**5.1 Materials and methods**

**5.1.1 Photocatalytic system preparation.** The catalyst synthesis and the photocatalytic system solution preparation protocols are detailed in [18, 21].

**5.1.2 Radiative properties.** Eosin Y UV-visible extinction cross section $E_\lambda$ and catalyst UV-visible absorption cross section $E_{cat,\lambda}$ spectra were measured with a UV-160A Shimadzu spectrophotometer (see Fig 5a) [18]. The Eosin Y fluorescent emission spectrum was measured using a Shimadzu RF-1501 spectrofluorimeter and is presented in Fig 5a. The Eosin Y overall fluorescence quantum yield $\Phi = 0.35$ was identified by minimising the mean squared errors on transmittance between measurements and Monte Carlo algorithm results [54].

**5.1.3 Photoreactor and incident light source.** The reaction was implemented in a laboratory-scale photoreactor extensively described and characterized elsewhere [18, 19, 55]. The gas tight photoreactor of 190 mL is a square-section torus of thickness L = 2.5 cm with two translucent glass faces at the front and the rear (see Fig 5d). It was filled with a liquid volume $V_L = 160$ mL; consequently the headspace gas volume $V_G$ was equal to 30 mL. The liquid photocatalytic system was composed of fluorescent Eosin Y at concentration C and the catalyst at concentration $C_{cat}$, in a mixture of triethylamine (10% vol.) and water [18]. Eosin Y and the catalyst absorb radiation (see Fig 5a). The photoreactor was perfectly mixed and the temperature was controlled at 25.0 ± 0.1°C by water circulation in the photoreactor vessel jacket [55].

Incident radiation was obtained from a 25-LED panel positioned at a distance of 15 cm from the reactor, providing quasi-collimated blue light with a spectrum $p'(\lambda)$ centered at a wavelength of 457 nm (see Fig 5c). The photon flux density entering the medium at the front face of the photoreactor, accurately controlled and easily modifiable, was measured via actinometry experiments, as described in [19].

In such a configuration, the radiative transfer theory can be approximated as a one-dimensional problem.

**5.1.4 Spectral light flux density measurements at the rear of the photoreactor.** To carry out the radiative analysis, we measured the mean photon flux density exiting the reactor $q_r$ measured at the rear glass window using a LI-COR quantum sensor (LI-190) connected to a LI-189 portable LI-COR Quantum meter/Radiometer/Photometer. This mean photon flux density at the rear of the photoreactor was obtained by measuring the photon flux density at 10 different positions. However, this device counts only the total hemispherical photon flux density and was not able to determine spectral distribution. Thus our experimental setup was complemented with a spectrometer (Ocean Optics USB 2000+) equipped with an optical fiber (QP 400-2-SR) and a cosine corrector (Ocean Optics CC-3) coupled to the fiber. The spectrometer was linked via USB connection to a computer running Oceanview software. Spectral distribution at the rear of the reactor $p'(\lambda)$ was thus obtained (see Fig 5e).

**5.1.5 Hydrogen reaction rate measurement.** Hydrogen was produced in the gas tight reactor under irradiation; this was indicated by an increase in pressure (see Fig 5b) measured using a pressure sensor (Keller PA 33X) located in the headspace of the reactor. To analyze this phenomenon, a complete mass analysis of the hydrogen was performed, taking into account a transient regime followed by a linear regime. During the linear regime, the mean volumetric rate of hydrogen production $\langle r_{H_2} \rangle$ can be calculated from the linear variation in pressure P with time t on the basis of the slope $\frac{dp}{dt}$ [55]. A complete analysis of the gas phase was routinely performed with a micro gas chromatograph to demonstrate that only hydrogen is produced.

**5.1.6 A typical experiment.** For a given photocatalytic system (Eosin Y and catalyst concentrations), the incident photon flux density $q_0$ in front of the photoreactor is fixed. The photon flux exiting at the rear of the photoreactor as well as the rear emission spectrum and the
pressure time course are measured. These measurements are repeated for different $q_0$ values. This experiment is implemented for different Eosin Y and catalyst concentrations.

## 5.2 Results

We demonstrated that the photocatalytic system follows a linear thermokinetic coupling law [18]. This means that a plot of the hydrogen production rate $\langle r_{H_2} \rangle$ versus MVRPA $\langle A \rangle^{(*)}$ is a straight line, the slope of which being the overall quantum yield $\varphi^{(*)}$ (see Eq 2):

$$ \varphi^{(*)} = \frac{\langle r_{H_2} \rangle}{\langle A \rangle^{(*)}} \qquad (63) $$

From the experimental hydrogen production rate $\langle r_{H_2} \rangle$, the overall quantum yield is identified by Eq 63 according to several methods used to obtain MVRPA, which are:

**5.2.1 Measure neglecting fluorescence (notation (•)≡ (exp, 0)).** We published in [18] experimental MVRPA results that were used to estimate $\varphi^{(\text{exp},0)}$. Those results were obtained from the difference between the incident photon flux density entering the front of the reactor $q_{0,\lambda} = q_0 p^i(\lambda)$ and the measured hemispherical photon flux density exiting the reactor at the rear. Photon flux at the rear of the photoreactor is composed of transmitted ballistic (blue) photons generated by the LED panel in spectral interval $\lambda \in [400 \text{ nm}; 490 \text{ nm}]$ and scattered (fluorescent) photons emitted by Eosin Y in interval $\lambda \in [490 \text{ nm}; 630 \text{ nm}]$, as presented in Fig 5e. The fact that these spectral intervals do not overlap allows us to distinguish between ballistic and fluorescence radiation at the rear to obtain the radiative balance on ballistic radiation only:

$$ \langle A \rangle^{(\text{exp},0)} = \frac{1}{L} \int_{400 \text{ nm}}^{490 \text{ nm}} d\lambda (q_{0,\lambda} - q_{r,\lambda}) f_{\lambda} \qquad (64) $$

where we introduced a factor $f_\lambda$ to account for catalyst absorption (see Eq 65).

Estimating the total MVRPA including fluorescence radiation would require measuring the outward hemispherical photon flux density exiting the front of the photoreactor. Indeed, ballistic radiation exits the medium at the rear only but fluorescence exits at the rear and front faces. However our device does not allow such measurements, and with luminescence the calculation of MVRPA from models is required.

**5.2.2 Calculation of MVRPA from models.** MVRPA is calculated with Eq 23, from the following absorbance models that have been presented in previous sections:

* Neglecting fluorescence (see Section 4): notation (•) ≡ ($\Phi = 0$)

* Reference model taking fluorescence into account (see Section 2.5): notation where superscript (•) is dropped

* Gray Single-scattering analytical approximation (see Section 4.3): notation (•) ≡ ($\overline{\text{SS}}$)

Absorption by the catalyst is simply added to the models established in previous sections by modifying the radiative properties as follows:

* $k_\lambda \rightarrow k_\lambda + k_{cat,\lambda}$ with the extinction coefficient of the catalyst $k_{cat,\lambda} = C_{cat} E_{cat,\lambda}$

* $\Phi \rightarrow \alpha_{s,\lambda} = \Phi \frac{k_\lambda}{k_\lambda + k_{cat,\lambda}}$

In addition, the calculation of MVRPA used for the formulation of thermokinetic coupling should not include the radiation absorbed by the catalyst since it does not lead to hydrogen production. To perform this calculation the integrand in Eq 24 (and the weight of the Monte
pressure time course are measured. These measurements are repeated for different $q_0$ values. This experiment is implemented for different Eosin Y and catalyst concentrations.

## 5.2 Results

We demonstrated that the photocatalytic system follows a linear thermokinetic coupling law [18]. This means that a plot of the hydrogen production rate $\langle r_{H_2} \rangle$ versus MVRPA $\langle A \rangle^{(*)}$ is a straight line, the slope of which being the overall quantum yield $\varphi^{(*)}$ (see Eq 2):

$$ \varphi^{(*)} = \frac{\langle r_{H_2} \rangle}{\langle A \rangle^{(*)}} \qquad (63) $$

From the experimental hydrogen production rate $\langle r_{H_2} \rangle$, the overall quantum yield is identified by Eq 63 according to several methods used to obtain MVRPA, which are:

**5.2.1 Measure neglecting fluorescence (notation (•)≡ (exp, 0)).** We published in [18] experimental MVRPA results that were used to estimate $\varphi^{(\text{exp},0)}$. Those results were obtained from the difference between the incident photon flux density entering the front of the reactor $q_{0,\lambda} = q_0 p^i(\lambda)$ and the measured hemispherical photon flux density exiting the reactor at the rear. Photon flux at the rear of the photoreactor is composed of transmitted ballistic (blue) photons generated by the LED panel in spectral interval $\lambda \in [400 \text{ nm}; 490 \text{ nm}]$ and scattered (fluorescent) photons emitted by Eosin Y in interval $\lambda \in [490 \text{ nm}; 630 \text{ nm}]$, as presented in Fig 5e. The fact that these spectral intervals do not overlap allows us to distinguish between ballistic and fluorescence radiation at the rear to obtain the radiative balance on ballistic radiation only:

$$ \langle A \rangle^{(\text{exp},0)} = \frac{1}{L} \int_{400 \text{ nm}}^{490 \text{ nm}} d\lambda (q_{0,\lambda} - q_{r,\lambda}) f_{\lambda} \qquad (64) $$

where we introduced a factor $f_\lambda$ to account for catalyst absorption (see Eq 65).

