Hydrothermal synthesis, structural characterization and photocatalytic properties of $\beta$ -Ag₂MoO₄ microcrystals: Correlation between experimental and theoretical data

2.1 Synthesis of β-Ag₂MoO₄ microcrystals by the CH method

The synthesis of β-Ag₂MoO₄ microcrystals is described as follows: 1 × 10⁻³ mols of molybdate sodium dihydrate (Na₂MoO₄·2H₂O; 99.5% purity, Sigma-Aldrich) and 2 × 10⁻³ mols of silver nitrate (AgNO₃; 99.8% purity, Sigma-Aldrich) were separately dissolved 45 mL of deionized water for each salt, in two plastic tubes (Falcon - capacity of 50 mL). These two solutions were then transferred into a Teflon autoclave (capacity of 150 mL) in a conventional hydrothermal (CH) system, which remained under magnetic stirring for 10 min, finally obtaining a light beige suspension. CH reactions were performed at different temperatures (100, 120, 140, and 160 °C) for 2 h. In principle, β-Ag₂MoO₄ microcrystals were obtained by the reaction between 2Ag⁺ ←: $\mathit{Mo}{O}_4^{2-}$ ions as described in Eqs. (1)–(3):

The obtained light beige suspension was washed with deionized water several times to remove any remaining Na⁺ and ${\mathit{NO}}_3^{-}$ ions. These crystals were separated by means of centrifugations at 4000 rpm for 10 min for seven times. Finally, these powders were oven dried at 75 °C for 12 h, then stored for further characterization and catalytic testing.

2.2 Characterization of β-Ag₂MoO₄ microcrystals

β-Ag₂MoO₄ microcrystals were structurally characterized by XRD patterns using a LabX XRD-6000 diffractometer (Shimadzu®, Japan) with Cu-Kα radiation (λ = 0.15406 nm) in the 2θ range from 10° to 80° with a scanning velocity of 2°/min. Rietveld analysis was conducted in the same range with a scanning velocity of 1°/min and a step of 0.02°. The semi-quantitative analyzes were obtained by X-ray fluorescence (XRF) using PANalytical, Epsilon 3^x (Netherlands) equipment, with Rhodium (Rh) silver (Ag) and molybdenum (Mo) excitation tubes operating under Voltage of 40 KV and current of 3 mA. The methodology adopted includes the “loose powder”, accommodating about 1.5 g of the microcrystals in polyethylene sample holders with dimensions of 2.5 cm (diameter) by 4 cm (height), sealed with films of propylene. M-Raman spectra were recorded using a SENTERRA spectrometer (Bruker®, Germany) equipped with He–Ne laser ( $\lambda$  = 532 nm) and CCD operating from 50 cm⁻¹ to 950 cm⁻¹. The incident laser beam power on the sample was kept at 0.2 mW. For the region located between 85 cm⁻¹ and 1000 cm⁻¹, 100 scans were completed with a spectral resolution of 4 cm⁻¹. A 50 μm lens was used to prevent overheating of the sample. Fourier Transform infrared (FT-IR) spectra were performed from 400 cm⁻¹ to 1100 cm⁻¹ with a spectrophotometer (Varian®, USA) operated in transmittance mode (model IR 660). UV–vis spectra were taken using a UV-2600 spectrophotometer (Shimadzu®, Japan) in diffuse-reflectance mode, using Barium sulfate (BaSO₄, 99.999%, Shimadzu) as analytical standard. Morphological aspects of the microcrystals were verified with FE-SEM using a Quanta SEM 250 microscope (FEI®Company, Netherlands) operated at 15 kV.

2.3 Photocatalytic activity measurement of β-Ag₂MoO₄ microcrystals

The PC properties of β-Ag₂MoO₄ microcrystals for the degradation of RhB [C₂₈H₃₁ClN₂O₃] (99.5% purity, Mallinckrodt, dissolved in water were tested under ultraviolet (UV) light illumination. In this case, 50 mg of catalyst crystals were placed in two beakers (maximum capacity of 250 mL) and then 50 mL of RhB solution (1 $\times$ 10⁵ mol L⁻¹) with pH = 4 were added. Before UV illumination, these suspensions were stirred by sonication during 10 min with an ultrasound bath model 1510 (Branson, USA) (frequency of 42 kHz) and stored in the dark for 5 min in order to allow the absorption of RhB dye on the catalysts. In the sequence, the beakers were placed inside a photo-reactor at 25 °C (Lianyungang Hongkang Quartz products Co., Ltd, China) and illuminated by six UV lamps (TUV Osram, 15 W, maximum intensity at λ = 254 nm). At ten-twenty-minute intervals, 3 mL aliquots of these solutions were removed of the PC system, placed in plastic tubes (Falcon) and centrifuged at 4000 rpm for 5 min to separate the crystals from the liquid phase. Finally, variations of the absorption band maximum at λ = 554 nm of RhB dyes were monitored by UV–Vis absorbance measurements using a double-beam spectrophotometer with a double monochromator and a photomultiplier tube detector of Thermo Scientific Instruments (Model Genesys 10S, USA). The stability of the β-Ag₂MoO₄ was investigated by reusing the microcrystals over three consecutive catalytic cycles initially using 50 mL of 5 mg L⁻¹ RhB dye solutions together with 50 mg of the catalyst.

2.4 Computational method for β-Ag₂MoO₄ microcrystals

Silver molybdate (β-Ag₂MoO₄) was modeled in bulk phase taking into account the internal and translational symmetry in order to well represent periodic 3D conditions. This allows evaluating ubiquitous long-range properties in the solid, more realistic and consistent with the obtained micrometric material. This β-Ag₂MoO₄ have cubic structure, with space group (Fd3′m), defined by one lattice parameter (a = 9.26 Å; V = 794.0 ų) and three non-equivalent atoms (1 Ag, 1 Mo, and 1O). This structure can be described by molybdenum tetrahedron [MoO₄] clusters linked to silver octahedron [AgO₆] clusters by the vertex. The shared oxygens are also on tetrahedron coordination O_(4c) bonded to three Ag and one Mo, forming the [OAg₃Mo] polyhedron (Andrés et al., 2015; Piasecki et al., 2015). The graphical manipulations were performed using the molecular graphics programs XCRYSDEN and VESTA version 3.4.0 for Linux (Kokalj 2003, Momma and Izumi, 2011; Gouveia et al., 2014; Kassou et al., 2016).

