Determination of the orientation distribution function in aligned single wall nanotube polymer nanocomposites by polarized Raman spectroscopy

Abstract

This work focuses on the derivation of the orientation distribution function (ODF) for a uniaxial-axially symmetric system using polarized Raman spectroscopy. A numerical methodology is proposed to determine the ODF that is formulated in terms of Legendre polynomials and the principle of maximum information entropy. The ultimate goal is to quantify the alignment of single wall nanotubes (SWNTs) in a polymer matrix using the experimental information from the Raman intensity. Some of the mathematical and numerical steps in the determination of ODF, not clarified in the current literature, are shown in this work. The proposed numerical methodology to obtain the ODF is illustrated using an experimental case. Electric field–aligned SWNT-urethane dimethacrylate/1,6-hexanediol dimethacrylate nanocomposites are investigated at different processing conditions to bring forward factors that may enhance the alignment of SWNT inclusions in the polymer. ODF results indicate that the higher electric field frequencies produce a good alignment of the SWNT inclusions; a result also corroborated by optical microscopy imaging and electrical conductivity measurements.

Raman tensor analysis of ultra-high molecular weight polyethylene and its application to study retrieved hip joint components

“…The orientation of a polyethylene molecule lying in the Raman probe volume can be expressed in terms of Wigner functions that are reduced to Legendre polynomials after symmetry considerations [19][20][21]28]. For doing so, an additional set of Euler's angles (a, b, c) is needed, whose spatial position is relative to a righthanded macroscopic system of Cartesian axes (xyz) lab (cf.…”

Direct Deposition of Aligned Single Walled Carbon Nanotubes by Fountain Pen Nanolithography

“…Further information about the alignment of the SWNTs in the deposited line can be obtained by applying the principle of maximum information entropy to define an orientation distribution function (ODF), as discussed by Perez et al 10 In general, the probability of finding a particle with orientations between angles ( , , ) and ( +d , +d , + d ) is:…”

“…The averaged Legendre polynomial, P 2 cos , is sometimes known as Herman's orientation parameter. 10 The angle is the angle that a nanotube makes with respect to the deposited line direction. In completely random arrangements Herman's orientation parameter would have θ / degrees Raman intensity / a.u.…”

“…To obtain quantitative information about the distribution of orientations of SWNTs in the deposited line, the analysis method of Liu and Kumar, 9 described by Perez et al, 10 was applied. The following equation was applied to the experimental data for the parallel arrangement and the order parameters P 2 cos and P 4 cos were obtained from a least-squares fit, as shown in Figure 5.…”

Precise determination of phonon constants in lead-free monoclinic (K0.5Na0.5)NbO3 single crystals

“…Although Raman spectroscopy cannot distinguish between 180 oriented domains, and thus cannot detect their polarization, it can obtain a domain orientation distribution function (ODF) from in-plane rotation measurements under fixed light polarization. [28][29][30] Raman measurements in polycrystalline KNN requires obtaining reliable single crystal data, since the scattering efficiency depends on the unit-cell Raman tensor parameters. 31 Not much Raman spectroscopy data are available for the KNN system and more specifically for KNN single crystals.…”