The Effect of Electronic and Optical Properties on the Kinetic Photocatalytic Model of Methyl Blue Degradation

Abstract

The photocatalytic activity as a function of effective irradiance, photocatalytic quantum yield and reactant coverage was thoroughly assessed for the proper photoreactor (PhR) selection. The emitted wavelength and effective irradiance of several PhRs, equipped with fluorescent and light-emitting diode (LED) lamps, were tested in the photodegradation of methylene blue (MB) in the solid phase using an AgTiO₂ photocatalyst. Among all tested PhRs, the one equipped with the low-pressure Hg lamp enhanced the photodegradation of MB, as the Hg lamp emitted UV-type radiation, which promotes the simultaneous photoactivation of the TiO₂ and the surface plasmon resonance phenomenon of the Ag nanoparticles. It was determined that high values of effective irradiance promoted photocatalytic activity because of the greater amount of photogenerated species [e⁻/h⁺]. Also, it was determined that the effective irradiance used in the photocatalytic process slows down the recombination rate of the [e⁻/h⁺] into photocatalytic material. A kinetic photocatalytic model (KPM) was proposed to describe photocatalytic reactions as a function of effective irradiance, photocatalytic quantum yield and reactant coverage, considering photocatalytic pseudo-steady state according to the reactant equilibrium coverage (Langmuir isotherm) and the transfer processes of the photoinduced charge carrier species.

1. Introduction

Photocatalysis is a cutting-edge research topic since toxic polluting compounds such as dyes, recalcitrant organic compounds, pharmaceuticals, and herbicides, among others, can be removed from wastewater using this technology [1,2,3,4,5,6,7]. The photocatalytic process can be described in three stages: (1) the semiconductor material (photocatalyst) is exposed to an irradiation source (photons); (2) if the incident energy of the photons is equal to or greater than the bandgap (E_g) of the photocatalyst, photogeneration of charge carrier species occurs; that is, electrons [e⁻] from the valence band are photoemitted to the conduction band, and consequently, holes [h⁺] are generated in the valence band; and (3) the charge carrier species are distributed to the surface of the photocatalyst by interfacial electron transfer processes to carry out the corresponding redox reactions to generate reactive oxygen species (ROS), which oxidize the toxic polluting compounds in a non-selective manner [8,9,10,11,12]. TiO₂ is a semiconductor material widely used in photocatalytic processes due to characteristics such as chemical and photochemical stability, low toxicity, wide availability and affordable cost [13,14]. However, the TiO₂ can only be photoactivated using high-energy photons, which corresponds to ultraviolet (UV) irradiation, due to the moderate E_g value of TiO₂ (~3.2 eV). Therefore, to reach photocatalytic activity under visible (vis) irradiation, TiO₂ should be doped or used to form composite materials with the aim of reducing the E_g value and therefore achieving the photoactivation of TiO₂ using low-energy photons. Among these strategies, the incorporation of nanoparticles (NPs) of plasmonic metals into TiO₂ allows the absorption of photons from the visible spectrum [15,16]. It has been reported that NPs of plasmonic Au and Ag incorporated into TiO₂ promote the absorption of low-energy photons due to the localized surface plasmon resonance (LSPR) phenomenon, if the photon wavelength is slightly larger than the size of the plasmonic NPs [17,18].

The proper assessment of photocatalytic processes is a determinant for the adequate synthesis of photocatalytic materials, as well as for the design of processes that involve photocatalytic or photo-assisted methods such as the degradation of pollutants, H₂ production or the synthesis of specific compounds. Thus, a photochemical reactor (PhR) is a crucial component of photocatalytic processes assessment, since it has an irradiation source (fluorescent lamps or light-emitting diodes), a photocatalyst and reactants for the photocatalytic processes [19]. The choice of specific radiation (UV or vis) must be made to promote the photoactivation of the photocatalysts to generate the charge carrier species [e⁻/h⁺], as well as for the selection of the geometrical space to provide adequate and uniform lighting to the reaction system [20,21,22]. It is important to note that the walls of the vessel, where the photocatalytic reaction is carried out, must be transparent to the radiation used for the photoactivation of the photocatalyst. Irradiance is another critical parameter in PhR design and selection. Irradiance (also called photon flux) indicates the energy per unit area, that is, the energy of the incident photons on the catalyst’s surface. Irradiance is expressed in watts per square meter; this magnitude represents the power density at which photons arrive at the surface, considering the individual energies of the incident photons [23].

In liquid photocatalytic reactions, non-homogeneous distribution of the incident light can occur due to the opacity and turbidity of the liquid medium. Therefore, photocatalytic reactions carried out in the liquid phase could be carried out unsatisfactorily. In addition, the nature of the liquid medium could promote or hinder the photocatalytic reactions, since the penetration of the radiation to the surface of the photocatalytic material cannot be reached properly [19,24]. Therefore, the selection of the liquid phase (solvent) is another important parameter in the design of photocatalytic processes, as the selected solvent can potentially impact the pathway of the incident light. Thus, the recombination rate of the photogenerated charge carrier species [e⁻/h⁺] can be increased or slowed, diminishing or promoting the photocatalytic activity, respectively [10,25,26].

