Speckle Visibility Spectroscopy and Variable Granular Fluidization

We introduce a dynamic light scattering technique capable of resolving motion that changes systematically, and rapidly, with time. It is based on the visibility of a speckle pattern for a given exposure duration. Applying this to a vibrated layer of glass beads, we measure the granular temperature and its variation with phase in the oscillation cycle. We observe several transitions involving jammed states, where the grains are at rest during some portion of the cycle. We also observe a two-step decay of the temperature on approach to jamming.

Figure 1

(a) Theoretical position of the cell, bead layer, and their difference ($5\times$) vs phase $\varphi$ for a sinusoidally oscillated cell. The particular curves are for $\Gamma =1.25$ and $f=10\mathrm{H}\mathrm{z}$. For intuition, the bead layer is modeled as a slab with an effective coefficient of restitution; a value of $ϵ=0.5$ is used for the data shown. The inset shows the optical geometry. (b) A representative speckle image vs $\varphi$ gathered by the linear CCD for $\Gamma =1.25$, $f=10\mathrm{H}\mathrm{z}$, and $T=100\mu \mathrm{s}$. For clarity, only 200 of 1024 pixels are shown. (c) The averaged normalized variance $V_2$ of 300 full images versus $\varphi$ for the above settings.

Figure 2

(a) Variance $V_2$ vs exposure time $T$ for $\Gamma =1.25$ and $f=10\mathrm{H}\mathrm{z}$ at three phases $\varphi$. The symbols correspond to the markers in Fig. 1c. The solid and dashed curves are single parameter fits to the data denoted by the larger symbols, using Eq. (3). The data fit by the dashed curve represent the effect of speckle wash alone, since the grains are at rest on the plate. The dotted curve assumes diffusive grain motion, $g_1(t)=\exp (-\sqrt{\gamma t})$, and gives a poor fit. (b) Fitted rates vs phase.

Figure 3

Gray scale plot of decay rate $\gamma$, based on fits to Eq. (2), as a function of both peak acceleration $\Gamma$ and phase $\varphi$ in the oscillation cycle; $\mathrm{\text{black}}=0$, $\mathrm{\text{white}}=120\mathrm{k}\mathrm{H}\mathrm{z}$. The solid white curve indicates where the grains are launched, at an instantaneous acceleration of $-g$. The dotted white curve indicates when a free-fall object would land.

Figure 4

Average fluctuation speed vs time after impact, extracted from Fig. 3, for different accelerations $\Gamma$. For the solid curves, all below the transition to continuous fluidization, $\delta v$ increases monotonically with driving amplitude. Above the transition, $\delta v$ never goes to zero and does not spike as strongly when the grains impact the plate.