Coalescence of Liquid Drops: Different Models Versus Experiment

The process of coalescence of two identical liquid drops is simulated numerically in the framework of two essentially different mathematical models, and the results are compared with experimental data on the very early stages of the coalescence process reported recently. The first model tested is the conventional' one, where it is assumed that coalescence as the formation of a single body of fluid occurs by an instant appearance of a liquid bridge smoothly connecting the two drops, and the subsequent process is the evolution of this single body of fluid driven by capillary forces. The second model under investigation considers coalescence as a process where a section of the free surface becomes trapped between the bulk phases as the drops are pressed against each other, and it is the gradual disappearance of this internal interface’ that leads to the formation of a single body of fluid and the conventional model taking over. Using the full numerical solution of the problem in the framework of each of the two models, we show that the recently reported electrical measurements probing the very early stages of the process are better described by the interface formation/disappearance model. New theory-guided experiments are suggested that would help to further elucidate the details of the coalescence phenomenon. As a by-product of our research, the range of validity of different `scaling laws’ advanced as approximate solutions to the problem formulated using the conventional model is established.

💡 Research Summary

The paper investigates the very early stages of coalescence between two identical liquid drops by comparing two fundamentally different mathematical models with recent high‑resolution electrical measurements. The first, “conventional” model assumes that as soon as the drops touch a smooth liquid bridge appears instantaneously, after which the evolution of a single fluid body is driven solely by capillary forces. This framework leads to classic scaling laws such as bridge radius r ∝ t¹ᐟ² (or r ∝ t¹ᐟ³ under certain approximations) derived from reduced forms of the Navier‑Stokes equations and the Laplace pressure balance. However, electrical resistance data obtained in the nanosecond regime show a much steeper drop in resistance than the conventional model predicts, suggesting that the instantaneous‑bridge assumption may be incomplete.

The second model, termed the “interface formation/disappearance” (IFD) model, treats coalescence as a two‑step process. When the drops are pressed together, a portion of the free surface becomes trapped between the bulk phases, creating an internal interface that separates the two drops despite their apparent contact. This internal interface possesses its own surface tension and evolves according to additional interfacial transport equations (including interface creation/annihilation rates, surface density evolution, and possible variations of surface tension). Only after this internal interface gradually disappears does the system transition to the conventional single‑bridge regime. Numerically, the authors solve the full Navier‑Stokes equations together with the interfacial balance laws on a time‑adaptive mesh that resolves both the bulk flow and the thin internal layer. Electrical resistance is computed by coupling the fluid conductivity field with the evolving geometry of the bridge.

Simulation results reveal stark differences in the predicted resistance curves. The IFD model reproduces the experimentally observed rapid resistance decline and predicts a bridge growth law closer to r ∝ t¹ᐟ³ during the first few hundred nanoseconds, whereas the conventional model underestimates the resistance drop and follows the r ∝ t¹ᐟ² law only after the bridge radius exceeds a critical size (≈10⁻⁶ m). By systematically varying parameters, the authors delineate the range of validity of the various scaling laws derived from the conventional model, showing that they are applicable only for times larger than ~10⁻⁶ s and for bridge radii where the internal interface has already vanished.

Beyond validation, the paper proposes new experiments to further discriminate between the models. Suggested approaches include: (i) varying the electrode spacing and pulse duration to resolve the resistance signature of the trapped internal interface; (ii) synchronizing ultra‑high‑speed imaging with electrical measurements to directly observe the disappearance of the internal surface; and (iii) introducing surfactants or temperature gradients to modify interfacial properties and assess their impact on the early‑time dynamics.

In conclusion, the study demonstrates that accounting for the formation and gradual disappearance of an internal interface is essential for accurately describing the earliest moments of drop coalescence. The conventional instantaneous‑bridge picture, while useful at later times, fails to capture the nanosecond‑scale physics revealed by modern electrical diagnostics. By establishing the limits of traditional scaling laws and offering concrete experimental pathways, the work provides a solid foundation for future research in microfluidics, additive manufacturing, and any technology where precise control of liquid merging at microscopic scales is required.