Effect of particle size on energy dissipation in viscoelastic granular collisions

We analyze the scaling properties of the Hertz-Kuwabara-Kono (HKK) model, which is commonly used in numerical simulations to describe the collision of macroscopic noncohesive viscoelastic spherical particles. Parameters describing the elastic and viscous properties of the material, its density, and the size of the colliding particles affect the restitution coefficient $\varepsilon$ and collision time $\tau$ only via appropriate rescaling but do not change the shape of $\varepsilon \left(v\right)$ and $\tau \left(v\right)$ curves, where $v$ is the impact velocity. We have measured the restitution coefficient experimentally for relatively large (1 cm) particles of microcrystalline cellulose to deduce material parameters and then to predict collision properties for smaller microcrystalline cellulose (MCC) particles by assuming the scaling properties of the HKK model. In particular, we demonstrate that the HKK model predicts the restitution coefficient of microscopic particles of about 100 $\mu$m to be considerably smaller than that of the macroscopic particles. In fact, the energy dissipation is so large that only completely inelastic collisions occur for weakly attractive particles. We propose a straightforward self-consistent extension to the Johnson-Kendall-Roberts (JKR) model to include dissipative forces and discuss the implications of our findings for the behavior of experimental powder systems.

Figure 1

The deformation, the rate of the deformation, and viscoelastic force during a collision of two viscoelastic particles characterized by a restitution coefficient of 0.5. All quantities are normalized by their maximum values.

Figure 2

Calculated restitution coefficient of two identical 100 $\mu$m spheres with MCC material properties as a function of the impact velocity calculated according to two different criteria to define the end of the collision: correct $\ddot{\delta }=0$ (solid red line) and incorrect $\delta =0$ (dashed green line). The dotted lines are asymptotes calculated numerically with their gradients shown.

Figure 3

Collision time of two identical 100 $\mu$m spheres with MCC material properties as a function of the impact velocity calculated according to two different criteria to define the end of the collision: correct $\ddot{\delta }=0$ (solid red line) and incorrect $\delta =0$ (dashed green line). The black dotted line is the asymptote calculated using Eq. (4) with the gradient of $-1/5$ as indicated. The red dot-dashed line is obtained for a material with Young's modulus 10 times greater than that of the MCC by shifting the solid line according to the rules given in Table .

Figure 4

Restitution coefficient measured for an axisymmetric biconvex MCC tablet (gray squares with the error bars indicating standard deviation) and fitted using HKK theory (blue solid line). The $\varepsilon \left(v\right)$ curves for a sphere-plane and a free collision of two spheres with the same curvature radius (shown as the green and red dashed lines, respectively) were obtained by applying the indicated shifts along the $x$ axis as described in the text. The vertical line indicates the collision velocity $v=\sqrt{2gH}=4.43$ m s${}^{-1}$ corresponding to the drop height of $H=1$ m.

Figure 5

Calculated restitution coefficient for a viscoelastic sphere with material parameters of MCC bouncing off a plane (a) with and (b) without the effect of the gravitational field. The circles correspond to consecutive rebounds of a sphere dropped from a height of $100d$ in a computer simulation with gravity. All data were obtained by solving Eq. (7) numerically for the different impact velocities $v$.

Figure 6

Maximum restitution coefficient predicted for spherical MCC granules as a function of their surface energy for different particle diameters, as indicated.