Synthesis of copper nano/microparticles via thermal decomposition and their conversion to copper oxide film

Copper nano/microparticles were synthesized in octadecene at 290 °C by thermal decomposition method. Copper acetate monohydrate, stearic acid, and 1-octadecanol were used as copper precursor, capping and mild reducing agents, respectively, at the synthesis. Borosilicate glass substrates submerged into the reaction solution during the synthesis were coated by copper nano/microparticles. Thermal decomposition and coating techniques were combined in this study. Copper nano/microparticles were thoroughly characterised via X-ray powder diffraction, X-ray photoelectron, Raman, and attenuated total reflectance-Fourier transform infrared spectroscopies, and scanning and transmission electron microscopies. The average minimum Feret’s diameter of these synthesized copper particles was measured as ~87 ± 19 nm. Copper nano/microparticles were converted to copper oxide nano/microparticles by applying heat treatment at 250 °C. The phase composition of copper oxide nano/microparticles was determined by reference intensity ratio analysis. The energy gap of copper oxide nano/microparticles was determined as ~2.33 eV by using Tauc’s method. Their band gap PL emission was observed at ~2.15 eV.

in 1-octadecene under pure argon. 1-octadecanol has been used as both organic dispersant and mild reducing agent at the synthesis. Copper nano/microparticles have been accumulated on borosilicate glass discs submerged in 1-octadecene during the synthesis. Thus, for the first time, both syntheses of copper nano/microparticles and their coating on glass have been achieved in one-pot without needing to use time-consuming processes or expensive and complex systems. Nonfused pure copper nano/microparticles have been observed under a magnification of 100,000×. Copper nano/microparticles have been converted to copper oxide nano/microparticles by the heat treatment process. The crystalline phase, forbidden energy gap, and band gap luminescence of these copper oxide nano/microparticles have been investigated by means of X-ray powder diffraction, optical transmission, and photoluminescence spectroscopies, respectively.

Methods
A schematic representation of the formation of copper nano/microparticles is shown in Figure 1. Thermal decomposition method was utilized to synthesize copper nano/microparticles. 0.1996 g copper(II) acetate monohydrate, 0.5984 g stearic acid, 1.3560 g 1-octadecanol, and 25.0760 g 1-octadecene were put into a three-neck round-bottom flask. This flask was placed on a heating mantle with magnetic stirrer. The glass cover was dipped into 1-octadecene and its position was fixed. The glass cover was gently pressed against the flask wall by using a borosilicate glass rod, and thus held steady during the synthesis. A glass reflux condenser was inserted into the middle neck of the flask. A glass coated thermocouple was immersed into 1-octadecene passing through one of the side necks to be able to monitor the real temperature of the mixture. High purity argon gas (99.999%) flowed through the flask throughout the synthesis. This mixture was started to be stirred up at ~754 rpm and heated to 150 °C at 50 min. Care was taken not to splash any drop from the mixture to the flask wall while it was stirring vigorously.
The temperature of the mixed solution was raised as a quadratic function of time between 25-150 °C. The colour of the solution turned from green to blue in this temperature range. Then, the temperature was elevated 10 °C per min up to 290 °C and reduced 7 °C per minute until 172 °C. Colourless solution initially appeared at about 231 °C. Yellowish and afterwards reddish-brown colour was observed above 255 °C. The reaction solution was immediately transferred into a glass vial when the temperature was dropped to 172 °C (see Figure 2a). Both surfaces of cover glass were coated by copper nano/microparticles during the synthesis (see Figure 2c). The coated cover glass was placed in a glass vial without touching its flat surfaces and stored in 1-octadecene against any oxidation of copper nano/microparticles (see Figures 2d and 2e). The interior wall of the flask was also coated with copper nano/microparticles during the synthesis (see Figure 2f). The cover glass containing copper nano/microparticles on both surfaces was washed with copious amounts of methanol and toluene. The reaction solution was centrifuged thrice at 2500 rpm for 10 min with methanol and toluene to collect copper nano/microparticles from the solution (see Figure 2b). This experiment explained above was repeated many times and a lot of samples were produced and prepared for further characterisations in this way. Some of the copper nano/microparticle coated glass covers were heat treated in a drying and heating oven (Binder FD 115) at 250 °C for 2 h to convert copper nano/microparticles to copper oxide nano/microparticles (see Figures 2g and 2h).

