Molecular Orientation Distribution of Regenerated Cellulose Fibers

Article Cite This: Macromolecules 2019, 52, 3918−3924

pubs.acs.org/Macromolecules

Molecular Orientation Distribution of Regenerated Cellulose Fibers Investigated with Polarized Raman Spectroscopy Leo Svenningsson, Yuan-Chih Lin, Maths Karlsson, Anna Martinelli, and Lars Nordstierna* Department of Chemistry and Chemical Engineering, Chalmers University of Technology, Göteborg 412 96, Sweden

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S Supporting Information *

ABSTRACT: The molecular orientation distribution of polymeric fibers influences physical properties. We present a novel method of analyzing polarized Raman experiments to determine molecular orientation, which is based on exchanging the Legendre polynomial approach with a wrapped Lorentzian function, as determined from a prescreening of X-ray scattering patterns. This method removes the need for performing right angle scattering experiments while avoiding common approximations. The molecular orientation of regenerated cellulose fibers, using the presented method, is shown to correlate well with X-ray scattering and an analogous experiment using solid-state NMR spectroscopy. Challenges of quantitatively measuring molecular anisotropy occur with semicrystalline, partially modified, or composite materials. As such, a plethora of techniques, each with a unique chemical selectivity, is paramount for material characterization.

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spinning9−11 (ROSMAS) or DECODER NMR spectroscopy,12,13 wide angle X-ray scattering (WAXS),14 infrared (IR) dichroism,15 birefringence,3 and polarized Raman spectroscopy,1,16−20 each method having advantages and limitations. The NMR techniques have the potential to measure all structural contributions of the material as well as high moment order parameters, but require the use of a bundle of fibers that may introduce a macroscopic orientation distribution and hence obscure the uniaxial fiber symmetry. WAXS not only reveals order parameters but also provides the entire orientation distribution, but usually only from the purely crystalline part of the cellulose fiber. IR dichroism and birefringence methods are limited to P2 estimations only. For comparison, polarized Raman spectroscopy is unique in that it can measure both P2 and P4 on a single fiber. Polarized Raman spectroscopy, using the so-called complete method, has previously been used to investigate high-density polyethylene fibers.17 A similar study has been carried out on poly(ethylene terephthalate)1,18 (PET) using an equivalent but more efficient experiment, referred to as the complete (tilt) method. With the aim to remove tedious right angle scattering experiments, where light is collected orthogonal to the incident angle, Frisk et al. compared the results obtained by the complete method to those achieved by assuming cylindrical symmetry for the Raman tensor and using an isotropic sample.20 Later, RichardLacroix and Pellerin19,21,22 presented a model based on Bower’s most probable distribution23 while still relying on cylindrically symmetric tensors. It has already been shown that

INTRODUCTION Macroscopic properties of polymeric fibers are known to greatly depend on the molecular orientation, which is usually induced by mechanical stretching along the fiber axis.1−4 Among cellulose-based materials, one fiber of specific industrial interest is the regenerated cellulose textile fiber that can be produced by the well-established Lyocell process or by more recently developed dissolution methods.2,5,6 Both production processes strive to refine cellulose fibers from wood in contrast to traditional cotton. The cellulose fiber is distinguished by significant proportions of both crystalline and amorphous phases as well as a large internal surface area, with crystallite dimensions from ten to a few hundred nanometers long.7,8 Due to this microstructural heterogeneity, it is important to be aware of what part, or parts, of the cellulose material is indeed responsible for a certain experimental observation. In other words, the choice of the analytical method to, e.g., determine the molecular orientation distribution of the fiber, may have a significant impact on the experimental interpretation. This study is also motivated by an interest in providing a plethora of reliable quantitative methods for the characterization of molecular orientation in cellulosic and other polymeric fibers. Mathematically, the molecular orientation distribution of a uniaxial fiber can be described by a weighted sum of Legendre polynomials, where the weighting factor is called the order parameter, referred to as PS for even integers of the summation index S . In general, P2 = 1 for a perfect molecular orientation, whereas P2 = 0 describes a sample for which the molecules are totally disordered. Potentially, more structural information could be achieved if several order parameters are known. A handful of experimental methods exist to investigate molecular orientations, such as rotor synchronized magic angle © 2019 American Chemical Society

