Capillary-wave-driven protrusion formation and satellite-droplet ejection in collisions of droplets with different surface tensions: quantitative theory and regime boundaries (ethanol–water)

When micrometer-sized ethanol and water droplets collide, Kohno, Kobayashi and Suzuki (2013) observed with microsecond time resolution that a deformation born at the contact region propagates along the water-droplet surface as a capillary wave (measured speeds 5.2–6.2 m/s, identified as a mode $l\approx 5$ capillary oscillation) and constructively interferes at the pole opposite the contact region to form a protrusion, interpreted as a nascent satellite droplet of the kind observed at millimeter scale by Gao et al. (2005). The account was qualitative and narrow: an inviscid, small-amplitude mode analysis on an assumed spherical droplet; only head-on and near-head-on collisions of roughly 38–78 μm droplets at 2.5–3.1 m/s; a single liquid pair; and no prediction of when the protrusion detaches as a satellite droplet rather than re-merging. Later work resolved parts of this question: the protrusion is water-rich rather than a mixing product (Suzuki and Kohno 2014), and the asymmetry of coalescing droplets with different surface tensions is caused by the distinct capillary waves of the two interfaces, with Marangoni stresses actually reducing the asymmetry (Hack et al. 2021); numerical regime maps exist for equal-sized head-on ethanol–water collisions (Singh and Saha 2024). What remains open is a quantitative, validated account that predicts, from collision and fluid parameters, the propagation and viscous damping of the collision-excited capillary waves, the amplitude and timing of the interference-driven protrusion, and the critical conditions (Weber number, impact parameter, size ratio, surface-tension mismatch) under which the protrusion detaches as a satellite droplet.

Background

Binary collisions of liquid droplets govern the drop-size distributions of sprays, clouds, inkjet printing, and microfluidic systems. For droplets of the same liquid, the collision outcome is conventionally classified by the Weber number $We=\rho v^2 d/\sigma$ (the ratio of inertial stress to surface-tension stress, with $\rho$ the density, $v$ the relative velocity, $d$ the droplet diameter, and $\sigma$ the surface tension) and by the impact parameter $B$ (the dimensionless lateral offset of the droplet centers in the plane perpendicular to $v$ , with $B=0$ denoting a head-on collision), into the regimes of bouncing, coalescence, reflexive separation (the flattened merged mass recoils along the impact axis and breaks apart), and stretching separation (an off-center merged mass elongates under shear and splits) (Ashgriz and Poo 1990; Qian and Law 1997). These regime maps were built for identical liquids and say nothing about collisions of two different liquids. For a pair of unequal liquids, for which the surface tension entering $We$ is not unique, $We$ is evaluated with a volume-averaged effective density and the mean droplet diameter, and the convention chosen for the effective surface tension (together with these effective quantities) is stated explicitly.

When the colliding droplets are miscible liquids with very different surface tensions — for ethanol and water, about 22 mN/m and 72 mN/m — the dynamics become asymmetric. Gao, Chen, Pu and Lin (2005) studied millimeter-scale ethanol–water drop collisions and reported an 'unbalanced-surface-force' deformation on the water (higher- $\sigma$ ) drop, which can squeeze out small satellite droplets at low impact parameter or split the water drop at high impact parameter. The mechanism producing this asymmetric deformation and the satellites was not resolved.

Kohno, Kobayashi and Suzuki (2013) built a stroboscopic apparatus (two piezo-driven droplet nozzles — nozzles squeezed by a piezoelectric element so as to eject uniform, repeatable droplets — with pulsed-LED back-illumination through a microscope, about 1 μs time resolution) and directly observed the early dynamics in head-on and near-head-on collisions of ethanol and water droplets of roughly 38–78 μm diameter at 2.5–3.1 m/s. A shoulder-shaped deformation appears at the contact region within about 5 μs, propagates along the water-droplet surface toward the counter pole (the point opposite the contact region), and is followed there by growth of a protrusion, which reaches maximum elongation about 30 μs after contact before merging back into the drop. Tracking the shoulder gave propagation speeds of 6.2 ± 0.8 m/s (ethanol–water) and 5.8 ± 0.2 and 5.2 ± 0.2 m/s (water–water), nearly identical despite the different origins of the deformation (surface-tension release versus inertia), implying propagation governed by the surface tension of water alone. Interpreting the shoulder as a capillary wave — an interfacial wave whose restoring force is surface tension — the authors applied the classical inviscid, small-amplitude capillary-oscillation frequency of a spherical droplet (Lamb's formula)