Estimating the total MVRPA including fluorescence radiation would require measuring the outward hemispherical photon flux density exiting the front of the photoreactor. Indeed, ballistic radiation exits the medium at the rear only but fluorescence exits at the rear and front faces. However our device does not allow such measurements, and with luminescence the calculation of MVRPA from models is required.

**5.2.2 Calculation of MVRPA from models.** MVRPA is calculated with Eq 23, from the following absorbance models that have been presented in previous sections:

* Neglecting fluorescence (see Section 4): notation (•) ≡ ($\Phi = 0$)

* Reference model taking fluorescence into account (see Section 2.5): notation where superscript (•) is dropped

* Gray Single-scattering analytical approximation (see Section 4.3): notation (•) ≡ ($\overline{\text{SS}}$)

Absorption by the catalyst is simply added to the models established in previous sections by modifying the radiative properties as follows:

* $k_\lambda \rightarrow k_\lambda + k_{cat,\lambda}$ with the extinction coefficient of the catalyst $k_{cat,\lambda} = C_{cat} E_{cat,\lambda}$

* $\Phi \rightarrow \alpha_{s,\lambda} = \Phi \frac{k_\lambda}{k_\lambda + k_{cat,\lambda}}$

In addition, the calculation of MVRPA used for the formulation of thermokinetic coupling should not include the radiation absorbed by the catalyst since it does not lead to hydrogen production. To perform this calculation the integrand in Eq 24 (and the weight of the Monte
<table><caption>**Table 4.** Overall quantum yields of the homogeneous photoreactive system at different catalyst (C<sub>cat</sub>) and Eosin Y (C) concentrations in mol.m<sup>-3</sup>. Results obtained from Eq 63, using several methods to estimate ⟨A⟩: φ<sup>(exp,0)</sup> uses measurements on ballistic radiation (fluorescence is neglected), φ uses the reference Monte Carlo calculation, φ<sup>(Φ=0)</sup> uses model results when fluorescence is neglected, φ<sup>(SS)</sup> uses the gray single-scattering analytical approximation presented in Section 4.3. Relative error δ with respect to the reference value φ is provided (see Eq 66). Optical thicknesses τ<sub>A</sub><sup>i</sup> and τ<sub>S</sub><sup>L</sup> are calculated from Eqs 4 and 6, with E&#773;<sup>i</sup> = 2556 m<sup>2</sup>.mol<sup>-1</sup> for the LED emission spectrum.</caption><thead><tr><th>C<sub>cat</sub></th><th>C</th><th>τ<sub>A</sub><sup>i</sup> | τ<sub>S</sub><sup>L</sup></th><th>Reference</th><th colspan="2">Neglecting luminescence</th><th>Approximation</th></tr><tr><th></th><th></th><th></th><th>φ</th><th>φ<sup>(exp,0)</sup></th><th>φ<sup>(Φ=0)</sup> | δ(Φ = 0)</th><th>φ<sup>(SS)</sup> | δ(ΣS)</th></tr></thead><tbody><tr><td>0.075</td><td>0.05</td><td>2.22 | 1.19</td><td>0.76 ± 0.08</td><td>0.61 ± 0.08</td><td>0.62 ± 0.08 | -20%</td><td>0.74 ± 0.08 | -3%</td></tr><tr><td>0.1</td><td>0.2</td><td>8.87 | 4.77</td><td>0.47 ± 0.04</td><td>0.44 ± 0.04</td><td>0.41 ± 0.04 | -13%</td><td>0.48 ± 0.04 | 2%</td></tr><tr><td>0.1</td><td>0.45</td><td>19.95 | 10.74</td><td>0.25 ± 0.02</td><td>0.22 ± 0.02</td><td>0.22 ± 0.02 | -12%</td><td>0.26 ± 0.02 | 4%</td></tr></tbody></table>

https://doi.org/10.1371/journal.pone.0255002.t004

Carlo algorithm) are multiplied by the probability f<sub>λ</sub> that a photon absorbed is absorbed by Eosin Y molecules (rather than by the catalyst):

$$f_{\lambda} = \frac{k_{\lambda}(1 - \Phi)}{k_{\lambda}(1 - \Phi) + k_{cat,\lambda}} \qquad (65)$$

where Φ is equal to 0 when luminescence is neglected. Note that, integrated over the spectral range from 400 nm to 490 nm, factor f varies between 79% and 96% for the concentration range studied (see Table 4). Indeed, absorption by the catalyst is unfavorable for hydrogen production and should remain low.

The resulting formulation of absorptance when accounting for catalyst absorption is provided in S3 Appendix.

Reflection and refraction at the rear of the photoreactor are not included in the models, since the transmission of the water / glass / air interface is ≈ 90% for our spectral range of work (value obtained from both measurements and geometrical optics calculations). Moreover, less than 10% of the incident radiation reaches the rear glass for the absorption optical thicknesses studied in Table 4 (e<sup>-τ<sub>A</sub><sup>i</sup></sup> < 0.1). Therefore, only 1% of the incident radiation is reflected back into the medium.

### 5.2.3 Results gathered in Table 4 also include.

* Absorption and scattering optical thicknesses defined in Eqs 4 and 6, with E&#773;<sup>i</sup> = 2556 m<sup>2</sup>.mol<sup>-1</sup> for the LED emission spectrum (E&#773;<sup>L</sup> = 2728 m<sup>2</sup>.mol<sup>-1</sup> is unchanged, see Table 1). Absorption by the catalyst has very little impact on photon transport analysis and is therefore not included in these optical thickness calculations.

* The relative difference

$$\delta(X) = \frac{\varphi^{(X)} - \varphi}{\varphi} \qquad (66)$$

between the reference overall quantum yield φ (obtained with the reference MVRPA given by the Monte Carlo method) and the overall quantum yields φ<sup>(X)</sup> obtained from the MVRPA models denoted (X) above.

## 5.3 Discussion

The overall quantum yield φ in Table 4 is expected to vary as a function of Eosin Y concentration. The increase in φ when Eosin Y concentration decreases could be explained by quenching phenomena: collisions between Eosin Y excited states and other molecules (quenchers)
<table><caption>**Table 4.** Overall quantum yields of the homogeneous photoreactive system at different catalyst (C<sub>cat</sub>) and Eosin Y (C) concentrations in mol.m<sup>-3</sup>. Results obtained from Eq 63, using several methods to estimate ⟨A⟩: φ<sup>(exp,0)</sup> uses measurements on ballistic radiation (fluorescence is neglected), φ uses the reference Monte Carlo calculation, φ<sup>(Φ=0)</sup> uses model results when fluorescence is neglected, φ<sup>(SS)</sup> uses the gray single-scattering analytical approximation presented in Section 4.3. Relative error δ with respect to the reference value φ is provided (see Eq 66). Optical thicknesses τ<sub>A</sub><sup>i</sup> and τ<sub>S</sub><sup>L</sup> are calculated from Eqs 4 and 6, with E&#773;<sup>i</sup> = 2556 m<sup>2</sup>.mol<sup>-1</sup> for the LED emission spectrum.</caption><thead><tr><th rowspan="2">C<sub>cat</sub></th><th rowspan="2">C</th><th rowspan="2">τ<sub>A</sub><sup>i</sup> | τ<sub>S</sub><sup>L</sup></th><th colspan="2">Neglecting luminescence</th><th>Approximation</th></tr><tr><th>Reference</th><th>φ</th><th>φ<sup>(exp,0)</sup></th><th>φ<sup>(Φ=0)</sup> | δ(Φ = 0)</th><th>φ<sup>(SS)</sup> | δ(ΣS)</th></tr></thead><tbody><tr><td>0.075</td><td>0.05</td><td>2.22 | 1.19</td><td>0.76 ± 0.08</td><td>0.61 ± 0.08</td><td>0.62 ± 0.08 | -20%</td><td>0.74 ± 0.08 | -3%</td></tr><tr><td>0.1</td><td>0.2</td><td>8.87 | 4.77</td><td>0.47 ± 0.04</td><td>0.44 ± 0.04</td><td>0.41 ± 0.04 | -13%</td><td>0.48 ± 0.04 | 2%</td></tr><tr><td>0.1</td><td>0.45</td><td>19.95 | 10.74</td><td>0.25 ± 0.02</td><td>0.22 ± 0.02</td><td>0.22 ± 0.02 | -12%</td><td>0.26 ± 0.02 | 4%</td></tr></tbody></table>

https://doi.org/10.1371/journal.pone.0255002.t004

Carlo algorithm) are multiplied by the probability f<sub>λ</sub> that a photon absorbed is absorbed by Eosin Y molecules (rather than by the catalyst):

$$f_{\lambda} = \frac{k_{\lambda}(1 - \Phi)}{k_{\lambda}(1 - \Phi) + k_{cat,\lambda}} \qquad (65)$$

where Φ is equal to 0 when luminescence is neglected. Note that, integrated over the spectral range from 400 nm to 490 nm, factor f varies between 79% and 96% for the concentration range studied (see Table 4). Indeed, absorption by the catalyst is unfavorable for hydrogen production and should remain low.

