Random packing fraction of binary hyperspheres with small or large size difference: A geometric approach

Area of Science:

• Physics
• Materials Science
• Geometry

Background:

• Understanding random packing fraction is crucial for materials science and physics.
• Previous models, like Onsager's excluded volume model, laid groundwork for binary particle packing.
• Geometric approaches offer insights into space-filling properties of particles.

Purpose of the Study:

• To investigate the random packing fraction of binary particles in D-dimensional Euclidean space.
• To develop and validate models for binary packing with both small and large size differences.
• To explore the relationship between particle size ratio, space dimension, and packing fraction.

Main Methods:

• Utilized a geometric approach based on excluded volume concepts.
• Applied a recently developed model for small size differences, comparing predictions with computational data for disks (D=2) and hyperspheres (D→∞).
• Employed Furnas's theory for large size differences, adapting it for hyperspheres and comparing with computational results.

Main Results:

• The packing fraction increase due to bidispersity is proportional to (1-f)(u^D-1)^2 for small size differences, showing good agreement with computational data.
• Models for large size differences were successfully compared with computational results for hyperspheres in the large-dimension limit.
• An asymptotic approximation for large size ratios was derived, showing a first-order variation proportional to (2-f)u^-1.

Conclusions:

• Geometric and space-filling theories for simple hard spheres are valuable for studying hypersphere random packing and amorphization.
• The developed models accurately predict binary packing fractions across different dimensions and size ratios.
• A normalized D-dimensional binary packing graph provides a simplified phase diagram for amorphous assemblies.