Mapping carbon nanotube orientation by fast fourier transform of scanning electron micrographs

A novel method of applying a two-dimensional Fourier transform (2D-FFT) to SEM was developed to map the CNT orientation in pre-formed arrays. Local 2D-FFTs were integrated azimuthally to determine an orientation distribution function and the associated Herman parameter. This approach provides data rapidly and over a wide range of lengthscales. Although likely to be applicable to a wide range of anisotropic nanoscale structures, the method was specifically developed to study CNT veils, a system in which orientation critically controls mechanical properties. Using this system as a model, key parameters for the 2D-FFT analysis were optimised, including magnification and domain size; a model set of CNT veils were pre-strained to 5%, 10% and 15%, to vary the alignment degree. The algorithm confirmed a narrower orientation distribution function and increasing Herman parameter, with increasing pre-strain. To validate the algorithm, the local orientation was compared to that derived from a common polarised Raman spectroscopy. Orientation maps of the Herman parameter, derived by both methods, showed good agreement. Quantitatively, the mean Herman parameter calculated using the polarised Raman spectroscopy was 0.42 – 0.004 compared to 0.32 – 0.002 for the 2D-FFT method, with a correlation coefficient of 0.73. Possible reasons for the modest and systematic discrepancy were discussed.

However, in many cases, the observed performance is lower than expected, due in large part to the challenge of aligning high loading fractions of high quality, high aspect ratio CNTs [1,2,3]. One particularly promising approach relies on the impregnation of pre-formed CNT M A N U S C R I P T

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arrays, such as veils, ribbons and ropes, with a polymer matrix. Aligning carbon nanotubes within such preformed arrays can improve mechanical [4,5,6,7], thermal [8,9,10,11] and electrical [12,13,14] properties. A number of studies have shown an linear increase in tensile modulus and strength with an increase in orientation ( Figure 1). [4,5,6,15,16] CNTs in these preformed arrays can be orientated by plastic deformation either when dry or following infusion with a resin for lubrication and cohesion. Improved alignment applies the intrinsically excellent axial properties of the CNTs in a desired direction, as well as promoting higher loading fractions and improved van der Waals intertube interactions [4].

Figure 1 : Summary of current literature showing the relationship between the orientation parameter and (a) Tensile Strength and (b) Young's Modulus
Since orientation has a large influence on the physical and mechanical properties of CNT veils and their composites, it is important to quantify the alignment accurately. In particular, it is interesting to map the degree of alignment across the structure to understand uniformity and localisation effects that may limit drawing. Quantified alignment helps to interpret performance, optimise processing, and can be implemented in computational models or empirical calculations of veil composites.

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A C C E P T E D ACCEPTED MANUSCRIPT The alignment of an individual carbon nanotube can be defined by the Euler angles, , and , where is the polar angle of the CNT tube from the Z-axis of the veil, i.e. alignment direction, is the second rotation about X-axis and is the angle of rotation about the former Z-axis (now z′ [20]. Within a veil, or other construct or composite, the orientation distribution function can be defined, , , . A spherical mapping is typically used to project these 3D spherical coordinates into a scalar value, such as the Herman's parameter in order to quantify the orientation between the nanotube and a defining axis.
The most common method used to measure CNT alignment is polarised Raman spectroscopy [17,18], using the two characteristics peaks, the D-band at 1350cm -1 and the G-band at 1590cm -1 [19]. The intensity of the G-band Raman scattering is sensitive to the orientation of carbon nanotubes when the excitation laser beam is linearly polarised. For an individual carbon nanotube, the polarised Raman intensity is proportional to cos , where is the angle between the polarised Raman direction and the carbon nanotube axis [20].
determine orientation [21]. Raman intensities measured at regular angles (between the polarised Raman direction and the CNT preferred axis, ϕ) can be fitted to an orientation distribution [5,21]. However, the majority of researchers have used a simple dichroic ratio Equation (1), for simplicity and ease of data collection: where is the ratio between the Raman scattering parallel to the carbon nanotube direction and the Raman scattering perpendicular to the carbon nanotube direction. For highly aligned veils, in the parallel direction (∥), the ratio, , tends to infinity and for highly aligned samples in the perpendicular direction (⊥) the ratio, , tends to 0. For a sample with no overall alignment is equal to 1. Although this ratio is widely used throughout literature, it only gives an indicative estimate of alignment [21]. One way to generate a more meaningful indication of alignment is to determine a general orientation parameter, also called the Herman parameter, [22]. Although these methods were developed for single-walled nanotubes, similar behaviour been observed for multi-walled carbon nanotubes, since the diameters remain much smaller than the wavelength of the incident light [23].
In this work, uniaxial orientation is assumed, in such a case the orientation distribution function (ODF) is independent of and, since nanotubes are cylindrically symmetrical, the ODF is also independent of . For such a case, the ODF can be modelled as an expanded series of generalised spherical functions, the Legendre polynomial functions (Equations (2) and (3)) [20,24,25]: The signal of a carbon nanotube veil can be related to the intensities of individual CNTs, by summation (Equation (7)), for example: Analogous equations exist for , -. and , .. [20].
By substituting the uniaxial orientation distribution expanded to the fourth polynomial into Equation (7), and its analogous forms, the Herman's parameter can be obtained (Equation Raman spectra are usually obtained from a laser spot diameter of approximately 1 µm, with an acquisition time on the order of 10 s or more. Raman point spectra can be collected systematically to produce a large orientation map; however, acquiring a large map typically takes several hours. Whilst an optical microscopy image of the sample can be correlated with the map, the optical resolution limits the ability to correlate the observed spectra with specific structural arrangements of nanoscale objects.

