Criterion for particle rebound during wet collisions on elastic coatings

We combine lubrication theory and solid linear elasticity to model the elastohydrodynamic deformation of a rigid, spherical particle impacting a stratified surface—a soft incompressible coating atop a rigid substrate. The model depends on three dimensionless parameters: Stokes number, elasticity parameter, and thickness parameter. These parameters collectively determine whether the particle will stick to or rebound against the stratified surface. We show that coating thickness moderates the degree to which elasticity affects the rebound criteria (and vice versa) in a fashion distinct from the effect of the elasticity or of the Stokes number. We also demonstrate the possibility of approximating a stratified surface as an elastic half space with an elasticity between the rigid substrate and soft coating. This “effective elasticity” of the half space couples both the elasticity parameter and thickness parameter of the stratified surface condition while maintaining the same sticking criteria. Lastly, we discuss the limitations of this effective elasticity, such as in thin or more-rigid coatings.

Figure 1

Schematic diagram showing the (a) approach and (b) bouncing of a spherical particle against a coated substrate. Important variables include the fluid gap width $\left(h\right)$, the undeformed separation $\left(x\right)$, the deformation $\left(w\right)$, particle velocity $\left(v\right)$, and coating thickness $\left(\delta \right)$. The particle is immersed in a Newtonian fluid with viscosity $\left(\eta \right)$. In (a) as the particle approaches the soft coating, fluid is drained out of the gap. In (b) fluid is infused into the gap as the particle rebounds away from the coating. The figure is scaled such that the horizontal axis (radial direction) is compressed 50 times relative to the vertical axis $(z$ direction), which results in the nonspherical appearance of the particle.

Figure 2

From top to bottom: (a) Central fluid gap width $\widehat{h}(0,\widehat{t})$, (b) central coating deformation $\widehat{w}(0,\widehat{t})$, and (c) particle velocity $\widehat{v}\left(\widehat{t}\right)$ during the collision of a spherical particle against a stratified, flat surface. Stokes number and elasticity of the film are held constant at $\mathrm{St}=5$ and $\varepsilon =0.01$. The four chosen thickness parameters encompass the thin $\left(T=0.05\right)$, intermediate $\left(T=0.5\right)$, and thick $(T=5$, 50) regimes for coating thicknesses. The DSH and Reynolds limit for a half space and foundation, respectively, are included as dashed lines. Arrows in (a) mark the first transition from fluid drainage to infusion at the center, while arrows in (c) mark the first transition from the particle's approach to rebound. Inset in (a): Pathway for fluid infusion into the central gap for intermediate and thick coatings, $T=0.5$ (left) and $T=50$ (right), respectively. When the deformation of the coating adopts a curvature similar to that of the particle, the narrowness of the fluid gap at the sides determines the rate of fluid infusion into the gap.

Figure 3

(a) Comparison between the dimensionless velocity (solid line) of the sphere and the dimensionless central gap (dashed line) for $\mathrm{St}=5,\varepsilon =0.01$, and $T=50$. The lag region (gray) is between ${\widehat{t}}_{\nu =0}$ and ${\widehat{t}}_{hmin}$. Inset: ${\widehat{t}}_{\nu =0}$ and ${\widehat{t}}_{hmin}$ for $\mathrm{St}=5,\varepsilon =0.01$, and different thickness parameters. (b) Comparison between the velocity of the sphere, the central coating deformation, and the central undeformed gap for $\mathrm{St}=5,\varepsilon =0.01$, and $T=0.05$. The undeformed gap is adjusted by subtracting ${\widehat{x}}_{\mathrm{eq}}$ so that it oscillates around the $x$ axis. Intersections between the central coating deformation (blue) and the $x$ axis (black line) are marked with $r_i$. Inset: The first through sixth time $\widehat{t}$ each of the curves crossed the $x$ axis.

Figure 4

Hydrodynamic forces during the sphere's approach and rebound. Stokes number and film elasticity are held constant at $\mathrm{St}=5$ and $\varepsilon =0.01$ while the thickness is changed. Inset: The peak force and the peak stored elastic energy as a function of the coating thickness $T$.

Figure 5

(a) The particle's kinetic energy (blue dashed line), the coating's stored elastic potential energy (red dot dashed line), and cumulative energy dissipated into the fluid (black solid line) for $T=50$. The gray and white areas denote the viscous- and elastic-dominated regime, respectively. The Stokes number and elasticity are fixed at $\mathrm{St}=5$ and $\varepsilon =0.01$. (b) The rate of energy dissipation into the fluid as a function of fluid gap for different thickness parameters. Inset: The time evolution of dissipated energy for different coating thicknesses.

Figure 6

(a) Comparison between the rebound criterion of Davis et al. [20]. (horizontal dotted line) and our own (vertical dashed line). Data points are selected from simulation results with $\mathrm{St}=1-7,\varepsilon =0.0001-0.01,T=0.05-50$ close to the rebound criteria and plotted against the ratio of central undeformed gap $\left(\widehat{x}\right)$ to deformation $\left(\widehat{w}\right)$ and the maximum rebound velocity $\left({\widehat{v}}_r\right)$. Red markers indicate small thickness parameters $\left(T<1\right)$ while blue markers indicate larger thickness parameters. Based on the criterion of Davis et al., quadrants I and II result in sticking while III and IV result in rebound. Based on our criterion, quadrants II and III result in stick while I and IV are rebound. Inset of (a): Cross section of coating deformation for different thicknesses parameters. The time when the deformation is captured is selected specifically to ensure the magnitude of deformation at the center is similar for all $T$. (b) The maximum rebound velocity $\left({\widehat{v}}_r^{}\right)$ for various Stokes numbers and thickness parameters. Here ${\widehat{v}}_r^{}=v_r/v_0$ can be interpreted as a wet coefficient of restitution. From bottom to top: $T=0.1$, 0.5, 1.0, 5.0, 10, and 50. Elasticity parameter was constant at 0.01. Condition for rebound was set to be a maximum rebound velocity of 0.05 (blue dashed line).

Figure 7

(a) Critical Stokes $\mathrm{S}{\mathrm{t}}_c$ as a linear function of $ln(1/\varepsilon )$ for different thickness parameters. From top to bottom, $T=0.1$, 0.5, 1, 5, 10, 50. Also shown are the predictions from Davis et al. [18] and Lian et al. [37] for comparison. Dashed lines are extrapolated from a linear fit of the data at $ln(1/\varepsilon )\ge 2$. (b) Slope $\alpha$ [Eq. (16)] and $y$ intercept $\beta$ [Eq. (17)] of the $\mathrm{S}{\mathrm{t}}_c$ equation [Eq. (15)], as functions of the thickness parameter $T$. The circular and triangular markers are values of slope and $y$ intercept, respectively, from Fig. 8, while the dotted lines are curve fits generated using matlab cftool.

Figure 8

(a) Contour plot showing the effective elasticity $\left({\varepsilon }^{\prime }\right)$ for different $T$ and $\varepsilon$ values. Every point on the same contours indicates that this combination of $T$ and $\varepsilon$ yields the same effective elasticity value, indicated in red at the upper portion of the figure. (b) Verification of proposed effective elasticity formula. Results from trials with different combinations of Stokes number and effective elasticity are shown with different markers. The red filled circular markers and the red empty circular markers denote sticking and rebound outcomes, respectively, in the data sets, both of which match our rebound criterion. The blue “×” markers denote data that do not fit the stick-rebound criterion proposed.