# American Institute of Mathematical Sciences

March  2018, 13(1): 1-26. doi: 10.3934/nhm.2018001

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

 1 Università di Cassino e del Lazio meridionale, Dipartimento di Ingegneria Elettrica e dell’Informazione, via Di Biasio 43, Cassino (FR), 03043, Italy 2 Università di Napoli Federico Ⅱ, Dipartimento di Matematica e Applicazioni, via Cintia, Monte S. Angelo, Napoli, 80126, Italy 3 University of Sussex, Department of Mathematics, Pevensey 2 Building, Falmer Campus, Brighton, BN1 9QH, United Kingdom

Received  February 2017 Revised  November 2017 Published  March 2018

Fund Project: RA and GL received funding from the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). When this research was carried out, GL was affiliated to SISSA and supported by the European Research Council under the Grant No. 290888 "Quasistatic and Dynamic Evolution Problems in Plasticity and Fracture"; he was then affiliated to the University of Vienna and supported by the Austrian Science Fund (FWF) project P27052. MP was partially supported by the EU Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie project ModCompShock agreement No. 642768.

We study continuum limits of discrete models for (possibly heterogeneous) nanowires. The lattice energy includes at least nearest and next-to-nearest neighbour interactions: the latter have the role of penalising changes of orientation. In the heterogeneous case, we obtain an estimate on the minimal energy spent to match different equilibria. This gives insight into the nucleation of dislocations in epitaxially grown heterostructured nanowires.

Citation: Roberto Alicandro, Giuliano Lazzaroni, Mariapia Palombaro. Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours. Networks and Heterogeneous Media, 2018, 13 (1) : 1-26. doi: 10.3934/nhm.2018001
##### References:
 [1] R. Alicandro, A. Braides and M. Cicalese, Continuum limits of discrete thin films with superlinear growth densities, Calc. Var. Partial Differential Equations, 33 (2008), 267-297.  doi: 10.1007/s00526-008-0159-4. [2] R. Alicandro, G. Lazzaroni and M. Palombaro, On the effect of interactions beyond nearest neighbours on non-convex lattice systems, Calc. Var. Partial Differential Equations, 56 (2017), Art. 42, 19 pp. [3] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. [4] A. Braides, $Γ$ -convergence for Beginners, Oxford University Press, Oxford, 2002. [5] A. Braides and M. Cicalese, Surface energies in nonconvex discrete systems, Math. Models Methods Appl. Sci., 17 (2007), 985-1037.  doi: 10.1142/S0218202507002182. [6] A. Braides and M. Solci, Asymptotic analysis of Lennard-Jones systems beyond the nearest-neighbour setting: A one-dimensional prototypical case, Math. Mech. Solids, 21 (2016), 915-930.  doi: 10.1177/1081286514544780. [7] M. Charlotte and L. Truskinovsky, Linear elastic chain with a hyper-pre-stress, J. Mech. Phys. Solids, 50 (2002), 217-251.  doi: 10.1016/S0022-5096(01)00054-0. [8] G. Dal Maso, An Introduction to $Γ$ -convergence, Birkhäuser, Boston, 1993. [9] E. Ertekin, P. A. Greaney, D. C. Chrzan and T. D. Sands, Equilibrium limits of coherency in strained nanowire heterostructures, J. Appl. Phys., 97 (2005), 114325. doi: 10.1063/1.1903106. [10] S. Fanzon, M. Palombaro and M. Ponsiglione, A variational model for dislocations at semi-coherent interfaces, J. Nonlinear Sci., 27 (2017), 1435-1461.  doi: 10.1007/s00332-017-9366-5. [11] I. Fonseca, N. Fusco, G. Leoni and M. Morini, A model for dislocations in epitaxially strained elastic films, J. Math. Pures Appl., to appear (2018). [12] G. Friesecke, R. D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., 55 (2002), 1461-1506.  doi: 10.1002/cpa.10048. [13] K. L. Kavanagh, Misfit dislocations in nanowire heterostructures, Semicond. Sci. Technol., 25 (2010), 024006. doi: 10.1088/0268-1242/25/2/024006. [14] G. Lazzaroni, M. Palombaro and A. Schlömerkemper, A discrete to continuum analysis of dislocations in nanowire heterostructures, Commun. Math. Sci., 13 (2015), 1105-1133.  doi: 10.4310/CMS.2015.v13.n5.a3. [15] G. Lazzaroni, M. Palombaro and A. Schlömerkemper, Rigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 119-139. [16] M. G. Mora and S. Müller, Derivation of a rod theory for multiphase materials, Calc. Var. Partial Differential Equations, 28 (2007), 161-178. [17] S. Müller and M. Palombaro, Derivation of a rod theory for biphase materials with dislocations at the interface, Calc. Var. Partial Differential Equations, 48 (2013), 315-335.  doi: 10.1007/s00526-012-0552-x. [18] B. Schmidt, On the passage from atomic to continuum theory for thin films, Arch. Ration. Mech. Anal., 190 (2008), 1-55.  doi: 10.1007/s00205-008-0138-0. [19] V. Schmidt, J. V. Wittemann and U. Gösele, Growth, thermodynamics, and electrical properties of silicon nanowires, Chem. Rev., 110 (2010), 361-388.  doi: 10.1021/cr900141g.

