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Explicit Analytical Solution for Random Close Packing in d=2 and d=3.

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|January 28, 2022
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We analytically derived random close packing (RCP) volume fractions for hard spheres in 3D and 2D. Our findings suggest RCP marks the onset of mechanical rigidity and the maximally random jammed state.

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Area of Science:

  • Physics
  • Materials Science
  • Statistical Mechanics

Background:

  • Random close packing (RCP) describes the densest possible arrangement of disordered spheres.
  • Determining the precise volume fraction of RCP has been a long-standing challenge, with various experimental and simulation methods yielding a range of values.
  • Understanding RCP is crucial for fields ranging from granular materials to condensed matter physics.

Purpose of the Study:

  • To analytically derive the volume fractions for random close packing (RCP) in both three (d=3) and two (d=2) dimensions.
  • To establish a unified methodology for calculating RCP volume fractions based on nearest neighbor statistics.
  • To explore the relationship between RCP, mechanical rigidity, and the maximally random jammed state.

Main Methods:

  • Utilized modified nearest neighbor statistics for hard spheres.
  • Developed an analytical derivation applicable to multiple dimensions.
  • Compared derived values with existing experimental and numerical simulation data.

Main Results:

  • Derived RCP volume fractions: ϕ_RCP = 0.65896 in d=3 and ϕ_RCP = 0.88648 in d=2.
  • Obtained values are consistent with literature findings from diverse methodologies.
  • The derivation supports the link between RCP, mechanical rigidity, and the coordination number (z).

Conclusions:

  • Random close packing (RCP) corresponds to the onset of mechanical rigidity and the maximally random jammed state.
  • The coordination number (z) dictates the RCP volume fraction, with z=12 representing a boundary condition at the face-centered cubic (fcc) limit.
  • Packings denser than RCP are achievable but require introducing some degree of order.