Classifying the Shape of Colloidal Nanocrystals by Complex Fourier

Dec 14, 2011 - These results suggest that classification of shape by Fourier descriptor analysis may in the near future be a powerful tool for continu...
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Classifying the Shape of Colloidal Nanocrystals by Complex Fourier Descriptor Analysis Katherine P. Rice, Aaron E. Saunders, and Mark P. Stoykovich* Department of Chemical and Biological Engineering, University of Colorado at Boulder, Boulder, Colorado 80309, United States S Supporting Information *

ABSTRACT: The optical, electrical, magnetic, and catalytic properties of colloidal nanocrystals are intimately tied to their form, in particular their physical size and shape. Synthetic techniques have been developed to produce metallic and semiconducting nanomaterials with well-controlled forms; however, characterization tools for describing shape have remained limited to small samples and lack the quantitative rigor necessary for a universal classification scheme. Here complex Fourier descriptors are shown to be a quantitative and high-throughput approach for classifying the shape of colloidal nanocrystals. Large, monodisperse, and polydisperse ensembles of CdSe nanocrystals are characterized with respect to shape and categorized as circles, triangles, squares, rods, and pentagonal or hexagonal platelets. These results suggest that classification of shape by Fourier descriptor analysis may in the near future be a powerful tool for continuous monitoring of synthesis, purification, or packaging/integration processes during industrialscale production of nanomaterials.

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nucleation,13 and the use of growth directing ligands14,15 that can block precursor addition to specific crystal facets. The characterization of the form of nanocrystals remains challenging, however, with the shape in particular of such materials being poorly defined. Morphometric characterization of nanocrystals, that is, a quantitative description of size, composition, and interior character, is currently accomplished through the combined application of direct imaging by electron microscopy, optical spectroscopy, and X-ray scattering and diffraction characterization, along with other more specialized techniques. The shape of nanoparticulates can be assigned via direct imaging techniques, but thus far this approach has not been quantitative, is subjective with respect to the operator and their classification scheme, and has been applied to exceedingly small samples of the overall nanostructure population. A rigorous and quantitative approach for classifying the shapes of colloidal nanocrystals could therefore impact many areas of nanocrystal synthesis and processing. In this article, we detail the application of a morphometric method based on complex Fourier descriptors to automate the classification of colloidal nanocrystals by their shape. This approach analyzes images obtained through transmission electron microscopy (TEM) and fits the boundary data to a curve defined by a convergent Fourier series. The Fourier descriptors or coefficients of lower order successfully describe the triangularity, squareness, and elongation that distinguish common shapes observed in nanocrystal systems including spheres, ellipsoids, tetrahedrons, cubes, rods, and hexagonal platelets.16 The classification of

ingle-crystalline particulates between 1 and 100 nm in size, or nanocrystals, are of broad technological interest in electronics,1 optics,2 sensors,3 catalysis,4 and other potential application areas because they exhibit material properties that are intimately linked to their “form”. Nanomaterials have relatively large surface to bulk ratios, and consequently many aspects of their “form”, in addition to size, can influence the surfaces and overall physical, electronic, optical, and chemical properties. Catalytic activity,4 chemical reactivity,5 and optical properties of both metallic6,7 and semiconducting8,9 nanocrystals, for example, have been observed to be dependent on their shape. The “form” of a nanocrystal, and of any multidimensional object in general, encompasses the size, shape, orientation, surface texture and color, interior character, and composition attributes of that material.10 The development of (1) synthetic techniques to control nanocrystal form and (2) tools to characterize these forms will become of increasing importance as the nanocrystal field transitions from discoveryscale experiments to large-scale production and materials applications.11 Of the standard attributes of form, nanocrystal size and shape are the most easily controlled by the chemist through the synthetic conditions, whereas composition and crystal structure are primarily determined by the crystalline material that is being grown. The size of nanocrystals can be controlled by the rational selection of the reaction time, reaction temperature, and precursor and capping ligand concentrations. Control over nanocrystal shape has also been achieved, with advances in synthetic methods enabling diverse shapes including spheres, rods, wires, cubes, hexagonal platelets, tetrapods, and other highly faceted particles of semiconductors and metals. Shape can be controlled during nanocrystal synthesis by a variety of methods, some of which include sequential injections of precursor,12 seed particle induced © 2011 American Chemical Society