f = \sqrt{\frac{2\,l(l-1)(l+2)\,\sigma}{\pi^2\,\rho\,D_m^3}}

where $l$ is the spherical-harmonic mode number and $D_m$ the merged-droplet diameter, together with the wave-speed relation $v=\pi D_m f/2$ , and identified the wave as a mode $l\approx 5$ oscillation, with an observed wavelength of about one fifth of the droplet's great circle. The wavelength and the speed relation are separate constraints carried by the source, and a complete account must reproduce both the observed wavelength and the observed propagation speed within one self-consistent picture. They attributed the counter-pole protrusion to constructive interference of the capillary wave and interpreted it as a nascent satellite droplet of the kind reported by Gao et al. The mechanism, however, was established only qualitatively: the mode analysis assumes an inviscid, near-spherical, isolated droplet, whereas the actual droplet is strongly deformed, merging, and laterally heterogeneous; the observations covered only one liquid pair in a narrow size–velocity–impact-parameter window; the possible role of ethanol–water mixing and the associated Marangoni stresses (tangential stresses driven by surface-tension gradients along the interface) was not addressed; and the conditions under which the protrusion actually detaches were not determined.

Problem Statement

Develop a quantitative, validated account of protrusion formation in binary collisions of miscible droplets with different surface tensions — the ethanol–water system of Kohno, Kobayashi and Suzuki (2013) being the reference case — and of its transition to satellite-droplet ejection. Specifically, given the collision and fluid parameters (droplet diameters and size ratio, relative velocity or Weber number $We$ , impact parameter $B$ , the surface-tension mismatch $\tilde\gamma=(\gamma_2-\gamma_1)/\gamma_1$ between the two droplets, the viscosities, and the composition-dependent surface tension of the miscible liquids), the account should predict: (i) the propagation of the collision-excited capillary wave on the higher-surface-tension droplet — its speed, mode content, and viscous damping — superseding the inviscid, small-amplitude, spherical-droplet mode analysis (Eq. 6 of the 2013 paper) that neglects both viscosity and the strong deformation of the merging droplets; (ii) the amplitude and timing of the protrusion formed at the counter pole by constructive interference of the capillary wave (in the reference experiment, a 78 μm water droplet struck by a 75 μm ethanol droplet at 2.5 m/s produced a protrusion reaching maximum elongation about 30 μs after contact, after a wave traveling at about 6 m/s); and (iii) the critical combinations of $We$ , $B$ , size ratio, and $\tilde\gamma$ at which the protrusion detaches as a satellite droplet rather than re-merging, for off-center as well as head-on collisions, at the micrometer and microsecond scales of the original observations. Validation is against the published microsecond-resolved observations of this system — the capillary wave propagating over the water-droplet surface and the water-composed protrusion formed at its antipode by constructive interference (Kohno et al. 2013; Suzuki and Kohno 2014) — and against the established qualitative mechanism whereby the intrinsic capillary waves associated with the two surface tensions cause the asymmetry while Marangoni stresses reduce it (Hack et al. 2021). Parts (i) and (ii) are constrained by the reference microsecond-resolved observations, whereas the detachment boundary of part (iii) is not fixed by any prior observation.