Structural, vibrational, and electronic properties were investigated by means of periodic first-principles calculations in the framework of density functional theory (DFT) with the B3LYP hybrid functional (Lee, Yang and Parr, 1988; Becke, 1993) using the CRYSTAL 17 package (Dovesi et al, 2017; Erba et al., 2017). The Grimme D3 potential (Grimme et al., 2010) was included to correct dispersion interactions, which may improve the structural and vibrational description. B3LYP functional has been successfully employed to describe many properties in several strongly correlated systems, serving as reference on the computational calculations of solids in periodic systems (Longo et al., 2014). CRYSTAL 17 program uses Gaussian-type basis set to represent crystalline orbitals as a linear combination of Bloch functions defined in terms of local functions. The all-electron triple-zeta 6211/411/1 (s/p/d) basis set was used for oxygens, and the Hay and Wadt small core (HAYWSC) pseudopotentials 311/31 (sp/d) for Ag and Mo, as available within the Crystal Basis Set Library (Dovesi et al., 2017). The β-Ag₂MoO₄ is a closed-shell system and was treated under restricted Kohn-Sham formalism. The accuracy for the Coulomb and exchange series was controlled by five tight thresholds parameters set to (10⁻¹⁰, 10⁻¹⁰, 10⁻¹⁰, 10⁻¹⁰, 10⁻²⁰). The shrinking factor (Pack–Monkhorst and Gilat net) was set to 4, corresponding to 40 independent k-points in the irreducible part of the Brillouin zone integration in the primitive cell.

The SCF criteria convergence was governed by a threshold on energy of 10⁻¹⁰ and 10⁻¹¹ Hartree for geometry optimizations and vibrational calculations, respectively. All structures were fully optimized (atomic positions and lattice parameters) with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm for Hessian updating, taking very tight criteria for convergence on gradient (0.0001 a.u.) and nuclear displacements (0.0004 a.u.).

Infrared and Raman normal modes and their correspondent harmonic frequencies were obtained from tightly optimized bulks. The frequencies were computed at the G point by diagonalizing the mass-weighted Hessian matrix. The Raman spectra with relative intensities of the peaks were computed analytically by exploiting a scheme based on the solution of first- and second-order Coupled-Perturbed-Hartree-Fock/Kohn-Sham (CPHF/KS) equations (Ferrero et al., 2008a,b), recently implemented in CRYSTAL. To the best of our knowledge, this is the first report describing the Raman spectra of β-Ag₂MoO₄ in this way. The code enables to simulate the final spectrum of powder samples taking into account all possible orientations of solid, providing a fingerprint by which the contribution of each cluster vibration can be detailed described. The modes were visualized through J-ICE program (Canepa et al., 2011).

The electronic structure was studied through atomic charges, band structure and its density of states (DOS), analyzed using the Properties 17 routine of the CRYSTAL code, considering the same k-point sampling as that used during the diagonalization of the Fock matrix. Electronic band structure was obtained at the appropriate high-symmetry path in the first Brillouin zone, and the density of states (DOS) was calculated according to the Fourier-Legendre technique, with polynomial degree equal to 12.

3.1 XRD, Rietveld refinement data, XRF and unit cell representation of β-Ag₂MoO₄ crystals analysis

The degree of structural order-disorder at long-range or the periodicity of crystalline lattice for β-Ag₂MoO₄ microcrystals was verified by the XRD technique. Fig. 1(a) and (b) shows XRD patterns of β-Ag₂MoO₄ microcrystals prepared at different temperatures (100, 120, 140 and 160 °C) for 2 h by the CH method and illustrates theoretical XRD profile with their specific lines position of the optimized cubic structure, respectively.

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Fig. 1 (a) XRD patterns of β-Ag₂MoO₄ microcrystals synthesized by the CH method at different temperatures (100, 120, 140 and 160 °C) for 2 h and (b) XRD patterns theoretically calculated, respectively. The vertical red lines indicate the position and relative intensity of the respective ICSD Card N^o. 36187.

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According to the XRD patterns analysis illustrated (Fig. 1a), all XRD peaks can be indexed perfectly to a spinel-type cubic structure with the space group (Fd3′m) and point-group symmetry ( $O_h^7$ ) (Wyckoff, 1992; Cunha et al., 2015). These crystals have sharp and well-defined diffraction peaks with a good degree of structural order at long range, and any diffraction peaks were observed to silver oxide (Ag₂O) or reduced silver (Ag⁰) nanoparticles by means of XRD measurements (Cavalcante et al., 2012a,b,c,d). Moreover, respective positions of all diffraction peaks in these XRD diffractograms are in good agreement with results reported in the Inorganic Crystal Structure Data (ICSD) base N^o. 36187 (Wyckoff, 1922) and the literature (Cao et al., 2017; Zhang & Ma, 2017a,b). The experimental lattice parameters, unit cell volume and atomic positions of β-Ag₂MoO₄ microcrystals were calculated using the Rietveld refinement method (Rietveld, 1967) with the FULLPROF software version 2016 (Rodriguez-Carvajal 2010, Som & Sharma, 2012). The experimental results obtained from Rietveld refinement were optimized by theoretical calculations. The theoretical lattice parameters and atomic positions were used to model theoretical XRD profile with their specific lines position (Fig. 1b).

Structural refinements using the Rietveld method (Rietveld, 1967) confirmed that all β-Ag₂MoO₄ microcrystals have a spinel-type cubic structure without secondary phases (see Fig. 2a–d).

