A matter of shape: contact area optimization in soft lubrication

Abstract

We study the fluid-mediated approach of a deformable axisymmetric object towards a rigid substrate, focusing on how its shape influences contact formation. For low approach velocities and large Stokes numbers, we show that sharper profiles (e.g., conical) maximize contact at the center and avoid fluid entrapment, while blunter ones form central dimples that trap bubbles. We also find that the resulting pressure distributions in the presence of thin viscous films can be predicted remarkably well by classical (dry) contact mechanics. These findings reveal shape as a design parameter for contact optimization in soft matter, adhesion, and elastohydrodynamics. Finally, we also theorize the possibility of a mechanical equivalence between shape and approach velocity.

Figures (3)

• Figure 1: Schematic of the axisymmetric approach problem. (a) A deformable elastic indenter approaches a rigid substrate through a viscous fluid film. moving at initial velocity $V$. (b) Equivalent dry contact setup: a rigid indenter compressing an elastic half-space, equivalent to (a) in the limit $\mathrm{St} \to \infty$. (c) Role-reversed system: elastic indenter and rigid substrate without the fluid layer, it develops similar interface tractions as (b) but the deformation is not flattening of the indenter but indetation of the half-space.
• Figure 2: Comparison of fluid-mediated deformation and pressure distribution for different shape exponents $n$. Each column shows the final deformed tip profile (dimensionless results) at the top (with a zoom at the tip in the inset), and the corresponding radial pressure distribution at selected times at the bottom. Vertical axes correspond to dimensionless height (top row) and dimensionless pressure (bottom row); horizontal axes are always dimensionless radial coordinate. Recall that the characteristic values $\mathcal{H}, \mathcal{L}, \mathcal{P}$ depend on $n$ (see Supplementary Material). Simulation results are in color and dry contact mechanics (DCM) approximation, based on numerical results of deformation at the tip, are in black dashed lined. The latter is computed using the impactor deformation at each time $t$, $\delta_{\text{num}}$, taken to be the value on the axis, i.e., $\delta_{\text{num}} = w(0,t)$. For instance: in the parabolic case $f(r) = r^2/2R$, the pressure distribution predicted with DCM is $p(r,t) = {2 E^* / \pi} (R \delta_{\text{num}}(t) - r^2)^{1/2}$ . Pressure results correspond to maximum pressure (no matter where the peak happens) in all cases but $n=1$, because the pressure is singular there; the finite pressure at $r = 0.04 \mathcal{L}$ is monitored instead. The dimensionless time is shifted so that $t/\tau = 0$ would correspond to the moment in which the solid would grace the substrate if there was no mediating fluid. The times corresponding to $n=1, 2$ are in the leftmost panel, the ones for $n=3, 4$ are in the rightmost one. Sharper tips ($n = 1$) remain convex and make central contact, while blunter shapes deform into dimples that trap air and shift the pressure peak outward.
• Figure 3: Three indenter profiles match at $r = R / \sqrt{5}$ and $z = h(r) = R/10$. Note: the three insets are at the same scale.