Coefficient of restitution for bouncing nanoparticles

We demonstrate the measurement of the coefficient of restitution, $e$, for nanoparticles, through observations of the final distribution of bismuth particles that have bounced within silicon V-grooves. The experiments, taken together with complementary molecular-dynamics simulations, show that $e$ is generally smaller for liquid than for solid nanoparticles, and that macroscopic theories underestimate the velocity dependence of $e$. Hence, while nanoparticles are harder than bulk materials, once they have begun to yield the rate of increase of the inelastic deformation is greater.

Figure 1

(Color online) Schematic of the particle bouncing process. Particles are incident normal to the surface of the silicon wafer, which results in an angle of incidence ${\theta }_i=35°$ relative to the surface of V-grooves etched into the wafer. (This angle is defined by the angle between the (100) and (111) planes of Si—see Ref. 5.) The particles bounce from one face of the V-groove (reflection angle ${\theta }_r$) and adhere to the opposite face. When the sample is tilted, the angle of incidence of the particles is different for the two sides of the V-groove, and the width of the particle distribution on the opposite face can be used to determine ${\theta }_r$ as a function of ${\theta }_i$.

Figure 2

(Color online) (a)–(e) Distribution of bismuth clusters in silicon V-grooves, which can be used to define the reflection angle ${\theta }_r$. (a) and (b) Bi cluster distributions in $\sim 5\mu \text{m}$ V-grooves for $f_{\text{Ar}}=100\text{sccm}$ and source and substrate temperatures $T_s=T_{sub}=290\text{K}$. The clusters have bounced so that the upper V-groove walls support only a few clusters. The nominal mean cluster thickness was 2.4 nm in (a) and 7.2 nm in (b). The solid and dashed red lines mark the edge of the cluster distribution and the apex of the V-groove, respectively. (c) and (d) Bi cluster distributions deposited using a mixture of Ar and He inert gases at constant total flow rate: $f_{\text{He}}/f_{\text{Ar}}$ is 10% in (c) and 50% in (d). Again $T_s=T_{sub}=290\text{K}$. The nominal mean cluster thickness was 5 nm. As discussed in the text, the increase in width of the cluster distribution corresponds to an increase in ${\theta }_r$, which is due to a change in state of the clusters, from liquid to solid. (e) Cluster distribution when the sample was tilted by $15°$ in order to change the angle of incidence of the clusters. Deposition conditions were as for (a) and (b) but the nominal mean cluster thickness was 0.2 nm. The asymmetrical cluster distribution results from different angles of reflection on the two sides of the V-groove. (f) Mean cluster size $\left(D_{av}\right)$ as a function of $f_{\text{Ar}}$. Squares: $T_s=290\text{K}$; triangles: $T_s=77\text{K}$; and circles: varying mixtures of Ar and He gas $\left(f_{\text{He}}+f_{\text{Ar}}=180\text{sccm}\right)$ and $T_s=290\text{K}$. Note that the circles are scattered and do not show a monotonic dependence on $f_{\text{He}}/f_{\text{Ar}}$.

Figure 3

(Color online) Rebound angles, ${\theta }_r$, measured from depositions of Bi nanoparticles on etched silicon V-grooves. ${\theta }_r$ is estimated from the geometry of the V-groove and the width of the distribution of nanoparticles in the V-grooves. In each case, the nominal mean thickness of the deposition was 2 nm. (a) Squares represent depositions with $T_s=290\text{K}$; the different colored squares are for different substrate temperatures, $T_{sub}$: green square 290 K; purple square 373 K; and red square 473 K. Triangles represent depositions with $T_s=77\text{K}$. (b) The circles indicate ${\theta }_r$ for varying mixtures of Ar and He gases in the source chamber with total flow rate $f_{\text{He}}+f_{\text{Ar}}=180\text{sccm}$ and $T_s=T_{sub}=290\text{K}$.

Figure 4

(Color online) Snapshots from MD simulations of collisions of (a) solid and (b) liquid nanoparticles with a flat surface, at ${\theta }_i=35°$ and $v_0=2.0v^{\ast }$. The reflection angle, ${\theta }_r$, is smaller for the liquid particle because more energy is dissipated in the deformation of the particle and the deformed liquid particles adhere to the substrate more strongly. (c) and (d) normal coefficient of restitution, $e_z$, for simulated solid and liquid LJ nanoparticles, respectively (note the different vertical scales). Collisions were simulated for a wide range of angles of incidence and incident velocities (each symbol type represents a different ${\theta }_i$), and $e_z$ is calculated from $v_{fz}/v_{0z}$. The solid lines in (c) and (d) are power laws with a $v_{0z}^{-1}$ dependence. Deviations from this dependence for $v_{0z}>2$ are due to an increasing degree of fragmentation.

Figure 5

(Color online) Experimentally determined normal coefficient of restitution, $e_z$, (normalized to the tangential coefficient of restitution, $e_y=0.8$—see text) for Bi nanoparticles, as a function of the normal component of the incident velocity, $v_{0z}$ (where $v_0^{+}=48\text{m}/\text{s}$) (Ref. 23). The data were obtained with Ar as the inert gas and $T_s=290\text{K}$, i.e., for liquid clusters. At small $v_{0z}$, it is difficult to estimate the position of the edge of the cluster distribution precisely, and so the uncertainty in $e$ is substantial. At higher velocities, much of the data lies close to the solid blue curve, which is a $v_{0z}^{-1}$ dependence. Circle data are for particles deposited at varying angles of incidence [see Fig. 2e]. In each case, the nominal mean thickness of the deposition was 0.2 nm. Square data are obtained from the reflection angle data in Fig. 3a for various $v_0$ and ${\theta }_i=35°$. Inset: snapshot showing extensive plastic deformation in a simulated solid LJ particle with $v_0=3.8v^{\ast }$ and ${\theta }_i=35°$.