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Small angle x-ray scattering (SAXS)
CHAPTER 3.2.2
Small angle x-ray scattering (SAXS) Michael Krumrey Physikalisch-Technische Bundesanstalt (PTB), Berlin, Germany
Abbreviations ASAXS CCD IFT GISAXS ISO NP SAXS SR WAXS
anomalous small angle x-ray scattering charge-coupled device indirect Fourier transform method grazing incidence small angle x-ray scattering International Organization for Standardization nanoparticle small angle x-ray scattering synchrotron radiation wide angle x-ray scattering
Introduction Small angle x-ray scattering (SAXS) is a well-established technique to investigate structural properties of materials at the nanoscale. One of the first textbooks on SAXS was already published in the 1950s [1] and another general description in the early 1980s [2]. However, due to the limited intensity of x-rays produced by conventional x-ray tubes, the technique only became more widespread with the development of synchrotron radiation (SR) facilities. Nowadays, a dedicated SAXS beamline is operated at most of these about 50 facilities around the world. But also in the laboratory, the development of powerful microfocus x-ray tubes and liquid-metal jets in combination with dedicated x-ray optics has greatly enhanced the capabilities. Both laboratory and SR installations profit also from advanced large-area detectors. This chapter provides an overview on the basic principles of SAXS and applications to nanoparticle characterization with respect to the mean particle size, the size distribution width, and the particle concentration. These and further aspects of nanoparticle (NP) characterization with SAXS are also described in recent extensive reviews [3,4].
Basic principles of SAXS for nanoparticle characterization Due to the short wavelength that is typically well below 1 nm, x-rays are perfectly suited for the investigation of nanoparticles. As an ensemble method, SAXS provides Characterization of Nanoparticles https://doi.org/10.1016/B978-0-12-814182-3.00011-0
© 2020 Elsevier Inc. All rights reserved.
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information already averaged over a large number of NPs, and there is no need to analyse many images as in microscopy to obtain statistically relevant results. On the other hand, there is no directly visible real space image from SAXS: The scattering information has to be analysed in reciprocal space, and modelling or fitting is required to extract the information. One of the great advantages of SAXS is that it can be applied directly to the NPs in suspension without extensive sample preparation. If an almost parallel and monochromatic x-ray beam impinges on material with density inhomogeneities on the nanoscale like NPs in a suspension, a small fraction of the radiation is scattered in forward direction around the transmitted direct beam (Fig. 1). This effect is called small angle x-ray scattering (SAXS) as the scattering angles are typically below about 5 degrees. For randomly distributed particles, the scattering pattern is independent of the azimuthal angle; thus, the scattering image can be circularly integrated to obtain the scattered intensity as function of the scattering angle. As the scattering depends also on the wavelength λ (or photon energy EPh) of the x-rays, the scattered intensity is usually analysed as function of the momentum transfer q, which is given by q ¼ 4 π=λ sin Θ ¼ 4 π=ðh c Þ EPh sin Θ
(1)
where Θ is half of the scattering angle, h the Planck constant, and c the speed of light. If the nanoparticles are sufficiently monodisperse, the scattering image consists of concentric rings.
Instrumentation SAXS measurements require intense, monochromatic x-rays of low divergence. Synchrotron radiation is therefore best suited, but laboratory sources have also been improved to achieve these requirements. In contrast to conventional sealed tubes or rotating anode tubes, microfocus tubes combine small source sizes below 100 μm and high photon fluxes. X-ray tubes with copper or molybdenum anodes provide characteristic x-rays with photon energies of 8.0 keV and 17.5 keV, respectively, as the dominating lines, while liquid-metal jet sources are operated with gallium or indium
Fig. 1 Principle of a SAXS measurement on nanoparticles in suspension.
Small angle x-ray scattering (SAXS)
gallium alloys, resulting in characteristic lines at 9.2 keV and around 24.2 keV. By using appropriate optics, other emission lines and the Bremsstrahlung background can be discriminated, and the monochromatic radiation can be collimated, for example, by Montel optics. Laboratory installations not only can have almost table size but also can be several metres long. At synchrotron radiation (SR) beamlines, either the broad spectrum of dipole radiation from a bending magnet or already quasi-monochromatic undulator radiation is monochromatized and collimated or even slightly focused on the sample. In principle, any photon energy from the source can be selected by a crystal monochromator and even adapted for the sample to be investigated, depending on the sample thickness and the involved chemical elements. Photon fluxes can be many orders of magnitude higher compared with laboratory instruments so that time-resolved measurements are possible. However, the installations are much more complex, the beamlines are typically between 30 m and about 100 m long, and access for beamtime can be difficult. For laboratory instruments and SR beamlines, the beam has to be well collimated, requiring also the use of slits or pinholes. If sufficient photon flux is available, point collimation with pinholes is preferred as it avoids most slit-smearing effects. As scattering from slits and pinhole can produce additional background, efforts have been made by manufactures to develop scatterless slits, based on silicon, germanium, or tantalum single crystals. These slits exhibit less scattering, but the use of a guard slit just in front of the sample can still further reduce the background. The sample environment can be very sophisticated, but for the characterization of NPs in liquid suspension, mainly thin-walled glass capillaries are used in transmission mode. While laboratory instruments operate usually in air, the capillaries at SR beamlines are often in vacuum that makes the use of flow-through capillaries more difficult. One of the most important components in a SAXS experiment is the large-area detector to record the scattering pattern. While its distance from the sample can be below 1 m in small table top instruments, distances up to 30 m are used at SR beamlines. Wire detectors and image plates that were used in former times are now replaced by semiconductor detectors, either CCDs—often in combination with a fibre-optic taper to increase the pixel size—or hybrid-pixel detectors. The most widespread detector is now the PILATUS photon-counting hybrid-pixel detector, which has a pixel size of 172 μm [5]. A module of this detector has 105 pixels, and arrangements of up to 60 modules are commercially available, covering an area of more than 40 40 cm2. The direct transmitted beam is more intensive than the scattered radiation by several orders of magnitude and must therefore be shielded by means of a beamstop not to overload the area detector. A photodiode incorporated in the beamstop might be useful to monitor the transmitted beam. For randomly distributed scatterers like NPs in suspension, the obtained scattering pattern is centrosymmetric; thus, azimuthal integration can be used to convert the pattern to the scattered intensity as function of q as shown in Fig. 2. For the determination of the
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Measured
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Fig. 2 Scattered intensity as measured, contribution from the suspending medium in the sample cell, and difference only due to particle scattering.
