A scheme for the passage from atomic to continuum theory for thin films, nanotubes and nanorods
Introduction
The ability to reproducibly synthesize structures having an atomic scale dimension, and the subsequent investigation of the unusual properties and possible applications of these nanoscale structures, has given rise to the field of nanotechnology. Properties that are known to be structure-sensitive on the macroscale, such as strength, plasticity, magnetic hysteresis, superconductivity, and properties altered by phase transformation are expected to exhibit unusual behavior on scales that are smaller than the typical microstructural feature that is responsible for the sensitivity.
When all of the dimensions of a nanoscale object are small, it is possible to use purely atomic scale computational methods to understand its behavior. But when at least one dimension of the object is much larger than atomic scale, purely atomistic methods become time consuming, and they also suffer from the drawback of offering only one-configuration one-computation; that is, they sometimes do not give insight into the variety of possible behaviors of the structure.
Lacking a suitable alternative, the current practice is to transport macroscale continuum theories down to the nanoscale. For example, the motion of nanoscale cantilevers is often analyzed using macroscale beam theory (e.g., Wong et al., 1997). The moment–curvature relation in these beam theories is given as the product EI of Young’s modulus and moment of inertia. For a nanotube or nanorod, it is not clear what one should use for the moment of inertia I of the cross-section. One could imagine that it represents a moment of inertia of a kind of cross-sectional envelope of the atoms. But cuts at different locations along the rod give very different sets of atoms. Furthermore, the origins of the term EI — the comparison of a solution in Euler beam theory with a similar solution of linear elasticity — indicate that the presence of this product is associated with a linear variation of stress across the cross-section, a situation that is difficult to imagine on the nanoscale. More than that, even the concept of stress (or, in electromagnetism, the concept of the electromagnetic field in a polarizable or magnetizable medium) is essentially macroscopic and may not be a reasonable concept to use at the nanoscale.
The existence of a nanoscale dimension is in the authors’ opinion not a reason by itself to reject continuum theory. Roughly, continuum theories should arise when there are one or more dimensions much larger than atomic scale and there is some macroscopic homogeneity. In statics, the problem of deriving continuum theory is essentially a problem of partitioning the configurational variables into two sets: one set is treated by atomic methods and the other set furnishes the variables of the continuum theory. Along the way, a successful method of partitioning achieves simplification. The opportunity for simplification arises from the fact that for an extended film large classes of nanoscale configurations have nearly the same energy, as compared to a typical change of energy of the whole film.
We propose the following method for deriving continuum theory from atomic theory. Consider, for example, a film having ν atomic layers and lateral dimensions k×k, where k is much larger than atomic scale. First, to limit the number of possible states, we assume that the distortion of the film is bounded; that is, we assume that the affine deformation experienced by every triple of atoms has a bounded gradient. This immediately implies that there exist some well-defined macroscale averages. Then the method is described as follows:
- 1.
Calculate all averages of atomic positions that are well-defined in the limit k→∞. In mathematical terms, for each fixed k rescale the deformation to a fixed domain, and calculate all weak limits of the sequence of rescaled deformations. Use these averages as the variables of continuum theory.
- 2.
Find a shrinking neighborhood of these averages having the property that, if a sequence of deformations lies in the neighborhood for each k, then they converge in the sense found above to these averages.
- 3.
Minimize the atomic degrees of freedom in the neighborhood for each k.
- 4.
Calculate the energy per unit area, pass to the limit k→∞ and extract a continuum theory.
To develop this procedure, there remains a lot of work to do. We only carry out the argument at a formal level for the thin film case, and indicate briefly how it works for nanotubes and rods. Second, as presented and expected, the scheme only delivers the membrane energy. The bending energy occurs at higher order in 1/k. One could think of applying a similar procedure to the difference between the original energy and the membrane energy, multiplied by a suitable power of k (k3 is expected classically), but we have no results in this direction. Third, the method is explicitly static. Lattice vibrations could be added by superimposing the formulas of statistical mechanics on the atomic scale relaxation as discussed by many authors, but full-scale dynamic motions are not treated by this method. Fourth, the continuum theory that emerges needs to be further simplified to be widely applicable. For example, the linearized theory needs to be explored, and any possible separation between the effects of geometry and atomic forces (such as embodied in expressions like EI) should be exposed.
Section snippets
Kinematics
Consider a single crystal film having ν monolayers. For simplicity we picture the nuclei as occupying a reference configuration on a Bravais latticeand a film of thickness and planar section , so that the nuclei in the reference configuration of the film occupy the positions,The theory is intended to apply to films that have been released from the substrate, or else to films attached to substrates that are also very thin. In other words,
Weak neighborhoods
Having identified and the functions as the basic functions of the continuum theory, we now face the task of designing a suitable shrinking neighborhood of these functions, which we denote by . The subscript ε refers to the a mesoscopic length scale ε=εk. All deformations inside the neighborhood should be allowed to compete in the atomic scale relaxation. Thus, the problem of energy minimization is reduced to a problem of, first, minimizing over deformations in the
Interpretation of the weak neighborhood
Since and therefore are entirely determined by the nuclear positions, any condition on these deformations is interpretable in terms of nuclear positions. In particular, it is interesting to interpret the weak neighborhood (17) in this way. We begin with Eq. (13), and rewrite this condition in terms of by using Eq. (6):Hence, the actual atomic positions are uniformly close to a scaled family of larger and larger surfaces of the form , related by
Passage to continuum theory
The atomic level theory shall rest on the assumption that the energy is expressible as a function of the nuclear positions. This is the fundamental setting delivered by quantum mechanics in the Born–Oppenheimer approximation after minimization over electronic states. (More precisely, the energy depends on the nuclear positions and their charges, i.e. the atomic species involved, but for the sake of simplicity we focus on a single species here.) In the present framework, in original and rescaled
Comments on the implied atomic scale calculation
Now we discuss briefly the atomic scale calculation implied by the right-hand side of Eq. (37). We work in original variables, this being the natural setting for atomic scale. Technically this calculation takes place near a deformed triangle but the shape of the set likely does not matter with suitable decay properties of E. First we note that if E is rotation and translation invariant, i.e.,then ϕ is similarly invariant:
Nanotubes
We make some brief remarks about the application of the scheme to other geometries. It can be seen from the above that the exact nature of the independent variables of the continuum theory will depend on the atomic structure of the object from which the condition of limited distortion is imposed. For definiteness we focus on a particular geometry. The most well-studied nanotube is certainly the carbon nanotube (Iijima, 1991), whose structure has been carefully elucidated by Amelinckx et al. (
Discussion
We have given a scheme for the direct passage from atomic to continuum theory applicable to cases in which one or more dimensions of the body are large. Of course, the scheme could be applied to the case in which all dimensions are large, but in this paper we discussed only the lower dimensional cases which have more interesting kinematics. The scheme is very much influenced by geometry, and the small parameter that leads to simplification is taken to be the ratio of length scales.
One could
Acknowledgements
This work was supported by AFOSR/MURI F49620-98-1-0433. The research also benefitted from the support of ARO DA/DAAG55-98-1-0335, NSF DMS-9505077 and ONR/DARPA N00014-95-1-1145 and -91-J-4034.
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