Received: March 17, 2019 Revised: May 4, 2019 Published: May 15, 2019 3918

DOI: 10.1021/acs.macromol.9b00520 Macromolecules 2019, 52, 3918−3924

Article

Macromolecules cylindrically symmetric Raman tensor approximation impacts the quantification of the molecular orientation in PET, i.e., while investigating the commonly used 1614 cm−1 Raman feature.20 In this work, we exploit the unique properties of polarized Raman spectroscopy to investigate oriented regenerated cellulose fibers. In particular, we investigate the 1096 cm−1 Raman active vibrational mode, which is assigned to the symmetric stretching of the glycosidic C−O−C bond of regenerated cellulose,24 Figure 1. This feature is particularly

BS BS 2 I33 (90) = I11 (0) = I0 ∑ α22 = I0 ∑ α112

(2)

BS BS 2 I31 (0) = I31 (90) = I0 ∑ α31 = I0 ∑ α132

(3)

BS I31 (45) =

1 1 I0 ∑ α112 − I0 ∑ α11α33 4 2 1 2 + I0 ∑ α33 4

(4)

In the notation IBS ij (β) used through eqs 1−4, BS refers to laser backscattering in the x2 direction in Figures 2 and 3, the

Figure 1. Cellulose repeating unit with two anhydroglucose groups connected by the glycosidic C−O−C bond.

useful since the principal axis Raman tensor component α33 can be approximated as parallel to the chain axis, thus eliminating the need for further coordinate transformations.6 Complementary WAXS data motivates an exchange of the Legendre polynomial model with the observed orientation distribution function (ODF). Our proposed polarized Raman methodology instead calculates the entire orientation distribution, which can be converted to any number of Legendre polynomial distributions. Our method is also featured by the removal of the tedious right-angle scattering measurements while avoiding otherwise spurious cylindrical Raman tensor symmetry and other aforementioned assumptions. The proposed method is used to establish a common ground for molecular anisotropy analysis of polarized Raman spectroscopy, X-ray scattering, and ROSMAS NMR spectroscopy. These techniques are essential for determining orientation distribution of complex, partially modified, and composite polymers that require a high degree of chemical selectivity. To increase user accessibility, we distribute a free online script for our proposed method and the complete (tilt) method1,18 in Matlab format (github.com/LeoSvenningsson/Raman-ODF).

Figure 2. Schematic of the backscattering experiment with the incident laser in the x2 direction and signal collection in the x2 direction. The fiber is positioned in the x1−x3 plane with an angle β and the fiber is parallel to x3 when β = 0.

subscripts i and j refer to the polarization of the analyzer (i) and of the incident light (j), and β is the tilting angle of the fiber in the x1−x3 plane. The summation is intended to be overall molecules captured within the focal volume of the incident laser, and the α values are the Raman tensor coefficients. I0 is the intensity of the incident laser, which must be constant throughout the different experiments. The complete method also requires at least one right angle scattering measurement, not shown here, though our suggested method removes the need for it. Wrapped Lorentzian-Based Analysis of Polarized Raman Scattering. The Legendre polynomial decomposition of the ODF is valuable since it produces linear contributions of P2 and P4, but is arguably a weak statement if a proper ODF is already known. The WAXS patterns recorded by us from a single fiber can in fact be well fitted with a Lorentzian function, to which we imply the periodical conditions i.e., the wrapped Lorentzian function f wL(θ)

