Droplet-substrate timescale matching modulates impact outcomes on flexible substrates.

Abstract

Droplet impact on flexible substrates occurs widely in natural and engineering systems. Although substrate deformation and vibration affect impact dynamics, a quantitative framework linking substrate vibration to distinct impact outcomes remains limited. Here, systematic experiments are conducted on cantilever substrates with tunable stiffness (<i>K</i> ∈ [0.4, ∞)) over We ∈ [2, 170]. The results show that We primarily determines the overall impact outcome, whereas substrate vibration modulates rebound and splashing through stage-dependent coupling with droplet dynamics. To quantify this modulation, we establish a unified stage-resolved timescale-coupling framework, with the coupling quantified by <i>Ω</i> = <i>f</i>_s<i>t</i>*, where <i>f</i>_s is the substrate vibration frequency and <i>t</i>* is the characteristic timescale of the corresponding stage. For rebound (We ∈ [2, 100]), <i>t</i>* is taken as the Rayleigh timescale <i>t</i>_R, reflecting inertial-capillary recovery of the deformed droplet. When <i>Ω</i> ≈ 1, upward substrate motion coincides with droplet recoil and assists lift-off, yielding the shortest contact time and enhanced rebound. For splashing, <i>t</i>* is defined as the spreading timescale <i>t</i>_S associated with lamella-instability development. At We ∈ (100, 150], when <i>Ω</i> approaches unity, substrate vibration is favorably coupled with early spreading and lamella-instability development. This favorable coupling enhances relative droplet-substrate motion, increases the relative-motion-based effective Weber number We_eff, reduces the splashing time <i>t</i>_sp, and alters the splashing mode. At higher We ∈ (150, 170], this coupling weakens, and splashing is dominated by intrinsic inertial-capillary dynamics. These findings provide a unified quantitative basis for interpreting and predicting impact outcomes on flexible substrates.