Determining the equivalent packing diameter of two-dimensional shapes

How to quantify the effective size of particles in a mixture, with different shapes and size dispersity, remains ambiguous. Recently, this issue has been extensively studied for disordered jammed packings in three dimensions, with particular emphasis on the concept of equivalent packing diameter (De). However, this approach cannot be simply generalized to two dimensions (2D). In this paper, we address this issue by calibrating De via quaternary mixtures in 2D, which effectively suppress the ordering in packings. The quaternary mixture consists of two bi-disperse systems of certain shapes with invariant size ratio 1.4. We verify the linear superposition of the specific volume of a shape-mixed disordered packing under identical De together with the transferability of the equivalent packing size ratio between different 2D shapes. We further devise a concise predictive model to estimate De for ellipses and superdisks. These results should constitute the foundation of 2D polydisperse packings.