Skip to main content

Nanograins: I. Elasticity and Compressibility

  • Chapter
  • First Online:
Relaxation of the Chemical Bond

Part of the book series: Springer Series in Chemical Physics ((CHEMICAL,volume 108))

  • 2174 Accesses

Abstract

Skin-resolved energy density determines the elasticity of nanograins. Skin-resolved cohesive energy loss depresses the thermal stability. Elasticity of a specimen may rise or drop with size reduction, depending on the separation between the melting point and the temperature of operation, Tm–T. Heating lengthens and softens the representative bond and hence lowers the energy density and elasticity. Compression shortens and stiffens the representative bond and raises the elasticity.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 219.99
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. G. Ouyang, X.L. Li, X. Tan, G.W. Yang, Size-induced strain and stiffness of nanocrystals. Appl. Phys. Lett. 89(3), 031904 (2006)

    ADS  Google Scholar 

  2. G. Ouyang, X. Tan, G.W. Yang, Thermodynamic model of the surface energy of nanocrystals. Phys. Rev. B 74(19), 195408 (2006)

    ADS  Google Scholar 

  3. Y.L. Liu, Y. Zhang, H.B. Zhou, G.H. Lu, M. Kohyama, Theoretical strength and charge redistribution of fcc Ni in tension and shear. J. Phys.Condens. Matter 20(33), 335216 (2008)

    ADS  Google Scholar 

  4. U. Gysin, S. Rast, P. Ruff, E. Meyer, D.W. Lee, P. Vettiger, C. Gerber, Temperature dependence of the force sensitivity of silicon cantilevers. Phys. Rev. B 69(4), 045403 (2004)

    ADS  Google Scholar 

  5. J.B. Wachtman, W.E. Tefft, D.G. Lam, C.S. Apstein, Exponential temperature dependence of youngs modulus for several oxides. Phys. Rev. 122(6), 1754 (1961)

    ADS  Google Scholar 

  6. G.A. Alers, D.L. Waldorf, Variation of the elastic moduli at the superconducting transition. Phys. Rev. Lett. 6(12), 677–679 (1961)

    ADS  Google Scholar 

  7. A.K. Swarnakar, L. Donzel, J. Vleugels, O. Van der Biest, High temperature properties of ZnO ceramics studied by the impulse excitation technique. J. Eur. Ceram. Soc. 29(14), 2991–2998 (2009)

    Google Scholar 

  8. R. Chang, L.J. Graham, Low-temperature elastic properties of ZrC and TiC. J. Appl. Phys. 37(10), 3778–3783 (1966)

    ADS  Google Scholar 

  9. O.L. Anderson, Derivation of Wachtman’s equation for the temperature dependence of elastic moduli of oxide compounds. Phys. Rev. 144(2), 553–557 (1966)

    ADS  Google Scholar 

  10. J. Garai, A. Laugier, The temperature dependence of the isothermal bulk modulus at 1 bar pressure. J. Appl. Phys. 101(2), 2424535 (2007)

    Google Scholar 

  11. S. Rast, C. Wattinger, U. Gysin, E. Meyer, Dynamics of damped cantilevers. Rev. Sci. Instrum. 71(7), 2772–2775 (2000)

    ADS  Google Scholar 

  12. U. Rabe, K. Janser, W. Arnold, Vibrations of free and surface-coupled atomic force microscope cantilevers: Theory and experiment. Rev. Sci. Instrum. 67(9), 3281–3293 (1996)

    ADS  Google Scholar 

  13. J. Liu, Y.K. Vohra, Raman modes of 6H polytype of silicon carbide to ultrahigh pressures: a comparison with silicon and diamond. Phys. Rev. Lett. 72(26), 4105 (1994)

    ADS  Google Scholar 

  14. Z.W. Chen, C.Q. Sun, Y.C. Zhou, O.Y. Gang, Size dependence of the pressure-induced phase transition in nanocrystals. J. Chem. Phys. C 112(7), 2423–2427 (2008)

    Google Scholar 

  15. S. Desgreniers, High-density phases of ZnO: Structural and compressive parameters. Phys. Rev. B 58(21), 14102 (1998)

    ADS  Google Scholar 

  16. H. Sowa, H. Ahsbahs, High-pressure X-ray investigation of zincite ZnO single crystals using diamond anvils with an improved shape. J. Appl. Crystallogr. 39(2), 169–175 (2006)

    Google Scholar 

  17. F. Decremps, F. Datchi, A.M. Saitta, A. Polian, S. Pascarelli, A. Di Cicco, J.P. Iti, F. Baudelet, Local structure of condensed zinc oxide. Phys. Rev. B. 68(10), 015502 (2003)

    Google Scholar 

  18. H. Karzel, W. Potzel, M. Köfferlein, W. Schiessl, M. Steiner, U. Hiller, G.M. Kalvius, D.W. Mitchell, T.P. Das, P. Blaha, K. Schwarz, M.P. Pasternak, Lattice dynamics and hyperfine interactions in ZnO and ZnSe at high external pressures. Phys. Rev.B 53(17), 11425–11438 (1996)

    ADS  Google Scholar 

  19. J.W. Li, S.Z. Ma, X.J. Liu, Z.F. Zhou, C.Q. Sun, ZnO meso-mechano-thermo physical chemistry. Chem. Rev. 112(5), 2833–2852 (2012)