mean particle diameter and the size distribution, only the q-axis needs to be calibrated; the scattering cross section is not required on an absolute scale. As shown earlier, q depends only on the scattering angle and the photon energy EPh (or the wavelength λ). Calibration of the q-axis can therefore be performed in a traceable way to the SI unit metre by accurate measurements of pixel size and sample detector distance and the wavelength, which can be done at SR beamlines, for example, by backreflection from a silicon crystal [6]. A widespread material to check the q calibration is silver behenate [6a]. Absolute scattering cross sections, thus scattered intensities normalized to the incident intensity, are required for more careful work, such as NP concentration determination. Here again, different options are possible: (i) using calibrated detectors (e. g. semiconductor photodiodes or ion chambers) to measure the incoming photon flux and an area detector with known quantum efficiency to register the scattering pattern [7], (ii) using, for example, glassy carbon as a reference material for scattered intensity calibration [8] or (iii) using an area detector with a high dynamic range like a photoncounting hybrid-pixel detector to register both the scattering pattern and, with reduced counting time, the incoming direct beam.
Mean particle size determination with SAXS Since 2015, the International Standard ISO 17867 is available where two methods for the determination of the mean particle diameter from SAXS measurements are described [9]: the Guinier approximation and the model-based data fitting. For the Guinier approximation, the scattered intensity is plotted as function of q2 (Guinier plot), and the radius of gyration can be obtained from the slope at very low q-values. Under certain
Small angle x-ray scattering (SAXS)
conditions, this can provide an estimate on the mean particle diameter, but the method has many limitations. A more accurate method to obtain the mean diameter is the model fitting for which software implementations are available, for example [10]. The scattering cross section of spherical particles with radius R as function of the momentum transfer q can be generally expressed as ð∞ dΣ gðRÞ Sðq, RÞ jf ðq, RÞj2 dR + BG: (2) ðqÞ ¼ re2 C Δρ2e dΩ 0 where re is the electron radius, C the concentration, thus the number of scatterers (NPs) per volume, Δρe the electron density difference between the particles and the surrounding medium, g(R) the size distribution function, S(q,R) the structure factor, f(q,R) the form factor, and BG(q) a background contribution. The structure factor S depends on the arrangement of the scattering objects and simplifies to unity in a dilute dispersion where the interaction between the particles and multiple scattering can be neglected. For higher particle concentrations and agglomerated particles, the structure factor becomes very relevant and describes the probability distribution of interparticle distances. The form factor f depends only on particle size and shape. For spherical particles, it is ! 4 3 sin ðqRÞ qR cos ðqRÞ f ðq, RÞ ¼ πR 3 (3) 3 ðqRÞ3 The size distribution of the particles is given by g(R). Different size contributions can be assumed; most widely used assumptions are Gaussian distribution, Zimm–Schultz distribution (often used for polymerization), or log-normal distribution that is, in principle, a Gaussian distribution on a logarithmic scale. Especially for the determination of the size distribution width, a correct determination of the background is crucial. Detailed recommendations for data correction are available [11] and will partly be included in the next edition of the ISO standard together with additional methods for data analysis like Monte-Carlo based fitting [12] and the indirect Fourier transform (IFT) method [13]. The comparability of these approaches has been shown in an interlaboratory comparison where small silver NPs were measured on 24 instruments. The mean radii obtained with model fitting, IFT, and a Monte-Carlo method were 2.80 nm, 2.82 nm, and 2.67 nm, with standard deviations of 0.07 nm, 0.04 nm, and 0.16 nm, respectively [14]. The periodicity of the observed oscillations can be used for the size determination, and the exact knowledge of the wavelength is exploited, which, contrary to light scattering in the visible range, is considerably smaller than the objects under investigation. Examples for the traceable determination of the mean particle diameter using SAXS and model fitting are shown in Fig. 3 for PMMA particles. The periodicity of the oscillation on the q-axis is inversely proportional to the mean diameter. For the more complex case of a bidisperse size
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Fig. 3 Measured and fitted scattered intensity for PMMA particles with nominal diameters of 108 nm (upper left) and 192 nm (upper right). The determined diameters are 109.0 nm and 188.0 nm, respectively. The scattered intensity for a mixture of both particle suspensions is shown in (lower left) together with the separated contributions, which correspond to the number weighted size distribution in (lower right). (Credit: From G. Gleber, L. Cibik, S. Haas, A. Hoell, P. M€ uller, M. Krumrey, Traceable size determination of PMMA nanoparticles based on small angle X-ray scattering (SAXS), J. Phys. Conf. Ser. 247 (2010) 012027.)
distribution, the scattering curve is a superposition of both contributions. The mean diameters were determined in agreement with the results obtained separately with each of the two components with a relative uncertainty of better than 1% [6].