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##### References:
 [1] R. Alicandro, A. Braides and M. Cicalese, Continuum limits of discrete thin films with superlinear growth densities, Calc. Var. Partial Differential Equations, 33 (2008), 267-297.  doi: 10.1007/s00526-008-0159-4. [2] R. Alicandro, G. Lazzaroni and M. Palombaro, On the effect of interactions beyond nearest neighbours on non-convex lattice systems, Calc. Var. Partial Differential Equations, 56 (2017), Art. 42, 19 pp. [3] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. [4] A. Braides, $Γ$ -convergence for Beginners, Oxford University Press, Oxford, 2002. [5] A. Braides and M. Cicalese, Surface energies in nonconvex discrete systems, Math. Models Methods Appl. Sci., 17 (2007), 985-1037.  doi: 10.1142/S0218202507002182. [6] A. Braides and M. Solci, Asymptotic analysis of Lennard-Jones systems beyond the nearest-neighbour setting: A one-dimensional prototypical case, Math. Mech. Solids, 21 (2016), 915-930.  doi: 10.1177/1081286514544780. [7] M. Charlotte and L. Truskinovsky, Linear elastic chain with a hyper-pre-stress, J. Mech. Phys. Solids, 50 (2002), 217-251.  doi: 10.1016/S0022-5096(01)00054-0. [8] G. Dal Maso, An Introduction to $Γ$ -convergence, Birkhäuser, Boston, 1993. [9] E. Ertekin, P. A. Greaney, D. C. Chrzan and T. D. Sands, Equilibrium limits of coherency in strained nanowire heterostructures, J. Appl. Phys., 97 (2005), 114325. doi: 10.1063/1.1903106. [10] S. Fanzon, M. Palombaro and M. Ponsiglione, A variational model for dislocations at semi-coherent interfaces, J. Nonlinear Sci., 27 (2017), 1435-1461.  doi: 10.1007/s00332-017-9366-5. [11] I. Fonseca, N. Fusco, G. Leoni and M. Morini, A model for dislocations in epitaxially strained elastic films, J. Math. Pures Appl., to appear (2018). [12] G. Friesecke, R. D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., 55 (2002), 1461-1506.  doi: 10.1002/cpa.10048. [13] K. L. Kavanagh, Misfit dislocations in nanowire heterostructures, Semicond. Sci. Technol., 25 (2010), 024006. doi: 10.1088/0268-1242/25/2/024006. [14] G. Lazzaroni, M. Palombaro and A. Schlömerkemper, A discrete to continuum analysis of dislocations in nanowire heterostructures, Commun. Math. Sci., 13 (2015), 1105-1133.  doi: 10.4310/CMS.2015.v13.n5.a3. [15] G. Lazzaroni, M. Palombaro and A. Schlömerkemper, Rigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 119-139. [16] M. G. Mora and S. Müller, Derivation of a rod theory for multiphase materials, Calc. Var. Partial Differential Equations, 28 (2007), 161-178. [17] S. Müller and M. Palombaro, Derivation of a rod theory for biphase materials with dislocations at the interface, Calc. Var. Partial Differential Equations, 48 (2013), 315-335.  doi: 10.1007/s00526-012-0552-x. [18] B. Schmidt, On the passage from atomic to continuum theory for thin films, Arch. Ration. Mech. Anal., 190 (2008), 1-55.  doi: 10.1007/s00205-008-0138-0. [19] V. Schmidt, J. V. Wittemann and U. Gösele, Growth, thermodynamics, and electrical properties of silicon nanowires, Chem. Rev., 110 (2010), 361-388.  doi: 10.1021/cr900141g.
The six tetrahedra in the Kuhn decomposition of a three-dimensional cube
Two possible recovery sequences for the profile at the centre of the figure. Here we picture only a part of the wire containing just one species of atoms, therefore the transition at the interface is not represented. A kink in the profile may be reconstructed by folding the strip, i.e., mixing rotations and rotoreflections (left); or by a gradual transition involving only rotations or only rotoreflections (right). In the limit, the former recovery sequence gives a positive cost, while the latter gives no contribution. If the stronger topology is chosen, the appropriate recovery sequence will depend on the value of the internal variable, which defines the orientation of the wire
Lattices with dislocations: choice of the interfacial nearest neighbours in $\mathcal{L}_\varepsilon(\rho, k)$ and $H \mathcal{L}_\varepsilon(\rho, k)$ for a Delaunay triangulation
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