The aim of this study is to determine the influence of electronic and optical properties of a composite of Ag-TiO₂/cotton fabric (AgTiC) on photocatalytic activity. To determine the effect of wavelength and irradiance on the photocatalytic activity, several photoreactors, equipped with different radiation sources, were assessed for the photodegradation of methylene blue (MB). The photodegradation of MB was carried out in the solid phase to avoid non-homogeneous distribution of light. MB impregnated on AgTiC was exposed to different photoreactor configurations to select an appropriate photoreactor. A kinetic photocatalytic model was proposed to relate the optical and electronic properties of the photocatalyst with the photocatalytic activity.

2. Results and Discussion

This section presents the characterization results, the electro-optical properties of the AgTiC and the emitted wavelength and irradiance of the radiation sources of the different PhRs used. A kinetic photocatalytic model was proposed considering the reactant equilibrium coverage (Langmuir isotherm), the pseudo-photocatalytic steady state and the transfer processes of the photoinduced charge carrier species. Finally, the photocatalytic activity results for the degradation of MB are discussed and linked to the parameters mentioned above.

2.1. Scanning Electron Microscopy (SEM) and Transmission Electron Microscopy (TEM) Results

Figure 1 shows SEM and TEM micrographs of AgTiC. The presence of NPs on the fabric was observed in the SEM micrograph, indicating successful synthesis of the TiO₂ and Ag. The EDS elemental analysis revealed an average Ti and Ag loading of 1.36 and 1.38 wt.%, respectively. The TEM micrograph shows quasi-spherical particles, dark and light; the largest and darkest particles in the micrograph show the presence of Ag NPs, while the smallest and lightest particles correspond to TiO₂ NPs. The average particle size calculated by ImageJ software (1.54k version) for Ag and TiO₂ (anatase) NPs was 21 nm and 9.6 nm, respectively.

2.2. UV-Vis Diffuse Reflectance Spectroscopy (UV-Vis-DRS), Photoluminescence Spectroscopy (PL) and UV-Vis Spectroscopy (UV-Vis) of AgTiC

The bandgap energy (E_g) of AgTiC was determined using the Kubelka–Munk function and the Tauc plot, assuming an indirect electronic transition from the valence band to the conduction band. The E_g value determined was 3.55 eV; therefore, the minimum required wavelength for the photoactivation of AgTiC was ~360 nm. The E_g values of the reference materials were 3.2 eV and 3.0 eV for TiO₂-anatase and Degussa P25, respectively. The increase in the Eg value for AgTiC is attributed to the smaller size of TiO₂ NPs (9.6 nm) in the nanocomposite compared to the size of the TiO₂ particles in bulk TiO₂-anatase (>50 nm) or Degussa P25 (~25 nm) [27]. It is important to note that although the E_g value indicates that AgTiC should be photoactivated by high-energy photons, from the UV spectrum, the presence of the nanoplasmonic Ag NPs could promote the absorption of lower-energy photons from vis radiation (>400 nm) due to the localized surface plasmon resonance phenomenon (LSPR). Ag NPs promote the electron flow from the nanoplasmonic NPs toward the conduction band of the TiO₂, which should enhance photocatalytic activity due to charge carrier species remaining on the surface of the photocatalyst in a Z-type scheme, as has been reported for several semiconductors in photocatalytic processes [1,2,28].

The fluorescence 3D excitation–emission matrix for AgTiC indicated that at an excitation wavelength (λ_ex) of 350 nm, the intensity of the emitted fluorescence (λ_em = 442 nm) was maximum [29]. The emitted band, at 442 nm, is attributed to the excitonic emission from TiO₂ due to vacancies on the surface [30]. Therefore, it is to be assumed that the optimal wavelength for the photoactivation of the AgTiC nanocomposite should be 350 nm, which corresponds to high-energy photons from the UV spectrum, as has also been suggested by the determined E_g value.

Since fluorescence radiation is caused by the fall of electrons from the conduction band to the valence band, the intensity of the emitted fluorescence band at 442 nm can be used to determinate the [e⁻/h⁺] recombination rate. Therefore, a higher intensity of the band at 442 nm indicates greater electron recombination. It can be observed in Figure 2 that the fluorescence band of AgTiC is at 442 nm. For comparison purposes, the fluorescence band at 442 nm of reference compounds such as bare cotton and TiO₂/cotton are also shown. The intensity of this band was lower than that of the reference compounds, which suggests that the recombination rate of [e⁻/h⁺] of AgTiC was very low.

AgTiC promoted better separation of the charge carrier species [e⁻/h⁺], boosting the charge carrier migration toward the material surface for the interfacial electron transfer processes for the generation of the reactive oxygen species (ROS). Therefore, the simultaneous incorporation of NPs (Ag and TiO₂) on cotton fabrics promotes the greater predominance of charge carrier species on the surface of the nanocomposite.

Figure 3 shows the UV-vis spectrum of AgTiC, which presents absorption bands of photons from 200 to 650 nm. The band at 442 nm corresponds to the wavelength of the maximal phosphorescence emission (see Figure 2), which is attributed to the incorporation of Ag NPs, which extended the absorption of lower-energy photons. Therefore, AgTiC can be used as a visible light-activated photocatalyst. Four intervals can be observed in Figure 3 to identify the type of irradiation required to activate AgTiC: UVC (<280 nm), UVB (280–320 nm), UV-A (320–400 nm) and visible (>400 nm).

It is important to note that generated charge carrier species could result in three possible phenomena: (1) radiative recombination, (2) non-radiative recombination or (3) separation and charge species transfer. Radiative recombination phenomena are caused by relaxation processes of the [e⁻/h⁺], where [e⁻] from the conduction band fall to the valence band and occupy the [h⁺] sites. Non-radiative recombination phenomena refer to the [e⁻] in the conduction band that fall from a higher-energy electronic state to a lower-energy electronic state within the conduction band, and the energy is lost as heat. Separation and charge species transfer refers to [e⁻/h⁺] species that are maintained on the surface of the semiconductor material and, therefore, are involved in photocatalytic processes.

