Theoretical modeling of the maximum spreading factor of an impacting droplet on cantilever beams

This study systematically investigates the dynamic behaviors of a droplet impinging on cantilever beams via experimental tests and theoretical analysis, with the aim of establishing a quantitative model for the maximum spreading factor of the droplet. The effects of the Weber number (We), surface wettability (θ_adv), and substrate stiffness (K_b) on droplet impact dynamics are comprehensively explored. The results reveal that the Weber number dominates the variation in the maximum spreading factor, whereas surface wettability and substrate stiffness exert relatively insignificant effects. Based on the energy conservation principle, a predictive model for the maximum spreading factor of a droplet impacting cantilever beams is developed. The proposed model well characterizes the competitive and synergistic coupling effects of inertial force, capillary force, and structural deformation under diverse wettability and stiffness conditions. Moreover, the model is extended to adapt to wider liquid viscosity ranges, and its predictive generality is validated within the parameter ranges of We = 20.76-51.91, θ_adv = 126°-147°, K_b = 0.63-11.85 N m^-1, Re = 327-650, and σ = 0.0695-0.0720 N m^-1. This study introduces the first theoretical model of the maximum spreading factor that explicitly incorporates the influence of surface elastic forces into the classical framework, thereby enhancing predictive accuracy across a broad spectrum of elastic substrates.