Abstract
We consider electrostatic interactions in two classes of nanostructures embedded in a three dimensional space: (1) helical nanotubes, and (2), thin films with uniform bending (i.e., constant mean curvature). Starting from the atomic scale with a discrete distribution of dipoles, we obtain the continuum limit of the electrostatic energy; the continuum energy depends on the geometric parameters that define the nanostructure, such as the pitch and twist of the helical nanotubes and the curvature of the thin film. We find that the limiting energy is local in nature. This can be rationalized by noticing that the decay of the dipole kernel is sufficiently fast when the lattice sums run over one and two dimensions, and is also consistent with prior work on dimension reduction of continuum micromagnetic bodies to the thin film limit. However, an interesting contrast between the discrete-to-continuum approach and the continuum dimension reduction approaches is that the limit energy in the latter depends only on the normal component of the dipole field, whereas in the discrete-to-continuum approach, both tangential and normal components of the dipole field contribute to the limit energy.
Similar content being viewed by others
Notes
That is, it is convergent, while the sums of only the positive terms and only the negative terms diverge, respectively, to \(\pm \infty \).
References
Alicandro, R., Braides, A., Cicalese, M.: Continuum limits of discrete thin films with superlinear growth densities. Calc. Var. Partial. Differ. Equ. 33(3), 267–297, 2008
Alicandro, R., Cicalese, M., Gloria, A.: Variational description of bulk energies for bounded and unbounded spin systems. Nonlinearity 21(8), 1881, 2008
Alicandro, R., Cicalese, M., Gloria, A.: Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity. Arch. Ration. Mech. Anal. 200(3), 881–943, 2011
Aghaei, A., Dayal, K., Elliott, R.S.: Anomalous phonon behavior of carbon nanotubes: first-order influence of external load. J. Appl. Phys. 113(2), 023503, 2013
Aghaei, A., Dayal, K., Elliott, R.S.: Symmetry-adapted phonon analysis of nanotubes. J. Mech. Phys. Solids 61(2), 557–578, 2013
Ahmadpoor, F., Deng, Q., Liu, L.P., Sharma, P.: Apparent flexoelectricity in lipid bilayer membranes due to external charge and dipolar distributions. Phys. Rev. E 88(5), 050701, 2013
Alicandro, R., Lazzaroni, G., Palombaro, M.: Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours. Netw. Heterog. Media 13(1), 1, 2018
Bach, A., Braides, A., Cicalese, M.: Discrete-to-continuum limits of multibody systems with bulk and surface long-range interactions. SIAM J. Math. Anal. 52(4), 3600–3665, 2020
Blanc, X., Le Bris, C., Lions, P.-L.: From molecular models to continuum mechanics. Arch. Ration. Mech. Anal. 164(4), 341–381, 2002
Blanc, X., Le Bris, C., Lions, P.-L.: Atomistic to continuum limits for computational materials science. ESAIM Math. Model. Numer. Anal. 41(2), 391–426, 2007
Brown, W.F.: Micromagnetics. Number 18. Interscience publishers, 1963
Carbou, G.: Thin layers in micromagnetism. Math. Models Methods Appl. Sci. 11(09), 1529–1546, 2001
Cicalese, M., DeSimone, A., Zeppieri, C.I.: Discrete-to-continuum limits for strain-alignment-coupled systems: magnetostrictive solids, ferroelectric crystals and nematic elastomers. Netw. Heterog. Media 4(4), 667, 2009
Chacouche, K., Hadiji, R.: Ferromagnetic of nanowires of infinite length and infinite thin films. Z. Angew. Math. Phys. 66(6), 3519–3534, 2015
Dumitrică, T., James, R.D.: Objective molecular dynamics. J. Mech. Phys. Solids 55(10), 2206–2236, 2007
Dobson, M., Luskin, M., Ortner, C.: Sharp stability estimates for the force-based quasicontinuum approximation of homogeneous tensile deformation. Multiscale Model. Simul. 8(3), 782–802, 2010
Friesecke, G., James, R.D.: A scheme for the passage from atomic to continuum theory for thin films, nanotubes and nanorods. J. Mech. Phys. Solids 48(6–7), 1519–1540, 2000
Grasinger, M., Dayal, K.: Architected elastomer networks for optimal electromechanical response. J. Mech. Phys. Solids 146, 104171, 2020
Grasinger, Matthew, Dayal, K.: Statistical mechanical analysis of the electromechanical coupling in an electrically-responsive polymer chain. Soft Matter 16, 6265–6284, 2020
Gaudiello, A., Hamdache, K.: A reduced model for the polarization in a ferroelectric thin wire. Nonlinear Differ. Equ. Appl. 22(6), 1883–1896, 2015
