Cellulose

, Volume 20, Issue 6, pp 2703–2718

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

Authors

  • Fernando L. Dri
    • School of Civil EngineeringPurdue University
  • Louis G. HectorJr.
    • Chemical Sciences and Materials Systems LaboratoryGeneral Motors Research and Development Center
  • Robert J. Moon
    • School of Materials Engineering and Birck Nanotechnology CenterPurdue University
    • USDA Forest ServiceForest Products Laboratory
    • School of Civil EngineeringPurdue University
Original Paper

DOI: 10.1007/s10570-013-0071-8

Cite this article as:
Dri, F.L., Hector, L.G., Moon, R.J. et al. Cellulose (2013) 20: 2703. doi:10.1007/s10570-013-0071-8

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.

Keywords

Crystalline celluloseCellulose IβDensity functional theoryYoung’s modulus

Introduction

Mechanical properties of cellulose nanocrystals (CNCs) are difficult to experimentally characterize owing largely to extreme anisotropy and uncertainties about the structure of these materials. For example, reported experimental values for the Young modulus of cellulose Iβ show a wide variation that is hard to explain considering the defect-free crystalline structure typically observed in CNCs (Sakurada et al. 1962, 1964; Matsuo et al. 1990; Roberts et al. 1994; Nishino et al. 1995; Ishikawa et al. 1997; Diddens et al. 2008; Wagner et al. 2011; Pakzad et al. 2012; Lahiji et al. 2010). Unfortunately, there is no standardization of the coordinate system and nomenclature used to measure the elastic moduli making quantitative comparisons of the elastic behavior between experiments and theory rather difficult. Most authors agree on defining both an axial or longitudinal Young modulus, EA, which is aligned with the longitudinal axis of the cellulose Iβ unit cell, and an additional Young modulus, perpendicular to the longitudinal axis, usually reported as the transverse modulus, ET. Early studies using X-ray diffraction (Sakurada et al. 1962, 1964; Matsuo et al. 1990; Nishino et al. 1995; Ishikawa et al. 1997) measured values of EA ranging from 90 to 138 GPa. Recently, Diddens et al. (2008) reported values of EA = 220 ± 50 GPa and ET = 15 ± 1 GPa using inelastic X-ray scattering (IXS). Diddens et al. (2008) claimed that IXS was not affected by the amorphous zones occurring in natural cellulose, and the elastic behavior was mostly related to the highly crystalline regions. Alternatively, larger uncertainties have been reported from atomic force microscopy (AFM) measurements of cellulose Iβ elastic properties. For example, Lahiji et al. (2010) and Wagner et al. (2011) reported ET = 8.1 GPa with a 95 % confidence, which translates into an interval ranging from 2.7 to 20 GPa.

A typical AFM indentation test of a CNC particle on a hard substrate is shown in Fig. 1a. As described by Lahiji et al. (2010), this test is used to measure ET. Figure 1b shows schematics of the direction of the load with respect to the expected crystallographic directions (a, b and c). Relating these experimental measurements to a specific crystallographic orientation is challenging since it is difficult to estimate the alignment of the CNC with respect to the substrate (Lahiji et al. 2010). Moreover, with no additional information on the direction assigned to ET, it may not be possible to properly interpret the experimental data and, therefore, any attempt to compare experiments with computational predictions is difficult. Uncertainties about the shape of the CNC after sample preparation and the determination of the crystallographic planes during the experiments contribute to this important challenge.
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Fig. 1

a Atomic force microscope topography image of a tunicate CNC showing its rod-like shape (on a mica substrate) (Wagner et al. 2011). The blue cross near the bottom of the figure denotes a location for AFM tip indentation. b Schematic of a CNC particle during AFM indentation for illustration purposes. Here, the cellulose chains are represented by straight ribbons, and the crystallographic directions indicated as a, b and c. The inset shows details of the layered cellulose structure where red spheres denote oxygen ions, gray spheres represent carbon ions and white spheres represent hydrogen ions. The red dotted lines indicate the repeating unit cell. (Color figure online)

In the present study, we quantify the anisotropy of the Young’s modulus and Poisson’s ratio of monoclinic cellulose Iβ using ab initio first principles density functional theory (DFT) (Kohn and Sham 1965) with a semi-empirical correction for van der Waals interactions (Bučko et al. 2010). The least squares fitting method of Le Page and Saxe (2002) is used to compute all unique components of the elastic stiffness and compliance matrices using the VASP code (Kresse and Hafner 1994; Kresse and Furthmuller 1996a, b; Hafner 2008) as the DFT computational engine. This information enables analysis of the variations of the Young’s modulus and Poisson’s ratio with crystallographic orientation. These variations are displayed as surfaces which are color contours showing crystallographic dependence of these properties. Polar plots, which show the Young’s modulus and Poisson’s ratio variation with angular orientation in a given plane within the cellulose Iβ crystal structure, are used to: (1) explain the substantial variability in the literature experimental data of the Young’s modulus for cellulose, and (2) understand the Poisson’s effect in selected planes.

