Polarized Raman Spectroscopy Strategy for Molecular Orientation of Polymeric Fibers with Raman Tensors Deviating from the Molecular Frame

: Polarized light is frequently used to identify molecular anisotropy in polymers, biological systems, and other materials. The in ﬂ uence of the Raman tensor by polarized light reveals not only the chemical structure but also information on the morphology of polymers. The orientation distribution of molecules in polymers has previously been determined for systems with the principal axis components of the Raman tensor parallel to the molecular frame. In many cases, the Raman tensor principal axis is not parallel to the molecular chain axis. Therefore, the orientation of the Raman tensor, relative to the molecular chain axis, is crucial if accurate information about the molecular orientation distribution is sought for. This work presents a strategy for separating the Raman tensor orientation angles from the molecular orientation angles for polymeric samples with ﬁ ber symmetry. Composite polymeric materials often experience signal overlap in the X-ray scattering wide-angle region, where the anisotropy is often resolved. While X-ray scattering investigates intermolecular distances, Raman spectroscopy resolves chemical information, and anisotropy, by the in ﬂ uence of Raman scattering. The quantitative principles presented here may aid in the evaluation of anisotropy in such composite materials.


■ INTRODUCTION
Polarized Raman spectroscopy is an essential technique to measure molecular anisotropy in materials, with the most accessible method being the measurement of fractional scattered intensity I ⊥ /I ∥ .−9 These methods are all based on Bower's seminal work (from 1972) on molecular orientation distribution measurements and all of which share one important limitation.The underlying model assumes that the Raman scattering tensor is parallel to the molecular frame (MF). 1 One such example is illustrated in Figure 1.
Many Raman scattering modes in polymers do not have their respective principal vibration in the molecular chain direction of the polymer.A set of reference frames and their relations are visualized in Figure 2 with the PAS of the Raman scattering tensor, an arbitrarily chosen MF, the DF of the fiber, and the laboratory frame (LAB).
This work presents a general model to determine the molecular orientation distribution when the Raman scattering tensor does not coincide with the MF.Such cases are useful when analyzing the crowded spectra, which is typically a situation for composite materials.More importantly, Raman spectroscopy can resolve anisotropy in the cases where the X-ray scattering pattern of two polymers with similar intermolecular distances overlap. 10,11In addition, the proposed method has the potential to follow changes in the Raman tensor orientation relative to the MF during stretching of polymeric fibers.The NMR spectroscopy chemical shielding/ shift anisotropy (CSA) orientation relative to the MF has previously been observed to change during stretching of polyamide-6 yarns. 12Both types of tensor interactions, CSA and Raman scattering, respectively, are strongly correlated with the underlying chemical bonds, which indicates that changes in the Raman scattering orientation relative to the MF may occur if the polymer is stretched. 13Fourier-transform infrared spectroscopy and X-ray scattering have also previously been used to study the changes in the angles of the glycosidic linkage and hydrogen bonding of cellulose, closely related to the corresponding Raman modes. 14The Raman tensor orientation relative to the MF has also been studied by Jasse and Koenig by importing molecular information on polystyrene from IR and birefringence measurements. 15The presented model in this work also has its use for structure illumination, drawing tangents to the work of Tsuboi et al. on biological systems. 16Traditionally, orientation has been studied to determine its influence on tensile stress and creep resistance. 17−20 ■ RESULTS AND DISCUSSION Bower's classical approach allows polarized light to interact with the Raman vibrational tensor and is subsequently analyzed through a polarization filter.The observed scattered intensity I s has the form where ′ i is the scattered light vector through a polarization filter, j is the incident polarized light vector, and α is the Raman tensor.I 0 is dependent on the instrumental factors and the incident light intensity, and should be kept as a constant.The outside summation signifies the sum of a single vibrational mode from the contributions of all of the molecules in the light focus volume.When the scattered and incident light can come from all directions in the lab frame, the resulting quadratic expression can be written in the form i k j j j j j j j y { z z z z z z z i k j j j j j j j y { z z z z z z z (2) where p and q are used to separate the coefficients of the quadratic expression.To reiterate on previous work, Bower showed that the scattered intensity can be written in the form of eq 3.
By using the tensor identity where a ik is a component of a Euler rotational matrix, and α kk PAS is a principal axis component of the Raman tensor in eq 5. i k j j j j j j j j j j j j j j y The function f(θ) is the implied uniaxial orientation distribution function (ODF) of the polymer in a fiber.The constant N 0 is the number of polymer chain elements illuminated by the laser, where I 0 is the intensity of the laser.The ODF is only dependent on θ because there is uniaxial cylindrical symmetry around the angle ϕ.In addition, because the sample is assumed to contain many molecular chain elements, all of which have no rotational preference around the polymer's own chain axis, there is a rotational symmetry around the molecular chain axis with its Euler rotation as ψ.Polymers tend to contain local crystalline structures, which may have an orientation preference locally, but in a macroscopic sample, these crystallites are also uniformly distributed around ϕ and ψ.−8 Hence, the Euler angles (ϕ, θ, and ψ), relevant to the polarized Raman experiments, describe the transformation directly from the PAS to the fiber DF.The molecular symmetries using Euler rotation are shown in Figure 3.
