Droplet impact on elastic substrates: force scaling crossover

TL;DR

This study reveals a force scaling crossover during droplet impacts on elastic substrates, shifting from inertial to Hertzian scaling depending on substrate softness and impact conditions, with implications for material design.

Contribution

It introduces a new understanding of force scaling laws during droplet impacts on elastic substrates, highlighting a crossover governed by a similarity parameter.

Findings

Identified a force scaling crossover from inertial to Hertzian impact.

Captured high-resolution stress fields using photoelastic tomography.

Defined a similarity parameter controlling impact force behavior.

Abstract

Droplet impacts are fundamental to fluid-structure interactions, shaping processes from erosion to bioprinting. While previous scaling laws have provided insights into droplet dynamics, force scaling laws remain insufficiently understood, particularly for soft substrates where both the droplet and substrate deform significantly. Here, we show that droplet impacts on elastic substrates exhibit a scaling crossover in maximum impact force, transitioning from inertial force scaling, typical for rigid substrates under high inertia, to Hertzian impact scaling, characteristic of rigid spheres on elastic substrates. Using high-speed photoelastic tomography, we captured high-resolution dynamic stress fields and identified a similarity parameter governing the interplay between droplet inertia, substrate elasticity, and deformation time scales. Our findings redefine how substrate properties influence impact forces, demonstrating that droplets under high inertia -- long thought to follow inertial force scaling -- can instead follow Hertzian impact scaling on soft substrates. This framework provides practical insights for designing soft, impact-resistant materials.