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An Effective Model for Droplet Impact Dynamics on Micro-Structured Surfaces: Nonlocal Theory and SPH Simulation of Pancake Bouncing

Zhonghua Qiao, Zuankai Wang, Yifan Wei

TL;DR

This work develops a rigorous nonlocal SPH framework to simulate 3D pancake bouncing of droplets on superhydrophobic microcone arrays. By deriving a theoretical link between cohesive interparticle forces and macroscopic surface tension, the authors define $A_\sigma = \frac{4\sigma}{\rho_0 \varepsilon}$ so that the cohesive forces generate a surface tension $\sigma$ without empirical tuning, and introduce a nonlocal boundary repulsion with a Wendland kernel to stabilize complex microstructured boundaries. The model is implemented in SPH and validated against experimental data across a range of Weber numbers, including horizontal and inclined impacts and Worthington jet–induced satellite droplets, demonstrating accurate shapes, timing, and pancake-like detachment. These results highlight the importance of nonlocal interactions in capturing microstructure–fluid coupling and offer a robust computational tool for designing anti-icing, self-cleaning, and microfluidic surfaces where precise droplet control is essential.

Abstract

The accurate mathematical modeling of droplet impact dynamics on micro-structured surfaces is fundamental to understanding and predicting complex fluid behaviors relevant to a wide range of engineering and scientific applications. In particular, the pancake bouncing phenomenon--systematically studied by Liu et al. (Nature Physics, 2014)--on superhydrophobic micro-structured substrates presents significant theoretical challenges. Central to these challenges is the need to construct effective mathematical models that capture the intricate influence of substrate micro/nanostructures on droplet dynamics. This requires the development of robust formulations for surface tension, contact line dynamics, and the interaction forces between fluid and solid structures. In this work, we formulate a nonlocal mathematical framework for the simulation of 3D pancake bouncing on superhydrophobic micro-cone arrays. The model incorporates intermolecular attractive forces to represent droplet surface tension, and we provide a strict theoretical derivation linking these forces quantitatively to the macroscopic surface tension coefficient, thereby circumventing the reliance on empirical parameter tuning. The complex geometry of micro-cone arrays introduces fundamental difficulties in defining local normal directions for contact algorithms. To overcome this, we develop a nonlocal contact repulsion force model that governs fluid-solid interactions and ensures numerical stability under high Weber number conditions. Based on this mathematical foundation, we implement the model using smoothed particle hydrodynamics (SPH), enabling high-precision 3D simulations. Computational experiments, validated against empirical data, confirm the model's accuracy and robustness, while underscoring the key role of numerical simulation in elucidating droplet-microstructure interactions.

An Effective Model for Droplet Impact Dynamics on Micro-Structured Surfaces: Nonlocal Theory and SPH Simulation of Pancake Bouncing

TL;DR

This work develops a rigorous nonlocal SPH framework to simulate 3D pancake bouncing of droplets on superhydrophobic microcone arrays. By deriving a theoretical link between cohesive interparticle forces and macroscopic surface tension, the authors define so that the cohesive forces generate a surface tension without empirical tuning, and introduce a nonlocal boundary repulsion with a Wendland kernel to stabilize complex microstructured boundaries. The model is implemented in SPH and validated against experimental data across a range of Weber numbers, including horizontal and inclined impacts and Worthington jet–induced satellite droplets, demonstrating accurate shapes, timing, and pancake-like detachment. These results highlight the importance of nonlocal interactions in capturing microstructure–fluid coupling and offer a robust computational tool for designing anti-icing, self-cleaning, and microfluidic surfaces where precise droplet control is essential.

Abstract

The accurate mathematical modeling of droplet impact dynamics on micro-structured surfaces is fundamental to understanding and predicting complex fluid behaviors relevant to a wide range of engineering and scientific applications. In particular, the pancake bouncing phenomenon--systematically studied by Liu et al. (Nature Physics, 2014)--on superhydrophobic micro-structured substrates presents significant theoretical challenges. Central to these challenges is the need to construct effective mathematical models that capture the intricate influence of substrate micro/nanostructures on droplet dynamics. This requires the development of robust formulations for surface tension, contact line dynamics, and the interaction forces between fluid and solid structures. In this work, we formulate a nonlocal mathematical framework for the simulation of 3D pancake bouncing on superhydrophobic micro-cone arrays. The model incorporates intermolecular attractive forces to represent droplet surface tension, and we provide a strict theoretical derivation linking these forces quantitatively to the macroscopic surface tension coefficient, thereby circumventing the reliance on empirical parameter tuning. The complex geometry of micro-cone arrays introduces fundamental difficulties in defining local normal directions for contact algorithms. To overcome this, we develop a nonlocal contact repulsion force model that governs fluid-solid interactions and ensures numerical stability under high Weber number conditions. Based on this mathematical foundation, we implement the model using smoothed particle hydrodynamics (SPH), enabling high-precision 3D simulations. Computational experiments, validated against empirical data, confirm the model's accuracy and robustness, while underscoring the key role of numerical simulation in elucidating droplet-microstructure interactions.

Paper Structure

This paper contains 21 sections, 2 theorems, 49 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Governing equations
  3. Modeling of droplet cohesion
  4. Modeling of substrate contact repulsion
  5. Governing equations with nonlocal interface and boundary interactions
  6. Numerical Methods
  7. Foundations of SPH Methodology
  8. SPH formulations
  9. Time integration
  10. Numerical Validation
  11. Geometric Configuration
  12. Computational Parameters
  13. Droplet Bouncing on Horizontal Microcone Arrays
  14. Case 1: Low-Weber-Number Impact ($\textit{We} = 7.1$)
  15. Case 2: Moderate-Weber-Number Impact ($\textit{We} = 14.1$)
  16. ...and 6 more sections

Key Result

Lemma 2.1

Assume $R_0>0$. Denote There holds Furthermore, if $f(\|\mathbf{x}\|)=\partial_{\|\mathbf{x}\|}W(\mathbf{x},R_0)$ and $W$ is any kernel function, there holds

Figures (10)

  • Figure 1: Droplet impact on microcone arrays: experimental observation (left; Copyright 2014 Springer Nature) and SPH simulation (right)
  • Figure 1: Schematic of cohesion at the gas-liquid contact surface
  • Figure 1: Comparison of simulated (bottom) and experimental (top; Copyright 2014 Springer Nature) droplet shapes during droplet bouncing at $\textit{We} = 7.1$. Time instances show initial impact at $t = 0$ ms, maximal spreading at $t = 4.8$ ms, and complete rebound at $t = 16.5$ ms.
  • Figure 2: Schematic diagram of a droplet with a substrate
  • Figure 2: Diameter and hight of droplet with $\textit{We} = 7.1$ and $\textit{We} = 14.1$
  • ...and 5 more figures

Theorems & Definitions (5)

  • Lemma 2.1
  • Theorem 2.2: Effective surface tension generation
  • Proof 1
  • Remark 2.3
  • Proof 2