A 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 l evel. The scheme is applicable to geometries, like films, rods and tubes, i n 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 gove rned by a variational principle resting on the Born-Oppenheimer approximati on. The atomic level energy is further assumed to satisfy certain decay pro perties when evaluated for disjoint sets of atoms. The scheme is based on t wo 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 emer ges is a Cosserat membrane theory with (v - 1) Cosserat vectors, v being th e number of atomic layers in the film. The arguments presented are not math ematically rigorous. One difficulty is that it is not clear under which cir cumstances our decay hypothesis on interatomic interactions is consistent w ith quantum mechanics or density functional theory. (C) 2000 Elsevier Scien ce Ltd. All rights reserved.