Condensed Matter Seminar: Exact solution of the random close packing problem in two dimensions

"Predicting the densest random disc packing fraction is an unsolved paradigm problem relevant to a number of disciplines and technologies. One difficulty is that it is ill-defined in the absence of a disorder criterion.

Another is that the density depends on the packing protocol and the multitude of possible protocol parameters has so far hindered a general solution. I will present a new approach. I will first formulate the problem in a well-posed form for planar packings of discs. Then I will present a way to parametric all the infinitely possible packing protocols by the distribution of one structural quantity — the cell order. I will then describe a systematic criterion that limits both crystalline hexagonal order and further topological order, which is hardly discussed in the literature. Finally, I show that these lead to a derivation of an exact value for the random close packing fraction: φRCP = 0.852514…. Thus, the method (i) yields directly the packing fraction; (ii) parametrises all possible packing protocols; (iii) makes it possible to define and limit  all topological disorder; (iv) is further useful for predicting the highest packing fraction in specific protocols, which I illustrate for a family of simply sheared packings that generate maximum-entropy cell order distributions. Time permitting, I will describe current work on the extension of the method to random close packing of spheres in three dimensions. Reference:1. R. Blumenfeld, Disorder Criterion and Explicit Solution for the Disc Random Packing Problem, Phys. Rev. Lett. 127, 118002 (2021)".