Quantifying alignment in carbon nanotube yarns and similar two-dimensional anisotropic systems

The uniaxial orientational order in a macromolecular system is usually specified using the Hermans factor which is equivalent to the second moment of the system's orientation distribution function (ODF) expanded in terms of Legendre polynomials. In this work, we show that for aligned materials that are two-dimensional (2D) or have a measurable 2D intensity distribution, such as carbon nanotube (CNT) textiles, the Hermans factor is not appropriate. The ODF must be expanded in terms of Chebyshev polynomials and therefore, its second moment is a better measure of orientation in 2D. We also demonstrate that both orientation parameters (Hermans in three dimensional (3D) and Chebyshev in 2D) depend not only on the respective full-width-at-half-maximum of the peaks in the ODF but also on the shape of the fitted functions. Most importantly, we demonstrate a method to rapidly estimate the Chebyshev orientation parameter from a sample's 2D Fourier power spectrum, using an analysis program written in Python which is available for open access. As validation examples, we use digital photographs of dry spaghetti as well as scanning electron microscopy images of direct-spun carbon nanotube fibers, proving the technique's applicability to a wide variety of fibers and images.

Automated summary

This study examines the measurement of uniaxial orientational order in macromolecular systems and demonstrates the applicability of Chebyshev polynomials for analyzing orientation in two-dimensional materials. Additionally, a method for rapidly estimating the orientation parameter from a sample's 2D Fourier power spectrum using an analysis program written in Python is presented.