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An Efficient Coupled Dipole Method for the Accurate Calculation of van der Waals Interactions at the Nanoscale

  • Hye-Young Kim
Chapter
Part of the Progress in Optical Science and Photonics book series (POSP, volume 2)

Abstract

The van der Waals (VDW) force arises from purely quantum mechanical charge fluctuations and is variously called a dispersion or London or Casimir force. This often considered as weak, yet ubiquitous, attractive interaction is important in many nanoscale systems. This chapter provides an overview of the Coupled Dipole Method (CDM), an atomistic and accurate computational method widely adopted to predict the VDW forces between dielectric nanomaterials. There is a concern about the burden of memory and computing time needed to solve eigenvalue problems by either diagonalization or iteration, which have hindered the implementation of CDM for large systems. Here, an efficient way, named trace-CDM (TCDM), is presented. TCDM uses the simple fact that the trace of a square matrix is equal to the sum of its eigenvalues and thus calculates the accurate VDW energies without solving for the eigenvalues. Four examples are solved to demonstrate the advantages of the method.

Keywords

van der Waals (VDW) forces and interactions Dispersion force London force Casimir force Coupled dipole method (CDM) Trace-CDM (TCDM) Nanoscale Dielectric nanomaterials Diagonalization Iteration VDW energies 

Notes

Acknowledgments

I am grateful to all my collaborators in research projects involving CDM—Milton Cole, Darrell Velegol, Amand Lucas, Jorge Sofo, and Paul Kent. I most gratefully acknowledge the significant contribution of Amand Lucas in guiding me through the initial formulations of CDM and TCDM. I thank the Louisiana Board of Regents-RCS grant (LEQSF(2012-15)-RD-A-19) for support.

