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

TL;DR

The paper investigates fractal properties of dense polydisperse disk packings generated by DT, CP, and GAP protocols across varying size ratios $R/a$, focusing on the mass-radius exponent $D_f$, structure-factor exponent $\alpha$, and size-distribution exponent $D$. Using a pore-analysis framework, it shows that for DT (and GAP) the exponents satisfy $D_f=\alpha=D$ when $R/a$ is sufficiently large, while CP packings exhibit non-universal exponents due to large cavities that reduce randomness; increasing $R/a$ gradually aligns CP exponents with the common value, albeit with significant computational costs. A novel algorithm to approximate pore filling demonstrates that the presence and statistics of cavities strongly influence $\alpha$ and the combined exponent $\alpha_c$, tying fractal scaling to configurational randomness and pore structure. Overall, the work argues that apparent protocol dependence is a finite-size and pore-structure effect, with all exponents converging to $D$ in the infinite size-ratio limit, thereby unifying the fractal description of dense polydisperse packings across packing schemes.

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 greater the ratio, the less the finite size effects are pronounced 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 big 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 configuration 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 infinite size ratio.

Figures (8)

• Figure 1: Comparison of spatial distributions of $N = 125000$ disks obtained from different protocols: (a) DT with zero boundary conditions (packing fraction 0.983). (b) CP with periodic boundary conditions (packing fraction 0.967). The disk borders and pores are highlighted in orange and white, respectively. One can notice some inhomogeneities and voids in the configuration generated with the CP protocol, see the discussion in Sec. \ref{['sec:CPP300']} below. (c) GAP (packing fraction 0.943) and N = 52366. In all cases the ratio between the largest and smallest radii is 1575.
• Figure 2: The smoothed structure factor (a) and mass-radius relation (b) for GAP when the ratio between the largest and smallest radii is 1575. Vertical dotted lines denote the borders of the range over which the fit has been performed
• Figure 3: The smoothed structure factor (a) and mass-radius relation (b) for different protocols when the ratio between the largest and smallest radii is 292. The number of particles is $N = 125000$ for each protocol. The results for the DT and CP protocols are represented in solid black and blue lines, respectively. The dashed orange lines show the combination of the DT and CP protocols (see Sec. \ref{['sec:CPP300']}), the corresponding curves are shifted vertically by a factor of 10 for better visualization. Vertical dotted lines denote the borders of the range over which the fit has been performed. Red curves represent errors after averaging over 20 trials.
• Figure 4: The smoothed structure factor (a) and mass-radius relation (b) for different protocols when the ratio between the largest and smallest radii is 1575. The results for the boundary-trimmed construction (see Fig. \ref{['figA1']}) is shown in solid green lines, and for the DT and CP protocols in solid black and blue lines, respectively. The corresponding curves are shifted vertically by a factor of 10 for better visualization. Vertical dotted lines denote the borders of the range over which the fit has been performed. Red curves represent errors after averaging over 20 trials.
• Figure 5: The smoothed structure factor (a) and mass-radius relation (b) for DT protocol with periodic boundary conditions when the ratio between the largest and smallest radii is 2500. Notations are the same as in Fig. \ref{['fig:DT_CP_300']}.