## A Model for the Reduction of Specific Surface Area of Powders with Age

A problem brought to the 85th European Study Group with Industry by AWE.

### Problem Description

High surface area powders are required in a number of technology areas. For example, the efficiency of catalysts is intimately linked to the surface area of the powder. In our context we are interested in explosives that are in the form of a powder. High surface area, small diameter powders tend to have a high Gibbs surface energy and tend to coarsen to reduce it.

To date the mathematical modelling of the coarsening process, sometimes referred to as .Ostwald Ripening., has usually made the assumptions that the explosives comprise a collection of detached spherical particles of differing radii and that mass transfer occurs via the Gibbs-Thompson effect.

The main aims of the proposed study are to investigate the physical modelling of the atomic-level mass-transfer processes causing coarsening of solid powders, and to model the evolution of the statistical distribution of particles.

### Study Group Report

Various modelling issues and unusual features of the measurement data were discussed. Four models of important processes were developed, and are reported here. Model (i) addresses the fundamental physics associated with the transport of mass by sublimation, difusion and condensation. Model (ii) uses chemical kinetics to develop a system of ordinary differential equations (ODEs) for the time-evolution of the frequencies of particle sizes. Model (iii) extends Model (ii) to a continuum particle size distribution. Lastly, Model (iv) considers the growth of particles as described by Cahn-Hilliard equations for the inter-particle transport of matter in Ostwald Ripening.

Models (i) and (iv) include the complex geometry and thermodynamics of the problem. By contrast, Models (ii) and (iii) focus on the time evolution of the PSD, but they are more difficult to associate with controllable variables, such as ambient temperature. Our discussions of models (ii) and (iii) suggest we can choose mass-transfer rate constants that reproduce the kind of observed evolution to a bimodal PSD. But more investigation is needed to determine how the rate constants may be associated with the particles' geometry and the thermodynamics of the mass transport processes.