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Determination of Nanocrystal Sizes: A Comparison of TEM, SAXS, and XRD Studies of Highly Monodisperse CoPt3 Particles Holger Borchert,*,†,‡ Elena V. Shevchenko,†,§ Aymeric Robert,| Ivo Mekis,† Andreas Kornowski,† Gerhard Gru¨bel,⊥ and Horst Weller† Institute of Physical Chemistry, Grindelallee 117, 20146 Hamburg, Germany, ESRF, 6 rue Jules Horowitz, 38043 Grenoble, France, and HASYLAB at DESY, Notkestr. 85, 22603 Hamburg, Germany Received September 13, 2004. In Final Form: December 8, 2004 One of the most fundamental tasks in nanoscience is the accurate determination of particle sizes. Various methods have been developed to elucidate the mean particle diameter and the standard deviation for an ensemble of nanocrystals. However, good agreement between the results from different methods is not always encountered in the literature. In this study, we investigate colloidally prepared, highly monodisperse CoPt3 nanoparticles by transmission electron microscopy (TEM), small-angle X-ray scattering (SAXS), and powder X-ray diffraction (XRD). The results are compared in order to examine to which extent agreement is obtained by the different techniques when applied to small nanocrystals in the size range below 10 nm. In particular, the applicability of the simple Scherrer formula for size determination from the broadening of XRD reflections is checked. When the different techniques are correctly applied, the results from all methods are in good agreement.
1. Introduction The accurate measurement of particle sizes has always been a fundamental task in nanoscience and became even more crucial with the discovery of the quantum size effect, that is, the phenomenon that in the range below 10 nm the band gap of semiconductor nanocrystals depends on their diameter.1-4 The experimental verification of theoretical models describing the quantum size effect is just one basic example where the measurement of particle sizes is required with high precision. Various techniques have been developed to determine the diameter of nanoparticles. Among the most commonly used methods certainly range electron microscopy, X-ray diffraction (XRD), small-angle X-ray scattering (SAXS), dynamic light scattering (DLS), and others. Although various methods are available, the accuracy of the results for the particle size is not always clear because of possible systematic errors in the analysis. All of the techniques have their own advantages and disadvantages. Transmission electron microscopy (TEM) is certainly the most direct method, providing real images of the particles. TEM can be considered as an essential tool to get an impression of the homogeneity of a given sample. However, the elaboration of size distribution curves is limited to the consideration of typically several hundreds of particles. Furthermore, lack of contrast or overlap of particles * Corresponding author. E-mail:
[email protected]. † Institute of Physical Chemistry. ‡ Present address: Department of Pure and Applied Chemistry, Physical Chemistry 1, Carl-von-Ossietzky-Str. 9-11, 26129 Oldenburg, Germany. § Present address: Department of Applied Physics & Applied Mathematics, Columbia University, 200 SW Mudd Building, 500 West 120th Street, New York, New York, 10027. | ESRF. ⊥ HASYLAB at DESY. (1) Brus, L. E. J. Chem. Phys. 1983, 79, 5566. (2) Brus, L. E. J. Chem. Phys. 1984, 80, 4403. (3) Weller, H. Angew. Chem., Int. Ed. Engl. 1993, 32, 41. (4) Alivisatos, A. P. J. Phys. Chem. 1996, 100, 13226.