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Fig. 2 Rietveld refinement plot for β-Ag₂MoO₄ microcrystals synthesized by the CH method at different temperatures: (a) 100 °C, (b) 120 °C, (c) 140 °C and (d) 160 °C for 2 h, respectively.

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From the standard diffraction data obtained by the XRD characterization, as well as the information extracted from the ICSD Card N^o. 36187, a more detailed study of the microstructures was made using the Rietveld method (Araújo Júnior et al., 2017). In Fig. 2(a)–(d) and Table 1, we find the information experimental extracted from the structural refinement and theoretical obtained ly calculated from DFT. The Rietveld refinement data are consisted in evaluating the agreement of the lattice parameters and internal angles (a, b, c, α, β and γ), unit cell volume (V) observed parameters (Y_Obs), calculated parameters (Y_Cal) for the residual line profile (Y_Obs − Y_Cal) and the R-parameters of (R_exp, R_wp, R_b,χ² and S), using the free availability software Fullprof, July 2016 version (Som & Sharma, 2012).

The X-ray fluorescence (XRF) data is shown in Table 1, which was used to verify the percentage composition of metals silver (Ag) and molybdenum (Mo) in β-Ag₂MoO₄ microcrystals synthesized at different temperatures by the CH method for 2 h, showing that the percentages of silver (Ag = 60%) and molybdenum (Mo = 40%) did not undergo stoichiometric variations of these in the composition.

The experimental, theoretical and residual graphic profile implied a good quality of the structural refinement, corroborating with the pertinent information to the parameters R in good concordance (see Table 1) presents a good agreement with the percentage composition of the metals employed indicating that no leaching occurred during the CH synthesis. The atomic coordinates (x, y. and z) of the silver (Ag) and molybdenum (W) atoms did not result in significant variations in their positions in the unit cell, remaining fixed in the same ones arranged in ICSD Card N^o. 36187. However, the atomic positions for the present oxygen atoms have undergone considerable variations, resulting from a high degree of distortions of the octahedral [AgO₆] clusters, tetrahedral [MoO₄] clusters, strain, oxygen atom vacancy ( $V_0^x$ ) as a function of nucleation process and growth of β-Ag₂MoO₄ microcrystals by the CH method employed (Gouveia et al., 2014).

The average size of the microcrystals ( $D_{\mathit{hkl}}^{\prime }$ ) was calculated using the Debye-Scherrer equation (Bindu & Thomas, 2014) presented in Eq. (4), where k corresponds to a constant related to the crystalline structure, being attributed to the present study k = 0.9 (approximately spherical) since silver molybdate (β-Ag₂MoO₄) microcrystals exhibit cubic structure, and λ is the wavelength of the radiation used in the measurement (Cu-Kα, λ = 0.1540 nm).

In the presented equation, ${\beta }_{\mathit{hkl}}$ is corresponds to the full width at half height (FWHM) for the corrected diffraction planes, obtained by Eq. (5), where ${\beta }_{\mathit{sample}}$ refers to FWHM of the diffraction peaks of the samples and ${\beta }_{\mathit{instrumental}}$ to FWHM obtained from the diffraction pattern and Rietveld's refinement, in this work, pure silicon.

The results obtained using the Eqs. (4) and (5) for the microcrystals synthesized at 100, 120, 140 and 160 °C are 99.606(4), 96.808(4), 99.123(4) and 94.285(2) nm, respectively.

As it can be noted, no linearity of the average microcrystals size was observed with the raise in hydrothermal synthesis temperature for 2 h. However, it was verified an increase in the volume of the unit cell (V), when the synthesis temperature was raised from 100 to 120 °C and from 120 to 140 °C. On the other hand, there was a decrease in average crystals size at temperatures from 140 to 160 °C.

Fig. 3(a) and (b) illustrates a schematic representation of a cubic unit cell for β-Ag₂MoO₄ structure modeled from Rietveld refinement data and structure modeled from theoretical calculations, respectively.

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Fig. 3 Schematic representation of the cubic unit cells corresponding to β-Ag₂MoO₄ microcrystals projected at same axis: (a) experimental and (a) theoretical, respectively.

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Lattice parameters and atomic positions obtained from Rietveld refinements were used to model this structure by the Visualization for Electronic and Structural Analysis (VESTA) program (version 3.4.0 for Windows) (Momma & Izumi, 2011). The spinel-type cubic structure of β-Ag₂MoO₄ microcrystals is characterized by the space group (Fd3′m) with eight molecular formula per unit cell (Z = 8) (Wyckoff, 1992). In these structures, silver atoms are coordinated to six oxygen atoms which form distorted octahedral [AgO₆] clusters. These polyhedra have a symmetry group (O_h) with six vertices, eight faces and twelve-edges (Cunha et al., 2015) (see Fig. 3a and b). In this same figure, molybdenum atoms are coordinated to four oxygen atoms which result in tetrahedral [MoO₄] clusters. In this case, these tetrahedra are related to the T_d symmetry group with four vertices, four faces and six edges (Cunha et al., 2015). These [MoO₄] clusters are slightly distorted in the lattice and exhibit a particular characteristic related to differences in O—Mo—O bond angles. These characteristics presented by these β-Ag₂MoO₄ microcrystals obtained at different temperatures present direct relations with the crystalline defects and interatomic distances caused by the distortions in the angles between the chemical bonds (Mo—O—Mo), (Ag—O—Ag) or (Mo—O—Ag) presents into the crystalline lattice. Moreover, it was verified an increase in the unit cell volume (Table 1) as a function of crystal growth mechanism with the modifications in the thermodynamic variables (temperature and pressure). In our case, the intrinsic defects can be increased with the temperature rise. Therefore, we presume that experimental conditions employed in the synthesis as well as the influence of the hydrothermal processing to crystals growth are key factors which caused local distortions in both octahedral [AgO₆] and tetrahedral [MoO₄] clusters.