The resulting formulation of absorptance when accounting for catalyst absorption is provided in S3 Appendix.

Reflection and refraction at the rear of the photoreactor are not included in the models, since the transmission of the water / glass / air interface is ≈ 90% for our spectral range of work (value obtained from both measurements and geometrical optics calculations). Moreover, less than 10% of the incident radiation reaches the rear glass for the absorption optical thicknesses studied in Table 4 (e<sup>-τ<sub>A</sub><sup>i</sup></sup> < 0.1). Therefore, only 1% of the incident radiation is reflected back into the medium.

### 5.2.3 Results gathered in Table 4 also include.

* Absorption and scattering optical thicknesses defined in Eqs 4 and 6, with E&#773;<sup>i</sup> = 2556 m<sup>2</sup>.mol<sup>-1</sup> for the LED emission spectrum (E&#773;<sup>L</sup> = 2728 m<sup>2</sup>.mol<sup>-1</sup> is unchanged, see Table 1). Absorption by the catalyst has very little impact on photon transport analysis and is therefore not included in these optical thickness calculations.

* The relative difference

$$\delta(X) = \frac{\varphi^{(X)} - \varphi}{\varphi} \qquad (66)$$

between the reference overall quantum yield φ (obtained with the reference MVRPA given by the Monte Carlo method) and the overall quantum yields φ<sup>(X)</sup> obtained from the MVRPA models denoted (X) above.

## 5.3 Discussion

The overall quantum yield φ in Table 4 is expected to vary as a function of Eosin Y concentration. The increase in φ when Eosin Y concentration decreases could be explained by quenching phenomena: collisions between Eosin Y excited states and other molecules (quenchers)
such as Et<sub>3</sub>N, H<sub>2</sub>O, or Eosin Y itself, for instance, limits the singlet and triplet state yield required for hydrogen production [18]. This effect is out of the scope of this article and we focus rather on the difference between φ values obtained using different MVRPA estimation methods.

**5.3.1 Comparison between measurements and model.** First, we compare the overall quantum yields obtained when luminescence is neglected, using both experimental MVRPA measurements in φ<sup>(exp,0)</sup> and model results in φ<sup>(Φ=0)</sup>. The results in Table 4 show no significant differences, validating the radiative properties of the model.

**5.3.2 Overall fluorescence effect.** Based on the results presented in Section 3, the analysis of optical thicknesses enables us to anticipate the error δ(Φ = 0) observed when fluorescence is neglected:

* For the scattering optical thicknesses $\tau_s^l > 1$ investigated in Table 4, we expect significant effects of fluorescence, leading to absorptance overestimation. According to Eq 66, we therefore expect the overall quantum yield φ to be underestimated when fluorescence is neglected (φ is inversely proportional to absorptance). Indeed, we verify that $\delta(\Phi = 0) < 0$ in Table 4.

* The first row in Table 4 addresses a configuration where $\tau_A^i = 2.22$ and $\tau_S^l = 1.19$, that is intermediate between the third and fourth columns in Table 2, for which we recorded errors $\Delta(\Phi = 0)$ in MVRPA estimation equal to 29.5% and 20.5% respectively. We therefore expect $\delta(\Phi = 0) \simeq 25\%$ in that configuration, which is confirmed.

* The second and third rows in Table 4 address situations with higher optical thicknesses. As discussed in Section 3.1, the error $\Delta(\Phi = 0)$ on MVRPA decreases as the absorption optical thickness $\tau_A^i$ increases, and indeed the error $|\delta(\Phi = 0)|$ decreases in Table 4. However, $\delta(\Phi = 0)$ values in the second and third rows are very close, despite the fact that optical thicknesses are twice as high in the third row. Indeed, we observe that the error on MVRPA decreases, without approaching zero since fluorescence radiation cannot be completely absorbed, even for very high concentrations. Here, we are approaching the asymptote.

These results confirm that fluorescence must be taken into account to accurately estimate the overall quantum yield; otherwise we are subject to a relative error between 12% and 20% for the experimental cases presented in this paper.

**5.3.3 Gray single-scattering analytical approximation.** Finally, in order to easily take fluorescence into account when estimating the overall quantum yield (without using an advanced numerical method such as Monte Carlo), we developed convenient analytical approximation solutions in Section 4. The validity of these approximations depends on the scattering optical thickness $\tau_S^l$ (see the summary in Conclusions and perspectives). The gray single-scattering approximation $\overline{SS}$ presented in Section 4.3 is the most relevant here, since $\tau_S^l > 1$ and spectral variations of Eosin Y are important. This approximation is remarkably accurate, leading to a relative error $\delta(\overline{SS})$ for the overall quantum yield estimation of less than 4% in every tested situation.

# 6 Conclusions and perspectives

We have presented a radiative model for luminescence radiation which can be used to study any photosensitized reactive system for photoreactor engineering applications. The analysis of luminescence radiation transport is approached from the multiple inelastic-scattering point of view, using Monte Carlo simulations, successive orders of scattering expansion formalism and five physical approximations that are combined to provide five convenient analytical approximate solutions. The study of four photosensitizers that are representative of photocatalytic
such as Et<sub>3</sub>N, H<sub>2</sub>O, or Eosin Y itself, for instance, limits the singlet and triplet state yield required for hydrogen production [18]. This effect is out of the scope of this article and we focus rather on the difference between φ values obtained using different MVRPA estimation methods.

**5.3.1 Comparison between measurements and model.** First, we compare the overall quantum yields obtained when luminescence is neglected, using both experimental MVRPA measurements in φ<sup>(exp,0)</sup> and model results in φ<sup>(Φ=0)</sup>. The results in Table 4 show no significant differences, validating the radiative properties of the model.

**5.3.2 Overall fluorescence effect.** Based on the results presented in Section 3, the analysis of optical thicknesses enables us to anticipate the error δ(Φ = 0) observed when fluorescence is neglected:

* For the scattering optical thicknesses $\tau_s^L > 1$ investigated in Table 4, we expect significant effects of fluorescence, leading to absorptance overestimation. According to Eq 66, we therefore expect the overall quantum yield φ to be underestimated when fluorescence is neglected (φ is inversely proportional to absorptance). Indeed, we verify that $\delta(\Phi = 0) < 0$ in Table 4.

* The first row in Table 4 addresses a configuration where $\tau_A^i = 2.22$ and $\tau_S^L = 1.19$, that is intermediate between the third and fourth columns in Table 2, for which we recorded errors $\Delta(\Phi = 0)$ in MVRPA estimation equal to 29.5% and 20.5% respectively. We therefore expect $\delta(\Phi = 0) \simeq 25\%$ in that configuration, which is confirmed.

* The second and third rows in Table 4 address situations with higher optical thicknesses. As discussed in Section 3.1, the error $\Delta(\Phi = 0)$ on MVRPA decreases as the absorption optical thickness $\tau_A^i$ increases, and indeed the error $|\delta(\Phi = 0)|$ decreases in Table 4. However, $\delta(\Phi = 0)$ values in the second and third rows are very close, despite the fact that optical thicknesses are twice as high in the third row. Indeed, we observe that the error on MVRPA decreases, without approaching zero since fluorescence radiation cannot be completely absorbed, even for very high concentrations. Here, we are approaching the asymptote.

These results confirm that fluorescence must be taken into account to accurately estimate the overall quantum yield; otherwise we are subject to a relative error between 12% and 20% for the experimental cases presented in this paper.

**5.3.3 Gray single-scattering analytical approximation.** Finally, in order to easily take fluorescence into account when estimating the overall quantum yield (without using an advanced numerical method such as Monte Carlo), we developed convenient analytical approximation solutions in Section 4. The validity of these approximations depends on the scattering optical thickness $\tau_S^L$ (see the summary in Conclusions and perspectives). The gray single-scattering approximation $\overline{SS}$ presented in Section 4.3 is the most relevant here, since $\tau_S^L > 1$ and spectral variations of Eosin Y are important. This approximation is remarkably accurate, leading to a relative error $\delta(\overline{SS})$ for the overall quantum yield estimation of less than 4% in every tested situation.

