, Volume 20, Issue 6, pp 2703–2718 | Cite as

Anisotropy of the elastic properties of crystalline cellulose Iβ from first principles density functional theory with Van der Waals interactions

  • Fernando L. Dri
  • Louis G. HectorJr.
  • Robert J. Moon
  • Pablo D. ZavattieriEmail author
Original Paper


In spite of the significant potential of cellulose nanocrystals as functional nanoparticles for numerous applications, a fundamental understanding of the mechanical properties of defect-free, crystalline cellulose is still lacking. In this paper, the elasticity matrix for cellulose Iβ with hydrogen bonding network A was calculated using ab initio density functional theory with a semi-empirical correction for van der Waals interactions. The computed Young’s modulus is found to be 206 GPa along [001] (c-axis), 98 GPa along [010] (b-axis), and 19 GPa along [100] (a-axis). Full compliance matrices are reported for 1.0, 1.5 and 2.0 % applied strains Color contour surfaces that show variations of the Young’s modulus and average Poisson’s ratio with crystallographic direction revealed the extreme anisotropies of these important mechanical properties. The sensitivity of the elastic parameters to misalignments in the crystal were examined with 2D polar plots within selected planes containing specific bonding characteristics; these are used to explain the substantial variability in the reported experimental Young’s moduli values. Results for the lattice directions [001], [010] and [100] are within the range of reported experimental and other numerical values.


Crystalline cellulose Cellulose Iβ Density functional theory Young’s modulus 



The authors wish to acknowledge the staff of the High Performance Computing Center at General Motors. Additional computational resources, networking, and support were provided by GM Information Systems and Services. R.J.M. and P.D.Z. are also grateful to financial support by the Forest Products Laboratory under USDA Grants: 11-JV-11111129-086, 07-CR-11111120-093, the Purdue Research Foundation and National Science Foundation through Grant No. CMMI-1131596.


