## 4 Conclusions  
This study investigates numerically the dynamic process of micrometer-sized droplets impacting horizontally moving micro-pillar surfaces. The dynamic behavior and key parameters are discussed based on the cases with different \(We_{n}\) and \(We_{t}\) values. This study enriches the understanding of the dynamic behavior of droplets impacting horizontally moving superhydrophobic surfaces. This will help to improve the control of wettability on horizontally moving microstructured surfaces. Future development of this research could focus on extending numerical models to investigate dynamic surface topology, coupled multiphysics field effects, and droplet dynamic behavior in extreme environments. The main conclusions are as follows:  
1. Under the operational conditions of this study on a stationary surface, unlike millimeter-sized droplets, micrometer-sized droplets cannot detach from the micro-pillar surface in either the Cassie or Wenzel states, achieving surface detachment only in the partial penetration state. Based on numerical simulation results, a dimensionless maximum spreading diameter formula for droplets impacting stationary micro-pillar surfaces is obtained, which can be expressed as \[D_{0}^{*}=(0.4\mathcal{I}We_{n})^{0.15}.\] (17)
2. In the process of droplets impacting horizontally moving micro-pillar structured surfaces, significant rolling phenomena occur when droplets are in the Cassie state, which leads to an increase in viscous dissipation and forms a competitive mechanism with the asymmetric spreading-retraction process of the droplets. This compromise effect influences the contact time and velocity restitution coefficient of the droplets. Moreover, the changes in contact time and velocity restitution coefficient exhibit distinct two-stage characteristics, providing new insights into understanding the dynamic behavior of droplets on microstructured surfaces.
3. Through systematic parameter studies, it can be concluded that when micrometer-sized droplets impact horizontally moving micro-pillar structured surfaces, the dimensionless maximum spreading diameter is linearly proportional to \(We_{t}\). Based on extensive numerical simulation data, a new correlation is fitted as \[D_{\max}^{*}=D_{0}^{*}+0.014\cdot We_{t}We_{n}^{-0.45}.\] (18) This model provides a theoretical basis for predicting the dynamic behavior of micrometer-sized droplets on microstructured surfaces.  
Conceptualization, H.Z. and M.W.; methodology, H.Z. and F.J.; software, H.Z.; validation, H.Z., F.J., and M.W.; formal analysis, F.J.; investigation, M.W.; resources, M.W.; data curation, F.J.; writing--original draft preparation, H.Z.; writing--review and editing, M.W.;visualization, H.Z. and F.J.; supervision, M.W.; project administration, M.W.; funding acquisition, M.W. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (Grant No. 52176080) and Natural Science Foundation of Shandong Province, China (Grant Nos. ZR2019 BEE012, ZR2022 ME003, and ZR2023 ME187).  
Not applicable.  
The data will be made available upon request.  
The authors declare no conflicts of interest.