# 6 Conclusions and perspectives

We have presented a radiative model for luminescence radiation which can be used to study any photosensitized reactive system for photoreactor engineering applications. The analysis of luminescence radiation transport is approached from the multiple inelastic-scattering point of view, using Monte Carlo simulations, successive orders of scattering expansion formalism and five physical approximations that are combined to provide five convenient analytical approximate solutions. The study of four photosensitizers that are representative of photocatalytic
systems for solar fuel production by artificial photosynthesis leads us to the following conclusions (see Table 2):

* the proportion of incident radiation that is absorbed within photoreactive media is reduced in the presence of luminescence because 1) part of the incident radiation is backscattered and leaves the medium and 2) part of the spectral distribution is shifted to a non-absorbing spectral range;

* these effects are substantial and cannot be totally neglected when studying photoreactors;

* the zero-order scattering approximation in Eq 43 is a relevant approximation that is straightforward to implement and that improves the description of photon transport considerably compared to simply neglecting luminescence. The gray single-scattering approximation $\overline{SS}$ in Eq 59 is more accurate: it estimates absorptance with a relative error below 6% in every tested situation. In addition, it only requires the exponential integral function to be computed (available in most scientific computing tools and applications);

* when absorption optical thickness $\tau_{\lambda}^j$ is adjusted such that incident radiation is absorbed as much as possible and converted into solar fuels or valuable products, scattering optical thickness $\tau_S^L$ often exceeds 0.1 (see Eq 6). In this case, two major conceptual and practical difficulties must be tackled: inelastic scattering and multiple scattering;

* complex effects of inelastic scattering are due to non-gray extinction cross section $E_{\lambda}$ in the luminescence spectral range $[\lambda_{min}^L, \lambda_{max}^L]$ (see Fig 2): the greater the spectral variations and $\tau_S^L$, the more difficult these effects are to grasp;

* of course, the effect of multiple scattering increases with optical thickness: the contribution of scattering orders $j$ decreases with $j$, and the higher $\tau_S^L$ is, the larger the contribution of high scattering orders becomes. We record significant contributions up to $j = 3$ in the cases investigated;

* the following analytical approximate solutions can be used to describe photon absorption with an error below 5%:

* zero-order scattering approximation in Eq 43 when $\tau_S^L < 0.5$; it provides accurate results for $Ru[Bpy]_{3}^{2+}$ and TATA$^+$, representing half of the test cases investigated in this paper,

* inelastic single-scattering approximation in Eq 55 and $\overline{\text{P1}}$ approximation in Eq 53 when $\tau_S^L < 1$ and spectral variations in $E_{\lambda}$ in $[\lambda_{min}^L, \lambda_{max}^L]$ are significant; they provide accurate results for $Ru[Bpy]_{3}^{2+}$, TATA$^+$ and Eosin Y, but require double numerical integration;

* gray $\overline{\text{P1}}$ approximation in Eq 44 when spectral variations in $E_{\lambda}$ in $[\lambda_{min}^L, \lambda_{max}^L]$ are small (almost gray, which is unlikely) and $\tau_S^L > 0.5$; none of the studied photosensitizers corresponds to this case;

* gray single-scattering approximation $\overline{SS}$ in Eq 59 when spectral variations in $E_{\lambda}$ in $[\lambda_{min}^L, \lambda_{max}^L]$ are large, whatever $\tau_S^L$; it provides remarkably accurate results in every tested configuration, but this accuracy arises from the compensation of two errors. Therefore, even if this approximation is of great interest for the study of photocatalytic systems, its use requires some precautions.

* Cases with high scattering optical thickness combined with strong spectral variations in $E_{\lambda}$ in $[\lambda_{min}^L, \lambda_{max}^L]$, as for Rhodamine B, can hardly be treated with confidence in any way that
systems for solar fuel production by artificial photosynthesis leads us to the following conclusions (see Table 2):

* the proportion of incident radiation that is absorbed within photoreactive media is reduced in the presence of luminescence because 1) part of the incident radiation is backscattered and leaves the medium and 2) part of the spectral distribution is shifted to a non-absorbing spectral range;

* these effects are substantial and cannot be totally neglected when studying photoreactors;

* the zero-order scattering approximation in Eq 43 is a relevant approximation that is straightforward to implement and that improves the description of photon transport considerably compared to simply neglecting luminescence. The gray single-scattering approximation $\overline{SS}$ in Eq 59 is more accurate: it estimates absorptance with a relative error below 6% in every tested situation. In addition, it only requires the exponential integral function to be computed (available in most scientific computing tools and applications);

* when absorption optical thickness $\tau_{\lambda}^j$ is adjusted such that incident radiation is absorbed as much as possible and converted into solar fuels or valuable products, scattering optical thickness $\tau_S^L$ often exceeds 0.1 (see Eq 6). In this case, two major conceptual and practical difficulties must be tackled: inelastic scattering and multiple scattering;

* complex effects of inelastic scattering are due to non-gray extinction cross section $E_{\lambda}$ in the luminescence spectral range [$\lambda_{min}^L, \lambda_{max}^L$] (see Fig 2): the greater the spectral variations and $\tau_S^L$, the more difficult these effects are to grasp;

* of course, the effect of multiple scattering increases with optical thickness: the contribution of scattering orders $j$ decreases with $j$, and the higher $\tau_S^L$ is, the larger the contribution of high scattering orders becomes. We record significant contributions up to $j = 3$ in the cases investigated;

* the following analytical approximate solutions can be used to describe photon absorption with an error below 5%:

* zero-order scattering approximation in Eq 43 when $\tau_S^L < 0.5$; it provides accurate results for $Ru[Bpy]_{3}^{2+}$ and TATA$^+$, representing half of the test cases investigated in this paper,

* inelastic single-scattering approximation in Eq 55 and $\overline{\text{P1}}$ approximation in Eq 53 when $\tau_S^L < 1$ and spectral variations in $E_{\lambda}$ in [$\lambda_{min}^L, \lambda_{max}^L$] are significant; they provide accurate results for $Ru[Bpy]_{3}^{2+}$, TATA$^+$ and Eosin Y, but require double numerical integration;

* gray $\overline{\text{P1}}$ approximation in Eq 44 when spectral variations in $E_{\lambda}$ in [$\lambda_{min}^L, \lambda_{max}^L$] are small (almost gray, which is unlikely) and $\tau_S^L > 0.5$; none of the studied photosensitizers corresponds to this case;

* gray single-scattering approximation $\overline{SS}$ in Eq 59 when spectral variations in $E_{\lambda}$ in [$\lambda_{min}^L, \lambda_{max}^L$] are large, whatever $\tau_S^L$; it provides remarkably accurate results in every tested configuration, but this accuracy arises from the compensation of two errors. Therefore, even if this approximation is of great interest for the study of photocatalytic systems, its use requires some precautions.

* Cases with high scattering optical thickness combined with strong spectral variations in $E_{\lambda}$ in [$\lambda_{min}^L, \lambda_{max}^L$], as for Rhodamine B, can hardly be treated with confidence in any way that
does not involve advanced numerical methods such as Monte Carlo. Even if the $\overline{\text{SS}}$ approxi-
mation provides surprisingly accurate results for Rhodamine B, we cannot ensure *a priori*
that this is the case for every photosensitizer.

The work presented in this article also allows us to take another look at overall quantum
yield estimation. The overall quantum yield of a photoreactive system composed of Eosin Y as
the photosensitizer and a bio-inspired catalyst for H₂ production was initially obtained without
accounting for the fluorescence of Eosin Y in the treatment of experimental results [18]. We
revisited these results here and show that:

* neglecting fluorescence on absorptance clearly affects the results of overall quantum yields, which are underestimated since absorptance is overestimated,

* according to the optical thickness of the experimental configuration, the gray single-scattering analytical approximation was selected. It provides results with an error of less than 4% on overall quantum yield estimation, enabling fluorescence to be taken into account without having to implement an advanced numerical method such as Monte Carlo.

Some interesting perspectives can also be drawn up. First of all, this work could be extended
to the evaluation of the mean volumetric rate of photon absorption $\langle A \rangle$, and to the estimation
of transmittance in a 1D Cartesian system (photoreactors or spectrophotometer cuvette). This
should not require too much additional work, since all the formal work presented in this article
is performed at local intensity, from which it is easy to evaluate transmittances (cosine-
weighted integration over the outgoing hemisphere) instead of absorptance. For the same rea-
sons, the assessment of the local rate of photon absorption can be implemented without major
difficulty: the final integration over reaction volume is simply suppressed in the formulations
of absorptance presented in this paper. This is necessary when dealing with a non-linear ther-
mokinetic coupling law [16, 50]. This work could also be extended by considering reflection at
the interfaces at both sides of the slab or using uncollimated incident radiation. An extension
of the 1D algorithm toward any complex 3D geometry for photoreactive applications is also at
hand thanks to the computational tools for ray tracing in complex geometry that have been
developed over the last twenty years by the computer graphics research community in the
Monte Carlo framework [47, 56].

Finally, many of the tools presented and discussed in this work could find interesting appli-
cations for the quantitative use of Fast Fluorescence Rate (FFR) or Pulse Amplitude Modula-
tion (PAM) fluorescence measurements using microalgae in the field of photobioreactor
engineering [57], as well as remote sensing [58, 59]. To do so, elastic scattering by particles will
have to be included in the models.

