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Feb 5, 2018 - approximate solutions are applicable to determine particles size .... The anatase nanoparticles of cubic shape expose (001) and (100) fa...
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Influence of Nanoparticles Size on XRD Patterns for Small Monodisperse Nanoparticles of Cu0 and TiO2 Anatase Alexander V. Vorontsov*,†,‡ and Sergei V. Tsybulya§ †

Altai State University, Prospekt Lenina 61, Barnaul 656049, Russia N. N. Vorozhtsov Novosibirsk Institute of Organic Chemistry, Prospekt Akademia Lavrentyeva 9, Novosibirsk 630090, Russia § Boreskov Institute of Catalysis, Prospekt Akademia Lavrentyeva 5, Novosibirsk 630090, Russia ‡

S Supporting Information *

ABSTRACT: The effect of nanoparticle size and structure on XRD and SAXS patterns was investigated using modeling with the Debye scattering equation for a series of nanoparticles (NP) with the positions of atoms kept according to the bulk lattice and after structure relaxation. The purpose of research was to determine if the changes in XRD peak positions for NP are entirely due to the well-known effects of lattice parameter change or if additional effects can arise from the size itself. It was found that for very small NPs with sizes below 5 nm, the size itself influences the XRD patterns. This effect can be caused by interference fringes and is not taken into account when considering XRD patterns in standard software. The research demonstrates that new methods for XRD pattern treatment of very small nanoparticles should be developed.

1. INTRODUCTION X-ray diffraction (XRD) is a materials characterization technique that is widely used for all kinds of solid particles including nanoparticles of all sizes.1 The theory of XRD is welldeveloped for single crystals and powders of solids with a large number of crystalline planes. The positions of the diffraction peaks is described by Braggs’ law and the width by the Scherrer equation,2 with deviations from these equations being attributed to structure distortions and microstrain. In the range of small diffraction angles of 2θ typically below 20°, small-angle X-ray scattering (SAXS) is observed, and the SAXS peaks are attributed to scattering on particles and superlattices. The theory of SAXS is not as simple as for XRD and approximate solutions are applicable to determine particles size using assumptions on the particles shape.3 Nanosized particles have one, two or three dimensions below 100 nm. The most interesting properties are observed for the sizes below 10 nm. For such small nanoparticles, very interesting effects are observed associated with predominance of surface phenomena over bulk ones and changes in electronic properties as a function of size called quantum size effect. XRD is used for characterization of nanopowders of any sizes, and the observed changes in positions of diffraction peaks are used to make conclusions on how crystal structure and cell parameters changes with the change in nanoparticles size and shape. It should be noted, however, that the traditionally used XRD theory was developed for relatively large particles with large number of diffracting planes and for which surface effects on XRD are negligibly small. It is not clear if traditionally used © XXXX American Chemical Society

methods of XRD treatment can be applied for very small nanoparticles. Among a variety of nanoscale objects, metallic and metal oxide nanoparticles attract a great deal of attention of researchers. These nanosystems are important for catalysis, photocatalysis, sensors, fuel cells, electronics, biomedical, and other applications. The objects of the present investigation are metallic copper and TiO2 anatase nanoparticles. Copper nanoparticles are important component of many photocatalytic and catalytic systems.4−9 Titanium dioxide in anatase phase is very active, stable photocatalyst,6,10−22 active component and support for catalysts of partial and complete oxidation and reduction, component of dye sensitized solar cells, lithium batteries, sensors, and electronic devices.23,24 Most interesting physical-chemical and functional properties of copper and anatase nanoparticles are observed at a size of few nanometers. Parameters of anatase unit cell are changed when the size of nanoparticles changes as XRD studies demonstrated. When the anatase nanoparticles size increased from 8 to 30 nm, the elementary cell volume increased from about 134.7 to 136 Å3.1 Crystallization of TiO2 nanoparticles resulted in an increase of cell parameter c from 9.490 to 9.510 Å for temperature 100 °C and from 9.530 to 9.540 for 200 °C that corresponded to primary particles size increase from 3 to 15 nm, the cell volume Received: Revised: Accepted: Published: A

October 31, 2017 December 31, 2017 February 5, 2018 February 5, 2018 DOI: 10.1021/acs.iecr.7b04480 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

TiO2 anatase nanoparticles of the present study were of cubic and decahedral shape. The anatase nanoparticles of cubic shape expose (001) and (100) facets that have different atomic structure. Decahedral anatase nanoparticles have exposed facets (001) and (101). Cell parameters for cubic nanoparticles were taken from accepted standard anatase XRD card JCPDS 21− 1272 that are a = b = 3.785, c = 9.513 Å. The cubic anatase nanoparticles were used for XRD patterns calculation without geometry relaxation. Only the effect of size was studied for cubic copper and anatase nanoparticles. Decahedral anatase nanoparticles were cut from a larger anatase nanoparticle that was optimized using periodic boundary conditions (PBC) and scc-dftb method. In order to obtain the cell parameters for PBC scc-dftb model, the supercells with size 3 × 3 × 1, 2 × 2 × 2, 3 × 3 × 2, 3 × 3 × 3, 4 × 4 × 2, 5 × 5 × 2, and 6 × 6 × 3 elementary cells were constructed and optimized. It turned out that the cell parameters are stabilized for the supercell of size 5 × 5 × 2 and beyond, and they are a = b = 3.798, c = 9.379 Å. These cell parameters deviate from cell parameters in JCPDS 21−1272 by +0.34 and −1.41%, respectively. The number of cell units in constructed decahedral anatase nanoparticles was from 2 to 5 in directions a and b. The number of layers of titanium atoms in the c direction was selected in such a way so that the thickness of nanoparticles be approximately equal in directions a and c. It has been demonstrated earlier that the nanoparticles of anatase obtained by simple cutting from the bulk anatase are nonstoichiometric and have deficit of oxygen.11,37 Therefore, decahedral anatase nanoparticles were augmented by hydroxyl groups and oxygen atoms for attaining charge neutrality and valence state of all titanium atoms equal to +4 and for all oxygen atoms equal to −2. Anatase structure is represented by distorted octahedrons of oxygen atoms surrounding the titanium atom. The coordination spheres of titanium atoms with formal charge of titanium lower than +4 were found with the help of the following equation.