3.2 Experimental and theoretical micro-Raman and infrared spectroscopy analysis

β-Ag₂MoO₄ microcrystals obtained in this work exhibit a spinel-type cubic structure with a space group (Fd3′m), a point-group symmetry ( $O_h^7$ ) and eight molecules per unit cell (Z = 8) (Wyckoff, 1992). According to group theory calculations, β-Ag₂MoO₄ microcrystals exhibits a total irreducible representation of vibrational modes or optical phonons is shown in Brillouin zone points as described by Eq. (6) (Beltrán et al., 2014):

Fig. 4(a) shows the m-Raman spectra of β-Ag₂MoO₄ microcrystals prepared at different temperatures for 1 h by the CH method, Fig. 4(b) illustrates theoretical Raman spectrum with their specific lines position of optimized cubic structure and Fig. 4(c) is illustrated the experimental and theoretical positions of these Raman-active vibrational modes, respectively.

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Fig. 4 (a) m-Raman spectra experimental from 50 to 1000 cm⁻¹ of β-Ag₂MoO₄ microcrystals obtained by the hydrothermal method at different temperatures for 2 h, (b) Theoretical Raman spectrum by DFT/B3LYP-D3 and (c) Comparing of position of the experimental and theoretical Raman-active modes. The vertical lines indicate the positions of the experimental/experimental Raman modes.

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In m-Raman spectra displayed in Fig. 4(a) were identified five Raman-active modes located between 50 cm⁻¹ and 1000 cm⁻¹. All these Raman modes are intense and well defined, it is possible to conclude that all β-Ag₂MoO₄ microcrystals are structurally ordered at short-range, and there are no active modes related to secondary phases, corroborating with the information obtained by DRX patterns and Rietveld refinement data (see Figs. 1a and 2a–d). The T_2g mode located at 86 cm⁻¹ is due to external modes of oxygen atoms with symmetric bending vibrations presents in tetrahedral [MoO₄] clusters. The E_g mode located at 277 cm⁻¹ is due to external structure vibrations of octahedral [AgO₆] clusters, while T_2g modes found at 353 cm⁻¹ and 761 cm⁻¹ are related to torsional vibrations of oxygen atoms and molybdenum atoms in O—Mo—O bonds inside tetrahedral [MoO₄] clusters. The A_1g mode situated at 872/873 cm⁻¹ is ascribed to symmetric stretching vibrations of oxygen atoms in [←O ← Mo → O→] bonds (Gouveia et al., 2014; Moura et al., 2017; Kroumova et al., 2003; Zhou et al., 2017).

The computational theoretical study of the frequencies and intensity of the Raman-active modes for the microcrystals of β-Ag₂MoO₄ was performed at the DFT/B3LYP-D3 level for solving the equations of first and second order (Coupled-Perturbed-Hartree-Fock/Kohn-Sham, CPHF/KS) to obtain the theoretical Raman spectrum shown in Fig. 4(b). Moreover, the relative experimental and theoretical positions of these vibrational modes displayed in Fig. 4(c) indicates a good agreement between experimental and theoretical Raman modes of β-Ag₂MoO₄ microcrystals.

The Table 2 presents show a comparative between the relative positions of Raman-active modes for the β-Ag₂MoO₄ microcrystals in this work with those reported in the literature (Gouveia et al., 2014; Moura et al., 2017; Zhou et al., 2017).

Table 2 Experimental and theoretical relative position of Raman-active vibrational modes for β-Ag₂MoO₄ microcrystals compared with those reported in the literature.

Methods	Temperature (°C)		Raman active modes				Ref.
		T2g	Eg	T2g	T2g	A1g
This work	100	86	277	353	761	872	⊗
	120	86	277	353	761	873	⊗
	140	86	277	353	761	873	⊗
	160	86	277	353	761	872	⊗
Theoretical (DFT/B3LYP-D3)	–	102	278	361	794	889
MH	140	–	277	348	756	867
PC	300	102	296	372	779	893	♠
HC	160	91	277	353	761	873

Legend: ⊗ = this work.  = Moura et al. (2017); ♠ = Fabro et al., 2015;  = Moura et al. (2016).

A detailed analysis of the results reported in Table 2 revealed some small variations in the relative positions corresponding to the Raman-active modes. In fact, it is acceptable that the presence of defects at short-range, interaction force between the clusters as well as the degree of structural order-disorder into the lattice can be responsible for this behavior.

Fig. 5(a) shows FT-IR spectra in the range from 400 to 11,000 cm⁻¹ of β-Ag₂MoO₄ microcrystals prepared at different temperatures for 1 h by the CH method and Fig. 5(b) is illustrated the experimental and theoretical positions of these IR-active vibrational modes, respectively.

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Fig. 5 (a) FT-IR spectra experimental from 400 to 1100 cm⁻¹ of β-Ag₂MoO₄ microcrystals obtained by the hydrothermal method at different temperatures for 2 h, and (b) Comparing of position of the experimental and theoretical IR-active modes. The vertical lines indicate the positions of the experimental/experimental IR modes.

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In infrared spectra are expected ten infrared vibrational modes for $\beta$ -Ag₂MoO₄ crystals, excluding the six Raman vibrational modes, as presented in Eq. (8) (Gouveia et al., 2014; Moura et al., 2017; Zhou et al., 2017):

However, the 2A_2u, 2E_u and 2T_2u are acoustic vibrations modes, i.e., infrared-inactive modes, while the others 4T_1u IR-active vibrational modes. Therefore, only 4 infrared-active vibrational modes remain, as presented in Eq. (9) (Gouveia et al., 2014; Moura et al., 2017; Kroumova et al., 2003; Zhou et al., 2017).

In our FT-IR spectra illustrated in Fig. 5(a), only one of the four IR-active modes were verified. The 1T_1u modes may not have been detected due to limitations imposed by the FT-IR equipment. As was previously described, the $\beta$ -Ag₂MoO₄ crystals with spinel-type cubic structure with four stretching and bending vibrational modes. The IR-active modes related to 1T_1u located at 834 cm⁻¹ is ascribed to anti-symmetric stretch (→O → Mo → O→) of distorted tetrahedral [MoO₄] clusters. Typical theoretical ( ) and experimental positions (♦) of IR-active modes are shown in Fig. 5(b). Moreover, their respective experimental and theoretical IR-active modes values are listed in Table 3 and compared to those values already reported in the literature.