The requirement of the PAS parallel to MF limits the number of vibrational modes that can be quantitatively analyzed.The polarized Raman experiment can be rewritten in a more general form that allows for a Raman tensor orientation, which is not parallel to the MF.The series of Euler rotations that describe the Raman scattering tensor in the experimental reference frames, shown in Figure 2, are described by forward/active Z−Y−Z rotations, following Bower's notation.To recall, an active Z−Y−Z rotation describes the rotation of an object first by a rotation of ϕ around the Z-axis, second by a rotation of θ around the new Yaxis, and third by a rotation of ψ around the new Z-axis.Conversely, an identical complete rotation can be seen as a rotation of the coordinate system by first a rotation of ψ around the Z-axis, second a rotation of θ around the new Yaxis, and third by a rotation of ϕ around the new Z-axis.The active Z−Y−Z Euler rotation matrix is described using eq 6. i k j j j j j j j j j j j y { z z z z z z z z z z z i k j j j j j j j j j y { z z z z z z z z z i k j j j j j j j j j j y { z z z z z z z z z z The Raman scattering tensor in the reference frames in Figure 2 is described as By inserting eqs 8 into 9, we find a rotational equality between the PAS, MF, and DF systems , ) ( , , ) with α ij DF describing a direct Euler rotation from the PAS to the DF, such as The new perspective of eq 12 integrates over all molecular angles (ϕ, θ, ψ) relative to the DF to accommodate the molecular orientation distribution, as previously.However, the Raman tensor is described in the DF through an Euler transformation from the PAS.The Raman tensor components in the DF are then By expressing α DF in terms of its equivalent rotation from eq 10, we get The Raman tensor rotations are now divided into a rotation from PAS to MF (α 0 , β 0 , γ 0 ), and a rotation from MF to DF (ϕ, θ, ψ), which are the desired molecular orientation angles.The angle α 0 coincides with molecular rotational symmetry.For a sample with many semicrystalline molecular chain elements, as in a polymer, the angle α 0 can be chosen freely.In usual circumstances, it is convenient to set α 0 to 0 for computations.The unknown Euler angles β 0 and γ 0 can be determined either from experimental observation or computational modeling.Finally, the fiber sample is rotated around the lab frame in the direction of the incident light, which we choose to be the Y-axis, or the x 2 -axis in the LAB frame.The polarized Raman spectroscopy setup is shown in Figure 4 for both the back scattering and the right angle scattering experiments.
For most experimental setups, the incident light and detection directions are fixed to the LAB frame x 1 , x 2 , and x 3 directions.The resulting possible experimental setups can then be simplified to eq 14 The general expression for both the back scattering and right angle scattering can be expressed as from which all of the experimental scattering geometries can be derived.The angle β 2 corresponds to the Y-rotation of the fiber in the x 1 −x 3 plane, with β 2 = 0 for a fiber in the x 3 or Zdirection.For the three back scattering polarization configurations, corresponding to the matrix indices, eq 16 through 20 are obtained. ∑ ∑ BS and RAS refer to back scattering and right angle scattering experiments, respectively.The subscripts in I ij BS (β) again refer to the polarization of the analyzer (i) and of the incident light (j), respectively.A common approximation to the Raman tensor sets the Raman tensor components to α 11 PAS = α 22 PAS , called the cylindrical tensor, which after careful consideration for certain molecular features can be introduced to reduce the number of unknown variables, thus decreasing the number of required experimental measurements. 3The inclusion of a cylindrically symmetric tensor coincidentally also removes the need for a γ 0 rotation because it is a rotation around the α 33 PAS axis.The choice of the ODF model f(θ) will affect the interpretation of the molecular order observed from experiments.The most commonly used ODF for fiber symmetries is the Legendre polynomials, for which the integral in eq 13 has an analytical solution.The Legendre polynomials, normalized over all Euler angles, are given using eq 21.
Another approach is to use an ODF, which fits well for the specific material from other experimental observations, such as wide angle X-ray scattering and solid-state NMR spectroscopy. 9,13In general, oriented materials do not adhere to a specific molecular orientation function, though in some cases a good enough approximation can be found.
Finally, the model provided by eq 13 was tested using the previously published data. 9To make these newly added variables accessible for experimental studies, the test assumes cylindrically symmetric tensor α 11 PAS = α 22 PAS , which, as explained earlier, reduces the number of unknown variables to: α 11 PAS , α 33 PAS , β 0 , and the ODF, f(θ).−6 The simplest experimental setup avoids the right angle scattering, hence requires that only 4 unknown variables are used with the complete (tilt) method.Notably, it is not possible to approximate the Legendre polynomials solely by P 2 using the complete (tilt) method because it leads to degenerate solutions. 9Therefore, we have used data from a regenerated cellulose material, where the ODF has previously been well approximated by a periodic Lorentzian function instead of the Legendre polynomials. 9When the set of equations could be solved with a minimized square sum, the angle β 0 = 0, and the order parameter P 2 = 0.41 after transforming the Lorentzian function to Legendre polynomials.The vibrational angle for the PAS to the MF β 0 suggests that the glycosidic C−O−C vibration is parallel to the MF.However, it should be noted that the effect of the cylindrically symmetric tensor approximation affects the measured order parameter for this selected sample with a change in the P 2 order parameter, from P 2 = 0.50 to P 2 = 0.42 in previous studies, and here obtained as P 2 = 0.41.

