Heterogeneous Structure of Granular Aggregates with Capillary Interactions

We investigate the spatial structure of cohesive granular matter with spheres floating at an air-liquid interface that form disordered close packings with pores in between. The interface is slowly lowered in a conical container to uniformly compress and study the system as a function of area fraction $\varphi$. We find that the free area distributions associated with Voronoi cells show significant exponential tails indicating greater heterogeneity compared with random distributions at low $\varphi$ with a crossover towards a $\Gamma$ distribution as $\varphi$ is increased. Further, we find significant short range order as measured by the radial correlation function and the orientational order parameter even at low and intermediate $\varphi$, which is absent when particles interact only sterically.

Figure 1

(a) Schematic diagram of the experimental apparatus. The liquid level inside the funnel is decreased to increase the area fraction $\varphi$ of the floating spheres. (b) The meniscus around floating spheres leads to attraction. (c) Mean number of individual aggregates versus $\varphi$. Only one aggregate remains on average above ${\varphi }_c=0.44$. Sample images of floating spheres ($N=688$) observed at $\varphi =0.33$ (d), $\varphi =0.50$ (e), and $\varphi =0.65$ (f), and corresponding Voronoi cells inside the dashed circle (g), (h), and (i). Cells corresponding to tightly packed regions using criteria $d_v/d<0.1$ and $0.1<d_v/d<0.5$ are identified with dark gray (red) and gray (green) markers, where $d_v$ is the distance of the particle from the center of the polygonal Voronoi cell.

Figure 2

Probability distribution function (PDF) of the free area $A$ normalized by the median value of $A$ is observed to deviate from a $\Gamma$ distribution ($\nu =3.6$) postulated for random distribution of points. The deviation is strongest at low $\varphi$ with significant probability of finding large cells. (b)–(d) The tail of the free area PDF with all the cells can be fitted by an exponential function with a decay parameter $\alpha =0.484$ (b), $\alpha =1.11$ (c), and $\alpha =2.82$ (d). When Voronoi cells are selected with $d_v/d<0.5$ the exponential tail decreases, and the PDF approaches a $\Gamma$ distribution ($\nu =3.6$) for $d_v/d<0.1$.

Figure 3

(a) The radial correlation function $g(r)$ for $\varphi =0.33$, $\varphi =0.50$, and $\varphi =0.65$ (top to bottom). Experimental results are compared with theoretical calculation for hard disks given by the Percus-Yevick equation. (b) $g(r=d)$ (amplitude of the first peak of this function) versus $\varphi$ for experiments with attractive particles and for solution of the Percus-Yevick equation.

Figure 4

(a) Average number of contacts inside capillary aggregates as a function of $\varphi$. Error bars are estimated from statistical dispersion and accuracy of measurement. (b) PDF of the local bond order parameter $|{\psi }_6|$, with a peak near 1 indicating strong hexagonal order.