frequently complicates the analysis of TEM images, because the particle boundaries cannot always be seen precisely. SAXS and XRD are indirect methods, but they provide more reliable information from the statistical point of view. An advantage of X-ray diffraction is also that this method provides a very simple possibility for estimating the particle size from the broadening of the XRD reflections by means of the so-called Scherrer formula:5-7
d)
Kλ w cos θ
(1)
where d is the particle size, λ is the wavelength of the radiation, θ is the angle of the considered Bragg reflection, w is the width on a 2θ scale, and K is a constant close to unity. Values for the coefficient “K” depend on factors such as the geometry of the crystallites and are, unfortunately, not always consistently used in the literature. Moreover, powder X-ray diffraction is sensitive to the size of coherent scattering domains which can significantly differ from the particle size in the case of eventually present lattice defects or amorphous surface layers. Uncertainties in the determination of particle size are a well-known and frequently encountered difficulty in nanoscience. A good example reflecting those uncertainties is the establishment of so-called “sizing curves” for semiconductor nanocrystals where the band gap determined by optical spectroscopy is plotted against the particle size. Data points can be considerably spread around the average values.8,9 From such difficulties arises the devotion to compare the various available methods in (5) Scherrer, P. Nachr. Ges. Wiss. Go¨ ttingen 1918, 2, 98. (6) Klug, H. P.; Alexander, L. E. X-ray Diffraction Procedures for Polycrystalline and Amorphous Materials, 2nd ed.; John Wiley & Sons: New York, 1974. (7) Lauriat, J.-P. Introduction a` la cristallographie et a` la diffraction Rayons XsNeutrons, Paris Onze e´dition N° K 150; Universite´ de ParisSud: Orsay, France, 1998. (8) Guzelian, A. A.; Banin, U.; Kadavanich, A. V.; Peng, X.; Alivisatos, A. P. Appl. Phys. Lett. 1996, 69, 1432. (9) Guzelian, A. A.; Katari, J. E. B.; Kadavanich, A. V.; Banin, U.; Hamd, K.; Juban, E. Alivisatos, A. P. J. Phys. Chem. 1996, 100, 7212.
10.1021/la0477183 CCC: $30.25 © 2005 American Chemical Society Published on Web 01/29/2005
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order to elaborate more precise techniques for the analysis of particle sizes. A considerable amount of work has been published on this topic in the past years.10-18 To give a few examples, Wagner et al. compared results from SAXS and DLS of ∼100 nm large perfluorocolloids.10 In the size range 100-400 nm, also SAXS and TEM have been carefully compared by Rieker et al. in a study of highly monodisperse silica spheres.11 Hall et al. studied ∼4 nm gold nanoparticles by TEM and X-ray diffraction and suggested a Fourier analysis method instead of the widely used Scherrer formula to determine the size from X-ray diffraction data.12 Kecskes et al. studied Fe powders with grain sizes of ∼50 nm using a variety of techniques for size measurement, but in their case, the lack of homogeneity of the samples complicates detailed comparison of the different methods.14 The last example points out that it is also not evident to find a good model system for thorough comparison. Despite all efforts reported in the literature, there is still a need to compare results from different methods in order to check their applicability to colloidally prepared nanocrystals, especially in the size range below 10 nm where quantization effects crucially depend on the particle dimensions. Recent progress in colloidal chemistry makes it possible to synthesize nanocrystals of high quality, that is, nearly monodisperse, highly crystalline nanoparticles with controllable size and shape. In the case of such high quality nanocrystals, the comparison of different methods of their investigation becomes more precise and meaningful. In the present article, we investigate samples of highly crystalline CoPt3 nanoparticles with a controllable degree of monodispersity