Dielectric breakdown in solids modeled by DBM and DLA

Using numerical simulation, two stochastic models of electrical treeing in solid dielectrics are compared. These are the diffusion-limited aggregation (DLA) model and the dielectric breakdown model (DBM or η-model). On a linear two-dimensional geometry, the relationship between both models, when the size of the structures is of the order of the experimental samples (the electrode gap is 100 times the length of the discharge channel), is explored by statistical methods. Although there is a one-to-one correspondence between DBM with η=1 and the DLA model when the structure size is very large, the case of rather smaller structures is not well known. From a fractal analysis, employing the method of the correlation function C(r), it follows that average fractal dimension of electrical trees, generated with the DLA or with the DBM (η=1), collapse (up to the numerical uncertainty), on a single curve that “universally” accounts for finite size effects. Even more, from this analysis we conclude that the two curves obtained for DLA and DBM (η=1) cannot be distinguished if one takes into account the error bars. This means that finite size effects in the fractal analysis of DLA and DBM (η=1) are quite the same (despite the differences in the algorithms respectively used to generate the electrical trees). To our knowledge no comparison has ever been made between the similarities and differences of the DBM and DLA approach on a geometry other than the open-planar geometry.