Scheme for the passage from atomic to continuum theory for thin films, nanotubes and nanorods

We propose a scheme for the direct passage from atomic level to continuum level. The scheme is applicable to geometries, like films, rods and tubes, in which one or more dimensions are large relative to atomic scale but other dimensions may be of atomic scale. The atomic theory is assumed to be governed by a variational principle resting on the Born-Oppenheimer approximation. The atomic level energy is further assumed to satisfy certain decay properties when evaluated for disjoint sets of atoms. The scheme is based on two hypotheses: (1) distortions are limited, and (2) there are many atoms in certain directions. The scheme produces in a natural way the variables of the continuum theory. In the case of a film, the continuum theory that emerges is a Cosserat membrane theory with (v-1) Cosserat vectors, v being the number of atomic layers in the film. The arguments presented are not mathematically rigorous. One difficulty is that it is not clear under which circumstances our decay hypothesis on interatomic interactions is consistent with quantum mechanics or density functional theory.