In this sense, the efficiency of the photocatalytic processes can be estimated by the photocatalytic quantum yield (${\Phi }_{\mathrm{p}\mathrm{c}}$), which can be interpreted as a parameter that indicates the number of charge carrier species directly involved in the photocatalytic process with respect to the total number of irradiated photons. Complementarily, ${\Phi }_{\mathrm{p}\mathrm{c}}$ can be estimated by means of the photoluminescence quantum yield (${\phi }_{\mathrm{p}\mathrm{l}}$), which can be interpreted as the counterpart parameter. ${\phi }_{\mathrm{p}\mathrm{l}}$ is a parameter of semiconductor materials that quantifies the capacity of the material to emit photons after it is irradiated with photons that exceed the corresponding wavelength to its E_g. ${\phi }_{\mathrm{p}\mathrm{l}}$ is defined as the ratio of emitted photons to absorbed photons [31,32]. Therefore, the intensity of the emitted fluorescence band at 350 nm (see Figure 2) can be used to estimate the ${\phi }_{\mathrm{p}\mathrm{l}}$, comparing emitted fluorescence intensities at 350 nm of AgTiC and the excitation light. Then ${\Phi }_{\mathrm{p}\mathrm{c}}$ can be estimated from the ${\phi }_{\mathrm{p}\mathrm{l}}$ and considering non-radiative recombination processes [8,9,10].

A high value of ${\phi }_{\mathrm{p}\mathrm{l}}$ indicates that a significant amount of photogenerated [e⁻/h⁺] recombine through radiative processes, releasing energy as light by fluorescence or phosphorescence phenomena. Conversely, a low ${\phi }_{\mathrm{p}\mathrm{l}}$ could be interpreted as charge carrier species probably involved in photocatalytic processes. Therefore, desirable photocatalytic materials must have lower ${\phi }_{\mathrm{p}\mathrm{l}}$ values, which suggests that radiative recombination phenomena are minimized, and separation and transport of charge carrier species to the photocatalyst surface most likely occurs. ϕ_pl for AgTiC was 0.0175, and for TiO₂/cotton was 0.21. AgTiC absorbs more photons than TiO₂/cotton, decreasing radiative recombination processes, suggesting that photocatalytic activity probably results from the greater amount of photogenerated charge carrier species on the surface of the photocatalytic material.

2.3. Irradiance and Emission Spectra of the Photoreactors (PhRs)

Figure 4 shows the irradiance as a function of the radiation wavelength for the tested PhRs. Figure 4a shows the PhR-T5 spectrum, where emission of UV-A radiation, and low contribution of visible radiation (400–420 nm) is observed. The irradiance recorded from 306 to 425 nm was 25.1 W/m². The visible radiation (corresponding to violet color) represents less than 10% of the total emission spectrum; therefore, PhR-T5 was classified as a source of UV-A radiation. The lower wavelengths (more energetic) of the emitted UV-A radiation from PhR-T5 surpassed the wavelength required by the E_g of TiO₂, but larger wavelengths could not activate the LSPR of the Ag NPs, although it is important to note that the irradiance value recorded was quite low.

The PhR-B spectrum (Figure 4b) shows emissions in 400–500 nm, and the maximum emission was at ~450 nm. The irradiance was 2360.4 W/m². Despite the greater irradiance value, the minimum wavelength required for the photoactivation of AgTiC was not achieved with PhR-B. However, the irradiated photons in this interval could activate the LSPR of Ag NPs.

The PhR-Hg spectrum (Figure 4c) shows radiation emitted from the UV (200–400 nm) and visible (>400 nm) intervals. The photons emitted from the visible spectrum correspond to a significant part of the blue spectrum. The irradiance corresponding to the emitted UV radiation (200–400 nm) was 41.1 W/m², and the emitted visible radiation at 400–500 nm generated an irradiance of 60.2 W/m². The irradiance at wavelengths greater than 500 nm was 125.5 W/m². According to previous sections, the photons with wavelengths below 360 nm could photoactivate the AgTiC, although it is likely that photons with wavelengths greater than 360 nm could photoactivate the LSPR of Ag NPs, which should trigger the photocatalytic activity of the AgTiC.

The PhR-I spectrum (see Figure 4d) shows emission from 420 to 780 nm, corresponding to the visible spectrum. The irradiance in 400–500 nm was 25.2 W/m². Meanwhile, the irradiance increased to 35.9 W/m² at wavelengths from 500 to 700 nm. Low irradiances were obtained by PhR-G (see Figure 4e). An irradiance of 1.3 W/m² at 420–500 nm was recorded, although a slight increase in the total irradiance (5.7 W/m²) was observed at 500–700 nm. The emission spectra of PhR-I and PhR-G (Figure 4d,e) confirm that these radiation sources do not emit UV radiation and provide only visible radiation. For these last irradiation sources, it is likely that the photons emitted between 420 and 500 nm could provide the necessary energy to activate the LSPR of Ag NPs in AgTiC. For comparison purposes, the irradiance for solar radiation (winter in Mexico City) is shown in Figure 4f. The irradiances determined were significantly greater than the other radiation sources used in this study. The irradiance was 30.5 W/m² in 200–400 nm, and a higher value was recorded (59.7 W/m²) in 400 to 500 nm. A significantly higher irradiance of 163.2 W/m² was observed at wavelengths greater than 500 nm.

