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Lepidocrocite-Type Titanate Formation from Isostructural Prestructures under Hydrothermal Reactions: Observation by Synchrotron X-ray Total Scattering Analyses

The formation of titanium dioxides, such as rutile and anatase, is known to proceed through the formation of a lepidocrocite-type layered structure under hydrothermal conditions, but the nucleation of this intermediate is still not understood well. Here, the nucleation of lepidocrocite-type layered titanates under hydrothermal conditions is observed by tracking the structural changes by in situ time-resolved pair distribution function analyses. We found that titanate clusters or corrugated layered prestructures having <1 nm domains with lepidocrocite-type connectivity were formed even before thermal treatment in alkaline aqueous solution. Upon thermal treatment, a two-dimensional layered structure grew directly from the prestructure, not from the amorphous polymeric hydroxide dissolved in the solution. Thus, we conclude that the formation of the lepidocrocite-like prestructure is the key for forming a layered titanate under hydrothermal conditions.

beam position is at the bottom left. The vertical black lines are dead pixels, which were excluded using the PIXIA program. (c) One-dimensional data with and without noise filtering. The noise was removed as described in the analytical details. This data was obtained by our first in situ measurement under the hydrothermal conditions at BL08W, SPring-8. We then focused on reducing the noise, and thus the other diffraction image data reported in this paper had less noise.

Figure S6
In situ X-ray total scattering experiments on the solution with titanate (a) and without titanate (b) (left: two theta; middle: relative PDF; right: temperature and pressure data). (c) Total scattering patterns at different temperatures (solid curves: data with titanate; dotted curves: data without titanate).  PDF data. The scattering image data for the solution containing titanate (time = 0, RT) was subtracted from the other data and then normalized by a form factor calculated for the chemical composition of TiO 2 (Q range 0.8-15.0 Å -1 ). This shows the structural change from the reference data. (c) Selected relative PDFs (Q max = 15.2 Å -1 ). This relatively low Q max was only used for these relative PDFs.
The X-ray total scattering image data for the solvent only was subtracted from the image data for the solution containing titanate at each temperature, and then a 1D scattering pattern was obtained. After the data corrections (e.g., Compton scattering subtraction), the intensity data was normalized by a form factor calculated for the chemical composition of TiO 2 (Q range 0.0-18.0 Å -1 ). Because these processes were automated and not optimized at each temperature, some of the G(r) data have noise (or termination ripples) in the low r range. Thus, we could access these data after all the measurements. To monitor the structural changes during the synchrotron X-ray experiments, we used relative PDFs.
The relative PDFs were calculated by subtracting the initial total scattering data from the subsequent data and then normalized by the form factor calculated for the chemistry of TiO 2 . The negative (darker) regions represent a decreased number of atom pairs compared with the initial data (time = 0 min), while the positive (brighter) regions mean an increased number of atom pairs. The positive regions S10 in the ranges of 3-4 Å and 5-6.5 Å suggest the formation of more Ti-Ti pairs in a layered structure, as found in the longer-range region as the thermal treatment proceeds. For example, even at 60°C, the features above 7 Å are almost the same as those in the PDF of the lepidocrocite structure, indicating the growth of lepidocrocite layered titanate.

Figure S9
Fourier-transform infrared spectroscopy spectrum of the as-prepared solution compared with that of a solution of tmah in water. This data was collected in the attenuated total reflection configuration. The peaks observed in the solution containing the Ti reagent (Ti isopropoxide) are assignable to isopropanol. S12 Figure S10 PDF curve-fitting results of the (a) titanate after the reaction measured at RT (fitting parameters were two lattice constants (a and b), the scale factor, the diameter, the sratio parameter for simulating short-range order within 3.46 Å, three parameters for the atomic coordinates of Ti and O atoms under symmetry constraints (Pmmn) and three U aniso parameters for each site of one Ti site and two O sites); (b) titanate in the as-prepared solution simulated using the smaller model (fitting parameters were two lattice constants (a and b), the scale factor, the diameter, three parameters for the atomic coordinates of Ti and O atoms under symmetry constraints (Pmmn) and two U iso parameters for Ti and O; and (c) titanate in the as-prepared solution simulated using the larger model (P1 space group, 2×2×1 supercell) (fitting parameters were two lattice constants (a and b), the scale factor, the diameter (fixed at 14.0 Å), two U iso parameters for Ti (fixed at 0.0021 Å 2 ) and O (fixed at 0.0020 Å 2 ) and atomic coordinates for eight Ti sites and 16 O sites (changed under the restraints described in the main text)). The background curves or baselines were simulated using an empirical model described in the later section, "simulation of PDF using isolated cluster/nanosheet models"

Details of data processing
Conversion of 2D data into 1D data using PIXIA program.
The conversion from 2D image data to a 1D scattering profile was performed using the PIXIA program. This program is being developed and will be available free of charge from National Institute for Materials Science (NIMS) (https://samurai.nims.go.jp/profiles/tominaka_satoshi?locale=en). The program is written in Python and has a graphical user interface prepared using the wxPython module. The current version of PIXIA is being tested by beta testers and will be publicly available soon. Figure S12 shows the user interface. Details of the mathematical treatments in this program will be available in future, and here we summarize the key concepts and equations required for the work reported in this paper.

