XRD ANISOTROPIC BROADENING OF NANO-CRYSTALLITES

91 92 XRD ANISOTROPIC BROADENING OF NANO-CRYSTALLITES Yu Wang 1, Sammy Lap Ip Chan 2, Rose Amal 3, Yan Rong Shen 2, Kunlanan Kiatkittipong 3 ABSTRACT A physical ellipsoid model has been established to address the influence of nano-crystallite size on XRD anisotropic peak broadening. The model shows that, for non-spherical nanocrystallite, its XRD peak broadening becomes anisotropic and orientation (hkl) dependent. In such a case, the crystallite size must be described by multi-dimensional lengths, possible with a polar orientation. Experiments on -Ni(OH) 2, -TiO 2 and anatase were carried. Results show that the model can explain the characteristics of nano-crystallies with different sizes and shapes. I I. INTRODUCTION Nano-crystallites have recently become the focus of attention in many applications due to their specific properties. This is particularly true for non-spherical crystallites, which possess high specific surface area. X-ray diffraction and Rietveld refinement are major methods used to characterise these crystallites. However, anisotropic peak broadening, which the peak width cannot be described as one polynomial function of 2tan(2 (Cagliotti formula), has been noticed in some occasions. Without proper rectification of the anisotropic peak broadening, Rietveld refinement and crystallite size analysis will not be able to accurately determine the actual size of the crystallites. In 1939 Patterson has calculated Scherrer constant (K) for all particle shapes along various directions with rigorous kinematical diffraction theory (Patterson, 1939). Recently there have been many attempts to dealing explicitly with nanoparticle diffraction (Scardi and Snyder et al., 2010; Ungar, 2003). Two Mark Wainwright Analytical Centre, University of New South Wales, Australia; School of Materials Science, University of New South Wales, Australia; School of Chemical Science and Engineering, University of New South Wales, Australia

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Copyright JCPDS-International Centre for Diffraction Data 2011 ISSN 1097-0002 92 93 phenomenological models are weell-known to treat anisotropic peak broadening iin Rietveld refinement. Stephens model (Stepphens, 1999) corrects metric parameters of recipprocal lattice, based on the fact that each crystaallite has its individual parameters, with multi-diimensional distribution. The spherical harmo onics model (Popa, 1992, 1998) is base on the voolumeaverage-column length, which is invariant to the Laue group operations, and thenn it can be expanded to a series of symmetriised spherical harmonics (h). Both models are comprehensive to handling the caase. However the physical interpretation can be varied, such as crystallite size or microstrain. II. ELLIPSOID MODEL OF NANO-CRYSTALLITE N The physical ellipsoid model illuustrates the influence of crystallite size on anisottropic peak broadening. It intends to interpreet the physical nature without complicated matheematic expression. Assuming there is on ne crystallite with ellipsoidal shape, where in sphherical coordinates this crystallite is exp pressed mathematically as: x a sin cos y b sin sin z c cos (1) Here is the azimuth angle, iss the inclination angle; a, b and c are dimensionaal radii in orthogonal directions. To simpliffy the discussion, the inclination angle () is set to zero and a b. Thus the ellipsoid can be draawn as Fig 1 (a). In Fig. 1 the bold black lines reprresent a and c lengths. Under diffraction plane, tthe ellipsoidal polar axis is aligned w with the normal of the plane. The thin lines paralllel to diffraction plane represent crystaal atomic planes with particular Miller index. Thhe correlation of a, b and c govern the shape off the ellipsoid, that:

93 94 when a b; when a b; when a b; c a c a c a needle sphere platelet (2) If a, b and c are completely diverse, it produces a large variety of the crystallite shapes. Once a dimensional radius is significantly larger or small than the others, the direction is defined as the polar orientation of the crystallite, and indicates that the crystallite possesses a high aspect shape ratio. The ellipsoid model represents polycrystalline status by rotating the ellipsoid in plane or out plane, asshowsinfig.1(b). When another (hkl) plane meets the Bragg condition, there will be another set of d-spacing and With the same operation, all (hkl) diffractions can be obtained. Rotating angles ( and ) are calculated from crystallographic data. Under this status, different dimensional radii represent their respective overall average. XRD diffraction peak broadening is well known from Scherrer formula: B size L size K cos. The peak width (B size ), the contribution from size broadening, is inversely proportional to crystallite size (L size ). The L size is physically the diameter of ellipsoidal crystallite along the normal of the diffraction plane; it is called dimensional length, as shown the bold blue line in Fig. 1 (a) & (b). In the ellipsoid model, this length is expressed as: L size 2 2 2 2 2 1/ 2 x y z 2 2 2 2 2 2 2 2 a sin cos b sin sin c cos 1/ 2 (3) Therefore the Scherrer formula can be rewritten as: B sizehkl 2 K 2 2 2 2 2 2 2 2 2 a sin cos b sin sin c cos 1/ cos (4) Here the modified equation reveals that peak broadening is not only dependent on the incident angle but also on the dimensional radii (a, b and c) and orientation of crystallite ( and ), which are related to (hkl), as illustrated by the difference of the bold blue lines in Figs. 1 (a) &(b). From the ellipsoid model we can derive that: 1. Size analysis of nonspherical nano-crystallites must determinate three dimensional lengths (diameters of the ellipsoid) in orthogonal directions, which fully describe the size and shape of the crystallites;

