2.1 Radiative configuration

Photon transport was modeled in a one dimensional slab of thickness L with incident-normal collimated radiation on the left-hand side (see Fig 1). The incident monochromatic photon flux density inside the medium, at x = 0, is q_0,λ = q₀ pⁱ(λ), where is the total photon flux density and pⁱ(λ) is the incident spectral probability density function, hereafter called incident spectrum: . The interfaces at both sides of the slab are assumed to be transparent in order to analyze the roles of luminescence and separate them from the impact of reflection-refraction at the boundaries.

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Fig 1. Radiative configuration.

(a) Incident-normal collimated radiation at x = 0 with spectrum pⁱ(λ) and surface flux density q₀. (b) Slab of thickness L containing a luminescent photosensitizer solution at concentration C. The phase space (r, u), with u the propagation direction, can be reduced to (x, μ) in such a one-dimensional slab. In this case, the isotropic distribution becomes as with μ = cos θ [35]. (c) Examples of molar extinction cross-section E_λ and luminescence spectra p^L(λ) of the photosensitizer.

https://doi.org/10.1371/journal.pone.0255002.g001

The slab is filled with a homogeneous photocatalytic system solution including luminescent photosensitizer at concentration C (there is no particle leading to elastic scattering in such systems). In the present configuration the photosensitizer is the only molecule that interacts with radiation in the spectral range of interest (UV-visible). Some catalysts also absorb radiation and we will show in section 5 that the extension to this situation is straightforward.

Luminescence can either be interpreted as an emission source in the volume or as an inelastic scattering phenomenon, as will be discussed in Section 2.2. In this work, the luminescence of photosensitizers is approached from the multiple inelastic-scattering point of view, because of the intuitive physical pictures it provides when describing the absorption of luminescence radiation. There is therefore an important distinction between 1) extinction cross-section E_λ, which characterizes all interactions with radiation, 2) scattering cross-section E_S,λ, which characterizes interactions leading to luminescence emission only and 3) absorption cross-section E_A,λ, which characterizes interactions leading to other phenomena, including photocatalytic reactions (and other relaxation phenomena), with E_λ = E_S,λ + E_A,λ. The luminescence quantum yield Φ_λ is the proportion of interactions that leads to luminescence emission, and therefore Φ_λ = E_S,λ/E_λ [36–38]. From the photon transport point of view, Φ_λ is the single-scattering albedo: if a photon interacts with a photosensitizer molecule, it is scattered with probability Φ_λ or absorbed with probability (1 − Φ_λ). In the following, the radiative properties of photosensitizers are provided by the molar extinction cross-section E_λ and luminescence quantum yield Φ_λ, since absorption and scattering cross-sections are easily deduced from these quantities: E_A,λ = (1 − Φ_λ)E_λ and E_S,λ = Φ_λ E_λ. In the same way, the extinction coefficient of the solution is k_λ = CE_λ, the absorption coefficient is (1 − Φ_λ)k_λ and the scattering coefficient is Φ_λ k_λ. Scattering events redistribute propagation directions according to isotropic phase function [38] and wavelengths according to the luminescence spectral probability density function p^L(λ|λ′), hereafter called luminescence spectrum: . The luminescence spectrum is the third radiative property of the photosensitizer, together with E_λ and Φ_λ, that may vary depending on the photocatalytic system. In the following, Kasha’s rule is assumed, and therefore the luminescence quantum yield Φ and spectrum p^L do not depend on the excitation radiation wavelength: Φ_λ ≡ Φ and p^L(λ|λ′) ≡ p^L(λ) [39]. In other words, the spectral distribution of scattered photons is independent of the spectral distribution of photons incident on the molecule.

For the purpose of radiative analysis, we define the following dimensionless optical thicknesses:

• spectral extinction optical thickness (3)
• mean absorption optical thickness for incident radiation (4) with the mean molar extinction cross-section over incident spectrum (gray approximation) (5)
• mean scattering optical thickness for luminescence radiation (6) with the mean molar extinction cross-section over luminescence spectrum (gray approximation) (7)

The extinction optical thickness for luminescence radiation will be also used to formulate approximations in Section 4.

