Droplet impact dynamics on solid surfaces, which are ubiquitously present in aerospace engineering, energy systems, agricultural production, etc., involve complex fluid-structure interactions. Herein, we employ a single-camera high-speed three-dimensional digital image correlation system to quantify the full-field deformations of flexible thin films during droplet impact dynamics. Experimental results revealed that the substrate flexibility not only reduces the maximum spreading diameter by 10% but also modulates rebound dynamics via energy competition between kinetic energy and surface adhesion energy, suggesting that coupled deformation of the solid-fluid interface plays an important role in the dynamic progress. We propose the structure-coupled response number (Sn), a governing dimensionless parameter unifying droplet spreading on both rigid and flexible films, validated by a universal 1/2 scaling law. A theoretical criterion for droplet rebound on hydrophobic flexible thin films is derived and experimentally demonstrated, which achieves the precise control of droplet rebound/non-rebound mode. This work bridges the theories of droplet impact dynamics on rigid and flexible substrates, offering a robust strategy to govern the droplet impact behaviors.
| [1] |
Josserand C, Thoroddsen ST. Drop impact on a solid surface. Annu Rev Fluid Mech. 2016; 48: 365-391.
|
| [2] |
Richard D, Clanet C, Quéré D. Contact time of a bouncing drop. Nature. 2002; 417: 811.
|
| [3] |
de Ruiter J, Soto D, Varanasi KK. Self-peeling of impacting droplets. Nat Phys. 2018; 14: 35-39.
|
| [4] |
Chandra S, Avedisian CT. On the collision of a droplet with a solid surface. Proc R Soc Lond Series A: Math Phys Sci. 1997; 432: 13-41.
|
| [5] |
Bergeron V, Bonn D, Martin JY, Vovelle L. Controlling droplet deposition with polymer additives. Nature. 2000; 405: 772-775.
|
| [6] |
Song M, Hu D, Zheng X, et al. Enhancing droplet deposition on wired and curved superhydrophobic leaves. ACS Nano. 2019; 13: 7966-7974.
|
| [7] |
Cui Y, Wang S, Li W, et al. Research progress in droplet deposition on superhydrophobic plant leaves. Chem J Chin Univ. 2021; 42: 1061-1073. (in Chinese)
|
| [8] |
Shiri S, Bird JC. Heat exchange between a bouncing drop and a superhydrophobic substrate. Proc Natl Acad Sci U S A. 2017; 114: 6930-6935.
|
| [9] |
Mukherjee R, Berrier AS, Murphy KR, Vieitez JR, Boreyko JB. How surface orientation affects jumping-droplet condensation. Joule. 2019; 3: 1360-1376.
|
| [10] |
Jiang M, Wang Y, Liu F, et al. Inhibiting the Leidenfrost effect above 1,000 °C for sustained thermal cooling. Nature. 2022; 601: 568-572.
|
| [11] |
Kim J. Spray cooling heat transfer: the state of the art. Int J Heat Fluid Flow. 2007; 28: 753-767.
|
| [12] |
Zhong Y, Fang H, Ma Q, Dong X. Analysis of droplet stability after ejection from an inkjet nozzle. J Fluid Mech. 2018; 845: 378-391.
|
| [13] |
Hu G, Yang L, Yang Z, et al. A general ink formulation of 2D crystals for wafer-scale inkjet printing. Sci Adv. 2020; 6:eaba5029.
|
| [14] |
Tian Y, Liu Y, Peng Z, et al. Air entrapment of a neutral drop impacting onto a flat solid surface in electric fields. J Fluid Mech. 2022; 946: A21.
|
| [15] |
Li H, Fang W, Zhao Z, et al. Droplet precise self-splitting on patterned adhesive surfaces for simultaneous multidetection. Angew Chem Int Ed. 2020; 59: 10535-10539.
|
| [16] |
Xu W, Zheng H, Liu Y, et al. A droplet-based electricity generator with high instantaneous power density. Nature. 2020; 578: 392-396.