Size distribution width determination with SAXS According to Eq. (3), perfectly monodisperse spheres would lead to oscillation of the scattered intensity with sharp minima as shown in Fig. 4. With increasing distribution widths, the minima smear out. Together with the size distribution widths, not only the assumed distribution (e.g. Gaussian or log-normal) in case of the model fitting has to be indicated, but it is even more important whether the number or volume weighted distribution is provided. While for almost monodisperse particles with a low polydispersity p ¼ σ d/d, where σ d is the standard deviation of the size distribution and d is the mean diameter,
Small angle x-ray scattering (SAXS)
Fig. 4 Influence of the polydispersity on the amplitude of the oscillation for particles with 50 nm diameter and different polydispersity.
Fig. 5 Comparison of different size distributions for nanoparticles with a diameter of 50 nm and a polydispersity of 0.06 (left) and of 0.24 (right). Log-normal and Gaussian distributions are shown; solid lines indicate the number weighted distributions, while dashed lines are for volume weighted. While there is almost no difference for narrow size distributions, significant differences are visible for larger distribution widths.
all distributions and weighting have similar results; there are large differences for broader size distributions. This effect is shown in Fig. 5 for p ¼ 0.06 and p ¼ 0.24. To obtain the correct size distribution, the shape of the particles has to be known as, for example, a larger distribution width of spherical particles can lead to a similar smearing of the scattering curve as, for example, elongated ellipsoidal particles. In principle, the particle shape can also be obtained from the slope of the intensity decay as function of q, but a simultaneous size and shape determination will not always lead to a unique solution. Microscopic techniques that provide direct imaging of particles should be used to investigate whether, for example, the assumption of a spherical shape is justified.
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Nanoparticle concentration determination with SAXS While only the q-axis needs to be calibrated for the size and size distribution determination, absolute scattering intensities are essential to determine the NP concentration. Due to the straightforward scattering theory, the scattered intensity can be calculated from the model fitting as well. The mean diameter and the size distribution are obtained from the oscillations as shown in Fig. 6, and the particle number concentration C, thus the number of particles per volume, is then finally a scaling parameter in Eq. (2)[14a]. However, the irradiated volume is not always well defined due to the profile of the incoming x-ray beam. For a capillary with a rectangular profile, only the sample thickness in beam direction needs to be determined. Also, obvious from Eq. (2) is the fact that the scattered intensity also scales with Δρ2e , thus even the square of the electron density difference between particle and the suspending medium. While the electron density of, for example, water is well known (333 nm3), the electron density of NPs is not always identical to the corresponding bulk material. This uncertainty contribution from the particle electron density is often underestimated.
Core–shell nanoparticles Most nanoparticles are not simple solid objects but have an inner morphology, which, in the easiest case, can be described by a core–shell structure. This not only can be intentionally produced on manufactured NPs, for example, for surface functionalization, but also might be the result of capping or coverage of the surface by organic material in the suspending medium. If the core is made from a high-density (inorganic) material, its scattering will dominate the observed scattering so that no information on a possible organic layer on the surface can be obtained. If the shell or both, core and shell, contain elements with a higher atomic number, anomalous SAXS (ASAXS) can be used to distinguish
Fig. 6 Scattering cross section in absolute units of gold nanoparticles with (27.2 0.7) nm diameter and a particle number concentration of (1.85 0.13) ∗ 1011 mL1 as determined from the measured data (points) and model fitting (solid red line).
Small angle x-ray scattering (SAXS)
between core and shell. Here, the change of the effective electron density in the vicinity of absorption edges of the involved chemical elements is exploited. This technique is only possible with synchrotron radiation as the photon energy needs to be tuned. Typically, measurements are performed at several photon energies in the range between a few hundred eV and a few eV below a relevant absorption edge, for example, the Au L3 edge for Ag/Au nanoparticles [15]. As only the contribution from this specific element is varying, the changes of the scattering curves can be associated with the presence of the element in core or shell. For lighter elements like silicon, phosphorus, sulphur, or calcium, the K absorption edges are in the so-called tender x-ray range where measurements have to be performed in vacuum and dedicated beamlines and detectors are required [16]. Mainly for organic systems with densities close to the density of water, the contrast can also be varied by changing the electron density of the suspending medium, for example, by adding sucrose. An elegant way is the continuous contrast variation as shown in Fig. 7, where a gradient of the electron density is created in a capillary. By moving the capillary through the x-ray beam, scattering curves can then be recorded at different positions and thus different electron density differences [17]. The scattering at low q-values has a minimum where the electron density of the suspending medium equals the mean electron density of the particles. However, for core–shell particles, not the entire scattering curves are shifted in intensity, but the scattered intensity remains constant at specific q-values. From these isoscattering points q∗, the outer particle radius can be calculated according to
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Fig. 7 Continuous contrast variation to reveal a core–shell structure of polystyrene nanoparticles. Scattering curves measured at different positions in the gradient capillary, corresponding to suspending medium electron densities (left) and selected scattering curves fitted with a core–shell model where the shell has a higher electron density than the core (right). (From R. Garcia-Diez, C. Gollwitzer, M. Krumrey, Nanoparticle characterization by continuous contrast variation in small-angle X-ray scattering with a solvent density gradient, J. Appl. Crystallogr. 48 (2015) 20–28.)
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tan ðq∗RÞ ¼ q∗R
(4)
The particle radius is also available from the model fitting using the form factor for spherical core–shell particles. For the polystyrene particles shown as example in Fig. 7, the outer radii obtained with both methods were (50.4 2.8) nm and (49.7 2.8) nm, respectively. The core–shell structure of these particles is due to the manufacture process, which is based on a styrene, methacrylic acid, and methyl methacrylate monomer mix, resulting in a thin shell of higher density.