increased by 0.8%, parameter a increased but the values were not reported.25 For anatase nanoparticles prepared from titanium isopropoxide, parameter c increased from about 9.45 to 9.78 but parameter a decreased from 3.796 to 3.785 when the size increased from 4.5 to 13.5 nm, and cell volume formed a maximum for size about 7 nm.26 Similar tendencies in relation of cell parameters and cell volume with nanoparticles size were reported for the size range 2−66 nm.27 Luca reported decrease in cell parameter a from 3.794 to 3.785 Å and a complex behavior in parameter c that first decreased from 9.498 to9.487 Å at size 9 nm and increased to 9.515 Å during particles size increase from 2 to 135 nm, and the cell volume decreased continuously from 136.7 to 136.3 Å3 for anatase nanoparticles prepared by sol−gel synthesis from Ti(OiPr)4.28 Gray and Wilson reported decrease in parameter a from 3.794 to 3.782 and increase in parameter c from 9.467 to 9.515 as the size of anatase nanoparticles increased from 4 to 25 nm.29 Titanium vacancies were reported to present in as high fraction as 0.2 at size 4 nm but they almost disappear at size above 9 nm. Therefore, contradictory trends of the influence of anatase nanoparticles size on its cell parameters are reported. The observed discrepancies can be attributed, at least partially, to the utilization of traditional line profile analysis instead of ab initio methods like Debye equation.30−32 It is also possible that different surface compositions of nanoparticles and extensive defects create different changes in the cell parameters and the cell volume of anatase nanoparticles. Development of computational power in recent years enabled researchers to perform facile modeling of objects, which was not reachable before. SAXS and XRD patterns can be obtained from first physical principles using Debye scattering equation (DSE).32,33 This equation was derived by averaging scattering from each atom in a nanoparticle over all orientations of the nanoparticle in space. Thus, DSE is applicable to systems of dilute nanoparticles with interparticle distance much larger than X-ray wavelength.3 The structure of nanoparticles can be obtained utilizing quantum computations with robust modern approximate methods such as density functional tight binding (dftb).34−36 Combination of both approaches enable studies of size effects in nanoparticles XRD and SAXS patterns with implications for real world characterization problems. In the present work, size effects are investigated for cubic metallic copper and cubic as well as decahedral TiO2 anatase nanoparticles with intent to answer the following two questions. (1) Does change in the size of small Cu 0 and TiO2 nanoparticles with ideal crystal structure influence the position of XRD peaks? (2) What is the influence of structure distortions of small decahedral anatase nanoparticles resulting from their small size and surface structure on their XRD and SAXS patterns?

Z = Z Ti +

∑

ZO N

(1)

where Z is effective charge of the titanium atom and its first coordination sphere, ZTi is the desired titanium formal charge equal to +4, ZO is the formal charge of oxygen atoms equal to −2, and N is the number of atoms in the first coordination sphere of a given oxygen atom; summation is carried out over all oxygen atoms in the first coordination sphere of a given titanium atom. If the value of Z is different from zero, charge compensation groups need to be attached to this titanium atom to obtain charge neutrality. Titanium atoms with the value of Z = +0.66(6) are present in vertices between four (101) planes and in some edges between (001) and (101) planes. There are many ways for distributing compensating charge hydroxyl groups and oxygen atoms over titanium atoms with oxygen deficit. To explore the effects of locating compensating charge atoms and groups in different patterns on structure and properties of decahedral anatase nanoparticles, we constructed a variety of clusters. The clusters are designated as TiMMrX, where M is the number of unit cells in directions a and b in the cluster, X is the number of the cluster version. The following clusters were obtained: Ti22r1, Ti22r2, Ti22r3, Ti33r1, Ti33r2, Ti33r3, Ti33r4, Ti33r5, Ti33r6, Ti33r7, Ti44r1, Ti44r2, Ti44r3, Ti44r4, Ti44r5, Ti44r6, Ti44r7, Ti55r2, Ti55r3, Ti55r4, and

2. EXPERIMENTAL SECTION Copper nanoparticles of cubic shape with face-centered crystal structure were generated using cell parameter a = 3.6149 Å.9 The length of the cube edge was from 1.5a to 14.5a and the number of copper nanoparticles structures modeled was 14. The length of the edges were from 0.54 to 5.24 nm. Copper nanoparticles were used for obtaining XRD patterns without geometry relaxation. B

DOI: 10.1021/acs.iecr.7b04480 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 1. XRD patterns for cubic Cu nanoparticles with edge length from 1.5a to 14.5a shown in Figure S1: (A) for edge length from 1.5a to 5.5a; (B) edge from 5.5a to 14.5a.