**Supporting information**

S1 Appendix. P1 approximation.
(ZIP)

S2 Appendix. Single-scattering approximation.
(ZIP)

S3 Appendix. Monte Carlo algorithm and extended models with catalyst absorption.
(ZIP)

S1 Nomenclature.
(PDF)
does not involve advanced numerical methods such as Monte Carlo. Even if the SS approximation provides surprisingly accurate results for Rhodamine B, we cannot ensure *a priori* that this is the case for every photosensitizer.

The work presented in this article also allows us to take another look at overall quantum yield estimation. The overall quantum yield of a photoreactive system composed of Eosin Y as the photosensitizer and a bio-inspired catalyst for H₂ production was initially obtained without accounting for the fluorescence of Eosin Y in the treatment of experimental results [18]. We revisited these results here and show that:

* neglecting fluorescence on absorptance clearly affects the results of overall quantum yields, which are underestimated since absorptance is overestimated,

* according to the optical thickness of the experimental configuration, the gray single-scattering analytical approximation was selected. It provides results with an error of less than 4% on overall quantum yield estimation, enabling fluorescence to be taken into account without having to implement an advanced numerical method such as Monte Carlo.

Some interesting perspectives can also be drawn up. First of all, this work could be extended to the evaluation of the mean volumetric rate of photon absorption $\langle A \rangle$, and to the estimation of transmittance in a 1D Cartesian system (photoreactors or spectrophotometer cuvette). This should not require too much additional work, since all the formal work presented in this article is performed at local intensity, from which it is easy to evaluate transmittances (cosine-weighted integration over the outgoing hemisphere) instead of absorptance. For the same reasons, the assessment of the local rate of photon absorption can be implemented without major difficulty: the final integration over reaction volume is simply suppressed in the formulations of absorptance presented in this paper. This is necessary when dealing with a non-linear thermokinetic coupling law [16, 50]. This work could also be extended by considering reflection at the interfaces at both sides of the slab or using uncollimated incident radiation. An extension of the 1D algorithm toward any complex 3D geometry for photoreactive applications is also at hand thanks to the computational tools for ray tracing in complex geometry that have been developed over the last twenty years by the computer graphics research community in the Monte Carlo framework [47, 56].

Finally, many of the tools presented and discussed in this work could find interesting applications for the quantitative use of Fast Fluorescence Rate (FFR) or Pulse Amplitude Modulation (PAM) fluorescence measurements using microalgae in the field of photobioreactor engineering [57], as well as remote sensing [58, 59]. To do so, elastic scattering by particles will have to be included in the models.

**Supporting information**

**S1 Appendix. P1 approximation.**
(ZIP)

**S2 Appendix. Single-scattering approximation.**
(ZIP)

**S3 Appendix. Monte Carlo algorithm and extended models with catalyst absorption.**
(ZIP)

**S1 Nomenclature.**
(PDF)
**Acknowledgments**

We gratefully acknowledge the CNRS research federation FedEsol, where fruitful discussions and debates have been taking place every year since 2012.

**Author Contributions**

**Conceptualization:** Caroline Supplis.

**Formal analysis:** Caroline Supplis, Jérémi Dauchet, Jean-François Cornet.

**Investigation:** Caroline Supplis, Jean-François Cornet.

**Methodology:** Caroline Supplis, Jérémi Dauchet, Thomas Vourc'h, Jean-François Cornet.

**Supervision:** Caroline Supplis, Jean-François Cornet.

**Validation:** Caroline Supplis, Victor Gattepaille, Fabrice Gros, Thomas Vourc'h.

**Writing – original draft:** Caroline Supplis, Jérémi Dauchet, Jean-François Cornet.

**References**

1. Braun AM, Maurette MT, Oliveros E. Photochemical Technology. John Wiley & Sons Ltd; 1991.

2. Albini A, Fagnoni M. Green chemistry and photochemistry were born at the same time. Green Chemistry. 2004; 6:1–6. https://doi.org/10.1039/b309592d

3. Oelgemoller M, Jung C, Mattay J. Green photochemistry: Production of fine chemicals with sunlight. Pure and Applied Chemistry. 2007; 79(11):1939–1947. https://doi.org/10.1351/pac200779111939

4. Tachibana Y, Vayssieres L, Durrant JR. Artificial photosynthesis for solar water-splitting. Nature Photonics. 2012; 6(8):511–518. https://doi.org/10.1038/nphoton.2012.175

5. Corredor J, Rivero MJ, Rangel CM, Gloaguen F, Ortiz I. Comprehensive review and future perspectives on the photocatalytic hydrogen production. Journal of Chemical Technology & Biotechnology. 2019; 94 (10):3049–3063. https://doi.org/10.1002/jctb.6123

6. Lan Y, Chuang H, Min-Quan Y, Yi-Jun X. Photocatalytic water splitting for solar hydrogen generation: fundamentals and recent advancements. International Reviews in Physical Chemistry. 2016; 35(1):1–36. https://doi.org/10.1080/0144235X.2015.1127027

7. Berardi S, Drouet S, Francas L, Gimbert-Surinach C, Guttentag M, Richmond C, et al. Molecular artificial photosynthesis. Chemical Society Reviews. 2014; 43:7501–7519. https://doi.org/10.1039/C3CS60405E PMID: 24473472

8. Izumi Y. Recent advances in the photocatalytic conversion of carbon dioxide to fuels with water and/or hydrogen using solar energy and beyond. Coordination Chemistry Reviews. 2013; 257(1):171–186. https://doi.org/10.1016/j.ccr.2012.04.018

9. Roy SC, Varghese OK, Paulose M, Grimes CA. Toward Solar Fuels: Photocatalytic Conversion of Carbon Dioxide to Hydrocarbons. American Chemical Society Nano. 2010; 4(3):1259–1278. PMID: 20141175

10. Kudo A, Miseki Y. Heterogeneous photocatalyst materials for water splitting. Chemical Society Reviews. 2009; 38:253–278. https://doi.org/10.1039/B800489G PMID: 19088977

11. Li J Xand Yu, Low J, Fang Y, Xiao J, Chen X. Engineering heterogeneous semiconductors for solar water splitting. Journal of Materials Chemistry A. 2015; 3:2485–2534. https://doi.org/10.1039/C4TA04461D

12. Chen S, Takata T, Domen K. Particulate photocatalysts for overall water splitting. Nature Reviews Materials. 2017; 2(10):17050. https://doi.org/10.1038/natrevmats.2017.50

13. Zhang T, Lin W. Metal–organic frameworks for artificial photosynthesis and photocatalysis. Chemical Society Reviews. 2014; 43:5982–5993. https://doi.org/10.1039/C4CS00103F PMID: 24769551

14. Wang F, Wang WG, Wang HY, Si G, Tung CH, Wu LZ. Artificial Photosynthetic Systems Based on [FeFe]-Hydrogenase Mimics: the Road to High Efficiency for Light-Driven Hydrogen Evolution. American Chemical Society Catalysis. 2012; 2(3):407–416. https://doi.org/10.1021/cs200458b

15. Simmons T, Berggren G, Bacchi M, Fontecave M, Artero V. Mimicking hydrogenases: From biomimetics to artificial enzymes. Coordination Chemistry Reviews. 2014; 270-271:127–150. https://doi.org/10.1016/j.ccr.2013.12.018
**Acknowledgments**

We gratefully acknowledge the CNRS research federation FedEsol, where fruitful discussions and debates have been taking place every year since 2012.

**Author Contributions**

**Conceptualization:** Caroline Supplis.

**Formal analysis:** Caroline Supplis, Jérémi Dauchet, Jean-François Cornet.

**Investigation:** Caroline Supplis, Jean-François Cornet.

**Methodology:** Caroline Supplis, Jérémi Dauchet, Thomas Vourc'h, Jean-François Cornet.

**Supervision:** Caroline Supplis, Jean-François Cornet.

**Validation:** Caroline Supplis, Victor Gattepaille, Fabrice Gros, Thomas Vourc'h.

**Writing – original draft:** Caroline Supplis, Jérémi Dauchet, Jean-François Cornet.

**References**

1. Braun AM, Maurette MT, Oliveros E. Photochemical Technology. John Wiley & Sons Ltd; 1991.

2. Albini A, Fagnoni M. Green chemistry and photochemistry were born at the same time. Green Chemistry. 2004; 6:1–6. https://doi.org/10.1039/b309592d

3. Oelgemoller M, Jung C, Mattay J. Green photochemistry: Production of fine chemicals with sunlight. Pure and Applied Chemistry. 2007; 79(11):1939–1947. https://doi.org/10.1351/pac200779111939

4. Tachibana Y, Vayssieres L, Durrant JR. Artificial photosynthesis for solar water-splitting. Nature Photonics. 2012; 6(8):511–518. https://doi.org/10.1038/nphoton.2012.175

5. Corredor J, Rivero MJ, Rangel CM, Gloaguen F, Ortiz I. Comprehensive review and future perspectives on the photocatalytic hydrogen production. Journal of Chemical Technology & Biotechnology. 2019; 94 (10):3049–3063. https://doi.org/10.1002/jctb.6123

