Local origins of volume fraction fluctuations in dense granular materials

Fluctuations of the local volume fraction within granular materials have previously been observed to decrease as the system approaches jamming. We experimentally examine the role of boundary conditions and interparticle friction $\mu$ on this relationship for a dense granular material of bidisperse particles driven under either constant volume or constant pressure. Using a radical Voronoï tessellation, we find the variance of the local volume fraction $\varphi$ monotonically decreases as the system becomes more dense, independent of boundary condition and $\mu$. We examine the universality and origins of this trend using experiments and the recent granocentric model [M. Clusel, E. I. Corwin, A. O. N. Siemens, and J. Brujić, Nature (London) 460, 611 (2009); E. I. Corwin, M. Clusel, A. O. N. Siemens, and J. Brujić, Soft Matter 6, 2949 (2010)], modified to draw particle locations from an arbitrary distribution $\mathcal{P}(s)$ of neighbor distances $s$. The mean and variance of the observed $\mathcal{P}(s)$ are described by a single length scale controlled by $\mathrm{\varphi ̄}$. Through the granocentric model, we observe that diverse functional forms of $\mathcal{P}(s)$ all produce the trend of decreasing fluctuations, but only the experimentally observed $\mathcal{P}(s)$ provides quantitative agreement with the measured $\varphi$ fluctuations. Thus, we find that both $\mathcal{P}(s)$ and $\mathcal{P}(\varphi )$ encode similar information about the ensemble of observed packings and are connected to each other by the local granocentric model.

Figure 1

(a) Photograph (left) and schematic (right) of the apparatus, showing confining wall in constant pressure (CP) configuration, with weights $m$ suspended from a pulley via a monofilament line. Constant volume (CV) is obtained by fixing the wall to the surface of the table. (b) Equation of state $\overline{\varphi }(\mathrm{P̃})$ for CP experiments at ${\mu }_1$ ($+$), ${\mu }_2$ ($°$), and ${\mu }_3$ ($\times$) on linear axes and inset with semilog axes.

Figure 2

(a) Representative $\mathcal{P}(\varphi )$ for three experimental runs at $\overline{\varphi }~0.780$: $+$ for CP and ${\mu }_2$, $•$ for CP and ${\mu }_3$, and $\times$ for CV and ${\mu }_2$. For the run at CP and ${\mu }_3$, dashed and dotted lines show ${\mathcal{P}}_{L,S}(\varphi )$ for large and small particles, respectively. All $\mathcal{P}(\varphi )$, whether or not they distinguish large and small particles, are normalized by the total number of measurements, so that $\mathcal{P}(\varphi )={\mathcal{P}}_L(\varphi )+{\mathcal{P}}_S(\varphi )$. (b) Mean $\overline{\varphi }$ and variance $\sigma {}^2$ of $\mathcal{P}(\varphi )$ measured from individual Voronoï cells. Each point is for a single experimental run with $\gtrsim {10}^4$ configurations. Interparticle friction is denoted by shape (${\mu }_1$, $°$; ${\mu }_2$, $▽$; ${\mu }_3$, $\square$) and boundary condition by open or filled markers (CP, open; CV, filled). The line is a least-squares fit with $\varphi$ intercept at $\varphi =0.842\pm 0.002$.

Figure 3

(a) Schematic of the central particle and one neighbor with radii $r_c$ and $r$, respectively. The shortest distance between the edges of the two particles is $s$. The shaded region is the contribution $V^{*}$ from this neighbor to the total radical Voronoï volume. (b) ${\mathcal{P}}^{\text{expt}}(s)$ (dotted) measured for two experimental runs at CV and ${\mu }_3$, with $\overline{\varphi }=0.747$ (thick) and $\overline{\varphi }=0.792$ (thin). For comparison, exponential distribution ${\mathcal{P}}^{\lambda }(s)=\frac{1}{\lambda }{e}^{-s/\lambda }$ (dashed) with $\lambda \equiv \overline{s}$ and $\delta$-function distribution ${\mathcal{P}}^{\delta }(s)=\delta (s-\overline{s})$ (solid). (c) Scaled mean ($△$) and standard deviation ($\times$) of $s$, for all boundary condition (CP and CV) and $\mu$, as a function of the local volume fraction $\overline{\varphi }$.

Figure 4

(a–c) Measured $\mathcal{P}(\varphi )$ (dashed line) and calculated granocentric prediction ${\mathcal{P}}^g(\varphi )$ ($○$). The granocentric prediction is calculated using (a,b) ${\mathcal{P}}^{\mathrm{expt}}(s)$ from Fig. 3b and (c) ${\mathcal{P}}^{\text{sim}}(s)$ for a simulated packing with $N={10}^4$ particles shown in (d).

Figure 5

(a) Mean and variance of $\mathcal{P}(\varphi )$ for all experiments ($\blacksquare$, same data as Fig. 2), compared to granocentric predictions drawn from several different $s$ distributions: the experimentally measured distribution ${\mathcal{P}}^{\mathrm{expt}}(s)$ ($•$), the $s<r_S$ portion of the experimentally measured distribution ${\mathcal{P}}^{\text{expt},\text{cut}}(s)$ ($+$), exponential distribution ${\mathcal{P}}^{\lambda }(s)=\frac{1}{\lambda }{e}^{-s/\lambda }$ ($▴$) where $\lambda ={\overline{s}}_{\mathrm{expt}}$, and ${\mathcal{P}}^{\delta }(s)=\delta (s-{\overline{s}}_{\mathrm{expt}})$ ($⧫$). The filled markers represent distribution in the experimentally observed range of $0.154<{\overline{s}}_{\mathrm{expt}}/r_S<0.339$. The open markers, $⧫$ and $△$ are ${\mathcal{P}}^{\delta }(\varphi )$ and ${\mathcal{P}}^{\lambda }(\varphi )$ for $\overline{s}<{\overline{s}}_{\mathrm{expt}}$. (b) The mean number of neighbors versus $\overline{\varphi }$ for ${\mathcal{P}}^{\lambda }(s)$ and ${\mathcal{P}}^{\delta }(s)$, with the same markers as in (a).