How can one optimize a material’s performance when faced with a myriad of choices and no established rules for guidance? This problem is becoming increasingly important in physics, chemistry, materials science and engineering. A prototypical example is the “inverse packing problem”: instead of finding the properties of a packing of particles with given shape, we ask which of many possible shapes produces packings with a given desired property.  We introduce an efficient new approach that takes a cue from biology and on the computer artificially mutates shapes to evolve them into the fittest performers. We then verify the results in experiments.

Despite research dating over 200 years, a general connection between the mechanical response of a granular material and the constituents' shape remains unknown. The key difficulty in articulating this relationship is that shape is an inexhaustible parameter. Thus systematically exploring its role has been infeasible.  Here we demonstrate that the role of particle shape can, however, be explored efficiently when granular design is viewed in a fresh context: artificial evolution. By introducing a mutable representation for particle shapes, we show how simulations allow shapes to be evolved, and how 3D-printing allows the results to be experimentally verified.  As proof of principle, we find predictive motifs linking shape to packing stiffness and discover a particle that produces aggregates that stiffen, rather than weaken, under compression. More generally, our approach facilitates exploring the role of arbitrary particle geometry in jammed systems and invites the discovery and design of granular matter with optimized properties.

1. •Marc Z. Miskin and Heinrich M. Jaeger, “Adapting granular materials through artificial evolution”, Nature Materials 12, 326-331 (2013). link to article  video
   

2. •News & Views article by Corey O’Hern and Mark Shattuck
   

3. •Article about our work on the Stratasys blog with more details about our type of 3D-printer
   

The image above shows 3D-printed ‘granular molecules’ that were identified by our algorithm as optimal shapes for specific tasks. These compound particles are composed of rigidly connected spheres, with sphere size and bond angles optimized by the algorithm, and are printed as single pieces.

Below we show two examples of the evolution of particle shapes. Starting from a random configuration on the left in panel (a), the evolutionary algorithm mutates the shape and sends the information to a DEM simulation, which calculates the performance of a random packing of multiple copies of that shape (typically ~2,500) under compressive stress. The results from that DEM simulation are then used to produce a next generation of improved shapes, and the algorithm iteratively leads to an optimized shape for given boundary conditions.  For the specific examples shown, the goals were to identify the shape of granular molecules composed from N=4 spheres that produce the packings with the largest (top) or smallest (bottom) Young’s modulus E (i.e., largest or smallest initial slope of the stress-strain curve). The results are a linear molecule for the softest packings and a compact, ‘boat-shaped’ (i.e., slightly bent out-of-plane) molecule for the stiffest packings.

Panel b, bottom left, shows the predicted (solid lines) and experimentally measured stress-strain curves for the optimized shapes. Panel c, bottom right, gives the probability of finding a particular modulus for the case of random mutations.  Note that the moduli of the optimized shapes are four to six standard deviations away from the mean for the random case.

We believe that this approach can be adapted to optimize the properties any material composed of identical objects left to self-assemble. As discussed so far, only steric interactions between athermal granular molecules were considered, but it would be straightforward to use simulations that include attractive interactions and/or kinetic energy. For example, additional magnetic or electrostatic interactions could be explored by incorporating site-specific “functional groups” into the molecules.  Since our evolution method is essentially black box, any type of simulation that can take shapes modeled by bonded spheres as an input can be used as an engine for optimization.  Within this framework, the limits to granular materials design are then simply the limits to computation.