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THEORY The derivation of Bower’s seminal work employs a classical interpretation of the scattered polarized Raman intensity.16 The theory describes the measured Raman intensities as a function of the Raman tensor values in its principal axis system, α11, α22, and α33, weighted by the order parameters P2 and P4. Bower found that five independent measurable quantities determine the principal Raman tensor values and the order parameter for the first two Legendre polynomials. Higher order parameters (i.e., P6, P8,...) are discarded since they typically have decreasing contributions, while odd order parameters vanish because of fiber symmetry. Our proposed approach is partly based on the complete (tilt) method, which includes an additional measurement with a tilted fiber.1,18,25 More precisely, Raman intensities must be measured at different experimental configurations as described below BS 2 I33 (0) = I0 ∑ α33

wrapped Lorentzian fwL (θ) =

sinh γ 1 · π cosh γ − cos 2θ

(5)

where θ is the molecular orientation from the fiber direction and γ represents the width of the wrapped Lorentzian. Inserting the wrapped Lorentzian function of eq 5 into the polarized Raman model of eq 6 and using the tensor identity as in eq 7, where the aik elements of the Euler rotation matrix relate the Raman tensor to our fiber axis, it is possible to calculate the Raman principal values and the width (γ) of the wrapped Lorentzian function.

∑ αijαpq = N0∫

0

2π

2π

∫0 ∫0

π

f (θ )αij(ψ , θ , ϕ) (6)

αpq(ψ , θ , ϕ)dθ dψ dϕ

(1) 3919


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BS Figure 3. Schematic of the IBS 31 (0) and I33 (90) configurations, with polarized laser in the x2 direction.

■

3

αij =

∑ aik(ψ , θ , ϕ)ajk(ψ , θ , ϕ)αk

Polarized Raman Spectroscopy. A Lyocell regenerated cellulose fiber (17 μm in diameter), kindly provided by RISE IVF, was fastened over the cross-section of a metal slit. Raman spectroscopy measurements were performed with an InVia Reflex spectrometer from Renishaw using the 785 nm line of a diode laser as the excitation source, which together with a 1200 lines/mm grating gives a spectral resolution of ca 2 cm−1. The laser power on the sample was set to be in the range of 15 mW, and spectra were measured for 30 s averaged over 60 accumulations. All Raman spectra were recorded at room temperature keeping the fiber lightly stretched. The spectrometer was equipped with a polarizer, adjustable at desired angles between 0 and 180°, and an analyzer arranged to let light parallel to x1 or parallel to x3 pass through. Five independent spectra using the 785 nm excitation line were BS BS BS collected in the IBS 33 (0), I33 (90), I31 (0), and I31 (45) configurations, while an additional spectrum for normalization purposes was also collected in the IBS 31 (90) configuration. The instrumental factor of the depolarization ratio F(ν) was determined using butyl benzoate, 2butanone, and CCl4 as reference materials, proposed by Yang and Michielsen.1,25 Normalization of the measured integrated intensity is described by

(7)

k=1

Where f(θ) is the molecular ODF when the Raman tensor is parallel to the molecular frame. A technical detail is that the integrals of the Euler angles ψ and ϕ can be calculated analytically, but θ cannot. The new systems of equations can be solved with iterative methods and the Legendre order parameters can thereafter be calculated for any even value of S directly from the wrapped Lorentzian distribution using eq 8. π

⟨PS⟩ =

∫0 PS(cos θ)·f (θ)·sin θ dθ

(8)

π

∫0 f (θ)·sin θ dθ

ODF Reconstruction. The orientation distribution function can be described as an infinite series of Legendre polynomials 1 1 ∑ ijjjS + yzzz·⟨PS⟩·PS(cos θ) 2π S = 0 k 2{ ∞

f (θ ) =

(9)

BS BS I33 (0) = I33 (0)meas

The model in eq 9 is flawed when directly plotting the contributions with a low number of order parameters, e.g., P2 and P4 only, as with most of the Raman spectroscopy methods, since it allows for nonphysical negative probability densities. Instead, we fit a suitable model function with the available order parameters. The three normalized orientation distribution functions considered in this work are described by eqs 5, 10, and 11 where γ, m, ϕ, λ2, and λ4 are fitting parameters. Gaussian fGauss (θ ) =