94 95 2. Due to non-spherical shape, their XRD peak broadening appear to be anisotropic and (h kl) dependent; 3. Characterisation of non-spherical nano-crystallies requires a full XRD pattern analysis, measuring at least three FWHMs of the peaks from orthogonal directions of the crystal structure. In Rietveld refinement, a (hkl) dependent rule has to be employed, apart from the Cagliotti formula. III. EXPERIMENTS Three types of specimens were prepared with diverse synthetic process in order to obtain different microstructure characteristics: nickel hydroxide -Ni(OH) 2 and titanium dioxide (- TiO 2 and anatase). Nickel hydroxide was synthesized by direct precipitation from 1 M solution of NiSO 4 6H 2 Owith2MNaOHand25%NH 4 OH at 70, washed and dried out in oven. For synthetic TiO 2 samples, a commercial titanium dioxide P25 was mixed with 10 M NaOH. The slurry was transferred to a Teflon-lined autoclave, and then hydrothermally treated at 200 HCl. After centrifuging and drying, the samples were annealed at a temperature of 500 or 700 Heating and cooling have been well controlled to minimise stresses induced. XRD patterns were collected using Panalytical MPD with Cu K=1.5418 ) radiation and Pixcel detector. Measurement conditions were 45 kv and 40 ma, step size (2) 0.013 Instrument broadening was determined using standard silicon (NIST SRM 640c). Due to severe peak overlapping, Rietveld refinement, with program of Brukers Topas-4, was used for XRD pattern analysis. Topas-4 provides two refinement modes, GUI and Launch, the latter allows user to input his own programming codes for specific refinement purpose. Initial refinement was run in GUI mode to refine lattice parameters and preferred orientation and to correct instrument broadening. After this the result was output to a x.inp file. The jedit program was used to add extra codes for rectification of the anisotropic peak broadening and characterization of nano-crystallites, where codes from Stephenwere employed. The volume weighted column height (LVol-IB), which is predefined in Topas-4, was used as the dimensional length. Finally the new x.inp file was run in the Launch mode. To verify the Rietveld refinement results, Hitachi 4500II scanning electron microscopy (SEM) were used to observe the morphology of different specimens. Philips CM200 field emission gun

95 96 transmission electron microscope (TEM), operating at 200kV, was used to measure dimensional lengths of crystallites and determine their orientations (electron diffraction). VI. RESULTS AND DISCUSSION 4.1 Nickel Hydroxide Ni(OH) 2 Fig 2 shows the Rietveld refinement result for -Ni(OH) 2, where the blue dot line is XRD data and red line is Rietveld fit profile. Criteria of the fit, obtained from the refinement, are 80,000 Rexp 0.56, Rwp = 1.73; and GOF=3.09. Without Theophrastite 100.00 % 70,000 anisotropic broadening rectification, the data are 60,000 Rexp 0.57, Rwp 7.98 and GOF 14.25. According 50,000 to ellipsoid model, the dimensional lengths 40,000 30,000 (LVol-IB) in orthogonal directions, (h 00)(0k0) 20,000 and (0 0 l), were calculated. Results are listed in 10,000 Table 1, where the standard deviation ( 0 collected from 6 repeated calculations with 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 different initial values of Stephen Fig. 2 Ni(OH) 2 Rietveld Result parameters and the influence of microstrain was not considered. Directions (h 00)(nm) (0k 0) (nm) (0 0 l)(nm) LVol-IB 149 184 12.4 4.8 4.5 9.2 The length of the (0 0 l) direction is at least 90% shorter than the other directions and there is a limited difference between the other two dimensions. It indicates that the Ni(OH) 2 possesses platelet shape with (0 0 1) as the polar orientation (in ellipsoid model, 2c=12.4 nm). TEM image (Fig. 3) reveals Ni(OH) 2 crystals as hexagonal platelets, not round shape. Electron diffraction patterns of Ni(OH) 2 in Fig. 4 show (0 0 1) direction even if there is interference from polycrystalline Laue ring, which conform the findings in XRD Rietveld refinement. This study also agrees with previous work (Casas-Cabanas et al., 2006)