Characterisations
Structural and morphological properties of copper nano/microparticles were examined by using X-ray powder diffraction (XRD), X-ray photoelectron spectroscopy (XPS), Raman spectroscopy, scanning electron microscopy (SEM), transmission electron microscopy (TEM), and attenuated total reflectance-Fourier transform infrared spectroscopy (ATR-FTIR). XRD spectra of the samples were measured by using Rigaku SmartLab or Rigaku Ultima-IV diffractometer. Copper K-alpha (1.54 Å) radiation was used for XRD analysis. XPS measurements were taken by PHI 5000 VersaProbe with monochromatized Al K-alpha (1486.6 eV) radiation. Raman spectra were obtained via Renishaw inVia Raman microscope. The samples were excited with a 633 nm wavelength of HeNe laser. Their SEM photos were taken with FEI Quanta 400F field emission scanning electron microscope at 20 kV acceleration voltage. Energy dispersive X-Ray analysis (EDX) was made during SEM. ATR-FTIR spectra of the samples were recorded between 400-4000 cm -1 by Perkin Elmer Spectrum 400. Optical transmittance and photoluminescence (PL) spectra of the samples were acquired with optical measurement systems constructed on a scientific grade optical table. Oriel 74000 Cornerstone 1/8 m monochromator including a grating of 1200 lines/mm and a silicon detector attached to this monochromator were used to detect the intensity of light passing through the sample in the visible and near-infrared range of 400-900 nm (corresponding to ~1.38-3.10 eV). The samples were excited with a wavelength of 442 nm of HeCd laser at PL measurement. PL of the samples was collected and focused into the entrance slit of Newport MS257 ¼ m imaging spectrograph with a charge-coupled device detector. The PL spectra were recorded between 450-770 nm (corresponding to ~1.61-2.76 eV). Thermogravimetric (TGA) analysis of some precursors was made via Mettler Toledo TGA/DSC 3+. The temperature of the precursor was raised 10 °C (10 K) per minute under 40 mL/min nitrogen flow between 25-655 °C. TEM photos of the sample which was produced from centrifugation of reaction liquid were taken with JEOL JEM-2100F. JEOL JEM-2100F was operated at an acceleration voltage of 200 kV. The average thickness of coatings composed of copper or copper oxide nano/microparticles on glass was measured by Filmetrics Profilm3D optical profilometer.

Results and discussion
The reaction mechanism can be understood by tracing the colour change of the reaction solution. The solution has acquired blue colour when the temperature increases from 25 °C to 150 °C. As the temperature increases, copper acetate monohydrate initially loses its water (see supplementary Figure S1). Then, copper acetate leaves its acetate anion and gets stearate anion of stearic acid. Thereby copper stearate occurs [24] and it gives blue colour to the solution. The chemical reaction between copper acetate monohydrate and stearic acid can be explained by Equation (1). (1) In Equation (1), Cu(CO 2 CH 3 (CH 2 ) 16 ) 2 , CH 3 COOH and H 2 O are copper stearate, acetic acid and water products, respectively. The colour of the solution has turned from homogeneous blue to transparent at about 231 °C. It means that copper stearate is thermally decomposed by the help of 1-octadecanol. Thermal decomposition of copper stearate between Figure 2. Photographs of samples. a) Copper nano/ microparticles synthesized in octadecene, b) Precipitated copper nano/microparticles in toluene, c) Cover glass coated with copper nano/microparticles in flask, d) 10 mm diameter cover glass coated with copper nano/microparticles, which is saved in octadecene against oxidation, e) 24 mm diameter cover glass coated with copper nano/microparticles, which is saved in octadecene against oxidation, f) Copper nano/ microparticles-coated flask wall, g) 10 mm diameter cover glass coated with copper nano/microparticles. This sample (shown at g) is called as "Cu n/m-particles on cover glass" in the paper, and h) 10 mm diameter cover glass coated with copper oxide nano/microparticles. This sample (shown at h) is called as "(86% Cu 2 O+14% CuO) n/m-particles on cover glass" in the paper.
200-300 °C under N 2 gas was investigated by Yuan et al. [25]. They showed that the main mass loss of copper stearate in this temperature range was due to releasing of stearic acid. After this thermal decomposition, the colour of the solution has changed from clear to reddish-brown above 255 °C and the solution has reddish-brown colour at about 264 °C. This colour change is an indicator of the copper particles nucleated and growth in the solution. The oxidation of copper nanoparticles is generally an important problem [26]. The synthesis of copper particles has been done under argon gas for this reason.