Received: September 3, 2011 Revised: December 12, 2011 Published: December 14, 2011 825

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frequently arises when isolating the objects from the image background. The edge coordinates are subsequently upsampled to M = 128 coordinates (x, y) per edge that are equally spaced along the closed boundary, that is, points moving at a constant velocity along the contour or edge. The complex Fourier descriptors (an, bn) are then calculated using the following mathematical formulation:16

shape is demonstrated for a set of images of real colloidal CdSe nanocrystals having monodisperse and polydisperse forms, and used to calculate distributions of nanocrystal shapes. The classification process by complex Fourier descriptors is robust for a wide range of shapes, including many regular or highly faceted shapes beyond those demonstrated here, and is proved to be only weakly influenced by the surface texture and the smoothing of particulate corners that commonly arise from wet chemical synthesis and TEM imaging. Currently there is no standard methodology to quantify object shape, although the primary approaches have been the conventional metrical approach,10 point homology,17 and Fourier descriptors.18,19 The metrical approach relies upon distinct and regular features to define the shape using a set of linear distances, angles, and ratios, and works effectively for man-made shapes and natural materials that display regularity. This comparative analysis becomes challenging, however, for more complex, noisy, and varied shapes, and is itself a subjective approach. Point homology requires a set of welldefined landmarks within the object that can be mapped in a one-to-one relationship across all shapes of the same family. Colloidal nanocrystals lack such obvious landmarks and cannot easily be characterized by homological techniques. The Fourier descriptor analysis, on the other hand, enables the quantification of shapes that are asymmetric or irregular, and has been applied in engineering, the biological sciences, medical sciences, geological and soil sciences, and social sciences to analyze shape from an overall morphology.10 This approach has successfully described shape in diverse systems including hominid skull remains,20 growth and development of the human body,21−24 species classification,25 clouds,26 flowers,27 snowflakes,28 grains of sand and soil,16 and the identification of handwriting29 and speech.30 Individual Fourier descriptors are not used in these complex systems to determine a specific shape, but rather a set of desciptors are compared against indexed libraries or databases to isolate the family of shapes to which the system most closely belongs. Fourier descriptor analysis includes a number of related techniques including conventional Fourier descriptors with either center-based19 or angular function algorithms,31 elliptical Fourier functions,32 and complex Fourier descriptors. The conventional Fourier descriptors suffer from a number of complications, including that the edge points must be recast from Cartesian coordinates into polar form, the angles between edge defining points must be equal to avoid weighting the Fourier descriptors, and the function that describes the edge must be single valued. The latter constraint is important because it limits conventional Fourier analyses to simple shapes without edges that cross or that are concave, and thus precludes the characterization of some nanocrystal shapes, for example, multiarmed nanocrystals. Complex and elliptical Fourier analyses are not subject to these complications including that of re-entrant angles, but tend to be more computationally expensive.

+M /2

an + ibn =

⎛ 2πnm −i (xm + iym )⎜cos ⎝ N m =−M /2

∑

sin

2πnm ⎞⎟ N ⎠

(1)

