Chapter 9: Quantification of dispersion and distribution of carbon nanotubes in polymer composites using microscopy techniques – Polymer-Carbon Nanotube Composites

9

Quantification of dispersion and distribution of carbon nanotubes in polymer composites using microscopy techniques

S. Pegel, T. Villmow and P. Pötschke,     Leibniz Institute of Polymer Research Dresden, Germany

Abstract:

Different stereological approaches have been applied to micrographs of light microscopy and transmission electron microscopy to estimate the degree of carbon nanotube dispersion in polymer composites at different length scales. Quantitative information about the spatial relationships between individual carbon nanotubes has been obtained by means of the spherical contact distribution function and a Boolean fibre model. The correlation function has been used to gain an orientation factor as a measure of carbon nanotube alignment. The corresponding theoretical foundations and suitable digital image processing techniques are introduced and illustrated with some specific examples.

Key words

carbon nanotube composites

dispersion analysis

image processing

9.1 Introduction

The quantification of carbon nanotube (CNT) dispersion, distribution, and alignment within polymeric matrices can help gain a better understanding of structure–property relationships, to perform quality controls, or to optimize processing conditions. The aim of this chapter is to introduce some basic microscopy-based techniques to gain such morphological information.

Regarding polymer–CNT composites, the dispersion process refers to the individualization or spreading of the CNTs within the polymer matrix. Thus, the state of dispersion is commonly understood to describe the relationship between dispersed and not dispersed CNTs. For the purpose of dispersion quantification, the general degree of dispersion D can be defined straightforwardly as the ratio between the volume fraction of dispersed nanotubes VVD and the total volume fraction VVT of CNTs:

[9.1]

The quantity VVT is given by the composition of the mixture and can be estimated from the mass fraction w = (mCNT/(mCNT + mM) and the density ratio of matrix v = ρM/ρCNT of matrix and carbon nanotube material, respectively, using Equation 9.2.

[9.2]

In contrast to VVT, the estimation of VVD is more difficult. Since only isolated CNTs (not in touch with other CNTs) can be considered as dispersed, the connectivity of every single CNT to its next neighbour has to be investigated. This can be managed practically only by using very elaborate techniques such as TEM tomography which offers nanometre resolution. However, in order to gain meaningful information about the composite morphology using simpler and more accessible methods, the definition of the term dispersion has to be reduced or differentiated.

In the case of light microscopy (LM), agglomerates larger than 1 pm can be detected. The corresponding agglomerate area fraction AA is the ratio of the cumulative area fraction of all agglomerates within a micrograph and the total area of the micrograph, and therefore is sometimes referred to as the area ratio. This quantity is related to the amount of non-dispersed CNT and has been used as a simple measure for dispersion (the higher the AA, the worse the dispersion, Kasaliwal et al., 2010; Krause et al., 2010). Additionally, the agglomerate size distribution, number density (Villmow et al., 2008b; Kasaliwal et al., 2010) as well as spatial relationships between the agglomerates have been evaluated to gain information about processing–structure–property relationships (Kashiwagi et al., 2007; Li et al., 2007).

Next to scanning electron microscopy (SEM) and atomic force microscopy (AFM), transmission electron microscopy (TEM) is used to gain information about CNT dispersion in the submicrometre range (Pötschke et al., 2004; Pötschke et al., 2008). In all cases it is commonly understood that the quality of dispersion increases with increasing spatial separation between neighbouring CNTs. Nevertheless, quantitative investigations concerning spatial relationships between individual CNTs are rarely performed. Bellayer et al. (2005), for instance, evaluated the nearest neighbour distance distribution of the nanotube centres within TEM micrographs to access a quantity for CNT dispersion. However, it has to be considered that the 2D micrographs only partially reflect the 3D structure of the material, i.e. the 3D nearest neighbour distance distribution cannot be evaluated by means of 2D micrographs. Thus the ‘true’ dispersion situation cannot be resolved.

Generally, quantitative morphological investigations based on classical microscopic techniques have to be carried out carefully. Next to the corresponding 3D morphology, the content of nanotubes or agglomerates observed in a micrograph depends on the section thickness t as schematically shown in Fig. 9.1. It can easily be seen that the number of objects involved in the projection increases with t. In the case of opaque spheres embedded in a transparent medium this means an increase of the AA with increasing t. Additionally, the 2D, particle size distribution (PSD) is dependent on t. This is also known as the Wicksell corpuscle problem (Wicksell, 1925).