BS BS BS BS I33 (90) = I31 (0)meas · I33 (90)meas /I31 (90)meas BS BS I31 (0) = F(ν)· I31 (0)meas

The superscripts meas and approx are the measured and approximated intensities, respectively. The 1000−1200 cm−1 spectral region was analyzed using Igor software v6.37, fitting the experimentally measured spectra with Gaussian functions and a linear background. Matlab R2016b was used to calculate the order parameters and orientation distributions. Wide Angle X-ray Scattering. A WAXS measurement was performed on a single fiber at Chalmers University of Technology with a 0.9 mm beam diameter Rigaku 003+ high brilliance microfocus Cu-radiation source at a distance of 130 mm. An experiment time of 22.5 h was needed due to the low scattering intensity from a single fiber.

(10)

with the constraints m ≥ 0 and 0 ≤ ϕ ≤ π/2, and most probable fmp (θ ) =

e λ2·P2(cos θ) + λ4·P4(cos θ) π λ ·P (cos θ ) + λ ·P (cos θ ) 2 2 4 4

∫0 e

(11)

■

dθ

RESULTS AND DISCUSSION Figure 4 shows details of the curve fitting used for the 1000− 1200 cm−1 spectral region of a Raman spectrum collected in the I33(0) configuration. Peak positions and assignments are taken from ref 24, although two unknown peaks are found at approximately 1082 and 1168 cm−1, as can be seen in the I31 spectra in Figure 5 (see asterisks). In addition, the peaks are

The fitting parameters can be calculated numerically by minimizing the function π ij ∫0 PS(cos θ)·f (θ)·sin θ dθ yzzz jj zz = 0 ∑ jjjj⟨PS⟩ − π zz j z ∫0 f (θ)·sin θ dθ k {

(13)

BS BS I31 (45)approx = F(ν)· I31 (45)meas

2

m /π · e−m·(θ − ϕ)

EXPERIMENTAL ASPECTS

2

(12)

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and a baseline to a thin strip between 1.58 and 1.63 Å of the azimuthal angle. WAXS images are available in the Supporting Information. The wrapped Lorentzian handles the periodic nature of the ODF, unlike a typical Lorentzian, although the difference would be minimal for narrow distributions. Lafrance14 and Krässig8 pointed out that the WAXS analysis is sensitive to errors related to data processing methods and this is indeed the case, particularly when considering that the 021 reflection hinders the analysis on the azimuthal angle. To reduce the impact of the 021 reflection,26 curve fitting of the 002 reflection is performed using data from 0 to 45° only, as shown in Figure 6. Figure 4. Peak fitting of the 1000−1200 cm−1 region with 785 nm laser polarization, analyzer filter, and fiber in parallel, I33(0), for a Lyocell fiber Raman spectrum.

Figure 6. Wrapped Lorentzian azimuthal angle curve fitting (dashed) of the 002 WAXS reflection (dots).

The wrapped Lorentzian-based experiment is evaluated by minimizing the rhs sum of square functions of eq 6, subtracted by its corresponding experimental intensity, with respect to γ and the Raman tensor principal axis components. The resulting wrapped Lorentzian ODF is sequentially converted to order parameters PS from eq 8 for a direct comparison between methods. Similarly, WAXS is also evaluated by eq 8 from wrapped Lorentzian ODF fitting of the 002 pattern. Most notably is that the results from WAXS and the 785 nm wrapped Lorentzian-based experiment in Table 1 are similar and within error margins, which suggests that our approach yields quantifiable results across these two techniques. The relative difference of the orientation parameters between fibers is estimated to be 2% as measured by birefringence. The

Figure 5. Lyocell fiber Raman spectra of the 1000−1200 cm−1 region for the I33(90), I31(90), I31(0), and I31(45) configurations using the 785 nm laser. The two unknown peaks are marked with asterisks.

