Structure of marginally jammed polydisperse packings of frictionless spheres

We model the packing structure of a marginally jammed bulk ensemble of polydisperse spheres. To this end we expand on the granocentric model [Clusel et al., Nature (London) 460, 611 (2009)], explicitly taking into account rattlers. This leads to a relationship between the characteristic parameters of the packing, such as the mean number of neighbors and the fraction of rattlers, and the radial distribution function $g\left(r\right)$. We find excellent agreement between the model predictions for $g\left(r\right)$ and packing simulations, as well as experiments on jammed emulsion droplets. The observed quantitative agreement opens the path towards a full structural characterization of jammed particle systems for imaging and scattering experiments.

Figure 1

Granocentric model. Modeling the shell of first neighbors of a central particle with radius $a_c$ using a one-dimensional statistical approach. Particles with radius $a$ are drawn randomly from a distribution $p_2\left(a\right)$ and added one after another as explained in the text. The solid angles occupied by the already added particles are shown as colored blocks. The remaining accessible solid angle for subsequent additions is denoted ${\Omega }_a$. The dashed line indicates the threshold solid angle ${\Omega }_{\text{max}}$.

Figure 2

Three-dimensional imaging of jammed emulsions droplets with an average droplet radius $\overline{a}=1.05\mu \mathrm{m}$. (a) Raw image of a plane in the bulk of the sample obtained by laser-scanning confocal microscopy of light emitted by the fluorescent dye Nile red at $\lambda =595$ nm. The droplets are marginally jammed and the volume fraction is $\varphi \simeq 0.646\pm 0.014$. (b) 3D reconstruction of the droplet positions using the sphere-matching method (SMM). The lines show the Voronoi radical tessellation around the droplet centroids. The total dimensions are $51.2\times 51.2\times 20.1\mu \mathrm{m}$. One corner is cut out to reveal the internal structure of the jammed system.

Figure 3

(a) Average number of first-shell neighbors $\overline{N}$ of jammed particles (full symbols) and rattlers (open symbols) versus polydispersity (PD). Squares denote the results from the extended granocentric model (eGCM) and circles shows the data from numerical simulations. The dashed line indicates a constant value of ${\overline{N}}_J=14.3$. (b) Fraction of rattlers $q$ predicted by the eGCM (open squares) and from simulations (open circles). Inset: Predicted jamming volume fraction ${\varphi }_c$ for all particles as a function of polydispersity. Dashed line: parabolic fit ${\varphi }_c=(0.634+0.0278)\mathrm{PD}+\left(0.196\right){\mathrm{PD}}^2$. Solid line: jamming volume fraction for particles in contact ${\varphi }_c^J=0.62$.

Figure 4

Probability distribution functions as a function of (a) number of nearest neighbors $N$ and (b) number of contacts $Z$ for a polydispersity $\mathrm{PD}=0.15$. Solid line: eGCM; squares: simulations; bars: experiments on emulsion droplets with an average droplet radius $\overline{a}=1.05\mu \mathrm{m}$. The eGCM and the simulations assume a log-normal size distribution. The size distribution of the emulsion is also close to log-normal with $\text{PD}\simeq 0.15\pm 0.01$.

Figure 5

(a) Polydispersity dependence of the first peak value $g_1$ of the radial distribution function. The dashed line shows the prediction for $g_1$ obtained from packings numerically derived using the original GCM (neglecting rattlers) [26]. The solid line shows the prediction by Eq. (2). Diamonds: experimental results for polydisperse emulsions. Full circles: packing simulations. Star: experimental results for microgel particles, Ref. [29]. Inset: The first maximum of the radial distribution function $g\left(r\right)$ for a polydispersity $\mathrm{PD}=0.05$ (left) and $\mathrm{PD}=0.15$ (right). Prediction by Eq. (2) (solid line), the eGCM (open black squares), simulations (full red circles), and the emulsion experiments (open blue diamonds), $\text{PD}\simeq 0.15\pm 0.01$. (b) Normalized FWHM.

Figure 6

Critical volume fraction ${\varphi }_c$ for binary particle size distributions with different polydispersity (PD), $\delta a/\overline{a}$, and skewness $S$. The lines denote predictions by the eGCM, and the full symbols represent the results from packing simulations taken from Ref. [37].