9.1 Schematic of the relationship between section thickness and features appearing in projections or micrographs.

The former considerations show that morphological quantities of the 3D structures, e.g. the volume content of a certain phase, can only be estimated by means of suitable stereological approaches. In Section 9.2.1, a germ grain model will be applied to LM micrographs to get an unbiased estimate of the CNT agglomerate volume fraction in polymer CNT composites. It can be used to estimate the light microscopy degree of dispersion DLM. In Section 9.3.1, a stereological approach for fibre processes will be used to get an unbiased estimate of VVD directly from TEM micrographs. Accordingly, the TEM degree of dispersion DTEM will be derived.

Next to the dispersion, it can be important to obtain more detailed information about the spatial distribution of individual CNTs within polymer matrices. Principally, morphologies with ‘correlated’ and ‘uncorrelated’ covering probabilities can be distinguished. In the case of ‘uncorrelated’ covering probability, the CNTs are randomly distributed in space (Poisson distribution) as shown exemplarily by means of the 2D point pattern of Fig. 9.2 (a). In contrast, Figs 9.2 (b) and 9.2 (c) show examples with ‘correlated’ probabilities. For ‘hard core’ processes, the covering probability near an object is smaller as compared to the left space (Fig. 9.2 (b)). For ‘cluster’ processes the covering probability near an object is larger, compared to the left space (Fig. 9.2 (c)). In Section 9.3.2, the distribution coefficient QP is introduced as a measure for the tendency of CNT cluster formation.

9.2 Different spatial distributions of random point pattern: (a) Poisson distribution; (b) hard core; (c) cluster process.

CNT alignment is another important aspect of CNT microstructure. Principally, it can be quantified by means of X-ray (Jin et al., 1998; Fischer et al., 2003; Sandler et al., 2004; Du et al., 2005; Fornes et al., 2006) or Raman (Wood et al., 2001; Bhattacharyya et al., 2003; Fischer et al., 2003; Pötschke et al., 2005; Fornes et al., 2006) scattering experiments. However, due to the limitations in focusing the X-ray or laser beam, the spatial resolution is limited for these methods. Furthermore, low filler contents cause high signal-to-noise ratios. The statistical analysis of suitable micrographs can provide information about CNT alignment at high resolutions and low filler contents. Shaffer et al. (1998) have already demonstrated the use of the power spectrum to evaluate local order parameters for different CNT films by means of scanning electron micrographs. In Sections 9.3.3 and 9.3.4, the similar covariance (Stoyan et al., 1987) or two point probability function (Torquato, 2006) is used to derive an orientation factor for CNTs.

9.2 Light microscopy

It is recommended to always start morphological investigations on polymer CNT composites with light microscopy, as the existence of remaining primary agglomerates can easily be seen. LM is a fast and comparatively simple method capable of investigating relatively large sample volumes.

9.2.1 Degree of dispersion

Due to the resolution limit of light microscopy, agglomerates in the micrometre range and below (typically < 1 μm diameter) are considered to belong to the fraction of dispersed fillers. The volume fraction of non-dispersed CNTs is given by the product: f · VVA, where VVA is the volume fraction of agglomerates and f the packing density or volume fraction of CNT within the agglomerates. Thus, the volume fraction of dispersed nanotubes is given by: VVD = VVT – f · VVA. Together with Equation 9.1, this leads to the light microscopic degree of dispersion:

[9.3]

This calculation is similar to the macroscopic dispersion index used for rubber carbon black compounds (ASTM, 2008) and has been successfully applied also to rubber CNT composites by Le et al. (2009). For the packing density f, Le used a value of 0.25 without further explanation. The investigations were performed on planar sections (t = 0) and thus DLM could be estimated quite easily by means of the stereological formula VVA = AA.

However, sometimes it is not possible to obtain suitable planar micrographs which can be analysed properly, for instance, due to bad contrast between matrix and agglomerates or artefacts resulting from sample preparation (e.g. scratches on the sample surface). In such cases, the slightly more elaborated evaluation of thin sections can be helpful. If the sections are thin enough, the agglomerates do not overlap in the projection and AA depends linearly on t (Stoyan et al., 1987):

[9.4]

The constant of proportionality is given by a quarter of the volume specific surface of the agglomerates SVA. The agglomerate volume fraction can be obtained by analysing series of sections with different thicknesses and extrapolation of AA towards t = 0.

