Fig. 5: Normalized scaling of Z, μ and shape descriptors.
Relationships between the normalized mean contact number, (Z − Zc)/ZΦ=1, and: (a) the normalized particle surface area, (A − Ac∣X)/AΦ=1, (equivalent to 1/Φ − 1/Φc∣Z), and (c) the difference between the aspect ratio, λ, and the crossover value, λc. Relationships between normalized apparent friction coefficient, (μ − μc)/μΦ=1, and: (b) 1/Φ − 1/Φc∣Z, and (d) λ − λc. In all subfigures, the location (0, 0) corresponds to the crossover point that determines the crossover values of X = Xc, Φ = Φc∣X, and λ = λc∣X. The solid lines show the analytical results of Eq. (1) with βZ = 2 in a and those with βμ = 20 in (b), as well as the analytical results of Eq. (2) with βλ = 13 and βZ = 2 in (c), and those with βλ = 13 and βμ = 20 in (d), respectively. The error bars denote ± 1 the standard deviation of the mean. See respective plots in Supplementary Fig. 5.