Conclusion SAXS has proven to be a very powerful ensemble technique for the characterization of nanoparticles in suspension without the need for drying or other sample preparation and thus in their native environment. For sufficiently monodisperse NPs, a traceable size and size distribution determination is possible as well as the concentration determination with known particle density. Additional information is available from related techniques: Wide angle x-ray scattering (WAXS) extends the accessible feature size range to 1 nm and below, ASAXS allows to obtain the mentioned element-specific information, and grazing incidence SAXS (GISAXS) can be used for the characterization of nanostructured surfaces or nanoparticles on surfaces in reflection geometry. SAXS and related techniques can be used for many different applications, but a great part of them is currently in the fields of particle synthesis including growth control [18] and catalysis [19] and in operando fuel cell [20] or battery characterization [21].
References [1] A. Guinier, G. Fournet, Small-Angle Scattering of X-Rays, John Wiley, New York, 1955. [2] O. Glatter, O. Kratky, Small-Angle X-Ray Scattering, Academic Press, 1982. [3] T. Li, A.J. Senesi, B. Lee, Small angle X-ray scattering for nanoparticle research, Chem. Rev. 116 (2016) 11128–11180. [4] B. Ingham, X-ray scattering characterisation of nanoparticles, Cryst. Rev. 21 (2015) 229–3033. [5] P. Kraft, A. Bergamaschi, C. Broennimann, R. Dinapoli, E.F. Eikenberry, B. Henrich, I. Johnson, A. Mozzanica, C.M. Schlep€ utz, P.R. Willmott, B. Schmitt, Performance of single-photon-counting PILATUS detector modules, J. Synchrotron Radiat. 16 (2009) 368–375. [6] G. Gleber, L. Cibik, S. Haas, A. Hoell, P. M€ uller, M. Krumrey, Traceable size determination of PMMA nanoparticles based on small angle X-ray scattering (SAXS), J. Phys. Conf. Ser. 247 (2010) 012027. [6a] T.N. Blanton, C.L. Barnes, M. Lelental, Preparation of silver behenate coatings to provide low- to midangle diffraction calibration, J. Appl. Crystallogr. 33 (2000) 172–173. [7] J. Wernecke, C. Gollwitzer, P. M€ uller, M. Krumrey, Characterization of an in-vacuum PILATUS 1M detector, J. Synchrotron Rad. 21 (2014) 529–536. [8] A.J. Allen, F. Zhang, R.J. Kline, W.F. Guthrie, J. Ilavsky, NIST standard reference material 3600: absolute intensity calibration standard for small-angle X-ray scattering, J. Appl. Crystallogr. 50 (2017) 462–474.
Small angle x-ray scattering (SAXS)
[9] ISO 17867, Particle Size Analysis – Small-Angle X-Ray Scattering, ISO International Organization for Standardization, Geneva, 2015. [10] I. Breßler, J. Kohlbrecher, A.F. Th€ unemann, SASfit: a tool for small-angle scattering data analysis using a library of analytical expressions, J. Appl. Crystallogr. 48 (2015) 1587–1598. [11] B.R. Pauw, Everything SAXS: small-angle scattering pattern collection and correction, J. Phys.: Condens. Matter 25 (2013) 383201. [12] B.R. Pauw, J.S. Pedersen, S. Tardif, M. Takata, B.B. Iversen, Improvements and considerations for size distribution retrieval from small-angle scattering data by Monte Carlo methods, J. Appl. Crystallogr. 46 (2013) 365–371. [13] O. Glatter, A new method for the evaluation of small-angle scattering data, J. Appl. Crystallogr. 10 (1977) 415–421. [14] B.R. Pauw, C. K€astner, A.F. Th€ unemann, Nanoparticle size distribution quantification: results of a small-angle X-ray scattering inter-laboratory comparison, J. Appl. Crystallogr. 50 (2017) 1280–1288. [14a] A. Schavkan, C. Gollwitzer, R. Garcia-Diez, M. Krumrey, C. Minelli, D. Bartczak, S. Cuello– Nun˜ez, H. Goenaga-Infante, J. Rissler, E. Sj€ ostr€ om, G.B. Baur, K. Vasilatou, A.G. Shard, Number concentration of gold nanoparticles in suspension: SAXS and spICPMS as traceable methods compared to laboratory methods, Nanomaterials 9 (2019) 502. [15] J. Haug, H. Kruth, M. Dubiel, H. Hofmeister, S. Haas, D. Tatchev, A. Hoell, ASAXS study on the formation of core–shell Ag/Au nanoparticles in glass, Nanotechnology 20 (2009) 505705. [16] A. Hoell, Z. Varga, V.S. Raghuwanshi, M. Krumrey, C. Bocker, C. R€ ussel, ASAXS study of CaF2 nanoparticles embedded in a silicate glass matrix, J. Appl. Crystallogr. 47 (2014) 60–66. [17] R. Garcia-Diez, C. Gollwitzer, M. Krumrey, Nanoparticle characterization by continuous contrast variation in small-angle X-ray scattering with a solvent density gradient, J. Appl. Crystallogr. 48 (2015) 20–28. [18] J. Polte, T. Ahner, F. Delissen, S. Sokolov, F. Emmerling, A.F. Th€ unemann, R. Kraehnert, Mechanism of gold nanoparticle formation in the classical citrate synthesis method derived from coupled in situ XANES and SAXS evaluation, J. Am. Chem. Soc. 132 (2010) 1296–1301. [19] H. Song, R.M. Rioux, J.D. Hoefelmeyer, R. Komor, K. Niesz, M. Grass, P. Yang, G.A. Somorjai, Hydrothermal growth of mesoporous SBA-15 silica in the presence of PVP-stabilized Pt nanoparticles: synthesis, characterization, and catalytic properties, J. Am. Chem. Soc. 128 (2006) 3027–3037. [20] M. Povia, J. Herranz, T. Binninger, M. Nachtegaal, A. Diaz, J. Kohlbrecher, D.F. Abbott, B.-J. Kim, T.J. Schmidt, Combining SAXS and XAS to study the operando degradation of carbon-supported Pt-nanoparticle fuel cell catalysts, ACS Catal. 8 (2018) 7000–7015. [21] G.O. Park, J. Yoon, E. Park, S.B. Park, H. Kim, K. Ho Kim, X. Jin, T.J. Shin, H. Kim, W.-S. Yoon, M.K. Ji, In operando monitoring of the pore dynamics in ordered mesoporous electrode materials by small angle X-ray scattering, ACS Nano 9 (2015) 5470–5477.