which form stable colloidal solutions in nonpolar solvents. Because of their high quality, these nanocrystals are a suitable model system to compare different methods for particle size analysis. The techniques compared in the present study are transmission electron microscopy, small-angle X-ray scattering, and powder X-ray diffraction. A major issue of the present study is the comparison of results for the size distribution as obtained from TEM and SAXS, respectively. The XRD data have been evaluated using simply the Scherrer formula, and an additional aim of our work is to check experimentally in which form this “simple” formula has to be applied in order to yield results consistent with more sophisticated methods such as TEM or SAXS. 2. Experimental Section Two samples of CoPt3 nanocrystals, named A and B in the following, have been synthesized under Schlenk-line.19 Briefly, 0.25 g of adamantanecarboxylic acid, 4.0 g of hexadecylamine, 0.0328 g of platinum(II) acetylacetonate, 0.13 g of 1,2-hexa(10) Wagner, J.; Ha¨rtl, W.; Hempelmann, R. Langmuir 2000, 16, 4080. (11) Rieker, T.; Hanprasopwattana, A.; Datye, A.; Hubbard, P. Langmuir 1999, 15, 638. (12) Hall, B. D.; Zanchet, D.; Ugarte, D. J. Appl. Crystallogr. 2000, 33, 1335. (13) Zanchet, D.; Hall, B. D.; Ugarte, D. J. Phys. Chem. B 2000, 104, 11013. (14) Kecskes, L. J.; Woodman, R. H.; Trevino, S. F.; Klotz, B. R.; Hirsch, S. G.; Gersten, B. L. Kona 2003, 21, 143. (15) Natter, H.; Krajewski, T.; Hempelmann, R. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 55. (16) Natter, H.; Schmelzer, M.; Hempelmann, R. J. Mater. Res. 1998, 13, 1186. (17) Zhang, Z.; Zhou, F.; Laverinia, E. J. Metall. Mater. Trans. A 2003, 34A, 1349. (18) Marinkovic, B.; de Avillez, R. R.; Saavedra, A.; Assuncao, F. C. R. Mater. Res. 2001, 4, 71. (19) Shevchenko, E. V.; Talapin, D. V.; Schnablegger, H.; Kornowski, A.; Festin, O.; Svedlindh, P.; Haase, M.; Weller, H. J. Am. Chem. Soc. 2003, 125, 9090.
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Figure 1. TEM images of the two CoPt3 nanocrystal samples. For both samples, in some regions on the grid, self-assembly into well-ordered 2D superstructures is observed (parts b and d). decanediol, and 3 mL of diphenyl ether were mixed and heated to 65 °C. A 0.043 g portion of Co2(CO)8 dissolved in 0.6 mL of 1,2-chlorobenzene was injected into the reaction mixture either at 195 °C (sample A) or at 145 °C (sample B). After injection, the reaction mixture was heated at the injection temperature for 20 min before the temperature was increased up to 270 °C for 40 min in order to improve the crystallinity of the nanoparticles. Finally, several postpreparative steps were applied to obtain colloidal solutions of CoPt3 nanocrystals in toluene.19 It is noted for this system that the average composition as determined by energy-dispersive X-ray analysis (EDX) and chemical analysis of digested nanocrystals is very close to the ratio given by the CoPt3 formula.19 The average composition was found to be even independent of various synthesis parameters such as the reaction temperature or the molar ratio of the precursors.19 This suggests that a thermodynamically stable phase with well-defined composition is formed. TEM images have been acquired with a Philips CM 300 microscope operating at 300 kV. For these measurements, droplets of colloidal solution in toluene were deposited onto carbon-coated copper grids. The excess of solvent was wicked away by filter paper, and the samples were dried under air. SAXS measurements were performed on the Troika Beamline ID10A of the European Synchrotron Radiation Facility (ESRF) in Grenoble, France, using an 8 keV monochromatic beam in transmission geometry. Concentrated solutions of colloidal CoPt3 nanocrystals in toluene were filled into quartz capillaries with diameters of 0.5 or 1 mm. Powder X-ray diffractograms have been measured with a Philips X’Pert device. Samples were prepared simply by dropping colloidal solution on standard single-crystal Si supports and letting the solvent evaporate.
