Fractal representation of the Debye theory for studying the heat capacity of macro-and nanostructures

Abstract

The traditional theory of Debye heat capacity with a single free parameter (characteristic temperature θD) is extended to fractal spaces taking into account two more “latent” parameters contained in it, viz., the phonon spectrum dimension df and dimension d determining the geometry of the skeleton of the structure under investigation. In the classical version of the Debye theory, df = d = 3. In the case under investigation, these parameters can assume arbitrary (including fractional) values, which is typical of materials such as polymers, colloid aggregates, and various porous structures and nanostructures, as well as materials with a complex chemical composition. The application of a fractal approach makes it possible to substantially extend the class of materials with a heat capacity described by the continual Debye approximation.