Table 3 Comparative results between the experimental and theoretical IR-active modes of β-Ag₂MoO₄ microcrystals obtained in this work with those published in the literature.

Methods	Temperature (°C)		Raman active modes				Ref.
		F1u	F1u	F1u	F1u	F1u
This work	100					834	⊗
	120					834	⊗
	140					834	⊗
	160					834	⊗
Theoretical (DFT/B3LYP-D3)		0	76	144	280	817	⊗
CH	120					891

Legend: CH = Conventional hydrothermal ⊗ = this work.  = Kumar et al. (2016).

This table verifies that some IR-active-mode relative positions have small shifts, which can be caused by different factors such as the preparation methods, average crystal size, distortions on O—Mo—O/O—Ag—O bonds, interaction forces between [AgO₆]−[MoO₄] clusters, and/or different degrees of structural order-disorder in the lattice at short range (Gouveia et al., 2014; Moura et al., 2017; Zhou et al., 2017).

3.3 UV–vis diffuse reflectance spectroscopy of β-Ag₂MoO₄ microcrystals

The optical band gap energy (E_gap) was calculated by the Kubelka-Munk equation (Kubelka and Munk, 1931) which is based on the transformation of diffuse reflectance measurements to estimate E_gap values with good accuracy (Morales et al., 2007). Particularly, it is used in limited cases of infinitely thick samples. The Kubelka–Munk Eq. (10) for any wavelength is described by:

Therefore, finding the F(R_∞) value from and plotting a graph of [F(R_∞)hν]² against hν, E_gap values were calculated for β-Ag₂MoO₄ microcrystals by extrapolating the linear portion of UV–vis curves.

Fig. 6(a)–(d) illustrate UV–vis spectra of β-Ag₂MoO₄ microcrystals prepared at different temperatures by the CH method, and the optical band gap values obtained as a function of processing temperature are shown in Fig. 6(e), respectively.

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Fig. 6 UV–vis absorbance spectra of β-Ag₂MoO₄ microcrystals synthesized by CH method at different temperatures: (a) 100 °C, (b) 120 °C, (c) 140 °C, (d) 160 °C for 2 h and (e) optical band gap (E_gap) values as a function of temperature.

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The E_gap is of fundamental importance in the discussions about orders/disorders, crystalline defects and vacancies in the long, medium and short range in the crystalline structures, implying the optical and catalytic properties. From the E_gap values, it becomes possible to establish the photon energy (hν), i.e. the wavelength of the radiation to be used to excite electrons from the VB) to the CB. Since that the photon energy must be of magnitude equal to or greater than the energy of E_gap − hν ≥ E_gap (Mith and Nie, 2010).

As it can be observed in Fig. 6(a)–(e), its noted a slight decrease in the E_gap values with an increase in the processing temperature. In principle, we believe that this behavior is related to the appearance of intermediary energy levels between the VB and CB, since the exponential optical absorption edge and E_gap are controlled by the degree of structural order-disorder in the lattice (Cavalcante et al., 2012a,b,c,d). The decrease in E_gap values can be attributed to structural defects at medium range and local bond distortions which yield localized electronic levels within the forbidden band gap (Cavalcante et al., 2013). A smaller E_gap was detected for β-Ag₂MoO₄ microcrystals processed at 160 °C for 2 h (see Fig. 6a) which suggests a high concentration of defects in the lattice due to high temperature employed in hydrothermal synthesis (Cavalcante et al., 2012a,b,c,d). Moreover, the obtained results and illustrated in Fig. 6(e) are listed in Table 4.

Table 4 Comparative results of the optical band gap energy (E_gap) values of β-Ag₂MoO₄ microcrystals obtained in this work with those reported in the literature.

Methods	Temperature (°C)	Time (min)	Egap (eV)	Ref.
This work	100	120	3.31	⊗
	120	120	3.30	⊗
	140	120	3.29	⊗
	160	120	3.26	⊗
MAH	60	480	3.32
CP	90	10	3.32	♠
CH	160	60	3.31

Legend: MAH = Microwave-Assisted Hydrothermal; CP = Controlled Precipitation; CH = Conventional hydrothermal; ⊗ = this work;  = Cunha et al., 2015; ♠ = Fabbro et al., 2016;  = Gouveia et al., 2014.

As it can be noted in this table the comparative between the E_gap values of β-Ag₂MoO₄ microcrystals obtained in this work and theoretically calculated by the DFT method and those reported in the literature exhibit a very good agreement. However, the presence of k-points between the VB and CB and electronic levels in these crystals can be achieved only by theoretical calculations, which will be shown below.

3.4 Band structures, density of states and map of charge density for β-Ag₂MoO₄ crystals

The lattice parameters (a = 9.345 Å; V = 816.2 ų) and atomic positions of β-Ag₂MoO₄ crystals were obtained in close agreement with experimental data. An accurate description of structural parameters is the first step to obtain reliable results on frequencies and electronic structure.

As it can be observed in Fig. 7(a), is shown the electronic band structure have revealed an indirect band gap ( $\mathrm{\Gamma }$  ← X) = 3.94 eV, close to experimental value, and similar to previous studies (Gouveia et al., 2014) using plane waves basis set. The lower direct ( $\mathrm{\Gamma }$  ←  $\mathrm{\Gamma }$ ) transition is 4.04 eV. It is well described that optical gaps can be empirically adjusted by modifying the amount of exact Hartree-Fock exchange included in the hybrid functional, although this may also worsen the description of the reduced cations. Besides, it should lead to results similar to those obtained with the more sophisticated, but still semiempirical, hybrid functional approach.

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Fig. 7 (a) Electronic band structure showing and indirect band gap, and (b) the projected DOS for β-Ag₂MoO₄ crystal revealing the contribution of each atom to valance and conduction bands. In both panels the top of VB was aligned to Fermi level.