183
Small angle x-ray scattering (SAXS) Michael Krumrey Physikalisch-Technische Bundesanstalt (PTB), Berlin, Germany
Abbreviations ASAXS CCD IFT GISAXS ISO NP SAXS SR WAXS
anomalous small angle x-ray scattering charge-coupled device indirect Fourier transform method grazing incidence small angle x-ray scattering International Organization for Standardization nanoparticle small angle x-ray scattering synchrotron radiation wide angle x-ray scattering
Introduction Small angle x-ray scattering (SAXS) is a well-established technique to investigate structural properties of materials at the nanoscale. One of the first textbooks on SAXS was already published in the 1950s [1] and another general description in the early 1980s [2]. However, due to the limited intensity of x-rays produced by conventional x-ray tubes, the technique only became more widespread with the development of synchrotron radiation (SR) facilities. Nowadays, a dedicated SAXS beamline is operated at most of these about 50 facilities around the world. But also in the laboratory, the development of powerful microfocus x-ray tubes and liquid-metal jets in combination with dedicated x-ray optics has greatly enhanced the capabilities. Both laboratory and SR installations profit also from advanced large-area detectors. This chapter provides an overview on the basic principles of SAXS and applications to nanoparticle characterization with respect to the mean particle size, the size distribution width, and the particle concentration. These and further aspects of nanoparticle (NP) characterization with SAXS are also described in recent extensive reviews [3,4].
Basic principles of SAXS for nanoparticle characterization Due to the short wavelength that is typically well below 1 nm, x-rays are perfectly suited for the investigation of nanoparticles. As an ensemble method, SAXS provides Characterization of Nanoparticles https://doi.org/10.1016/B978-0-12-814182-3.00011-0
© 2020 Elsevier Inc. All rights reserved.
173
174
Characterization of nanoparticles
information already averaged over a large number of NPs, and there is no need to analyse many images as in microscopy to obtain statistically relevant results. On the other hand, there is no directly visible real space image from SAXS: The scattering information has to be analysed in reciprocal space, and modelling or fitting is required to extract the information. One of the great advantages of SAXS is that it can be applied directly to the NPs in suspension without extensive sample preparation. If an almost parallel and monochromatic x-ray beam impinges on material with density inhomogeneities on the nanoscale like NPs in a suspension, a small fraction of the radiation is scattered in forward direction around the transmitted direct beam (Fig. 1). This effect is called small angle x-ray scattering (SAXS) as the scattering angles are typically below about 5 degrees. For randomly distributed particles, the scattering pattern is independent of the azimuthal angle; thus, the scattering image can be circularly integrated to obtain the scattered intensity as function of the scattering angle. As the scattering depends also on the wavelength λ (or photon energy EPh) of the x-rays, the scattered intensity is usually analysed as function of the momentum transfer q, which is given by q ¼ 4 π=λ sin Θ ¼ 4 π=ðh c Þ EPh sin Θ
(1)
where Θ is half of the scattering angle, h the Planck constant, and c the speed of light. If the nanoparticles are sufficiently monodisperse, the scattering image consists of concentric rings.
Instrumentation SAXS measurements require intense, monochromatic x-rays of low divergence. Synchrotron radiation is therefore best suited, but laboratory sources have also been improved to achieve these requirements. In contrast to conventional sealed tubes or rotating anode tubes, microfocus tubes combine small source sizes below 100 μm and high photon fluxes. X-ray tubes with copper or molybdenum anodes provide characteristic x-rays with photon energies of 8.0 keV and 17.5 keV, respectively, as the dominating lines, while liquid-metal jet sources are operated with gallium or indium
Fig. 1 Principle of a SAXS measurement on nanoparticles in suspension.
