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In this issue Depth-gradient measurements Energy-dispersive diffractionmapping Intergrowth Microdomains Nanostructures Powder diffraction Rietveld methods Self-assembled droplets Small-angle scattering Texture Thermal diffuse scattering

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Robust structure and morphology parameters for CdS nanoparticles by combining small-angle X-ray scattering and atomic pair distribution function data in a complex modeling framework Christopher L. Farrow, Chenyang Shi, Pavol Juhás, Xiaogang Peng and Simon J. L. Billinge

J. Appl. Cryst. (2014). 47, 561–565

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Many research topics in condensed matter research, materials science and the life sciences make use of crystallographic methods to study crystalline and non-crystalline matter with neutrons, X-rays and electrons. Articles published in the Journal of Applied Crystallography focus on these methods and their use in identifying structural and diffusioncontrolled phase transformations, structure-property relationships, structural changes of defects, interfaces and surfaces, etc. Developments of instrumentation and crystallographic apparatus, theory and interpretation, numerical analysis and other related subjects are also covered. The journal is the primary place where crystallographic computer program information is published.

Crystallography Journals Online is available from journals.iucr.org J. Appl. Cryst. (2014). 47, 561–565

Christopher L. Farrow et al. · Robust structure and morphology parameters for CdS

research papers Journal of

Applied Crystallography ISSN 1600-5767

Robust structure and morphology parameters for CdS nanoparticles by combining small-angle X-ray scattering and atomic pair distribution function data in a complex modeling framework

Received 21 September 2013 Accepted 17 December 2013

Christopher L. Farrow,a Chenyang Shi,b Pavol Juha´s,b Xiaogang Pengc and Simon J. L. Billingeb,c* a Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA, bCondensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, NY 11973, USA, and cDepartment of Chemistry, Zhejiang University, Hangzhou, Zhejiang 310027, People’s Republic of China. Correspondence e-mail: [email protected]

# 2014 International Union of Crystallography

In this work, the concept of complex modeling (CM) is tested by carrying out a co-refinement of the atomic pair distribution function and small-angle X-ray scattering data from CdS nanoparticles. It is shown that, compared with either single technique alone, the CM approach yields a more accurate and robust structural insight into the atomic structure and morphology of nanoparticles. This work opens the door for the application of CM to a wider class of nanomaterials and for the incorporation of additional experimental and theoretical techniques into these studies.

1. Introduction The atomic pair distribution function (PDF) of X-ray and neutron powder diffraction is a powerful tool for the analysis of the structure of nanoparticles (Egami & Billinge, 2013; Du et al., 2012; Juha´s et al., 2006; Gilbert et al., 2004; Masadeh et al., 2007). With the advent of synchrotron X-ray techniques and advanced computing power, the PDF yields structural information at the a˚ngstro¨m level, which is particularly important for nanocrystalline and poorly crystallized materials and is gaining popularity in the scientific community (Shyam et al., 2012; Bozˇin et al., 2010; Dykhne et al., 2011; Michel et al., 2007; Page et al., 2008; Polking et al., 2012). However, in many practical cases, the PDF may not contain enough information to constrain the inverse problem resulting in a unique nanoparticle structure. In this case, a ‘complex modeling’ (CM) approach (Billinge & Levin, 2007) is needed that incorporates extra information in the form of both theory and multiple complementary experimental data sets, such as from transmission electron microscopy (TEM), extended X-ray absorption fine structure (EXAFS), small-angle X-ray scattering (SAXS) and Raman spectroscopy, in a combined (or complexed) optimization. There are some steps being taken towards this goal with attempts to add diffraction (Tucker et al., 2007) and EXAFS (Krayzman et al., 2008) to PDF data in a reverse Monte Carlo style ‘big-box’ modeling scheme. To do this systematically and flexibly is a theoretical and computational challenge but is necessary in problems at the frontier of materials complexity (Bennett et al., 2010; Cliffe et al., 2010; Hwang et al., 2012; Corr et al., 2010; Goodwin et al., 2010). Here we present a first step towards a combined refinement of J. Appl. Cryst. (2014). 47, 561–565