XRD spectra of five samples are shown in Figure 3. Three of the samples were bought from Merck. These are powders of copper, copper(II) oxide, and copper(I) oxide. XRD spectra of these powders are labeled as "Cu powder", "CuO powder", and "Cu 2 O powder", respectively, in Figure 3. The remaining two samples are the samples obtained via the synthesis explained above. These synthesis samples are copper nano/microparticles deposited on the cover glass (see Figure 2g) and copper oxide nano/microparticles formed on the cover glass by the heat-treatment (see Figure 2h). XRD spectra of these synthesis samples are labeled as "Cu n/m-particles on cover glass" and "(86% Cu 2 O+14% CuO) n/m-particles on cover glass", respectively, in Figure 3. Here, XRD spectra of Cu, CuO, and Cu 2 O powders are used as references to identify phases in the synthesis samples. XRD peaks of Cu n/m-particles on cover glass correspond to those of Cu powder between 0-100°. Therefore, it has been understood that sample Cu n/m-particles on cover glass is composed of pure copper particles on glass. The copper phase is marked with black dots () in this figure. XRD peaks of (86% Cu 2 O+14% CuO) n/m-particles on cover glass are compatible with those of CuO and Cu 2 O powders. Black stars () and diamonds (♦) show CuO and Cu 2 O crystalline phases, respectively. When the reference intensity ratio (RIR) method, which is a quantitative phase analysis used for XRD, is applied to the sample called as (86% Cu 2 O+14% CuO) n/m-particles on cover glass, it has been found that this sample consists of 86 wt% Cu 2 O and 14 wt% CuO. For this reason, this sample has been named as (86% Cu 2 O+14% CuO) n/m-particles on cover glass in the figures and in the text. Comparison of XRD spectra of the cover glass and its Cu n/m-particles coated version is shown at supplementary Figure S2. XRD peak of amorphous silicon dioxide (silica) of cover glass appears around 24°. Thus, the wide XRD peaks seen between 18-24° in Figure 3 have been attributed to the cover glass.
XRD spectra of the synthesis samples have been fitted by using Gaussian-Lorentzian curves and a constant baseline (background). Used Gaussian-Lorentzian function for the XRD curve-fitting is written at supplementary Equation (S1). Thus, positions, full width at half maxima and relative intensities of XRD peaks have been extracted from the spectra. At Figure 4, XRD spectra of the synthesis samples and their fitted peaks are indicated. XRD spectrum of Cu n/m-particles on cover glass is fitted in Figure 4a and XRD spectrum of (86% Cu 2 O+14% CuO) n/m-particles is fitted in Figure 4b.  two synthesis samples are given in Table 1. 2Θ and β indicate peak position and full width at half maximum (FWHM), respectively, in the table. Grain size (L) for each sample has been determined by Scherrer equation (L = Kλ/(βcosΘ)). As seen in Table 1, average grain sizes have been calculated as ~30.23 nm for Cu n/m-particles and ~13.88 nm for (86% Cu 2 O+14% CuO) n/m-particles. Lattice constant and relative error of lattice constant have also been found for these two samples. Calculated ~0.3% and ~0.06% relative errors confirm copper phase for Cu n/m-particles and copper oxide phase for (86% Cu 2 O+14% CuO) n/m-particles (See Table 1).