    Google Scholar 

  20. P.R. Couchman, F.E. Karasz, Effect of particle-size on Debye temperature. Phys. Lett. A 62(1), 59–61 (1977)

    ADS  Google Scholar 

  21. A. Balerna, S. Mobilio, Dynamic properties and debye temperatures of bulk au and au clusters studied using extended x-ray-absorption fine-structure spectroscopy. Phys. Rev. B 34(4), 2293–2298 (1986)

    ADS  Google Scholar 

  22. Y.H. Zhao, K. Lu, Grain-size dependence of thermal properties of nanocrystalline elemental selenium studied by x-ray diffraction. Phys. Rev.B 56(22), 14330–14337 (1997)

    ADS  Google Scholar 

  23. C.C. Yang, M.X. Xiao, W. Li, Q. Jiang, Size effects on Debye temperature, Einstein temperature, and volume thermal expansion coefficient of nanocrystals. Solid State Commun. 139(4), 148–152 (2006)

    ADS  Google Scholar 

  24. M.A. Shandiz, Effective coordination number model for the size dependency of physical properties of nanocrystals. J. Phys. Condens. Matter 20(32), 325237 (2008)

    Google Scholar 

  25. R.C.G. Killean, E.J. Lisher, Debye temperatures of cubic elements and their relationship to melting points. J. Phys. C-Solid State Phys. 8(21), 3510–3520 (1975)

    ADS  Google Scholar 

  26. S.A. Peng, G. Grimvall, Bonding and Debye temperatures in alkali earth-metal halides. J. Phys. Chem. Solids 55(8), 707–710 (1994)

    ADS  Google Scholar 

  27. C.J. Martin, D.A. Oconnor, Experimental test of Lindemanns melting law. J. Phys. C-Solid State Phys. 10(18), 3521–3526 (1977)

    ADS  Google Scholar 

  28. F.A. Lindemann, The calculation of molecular natural frequencies. Physikalische Zeitschrift 11, 609–612 (1910)

    MATH  Google Scholar 

  29. V. Novotny, J.H.P. Watson, P.P.M. Meincke, Effect of size and surface on specific-heat of small lead particles. Phys. Rev. Lett. 28(14), 901 (1972)

    ADS  Google Scholar 

  30. Q.L. Song, Z. Cui, S.H. Xia, S.F. Chen, An ac microcalorimeter for measuring specific heat of thin films. Microelectron. J. 35(10), 817–821 (2004)

    Google Scholar 

  31. Y. Lu, Q.L. Song, S.H. Xia, Calculation of specific heat for aluminium thin films. Chin. Phys. Lett. 22(9), 2346–2348 (2005)

    ADS  Google Scholar 

  32. J. Yu, Z.A. Tang, F.T. Zhang, G.F. Wei, L.D. Wang, Investigation of a microcalorimeter for thin-film heat capacity measurement. Chin. Phys. Lett. 22(9), 2429–2432 (2005)

    ADS  Google Scholar 

  33. K. Lu, Nanocrystalline metals crystallized from amorphous solids: Nanocrystallization, structure, and properties. Mater. Sci. Eng. R-Rep. 16(4), 161–221 (1996)

    Google Scholar 

  34. R.S. Prasher, P.E. Phelan, Non-dimensional size effects on the thermodynamic properties of solids. Int. J. Heat Mass Transf. 42(11), 1991–2001 (1999)

    MATH  Google Scholar 

  35. M.X. Gu, C.Q. Sun, Z. Chen, T.C.A. Yeung, S. Li, C.M. Tan, V. Nosik, Size, temperature, and bond nature dependence of elasticity and its derivatives on extensibility, Debye temperature, and heat capacity of nanostructures. Phys Rev B 75(12), 125403 (2007)

    ADS  Google Scholar 

  36. X.X. Yang, J.W. Li, Z.F. Zhou, Y. Wang, L.W. Yang, W.T. Zheng, C.Q. Sun, Raman spectroscopic determination of the length, strength, compressibility, Debye temperature, elasticity, and force constant of the C–C bond in graphene. Nanoscale 4(2), 502–510 (2012)

    ADS  Google Scholar 

  37. C.Q. Sun, Thermo-mechanical behavior of low-dimensional systems: The local bond average approach. Prog. Mater Sci. 54(2), 179–307 (2009)

    Google Scholar 

  38. M.A. Haque, M.T.A. Saif, Thermo-mechanical properties of nano-scale freestanding aluminum films. Thin Solid Films 484(1–2), 364–368 (2005)

    ADS  Google Scholar 

  39. Y.H. Wu, F.J. McGarry, B.Z. Zhu, J.R. Keryk, D.E. Katsoulis, Temperature effect on mechanical properties of toughened silicone resins. Polym. Eng. Sci. 45(11), 1522–1531 (2005)

    Google Scholar 

  40. B. Chen, D. Penwell, L.R. Benedetti, R. Jeanloz, M.B. Kruger, Particle-size effect on the compressibility of nanocrystalline alumina. Phys. Rev.B 66(14), 144101 (2002)

    ADS  Google Scholar 

  41. S. Vennila, S.R. Kulkarni, S.K. Saxena, H.P. Liermann, S.V. Sinogeikin, Compression behavior of nanosized nickel and molybdenum. Appl. Phys. Lett. 89(26), 261901 (2006)