Background

Cellulose [[C6H10O5]n, n = 10,000–15,000, where n depends upon the cellulose source material (Azizi Samir et al. 2005)] is an organic compound that can be described as a linear chain of glucose rings with a flat ribbon-like conformation. Each chain is formed by one-hundred to over ten-thousand β (1 → 4) linked d-glucose units; van der Waals (vdW) and intermolecular hydrogen bonds promote parallel stacking of multiple cellulose chains within the crystal (Nishiyama et al. 2008; Wada et al. 2008) forming the characteristic layered structure along the a crystallographic direction (see Fig. 2a).
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Fig. 2

Expanded views of the P21 unit cell structure of the cellulose Iβ network A showing the characteristic layered conformation (Zuluaga et al. 2013b). Experimental (room temperature) lattice parameters a, b, c, from Nishiyama et al. (2002) are shown. Red spheres denote oxygen ions, gray spheres represent carbon ions and white spheres represent hydrogen ions. Dotted blue lines denote the unit cell. a View along the c-axis (perpendicular to the page). Layers of Iβ are stacked along the a-axis. b View along the a-axis direction. Atomic coordinates were obtained after applying symmetry operations to the original structure reported by Nishiyama et al. (2002). Intra- and intermolecular hydrogen bonds are depicted in green and orange respectively, according to the hydrogen bond network A pattern reported in Refs. Nishiyama et al. (2008) and Šturcová et al. (2004). The symbol // in this figure means “parallel to.” For example, a//[001]//1 means that the crystallographic direction a, with Miller indices [001], is parallel to the Cartesian axis 1. (Color figure online)

The most basic classification method divides crystalline cellulose types into 4 basic polymorphs that are identified as I, II, III or IV, each one having its own subtype (Moon et al. 2011). Cellulose I, also called native cellulose, has a mix of two polymorphs, viz., cellulose Iα, which has a triclinic P1 (#1) structure, and Iβ, which has a monoclinic P21 (#4) structure, that coexist in various proportions depending on the source of the CNC (Nishiyama et al. 2002, 2003). The Iα structure is the dominant polymorph in most algae and bacteria, whereas Iβ is the dominant polymorph for higher plant cell wall cellulose and in tunicates (Parthasarathi et al. 2011; Moon et al. 2011). A further classification of cellulose I can be based on the hydrogen bond network patterns, A and B, proposed by Nishiyama et al. (2002). The relative occupancies of the two networks are different according to the polymorph: network A occupies ~70–80 % of all the chain positions in Iβ, but only ~55 % in Iα (Nishiyama et al. 2003, 2008). This study focuses on cellulose Iβ with network A since it is the most commonly occurring polymorph in higher plant cell wall cellulose and in tunicates.

The crystal structure and the hydrogen bond system in cellulose Iβ have been characterized by Nishiyama et al. (2002, 2003), Langan et al. (2005), Nishiyama et al. (2008, 2010). Here, we adopt the atomic coordinates for the cellulose Iβ network A reported by Nishiyama et al. (2002). To account for the atomic positions inside the unit cell, we take advantage of the symmetry and antisymmetry operations provided by the crystallographic space group, commonly accepted to be monoclinic P21 (Sugiyama et al. 1991). Each crystallographic cell contains two molecular chains with a total of 42 ions per chain (84 ions per crystallographic cell). Note that the formula unit (f.u.) for this structure is C6O5H10; there are two f.u.’s in the primitive cell and four in the crystallographic cell. Figure 2 depicts the crystalline structure reported by Nishiyama et al. (2002) after the symmetry operations are applied to the atomic coordinates. The represented structure was constructed using the Crystalline celluloseatomistic toolkit (Zuluaga et al. 2013b). Intra- and inter-molecular hydrogen bonds are depicted in Fig. 2b following the hydrogen bond network A pattern reported in (Nishiyama et al. 2008; Šturcová et al. 2004). Cellulose chains are organized in hydrogen bonded planes (in the bc plane) that are stacked together and held in position primarily by weak vdW interactions. Out-of-plane intermolecular hydrogen bonds, connecting cellulose chains in different planes, have also been reported (Matthews et al. 2012). Lattice parameters and crystallographic directions are superimposed in Fig. 2a, b for reference purposes. To facilitate our predictions of the anisotropy of the Young’s modulus and Poisson’s ratio of monoclinic cellulose Iβ, we define a Cartesian system of coordinates 1, 2 and 3. Direction 1 is chosen to be parallel to a ([001]), and direction 3 is parallel to c ([001]). For the monoclinic P21 structure, b is not orthogonal to a. Therefore, direction 2 is chosen such that it is orthogonal to directions 1 and 3 as shown in Fig. 2a, b.

The lattice parameters for cellulose have been measured by several authors (Langan et al. 2005; Nishiyama et al. 2002, 2003, 2008, 2010; Sakurada et al. 1962, 1964; Matsuo et al. 1990; Sugiyama et al. 1991; Finkenstadt and Millane 1998) using different experimental techniques and crystal sources. For the cellulose Iβ network A structure, Nishiyama et al. (2002) reported: a = 7.784 Å, b = 8.201 Å, c = 10.380 Å, α = 90°, β = 90°, γ = 96.55°, with a 658.3 Å3 volume at 293 K. Most of the measured lattice parameters exhibit variations around 1 % over a wide range of temperatures and crystalline sources, except for the lattice parameter a. As cellulose Iβ is cooled or heated, the lattice remains remarkably constant in the directions within the hydrogen bonded planes containing the chains (i.e. b and c); the same is not true along the a axis direction where the contractions or expansions are controlled primarily by weak vdW interactions and interplanar hydrogen bonding (Wada et al. 2008; Langan et al. 2005). Nishiyama et al. (2008) reported a change from 7.64 to 7.76 Å in the a lattice parameter when the temperature was raised from 15 to 295 K. Langan et al. (2005) reported a value of 7.83 Å for the same parameter at 298 K. Although hydrogen bond interactions are present along the stacking direction (a-axis), they apparently do not prevent thermal expansion at temperatures up to the transition to a high-temperature phase at ~200 °C (Wada 2002; Bergenstråhle et al. 2007). This thermal sensitivity of the lattice parameter a should be kept in mind when comparing 0 K ab initio calculations with experimental values acquired at temperatures above 0 K. The temperature variation of the cellulose Iβ structure is outside the scope of the current study, but it is reported elsewhere (Dri et al. 2013). Experimental lattice parameter and cell volume values, as well as ab initio simulation results, are summarized in Table 1.
Table 1

Ab initio and experimental lattice parameters (a, b, c, and γ) of cellulose Iβ from different sources are listed