Next to VVA, an estimate of f is needed to evaluate DLM. Usually, the infiltration method can be used to get a rough estimate of pure CNT materials. For this purpose the nanotube material is infiltrated with a low viscosity epoxy resin. After curing of the resin, VVA is estimated for the infiltrated material by means of LM and the stereological approach as described before. If no swelling takes place, the infiltrated material can be considered as a composite with DLM = 0 and together with VVA and the known value of VVT, Equation 9.3 can be solved for f. Thus, f is basically given by the ratio of VVT and VVA (f = VVT/VVA). If it is not possible to estimate the packing density of a CNT material with the infiltration method, it is appropriate to obtain a lower bound fmin using the nanotube powder density ρP. Due to the empty spaces between the agglomerates, the packing density must be larger than fmin = ρP/ρCNT. An upper bound of the packing density fmax can be estimated by uniaxial compression tests of the crude CNT material up to pressures expected in the processing equipment, for instance.

Since the three quantities VVA, VVT and f are experimentally determined, they contain uncertainties. The propagation of uncertainty leads to:

[9.5]

with the uncertainties uDLM, uf, uVA and uVT of the degree of dispersion, based on light microscopy packing density, agglomerate volume fraction and total volume fraction of incorporated nanotubes.

9.2.2 Digital image processing and specific examples

For the evaluation of the agglomerate area fraction AA, the light microscopic micrographs have to be segmented adequately. This means that the content of the digital image has to be assigned either to the matrix or the agglomerates. Usually, the pixels are simply reassigned to black and white (binarization). In the present case, the binarization can be carried out by application of a global threshold to the intensity of background corrected grey scaled images.

The background correction is applied to obtain an even background illumination and to improve the contrast of the agglomerates. Most of the commercially and freely available image processing systems, e.g. ImageJ (Abramoff et al., 2004; Rasband, 2008) provide more or less suitable functions for this purpose. Comparatively good results can be obtained with the following procedure: A mean filter is applied to a copy of the original image. To avoid artefacts, the filter size has to be in the range of the largest agglomerates. A new image is created whereby the filtered image is compared with the original and the brighter pixels have been taken over. The mean filter is applied to the result again. Finally, each pixel is divided by the corresponding value of the original. In Figs 9.3 (a) and 9.3 (b) this is demonstrated for a sample with 1 wt% multi-walled CNTs (MWCNTs) in polycarbonate (PC).

9.3 Application of a background correction procedure to an LM micrograph: (a) original; (b) filtered image.

To determine adequate threshold values, some automated algorithms are available (Sahoo et al., 1988; Russ and Woods, 1995; Sezgin and Sankur, 2004). Most of these techniques evaluate the distribution of grey values (Otsu, 1975; Kapur et al., 1985). However, in the case of the presence of many small agglomerates with weak contrasts to the background, the corresponding histograms do not provide sufficient information and such algorithms are not applicable (Figs 9.4 (a)–(e)). Comparatively good results can be obtained with the feature-based algorithm described as follows: The correlation between the threshold and number of individual objects with roundness less than 0.5 is determined (Fig. 9.4 (e)). Generally, the number of detected objects increases with the threshold until a strong maximum is reached. This can be attributed to the inclusion of more and more background pixels which form objects with very irregular shapes. On exceeding the maximum, the individual objects aggregate and thus the total amount decreases. The binarization threshold is given by the longest distance between the curve and a straight line between the maximum and the origin of the diagram. To the binarized image, the morphological filter ‘open’ is applied to remove smaller objects (Fig. 9.4 (f)). The area fraction of agglomerates is given simply by the ratio of the black and total amount of pixels, which can be accessed by the histogram of the binary image.

9.4 Feature-based algorithm for segmentation of grey scaled light microscopy micrographs with carbon nanotube agglomerates: (a) detail of Fig. 9.5, this is binarized with grey scale thresholds of 100, (c) 138, and (d) 175. With the increasing threshold value, the number of agglomerates and thus AA increases. The corresponding grey scale histogram (e) cannot be used to determine an accurate threshold value. In contrast, the object count method (f) delivers a reliable threshold value of 138 with the corresponding binary image (c).

As an example, two composites with 1 wt% MWCNT (Nanocyl 7000, from Nanocyl S.A., Belgium) in PC processed under different melt mixing conditions are considered. The samples were processed with a small-scale batch compounder (such as described in Chapter 4) at 240 °C and 50 rpm. Sample 1 was mixed for 30 min and sample 2 for 5 min. Light microscopy (t = 25 μm) reveals a better filler dispersion for sample 1 as compared to sample 2 as indicated by the lower amount of visible agglomerates (Figs 9.5 (a) and 9.5 (b)). The evaluation of several digital micrographs as previously described leads to the relationship between the section thickness and the agglomerate area fraction AA(t), as shown in Figs 9.6 (a) and 9.6 (b). Linear fits with instrumental weighting result in agglomerate volume fractions of 0.6 ± 0.1 and 2.80 ± 0.05 vol.% for sample 1 and 2, respectively. The infiltration method yields an estimation of the packing density for the pure nanotube material of approximately 0.07. This is a comparatively low value and it cannot be ruled out that the CNT material is further compressed during processing. Therefore, uniaxial compression tests have been carried out to obtain an upper bound of approximately 0.25. Thus, the mean packing density is given by 0.16 with an uncertainty of 0.09.