XPS spectra of Cu n/m-particles on cover glass are given at Figure 5. XPS peaks of Cu, Si, C, and O have been detected in Figure 5a. Photoelectron and Auger transitions corresponding to these XPS peaks are written in the figure. XPS peaks of Si are due to the bulk of the glass. As seen in Figure 5b, XPS spectrum of the sample was scanned between 926-976 eV. After fitting this XPS spectrum, Cu 2p 3/2 and Cu 2p 1/2 XPS peaks have been observed at 933.14 eV and 952.99 eV, respectively. This finding is compatible with XPS data of the literature (See Table 2) [28,29]. Strong shake-up satellite peaks have not seen in two binding energy ranges of 938-946 eV and 958-966 eV for this sample [30][31][32][33][34]. This indicates that there is no CuO in the sample.  Table 1. Results of XRD measurements of Cu n/m-particles and (86% Cu 2 O+14% CuO) n/m-particles. In the first column, K,λ, β, 2Θ, L, L avg , d, (h,k,l), a, a avg , a bulk and relative error of a are shape factor, wavelength of X-ray, full width at half maximum, double Bragg angle, grain size, average grain size, distance between adjacent crystal planes, Miller  In Figure 6, Raman spectra of stearic acid, 1-octadecanol, cover glass, Cu n/m-particles on cover glass in octadecene and Cu n/m-particles on cover glass are shown from bottom to top. Raman peak of Si due to the cover glass is seen at 520 cm -1 . Stars () show these Si Raman peaks in the figure. According to the literature, strong A g and B 1g Raman phonon modes are observed at ~297 cm -1 and ~345 cm -1 for CuO and strong 2E u Raman mode is observed at ~220 cm -1 for Cu 2 O [35][36][37][38]. These Raman modes have not seen for Cu n/m-particles on cover glass. The absence of A g , B 1g and 2E u Raman modes in the spectrum of Cu n/m-particles on cover glass is other indication of oxide-free copper in this sample. Above 606 cm -1 , intensity of Raman signal of this sample sharply increases. There is a high probability that PL signal may be  coming from the sample above 606 cm -1 . Hence, its PL will overlap the Raman signal of the sample itself. Some PL peaks are indicated with hexagons () in the figure. Based on these measurements, it has been concluded that the synthesized copper nano/microparticles do not contain any oxide phase. SEM photos of Cu n/m-particles on cover glass are shown in Figure 7. These SEM photos were taken at between 25,000 X -100,000 X magnification. In Figure 7a, it is seen that the cover glass is covered by copper particles. Cu nano/ microparticles are clearly observed on the cover glass when it is zoomed into the sample from Figure 7a to Figure 7d. As understood from the SEM photos, the sample contains round and nonround nano and micron sized particles, such as rods. Copper seeds in solution growing equally in all directions lead to round particles. However, stearic acid and octadecanol molecules can induce different growth rates along the crystal facets of copper seeds. This might be the reason why some copper particles are rod-shaped. It has been looked at shortest and longest dimensions of the particles via image processing because of the particle shape diversity in the sample. These sizes are known as minimum and maximum Feret's diameters. The overlapped particles and the particles located at the edges of the image have not been considered during this image processing. One hundred and sixty-two particles have been examined in Figure 7d. Number of particles versus their shortest and longest size has been plotted in Figure 8a and 8b, respectively. Best distribution describing particle size distribution is also shown in these figures. Black curve shows particle size distribution in Figure 8a and 8b. The size distribution is Gaussian (normal) and Burr type XII in Figure 8a and 8b, respectively. The equations of these distributions are written at supplementary Equations (S2) and (S3). Median and average particle sizes for the particle shortest dimension will be same due to the normal distribution and it has been determined to be ~87 ± 19 nm. Moreover, median particle size (it means 50% of the particle sizes is below of this value) considering longest dimension has been found to be ~142 nm. SEM-EDX elemental analysis of Cu n/m-particles on cover glass is shown in supplementary Figure S3. The particle sizes of copper nano/microparticles are greater than their grain sizes. This shows that synthesized copper nano/microparticles have polycrystalline structure.
Surface caps of Cu n/m-particles have been investigated by ATR-FTIR spectroscopy. ATR-FTIR transmittance spectra of stearic acid, 1-octadecanol and Cu n/m-particles are shown in Figure 9. The transmittance spectrum of Cu n/m-particles was 8 times intensified by multiplying it with 8 for a better comparison. Vibrational modes of stearic acid and 1-octadecanol  are indicated in the figure by using the literature FTIR data [39][40][41]. Symmetric and antisymmetric vibrations of methyl groups (ν s (CH 3 ) and ν as (CH 3 )), scissoring bending vibration of methylene group (δ s (CH 2 )), rocking vibration of methylene group (ρ(CH 2 )), and bending vibration of COO (δ(O-C=O)) have been seen at the spectrum of Cu n/m-particles (See Table 3). In addition, symmetric vibration of COO -(ν s (COO -)) has also been ascertained. However, strong vibrational modes of stearic acid and 1-octadecanol such as symmetric and antisymmetric vibrations of methylene groups (ν s (CH 2 ) and ν as (CH 2 )) and stretching vibration of CO (ρ(C=O)) have not been observed in the spectrum of Cu n/m-particles. Therefore, it may be inferred that Cu n/m-particles are weakly surrounded by remnant organic molecules after the thermal decomposition. A TEM photo of a Cu n/m-particle precipitated from the reaction solution has been taken (See Figure S4). Surface cap layer on this particle has not been seen. This observation supports the assumption of weak capping. Strong Figure 8. Number of particles versus particle size. a) Versus particle shortest dimension and b) Versus particle longest dimension. D avg,short , D avg,long , D median, and D mode are average particle size derived by considering shortest length of particles (minimum Feret's diameter), average particle size derived by considering longest length of particles (maximum Feret's diameter), median particle size and mode particle size, respectively. Black curves correspond to best curve fits. These are Gaussian and Burr type XII curves for a) and b), respectively. Copper nano/microparticles seen at Figure 7d have been counted and examined to plot Figures 8a and 8b. FTIR peaks for CuO and Cu 2 O particles have been reported at around 530 cm -1 and 623 cm -1 , respectively [42][43][44][45][46][47]. As expected, such peaks have not seen in the spectrum of Cu n/m-particles. Average film thicknesses of Cu n/m-particles and (86% Cu 2 O+14% CuO) n/m-particles have been measured from glass level as ~421.3 nm and ~384.4 nm, respectively (See supplementary Figure S5). The surface roughness of both films has been found to be ~70 nm.