■ CONCLUSIONS
The separation of the Raman scattering tensor PAS and the MF allows additional Raman modes to be used to determine the molecular anisotropy in polymers.A plenitude of useable signals are needed when spectral features overlap.However, more importantly, Raman spectroscopy has the possibility to determine molecular anisotropy in the cases where X-ray scattering patterns of the composite materials overlap.The proposed strategy also allows monitoring the changes in the Raman tensor orientation relative to the MF.Such changes may occur in polymers during stretching from variations in the polymer bond angles and should in principle be accounted for in quantitative analysis.

Figure 1 .
Figure1.Visualization of the Raman tensor-related reference frames in a fiber with a short cellulose chain segment shown at an arbitrarily chosen angle deviation from the fiber axis, that is, director frame (DF).(Left) A Raman vibrational mode with the Raman principal axis system (PAS) tensor parallel to the MF, hence β 0 = 0. (Right) A Raman vibrational mode that is not parallel to the MF, hence β 0 ≠ 0.

Figure 2 .
Figure 2. Notation of the reference frames for the polarized Raman experiment.

Figure 3 .
Figure 3. Visualization of the three Euler rotations between the MF and the DF.The polymer in the cylindrical fiber is in this case, the molecular structure of cellulose.

Figure 4 .
Figure 4. Schematic of the polarized Raman experiment on fibers with the laser entering in the x 2 direction and exiting in the x 1 direction for the right angle scattering geometry, while the laser is entering in the x 2 direction and exiting at x 2 for the back scattering geometry.The fiber is positioned in the x 1 −x 3 plane with an angle β 2 that is parallel to x 3 when β 2 = 0.