Table 1 A summary of fundamental wettability equations.

Young theory⁷

\({\gamma }_{SG}={\gamma }_{SL}+{\gamma }_{LG}\cdot \text{cos}\theta\) (1)

γ denotes the surface tension coefficient. The subscripts SG, SL, and LG stand for solid-gas, solid-liquid, and liquid–gas interfaces, respectively

The equilibrium contact angle, θ, corresponds to the minimal energy state among the three phases.

For ideally smooth and homogeneous surfaces

This theoretical relationship can’t measure the contact angle of rough surfaces

Wenzel’s theory^6,10

\(\text{cos}\left({\theta }_{A}\right)=r\cdot \text{cos}\theta\) (2)

θ_A is an apparent contact angle and θ is the angle corresponding to the ideal surface. r is the ratio of the real rough surface area to the projected ideally smooth surface

For rough surfaces

This relationship is not suitable for heterogeneous and non-uniform rough surfaces

Cassie and Baxter’s theory^8,9

90° < θ < 180°

\(\text{cos}\left({\theta }_{A}\right)={{\Phi }}_{1}\cdot \text{cos}\left({\theta }_{1}\right)+{{\Phi }}_{2}\cdot \text{cos}\left({\theta }_{2}\right)\) (3)

θ₂ = 180

\(\text{cos}\left({\theta }_{A}\right)={{\Phi }}_{LS}\cdot \left[\text{cos}\left(\theta \right)+1\right]-1\) (4)

Φ₁ specifies the fraction of interface length and θ₁ stands for the contact angle for the first component. Φ₂ and θ₂ denote the associated values for the second component

Modeling the surface roughness effects

Defines the apparent contact angle based on the fraction of interface length in the different components

This relationship is not suitable for heterogeneous and non-uniform rough surfaces