Figure S12
User interface of PIXIA program.
PIXIA stands for "Pixel-based Image Analyzer", and is dedicated to image processing in X-ray scattering experiments. Briefly, the program can extract scattering data through the noise filtration and data corrections required for X-ray scattering analysis such as polarization correction. We assume a 2D detector as an array of square-pillar detectors, and the coordinates of each pixel are defined as those at its center as shown in Figure S13. The 2D data is a function of the physical space coordinates (x, y, z) or X-ray scattering coordinates (θ, β), where β is the angle in the circumferential direction (anticlockwise). After image processing and correction for the intensities recorded at each pixel, a data list of the corrected intensity, i cor , vs θ is obtained. The histogram of i cor vs θ has many data points (e.g., one million points for 1000×1000 pixels). A mathematical filter is used to obtain the scattering signal by removing the noise (note that current version of PIXIA uses a Savitzky-Golay filter).

Figure S13
Concept of data processing in PIXIA.
In the data correction, the recorded intensities, I rec , are corrected by removing offset values, I ofs , and then normalized by the gain values G, to obtain observable intensities, I ob , (S1.1) The polarization effect, P, is corrected in accordance with the original equation written by Kahn et al., 2 which requires more calculations than the modified equation used for some programs dedicated to XRD within the small-Q range. Then, the intensity of each pixel is normalized by the proportion of Debye-Scherrer rings recorded on the projected area of each pixel, D. The intensity of each pixel is further corrected for the oblique incidence of the X-ray to the detector having thickness h and absorption coefficient µ. Details of the mathematics will be available in future (e.g., in a manual).
The intensities still contain unavoidable noise, I n , which contains statistical noise and other noises. The noise is not a function of the physical space coordinates (x, y, z) or the X-ray scattering coordinates (θ, β); thus, we can remove the noise from both coordinates. We excluded pixels (i) where G values are not constant (e.g., within 10% error) at different exposure times, (ii) where the intensities are markedly different from those of the surrounding pixels in the physical space coordinates (e.g., 50% error) and then (iii) where there are apparently different intensities (e.g., 50% error) compared with averaged intensities at each 2θ angle in the I vs 2θ graph. Then, intensity vs 2θ data is obtained using all the remaining pixel data.

Figure S14
shows an example of data conversion. Figure S14a shows the 2D data (NIST CeO 2 ) after subtracting the air background recorded under the same integration conditions. Parameters including angles, polarization and area correction factors were calculated as shown in Fig. S14b. Then, after masking the beam-stop region and dead pixels, the 2D data shown in Fig. S14a was converted into 1D profile shown in Fig. S14c. In the case of XRD, the profile shown in Fig. S14c is sufficiently accurate for data analysis. However, in the case of PDF analysis, the data still contains the fluorescence intensity and Compton scattering intensity as well as the effect of the modulated offset due to the exposure. As described above, the conversion of a 2D image into a 1D profile requires a complicated mathematical treatment, and thus even constant offset values remain in the data signals and the resultant 1D profile has a curved offset shape. For example, a dummy image having 100 counts at each pixel was converted into the 1D profile shown in Fig. S14d using the same parameters as those for CeO 2 data conversion. Thus, for the PDF analysis, we converted dummy image data having a constant intensity at each pixel into a 1D profile using the same conversion parameters as those for the total scattering experimental data. In the PDF conversion process using the MaterialsPDF program, we subtracted the curved dummy profile from the experimental data as shown in Fig. S15. The intensity in the dummy profile is not significant but its effect on the profile at high-Q values is clear.

Conversion of 1D data into PDF using MaterialsPDF program
The data conversion from a 1D scattering profile to a PDF was performed using the MaterialsPDF program (Fig.   S16). MaterialsPDF and PIXIA are parts of the Orochi program, with which X-ray scattering data can be seamlessly processed and analyzed on the same platform. "Orochi" is named after a legendary eight-headed dragon in Japanese mythology, "Yamata no Orochi". All the programs in Orochi are written in Python and connected to work concertedly. Thus, the data processed by the PIXIA program can be immediately handled in MaterialsPDF.

Figure S15
User interface of MaterialsPDF program.
The 1D profile for the CeO 2 data shown in

S21
Data is processed by background subtraction (or capillary and air intensity subtraction), fluorescence subtraction and Compton scattering subtraction by simulating the scattering profile based on the database to obtain the coherent scattering intensity, I coh , 8 where I com is the Compton scattering intensity, I flu is the fluorescence intensity, and I bg is the background intensity.
I bg is often removed by subtracting the intensity obtained for a blank container. For the data processing of a 1D profile obtained by a 2D detector, the corrections of intensity using the profile obtained from a dummy image, as described in the previous sections are also very important as shown in Fig. S15.
Then, the data is normalized with a form factor based on the Faber-Ziman formalism calculated using atomic scattering factors 9-10 to obtain structure function, S(Q), where L(Q) is the Laue scattering term, N is the number of scattering atoms and f(Q) is the atomic form factor. where F(Q) is named the reduced structure function.
In the MaterialsPDF program, these corrections can be performed as an interactive computation by checking the final PDF data. The obtained G(r) is consistent with that obtained by other programs and can be well simulated (Fig. S17). We also verified the quality of G(r) for SiO 2 glass as a standard amorphous material.