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Despite the vast range of electron micrographs of CNT veils and arrays, routinely used to examine the structures qualitatively, the alignment is rarely quantified. However, in other research areas, Fourier transforms are often used to analyse microscope images. One particularly relevant example is the use of fast Fourier transforms (2D-FFT) to measure fibre misalignment in conventional carbon fibre composites (fibre diameter on the order of 7 µm) [26]. On a smaller scale, 2D-FFT methods have been successfully applied to determine the alignment of electrospun scaffolds [27] and analyse elastin networks [28]. Using these examples as inspiration, a method was developed to measure the orientation of CNT veils, via image processing of electron micrographs via 2D-FFT. In particular, the methodology in this paper uses a similar algorithm to that applied to carbon fibre composites in [26] for performing a 2D-FFT but is developed further to calculate orientation parameters from the scanning electron micrographs. This paper presents the development and evaluation of this method, parameter optimisation for the algorithm, and the validation of the method by comparison to Raman results, for initial application to neat CNT veils.

Experimental
CNT veils were kindly provided by Suzhou Institute of Nano-Tech and Nano-Bionics (Chinese Academy of Sciences) and used as received. The CNTs in the veil were multiwalled CNTs with a diameter of approximately 10-30 nm, 10-15% iron catalyst, and density 0.5 g/cm 3 . The veils were produced from vertically aligned CNT forests that had been directly grown on a predeposited catalyst film. The thickness of the veil sheet was measured to be 20 µm using a micrometer (

Fast Fourier Transform Methodology
The Fourier transform picks out repetitive elements in the original spatial domain image, and manifests them in the frequency domain [30]. In this case, the diameters of individual and bundled CNTs are the features of interest. The 2D-Fourier transform was computed using the inbuilt Matlab function FFT2, which returns the two-dimensional discrete Fourier transform (DFT) of the image. The DFT was computed with a fast Fourier transform (FFT) algorithm.
The result of the transformation was a matrix of the same size as the original image [31]. The resulting information was in the frequency domain denoted F G, H .
The 2D-Fourier transform is defined as: where I and J are the spatial domain dimensions and G and H are the spatial frequencies. In general, the solution to the 2D-FFT is complex; where the subscript and , indicate the real and imaginary parts of the solution respectively.
To interpret the FFT, the magnitude at each pixel can be calculated, to give the power spectrum. The power spectrum contains no phase information but provides the total amount of information at a given frequency. The power spectrum is computed by: The dynamic range of the power spectrum is often mapped to an 8-bit greyscale using logarithmic transformation [26]: The output of the above Matlab functions were translated to centre (zero-frequency component)[32] using the Matlab fftshift function, to simplify visualisation.

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The mean fibre direction and orientation distribution function were calculated from the 2D-FFT power spectrum (see Figure 1). In the 2D-FFT power spectrum (Figure 3b), the azimuthal distribution of intensity relates to the orientations of the contributing CNTs ( Figure   3a). The total intensity for each angle θ was calculated by summing the intensity radially (shown as [ in Figure 3c and Equation (13)): The azimuthal summation was performed across the whole FFT. It could be argued that the radial range should be selected to focus on the lengthscale of individual CNTs. However, it was found that the result was not significantly changed by introducing radial filtering ranges.
The most likely explanation is that the short lengthscale detail is already averaged out by the magnification selected (see section 4.1). It is more efficient to collect only the data required rather than discard it digitally later.
To calculate the orientation distribution function, the summed intensity was fitted to a function (Equation (14)) based on a normal distribution. The angle at which this orientation distribution function reached a maximum was interpreted as the local mean CNT direction, or director. , where a is the predominant direction. The inbuilt Matlab function 'fit' was used to determine the coefficients ^, b, and d, and hence obtain the orientation distribution function (ODF) Finally, this normalised orientation distribution was converted into the Herman's parameter, , using Equation (15), which represents the P 2 term in Equation (3), and Equation (16); where where , is the intensity of the orientation distribution function as shown in Equation (14).

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Results and Discussion
The CNT veils were readily imaged by FEG-SEM, without additional coating, allowing the underlying structure to be visualised at a range of resolutions. The approach described above, and summarised in Figure 3, was used successfully to calculate the local order parameters.
These CNT arrays were used as a case study in order to identify the best imaging magnification and domain size for the analysis.

