Volume 2 Number 9 Dennis R. Dinger 1 July 2004
An Update
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This topic was suggested by a reader.
Using DRD Add-In
Functions to Adjust Fired Shrinkages
and Water Absorptions
of Extruded Products
The assumption behind the original question was that all other influences on fired shrinkages and water absorptions are under control. In this article, we will consider both how to apply calculated minimum porosity values from the add-in functions, as well as other influences on fired shrinkages and water absorptions.
Factors Affecting Fired Shrinkages and Water Absorptions
Fired shrinkages and water absorptions are obviously related. As interparticle porosity in a compact is eliminated less open porosity remains in the fired ware to absorb water. One fundamental factor that is related to fired shrinkages and water absorptions is interparticle porosity in the green and dry states. This is the control factor that prompted the question leading to this topic. How much porosity is contained in the extruded product? How does one control the porosity in extruded products? How does interparticle porosity relate to the particle size distribution of the powders in the ware?
The other fundamental factor that relates interparticle porosity, fired shrinkage, and water absorption is firing behavior. Is the goal to fire and sinter ware to zero porosity? Does the body have a liquid phase that binds all particles together? How far has the body actually matured during firing?
When all wares are fired to zero porosity, fired shrinkages will be directly related to the porosities of the dry, unfired wares. When firing does not remove all porosity and water absorptions are finite positive values, or firing produces water absorptions that vary from day to day over a range of values, then fired shrinkages and porosities will not closely correlate.
To simplify the organization of this article, we will assume that each product has a fixed, final, desired water absorption that is routinely achieved. In other words, we will assume that the personnel at each production facility know how to monitor and control firing of their products to achieve constant water absorptions at the target values, and that they routinely achieve the target values. With these firing behavior restrictions imposed, we can now more easily consider factors affecting fired shrinkages of extruded wares.
Porosity Effects on Fired Shrinkage
If a compact contains 40% interparticle porosity, and firing removes all of the porosity, fired shrinkage will be fairly large. If the interparticle porosity is known, firing shrinkage can be calculated. A 1000cm3 piece of dry ware which contains 40vol% interparticle porosity will fire to zero porosity and zero water absorption at a volume of 600cm3 (the original volume minus the porosity). If the interparticle porosity in the dry ware is larger than 40%, shrinkage during firing to zero porosity will be even greater. If the interparticle porosity in the dry ware is less than 40%, shrinkage during firing to zero porosity will be reduced. For example, when the 1000cm3 ware contains only 10% porosity, the fired ware will be 900cm3.
This shows that the final (fired) size of ware produced in a single die will vary as the interparticle porosity in the green and dry ware varies, and the interparticle porosity will vary as particle size distribution (PSD) varies. The interparticle porosity can be estimated from the PSD using the DRD Add-In functions. Actual interparticle porosities will never be as low as calculated values. In deflocculated systems, interparticle porosity will track with and correspond to calculated minimum porosities: the lower the calculated value, the lower the actual value. For this reason, altering the PSD to reduce the calculated minimum porosity should produce better packing suspensions and forming bodies.
But when suspensions and bodies are flocculated, or even just partially-flocculated, actual porosities may have no correspondence at all to calculated minimum porosities of the PSDs. Viscous and rheological properties of flocculated suspensions will almost always benefit from PSDs with reduced calculated minimum porosities, but interparticle porosities in the green and dry states of flocculated bodies will be controlled by the flocculation, not by PSD.
When PSDs are excellent from a packing point-of-view, viscosities in deflocculated systems will be minimized at each particular solids content. If you need a lower viscosity at a higher solids content, improve the PSD of the particles (that is, improve the PSD to achieve a lower calculated minimum porosity), deflocculate the system, and raise the solids content. These five parameters are closely related. Each PSD will have a maximum solids content at which it can be practically used. To move to a lower solids content, however, suspensions and bodies only need to be diluted and flocculated. No changes need be made to PSD.
In a deflocculated, high solids suspension of a good PSD, interparticle porosity will be minimal, drying shrinkages will be minimal, and fired shrinkages will be minimal as well. Firing shrinkage will correlate with actual interparticle porosity, calculated interparticle porosity, and PSD in the deflocculated body. In a flocculated, low solids suspension, interparticle porosity will be high and fired shrinkages will be high. Firing shrinkage will correlate with actual interparticle porosity in the flocculated body, but it will not correlate to calculated interparticle porosity nor to PSD in the flocculated body.