References

Methodology

Type of cellulose

a

[Å]

b

[Å]

c

[Å]

γ

[Deg]

Volume

3]

Temp [K]

Sakurada et al. (1962)

X-Ray diffraction (XRD)

Bleached ramie

(cellulose I)

7.9

8.35

10.3

96

675.7a

Sugiyama et al. (1991)

Electron diffraction

Green alga

8.01

8.17

10.36

97.3

672.5a

293

Finkenstadt and Millane (1998)

XRD

Valonia

(cellulose Iβ)

7.85

8.27

10.38

96.3

669.8a

7.82

8.16

10.32

97.5

652.9a

Nishiyama et al. (2002)

XRD and neutron fiber diffraction (NFD)

Tunicate

(cellulose Iβ)

7.784

8.201

10.38

96.5

658.3

293

Langan et al. (2005)

XRD

Tunicate

(cellulose Iβ)

7.76

8.19

10.38

96.51

655.4

100

7.83

8.19

10.38

96.55

661.3

298

Nishiyama et al. (2008)

XRD and NFD

Tunicate

(cellulose Iβ)

7.64

8.18

10.37

96.54

643.9

15

7.76

8.20

10.37

96.62

655.5

295

Bučko et al. (2010)

PBE

Cellulose Iβ

8.70

8.23

10.46

95.5

744.9

0

PBE-D2

7.65

8.14

10.39

96.8

642.5

Bučko et al. (2011)

PBE

Cellulose Iβ network A

8.37

8.23

10.45

96.0

716

0

PBE-D2

7.57

8.14

10.39

96.5

636

PBE-D2

Cellulose Iβ network B

7.51

8.55

10.30

98.2

655

Li et al. (2011)

LDA

Cellulose Iβ

7.41

7.94

10.24

96.2

599.0a

0

PBE

Not binding

8.27

10.54

94.7

PBE-D

7.85

8.18

10.47

96.6

667.9a

This study

PBE-D2

Cellulose Iβ network A

7.56

8.13

10.39

96.7

635

0

The double lines separate experimental results from simulations. Exchange–correlation functionals used in the ab initio simulations are: LDA (local density approximation), PBE (the generalized gradient approximation of Perdew–Burke–Ernzerhof or GGA-PBE) (Perdew et al. 1996); PBE-D and PBE-D2 (GGA-PBE with van der Waals corrections) (Kresse and Hafner 1994; Kresse and Furthmuller 1996a, b). See the “Computational methodology” section for additional details. The temperature at which each experiment was performed is reported in the last column. The symbol “–” means that the temperature was not reported

aNot reported, computed based on the lattice parameters

Elastic stiffness matrix Cij and compliance matrix Sij for crystalline cellulose Iβ

Materials that exhibit directional independence of their mechanical properties are referred to as isotropic. They have the advantage that their elastic response is characterized by only two parameters: the Young modulus (E) and the Poisson’s ratio (ν). In addition, the stress–strain tensile behavior will be completely independent of the relative orientation of the specimen with respect to the loading direction. Anisotropic materials, however, exhibit variations in their mechanical properties with respect to an intrinsic direction within the material. One of the most common examples of a naturally-occurring anisotropic material is wood, were the grain is oriented along a particular direction. This significantly affects the mechanical response of the material. Hence, it is necessary to define a more general stress–strain relation for anisotropic solids that will account for mechanical property variations with direction within the material. The most general linear stress–strain relation for anisotropic materials is Hooke’s Law, which has the form: σij = Cijklεkl, where σij and εkl represent the components of the second-order stress and strain tensors, respectively, and Cijkl are the components of the fourth-order elasticity tensor with 81 components (Bower 2011). The inverse relation can also be written as εij = Sijklσkl, where Sijkl are the components of the elastic compliance tensor. These relationships can be converted to a matrix form based upon symmetries in Cijkl and Sijkl, leading to a stiffness matrix with components Cij and a compliance matrix with components Sij, both with 21 independent components (Bower 2011).

The monoclinic space group of cellulose Iβ dictates that the unit cell has a symmetry plane that is defined by the c-axis. The presence of a symmetry plane has a direct consequence over the Cij and Sij in that it reduces the number of components necessary to represent the system. This leaves only 13 independent constants in each of Cij and Sij. Lastly, the matrix representation of the stress–strain relation must follow a coordinate system that allows a direct relationship between the matrix components and the cellulose Iβ structure. Figure 3a shows a schematic representation of the main directions associated with the cellulose Iβ unit cell and Sij associated with the Cartesian system of coordinates denoted by axes 1, 2 and 3.
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Fig. 3

a Schematic representation of the cellulose Iβ monoclinic (P21) unit cell aligned with the Cartesian coordinate system used in this study (red solid lines). A rectangular prism cell (black dashed lines) is used to help visualize the orthogonally between the ac and bc axes, highlighting the non-orthogonal relationship between a and b. b Relationship between the strain and stress vectors via Sij (Jones 1975) (εij = Sijklσkl) for the P21 cellulose Iβ unit cell. Eii is the Young’s modulus in the i-direction, Gij is the shear modulus in the ij plane, νij is the Poisson’s ratio which quantifies the contraction in the j-direction due to uniaxial loading in the i-direction, ηij,k are the coefficients of mutual influence of the first kind which characterize normal strain in the k-direction due to shear stress in the ij plane, and μij,kl are Chentsov’s coefficients which characterize shear strain in the kl plane due to shear stress in the ij plane. (Color figure online)

Figure 3b shows Sij in terms of the Young’s Modulus and Poisson’s ratio in each direction for the P21 cellulose Iβ unit cell (also appropriate for any monoclinic space group) (Jones 1975). With this information, three-dimensional surfaces can be constructed that are color contours of the Young’s Modulus and Poisson’s ratio. These contours follow variations of these properties with crystallographic direction. Additionally, 2D polar plots of the Young’s Modulus and Poisson’s ratio can be constructed which enable analysis of the variations of the Young’s Modulus and Poisson’s ratio with respect to a particular orientation angle within a given plane in the cellulose Iβ unit cell.