Ti55r5. All the clusters have formulas (TiO2)n(H2O)m, where n and m are integer numbers. XRD patterns of nanoparticles were computed using Debye scattering equation30 N

I(q) =

N

∑ ∑ fi (q)f j (q) i=1 j=1

inversely related to their particles size. The smallest Cu nanoparticles with size of few nanometers are expected to possess the highest activity. Characterization of such very small nanoparticles can meet difficulties because XRD patterns can deviate from those of bulk copper significantly. Figure S1 shows small cubic copper nanoparticles with edge length from 1.5a to 14.5a used in the present study. Cubic copper nanoparticles seem to be one of the most stable shapes for solvothermal synthesis of monodisperse particles with capping agents addition.41 Metallic copper has face-centered cubic crystal lattice with cell parameter a = 3.6149 ± 0.0002 Å.9 Interplanar spacing for cubic lattice are given by the following formula. a dHKL = 2 (4) H + K 2 + L2

sin rijq rijq

(2)

where q is scattering vector absolute value given by equation

q=

4π sin θ λ

(3)

θ is diffraction angle, λ is X-ray wavelength, rij is the distance between atoms i and j, f i(q) is atomic scattering factor available in International tables for crystallography for atoms, N is the number of atoms in the nanoparticle. XRD patterns were calculated in the diffraction angle range 2θ = 2−90° with 0.1° step and therefore they cover the small-angle X-ray scattering range. CuKα radiation wavelength of 1.54184 Å was used for computing all the diffractograms. The XRD patterns were not normalized. The structure of decahedral anatase nanoparticles was optimized with method scc-dftb using program dftb+38 using Slater-Koster files tiorg-0−139 and mio-1−1.40 The selfconsistent charge dftb (scc-dftb) method is based on Taylor expansion of density functional theory energy on charge density up to the second term. Criterion for finishing optimization was maximal force on atoms less than 0.05 kcal/(mol Å). The sccdftb method was demonstrated to produce reasonable results concerning TiO2 structure and electronic properties.39 Radial distribution of interatomic distances was computed from the coordinates of unoptimized and fully optimized with scc-dftb decahedral anatase nanoparticles with different distribution of compensating charge attached hydroxyl groups and oxygen atoms with the interval of 0.01 Å.

where H, K, and L are the Miller indices. For face-centered cubic lattice, only reflections with all even or all odd indices are allowed. Thus, reflections of planes (111), (200), and (220) are observed in the 2θ range from 0 to 90°. Corresponding Interplanar distances are 2.0871, 1.8075, and 1.2781 Å. XRD peaks for CuKα radiation are located at 2θ = 43.35, 50.49, and 74.20°, correspondingly. Figure 1 demonstrates XRD patterns computed for cubic copper nanoparticles of different size. For very small cubic nanoparticles with edge length below 2 nm, very wide XRD peaks are observed as Figure 1A shows. In the range of diffraction angles 2θ below 40°, small-angle X-ray scattering peaks are seen. For the smallest Cu nanoparticles with edge of 1.5a, only one SAXS peak is observed and XRD peaks for planes (111) and (220) are seen while the peak for (200) merges with the (111) peak. Cu nanoparticles with next size of edge equal to 0.90 nm (2.5a) produce two SAXS peaks and also only two XRD peaks corresponding to (111) and (220). Nanoparticles with size of 1.27 nm (3.5a) and larger demonstrate all three XRD peaks and an increasing number of SAXS peaks. The interpeak distance of SAXS peaks decreases as the nanoparticles size increases. Figure 1B shows SAXS and XRD patterns for larger cubic Cu nanoparticles with edge length up to 5.24 nm (14.5a). One can observe that besides the SAXS and XRD peaks, the diffractograms also have such

3. RESULTS AND DISCUSSION Metallic copper nanoparticles are important component of many catalytic8 and photocatalytic systems.7 Functional properties of these technologically important nanoobjects are C

DOI: 10.1021/acs.iecr.7b04480 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 2. (A) SAXS interpeak distance as a function of edge length of cubic metallic copper nanoparticles. The approximating curve is y = a/x, a = 79.9 ± 1.1. (B) First SAXS peak position versus copper particles edge, curve y = a/x + b, parameters a = 95.6 ± 2.0, b = 0.83 ± 0.13.

