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Disorder Criterion and Explicit Solution for the Disc Random Packing Problem.

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  • 1Gonville & Caius College, University of Cambridge, Trinity Street, Cambridge CB2 1TA, United Kingdom.

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|September 24, 2021
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Summary
This summary is machine-generated.

Researchers solved the densest random disc packing problem, deriving the maximum packing fraction (ϕ_RCP) of 0.852525. This new method defines disorder and protocol independence for planar disc packings.

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Area of Science:

  • Physics
  • Materials Science
  • Applied Mathematics

Background:

  • Predicting the densest random disc packing fraction is a long-standing challenge across multiple scientific and technological fields.
  • Existing methods are hindered by ill-defined criteria for disorder and dependence on specific packing protocols.
  • A general, protocol-independent solution has remained elusive.

Purpose of the Study:

  • To formulate a well-posed, protocol-independent problem for planar disc packings.
  • To establish a systematic criterion for defining and limiting topological disorder, avoiding crystalline or ordered states.
  • To derive the exact highest possible random packing fraction.

Main Methods:

  • Formulation of a general, protocol-independent problem for planar disc packings.
  • Introduction of a systematic criterion to define and limit topological disorder.
  • Development of the cell order distribution as a central analytical tool.

Main Results:

  • The highest possible random packing fraction (ϕ_RCP) was derived exactly as 0.852525.
  • The cell order distribution was shown to directly yield the packing fraction.
  • This distribution parametrizes all possible packing protocols and quantifies topological disorder.

Conclusions:

  • A novel, exact solution for the densest random disc packing fraction is presented.
  • The cell order distribution provides a universal framework for analyzing random packings.
  • The method is applicable to predicting maximum packing fractions in specific protocols, such as sheared packings.