All PhRs emitted UV and visible radiation; however, the irradiances recorded were different. The UV irradiance trend among the tested PhRs was: solar radiation > PhR-Hg > PhR-T5 > PhR-I > PhR-G. Meanwhile, the visible irradiance trend of the PhRs was: PhR-B > solar radiation > PhR-Hg > PhR-I > PhR-G > PhR-T5. The UV irradiance of PhR-Hg was 2-fold, 33-fold and 1.6-fold greater than the UV irradiance for PhR-I, PhR-G and PhR-T5, respectively. In a similar sense, the visible irradiance of PhR-Hg was 3.5-fold and 22-fold greater than the visible irradiance for PhR-I and PhR-G, respectively. It is important to note that the visible irradiation was not significant using PhR-T5. Also, the visible irradiance for solar radiation was higher than for other radiation sources.

Based on the above results, the selected PhRs for the photocatalytic degradation of MB were those with high irradiance, mainly in the UV spectrum. Thus, the PhR-Hg and PhR-T5 were used as radiation sources for the assessment of photocatalytic activity using AgTiC. In

Table 1. Irradiance from PhR-Hg and PhR-T5 at different configurations.

		Irradiance
Photoreactor	Photoreactor Configuration	UV (W/m2)	Visible (W/m2)
PhR-Hg	75 V a	41.1	60.2
	90 V a	50.7	73.0
PhR-T5	11.5 cm b	7.0	0.3
	4.5 cm b	9.0	0.5
	0 cm b	22.6	2.5

^a Supplied voltage. ^b Height of the lamp with respect to the reaction system.

the irradiances are reported for the corresponding UV and visible intervals as established in Figure 4. The irradiance of the PhR-Hg was obtained by switching the supplied voltage (75 or 90 V), while for PhR-T5 it was obtained by placing the lamp at different heights with respect to the reaction system. The recorded irradiance from the PhRs is shown for UV and visible irradiation at the corresponding photoreactor setups. The irradiance of PhR-Hg was significantly greater than that of PhR-T5 for both types of radiation.

2.4. Kinetic Photocatalytic Model (KPM)

A model that encompasses all aspects involved in the overall photocatalytic process is required for the kinetic description of the photocatalytic reactions. According to [33,34,35], the photocatalytic reaction rate must consider two fundamental aspects. The first one involves the heterogeneous reaction on the photocatalyst surface. The second one considers the optical and electronic processes that take place throughout the irradiation process. The photocatalytic reaction rate can be defined according to the photoinduced charge carrier transfer processes [36,37] as:

$$r_A=-{\displaystyle \frac{{dC}_A}{dt}}=f \left[{\theta }_A, {\Phi }_{pc}, I\right]$$

(1)

where θ_A corresponds to the occupied surface fraction of the adsorbed reactant A (A*), Φ_pc is the photocatalytic quantum efficiency and I is the irradiance. It is important to note that the θ_A parameter involves the reactant equilibrium coverage in the same sense as the Langmuir isotherm. However, for photocatalytic reactions, the θ_A parameter is not the same as that stated by Langmuir.

Scheme 1 is a schematic representation of the photocatalytic process, also showing the recombination pathways of the charge carrier species.

According to Scheme 1, step 1, reactant A must be absorbed on the photocatalytic surface:

$$A+\left(\ast \right) \begin{array}{c}{ k}_{a }\\ \overrightarrow{\overleftarrow{{ k}_{d }}}\end{array} A*$$

(2)

where (*) is the adsorption site and A* corresponds to the adsorbed reactant. k_a corresponds to the adsorption constant of reactant A and k_d corresponds to the desorption constant of reactant A. In agreement with Ollis D.F. photocatalytic reactions involve the generation of reactive intermediates such as free electrons [e⁻], holes [h⁺] and hydroxyl radicals [•OH⁻] [33]. The adsorption–desorption equilibrium of reactant A could not be reached due to the reactivity of the A* with the [•OH⁻] adsorbed on the surface. Thus, the slow step (single reactant adsorption) stated in the Langmuir–Hinshelwood model (LHM) is not fulfilled in photocatalysis, and a photocatalytic pseudo-steady state approach must be utilized in the kinetic equation for reactant and surface radicals. This model assumes that the reaction of adsorbed reactant with surface [•OH⁻] occurs,

$$A*+ {• OH}^{-}\underset{{ k}_{s }}{\to } P$$

(3)

where k_s refers to the surface reaction constant between adsorbed reactant A* and the surface [•OH⁻].

Therefore, the surface reaction rate must be as follows:

$$rate=k_s{\theta }_A$$

(4)

In agreement with Ollis D.F., the pseudo-steady-state hypothesis applied to θ_A assumes that [33]:

$$r_A=-{\displaystyle \frac{d{\theta }_A}{dt}}=k_a{C}_A\left(1-{\theta }_A\right)-k_d\left({\theta }_A\right)-k_s\left({\theta }_A\right)$$

(5)

In consequence, if the adsorption equilibrium constant $\left(K_A={\displaystyle \raisebox{1ex}{$k_a$}\!\left/ \!\raisebox{-1ex}{$k_d$}\right.}\right)$ is irreversible k_d << k_a, then for a photocatalytic pseudo-steady state, the reactant coverage is:

$${\theta }_A={\displaystyle \frac{K_0{C}_A}{1+K_0{C}_A}}$$

(6)

where K₀ corresponds to the adsorption equilibrium constant of reactant A, which is a function of the irradiance intensity and the absorbed concentration of •OH⁻. This photocatalytic pseudo-steady state model is mathematically analogous to the LHM, but with the adsorption constants conceptually different.