Small angle x-ray scattering (SAXS)
gallium alloys, resulting in characteristic lines at 9.2 keV and around 24.2 keV. By using appropriate optics, other emission lines and the Bremsstrahlung background can be discriminated, and the monochromatic radiation can be collimated, for example, by Montel optics. Laboratory installations not only can have almost table size but also can be several metres long. At synchrotron radiation (SR) beamlines, either the broad spectrum of dipole radiation from a bending magnet or already quasi-monochromatic undulator radiation is monochromatized and collimated or even slightly focused on the sample. In principle, any photon energy from the source can be selected by a crystal monochromator and even adapted for the sample to be investigated, depending on the sample thickness and the involved chemical elements. Photon fluxes can be many orders of magnitude higher compared with laboratory instruments so that time-resolved measurements are possible. However, the installations are much more complex, the beamlines are typically between 30 m and about 100 m long, and access for beamtime can be difficult. For laboratory instruments and SR beamlines, the beam has to be well collimated, requiring also the use of slits or pinholes. If sufficient photon flux is available, point collimation with pinholes is preferred as it avoids most slit-smearing effects. As scattering from slits and pinhole can produce additional background, efforts have been made by manufactures to develop scatterless slits, based on silicon, germanium, or tantalum single crystals. These slits exhibit less scattering, but the use of a guard slit just in front of the sample can still further reduce the background. The sample environment can be very sophisticated, but for the characterization of NPs in liquid suspension, mainly thin-walled glass capillaries are used in transmission mode. While laboratory instruments operate usually in air, the capillaries at SR beamlines are often in vacuum that makes the use of flow-through capillaries more difficult. One of the most important components in a SAXS experiment is the large-area detector to record the scattering pattern. While its distance from the sample can be below 1 m in small table top instruments, distances up to 30 m are used at SR beamlines. Wire detectors and image plates that were used in former times are now replaced by semiconductor detectors, either CCDs—often in combination with a fibre-optic taper to increase the pixel size—or hybrid-pixel detectors. The most widespread detector is now the PILATUS photon-counting hybrid-pixel detector, which has a pixel size of 172 μm [5]. A module of this detector has 105 pixels, and arrangements of up to 60 modules are commercially available, covering an area of more than 40 40 cm2. The direct transmitted beam is more intensive than the scattered radiation by several orders of magnitude and must therefore be shielded by means of a beamstop not to overload the area detector. A photodiode incorporated in the beamstop might be useful to monitor the transmitted beam. For randomly distributed scatterers like NPs in suspension, the obtained scattering pattern is centrosymmetric; thus, azimuthal integration can be used to convert the pattern to the scattered intensity as function of q as shown in Fig. 2. For the determination of the
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mean particle diameter and the size distribution, only the q-axis needs to be calibrated; the scattering cross section is not required on an absolute scale. As shown earlier, q depends only on the scattering angle and the photon energy EPh (or the wavelength λ). Calibration of the q-axis can therefore be performed in a traceable way to the SI unit metre by accurate measurements of pixel size and sample detector distance and the wavelength, which can be done at SR beamlines, for example, by backreflection from a silicon crystal [6]. A widespread material to check the q calibration is silver behenate [6a]. Absolute scattering cross sections, thus scattered intensities normalized to the incident intensity, are required for more careful work, such as NP concentration determination. Here again, different options are possible: (i) using calibrated detectors (e. g. semiconductor photodiodes or ion chambers) to measure the incoming photon flux and an area detector with known quantum efficiency to register the scattering pattern [7], (ii) using, for example, glassy carbon as a reference material for scattered intensity calibration [8] or (iii) using an area detector with a high dynamic range like a photoncounting hybrid-pixel detector to register both the scattering pattern and, with reduced counting time, the incoming direct beam.
Mean particle size determination with SAXS Since 2015, the International Standard ISO 17867 is available where two methods for the determination of the mean particle diameter from SAXS measurements are described [9]: the Guinier approximation and the model-based data fitting. For the Guinier approximation, the scattered intensity is plotted as function of q2 (Guinier plot), and the radius of gyration can be obtained from the slope at very low q-values. Under certain
Small angle x-ray scattering (SAXS)
conditions, this can provide an estimate on the mean particle diameter, but the method has many limitations. A more accurate method to obtain the mean diameter is the model fitting for which software implementations are available, for example [10]. The scattering cross section of spherical particles with radius R as function of the momentum transfer q can be generally expressed as ð∞ dΣ gðRÞ Sðq, RÞ jf ðq, RÞj2 dR + BG: (2) ðqÞ ¼ re2 C Δρ2e dΩ 0 where re is the electron radius, C the concentration, thus the number of scatterers (NPs) per volume, Δρe the electron density difference between the particles and the surrounding medium, g(R) the size distribution function, S(q,R) the structure factor, f(q,R) the form factor, and BG(q) a background contribution. The structure factor S depends on the arrangement of the scattering objects and simplifies to unity in a dilute dispersion where the interaction between the particles and multiple scattering can be neglected. For higher particle concentrations and agglomerated particles, the structure factor becomes very relevant and describes the probability distribution of interparticle distances. The form factor f depends only on particle size and shape. For spherical particles, it is ! 4 3 sin ðqRÞ qR cos ðqRÞ f ðq, RÞ ¼ πR 3 (3) 3 ðqRÞ3 The size distribution of the particles is given by g(R). Different size contributions can be assumed; most widely used assumptions are Gaussian distribution, Zimm–Schultz distribution (often used for polymerization), or log-normal distribution that is, in principle, a Gaussian distribution on a logarithmic scale. Especially for the determination of the size distribution width, a correct determination of the background is crucial. Detailed recommendations for data correction are available [11] and will partly be included in the next edition of the ISO standard together with additional methods for data analysis like Monte-Carlo based fitting [12] and the indirect Fourier transform (IFT) method [13]. The comparability of these approaches has been shown in an interlaboratory comparison where small silver NPs were measured on 24 instruments. The mean radii obtained with model fitting, IFT, and a Monte-Carlo method were 2.80 nm, 2.82 nm, and 2.67 nm, with standard deviations of 0.07 nm, 0.04 nm, and 0.16 nm, respectively [14]. The periodicity of the observed oscillations can be used for the size determination, and the exact knowledge of the wavelength is exploited, which, contrary to light scattering in the visible range, is considerably smaller than the objects under investigation. Examples for the traceable determination of the mean particle diameter using SAXS and model fitting are shown in Fig. 3 for PMMA particles. The periodicity of the oscillation on the q-axis is inversely proportional to the mean diameter. For the more complex case of a bidisperse size
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Fig. 3 Measured and fitted scattered intensity for PMMA particles with nominal diameters of 108 nm (upper left) and 192 nm (upper right). The determined diameters are 109.0 nm and 188.0 nm, respectively. The scattered intensity for a mixture of both particle suspensions is shown in (lower left) together with the separated contributions, which correspond to the number weighted size distribution in (lower right). (Credit: From G. Gleber, L. Cibik, S. Haas, A. Hoell, P. M€ uller, M. Krumrey, Traceable size determination of PMMA nanoparticles based on small angle X-ray scattering (SAXS), J. Phys. Conf. Ser. 247 (2010) 012027.)
