It is well established that roughness and chemistry play a crucial role in the wetting properties of a substrate. Yet, few studies have analyzed systematically the effect of the non-uniformity in the distribution of texture and surface tension of substrates on its wetting properties. In this work we investigate this issue theoretically and numerically. We propose a continuous model that takes into account the total energy required to create interfaces of a droplet in two possible wetting states: Cassie-Baxter (CB) with air pockets trapped underneath the droplet; and the other characterized by the homogeneous wetting of the surface, called the Wenzel (W) state. To introduce geometrical non-regularity we suppose that pillar heights and pillar distances are Gaussian distributed instead of having a constant value. Similarly, we suppose a heterogeneous distribution of Young's angle on the surface to take into account the chemical non-uniformity. This allows to vary the "amount" of disorder by changing the variance of the distribution. We first solve this model analytically and then we also propose a numerical version of it, which can be applied to study any type of disorder. In both versions, we employ the same physical idea: the energies of both states are minimized to predict the thermodynamic wetting state of the droplet for a given volume and surface texture. We find that the main effect of disorder is to induce the stability of both wetting states on the same substrate. In terms of the influence of the disorder on the contact angle of the droplet, we find that it is negligible for the chemical disorder and for pillar-distance disorder. However, in the case of pillar-height disorder, it is observed that the average contact angle of the droplet increases with the amount of disorder. We end the paper investigating how the region of stability of both wetting states behaves when the droplet volume changes.

The relation between wetting properties and geometric parameters of fractal surfaces are widely discussed on the literature and, however, there are still divergences on this topic. Here we propose a simple theoretical model to describe the wetting properties of a droplet of water placed on a hierarchical structured surface and test the predictions of the model and the dependence of the droplet wetting state on the initial conditions using simulation of the 3-spin Potts model. We show that increasing the auto-similarity level of the hierarchy – called n – does not affect considerably the stable wetting state of the droplet but increases its contact angle. Simulations also explicit the existence of metastable states on this type of surfaces and shows that, when n increases, the metastability becomes more pronounced. Finally we show that the fractal dimension of the surface is not a good predictor of the contact angle of the droplet.