where N is the total number of descriptors, n is the descriptor number, M is the total number of points that define the edge, and m is the index number of a point on the edge. The total number of Fourier descriptors that have been calculated for all cases is N = 127, which corresponds to n ranging from −63 to +63. The up-sampling step is unnecessary for the calculation of the Fourier descriptors as it generates a set of redundant edges; however, it has been employed nonetheless to provide flexibility to the program such that N = 127 can always be used for the calculation of the discrete Fourier transform regardless of the extent of down-sampling (to ≤128 coordinates). The classification of nanocrystal shapes was performed using a subset of 15 descriptors from −7 ≤ n ≤ 7 that are below the Nyquist frequency in order to avoid issues with aliasing. Complex Fourier descriptors of each order contain information on magnitude and phase that correlates to shape, but only the magnitudes f n = (an2 + bn2)1/2 were considered here. The Fourier descriptors were subsequently normalized to f 0 to enable direct comparison been the spectra of nanocrystals with different shapes, sizes, and locations in the TEM images. Figure 1 shows plots of the magnitude of the Fourier descriptors f n for many of the common shapes observed thus far in colloidal nanocrystals, including circles, triangles, squares, pentagons, hexagons, octagons, and elongated variants thereof. Circles for example have relatively small intensities (≤0.1) of the normalized descriptors for all but n = −3 and 4, which can be used to distinguish circles from triangles that exhibit a large peak magnitude ( f−2 ≥ 10) for n = −2 and overall higher intensities (≈1). Squares are characterized by peak intensities at n = −3 and 4, pentagons by peaks at n = −4 and 5, hexagons by peaks at n = −5, 2, 4, and 6, and octagons by peaks at n = −7, −3, and 4. Elongated version of these shapes have large descriptor magnitudes at n = −1, the magnitude of which depends directly on the aspect ratio of the shape. Table 1 highlights the characteristic signatures for each of these regular shapes in terms of the magnitudes of the complex Fourier descriptors. Some of the descriptors clearly correlate to distinct aspects of shape. Descriptors n = −3, −2, and −1 represent squareness, triangularity, and elongation, respectively. More broadly, polygons with P-sides have significant descriptor magnitudes for the descriptors n = P and n = −1*(P − 1); that is, f5 and f−4 are measures of pentagonality and f6 and f−5 are measures of hexagonality. Figure 2 demonstrates the application of Fourier descriptors to the accurate classification of squares, circles, and triangles in geometric artwork. The software determines only the exterior boundary of each object and fills the remainder of the object even if interior black-and-white boundaries are present; thus,

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RESULTS AND DISCUSSION We have opted here to use complex Fourier descriptors to classify nanocrystal shape. The boundaries of colloidal nanocrystals are traced from black-and-white transmission electron micrographs using only the 4-nearest neighbor pixels to define the boundary, and these edge coordinates (x′, y′) are tabulated. The edges are then down-sampled to 16 equally spaced coordinates to reduce roughness or noise in the boundary that 826

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boundaries, particularly those that are light blue, gray, and yellow, are not characterized because they are lost upon conversion of the grayscale image (Figure 2b) to the binary image (not shown). In this case, pixels that had gray values less than 50% of the black intensity were thresholded to white, but this thresholding level can be adjusted in order to analyze lighter objects as well. It is important to note that the software also does not characterize any object that intersects the maximal boundary of the image because these fixed edges may influence the shape assignment. Figure 3 highlights the steps important for the characterization of an individual nanocrystal shape by complex Fourier descriptor analysis. The original TEM image in grayscale (Figure 3a) is first converted to a binary image (Figure 3b,c), and subsequently noise is removed via a despeckle process (Figure 3d). The continuous boundary of the nanocrystal is extracted (red curve in Figure 3e) and down-sampled to 16 segments (black curve in Figure 3e). The down-sampled boundary reduces the high-frequency roughness of the edge that arises either from noise in the TEM images or difficulties in defining the perimeter due to poor contrast between the nanocrystal and the background. The boundary is subsequently up-sampled to 128 edges, and the complex Fourier descriptors (eq 1) are calculated, from which the nanocrystals can be classified to a shape category and appropriately colorized (Figure 3f). The shape of colloidal nanocrystals having diverse forms can be classified by Fourier descriptor analysis. Figure 4 illustrates TEM images and the corresponding colormaps of shape for CdSe nanocrystals that were classified to be predominantly rods (Figure 4a,d), cubes (Figure 4b,e), and hexagonal platelets (Figure 4c,f). The sample in Figure 4a, with a total of 96 nanocrystals that were analyzed, was calculated to consist of 87.5% rods (84 out of 96), 7.3% squares (7 out of 96), 4.2% triangles (4 out of 96), and 1.0% hexagons (1 out of 96). Overall this classification of shape matches very well to a visual evaluation of shape; however, the Fourier descriptor analysis shows great sensitivity in highlighting nanocrystal shapes that do not conform to the majority shape in the sample. For example, the squares are four-sided shapes that lack elongation and thus have smaller magnitudes for the n = −1 descriptor relative to the rods (see Supporting Information, Figure S1 for plots of the descriptor magnitudes). The criteria for distinguishing between rods and squares in Figure 4 was arbitrarily fixed to be an aspect ratio of 1.3, with the nanocrystals with an aspect ratio greater than and less than 1.3 being assigned as rods and