considerable superposition effect on lattice XRD peaks. This readily explains the periodic changes of the XRD peaks position with an increase in the nanoparticles size. The peaks of diffraction fringes can possibly be utilized for determination of nanoparticles size. Figure 2 shows the dependence of SAXS interpeak separation and first peak diffraction angle on cubic copper nanoparticles edge length. The both dependences are well approximated by reciprocal function y = a/x + b with parameters listed in the caption to Figure 2. Three types of fitting curves are used for the dependence of first SAXS peak on particles size, single exponential decay, reciprocal, and double exponential decay. The sum of squared deviations decreases in the row of these functions from 1.75 to 1.26 to 0.016 demonstrating that double exponential decay fits data very. However, double exponential decay is difficult to implement for obtaining particles size from SAXS data, because there is no reverse analytical function to double exponential decay for obtaining particles size from SAXS diffraction peak position or SAXS interpeak spacings. Copper nanoparticles are a convenient object for the study of size effects in SAXS and XRD patterns because their diffractograms have only three peaks that are well separated. TiO2 anatase represents a considerably more complex system for such studies because its diffractogram in 2θ range from 20 to 90° contains over 10 peaks many of which overlap. At the first stage of the study on size effects in TiO2 anatase XRD patterns, cubic anatase nanocrystalls with exposed facets (001) and (100) were constructed using published cell parameters according to JCPDS 21−1272. Figure 3 shows the five anatase nanoclusters of the present study with approximately same thickness in directions a, b, and c. Figure 4 demonstrates XRD patterns of anatase nanoparticles of cubic shape. For the smallest anatase nanocrystal Ti331c with size of just 0.91 × 0.95 nm and diagonal of 1.62 nm, the peaks are very wide and many peaks are missing because they merge with neighbor peaks. Only one peak is present in SAXS angle range. The peak of the largest intensity (101) is observed at 2θ = 24.3°. Two other prominent peaks are (004) at 36.6° and peak resulting from merging peaks of (200), (105), and (211) at 51.1°.

specific features as satellite peaks located equidistantly at larger and smaller diffraction angles around XRD peaks. The satellite peaks are superimposed with size-dependent diffraction fringes of the XRD peak. Table S1 lists positions of SAXS and XRD peaks for cubic copper nanoparticles as well as interpeak distance for small angle diffraction. The position of the (111) peak changes irregularly with an increase in Cu nanoparticles size: first it decreases from 43.5 to 42.9° for nanoparticle with edge of 3.5a and then it increases up to the steady value of 43.3° reached for nanoparticle with edge of 5.5a = 1.99 nm. The final value of 2θ for (111) peak agrees well with the theoretically calculated value of 43.35°. The initial value of 2θ = 43.5° exceeding the theoretical value obtained for (111) interplanar spacing could be explained by superposition of (111) and (200) peaks. However, the decrease of 2θ below theoretical and its further increase cannot be connected with XRD peaks superposition. The peak for planes (200) appears only at size of 3.5a. This peak shifts from 50.0 to higher angles to reach its final value of 50.5° for edge length of 7.5a = 2.71 nm. This shift is consistent with superposition of (111) and (200) peaks. The final steady value of 2θ agrees with the theoretical value of 50.49° within the error of computation. The diffraction angle for the (220) peak undergoes fluctuations when the cubic copper nanoparticles size increases: it increases from 72.9 to 74.1°, then decreases to 74.0, increases to 74.2, decreases to 74.1 and reaches its final steady value of 74.2° for nanoparticles with edge length of 6.5a = 2.35 nm. These periodic fluctuations of the XRD peak position are inconsistent with the traditionally used Bragg’s and Scherrer equations. To get an idea for the possible causes of XRD peak position fluctuations, diffraction fringes are considered. These fringes are observed for small angle range as well as around each of lattice diffraction peaks. The interpeak distance for the diffraction fringes is approximately same for the small angle range and around each of lattice diffraction peaks. Table S1 gives the positions of the first four SAXS peaks. The distance between the neighbor SAXS peaks is approximately same for all SAXS peaks of the given sample. The interpeak separation decreases steadily as the particles size increases. For very small copper nanoparticles, these peaks of diffraction fringes give a D

DOI: 10.1021/acs.iecr.7b04480 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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peak is at 25.0°. Only one peak related to diffraction fringes from main XRD peaks is observed and it is located between (101) and (004) peaks where there is enough space for this peak to appear. XRD patterns of nanoparticles Ti883c with size 2.84 × 2.81 nm and main diagonal of 4.90 nm demonstrate all major XRD peaks, multiple SAXS peaks and diffraction fringes peaks. These SAXS peaks and peaks around main XRD peaks are called thickness fringes. The interpeak distance of these multiple small peaks is related with the particles thickness: the smaller the thickness, the larger the interpeak distance. Diffractograms of the larger anatase nanoparticles Ti10104c and Ti13135 continue to demonstrate increased peak sharpness due to the increase in size. An increased number of small peaks in SAXS range and in thickness fringes around XRD peaks are observed. Table S2 summarizes peak positions in diffraction patterns of cubic anatase nanoparticles. There is a significant difference in diffraction angles of all major XRD peaks as the size of anatase nanoparticles increases. The largest change is observed for peaks (101), for which the diffraction angle increases from 24.3 to 25.3°, (004) with an increase from 36.6 to 37.8°, (204) with a decrease from 63.4 to 62.7°, and (215) with an increase from 71.8 to 75.2°. It should be emphasized that these differences are not related to any changes in cell or lattice parameters or any defects because all the nanoparticles have ideal lattice and no defects. For large diffraction angle peaks, the change in peak diffraction angle can be related to different overlap of the merging neighbor peaks because large angle peaks all have neighbor peaks or are formed by merging two or three peaks of similar intensity. However, peaks (101) and (004) stay separately from other peaks and therefore cannot change their location because of overlap with other diffraction peaks. A possible cause for the shift of peak positions can be overlap with thickness fringes peaks of the neighbor XRD peaks. This effect was clearly observed for diffraction patterns of copper nanoparticles, which have a large separation of XRD peaks. Gray and Wilson also observed shift of anatase diffraction peaks to lower angles when the size of anatase nanoparticles decreased from 4 to 1 nm.29 Rietveld refinement of these

Figure 3. External view of TiO2 anatase nanoparticles of cubic shape made with experimental anatase cell parameters designated as Ti331c, Ti552c, Ti883c, Ti10104c, and Ti13135c.

