Collision dynamics of granular particles with adhesion

We investigate the collision of adhesive viscoelastic spheres in quasistatic approximation where the adhesive interaction is described by the Johnson, Kendall, and Roberts (JKR) theory. The collision dynamics, based on the dynamic contact force, describes both restitutive collisions quantified by the coefficient of restitution $\epsilon$ as well as aggregative collisions, characterized by the critical aggregative impact velocity $g_{\mathrm{cr}}$. Both quantities $\epsilon$ and $g_{\mathrm{cr}}$ depend sensitively on the impact velocity and particle size. Our results agree well with laboratory experiments.

Figure 1

Sketch of a collision of adhesive particles. The left column represents the type-I and the right column the type-II initial condition: (a) The collision starts prior to jumping at time $t=-0$ when the distance $R_1+R_2$ corresponds to the compression $\xi =0$ at relative velocity $g_{\mathrm{imp}}$. (b) Right after jumping at time $t=+0$, a finite contact area of radius $a_{\mathrm{init}}$ is formed (type I: $a_{\mathrm{init}}=a_0\leftrightarrow \xi =0$, left; type II: $a_{\mathrm{init}}=a_{\mathrm{eq}}\leftrightarrow {\xi }_{\mathrm{eq}}>0$, right). In both cases the impact velocity $g_{\mathrm{imp}}$ is preserved, see Eqs. (35, 37). (c) Subsequently, both particles further deform each other until the maximum compression is reached. (d), (e) Then, they depart and reach $\xi =0$ with a nonzero contact radius $a$. (f) The adhesive force leads to the formation of a neck and although $\xi <0$ the contact is not broken. (g) Finally, at ${\xi }_{\mathrm{sep}}$ the particles lose contact and the bond is broken. The energy stored in the neck is dissipated by subsequent viscous oscillations of the particles.

Figure 2

Coefficient of restitution $\epsilon$ as a function of the impact rate $g_{\mathrm{imp}}$ for same-sized icy particles of radius $R=2\mathrm{cm}$ $(R_{\mathrm{eff}}=1\mathrm{cm})$ for different model realizations. As expected and in general agreement with experiments [40, 41], adhesive collisions are more dissipative than purely viscoelastic ones (cf. the dashed black and the solid and dashed-dotted lines). Moreover, below a certain impact speed, here $g_{\mathrm{cr}}\approx 2\mathrm{mm}∕\mathrm{s}$, particles stick together instead of rebouncing in the case of restitution for $g_{\mathrm{imp}}>g_{\mathrm{cr}}$. Results of [39] are shown as an exemplary comparison.

Figure 3

The coefficient of restitution $\epsilon$ as a function of the effective particle radius $R_{\mathrm{eff}}$ for icy particles colliding at a fixed impact rate $g_{\mathrm{imp}}$. Restitution becomes dominant with increasing effective particle radius. Slower impacts and size ratios different from one favor aggregation. The limit of purely elastic collision for large particles is only reached in case of nonadhesive collision. Thus, an adhesive collision is always dissipative.

Figure 4

Critical velocity $g_{\mathrm{cr}}$ as a function of the effective particle radius for the collision of same-sized icy particles. The analytical estimate given by Eqs. (44, 46) is compared to the exact numerical results given by Eq. (34). The critical impact rate distinguishes restitutive from aggregative collisions, where the domain of restitution is above the respective and the domain of aggregation below each respective curve. The results from Refs. 23, 37 show a qualitatively similar particle size dependence of $g_{\mathrm{cr}}$.

Figure 5

Illustrates the geometry of the contact formation for the initial conditions (35). For simplicity the collision of a sphere with a plane is considered. The radius of the sphere is $R$ and the current radius of the contact area is $a$. The displacement of the sphere’s surface is zero at the center of the contact zone, $u\left(0\right)=0$, while at the boundary of the contact zone, that is, on the distance $a$ from the center $u\left(a\right)=a^2∕R$. This yields the estimate for the characteristic strain, ${\overline{u}}_{ij}\sim (a^2∕R)∕a\sim a∕R$. The shadowed region corresponds to the domain of the most intensive deformation. Its volume is equal to the volume difference of the truncated cone with basis radii $2a$ and $a$ and height $h=a\sin \beta$, and the spherical cap of the same height $h$ and basis radius $2a$. For $a∕R⪡1$, from simple geometry, follows $\tan (\beta ∕2)\simeq \beta ∕2=a∕R$ or $\tan \beta \simeq \sin \beta \simeq 2a∕R$. The velocity of a material point on the sphere’s surface ${\dot{u}}_z\left(a\right)$ is related to the rate of the contact enlargement $\dot{a}$ (which is the speed of a geometric point) as ${\dot{u}}_z\left(a\right)=\dot{a}\tan \beta \simeq \dot{a}(2a∕R)$.