Figure 1. Magnitude ( fn) of complex Fourier descriptors ranging from n = −7 to +7 for idealized shapes: (a) circles, (b) triangles, (c) rectangles, and (d) regular polygons.

nested objects are eliminated and only the shape of the exterior object is classified. Some of the lighter colored objects and

Table 1. Characteristic Signatures of Idealized Shapes Based on Complex Fourier Descriptorsa descriptor number, n −7 circle triangle square pentagon hexagon octagon ellipse rod

+ − + +

−6

−5

−4

−3

−2

−1

−

− + − + + −

−

− ++

−

− +

+ + + +

− − −

+ + +

− − − − −

+

1

2

3

4

5

6

7

− + −

+ + + +

−

− + −

−

+

− + − +

− − − −

+ + −

++ ++

− −

+ +

+ + +

− + − − − −

+ − +

− − − − −

a Each box in the table illustrates the relative magnitude of the Fourier descriptor, f n, for descriptors numbered n = −7 to 7. Descriptor n = 0 is not included because f 0 was used to normalize the magnitudes of all Fourier descriptors. The cutoffs between magnitudes have been coarsely and arbitrarily assigned as follows: ++ for f n > 10; + for 1 < f n ≤ 10; empty for 0.1 < f n ≤ 1; and − for f n ≤ 0.1.

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Figure 2. Classification of shapes in artwork containing geometric objects. The (a) original color artwork is converted to (b) grayscale and the assigned shapes are visualized with (c) a colorized map of the objects. The software accurately assigns the shapes of the objects as circular (green), triangular (pink), or square (blue) based on the magnitude of the Fourier descriptors.

weighted toward aspect ratios between 1 and 1.3 (see Supporting Information, Figure S2b). It is important to note that the Fourier descriptor analysis accurately assigns shapes to nanocrystals in TEM images acquired at a wide range of magnifications, such that the features of interest can be small (e.g., the cubes in Figure 4b) or large (e.g., the rods in Figure 4a). Figure 4c,f shows the successful classification of nanocrystal shapes by Fourier descriptor analysis for a sample that is polydisperse in both shape and size. The majority of the 258 nanocrystals that were analyzed in the image were determined to be 54.3% hexagons, 16.3% triangles, and 15.9% squares. In general the nanocrystals that were assigned as triangular shapes have six sides. These nanocrystals were not assigned as hexagons, however, because they are not regular hexagons with sides of equal length. The presence of three short sides and three long sides provides the nanocrystal with an overall triangular character that is distinct from a regular hexagon and is likely a direct consequence of the mechanism by which the platelets are grown. One possible explanation is that certain equivalent facets of the growing hexagonal crystallites are slightly more reactive and add precursor reactants in a statistically preferential manner. A quantitative examination of the relative lengths of the short sides versus the long sides of the hexagonal and triangular nanocrystals could be performed and used to measure the distribution of growth rates on each type of crystal facet. Future efforts in classification by Fourier descriptors may therefore benefit from the assignment of shapes into subcategories. Triangles, for example, could be further classified as regular triangles, elongated triangles, triangles with rounded corners or bullet-shaped particles (Figure 4a), and triangles with truncated corners (Figure 4c). Although the Fourier descriptor analysis has been demonstrated to be robust at classifying the shapes of colloidal nanocrystals using TEM images, this characterization approach is subject to a few limitations. Foremost among the limitations is the requirement for high-quality images. The images, regardless of whether they are acquired by TEM, scanning electron microscopy, atomic force microscopy, or another nanoscale imaging technique, must be of sufficient quality such that distinguishing the nanocrystals from each other and from the background can be achieved. The TEM images presented here are of suitable quality for analysis but are improved upon postprocessing using image filters and watershedding to separate particles. It is important to note that, even with such high-quality images and image processing techniques, it can still be difficult to isolate nanocrystals as a result of poor contrast or

Figure 3. Shape classification for a colloidal CdSe nanocrystal. The (a) original grayscale TEM image is preprocessed in ImageJ to a blackand-white image by either (b) enhancing brightness/contrast and thresholding or (c) automated binary conversion. (d) Noise is eliminated with the despeckle function in ImageJ. (e) The particle boundary is extracted in MATLAB (red curve) and down-sampled to a set of 16 linear edges (black curve). (f) The final assignment of shape is performed by complex Fourier descriptor analysis in MATLAB, and a colorized map of the particle shapes is generated. A red color indicates that this particle was correctly classified as having a hexagonal shape.

