We firmly proposed that the macroscopic contact angle (MCA) of any point on a general substrate can be described by a continuous function with respect to the coordinate. Based upon variational theory of the total potential functional dealing with the movable boundary condition, we proved that the MCA equation is actually the transversality condition. It was found that the contact angle depends only on the chemical and geometric property at the triple contact line (TCL), and is not affected by the gravity of and contact area beneath droplet. The analysis results on several typical substrates with special hydrophilicities and geometric topologies show that our mechanism-based model and the classical Wenzel & Cassie Models are fundamentally different. In use of this idea, we explored the pinning effect on a sharp wedge or the interface between two different phases. Our study revisits the fundamentals of wetting on rough and heterogeneous substrates, which will help designing super-hydrophobic materials and provide the prediction required to engineer novel micro-fluidic devices.
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