Figure 4. XRD patterns of cubic TiO2 anatase nanoparticles shown in Figure 4. From bottom to top: Ti331c, Ti552c, Ti883c, Ti10104c, and Ti13135c.

For the next nanocluster Ti552c with size of 1.70 × 1.86 and main diagonal 3.04 nm, already all major XRD peaks are present and three SAXS peaks are observed. The XRD peaks are wide and many peaks partially overlap. The tallest (101)

Figure 5. (A) SAXS interpeak spacing as a function of Ti−Ti RDF diameter of cubic TiO2 anatase nanoparticles. Best approximating curve y = a/x + b, a = 122 ± 4, b = 0.31 ± 0.09. (B) First SAXS peak position against cubic anatase nanoparticles Ti−Ti RDF diameter. Approximating curve y = a/x + b, a = 157 ± 6, b = 0.86 ± 0.21. E

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is maintained by additions of hydroxyl groups or oxygen atoms to titanium atoms having uncompensated charge. For the smallest decahedral anatase nanoparticles Ti22rX, only addition of hydroxyl groups was used because oxygen atoms would create too strongly distorted structures. Hydroxyl groups are either attached to the edges between facets (001) and (101) only to give Ti22r1 cluster, or to both edges (001)/ (101) and vertices between four adjacent (101) facets to give Ti22r2 and Ti22r3 clusters. Ti33rX nanoparticles have hydroxyl groups attached to (001)/(101) edges only for Ti33r1, or hydroxyl groups attached to both (001)/(101) edges and (101)/(101) vertices for Ti33r2 and Ti33r3. Oxygen atoms are used for (001)/(101) edges only in Ti33r4 and for all (101)/(101) vertices only in Ti33r5 cluster. Ti44rX nanoclusters have seven different versions of distribution of hydroxyl groups and oxygen atoms. For Ti44r1, hydroxyl groups are attached to (001)/(101) edges only, for Ti44r2 and Ti44r3, hydroxyl groups are placed in all (101)/(101) vertices and to different Ti atoms at (001)/(101) edges. In Ti44r4 cluster, four hydroxyl groups are placed in (101)/(101) vertices, whereas four oxygen atoms are at Ti atoms of (001)/(101) edges. Six oxygen atoms are attached to different titanium atoms of (001)/(101) edges and (101)/ (101) vertices for clusters Ti44r5, Ti44r6, and Ti44r7. Figure 6 demonstrates X-ray diffractograms for unoptimized decahedral anatase nanoparticles that are shown in Figure S2.

DSE generated diffractograms produced an increasing trend in cell parameters a and c as the size of NP decreased with a = 3.89 and c = 9.605 for 1 nm NP. Good fitting could be obtained for 4 nm NP and Rietveld generated curves for 1−3 nm NP deviated from DSE generated patterns. Obviously, cell parameters produced by Rietveld refinement for NP with size below 4 nm deviates from true values and needs to be corrected by some theoretical models. The change in XRD peaks positions is observed for cubic anatase nanoparticles of all sizes and the peaks diffraction angles become equal to those of bulk anatase only for Ti13135c nanoparticles with size of 4.73 × 4.72 nm and main diagonal of 8.19 nm in agreement with published data.29 The observed changes in diffraction angle should be kept in mind when reporting structure changes for small nanoparticles deduced from changes in XRD peaks positions for small nanoparticles. Such changes in peak positions may be not due to structure change but mainly due to diffraction effects of small nanoparticles. The number and location of small-angle X-ray scattering peaks changes as the size of cubic anatase nanoparticles increases. It can be noted that the interpeak spacing of SAXS peaks is approximately same for all SAXS peaks of a given sample. Therefore, interpeak distance can be averaged for all the SAXS peaks and the mean value be used for estimating particles size. Figure 5A shows SAXS interpeak spacing as a function of cubic anatase nanoparticles diagonal length. The data points are well approximated by the reciprocal function y = a/x + b that can be used to estimate the size of nanoparticles from experimental values of SAXS interpeak spacing. Parameters of the reciprocal function are given in caption to Figure 5A. The position of the first SAXS peak also undergoes shift when the particles size changes. Figure 5B shows first SAXS peak angle as a function of particles diagonal length for different cubic anatase nanoparticles. The data points are well-fitted to the reciprocal function, the parameters of which are given in the Figure 5B caption. Because very small anatase nanoparticles display only one SAXS peak, its position can be used for estimating particles size from experimentally observed first peak position for the smallest anatase nanoparticles. Real anatase crystals expose (001) and (101) facets if special surface energy modification agents or procedures were not used in their preparation. The influence of both factors, size of anatase nanoparticles under constant ideal lattice structure and structure relaxation, are investigated using a series of decahedral anatase nanoparticles. The number of atoms in such nanoparticles is limited by the available computational capabilities to approximately 900 because larger nanoparticles structure optimization takes more than 2 days and ia subject to errors accumulated during such long runs. Figure S2 shows decahedral anatase nanoparticles constructed using cell parameters obtained with scc-dftb method. The cell parameters of real anatase were not used because this would preclude the study of structure relaxation effect on XRD and SAXS patterns. The details of the clusters creation procedure are given in Experimental section. Four sizes of nanoparticles are investigated with 2 × 2, 3 × 3, 4 × 4 and 5 × 5 anatase cells in directions a and b. These decahedral nanoparticles contain 5, 7, 7, and 9 layers of Ti atoms in anatase lattice direction c, so that approximately equal size of nanoparticles in all directions is attained. Charge neutrality and stoichiometry of nanoparticles