Scheme 1 shows a zoomed-in view of a volume-element of photocatalytic surface. The optical and electronic processes, due to the irradiation process on the photocatalytic material, can be described as charge transfer processes [34]. The scheme shows the classification of the charge carrier transfer processes as radiative, non-radiative and photocatalytic. To clarify, it is important to note that in an irradiation process, the emitted photons with a wavelength greater than or equal to that required by the E_g of the photocatalyst promote the generation of the [e⁻/h⁺] pair. Radiative processes refer to the recombination of the charge carrier species, where the [e⁻] in the conduction band (CB) fall to the valence band (VB), and therefore the [h⁺] are occupied. The radiative processes are characterized by the radiative optical constant (k_r). In radiative processes the phenomena of fluorescence and phosphorescence could happen. Fluorescence implies that [e⁻] falls from the singlet state in the CB to the singlet state in the VB; meanwhile, in the phosphorescence the [e⁻] falls from the triplet state in the CB to the singlet state in the VB.

In non-radiative processes, the charge carrier species do not recombine; that is, the [e⁻] in the CB do not fall to the VB. The [e⁻] held in the CB could transfer energy as heat in energy decay processes within the CB. These non-radiative processes are characterized by the non-radiative optical constant (k_nr). If the [e⁻] does not transfer energy as heat, the charge carriers are kept on the surface of the photocatalytic material, and the generation of reactive oxygen species (ROS) occurs. Thus, [e⁻/h⁺] pairs that do not recombine are the charge carrier species involved in the photocatalytic degradation of the adsorbed reactants. These charge carrier transfer processes are characterized by the photocatalytic optical constant (k_p). The intrinsic electro-optical processes can be quantified by the photocatalytic quantum efficiency (Φ_pc). This characteristic parameter of the semiconductor materials allows the evaluation of the effectiveness of a photocatalyst for the generation of charge carrier species [e⁻/h⁺] that are maintained on the surface of the material, through non-recombination processes. Therefore, Φ_pc indirectly assesses the effective generation of charge carrier species, which could very likely be involved in the improvement of the photocatalytic activity.

The intrinsic optical and electronic processes are triggered by the irradiance (I) intensity on the photocatalyst. That is, high irradiance values promote photocatalytic activity, as stated in [33]. The irradiance must be raised to an exponent that refers to the magnitude of the irradiance on the catalyst. The value of this exponent is less than one when irradiance is low, demonstrating a significant dependence of the photocatalytic activity on the irradiance. In this sense, in the present work the effective irradiance (I_e) can be defined as the flux of photons with energy equal to or higher than the E_g of the semiconductor material, which is necessary to generate the [e⁻/h⁺] pairs; that is, I_e must correspond to the minimum wavelength of the incident photons, which photoactivate the semiconductor material. Thus, in this work, the kinetic photocatalytic model (KPM) is proposed, which encompasses the surface reaction contributions (Equations (1)–(6)), the concurrent electronic and optical processes and the effective irradiance dependence as follows:

$$r_A=-{\displaystyle \frac{{dC}_A}{dt}}=\left({\Phi }_{pc}\right)\left(I_e^{\beta }\right)\left({\displaystyle \frac{K_0{C}_A}{1+K_0{C}_A}}\right)$$

(7)

If

$$k=\left({\Phi }_{pc}\right)\left(I_e^{\beta }\right)\left(K_0\right)$$

(8)

The KPM is:

$$r_A=-{\displaystyle \frac{{dC}_A}{dt}}={\displaystyle \frac{k{C}_A}{1+K_0{C}_A}}$$

(9)

2.5. Photocatalytic Activity Results for MB Degradation

Photocatalytic activities were determined using PhR-Hg, PhR-T5 and PhR-B since these photoreactor configurations reached an effective irradiance for activating the photocatalyst. It is important to note that photoactivation of the LSPR phenomenon can be achieved under these PhRs, contributing to the photocatalytic activity. Figure 5 shows the UV-vis DRS spectra of MB at different exposition times during the photocatalytic reaction. The photodegradation of MB after 90 min using PhR-Hg at 75 V can be observed, where an irradiance of 41.1 W/m² was emitted.

It can be observed in Figure 5 that under PhR-Hg at 75 V (41.1 W/m²), the photoactivation of the AgTiC was achieved since the adsorption band of MB (λ_max = 672 nm) decreases with the reaction time. MB was degraded almost 88% after 90 min using PhR-Hg. Similar results were obtained by PhR-T5 and solar radiation for MB degradation. With PhR-B the MB degradation reached was nearly 33%. For comparison purposes, the MB degradation was carried out on bare cotton fabrics in the PhRs tested. MB degradation by photolysis was less than 5% at 90 min, as has also been reported by [38]. The photocatalytic efficiency under the different PhRs can be directly correlated with the irradiance spectra shown in Figure 4. For PhR-T5, the emitted radiation was mainly UV-A; therefore, the wavelength was enough for the photoactivation of AgTiC, which allowed for the generation of the [e⁻/h⁺] pair from TiO₂, although the LSPR effect did not occur under this radiation source because its emitted wavelength was not enough for LSPR photoactivation. According to Figure 4, PhR-B cannot photoactivate AgTiC due to the emitted low-energy wavelength (450 nm). In this sense, PhR-Hg and solar radiation can achieve photoactivation of AgTiC and LSPR phenomena more than the other PhRs used in this study.