    ADS  Google Scholar 

  42. F. Szuecs, M. Werner, R.S. Sussmann, C.S.J. Pickles, H.J. Fecht, Temperature dependence of Young’s modulus and degradation of chemical vapor deposited diamond. J. Appl. Phys. 86(11), 6010–6017 (1999)

    ADS  Google Scholar 

  43. N. Ono, R. Nowak, S. Miura, Effect of deformation temperature on Hall-Petch relationship registered for polycrystalline magnesium. Mater. Lett. 58(1–2), 39–43 (2004)

    Google Scholar 

  44. D.G. Eskin, A. Suyitno, L. Katgerman, Mechanical properties in the semi-solid state and hot tearing of aluminium alloys. Prog. Mater Sci. 49(5), 629–711 (2004)

    Google Scholar 

  45. S.B. Qadri, J. Yang, B.R. Ratna, E.F. Skelton, J.Z. Hu, Pressure induced structural transitions in nanometer size particles of PbS. Appl. Phys. Lett. 69(15), 2205–2207 (1996)

    ADS  Google Scholar 

  46. L. Lu, M.L. Sui, K. Lu, Superplastic extensibility of nanocrystalline copper at room temperature. Science 287(5457), 1463–1466 (2000)

    ADS  Google Scholar 

  47. K.S. Siow, A.A.O. Tay, P. Oruganti, Mechanical properties of nanocrystalline copper and nickel. Mater. Sci. Technol. 20(3), 285–294 (2004)

    Google Scholar 

  48. W.W. Gerberich, W.M. Mook, C.R. Perrey, C.B. Carter, M.I. Baskes, R. Mukherjee, A. Gidwani, J. Heberlein, P.H. McMurry, S.L. Girshick, Superhard silicon nanospheres. J. Mech. Phys. Solids 51(6), 979–992 (2003)

    ADS  Google Scholar 

  49. C.Q. Chen, Y. Shi, Y.S. Zhang, J. Zhu, Y.J. Yan, Size dependence of Young’s modulus in ZnO nanowires. Phys. Rev. Lett. 96(7), 075505 (2006)

    ADS  Google Scholar 

  50. X.X. Li, T. Ono, Y.L. Wang, M. Esashi, Ultrathin single-crystalline-silicon cantilever resonators: Fabrication technology and significant specimen size effect on Young’s modulus. Appl. Phys. Lett. 83(15), 3081–3083 (2003)

    ADS  Google Scholar 

  51. C.Q. Sun, B.K. Tay, S.P. Lau, X.W. Sun, X.T. Zeng, S. Li, H.L. Bai, H. Liu, Z.H. Liu, E.Y. Jiang, Bond contraction and lone pair interaction at nitride surfaces. J. Appl. Phys. 90(5), 2615–2617 (2001)

    ADS  Google Scholar 

  52. Q.F. Gu, G. Krauss, W. Steurer, F. Gramm, A. Cervellino, Unexpected high stiffness of Ag and Au nanoparticles. Phys. Rev. Lett. 100(4), 045502 (2008)

    ADS  Google Scholar 

  53. M.X. Gu, Y.C. Zhou, C.Q. Sun, Local bond average for the thermally induced lattice expansion. J. Phys. Chem. B 112(27), 7992–7995 (2008)

    Google Scholar 

  54. F.H. Streitz, R.C. Cammarata, K. Sieradzki, Surface-stress effects on elastic properties 1 Thin metal-films. Phys. Rev. B 49(15), 10699–10706 (1994)

    ADS  Google Scholar 

  55. M.T. McDowell, A.M. Leach, K. Gaill, On the elastic modulus of metallic nanowires. Nano Lett. 8(11), 3613–3618 (2008)

    ADS  Google Scholar 

  56. J. Li, Y.X. Li, X. Yu, W.J. Ye, C.Q. Sun, Local bond average for the thermally driven elastic softening of solid specimens. J. Phys. D-Appl. Phys. 42(4), 045406 (2009)

    ADS  Google Scholar 

  57. C.D. Martin, S.M. Antao, P.J. Chupas, P.L. Lee, S.D. Shastri, J.B. Parise, Quantitative high-pressure pair distribution function analysis of nanocrystalline gold. Appl. Phys. Lett. 86(6), 061910 (2005)

    ADS  Google Scholar 

  58. K. Syassen, W.B. Holzapfel, Isothermal compression of Al and Ag to 120 KBar. J. Appl. Phys. 49(8), 4427–4430 (1978)

    ADS  Google Scholar 

  59. D.L. Heinz, R. Jeanloz, The equation of state of the gold calibration standard. J. Appl. Phys. 55(4), 885–893 (1984)

    ADS  Google Scholar 

  60. J. Menendez, M. Cardona, Temperature-dependence of the 1st-order raman-scattering by phonons in Si, Ge, and a-Sn-anharmonic effects. Phys. Rev. B 29(4), 2051–2059 (1984)

    ADS  Google Scholar 

  61. M.E. Fine, Elasticity and thermal expansion of germinium between 195 deg-C 275 deg-C. J. Appl. Phys. 24(3), 338–340 (1953)

    ADS  Google Scholar 

  62. T.R. Hart, R.L. Aggarwal, B. Lax, Temperature dependence of Raman scattering in silicon. Phys. Rev.B-Solid State 1(2), 638 (1970)

    ADS  Google Scholar 

  63. M. Balkanski, R.F. Wallis, E. Haro, Anharmonic effects in light-scattering due to optical phonons in silicon. Phys. Rev. B 28(4), 1928–1934 (1983)

