The influence of packing protocol, size ratio, and pore structure on fractal exponents in dense polydisperse packings

Abstract

We study fractal properties of systems of densely and randomly packed disks, obeying a power-law distribution of radii, which is generated by using various protocols: Delaunay triangulation (DT) and constant pressure (CP) protocols and the generalized Apollonian packing. The power-law exponents of the mass-radius relation and structure factor are obtained numerically for various values of the size ratio of the distribution, defined as the largest-to-smallest radius ratio. We show that the size ratio is an important control parameter responsible for the consistency of the fractal properties of the system: the larger the ratio, the less pronounced the finite-size effects and the better the agreement between the exponents. For the DT protocol, all three exponents coincide even at moderate values of the size ratio. For the CP protocol, the exponents are different for both moderate and large size ratios. The suppression of the exponent of the structure factor in the CP packing is explained by the specific behaviour of pores, which contain relatively large cavities. We develop an algorithm for calculating the pore size distribution and show that it is related to the exponent of the structure factor. We argue that the presence of the cavities lowers the configurational entropy and thus reduces the randomness of the CP packing. Thus the cavities reduce both packing fraction and randomness of the CP packings. Nevertheless, there is a tendency for the exponents to converge as the size ratio increases, suggesting that all the exponents become equal in the limit of an infinite size ratio.