9.5 Micrographs of differently melt processed composites (small-scale batch compounder such as used in Chapter 4) with 1 wt% MWCNT in PC obtained by light microscopy (t = 25 μm): (a) sample 1 mixed at 240 °C and 50 rpm for 30 min, DLM = 0.87 ± 0.08; (b) sample 2 mixed at 240 °C and 50 rpm for 5 min, DLM = 0.33 ± 0.40.

9.6 Evaluation of AA for different section thickness: (a) sample 1 and (b) sample 2. The extrapolation t → 0 yields agglomerate volume fractions of 0.6 ± 0.1 and 2.80 ± 0.05 vol.%, respectively.

Accordingly, the degree of dispersion is DLM = 0.87 ± 0.08 for sample 1 and DLM = 0.33 ± 0.40 for sample 2. The comparatively large uncertainty of DLM of sample 2 is caused by the large uncertainty of packing density and the large agglomerate volume fraction. For low agglomerate volume fractions, the uncertainty of packing density is insignificant (sample 1). Generally, the light microscopic degree of dispersion exhibits a good dispersion measure, especially for small agglomerate volume fractions.

9.2.3 Further remarks

The specific examples illustrate that an unbiased estimate of the agglomerate volume fraction can be obtained with a series of thin sections. However, the determination of DLM can be inaccurate for samples with high agglomerate volume fractions and CNT materials with low packing densities f. Due to the possible densification of CNT agglomerates during processing, the estimate for f is afflicted with a comparatively high uncertainty. If highly packed CNT materials (see also Chapter 4) are used, the uncertainty of f and thus DLM is comparatively small since the agglomerates cannot be further compressed during processing.

9.3 Transmission electron microscopy

In contrast to LM, micrographs with nanometre resolution can be obtained by means of transmission electron microscopy and thus individual CNTs can be recognized. This leads to the possibility of estimating the volume fraction of dispersed CNTs directly and of evaluating spatial relationships between individual CNTs. However, to ensure a sufficient confidence level, a reasonable number of micrographs have to be evaluated.

9.3.1 Degree of dispersion

To estimate the volume fraction of dispersed CNTs, they are treated as a 3D fibre system. In 3D, this can be characterized by the total fibre length per unit volume Jv. Accordingly, a 2D fibre system can be characterized by the total fibre length per unit area LA. For a projection of an isotropic 3D fibre system on a plane, the intensities JV and LA are connected to the section thickness t by:

[9.6]

LA can be obtained with a system of test lines as schematically shown in Fig. 9.7. With a sufficient total length of the test system L, an unbiased estimate for LA can be evaluated by means of the number of intersections with the fibre system N and Equation 9.7.

9.7 Schematic for the determination of the volume specific fibre length by means of planar projection and a system of test lines.

[9.7]

The volume content of dispersed CNTs is simply given by: VVD = JV · A0, where A0 denotes the mean cross-sectional area of the nanotubes (Pegel et al., 2009). Thus, the degree of dispersion DTEM is given by Equation 9.8.

[9.8]

Finally, the Equations 9.7 and 9.8 as well as the law of the propagation of uncertainty lead to:

[9.9]

where uDTEM, uLA, ut, uVT and uA0 are the uncertainties of DTEM, LA, t, VVT and uA0, respectively.

9.3.2 Distribution coefficient

To investigate the distribution characteristics of individual nanotubes, the spherical contact distribution function (SCDF) can be employed. Mathematically, the SCDF HS(r) basically describes the distribution of the shortest distance between a random closed set (RCS) Ξ and a random point which is located outside Ξ. For the evaluation of a binarized TEM micrograph the pixels which are assigned to carbon nanotubes are treated as realization of a RCS. The corresponding empirical SCDF can be derived by dilation with spherical structure elements of different radii as demonstrated in Fig. 9.8. With increasing radius r of the structure element the area fraction of a dilated set TΞ(r) increases as well. For r →  the function TΞ(r) approaches the limit one. can be estimated from TΞ(r) by means of Equation 9.10.

