Superhydrophobicity of biological surfaces has recently been studied intensively with the aim to design artificial surfaces. It has been revealed that nearly all of the superhydrophobic surfaces consist of the intrinsic hierarchical structures. However, the role of such structures has not been completely understood. In this study, different scales of hierarchical structures have been thermodynamically analyzed using a 2-D model. In particular, the free energy (FE) and free energy barrier (FEB) for the composite wetting states are calculated, and the effects of relative pillar height (hr) and relative pillar width (ar) on contact angle (CA) and contact angle hysteresis (CAH) have been investigated in detail. The results show that if the geometrical parameter ratio is the same (e.g., a:b:h = 2:2:1), the equilibrium CA for the composite of the three-, dual-, and single-scale roughness structures is 159.8°, 151.1°, and 138.6°, respectively. Furthermore, the nano-to microstructures of such surfaces can split a large FEB into many small ones and hence can decrease FEB; in particular, a hierarchical geometrical structure can lead to a hierarchical "FEB structure" (e.g., for a dual-scale roughness geometrical structure, there is also a dual-scale FEB structure). This is especially important for a droplet to overcome the large FEBs to reach a stable superhydrophobic state, which can lead to an improved self-cleaning property. Moreover, for extremely small droplets, the secondary or third structure (i.e., submicrostructure or nanostructure) can play a dominant role in resisting the droplets into troughs, so that a composite state can be always thermodynamically favorable for such a hierarchical structured system.
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