Skip to main content
SpringerLink
Log in

Various Modeling Techniques for Nanostructures

  • Chapter
  • First Online:
Wave Propagation in Nanostructures

Part of the book series: NanoScience and Technology ((NANO))

  • 1718 Accesses

Abstract

Mathematical modeling of structures at the micro and macroscales are quite well known and the methods have been well established. The laws of physics, which is fundamental to any modeling, is pretty well understood at these scales. At the nanometer levels, we need to deal with atoms, molecules, and their interactions. The laws of physics, at these scales, are not that well understood. The main difference lies in representing the models in different scales. That is, the philosophy of modeling at different scales are different.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. S. Yip (ed.), Handbook of Materials Modeling (Springer, Netherlands, 2005)

    Google Scholar 

  2. P. Hohenberg, W. Kohn, Inhomogeneous electron gas. Phys. Rev. B 136, 864–871 (1963)

    Article  MathSciNet  Google Scholar 

  3. W. Kohn, L.J. Sham, Self-consistent equations including exchange and correlation effects. Phys. Rev. B 140, 1133–1138 (1965)

    Article  MathSciNet  ADS  Google Scholar 

  4. M.C. Payne, M.P. Teter, D.C. Allan, T. Arias, J.D. Joannopoulos, Iterative minimization techniques for abinitio total-energy calculations: molecular dynamics and conjugate gradients. Rev. Mod. Phys. 64, 1045–1097 (1992)

    Article  ADS  Google Scholar 

  5. M.T. Yin, M.L. Cohen, Theory of Ab initio pseudopo-tential calculations. Phys. Rev. B 25, 7403–7412 (1982)

    Article  ADS  Google Scholar 

  6. J.M. Haile, Molecular Dynamics Simulation (John Wiley, New York, 1992)

    Google Scholar 

  7. D.C. Rapaport, The Art of Molecular Dynamics Simulation (Cambridge University Press, Cambrigde, 1992)

    Google Scholar 

  8. L. Verlet, Computer experiments on classical fluids: I. Thermodynamical properties of Lennard-Jones molecules. Phys. Rev. B 159, 98–103 (1967)

    Article  ADS  Google Scholar 

  9. L. Verlet, Computer experiments on classical fluids: II. Equilibrium correlation functions. Phys. Rev. B 165, 201–214 (1968)

    Article  ADS  Google Scholar 

  10. C.W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, 1971)

    MATH  Google Scholar 

  11. S. Plimpton, Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117, 1–19 (1995)

    Article  ADS  MATH  Google Scholar 

  12. S. Ogata, T.J. Campbell, R.K. Kalia, A. Nakano, P. Vashishta, S. Vemparala, Scalable and portable implementation of the fast multipole method on parallel computers. Comput. Phys. Commun. 153, 445–461 (2003)

    Article  ADS  Google Scholar 

  13. J.E. Jones, On the determination of molecular fields: II. From the equation of state of a gas. Proc. R. Soc. A 106, 463–477 (1924)

    Article  ADS  Google Scholar 

  14. P.M. Morse, Diatomic molecules according to the wave mechanics: II. Vibrational levels. Phys. Rev. B 34, 57–64 (1929)

    Article  ADS  MATH  Google Scholar 

  15. F.H. Stillinger, T.A. Weber, Computer simulation of local order in condensed phases of silicon. Phys. Rev. B 31, 5262–5271 (1985)

    Article  ADS  Google Scholar 

  16. J. Tersoff, New empirical model for the structural properties of silicon. Phys. Rev. Lett. 56, 632–635 (1986)

    Article  ADS  Google Scholar 

  17. D.W. Brenner, Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. Phys. Rev. B 42, 9458–9471 (1990)

    Article  ADS  Google Scholar 

  18. M.S. Daw, M.I. Baskes, Semiempirical, quantum mechanical calculation of hydrogen embrittlement in metals. Phys. Rev. Lett. 50, 1285–1288 (1983)

