Coalescence of viscoelastic sessile drops: the small and large contact angle limits

TL;DR

This work demonstrates that viscoelastic effects in coalescing sessile drops depend critically on the contact angle. In the thin-film, small-angle limit, lubrication analysis shows the Deborah number scales as $De_{\theta} \sim \theta^3$, rendering polymer stresses negligible and yielding near-Newtonian coalescence consistent with the classical viscous scaling $h_0 \propto t^{1}$ (with $h_0 \approx v t$ and $v$ set by capillary and viscous parameters). In contrast, the large-angle regime where inertia dominates reveals pronounced elastic effects, captured by the Oldroyd-B model, with the dynamics governed by $Oh$, $De$, and $Ec$; polymer stresses localize near the coalescence point and can slow bridge growth, altering the $h_0 \propto t^{2/3}$ scaling. The combination of 3D FS-SS imaging and 2D Basilisk simulations maps out the transition, showing when elasticity matters in capillary-driven coalescence and providing insight into designing thin-film flows of polymeric liquids for controlled outcomes.

Abstract

The coalescence and breakup of drops are classic examples of flows that feature singularities. The behavior of viscoelastic fluids near these singularities is particularly intriguing - not only because of their added complexity, but also due to the unexpected responses they often exhibit. In particular, experiments have shown that the coalescence of viscoelastic sessile drops can differ significantly from their Newtonian counterparts, sometimes resulting in a sharply defined interface. However, the mechanisms driving these differences in dynamics, as well as the potential influence of the contact angle are not fully known. Here, we study two different flow regimes effectively induced by varying the contact angle and demonstrate how that leads to markedly different coalescence behaviors. We show that the coalescence dynamics is effectively unaltered by viscoelasticity at small contact angles. The Deborah number, which is the ratio of the relaxation time of the polymer to the timescale of the background flow, scales as $θ^3$ for $θ\ll 1$, thus rationalizing the near-Newtonian response. On the other hand, it has been shown previously that viscoelasticity dramatically alters the shape of the interface during coalescence at large contact angles. We study this large contact angle limit using experiments and 2D numerical simulations of the equation of motion. We show that the departure of the coalescence dynamics from the Newtonian case is a function of the Deborah number and the elastocapillary number, which is the ratio between the shear modulus of the polymer solution and the characteristic stress in the fluid.

Figures (12)

• Figure 1: The shape of the interface during the coalescence of Newtonian and polymeric drops. (a) Schematic of the side view profile of two drops during a typical coalescence experiment. Experimental images of large $\theta$ coalescence of (b) water and (c) 0.5% PEO drops show a significant difference in the shape of the interface. On the other hand, small $\theta$ coalescence of (d) 1000 cSt silicone oil and (e) 0.5% PEO drops show similar shape of the interface. Scale bars represent 0.1 mm.
• Figure 2: Three-dimensional reconstruction of the shape of the interface using Free-Surface Synthetic Schlieren imaging. (a) Schematic showing the experimental setup and the drop geometry. (b) Sequence of experimental images showing a reference frame taken before the drop appeared, and two time steps during the spreading and coalescing of 1 wt% PEO drops. (c) The 3D reconstruction of the interface shape corresponding to the $t=1$ s after coalescence.
• Figure 3: The time evolution of the height of the interface $h_0(t)$ at the initial coalescence point. (a) Raw data showing $h_0(t)$ from experiments using three different polymer concentrations, spanning the ranges $De_\theta = [0.002,~0.06]$ and $Ec_\theta = [0.07,~1.7]$. (b) Average power-law exponent $\alpha$ from fitting the data for the different polymer concentrations. (c) The $h_0$ versus $t$ data rescaled according to the Newtonian viscous scaling. Rescaling reasonably collapses the data, and the black line has a power-law exponent $\alpha=1$ and a prefactor $A=0.818$, predicted by the viscous theory.
• Figure 4: The interface profiles along the $x$ and $y$ axes from the coalescence of 0.5% PEO drops with $\theta \approx 8.1^{\circ}$, corresponding to $De_\theta = 0.009$ and $Ec_\theta = 0.74$. (a) Schematic of the interface in the $xz$-plane where the height $h_0(t)$ at the coalescence point is labelled. (b) Experimental data showing the dynamic shape of the interface in this plane. Notice that the darker colored markers correspond to earlier times and the lighter colored ones to the later times. Markers are connected by a faint line that is intended to only serve as a guide for the eyes. (c) The interface profiles rescaled with $h_0(t)$. The black line is the self-similar profile in the $xz$-plane. (d) Schematic of the interface in the $yz$-plane, where $a$ is the radius of a spherical cap. (e) Experimental data showing the dynamic shape of the interface in the $yz$-plane. (f) The rescaled interface profiles with $a=2.7$ mm.
• Figure 5: The Newtonian three-dimensional self-similarity also describes the coalescence of semi-dilute polymeric drops at small $\theta$. (a) Experimental data from the coalescence of 0.5% PEO drops with $\theta \approx 8.1^{\circ}$, corresponding to $De_\theta = 0.009$ and $Ec_\theta = 0.74$, showing the three-dimensional shape of the interface near the coalescence point at early times ($t= 0.05,~0.15,~0.22$ s). The darker colored markers correspond to earlier times and the lighter colored ones to the later times. (b) Experimental data from the coalescence of 0.1%, 0.5%, and 1.0% PEO drops at 4 different times and 3 different $yz$-planes (total of 36 curves) rescaled according to the similarity solution. The rescaled data collapses onto the universal self-similar curve (black line).