Disorder Criterion and Explicit Solution for the Disc Random Packing Problem

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.