Improving the PSD (from a packing point-of-view) of any body will help viscous and rheological properties, but it will not necessarily help interparticle porosities nor fired shrinkages. If the goal is a really dense body at high solids content, then PSD must be improved and the body will usually need to be at least partially deflocculated to achieve usable viscosities at the high solids contents. Formed bodies will then be fairly dense and dry and fired shrinkages will be relatively small. When dense bodies are not a priority, PSD is not critical, solids content is not critical, bodies and suspensions are frequently (and desirably) flocculated, and fired shrinkages will have little relationship to calculated interparticle porosities.
This explanation has been given to show that fired shrinkages relate to PSD of powders in some systems, but not in all systems. It depends which body properties are important and how bodies are controlled that determines whether or not PSD and calculated minimum porosities are related to firing shrinkages.
Particle Size Distribution Effects on Porosity
In those cases when PSD does affect porosity, how are the two related? The particle size distribution of body powders is the major factor that ultimately controls the porosity in such compacts. A monodispersion of particles (all particles are one identical size) will produce porosities of about 40%. A broad distribution designed to achieve perfect packing may define, for example, an ultimate porosity of 5%. In each of these two systems, when mixing is perfect, minimum calculated porosities (40% and 5%) can be achieved.
How does one reduce minimum calculated porosities? One must improve the packing capabilities of the particle size distribution. How? The particle size distribution equation to produce maximum packing is:
CPFT/100 = (Dn - DSn) / (DLn - DSn) (1)
where CPFT = cumulative percent finer than, D = particle size, DL = largest particle size, Ds = finest particle size, and n = distribution modulus.
To improve packing and to decrease porosity requires that the distribution be as broad as possible (that is, if covers the broadest possible range of particle sizes), and that it fits this equation with a distribution modulus of n = 0.37. Actually, it won't hurt much (in terms of calculated porosities) to target distribution moduli from n=~0.37 down to n = ~0.2. Reducing the distribution modulus doesn't hurt porosity much, but it frequently includes more fines in the distribution which helps viscous and rheological properties.
From a practical point of view, what does this mean? A distribution that can pack really well has the right size distribution so finer particles pack into all available pores created by the packing of coarser particles. One way to picture this is to imagine a monodispersion with its 40% pores. Figure out which particle sizes can be packed into each of the pores and add them to the distribution. This process creates more, smaller pores. Figure out which particle sizes can then be packed into each of these new pores and add them to the distribution. Continue this process until you run out of finer particles to be packed. This is one way to create a dense packing PSD. This explanation does not assume continuous distributions of particles. But the idea is always the same. When pores occur, appropriately sized particles need to be added to the distribution to fill those pores. As this is successfully achieved, porosities will decrease.
Equation 1 defines PSDs that are consistent with this explanation, but the equation assumes that the distribution of particle sizes is continuous -- that is, all possible particle sizes are present in the powder. The equation also assumes similarity of packing by all particles -- that is, boulders are surrounded by finer particles in similar manner to the way sand grains are surrounded by finer particles. When the packing surrounding each particle size is similar to the packing surrounding all other particle sizes, similarity of packing exists. And the similarity condition is a feature of equations of the form of Equation 1.
Mixing Effects
It's not just a question of PSD, however. It's not that simple. Perfect particle size distributions won't pack perfectly if the particles are not properly (homogeneously) mixed. Consider a particle size distribution that perfectly fits Equation 1. It is a broad distribution of particles which includes really coarse through really fine powders and the distribution modulus is perfect (n=0.37). This perfect distribution can pack and produce porosities from 40% to the minimum (perfect packing) porosity and anywhere in between.
How can a perfect distribution pack to 40% porosity? When the distribution is totally unmixed, and all particles are segregated by size so they all pack only with others of their own size, packing will be resemble that in a monodisperse system. Let's assume that the system contains a perfect distribution of basketballs, soccer balls, volleyballs, softballs, baseballs, golf balls, and marbles. This distribution may be capable of packing to 80% packing factor (PF) which is 20% porosity. But how can it pack to 40% porosity?
If all the basketballs are packed together, they will pack to 40% porosity. If the next layer on top of them are the soccer balls, they also pack to 40% porosity. Continuing in this way by adding a layer of volleyballs (at 40% porosity), a layer of softballs (at 40% porosity), etc., the whole system will also define a porosity of ~40%. Even though it may be a perfect distribution, it can still pack to 40% porosity.