3. Results and Discussion Figure 1 shows TEM images of the two CoPt3 samples. In some regions on the TEM grids, the highly monodisperse nanocrystals form self-assembled 2D superstructures (Figure 1b and d). The particle size distributions have been elucidated with the help of the commercial software NIH Image which measures the areas of the particles on the images. For about 3000 particles in the case of sample A and about 1000 particles in the case of sample B, the particle diameter, d, has been calculated from the projected
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Table 1. Summary of the Results for the Mean Particle Diameter and the Standard Deviation Obtained from the Different Methods of Investigation (TEM, SAXS, and XRD) sample A method TEM
number av volume av SAXS XRD reflctns (111) (200) (220) (311) XRD av
sample B
diameter (nm) and diameter (nm) and standard deviation standard deviation 4.86 ( 0.45 (9%) 4.98 4.97 ( 0.40 (8%) 5.09 4.92 5.33 4.98 5.1 ( 0.2
8.50 ( 0.59 (7%) 8.66 8.11 ( 0.57 (7%) 8.40 8.52 8.21 8.4 ( 0.2
area, A, on the images as d ) 2(A/π)1/2; that is, the particle diameter has been defined as the diameter of a sphere of equivalent cross-sectional area on the images. Common Gaussian statistics yields values of 4.9 and 8.5 nm for the mean diameter (number average diameter) and 9 and 7% for the standard deviation for samples A and B, respectively. The results are summarized in Table 1. Figure 2 shows histograms of the particle size distributions together with the Gaussian profiles (solid lines) corresponding to the above-calculated parameters for the mean value and standard deviation. While the Gaussian profile matches the distribution as visualized by the histogram very well for sample A (Figure 2a), there are some deviations observed for sample B (Figure 2b). This means that the size distribution is not really Gaussian in the latter case. Mathematically, this is reflected by the skewness and the excess kurtosis which are defined as follows: N
skewness )
(di - dmean)3 ∑ i)1
(2)
(N - 1)s3 N
excess kurtosis )
(di - dmean)4 ∑ i)1 (N - 1)s4
-3
(3)
where N is the number of particles in the ensemble, di are the individual diameters, dmean is the average diameter, and s is the standard deviation. The skewness is a measure for the degree of asymmetry of a distribution and is zero in the case of a Gaussian distribution. The excess kurtosis describes how sharp the maximum is. It is zero for a Gaussian distribution. Positive and negative values indicate a sharper or flatter distribution, respectively. Results for these parameters are given in Figure 2. In the case of sample A, the skewness and excess kurtosis are close to the theoretical values of a Gaussian distribution. Therefore, the distribution is very well described by a Gaussian profile with the mean diameter and the standard deviation as calculated from the data set of about 3000 particles. In the case of sample B, the high excess kurtosis and the positive value of the skewness indicate that the distribution is sharper than a Gaussian distribution and that the ensemble of particles contains some excess of larger particles. Consequently, the Gaussian profile as determined by the mean value and standard deviation of the data set does not perfectly describe the distribution as visualized in the histogram. The origin of the deviations from a Gaussian profile is the presence of
Figure 2. Histograms of the particle size distribution (number distribution) as determined by TEM for the two CoPt3 nanocrystal samples. The solid lines are the Gaussian profiles obtained with the mean diameter and the standard deviation as calculated from the data sets of 2824 and 1156 particles, respectively.
a quite small number of particles with considerably larger diameters: within the ensemble of 1156 particles, only 12 particles with diameters larger than 10 nm were found. Nevertheless, their presence leads to a considerable increase of the standard deviation. The example shows that the rigorous description of a particle size distribution may require not only the mean value and standard deviation but also the statistical measures of skewness and excess kurtosis. The last two parameters should be taken into account especially in the case of nonsymmetric size distribution profiles. Since SAXS and XRD provide information on the particle size in terms of a volume average, it is advisable to convert the number distribution as revealed by TEM into a volume distribution before comparison. The volume distribution function is obtained by weighting the number distribution function with the cube of the particle diameter.20 If f(d) designs the normalized number distribution function, then the corresponding volume distribution function, fV(d), is given by the following equation:
fV(d) )
d3 f(d)
∫d3 f(d) dd
(4)
In the case of a Gaussian number distribution function, g(d), centered at dmean and with the standard deviation σ, one easily calculates that the corresponding volume distribution function, gV(d), is given as follows:
gV(d) )
d3 g(d) 3dmeanσ2 + d3mean
(20) Krill, C. E.; Birringer, R. Philos. Mag. A 1998, 77, 621.