|
| [17] |
Zhang Q, Li Y, Cai H, et al. A single-droplet electricity generator achieves an ultrahigh output over 100 V without pre-charging. Adv Mater. 2021; 33:2105761.
|
| [18] |
Wang W, Zhang L, Wang H, et al. High-output single-electrode droplet triboelectric nanogenerator based on asymmetrical distribution electrostatic induction enhancement. Small. 2023; 19:2301568.
|
| [19] |
Gao Y, Chen C, Wang F, Li M, Sun C. Characterization and removal of contaminants in lithography. Sci China: Phys Mech Astron. 2025; 68:294702.
|
| [20] |
Sikalo S, Ganic EN. Phenomena of droplet-surface interactions. Exp Therm Fluid Sci. 2006; 31: 97-110.
|
| [21] |
Riboux G, Gordillo JM. Experiments of drops impacting a smooth solid surface: a model of the critical impact speed for drop splashing. Phys Rev Lett. 2014; 113:024507.
|
| [22] |
Mongruel A, Daru V, Feuillebois F, Tabakova S. Early post-impact time dynamics of viscous drops onto a solid dry surface. Phys Fluids. 2009; 21:032101.
|
| [23] |
Yarin AL. Drop impact dynamics: splashing, spreading, receding, bouncing. Annu Rev Fluid Mech. 2006; 38: 159-192.
|
| [24] |
Wang F, Yuan Q, Shi X. Instability-induced crystal self-assembly in film-substrate system for the construction of large-area micro- and nano-chiral structures. Nat Commun. 2025; 16: 5463.
|
| [25] |
Liu Y, Andrew M, Li J, Yeomans JM, Wang Z. Symmetry breaking in drop bouncing on curved surfaces. Nat Commun. 2015; 6:10034.
|
| [26] |
Yuan Q, Huang X, Zhao Y-P. Dynamic spreading on pillar-arrayed surfaces: viscous resistance versus molecular friction. Phys Fluids. 2014; 26:092104.
|
| [27] |
Chen E, Yuan Q, Huang X, Zhao Y-P. Dynamic polygonal spreading of a droplet on a lyophilic pillar-arrayed surface. J Adhes Sci Technol. 2016; 30: 2265-2276.
|
| [28] |
Wang L-Z, Huang X, Yuan Q, Chen L, Yu Y-S. Dilute sodium dodecyl sulfate droplets impact on micropillar-arrayed non-wetting surfaces. Phys Fluids. 2021; 33:107103.
|
| [29] |
Wang S-Z, Huang X, Chen L, Yu Y-S. Bouncing droplets on micro-grooved non-wetting surfaces. Phys Fluids. 2023; 35:027118.
|
| [30] |
Wang D, Sun Q, Hokkanen MJ, et al. Design of robust superhydrophobic surfaces. Nature. 2020; 582: 55-59.
|
| [31] |
Hu Z, Chu F, Shan H, Wu X, Dong Z, Wang R. Understanding and utilizing droplet impact on superhydrophobic surfaces: phenomena, mechanisms, regulations, applications, and beyond. Adv Mater. 2024; 36:2310177.
|
| [32] |
Gauthier A, Symon S, Clanet C, Quéré D. Water impacting on superhydrophobic macrotextures. Nat Commun. 2015; 6: 8001.
|
| [33] |
Liu Y, Moevius L, Xu X, Qian T, Yeomans JM, Wang Z. Pancake bouncing on superhydrophobic surfaces. Nat Phys. 2014; 10: 515-519.
|
| [34] |
Li H, Fang W, Li Y, et al. Spontaneous droplets gyrating via asymmetric self-splitting on heterogeneous surfaces. Nat Commun. 2019; 10: 950.
|
| [35] |
Karim AM. Physics of droplet impact on flexible materials: a review. Adv Mech Eng. 2022; 14:16878132221137237.
|
| [36] |
Ma Y, Huang H. Scaling maximum spreading of droplet impacting on flexible substrates. J Fluid Mech. 2023; 958: A35.
