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Excluded volume geometry and packing fraction in binary convex hyperparticle mixtures.
1Eindhoven University of Technology, Department of the Built Environment, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.
Physical Review. E
|March 20, 2026
Summary
This study explores the excluded volume of similar hyperparticles in D-dimensional spaces using statistical geometry. Findings provide new equations for packing fractions of binary convex particles.
Area of Science:
- Statistical geometry
- Geometric measure theory
- Computational physics
Background:
- Understanding particle packing is crucial in materials science and statistical physics.
- Previous models often simplified particle shapes and dimensionality.
- The excluded volume concept is fundamental to predicting macroscopic properties from microscopic interactions.
Purpose of the Study:
- To calculate the excluded volume of binary similar hyperparticles in D-dimensional Euclidean spaces (R², R³, R⁴).
- To derive explicit equations for bidisperse and random packing fractions using two distinct statistical geometry approaches.
- To investigate the role of geometric measures in determining packing properties.
Main Methods:
- Utilized orientation geometry to analyze the excluded volume of rectangles in R².
- Employed integral geometry to determine excluded volumes for convex particles in D=2, 3, and 4.
- Derived closed-form expressions for packing fractions based on particle volume, surface area, mean curvature, and the second quermassintegral.
Main Results:
- Derived an explicit equation for the bidisperse packing fraction of rectangles in R², consistent with prior work.
- Presented excluded volumes for pairs of convex particles in R², R³, and R⁴ using integral geometry.
- Demonstrated that orientation geometry-based excluded volumes incorporate key geometric measures, enabling generalized packing fraction derivations.
Conclusions:
- The study successfully calculates excluded volumes and packing fractions for hyperparticles in various dimensions.
- The employed statistical geometry methods provide a unified framework for understanding particle packing.
- The derived expressions offer valuable tools for predicting the behavior of complex particulate systems.
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