder for a -200 mesh powder? Are they equivalent? No, they're not. But a -325 mesh powder does satisfy the -200 mesh requirement (or even a -100 mesh requirement).
Our experience has shown that corrections to the coarse end of a distribution have the most influence on the packing of powders. Separating coarse fractions and adding appropriate amounts of each of them to distributions to improve their packing is much easier to perform than trying to separate fines and/or colloid fractions for the same purpose. Not only is it easier to perform the separations, but additional coarse particles have greater effects on improved packing than fines.
But it should also be remembered that coarse particles (compared to fines) have greater influences on dilatancy when solids contents and shear rates are high. A suspension containing a bell-shaped histogram with few coarse particles, lots of middles, and few fines, will perform better at high shear rates than a suspension consisting of lots of coarse particles and more optimized packing. There are always trade-offs and this is one of them.
Smallest Particle Size, Ds
The smallest particle in the size distribution, Ds, has seldom, if ever, been considered as a control parameter because it was not available until the advent of the PCI calculation. If the entire particle size distribution is coarse, then the smallest particle may be important because it, too, will be coarse. One way to think of suspensions is to consider them to be coarse particles suspended in fluids. When considered in this way, the fluid fraction usually includes the colloids. If there are no colloids, dilatancy can be exacerbated because colloids tend to help cushion the collisions between coarse particles.
Too many colloids, or all colloids, can cause dilatancy simply because the colloids can be crowded, and collision energies can be excessive. Just because colloids tend to cushion other collisions, and they are easily accelerated out of the path of larger particles, does not mean they will never colloid.
For example, one processing approach said that all particles in a body should be in the colloidal size range because then the grain sizes during sintering would be uniformly small. This may have been great from a fired property point of view, but it was (and is) a disaster from rheological and forming properties points of view. Such systems will typically be severely dilatant and forming processes will be nigh unto impossible.
A Viscometer Program to Measure Dilatancy
When process bodies show evidence of dilatant character, it would be handy to have a routine test that can demonstrate (prior to the forming process) whether or not the slips are nearing the onset of dilatancy. In this article, I will describe a simple rheometer program that can be used to measure whether a slip is pseudoplastic, whether it is approaching its onset of dilatancy, or whether it is actually dilatant.
The 20 minute gelation test discussed in the previous issue can give an indication of the rheological properties of a suspension, but it was designed to give more details about the gelation process than about the dilatant character of suspensions.
All Suspensions Are Yield-Dilatant
Those of you who are familiar with our green PPC book, or with my new Rheology book, know that all ceramic suspensions are yield-dilatant. (If you didn't know that, I'm making that statement here and now.) Because of the gelation behavior, ceramic slips are pseudoplastic (the time-independent shear-thinning rheology) and usually also thixotropic (the time-dependent shear-thinning rheology.) Where does yield-dilatancy enter the picture?
Yield-dilatancy becomes important as shear rates increase. Dilatancy is produced by collisions between particles in suspension as they are sheared. As a gelled suspension is exposed to shear, the gel structure breaks down, and apparent viscosities decrease. But eventually, all of the gel structure will be destroyed and all particles will be traveling as independent, free particles. Beyond that point of shear, viscosities will no longer decrease. As shear rates continue to increase, particle-particle collisions will become more and more intense and viscosities will begin to increase again.
The shear rate at the point of minimum viscosity is defined as the onset of dilatancy. Beyond that shear rate, viscosities will climb (which by definition is dilatancy.) All suspensions will have an onset of dilatancy shear rate. The only question that needs to be answered in each case is this: What shear rate defines the onset of dilatancy for each suspension? When solids contents are low, which means relatively few particles are in suspension, onsets of dilatancy will be high. Frequently, they will be so high that they are of no practical concern for the process.
But as suspension solids contents increase, and/or as particle size distributions change, the onset of dilatancy can decrease to levels that are within the applied shear rate range of the process. Since many common viscometers are limited to low shear rates, the onset of dilatancy may be beyond the shear rate range of the viscometer, but within the shear rate range of the process.
We want a test that can either directly measure and identify the onset of dilatancy, or one that at least shows when the slip is approaching the onset of dilatancy at the higher viscometer shear rates.
