Diffusion model for the treeing process of electrodeposition experiments

We simulate a model that captures all the features of the silver electrodeposition experiment in a rectangular cell. We study the bulk of the aggregates on the basis of a treeing process (M. Matsushita, P. Meakin, Cluster size distribution of self-affine fractals, Phys. Rev. A 37 (1988) 3645; F. Romá, C.M. Horowitz, E.V. Albano, Numerical study of the development of bulk scale-free structures upon growth of self-affine aggregates, Phys. Rev. E 66 (2002) 066115). The model proposed is a diffusion limited process in 1+1 dimension, where one dimension is the linear size L and the other the height. In our model the particles are dropped from the top of a rectangular lattice and are allowed to diffuse. The diffusion upwards is forbidden, whereas in the other directions the particles are allowed to diffuse with probability 1-p to the lateral nearest neighbors positions and with probability p downwards. Here p takes into account the strength of the electric field. When a newly deposit particle has a nearest neighbor which belongs to only one tree, it sticks to that tree. If the particle has more than one nearest neighbor that belongs to different trees one of them is selected at random and the particle sticks to the chosen tree. We compute the r.m.s height hs, the r.m.s width ws and the size distribution of the trees Ns as function of the mass s of the “frozen” trees for different values of p. We found that the scaling behavior with s of hs and Ns depends on p, while ws does not depend on p. In the limit p→1, the values obtained for the exponents, that characterize the scaling behavior of the magnitudes studied here, are close between the error bars, with the one found in the experiment of silver electrodeposition (C.M. Horowitz, M.A. Pasquale, E.V. Albano, A.J. Arvia, Experimental evidence of the development of scale invariance in the internal structure of self-affine aggregates, Phys. Rev. B 70 (2004) 033406 ; E.V. Albano, R.C. Salvarezza, L.Vázquez, A.J. Arvia, Validity of the Kardar–Parisi–Zhang equation in the asymptotic limit of metal electrodeposition, Phys. Rev. B 59 (1999) 7354), that was suggested to belong to the Kardar–Parisi–Zhang (KPZ). For finite p, the results suggest that our model may belong to another universality class. We also studied the finite size effect in the values obtained for the exponents and found that the p dependence is not due to finite size effects.