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Universal mapping of drop impact spreading from wetting to Leidenfrost regimes

This paper establishes a universal energy-dissipation-based mapping that accurately predicts the maximum spreading of droplets across both wetting and Leidenfrost regimes over a wide range of Ohnesorge and Weber numbers, enabling the conversion of wetting impact predictions to their Leidenfrost counterparts.

Original authors: Shushan Hu, Nan Hu, Zirui Li, Yifei Sun, Xiang Gao, Liwu Fan

Published 2026-08-31

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CC BY 4.0

Original authors: Shushan Hu, Nan Hu, Zirui Li, Yifei Sun, Xiang Gao, Liwu Fan

Original paper licensed under CC BY 4.0 (https://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

When a drop of liquid hits a surface, it does not simply stop; it flattens, spreads out, and then often recoils. This everyday event, from rain hitting a window to ink striking paper, is governed by a tug-of-war between the drop's forward momentum, the stickiness of its own surface, and the internal friction of the liquid. Scientists have long understood how this plays out when the liquid touches the surface directly, a state known as wetting. However, a different and more dramatic scenario occurs when the surface is extremely hot. In this case, the liquid does not touch the solid at all. Instead, it levitates on a thin, invisible cushion of its own vapor, a phenomenon known as the Leidenfrost effect. This state is crucial for technologies ranging from cooling systems to fuel injection, yet predicting exactly how far a drop will spread in this floating state has remained a difficult puzzle, largely because the physics of a floating drop differs significantly from one that is stuck to the ground.

Researchers at Zhejiang University and Princeton University have now connected these two worlds. By combining high-speed experiments with detailed computer simulations, they established a universal rule that translates the spreading behavior of a wetting drop into the behavior of a Leidenfrost drop. Their work reveals that the key difference between the two states lies not in the surface tension or the speed of the drop, but in how the liquid loses energy. When a drop hits a wet surface, the friction against the wall acts like a brake, slowing the spread and dissipating energy near the bottom. When the drop is floating on vapor, that wall friction disappears, allowing the drop to spread much farther. The team found that this extra distance is not random; it is directly linked to the amount of energy saved by avoiding that wall friction.

To uncover this connection, the scientists watched millimeter-sized drops of water and glycerol mixtures strike a polished sapphire disk. In some trials, the disk was at room temperature, allowing the drops to wet the surface. In others, the disk was heated to 550 degrees Celsius, creating the vapor cushion. They recorded the impacts with cameras capturing 4,000 frames per second, tracking exactly how wide the drops became before they stopped expanding. They paired these observations with computer models that could calculate the invisible flow of energy inside the drop. The simulations showed that while the floating drops indeed avoided the heavy braking of the wall, the liquid inside them began to churn more vigorously, creating a new kind of internal friction that partially offset the gain. Despite this internal compensation, the total energy saved by the floating drop was substantial and predictable.

The researchers discovered that this energy saving follows a precise pattern across a vast range of conditions. They tested drops with different thicknesses and speeds, covering a span where the liquid's resistance to flow varied by a factor of one hundred and the impact force varied by ten times. In every case, the difference in how far the drop spread between the wet and floating states could be calculated simply by looking at how much energy was lost to friction in the wet case. This finding allows scientists to take any existing prediction for a wet drop and instantly convert it to an accurate prediction for a floating drop, without needing to build a new, complex model from scratch.

This mapping holds true even when the drops are very viscous or moving very fast, provided they do not shatter upon impact. The study confirms that the "one-half rule," a previous idea suggesting that floating drops always convert half their energy into surface area, is only accurate for very fast, thin drops. For thicker or slower drops, the internal friction becomes significant, and the new mapping accounts for this by showing exactly how much less the drop spreads than the idealized limit. The work provides a unified way to understand droplet impact, bridging the gap between the familiar splash of a wet surface and the mysterious glide of a Leidenfrost drop, offering a clearer path to controlling these impacts in industrial and thermal applications.