If the whole volume in this example becomes well-mixed (and mono-sized layers are no longer present), the porosity of such a distribution will decrease well below 40%. If you still don't believe this, an experiment can be done which will prove the point. The experiment is this: define an excellent-packing distribution consistent with Equation 1; separate all particle sizes using sieves; and pack the powders into a volumetric cylinder in order of size -- coarse first, finest last. Use no vibrations to help pack particles into the cylinder -- simply pour each size class of powder one-at-a-time, coarse first, fine last, into the cylinder. When all powders have been poured into the cylinder, note the total volume occupied. Then, pour the powders into a bowl; stir and mix well; and pour the mixture back into the cylinder. The result should be a much-improved pack that fits into a smaller volume in the cylinder.
What is the point? Perfect particle size distributions, when not mixed well, will not produce expected porosities. Mixing, therefore, is important to successfully achieving controlled porosities.
Additive Effects
Is it possible for distributions to pack worse than ~40%? Yes. In the previous example, assume that all of the basketballs, soccer balls, volleyballs, etc., have been painted with contact cement before being dumped into the container. When such coated particles are poured into a container, porosities can be 50%, and even worse. When particles cannot slide against one another, but they stick to each other at initial contact, high porosities can be defined. Maybe you aren't using contact cement in your body formulation! But many extruded products use binders, which are distant cousins of contact cements. Some bodies use exceptionally rough particles, and extremely fine (colloidal) particles with especially high surface areas. They behave similarly.
Many forming bodies are flocculated, and flocculation is also a distant cousin of contact cement behavior as well. Flocculation causes particles to come together, to form chains, and then to define 3-D gel structures with pore channels throughout. Many pore channels are quite large. A perfect packing particle size distribution will never pack densely if it is used in a state of flocculation. For perfect packing to be visible, systems must be extremely deflocculated. But extreme deflocculation is bad rheologically, so it is not desirable.
Another additive consideration is mixing. It is relatively easy to mix deflocculants into bodies to achieve homogeneity; it is more difficult to mix flocculants to achieve homogeneity; and it is most difficult to mix binders into bodies to achieve homogeneity. The level of mixing difficulty increases as solids contents increase -- and extrusion bodies are normally high solids contents. This means that it will be most difficult to mix additives uniformly into high solids content extrusion bodies.
How to Use the DRD Add-In Functions to Affect Shrinkage?
The DRD Add-In function required is the DRD_Porosity function which calculates minimum expected porosities.
Fired Shrinkages
Let's say that the calculated minimum porosity (using the DRD_Porosity function) is 20% for a production body, and the fired shrinkage is too high. Assuming that firing is under control and achieves the same final state of water absorption, a firing shrinkage that is too high says that the body is shrinking too much, and fired products are too small. This means that the ware's dry porosity must be decreased and the PSD must be improved from a packing point of view. To improve packing, one must broaden the distribution, improve the distribution modulus in Equation 1, and/or generally improve the fit of the actual particle size distribution towards the theoretically best distribution. For this type of PSD improvement to occur, the minimum expected porosity, calculated using the DRD_Porosity function, must be decreased.
To adjust calculated porosities of a production body, one usually has several PSDs available to be mixed into the body, and one simply tries different compositions of those candidate distributions to achieve a variety of calculated porosities.
On the other hand, if the firing shrinkage is too low, the fired ware will be too large, and one needs to add more porosity to the dry body. In this case, the distribution must be made worse from a packing point of view, and the minimum expected porosity, calculated using the DRD_Porosity function, must be increased. To worsen a distribution's packing capabilities, one must make it narrower (a smaller range of size classes), make the distribution modulus higher, and/or put too many particles into any one size class. This is usually fairly easy to do because normal ball milling produces distributions with relatively high distribution moduli (~0.6 and higher.)
Once again, several powder distributions should be available to be mixed into the production body. Different compositions of candidate powders will again allow adjustment of calculated porosities.
Out-of-Spec Fired Sizes
Let's take this one step further. When die tooling is a fixed size, assumes a particular fired shrinkage, and final ware sizes are out-of-spec, this problem (firing shrinkage) applies. When fired ware sizes are out-of-spec and too large, firing shrinkage is too small, dry porosities are too low, and distributions have improved from a packing point of view. When the fired ware sizes are out-of-spec and too small, firing shrinkage is too large, dry porosities are too high, and distributions have worsened from a packing point of view.
When ware sizes are out-of-spec, it can be an indication of forming problems in the process. But it can also be a sign that particle size distributions have changed over time.
Mixing in Extruders
It is also highly probable that firing shrinkage variations have occurred due to mixing problems -- especially in extrusion processes. In many extruders, the quest for throughput speed has reduced the level of mixedness of bodies. Bodies should be well-mixed before they are fed into extruders. The desire for speed causes inadequate mixing of additives, water, and powdered ingredients in bodies fed into extruders. Poor mixing causes the rheological problems known as dilatancy and dilatant blockages. High process speeds exacerbate these two rheological problems.
