Dense disordered jammed packings of hard very elongate particles: A new derivation of the random contact equation

This work describes a derivation of the random contact equation that predicts the packing fraction ϕ_MRJ hr of a dense disordered (maximally random) jammed state of hard, very elongate particles. This derivation is based on (i) the compressibility equation connecting the compressibility of a uniform system with its pair-correlation function: it is assumed equal to zero at jamming; (ii) the pair-correlation function of the interparticle distance scaled with respect to the orientationally dependent contact distance: it is assumed equal to the sum of a delta function and a unit-step function at jamming, where the former function accounts for the interparticle contacts, while the latter function accounts for the background. On assuming that the hard, very elongate particles are cylindrically symmetric with a length L and a diameter D and isostaticity occurs at jamming, the prediction, in particular that, in the limit of L/D → +∞, ϕ_MRJ hr L/D = (10 + 1)/2, is compared to the available experimental data.