Forbidden energy gap of (86% Cu 2 O+14% CuO) n/m-particles on cover glass has been determined by using Tauc's method [48]. Tauc's plot is shown for (86% Cu 2 O+14% CuO) n/m-particles in Figure 10. Herein, optical absorption of cover glass has been subtracted from optical absorption of (86% Cu 2 O+14% CuO) n/m-particles on cover glass and so merely optical absorption from copper oxide nano/microparticles has been considered. Accepting that n value equals ½ due to the direct transition for copper oxide [49], (αhn) 1/n is plotted against photon energy hν and then linear section of the curve is extrapolated. Interception point of the extrapolated line with hn axis gives the forbidden energy gap. The Figure 9. ATR-FTIR transmittance spectra of stearic acid powder, 1-octadecanol powder, and copper nano/microparticles. Transmittance spectrum of copper nano/microparticles is multiplied by 8 ( x 8). ν s , ν as , ν, δ s , δ, ρ, and r indicate symmetric stretching, asymmetric stretching, stretching, scissoring, bending, out of plane bending, and rocking vibrational modes of functional groups such as CH 3 , CH 2 , C=O etc. Star () shows asymmetric vibration mode of COO -.  energy gap of (86% Cu 2 O+14% CuO) n/m-particles has been measured ~2.33 eV by this method. This value is acceptable because the energy gaps of CuO and Cu 2 O (determined from Tauc's plot) range between ~1.31-2.00 eV and ~1.98-2.58 eV, respectively, in the literature [50][51][52][53][54][55][56]. PL measurements of cover glass and (86% Cu 2 O+14% CuO) n/m-particles on cover glass are shown in Figure 11. In this figure, PL spectrum of cover glass is given below and PL spectrum of (86% Cu 2 O+14% CuO) n/m-particles on cover glass is given above. Gaussian peaks have been used for curve fittings of PL spectra. Black squares () and black dots () indicate these Gaussian peak positions. In other words, they correspond to PL peak positions. Red lines show the curve fittings. PL peak positions of cover glass have been found at ~1.82 eV, ~2.34 eV, and ~2.74 eV (See Table 4). At this point, it should be stated that a soda-lime glass was excited with 488 nm at 77 K by Martínez-Saucedo et al. [51] and they observed PL peaks of a soda-lime glass at energies close to those of the cover glass. Six peak positions have been determined for (86% Cu 2 O+14% CuO) n/m-particles on cover glass. These peaks are at ~1.80 eV, ~1.98 eV, ~2.12 eV, ~2.15 eV, ~2.31 eV, and ~2.74 eV. According to the electronic band structure of Cu 2 O, band gap PL ( transition) appears at ~2.17 eV and excitonic transitions of Cu 2 O due to yellow, green, blue, and indigo series are in the range of ~2.17-2.75 eV [57,58]. The PL peak at ~2.15 eV has located close to the transitions of yellow series excitons, and it is slightly below the optical energy gap of ~2.33 eV. For these reasons, it can be accepted as room temperature band gap PL of (86% Cu 2 O+14% CuO) n/m-particles.