Viscometer Limitations
Many common, rotational viscometers are limited to relatively low shear rates. The 20-minute gelation test is designed to use low shear rates to measure gel properties. A dilatancy test, however, should utilize the full range of viscometer shear rates and it should especially use the highest possible viscometer shear rates.
Some of the older viscometers were limited to 60 rpm maximum. Shear rates at 60 rpm are low, but if that's the highest shear rate a viscometer can achieve, then it must be used. Newer viscometers can achieve 100 rpm, 400 rpm, and even 1000 rpm. The higher the rpm and the higher the imposed shear rates, the more accurately one can measure the onset of dilatancy.
If a viscometer is programmable, a dilatancy test should be programmed into it so it can be run automatically. If a viscometer is not programmable, dilatancy tests can still be run, although it will be more difficult to achieve consistency from test to test.
The Test -- Use Several Shear Rates
A simple dilatancy test uses several shear rates, with a one minute hold, or a 30 second hold, (or some other fixed time) at each shear rate. As many shear rates as possible should be used. A practical number to use is 10 different shear rates, although more is better. Even the older Brookfield viscometers which were limited to 60 rpm maximum, had eight speeds. To perform a good dilatancy test on such an instrument, all of the speeds should be used.
Because shear-thinning rheologies are generally described using power law equations, dilatancy test results should be plotted on log-log axes. The X-axis should be a log axis of shear rate (or rpm) and the Y-axis should be a log axis of viscosity. When 10 different shear rates are used, the shear rates should define uniform increments along a log axis.
The test procedure to use (for example) would be to:
1. Subject the suspension to HID (high intensity dispersion) for one minute prior to the dilatancy test.
2. Allow a fixed amount of time after removal from HID to move the sample to the rheometer, to insert the spindle into the sample, and to get ready to start the test. For example, allow 30 seconds to do this.
3. Start the automatic dilatancy test, which does the following:
a. Measures the viscosity at the lowest shear rate for the selected time duration (e.g., 30 seconds.)
b. At the end of the 30 seconds, increase to the next higher shear rate and measure viscosity there for another 30 seconds.
c. Repeat 3b until all shear rates have been used and viscosities have been measured for the same length of time at each shear rate.
4. Plot the results on a log-log chart.
Figure 1 shows results from dilatancy tests performed on two suspensions. In this case, the viscometer was not constantly monitoring viscosity. The viscometer program was set to measure the viscosity at each shear rate at the end of each 30 second measurement interval before changing to the next shear rate.
Figure 1. Dilatancy Test Covering
the 1-250 rpm Range
Blue is more dilute. Green is
higher solids content and more deflocculated.
The viscometer used for these tests was limited to integer rpm values from 1 to 250. As a result, the rpms selected were those that would best produce uniform log increments. Rpms were 1, 2, 3, 4, 5, 7, 10, 14, 19, 27, 37, 52, 72, 100, 139, 193, and 250. The procedure used to calculate these rpm values is one of the demonstrated calculations in the Particle Calculations for Ceramists book. The greatest deviations between calculated and actual rpm values occurred in the 1 - 4 rpm range due to viscometer limitations. Rpms should have been 1.4, 1.9, 2.7, and 3.7. We could not hit those rpms, so we rounded off to the nearest integer rpm values and proceeded.
The two samples in Figure 1 were the same basic suspension that were set at different solids contents and then tuned with deflocculant. The lower (green) curve corresponds to a more deflocculated, higher solids content sample than the upper (blue) curve.
The more dilute sample (blue) deviates from a straight line at the first few (1, 2, 3, and 4 rpm) points. This is due to the use of a 30 second time duration. All values would have changed slightly had we used a longer measurement time interval, and we would expect the first four points to better fit the expected linear behavior had we done so.. A 60 second time interval would have doubled the overall length of the test, so we opted for shorter intervals and a shorter overall test.
Interpretation of Results
Figure 1 shows one suspension that is not dilatant within the measurement range, and one that is. Shear rates corresponding to 250 rpm viscometer speeds can occur in some pipes, pumps, and other process devices, so the fact that the test demonstrated dilatancy in one suspension should be considered a red flag regarding that suspension's process performance.