9.8 Two realizations of an RCS with differently distributed line segments (similar mean fibre length and intensity, image size 500 × 500 pixels) dilated with circles of 25, 50 and 75 pixels radius (a). In the diagram the SCDF of those sets as well as the corresponding Boolean model is shown (b).

[9.10]

For planar fibre processes with randomly distributed straight line segments, i.e. the covering probability for the line segments is uniformly distributed in space, the SCDF can be estimated with Equation 9.11 (Boolean model). Hereby, the number of fibre segments per unit area is denoted by λ (intensity of the fibre process) and the mean fibre length by m1.

[9.11]

A deviation of the SCDF of a RCS with line segments of length m1 and intensity λ from the corresponding Boolean model can be attributed either to attractive (cluster process) or repulsive (hard core process) interactions between the fibre segments. Fibre processes with the tendency to maximize the distance between the individual fibres are characterized by comparatively fast convergence of TΞ.(r) and thus against the limit of one. For cluster processes, this function approaches the limit of one comparatively slowly due to the larger spacing between the fibre segments. Figure 9.8 shows a schematic to illustrate the differences.

To obtain a metric for the deviation of the empirical data from the corresponding Boolean model the distribution coefficient QP is introduced:

[9.12]

Since the integral ∫ 01 − F(X) results in the mean value of random variable X with the cumulative distribution function F(X), the intensity QP can be considered the ratio of the mean values of the empirical SCDF and the corresponding Boolean model. Due to the larger gaps between fibre clusters, QP is larger than 1 for this kind of processes. Values of QP smaller than 1 indicate a hard core process.

The uncertainty of QP depends mainly on the uncertainties of m1 and λ. Especially for systems with strongly clustered filler particles, it can be problematic to get an unbiased estimate due to the overlap of neighbouring CNTs. Additionally, it has to be considered that only the distribution of the projected CNT system is analysed. Thus, it has to be assumed that the distribution state of the 3D system is somehow connected to the distribution of the projected 2D system. Furthermore, only a limited section of the entire CNT system can be evaluated due to the limits in resolution and data processing capability, i.e. a holistic investigation on several length scales is not possible.

9.3.3 Orientation factor

The evaluation of the covariance (Stoyan et al., 1987) or the two-point probability function (Torquato, 2006) is a comparatively convenient way to gain information about the state of CNT orientation. The covariance C(r) of a RCS Ξ is defined by the probability P that a randomly chosen point x and a second point x + r both belong to Ξ. If Ξ2 holds, this can also be expressed as the area fraction of the intersection between Ξ and Ξ shifted by the vector r (Equation 9.13).

[9.13]

To elucidate some important properties of C(r) in Fig. 9.9(a) an example with four parallel bars is considered. For small r (||r|| in the range of the bar width), it can easily be seen that the intersection area of the original and shifted set of bars is strongly related to the orientation of the vector r (Fig. 9.9 (b)). The corresponding 2D covariance (Fig. 9.9 (c)) exhibits a kind of averaged structure whereby the alignment of the bars is reflected by the distortion of the central pattern. Furthermore, the uniform lateral distance between the bars results in periodic pattern of C(r) in the x-direction.

9.9 (a) a 2D realization of a RCS Ξ with the shape of four parallel bars; (b) the intersection of Ξ and Ξ shifted by the vector r depends on the modulus and the orientation of r; (c) 2D covariance as grey scaled image; (d) 1D correlation function derived from the covariance along the x- and y-direction. Adapted from (Pegel et al., 2009)

The normalization of the covariance leads to the correlation function κ(r) where the influence of area fraction AA of the RCS is eliminated:

[9.14]

For stationary (invariance of translation) and isotropic (invariance of rotation) RCS, the correlation function as well as the covariance depends only on the distance r = ||r||. In this case, the correlation function holds the properties: κ(0) = 1 and κ() = 0. As in the case of the covariance, for anisotropic RCS, the curve shape of the correlation function is direction-dependent.

The example with the four oriented bars reveals that the drop of the correlation function in the direction coinciding with the main bar orientation κy(r) is much less pronounced as compared to the direction perpendicular to the main bar orientation κx(r) (Fig. 9.9 (d)). To obtain a measure for orientation, different features of κx(r) and κy(r) can be compared. The orientation factor Or is a relatively simple measure (Equation 9.15), whereby the ratio κx(r)/κy(r) is considered at a given distance r (practically r has to be in the range of the width of the elongated objects).

[9.15]

With increasing anisotropy of the investigated system the orientation factor approaches one. For isotropic systems, the orientation factor equals zero, since κx(r) and κy(r) overlap.