  1. Antony J, Grimme S (2006) Density functional theory including dispersion corrections for intermolecular interactions in a large benchmark set of biologically relevant molecules. Phys Chem Chem Phys 8(45):5287–5293. doi: 10.1039/b612585a CrossRefGoogle Scholar
  2. Azizi Samir MAS, Alloin F, Dufresne A (2005) Review of recent research into cellulosic whiskers, their properties and their application in nanocomposite field. Biomacromolecules 6(2):612–626CrossRefGoogle Scholar
  3. Bergenstråhle M, Berglund LA, Mazeau K (2007) Thermal response in crystalline Iβ cellulose: a molecular dynamics study. J Phys Chem B 111(30):9138–9145. doi: 10.1021/jp072258i CrossRefGoogle Scholar
  4. Blöchl PE (1994) Projector augmented-wave method. Phys Rev B 50(24):17953–17979CrossRefGoogle Scholar
  5. Bower AF (2011) Applied mechanics of solids. CRC press, Boca Raton, FLGoogle Scholar
  6. Bučko T, Hafner J, Lebègue S, Ángyán JG (2010) Improved description of the structure of molecular and layered crystals: ab initio DFT calculations with van der Waals corrections. J Phys Chem A 114(43):11814–11824. doi: 10.1021/jp106469x CrossRefGoogle Scholar
  7. Bučko T, Tunega D, Ángyán JG, Hafner J (2011) Ab initio study of structure and interconversion of native cellulose phases. J Phys Chem A 115(35):10097–10105. doi: 10.1021/jp205827y CrossRefGoogle Scholar
  8. Diddens I, Murphy B, Krisch M, Müller M (2008) Anisotropic elastic properties of cellulose measured using inelastic X-ray scattering. Macromolecules 41(24):9755–9759. doi: 10.1021/ma801796u CrossRefGoogle Scholar
  9. Dri F, Shang S, Hector LG Jr, Zi-Kui Liu, Moon RJ, Zavattieri PD (in preparation, 2013) Study of thermodynamic and mechanical properties of crystalline cellulose Google Scholar
  10. Eichhorn SJ, Davies GR (2006) Modelling the crystalline deformation of native and regenerated cellulose. Cellulose 13(3):291–307. doi: 10.1007/s10570-006-9046-3 CrossRefGoogle Scholar
  11. Finkenstadt VL, Millane RP (1998) Crystal structure of valonia cellulose Iβ. Macromolecules 31(22):7776–7783. doi: 10.1021/ma9804895 CrossRefGoogle Scholar
  12. Grimme S (2006) Semiempirical GGA-type density functional constructed with a long-range dispersion correction. J Comput Chem 27(15):1787–1799. doi: 10.1002/jcc.20495 CrossRefGoogle Scholar
  13. Hafner J (2008) Ab-initio simulations of materials using VASP: density-functional theory and beyond. J Comput Chem 29(13):2044–2078CrossRefGoogle Scholar
  14. Hector LG Jr, Herbst JF (2004) Electronic and elastic properties of RCo5 and RCo5Hn (R = La, Ce, Pr). J Alloy Compd 379(1–2):41–53. doi: 10.1016/j.jallcom.2004.02.042 CrossRefGoogle Scholar
  15. Hector LG Jr, Herbst JF, Capehart TW (2003) Electronic structure calculations for LaNi5 and LaNi5H7: energetics and elastic properties. J Alloy Compd 353(1–2):74–85. doi: 10.1016/s0925-8388(02)01324-5 CrossRefGoogle Scholar
  16. Hector L Jr, Herbst J, Wolf W, Saxe P, Kresse G (2007) Ab Initio thermodynamic and elastic properties of alkaline-earth metals and their hydrides. Phys Rev B 76(1):014121CrossRefGoogle Scholar
  17. Heyd J, Scuseria GE, Ernzerhof M (2003) Hybrid functionals based on a Coulomb potential. J Chem Phys 118:8207CrossRefGoogle Scholar
  18. Heyd J, Scuseria GE, Ernzerhof M (2006) Erratum:“Hybrid functionals based on a screened Coulomb potential”[J. Chem. Phys. 118, 8207 (2003)]. J Chem Phys 124:219906CrossRefGoogle Scholar
  19. Ishikawa A, Okano T, Sugiyama J (1997) Fine structure and tensile properties of ramie fibres in the crystalline form of cellulose I, II, IIII and IVI. Polymer 38(2):463–468. doi: 10.1016/S0032-3861(96)00516-2 CrossRefGoogle Scholar
  20. Jones RM (1975) Mechanics of composite materials, vol 2. Taylor & Francis, LondonGoogle Scholar
  21. Kohn W, Sham LJ (1965) Self-consistent equations including exchange and correlation effects. Phys Rev 140(4A):A1133CrossRefGoogle Scholar
  22. Kresse G, Furthmuller J (1996a) Efficiency of ab initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput Mater Sci 6(1):15–50. doi: 10.1016/0927-0256(96)00008-0 CrossRefGoogle Scholar
