(216e) Breakage of Structured Particles through Crack Propagation Phase-Field Model (PFM) Prediction Coupled with Population Balance Modeling

Population balance models are powerful tools that relate the breakage observed to the probability that stress-stain events may induce the breakage of a given size particle. Population balance models relate the observed breakage to how a particular particle might break. They help determine how, or how much of, a particle of size bin 1 will break into size bin 2, size bin 3, size bin 4, etc. The trouble is that these population balance models are not predictive. They can analyze a set of batch breakage data and determine how and at what rate the various-sized particles might break, but they cannot look at a set of particles interacting and determine what particle shapes might result in the most probable breakage scenario. That scenario depends on the fracture pattern that any particle might have. The fracture pattern depends on the shape of the particle that breaks, how the particle interacts with other particles during stress-strain events, and the texture of the particle surface as well as the internal structure of the particle.

Consider an oval tablet made from five or more components. The fact that the particle is oval suggests that the particle may self-align with other oval particles to minimize the shear during a stress stain event. This limits the way that particle-particle interactions happen during shear. Next, consider what might happen if the particle shape were a series of crosses. As the series of cross-shaped particles interact, where would the preferred interaction happen and, more important, what stresses on this odd particle shape would induce a preferred breakage pattern? What if the particles are long rods – what would be the preferred breakage pattern? Or, suppose the particles are shaped like pretzels or curved potato chips with ridges. What if the particle of interest is a larger particle made of seeds of all sizes and shapes glued together with some form of binding agent? What if the particle of interest is a ceramic part designed with heat and cooling fins and a series of holes spaced in the particle so it can be attached to some form of electronic device? How would those particles interact with a similarly-shaped particle, and what would the preferred breakage planes be during stress-strain events?

Almost all materials subject to breakage have some form of structure associated with them. It is, therefore, reasonable to suggest that the breakage selectivity in the population balance model could be predicted from first principles if we could define a mathematical structure to capture the shape of the particles of interest and the internal structure of what makes up the particle. Here is where finite element and phase-field modeling come into play. The finite element method can compute the expected forces and stresses on any complex shape based on simple interactions on the surface of the particles. In a very simple particle example, consider a long slender rod-shaped particle that is supported between two contact points. As an additional particle presses on the rod it will break at the prescribed location where stresses are greatest predicted by FEM analysis. Very complex structures can be examined by finite element methods to predict stresses on the particle. A crack will begin to form when the local stresses approach the yield stress. However, the problem is that FEM analysis requires a continuum to compute the stress and strains, while the act of breakage induces gaps in this continuum, thereby making a FEM analysis difficult to do without complex and continual regeneration of meshes as new particles are fractured or as cracks grow in the particle of interest.

This complexity can be modeled using a phase-field model (PFM) to represent a crack geometry in a diffusive way without introducing sharp discontinuities. This enables PFM to model crack propagation using standard FEM. Thus, how a crack will form and propagate in a particle of a given configuration can be predicted. Applying a series of allowable contact points and positions to structured particles, and then using PFM to determine the location of the crack, allows engineers to develop a model that relates shape, powder properties, and potential particle-particle contact patterns to the breakage of structured particles. The powder flow properties help predict the breakage rates in the population balance model while the PFM crack propagation pattern helps predict the breakage stoichiometry. Thus, this approach allows the population balance model to be predictive from first principles.