PDF and SAXS data from CdS nanoparticles using a recently written CM framework, SrFit (C. L. Farrow, P. Juhas & S. J. L. Billinge, 2010, unpublished). We show that indeed more robust structural parameters are obtained this way, demonstrating the viability of this approach. In a bulk material one does not consider the sample size or shape because of its macroscopic size in all directions. Crystallography reveals information about the intrinsic atomic geometry. However, in a nanomaterial knowledge of both atomic structure and also particle shape and size are important. The PDF contains information about particle size and shape as well as its internal structure (Egami & Billinge, 2013; Masadeh et al., 2007). Practically, however, the PDF is insensitive to information about morphology. The PDF yields the range of coherence of the structure which, in general, may be smaller than the actual particle size (Egami & Billinge, 2013; Zhang et al., 2003). The refinement of particle size and shape from the PDF may also not be very robust owing, for example, to parameter correlations and a lack of information about particle size distributions (Shi et al., 2013). On the other hand, SAXS gives the shape and size of nanoparticles with good precision but contains no information at the atomic scale (Glatter & Kratky, 1982). In order to obtain precise information for both, it is preferable to use a CM approach such as a combination of SAXS and PDF. Recently the theoretical relationship between the two techniques has been established (Farrow & Billinge, 2009). Estimates of nanoparticle shape and size from SAXS data have been compared with PDF data to provide insight into nanoparticle disorder (Gilbert et al., 2004), but SAXS is not commonly used to corroborate PDF doi:10.1107/S1600576713034055

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research papers data and has never been co-refined with PDF data in a CM scenario, before this study. However, with the advent of equipment that facilitates the collection of SAXS with wideangle X-ray scattering data suitable for PDF work (Daniels et al., 2010) this approach can be expected to become more common in the future. This software development will increase the power of such measurements.

2. Experimental methods Here we study the structure of CdS nanoparticles that were synthesized and purified using the method reported by Yu & Peng (2002). A TEM image of the nanoparticles is shown in Fig. 1. The CdS nanoparticles are evident as darker regions in some of which it is possible to resolve lattice fringes. From the image, they have a rather uniform size distribution and appear to have a slightly spheroidal shape with the largest dimension around 3 nm and an aspect ratio between 1:1 (spherical) and 2:1 (slightly elongated). X-ray PDF data were collected at beamline 11-ID-C of the Advanced Photon Source at Argonne National Laboratory. The PDF data were measured in a flowing nitrogen cryocooler at a temperature of 100 K using the rapid acquisition pair distribution function (RaPDF) method (Chupas et al., 2003) ˚ and a sample-towith an X-ray wavelength of 0.10798 A detector distance of 391.1 mm. A large area two-dimensional fast readout Perkin Elmer image plate detector was used. The raw two-dimensional data were azimuthally integrated and converted to one-dimensional intensity versus 2 using FIT2D (Hammersley et al., 1996). An empty capillary was measured

for background subtraction. PDFgetX3 (Juhas et al., 2013) was used to correct and normalize the diffraction data and to Fourier transform R Qmax them to obtain the PDF, G(r), according to Q½SðQÞ 1 sin Qr dQ. Here Q is the GðrÞ ¼ ð2=Þ Qmin magnitude of the momentum transfer on scattering and S(Q) is the properly corrected and normalized powder diffraction intensity measured from Qmin to Qmax (Egami & Billinge, 2013). G(r) gives the probability of finding a pair of atoms separated by a distance r, GðrÞ ¼ 4r½ðrÞ 0 0 ðrÞ;

where ðrÞ is the atomic pair density, 0 is the average atomic number density and 0 ðrÞ is the characteristic function for the sample shape (Farrow & Billinge, 2009). The characteristic function is the autocorrelation of the shape function of the sample, which is a geometric function that has value unity inside the (average) nanoparticle and value zero outside. For isotropically scattering samples such as powders of nanoparticles, we use the orientationally averaged characteristic function (Gilbert, 2008; Farrow & Billinge, 2009). This quantity is obtained directly from SAXS (Guinier, 1963) and in the PDF it describes the attenuation of a nanoparticle PDF with increasing r (Masadeh et al., 2007; Gilbert, 2008; Farrow & Billinge, 2009). Before we can extract accurate values for the particle size it is necessary to correct for the finite resolution of the measurement. Both standard nickel and cerium oxide samples were measured to extract Qdamp, which is the parameter that corrects the PDF envelope function for instrument resolution effects (Proffen & Billinge, 1999; Farrow et al., 2007). The standard approach was used based on the knowledge that the calibration samples are crystalline and so any fall-off in PDF peak intensity with increasing r from these samples is due to the experimental resolution. The particle size in the fit is set to infinity and the Qdamp resolution parameter is ˚ 1 is then fixed in varied. The refined value of Qdamp = 0.0396 A model refinements to the nanoparticle data. The PDF of the CdS nanoparticle was calculated theoretically from atomic models by multiplying the bulk Gbulk(r) by a spheroidal characteristic function (r) (Gilbert, 2008; Farrow & Billinge, 2009; Masadeh et al., 2007; Yang et al., 2013): Gnano ðrÞ ¼ ðrÞGbulk ðrÞ:

Figure 1 (a) A TEM image of a representative region of the CdS nanoparticle sample. (b), (c) Spheroidal morphology and atomic structure of CdS nanoparticles obtained (b) from a PDF fit and (c) from the PDF–SAXS co-refinement. Cd is shown as gray and S as blue in the image.

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ð1Þ

ð2Þ

Here Gbulk(r) is a PDF calculated using periodic boundary conditions and the model used for the bulk structure was the one appropriate for II–VI nanoparticles that contain significant numbers of stacking faults. As discussed by Yang et al. (2013), the local structure in these compounds is well explained by a model that mixes wurtzite and zinc blende PDFs with a mixing fraction that may be refined, but which depends on the stacking fault density. In this approach isotropic atomic displacement parameters are used and the lattice parameters of the hexagonal wurtzite and cubic zinc blende structures are constrained to be consistent. The modeling was carried out using the SrFit program. The SAXS data of the same samples were collected at beamline X10A of the National Synchrotron Light Source (NSLS) at Brookhaven National Laboratory. A Bruker

Robust structure and morphology parameters for CdS

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J. Appl. Cryst. (2014). 47, 561–565

research papers SMART 1500 CCD detector (10 10 cm, 1024 1024 pixels) was placed 695.8 mm from the sample. The incident beam had ˚ . The samples were exposed for 10 min a wavelength of 1.095 A at room temperature to obtain a two-dimensional diffraction image. Similar to the PDF, the raw two-dimensional data were azimuthally integrated and converted to one-dimensional intensity versus 2 curves using FIT2D. The empty capillary was measured to determine the background signal. The SAXS data are shown in Fig. 2. In order to ensure that we measured the same sample under the same conditions, the sample was measured as a solid aggregate. As a result there is a pronounced interparticle interference function, Sp(Q), that results in a well defined correlation peak in the data at around ˚ 1. The actual measured signal, I(Q), is then the 0.18 A product of Sp(Q) and P(Q), where Sp(Q) is an analytic function appropriate for a disordered hard-sphere structure model and P(Q) is the Fourier transform of the orientationally averaged characteristic function (Glatter & Kratky, 1982). In detail, IðQÞ ¼ PðQÞSp ðQÞ þ BEðQÞ, where EðQÞ is the experimentally measured empty capillary SAXS profile that was scaled by a prefactor B during the fit to the SAXS data. One should note that the strong interparticle correlation, Sp(Q), makes the precise determination of the SAXS form factor fit difficult. This is a drawback of measuring the SAXS data on the same sample as the PDF was measured on. One motivation was to test the principle that the PDF and SAXS data could be collected from the same sample and combined in a complex. The approach establishes a proof of principle for this method, but SAXS data from dilute samples are preferred for the most accurate SAXS information.

3. Complex modeling of PDF and SAXS data Co-refinement of the PDF and SAXS data is carried out by constructing a ‘complex’ in the SrFit CM framework, as shown