Table 1. Order Parameters Calculated from the Wrapped Lorentzian Raman Spectroscopy Experiment along with WAXS Measurementsa

likely to have a mixture of Gaussian and Lorentzian line shapes. Since this mixture is not exactly known, we resort to exclusively fit Gaussian line shapes in the 1000−1200 cm−1 region. The calculated standard deviation of P2 and P4 reflects only uncertainties from the curve fitting. The instrument-dependent depolarization factor, F(1096 cm−1), was determined to be 0.91 when using the 785 nm laser. WAXS-based molecular orientation measurements are typically performed on fiber bundles. However, it can be easily shown that the resulting orientation distribution is not uniquely dependent on molecular orientation but also on the fiber bundle distribution. This is why we quantitatively analyze a single fiber only limiting the fit with a wrapped Lorentzian

wL Raman spectroscopy (single fiber) WAXS (single fiber) WAXS (bundled fibers) ROSMAS (bundled fibers) wL ROSMAS (bundled fibers)

P2

σP2

P4

σP4

0.50 0.48 0.46 0.45 0.44

0.03 0.03 0.02 0.02 0.01

0.32 0.29 0.28 0.27* 0.25

0.03 0.02 0.03 0.01

a

Rotor synchronized MAS solid-state NMR experiments by Svenningsson et al.11 P4* is implicit by assuming a wrapped Lorentzian ODF.

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Gaussian, Lorentzian, most probable, and Pearson VII function. However, a direct comparison to WAXS data on a single fiber has never been presented. Figure 7 shows the ODF

Raman tensor principal axis components where calculated to be α1 = −0.098, α2 = 0.172, and α3 = 0.458 for the 785 nm wL experiment. The 1096 cm−1 Raman mode is likely to contain both crystalline and noncrystalline cellulose, which could affect how orientation should be interpreted. However, we have recently found, using ROSMAS solid-state NMR spectroscopy, no or negligible differences in orientation for these two phases.11 In the same study, a wrapped Lorentzian-based ROSMAS method was also successfully applied, with the results summarized in Table 1, which shows that the wrapped Lorentzian model is a good approximation also for fiber bundles. Consequently, we are confident in claiming to have constructed a methodology that yields quantitative results across three techniques, i.e., Raman spectroscopy, NMR spectroscopy, and WAXS, based on different physical phenomena. We have also explored approximations used by other researchers. When cylindrical symmetric tensor is implied, then P2 = 0.42 and α11 = α22 = 0.038 and α33 = 0.424. We therefore conclude that the cylindrical symmetric tensors are a poor approximation for regenerated cellulose fibers. However, for materials where cylindrical symmetry can be applied, it is possible to remove an additional measurement, such as I31(45). Richard-Lacroix and Pellerin19 use the most probable function to predict P4 from P2, however, this method underestimates P4 when compared to our WAXS study (P2 = 0.48 → P4 = 0.15), which sequentially leads to a few unlikely solutions when their P4 prediction is applied as a condition to remove RAS from the complete (tilt) method. The noncomplex solution pairs are then: P2 = 0.15 with P4 = 0.01 and P2 = 0.72 with P4 = 0.37. The use of a supplementary isotropic sample may violate the assumption of a Raman vibrational mode being parallel to the molecular frame. In addition, the use of isotropic samples assumes that the Raman tensor principal axis components also remain unchanged. A change in the electronic structure for the underlying chemical shielding anisotropy was observed by Svenningsson et al.11 when comparing ROSMAS solid-state NMR spectra of regenerated cellulose fibers with density functional theory models of the isotropic crystalline structure cellulose II found in regenerated cellulose. The electronic structure findings are likely to carry over to Raman vibrations of the 1096 cm−1 Raman mode. In addition, an isotropic sample of regenerated cellulose is inconvenient and difficult to compare with conditions from a spun fiber. The wL model, as presented in this work, can most possibly be applied to other polymers as long as the vibration mode is parallel to the molecular frame, otherwise this difference in the vibration angle has to be compensated for with a Euler transform. A WAXS measurement can be conducted to confirm that the wL model is an appropriate ODF for the polymer of interest. For noncrystalline or otherwise WAXSinsensitive materials, we cannot directly confirm or deny the choice of ODF with the method used in this work. However, both polarized Raman spectroscopy and ROSMAS NMR spectroscopy offer indirect methods to confirm a choice of ODF. The proposed method would calculate order parameters with the Legendre polynomial model and a chosen ODF, for example, the wL ODF, using the same experimental data set. The correct ODF is likely found when the respective order parameter is equal between the Legendre polynomial model and the chosen ODF model. ODF Reconstruction. In previous studies,1,14,18,23,27,28 attempts have been made to reconstruct the ODF with a