distribution, the scattering curve is a superposition of both contributions. The mean diameters were determined in agreement with the results obtained separately with each of the two components with a relative uncertainty of better than 1% [6].
Size distribution width determination with SAXS According to Eq. (3), perfectly monodisperse spheres would lead to oscillation of the scattered intensity with sharp minima as shown in Fig. 4. With increasing distribution widths, the minima smear out. Together with the size distribution widths, not only the assumed distribution (e.g. Gaussian or log-normal) in case of the model fitting has to be indicated, but it is even more important whether the number or volume weighted distribution is provided. While for almost monodisperse particles with a low polydispersity p ¼ σ d/d, where σ d is the standard deviation of the size distribution and d is the mean diameter,
Small angle x-ray scattering (SAXS)
Fig. 4 Influence of the polydispersity on the amplitude of the oscillation for particles with 50 nm diameter and different polydispersity.
Fig. 5 Comparison of different size distributions for nanoparticles with a diameter of 50 nm and a polydispersity of 0.06 (left) and of 0.24 (right). Log-normal and Gaussian distributions are shown; solid lines indicate the number weighted distributions, while dashed lines are for volume weighted. While there is almost no difference for narrow size distributions, significant differences are visible for larger distribution widths.
all distributions and weighting have similar results; there are large differences for broader size distributions. This effect is shown in Fig. 5 for p ¼ 0.06 and p ¼ 0.24. To obtain the correct size distribution, the shape of the particles has to be known as, for example, a larger distribution width of spherical particles can lead to a similar smearing of the scattering curve as, for example, elongated ellipsoidal particles. In principle, the particle shape can also be obtained from the slope of the intensity decay as function of q, but a simultaneous size and shape determination will not always lead to a unique solution. Microscopic techniques that provide direct imaging of particles should be used to investigate whether, for example, the assumption of a spherical shape is justified.
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Nanoparticle concentration determination with SAXS While only the q-axis needs to be calibrated for the size and size distribution determination, absolute scattering intensities are essential to determine the NP concentration. Due to the straightforward scattering theory, the scattered intensity can be calculated from the model fitting as well. The mean diameter and the size distribution are obtained from the oscillations as shown in Fig. 6, and the particle number concentration C, thus the number of particles per volume, is then finally a scaling parameter in Eq. (2)[14a]. However, the irradiated volume is not always well defined due to the profile of the incoming x-ray beam. For a capillary with a rectangular profile, only the sample thickness in beam direction needs to be determined. Also, obvious from Eq. (2) is the fact that the scattered intensity also scales with Δρ2e , thus even the square of the electron density difference between particle and the suspending medium. While the electron density of, for example, water is well known (333 nm3), the electron density of NPs is not always identical to the corresponding bulk material. This uncertainty contribution from the particle electron density is often underestimated.
Core–shell nanoparticles Most nanoparticles are not simple solid objects but have an inner morphology, which, in the easiest case, can be described by a core–shell structure. This not only can be intentionally produced on manufactured NPs, for example, for surface functionalization, but also might be the result of capping or coverage of the surface by organic material in the suspending medium. If the core is made from a high-density (inorganic) material, its scattering will dominate the observed scattering so that no information on a possible organic layer on the surface can be obtained. If the shell or both, core and shell, contain elements with a higher atomic number, anomalous SAXS (ASAXS) can be used to distinguish
Fig. 6 Scattering cross section in absolute units of gold nanoparticles with (27.2 0.7) nm diameter and a particle number concentration of (1.85 0.13) ∗ 1011 mL1 as determined from the measured data (points) and model fitting (solid red line).
Small angle x-ray scattering (SAXS)
between core and shell. Here, the change of the effective electron density in the vicinity of absorption edges of the involved chemical elements is exploited. This technique is only possible with synchrotron radiation as the photon energy needs to be tuned. Typically, measurements are performed at several photon energies in the range between a few hundred eV and a few eV below a relevant absorption edge, for example, the Au L3 edge for Ag/Au nanoparticles [15]. As only the contribution from this specific element is varying, the changes of the scattering curves can be associated with the presence of the element in core or shell. For lighter elements like silicon, phosphorus, sulphur, or calcium, the K absorption edges are in the so-called tender x-ray range where measurements have to be performed in vacuum and dedicated beamlines and detectors are required [16]. Mainly for organic systems with densities close to the density of water, the contrast can also be varied by changing the electron density of the suspending medium, for example, by adding sucrose. An elegant way is the continuous contrast variation as shown in Fig. 7, where a gradient of the electron density is created in a capillary. By moving the capillary through the x-ray beam, scattering curves can then be recorded at different positions and thus different electron density differences [17]. The scattering at low q-values has a minimum where the electron density of the suspending medium equals the mean electron density of the particles. However, for core–shell particles, not the entire scattering curves are shifted in intensity, but the scattered intensity remains constant at specific q-values. From these isoscattering points q∗, the outer particle radius can be calculated according to
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Fig. 7 Continuous contrast variation to reveal a core–shell structure of polystyrene nanoparticles. Scattering curves measured at different positions in the gradient capillary, corresponding to suspending medium electron densities (left) and selected scattering curves fitted with a core–shell model where the shell has a higher electron density than the core (right). (From R. Garcia-Diez, C. Gollwitzer, M. Krumrey, Nanoparticle characterization by continuous contrast variation in small-angle X-ray scattering with a solvent density gradient, J. Appl. Crystallogr. 48 (2015) 20–28.)
