The colliding-droplet microreactor initiates chemistry by colliding two airborne, monodisperse reactant droplets, achieving sub-millisecond mixing (from ~900 microseconds at 0.1 m/s to under 200 microseconds at ~6 m/s for head-on impacts of 40-micrometer droplets). Its empirical characterization left a predictive gap: the measured mixing-time dependence on collision geometry lacked a first-principles description, the droplet-size scaling (extrapolated to ~20 microsecond mixing for ~10 micrometer droplets at ~14 m/s) was never validated, mixing was characterized only for uncharged ~40 micrometer droplets, and the free-flight reactor was never coupled to mass spectrometric product identification. This problem asks for a quantitative, experimentally validated theory or simulation of inertial mixing in free-flight binary microdroplet collisions covering the impact-parameter and velocity dependence, the size scaling down to ~10 micrometers, and extension to highly charged droplets, together with product-identifying detection integrated into the free-flight platform.

Background

Studying fast aqueous reactions requires mixing reactants faster than the reaction itself. For microdroplet chemistry — the observation that reactions in micron-scale aqueous compartments can differ in rate and mechanism from bulk solution — an on-demand route is to collide two airborne droplets, each carrying one reactant, so that the reaction starts at a defined time and place. Davis, Jacobs, Houle and Wilson (Anal. Chem. 2017, 89, 12494) built such a colliding-droplet microreactor: two monodisperse streams of ~40 micrometer droplets collide in free flight, and mixing inside the merged droplet is quantified by streak-based fluorescence microscopy, in which the fluorescence of the fast-moving merged droplet, imaged as a streak, reports how completely and how quickly the two liquids homogenize. They measured mixing times from ~900 microseconds at a collision velocity of 0.1 m/s to below 200 microseconds at ~6 m/s for head-on collisions. The geometry matters: the off-centeredness of a binary collision is measured by the impact parameter X (X = 0 for a head-on hit, X approaching 1 for a grazing miss). For highly off-center, high-velocity collisions the mixing time grew by up to a factor of ~6 (to ~1200 microseconds at ~6 m/s), and above ~7 m/s the merged droplet separated and fragmented, leaving mixing incomplete. The droplet collision regime is conventionally parameterized by the Weber number, the ratio of collision kinetic energy to surface-tension energy, and the Ohnesorge number, the ratio of viscous to capillary slowing. The paper benchmarked the reactor on the iron-catalyzed Fenton reaction (hydroxyl-radical production from hydrogen peroxide) and showed bulk kinetics reproduce the observations, establishing merged airborne droplets as quantitative microsecond reactors. What the source paper itself flagged as unresolved anchors this problem: the link between the internal flow patterns generated by a collision (for example, the unidirectional fluid motion of off-center impacts) and the measured mixing-time variation with X was speculative, with no first-principles description; the size scaling of mixing dynamics was an untested extrapolation; and the characterization applied only to uncharged droplets near 40 micrometers in diameter. Related but distinct platforms frame what has and has not been settled: the same group's branched quadrupole trap (an electrodynamic balance that levitates charged droplets and lets two of them collide and merge gently) couples droplet collisions to single-droplet paper-spray mass spectrometry, in which a droplet deposited on paper is ionized by an applied voltage for mass analysis; and volume-of-fluid direct numerical simulations, which track the liquid-gas interface and the interpenetration of the two liquids (often scored by a segregation parameter that is zero when the liquids are fully mixed), have analyzed mixing in binary droplet collisions in slow-coalescence regimes relevant to inkjet printing. Whether a validated quantitative account exists for the free-flight inertial regime of the colliding-droplet microreactor — and how far its empirical characterization extends — is the subject of this problem.

Problem Statement

Establish a quantitative, experimentally validated first-principles description of inertial mixing in free-flight (unconfined, airborne) binary collisions of aqueous microdroplets — the regime of the colliding-droplet microreactor of Davis et al. (2017) — and extend its empirical characterization. The question has four linked parts, each calling for a falsifiable quantitative result — a derived or simulated result, compared against existing data where such data exist, and, where no such data exist, the prediction that a proposed measurement would test — obtained by theory or simulation rather than executed laboratory work: (1) connect the internal flow patterns generated by head-on and off-center collisions to the measured mixing-time dependence on the impact parameter X and collision velocity, reproducing the fluorescence-streak data for ~40 micrometer droplets from 0.1 to ~7 m/s, including the up-to-6x slower mixing of highly off-center high-velocity collisions and the droplet-separation/fragmentation onset above ~7 m/s; (2) predict the droplet-size scaling of the mixing dynamics, including whether ~10 micrometer droplets colliding at ~14 m/s mix in ~20 microseconds as the source extrapolation suggests, and specify the measurement that would confirm or refute it; (3) characterize, as quantitative predictions, mixing behavior in conditions far from the characterized uncharged ~40 micrometer regime, in particular for highly charged droplets charged to a substantial fraction of the Rayleigh limit, such as those produced in electrospray plumes, and specify the measurement that would test them; and (4) specify and design a product-identifying detection to be integrated with the free-flight colliding-droplet reactor itself, since existing droplet-collision mass spectrometry operates only on levitated or guided droplets at low collision velocity.

