Dense random packing with a power-law size distribution: The structure factor, mass-radius relation, and pair distribution function.

We consider a dense random packing of disks with a power-law distribution of radii and investigate their correlation properties. We study the corresponding structure factor, mass-radius relation, and pair distribution function of the disk centers. A toy model of dense segments in one dimension (1D) is solved exactly. It is shown theoretically in 1D and numerically in 1D and 2D that such a packing exhibits fractal properties. It is found that the exponent of the power-law distribution and the fractal dimension coincide. An approximate relation for the structure factor in arbitrary dimensions is derived, which can be used as a fitting formula in small-angle scattering. These findings can be useful for understanding the microstructural properties of various systems such as ultra-high performance concrete, high-internal-phase-ratio emulsions, or biological systems.

[1]  R. Botet,et al.  Filling space with polydisperse spheres in a non-Apollonian way , 2021, Journal of Physics A: Mathematical and Theoretical.

[2]  R. Botet,et al.  Percus–Yevick structure factors made simple , 2020, 2007.06779.

[3]  R. Botet,et al.  Apollonian packing in polydisperse emulsions. , 2019, Soft matter.

[4]  Simon Unwin.  Circle , 2019, Children.

[5]  E. M. Anitas,et al.  The structure of deterministic mass and surface fractals: theory and methods of analyzing small-angle scattering data. , 2019, Physical chemistry chemical physics : PCCP.

[6]  M. Setou,et al.  A power law distribution of metabolite abundance levels in mice regardless of the time and spatial scale of analysis , 2018, Scientific Reports.

[7]  S. Torquato.  Basic Understanding of Condensed Phases of Matter via Packing Models , 2018, 1805.04468.

[8]  E. Anitas,et al.  Small-angle scattering from the Cantor surface fractal on the plane and the Koch snowflake. , 2016, Physical chemistry chemical physics : PCCP.

[9]  Saulo D. S. Reis,et al.  How dense can one pack spheres of arbitrary size distribution? , 2011, 1109.0966.

[10]  E. Anitas,et al.  Prospects for investigating deterministic fractals: Extracting additional information from small angle scattering data , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  C. Sorensen.  The Mobility of Fractal Aggregates: A Review , 2011 .

[12]  Gianni Schena,et al.  X-ray tomography measurements of power-law cluster size distributions for the nonwetting phase in sandstones. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  F. Stillinger,et al.  Jammed hard-particle packings: From Kepler to Bernal and beyond , 2010, 1008.2982.

[14]  Robert S Farr,et al.  Close packing density of polydisperse hard spheres. , 2009, The Journal of chemical physics.

[15]  M. E. El Youssoufi,et al.  Space-filling properties of polydisperse granular media. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Riyadh I. Al-Raoush,et al.  Simulation of random packing of polydisperse particles , 2007 .

[17]  E. Santiso,et al.  Dense packing of binary and polydisperse hard spheres , 2002 .

[18]  J. S. Pedersen.  Analysis of small-angle scattering data from micelles and microemulsions: free-form approaches and model fitting , 1999 .

[19]  T. Hales.  Sphere packings, I , 1998, Discret. Comput. Geom..

[20]  Lado.  Integral equation theory of polydisperse colloidal suspensions using orthogonal polynomial expansions. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[21]  Aste.  Circle, sphere, and drop packings. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[22]  R. Peikert,et al.  THE FRACTAL DIMENSION OF THE APOLLONIAN SPHERE PACKING , 1994 .

[23]  S. S. Manna,et al.  Precise determination of the fractal dimensions of Apollonian packing and space-filling bearings , 1991 .

[24]  J. Teixeira,et al.  Small‐angle scattering by fractal systems , 1988 .

[25]  G. Stell,et al.  Polydisperse systems: Statistical thermodynamics, with applications to several models including hard and permeable spheres , 1982 .

[26]  B. Widom,et al.  Random Sequential Addition of Hard Spheres to a Volume , 1966 .

[27]  S. Torquato,et al.  Dense packings of the Platonic and Archimedean solids , 2009 .