Rigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires

Abstract

In the context of nanowire heterostructures we perform a discrete to continuum limit of the corresponding free energy by means of Γ-convergence techniques. Nearest neighbours are identified by employing the notions of Voronoi diagrams and Delaunay triangulations. The scaling of the nanowire is done in such a way that we perform not only a continuum limit but a dimension reduction simultaneously. The main part of the proof is a discrete geometric rigidity result that we announced in an earlier work and show here in detail for a variety of three-dimensional lattices. We perform the passage from discrete to continuum twice: once for a system that compensates a lattice mismatch between two parts of the heterogeneous nanowire without defects and once for a system that creates dislocations. It turns out that we can verify the experimentally observed fact that the nanowires show dislocations when the radius of the specimen is large. 1

On the effect of interactions beyond nearest neighbours on non-convex lattice systems

“…Applying the transformation H, the fourteen bonds in B 1 are mapped into eight vectors of length √ 3 2 and six of length one; these correspond exactly to the nearest neighbour interactions in the body-centred cubic lattice, if the definition of the neighbours is based on a Delaunay triangulation, see [20] for details. The twelve bonds in B 2 are in bijection with vectors corresponding to the next-to-nearest neighbour interactions for that triangulation.…”

“…Key mathematical tools in its proof are the well-known rigidity estimate of Friesecke, James, and Müller [17] and the piecewise rigidity result proven by Chambolle, Giacomini, and Ponsiglione in [12]. The compactness result provided by Theorem 3.1 has applications also in problems of dimension reduction, for example it is used in [2] to prove scaling properties of energies in nanowires, in particular it allows to extend the results of [19,20] by removing the positive-determinant constraint.…”

Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours

“…The above sum is taken over a "thin" domain, i.e., a domain consisting of a few lines of atoms (for the precise formula see (1.4)); as the lattice distance converges to zero, we perform a discrete to continuum limit and a dimension reduction simultaneously. This model was first studied in [14,15] under the assumption that the admissible deformations satisfy the non-interpenetration condition, namely, that the Jacobian determinant of a suitably defined piecewise affine interpolation of u is positive. Here we remove such assumption and we show that, by incorporating into the energy the effect of interactions in a certain finite range, one can recover the results of [14,15] and get even further insight into the problem.…”

“…This model was first studied in [14,15] under the assumption that the admissible deformations satisfy the non-interpenetration condition, namely, that the Jacobian determinant of a suitably defined piecewise affine interpolation of u is positive. Here we remove such assumption and we show that, by incorporating into the energy the effect of interactions in a certain finite range, one can recover the results of [14,15] and get even further insight into the problem. More precisely, we obtain an effective energy that accounts for the effects of changes of orientation in the lattice.…”

“…We also point out that the effects of long range interactions in non-convex lattice systems have already been analysed in [7,5,6] in the one-dimensional case.For the scaling of (0.1), we obtain a complete description of the Γ-limit with respect to two different topologies (Theorems 5.1 and 5.4). It turns out that the Γ-limit with respect to the topology used in [14,15] is trivial (see Remark 4), that is, one can exhibit recovery sequences for which the gradient always lies in the same energy well up to an asymptotically vanishing correction. In order to see the effects of changes of orientation in the nanowire, we introduce a stronger topology which is sensitive to them.…”