6. Lan Y, Chuang H, Min-Quan Y, Yi-Jun X. Photocatalytic water splitting for solar hydrogen generation: fundamentals and recent advancements. International Reviews in Physical Chemistry. 2016; 35(1):1–36. https://doi.org/10.1080/0144235X.2015.1127027

7. Berardi S, Drouet S, Francas L, Gimbert-Surinach C, Guttentag M, Richmond C, et al. Molecular artificial photosynthesis. Chemical Society Reviews. 2014; 43:7501–7519. https://doi.org/10.1039/C3CS60405E PMID: 24473472

8. Izumi Y. Recent advances in the photocatalytic conversion of carbon dioxide to fuels with water and/or hydrogen using solar energy and beyond. Coordination Chemistry Reviews. 2013; 257(1):171–186. https://doi.org/10.1016/j.ccr.2012.04.018

9. Roy SC, Varghese OK, Paulose M, Grimes CA. Toward Solar Fuels: Photocatalytic Conversion of Carbon Dioxide to Hydrocarbons. American Chemical Society Nano. 2010; 4(3):1259–1278. PMID: 20141175

10. Kudo A, Miseki Y. Heterogeneous photocatalyst materials for water splitting. Chemical Society Reviews. 2009; 38:253–278. https://doi.org/10.1039/B800489G PMID: 19088977

11. Li J Xand Yu, Low J, Fang Y, Xiao J, Chen X. Engineering heterogeneous semiconductors for solar water splitting. Journal of Materials Chemistry A. 2015; 3:2485–2534. https://doi.org/10.1039/C4TA04461D

12. Chen S, Takata T, Domen K. Particulate photocatalysts for overall water splitting. Nature Reviews Materials. 2017; 2(10):17050. https://doi.org/10.1038/natrevmats.2017.50

13. Zhang T, Lin W. Metal–organic frameworks for artificial photosynthesis and photocatalysis. Chemical Society Reviews. 2014; 43:5982–5993. https://doi.org/10.1039/C4CS00103F PMID: 24769551

14. Wang F, Wang WG, Wang HY, Si G, Tung CH, Wu LZ. Artificial Photosynthetic Systems Based on [FeFe]-Hydrogenase Mimics: the Road to High Efficiency for Light-Driven Hydrogen Evolution. American Chemical Society Catalysis. 2012; 2(3):407–416. https://doi.org/10.1021/cs200458b

15. Simmons T, Berggren G, Bacchi M, Fontecave M, Artero V. Mimicking hydrogenases: From biomimetics to artificial enzymes. Coordination Chemistry Reviews. 2014; 270-271:127–150. https://doi.org/10.1016/j.ccr.2013.12.018
16. Cassano AE, Martin CA, Brandi RJ, Alfano OM. Photoreactor Analysis and Design: Fundamentals and Applications. Industrial & Engineering Chemistry Research. 1995; 34(7):2155-2201. https://doi.org/10.1021/ie00046a001

17. Spadoni G, Bandini E, Santarelli F. Scattering effects in photosensitized reactions. Chemical Engineering Science. 1978; 33(4):517-524. https://doi.org/10.1016/0009-2509(78)80012-8

18. Supplis C, Gros F, Dahi G, Dauchet J, Roudet M, Gloaguen F, et al. Spectral radiative analysis of bio-inspired H₂ production in a benchmark photoreactor: A first investigation using spatial photonic balance. International Journal of Hydrogen Energy. 2018; 43(17):8221-8231. https://doi.org/10.1016/j.ijhydene.2018.03.097

19. Rochatte V, Dahi G, Eskandari A, Dauchet J, Gros F, Roudet M, et al. Radiative transfer approach using Monte Carlo Method for actinometry in complex geometry and its application to Reinecke salt photodissociation within innovative pilot-scale photo(bio)reactors. Chemical Engineering Journal. 2017; 308:940-953. https://doi.org/10.1016/j.cej.2016.08.112

20. Cornet JF, Dussap CG, Gros JB. Conversion of radiant light energy in photobioreactors. American Institute of Chemical Engineers Journal. 1994; 40(6):1055-1066. https://doi.org/10.1002/aic.690400616

21. Orain C, Quentel F, Gloaguen F. Photocatalytic Hydrogen Production Using Models of the Iron-Iron Hydrogenase Active Site Dispersed in Micellar Solution. ChemSusChem. 2014; 7(2):638-643. https://doi.org/10.1002/cssc.201300631 PMID: 24127363

22. Wang XB, Zheng HQ, Rao H, Yao HC, Fan YT, Hou HW. Synthesis of a new iron-sulfur cluster compound and its photocatalytic H₂ evolution activity through visible light irradiation. Applied Organometallic Chemistry. 2016; 30(8):638-644. https://doi.org/10.1002/aoc.3481

23. Hui-Qin Z, Heng R, Jun W, Yao-Ting F, Hong-Wei H. Synthesis and photo-catalytic H₂ evolution of three novel biomimetic photocatalysts based on [FeFe]-Hases model compound. Journal of Power Sources. 2015; 273:1038-1047. https://doi.org/10.1016/j.jpowsour.2014.09.172

24. Hui-Qin Z, Xiao-Bo W, Ji-Yong H, Jin-An Z, Chen-Xia D, Yao-Ting F, et al. Photo-catalytic H₂ evolution, structural effect and electron transfer mechanism based on four novel [Fe₂S₂] model complexes by photochemical splitting water. Solar Energy. 2016; 132:373-385. https://doi.org/10.1016/j.solener.2016.03.010

25. Gueret R, Poulard L, Oshinowo M, Chauvin J, Dahmane M, Dupeyre G, et al. Challenging the [Ru(bpy)₃]²⁺ Photosensitizer with a Triazatriangulinium Robust Organic Dye for Visible-Light-Driven Hydrogen Production in Water. American Chemical Society Catalysis. 2018; 8(5):3792-3802. https://doi.org/10.1021/acscatal.7b04000

26. Karstens T, Kobs K. Rhodamine B and rhodamine 101 as reference substances for fluorescence quantum yield measurements. The Journal of Physical Chemistry. 1980; 84(14):1871-1872. https://doi.org/10.1021/j100451a030

27. McCormick TM, Calitree BD, Orchard A, Kraut ND, Bright FV, Detty MR, et al. Reductive Side of Water Splitting in Artificial Photosynthesis: New Homogeneous Photosystems of Great Activity and Mechanical Insight. Journal of the American Chemical Society. 2010; 132(44):15480-15483. https://doi.org/10.1021/ja1057357 PMID: 20945839

28. Quaranta A, Lachaud F, Herrero C, Guillot R, Charlot MF, Leibl W, et al. Influence of the Protonic State of an Imidazole-Containing Ligand on the Electrochemical and Photophysical Properties of a Rutheinium(II)-Polypyridine-Type Complex. Chemistry—A European Journal. 2007; 13(29):8201-8211. https://doi.org/10.1002/chem.200700185 PMID: 17639540

29. Herrero C, Hughes JL, Quaranta A, Cox N, Rutherford AW, Leibl W, et al. Intramolecular light induced activation of a Salen-Mn¹¹¹ complex by a ruthenium photosensitizer. Chemical Communications. 2010; 46(40):7605. https://doi.org/10.1039/c0cc01710h PMID: 20835470

30. Klose AD, Hielscher AH. Fluorescence tomography with simulated data based on the equation of radiative transfer. Optics Letters. 2003; 28(12):1019. https://doi.org/10.1364/OL.28.001019 PMID: 12836765

31. Klose AD, Ntziachristos V, Hielscher AH. The inverse source problem based on the radiative transfer equation in optical molecular imaging. Journal of Computational Physics. 2005; 202(1):323-345. https://doi.org/10.1016/j.jcp.2004.07.008

32. Cerezo E, Seron FJ. Inelastic Scattering and Participating Media. Application to the ocean. In: Eurographics 2003—Short Presentations. Eurographics Association; 2003.