    Article  ADS  Google Scholar 

  19. K.W. Jacobsen, J.K. Norskov, M.J. Puska, Interatomic interactions in the effective-medium theory. Phys. Rev. B 35, 7423–7442 (1987)

    Article  ADS  Google Scholar 

  20. M.W. Finnis, J.E. Sinclair, A simple empirical N-body potential for transition metals. Philos. Mag. A 50, 45–55 (1984)

    Article  ADS  Google Scholar 

  21. R.E. Cohen, M.J. Mehl, D.A. Papaconstantopoulos, Tight-binding total-energy method for transition and noble metals. Phys. Rev. B 50, 14694–14697 (1994)

    Article  ADS  Google Scholar 

  22. D.G. Pettifor, New many-body potential for the bond order. Phys. Rev. Lett. 63, 2480–2483 (1989)

    Article  ADS  Google Scholar 

  23. J.A. Moriarty, Density-functional formulation of the generalized pseudopotential theory: III. Transitionmetal interatomic potentials. Phys. Rev. B 38, 3199–3231 (1988)

    Article  ADS  Google Scholar 

  24. M. Schneider, A. Rahman, I.K. Schuller, Role of relaxation in epitaxial growth-a molecular dynamics study. Phys. Rev. Lett. 55, 604–606 (1985)

    Article  ADS  Google Scholar 

  25. A.F. Voter, F. Montalenti, T.C. Germann, Extending the time scales in atomistic simulation of materials. Ann. Rev. Mater. Res. 32, 321–346 (2002)

    Article  Google Scholar 

  26. C.Y. Li, T.W. Chou, Single-walled carbon nanotubes as ultrahigh frequency nanomechanical resonators. Phys. Rev. B 68, 073405 (2003)

    Article  ADS  Google Scholar 

  27. C.Y. Li, T.W. Chou, Vibrational behaviors of multiwalled-carbon-nanotube-based nanomechanical resonators. Appl. Phys. Lett. 84, 121–123 (2003)

    Article  ADS  Google Scholar 

  28. G.X. Cao, X. Chen, J.W. Kysar, Thermal vibration and apparent thermal contraction of single-walled carbon nanotubes. J. Mech. Phys. Solids 54, 1206–1236 (2006)

    Article  ADS  MATH  Google Scholar 

  29. C.Y. Li, T.W. Chou, Strain and pressure sensing using single-walled carbon nanotubes. Nanotechnology 15, 1493–1496 (2004)

    Article  ADS  Google Scholar 

  30. V. Sazonova, Y. Yaish, H. Ustunel, D. Roundy, T.A. Arias, P.L. McEuen, A tunable carbon nanotube electromechanical oscillator. Nature 431, 284 (2004)

    Article  ADS  Google Scholar 

  31. G.X. Cao, X. Chen, J.W. Kysar, Strain sensing of carbon nanotubes: numerical analysis of the vibrational frequency of deformed sing-wall carbon nanotubes. Phys. Rev. B 72, 195412 (2005)

    Article  ADS  Google Scholar 

  32. Q. Zhao, Z.H. Gan, O.K. Zhuang, Electrochemical sensors based on carbon nanotubes. Electroanalysis 14, 1609–1613 (2002)

    Article  Google Scholar 

  33. Y.Y. Zhang, C.M. Wang, V.B.C. Tan, Assessment of Timoshenko beam models for vibrational behavior of single-walled carbon nanotubes using molecular dynamics. Adv. Appl. Math. Mech. 1(1), 89–106 (2009)

    MathSciNet  Google Scholar 

  34. D.W. Brenner, O.A. Shenderova, J.A. Harrison, S.J. Stuart, B. Ni, S.B. Sinnott, A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons. J. Phys.: Condens. Matter 14, 783–802 (2002)

