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Inelastic spreading of viscoelastic drops

This study experimentally demonstrates that the elastic properties of Boger fluid drops do not influence their maximum spreading radius upon impact, as elastic effects are energetically negligible compared to inertial, capillary, and viscous forces despite high Weissenberg and Deborah numbers.

Original authors: Mete Abbot, Thijs Varkevisser, Manglesh Singh, Antoine Gaillard, Devranjan Samanta, Daniel Bonn

Published 2026-08-12

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CC BY 4.0

Original authors: Mete Abbot, Thijs Varkevisser, Manglesh Singh, Antoine Gaillard, Devranjan Samanta, Daniel Bonn

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Splash Zone: When Drops Hit the Floor

Imagine you are watching a raindrop hit a puddle, or a droplet of paint splatter onto a canvas. In the world of physics, this is a high-speed drama played out by fluid dynamics. It's the study of how liquids move, stretch, and crash. When a drop hits a surface, it doesn't just stop; it flattens out like a pancake, reaching a maximum size before either bouncing back up or spreading forever. Scientists have long known that two main forces are fighting this battle: inertia (the drop's momentum, like a car speeding down a hill) and viscosity (the liquid's internal "stickiness" or resistance to flow, like honey versus water).

But there's a third player that often enters the chat: elasticity. Think of elasticity as a liquid's ability to act like a rubber band. If you stretch a rubber band, it wants to snap back. In liquids, this happens when long, stringy molecules (polymers) are dissolved in the water. You might expect that if you add these stretchy strings to a drop, the drop would act like a bouncy ball, resisting the flattening and maybe even shrinking back up faster. This is the big question that has puzzled scientists for years: If you make a liquid stretchy, does it stop a drop from spreading as wide as it normally would?

The Great Elasticity Paradox

For a long time, researchers noticed a strange contradiction. When they added tiny amounts of stretchy polymers to water, the drops behaved very strangely after they hit the ground. They would flatten out, but then, instead of bouncing back up like a normal drop, they would stay stuck to the surface, refusing to retract. It seemed like the "rubber band" effect was strong enough to hold the drop down.

However, here is the twist: while the polymers stopped the drop from bouncing back, they did not stop the drop from spreading out in the first place. No matter how stretchy the liquid was, the drop would flatten to the exact same maximum size as a plain water drop. It was as if you added super-stretchy rubber bands to a car, but when the car hit a wall, it still crumpled to the exact same size as a car without rubber bands. This was a paradox. Classical physics suggested that if a material is stretchy and you hit it fast enough, it should act like a solid and resist the impact. But the drops said, "Nope, we're still spreading just like water."

The Energy Detective Work

In this new study, a team of scientists set out to solve this mystery. They wanted to know: Is the elasticity actually there, but just too weak to matter? Or is it hiding somewhere else? To find out, they used special liquids called Boger fluids. These are tricky mixtures designed to be perfectly sticky (viscous) like water but incredibly stretchy (elastic) like rubber. By using these, the scientists could test the "stretchiness" without the confusion of the liquid getting thinner and runnier (a common side effect called shear-thinning).

They dropped these special liquids onto surfaces at high speeds and measured exactly how wide they got. The results were clear and surprising: The stretchiness made absolutely no difference to the maximum size of the splash. Even with high concentrations of polymers (up to 1000 parts per million), the drops spread just as far as plain water drops.

The "Energy Budget" Explanation

So, why didn't the rubber bands stop the spread? The authors created a new way of looking at the problem using an "energy budget." Imagine the drop has a certain amount of energy when it hits the floor. It has to spend this energy to do three things:

Spread out (creating new surface area).
Fight friction (viscosity, or the internal stickiness).
Stretch the rubber bands (elasticity).

The scientists calculated how much energy it would take to stretch the polymers versus how much energy was lost to friction. They found that during the split-second of impact, the liquid is moving so fast and shearing so hard that the energy required to stretch the polymers is tiny compared to the energy lost to friction.

They introduced a new number, called Gamma (Γ), to measure this balance. If Gamma is big (greater than 1), the stretchiness should matter. But in every single experiment they ran, Gamma was tiny (much less than 1). This means the "rubber band" effect was energetically negligible. The drop was moving so fast that the polymers didn't have time to act like strong springs; they just got dragged along, and the drop's spread was controlled entirely by inertia and friction.

Why It Matters (and What It Isn't)

The study also looked at what happens with very thick, concentrated polymer solutions. In those cases, the drops actually spread further than water. It might seem like the stretchiness is helping, but the paper argues the opposite. The extra spreading isn't because the polymers are pushing the drop out; it's because the polymers make the liquid thinner (less sticky) when it moves fast. This "shear-thinning" lets the drop slide out further, overpowering any tiny resistance the elasticity might have offered.

The authors are very careful to state that while elasticity is famous for stopping drops from bouncing back (retraction), it simply doesn't have the power to stop them from spreading out in the first place. The "solid-like" resistance that physics textbooks predict for fast-moving elastic materials just doesn't show up in the energy balance of a splashing drop.

In short, if you want to control how far a drop spreads, adding stretchy polymers won't help you shrink the splash. The drop will spread just as wide as it would without them. The elasticity is there, but during the crash, it's too busy getting stretched to put up a fight. The real boss of the splash is still the drop's speed and the liquid's stickiness, not its rubberiness.