(5)
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Figure 4. Small-angle X-ray scattering curves for the two CoPt3 nanocrystal samples. The experimental data (dots) are plotted together with fitting curves (lines) assuming a Schultz-Flory distribution of the particle size.
Figure 3. Volume-weighted distributions of the particle size as generated from the histograms in Figure 2. The solid lines are the volume distribution functions corresponding to the Gaussian profiles in Figure 2. The dashed lines are SchultzFlory profiles determined by SAXS.
Thus, one can easily plot the volume distribution functions corresponding to the Gaussian number distribution functions which were shown in Figure 2. The histogram can also easily be converted. If h(di) designs the frequency for particles with diameters in the ith class of the number distribution (Figure 2), the corresponding frequency, hV(di), for the volume distribution is obtained as follows:
hV(di) ) N
d3i h(di)
∑i
d3i
(6)
h(di)
where N is the total number of particles. Transformed histograms reflecting the volume distribution are shown in Figure 3. The continuous distribution functions plotted as solid lines are profiles calculated according to eq 5 from the Gaussian number distributions in Figure 2. The volume average diameter, dmean,vol, can be calculated directly from the data set:13,21
dmean,vol )
∑i d4i ∑i
(7)
d3i
where di are the diameters measured for the individual particles. Since the size distributions are not perfectly monodisperse, the resulting values are slightly higher than the corresponding number average diameters (see Table 1). We also note that, in the case of sample B, calculation (21) Xu, Q.; Kharas, K. C. C.; Datye, A. K. Stud. Surf. Sci. Catal. 2001, 139, 157.
of the volume average diameter from the transformed Gauss distribution leads to a value of 8.62 nm, that is, slightly below the value 8.66 nm calculated directly from the data set; thus, the transformed Gaussian profile does not perfectly describe the volume-weighted size distribution in the case of the less homogeneous sample B. Figure 4 shows the measured SAXS scattering intensity for CoPt3 samples A and B. The scattering curves have been fitted by model calculations assuming spherical particles where a Schultz-Flory distribution has been used to take care of the polydispersity:22
f (d) ) 2 Z+1 Γ(Z + 1) dmean
( )
Z+1
(
dZ exp -
)
(Z + 1)d , Z > -1 (8) dmean
Herein, Γ(x) is the Gamma function, dmean is the mean particle diameter, and Z is a parameter describing the degree of polydispersity. The relative standard deviation is related to Z by the following equation:
〈(d - dmean)2〉 x 1 ∆d ) ) dmean dmean xZ + 1
(9)
The simulated curves fit the experimental data very well and yield particle diameters of 5.0 and 8.1 nm for samples A and B, respectively. The standard deviation is of the order of 8%. This corresponds to Z values of the order of 150-200. At such elevated Z values, the SchultzFlory distribution (which is highly asymmetric for low values of Z) is almost symmetric and closely matches a Gaussian distribution. Therefore, the results for the standard deviation are directly comparable with the standard deviations determined by TEM. In the case of sample A, the SAXS results are in excellent agreement with the TEM data (see Table 1). The Schultz-Flory distribution as determined by SAXS is shown in Figure 3a as a dashed line and matches the volume distribution as visualized by the histogram very well. In the case of sample B, the mean diameter found by SAXS is slightly lower than the corresponding value determined by TEM; the results agree within 7%. (22) Kotlarchyk, M.; Chen, S.-H. J. Chem. Phys. 1983, 79, 2461.
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Figure 5. Powder X-ray diffractogram of sample A. The four available reflections have been fitted with Gaussian profiles.