squares, respectively (see Supporting Information, Figure S2a). This type of assessment of shape based on quantitative parameters, as well as the number of sides, would be difficult and time-consuming to perform manually, and consequently has often been avoided in the literature. In addition, the nanocrystals assigned as triangles in Figure 4d have a definitive taper toward one end and a rounded short end. These elongated and tapered rods may be the result of the mechanism of CdSe nanocrystal growth or, in part, artifacts from the TEM imaging. The CdSe nanocrystals in Figure 4b are classified as having predominantly a square shape (cubes in three dimensions). A total of 181 nanocrystals were characterized with the shapes being assigned as 80.1% squares, 15.4% rods, 3.9% triangles, and 0.6% pentagons. The quantitative distinction between cubes and rods is again specified to be an aspect ratio of 1.3, but the distribution of aspect ratios for this sample is heavily 828

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Figure 4. Classification of shape for real samples of CdSe nanocrystals. The top row shows grayscale transmission electron micrographs for nanocrystals that are predominantly (a) rods, (b) cubes, and (c) hexagonal platelets. The bottom row (d−f) displays the corresponding colormap of nanoparticle shapes and a bar plot of the number percentages of each shape in the image.

physical overlap in the features. These “aggregates” of multiple nanocrystals should not be analyzed and thus have been eliminated based on their large size relative to the distribution of nanocrystal dimensions. On the left side of Figure 4c, for example, a few nanocrystals were impossible to isolate and were not characterized for their shape and size (visible as voids in the colormap of shape). Improvements to the nanoscale characterization technique also benefit the image processing and shape classification. Our images acquired by TEM could potentially have been improved by using thinner, amorphous carbon TEM grids to enhance the contrast between the nanocrystals and the background and to reduce the background noise, as well as by using a TEM tool with better resolution. In addition, the analysis is necessarily based upon a twodimensional representation of a three-dimensional object and does not take into account that anisotropic particles may be not be oriented in the same way on the substrate. In this case, the various orientations may give rise to different two-dimensional projections and would be classified as different shapes even though the parent three-dimensional objects may be identical. For example, a sample of nanorods could deposit onto a substrate with their long axis parallel to the substrate, or with the long axis perpendicular to the substrate. In this case, the software would classify the former as nanorods, and the latter as circles or hexagons (depending on the shape of the radial crosssection of the nanorods). Such effects may manifest themselves as a binary (or higher) distribution of shapes in the sample. This possibility appears to be minimized for widely separated particles; special care would need to be exercised if attempting to analyze assemblies of close-packed particles. However, this limitation also suggests that an extension of the approach presented here might enable the automated classification of the three-dimensional structure from the two-dimensional projections, for example, by reconstruction from a series of tilted TEM images or a careful analysis of the electron cross-sections in the TEM images.

Surface texture or noise in the nanocrystal boundary may be anticipated to cause difficulties in the assessment of nanocrystal shape by Fourier descriptors, but the approach presented here is found to be relatively immune to boundary roughness. Figure 5 shows that low-frequency and large amplitude roughness does not significantly influence the classification of circular, square, and hexagonal shapes. The roughness does result in an increase in the descriptor intensities overall, but the characteristic descriptors for each shape are clear (e.g., n = −3 and −5 for the square and hexagon, respectively) even though the contrast with the other descriptors is reduced. All of the shapes in Figure 5b,c are correctly classified as squares and hexagons, respectively. The circular shapes with roughness are more difficult to classify because the roughness adds straight edges that are not inherently characteristic of round shapes. In Figure 5a the Fourier descriptor analysis therefore assigns the original and low roughness objects to be circles, but the objects with medium and high roughness are classified as a hexagon and square, respectively. High-frequency and low amplitude noise, which is introduced by the imaging technique and the blackand-white conversion process, is eliminated by the downsampling of the nanocrystal boundary. This approach only works for boundaries defined by sufficient numbers of points such that down-sampling averages out the roughness or noise among many neighboring boundary positions.