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Fig. 7(a) and (b) illustrate the electronic band structure, and density of states for β-Ag₂MoO₄ crystals which were theoretically calculated from DFT method, respectively.

It is well described that optical gaps can be empirically adjusted by modifying the amount of exact Hartree-Fock exchange included in the hybrid functional, although this may also worsen the description of the reduced cations. Besides, it should lead to results similar to those obtained with the more sophisticated, but still semiempirical, hybrid functional approach. The density of states (DOS) projected on each atom is displayed in Fig. 7(b) and indicates an edge of the valence band (VB) composed mainly by Ag and O atoms, while the bottom of conduction band (CB) is composed by Ag, Mo and O atoms. The Fermi level is located at -5.26 eV, between the redox potential of H⁺/H₂ and O₂/H₂O. Any local disorder on O—Mo—O or O—Ag—O bond length or angles, originated by local stress, or edge and interfaces deformation are sufficient to break the symmetry of clusters and modify the band gap. If the local disorder is already present before light radiation, the polaronic deformation is facilitated and the material should present a better photocatalytic activity than the perfect single crystal. It is only one factor that can determine the generation of electrons (e′)/holes (h^●) pair. The crystalline orbitals related to the top of VB are hybridized Ag 4d orbitals and O 2p orbitals, while the bottom of CB is formed by Mo 4d orbitals and O 2p orbitals (Gouveia et al., 2014).

Fig. 8 shows the charge density difference ( ${\rho }_{\mathit{solid}}$  −  ${\rho }_{\mathit{atom}}$ ) on three selected planes across the O —Mo—O, O—Ag—O and Ag—Ag bonds.

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Fig. 8 Charge density difference across three selected bonds: Mo-O, Ag-O and Ag-Ag. Blue and red regions indicates the loss and gain of electrons with respect to isolated atom. The Mulliken population analyses shows a higher overlap inside tetrahedral [MoO₄] clusters when compared with octahedral [AgO₆] clusters, and negligible for Ag ↔ Ag iteration. The Mulliken charges follow the same trend of expected classical oxidation state for all atoms.

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The Mulliken electron distribution analysis reinforces the structural point of view in which the tetrahedral [MoO₄] clusters form chemical bonds more covalent than Ag—O bonds, which form the octahedral [AgO₆] clusters. Blue and red regions on maps show the loss and gain of electron, respectively, concerning isolated atom, exposing clearly the effect of the solid environment as shown in Fig. 8. Besides charge maps, the Mulliken atomic charges point out the fluctuation local in the electron density and electronic polarization on the metal-oxygen bonds.

3.5 FE-SEM images and growth process analyses of β-Ag₂MoO₄ microcrystals

Fig. 9(a)–(l) illustrate FE-SEM images of β-Ag₂MoO₄ microcrystals prepared at different temperatures: (a)–(c) 100 °C, (d)–(f) 120 °C (g)–(i) 140 °C and (j)–(l) 160 °C for 2 h by the CH method, respectively.

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Fig. 9 FE-SEM images and EDS for β -Ag₂MoO₄ microcrystals obtained from HC method in the temperatures: 100 °C (a, b and c); 120 °C (d, e and f); 140 °C (g, h and i); 160 °C (j, k and l) for 2 h.

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In CH processing stages at 100 °C for 2 h, the growth is promoted by interior part of the furnace through the thermal energy. In this case, the electrical resistances provide the thermal energy transferred to suspension, which the starting $\beta$ -Ag₂MoO₄ microcrystals from aqueous of their respective starting salts to accelerate solid particles to high velocities. These two phenomena induce a random aggregation between the small particles due to the increase in effective collisions which results in a system composed of several irregular β-Ag₂MoO₄ microcrystals (see Fig. 9a–c). The FE-SEM images illustrated in Fig. 9(c)–(i) confirms a mass transport between these particles in contact at temperatures of 120 °C and 140 °C, respectively. As a consequence of this mechanism, the formation of particles with well-defined shapes is impossible. Despite the polydisperse nature of these crystals, the raise in thermodynamic conditions promote a crystal growth process to formation of quasi spherical-shaped β-Ag₂MoO₄ microcrystals were found at 160 °C (see Fig. 9j–l). Recently, (Kumar et al., 2016) reported the formation of this same potato-like β-Ag₂MoO₄ microparticles synthesized by the hydrothermal at 120 °C for 8 h. According to these authors, the CH synthesis promoted to formation of bunches of potatoes-like with clean and fairly smooth surfaces. In our paper, we have observed to same crystal shape to β-Ag₂MoO₄ and at 160 °C the presence of several coral-like β-Ag₂MoO₄ crystals in similarity to the work recently reported by (Oliveira et al., 2017). In addition, these β-Ag2MoO₄ microcrystals exhibits a high level of aggregation and not its possible to measurement of the average size crystal to perform a distribution histogram. However, it is observed that they are in limits ranging from 0.1 to 17 μm. Therefore, these observations clearly indicate that the processing temperature is a key parameter in the morphologic control of β-Ag₂MoO₄ microcrystals.

Fig. 10 shows the illustrate growth process of β-Ag₂MoO₄ microcrystals obtained by te CH method at different temperature for 2 h.

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Fig. 10 Scheme of formation and growth process for β-Ag₂MoO₄ microcrystals synthesized by the CH method at different synthesis temperatures (100, 120, 140 and 160 °C) for 2 h, resulting in coral-like and potatos-like β-Ag₂MoO₄ microcrystals.

