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

Cellulose volume 20, pages 2703–2718 (2013)Cite this article

Abstract

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.

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References

  • 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

    Article  CAS  Google Scholar 

  • 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):612–626

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Blöchl PE (1994) Projector augmented-wave method. Phys Rev B 50(24):17953–17979

    Article  Google Scholar 

  • Bower AF (2011) Applied mechanics of solids. CRC press, Boca Raton, FL

    Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • 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

  • 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

    Article  CAS  Google Scholar 

  • Finkenstadt VL, Millane RP (1998) Crystal structure of valonia cellulose Iβ. Macromolecules 31(22):7776–7783. doi:10.1021/ma9804895

    Article  CAS  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • Hafner J (2008) Ab-initio simulations of materials using VASP: density-functional theory and beyond. J Comput Chem 29(13):2044–2078

    Article  CAS  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • 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):014121

    Article  Google Scholar 

  • Heyd J, Scuseria GE, Ernzerhof M (2003) Hybrid functionals based on a Coulomb potential. J Chem Phys 118:8207

    Article  CAS  Google Scholar 

  • 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 124:219906

    Article  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • Jones RM (1975) Mechanics of composite materials, vol 2. Taylor & Francis, London

    Google Scholar 

  • Kohn W, Sham LJ (1965) Self-consistent equations including exchange and correlation effects. Phys Rev 140(4A):A1133

    Article  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • 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):11169–11186

    Article  CAS  Google Scholar 

  • Kresse G, Hafner J (1994) Ab initio molecular-dynamics simulation of the liquid-metal–amorphous-semiconductor transition in germanium. Phys Rev B 49(20):14251–14269

    Article  CAS  Google Scholar 

  • Lahiji RF, Xu X, , Reifenberger R, Raman A, Rudie A, Moon RJ (2010) Atomic force microscopy characterization of cellulose nanocrystals. Langmuir,  26(6): 4480–4488

    Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • 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):104104

    Article  Google Scholar 

  • Leslie WC (1981) The physical metallurgy of steels. Hempisphere Publishing Corporation

  • 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

    Article  CAS  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • Moon RJ, Martini A, Nairn J, Simonsen J, Youngblood J (2011) Cellulose nanomaterials review: structure, properties and nanocomposites. Chem Soc Rev 40(7):3941–3994

    Article  CAS  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • Pakzad A, Simonsen J, Heiden PA, Yassar RS (2012) Size effects on the nanomechanical properties of cellulose I nanocrystals. J Mater Res 27(3):528–536

    Article  CAS  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • Perdew JP, Burke K, Ernzerhof M (1996) Generalized gradient approximation made simple. Phys Rev Lett 77(18):3865–3868

    Article  CAS  Google Scholar 

  • Qi Y, Hector LG Jr (2004) Adhesion and adhesive transfer at aluminum/diamond interfaces: a first-principles study. Phys Rev B 69(23):235401

    Article  Google Scholar 

  • Qi Y, Hector LG (2007) Planar stacking effect on elastic stability of hexagonal boron nitride. Appl Phys Lett 90(8):081922–081923

    Article  Google Scholar 

  • 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–A566

    Article  CAS  Google Scholar 

  • Roberts R, Rowe R, York P (1994) The Poisson’s ratio of microcrystalline cellulose. Int J Pharm 105(2):177–180

    Article  CAS  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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):246001

    Article  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • Šturcová A, His I, Apperley DC, Sugiyama J, Jarvis MC (2004) Structural details of crystalline cellulose from higher plants. Biomacromolecules 5(4):1333–1339

    Article  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • Wada M, Nishiyama Y, Chanzy H, Forsyth T, Langan P (2008) The structure of celluloses. Powder Diffr 23(2):92–95

    Article  CAS  Google Scholar 

  • Wagner R, Moon R, Pratt J, Shaw G, Raman A (2011) Uncertainty quantification in nanomechanical measurements using the atomic force microscope. Nanotechnology 22(45):455703

    Article  Google Scholar 

  • 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):045507

    Article  CAS  Google Scholar 

  • 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):296–310

    Article  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • Zuluaga MG, Dri FL, Moon RJ, Zavattieri PD (2013a) Anisotropy calculator—3D visualization toolkit. https://nanohub.org/tools/matrix2surface

  • Zuluaga MG, Dri FL, Moon RJ, Zavattieri PD (2013b) Crystalline cellulose—atomistic toolkit. https://nanohub.org/tools/ccamt

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Acknowledgments

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.

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Authors and Affiliations

  1. School of Civil Engineering, Purdue University, 550 Stadium Mall Drive, West Lafayette, IN, 47907-2051, USA

    Fernando L. Dri & Pablo D. Zavattieri

  2. Chemical Sciences and Materials Systems Laboratory, General Motors Research and Development Center, Warren, MI, 48090, USA

    Louis G. Hector Jr.

  3. School of Materials Engineering and Birck Nanotechnology Center, Purdue University, West Lafayette, IN, 47907, USA

    Robert J. Moon

  4. USDA Forest Service, Forest Products Laboratory, Madison, WI, 53726, USA

    Robert J. Moon

Authors
  1. Fernando L. Dri
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  2. Louis G. Hector Jr.
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  3. Robert J. Moon
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  4. Pablo D. Zavattieri
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Correspondence to Pablo D. Zavattieri.

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Dri, F.L., Hector, L.G., Moon, R.J. et al. Anisotropy of the elastic properties of crystalline cellulose Iβ from first principles density functional theory with Van der Waals interactions. Cellulose 20, 2703–2718 (2013). https://doi.org/10.1007/s10570-013-0071-8

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  • DOI: https://doi.org/10.1007/s10570-013-0071-8

Keywords

  • Crystalline cellulose
  • Cellulose Iβ
  • Density functional theory
  • Young’s modulus