Random-field Ising model for the hysteresis of the prewetting transition on a disordered substrate

We study the effect of substrate randomness on the dynamics of a prewetting transition. The prewetting system is mapped to a 2D-random-field Ising model (RFIM) in an external field H, in which the Ising spins ±1 are identified with a wet or nonwet patch of surface. The model is studied in a cylindrical geometry at zero temperature by simulations with a single-spin-flip rule for the dynamics. The total coverage C and the coverage Cb connected to an initially flipped border are both considered, which correspond to physical quantities actually measured in experiments. Ramping the external field H up and down produces smooth and hysteretic curves for C and Cb. The covering of the substrate appears to be a critical transition for Cb occurring at a nonzero critical field Hc. While the growth exponents are compatible with those of percolation, the thresholds Hc depend in a complex way on the disorder strength. Analytical expressions are proposed for Hc in the weak and strong disorder limits, corresponding to two different growth mechanisms.