Figure 7. WAXS data with order parameters P2 = 0.48 and P4 = 0.29 from Figure 6 displayed alongside ODF reconstruction using P2 = 0.50 and P4 = 0.31, experimentally received using the complete (tilt) method with a regenerated cellulose fiber. See the Supporting Information for experimental details.

reconstruction when P2 and P4, from the complete (tilt) method presented in the Supporting Information, are used as fitting parameters with eqs 5 and 12. It is evident that the wrapped Lorentzian better describes the ODF, as seen from the WAXS data in Figure 6. The closely matched correlation between Legendre polynomial modeling and WAXS fitting models is the base for a wrapped Lorentzian-based polarized Raman experiment. ODF reconstruction is important to transfer knowledge on molecular orientation distributions since the Legendre polynomial description is less theoretically accessible than noncomposite ODFs, such as the Lorentzian function. Our wrapped Lorentzian-based Raman spectroscopy order parameters are not used for reconstruction to avoid circular arguments, but it would obviously produce an almost identical curve set, since the order parameters received from the complete (tilt) method and the wrapped Lorentzian method are similar. The particular version of Gaussian reconstruction from eq 10 was chosen mostly for historical reasons,1 but this model breaks if ϕ ≠ 0 or when the distribution extends beyond the periodic limit θ = ±π/2. Additional Considerations. From what is mostly found in the literature about analysis of stretched polymers, P2 monotonically increases with mechanical stretching of the fiber.1,14,20 A wrapped Lorentzian follows a similar trend when γ decreases, as illustrated in Figure 8. Hypothetically, this tells that there is a one-to-one correspondence between P2 and γ, which suggests that it would be sufficient to rewrite the polarized Raman experiment only considering P2. It turns out that this is not the case if the goal is to avoid tedious rightangle scattering measurements from the complete (tilt) method. It can be shown with eq 6 that I31(45) and I31(0) are equal only when considering P2, as in eq 14. Therefore, our suggested model for polarized Raman spectroscopy is still a preferred option. 3922


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Article

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.9b00520.

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Complete (tilt) method experiments; WAXS patterns (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +46 (0)31 7722973. ORCID

Yuan-Chih Lin: 0000-0002-0028-7481 Maths Karlsson: 0000-0002-2914-6332 Lars Nordstierna: 0000-0002-6580-0610 Figure 8. Calculated P2 (green) and P4 (purple) as a function of the wrapped Lorentzian f(γ).

Notes

If the ODF is not known, then plots like Figure 8 offers another way to identify the true ODF. This can be done by either employing the complete method polarized Raman spectroscopy or ROSMAS NMR spectroscopy to see if both P2 and P4 are on the expected curve. If a chosen function produces PS curves that correlate with experimental data, then a possible true ODF candidate is found.

ACKNOWLEDGMENTS We would like to thank Ezio Zanghellini and Aleksandar Matic for useful discussions and spectrometer time, as well as Katarina Logg for WAXS measurement support. We are also grateful to RISE IVF for supplying the cellulose samples. This study has been carried out within the framework of AvancellCenter for Fiber Engineering. Financial support from the “Södra Skogsägarna Foundation for Research, Development, and Education” and “Stiftelsen Nils och Dorthi Troëdssons forskningsfond” is gratefully acknowledged.