When setting up a dilatancy test, the goal is to use shear rates at or near the viscometers' maximums. In this case, 250 rpm was the max for this viscometer. Had we been able to use higher rpms, we would have. But as you can see, this test utilized the full range of this viscometer's achievable shear rates.
There is no indication of dilatancy in the upper (blue) curve. Even at the maximum shear rate, the suspension's viscosity still appears to be decreasing linearly. The onset of dilatancy for this sample is difficult to predict because it occurs at a shear rate higher than the viscometer could measure. It could possibly even occur at shear rates greater than 1000 rpm (but we don't know that), which would suggest that dilatancy probably will not be any problem during normal processing of this suspension. The specific nature of the forming process in which this sample is to be used will help one make this judgement. A data base of dilatancy tests for this particular body, correlated with past process performances, will also give strong indications whether or not this is an acceptable rheogram. The blue rheogram is OK, especially when compared to the second suspension.
The higher solids content green curve, on the other hand, shows the onset of dilatancy at the shear rate corresponding to the 50 rpm measurement. The 50 rpm viscosity is the lowest of that test, and the viscosity increases from there until the 250 rpm measurement. Increasing viscosity with increasing shear rate is the definition of dilatancy. This second suspension should be expected to have problems during processing because most forming processes incorporate shear rates higher than that corresponding to a ~50 rpm measurement. Even if the viscometer used for this test could only achieve 100 rpm maximum, the onset of dilatancy could still have been measured for this sample.
This higher solids suspension, however, still may not experience any process problems. It depends how you're processing it. That's why you need a data base of dilatancy tests and notes of actual process performances of previous bodies. All of these details are needed before one can say with certainty that a suspension will have processing problems. Not knowing any better, however, I would suggest that if you can see dilatancy on a rheogram like this, the body will probably experience processing problems.
Suspensions that are intermediate to these two examples would produce curves that are more difficult to interpret. As the onset of dilatancy nears the maximum measured shear rate, rheograms will begin to flatten at the higher shear rates. Only when the maximum viscometer shear rate exceeds the onset shear rate will dilatancy actually be demonstrated by the test. When dilatancy test rheograms begin to flatten near the maximum measurement shear rate, the data base of rheograms and correlated process performances become very handy.
Another point to note on this figure is that both rheograms have the same viscosity at about 200 rpm. One is shear-thinning at this viscosity, and the other is dilatant at this viscosity. If one had simply measured the 200 rpm viscosities of these two suspensions, without measuring either at any of the other shear rates, they might appear to be identical. Rheologically, they are clearly NOT identical.
Summary
The two phenomena that occur in almost all suspensions are gelation and particle-particle collisions. Gelation causes shear-thinning behavior, while collisions cause dilatant behavior. The use of both the 20 minute gel test discussed in the previous issue of this E-zine, and this dilatancy test, allows one to measure both phenomena using the same suspension samples. One only needs to remember to HID the samples before the gelation test and again before the dilatancy test. The combined information from these two tests will be very useful for predicting process performance.
Miscellany
Any of the short courses I am offering in Clemson in June, or any other requested topics that are within my area of capability, can be taught on site. On-site fees start at a group size which includes up to 10 people. There is a fee for each additional person above 10, and the fee also includes the cost of my travel and expenses. This is an excellent way to enhance the education and understandings of larger numbers of employees in the fundamentals of ceramic processing. Plus, since I am the one doing the teaching, your employees would hear the various subjects from my point-of-view and in my teaching style, both of which I believe have been evident throughout these E-zines. If you would like any more information about on-site courses, please contact me.
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.
If you are trying to buy books or software or pay for short courses using PayPal, I recommend that you first set up an account with them, and then you return to the appropriate page on my web site and click one of the icons to purchase an item. PayPal uses two types of accounts: business and personal. To purchase items, you want a personal account. It appears that if you try to buy something without first having an account, the computer decides the type of account you need, and it sometimes selects business accounts. To circumvent this problem, you should set up an account with them first. The procedures are given on the Book Sales page and on the Short Course payment page on my web site. Following this procedure, you can set up domestic or international personal accounts (depending on your country of residence), and the problems some have been having with PayPal should disappear.
I look forward to hearing from you. See you next time. Thanks.
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103 Augusta Rd, Clemson, SC 29631 (864) 654-3155
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