Elasto-viscous regime in coalescence of viscoelastic droplets

TL;DR

The study reveals a regime transition in pendant–pendant coalescence of viscoelastic droplets, driven by elasticity and molecular relaxation. It combines long-timescale experiments and two-dimensional VO F simulations using the exponential Phan–Thien–Tanner (ePTT) model, validated against extensional rheology measurements from CaBER-DoS and a consumer product (shampoo). Two distinct coalescence regimes emerge at $c/c^* > 28$, with the neck radius following $R = a t^{b}$ and $b$ remaining in the sub-Newtonian range ($b<0.5$; Regime 1 ≈ $0.1$–$0.2$, Regime 2 lower), accompanied by significant axial-curvature deviations from Newtonian assumptions. This work highlights the breakdown of self-similarity due to elastic stresses, demonstrates the predictive power of the ePTT framework for viscoelastic droplet coalescence, and extends relevance to practical fluids like shampoo.

Abstract

We report a regime transition in the coalescence of concentrated polymeric droplets in a pendant-pendant configuration. While Newtonian droplet coalescence has been extensively studied with distinct identification of viscous and inertial regimes, the presence of polymers introduces additional regimes governed by elasticity and molecular relaxation effects. The coalescence process is typically characterized by the neck radius, $R$, of the liquid bridge connecting the two droplets, following a power-law relation with time: $R=at^{b}$. Most of the existing studies, including Newtonian and non-Newtonian fluids, report a unique value of $b$ for a given fluid. In contrast, our findings reveal that elasticity induces a temporal transition from one $b$ values to another, marking a shift in the coalescence regime. In particular, our measured $b$ value falls in the sub-Newtonian regime, highlighting the role of elasticity in governing the dynamics. We conducted two-dimensional simulations using a volume-of-fluid framework with the exponential Phan-Thien-Tanner model, which quantitatively reproduced Newtonian benchmarks and accurately captured viscoelasticity induced neck growth in close agreement with experiments. Furthermore, we determined the curvature experimentally, as the assumptions typically employed in the literature to approximate axial curvature are not universally valid.

Figures (9)

• Figure 1: (a) Power-law exponent corresponding to various coalescence regimes, (b) schematic of the pendent drop showing the calculation of initial droplet radius $R_0$, and (c) coalescence of droplets in pendent-pendent configuration showing the neck radius $R$ and radius of curvature $R_k$ in the axial direction.
• Figure 2: (a) Snapshots of droplet shapes displaying varying neck radius for DI Water, Glycerol, Honey and polymer solutions with different $c/c^*$ of 14, 28, 42 and 56, (b) bridge profiles at different times and their corresponding rescaled profiles according to the Newtonian scaling laws for (b, c) DI Water, and polymer solution with (d,e) $c/c^*$ = 28 and (f,g) $c/c^*$ = 56.
• Figure 3: Temporal evolution of neck radius for polymeric droplets with different $c/c^*$, (a) which do not exhibit a transition ($c/c^* < 28$), and (b) which do show a transition $c/c^* > 28$ with normalised time. Variation of dimensionless numbers (c) Weissenberg number, (d) Elastocapillary number, and (e) Capillary number with normalised time.
• Figure 4: Comparison of experimental and numerical results for the coalescence of viscoelastic droplets. (a) Neck radius evolution in Regime 1 ($c/c^{*} = 14, 19, 24, 28$). (b) Neck radius evolution in Regime 2 ($c/c^{*} = 32, 35, 42, 45$). (c) Power-law exponents $b$ extracted from experimental and numerical data across all concentrations.
• Figure 5: Variation of normalized neck radius of shampoo (Dove) with time.