Several relevant directions are depicted in Fig. 3a. For instance, E11, E22 and E33 are the Young’s moduli with respect to the directions 1, 2 and 3, respectively. It is important to note that E11 and E33 are the Young’s moduli defined as the slope of the stress–strain curve produced in simple tension when the load is applied parallel to axes along a and c, respectively. Alternatively, E22 is defined along a direction determined by the cross product between the a and c-axes (i.e. ac plane) (see Figs. 2 and 3a). In the ab plane, two additional values of interest are defined, viz., E[110] and E[010], in which the subscripted indices represent the crystallographic direction defined by the Miller indices. Note that E[110] is along the axis that runs through the center chain in the unit cell (see Fig. 2a), whereas E[010] is aligned with the b-axis. The Poisson’s ratio is also reported using a similar nomenclature; for example, ν12 is the Poisson’s ratio for the contraction in the 2-direction due to uniaxial loading in the 1-direction. It is common to report an average value of the Poisson’s ratio using the two perpendicular directions with respect to the loading direction. Hence, only one index is needed. For the direction 1, this is defined as
$$\bar{\nu }_{1} = \frac{{\nu_{12} + \nu_{13} }}{2}$$
(1)

Computational methodology

All calculations in this study were conducted with the Vienna Ab initio Simulation Package (VASP), a plane wave DFT code (Kresse and Hafner 1994; Kresse and Furthmuller 1996a, b; Hafner 2008). The electron–ion interactions were described by the full potential projector augmented wave (PAW) method (Blöchl 1994). Exchange–correlation was treated within the generalized gradient approximation of Perdew, Burke and Ernzerhof (GGA-PBE) (Perdew et al. 1996). Standard density functionals within the GGA or LDA cannot correctly describe vdW interactions resulting from dynamical correlations between fluctuating charge distributions (Bučko et al. 2010). This makes them intrinsically unsuitable for computing structural parameters of cellulose Iβ. Using PBE functionals, Bučko et al. (2010, 2011) reported a value of a that is overestimated by ~15 % compared to experimental results at 15 K (Nishiyama et al. 2008). Li et al. (2011) found that cellulose fails to retain its crystalline structure when using PBE functionals (see Table 1 for more information). In order to circumvent this problem, a semi-empirical correction for the vdW interactions [now incorporated as PBE-D and PBE-D2 in VASP (Kresse and Hafner 1994; Kresse and Furthmuller 1996a, b)] was proposed by Grimme and coworkers (Grimme 2006; Antony and Grimme 2006). Using this dispersion-corrected DFT method (PBE-D2), Bučko et al. (2010, 2011), Li et al. (2011) and Parthasarathi et al. (2011) showed that vdW and hydrogen bonding interactions play an equally important role in defining the final shape of the cellulose Iβ monoclinic (P21) structure and hence they cannot be neglected. Therefore, all calculations in this study are conducted using the dispersion corrected PBE-D2 in VASP (Bučko et al. 2010, 2011).

Structural parameters and mechanical property calculations were computed by simultaneously minimizing all atomic forces and stress tensor components via a conjugate gradient method. The results presented in this work were obtained using a simulation cell corresponding to one crystallographic cell of cellulose Iβ, following the structure reported by Nishiyama et al. (2002). Three successive full-cell optimizations were conducted (adapting basis vectors and computational grids to the cell parameters) to ensure convergence of cell energies and structural parameters. Total energies were calculated for the relaxed cellulose Iβ structure by integrating over a Monkhorst–Pack mesh of k-points in the Brillouin zone with the linear tetrahedron method with Blöchl corrections. The plane wave cutoff energy for all calculations was 500 eV. The total energy was converged to 10−7 eV/cell and the force components were relaxed to at least 10−4 eV/Å. For all calculations (i.e. structural and elastic properties), a 7 × 7 × 7 k-point mesh, corresponding to a k-point spacing of 0.110 × 0.086 × 0.110 per Angstrom, was used.

