  It is 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, 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.