Figure 6. XRD patterns for unoptimized decahedral anatase TiO2 nanoparticles. Coinciding diffractograms from bottom to top of increasing intensity are Ti22r1 − Ti22r2, Ti33r1 − Ti33r5, Ti44r1 − Ti44r7, and Ti55r2 − Ti55r5.

These nanoparticles have lattice parameters obtained from sccdftb method optimization of infinite periodic anatase supercells. Therefore, diffraction angles for peaks of these nanoparticles are somewhat different from those for real anatase. It can be noticed that diffraction patterns for different versions of nanoclusters with the same size mostly coincide. Only peaks corresponding to reflections (101), (004) and (105)+(211) are observed and the last two peaks merges almost completely for Ti22r1 and Ti22r2 nanoparticles. For nanoparticles Ti33rX and F

DOI: 10.1021/acs.iecr.7b04480 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 7. Decahedral TiO2 anatase nanoparticles fully optimized with scc-dftb. Designations of nanoparticles: (A) Ti22r1, (B) Ti22r2, (C) Ti33r1, (D) Ti33r2, (E) Ti33r3, (F) Ti33r4, (G) Ti33r5; (H) Ti44r1, (I) Ti44r2, (J) Ti44r3, (K) Ti44r4, (L) Ti44r5, (M) Ti44r6, (N) Ti44r7; (O) Ti55r2, (P) Ti55r3, (Q) Ti55r4, (R) Ti55r5.

Ti44rX, only peak corresponding to (116)+(220) reflections is not resolved, whereas for Ti55rX nanoparticles, all eight major XRD peaks are observed in the range of diffraction angles of 20 to 90°.

Table S3 summarizes peak positions in diffractograms of unoptimized decahedral nanoparticles shown in Figure S2. When considering changes in XRD peak positions among series of nanoparticles with the same size, it can be found out that the G

DOI: 10.1021/acs.iecr.7b04480 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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undergoes little deformation. This is in agreement with reported experimental results of crystalline core−amorphous shell structure of 2 nm TiO2 nanoparticles.42 The pronounced distortions of atomic layers near the surface of decahedral anatase nanoparticles is expected to give rise to changes in XRD and SAXS patterns of the nanoparticles. Figure 8 shows the diffractograms obtained using Debye equation for

largest differences are for Ti22rX nanoparticles that have 0.2 to 0.4° difference. For Ti33rX, the highest 0.3° difference is present for (204) peak. For other nanoparticles, 0.1 to 0.2° difference is present. Much larger differences in XRD peaks position can be observed for nanoparticles of different sizes. When going from Ti22rX to Ti33rX nanoparticles, + 0.7 to +0.9° change is present for (101) peak, −0.1 to −0.6° for (004) and +1.8 to +2.2° for (105)+(211) peak. Increase of nanoparticles size from Ti33rX to Ti44rX results in +0.4° shift of (101) peak, + 0.1 to +0.3° shift for (004), + 0.8 to 0.9° shift for (105)+(211), −0.2 to −0.4° shift for (204), and +0.9 to +1.2° shift for (215) peak. Smaller changes are observed for the transfer from Ti44rX to Ti55rX that are 0.1 to 0.2° mostly but reaching +0.5 to 0.7° for the (215) peak. Considering data in Table S3, it can be concluded that as large peak shifts are present as +1.5° for (101) and +1.8° for (215) peak when the size of decahedral anatase nanoparticles increases from 1.1 to 2.7 nm. No charge in the crystal structure is present for these nanoparticles. Therefore, care should be taken in assignment of changes in XRD peak position of anatase nanoparticles with nanoparticles size increase to changes in crystal structure cell parameters and defects. In a comparison of XRD peak position for decahedral anatase nanoparticles (Table S3) and cubic nanoparticles (Table S2), it can be noticed that equally strong shifts are observed. For cubic nanoparticles, the changes were +1.0° for (101), + 1.2° for (004), −0.7° for (204), and +3.4° for (215) peak when the diameter of nanoparticles changed from 1.6 to 8.2 nm. The largest shift is observed for (215) reflection for both cubic and decahedral anatase nanoparticles. Figure S3 shows distribution of interatomic distances for decahedral nanoparticles with unoptimized structure shown in Figure S2. The smallest interatomic distance of about 0.96 Å corresponds to O−H bonds. There are 13 prominent interatomic distances in the range from 0 to 5 Å. The O−O interatomic distances are 2.52, 2.77, 3.01, 3.57, 3.79, 4.00, 4.55, and 4.84 Å. Ti−Ti interatomic distances are 3.01, 3.79, and 4.84 Å. Finally, Ti−O distances 1.92−1.93 and 2.00 Å correspond to bonds in TiO6 distorted octahedrons, the rest of Ti−O distances are 3.79, 4.25, 4.55, and 4.79 Å. The interatomic distance peaks are sharp as expected for highly crystalline material. Optimization of decahedral anatase nanoparticles with sccdftb method results in nanoparticle structures depicted in Figure 7. Compared to unoptimized nanoparticles shown in Figure S2, the optimized nanoparticles have distortions of crystalline structure that are most pronounced for smallest nanoparticles of series Ti22rX and for atoms near (101)/(101) vertices if such vertices do not have charge compensating hydroxyl group or oxygen atom attached to titanium atom. The structure of (001) facets also undergoes distortion that results in compression of this facet in one direction while leaving it almost unchanged in the perpendicular direction. Strong lattice deformations are also observed at titanium atoms without needed compensating charge group that is observed for nanoparticles Ti22r2 (Figure 7B), Ti33r4 (Figure 7G), Ti44r4 (Figure 7L), Ti44r5 (Figure 7M), Ti44r6 (Figure 7N), Ti44r7 (Figure 7O), Ti55r2 (Figure 7P). The structure of (101) facets is also distrorted somewhat: there is a curvature that is most pronounced near the edges between adjacent (101) facets. It can be noticed that while several external layers of atoms undergo deformation, there is an internal core that