33. Sobouti Y. Fluorescent Scattering in Planetary Atmospheres. II. Coupling among Transitions. Astrophysical Journal. 1962; 135:938. https://doi.org/10.1086/147336

34. Haltrin VI, Kattawar GW. Self-consistent solutions to the equation of transfer with elastic and inelastic scattering in oceanic optics. I. Model. Applied Optics. 1993; 32(27):5356-5367. https://doi.org/10.1364/AO.32.005356 PMID: 20856345
16. Cassano AE, Martin CA, Brandi RJ, Alfano OM. Photoreactor Analysis and Design: Fundamentals and Applications. Industrial & Engineering Chemistry Research. 1995; 34(7):2155-2201. https://doi.org/10.1021/ie00046a001

17. Spadoni G, Bandini E, Santarelli F. Scattering effects in photosensitized reactions. Chemical Engineering Science. 1978; 33(4):517-524. https://doi.org/10.1016/0009-2509(78)80012-8

18. Supplis C, Gros F, Dahi G, Dauchet J, Roudet M, Gloaguen F, et al. Spectral radiative analysis of bio-inspired H₂ production in a benchmark photoreactor: A first investigation using spatial photonic balance. International Journal of Hydrogen Energy. 2018; 43(17):8221-8231. https://doi.org/10.1016/j.ijhydene.2018.03.097

19. Rochatte V, Dahi G, Eskandari A, Dauchet J, Gros F, Roudet M, et al. Radiative transfer approach using Monte Carlo Method for actinometry in complex geometry and its application to Reinecke salt photodissociation within innovative pilot-scale photo(bio)reactors. Chemical Engineering Journal. 2017; 308:940-953. https://doi.org/10.1016/j.cej.2016.08.112

20. Cornet JF, Dussap CG, Gros JB. Conversion of radiant light energy in photobioreactors. American Institute of Chemical Engineers Journal. 1994; 40(6):1055-1066. https://doi.org/10.1002/aic.690400616

21. Orain C, Quentel F, Gloaguen F. Photocatalytic Hydrogen Production Using Models of the Iron-Iron Hydrogenase Active Site Dispersed in Micellar Solution. ChemSusChem. 2014; 7(2):638-643. https://doi.org/10.1002/cssc.201300631 PMID: 24127363

22. Wang XB, Zheng HQ, Rao H, Yao HC, Fan YT, Hou HW. Synthesis of a new iron-sulfur cluster compound and its photocatalytic H₂ evolution activity through visible light irradiation. Applied Organometallic Chemistry. 2016; 30(8):638-644. https://doi.org/10.1002/aoc.3481

23. Hui-Qin Z, Heng R, Jun W, Yao-Ting F, Hong-Wei H. Synthesis and photo-catalytic H₂ evolution of three novel biomimetic photocatalysts based on [FeFe]-Hases model compound. Journal of Power Sources. 2015; 273:1038-1047. https://doi.org/10.1016/j.jpowsour.2014.09.172

24. Hui-Qin Z, Xiao-Bo W, Ji-Yong H, Jin-An Z, Chen-Xia D, Yao-Ting F, et al. Photo-catalytic H₂ evolution, structural effect and electron transfer mechanism based on four novel [Fe₂S₂] model complexes by photochemical splitting water. Solar Energy. 2016; 132:373-385. https://doi.org/10.1016/j.solener.2016.03.010

25. Gueret R, Poulard L, Oshinowo M, Chauvin J, Dahmane M, Dupeyre G, et al. Challenging the [Ru(bpy)₃]²⁺ Photosensitizer with a Triazatriangulinium Robust Organic Dye for Visible-Light-Driven Hydrogen Production in Water. American Chemical Society Catalysis. 2018; 8(5):3792-3802. https://doi.org/10.1021/acscatal.7b04000

26. Karstens T, Kobs K. Rhodamine B and rhodamine 101 as reference substances for fluorescence quantum yield measurements. The Journal of Physical Chemistry. 1980; 84(14):1871-1872. https://doi.org/10.1021/j100451a030

27. McCormick TM, Calitree BD, Orchard A, Kraut ND, Bright FV, Detty MR, et al. Reductive Side of Water Splitting in Artificial Photosynthesis: New Homogeneous Photosystems of Great Activity and Mechanical Insight. Journal of the American Chemical Society. 2010; 132(44):15480-15483. https://doi.org/10.1021/ja1057357 PMID: 20945839

28. Quaranta A, Lachaud F, Herrero C, Guillot R, Charlot MF, Leibl W, et al. Influence of the Protonic State of an Imidazole-Containing Ligand on the Electrochemical and Photophysical Properties of a Rutheinium(II)-Polypyridine-Type Complex. Chemistry—A European Journal. 2007; 13(29):8201-8211. https://doi.org/10.1002/chem.200700185 PMID: 17639540

29. Herrero C, Hughes JL, Quaranta A, Cox N, Rutherford AW, Leibl W, et al. Intramolecular light induced activation of a Salen-Mn¹¹¹ complex by a ruthenium photosensitizer. Chemical Communications. 2010; 46(40):7605. https://doi.org/10.1039/c0cc01710h PMID: 20835470

30. Klose AD, Hielscher AH. Fluorescence tomography with simulated data based on the equation of radiative transfer. Optics Letters. 2003; 28(12):1019. https://doi.org/10.1364/OL.28.001019 PMID: 12836765

31. Klose AD, Ntziachristos V, Hielscher AH. The inverse source problem based on the radiative transfer equation in optical molecular imaging. Journal of Computational Physics. 2005; 202(1):323-345. https://doi.org/10.1016/j.jcp.2004.07.008

32. Cerezo E, Seron FJ. Inelastic Scattering and Participating Media. Application to the ocean. In: Eurographics 2003—Short Presentations. Eurographics Association; 2003.

33. Sobouti Y. Fluorescent Scattering in Planetary Atmospheres. II. Coupling among Transitions. Astrophysical Journal. 1962; 135:938. https://doi.org/10.1086/147336

34. Haltrin VI, Kattawar GW. Self-consistent solutions to the equation of transfer with elastic and inelastic scattering in oceanic optics. I. Model. Applied Optics. 1993; 32(27):5356-5367. https://doi.org/10.1364/AO.32.005356 PMID: 20856345
35. Ishimaru A. Wave Propagation and Scattering in Random Media. John Wiley & Sons; 1978. Available from: http://www.sciencedirect.com/science/article/pii/B9780123747013500053.

36. Omary MA, Patterson HH. Luminescence, Theory. In: Lindon J, Tranter GE, Koppenaal DW, editors. Encyclopedia of Spectroscopy and Spectrometry (Third Edition). third edition ed. Oxford: Academic Press; 2017. p. 636–653.

37. Klan P, Wirz J. Techniques and Methods. In: Photochemistry of Organic Compounds. John Wiley & Sons, Ltd; 2009. p. 73–135. Available from: https://onlinelibrary.wiley.com/doi/abs/10.1002/9781444300017.ch3.

38. Lakowicz JR. Principles of Fluorescence Spectroscopy. Third ed. Springer, Boston, MA; 1983. Available from: https://doi.org/10.1007/978-1-4615-7658-7.

39. Jameson DM. Introduction to Fluorescence. First ed. CRC Press; 2014.

40. Myneni RB, Asrar GR, Kanemasu ET. Light scattering in plant canopies: The method of Successive Orders of Scattering Approximations (SOSA). Agricultural and Forest Meteorology. 1987; 39(1):1–12.
https://doi.org/10.1016/0168-1923(87)90011-6

41. Zhai PW, Hu Y, Chowdhardy J, Trepte CR, Lucker PL, Josset DB. A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface. Journal of Quantitative Spectroscopy and Radiative Transfer. 2010; 111(7):1025–1040. https://doi.org/10.1016/j.jqsrt.2009.12.005

42. Min Q, Duan M. A successive order of scattering model for solving vector radiative transfer in the atmosphere. Journal of Quantitative Spectroscopy and Radiative Transfer. 2004; 87(3):243–259. https://doi.org/10.1016/j.jqsrt.2003.12.019

43. Liou KN, Rao N. Radiative Transfer in Cirrus Clouds. Part IV: On Cloud Geometry, Inhomogeneity, and Absorption. Journal of the Atmospheric Sciences. 1996; 53(21):3046–3065. https://doi.org/10.1175/1520-0469(1996)053%3C3046:RTICCP%3E2.0.CO;2

44. Hou W, Yin Q, Xu H, Li L, Chen Z. A comparison of two stream approximation for the discrete ordinate method and the SOS method. In: 2010 IEEE International Geoscience and Remote Sensing Symposium; 2010. p. 3580–3583.

45. Van De Hulst HC. Multiple Light Scattering. Academic Press; 1980.

46. Wauben W, De Haan JF, Hovenier JW. Low orders of scattering in a plane-parallel homogeneous atmosphere. Astronomy and Astrophysics. 1993; 276:589.

47. Delatorre J, Baud G, Bézian JJ, Blanco S, Caliot C, Cornet JF, et al. Monte Carlo advances and concentrated solar applications. Solar Energy. 2014; 103:653–681. https://doi.org/10.1016/j.solener.2013.02.035

48. Zhou M, Robertson GP, Roovers J. Comparative Study of Ruthenium(II) Tris(bipyridine) Derivatives for Electrochemiluminescence Application. Inorganic Chemistry. 2005; 44(23):8317–8325. https://doi.org/10.1021/ic0510112 PMID: 16270970

49. Spectra Database hosted at the University of Arizona; http://www.spectra.arizona.edu [Accessed: 2018-08-28].

50. Dauchet J, Cornet JF, Gros F, Roudet M, Dussap CG. Photobioreactor Modeling and Radiative Transfer Analysis for Engineering Purposes. In: Advances in Chemical Engineering. vol. 48. Elsevier; 2016. p. 1–106. Available from: https://linkinghub.elsevier.com/retrieve/pii/S0065237715000162.

