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 C-b 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 C-b. The covering of the substrate appears to be a critical transition for C-b occurring at a nonzero critical field H-c. While the growth exponents are compatible with those of percolation, the thresholds H-c depend in a complex way on the disorder strength. Analytical expressions are proposed for H-c in the weak and strong disorder limits, corresponding to two different growth mechanisms.