The authors declare no competing financial interest.

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P2 P2 (45) = I31 (0) I31

■

= I0[14α12 + 14α22 + 14α32 − 14α1α2 − 14α1α3 − 14α2α3 + +

■

10α32

P2( − 5α12

−

5α22

REFERENCES

(1) Yang, S.; Michielsen, S. Orientation distribution functions obtained via polarized Raman spectroscopy of poly(ethylene terephthalate) fibers. Macromolecules 2003, 36, 6484−6492. (2) Sixta, H.; Michud, A.; Hauru, L.; Shirin Asaadi, Y. M.; King, A. W.; et al. Ioncell-F: A High-strength regenerated cellulose fibre. Nord. Pulp Pap. Res. J. 2015, 30, 43−57. (3) Developments in Polymer Characterization-4, 1st ed.; Dawkins, J., Ed.; Applied Science Publishers: London, 1983. (4) Hynynen, J.; Järsvall, E.; Kroon, R.; Zhang, Y.; Barlow, S.; Marder, S. R.; Kemerink, M.; Lund, A.; Müller, C. Enhanced Thermoelectric Power Factor of Tensile Drawn Poly(3- hexylthiophene). ACS Macro Lett. 2019, 8, 70−76. (5) Kosan, B.; Michels, C.; Meister, F. Dissolution and forming of cellulose with ionic liquids. Cellulose 2008, 15, 59−66. (6) Wanasekara, N. D.; Michud, A.; Zhu, C.; Rahatekar, S.; Sixta, H.; Eichhorn, S. J. Deformation mechanisms in ionic liquid spun cellulose fibers. Polymer 2016, 99, 222−230. (7) Klemm, D.; Heublein, B.; Fink, H.-P.; Bohn, A.; et al. Nanocelluloses: A New Family of Nature-Based Materials. Angew. Chem., Int. Ed. 2011, 50, 5438−5466. (8) Cellulose: Structure, Accessibility, and Reactivity; Krässig, H. A., Ed.; Gordon and Breach Science, cop.: Yverdon, Switzerland, 1993. (9) Harbison, G. S.; Vogt, V.-D.; Spiess, H. W. Structure and order in partially oriented solids: Characterization by 2D-magic-anglespinning NMR. J. Chem. Phys. 1987, 86, 1206−1218. (10) Titman, J. J.; de Lacroix, S. F.; Spiess, H. W. Structure and order in partially oriented solids: Characterization by 2D-magic-anglespinning NMR. J. Chem. Phys. 1993, 98, 3816−3826. (11) Svenningsson, L.; Sparrman, T.; Bialik, E.; Bernin, D.; Nordstierna, L. Molecular orientation distribution of regenerated cellulose fibers investigated with rotor synchronized solid state NMR spectroscopy. Cellulose 2019, 4681−4692.

(14)

+ 20α1α2 − 10α1α3 − 10α2α3)]

/105

CONCLUSIONS We have demonstrated that it is possible to determine the ODF of regenerated cellulose by the wrapped Lorentzianbased experiment, a result that is potentially useful for many types of other polymers. Using the analytical methods, described in this work, gives a direct comparison of molecular anisotropy measurements from WAXS, polarized Raman spectroscopy, and rotor synchronized NMR spectroscopy, which can be used for quantitative comparison of modified fibers or composites in a broader polymer setting. We suggest to always try to perform one or several WAXS measurements while investigating anisotropic polymers to establish a good ODF approximation. ODFs for noncrystalline or WAXSinsensitive polymers can be established by imposing a model function, for example, the wL function, while comparing the order parameters of a pure Legendre polynomial-based experiment. We have also shown that modeling an ODF from the order parameters P2 and P4 favors a deliberate choice of a model function, the wrapped Lorentzian, which was chosen because it best describes the WAXS data. The mathematical tools for molecular orientation calculations have been made public at github.com/LeoSvenningsson/ Raman-ODF. 3923


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