Considering the TEM images in Figure 1, especially in the case of sample B, one can notice some particles which are nonspherical. The variations of the particle shape make the analysis based on the assumption of spherical geometry less reliable and are probably the origin of the slight disagreement between the average diameters as determined by SAXS and TEM in the case of sample B. Concerning the standard deviation, variation of the particle shape affects the results from TEM and SAXS in a rather similar manner. The SAXS analysis does not separate the influence of particle shape and size distribution. Both effects are considered together and contribute to the standard deviation. When calculating particle sizes as the diameter of a sphere of equivalent cross-sectional area on the TEM images, there is also no separation of shape effects. For this reason, the standard deviations as determined by SAXS and TEM have the same meaning here, and rather similar results are obtained. Figures 5 and 6 show powder X-ray diffractograms of CoPt3 samples A and B, revealing chemically disordered face-centered cubic crystal structures for both samples. It is noted that the powder X-ray diffractogram of sample B presents some very weak additional reflections which are presumably due to the crystallization of an excess of stabilizing molecules during preparation of the samples for the XRD measurements, that is, when the solvent evaporates. Since SAXS experiments were performed on solutions and small crystals of organic stabilizers are not visible by TEM, the above-discussed SAXS and TEM results are not affected by this observation. All available reflections of the CoPt3 phase have been fitted with Gaussian line profiles using a linear background function. A very simple way to estimate the particle size is its calculation from the width of the Bragg reflections according to the Scherrer formula (eq 1).5-7 This formula describes quantitatively the phenomenon that the broadening of Bragg reflections is related to the number of parallel lattice planes giving rise to diffraction. Frequently, the Scherrer formula is simply used with K ) 1 to give a rough estimation of the particle size.
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Figure 6. Powder X-ray diffractogram of sample B. The three available reflections have been fitted with Gaussian profiles.
For more precise calculations, the Scherrer formula should however be adapted to the geometrical properties of the sample; that is, one should take into account factors such as the shape of the crystallites and the size distribution, because the number of parallel lattice planes responsible for the line broadening may locally vary in the sample. Initially, Scherrer developed his formula for cubic crystallites and gave a value of 2(ln 2/π)1/2 ) 0.94 for K, if d designs the length of the edges of the cubes and w the full width at half-maximum (fwhm).5 A simplified derivation of the Scherrer equation leading to a value of 0.89 for K is, for example, presented by Klug and Alexander.6 Values for K corresponding to various other situations and definitions can be found in a review by Langford and Wilson.23 The CoPt3 nanocrystals studied in the present work can be considered as nearly spherical particles, which is a frequently encountered situation in nanoscience. As mentioned above, the broadening of Bragg reflections is basically determined by the number of unit cells along columns perpendicular to the diffraction planes. For spherical particles of diameter d, the length of such columns of unit cells varies within a given particle. Consequently, calculation of the particle size with the Scherrer equation will lead to an effective diameter, deff, which is smaller than the geometric diameter. Detailed descriptions and comprehensive illustrations can, for example, be found in an article by Natter et al.24 A rigorous treatment taking into account also size distribution effects leads to the following relationship between the volumeweighted column length, deff, and the average grain diameter, d:20,24
3 deff ) d 4
(10)
With a value of 0.9 for K, the following equation may then (23) Langford, J. I.; Wilson, A. J. C. J. Appl. Crystallogr. 1978, 11, 102. (24) Natter, H.; Schmelzer, M.; Lo¨ffler, M.-S.; Krill, C. E.; Fitch, A.; Hempelmann, R. J. Phys. Chem. B 2000, 104, 2467.
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serve to estimate the diameter of spherical particles from the width (fwhm) of a given Bragg reflection:24
d)
4 0.9λ 3 w cos θ
(11)
In that form, the Scherrer equation has, for example, been used by Nanda et al. to evaluate the size of semiconductor nanocrystals,25,26 and we use it here to calculate the diameter of the nearly spherical CoPt3 nanocrystals. Sizes have been separately calculated for all pronounced reflections. The results are given in Table 1. The maximum deviation between the values obtained from the single reflections and the mean value is of the order of only 5%. The mean values are for both samples in good agreement with the results from TEM and SAXS. The differences are
