We present a phase field approach to wetting problems, related to
the minimization of capillary energy. We discuss in detail both
the Γ-convergence results on which our numerical algorithm
are based, and numerical implementation. Two possible choices of
boundary conditions, needed to recover Young's law for the contact
angle, are presented. We also consider an extension of the
classical theory of capillarity, in which the introduction of a
dissipation mechanism can explain and predict the hysteresis of
the contact angle. We illustrate the performance of the model by
reproducing numerically a broad spectrum of experimental results:
advancing and receding drops, drops on inclined planes and
superhydrophobic surfaces.