Components of the stiffness matrix, Cij, were computed from the first derivatives of the stresses computed in VASP, rather than from the second derivatives of the total energy with respect to strain, using the Le Page and Saxe least squares method (Le Page and Saxe 2002). This method avoids the numerical difficulties often encountered with evaluations of the latter and reduces the number of required VASP calculations. All Cij values were computed simultaneously rather than as independent sums. The Cij are sensitive to the k-point mesh, and this required a series of ancillary calculations to test k-point convergence of each of the 13 unique Cij for the monoclinic cellulose Iβ structure. In addition, it was determined that the application of four successive strains, viz., 0.05, 1.0 1.5, and 2.0 % was adequate to obtain ≤1.0 % statistical error in each Cij. The quality of the least squares fit, as gauged by the computed least squares residual, was ≤1.0 % for all calculations. The small residuals are consistent with negligible anharmonic effects in the computed Cij due to the applied strains. Note that the Le Page and Saxe method for computing elastic properties with DFT has been successfully used to compute elastic properties for a wide variety of materials, including hydrides (Hector et al. 2003, 2007; Hector and Herbst 2004), batteries (Qi et al. 2010; Shang et al. 2012), ceramics (Qi and Hector 2004; Qi and Hector 2007), metals (Shang et al. 2009; Wróbel et al. 2012) and defects (Woodward et al. 2008). Once the stiffness matrix was computed, it was subsequently inverted to obtain the compliance matrix, Sij. As explained in the previous section, the Cij and the Sij depend on the definition of the coordinate system chosen for the simulations. We use rotation techniques, such as those detailed in Bower (2011), which allow us to convert the computed compliance matrix to any desired orientation. The basis change follows from
$$S^{{\left( {1^{\prime } 2^{\prime } 3^{\prime } } \right)}} = K^{ - T} S^{{\left( {123} \right)}} K^{ - 1}$$
(2)
For the particular case of rotation through an angle θ in a counterclockwise sense about the 1, 2 and 3 axes, respectively, the rotation matrix K reduces to
$$\left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & {c^{2} } & {s^{2} } & {2cs} & 0 & 0 \\ 0 & {s^{2} } & {c^{2} } & { - 2cs} & 0 & 0 \\ 0 & { - cs} & {cs} & {c^{2} - s^{2} } & 0 & 0 \\ 0 & 0 & 0 & 0 & c & { - s} \\ 0 & 0 & 0 & 0 & s & c \\ \end{array} } \right]\quad \left[ {\begin{array}{*{20}c} {c^{2} } & 0 & {s^{2} } & 0 & {2cs} & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ {s^{2} } & 0 & {c^{2} } & 0 & { - 2cs} & 0 \\ 0 & 0 & 0 & c & 0 & { - s} \\ { - cs} & 0 & {cs} & 0 & {c^{2} - s^{2} } & 0 \\ 0 & 0 & 0 & s & 0 & c \\ \end{array} } \right]\quad \left[ {\begin{array}{*{20}c} {c^{2} } & {s^{2} } & 0 & 0 & 0 & {2cs} \\ {s^{2} } & {c^{2} } & 0 & 0 & 0 & { - 2cs} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & c & s & 0 \\ 0 & 0 & 0 & { - s} & c & 0 \\ { - cs} & {cs} & 0 & 0 & 0 & {c^{2} - s^{2} } \\ \end{array} } \right]$$
(3)
where c = cos(θ) and s = sin(θ). Clearly, applying the three rotations successively can produce an arbitrary orientation change. This provides the basis to construct the 3D surface contour plots and polar plots of the angular variation of the Young's Modulus and Poisson's ratio. .

Results and discussion

Results from VASP calculations with the semi-empirical correction for the vdW interactions were used to generate the surface contour plot of the Young’s modulus variation with crystallographic direction shown in Fig. 4a. This is based upon Eqs. (2) and (3) and the Sij computed from application of 1.0 % strain in the Le Page and Saxe method (Le Page and Saxe 2002). A post processing software package, the Anisotropy Calculator3D Visualization Toolkit, was specifically developed to generate the surface contour plot of the Young’s modulus based on Sij and is now publically available (Zuluaga et al. 2013a). Each point on the surface represents the magnitude of Young’s modulus in the direction of a vector from the origin of the surface (i.e. at the intersection of the 1, 2 and 3 axes in the interior of the surface) to a given point on the surface. The shape of this surface is indicative of the anisotropy of cellulose Iβ. For instance, the computed Young modulus surface for a linearly elastic isotropic material would be a perfect sphere with the same value in any direction. However, the cellulose Iβ surface in Fig. 4a exhibits extreme variations in the Young’s modulus, as denoted by the accentuated contour lobe along the 3-axis (i.e. along the cellulose chains) relative to the smaller lobes along the 1 and 2 directions. The largest values (red contours) are along the 3-axis, with the smallest values along the 1-axis. The greatest value of the Young’s modulus is 206 GPa, which is comparable to that of steel (~207 GPa) (Leslie 1981). Figure 4b, c and d show side views of the same surface in Fig. 4a to put in evidence the differences in Young’s moduli for directions lying on the 12, 13 and 23 planes in better perspective.
https://static-content.springer.com/image/art%3A10.1007%2Fs10570-013-0071-8/MediaObjects/10570_2013_71_Fig4_HTML.gif
Fig. 4

a Surfaces showing contours of computed Young’s modulus values for cellulose Iβ based on an Sij from application of a 1 % strain in the Le Page and Saxe method (Le Page and Saxe 2002). Each point on the surface represents the magnitude of the Young’s modulus in the direction of a vector from the origin of the surface to that point. The color contours help to identify the Young modulus variation of cellulose Iβ and emphasizes its extreme anisotropy (note the significant elongation of the surface along axis 3). Side views of the same surface are shown for the b12 plane, c13 plane, d23 plane. Note that axis 2 is not seen in these additional views. (Color figure online)

Polar plots of the angular variation of the Young's modulus within a given crystallographic plane of the monoclinic cellulose Iβ crystal structure are shown in Figs. 5 and 6. These plots can be used to help provide insight into the variability in the reported experimental values (ranging from 90 to 220 GPa) of the Young’s modulus. Figure 5 shows the angular variation of the Young’s modulus along the 13 plane (as shown by the gray plane in the inset on the upper left of the figure). Three axes are considered: direction 1 (which is the vertical axis), direction 3 (the horizontal axis) and the semi-circular line showing the angle with respect to the origin. These are the same axes defined in Figs. 2 and 3a. The scale of the vertical axis denotes the magnitude of E11, whereas the scale of the horizontal axis denotes the magnitude of E33. The inset in the bottom semi-circular part shows the orientation of the directions 1 and 3 with respect to the cellulose Iβ unit cell. Here, θ is the angle between the 3′-direction, along which the load is applied, and the 3-direction (in the 13 plane). The plot is generated by computing Sij for different angles using Eqs. (2) and (3), and extracting the Young modulus value in the 3′-direction from the rotated compliance matrix. Components of the Sij, computed in the Cartesian coordinate system shown in Fig. 3a were obtained following the Le Page and Saxe method (Le Page and Saxe 2002) for three values of applied strains, viz. 1.0 1.5, and 2.0 %. A change of only 10° in the longitudinal alignment (c-axis) reduces the DFT-predicted Young modulus from 206 to ~70 GPa. This considerable reduction is related to the deformation mechanism in which the cellulose Iβ structure is under simple tension in each of the 1, 2, 3 directions.
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Fig. 5