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tan ðq∗RÞ ¼ q∗R
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The particle radius is also available from the model fitting using the form factor for spherical core–shell particles. For the polystyrene particles shown as example in Fig. 7, the outer radii obtained with both methods were (50.4 2.8) nm and (49.7 2.8) nm, respectively. The core–shell structure of these particles is due to the manufacture process, which is based on a styrene, methacrylic acid, and methyl methacrylate monomer mix, resulting in a thin shell of higher density.
Conclusion SAXS has proven to be a very powerful ensemble technique for the characterization of nanoparticles in suspension without the need for drying or other sample preparation and thus in their native environment. For sufficiently monodisperse NPs, a traceable size and size distribution determination is possible as well as the concentration determination with known particle density. Additional information is available from related techniques: Wide angle x-ray scattering (WAXS) extends the accessible feature size range to 1 nm and below, ASAXS allows to obtain the mentioned element-specific information, and grazing incidence SAXS (GISAXS) can be used for the characterization of nanostructured surfaces or nanoparticles on surfaces in reflection geometry. SAXS and related techniques can be used for many different applications, but a great part of them is currently in the fields of particle synthesis including growth control [18] and catalysis [19] and in operando fuel cell [20] or battery characterization [21].
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[9] ISO 17867, Particle Size Analysis – Small-Angle X-Ray Scattering, ISO International Organization for Standardization, Geneva, 2015. [10] I. Breßler, J. Kohlbrecher, A.F. Th€ unemann, SASfit: a tool for small-angle scattering data analysis using a library of analytical expressions, J. Appl. Crystallogr. 48 (2015) 1587–1598. [11] B.R. Pauw, Everything SAXS: small-angle scattering pattern collection and correction, J. Phys.: Condens. Matter 25 (2013) 383201. [12] B.R. Pauw, J.S. Pedersen, S. Tardif, M. Takata, B.B. Iversen, Improvements and considerations for size distribution retrieval from small-angle scattering data by Monte Carlo methods, J. Appl. Crystallogr. 46 (2013) 365–371. [13] O. Glatter, A new method for the evaluation of small-angle scattering data, J. Appl. Crystallogr. 10 (1977) 415–421. [14] B.R. Pauw, C. K€astner, A.F. Th€ unemann, Nanoparticle size distribution quantification: results of a small-angle X-ray scattering inter-laboratory comparison, J. Appl. Crystallogr. 50 (2017) 1280–1288. [14a] A. Schavkan, C. Gollwitzer, R. Garcia-Diez, M. Krumrey, C. Minelli, D. Bartczak, S. Cuello– Nun˜ez, H. Goenaga-Infante, J. Rissler, E. Sj€ ostr€ om, G.B. Baur, K. Vasilatou, A.G. Shard, Number concentration of gold nanoparticles in suspension: SAXS and spICPMS as traceable methods compared to laboratory methods, Nanomaterials 9 (2019) 502. [15] J. Haug, H. Kruth, M. Dubiel, H. Hofmeister, S. Haas, D. Tatchev, A. Hoell, ASAXS study on the formation of core–shell Ag/Au nanoparticles in glass, Nanotechnology 20 (2009) 505705. [16] A. Hoell, Z. Varga, V.S. Raghuwanshi, M. Krumrey, C. Bocker, C. R€ ussel, ASAXS study of CaF2 nanoparticles embedded in a silicate glass matrix, J. Appl. Crystallogr. 47 (2014) 60–66. [17] R. Garcia-Diez, C. Gollwitzer, M. Krumrey, Nanoparticle characterization by continuous contrast variation in small-angle X-ray scattering with a solvent density gradient, J. Appl. Crystallogr. 48 (2015) 20–28. [18] J. Polte, T. Ahner, F. Delissen, S. Sokolov, F. Emmerling, A.F. Th€ unemann, R. Kraehnert, Mechanism of gold nanoparticle formation in the classical citrate synthesis method derived from coupled in situ XANES and SAXS evaluation, J. Am. Chem. Soc. 132 (2010) 1296–1301. [19] H. Song, R.M. Rioux, J.D. Hoefelmeyer, R. Komor, K. Niesz, M. Grass, P. Yang, G.A. Somorjai, Hydrothermal growth of mesoporous SBA-15 silica in the presence of PVP-stabilized Pt nanoparticles: synthesis, characterization, and catalytic properties, J. Am. Chem. Soc. 128 (2006) 3027–3037. [20] M. Povia, J. Herranz, T. Binninger, M. Nachtegaal, A. Diaz, J. Kohlbrecher, D.F. Abbott, B.-J. Kim, T.J. Schmidt, Combining SAXS and XAS to study the operando degradation of carbon-supported Pt-nanoparticle fuel cell catalysts, ACS Catal. 8 (2018) 7000–7015. [21] G.O. Park, J. Yoon, E. Park, S.B. Park, H. Kim, K. Ho Kim, X. Jin, T.J. Shin, H. Kim, W.-S. Yoon, M.K. Ji, In operando monitoring of the pore dynamics in ordered mesoporous electrode materials by small angle X-ray scattering, ACS Nano 9 (2015) 5470–5477.
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