New rigorous description of diffusion and reactions on arbitrary (grain boundary) networks by a 4th rank tensor theory

A new approach to describe rigorously diffusion and diffusion-related reactions on arbitrary networks consisting, for example, of grain boundaries in coarse materials. The method is based on density function approach to diffusion and on an adjacency matrix concept used in graph theory extended here to a 4th rank tensor algebra to describe arbitrary consecutive diffusional jumps on a 2D lattice. The application of the method to various network geometries reveals among other things a new phenomenon: In contrast to homogeneous medium, diffusion on microscopic networks leads to a concentration build-up at the network intersections instead of smoothing out. In columnar grain films, this phenomenon constitutes a new kind of segregation mechanism. Several applications relevant to semiconductor processes are rigorously evaluated and offer insight into their true nature for the first time.<>