  23. Kresse G, Furthmuller J (1996b) Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys Rev B 54(16):11169–11186CrossRefGoogle Scholar
  24. Kresse G, Hafner J (1994) Ab initio molecular-dynamics simulation of the liquid-metal–amorphous-semiconductor transition in germanium. Phys Rev B 49(20):14251–14269CrossRefGoogle Scholar
  25. Lahiji RF, Xu X, , Reifenberger R, Raman A, Rudie A, Moon RJ (2010) Atomic force microscopy characterization of cellulose nanocrystals. Langmuir,  26(6): 4480–4488Google Scholar
  26. Langan P, Sukumar N, Nishiyama Y, Chanzy H (2005) Synchrotron X-ray structures of cellulose Iβ and regenerated cellulose II at ambient temperature and 100 K. Cellulose 12(6):551–562. doi: 10.1007/s10570-005-9006-3 CrossRefGoogle Scholar
  27. Le Page Y, Saxe P (2002) Symmetry-general least-squares extraction of elastic data for strained materials from ab initio calculations of stress. Phys Rev B 65(10):104104CrossRefGoogle Scholar
  28. Leslie WC (1981) The physical metallurgy of steels. Hempisphere Publishing Corporation Google Scholar
  29. Li Y, Lin M, Davenport JW (2011) Ab Initio studies of cellulose I: crystal structure, intermolecular forces, and interactions with water. J Phys Chem C 115(23):11533–11539. doi: 10.1021/jp2006759 CrossRefGoogle Scholar
  30. Matsuo M, Sawatari C, Iwai Y, Ozaki F (1990) Effect of orientation distribution and crystallinity on the measurement by X-ray diffraction of the crystal lattice moduli of cellulose I and II. Macromolecules 23(13):3266–3275. doi: 10.1021/ma00215a012 CrossRefGoogle Scholar
  31. Matthews JF, Beckham GT, Bergenstråhle-Wohlert M, Brady JW, Himmel ME, Crowley MF (2012) Comparison of cellulose Iβ simulations with three carbohydrate force fields. J Chem Theory Comput 8(2):735–748. doi: 10.1021/ct2007692 CrossRefGoogle Scholar
  32. Moon RJ, Martini A, Nairn J, Simonsen J, Youngblood J (2011) Cellulose nanomaterials review: structure, properties and nanocomposites. Chem Soc Rev 40(7):3941–3994CrossRefGoogle Scholar
  33. Nakamura KI, Wada M, Kuga S, Okano T (2004) Poisson’s ratio of cellulose Iβ and cellulose II. J Polym Sci Part B Polym Phys 42(7):1206–1211. doi: 10.1002/polb.10771 CrossRefGoogle Scholar
  34. Nishino T, Takano K, Nakamae K (1995) Elastic modulus of the crystalline regions of cellulose polymorphs. J Polym Sci Part B Polym Phys 33(11):1647–1651. doi: 10.1002/polb.1995.090331110 CrossRefGoogle Scholar
  35. Nishiyama Y, Langan P, Chanzy H (2002) Crystal structure and hydrogen-bonding system in cellulose Iβ from synchrotron X-ray and neutron fiber diffraction. J Am Chem Soc 124(31):9074–9082. doi: 10.1021/ja0257319 CrossRefGoogle Scholar
  36. Nishiyama Y, Sugiyama J, Chanzy H, Langan P (2003) Crystal structure and hydrogen bonding system in cellulose Iα from synchrotron X-ray and neutron fiber diffraction. J Am Chem Soc 125(47):14300–14306. doi: 10.1021/ja037055w CrossRefGoogle Scholar
  37. Nishiyama Y, Johnson GP, French AD, Forsyth VT, Langan P (2008) Neutron crystallography, molecular dynamics, and quantum mechanics studies of the nature of hydrogen bonding in cellulose Iβ. Biomacromolecules 9(11):3133–3140. doi: 10.1021/bm800726v CrossRefGoogle Scholar
  38. Nishiyama Y, Langan P, Wada M, Forsyth VT (2010) Looking at hydrogen bonds in cellulose. Acta Crystallogr Sect D 66(11):1172–1177. doi: 10.1107/S0907444910032397 CrossRefGoogle Scholar
  39. Pakzad A, Simonsen J, Heiden PA, Yassar RS (2012) Size effects on the nanomechanical properties of cellulose I nanocrystals. J Mater Res 27(3):528–536CrossRefGoogle Scholar
  40. Parthasarathi R, Bellesia G, Chundawat SPS, Dale BE, Langan P, Gnanakaran S (2011) Insights into hydrogen bonding and stacking interactions in cellulose. J Phys Chem A 115(49):14191–14202. doi: 10.1021/jp203620x CrossRefGoogle Scholar
  41. Perdew JP, Burke K, Ernzerhof M (1996) Generalized gradient approximation made simple. Phys Rev Lett 77(18):3865–3868CrossRefGoogle Scholar
  42. Qi Y, Hector LG Jr (2004) Adhesion and adhesive transfer at aluminum/diamond interfaces: a first-principles study. Phys Rev B 69(23):235401CrossRefGoogle Scholar
  43. Qi Y, Hector LG (2007) Planar stacking effect on elastic stability of hexagonal boron nitride. Appl Phys Lett 90(8):081922–081923CrossRefGoogle Scholar
