Coarsening process in one-dimensional surface growth models |
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Authors: | Alessandro Torcini Paolo Politi |
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Institution: | (1) Dipartimento di Energetica “S. Stecco", Università di Firenze, Via S. Marta 3, 50139 Firenze, Italy, IT;(2) Istituto Nazionale per la Fisica della Materia, UdR Firenze, Via G. Sansone 1, 50019 Sesto Fiorentino, Italy, IT |
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Abstract: | Surface growth models may give rise to instabilities with mound formation whose typical linear size L increases with time (coarsening process). In one dimensional systems coarsening is generally driven by an attractive interaction
between domain walls or kinks. This picture applies to growth models for which 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
1/3) when noise is present. If the surface slope increases indefinitely, the deterministic equation looks like a modified Cahn-Hilliard
equation: here we study the late stages of coarsening through a linear stability analysis of the stationary periodic configurations
and through a direct numerical integration. Analytical and numerical results agree with regard to 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
α∼L) we find that L(t) ∼t
n: n is equal to for 1≤α≤2 and it decreases from to for α≥2, according to n = α/(5α - 2). On the other side, 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
1/4, 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.
Received 28 September 2001 and Received in final form 21 November 2001 |
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Keywords: | PACS 68 Surfaces and interfaces – 81 10 Aa Theory and models of film growth – 02 30 Jr Partial differential equations |
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