![]() This implies that no current flows across the boundary C, i.e., n The divergence of the last Maxwell equation shows that the current density is solenoidal, i.e., ∇ The remaining boundary of the metal is denoted C = ∂ M \ ( A ∪ B ), see Fig. The subset M ⊂ W is the space occupied by metal, its left boundary is denoted A and its right boundary is B. Thus the electric field E has a potential u, i.e., E = ∇ u. In the present case all time dependent terms can be dropped. Where E is the electric field, ρ is the charge density, B is the magnetic induction and j is the current density. In a composite it is even possible to obtain properties which never can be observed in one-phase materials. The latter is called microstructure and is exactly what makes composites so attractive: properties can be combined and tuned by altering the microstructure. They are determined by three factors: the properties of the phases, the properties of the interfaces, and the spatial arrangement of the phases. The properties observed at the macroscopic scale are called effective properties. 4 Therein a practical procedure for determining the effective properties of a composite with microstructure modeled by a random field is given. A rigorous study of the approximations adopted by this approach was given by Sab. This idea is closely related to the definition of a so-called representative volume element of a composite. This means that there is a length scale such that all samples of the composite larger than a square of this length have statistically the same properties. ![]() On the macroscopic scale such composites behave as if they were one-phase materials. ![]() The heterogeneous nature becomes apparent in the microscopic range. Typically the different phases are finely dispersed but the regions occupied by one phase are still large compared to the atomic length scale. We conclude that the backbone will be useful as a first ingredient for a geometric estimator of the effective conductivity of metal-insulator composites.Ĭomposite materials consist of a mixture of at least two immiscible phases. It is found that the backbone filter simplifies the geometry of complex microstructures significantly and at the same time preserves their electrical DC behavior. The change of both area fraction and effective conductivity induced by applying the backbone filter to various binary images and a two-parameter family of sets is assessed by numerical means. It is based on a sequence of image analysis operations defining the backbone in terms of an image filter. To expand the applicability of the concept, we present a purely geometric definition for the backbone of a two-dimensional percolating cluster. For structures with high degree of spatial correlation, as they are typical for porous thin films, trimming of the full structure to reveal the part determining the electrical conductivity is more subtle than the classic definition of the backbone. In percolation theory, the backbone is defined by chopping off dangling ends from the percolating cluster.
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