  44. Qi Y, Guo H, Hector LG, Timmons A (2010) Threefold increase in the Young’s modulus of graphite negative electrode during lithium intercalation. J Electrochem Soc 157(5):A558–A566CrossRefGoogle Scholar
  45. Roberts R, Rowe R, York P (1994) The Poisson’s ratio of microcrystalline cellulose. Int J Pharm 105(2):177–180CrossRefGoogle Scholar
  46. Sakurada I, Nukushina Y, Ito T (1962) Experimental determination of the elastic modulus of crystalline regions in oriented polymers. J Polym Sci 57(165):651–660. doi: 10.1002/pol.1962.1205716551 CrossRefGoogle Scholar
  47. Sakurada I, Ito T, Nakamae K (1964) Elastic moduli of polymer crystals for the chain axial direction. Die Makromolekulare Chem 75(1):1–10. doi: 10.1002/macp.1964.020750101 CrossRefGoogle Scholar
  48. Santiago Cintrón M, Johnson G, French A (2011) Young’s modulus calculations for cellulose Iβ by MM3 and quantum mechanics. Cellulose 18(3):505–516. doi: 10.1007/s10570-011-9507-1 CrossRefGoogle Scholar
  49. Shang S, Hector L Jr, Wang Y, Zhang H, Liu Z (2009) First-principles study of elastic and phonon properties of the heavy fermion compound CeMg. J Phys: Condens Matter 21(24):246001CrossRefGoogle Scholar
  50. Shang S-L, Hector LG Jr, Shi S, Qi Y, Wang Y, Liu Z-K (2012) Lattice dynamics, thermodynamics and elastic properties of monoclinic Li2CO3 from density functional theory. Acta Mater 60(13–14):5204–5216. doi: 10.1016/j.actamat.2012.06.006 CrossRefGoogle Scholar
  51. Šturcová A, His I, Apperley DC, Sugiyama J, Jarvis MC (2004) Structural details of crystalline cellulose from higher plants. Biomacromolecules 5(4):1333–1339CrossRefGoogle Scholar
  52. Sugiyama J, Vuong R, Chanzy H (1991) Electron diffraction study on the two crystalline phases occurring in native cellulose from an algal cell wall. Macromolecules 24(14):4168–4175. doi: 10.1021/ma00014a033 CrossRefGoogle Scholar
  53. Tashiro K, Kobayashi M (1991) Theoretical evaluation of three-dimensional elastic constants of native and regenerated celluloses: role of hydrogen bonds. Polymer 32(8):1516–1526. doi: 10.1016/0032-3861(91)90435-L CrossRefGoogle Scholar
  54. Wada M (2002) Lateral thermal expansion of cellulose Iβ and IIII polymorphs. J Polym Sci Part B Polym Phys 40(11):1095–1102. doi: 10.1002/polb.10166 CrossRefGoogle Scholar
  55. Wada M, Nishiyama Y, Chanzy H, Forsyth T, Langan P (2008) The structure of celluloses. Powder Diffr 23(2):92–95CrossRefGoogle Scholar
  56. Wagner R, Moon R, Pratt J, Shaw G, Raman A (2011) Uncertainty quantification in nanomechanical measurements using the atomic force microscope. Nanotechnology 22(45):455703CrossRefGoogle Scholar
  57. Woodward C, Trinkle D, Hector L Jr, Olmsted D (2008) Prediction of dislocation cores in aluminum from density functional theory. Phys Rev Lett 100(4):045507CrossRefGoogle Scholar
  58. Wróbel J, Hector L Jr, Wolf W, Shang S, Liu Z, Kurzydłowski K (2012) Thermodynamic and mechanical properties of lanthanum–magnesium phases from density functional theory. J Alloy Compd 512(1):296–310CrossRefGoogle Scholar
  59. Wu X, Moon R, Martini A (2013) Crystalline cellulose elastic modulus predicted by atomistic models of uniform deformation and nanoscale indentation. Cellulose 20(1):43–55. doi: 10.1007/s10570-012-9823-0 CrossRefGoogle Scholar
  60. Zuluaga MG, Dri FL, Moon RJ, Zavattieri PD (2013a) Anisotropy calculator—3D visualization toolkit.
  61. Zuluaga MG, Dri FL, Moon RJ, Zavattieri PD (2013b) Crystalline cellulose—atomistic toolkit.

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Fernando L. Dri
    • 1
  • Louis G. HectorJr.
    • 2
  • Robert J. Moon
    • 3
    • 4
  • Pablo D. Zavattieri
    • 1
    Email author
  1. 1.School of Civil EngineeringPurdue UniversityWest LafayetteUSA
  2. 2.Chemical Sciences and Materials Systems LaboratoryGeneral Motors Research and Development CenterWarrenUSA
  3. 3.School of Materials Engineering and Birck Nanotechnology CenterPurdue UniversityWest LafayetteUSA
  4. 4.USDA Forest ServiceForest Products LaboratoryMadisonUSA

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