Gioia, G., James, R.D.: Micromagnetics of very thin films. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 453(1956), 213–223, 1997
Hakobyan, Y., Tadmor, E.B., James, R.D.: Objective quasicontinuum approach for rod problems. Phys. Rev. B 86(24), 245435, 2012
James, R.D.: Objective structures. J. Mech. Phys. Solids 54(11), 2354–2390, 2006
Jha, P.: Coarse Graining of Electric Field Interactions with Materials. PhD thesis, Carnegie Mellon University, 2016
James, R.D., Müller, S.: Internal variables and fine-scale oscillations in micromagnetics. Contin. Mech. Thermodyn. 6(4), 291–336, 1994
Knap, J., Ortiz, M.: An analysis of the quasicontinuum method. J. Mech. Phys. Solids 49(9), 1899–1923, 2001
Kohn, R.V., Slastikov, V.V.: Another thin-film limit of micromagnetics. Arch. Ration. Mech. Anal. 178(2), 227–245, 2005
Kružík, M., Stefanelli, U., Zanini, C.: Quasistatic evolution of magnetoelastic plates via dimension reduction. Discrete Contin. Dyn. Syst. A 35(12), 5999, 2015
Li, X.H., Luskin, M., Ortner, C.: Positive definiteness of the blended force-based quasicontinuum method. Multiscale Model. Simul. 10(3), 1023–1045, 2012
Lazzaroni, G., Palombaro, M., Schlömerkemper, A.: A discrete to continuum analysis of dislocations in nanowire heterostructures. Commun. Math. Sci. 13(5), 2015
Lazzaroni, G., Palombaro, M., Schlömerkemper, A.: Rigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires. Discrete Contin. Dyn. Syst. 10(1), 119–139, 2017
Liu, L.P., Sharma, P.: Flexoelectricity and thermal fluctuations of lipid bilayer membranes: renormalization of flexoelectric, dielectric, and elastic properties. Phys. Rev. E 87(3), 032715, 2013
Marshall, J., Dayal, K.: Atomistic-to-continuum multiscale modeling with long-range electrostatic interactions in ionic solids. J. Mech. Phys. Solids 62, 137–162, 2014
Müller, S., Schlömerkemper, A.: Discrete-to-continuum limit of magnetic forces. C. R. Math. 335(4), 393–398, 2002
Miller, R.E., Tadmor, E.B.: The quasicontinuum method: overview, applications and current directions. J. Comput. Aided Mater. Des. 9(3), 203–239, 2002
Schlömerkemper, A.: Mathematical derivation of the continuum limit of the magnetic force between two parts of a rigid crystalline material. Arch. Ration. Mech. Anal. 176(2), 227–269, 2005
Schmidt, B.: A derivation of continuum nonlinear plate theory from atomistic models. Multiscale Model. Simul. 5(2), 664–694, 2006
Schmidt, B.: On the passage from atomic to continuum theory for thin films. Arch. Ration. Mech. Anal. 190(1), 1–55, 2008
Schmidt, B.: On the derivation of linear elasticity from atomistic models. Netw. Heterog. Media 4(4), 789, 2009
Schlömerkemper, A., Schmidt, B.: Discrete-to-continuum limit of magnetic forces: dependence on the distance between bodies. Arch. Ration. Mech. Anal. 192(3), 589–611, 2009
Steigmann, D.J.: Mechanics and physics of lipid bilayers. In: The Role of Mechanics in the Study of Lipid Bilayers, pp. 1–61. Springer, 2018
Sen, S., Wang, Y., Breitzman, T., Dayal, K.: Two-scale analysis of the polarization density in ionic solids. in preparation, 2021
Tadmor, E.B., Miller, R.E.: Modeling Materials: Continuum, Atomistic and Multiscale Techniques. Cambridge University Press, Cambridge, 2011
Torbati, M., Mozaffari, K., Liu, L., Sharma, P.: Coupling of mechanical deformation and electromagnetic fields in biological cells. Rev. Mod. Phys. 94(2), 025003, 2022
Tadmor, E.B., Ortiz, M., Phillips, R.: Quasicontinuum analysis of defects in solids. Philos. Mag. A 73(6), 1529–1563, 1996
Toupin, R.A.: The elastic dielectric. J. Ration. Mech. Anal. 5(6), 849–915, 1956
Xiao, Y.: The influence of oxygen vacancies on domain patterns in ferroelectric perovskites. Ph.D. thesis, California Institute of Technology, 2005
Acknowledgements
This paper draws from the doctoral dissertation of Prashant K. Jha at Carnegie Mellon University [24]. We thank Richard D. James for useful discussions; AFRL for hosting a visit by Kaushik Dayal; and NSF (2108784, 1921857), ARO (W911NF-17-1-0084), ONR (N00014-18-1-2528), and AFOSR (MURI FA9550-18-1-0095) for financial support.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by K. Bhattacharya.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Jha, P.K., Breitzman, T. & Dayal, K. Discrete-to-Continuum Limits of Long-Range Electrical Interactions in Nanostructures. Arch Rational Mech Anal 247, 29 (2023). https://doi.org/10.1007/s00205-023-01869-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00205-023-01869-6