51. Gloaguen F. Electrochemistry of Simple Organometallic Models of Iron–Iron Hydrogenases in Organic Solvent and Water. Inorganic Chemistry. 2016; 55(2):390–398. https://doi.org/10.1021/acs.inorgchem.5b02245 PMID: 26641526

52. Quentel F, Passard G, Gloaguen F. Electrochemical hydrogen production in aqueous micellar solution by a diiron benzenedithiolate complex relevant to [FeFe] hydrogenases. Energy Environ Sci. 2012; 5:7757–7761. https://doi.org/10.1039/c2ee21531d

53. Parker CA, Hatchard CG. Triplet-singlet emission in fluid solutions. Phosphorescence of eosin. Trans Faraday Soc. 1961; 57:1894–1904. https://doi.org/10.1039/tf9615701894

54. Supplis C. Modélisation et étude expérimentale de la production d'hydrogène solaire en photoréacteur [Theses]. Université Clermont Auvergne; 2020. Available from: https://tel.archives-ouvertes.fr/tel-03125605.

55. Dahi G, Eskandari A, Dauchet J, Gros F, Roudet M, Cornet JF. A novel experimental bench dedicated to the accurate radiative analysis of photoreactors: The case study of CdS catalyzed hydrogen production from sacrificial donors. Chemical Engineering and Processing: Process Intensification. 2015; 98:174–186. https://doi.org/10.1016/j.cep.2015.09.015

56. Dauchet J, Blanco S, Cornet JF, El Hafi M, Eymet V, Fournier R. The practice of recent radiative transfer Monte Carlo advances and its contribution to the field of microorganisms cultivation in photobioreactors. Journal of Quantitative Spectroscopy and Radiative Transfer. 2013; 128:52–59. https://doi.org/10.1016/j.jqsrt.2012.07.004
35. Ishimaru A. Wave Propagation and Scattering in Random Media. John Wiley & Sons; 1978. Available from: http://www.sciencedirect.com/science/article/pii/B9780123747013500053.

36. Omary MA, Patterson HH. Luminescence, Theory. In: Lindon J, Tranter GE, Koppenaal DW, editors. Encyclopedia of Spectroscopy and Spectrometry (Third Edition). third edition ed. Oxford: Academic Press; 2017. p. 636–653.

37. Klan P, Wirz J. Techniques and Methods. In: Photochemistry of Organic Compounds. John Wiley & Sons, Ltd; 2009. p. 73–135. Available from: https://onlinelibrary.wiley.com/doi/abs/10.1002/9781444300017.ch3.

38. Lakowicz JR. Principles of Fluorescence Spectroscopy. Third ed. Springer, Boston, MA; 1983. Available from: https://doi.org/10.1007/978-1-4615-7658-7.

39. Jameson DM. Introduction to Fluorescence. First ed. CRC Press; 2014.

40. Myneni RB, Asrar GR, Kanemasu ET. Light scattering in plant canopies: The method of Successive Orders of Scattering Approximations (SOSA). Agricultural and Forest Meteorology. 1987; 39(1):1–12. https://doi.org/10.1016/0168-1923(87)90011-6

41. Zhai PW, Hu Y, Chowdhardy J, Trepte CR, Lucker PL, Josset DB. A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface. Journal of Quantitative Spectroscopy and Radiative Transfer. 2010; 111(7):1025–1040. https://doi.org/10.1016/j.jqsrt.2009.12.005

42. Min Q, Duan M. A successive order of scattering model for solving vector radiative transfer in the atmosphere. Journal of Quantitative Spectroscopy and Radiative Transfer. 2004; 87(3):243–259. https://doi.org/10.1016/j.jqsrt.2003.12.019

43. Liou KN, Rao N. Radiative Transfer in Cirrus Clouds. Part IV: On Cloud Geometry, Inhomogeneity, and Absorption. Journal of the Atmospheric Sciences. 1996; 53(21):3046–3065. https://doi.org/10.1175/1520-0469(1996)053%3C3046:RTICCP%3E2.0.CO;2

44. Hou W, Yin Q, Xu H, Li L, Chen Z. A comparison of two stream approximation for the discrete ordinate method and the SOS method. In: 2010 IEEE International Geoscience and Remote Sensing Symposium; 2010. p. 3580–3583.

45. Van De Hulst HC. Multiple Light Scattering. Academic Press; 1980.

46. Wauben W, De Haan JF, Hovenier JW. Low orders of scattering in a plane-parallel homogeneous atmosphere. Astronomy and Astrophysics. 1993; 276:589.

47. Delatorre J, Baud G, Bézian JJ, Blanco S, Caliot C, Cornet JF, et al. Monte Carlo advances and concentrated solar applications. Solar Energy. 2014; 103:653–681. https://doi.org/10.1016/j.solener.2013.02.035

48. Zhou M, Robertson GP, Roovers J. Comparative Study of Ruthenium(II) Tris(bipyridine) Derivatives for Electrochemiluminescence Application. Inorganic Chemistry. 2005; 44(23):8317–8325. https://doi.org/10.1021/ic0510112 PMID: 16270970

49. Spectra Database hosted at the University of Arizona; http://www.spectra.arizona.edu [Accessed: 2018-08-28].

50. Dauchet J, Cornet JF, Gros F, Roudet M, Dussap CG. Photobioreactor Modeling and Radiative Transfer Analysis for Engineering Purposes. In: Advances in Chemical Engineering. vol. 48. Elsevier; 2016. p. 1–106. Available from: https://linkinghub.elsevier.com/retrieve/pii/S0065237715000162.

51. Gloaguen F. Electrochemistry of Simple Organometallic Models of Iron–Iron Hydrogenases in Organic Solvent and Water. Inorganic Chemistry. 2016; 55(2):390–398. https://doi.org/10.1021/acs.inorgchem.5b02245 PMID: 26641526

52. Quentel F, Passard G, Gloaguen F. Electrochemical hydrogen production in aqueous micellar solution by a diiron benzenedithiolate complex relevant to [FeFe] hydrogenases. Energy Environ Sci. 2012; 5:7757–7761. https://doi.org/10.1039/c2ee21531d

53. Parker CA, Hatchard CG. Triplet-singlet emission in fluid solutions. Phosphorescence of eosin. Trans Faraday Soc. 1961; 57:1894–1904. https://doi.org/10.1039/tf9615701894

54. Supplis C. Modélisation et étude expérimentale de la production d'hydrogène solaire en photoréacteur [Theses]. Université Clermont Auvergne; 2020. Available from: https://tel.archives-ouvertes.fr/tel-03125605.

55. Dahi G, Eskandari A, Dauchet J, Gros F, Roudet M, Cornet JF. A novel experimental bench dedicated to the accurate radiative analysis of photoreactors: The case study of CdS catalyzed hydrogen production from sacrificial donors. Chemical Engineering and Processing: Process Intensification. 2015; 98:174–186. https://doi.org/10.1016/j.cep.2015.09.015

56. Dauchet J, Blanco S, Cornet JF, El Hafi M, Eymet V, Fournier R. The practice of recent radiative transfer Monte Carlo advances and its contribution to the field of microorganisms cultivation in photobioreactors. Journal of Quantitative Spectroscopy and Radiative Transfer. 2013; 128:52–59. https://doi.org/10.1016/j.jqsrt.2012.07.004
57. Murphy TE, Prufert-Bebout LE, Bebout BM. A radiative transfer modeling approach for accurate interpretation of PAM fluorometry experiments in suspended algal cultures. Biotechnology Progress. 2016; 32(6):1601–1608. https://doi.org/10.1002/btpr.2394 PMID: 27801554

58. Meroni M, Rossini M, Guanter L, Alonso L, Rascher U, Colombo R, et al. Remote sensing of solar-induced chlorophyll fluorescence: Review of methods and applications. Remote Sensing of Environment. 2009; 113(10):2037–2051. https://doi.org/10.1016/j.rse.2009.05.003

59. Louis J, Ounis A, Ducruet JM, Evain S, Laurila T, Thum T, et al. Remote sensing of sunlight-induced chlorophyll fluorescence and reflectance of Scots pine in the boreal forest during spring recovery. Remote Sensing of Environment. 2005; 96(1):37–48. https://doi.org/10.1016/j.rse.2005.01.013
57. Murphy TE, Prufert-Bebout LE, Bebout BM. A radiative transfer modeling approach for accurate interpretation of PAM fluorometry experiments in suspended algal cultures. Biotechnology Progress. 2016; 32(6):1601–1608. https://doi.org/10.1002/btpr.2394 PMID: 27801554

58. Meroni M, Rossini M, Guanter L, Alonso L, Rascher U, Colombo R, et al. Remote sensing of solar-induced chlorophyll fluorescence: Review of methods and applications. Remote Sensing of Environment. 2009; 113(10):2037–2051. https://doi.org/10.1016/j.rse.2009.05.003

59. Louis J, Ounis A, Ducruet JM, Evain S, Laurila T, Thum T, et al. Remote sensing of sunlight-induced chlorophyll fluorescence and reflectance of Scots pine in the boreal forest during spring recovery. Remote Sensing of Environment. 2005; 96(1):37–48. https://doi.org/10.1016/j.rse.2005.01.013