In Section 3 and are used to characterize incident and luminescence photon transport, respectively. We choose to focus on the absorption of incident radiation because absorbing and converting incident radiation is the aim of photoreactive systems. On the other hand, we choose to focus on luminescence radiation scattering because the aim of this study is to analyze the impact of multiple inelastic scattering in such systems. However, note that it is straightforward to compute the scattering optical thickness for incident radiation and the absorption optical thickness for luminescence radiation from the data provided in Table 2, since the previous definitions of cross-sections lead to τ = τ_S + τ_A and Φ = τ_S/τ.

2.2 Radiative transfer equation

The Radiative Transfer Equation (RTE) for a one-dimensional slab filled with a luminescent medium is (8) where I_λ is the intensity, μ is the dot product between the propagation direction and e_x (see Fig 1). The extinction coefficient k_λ, the single-scattering albedo Φ (the luminescent quantum yield) and the luminescence spectrum p^L(λ) were defined in the previous paragraphs. The collision term in Eq 8 describes luminescence. When it is interpreted as an emission source in the volume, it can be read as follows:

• is the volumetric rate at which photons with all wavelength λ′, propagating in all directions μ′, are absorbed at location x,
• multiplying the above term by the luminescence quantum yield Φ leads to the rate of luminescence emission,
• this emission is spectrally distributed according to p^L(λ), with isotropic distribution of emission directions μ (see Fig 1) [35].

When it is interpreted as an inelastic scattering phenomenon, it can be read as follows:

• is the volumetric rate at which photons with wavelength λ′, propagating in all directions μ′, are scattered at location x,
• multiplying the above term by the isotropic phase function (see Fig 1) leads to the rate at which photons are scattered in direction μ (per unit dμ),
• dλ′p^L(λ) is the probability that a photon with wavelength in dλ′ around λ′ is scattered with wavelength in dλ around λ,
• Integration over [0, + ∞[ sums the contributions of all wavelengths λ′.

The analysis in the article will be mainly based on this second interpretation.

For our configuration with normal collimated incident radiation, boundary conditions for μ > 0 are: (9) (10)

2.3 Successive orders of scattering expansion

The analysis in this paper is based on expansion into successive orders of scattering (numerical solutions use the Monte Carlo method, see Section 2.5).

Intensity obeys the linear transport equation in Eq 8 and can therefore be expanded into successive orders of scattering [40–46]. The intensity I of the entire photon population is formulated as the sum of the intensities I^(j) corresponding to photons that have undergone j scattering events: (11)

Ballistic intensity is due to photons that come directly from the incident-normal collimated source at the boundary x = 0, and is only attenuated within the medium: (12) with the boundary condition (13) (14)

Order j intensity is due to photons that have undergone j scatterings before reaching phase-space location (x, μ): I⁽¹⁾ accounts for photons that have undergone one scattering event, I⁽²⁾ accounts for photons that have undergone two scattering events, and so on. Each intensity I^(j>0) obeys a radiative transfer equation of its own, in which the source term corresponds to lower-order photons I^(j−1) that are scattered locally and move from population (j − 1) to population (j): (15) with the boundary condition (16) since no photon having undergone j > 0 scattering events is incident on the slab.

Luminescence intensity sums the contributions j > 0 of all scattered photons: (17) leading to (see Eq 11) (18) obeys the following RTE, that is the sum of the RTEs in Eq 15 for all j > 0: (19) with the boundary condition (20) This is the same RTE as that for intensity I_λ except that the source is not at the boundary because luminescence emission stimulated by ballistic photons I⁽⁰⁾ within the volume is the source of scattered photons, as formulated in the last term of Eq 19.