Figure 8. XRD patterns for decahedral nanoparticles fully optimized with dftb. The almost coinciding diffractograms correspond to nanoparticles series from low to high intensity: Ti22rX, Ti33rX, Ti44rX, and Ti55rX.

all the optimized decahedral anatase nanoparticles. The almost coinciding patterns correspond to nanoparticles of the same size series. The diffraction patterns of the nanoparticles of series Ti22rX undergo strong changes as a result of their geometry relaxation. Diffraction peak of (004) disappears while the peak of (105)+(211) undergoes strong shift. A pronounced shoulder peak with 2θ ≈ 33° appear. Strong changes in XRD patterns appear also for Ti33rX nanoparticles. The peak for reflection (004) appear as a shoulder only. Peak of (105)+(211) is shifted by −0.4 to +0.2°, (204) by +0.8 to +1.4°, and (215) by −0.5 to −1.7°. For nanoparticles Ti44rX, the (004) peak is shifted by −0.4 to +0.2°, (200) by +0.3 to +0.5°, (204) by +0.6 to +0.7°, and for (215) by −3.1 to −3.4° compared to unoptimized nanoparticles. The changes become smaller for nanoparticles Ti55rX reaching −0.3° for peak (004), −0.4° for (215) and +0.5° for (224). Interestingly, the position of the peak (101) changes at most by +0.1 to +0.2° for all the optimized nanoparticles. Table S4 summarizes changes in positions of XRD peaks for all decahedral anatase nanoparticles of the present study. It is possible to compare the magnitude of effects of nanoparticles size and structural changes on XRD patterns using data in Tables S3 and S4. The position of the (101) peak changes by +1.4° as a result of the nanoparticle size increase, whereas the effect of crystalline lattice deformation gives no more than a +0.2° contribution. For the diffraction peak (004), the quantitative effect of nanoparticles size increase is from 0 to −0.4°, whereas the contribution of crystal structure deformation is −0.4 to +0.2°, which means that both factors give a small effect, with approximately the same magnitude of contributions H

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Figure 9. (A) Dependence of interpeak distance in SAXS peaks on Ti−Ti RDF diameter of decahedral TiO2 anatase nanoparticles that were not optimized and were fully optimized with dftb. Fitted curve y = a/x + b, a = 75.9 ± 1.5, b = 1.37 ± 0.08. (B) First SAXS peak against Ti−Ti RDF diameter of decahedral anatase nanoparticles. Fitted curve y = a/x + b, a = 116 ± 2, b = 2.4 ± 0.1.

Figure 10. Interatomic distance distribution function for fully optimized with scc-dftb decahedral anatase nanoparticles: (A) from bottom to top Ti22r1, Ti22r2, Ti33r1, Ti33r2, Ti33r3, Ti33r4, Ti33r5; (B) from bottom to top Ti44r1, Ti44r2, Ti44r3, Ti44r4, Ti44r5, Ti44r6, Ti44r7, Ti55r2, Ti55r3, Ti55r4, Ti55r5.

absence of a theory that can describe the size effects in a quantitative analytical form. SAXS peak positions are given in Tables S2 and S4 for unoptimized and optimized decahedral anatase nanoparticles. The change in SAXS peaks position is within ±0.1° as a result of surface groups, size and lattice deformation effects. Therefore, the data for unoptimized and optimized nanoparticles can be considered together for producing the correlations of SAXS peak position and particles size. Figure 9A shows how nanoparticle diameter changes interpeak spacing in SAXS range. The effect is described by reciprocal function y = a/x + b and the obtained parameters are listed in Figure 9 caption. The first SAXS peak position as a function of decahedral anatase nanoparticles size is plotted in Figure 9B. The points are well-fitted to reciprocal function that can be

for the (200) peak. The position of (105)+(211) peak changes by +3.1° as a result of size increase and by −0.4° as a result of structure change except for the cluster Ti22r2 that has +1.3° contribution from structure change. The position of (204) peak does not change significantly as a result of nanoparticles size increase but undergoes up to +1.4° shift as a result of crystal structure deformation. The position of peak (215) changes considerably following both size and structure change but this peak is strongly merged with peaks (116)+(220). Consideration of the peak position changes as a result of nanoparticles size increase and crystal lattice deformation leads to conclusion that both factors can give as large effect as peaks positions changes by a few degrees. Although the contribution of the structure change is discussed often in the literature, the contribution of the size to XRD peaks positions is not taken into account. This is due to the I

