Packing fraction of trimodal spheres with small size ratio: An analytical expression

Abstract

In previous papers analytical expressions were derived and validated for the packing fraction of bimodal hard spheres with small size ratio, applicable to ordered (crystalline) (2013)]. In the present paper the underlying statistical approach, based on counting the occurrences of uneven pairs, i.e., the fraction of contacts between unequal spheres, is applied to trimodal discretely sized spheres. The packing of such ternary packings can be described by the same type of closed-form equation as the bimodal case. This equation contains the mean volume of the spheres and of the elementary cluster formed by these spheres; for crystalline arrangements this corresponds to the unit cell volume. The obtained compact analytical expression is compared with empirical packing data concerning random close packing of spheres, taken from the literature, comprising ternary binomial and geometric packings; good agreement is obtained. The presented approach is generalized to ordered and disordered packings of multimodal mixes.

Packing fraction of particles with a Weibull size distribution

“…For the crushed glass the measured data can be compared using Eqs. (21) and (22). We can assert that the state of packing is similar to that of the beads; both particle types were compacted in the same way, so that B = 0.279 holds for beads and crushed glass alike.…”

“…In Eqs. (21) and (22), the effect of particle shape and densification on void fraction is accounted for by the monosized void fraction ϕ 1 . In both equations the monosized void fraction is found in both coefficients appearing on the right-hand sides.…”

“…In both equations the monosized void fraction is found in both coefficients appearing on the right-hand sides. One can see that for particles with a larger monosized void fraction, its void reduction by polydispersity is more pronounced as the second coefficient, the exponential function, of expressions (21) and (22), will then deviate more from unity. Applying less compaction to obtain a looser packing and achieve a larger monosized void fraction is, however, counterproductive, as then also a smaller B applies.…”

“…Positive values of β imply that the void fraction decreases (packing fraction increases) when some of the particles are larger (or smaller) relative to the others, so resulting in a void contraction. For these disordered packings it holds that, when particles of different sizes are combined, the void fraction is reduced (positive contraction) [19][20][21][22][23]. For ordered (crystalline) packings β is negative, featuring a void fraction increase (packing fraction reduction) when polydispersity is introduced in these packing types.…”

“…Another complication arises when ordered or disordered arrangements contain similar (isomorphous) particles of different sizes that are randomly placed. For mixes of discretely sized particles and continuous power-law and lognormal distributions, analytical expressions for the void fraction have been derived [19][20][21][22][23]. In Sec.…”