Relative pair distribution functions (PDFs)
For the time-dependent PDFs, the concept of relative PDFs described below is useful for monitoring the change in structures because they can extract the change in PDFs rather than show entire PDFs. For example, in the reaction we investigated in this work, the solution contained precursors, reacted titanium oxide and solvent molecules. The structure of the solvent molecules can be regarded as constant at a given temperature. The change in structure is accounted for by the structural transformation from unreacted molecules to titanite. Thus, if we can normalize the data using the initial structure data, structure information related to the bulk solvent and the unreacted molecules can be cancelled. Then, it is easy to find the start of reactions.
The difference between two reduced PDFs having the same atomic concentrations and scattering factors can be expressed as follows.

∆‫ܩ‬ሺ‫ݎ‬ሻ ൌ
Assuming that PDFs obtained from multiple components can be approximated as the weighted sum of partials, 6 we can normalize the scattering data (I tot ) with N<f> 2 for the changed components. Thus, the relative values ∆‫ܩ‬ሺ‫ݎ‬ሻ (or relative PDFs) are considered to be sensitive only to the changed components. However, the changed components change over time during in situ measurements and are difficult to know before detailed analyses; thus, we must assume or decide that the normalization function N<f> 2 has reasonable values for the changing components. Therefore, the heights of the relative PDFs may not be accurate, although the interatomic distance information is still accurate and useful. Considering the use of spline curves as the normalization function in extended X-ray absorption fine structure (EXAFS) analysis, this relative PDF is similarly reliable.
Because of the simple calculations and possible cancellation of constant errors in the data, the computation of relative PDFs is an easy and straightforward way to determine the start of reactions. Thus, relative PDFs can be obtained at the time of time-resolved measurements and are useful for monitoring the reactions. The dimension and fundamental mathematics of such a relative PDF are similar to those of differential PDFs, but we refer to it as "relative" to emphasize the difference in the normalization.

Treatments of small-angle scattering peaks
The small-angle X-ray scattering intensities were excluded by defining a suitable minimum Q value (Q min ) as reported in the literature. 12 In more detail, Q min = >1.0 Å -1 was used for the titanate reactions, and this limit is sufficiently low to exclude the effect of particles of radius >5 Å.

Simulation of PDF using isolated cluster/nanosheet models
In the simulation with the removal of the baseline and using isolated cluster/nanosheet models, we first removed the baseline in the PDF by estimating it using equation (S1.14). The reduced PDF, G(r), is expressed as follows: 6 ‫ܩ‬ሺ‫ݎ‬ሻ ൌ 4ߨߩ ሺ݃ሺ‫ݎ‬ሻ െ 1ሻ, (S1.10) where ρ 0 is the average number density and g(r) is the atomic PDF. This function, G(r), approaches the straight line of -4πρ 0 r as r approaches zero. According to the experimental data, G(r) decays with increasing r owing to the limited resolution in Q and the limited atomic correlation associated with the particle size. For example, a Gaussian envelope of the following form is used in the PDFfit2 (and PDFfit) program: 13 where Q damp is the damping constant. The spherical particle envelope is defined as follows: 14 where d is the particle diameter and H is the Heaviside step function. Thus, the experimentally obtained reduced PDF of spherical particles can be expressed as Thus, the baseline is -4πρ 0 r around r = 0 but it behaves as -4πρ 0 rAB. Assuming that we can appropriately model the structure as well as the size effect, we can simulate the PDF well. However, this does not hold true for the structures we analyzed in this work. This is because the materials are clusters (too small to be simulated as spherical particles) and 2D (not spheres) and, moreover, the density associated with the surrounding solvent molecules is uncertain. Thus, we simulated the baseline of the PDFs using the following function: , (S1.14) where x is a constant used to adjust the decay curves, s1 is the upper limit where the particle size envelope cannot be held and s2 is the lower limit where the envelope can be held. This is an empirical model, and thus we will evaluate its physical meaning in order to optimize it. The range s1 < r < s2 was connected using a spline function.

S26
This function is not definitive, but we consider that the structure analysis for clusters and nanosheets surrounded by solvent molecules requires a treatment removing such a baseline, and considering the uncertainty of the contribution of such a solvent to the baseline, our approach is still reasonable. Thus, we used this approach to obtain 4πρ 0 g(r)A, which can be simulated using the PDFfit program using a cluster or isolated nanosheet model as shown in Fig. 5, Fig. 6 and Fig. S10.

Structural information data
The followings are the structural information file exported from EXPO2014 (XRD data) and PDFgui (PDF fittings). Regarding the PDF fitting, the fittings were carried out under symmetry restraints though the exported structure ins P1 symmetry.