Accurate determination of the shapes of granular charge distributions

Experiments have shown that charge distributions of granular materials are non-Gaussian, with broad tails that indicate many particles with high charge. This observation has consequences for the behavior of granular materials in many settings, and may bear relevance to the underlying charge transfer mechanism. However, there is the unaddressed possibility that broad tails arise due to experimental uncertainties, as determining the shapes of tails is nontrivial. Here we show that measurement uncertainties can indeed account for most of the tail broadening previously observed. The clue that reveals this is that distributions are sensitive to the electric field at which they are measured; ones measured at low (high) fields have larger (smaller) tails. Accounting for sources of uncertainty, we reproduce this broadening in silico. Finally, we use our results to back out the true charge distribution without broadening, which we find is still non-Guassian, though with substantially different behavior at the tails and indicating significantly fewer highly charged particles. These results have implications in many natural settings where electrostatic interactions, especially among highly charged particles, strongly affect granular behavior.

Figure 1

(a) Experimental system. A high speed camera falls along low friction rails while watching a cofalling stream of granular particles. The particles are accelerated into/out of the page by an electric field across two copper plates (drawn outside chamber for clarity). (b) Examples of particle trajectories in the horizontal direction at an applied field of $150\mathrm{kV}/\mathrm{m}$. (c) Measured charge distribution for $\left|\mathbf{E}\right|=90\mathrm{kV}/\mathrm{m}$, with uncertainty bands obtained as mentioned in the text. The solid green curve is the Gaussian fit of the central peak. The solid red line is the phenomenological fit to Eq. (1). The inset shows the same data/fit on a double logarithmic scale.

Figure 2

(a) Charge PDFs for all data from the field-sweep experiments plotted together on a semilog scale. The non-Gaussian tails appear to change with the electric field, as well as the PDF maximum value (figure inset). (b) Charge PDF maximum value $F_o$, (c) characteristic distribution width ${\mathrm{\Delta }}_q$ and (d) limiting power law exponent $\alpha$, respectively, vs the electric field $\left|\mathbf{E}\right|$. All quantities tend to converge to constant values for the largest $\left|\mathbf{E}\right|$. The dashed lines show the averages of the last three values. The quantities $F_o$, ${\mathrm{\Delta }}_q$, and $\alpha$ are obtained from fits to Eq. (1), as shown in Fig. 1.

Figure 3

(a) Acceleration uncertainty, ${\sigma }_a$, plotted against lifetime, $\tau$, for all $\left|\mathbf{E}\right|$. The data have been windowaveraged in 39 bins. Error bars correspond to the standard deviation of the data of each bin. A decreasing power law behavior is obtained, $\langle {\sigma }_a\rangle =D{\tau }^{\epsilon }$. As there is no $\left|\mathbf{E}\right|$ dependence, we merge all the data together to fit and obtain $\epsilon =-2.50\pm 0.01$ and $D=(1.500\pm 0.015)\times {10}^{-6}\mathrm{m}$ ${\mathrm{s}}^{1/2}$, shown with the continuous black line. (b) Particle lifetime PDF, also for all fields. The peak at the maximum trajectory lifetime, ${\tau }_{\max }=0.3359$ s, corresponds to those trajectories that remain tracked throughout the complete video acquisition (490 frames at 1456 fps). Again there is no field dependence. (c) Tracked position residuals standard deviation, ${\sigma }_{\mathrm{res}}$, versus lifetime, $\tau$, for the highest field $\left|\mathbf{E}\right|=150\mathrm{kV}/\mathrm{m}$. We also checked that there is no field dependence. These residuals ultimately cause the acceleration uncertainties in (a), and they are also correlated to the lifetime through a power law. For the window-averaged data, we get $\langle {\sigma }_{\mathrm{res}}\rangle =C{\tau }^{\delta }$, with $\delta =0.47\pm 0.03$ and $C=(4.93\pm 0.04)\times {10}^{-5}\mathrm{m}$ ${\mathrm{s}}^{-\delta }$, shown with the continuous black line.

Figure 4

(a) Charge PDFs for the simulation data of an input Gaussian charge distribution, with ${\sigma }_q=0.45\times {10}^{6}e$, as the electric field is increased. The inset shows the detail around the distribution maxima for fields up to 150 kV/m, as in the experiment. In panels (b), (c), and (d) we plot the PDF maximum, width, and limiting power, respectively, similar to the experimental data of Figs. 2. The shapes of these distributions are not fitted well by Eq. (1), hence we use a slightly different approach to extract these values. For panels (b) and (c), we fit the central peak to a Gaussian, $A_G/\left(\sqrt{2\pi }{\sigma }_G\right)exp[-q^2/\left(2{\sigma }_G^2\right)]$, where the factor $A_G$ gives the PDF maximum and ${\sigma }_G$ characterizes the width. For the limiting power law behavior, we extend the field dependence up to 1000 kV/m and simply fit the tails to a power law. Contrary to experiments, $\alpha$ does not converge to a fixed value. This is consistent with the fact that the underlying PDF is a Gaussian. The dashed lines in panels (b) and (c) show the original charge distribution values for $A_G=1$ and ${\sigma }_q=0.45\times {10}^{6}e$.

Figure 5

(a) Objective function ${\chi }^2$ vs charge distribution width ${\sigma }_q$ in log-log scale. A clear minimum is observed at ${\sigma }_q=(0.45\pm 0.01)\times {10}^{6}e$. (b) Comparison of the best simulation charge PDF (red $\diamond$), obtained with zero mean and ${\sigma }_q=0.45\times {10}^{6}e$, with the experimental one (blue $\circ$), both for $\left|\mathbf{E}\right|=150\mathrm{kV}/\mathrm{m}$. The continuous red line shows the analytical charge PDF used in the simulation. As is clear, even this “best Gaussian” is wholly inadequate to explain experimental data. (c) Simulation (red $\diamond$) and experimental (blue $\circ$) charge PDF for $\left|\mathbf{E}\right|=150\mathrm{kV}/\mathrm{m}$. We generate the former by using experimental data at the highest fields as our input charge distribution and subjecting it to the noise sources as described in the text.