Conclusions
Copper nano/microparticles have been formed and growth in noncoordinating solvent octadecene by using one-pot thermal decomposition method. Borosilicate cover glasses have been immersed in octadecene during the synthesis for coating them with copper nano/microparticles. This could be a new approach to thermal decomposition because coating of cover glass with copper nano/microparticles occurs concurrently with thermal decomposition. Copper(II) acetate monohydrate and stearic acid have been reacted to form copper stearate complex and then this complex has been thermally decomposed into copper nano/microparticles at 290 °C under argon gas. 1-octadecanol has been used as a mild reducer in the synthesis. Unlike other thermal decomposition syntheses, copper acetate monohydrate, stearic acid, and 1-octadecanol have been used in one-pot. A thoroughly characterisation has been done for copper nano/microparticles coated cover glasses. Pure copper crystalline phase of nano/microparticles has been revealed by means of XRD, XPS, Raman, SEM-EDX, and ATR-FTIR measurements. Median copper particle size has been found as ~87 ± 19 nm for shortest length and ~142 nm for longest length by examining Feret's diameters of the particles with SEM, respectively. SEM images  have showed that synthesized copper particles are consists of nano and micron sized particles. Weakly capped surface of copper nano/microparticles with organic molecules arising from thermal decomposition has been indicated by ATR-FTIR and TEM. Copper nano/microparticles have been converted into copper oxide nano/microparticles after heat treatment at 250 °C in air atmosphere. Both CuO and Cu 2 O phases have been observed at XRD diffractogram of copper oxide nano/microparticles. Percent weight composition of copper oxide nano/microparticles has been evaluated by XRD-RIR analysis. It has been found that copper oxide nano/microparticles are composed of 14wt% CuO and 86wt% Cu 2 O. Average thickness of copper oxide film layer has been measured as ~384.4 nm by optical profilometer. Energy gap of these copper oxide nano/microparticles have been determined as ~2.33 eV by means of Tauc's plot method. PL emission slightly below the energy gap of copper oxide nano/microparticles has been observed at ~2.15 eV. After the characterisations explained above, antibacterial properties of these nano/microparticles will be studied in future.

Supplement
First remarkable mass loss range is observed at between ~126-159 °C due to releasing of water from copper(II) acetate monohydrate (see Figure S1). This temperature range has been determined by the extrapolation method. Figure S2. XRD spectra of cover glass and copper nano/microparticles (Cu n/m-particles) on cover glass. Black squares () indicate amorphous SiO 2 XRD peak of cover glass. Black dots () show pure copper XRD peaks of nano/ microparticles. Figure S1. TGA curve of copper(II) acetate monohydrate. 12.5580 mg copper(II) acetate monohydrate was analyzed. Temperature of this sample was increased from ~24 °C to ~654 °C. Heating rate was 10 °C/min in N 2 gas atmosphere. N 2 gas flow rate was 40 mL/min. XRD peak of amorphous SiO 2 has been appeared at ~24° (see Figure S2). Figure S4. TEM photos of a copper nano/microparticle. a) Scale bar is 100 nm, and b) Scale bar is 50 nm. This copper nano/microparticle was precipitated from the reaction solution by centrifugation process. Figure S3. SEM-EDX spectrum of copper nano/microparticles on cover glass. Cu peaks are seen.
Surface morphologies of films of copper and copper oxide nanoparticles are shown at Figure S5. Figure S5. Film thicknesses of copper and copper oxide nano/microparticles on cover glass. a) Film thickness of Cu n/m-particles, and b) Film thickness of (86% Cu 2 O+14% CuO) n/m-particles. Vertical colour bars given at the left show film heights and each colour corresponds to a different height. Surface morphologies of these deposited films (coatings) were measured by using Filmetrics Profilm3D optical profilometer. A Gaussian-Lorentzian function is described at Equation (S1). Here, this equation represents a XRD peak in the context of this paper. Where 2Θ is position in degrees, I G-L (2Θ) is intensity of XRD peak depending on 2Θ, a 0 is amplitude of XRD peak, a 1 is center of XRD peak, a 2 is FWHM of XRD peak, and a 3 is shape factor of XRD peak. Here, a 2 > 0 and 0 ≤ a 3 ≤1. A Gaussian and Burr type XII functions are given at Equations (S2) and (S3), respectively. Here, these functions represent particle size distributions. ! ( ) = $ "$.12 3"-! -" 4 " (S2) Where d is particle size, N G (d) is number of particles for any particle size of d in distribution, a 0 is maximum particle number for d = a 1 , a 1 is average particle size of particle size distribution and a 2 is standard deviation of particle size distribution. Here, a 2 > 0.
Where d is particle size and N B (d) is number of particles for any particle size of d in distribution. In (S3), a 0 , a 1, and a 2 are parameters which define mean value and standard deviation. Here, a 0 > 0, a 1 > 0, and a 2 > 0.