References

  1. 1.
    Rowlinson JS (2005) Cohesion: a scientific history of intermolecular forces. Cambridge University Press, CambridgeGoogle Scholar
  2. 2.
    Margenau H, Kestner NR (1972) Theory of intermolecular forces. Pergamon, OxfordGoogle Scholar
  3. 3.
    Langbein D (1974) Springer tracks in modern physics, vol 72. Springer, BerlinGoogle Scholar
  4. 4.
    Parsegian VA (2006) van der Waals forces: a handbook for biologists, chemists, engineers, and physicists. Cambridge University, New YorkGoogle Scholar
  5. 5.
    Israelachvili J (1991) Intermolecular and surface forces, 2nd edn. Academic, New YorkGoogle Scholar
  6. 6.
    Eisenschitz R, London F (1930) Über das Verhältnis der van der Waalsschen Kräfte zu den Homöopolaren Bindungskräften. Z Physik 60:491–527 (in German)CrossRefGoogle Scholar
  7. 7.
    London F (1930) Zur Theorie und Systematik der Molekularkräfte. Z Physik 63:245–2791930 (in German)Google Scholar
  8. 8.
    London F (1930) Über einige Eigenschaften und Anwendungen der Molekularkräfte. Z Phys Chem B11:222–251 (in German)Google Scholar
  9. 9.
    Hamaker HC (1937) The London-Van der waals attraction between spherical particles. Physica 4:1058–1072CrossRefGoogle Scholar
  10. 10.
    London F (1937) The general theory of molecular forces. Trans Faraday Soc 33:8–26CrossRefGoogle Scholar
  11. 11.
    Bishop KJM, Wilmer CE, Soh S, Grzybowski BA (2009) Nanoscale force and their use in self-assembly. Small X:1–31Google Scholar
  12. 12.
    Sernelius BE, Román-Velázquez CE (2008) Beyond the simple proximity force approximation: geometrical effects on the nonretarded Casimir interaction. Phys Rev A 78:032111CrossRefGoogle Scholar
  13. 13.
    Marom N, DiStasio Jr. RA, Atalla V, Levchenko S, Reilly AM, Chelikowsky JR, Leiserowitz L, Tkatchenko A (2013) Many-body dispersion interactions in molecular crystal polymorphism. Angew Chem Int Ed 52:1–5CrossRefGoogle Scholar
  14. 14.
    Bergström L (1997) Hamaker constants of inorganic materials. Adv Colloid Interface Sci 70:125–169CrossRefGoogle Scholar
  15. 15.
    Girard C, Bouju X (1991) Coupled electromagnetic modes between a corrugated surface and a thin probe tip. J. Chem. Phys. 95:2056–2064CrossRefGoogle Scholar
  16. 16.
    Nel AE, Mädler L, Velegol D, Xia T, Hoek EMV (2009) Understanding biophysicochemical interactions at the nano-bio interface. Nature Mat. 8:543–557CrossRefGoogle Scholar
  17. 17.
    Bruch LW, Diehl RD, Venables JA (2007) Progress in the measurement and modeling of physisorbed layers. Rev Mod Phys 79:1381–1454CrossRefGoogle Scholar
  18. 18.
    Sushkov AO, Kim WJ, Dalvit DAR, Lamoreaux SK (2011) Observation of the thermal Casimir force. Nature Phys. 7:230–233CrossRefGoogle Scholar
  19. 19.
    Berland K, Borck Ø, Hyldgaard P (2011) Van der Waals density functional calculations of binding in molecular crystals. Comput Phys Commun 182:1800–1804CrossRefGoogle Scholar
  20. 20.
    Liu R-F, Ángyán JG, Dobson J (2011) Dispersion interaction in hydrogen-chain models. J Chem Phys 134:114106CrossRefGoogle Scholar
  21. 21.
    Mo Y, Turner KT, Szlufarska I (2009) Friction laws at the nanoscale. Nature 457:1116–1119CrossRefGoogle Scholar
  22. 22.
    Velegol D, Holtzer GL, Radović-Moreno A, Cuppett JD (2007) Force Measurements between Sub-100 nm colloidal particles. Langmuir 12:1275–1280CrossRefGoogle Scholar
  23. 23.
    Pio JM, Taylor MA, van der Veer WE, Bieler CR, Cabrera JA, Janda KC (2010) Real-time dissociation dynamics of the Ne2Br2 van der Waals complex. J. Chem. Phys. 133:014305CrossRefGoogle Scholar
  24. 24.
    Wang ZH, Wei J, Morse P, Dash JG, Vilches OE, Cobden DH (2010) Phase transitions of adsorbed atoms on the surface of a carbon nanotube. Science 327:552–555CrossRefGoogle Scholar
  25. 25.
    Krim J, Solina DH, Chiarello R (1991) Nanotribology of a Kr monolayer: a quartz-crystal microbalance study of atomic-scale friction. Phys Rev Lett 66:181–184CrossRefGoogle Scholar
  26. 26.
    Stifter T, Marti O, Bhushan B (2000) Theoretical investigation of the distance dependence of capillary and van der Waals forces in scanning force microscopy. Phys Rev B62:667–673Google Scholar
  27. 27.
    Böttcher CJF (1973) Theory of electric polarization, 2nd edn, vol 1. Dielectrics in static fields, Chap. V. Elsevier Scientific Publishing Co. AmsterdamGoogle Scholar
  28. 28.
    Kim H-Y, Kent PRC (2009) van der Waals forces: accurate calculation and assessment of approximate methods in dielectric nanocolloids up to 16 nm. J Chem Phys 131:144705CrossRefGoogle Scholar
  29. 29.
    Margenau H (1938) Quadrupole contributions to London’s dispersion forces. J Chem Phys 6:896CrossRefGoogle Scholar
  30. 30.
    Hirschfelder JO, Meath WJ (2007) The nature of intermolecular forces (1967). Adv Chem. Phys.: Intermol Forces 12 (Published on line)Google Scholar
  31. 31.
    Axilrod BM, Teller E (1943) Interaction of the van der Waals type between three atoms. J Chem Phys 11:299–300CrossRefGoogle Scholar
  32. 32.
    Axilrod BM (1951) Triple-dipole interaction. J Chem Phys 19:719–724CrossRefGoogle Scholar
  33. 33.
    Muto Y (1943) Force between non-polar molecules. Proc Phys Math Soc Jpn 17:629–631 in JapaneseGoogle Scholar
  34. 34.
    Bade WL (1957) Drude-model calculation of dispersion forces. I General theory. J Chem Phys 27:1280–1284CrossRefGoogle Scholar
  35. 35.