A meaningful property is due to the fact that the expansion has a zero-order closure: has a closed-form expression, independent of the , and each higher order j > 0 is a function of only. Therefore, truncating the expansion at order q means that the contribution of photons that have undergone more than q scattering events is neglected. As a result, the q-th order expansion in successive orders of scattering systematically underestimates intensity.

2.4 Radiative quantities of interest

The Mean Volumetric Rate of Photons Absorbed (MVRPA) is the key radiative quantity in the study of photoreactive systems [16]. It is obtained by integrating the intensity I over directions, locations and wavelengths: (21) where superscript ^(•) is dropped in the formulation of the total rate of absorption ; otherwise, take • ≡ j for the rate at which photons that have undergone j scattering events are absorbed, and • ≡ S for the rate at which scattered (i.e. luminescence) photons are absorbed: (22)

Absorptance is the key quantity in our radiative analysis. It can be interpreted as the proportion of photons that are absorbed in the medium, leading in our 1D cartesian geometry to: (23) The maximum photon absorption rate is then obtained with , which is lower than 1 if any scattering phenomenon exists in the medium. would be only achieved if ballistic and scattered radiation could be completely absorbed. We chose to base the radiative analysis in Section 3 mainly on absorptance rather than MVRPA, because it enables us to clearly distinguish the impact of:

• photon transport, which is contained in only and depends on optical thicknesses,
• dimension L of the system, which depends on the experimental setup or the engineering specifications when designing units for industrial-scale production (note that for a given dimension L, optical thickness can be controlled by adjusting the photosensitizer concentration C),
• incident photon flux q₀, which depends on the source and fluctuates in solar conditions.

Substituting Eq 21 in Eq 23 leads to: (24) Overall, expansion into successive orders of scattering is: (25)

The distribution P^(j) of scattering orders within the expansion indicates the respective weight of each scattering order: (26) The cumulative is the proportion of absorption that is described by the q-th order expansion in successive orders of scattering. Alternatively, the complementary (tail) cumulative is the relative error on both absorptance and MVRPA when the expansion is truncated at the q-th order. Of course, and P^{(j > q)} = 1 − P^{(j ≤ q)} according to Eq 25. In addition, the contribution of luminescence radiation is , and we have P⁽⁰⁾ + P^(S) = 1. Note that P^(j) can also be interpreted as the proportion of photons absorbed after j scattering events with respect to all the absorbed photons, or as the probability that a photon is absorbed after j scattering events.

2.5 Monte Carlo algorithm

An analytical solution for the above multiple inelastic-scattering radiative model is not available. Our study will therefore use a Monte Carlo numerical solution, which has three main advantages: 1) the algorithms are intuitive and easily modified to evaluate physical hypotheses, as will be presented in Section 3.3; 2) it provides a reference solution including statistical error estimation and 3) even if the present work is conducted in a one-dimensional cartesian configuration, it is straightforward to extend implementation to complex geometry in subsequent studies [47].

The Monte Carlo algorithm presented below was used to estimate the absorptance and each scattering order for a given concentration C, slab thickness L and radiative properties Φ and E_λ. It consists in sampling N independent realizations w_i, i = 1, …N with the following sampling procedure:

• Step 1: Initialization of Monte Carlo weights w_i = 0 and for , scattering counter j = 0, emission location r = 0 and emission direction u = e_x (see Fig 1).
• Step 2: Wavelength λ is sampled according to the incident spectrum pⁱ(λ), and the extinction coefficient k_λ = CE_λ is interpolated from spectral database.
• Step 3: A first extinction length l is sampled over [0, +∞[ according to the Bouguer extinction probability density function and the location is updated: r = r + l u.
• while 0 ≤ r · e_x ≤ L (location is inside the slab) do
  Step 4: Bernoulli test: uniform sampling of a realization ξ over [0, 1]
  • if ξ < Φ (scattering event) then
    Step 4a.1: Scattering counter is incremented: j = j + 1.
    Step 4a.2: Scattering direction u is sampled according to the isotropic phase function.
    Step 4a.3: Wavelength λ is sampled according to the luminescence spectrum p^L(λ) and the extinction coefficient k_λ = CE_λ is interpolated from the spectral database.
    Step 4a.4: Extinction length l is sampled over [0, +∞[ according to the Bouguer extinction probability density function and the location is updated: r = r + l u.
    end
  • Else
    Step 4b.1: Path sampling is terminated due to absorption:
    • weights are computed:
    