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CONCLUSIONS In conclusion, we have applied complex Fourier descriptors to the classification of the shape of colloidal nanocrystals. This morphometric approach to classifying shape eliminates much of the ambiguity and subjectiveness that plagues current approaches to determining nanocrystal shape, and has been demonstrated here to provide quantitative information far beyond that accessible to manual characterization. Hundreds of nanocrystals can be simultaneously analyzed in a single TEM image, or thousands of particles can be analyzed using 829

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shape and material characterization will be reported in a future publication. Nanocrystal Imaging. Transmission electron microscopy (TEM, a Phillips CM100 TEM 100 kV operating at 80 kV was used to acquire real-space images of the nanocrystals. Samples for TEM imaging were prepared on Formvar-coated copper grids (200 mesh, Electron Microscopy Sciences) by the drop casting and evaporation of solvent from dilute nanocrystal solutions. Particles were purified from the reaction mixture before imaging by adding an antisolvent to the reaction mixture and centrifuging. The particles were then redispersed in an organic solvent, typically hexanes. The TEM images were acquired in the 8-bit tagged image file (TIF) format by the imaging program AMV600 and were processed prior to being analyzed by the Fourier descriptor program. Preprocessing in ImageJ34 adjusted the brightness/contrast levels of the grayscale images and reduced background noise resulting from the TEM grid by functions that performed despeckling and removed outliers. The grayscale images were subsequently converted to binary black-and-white images using either ImageJ or thresholding functions defined in the MATLAB code. Fourier Descriptor Analysis and Program for Shape Classification. The complex Fourier descriptor analysis and shape classification was implemented in MATLAB.35 The in-house software accepts grayscale images in TIF format and can perform a series of image enhancement processes to improve the definition of the nanocrystal boundaries and distinguish individual nanocrystals, for example, top-hat filtering, image flattening, and equalization. Thresholding is performed to convert the grayscale image to a binary image from which the nanocrystal boundaries are distinguished. The software then calculates the complex Fourier descriptors (eq 1) and outputs an image with the nanocrystals colorized by shape.

Figure 5. The impact of surface texture on the magnitude ( fn) of complex Fourier descriptors ranging from n = −7 to +7 for (a) circular, (b) square, and (c) hexagonal objects. The roughness in the boundaries was added to the ideal shapes using the ripple filter function in Adobe Photoshop36 with small (light gray), medium (dark gray), and large amplitudes (top, dark color).

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* Supporting Information The Fourier descriptor magnitudes and aspect ratio distributions of the real CdSe nanocrystals characterized in Figure 4. This information is available free of charge via the Internet at http://pubs.acs.org.

automated processing of several TEM images, from which statistically relevant distributions of size and shape can be calculated. We envision, therefore, that Fourier descriptors could be important for many future applications in nanomaterials. For example, the extensive characterization of shape as a function of synthetic conditions has been difficult, and only rough phase-diagrams can be outlined. Often nanocrystal syntheses do not produce ideal, monodisperse products with a single shape; thus a quantitative measure of morphology in diverse systems can more accurately characterize and describe shape in a phase diagram. Furthermore, optimizing a synthetic method or shape-selective purification process for the industrial-scale production of nanocrystals of a specific form may only be enabled by rapid, in-line methods to screen quantitative information about shape. The Fourier descriptor analysis detailed here can easily be implemented for in-line characterization in continuous processes. Efficient morphometric methods also enable the shape and form of nanocrystals to be monitored at times far removed from their initial synthesis. Nanocrystals are subject to many processes, including surface initiated reactions (e.g., oxidation, reduction, and sulfidation reactions), aggregation/coalescence, Ostwald ripening, and dissolution, that modify the nanocrystal shape and size, and consequently characterizing the long-term stability of such materials is critical for many applications.

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ASSOCIATED CONTENT

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

ACKNOWLEDGMENTS This work was supported by funding from the University of Colorado at Boulder and ConocoPhillips (K.P.R.). The authors would also like to thank Tom Giddings for helpful discussions.

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Nanocrystal Synthesis. CdSe nanocrystals of different shapes were synthesized using a seed-mediated growth procedure modified from Manna and co-workers.33 Specific details of the synthesis of each 830

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