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The mechanism to formation and growth process of β-Ag₂MoO₄ microcrystals is shown in the scheme outlined in Fig. 8, in which said process comprises the stages described by Ostwald ripening for growth formation of microcrystals (Vengrenovitch, 1982). Due to the high value of the water dielectric constant (ε = 80 D, 25 °C), the solubility and nucleation of the (Ag₊ and $\mathit{Mo}{O}_4^{2-}$ ions) that make up the starting salts (AgNO₃ and Na₂MoO₄·2H₂O) employed in synthesis precursors at room temperature under constant magnetic stirring. The small particles that underwent nucleation and coalescence in the first stage of the process when subjected to the hydrothermal synthesis at the temperatures assigned in the present study (100, 120 and 140 and 160 °C) underwent multiple rearrangements due to the decrease of the dielectric constant of the water, as well as, due to the increase of the synthesis temperature, thus seeking the equilibrium state of the particles in the interior and surface of the microcrystals, obtaining in the end, coral-like and potatoes-like microcrystals. According to (Gouveia et al., 2014) the formation of β-Ag₂MoO₄ microcrystals using the hydrothermal microwave method using sodium dodecyl sulfate as an anionic surfactant at different temperatures (100–160 °C) for 1 h promotes to growth of cube-like β-Ag₂MoO₄ crystals.

3.6 Photocatalytic activity of β-Ag₂MoO₄ microcrystals for the degradation of RhB dyes

The catalytic assays were performed to degradation of Rhodamine B (RhB) dye molecules using the β-Ag₂MoO₄ microcrystals obtained by hydrothermal synthesis at different temperatures for 1 h are shown in parts of Fig. 11(a)–(e), respectively.

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Fig. 11 Evolution of UV–vis absorption spectra from after 120 min of illumination for photodegradation of RhB dye solution: (a) photolysis and by the catalysts β-Ag₂MoO₄ microcrystals synthesized at the temperature of 100 (b), 120 (c), 140 (d) and (e) 160 °C, respectively.

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The wavelength of 554 nm was monitored in order to correspond to the absorption maximum for the RhB dye, with a gradual increase in the exposure time to the electromagnetic radiation (UV) in the presence of β-Ag₂MoO₄ microcrystals to reduction of this, implying continuous discoloration of the catalyzed solution. The effect of UV-light in the absence of the catalysts (photolysis) did not show significant degradation/discoloration, thus confirming the stability presented by the RhB molecules in aqueous solution (see Fig. 11a), as well as the relative efficiency of β-Ag₂MoO₄ microcrystals in the catalytic process as shown in Fig. 11(b)–(e). Therefore, we assumed that a high percentage of RhB was destroyed or photodegraded after 120 min under UV light. Moreover, we verified that our β-Ag₂MoO₄ microcrystal catalyst obtained by CH method at 140 °C for 2 h was the most efficient for the degradation of RhB dyes under UV-light.

The plot of the relative concentration of the solution [(C₀- C_n)/C₀] as a function of the exposure time (t/min) respectively. The results of the linearized relative concentration values - ln [(C₀ − C_n)/C₀] as a function of time (t/min) of exposure to UV-light are shown in Fig. 12(a) and (b).

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Fig. 12 (a) Evolution of the catalysis of the RhB dye as a function of time (min) of exposure to UV-light and (b) First-order kinetics: without catalysts and with catalysts β-Ag₂MoO₄ microcrystals synthesized at the temperatures (100, 120, 140 and 160 °C0 for 2 h, respectively.

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As it can be noted in Fig. 11(a), we have verified that the β-Ag₂MoO₄ microcrystals synthesized at temperatures of 120 and 140 °C presented the best catalytic performances among all four synthesized and photolysis samples, corresponding to 96.8 and 97.3%, respectively. The apparent velocity constant (k_app) for photolysis and solutions catalyzed by β-Ag₂MoO₄ microcrystals obtained by the HC method at different synthesis times for 2 h. The values of k_app resulted in the decreasing order k_{(120 °C/2h)} > k_{(140 °C/2h)} > _{k(160 °C/2h)} > k_{(100 °C/2h)} > k_(photolysis), corresponding to k_app values equal to: 25.7 × 10⁻³, 25.0 × 10⁻³, 13.5 × 10⁻³, 9.23 × 10⁻³ and 7.37 × 10⁻⁴, respectively.

The study of the kinetic model of degradation of the RhB dye relative to the catalytic profile of photolysis and microcrystals composed of β-Ag₂MoO₄ microcrystals, obtained by the HC method for 2 h, revealed that the experimental data were better adjusted when the pseudo primer-order model of Langmuir-Hinshelwood (Araújo Júnior et al., 2017) as shown in eq. (13), which best fit the experimental values obtained.

The results presented in this table for β-Ag₂MoO₄ microcrystals synthesized at 120 °C for 2 h is the lowest value of t_1/2. This being is related to 26.9 min, while for the photolysis to obtain the same percentage of degradation performed by the previously described sample, required 940.5 min, that is, 44.3 times less efficient. Because the β-Ag₂MoO₄ crystal undergoes a photoreduction on its surface by UV light, occurs an improvement in photocatalytic performance by means of formation of by metallic Ag nanoparticles at β-Ag₂MoO₄ surface and can occur from the surface plasmon resonance effect, which these metal Ag nanoparticles due Ag ionic mobility act as electron traps, scattering or absorbing.

The stability of the β-Ag₂MoO₄ was investigated by reusing the microcrystals over three consecutive catalytic cycles initially using 50 mL of 5 mg L⁻¹ RhB dye solutions together with 50 mg of the catalyst. In Fig. 13a the result for the three catalytic cycles performed with the microcrystals β-Ag₂MoO₄ synthesized at the temperature of 120 °C are presented. It is clear that the catalysts exhibited catalytic activity over the three consecutive cycles, resulting in the percentages of 97.2, 93.9 and 78.8% degradation of the RhB dye molecules for the first, second and third cycle, respectively. Thus, for the first and second catalytic cycle there were no significant differences in the catalytic performances. Therefore, there was a significant difference when the third cycle was performed. However, the percentage of molecules adsorbed during the 10 min in the absence of UV light is significantly reduced from 15 to 4% when comparing the first and second cycles. These observations can be justified by the mechanism of saturation of the adsorptive capacity of the catalysts in the presence of the RhB dye molecules. The diffraction pattern for the microcrystals β-Ag₂MoO4 collected after the three consecutive catalytic cycles is shown in Fig. 13b. In summary, it was noted that there were no peaks that suggested the formation of secondary phases throughout the photocatalytic process, implying, therefore, the high stability of the microcrystals against the photocatalytic tests.

