(194c) Design of Particles with a Prescribed Bulk Strength By Relating Particle Scale Forces to Bulk Cohesive Behavior in the Presence of Flow Aids

Sometimes in industrial applications, we need to design a powder of limited bulk strength to prevent cohesive hang-ups (i.e. arches and ratholes). Sometimes we need to design a powder with sufficient cohesion to form good tablets. Sometimes we need to design a powder free of segregation effects. Cohesion/adhesion between particles in a mixture can play a significant role in optimal product design to prevent segregation. In general, mixtures that are cohesive tend not to have severe segregation. Finally, the design of robust agglomerates requires that a powder have the right cohesive tendency within the bulk, and how much cohesion/adhesion exists between particles plays a significant role in that agglomerate product design. In short, we want to control the cohesion in the bulk by changing something about the particle scale world and the interaction of particles within the bulk.

Unconfined yield strength is defined as the major principle stress that will cause a bulk material to fail or yield in shear when placed in an unconfined state. Bulk powder strength is defined as a bulk property, but it is caused by particle scale interactions. Thus, we can also view the bulk cohesive strength from a particle scale approach. Three or more adjacent particles can come in contact with each other and undergo a shear event. During that event, these particles move past each other in the matrix of particles that make up the bulk. As this motion proceeds the force to resist this motion is comprised of sliding friction on one of the particles and the adhesion or pull-away forces present on the other particle(s). Both forces act on the particle moving and contribute to the work required to fail or yield the bulk material in shear. Thus, by adding up the contributions of the work of the adhesion forces and the frictional sliding forces we can relate the particle scale forces to the bulk cohesive behavior. It turns out that this analysis predicts a given dependence on the particle size, depending on what causes the adhesive forces between particles. Table I summarizes the adhesion mechanisms with the particle scale strength. For the case of narrow particle size ranges, the bulk unconfined yield strength is proportional to 1/D_p or 1/Dp² or 1/D_p^1/2 or 1/Dpⁿ where Dp is the particle size. The exact form of the relationship depends on the cause of the adhesive forces in the bulk material.

The relationship shown in Table I is a simplification since it applies to cases where the particle sizes are uniform and real systems rarely have particles of uniform size, but are made up of particle size distributions. However, looking at a unit cube consisting of eight random neighboring particles with void space in between allows us to determine the average friction and adhesion forces to move the top assembly of particles relative to the bottom assembly of particles one particle dimension. Therefore, it is possible to use the particle size distribution to generate a series of unit cubes that represent the particles in the bulk and then compute the average work to move these particles a distance of one unit cube.

Depending on what causes the adhesion in the bulk between adjacent particles, there will be different force-work behaviors with different dependence on the particle size. Also, it is completely probable that several force-work mechanisms will be active in the failure shear motion of the particles in this unit cube. Additionally, the void space in the interior of the unit cube can be filled with smaller-sized particles which also undergo shear motion during failure. However, not all the particles in this void space may undergo shear. Some of the finer particles in the void space may be tucked into the crevasses in the adjacent particles and simply translate with the moving particles. Taking into account all of these particle interactions on a unit cube level allows us to generate a mathematical model describing the unconfined yield strength as a function of the particle size distribution and the mechanism causing adhesion between adjacent particles.

The structure of the meso-scale (i.e. the unit cube) often dictates the bulk cohesive properties. In fact, the highest bulk strength is generally caused by the condition where the interior voids are just filled with smaller cohesive particles. This should not be a surprise since it is common knowledge that cement with a well-graded aggregate gives the strongest concrete structure. One final complication occurs in the micro-scale (i.e.in the contact zone between adjacent particles). In cases of caking where crystal growth between contact points of adjacent particles is one cause of strength, very fine flow-aids can either act to decrease the adhesion force between adjacent particles or act as an aggregate binder in the solidified solid bridges formed between adjacent particles during the caking event.

The basic approach of this work is to:

1. Sieve a given material to create a series of close-sized particle sizes.
2. Measure the bulk strength of various close-size particle distributions.
3. Plot this strength data as a function of the particle diameter.
4. Fit the resulting curves to a linear combination of the strength based on each mechanism to determine the mixing coefficients and the dependence on the particle diameter.
5. Generate a mathematical model using unit cubes produced from a preselected particle size of interest (i.e. a particle size for which you wish to estimate the bulk strength). This model will include the effect of filling voids between the particles (meso-scale) with finer particles (micro-scale) and the caking time effects at particle-particle contacts with hydrophobic and hydrophilic flow aids).
6. Compare the bulk strength computed for the model with the actual measured bulk strength of the real full-size distribution material to determine the accuracy of the model.

Two particle systems are considered. The first system is a non-caking system looking at a common drug product and will predict the strength of the drug based on knowledge of the particle size distribution. The second system is a salt subject to caking, but with the addition of both a hydrophobic flow aid and a hydrophilic flow aid. The net goal of the project is to demonstrate that knowledge of the bulk strength of any real system can be computed by knowing the particle size distribution and some basic information concerning what causes adhesion between particles. It was found that the non-caking system was very well predicted by knowing the particle size distribution and the mechanistic causes of caking in the system. Good agreement was also found between the particle size distribution model and the bulk strength of the caking material. However, the hydrophobic flow aid showed a significant decrease in bulk strength versus the hydrophilic flow aid. This shows the impact that micro-scale particle contacts have on the strength prediction models.

This approach can provide formulators with a means of predicting the bulk strength of any powder using particle size distribution data and some knowledge of the causes of adhesion between particles.