9.3.4 Digital image processing and specific examples

As in the case of LM analysis, the TEM micrographs have to be segmented at first. A Fourier band pass filter can be used to provide an even background illumination and to increase the contrast of nanotubes (Fig. 9.10). If necessary, further improvements can be achieved with the background correction used for the LM micrographs.

9.10 TEM micrograph of an injection-moulded PC sample with 2 wt% MWCNT near the mould wall: (a) original; (b) after filtering with a Fourier band pass filter.

For the binarization of the background-corrected TEM micrographs, an algorithm based on the scheme shown in Fig. 9.11 gives reasonable results. At first, the total number of objects depending on the threshold value is determined. In the next steps two binary images with two thresholds are created. The first threshold is chosen at the point when some background pixels are involved by the selection. This point 1 is determined by the longest distance between the particle threshold curve and the straight line between the maximum and the origin of the diagram. To remove the few background pixels or artefacts, the morphological filter ‘open’ is applied. The second binary image is created with the threshold corresponding to the maximum of detected particles (point 2 in Fig. 9.11). The two binary images serve as seed and mask for the final binary reconstruction. This means basically that disconnected regions of the second image are removed if they are not marked by the first one.

9.11 Scheme for the binarization of TEM micrographs.

As examples, TEM micrographs of sample 2 (Fig. 9.12 (a), see section 9.2.2) and another composite with 1 wt% MWCNTs in PC (sample 3, Fig. 9.12 (b)) are considered. As mentioned before, sample 2 was mixed with a small-scale batch compounder at 240 °C and 50 rpm for 5 min. In contrast, sample 3 (same composition) was mixed at 300 °C and 50 rpm for 5 min. The thin sections shown in Figs 9.12 (a) and 9.12 (b) have been prepared from the extruded strands (t ≈ 120 nm, estimated from interference colours). In sample 2, the amount of dispersed MWCNTs is obviously larger compared to sample 3. The non-dispersed MWCNTs of sample 3 remain in larger agglomerates which are not considered here. Furthermore, the MWCNTs seem to be more clustered in sample 3.

9.12 TEM micrographs of two composites with 1 wt % MWCNT in PC prepared by means of a small-scale batch compounder and different melt mixing conditions: (a) sample 2 mixed at 240 °C and 50 rpm for 5 min, DTEM = 0.77 ± 0.13; (b) sample 3 mixed at 300 °C and 50 rpm for 5 min and DTEM = 0.28 ± 0.06.

For the quantitative morphological analysis five TEM micrographs of each sample were binarized as described before. To obtain an estimate of LA, a set of horizontal and vertical test lines (line distance 10 pixels) was created. The number of intersections with the projected MWCNT system has been counted by evaluating the intensity profiles along each test line (number of peaks counted). The mean cross-sectional area A0 ≈ 81 ± 25 nm2 of the MWCNTs (Nanocyl 7000) was determined by means of higher magnified micrographs of the composite and an algorithm based on that reported by Gommes et al. (2003). Considering the uncertainty of the section thickness ut ≈ 10 nm (Sakai, 1980; Michler et al., 2004), the results obtained are DTEM = 0.77 ± 0.13 for sample 2 and DTEM = 0.28 ± 0.06 for sample 3. In contrast to light microscopy, the TEM investigations result in a comparative large degree of dispersion for sample 2. With regard to the statistical significance of DTEM (many TEM micrographs have been taken at different positions), the difference to DLM can be explained by the high uncertainty of the CNT packing density within the agglomerates.

To investigate the distribution state of the individual MWCNTs, the empirical SCDF was determined for both micrographs by dilating the binary images with disc-like structuring elements and evaluation of the area fraction. To obtain the SCDF of the corresponding Boolean model, the fibre ends were counted and the values for λ and m1 determined. The ends can be counted simply by performing the morphological operation of skeletization and determining the number of pixels with a connectivity number (connected pixels in the neighbourhood) of one. The mean fibre length m1 is given by LA/λ.