References

Jun-ya Kohno, Makoto Kobayashi, Tomoko Suzuki, 'Protrusion formation during the collisional process of ethanol and water droplets: Capillary wave propagation on the water droplet', Chemical Physics Letters 578, 15–20 (2013). DOI: 10.1016/j.cplett.2013.06.010, https://doi.org/10.1016/j.cplett.2013.06.010
T.-C. Gao, R.-H. Chen, J.-Y. Pu, T.-H. Lin, 'Collision between an ethanol drop and a water drop', Experiments in Fluids 38, 731–738 (2005). DOI: 10.1007/s00348-005-0952-1, https://doi.org/10.1007/s00348-005-0952-1
N. Ashgriz, J. Y. Poo, 'Coalescence and separation in binary collisions of liquid drops', Journal of Fluid Mechanics 221, 183–204 (1990). DOI: 10.1017/S0022112090003536, https://doi.org/10.1017/S0022112090003536
J. Qian, C. K. Law, 'Regimes of coalescence and separation in droplet collision', Journal of Fluid Mechanics 331, 59–80 (1997). DOI: 10.1017/S0022112096003722, https://doi.org/10.1017/S0022112096003722
Tomoko Suzuki, Jun-ya Kohno, 'Simultaneous Detection of Images and Raman Spectra of Colliding Droplets: Composition Analysis of Protrusions Emerging during Collisions of Ethanol and Water Droplets', The Journal of Physical Chemistry B 118, 5781–5786 (2014). DOI: 10.1021/jp503285w, https://doi.org/10.1021/jp503285w, PMID: 24806362
Kazuma Anahara, Jun-ya Kohno, 'Initial Collision Process of Two Miscible Droplets', The Journal of Physical Chemistry B 121, 9895–9901 (2017). DOI: 10.1021/acs.jpcb.7b08526, https://doi.org/10.1021/acs.jpcb.7b08526
Michiel A. Hack, Patrick Vondeling, Menno Cornelissen, Detlef Lohse, Jacco H. Snoeijer, Christian Diddens, Tim Segers, 'Asymmetric coalescence of two droplets with different surface tensions is caused by capillary waves', Physical Review Fluids 6, 104002 (2021). DOI: 10.1103/PhysRevFluids.6.104002, https://doi.org/10.1103/PhysRevFluids.6.104002, arXiv:2201.04201, https://arxiv.org/abs/2201.04201
Swati Singh, Arun K. Saha, 'Numerical study of two equal-sized miscible drops undergoing head-on collision', Physics of Fluids 36, 103624 (2024). DOI: 10.1063/5.0221949, https://doi.org/10.1063/5.0221949
Yu Nishio, Kohei Komori, Seiichiro Izawa, Yu Fukunishi, 'Simulation of collisions of two droplets containing two different liquids using incompressible smoothed particle hydrodynamics method', Theoretical and Computational Fluid Dynamics 34, 105–117 (2020). DOI: 10.1007/s00162-020-00514-2, https://doi.org/10.1007/s00162-020-00514-2
Daichi Hayakawa, Taichi Hirano, Shujiro Mitani, Keiji Sakai, 'Measurement of surface tension of liquid microdroplets through observation of droplet collision', Japanese Journal of Applied Physics 56, 07JB02 (2017). DOI: 10.7567/jjap.56.07jb02, https://doi.org/10.7567/jjap.56.07jb02
François Blanchette, Terry P. Bigioni, 'Partial coalescence of drops at liquid interfaces', Nature Physics 2, 254–257 (2006). DOI: 10.1038/nphys268, https://doi.org/10.1038/nphys268
Yupeng Leng, Chengming He, Qian Wang, Zhixia He, Nigel Simms, Peng Zhang, 'Symmetry-Breaking-Induced Internal Mixing Enhancement of Droplet Collision', Symmetry 16(1), 47 (2023). DOI: 10.3390/sym16010047, https://doi.org/10.3390/sym16010047
Chengming He, Peng Zhang, Chung K. Law, 'Dynamics of binary droplet collisions', Progress in Energy and Combustion Science 111, 101252 (2025). DOI: 10.1016/j.pecs.2025.101252, https://doi.org/10.1016/j.pecs.2025.101252