schematically in Fig. 3. The structural model is specified by parameters. Additional parameters that incorporate morphological, thermal and experimental effects are also included in the complex. The parameters are linked to variables which are passed to the optimizer. Constraints may be applied to individual variables and to connect different parameters and variables. Constraints may be applied within a single calculated function, or ‘leaf’, of the complex. Also, parameters in different leaves may be constrained together. For example, in this case the nanoparticle major and minor axes were constrained to be the same in the SAXS and PDF calculations. The SrFit implementation is highly modular and makes heavy use of object-oriented software design, which makes it highly flexible and easily extensible and thus ideal for an application such as the heterogeneous refinement described here. This example demonstrates the utility of the complex modeling approach for increasing the robustness of structure models. For this study it was necessary to implement a new SAXS function calculator in SrFit, which calculates the SAXS intensity given a morphological model that is shared between the SAXS and PDF leaves of the complex. The parameters that the SAXS leaf uses are the major and minor axes of the spheroidal nanoparticle shape, as well as mean particle separation and effective radius that goes into the Sp(Q) calculation. Additionally, there were independent scale factors for the sample and for the subtraction of the measured background. The spheroid parameters are shared with the PDF calculator but the particle packing information is not. SrFit was configured to use differential evolution (DE) as a regression algorithm. In DE, populations of candidate structure solutions are retained and combined with each other (the analog of reproduction) as well as mutated. In this case, we retained a population of 160 individuals (ten times the number of parameters) during the regression. All parameters, including the diameters, were varied when generating the population. After 105 function evaluations (roughly 625 generations) the minimization was stopped and a downhill simplex algorithm used to find the local minimum.

4. Results and discussion We first show the results that are obtained when the SAXS and PDF data are fitted separately. The structural results are

Figure 2 Spheroidal model fit to SAXS data. Intensity is calculated as Sp ðQÞPðQÞ þ BEðQÞ, where Sp(Q) (black dashed line) is the particle– particle interference function, P(Q) is the spheroidal shape factor (gray line) and E(Q) is the SAXS profile of the empty capillary (green dashed line) that was scaled by a constant B during the fit. BE(Q) is shown as a green curve. The SAXS data are shown as a thick blue line and the model fit is in red. J. Appl. Cryst. (2014). 47, 561–565

Figure 3 Schematic of the complex modeling setup in SrFit, described in the text, which combines PDF and SAXS data in a single optimization problem.


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research papers Table 1 Structural and shape parameters from spheroidal CdS nanoparticle model fits to the PDF and SAXS data. Results for the wurtzite phase are shown below [Cd at (1/3, 2/3, 0), S at (1/3, 2/ 3, z)].

Rw(PDF) Rw(SAXS) ˚) a (A ˚) c (A Sz ˚ 2) Cd Uiso (A ˚ 2) S Uiso (A Wurtzite fraction ˚) Equatorial radius (A ˚) Polar radius (A

SAXS

PDF

Complex

– 0.0148 – – – – – – 11.36 20.77

0.146 – 4.134 6.753 0.441 0.0094 0.0158 0.366 10.17 32.43

0.149 0.0158 4.135 6.752 0.434 0.0095 0.0173 0.376 11.38 23.45

shown in Table 1. The fits to the experimental PDFs are shown in Fig. 4 with the difference curve offset below. The fits are quite good. The high-r region is shown on an expanded y scale to highlight the small differences. This quality of fit would be considered to be quite satisfactory in a regular PDF refinement of nanoparticle samples (Masadeh et al., 2007; Yang et al., 2013; Du et al., 2012; Gilbert et al., 2004; Polking et al., 2012). The fits result in a nanoparticle aspect ratio that is 3.2:1 as is shown in Fig. 1(b). The fits to the SAXS data alone are shown in Fig. 2 as the solid red line, with Sp(Q) and P(Q) (Glatter & Kratky, 1982) shown separately, offset below. The best fit ellipsoid has an ˚, equatorial radius and polar radius of 11.36 and 20.77 A respectively, giving an aspect ratio of 1.8:1. It is striking that the best fit PDF model results in major and minor axes for the spheroid that are quite different from the SAXS-derived ones. Although the ‘size’ obtained from the PDF may be smaller than that obtained from SAXS because of disorder, it cannot ˚ be bigger, as we see here for the major axis which is 32.43 A ˚ from the PDF but only 20.77 A from SAXS. This indicates an aberration in either the PDF or the SAXS result, which may

Figure 4 Experimental (blue) and corresponding simulated PDFs (red) with difference curve (green) offset below for CdS nanoparticles (a) fitted using the PDF alone and (b) co-refined to PDF and SAXS data. Data in ˚ ) are magnified ten times for clarity. the high-r range (>20 A

