Structural and Mechanical Characteristics of the Disorded-Ordered Transition in Mechanically Stable Sphere Packings

Using the discrete element method, we generate mechanically stable sphere packings in three dimensions that span a wide range in structural order, ranging from fully amorphous through to (quasi) ordered structures, as characterized by the globally averaged bond orientational order parameter. While amorphous systems exhibit features consistent with hyperuniformity – suppressed density fluctuations in the long-wavelength limit – as the packing structure becomes more ordered, the low-wavenumber limit of the static structure factor grows with increasing order. As the packing pressure, p, is varied from the marginally rigidity (p~0) to more robust systems (p>>0), the packing coordination number, z, follows a familiar scaling relation with pressure, Δz = z-z₀ ~ p^1/2, where z₀ = z(p=0). While it has previously been noted that Δz is the control parameter that determines packing properties, here we show how packing structure plays an influential role on the mechanical properties of the packings. Specifically, we find that the elastic (bulk and shear) moduli, generically referred to as M, become functions of both Dz and structure, to the extent that, M – M₀ ~ Δz. Here, M₀ are the values of the elastic moduli in the zero-pressure limit, whose values depend on the structure of the packing.