[Paper Review] How geometry determines the coalescence of low-viscosity drops

Antonin Eddi, Koen G. Winkels+1 more|arXiv (Cornell University)|Jul 29, 2013

Electrohydrodynamics and Fluid Dynamics3 citations

TL;DR

This study reveals that the coalescence dynamics of low-viscosity drops on a substrate is governed by geometric constraints at contact, with a transition from $ h \sim t^{2/3} $ for contact angles $ \theta < 90^\circ $ to $ h \sim t^{1/2} $ at $ \theta = 90^\circ $. A geometric model unifies inertial coalescence of sessile and freely suspended drops by capturing the curvature-dependent bridge growth via a unified scaling law with no adjustable parameters.

ABSTRACT

The coalescence of water drops on a substrate is studied experimentally. We focus on the rapid growth of the bridge connecting the two drops, which very quickly after contact ensues from a balance of surface tension and liquid inertia. For drops with contact angles below $90^\circ$, we find that the bridge grows with a self-similar dynamics that is characterized by a height $h\sim t^{2/3}$. By contrast, the geometry of coalescence changes dramatically for contact angles at $90^\circ$, for which we observe $h\sim t^{1/2}$, just as for freely suspended spherical drops in the inertial regime. We present a geometric model that quantitatively captures the transition from 2/3 to 1/2 exponent, and unifies the inertial coalescence of sessile drops and freely suspended drops.

Motivation & Objective

• To understand how substrate geometry and contact angle influence the inertial coalescence dynamics of low-viscosity drops.
• To resolve the discrepancy between observed $ t^{2/3} $ scaling in experiments and the classical $ t^{1/2} $ law for freely suspended drops.
• To develop a geometric model that unifies inertial coalescence of sessile and freely suspended drops by accounting for curvature at contact.
• To quantify the transition in scaling exponent as contact angle approaches $ 90^\circ $, where the dynamics shifts from wedge-like to spherical behavior.

Proposed method

• Experiments using ultra-pure water drops on coated glass substrates with controlled advancing contact angles from $ 73^\circ $ to $ 90^\circ $, using slow needle approach to minimize dynamic effects.
• High-speed imaging at microsecond resolution to capture the initial bridge growth dynamics immediately after coalescence.
• Use of conical drop shapes to enhance geometric clarity and extend the range of observable self-similar scaling.
• Development of a geometric model based on intersecting circles to describe the meniscus width $ w $ as a function of height $ h_0 $, contact angle $ \theta $, and drop radius $ R $, leading to a unified scaling law.
• Application of similarity ansatz with rescaling of horizontal coordinate to test collapse of data under different geometric regimes.
• Theoretical derivation of the bridge dynamics equation: $ \frac{D_0^3 \gamma t^2}{\rho} = h_0^2 R \left[ \sin\theta - \sqrt{1 - \left( \frac{h_0}{R} + \cos\theta \right)^2 } \right] $, with $ D_0 = 0.89 $ fixed from prior calibration.

Experimental results

Research questions

• RQ1 How does the contact angle at coalescence influence the scaling exponent of bridge height growth in low-viscosity drops on a substrate?
• RQ2 Why does the coalescence dynamics transition from $ t^{2/3} $ to $ t^{1/2} $ scaling at $ \theta = 90^\circ $, and what geometric mechanism underlies this change?
• RQ3 Can a single geometric model unify the inertial coalescence of sessile drops and freely suspended drops by capturing the curvature of the interface at contact?
• RQ4 What is the role of initial geometry—specifically the shape of the liquid bridge at contact—in determining the dominant forces (capillary vs. inertial) during early coalescence?
• RQ5 How does the range of the $ t^{2/3} $ regime shrink as $ \theta \to 90^\circ $, and what determines the onset of the $ t^{1/2} $ scaling?

Key findings

• For contact angles below $ 90^\circ $, the bridge height grows as $ h_0 \sim t^{2/3} $, consistent with a wedge-like geometry and inertial dominance.
• At $ \theta = 90^\circ $, the scaling transitions to $ h_0 \sim t^{1/2} $, matching the behavior of freely suspended spherical drops.
• The transition is not abrupt but results from a geometric crossover: the meniscus width $ w $ evolves from $ w \sim h_0 $ (for $ \theta < 90^\circ $) to $ w \sim h_0^2 / R $ (for $ \theta = 90^\circ $), altering the pressure balance.
• The proposed geometric model, based on intersecting circles, quantitatively captures the $ \theta $-dependent scaling with no adjustable parameters, using $ D_0 = 0.89 $ from prior calibration.
• The $ t^{2/3} $ regime vanishes as $ \theta \to 90^\circ $, with the asymptotic range shrinking as $ h_0 / R \ll \pi/2 - \theta $, explaining the sudden transition in scaling.
• Experimental data for $ \theta = 90^\circ $ collapse under the same scaling as freely suspended drops, confirming the universality of the $ t^{1/2} $ law in the inertial regime.