References

  1. Ryan D. Davis, Michael I. Jacobs, Frances A. Houle, Kevin R. Wilson, 'Colliding-Droplet Microreactor: Rapid On-Demand Inertial Mixing and Metal-Catalyzed Aqueous Phase Oxidation Processes', Analytical Chemistry 2017, 89, 12494-12501, DOI 10.1021/acs.analchem.7b03601, https://doi.org/10.1021/acs.analchem.7b03601
  2. Michael I. Jacobs, James F. Davies, Lance Lee, Ryan D. Davis, Frances A. Houle, Kevin R. Wilson, 'Exploring Chemistry in Microcompartments Using Guided Droplet Collisions in a Branched Quadrupole Trap Coupled to a Single Droplet, Paper Spray Mass Spectrometer', Analytical Chemistry 2017, 89, 12511-12519, DOI 10.1021/acs.analchem.7b03704, https://doi.org/10.1021/acs.analchem.7b03704
  3. Élfego Ruiz-Gutiérrez, Josef Hasslberger, Markus Klein, Kenny Dalgarno, Nilanjan Chakraborty, 'Analysis and optimisation of mixing in binary droplet collisions', Journal of Fluid Mechanics 2023, 973, A38, DOI 10.1017/jfm.2023.749, https://doi.org/10.1017/jfm.2023.749
  4. Élfego Ruiz-Gutiérrez, Josef Hasslberger, Markus Klein, Kenny Dalgarno, Nilanjan Chakraborty, 'Binary Droplet Collisions in Bioprinting: Influence of Material Properties on Mixing and Repeatability', Flow, Turbulence and Combustion 2024, 114, 913-939, DOI 10.1007/s10494-024-00606-7, https://doi.org/10.1007/s10494-024-00606-7
  5. Emily K. Brown, Grazia Rovelli, Kevin R. Wilson, 'pH jump kinetics in colliding microdroplets: accelerated synthesis of azamonardine from dopamine and resorcinol', Chemical Science 2023, 14, 6430-6442, DOI 10.1039/d3sc01576a, https://doi.org/10.1039/d3sc01576a
  6. Kevin R. Wilson, Alexander M. Prophet, Grazia Rovelli, Megan D. Willis, Rebecca J. Rapf, Michael I. Jacobs, 'A kinetic description of how interfaces accelerate reactions in micro-compartments', Chemical Science 2020, 11, 8533-8545, DOI 10.1039/d0sc03189e, https://doi.org/10.1039/d0sc03189e
  7. Grazia Rovelli, Michael I. Jacobs, Megan D. Willis, Rebecca J. Rapf, Alexander M. Prophet, Kevin R. Wilson, 'A critical analysis of electrospray techniques for the determination of accelerated rates and mechanisms of chemical reactions in droplets', Chemical Science 2020, 11, 13026-13043, DOI 10.1039/d0sc04611f, https://doi.org/10.1039/d0sc04611f
  8. Megan D. Willis, Grazia Rovelli, Kevin R. Wilson, 'Combining Mass Spectrometry of Picoliter Samples with a Multicompartment Electrodynamic Trap for Probing the Chemistry of Droplet Arrays', Analytical Chemistry 2020, 92, 11943-11952, DOI 10.1021/acs.analchem.0c02343, https://doi.org/10.1021/acs.analchem.0c02343
  9. Jim S. Walker, Bryan R. Bzdek, 'Rapid and Sensitive Chemical Analysis of Individual Picolitre Droplets by Mass Spectrometry', Analytical Chemistry 2024, 97, 854-861, DOI 10.1021/acs.analchem.4c05458, https://doi.org/10.1021/acs.analchem.4c05458
  10. Michael I. Jacobs, Ryan D. Davis, Rebecca J. Rapf, Kevin R. Wilson, 'Studying Chemistry in Micro-compartments by Separating Droplet Generation from Ionization', Journal of the American Society for Mass Spectrometry 2019, 30, 339-343, DOI 10.1007/s13361-018-2091-y, https://doi.org/10.1007/s13361-018-2091-y
  11. Shuhei Suzuki, Aya Kamoshita, Jun-ya Kohno, 'Chemical Reactions at the Interface Periphery of Colliding Droplets Studied by Raman Image Analysis', The Journal of Physical Chemistry A 2021, 125, 9161-9166, DOI 10.1021/acs.jpca.1c06163, https://doi.org/10.1021/acs.jpca.1c06163
  12. Hao Deng, Jacek J. Jasieniak, Tuncay Alan, 'Precisely Timed On-Demand Droplet Ejection and Collision in Free Space for Controlled Reaction Initiation', Small 2026, e74957, DOI 10.1002/smll.74957, https://doi.org/10.1002/smll.74957
  13. Soichiro Tsujino, Yohei Sato, Shichao Jia, Michal W. Kepa, Sofia Trampari, Takashi Tomizaki, 'Inertial mixing of acoustically levitated droplets for time-lapse protein crystallography', Droplet 2024, 3, e132, DOI 10.1002/dro2.132, https://doi.org/10.1002/dro2.132
  14. R. Allen LaCour, Joseph P. Heindel, Ruoqi Zhao, Teresa Head-Gordon, 'The Role of Interfaces and Charge for Chemical Reactivity in Microdroplets', Journal of the American Chemical Society 2025, 147, 6299-6317, DOI 10.1021/jacs.4c15493, https://doi.org/10.1021/jacs.4c15493
  15. Taisuke Maki, Saki Takeda, Yosuke Muranaka, Kazuhiro Mae, 'Silver Nanoparticle Synthesis Using an Inkjet Mixing System', Frontiers in Chemical Engineering 2021, 3, 742322, DOI 10.3389/fceng.2021.742322, https://doi.org/10.3389/fceng.2021.742322