    Bade WL, Kirkwood JG (1957) Drude-model calculation of dispersion forces. II The linear lattice. J Chem Phys 27:1284–1288CrossRefGoogle Scholar
  36. 36.
    Bade WL (1958) Drude-model calculation of dispersion forces. III The fourth-order contribution. J Chem Phys 28:282–284CrossRefGoogle Scholar
  37. 37.
    Lucas AA (1967) Collective contributions to the long-range dipolar interaction in rare-gas crystals. Physica 35:353–368CrossRefGoogle Scholar
  38. 38.
    McLachlan AD (1964) Van der Waals forces between an atom and a surface. Mol Phys 7:381–388CrossRefGoogle Scholar
  39. 39.
    Barker JA (1976) Interatomic potentials for inert gases from experimental data. In: Klein ML, Venables JA (eds) Rare gas solids, vol 1, Chap. 4. Academic Press, New York, pp 212–264Google Scholar
  40. 40.
    Klein ML, Koehler TR (1976) Lattice dynamics of rare gas solids. In: Klein ML, Venables JA (eds) Rare gas solids. Academic Press, New YorkGoogle Scholar
  41. 41.
    Meath WJ, Aziz RA (1984) On the importance and problems in the construction of many-body potentials. Mol Phys 52:225–243CrossRefGoogle Scholar
  42. 42.
    Dzyaloshinskii IE, Lifshitz EM, Pitaevskii LP (1961) The general theory of van der Waals forces. Adv Phys 10:165–209CrossRefGoogle Scholar
  43. 43.
    Schmeits M, Lucas AA (1983) Physical adsorption and surface plasmons. Prog Surf Sci 14:1–52CrossRefGoogle Scholar
  44. 44.
    Hamaker HC (1937) The London-van der Waals attraction between spherical particles. Physica 4:1058–1072CrossRefGoogle Scholar
  45. 45.
    Hunter J (1986) Foundations of colloids science. In: White LR (ed) The theory of van der Waals forces, vol 1, Chap. 4. Clarendon/Oxford University, New YorkGoogle Scholar
  46. 46.
    Munday JN, Capasso F, Parsegian VA (2009) Measured long-range repulsive Casimir-Lifshitz force. Nature 457:170–173CrossRefGoogle Scholar
  47. 47.
    Kim H-Y, Sofo JO, Velegol D, Cole MW, Mukhopadhyay G (2005) Static Polarizabilities of dielectric nanoclusters. Phys Rev A 72:053201CrossRefGoogle Scholar
  48. 48.
    Kim H-Y, Sofo JO, Velegol D, Cole MW (2006) Fully retarded van der Waals interaction between dielectric nanoclusters. J Chem Phys 125:174303CrossRefGoogle Scholar
  49. 49.
    Kim H-Y, Sofo JO, Velegol D, Cole MW, Lucas AA (2006) van der Waals forces between nanoclusters: importance of many-body effects. J Chem Phys 124:074504CrossRefGoogle Scholar
  50. 50.
    Kim H-Y, Sofo JO, Velegol D, Cole MW (2007) Van der Waals dispersion forces between dielectric nanoclusters. Langmuir 23:1735–1740CrossRefGoogle Scholar
  51. 51.
    Cole MW, Velegol D, Kim H-Y, Lucas AA (2009) Nanoscale van der Waals interactions. Mol Simul 35:849–866CrossRefGoogle Scholar
  52. 52.
    Cole MW, Gergidis LN, McNutt JP, Velegol D, Kim H-Y, Bond ZK (2010) Many body VDW forces involving chains. SPIE J Nanophotonics 4:041560CrossRefGoogle Scholar
  53. 53.
    Hohm U, Kerl K (1987) Temperature dependence of measured mean molecular polarizability on account of translational motion. Mol Phys 61:1295–1298CrossRefGoogle Scholar
  54. 54.
    Bohren CF, Huffman DR (1983) Absorption and scattering of light by small particles. Wiley-VCH Verlag GmbH & Co, WeinheimGoogle Scholar
  55. 55.
    Born M, Wolf E (1975) Principles of optics: electromagnetic theory of propagation, interference and diffraction of light, 5th edn. Pergamon Press, OxfordGoogle Scholar
  56. 56.
    McLachlan AD (1963) Retarded dispersion forces between molecules. Proc Roy Soc A271:387–401CrossRefGoogle Scholar
  57. 57.
    McLachlan AD (1963) Retarded dispersion forces in dielectrics at finite temperatures. Proc Roy Soc A274:80–90CrossRefGoogle Scholar
  58. 58.
  59. 59.
  60. 60.
    Lamoreaux SK (2011) Casimir force, scholarpedia 6:9746. doi: 10.4249/scholarpedia.9746
  61. 61.
    Gobre VV, Tkatchenko A (2013) Scaling laws for van der Waals interactions in nanostructured materials. Nat Commun 4:2341. doi: 10.1038/ncomms3341
  62. 62.
    Tkatchenko A, Scheffler M (2009) Accurate Molecular Van Der waals interactions from ground-state electron density and free-atom reference data. Phys Rev Lett 102:073005. doi: 10.1103/PhysRevLett.102.073005
  63. 63.
    Tkatchenko A, DiStasio Jr. RA, Car R, Scheffler M (2012) Accurate and efficient method for many-body van der Waals interactions. Phys Rev Lett 108:236402. doi: 10.1103/PhysRevLett.108.235402
  64. 64.
    Tudor B, Space B (2013) Solving the many-body polarization problem in GPUs: application to MOF’s. J Comp Sci Educ 4:30–34Google Scholar
  65. 65.
    Massively Parallel Monte Carlo (MPMC) http://code.google.com/p/mpmc/
  66. 66.
    Saad Y (2011) Numerical methods for large eigenvalue problems. Society for Industrial and Applied Mathematics, http://www.siam.org/books/c166)
  67. 67.
    Lucas AA (1966) Contributions electrostatiques Collectives et corrections Radatives à l’Energie de Van der Waals des Cristaux Moléculaires. Thèse de doctorat. University of Liege (unpublished: Private communication)Google Scholar
  68. 68.
    Stone A (2013) The theory of intermolecular forces, 2nd edn. Oxford University Press, OxfordGoogle Scholar

Copyright information

© Springer Science+Business Media Singapore 2015

Authors and Affiliations

  1. 1.Department of Chemistry and PhysicsSoutheastern Louisiana UniversityHammondUSA

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