2.6. Kinetic Photocatalytic Model of Pseudo-First-Order (PFO-KPM)

Equation (9) can be simplified considering that the adsorption–desorption equilibrium of reactant A was not reached. Since the photocatalytic degradation of MB, using AgTiC, was carried out in the solid phase, in similar conditions when $K_0{C}_A\ll 1$.

Then the KPM can be simplified as follows:

$$r_A=-{\displaystyle \frac{{dC}_A}{dt}}=k{C}_A$$

(10)

Thus, the pseudo-first-order kinetic constant (k) can be determined as the slope of the integrated form of Equation (10) as follows:

$$\ln \left({\displaystyle \frac{C}{C_0}}\right)=-kt$$

(11)

Figure 6 shows the photocatalytic activity of AgTiC, represented as k and the characteristic degradation time (τ), which is the reciprocal of k. Therefore, lower τ implies higher MB degradation, and therefore, higher photocatalytic activity. Figure 6a shows the τ as a function of the total irradiance for PhR-Hg and PhR-T5; solar radiation was included for comparison. The total irradiance used is shown inside the bars. For these photocatalytic tests, total irradiance refers to the sum of the UV and visible irradiance reported in Table 1 for each PhR configuration.

The photocatalytic results obtained with PhR-B (k = 0.0047 min⁻¹, τ = 212.8 min) were excluded from the comparison due to their low relative photocatalytic activity. Therefore, despite the high irradiance of PhR-B in the blue region of the visible spectrum (2360.4 W/m²), the LSPR phenomenon from Ag NPs could not be activated by the low-energy wavelength. In this sense, the photocatalytic activities with PhR-Hg and solar radiation were greater than with PhR-B. In Figure 6a, two irradiance values are shown for PhR-Hg: 101 W/m² for 75 V and 143 W/m² for 90 V. For PhR-T5, three irradiances are shown: 7.3 W/m² at 11.5 cm, 9.5 W/m² at 4.5 cm and 25.1 W/m² at 0 cm. For all PhRs, the irradiated wavelengths were from 200 to 500 nm, which were able to photoactivate AgTiC with the highest energy wavelengths.

According to the results shown in Figure 6a, higher irradiance values promoted better photocatalytic activity. That is, for PhR-Hg, the irradiance of 143 W/m² enhanced the photocatalytic activity with regard to the photocatalytic activity obtained under the irradiance of 101 W/m². The same trend was observed for the photocatalytic activities obtained with different PhR-T5 configurations. Figure 6a shows that MB was degraded more effectively with PhR-Hg at 143 W/m²; meanwhile, the less effective PhR configuration was PhR-T5 at 7.25 W/m². That is, higher irradiance from PhR-Hg enhanced the photocatalytic MB degradation significantly with regard to the lowest irradiance from PhR-T5. It is important to note that solar radiation inefficiently degraded MB due to the low-energy photons contained in the UV spectrum of this radiation source.

Figure 6b shows k values with PhR-Hg and PhR-T5 and for solar irradiation as a function of the total irradiance. The comparison of the highest tested irradiance for PhR-Hg (143 W/m², at 90 V) with the lowest irradiance for PhR-T5 (7.25 W/m² at 11.5 cm) indicated that the irradiance was nearly 20 times higher for PhR-Hg. The corresponding k values for PhR-Hg (143 W/m², at 90 V) and PhR-T5 (7.25 W/m² at 11.5 cm) were 0.033 and 0.0036 min⁻¹, respectively. This reflects a nearly 10-fold improvement in the photocatalytic activity for the PhR providing the highest irradiance. That is, the photocatalytic activity was enhanced 10 times when the irradiance increased 20 times. It is important to note that the tested PhRs emitted photons with different wavelengths, with the photons corresponding to the effective irradiance being those that enhanced the photocatalytic activity, as observed for PhR-Hg. These results highlight that photocatalytic activity is a function of the intensity of the irradiance as well as the wavelength emitted by the PhR.

2.7. Photocatalytic Activity as a Function of the Effective Irradiance

The relationship of the photocatalytic activity with the irradiance could be determined based on the photocatalytic activity results (Section 2.4 and Section 2.6) and the KPM discussed in Section 3.4. According to Figure 6, photocatalytic activity was enhanced at higher irradiances and when the wavelength irradiated was equal to or lower than that required by the E_g of AgTiC. According to the Tauc plot analysis, the E_g of the AgTiC was 3.55 eV. Therefore, the required wavelength for the photoactivation of AgTiC should be equal to or lower than 360 nm. Thus, it is imperative to identify the specific wavelengths to activate the photocatalyst and therefore assess in photocatalytic activity as a function of irradiance in a proper way. In this sense, the PhR-Hg (Figure 4c) emitted radiation from 200 nm to 780 nm. Thus, only the UV irradiance, equal to or lower than 360 nm, is suitable for photoactivating AgTiC. As was mentioned previously, this specific irradiance at an adequate wavelength is named the effective irradiance (I_e). Also, for PhR-Hg, it is very likely that photons from 400 to 500 nm could activate the LSPR from Ag NPs. Conversely, the photons from the visible spectrum between 500 and 780 nm are not significantly absorbed by AgTiC and, therefore, do not contribute to the photocatalytic process.