Technical Summary: Universal Mapping of Drop Impact Spreading from Wetting to Leidenfrost Regimes

Problem Statement
The impact of a liquid drop onto a substrate is a canonical fluid dynamics problem governed by the interplay of inertia, capillarity, and viscous dissipation. While the maximum spreading factor ( $\beta_m = D_m/D_0$ ) for wetting impacts (where direct liquid-solid contact occurs) is well-characterized across broad ranges of Weber ( $We$ ) and Ohnesorge ( $Oh$ ) numbers, the Leidenfrost (LF) regime (where a stable vapor film levitates the drop) lacks a similarly universal predictive framework. Existing LF models often focus on specific liquids or limiting regimes. Notably, the widely used "1/2 rule" (Wildeman et al.) assumes inviscid flow and neglects bulk viscous dissipation, leading to systematic overpredictions of spreading when $Oh \gtrsim 0.05$ . A universal mapping that could convert established wetting predictions into their Leidenfrost counterparts has not yet been established.

Methodology
The authors establish a universal mapping based on an energy-dissipation perspective, relating the maximum spreading in the LF regime to that in the wetting regime through the difference in total viscous dissipation.

Experimental and Numerical Approach: The study combines high-speed imaging experiments with direct numerical simulations (DNS) using the open-source volume-of-ﬂuid solver Basilisk.
- Experiments: Millimetric drops of water and aqueous glycerol solutions were impacted onto a sapphire substrate under unheated (wetting) and heated ( $550^\circ$ C, LF) conditions.
- Simulations: Axisymmetric two-phase simulations were performed. The wetting case utilized a no-slip boundary condition, while the LF case employed a shear-free boundary condition to represent the suppression of near-wall shear by the vapor film.
Energy Budget Analysis: The study decomposes the total viscous dissipation ( $E_d$ ) into near-wall boundary layer dissipation ( $E_{BL}^d$ ) and bulk dissipation ( $E_{bulk}^d$ ). By comparing the energy balances of wetting and LF impacts under identical initial conditions, the authors isolate the reduction in total viscous dissipation ( $\Delta E_d = E_{d,w} - E_{d,LF}$ ) available for additional surface expansion in the LF regime.
Scaling Derivation: The authors derive a scaling law for the dissipation difference, showing that $\Delta E_d$ scales with the wetting boundary-layer contribution ( $E_{BL}^d \sim We^{-1/4}Oh^{1/2}\beta_{m,w}^2(\beta_{m,w}-1)^{1/2}$ ). This leads to a direct mapping equation relating $\beta_{m,LF}$ to $\beta_{m,w}$ .

Key Contributions and Results

Universal Mapping Equation: The paper derives a direct mapping (Eq. 4) that predicts the maximum Leidenfrost spreading factor from any prescribed wetting spreading law:
$\beta_{m,LF} = \beta_{m,w} \left[ 1 + \frac{7}{60} We^{3/4} Oh^{1/2} (\beta_{m,w} - 1)^{1/2} \right]^{1/2}$
This framework allows the conversion of established wetting models (e.g., those by Sanjay and Lohse, Liu et al., or Abbot and Bonn) into LF predictions without constructing a separate, independent LF model.
Validation: The mapping is validated against experimental data and simulations spanning $100 \le We \le 10^3$ and $10^{-3} \le Oh \le 0.2$ . The results cover two to three orders of magnitude in $We$ and two orders of magnitude in $Oh$ , incorporating data for water, glycerol-water mixtures, ethanol, and fluorinated liquids.
Physical Insight: The study demonstrates that while the vapor film in the LF regime strongly suppresses near-wall dissipation, bulk dissipation increases and partially compensates for this loss. However, the net reduction in total dissipation remains comparable to the wetting boundary-layer contribution, which sets the characteristic scale for the mapping.
Performance: The mapping successfully captures the enhanced spreading in the LF regime and the progressive deviation from the inviscid "1/2 rule" as viscosity increases. It remains robust even at low $We$ , where the correction term naturally vanishes as $\beta_{m,w} \to 1$ .

Significance and Claims
The authors claim to have established a unified energetic connection between wetting and Leidenfrost drop impacts. By demonstrating that the difference in total viscous dissipation between the two regimes is directly reflected in the difference in spreading factors, the paper provides a simple route to extend the extensive theoretical framework of wetting impacts to the Leidenfrost regime.

The work is presented as a practical tool for predicting and controlling droplet impact in applications such as thermal management, fuel injection, self-propulsion, and energy harvesting. The authors note that the framework is valid within the investigated parameter space; limitations may arise at extreme $We$ (due to splashing or vapor film destabilization) or very low $We$ (where weak deformation challenges geometric approximations), but these effects are shown to have minimal influence on the maximum spreading within the studied range. The mapping also offers a method to estimate the total dissipation difference from paired wetting and LF measurements, a quantity that is otherwise difficult to access directly.

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