Self-similar effective-to-apparent contact line length ratio in droplet friction on textured surfaces

The length of a three-phase contact line that effectively contributes to droplet pinning is one of the most important parameters in wetting. However, the definition, quantification, and utilization of the effective contact line remain unclear, thereby predicting the friction of droplets in the Cassie-Baxter state on superhydrophobic surfaces remains challenging and a fitting constant k is typically adopted. Here, we define a parameter termed effective-to-apparent contact line length ratio x, that is, the ratio of the effective contact line length that contributes to droplet pinning to the apparent length of the sliding contact line, to express the fitting constant k. By customizing a setup that simultaneously measures the droplet adhesion forces, contact line dynamics, and droplet shapes, the expression of contact line length ratio based on micropillar dimensions and surface intrinsic wettability is established. Then, a model predicting droplet friction is established and corroborated by the experimental data reported in this and prior studies. By simultaneously measuring the droplet adhesion force and contact line dynamics, this study introduces a parameter termed the effective-to-apparent contact line length ratio. This parameter leads to a model predicting droplet friction forces regardless of the variations in microstructure dimension, intrinsic wettability, and droplet volume.