Angular variation of the longitudinal Young’s modulus (EA) within the 13 plane for applied strains of 1.0, 1.5, and 2.0 %. The 2-axis (perpendicular to the page) is considered to be the rotation axis; the value of the longitudinal modulus for a given direction can be read directly from the figure by defining a straight line from the origin to the desired angle. It is important to notice how small misalignments between the cellulose Iβc-axis and the 3-direction will produce an important reduction in the interpretation of EA during experimental characterization. The inset in the bottom semi-circular part shows the orientation of the 1 and 3 directions with respect to the cellulose Iβ unit cell

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Fig. 6

Angular variation of the ET at applied strains of 1.0, 1.5, and 2.0 %. The 3-axis (pointing perpendicular to the page) is considered to be the rotation axis; the value of ET for a given direction can be read directly from the figure by defining a straight line from the origin to the desired angle. The intersection between the straight line and the curves for different strains provides the value of ET. Three critical directions (// to b, through the center chain and ⊥ to b) are marked for reference using Miller indices; the direction parallel to the a-axis coincides with the vertical axis of the figure. The superposition of the three strain curves allow us to concluded that the crystal behaves linearly for strains up to 2 %. The inset in the bottom semi-circular part shows the orientation of the 1 and 2 directions with respect to the cellulose Iβ unit cell

For instance, imposing a deformation perfectly aligned with the c-direction involves a series of very complex deformation mechanisms. Among them we found stretching of the covalent bonds between C and O ions that form the cellulose chain (shown as the ball and stick features Fig. 2), stretching of intra-molecular hydrogen bonds (shown as green dashed lines in Fig. 2), angle bending, rotations and distortions. This results in the highest value of the Young modulus (206 GPa) along the c-direction (the 3-direction). In the a-direction (which corresponds to the 1-direction), hydrogen bonded planes are stacked together and held in position primarily by weak vdW interactions. Any deviation of the stretching direction in the 13 plane will produce a rapid decrease in the Young’s modulus due to sliding between adjacent planes. In the b-direction, the intermolecular hydrogen bonds (See Fig. 2b) provide additional reinforcement to keep the cellulose chains from sliding. As a consequence, the effects of misalignments on the resulting Young’s modulus in any direction lying on the 23 plane are less severe compared with those in the 13 plane. Note that deviations between the curves from the applied strains, viz. 1.0, 1.5, and 2.0 % are very minimal, as it can be observed in Figs. 5 and 6.

Upon comparing our DFT-computed results with experimental data (Table 2), E33 can be regarded as the axial Young’s modulus (EA). However, the experimental value of ET can be interpreted as any of the Young’s moduli in any of the directions lying in the 12 plane. Figure 6 is polar plot that shows the variation of the Young’s modulus with angular orientation within the 12 plane. Here, θ is the angle between the 1′-direction, along which the load is applied, and the 1-direction (in the 12 plane). The plot is generated by computing Sij for different angles using Eqs. (2) and (3) and extracting the Young modulus value in the 1′-direction from the rotated Sij. Stretching in the b-direction implies separating chains in the hydrogen bonded planes which explains why E[010] has the highest value for ET (98 GPa). Perpendicular to the b-direction (marked as ⊥[010] in Fig. 6), non-bonded interactions have to be overcome to increase the distance between adjacent planes, producing a relative maximum (19 GPa) in the Young’s modulus. Another observation is that ET reaches its minimum (13 GPa) between the b-direction and the direction perpendicular to the b-direction (~30° in the plot). This could be caused by the relative sliding between hydrogen bonded planes. Based on Fig. 6, it can be estimated that there is a 65 % probability of measuring ET under 20 GPa during experimental measurements; misalignments in the c-direction could increase this probability. Our simulations at 0 K yielded values from 13 to 98 GPa, which are in good agreement with reported experimental results (Diddens et al. 2008; Wagner et al. 2011; Pakzad et al. 2012; Lahiji et al. 2010) and numerical simulations (Matsuo et al. 1990; Eichhorn and Davies 2006; Wu et al. 2013). A summary is shown in Table 2. The calculated Young’s moduli in selected directions are reported for different total strains [applied in the Le Page and Saxe method (Le Page and Saxe 2002)] in Table 3.
Table 2

Summary of Young’s modulus and Poisson’s ratio values from different crystalline sources and authors

References

Methodology

Type of cellulose

EA

[GPa]

ET

[GPa]

νt

Temp

[K]

Sakurada et al. (1962), (1964), Matsuo et al. (1990), Nishino et al. (1995), Ishikawa et al. (1997)

X-Ray diffraction (XRD)

Ramie

(cellulose I)

90–138

Nakamura et al. (2004)

XRD

Ramie fiber

(cellulose Iβ)

ν[200]/[004] = 0.377 ± 0.041b

ν[110]/[004] = 0.639 ± 0.338b

Room

Temp

Diddens et al. (2008)

Inelastic X-ray (IXR)

Flax fibers

(cellulose Iβ)

220 ± 50

14.8 ± 0.8

Wagner et al. (2011)

Atomic force  microscope (AFM)

Tunicate CNC

2.7–20

Pakzad et al. (2012)

AFM + finite elements

Wood CNC

24.8 ± 7.0

Cotton CNC

17.7 ± 5.0

Matsuo et al. (1990), Tashiro and Kobayashi (1991)

Theoretical derivation

Cellulose I

167.5

Et1 = 50.2a

Et2 = 10.6a

ν = 0.42a

Eichhorn and Davies (2006)

Molecular mechanics (MM) [COMPASS]

Cellulose Iβ—Nishiyama

149

Et1 = 46.8a

Et2 = 18.9a

ν = 0.439a

Minc

Cellulose Iβ—Finkenstadt

116

Et1 = 50.9a

Et2 = 11.9a

ν = 0.86a

Santiago Cintrón et al. (2011)

