Random Close Packing

Author

Eric W. Weisstein
January 18, 2022

This notebook downloaded from http://mathworld.wolfram.com/notebooks/ComputationalGeometry/RandomClosePacking.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/RandomClosePacking.html.

©2022 Wolfram Research, Inc. except for portions noted otherwise

2D

Zaccone, A.  "Explicit Analytical Solution for Random Close Packing in d == 2 and d == 3."  Phys. Rev. Lett. 128, 028002, pp. 1–5, 2022.

"very accurate" Carnahan-Starling (CS) expression:

γ2=.43599;

g[σ_,d_,ϕ_]:=((1-Switch[d,3,1/2,2,γ2]ϕ)/(((1-ϕ))^d))

gg0=g0/.With[{ϕ=(π/(2(Sqrt[3])))},Solve[8ϕ (1/σ)g0 g[σ,2,ϕ]==6,g0]][[1]]

Select[ϕRCP/.NSolve[(8/σ) gg0 ϕRCP ((1-γ2 ϕRCP)/(((1-ϕRCP))^2))==4,ϕRCP],#<1&]

3D

Zaccone, A.  "Explicit Analytical Solution for Random Close Packing in d == 2 and d == 3."  Phys. Rev. Lett. 128, 028002, pp. 1–5, 2022.

"very accurate" Carnahan-Starling (CS) expression:

g[σ,3,ϕ]

Exact analytical solutions for g(σ) of hard spheres in all odd space dimensions are available based on Percus-Yevick (PY) theory

Exact P-Y result (eqn. (7)):

((1+(ϕ/2))/(((1-ϕ))^2))

ϕRCP3D=((2(Sqrt[648+π(π(54-24(Sqrt[2])π+5(π^2))-108(Sqrt[2]))]))/(36(Sqrt[2])+π((Sqrt[2])π-36)))+((2(36(Sqrt[2])-48π))/(36(Sqrt[2])+π((Sqrt[2])π-36)))-3;

N[ϕRCP3D,10]

FullSimplify[ϕRCP3D]

Together[((2(Sqrt[648+π(π(54-24(Sqrt[2])π+5(π^2))-108(Sqrt[2]))]))/(36(Sqrt[2])+π((Sqrt[2])π-36)))+((2(36(Sqrt[2])-48π))/(36(Sqrt[2])+π((Sqrt[2])π-36)))-3]//FullSimplify

-3+((2 (36 (Sqrt[2])-48 π+(Sqrt[Expand[648+π (-108 (Sqrt[2])+π (54-24 (Sqrt[2]) π+5 (π^2)))]])))/(Expand[36 (Sqrt[2])+π (-36+(Sqrt[2]) π)]))

N[((2 (36 (Sqrt[2])-48 π+(Sqrt[648-108 (Sqrt[2]) π+54 (π^2)-24 (Sqrt[2]) (π^3)+5 (π^4)])))/(36 (Sqrt[2])-36 π+(Sqrt[2]) (π^2)))-3,20]

The same procedure can be done using the CS expression instead of the PY one, which yields (Subscript[ϕ, RCP]) == 0.677376, i.e., a higher value. This can be understood as a different “protocol” for implementing statistical strong correlations among particles