Application of relative configurational entropy as a measure of spatial inhomogeneity to Co/C film evolving along the temperature

Abstract

It is known that physical properties of a thin metallic film depend on spatial distribution of metal phase. Thermally treated thin film with metal particle clusters evolving along the temperature is a system complex enough system to attract scientific interest, see for example [1]. However, the simple visual study of a transformation process of heated thin metallic films on the basis of the transmission electron microscopy (TEM) micrographs cannot give valuable information. For instance, the important (in context of possible connection with effective properties of the whole structure) feature of a given microstructure is its spatial inhomogeneity depending by definition on the length scale considered. Thus, we need a tool for quantitative characterization of the spatial inhomogeneity. A simple mathematical approach [2] related to a point object distribution has been already used to analyze an inhomogeneity degree of TEMs of thin gold films [3] on the basis of Doremus’ micrographs [4] and polymer/carbon composites [5]. However, for digitized images with a pixel as natural unit of length this approach works only for the length scales commensurate with a side length of the image. Recently, a useful physical measure based on configurational entropy has been proposed [6] that overcomes this difficulty. The extension of this entropic measure for “finite-sized” objects [7] compared with the “normalized information entropy” [8] shows even more details for structures statistically self-similar. Among other entropic measures worked out to characterize random microstructures are the “local porosity entropy” [9] and the “configuration entropy” [10]. They are based on the adaptation of Shannon information entropy and were found to be rigorously connected [11]. In this letter we apply the simple in usage entropic measure for point objects [6]. A linear transformation f (S) of configurational entropy S with length scale dependent coefficients as a measure of spatial inhomogeneity has already been tested for computer generated pixel distributions. To check its behavior on real micrographs the Co/C film evolving along the temperature was chosen. The method used can be treated as one possible approach to finding a supposed connection between the changes in the structure of a thin film subject to heating conditions and its effective properties depending on temperature. To present the main idea of the method let us consider an electron micrograph of a thin metallic film. Its digitized binary image of size L × L can be treated as a set of χ = (L/k)2 lattice cells of size k× k in which n black pixels considered as point objects are distributed. These pixels represent small metal grains on a photographic negative. The pixel clusters certainly reflect less or more accurately (in dependence of the scanner resolution used) the two-dimensional area covered by a metallic phase. For each length scale k with a given object distribution (n1, . . . , ni , . . . , nχ ) one can associate a configurational entropy S= kB ln A, where the Boltzmann constant will be set to kB= 1 for convenience and A is the number of different ways of generating the fixed distribution of objects. To evaluate the spatial inhomogeneity as the average deviation from the most uniform object arrangement, defined by condition |ni − n j | ≤ 1 for each pair i 6= j , we use f (S)≡ (Smax− S)/χ . The highest possible value of configurational entropy Smax (at a given length scale) corresponds the most spatially uniform object configuration while S relates to the actual configuration. A more detailed description is given in the earlier paper [6]. Here the final formula is presented