Parameter optimisation
The sample contains structure at different characteristic lengthscales and the objects resolved normalised Herman's parameter falls consistently in a narrow range.. As the resolution increased further, the range of Herman parameters again broadened, due to the additional detail in the image (such as CNT surface texture, contaminating nanoparticles, or simply imaging noise) unrelated to the primary CNT orientation. Therefore, for all further analysis in this paper, a magnification of 1 to 3 pixels / CNT was used. A similar effect might be obtained by applying a radial filter to the FFT of high resolution images before azimuthal integration. However, it is more efficient to collect only the necessary data rather than discarding information digitally. Within the selected magnification range, further radial filtering had little effect. The selected magnification (and implied frequency range) effectively evaluates the waviness of the CNTs due to deformation or growth stresses (typically on lengthscales of 100 nm or more) which is particularly relevant to the composite performance. Crystalline 'quality' of the CNTs which governs intrinsic axial properties, is better assessed by Raman spectroscopy, high resolution TEM, or other established methods [19]. Our TEM measurement (ESI- Figure 1 and ESI- Figure 2) has confirmed the general crystalline structure of the CNT veil used in this research with a low incidence of particles and kinks. and ℎ e can be found. As is the case with one-dimensional Fourier Transforms, linearity applies to two dimensional transforms (Equation (17)) [33]. At small domain sizes, the spread of calculated Herman's parameters is large and the coefficient of best fit, R 2 , for fitting the 2D-FFT data to a Gaussian fit is poor (Figure 6b and CNT diameters, and this size was used for subsequent analysis.

CNT veil orientation measurement and validation
It has been shown previously, via polarised Raman spectroscopy, that straining veils induces CNT alignment [1,5]. Indeed, qualitatively, SEM (Figure 7a-c) confirms increasing alignment with increasing applied strain (5%, 10% and 15%). In this study, it was not possible to apply strains greater than ~15% without breaking the veils. Higher strains have been reported using a variety of methods [4,18], but the maximum is likely dependent on the initial orientation of the veil, and other processing factors such as rate and lubrication. Here, different strains are applied simply to demonstrate the analysis method, although in the medium term, the approach should help to optimise the drawing process.
Therefore, it is expected that the veils strained to 5%, 10% and 15% would have increasing degrees of alignment, as indeed can be seen qualitatively in the electron micrographs ( Figure   7a-c). The alignment is characterised in the 2D-FFT, and therefore the computed orientation distribution function.   More fundamentally, the FFT is sensitive to the topology of the CNTs, as revealed by the secondary electron generation. Raman is sensitive to local polarisability and crystallinity. Therefore, differences in crystallinity will affect the signals differently. Disruption to the crystalline quality of the CNTs or the orientation of the graphitic planes to the CNT axis will reduce the degree of orientation measured by Raman spectroscopy; however, the FFT will only be affected if there is gross distortion of the CNT shape. On the other hand, the FFTbased Herman's parameter will be more strongly reduced by CNT ends / discontinuities, and features related to intersecting CNTs, as well as lost information from unresolved parallel bundles. Since these CNT samples are relatively crystalline (I G /I D ratio of a sample area of pristine CNT veils varies from 8.2-13.6 (example spectre, ESI- Figure 4), and the Ramanderived Herman's parameter is higher, it seems likely that these latter effects may be the dominant source of divergence.

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A C C E P T E D ACCEPTED MANUSCRIPT Figure 11 : SEM micrograph of as-received CNT veil (x150,000).

Conclusions
A novel method was developed to quantify the alignment of carbon nanotubes within preformed arrays, specifically including veils. By taking a 2D fast Fourier transform of a scanning electron micrograph, the local orientation was converted into an an ODF and subsequently into a Herman's orientation parameter. The most effective magnification for this analysis was found to be around 1-3 pixel / CNT diameter, in order to resolve the principle features related to overall orientation. In addition, to successfully extract a statistically meaningful single director, a minimum region size of 200-500 CNT diameters was required, defining the resolution limit of the resulting orientation maps.
By quantifying orientation via image processing techniques, large scale high resolution orientation maps can be produced at much faster rates than current techniques. Such maps can resolve orientational inhomogenities which are in good qualitative agreement with orientation maps created using more common polarised Raman spectroscopy. Small deviations in the absolute Herman parameter extracted from the two methods may reflect the fundamental differences between the techniques;. It is not immediately obvious which absolute measurement is more relevant to the development of practical applications of such veils. Both approaches assume approximately planar arrangements; further refinement would be required to model more 3D distributions.

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Nevertheless, the 2D-FFT approach offers a significant advantage in speed as compared to Raman mapping methods. Large Raman maps of the type used in this study can easily take more than 12 hours to acquire, compared to less than an hour for stitched SEM images of a similar region, at the required resolution. The 2D-FFT method offers a simple, and easily understood, method to calculate the orientation of carbon nanotubes in pre-formed arrays, whilst directly observing the associated microstructural features. This type of information will assist the development of CNT orientation process and the associated performance of resulting composites.
Whilst illustrated here for CNT veils, the methodology should be widely applicable to other nanostructured systems. The dimensionless magnification and domain size guidelines should provide a useful reference in such cases. The approach is directly applicable to CNT veil composites for which analysing and optimising the orientation of veils, as produced, before infusion is a key step for enhancing performance. In principle the method may also be directly applicable to CNT veil composites impregnated with a beam stable resin, since