    Article  ADS  Google Scholar 

  35. D.W. Brenner, Empirical potential for hydrocarbons for use in simulation the chemical vapor deposition of diamond films. Phys. Rev. B 42, 9458 (1990)

    Article  ADS  Google Scholar 

  36. O.A. Shenderova, D.W. Brenner, A. Omeltchenko, X. Su, L.H. Yang, Atomistic modeling of the fracture of polycrystalline diamond. Phys. Rev. B 61, 3877 (2000)

    Article  ADS  Google Scholar 

  37. Y. Huang, J. Wu, K.C. Hwang, Thickness of graphene and single-wall carbon nanotubes. Phys. Rev. B 74, 245413 (2006)

    Article  ADS  Google Scholar 

  38. NanoHive-1, v. 1.2.0-b1., Nanorex Inc., http://www.nanorex.com (2005)

  39. F. Khademolhosseini, A.S. Phani, A. Nojeh, R.K.N.D. Rajapakse, Nonlocal continuum modeling and molecular dynamics simulation of torsional vibration of carbon nanotubes. IEEE Trans. Nanotechnol. 11(2), 34–43 (2011)

    ADS  Google Scholar 

  40. S.J. Stuart, A.B. Tutein, J.A. Harrison, A reactive potential for hydrocarbons with intermolecular interactions. J. Chem. Phys. 112, 6472–6486 (2000)

    Article  ADS  Google Scholar 

  41. M. Chen, J. Zang, D. Xiao, F. Liu, Mechanical wave propagation in carbon nanotubes driven by an oscillating tip actuator. J. Appl. Phys. 105, 026102 (2009)

    Article  ADS  Google Scholar 

  42. Y.G. Hu, K.M. Liew, Q. Wang, X.Q. He, B.I. Yakobson, Nonlocal shell model for elastic wave propagation in single-and double-walled carbon nanotubes. J. Mech. Phys. Solids 56, 3475–3485 (2008)

    Article  MATH  ADS  Google Scholar 

  43. C. Lin, H. Wang, W. Yang, Wave propagation and scattering in deformed single-wall carbon nanotubes. J. Comput. Theor. Nanosci. 8(10), 2019–2024 (2011)

    Article  Google Scholar 

  44. B. Arash, Q. Wang, K.M. Liew, Wave propagation in graphene sheets with nonlocal elastic theory via finite element formulation. Comput. Methods Appl. Mech. Eng. 223–224, 1–9 (2012)

    Article  MathSciNet  Google Scholar 

  45. D. Landau, K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, Cambridge, 2000)

    MATH  Google Scholar 

  46. D.D. Mc Cracken, The Monte Carlo method. Sci. Am. 192, 90–95 (1955)

    Article  ADS  Google Scholar 

  47. N. Metropolis, A.W. RosenbluthW, M.N. Rosenbluth, A.H. Teller, E. Teller, Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092 (1953)

    Article  ADS  Google Scholar 

  48. N.G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981)

    MATH  Google Scholar 

  49. Z. Wang, E.G. Seebauer, Estimating pre-exponential factors for desorption from semiconductors: consequences for a priori process modelling. Appl. Surf. Sci. 181, 111–120 (2001)

    Article  ADS  Google Scholar 

  50. G.K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, Cambridge, 1967)

    MATH  Google Scholar 

  51. J.N. Reddy, An Introduction to the Finite Element Method (McGraw-Hill, New York, 1993)

    Google Scholar 

  52. H.S. Park, W.K. Liu, An introduction and tutorial on multiple-scale analysis in solids. Comput. Methods Appl. Mech. Eng. 193, 1733–1772 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  53. A.F. Voter, Hyperdynamics: accelerated molecular dynamics of infrequent events. Phys. Rev. Lett. 78(20), 3908–3911 (1997)

    Article  ADS  Google Scholar 

  54. A.F. Voter, F. Montalenti, T.C. Germann, Extending the time scale in atomistic simulation of materials. Annu. Rev. Mater. Res. 32, 321–346 (2002)