Technical Summary: Inelastic Spreading of Viscoelastic Drops

Problem Statement
The impact of liquid drops on solid surfaces is a fundamental phenomenon in applications ranging from agricultural spraying to inkjet printing. While Newtonian spreading is well-understood as a balance between inertia, viscosity, and capillarity, the role of fluid elasticity remains a subject of debate. It is established that trace amounts of polymers can suppress drop rebound and splashing, yet paradoxically, the maximum spreading radius ( $D_{max}$ ) often remains identical to that of the pure Newtonian solvent, even when the fluid exhibits high elasticity.

Classical rheological scaling suggests that under drop impact conditions, the Weissenberg number ($Wi$) and Deborah number ($De$) are significantly greater than 1, predicting a "solid-like" elastic resistance that should inhibit spreading. However, experimental observations contradict this, showing no reduction in $D_{max}$ . The difficulty in resolving this paradox lies in isolating elastic resistance from the concurrent effects of shear thinning, which enhances spreading. Furthermore, standard rheological measurements often fail to capture the extreme shear rates ( $\gg 1000 \text{ s}^{-1}$ ) present during impact, and dynamic surface tension variations add further complexity.

Methodology
To decouple elastic effects from shear thinning, the authors conducted experiments using Boger fluids—solutions designed to possess high elasticity while maintaining constant viscosity. They utilized polyethylene oxide (PEO) and polyacrylamide (PAAM) dissolved in Newtonian solvents (water and glycerol-water mixtures) at concentrations up to 1000 ppm.

The study spanned a broad range of impact conditions:

Weber numbers ($We$): 155 to 763
Reynolds numbers ($Re$): 18 to 11,500
Ohnesorge numbers ($Oh$): 0.002 to 1

High-speed imaging (4000 fps) captured the spreading kinematics. To analyze the energy budget, the authors extended the classical energy balance framework to include an elastic term. They modeled the droplet as a linear elastic (Hookean) solid to derive a conservative upper bound for stored elastic energy, scaling it with the fifth power of the maximum spreading ratio ( $\beta_{max}^5$ ). This approach was chosen to ensure that if elasticity were negligible under this strong scaling assumption, it would be negligible under weaker scaling models (e.g., Neo-Hookean, Oldroyd-B).

The authors defined a new dimensionless energetic criterion, $\Gamma$ , representing the ratio of elastic energy to viscous energy during the spreading phase. They also explicitly defined the characteristic impact shear rate ( $\dot{\gamma}_s$ ) based on volume conservation and spreading kinematics, noting that traditional inertial estimates significantly underestimate the true shear rate.

Key Contributions and Results

Experimental Decoupling: The experiments demonstrated that for Boger fluids (constant viscosity, high elasticity), the maximum spreading ratio ( $\beta_{max}$ ) is identical to that of the Newtonian solvent. This confirms that polymer elasticity, even at high concentrations, does not resist the initial spreading phase.
The Energetic Criterion ( $\Gamma$ ): The authors formulated a dimensionless parameter $\Gamma$ to quantify the relative importance of elastic damping versus viscous damping:
$\Gamma \equiv \frac{\text{Elastic Energy}}{\text{Viscous Energy}} = \frac{8}{5\pi} \left( \frac{G D_0}{\mu_i V_0 \sqrt{E}} \right)$
Where $G$ is the elastic modulus, $\mu_i$ is the viscosity at the impact shear rate, and $E$ is the energy density.
Inelastic Spreading Regime: Across all tested conditions, the calculated $\Gamma$ values were found to be $\ll 1$ . This indicates that the elastic energy storage is energetically negligible compared to viscous dissipation during the spreading phase.
Failure of Classical Numbers: The study shows that high $Wi$ and $De$ numbers do not predict the suppression of spreading. These numbers rely on microscopic relaxation dynamics and polymer contributions to viscosity, whereas the spreading process is governed by macroscopic energy dissipation where the bulk apparent viscosity dominates.
Shear Thinning vs. Elasticity: For concentrated, shear-thinning fluids, the observed increase in spreading (deviation above the Newtonian baseline) is attributed to a reduction in apparent viscosity due to shear thinning, not a lack of elastic resistance. As polymer concentration increases, the zero-shear viscosity ( $\mu_0$ ) and impact viscosity ( $\mu_i$ ) increase faster than the elastic modulus ( $G$ ), causing the ratio $G/\mu_i$ and consequently $\Gamma$ to decrease further.

Significance and Claims
The paper claims to resolve the long-standing paradox of why viscoelastic drops spread identically to Newtonian drops despite high elasticity. The authors argue that the "solid-like" resistance predicted by classical rheology is never realized during the spreading phase because the extreme deformation rates of impact cause viscous dissipation to overwhelmingly dominate the structural elastic capacity of the polymer network.

The significance of this work lies in:

Establishing an energetic boundary condition ( $\Gamma < 1$ ) that defines the "inelastic" nature of drop spreading, regardless of the fluid's relaxation time or elastic modulus.
Demonstrating that the Weissenberg and Deborah numbers are insufficient for predicting maximum spreading in drop impact scenarios.
Providing a physical explanation that elastic effects are energetically negligible during the spreading phase, while acknowledging that elasticity becomes dominant only during the subsequent retraction phase (where contact line dynamics and normal stresses play a critical role).

The authors conclude that for dilute and semi-dilute polymer solutions, the maximum spreading is governed primarily by inertia, capillarity, and viscous dissipation, with elasticity playing no measurable role in limiting the spread.