Three-Dimensional Orientation of Nanofibrils in Axially Symmetric Systems Using Small-Angle X-ray Scattering

Abstract

The increased availability and brilliance of new X-ray facilities have in the recent years opened up the possibility to characterize the alignment of dispersed anisotropic nanoparticles in various microuidic applications, from hydrodynamic self assemblies to ows in complex geometries. In such applications, it is vital to study the alignment of the nanoparticles in the ow, as this in turn aects the nal properties of the self-assembled superstructures or those of the ow itself. Small angle X-ray scattering (SAXS) is a well-suited characterization technique for this, but typically provides the alignment in a projected plane perpendicular to the beam direction. In this work, we demonstrate a simple method to reconstruct the full three-dimensional (3D) orientation distribution function (ODF) from a SAXS-experiment through the assumption that the azimuthal angle of the nanoparticles around the ow direction is distributed uniformly; an assumption that is valid for a large range of nanoparticle ow processes.For demonstration purposes, the experimental results from previous works on hydrodynamic self-assembly of cellulose nanobrils (CNF) into laments have been revised, resulting in a small correction to the presented order parameters. The results are then directly compared with simple numerical models to describe the increased alignment of CNF both in the owing system and during the drying of the lament. The proposed reconstruction method will allow for further improvements of theoretical or numerical simulations and consequently open up new possibilities for optimizing assembly processes, which include ow-alignment of elongated nanoparticles.

Elucidating the Opportunities and Challenges for Nanocellulose Spinning

“…Only describing the alignment in the 2D plane can also lead to interesting implications even in a cylindrically symmetric system. [95] For example, in a system aligned through increasing uniaxial extension, the true 3D order parameter 3D S φ can show almost perfect alignment, even though the measured projected order parameter is 0.6 3D S ≈ χ . Both problems, i.e., the loss of information through dimensionality reduction and the dependence of viewing direction, must be properly considered whenever comparing the average orientations between different experiments, or ever comparing experiments to theoretical/numerical models.…”

“…The normalized scattering intensity in azimuthal direction at a given scattering angle (corresponding to length scales between particle length and width) represents the projected orientation distribution function (ODF) of the illuminated nanofibrils in the plane perpendicular to the beam. [95] An example of how this is done is illustrated in Figure 15. [90] The orientational information is not dependent on the exact cross-sectional shape of the nanofibrils and can thus be extracted without any fitting.…”

“…The projected ODF of nanofibrils can thus be derived through the normalized scattering intensity of these Bragg reflections in azimuthal direction. [95] Figure 16 shows an example of how WAXS was utilized to study in situ the nanofibril alignment during wet-stretching of CNF filaments. [34] All previous WAXS characterizations of nanocellulose are relying on measurements of almost dry samples as water has very strong scattering at these angles even for semidilute dispersions.…”

“…It is commonly determined from an anisotropic SAXS/WAXS detector image I(q), whereby subsequent conversion to polar coordinates, normalization, and averaging provides an orientation distribution function (ODF) Ψ χ of the projected angle χ on a plane perpendicular to the X-ray beam. [95] Most often, the degree of alignment is evaluated through an order parameter based on the 2nd Legendre polynomial The normalization with the sin χ-term is introduced to treat the projected angle on a 2D plane as a polar angle in spherical coordinates. [95] The true 3D order parameter 3D S φ based on the actual spherical polar angle φ can be calculated by knowing the ODF Ψ φ in the same way.…”

“…filament, the ODF Ψ φ can be calculated from Ψ χ . [95] A more suitable choice to describe alignment in the detector plane would be to use an order parameter based on the projected angle according to…”