The photocatalytic activity as a function of irradiance should be determined, and this relationship must be considered in the KPM. It is also important to consider the I_e instead of total irradiance. For this reason, experimental photocatalytic data (for PhR-Hg and PhR-T5) were fitted considering the effective irradiance to determine the dependence of the photocatalytic activity with the irradiance, as follows:

From Equation (8):

$$k=K_0\left({\Phi }_{pc}\right)\left(I_e^{\beta }\right)$$

Thus:

$$\ln \left(k\right)=\ln \left(K_0{\Phi }_{pc}\right)+\beta \ln \left(I_e\right)$$

(12)

where $K_0{\Phi }_{pc}$ is constant for a photocatalyst.

Figure 7 shows the plot of ln(k) as a function of ln(I_e) for the photocatalytic tests carried out. The photocatalytic results under UV and UV+visible irradiation can be observed. For only UV irradiation, the β coefficient, determined as the slope of the linear equation, was 0.986 (R² = 0.93). This suggests that irradiance intensity with a wavelength equal to or lower than the one required according to the E_g of AgTiC has a direct function in photocatalytic activity. For comparison purposes the photocatalytic activity under UV+visible irradiation was carried out considering the region equal to or lower than 500 nm. It is important to note that visible radiation could only activate the LSPR of a small fraction of the Ag NPs. The analysis of the data under UV + visible irradiance indicated a β of 0.61 (R² = 0.90). This value of β suggests the presence of intrinsic inefficiencies for the absorption of incident photons, both for the photoactivation of TiO₂ and Ag NPs, as had been previously assumed. The above results indicate that the type of radiation was preponderant to the photoactivation of the AgTiC, and therefore for the enhancement of the photodegradation of MB.

Therefore, a representative kinetic correlation for the description of the photoactivity of AgTiC must consider only the fraction of UV irradiance (I_e) that is effective for the photoactivation. Then, it can be stated that high effective irradiance values are directly proportional to photocatalytic activity. It is important to note that experimental photocatalytic data obtained were fitted well with Equation (10), where a pseudo-first-order photocatalytic kinetic model and β = 1 were considered:

$$r_A=-{\displaystyle \frac{{dC}_A}{d_t}}=\left({\Phi }_{pc}\right)\left(I_e\right)\left(K_0\right)\left(C_A\right)=k\left(C_A\right)$$

3. Materials and Methods

Commercial indiolino fabric (100% cotton, 175 g/m²) was used as a support. Titanium butoxide (97%), terbutanol (99%), acetic acid (99.7%), AgNO₃ (99%), KOH (99%), NaBH₄ (99%) and MB (95%) were provided by Merck (Darmstadt, Germany) and were used without further purification. TiO₂ anatase (analytical grade) and Degussa P25 (TiO₂ anatase/TiO₂ rutile, 80%/20%) were used as references.

3.1. Synthesis of the Photocatalyst (AgTiC)

Cotton fabrics were initially functionalized with a KOH solution (1 M) at room temperature (RT) under stirring for 10 min, according to [39]. Subsequently, the functionalized fabrics were washed with distilled water and dried. The anchoring of Ag and TiO₂ NPs on the fabric was carried out by successive impregnation as was reported in previous work [29]. Briefly, the functionalized cotton fabric was dipped into an aqueous solution of AgNO₃ (0.02 M). Afterward, the fabric was dried at 90 °C and then immersed in a NaBH₄ solution (0.1 M) to reduce surface Ag⁺ ions to Ag⁰ NPs. The cotton fabric with anchored Ag NPs was washed with distilled water and dried at 90 °C. TiO₂ was incorporated on the fabric by hydrothermal synthesis as was reported in previous work [40]. Then, cotton fabric was immersed in a mixture of titanium butoxide (1 wt.%), terbutanol and acetic acid (90/10 wt.%) for 12 h under continuous stirring and at RT. Next, the treated fabric was dried and subjected to hydrothermal treatment in an autoclave at 110 °C for 3 h. The obtained nanomaterial was dried at 50 °C and labeled as AgTiC.

3.2. Physicochemical Characterization of the AgTiC Photocatalyst

The morphology of the AgTiC was analyzed by scanning electron microscopy (SEM) using a JEOL JSM-5900 LV microscope (Akishima, Japan) coupled with an energy dispersive X-ray spectroscopy probe (EDS). The structure and particle size were analyzed by transmission electron microscopy (TEM) with a JEOL LEM-2010 microscope. The bandgap of the samples was determined by UV-vis diffuse reflectance spectroscopy (DRS) using a Perking Elmer Lambda 365 (Waltham, MA, USA). The photoluminescence quantum yield and the recombination rate of electron-hole pairs [e⁻/h⁺] were evaluated by photoluminescence spectroscopy (PL) using a Jasco FP-8550 (Hachioji, Japan) adapted with an integration sphere (ISF-134). Complementary physicochemical characterizations of AgTiC such as attenuated total reflectance infrared spectroscopy (ATR-FTIR), Raman spectroscopy, XRD, N₂ adsorption–desorption and mechanical resistance are described in our previous work [28].