    ADS  Google Scholar 

  64. M.S. Liu, L.A. Bursill, S. Prawer, R. Beserman, Temperature dependence of the first-order Raman phonon lime of diamond. Phys. Rev. B 61(5), 3391–3395 (2000)

    ADS  Google Scholar 

  65. J.B. Cui, K. Amtmann, J. Ristein, L. Ley, Noncontact temperature measurements of diamond by Raman scattering spectroscopy. J. Appl. Phys. 83(12), 7929–7933 (1998)

    ADS  Google Scholar 

  66. H. Herchen, M.A. Cappelli, 1 st-order Raman-spectrum of diamond at high-temperatures. Phys. Rev. B 43(14), 11740–11744 (1991)

    ADS  Google Scholar 

  67. E.S. Zouboulis, M. Grimsditch, Raman-scattering in diamond up to 1900-K. Phys. Rev. B 43(15), 12490–12493 (1991)

    ADS  Google Scholar 

  68. D.A. Czaplewski, J.P. Sullivan, T.A. Friedmann, J.R. Wendt, Temperature dependence of the mechanical properties of tetrahedrally coordinated amorphous carbon thin films. Appl. Phys. Lett. 87(16), 2108132 (2005)

    Google Scholar 

  69. M.X. Gu, Y.C. Zhou, L.K. Pan, Z. Sun, S.Z. Wang, C.Q. Sun, Temperature dependence of the elastic and vibronic behavior of Si, Ge, and diamond crystals. J. Appl. Phys. 102(8), 083524 (2007)

    ADS  Google Scholar 

  70. C. Kittel, Introduction to Solid State Physics, 8th edn. (Willey, New York, 2005)

    Google Scholar 

  71. C.O. Girit, J.C. Meyer, R. Erni, M.D. Rossell, C. Kisielowski, L. Yang, C.H. Park, M.F. Crommie, M.L. Cohen, S.G. Louie, A. Zettl, Graphene at the edge: Stability and dynamics. Science 323(5922), 1705–1708 (2009)

    ADS  Google Scholar 

  72. W.T. Zheng, C.Q. Sun, Underneath the fascinations of carbon nanotubes and graphene nanoribbons. Energy Environ. Sci. 4(3), 627–655 (2011)

    Google Scholar 

  73. X. Zhang, Y.G. Nie, W.T. Zheng, J.L. Kuo, C.Q. Sun, Discriminative generation and hydrogen modulation of the Dirac-Fermi polarons at graphene edges and atomic vacancies. Carbon 49(11), 3615–3621 (2011)

    Google Scholar 

  74. K.S. Novoselov, Z. Jiang, Y. Zhang, S.V. Morozov, H.L. Stormer, U. Zeitler, J.C. Maan, G.S. Boebinger, P. Kim, A.K. Geim, Room-temperature quantum hall effect in graphene. Science 315(5817), 1379 (2007)

    ADS  Google Scholar 

  75. K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Electric field effect in atomically thin carbon films. Science 306(5696), 666–669 (2004)

    ADS  Google Scholar 

  76. X.X. Yang, J.W. Li, Z.F. Zhou, Y. Wang, W.T. Zheng, C.Q. Sun, Frequency response of graphene phonons to heating and compression. Appl. Phys. Lett. 99(13), 133108 (2011)

    ADS  Google Scholar 

  77. Y. Wang, X.X. Yang, J.W. Li, Z.F. Zhou, W.T. Zheng, C.Q. Sun, Number-of-layer discriminated graphene phonon softening and stiffening. Appl. Phys. Lett. 99(16), 163109 (2011)

    ADS  Google Scholar 

  78. D. Graf, F. Molitor, K. Ensslin, C. Stampfer, A. Jungen, C. Hierold, L. Wirtz, Spatially resolved raman spectroscopy of single- and few-layer graphene. Nano Lett. 7(2), 238–242 (2007)

    ADS  Google Scholar 

  79. A. Gupta, G. Chen, P. Joshi, S. Tadigadapa, P.C. Eklund, Raman scattering from high-frequency phonons in supported n-graphene layer films. Nano Lett. 6(12), 2667–2673 (2006)

    ADS  Google Scholar 

  80. H. Wang, Y. Wang, X. Cao, M. Feng, G. Lan, Vibrational properties of graphene and graphene layers. J. Raman Spectrosc. 40(12), 1791–1796 (2009)

    ADS  Google Scholar 

  81. A.K. Gupta, T.J. Russin, H.R. Gutierrez, P.C. Eklund, Probing graphene edges via Raman scattering. ACS Nano 3(1), 45–52 (2008)

    Google Scholar 

  82. C. Thomsen, S. Reich, Double resonant Raman scattering in graphite. Phys. Rev. Lett. 85(24), 5214–5217 (2000)

    ADS  Google Scholar 

  83. T.M.G. Mohiuddin, A. Lombardo, R.R. Nair, A. Bonetti, G. Savini, R. Jalil, N. Bonini, D.M. Basko, C. Galiotis, N. Marzari, K.S. Novoselov, A.K. Geim, A.C. Ferrari, Uniaxial strain in graphene by Raman spectroscopy: G peak splitting, Gruneisen parameters, and sample orientation. Phys. Rev. B 79(20), 205433 (2009)