Mixing of high solids extrusion bodies prior to extrusion is difficult at best. The exception to this is when extrusion bodies are mixed as low solids content suspensions and they are filter pressed prior to extrusion. In many cases, bodies are mixed at the solids contents at which they are to be extruded. In cases when mixing prior to extrusion is poor, the best mixer for extrusion bodies may be the extruder itself. Why? I would be the first person to tell you that mixers are designed for mixing, and extruders are designed for extrusion, so use the right piece of equipment for the task for which it was designed. But extruders subject bodies to relatively high pressures at relatively low shear rates, which is an excellent set of conditions for mixing. The normal goal for mixing is to subject bodies to relatively low pressures at relatively high shear rates. Both modes work however, and the high pressure/low shear rate mode is especially good for extrusion bodies.
Experience has shown that to extrude some non-plastic powders (like alumina systems) with plasticizers and binders, the first pass through the extruder showed the results of poor mixing. For example, attempts to extrude 1/4" diameter rods produced 1/4" diameter pagoda shaped pieces. However, after passing the extruded products through the extruder several times, the binders and plasticizers were forced (by extrusion pressures) more uniformly throughout the bodies, and smooth 1/4" diameter rods were soon produced.
This should be a consideration for those who are extruding wares. Those who are making suspensions, filter pressing, and extruding may not see these problems. But those who are mixing only sufficient fluids and additives to dry powders to achieve extrusion body solids contents may experience mixing problems and corresponding extrusion problems. When this occurs, several slow passes through the extruder may help. Increasing the speed of the extruder, however, will usually make the problem worse, not better. Obviously this adds more steps to the processing of these wares, but it is a viable method to improve extrusion qualities.
Effects of PSD Variations
It is important to realize that process conditions may mask the effects of PSD variations. From a packing point of view, highly porous compacts will shrink well to produce zero porosity fired wares. Low porosity compacts (in which powders are already well-packed) will not shrink much.
But flocculation/deflocculation states during processing can mask the benefits of PSD control. To best see PSD effects, bodies should be deflocculated. But deflocculated bodies tend to be dilatant, which is bad, so they are not frequently used. In many bodies, deflocculants are added to reduce the state of flocculation, but most bodies are not intentionally adjusted to be highly deflocculated.
This does not mean that PSD adjustments will have no affects at all on processing properties. Particle size distributions that pack well usually produce excellent rheologies. Excellent rheologies are those in which dilatancy is not a problem. PSDs that pack poorly tend to be dilatant, even at low shear rates. Dilatancy from extremely narrow PSDs can be real processing nightmares.
So -- do not rule out the many other parameters that can and do affect firing shrinkages. But when adjustments to PSDs are required, it is simply a matter of raising or lowering calculated porosities for the total PSD of the body. If the calculated porosity is 5%, don't actually expect to achieve 5%. 10% or 15% porosities might be possible to achieve when the calculated porosity is 5%. The absolute value of the calculated porosity is not critical. When a calculated porosity value changes from 10% to 5% as the result of a PSD change, the relative (calculated) packing increase of 5% should produce a corresponding relative ~5% packing increase in the body. Similarly, a change in PSD which produces a change from 10% to 20% in the calculated porosity value should produce a corresponding ~10% increase in the actual body porosity.
Summary
Variations in fired shrinkage may result from particle size distribution variations, mixing inhomogeneities, additives and binders, and/or firing variations. When process, firing, mixing, and additive chemistry problems have been ruled out, particle size distributions can be adjusted to affect fired shrinkages.
Using the DRD_Porosity function, one can change particle size distributions to improve (decrease) the calculated porosity value to decrease fired shrinkages. The better the system can pack in green and dry states, the less porosity contained before firing, and the lower the firing shrinkage will be. One can also change the particle size distribution to worsen (increase) the calculated porosity value to increase fired shrinkages. The worse the packing in green and dry states, the more porosity contained before firing, and the higher the firing shrinkage will be.
The relative change of porosity from one PSD to another is the important value to monitor and control when using the DRD_Porosity function.
Remember that the benefits of adjusting particle size distributions to affect fired shrinkages may not be achieved when process suspensions and bodies are flocculated. Packing will vary consistent with calculated porosities when suspensions and bodies are deflocculated, but deflocculation favors the bad rheological property known as dilatancy. Higher solids contents, more deflocculated conditions, and slower extrusion speeds, however, can show particle size distribution adjustments in the wares' fired shrinkage values.
Miscellany
Please continue to send your ideas or questions for future topics. Thanks.
Until next time ...
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