3.3. Photocatalytic Test

The photocatalytic activity of AgTiC was evaluated for MB degradation in the solid phase. For this, 3 mL of an MB solution (5 ppm, pH = 6.5) were impregnated on 2.5 cm × 2.5 cm AgTiC squares. Then, the squares were exposed to radiation sources (UV and vis) within the photoreactor (Scheme 2). The solid samples were analyzed by UV-vis diffuse reflectance spectroscopy for solids (UV-vis DRS) at 0, 15, 30, 60 and 90 min. The MB degradation and kinetic analysis were determined by comparing the initial concentration of MB (C₀) at t = 0 min with regard to the concentration of MB at the reaction times sampled (C).

3.4. Characterization of Photoreactors

The photoreactors (PhRs) were equipped with a variety of irradiation sources, such as a low-pressure Hg lamp, an UV-A type radiation lamp (T5 lamp), an industrial LED luminaire lamp, a greenhouse full-spectrum LED lamp and a blue-radiation LED lamp. The irradiated area for all tests was 6.25 cm². The temperature was kept constant (T = 25 °C) in all tests by means of a fan located in each photoreactor. The PhRs were labeled according to the equipped lamp, and their technical specifications are shown in

Table 2. Technical characteristics and specifications of PhRs.

Photoreactor	Equipped Lamp	Energy Source	Power Consumed
PhR-Hg	Low pressure Hg (tubular bulb) (diameter: 5/8 in) (PL-VM-E93-250W)	Hg vapor	250 W
PhR-T5	UV-A type radiation-T5 (tubular bulb) (diameter: 5/8 in) (Tecnolite)	diodes	16 W
PhR-I	Industrial LED luminaire (LEDs array)	diodes Panel 1: 48 LEDs Panel 2: 48 LEDs	1.5 V–3.3 V
PhR-G	Greenhouse full-spectrum LED (LEDs array)	diodes Panel 1: 126 LEDs Panel 2: 164 LEDs	60 W 6000 Lm
PhR-B	Blue-radiation LED (LEDs array) (M2 photoreactor, MERK)	diodes Panel 1: 4 LEDs	12 V

. PhR-Hg, equipped with a fluorescent lamp, irradiates photons due to the interaction of the electric current with Hg vapor. Meanwhile, PhR-T5, PhR-I, PhR-G and PhR-B, equipped with diode lamps, emitted photons due to the interaction of electric current with semiconductor diodes. The power consumed by the PhRs is a function of the wavelength and the emitted irradiance.

The data shown in Table 2 is important in determining the quantity and arrangement of the lamps. For example, tubular bulbs will only illuminate uniformly from the front of the lamp. Otherwise, if the sample is somewhat far from the front, it will be necessary to place more lamps for adequate illumination of the sample. Meanwhile, integrated diodes arranged in a rectangular array could allow for a greater range of illumination, which could imply the use of fewer diodes. For the PhR-Hg, the irradiance was switched by supplying different voltages, while for PhR-T5, PhR-I, PhR-B and PhR-G, the irradiance was modified by changing the position of the reaction system to different heights (11.5, 4.5 and 0 cm) with regard to the irradiation source. The photocatalytic reactions were carried out under UV and vis radiation, and for comparative purposes solar radiation (winter in Mexico City) was also used. The emitted wavelength and the irradiance were recorded by an LMS-6000C (Lisun Instruments, Shanghai, China) and C-800 SEKONIC (Sekonic, Tokyo, Japan) spectrometers. Additionally, spot irradiance measurements were recorded using a digital UV photometer UV340B (Eujgoov, Shenzhen, China) and a wide-range lux meter EA30-EXTECH (Extech, Pittsburgh, PA, USA) for UV (UV-A + UVB) and visible wavelengths, respectively.

4. Conclusions

The photocatalytic activity as a function of the effective irradiance, photocatalytic quantum yield and reactant coverage was determined for the proposal of a kinetic photocatalytic model (KPM). The photocatalytic MB degradation in solid-state was performed using a photocatalyst based on Ag and TiO₂ nanoparticles on cotton fabric (AgTiC).

According to the results, AgTiC can be photoactivated with UV and vis radiation. However, the photocatalytic activity was preponderant using photoreactors equipped with lamps that emit UV radiation, including low-energy UV radiation near the visible spectrum. The above is most likely due to the presence of low-energy electronic states due to the simultaneous presence of TiO₂ and Ag nanoparticles, in addition to the photoactivation of the localized surface plasmon resonance (LSPR) phenomenon by the Ag nanoparticles.

It was stated that effective irradiance, interpreted as the specific wavelength (or higher) for the photoactivation of the nanophotocatalyst, is required for the enhancement of the photogeneration of charge carrier species [e⁻/h⁺] and the LSPR phenomenon.

The proposed KPM was developed considering the assumptions of the pseudo-steady state and the optoelectronic processes of the charge carrier species. KPM establishes that photocatalytic activity is proportional to the intensity of the irradiance only when effective irradiance is used. Also, it is elucidated that photocatalytic quantum efficiency tends to a maximum when effective irradiance is used. Therefore, both the high intensity of effective irradiance and high photocatalytic quantum efficiency trigger the photocatalytic activity for the MB degradation.

This study highlights the importance of carefully selecting effective irradiance in photoreactors, as well as high irradiance values, for proper evaluation of photocatalytic activity. It is important to note that the study carried out here on the photodegradation of MB in the solid phase is useful for discerning the real effect of irradiance and wavelength on photocatalytic activity using novel semiconductor materials. Further studies on the photocatalytic kinetic model for liquid phase photocatalytic reactions could differ somewhat from the photocatalytic kinetic model deduced in the solid state, if the charge-transfer effects and non-homogeneity effects of the light path are not considered.