    ADS  Google Scholar 

  84. F. Ding, H. Ji, Y. Chen, A. Herklotz, K. Dorr, Y. Mei, A. Rastelli, O.G. Schmidt, Stretchable Graphene: A close look at fundamental parameters through biaxial straining. Nano Lett. 10(9), 3453–3458 (2010)

    ADS  Google Scholar 

  85. C.Q. Sun, Y. Sun, Y.G. Nie, Y. Wang, J.S. Pan, G. Ouyang, L.K. Pan, Z. Sun, Coordination-resolved C–C bond length and the c 1s binding energy of carbon allotropes and the effective atomic coordination of the few-layer Graphene. J. Chem. Phys. C 113(37), 16464–16467 (2009)

    Google Scholar 

  86. M. He, E. Rikkinen, Z. Zhu, Y. Tian, A.S. Anisimov, H. Jiang, A.G. Nasibulin, E.I. Kauppinen, M. Niemela, A.O.I. Krause, Temperature dependent Raman spectra of carbon nanobuds. J. Phys. Chem. C 114(32), 13540–13545 (2010)

    Google Scholar 

  87. M. Matus, H. Kuzmany, Raman spectra of single-crystal C < sub > 60 </sub&gt. Appl. Phys. A Mater. Sci. Process. 56(3), 241–248 (1993)

    ADS  Google Scholar 

  88. I. Calizo, A.A. Balandin, W. Bao, F. Miao, C.N. Lau, Temperature Dependence of the Raman Spectra of Graphene and Graphene Multilayers. Nano Lett. 7(9), 2645–2649 (2007)

    ADS  Google Scholar 

  89. N.J. Everall, J. Lumsdon, D.J. Christopher, The effect of laser-induced heating upon the vibrational raman spectra of graphites and carbon fibres. Carbon 29(2), 133–137 (1991)

    Google Scholar 

  90. I. Calizo, F. Miao, W. Bao, C.N. Lau, A.A. Balandin, Variable temperature Raman microscopy as a nanometrology tool for graphene layers and graphene-based devices. Appl. Phys. Lett. 91(7), 071913–071916 (2007)

    ADS  Google Scholar 

  91. Z. Zhou, X. Dou, L. Ci, L. Song, D. Liu, Y. Gao, J. Wang, L. Liu, W. Zhou, S. Xie, D. Wan, Temperature dependence of the Raman spectra of individual carbon nanotubes. J. Phys. Chem. B 110(3), 1206–1209 (2006)

    Google Scholar 

  92. S. Chiashi, Y. Murakami, Y. Miyauchi, S. Maruyama, Temperature dependence of raman scattering from single-walled carbon nanotubes: Undefined radial breathing mode peaks at high temperatures. Jap. J. Appl. Phys. 47(4), 2010–2015 (2008)

    ADS  Google Scholar 

  93. H. Boppart, J. van Straaten, I.F. Silvera, Raman spectra of diamond at high pressures. Phys. Rev. B 32(2), 1423–1425 (1985)

    ADS  Google Scholar 

  94. A. Merlen, N. Bendiab, P. Toulemonde, A. Aouizerat, A. San Miguel, J.L. Sauvajol, G. Montagnac, H. Cardon, P. Petit, Resonant Raman spectroscopy of single-wall carbon nanotubes under pressure. Phys. Rev. B 72(3), 035409–035415 (2005)

    ADS  Google Scholar 

  95. J.E. Proctor, E. Gregoryanz, K.S. Novoselov, M. Lotya, J.N. Coleman, M.P. Halsall, High-pressure Raman spectroscopy of graphene. Phys. Rev. B. 80(7), 073408 (2009)

    ADS  Google Scholar 

  96. M. Hanfland, H. Beister, K. Syassen, Graphite under pressure: Equation of state and first-order Raman modes. Phys. Rev. B 39(17), 12598–12603 (1989)

    ADS  Google Scholar 

  97. X.X. Yang, Z.F. Zhou, Y. Wang, J.W. Li, N.G. Guo, W.T. Zheng, J.Z. Peng, C.Q. Sun, Raman spectroscopic determination of the length, energy, Debye temperature, and compressibility of the C–C bond in carbon allotropes. Chem. Phys. Lett. 575, 86–90 (2013)

    ADS  Google Scholar 

  98. F.C. Marques, R.G. Lacerda, A. Champi, V. Stolojan, D.C. Cox, S.R.P. Silva, Thermal expansion coefficient of hydrogenated amorphous carbon. Appl. Phys. Lett. 83(15), 3099–3101 (2003)

    ADS  Google Scholar 

  99. R. Kalish, Ion-implantation in diamond and diamond films: Doping, damage effects and their applications. Appl. Surf. Sci. 117, 558–569 (1997)

    ADS  Google Scholar 

  100. S. Reich, C. Thomsen, P. Ordejon, Elastic properties of carbon nanotubes under hydrostatic pressure. Phys. Rev. B 65(15), 153407–153411 (2002)

    ADS  Google Scholar 

  101. D.L. Farber, J. Badro, C.M. Aracne, D.M. Teter, M. Hanfland, B. Canny, B. Couzinet, Experimental evidence for a high-pressure isostructural phase transition in osmium. Phys. Rev. Lett. 93(9), 095502–095506 (2004)

    ADS  Google Scholar 

  102. W.R. Panero, R. Jeanloz, X-ray diffraction patterns from samples in the laser-heated diamond anvil cell. J. Appl. Phys. 91(5), 2769–2778 (2002)