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occur if the information content of the data that constrains that parameter is weak, if there are systematic errors unaccounted for in the model or if refinement parameters are significantly correlated (or some combination of these). For example, in the current case the information in the PDF is somewhat weak because an approximate Gaussian is used for the effects of the measurement resolution, as we discuss below, and the information content on the nanoparticle size and shape from the SAXS is weakened by the fact that it derives from a fit to data from a concentrated sample rather than a dilute nanoparticle dispersion optimized for nanoparticle form factor determination. The idea behind complex modeling is that it both pinpoints such systematic errors, which are sometimes unavoidable, and mitigates their effects. We now consider the results of the complex modeling approach where the information from PDF and SAXS is used together in the same refinement. The fit is shown in Fig. 4(b) and summarized in Table 1. A comparison of the morphology obtained by the PDF alone and the PDF–SAXS complex is shown in Figs. 1(b) and 1(c). The quality of the PDF fit, as measured by the agreement factor, Rw, is slightly worse in the complex. However, the addition of the SAXS data constrains the particle size and shape to values that better reflect the TEM image evident in Fig. 1(a). Thus, the complexed model has produced a more physically reliable result than the PDF refinement alone. There are corresponding small adjustments to the refined values of the wurtzite fraction and isotropic displacement parameters. The PDF-only and the complexed refinements converged to slightly different minima in the physical-parameter space. To explore the robustness of the refinements we tried varying starting values in an attempt to steer each refinement into the other parameter space minimum, without success. Further, we tried fixing the wurtzite fraction or atomic displacement parameters in the PDF-only model to those values refined in the complexed model, but in each case the PDF-alone refinement preferred the elongated nanoparticle shape and did not find the alternative minimum. This indicates that these are stable minima for each model and the problem is not due to a lack of model convergence. By fixing the particle shape parameters to those obtained from the complex, the PDFalone refinement did refine to the same minimum as the complex, albeit with a higher Rw. This suggests that there is some systematic error in the model used for the PDF calculation with respect to the particle shape, which may originate from an imperfect correction for measurement resolution effects or perhaps comes from inadequacies in the Morningstar–Warren approximation (Warren et al., 1936; Egami & Billinge, 2013). To test the former we allowed the Qdamp parameter to vary in the refinement rather than fixing it to the value obtained in the calibration measurement. In this case, the value refined for the short axis did not change much, but Qdamp and the long-axis parameters became highly correlated, resulting in a large increase in both Qdamp and the value of the ˚ . This shows that long-axis parameter, which went up to 127 A the values refined for the particle shape functions are rather highly correlated with Qdamp, which must be determined


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research papers carefully from calibration data for accurate results. Further, it shows that the stability of the refined particle size parameters decreases as the particle dimensions get larger. By introducing the information from small-angle scattering data in a modeling complex, the uncertainty on these parameters is greatly reduced, leading to both more physical refined structural parameters and a better understanding of the uncertainties and limitations of our models. For the most reliable corefinement, SAXS data from a dilute suspension are preferred to avoid uncertainties from the separation of the form factor and the structure factor in the SAXS. However, in some cases it may not be possible to make a separate dilute sample, and this proof of principle demonstration indicates that SAXS data from a concentrated sample may still add value in a complex modeling framework.

5. Conclusion Many experiments are being carried out to elucidate the structure of complex materials and the data being analyzed quantitatively using model refinements, whether it is Rietveld refinement of powder diffraction data or fits to PDF or EXAFS data, for example. The accuracy (as opposed to the precision) of the results is often limited by systematic errors or uncertainties in the measurement, model description or data processing, rather than the statistical uncertainty on the data. This study shows how complex modeling can help to test the reliability of physical parameter estimates as well as giving insight into the limitations of calculated and experimentally determined spectra by using multiple data/information sources. This is an important step towards obtaining more reliable quantitative estimates of structural parameters that will be used to understand the properties of complex materials. The SrFit complex modeling framework is designed to make this approach more straightforward to apply.

The SrFit framework was developed and the data were collected under the National Science Foundation (NSF) funded DANSE project (DMR-0520547). The samples were prepared at Zhejiang University funded by the Columbiabased US-DOE-BES-funded Energy Frontier Research Center under award No. DE-SC0001085. The data analysis was carried out and the project finalized as part of the Complex Modeling LDRD project at Brookhaven National Laboratory, which is funded by the US-DOE-BES under contract No. DE-AC02-98CH10886. Use of the National Synchrotron Light Source at Brookhaven National Laboratory is also funded under contract No. DE-AC02-98CH10886. Use of the Advanced Photon Source is supported by the UUSDOE-BES under contract No. DE-AC02-06CH11357.

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