Particle diffusion in slow granular bulk flows

We probe the diffusive motion of particles in slowly sheared three-dimensional granular suspensions. For sufficiently large strains, the particle dynamics exhibits diffusive Gaussian statistics, with the diffusivity proportional to the local strain rate —consistent with a local, quasistatic picture. Surprisingly, the diffusivity is also inversely proportional to the depth of the particles below the granular surface —at their free surface, the diffusivity thus appears to diverge and is ill defined. We find that the crossover to Gaussian displacement statistics is governed by the same depth dependence, evidencing a non-trivial strain scale in three-dimensional granular flows. editor’s choice Copyright c © EPLA, 2012 Flowing soft disordered solids, such as colloidal glasses and gels [1–3], foams and emulsions [4–6] or granular media [7–13], exhibit rich particle dynamics and complex rheology. In the last decade, major progress has been made in the understanding of the rheology of granular media and suspensions. Important developments include the description of rapid, “inertial” dry granular flows [7,8], and the observation of connections between this inertial rheology and the classical rheology of density-matched suspensions [14]. The most challenging regime is that of very slow flows, where the stresses become (approximately) rate independent [15] and non-local effects play an important role [16–18]. Little is known about the fluctuations of the microscopic grain motion, i.e., self-diffusion, in such flows. A recurring problem is that granular media are opaque, so only motion in two-dimensional (2D) model systems [9], near transparant walls [10,19] or at a free surface can be observed [11]. Walls lead to layering [20] and other artifacts [21], while at the free surface, the particles experience a different local geometry than in the bulk. Measurements in the bulk of a granular flow are thus essential. Recently, we have shown [13] that for sufficiently slow flows, dry granular media and non–density-matched, but optically matched suspensions exhibit identical flow patterns and rheology, opening up the possibility to probe 3D microscopic particle dynamics within slow granular flows. Here we experimentally probe the full 3D particle trajectories of a sheared granular suspension in a split bottom cell (fig. 1). We focus on sufficiently slow strain rates so that viscous effects are negligible: as detailed below, typical Reynolds and Stokes numbers are of order 10 and 10, respectively, implying that we are in the so-called free fall regime, relevant for dry granular flows [13,22]. In a quasi-static picture, the mean squared displacements should only depend on the strain, or equivalently, the diffusion coefficient should be proportional to the strain rate, as seen already in [9,23]. By extracting the particle diffusivity and local strain rates from our particle trajectories, we confirm that this is true in 3D also. The crucial point is that our 3D geometry allows us to probe the depth dependence of the diffusivity. In the simplest picture, one might expect the diffusion to the independent of depth. However, our main result is showing that the diffusion is inverse proportional to the depth, and thus diverges near the free surface: at the free surface, diffusivity is ill defined. To answer whether this depth dependence stems from a depth dependence of the mean squared displacements or of the characteristic strain, both of which can lead to a depth dependent diffusivity, we probe the displacement distributions. These evolve from very wide distributions at short time with (reduced) kurtosis exceeding 40, to Gaussian distributions at late times. We find that, for a given depth, the kurtosis scales as a power law with strain, and that we can collapse kurtosis data taken at different depths by introducing a depth