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Fig. 13 (a) Reusability and (b) diffraction pattern for β -Ag₂MoO₄ microcrystals the in three consecutive cycles.

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3.7 A possible photocatalytic mechanism for the degradation of RhB dyes by β-Ag₂MoO₄ microcrystals

Fig. 14 shows the proposed scheme for the excitation/recombination processes occurring in the band structures (BV and BC) of silver molybdenum microcrystals when irradiated with wavelength radiation in the UV region (252 nm) in aqueous medium.

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Fig. 14 Scheme of electron excitation/recombination process during, illumination with UV-light of β-Ag₂MoO₄ microcrystals in solutions of RhB dyes, band structures and mineralization of Rhodamine B (RhB) dye molecules.

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In these processes it is of fundamental importance to consider the mechanism of adsorption of the species present in the reaction medium, so that the period of 10 min under ultrasonic agitation in the absence of light is necessary to reach the equilibrium of adsorption of the dye with the microcrystals, as well as the water and oxygen molecules available in the system (Cavalcante et al., 2012a,b,c,d). The heterogeneous photocatalysis basically consists of the catalysis initiated by the oxidative processes results of the energy photon absorption mechanism with magnitude equal to or greater than the value of Egap (hν ≥ Egap) that limits the BV, where it is filled with electrons for the BC, electron-deficient region, with intermediate levels in this range (Feltrin et al., 2013). Electron excitation of the electrons (′) from BV to BC results in the formation of holes (●) in the structures, which act as oxidants to adsorbed water molecules on the surface of $\beta$ -Ag₂MoO₄ microcrystals, resulting in the formation of H⁺ ions and hydroxyl radicals (OH^•) (Roca et al., 2005) as shown in Fig. 13.

The unit cell for $\beta$ -Ag₂MoO₄ microcrystals is composed of distorted octahedral [AgO₆] clusters and tetrahedral [MoO₄] clusters, which depending on numerous factors may exhibit greater or lesser amounts of crystalline defects, of vacancy oxygen ( $V_0^x$ ), distortions in lattice related to phenomenon to order-disorder at medium range, generating species of complex clusters presentes into VB and complex clusters located main in BC (Cavalcante et al., 2012a,b,c,d). In addition, other factors may be considered essential in the catalytic process, highlighting the surface energy (Roca et al., 2015) presence of impurities (Vogt, Weckhuysen, and Ruiz-Martínez, 2017), particle size and shape and preferential orientation (Cheng, Li, and Schlaberg, 2016; Liu et al., 2010).

The process of excitation/recombination of electrons in the internal structures of β-Ag₂MoO₄ microcrystals is initiated by the absorption of the photon (hν) with energy of magnitude equal or superior to 4.92 eV (UV-light) (E_gap ≥ 3.35 eV), in the absence of this process, it is important to note that along the surface and internal structures of the microcrystals the defects and distortions caused by them imply polarization resulting in the ordered electronic transitions clutter in the clusters (Cavalcante et al., 2012a,b,c,d). The polarization processes occurring in the clusters present in the microstructures of silver molybdenites are shown in Eqs. (14) and (15) below:

After the absorption of electromagnetic radiation by the microcrystals, there is formation of the electron/holes pairs in the complex clusters present in the unit cell, resulting, therefore, in the pairs ${\left[Mo{O}_4\right]}_d-{\left[Mo{O}_4\right]}_o\mathrm{e}{\left[Ag{O}_6\right]}_d-{\left[{\mathit{AgO}}_6\right]}_o$ . In aqueous media, they adsorb the water molecules present on the surface of the microcrystals, which in the face of the formed holes, oxidize the water molecules to radicals ( $\mathit{HO}$ ) and hydrons (H⁺) as shown in Eqs. (16)–(21):

The electrons excited to the CB are captured by numerous species that are present in the reaction medium, thus avoiding the return of these to the VB (Schneider et al., 2014). Oxygen (O₂) is usually diluted in aqueous systems, which is considered to be one of the main electron capture species for the formation of the superoxide radical anion ( $O_2$ ), which once in the presence of hydronium (H⁺) react resulting in the radical hydroperoxide ( ${\mathrm{O}}_{2}H$ ), represented by Eqs (22)–(25):

The RhB dye molecules themselves absorb electromagnetic radiation, called photolysis, resulting in the breaking of the carbon chains generating by-products by the process described as mineralization (Tian et al., 2016). However, the oxidative processes generated by the $O_2{H}^{\bullet }$ and $\mathit{HO}$ radicals using $\beta$ -Ag₂MoO₄ microcrystals as catalyst are of higher evaluation, considering that in the present study the photolysis composed only 8.7% of the percentage of degradation during 2 h of catalysis, whereas for the microcrystals, there was for the sample synthesized at 140 °C, 97.3% degradation of RhB dye molecules for 2 h of exposure to UV-light, resulting in higher rate of radical formation $O_2{H}^{\bullet }$ and $\mathit{HO}$ radicals e por consequência, descoloração and degradation of the RhB dye molecules presentes in aqueloes solutions (Barreiros, David and David, 2006), can be observed in Eqs. (26) and (27) below:

Based on this photocatalytic mechanism, we assume that the defects on the crystal surface and the electronic structure of the complex distorted/disordered ${\left[Ag{O}_6\right]}_d$ and ${\left[Mo{O}_4\right]}_d$ complex clusters and ordered ${\left[{\mathit{AgO}}_6\right]}_o$ and ${\left[Mo{O}_4\right]}_o$ complex clusters display an important role in the production of $O_2{H}^{\bullet }$ and $\mathit{HO}$ radicals, which are the most oxidizing species in these chemical reactions for the degradation of the organic RhB dye in aqueous solution.