Quantum mechanics

[B3LYP/6-311G**+]

1,40-O-dimethyl-b-cellobioside

99.7

Wu et al. (2013)

Molecular dynamics (MD) [ReaxFF]

Cellulose Iβ

139.5 ± 3.5

Et1 = 28.8 ± 2.9

Et2 = 7.0 ± 1.7

300 K

The double line in the table separates experimental from theoretical results. The symbol “–” means that the corresponding value was not reported

aNot reported, computed based on reported Sij or Cij

bUpper limit of the standard deviation

cThe authors perform energy minimization without specifying any temperature

Table 3

Young’s moduli as a function of orientation for applied strains of 1.0, 1.5, and 2.0 % from DFT calculations in the present study

Young’s modulus

Total strain

Average

1.0 %

1.5 %

2.0 %

E33

206.7

206.7

206.2

206.5

E11

19.3

19.7

18.5

19.1

E⊥[010]

19.6

19.4

18.9

19.3

E[110]

14.8

16.3

15.0

15.3

E[010]

98.7

97.9

96.7

97.8

E22

75.4

79.2

76.4

77.0

This table summarizes the results shown in Fig. 6. Young's modulus are reported in GPa, computed at 0 K using DFT with van der Waals interactions (this work)

The variation of the average Poisson’s ratio \(\bar{\nu }_{{1^{\prime } }}\) with respect to any arbitrary direction 1′ was computed from the Sij by applying Eq. (1) to the rotated compliance matrix obtained using Eqs. (2) and (3). Here the 1′-direction indicates the orientation of the load (which is the orientation that is being evaluated). Figure 7a shows one view of the computed \(\bar{\nu }_{{1^{\prime } }}\) surfaces. Figure 7b–d show views of the same surface in the 12, 13, and 23 planes, respectively. These results clearly show the strong variations in \(\bar{\nu }_{{1^{\prime } }}\) with minimum values along the directions [100], [010] and [001]. As it can be observed in Fig. 7, the type of anisotropy in \(\bar{\nu }_{{1^{\prime } }}\) is different from that shown for the Young’s Modulus (Fig. 4). The details on how \(\bar{\nu }_{{1^{\prime } }}\) varies in some selected planes will be discussed next (Table 4).
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Fig. 7

Average Poisson’s ratio \(\bar{\nu }_{{1^{\prime } }}\) surfaces for cellulose Iβ computed using Eq. 1 based on Sij for 1.0 % of total deformation in the Le Page and Saxe method (Le Page and Saxe 2002). Each point on the surface represents the magnitude of Poisson’s ratio in the direction of a vector from the origin to that point. The color contours help to identify the Poisson’s ratio variation and emphasizes the extreme anisotropy of the system. Side views of the same surface are shown for the b12 plane, c13 plane, d23 plane. Note that axis 2 is not seen from this view. (Color figure online)

Table 4

Computed Poisson’s ratio values, as a function of the orientation with respect to the Cartesian system of coordinates 123, for different total strains

Poisson’s ratio

Total strain applied in the Le Page and Saxe method (Le Page and Saxe 2002)

Average over strains

1.0 %

1.5 %

2.0 %

\(\nu_{12}\)

0.143

0.130

0.148

0.140 ± 0.010

\(\nu_{13}\)

0.042

0.042

0.039

0.041 ± 0.002

\(\bar{\nu }_{1}\)

0.092

0.086

0.094

0.091 ± 0.005

\(\nu_{ \bot [010]/[010]}\)

0.107

0.111

0.114

0.111 ± 0.003

\(\nu_{ \bot [010]/[001]}\)

0.048

0.046

0.044

0.046 ± 0.002

\(\bar{\nu }_{ \bot [010]}\)

0.077

0.078

0.079

0.078 ± 0.001

\(\nu_{{[110]/ \bot [\bar{1}\bar{1}0]}}\)

0.715

0.714

0.683

0.704 ± 0.021

\(\nu_{[110]/[001]}\)

0.004

0.003

0.004

0.004 ± 0.001

\(\bar{\nu }_{[110]}\)

0.360

0.359

0.344

0.354 ± 0.010

\(\nu_{{[010]/ \bot [0\bar{1}0]}}\)

0.537

0.558

0.586

0.560 ± 0.026

\(\nu_{[010]/[001]}\)

0.024

0.025

0.026

0.025 ± 0.001

\(\bar{\nu }_{[010]}\)

0.280

0.291

0.306

0.292 ± 0.014

This table summarizes the results shown in Fig. 8

Figure 8 shows a polar plot of ν1′2′, ν1′3 and \(\bar{\nu }_{1}\) using the c-axis as the rotation axis, in which variations of these quantities are examined over the 12 plane (where θ is the angle between the 1′-direction, along which the load is applied, and the 1-direction in the 12 plane). In order to quantify lateral deformation, we define a local Cartesian system 1′2′ that rotates with respect to the 1 and 2 directions and remains in the 12 plane. First, ν1′2′ provides the lateral deformation in the 2′-direction as uniaxial stress is applied along the 1′-direction. It is important to remember that the 2′-direction rotates perpendicular to the 1′-direction in the 12 plane in Fig. 8. The smallest value of ν1′2′ ≈ 0.1 is found when the 1′-direction is aligned with the direction marked as ⊥[010] in the plot. This happens to be the direction perpendicular to the hydrogen bonded planes which contain the cellulose chains. This small value is expected since increasing the separation between hydrogen bonded planes, held together primarily by weak vdW interactions, has little effect over the arrangement of the cellulose chains inside the plane. The next local minimum is found to be in the [010]-direction where the opposite effect is observed. Trying to separate cellulose chains from each other has little effect over the arrangement of hydrogen bonded planes. The axial deformation of the system shows a different behavior. Here, ν1′3 measures the contraction in the 3-direction as uniaxial stress is applied in the 1′-direction (since the 1′ and the 2′ directions remain in the 12 plane, the 3′-direction coincides with the 3 direction). In Fig. 8, the 3-direction coincides with the rotation axis (perpendicular to the plane of the plot), meaning that this direction is always coincident with the cellulose chain regardless of the orientation angle. The small values of ν1′3 for all directions (ranging from 0 to 0.05) can be easily justified by the covalent bonds between carbon and oxygen ions that govern the mechanical response in the c-direction: these are largely unaffected by deformation in other directions. When the cellulose Iβ is deformed along a direction that passes near the center chain (around 45° in the plot), practically no Poisson effect (no lateral expansion in the 3-direction) is observed in the axial direction (Fig. 8).
https://static-content.springer.com/image/art%3A10.1007%2Fs10570-013-0071-8/MediaObjects/10570_2013_71_Fig8_HTML.gif
Fig. 8

