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

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

Roberto Alicandro 1, , Giuliano Lazzaroni 2, and  Mariapia Palombaro 3,

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.

Keywords: Nonlinear elasticity, discrete to continuum, geometric rigidity, next-to-nearest neighbours, Gamma-convergence, dimension reduction, heterostructures, dislocations.
Mathematics Subject Classification: Primary: 74B20; Secondary: 74K10, 74E15, 74G65, 49J45.
Citation: Roberto Alicandro, Giuliano Lazzaroni, Mariapia Palombaro. Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours. Networks & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

A. Braides, $Γ$ -convergence for Beginners, Oxford University Press, Oxford, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

G. Dal Maso, An Introduction to $Γ$ -convergence, Birkhäuser, Boston, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[13]

K. L. Kavanagh, Misfit dislocations in nanowire heterostructures, Semicond. Sci. Technol., 25 (2010), 024006. doi: 10.1088/0268-1242/25/2/024006.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[16]

M. G. Mora and S. Müller, Derivation of a rod theory for multiphase materials, Calc. Var. Partial Differential Equations, 28 (2007), 161-178.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

A. Braides, $Γ$ -convergence for Beginners, Oxford University Press, Oxford, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

G. Dal Maso, An Introduction to $Γ$ -convergence, Birkhäuser, Boston, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[13]

K. L. Kavanagh, Misfit dislocations in nanowire heterostructures, Semicond. Sci. Technol., 25 (2010), 024006. doi: 10.1088/0268-1242/25/2/024006.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[16]

M. G. Mora and S. Müller, Derivation of a rod theory for multiphase materials, Calc. Var. Partial Differential Equations, 28 (2007), 161-178.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The six tetrahedra in the Kuhn decomposition of a three-dimensional cube
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Figure 2.  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
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Figure 3.  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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