|
| [37] |
Xiong Y, Huang H, Lu X-Y. Numerical study of droplet impact on a flexible substrate. Phys Rev E. 2020; 101:053107.
|
| [38] |
Vasileiou T, Schutzius TM, Poulikakos D. Imparting icephobicity with substrate flexibility. Langmuir. 2017; 33: 6708-6718.
|
| [39] |
Vasileiou T, Gerber J, Prautzsch J, Schutzius TM, Poulikakos D. Superhydrophobicity enhancement through substrate flexibility. Proc Natl Acad Sci U S A. 2016; 113: 13307-13312.
|
| [40] |
Upadhyay G, Kumar V, Bhardwaj R. Bouncing droplets on an elastic, superhydrophobic cantilever beam. Phys Fluids. 2021; 33:042104.
|
| [41] |
Weisensee PB, Tian J, Miljkovic N, King WP. Water droplet impact on elastic superhydrophobic surfaces. Sci Rep. 2016; 6:30328.
|
| [42] |
Zhou X, Xu Y, Zhang Q, Zhang W, Peng Z-R. Droplet impact on a flexible disk. J Fluid Mech. 2025; 1002: A8.
|
| [43] |
Howland CJ, Antkowiak A, Castrejón-Pita JR, et al. It's harder to splash on soft solids. Phys Rev Lett. 2016; 117:184502.
|
| [44] |
Chen L, Bonaccurso E, Deng P, Zhang H. Droplet impact on soft viscoelastic surfaces. Phys Rev E. 2016; 94:063117.
|
| [45] |
Huang X, Dong H, Liu Z, Zhao Y-P. Probing micro-Newton forces on solid/liquid/gas interfaces using transmission phase shift. Langmuir. 2019; 35: 5442-5447.
|
| [46] |
Sun T-P, Alvarez-Novoa F, Andrade K, Gutierrez P, Gordillo L, Cheng X. Stress distribution and surface shock wave of drop impact. Nat Commun. 2022; 13: 1703.
|
| [47] |
Zhang B, Sanjay V, Shi S, et al. Impact forces of water drops falling on superhydrophobic surfaces. Phys Rev Lett. 2022; 129:104501.
|
| [48] |
Karim AM. Physics of droplet impact on various substrates and its current advancements in interfacial science: a review. J Appl Phys. 2023; 133:030701.
|
| [49] |
Fang W, Wang S, Duan H, et al. Target slinging of droplets with a flexible cantilever. Droplet. 2023; 2:e72.
|
| [50] |
Kim J-H, Rothstein JP, Shang JK. Dynamics of a flexible superhydrophobic surface during a drop impact. Phys Fluids. 2018; 30:072102.
|
| [51] |
Pan B, Yu L, Zhang Q. Review of single-camera stereo-digital image correlation techniques for full-field 3D shape and deformation measurement. Sci China: Technol Sci. 2018; 61: 2-20.
|
| [52] |
Yu L, Pan B. Single-camera stereo-digital image correlation with a four-mirror adapter: optimized design and validation. Opt Lasers Eng. 2016; 87: 120-128.
|
| [53] |
Zhou Y, Zhang C, Zhao W, Wang S, Zhu P. Suppression of hollow droplet rebound on super-repellent surfaces. Nat Commun. 2023; 14: 5386.
|
| [54] |
Xu H, Zhang B, Lv C. Impact forces of drops falling on inclined superhydrophobic surfaces. Appl Phys Lett. 2024; 125:091602.
|
| [55] |
Clanet C, BÉGuin C, Richard D, QuÉRÉ D. Maximal deformation of an impacting drop. J Fluid Mech. 2004; 517: 199-208.
|
| [56] |
Solav D, Silverstein A. DuoDIC: 3D digital image correlation in MATLAB. J Open Source Software. 2022; 7: 4279.
|
RIGHTS & PERMISSIONS
2026 The Author(s). Droplet published by Jilin University and John Wiley & Sons Australia, Ltd.
PDF