    ADS  Google Scholar 

  103. R. Agrawal, B. Peng, E.E. Gdoutos, H.D. Espinosa, Elasticity size effects in ZnO nanowires—a combined experimental-computational approach. Nano Lett. 8(11), 3668–3674 (2008)

    ADS  Google Scholar 

  104. J. Wang, A.J. Kulkarni, F.J. Ke, Y.L. Bai, M. Zhou, Novel mechanical behavior of ZnO nanorods. Comput. Meth. Appl. Mech. Eng. 197(41–42), 3182–3189 (2008)

    ADS  MATH  Google Scholar 

  105. W. Moon, H. Hwang, Atomistic study of structures and elastic properties of single crystalline ZnO nanotubes. Nanotechnology 19(22), 225703 (2008)

    ADS  Google Scholar 

  106. G. Cao, X. Chen, Energy analysis of size-dependent elastic properties of ZnO nanofilms using atomistic simulations. Phys. Rev.B 76(16), 165407 (2007)

    ADS  Google Scholar 

  107. G.X. Cao, X. Chen, Size dependence and orientation dependence of elastic properties of ZnO nanofilms. Int. J. Solids Struct. 45(6), 1730–1753 (2008)

    MATH  Google Scholar 

  108. J.S. Qi, D.N. Shi, J.M. Jia, First-principles studies of the electronic and mechanical properties of ZnO nanobelts with different dominant surfaces. Nanotechnology 19(43), 435707 (2008)

    ADS  Google Scholar 

  109. F. Decremps, J. Pellicer-Porres, A.M. Saitta, J.-C. Chervin, A. Polian, High-pressure Raman spectroscopy study of wurtzite ZnO. Phys. Rev.B 65(9), 092101 (2002)

    ADS  Google Scholar 

  110. J. Serrano, A.H. Romero, F.J. Manjon, R. Lauck, M. Cardona, A. Rubio, Pressure dependence of the lattice dynamics of ZnO: An ab initio approach. Phys. Rev.B 69(9), 094306 (2004)

    ADS  Google Scholar 

  111. H. Alawadhi, S. Tsoi, X. Lu, A.K. Ramdas, M. Grimsditch, M. Cardona, R. Lauck, Effect of temperature on isotopic mass dependence of excitonic band gaps in semiconductors: ZnO. Phys. Rev.B 75(20), 205207 (2007)

    ADS  Google Scholar 

  112. V.V. Ursaki, I.M. Tiginyanu, V.V. Zalamai, E.V. Rusu, G.A. Emelchenko, V.M. Masalov, E.N. Samarov, Multiphonon resonant Raman scattering in ZnO crystals and nanostructured layers. Phys. Rev.B 70(15), 155204 (2004)

    ADS  Google Scholar 

  113. S.H. Eom, Y.M. Yu, Y.D. Choi, C.S. Kim, Optical characterization of ZnO whiskers grown without catalyst by hot wall epitaxy method. J. Cryst. Growth 284(1–2), 166–171 (2005)

    ADS  Google Scholar 

  114. R. Hauschild, H. Priller, M. Decker, J. Bruckner, H. Kalt, C. Klingshirn, Temperature dependent band gap and homogeneous line broadening of the exciton emission in ZnO. Phys. Status Solidi C 3(4), 976–979 (2006)

    ADS  Google Scholar 

  115. Y. Fei, S. Cheng, L.B. Shi, H.K. Yuan, Phase transition, elastic property and electronic structure of wurtzite and rocksalt ZnO. J. Synth. Cryst. 38(6), 1527–1531 (2009)

    Google Scholar 

  116. E.S.F. Neto, N.O. Dantas, S.W.d Silva, P.C. Morais, M.A.P.d Silva, A.J.D. Moreno, V.L. Richard, G.E. Marques, C.T. Giner, Temperature-dependent Raman study of thermal parameters in CdS quantum dots. Nanotechnology 23(12), 125701 (2012)

    ADS  Google Scholar 

  117. J. Zhang, Z. Peng, A. Soni, Y. Zhao, Y. Xiong, B. Peng, J. Wang, M.S. Dresselhaus, Q. Xiong, Raman spectroscopy of few-quintuple layer topological insulator Bi2Se3 nanoplatelets. Nano Lett. 11(6), 2407–2414 (2011)

    ADS  Google Scholar 

  118. D.S. Chuu, C.M. Dai, Quantum size effects in CdS thin films. Phys. Rev. B 45(20), 11805–11810 (1992)

    ADS  Google Scholar 

  119. A. Tanaka, S. Onari, T. Arai, Raman scattering from CdSe microcrystals embedded in a germanate glass matrix. Phys. Rev. B 45(12), 6587 (1992)

    ADS  Google Scholar 

  120. Y.N. Hwang, S.H. Park, D. Kim, Size-dependent surface phonon mode of CdSe quantum dots. Phys. Rev. B 59(11), 7285 (1999)

    ADS  Google Scholar 

  121. Y. Kim, X. Chen, Z. Wang, J. Shi, I. Miotkowski, Y.P. Chen, P.A. Sharma, A.L.L. Sharma, M.A. Hekmaty, Z. Jiang, D. Smirnov, Temperature dependence of Raman-active optical phonons in Bi[sub 2]Se[sub 3] and Sb[sub 2]Te[sub 3]. Appl. Phys. Lett. 100(7), 071907 (2012)