The empirical SCDF of the TEM micrographs of Figs 9.12 (a) and 9.12 (b), as well as the SCDF of the corresponding Boolean models, are shown in Figs 9.13 (a) and 9.13 (b), respectively. Evidently, the MWCNTs of sample 2 are arranged with an uncorrelated covering probability. The curves of the SCDF of the empirical data and the Boolean fibre model with the same intensity and mean fibre length (λ ≈ 47.41 μm− 2 and m1 ≈ 0.12 μm) are very similar. Thus, the improper integrals and dr and ∫ 0 exp{− 47.41μm− 2 ⋅ r(2.0.12μm πr)} dr (this basically corresponds to the shaded area in Fig. 9.13 (a)) yield very similar values. The evaluation of five micrographs leads to a distribution coefficient of 1.04 ± 0.01 for sample 2. Considering the TEM micrograph of sample 3 shown in Fig. 9.12 (b), a totally different behaviour of the SCDF of empirical data and the corresponding Boolean fibre model (λ ≈ 17.82 μm− 2 and m1 ≈ 0.14 μm) can be observed. Due to the tendency of MWCNT cluster formation, the empirical SCDF approaches the limit only at comparatively large values of r (Fig. 9.13 (b)). The evaluation of five micrographs of sample 3 results in QP = 2.07 ± 0.12.

9.13 SCDF of the binarized TEM micrographs shown in Fig. 9.12: (a) sample 2, QP = 1.04 ± 0.01; (b) sample 3, QP = 2.07 ± 0.12.

In a second example, the morphology of two different injection-moulded plates here referred to as samples 4 and 5 are discussed. Both samples consist of a composite with 2 wt% MWCNTs in PC which was produced by means of masterbatch dilution using a co-rotating twin screw extruder (for details, see Villmow et al., 2008a). The samples have dimensions of 80 × 80 × 2 mm3 and were processed under the injection moulding conditions as listed in Table 9.1. The different injection moulding conditions resulted in electrical volume conductivities measured through the sample thickness of approximately 10− 13 and 10− 8 S/cm for samples 4 and 5, respectively.

Table 9.1

Injection moulding parameters of samples 4 and 5 (2 wt% MWCNT in PC)

TEM investigations near the mould wall resulted in the micrographs shown in Figs 9.14 and 9.15. In this case, the low resolution micrographs have been used to obtain an overview and to determine the distance to the mould wall for the high resolution micrographs. The high resolution images have been used for the detailed morphological studies as shown below.

9.14 TEM micrographs of the injection-moulded sample 4 at different positions to the mould wall. (adapted from Villmow et al., 2008a)

9.15 TEM micrographs of the injection-moulded sample 5 at different positions to the mould wall. (adapted from Villmow et al., 2008a)

To gain information about the state of MWCNT orientation, the orientation factor Or was evaluated as demonstrated by means of the two examples shown in Figs 9.16 (a) and 9.16 (b). These are two binarized micrographs of sample 4 taken at 2 and 42 μm distance to the mould wall. The grey scaled images shown in Figs 9.16 (c) and 9.16 (d) represent enlarged sections of the corresponding 2D correlation functions, which can easily be obtained by application of the fast Fourier transform (Wiener-Khinchin theorem) as well as normalization (see Equation 9.14). It can be seen that the distinct orientation of MWCNT at a distance of 2 μm to the mould wall (Fig. 9.16 (a)) leads to a strong elliptical distortion of the corresponding 2D correlation function (Fig. 9.16(c)). In contrast, the 2D correlation function of the micrograph taken at 42 μm distance to the mould wall (Figs 9.16 (b) and 9.16 (d)) exhibits a comparatively regular angular intensity distribution originating from the image centre.

9.16 Binarized TEM images of sample 4 at (a) 2 μm and (b) 42 μm distance to the mould wall, enlarged 2D correlation function of the binary images with elliptical fits (c) d = 2 μm and (d) d = 42 μm, 1D correlation function in the major (y-) and minor (x-) directions of MWCNT orientation; (e) d = 2 μm; (f) d = 42 μm. Panels (e) and (f) are adapted from Pegel et al. (2009).

To evaluate the mean direction of the MWCNT orientation, different thresholds were applied to the 2D correlation function to filter regions which correspond to small ||r|| (centre of image). By means of an elliptical fit to the selected set of pixels, the major and minor directions (respectively the y and x directions) were determined for the different thresholds. The major and minor directions were calculated and the corresponding radial intensity profiles extracted from the 2D correlation function (Figs 9.16 (e) and 9.16 (f)). To obtain a meaningful measure for the state of MWCNT alignment, the orientation factor Or was determined at ||r|| = 16 nm (correlations to neighbouring CNTs are not considered).

In Fig. 9.17, the dependency of O16nm from the distance d to the mould wall is shown for samples 4 and 5. It can be seen that both samples exhibit a significant MWCNT alignment in the proximity of the mould wall. In comparison to sample 5, the MWCNT alignment is more pronounced in sample 4. However, already at distances from the mould wall greater than 25 μm, no significant differences can be seen.

