Introduction

Over the last 16 years, many different engineers representing a wide range of ceramic industries have used our packing calculations to produce dense packing distributions and/or low viscosity/high solids content suspensions.  There are three major limitations that will prevent laboratory and plant suspensions from achieving the desired high density results.  In this article, we will briefly discuss these three limitations.

Limitation #1:  The Actual Distribution Is Not Continuous

The main assumption of the continuous approach to particle packing, a.k.a. the Andreasen approach or the Dinger-Funk approach, is that the distributions to be mixed are truly continuous.  When the "continuous" assumption does not apply, lab and plant results will not be consistent with calculated results using the minimum porosity calculations discussed in earlier E-zine articles.  This is not necessarily a bad thing -- but the gaps need to be thought through thoroughly to insure that they are properly placed for very specific reasons.

          Excel® Function Problems

At the extreme, a discontinuous distribution is actually a discrete distribution.  When some particle size classes contain particles and other size classes do not, body properties and calculation results will differ.  Any cells in the spreadsheet which use logarithm functions will produce calculation errors when histogram size class contents contain zeros.  This happens sometimes in very coarse sizes, all classes of which contain 100 Cumulative Percent Finer Than values.  It happens similarly in extremely fine sizes, all classes of which contain 0 Cumulative Percent Finer Than values. 

When there are gaps in the particle size distribution between the largest (D_L) and smallest (Ds) sizes, similar calculation errors will occur in the middle size classes. 

          Problems With The Continuous Theory

The Continuous Theory and the calculation algorithms are not designed to handle discrete or discontinuous distributions.  Other than the aforementioned calculation errors, the calculation algorithms will certainly produce results for discrete and/or discontinuous distributions, but such results should be questioned.

A perfect packing distribution with a distribution modulus of n=0.37 is a fractal distribution.  The fact that perfect distributions plot as straight-line histograms on log-log axes is the requirement that produces the fractal properties.  Gaps in the distribution change the nature of the packing assumptions which require all sizes to be represented.

The requirement which produces fractal properties is the similarity condition.  All particles in the distribution pack similarly (relative to surrounding particle sizes) as all other particles.  Consider a broad, continuous distribution that starts with basketball-sized and volleyball-sized particles and after several intermediate size classes also includes BBs that are 1/100 the size of the basketballs and buckshot that are 1/100 the size of the volleyballs.  Under ideal conditions, the volleyballs will pack in a certain way relative to the basketballs, and the similarity condition will cause the buckshot to pack in that same identical way relative to the BBs.  Each other size in the distribution will pack in that same identical way relative to its next larger neighbor size.  This is the 'similarity condition' that is required by the Continuous Packing Theory and by fractals.  Removing a few size classes will change packing arrangements in ways that are not accounted for by the continuous packing theory nor by fractal requirements. 

When a few continuous size classes are in a group, followed by a relatively large size gap (containing no particles), followed by several more continuous size classes forming a second group, the continuous approach's calculation routines will not produce proper results. 

The solution to this limitation is to use continuous packing algorithms only with truly continuous distributions -- not with discrete distributions or semi-continuous distributions.

Limitation #2:  The D_L of the Actual Distribution Is Very Small

On the coal slurry project when we developed these algorithms, D_L values were approximately 300μm and the smallest particles (Ds) were submicron particles.  These were very broad, continuous distributions.  Typical minimum porosities calculated for these distributions were less than 6%.  The actual powders were about 4 decades in width (from ~300μm to ~0.03μm.)  These slurries packed very densely and produced excellent viscosities at the high solids contents.  When the D_L values were reduced substantially, or when the extremely fine Ds values couldn't be achieved, desired dense packing values and low viscosities could not be achieved.  D_L values (and to some extent, Ds values) varied with milling variations.  Ds values primarily varied with coal types.  Some coals easily produced the desired colloidal sizes during milling; some did not.

In summary, when we could produce the full, broad range of particle sizes during milling, slurry viscosities were extremely low, and packing densities were extremely high.  When we could not, viscosities were higher and packing densities were lower.

Unlike the coal slurries, many ceramic bodies cannot contain such large particles.  In many ceramic systems,   300μm particles are literally boulders!  Typical D_L values in many ceramic systems are 10μm or smaller.  In such cases, the width of product distributions will be much narrower than the coal slurry distributions.  These suspensions will never pack as well as the coal slurry powders because they are much narrower.  When this occurs, there are simply not enough particle size classes available to achieve the high density systems. 

Limitation #3:  The Surfaces of Particles Are Very Rough OR All Particles Are Very Small

The third major limitation on achieving really dense packs of particles is when particle surfaces are particularly rough, or when all particles are particularly small. 

This is not to suggest that the particles don't pack according to the theory.  Rather, it is the case that the roughness of the surfaces and the extremely high surface areas don't allow the particles to slide into their densest, ideal positions.  The particles would pack perfectly if they could move into their ideal positions, but their surfaces prevent them from reaching those positions.

Tests were conducted in our laboratory on the packing of several fine alumina powders that were mixed in percentages consistent with the continuous theory.  Densities achieved were all over the map -- nothing was similar to the predictions from the packing algorithms.  We then plotted the packing densities from the alumina distributions as a function of surface area of each distribution.  The results lined up perfectly with surface areas:  the higher the surface area, the worse the packing; the lower the surface area, the better the packing. 

These results showed that packing densities are affected (controlled??) by surface roughnesses and surface areas.

Summary

These are the three main limitations that have hindered successful dense packing in ceramic systems.  Poorly mixed distributions and distributions that are only poor approximations to the desired ideal distributions can always hinder successful dense packing.  But when mixing is not an issue, and when the distributions are excellent fits to the ideal, target distributions, these three limitations are the three major problems:

           1 -  Powders which are discrete or discontinuous 'continuous' distributions don't pack as expected.

           2 -  Due to the necessity to meet other product specifications, D_L values which are particularly small and/or distributions which are particularly narrow won't pack well.

           3 -  When powders are particularly small, they have especially high surface areas, or their surface textures are extremely rough, they, too, won't pack well.

In such cases, it may simply not be possible to achieve truly high density packs. 

Miscellany

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

Thanks for signing up to receive Ceramic Processing E-zine
To unsubscribe, click the "Remove Me" link below.

Copyright © 2006  Dennis R Dinger

103 Augusta Rd, Clemson, SC 29631   (864) 654-5731

consulting@DingerCeramics.com

www.DingerCeramics.com

All Rights Reserved.

All Rights Reserved.