    ADS  Google Scholar 

  122. X.X. Yang, Z.F. Zhou, Y. Wang, R. Jiang, W.T. Zheng, C.Q. Sun, Raman spectroscopy determination of the Debye temperature and atomic cohesive energy of CdS, CdSe, Bi2Se3, and Sb2Te3 nanostructures. J. Appl. Phys. 112(8), 4759207 (2012)

    Google Scholar 

  123. J. Rockenberger, L. Tröger, A. Kornowski, T. Vossmeyer, A. Eychmüller, J. Feldhaus, H. Weller, EXAFS studies on the size dependence of structural and dynamic properties of CdS nanoparticles. J. Phys. Chem. B 101(14), 2691 (1997)

    Google Scholar 

  124. J.S. Dyck, W. Chen, C. Uher, Č. Drašar, and P. Lošt’ák, Heat transport in \( {\text{Sb}}\_\left\{ {2 - x} \right\}V\_\left\{ x \right\}T{\text{e}}\_\left\{ 3 \right\} \) single crystals. Phys. Rev. B. 66(12), 125206 (2002)

    Google Scholar 

  125. G.E. Shoemake, J.A. Rayne, R.W.J. Ure, Specific heat of n- and p-type Bi_{2}Te_{3} from 1.4 to 90°K. Phys. Rev. 185(3), 1046 (1969)

    ADS  Google Scholar 

  126. P. Perlin, A. Polian, T. Suski, Raman-scattering studies of aluminum nitride at high-pressure. Phys. Rev. B 47(5), 2874–2877 (1993)

    ADS  Google Scholar 

  127. M. Kuball, J.M. Hayes, A.D. Prins, N.W.A. van Uden, D.J. Dunstan, Y. Shi, J.H. Edgar, Raman scattering studies on single-crystalline bulk AlN under high pressures. Appl. Phys. Lett. 78(6), 724–726 (2001)

    ADS  Google Scholar 

  128. M. Ueno, A. Onodera, O. Shimomura, K. Takemura, X-ray-observation of the structural phase-transition of aluminium nitride under high-pressure. Phys. Rev. B 45(17), 10123–10126 (1992)

    ADS  Google Scholar 

  129. M. Kuball, J.M. Hayes, Y. Shi, J.H. Edgar, A.D. Prins, N.W.A. van Uden, D.J. Dunstan, Raman scattering studies on single-crystalline bulk AlN: temperature and pressure dependence of the AlN phonon modes. J. Cryst. Growth 231(3), 391–396 (2001)

    ADS  Google Scholar 

  130. M.P. Halsall, P. Harmer, P.J. Parbrook, S.J. Henley, Raman scattering and absorption study of the high-pressure wurtzite to rocksalt phase transition of GaN. Phys. Rev. B 69(23), 235207 (2004)

    ADS  Google Scholar 

  131. P. Perlin, C. Jauberthiecarillon, J.P. Itie, A. SanMiguel, I. Grzegory, A. Polian, Raman-scattering and x-ray-absorption spectroscopy in gallium nitride under high-pressure. Phys. Rev. B 45(1), 83–89 (1992)

    ADS  Google Scholar 

  132. S. Limpijumnong, W.R.L. Lambrecht, Homogeneous strain deformation path for the wurtzite to rocksalt high-pressure phase transition in GaN. Phys. Rev. Lett. 86(1), 91–94 (2001)

    ADS  Google Scholar 

  133. A. Link, K. Bitzer, W. Limmer, R. Sauer, C. Kirchner, V. Schwegler, M. Kamp, D.G. Ebling, K.W. Benz, Temperature dependence of the E-2 and A(1)(LO) phonons in GaN and AlN. J. Appl. Phys. 86(11), 6256–6260 (1999)

    ADS  Google Scholar 

  134. C. Pinquier, F. Demangeot, J. Frandon, J.C. Chervin, A. Polian, B. Couzinet, P. Munsch, O. Briot, S. Ruffenach, B. Gil, B. Maleyre, Raman scattering study of wurtzite and rocksalt InN under high pressure. Phys. Rev. B 73(11), 115211 (2006)

    ADS  Google Scholar 

  135. X.D. Pu, J. Chen, W.Z. Shen, H. Ogawa, Q.X. Guo, Temperature dependence of Raman scattering in hexagonal indium nitride films. J. Appl. Phys. 98(3), 2006208 (2005)

    Google Scholar 

  136. M.X. Gu, L.K. Pan, T.C.A. Yeung, B.K. Tay, C.Q. Sun, Atomistic origin of the thermally driven softening of Raman optical phonons in group III nitrides. J. Chem. Phys. C 111(36), 13606–13610 (2007)

    Google Scholar 

  137. G. Ouyang, C.Q. Sun, W.G. Zhu, Pressure-stiffened Raman phonons in group III nitrides: A local bond average approach. J. Phys. Chem. B 112(16), 5027–5031 (2008)

    Google Scholar 

  138. L.K. Pan, C.Q. Sun, C.M. Li, Elucidating Si–Si dimmer vibration from the size-dependent Raman shift of nanosolid Si. J. Phys. Chem. B 108(11), 3404–3406 (2004)