Poisson’s ratio as a function of the rotation angle for 1.0 % of total strain. The inset in the upper-left corner depicts the surface of the average Poisson's ratio as shown in Fig. 7 and the selected plane in which the polar plot lies. The 3-axis (pointing perpendicular to the page) is considered to be the rotation axis; the value of the Poisson’s ratio for a given direction can be read directly from the figure by defining a straight line from the origin to the desired angle. The intersection between the straight line and the curves provide the values of ν1′2′, ν1′3 and the average of them (\(\bar{\nu }_{{1^{\prime } }}\)) for that direction. Three critical directions (// to b, through the center chain and ⊥ to b) are marked for reference using Miller indices; the direction parallel to the a-axis coincides with the vertical axis of the figure

https://static-content.springer.com/image/art%3A10.1007%2Fs10570-013-0071-8/MediaObjects/10570_2013_71_Fig9_HTML.gif
Fig. 9

Elastic compliance (S) and stiffness (C) matrices for cellulose Iβ based upon the Nishiyama et al. (2002) initial structure with respect to the Cartesian system of coordinates shown in Fig. 3a. All components were computed using the Le Page and Saxe method with with the dispersion-corrected DFT method (PBE-D2) in VASP for three different total strains (1.0, 1.5 and 2.0 %). Values of S are given in [1/GPa] × 1000, and those for C are in [GPa]

The extreme anisotropies observed in cellulose Iβ can be once again evidenced in the high variations of the Poisson’s ratio. Depending on the selected direction, the Poisson’s ratio will range from almost 0 to 0.71. Comparison with previous publications (Matsuo et al. 1990; Eichhorn and Davies 2006; Nakamura et al. 2004) shows results that are in good agreement with the values reported in this study. Finally, our computed Cij and Sij, based on the Cartesian system shown in Fig. 2, are reported for each applied strain in Fig. 9.

Conclusion

The full elasticity tensor was computed for cellulose Iβ network A using DFT with van der Waals interactions using a least squares fitting method with VASP as the computational engine. Our results show a good agreement with previous experimental work, in particular, a remarkable agreement is found with the IXS experiments conducted by Diddens et al. (2008). Three dimensional surfaces, which are color contours showing the crytstallographic dependence of the Young's modulus and Poisson’s ratio, were computed to examine the extreme anisotropy of these important elastic properties. A clear correlation between the stiffness of the crystal and the different deformation mechanisms was noted. The largest Young’s modulus (206 GPa) was found to be aligned with the c-axis were covalent bonds dominate the mechanical response of the crystal. Perpendicular to the cellulose chain axis, the b-direction shows the next greatest value for the Young modulus (98 GPa); this can be explained by the presence of the hydrogen bond network linking the cellulose chains. Finally, a value for the Young modulus of only 19 GPa was computed along the direction perpendicular to the previous two, where weak vdW interactions play a dominant role in the mechanical response of the material. Based on our 0 K simulations with dispersion-corrected DFT in VASP, the transverse Young Modulus for crystalline cellulose can be defined in the range between 13 and 98 GPa, in good agreement with reported experimental results (Diddens et al. 2008; Wagner et al. 2011; Pakzad et al. 2012; Lahiji et al. 2010) and other numerical simulations (Matsuo et al. 1990; Eichhorn and Davies 2006; Wu et al. 2013).

It should be mentioned that the predicted values of Young’s modulus along the c-axis in this work is higher than the ~100–150 GPa range usually reported in other numerical works (Matsuo et al. 1990; Tashiro and Kobayashi 1991; Eichhorn and Davies 2006; Wu et al. 2013). While the origins of this discrepancy between previous molecular dynamics calculations and the present DFT results is not known, some additional comments on this issue are warranted. Molecular dynamics calculations are sensitive to the force field being used. These usually involve semi-empirical potentials for the bonding interactions between the elemental constituents. As such they do not explicitly account for electron exchange and correlation as DFT does. Alternatively, a potential contributing factor from DFT may be the choice of the exchange–correlation functional, for which we chose the gradient-corrected formalism of Perdew, Burke and Ernzerhof (GGA-PBE) (Perdew et al. 1996). However, the GGA approach is known to predict mechanical properties that are closer to those reported in experimental works than those predicted by local density formalisms (Hector et al. 2007). Moreover, future DFT studies of cellulose will focus on exploring HSE06 functionals (Heyd et al. 2003, 2006), which allow exact exchange, rather than the approximations involved with other DFT functionals. These recent functionals mix the DFT and Hartree–Fock approaches and have been shown to give more accurate predictions of band gaps, for example, relative to the local density or gradient corrected approximations. The extent to which HSE06 functionals influence predicted mechanical properties for materials such as cellulose relative to a gradient corrected functional, such as that used herein, is currently unknown.

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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© Springer Science+Business Media Dordrecht 2013