Coarsening process in one-dimensional surface growth models

Surface growth models may give rise to unstable growth with mound formation whose tipical linear size L increases in time (coarsening process). In one dimension coarsening is generally driven by an attractive interaction between domain walls or kinks. This picture applies to growth models where the largest surface slope remains constant in time (corresponding to model B of dynamics): coarsening is known to be logarithmic in the absence of noise (L(t) ∼ ln t) and to follow a power law (L(t) ∼ t) in the presence of it. If surface slope increases indefinitely, the deterministic equation looks like a modified Cahn-Hilliard equation whose late stages of coarsening we study here through a linear stability analysis of the stationary periodic configurations and through a direct numerical integration. Analytical and numerical results agree well in the conclusion that steepening of mounds makes deterministic coarsening faster : if α is the exponent describing the steepening of the maximal slope M of mounds (M ∼ L) we find that L(t) ∼ t: n is equal to 1 4 for 1 ≤ α ≤ 2 and it decreases from 1 4 to 1 5 for α ≥ 2, according to n = α/(5α − 2). Contrastingly, the numerical solution of the corresponding stochastic equation clearly shows that in the presence of shot noise steepening of mounds makes coarsening slower than in model B: L(t) ∼ t, irrespectively of α. Finally, the presence of a symmetry breaking term is shown not to modify the coarsening law of model α = 1, both in the absence and in the presence of noise. PACS. 68. Surfaces and interfaces – 81.10.Aa Theory and models of film growth – 02.30.Jr Partial differential equations