## 3 Results and Discussion  
In order to explain this formation of wrinkles, many theories and experiments have been made over the past 10 years [14; 23; 24; 25; 26; 27; 28; 29]. The observed wavelength is explained by the balance between the bending energy \(U_{b}\propto\)\(Eh^{3}\frac{\Lambda^{2}}{\lambda^{3}}\) of the stiff upper membrane and the stretching energy \(U_{s}\propto\)\(E_{s}A^{2}\) of the \"semi-infinite\" foundation. We thus obtain \(\lambda\propto\)\(h\left(\frac{E}{E_{s}}\right)^{\frac{1}{3}}\), where \(E\) and \(E_{s}\) are respectively the Young modulus of the upper membrane and of the foundation. The amplitude of the wrinkles is obtained from the inextensibility constraint imposed to the stiff upper membrane: the length of the membrane stays constant during the compression process. This geometric constraint leads to the relation \(A\propto\)\(\lambda\)\(\sqrt{\delta}\). For the small deformation (\(\delta<0.3\%\)), we observed that there is a continuous increase of the amplitude \(A\) of the wrinkles and a continuous decrease of the wavelength \(\lambda\) (\(\lambda\propto\)\(\lambda_{0}\)\((1-\delta)\) ) (Figure 2(a1), (a2), c). For the large deformation (\(\delta>0.3\)), the situation is different. The profile of the thin elastic membrane is no longer described by a simple sinusoidal. A more complex pattern emerges and is characterized by two amplitudes \(A_{1}\) and \(A_{2}\) (Figure 2(a3), b, c). The evolution of the amplitudes of the pattern presents a bifurcation above some critical threshold: some wrinkles grow in amplitude while others decrease. The new pattern emerging is now characterized by a wavelength which is double the initial one. The amplitude of the \(2\lambda\) mode increases with the compression ratio \(\delta\), while the amplitude of \(\lambda\) mode vanishes progressively. Figure 1: (**a**) Schematics illustrating the steps for fabricating micro-wrinkling pattern on the poly-dimethylsiloxane (PDMS) substrate; (**b**,**c**) Scanning Electron Microscope (SEM) images of wrinkling pattern on a thin stiff PDMS film resting on a thick soft PDMS foundation and obtained with mechanical compression of 28% and 38% respectively. Superhydrophobic surfaces are determined by both chemical composition and a dual roughness at both micrometer and nanometer scales. The roughness of hydrophobic surfaces might lead to two states: Wenzel and Cassie-Baxter states. The Wenzel state [30; 31] describes a liquid in contact with the whole structured solid surface and therefore displaying low contact angle and high hysteresis. This state assumed that the liquid completely wets the grooves of the rough surface. The Wenzel state describes homogenous wetting by the relation:  
\[\cos\theta_{\rm w}=\cos\theta_{\rm e} \tag{2}\]  
where \(\theta_{\rm w}\) and \(\theta_{\rm e}\) are the Wenzel contact angle and the Young contact angle, respectively, and r is the roughness factor defined as the ratio of the true area of the solid-liquid to its projected area. The Cassie-Baxter state describes a liquid which is resting partly on the features of a solid material and bridging air between these features [32; 33]. The relation between the apparent contact angle \(\theta_{\rm c}\) and the equilibrium angle \(\theta_{\rm e}\) is described as:  
\[\cos\theta_{\rm c}=-1+\Phi_{\rm s}\left(1+\cos\theta_{\rm e}\right) \tag{3}\]  
where \(\Phi_{\rm s}\) is the area fraction of the projected wet area. We used the wrinkled topography of the surfaces, as shown before, to study the wettability of surfaces where the wavelength and the amplitude of wrinkles can be conveniently regulated by varying the time of treatment with UV/ozone, the module of elasticity and the applied strain level. By adjusting these parameters, it is possible to get surfaces with very large roughness. We measured the angle of 1 \(\mu\)L water drop on the molded surface in different cases. First, on the smooth, non-structured surface of PDMS, the equilibrium contact angle was \(\theta_{\rm e}=110^{\circ}\) (Figure 3a).