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Industrial & Engineering Chemistry Research used for estimation of decahedral nanoparticles size from experimental position of the first SAXS peak. The cumulative changes in structure of decahedral anatase nanoparticles as a result of their size and surface groups change can be expressed as changes in interatomic distances distribution. Figure 10 shows number of interatomic distances per distance interval for all decahedral anatase nanoparticles. There are significant changes in interatomic distance distribution compared to anatase structure as comparison with Figure S3 reveals. The smallest interatomic distance 0.98 Å corresponds to O−H bonds in nanoparticles containing surface hydroxyl groups. Sharp peaks in the distribution function are seen for Ti−O distances in the range of 1.7 to 2.1 Å, whereas peaks with distance over 2.3 Å are smeared in Figure 10. The large “noise” in distribution function for distances above 2.5 Å is caused by a diversity of interatomic distances in relaxed nanoparticles. Structure changes take place in several layers of atoms closest to the surface of nanoparticles. Their contribution decreases when the size of nanoparticles increases and the “noise” becomes smaller in Figure 10B for larger nanoparticles. In initial scc-dftb anatase lattice, 1.92 Å is the length of lateral Ti−O bonds and 2.00 Å is the length of vertical Ti−O bonds. The 2:1 ratio in the number of distances is in accord with four oxygen atoms in lateral coordination locations and two in vertical. In the nanoparticles, these interatomic distances are also present in different proportions but never in a 2:1 ratio. Interatomic distance 1.8 Å appear in different intensity. This distance corresponds to Ti−O bonds in (101) and (001) surface and subsurface space. The number of these bonds is higher for nanoparticles with fully hydroxylated edges between facets (001) and (101): Ti33r1, Ti44r1, Ti55r4. There are also a small number of 2.12 Å Ti−O bonds present on the (101) surface. The location of charge-compensating groups has a marked influence on shift of XRD peaks for smaller decahedral anatase nanoparticles. For Ti33r1 nanoparticle with hydroxyl groups at (001)/(101) edge only, peaks (004) and (105)+(211) are shifted by −0.4°. For nanoparticles Ti33r2 and Ti33r3 with hydroxyl groups located at (001)/(101) edges and (101)/ (101) vertices, these peaks are shifted by +0.1 and 0.3°. Placement of oxygen atoms at (001)/(101) edges also results in the shift of (004) and (105)+(211) peaks to smaller angles, while placement of oxygen atoms at (101)/(101) vertices makes the shift of these peaks positive. The same tendencies in XRD peak shifts are observed for nanoparticles of series Ti44rX. For nanoparticles of the series Ti55rX, the shift in XRD peak positions becomes small and similar for all distributions of hydroxyl groups or oxygen atoms over the nanoparticles surface. Obviously, the influence of surface groups became small for these nanoparticles because the number of atoms over facets and in the bulk of nanoparticles is much higher than the number of edge atoms.

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2.7 nm. The changes are possibly due to the interference fringes summation with the main XRD peaks. 2. For cubic TiO2 anatase nanoparticles with ideal structure, XRD peak positions changes with an increase in size from 1.4 to 8.2 nm probably due to the peaks interference effect. 3. For decahedral TiO2 anatase nanoparticles with ideal structure, the XRD patterns change in the whole range of nanoparticles sizes studied of 1.1 to 2.7 nm. Structure changes resulting from size effect give rise to XRD peak shifts comparable in magnitude to peak shifts due to size itself. Structure changes resulting from different distribution of surface hydroxyl groups and oxygen atoms give rise to XRD peak shifts up to 1.4° but are generally less important factor than size itself and lattice deformation due to nanoparticle size. 4. SAXS interpeak spacing and first SAXS location are reciprocal function of nanoparticles size and can be used for estimation of nanoparticles size. The parameters of the reciprocal function are different for different materials and particles shape.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.7b04480. External view of metallic copper cubic nanoparticles (Figure S1), SAXS and XRD peak positions for cubic Cu0 nanoparticles (Table S1) and cubic anatase nanoparticles (Table S2), unoptimized decahedral anatase nanoparticles with different size and location of surface groups (Figure S2), peak positions in XRD and SAXS of decahedral unoptimized NP (Table S3) and optimized NP (Table S4), distribution of interatomic distances for decahedral unoptimized NP (Figure S3) (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel. 7 953 785 5528. ORCID

Alexander V. Vorontsov: 0000-0002-2791-3278 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The work was supported with RFBR grant 15-08-01936. REFERENCES

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4. CONCLUSIONS The effects of the size and structure on XRD and SAXS patters were investigated for cubic metallic copper nanoparticles, cubic TiO2 anatase nanoparticles and decahedral anatase nanoparticles with different distribution of surface hydroxyl groups and oxygen atoms. 1. For cubic copper nanoparticles with ideal structure, the positions or three XRD peaks (111), (200), and (220) changes as the size of nanoparticles increases from 0.5 to J

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