9.17 Morphological quantities of the injection moulded samples 4 and 5 with regard to mould wall distance: (a) orientation factor; (b) distribution coefficient. (adapted from Pegel et al., 2009)

To gain more information about the morphology, for these samples the distribution coefficient has also been evaluated (Fig. 9.17 (b)). It can be seen that the MWCNTs are almost randomly distributed within sample 4 (QP ≈ 1). In contrast, QP > 1.3 indicates a significant MWCNT cluster formation in sample 5. For both samples no clear correlation between d and QP can be seen.

The results illustrate that the higher electrical volume conductivity of sample 5 as compared to sample 4 can be attributed to the less pronounced MWCNT alignment within a surface zone of approximately 25 μm, combined with cluster formation through the whole sample.

9.3.5 Further remarks

Principally, the concepts of dispersion degree and distribution coefficient can be adapted to other microscopic methods such as SEM or AFM. In that case, a stereological approach for planar sections through isotropic fibre systems and the SCDF of point patterns has to be used.

The number of cross-section points per unit area PA of an isotropic 3D fibre system with random plane section is given by the simple Equation 9.16. With the cross-sectional area A0 of the CNT the volume fraction of dispersed CNT VVD and thus the degree of dispersion can be estimated as shown before.

[9.16]

The SCDF of a point pattern with Poisson distribution is given by Equation 9.17. To obtain a distribution coefficient according to Equation 9.12, the empirical contact distribution function has to be determined by dilation of the point pattern with spherical structure elements of different radii as described before.

[9.17]

9.4 Conclusion and future trends

LM and TEM can be used to gain a measure of the state of CNT dispersion in polymer CNT composites at different length scales. In the case of LM, the volume fraction of CNT agglomerates is used to estimate the fraction of dispersed CNT. However, for this purpose, the packing density of CNT within the agglomerates has to be known as closely as possible. At high agglomerate volume fractions, a high uncertainty of the packing density can cause significant uncertainties about DLM. In such cases, TEM investigations can provide further information since the uncertainty of DTEM is small for poorly dispersed CNT. Additionally, the TEM micrographs can be used to evaluate QP which serves as measure for CNT clustering.

The basic image processing algorithms which have been applied to some selected examples are usually available in every common image processing software. The single processing steps can easily be automated by means of macro programming. Thus, very large sample series can be evaluated in short time periods. In practice, the algorithms proved to be very reliable.

For the future it could be interesting to develop algorithms for statistical evaluation of 3D data with nanometre resolution. In this way for instance, percolation or agglomerate structures can be characterized in more detail.

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9.6 Appendix: list of abbreviations

AFM Atomic force microscopy
CNT Carbon nanotube
LM Light microscopy
MWCNT Multi-walled carbon nanotube
PC Polycarbonate
PSD Particle size distribution
RCS Random closed set
SCDF Spherical contact distribution function
SEM Scanning electron microscopy
TEM Transmission electron microscopy
A0 Cross-sectional area of the CNT
AA Area fraction of agglomerates or RCS
C(r) Covariance
D Degree of dispersion (general case)
d Distance to the mould wall (injection-moulded samples)
DLM Light microscopic degree of dispersion
DTEM Transmission electron microscopic degree of dispersion
f Packing density of CNT within the agglomerates
fmax Maximum packing density of CNT within agglomerates (estimated by means
of pressed tablets)
fmin Minimum packing density of CNT within agglomerates (estimated by means
of the powder density)
HS(r) Spherical contact distribution function
JV Volume specific fibre length
κ(r) Correlation function
λ Intensity (number of fibres per unit area)
LA Area specific fibre length (cumulative fibre length per unit area)
m1 Mean length of line segments
mCNT CNT mass within a composite
mM Matrix mass within a composite
N Number of intersections with a system of test lines
Or Orientation factor
PA Points per unit area
QP Distribution coefficient
r Radius, distance
ρCNT CNT density
ρM Matrix density
SVA Volume specific surface of agglomerates (agglomerate surface area per unit
volume)
t Section thickness
TΞ Area fraction of an RCS Ξ dilated with a disc-like structure element of radius r (capacity functional)
uA0 Uncertainty of CNT cross-sectional area
uDLM Uncertainty of DLM
uDTEM Uncertainty of DTEM
uf Uncertainty of f
uLA Uncertainty of LA
ut Uncertainty of t
uVA Uncertainty of VVA
uVT Uncertainty of VVT
V Ratio of CNT and matrix density
VVA Agglomerate volume fraction (agglomerate volume per unit volume)
VVD Volume fraction of dispersed CNT
VVT Total volume fraction of CNT (dispersed and non-dispersed)
w CNT mass fraction
Ξ Identifier for a realization of an RCS