Copyright JCPDS-International Centre for Diffraction Data 2011 ISSN 1097-0002 96 97! " # # $ %!!% $ 4.2 Beta-Titanium Dioxides (TiO2) Further study is on synthetic TiO2. When the annealing temperature was 500 the -TiO2 (PDF 35-0088) has been identified. The pattern shows only one sharp peak. The others are broadening and significantly overlapping (Fig. 5). The sharp peak is (0 2 0) diffraction. SEM reveals the crystallites are in nano range and appear to resemble rod shape, see Fig. 6. Beta-Tio2 100.00% 20,000 18,000 16,000 14,000 12,000 10,000 8,000 6,000 4,000 2,000 0-2,000-4,000 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 & '( ) %! * +, -! * + Again using the Rietveld refinement to characterise these nano-crystallites and the results are given in Table 2. The values of the dimensional length are completely different in the three directions. -. / % $ %! * + Directions LVol-IB! (h 0 0) (nm) 22 (0 k 0) (nm) 468 (0 0 l) (nm) 113 10.2 6.8 7.2

Copyright JCPDS-International Centre for Diffraction Data 2011 ISSN 1097-0002 97 98 Table 2 indicates that -TiO2 is in form of a thin rectangular sheet (or nano-ribbon). At the bottom left corner of TEM image (Fig. 7), the sheets split from a rod can be found. The electron diffraction, collected from a point on the middle rod, shows (2 0 0), that is the orientation of the shortest dimensional length. It says the rod in Fig. 6 may be the result of a bunch of the sheets stacked together. Fig. 7 TEM Image of Beta-TiO2 Fig. 8 (2 0 0) Electron Diffraction Pattern 4.3 Anatase (TiO2) When the annealing temperature was increased to 700 Fig. 9 gives XRD pattern of pure hexagonal anatase (TiO2). Its FWHM values of the peaks appear to be gradually increased via 2, no anisotropic broadening. SEM shows morphologic feature is still rod type, but the rod appears barrelled and sort of sphere-like. Fig. 9 XRD Pattern of Anatase Annealed at 700C Fig. 10 SEM Image of the Anatase

98 99 Table 3 Size Characterisation Result for Anatase Annealed 700C Directions (h 00)(nm) (0k 0) (nm) (0 0 l)(nm) LVol-IB 134 102 120 5.1 6.4 4.6 The result of dimensional lengths (Table 3) shows minor difference among the three. This indicates the anatase obtained from 700C annealing is spherical shape with no polar orientation. TEM image (Fig. 11) shows dark bands across the rod, which are the grain boundaries to break TiO 2 rods into spheres. This explains the inconsistence between Fig. 10 and characterisation result (Table 3). A comparison between crystallite sizes obtained from Fig. 11 TEM Image of the Anatase TEM and Table 3 has been carried out. With 232 TiO 2 particles measured from TEM images the average diameter of the TiO 2 was 110 nm. It was generally in fair agreement with those calculated in Table 3. Through three different samples, it is clear the ellipsoid model and program used are capable to characterise non-spherical nano-crystallites. It also shows that size/shape analysis on XRD peak broadening is very sensitive to crystallite in the nano-range size, even better than SEM observation alone. In principle it is not necessary to use Rietveld refinement for this analysis. With a full range XRD pattern, together with material crystallographic data, the formula (4) of the ellipsoid model can be used to calculate individual peak broadening and gives size and shape of the nano-crystallite. However, in most practice, Rietveld refinement was employed because of severe peak overlap and complex crystal structures. In case of the above three samples the dimensional lengths were calculated in (h 0 0), (0 k 0) and (0 0 l) directions. It does not mean orientation of the lengths is always aligned with the principle axis of the crystal unit cell. However, it is reasonable that they are normally with low order of Miller indexes. IV. CONCLUSION

100 99 The ellipsoid model shows that non-spherical nano-crystallite is a reason of XRD anisotropic peak broadening. It points out that size analysis of non-spherical nano-crystallite requires determination of at least three dimensional lengths in orthogonal directions. After rectification of anisotropic peak broadening, Rietveld refinement is valid to characterise size variety of non-spherical nano-crystallites. V. ACKNOWLEDGEMENT We are grateful to Prof. Stephens, P. W., Department of Physics and Astronomy, Stony Brook University, for providing some programming codes. We also thank Dr. Kong, C., Electron Microscopy Unit, University of New South Wales, for his assistance on SEM and TEM work. REFERENCES 1. Casas-Cabanas, M. et. al., (2006). characterisation of nickel hydroxides and correlation with electrochemical properties J. Mater. Chem, 16, 2925-2939. 2. Patterson, A. L. (1939). -rays by small crystalline particles Phys. Rev 56, 972-982. 3. Popa, N. C. (1992). Appl. Cryst 31, 176-180. 4. Popa, N. C. (1998). hkl) dependence of diffraction-line broadening caused by strain and size for all Laue group in Rietveld refinement, J.Appl.Cryst31, 176-180. 5. Scardi, P., Snyder, R. et. al., (2010). crystallites90, 3891-3905. 6. Stephens, P. W. (1999). model of anisotropic peak broadening in powder diffractionj. Appl. Cryst. 32 281-289. 7. Ungar, T., (2003). -ray diffraction peaks5. 323-329.