    Article  Google Scholar 

  55. P. Zhang, Y. Huang, P.H. Geubelle, P.A. Klein, K.C. Hwang, The elastic modulus of single-wall carbon nanotubes: a continuum analysis incorporating interatomic potentials. Int. J. Solids Struct. 39, 3893–3906 (2002)

    Article  MATH  Google Scholar 

  56. D.W. Brenner, O.A. Shenderova, J.A. Harrison, S.J. Stuart, B. Ni, S.B. Sinnott, A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons. J. Phys.: Condens. Matter 14(4), 783–802 (2002)

    Article  ADS  Google Scholar 

  57. W.A. Curtin, R.E. Miller, Atomistic/continuum coupling in computational materials science. Modell. Simul. Mater. Sci. Eng. 11(3), R33–R68 (2003)

    Article  ADS  Google Scholar 

  58. M. Arroyo, T. Belytschko, An atomistic-based finite deformation membrane for single layer crystalline films. J. Mech. Phys. Solids 50(9), 1941–1977 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  59. W.E.Z. Huang, Matching conditions in atomisticcontinuum modeling of materials. Phys. Rev. Lett. 87, 135501 (2001)

    Article  ADS  Google Scholar 

  60. G.J. Wagner, W.K. Liu, Coupling of atomistic and continuum simulations using bridging scale decomposition. J. Comput. Phys. 190, 249–274 (2003)

    Article  ADS  MATH  Google Scholar 

  61. V.B. Shenoy, R. Miller, E.B. Tadmor, D. Rodney, R. Phillips, M. Ortiz, An adaptive finite element approach to atomic-scale mechanics the quasicontinuum method. J. Mech. Phys. Solids 47(3), 611–642 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  62. L.E. Shilkrot, W.A. Curtin, R.E. Miller, A coupled atomistic/continuum model of defects in solids. J. Mech. Phys. Solids 50(10), 2085–2106 (2002)

    Article  ADS  MATH  Google Scholar 

  63. E.B. Tadmor, M. Ortiz, R. Phillips, Quasicontinuum analysis of defects in solids. Philos. Mag. A 73(6), 1529–1563 (1996)

    Article  ADS  Google Scholar 

  64. D. Rogula, Introduction to nonlocal theory of material media, in Nonlocal Theory of Material Media, ed. by D. Rogula, CISM Courses and Lectures, vol 268 (Springer, New York, 1982) pp. 125–222

    Google Scholar 

  65. Z.P. Bazant, M. Jirasek, Nonlocal Integral formulations of plasticity and damage: survey of progress. J. Eng. Mech. 128, 1119–1149 (2002)

    Article  Google Scholar 

  66. M. Jirasek, Nonlocal theories in continuum mechanics. Acta Polytech. 44, 5–6 (2004)

    Google Scholar 

  67. J. Fish, W. Chen, G. Nagai, onlocal dispersive model for wave propagation in heterogeneous media, Part 1: one-dimensional case. Int. J. Numer. Methods Eng. 54, 331–346 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  68. H. Askes, E. Aifantis, Gradient elasticity theories in statics and dynamicsa unification of approaches. Int. J. Fract. 139, 297–304 (2006)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Aerospace Engineering, Indian Institute of Science, Bangalore, 560012, India

    Srinivasan Gopalakrishnan

  2. Defence Research and Development Organisation (DRDO), Hyderabad, 110105, India

    Saggam Narendar

Authors
  1. Srinivasan Gopalakrishnan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Saggam Narendar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Srinivasan Gopalakrishnan .

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Gopalakrishnan, S., Narendar, S. (2013). Various Modeling Techniques for Nanostructures. In: Wave Propagation in Nanostructures. NanoScience and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-01032-8_3

Download citation

Publish with us

Policies and ethics

Search

Navigation