    Google Scholar 

  139. R.R. Reeber, K. Wang, Lattice parameters and thermal expansion of GaN. J. Mater. Res. 15(1), 40–44 (2000)

    ADS  Google Scholar 

  140. G.A. Slack, S.F. Bartram, Thermal expansion of some diamond-like crystals. J. Appl. Phys. 46(1), 89–98 (1975)

    ADS  Google Scholar 

  141. Y.L. Du, Y. Deng, M.S. Zhang, Variable-temperature Raman scattering study on anatase titanium dioxide nanocrystals. J. Phys. Chem. Solids 67(11), 2405–2408 (2006)

    ADS  Google Scholar 

  142. A.Y. Kuznetsov, R. Machado, L.S. Gomes, C.A. Achete, V. Swamy, Size dependence of rutile TiO2 lattice parameters determined via simultaneous size, strain, and shape modeling. Appl. Phys. Lett. 94, 193117 (2009)

    ADS  Google Scholar 

  143. V. Swamy, A. Kuznetsov, L.S. Dubrovinsky, R.A. Caruso, D.G. Shchukin, B.C. Muddle, Finite-size and pressure effects on the Raman spectrum nanocrystalline anatse TiO2. Phys. Rev. B 71(18), 184302 (2005)

    ADS  Google Scholar 

  144. V. Swamy, Size-dependent modifications of the first-order Raman spectra of nanostructured rutile TiO_ {2}. Phys. Rev. B 77(19), 195414 (2008)

    ADS  Google Scholar 

  145. S. Sahoo, A.K. Arora, V. Sridharan, Raman line shapes of optical phonons of different symmetries in anatase TiO2 nanocrystals. J. Phys. Chem. C 113(39), 16927–16933 (2009)

    Google Scholar 

  146. V. Swamy, D. Menzies, B.C. Muddle, A. Kuznetsov, L.S. Dubrovinsky, Q. Dai, V. Dmitriev, Nonlinear size dependence of anatase TiO2 lattice parameters. Appl. Phys. Lett. 88(24), 243103 (2006)

    ADS  Google Scholar 

  147. L. Dai, C.H. Sow, C.T. Lim, W.C.D. Cheong, V.B.C. Tan, Numerical Investigations into the Tensile Behavior of TiO2 Nanowires: Structural Deformation, Mechanical Properties, and Size Effects. Nano Lett. 9(2), 576–582 (2009)

    ADS  Google Scholar 

  148. J. Zhu, J.X. Yu, Y.J. Wang, X.R. Chen, F.Q. Jing, First-principles calculations for elastic properties of rutile TiO2 under pressure. Chin. Phys. B 17(6), 2216 (2008)

    ADS  Google Scholar 

  149. V. Swamy, A.Y. Kuznetsov, L.S. Dubrovinsky, A. Kurnosov, V.B. Prakapenka, Unusual Compression Behavior of Anatase TiO2 Nanocrystals. Phys. Rev. Lett. 103(7), 75505 (2009)

    ADS  Google Scholar 

  150. X.J. Liu, L.W. Yang, Z.F. Zhou, P.K. Chu, C.Q. Sun, Inverse Hall-Petch relationship of nanostructured TiO2: Skin-depth energy pinning versus surface preferential melting. J. Appl. Phys. 108, 073503 (2010)

    ADS  Google Scholar 

  151. X.J. Liu, L.K. Pan, Z. Sun, Y.M. Chen, X.X. Yang, L.W. Yang, Z.F. Zhou, C.Q. Sun, Strain engineering of the elasticity and the Raman shift of nanostructured TiO2. J. Appl. Phys. 110(4), 044322 (2011)

    ADS  Google Scholar 

  152. A.Y. Wu, R.J. Sladek, Elastic Debye temperatures in tetragonal crystals: Their determination and use. Phys. Rev. B 25(8), 5230 (1982)

    ADS  Google Scholar 

  153. B. Chen, H. Zhang, K. Dunphy-Guzman, D. Spagnoli, M. Kruger, D. Muthu, M. Kunz, S. Fakra, J. Hu, Q. Guo, J. Banfield, Size-dependent elasticity of nanocrystalline titania. Phys. Rev. B 79(12), 125406 (2009)

    ADS  Google Scholar 

  154. F. Birch, Finite elastic strain of cubic crystals. Phys. Rev. 71(11), 809–824 (1947)

    ADS  MATH  Google Scholar 

  155. F.D. Murnaghan, The compressibility of media under extreme pressures. Proc. Nat. Acad. Sci. 30(9), 244–247 (1944)

    ADS  MATH  MathSciNet  Google Scholar 

  156. N. Iles, A. Kellou, K.D. Khodja, B. Amrani, F. Lemoigno, D. Bourbie, H. Aourag, Atomistic study of structural, elastic, electronic and thermal properties of perovskites Ba(Ti, Zr, Nb)O3. Comput. Mater. Sci. 39(4), 896–902 (2007)

    Google Scholar 

  157. X.J. Liu, L.K. Pan, Z. Sun, X.H. Wang, J. Zhou, L.T. Li, C.Q. Sun, Mechanically stiffened and thermally softened bulk modulus of BaXO(3) (X = Ti, Zr, Nb) cubic perovskites. J. Appl. Phys. 109(3), 033511 (2011)

    ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Chang Q Sun .

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media Singapore

About this chapter

Cite this chapter

Sun, C.Q. (2014). Nanograins: I. Elasticity and Compressibility. In: Relaxation of the Chemical Bond. Springer Series in Chemical Physics, vol 108. Springer, Singapore. https://doi.org/10.1007/978-981-4585-21-7_27

Download citation

Publish with us

Policies and ethics