    w_i = 1
    • go to the end of Monte Carlo realization.
    end
  
  end

Absorptance is estimated as: (27) with standard error: (28) In Tables 2 and 3, the number N of Monte Carlo realizations has been adjusted to obtain a relative uncertainty which is always smaller than 1%, with defined as the 95% confidence interval: , where t_0.05,N−1 is the Student quantile (t_0.05,N−1 ≃ 2 for a typical number N of Monte Carlo realizations). Scattering orders are obtained in the same way, but replacing w_i by in Eqs 27 and 28.

3.1 Overall luminescence effect

Here we compare absorptance values obtained when luminescence is neglected, i.e. when Φ is set to zero in our model, with values obtained from the reference model. The results in Table 2 present the relative difference Δ(Φ = 0) computed with Eq 29, where X ≡ (Φ = 0).

Neglecting luminescence always leads to an overestimation of absorptance: Δ(Φ = 0)>0. Indeed when luminescence is not taken into account, each interaction with photosensitizers necessarily leads to absorption, whereas luminescence interaction has a probability Φ of leading to scattered photons that might exit the medium. We record Δ(Φ = 0) values between 10% and 20% for TATA⁺, between 20% and 50% for Eosin Y and larger than 100% for Rhodamine B. The only photosensitizer for which Δ(Φ = 0) is constant and lower than 10% is , but we will see in the next paragraphs and in Section 4 that strictly accounting for luminescence is very simple in this particular case. As the absorption optical thickness increases, absorptance given by both models increases (see values in Table 2) and the error Δ(Φ = 0) decreases. Nevertheless, Δ(Φ = 0) does not approach 0 when photosensitizer concentration goes to infinity. Indeed, reference approaches a limit value lower than that for , because luminescence radiation cannot be completely absorbed, for two reasons. The first reason is caused by inelastic scattering: part of the luminescence radiation belongs to a spectral range where the extinction cross section is equal to zero (see Fig 2 and Section 2.3) and therefore photons exit the medium without interacting with photosensitizer. The second is driven by multiple scattering: some photons are backscattered in the vicinity of x = 0 and exit the medium through the front face of the reactor (even when concentration goes to infinity, or equivalently, for a semi-infinite medium).

Overall, these results indicate significant luminescence effects that should not be neglected when studying photosensitized systems.

3.2 Successive orders of scattering analysis

Here we analyze the effect of multiple luminescence scattering. In the previous paragraphs we discussed the fact that due to luminescence, interaction has a probability Φ of leading to scattered photons that might exit the medium. Therefore is decreased compared to situations without luminescence, leading to a loss with respect to the photoreaction. However, those scattered photons emitted by luminescence do not necessarily exit the medium. They can interact (again) with photosensitizers and be absorbed—in this case they participate in the reaction—or scattered, and so on.

Table 3 presents the distribution P^(j) of scattering orders and its tail cumulative P^(j>q) (see Eq 26): P⁽⁰⁾ is the contribution of ballistic photons, P⁽¹⁾ is the contribution of photons absorbed after one scattering event (first order), P⁽²⁾ is the contribution of photons absorbed after two scattering events (second order), etc., with decreasing contributions P^(j) > P^(j+1) and normalization . The key parameter in analyzing this distribution is the scattering optical thickness . The absorption of luminescence radiation increases with optical thickness, leading to a higher luminescence contribution P^(j>0) = P^(S) and lower ballistic contribution P⁽⁰⁾ (the distribution is normalized). Furthermore, as regards luminescence contributions, the higher the , the larger the contribution of high scattering orders.

The following case-by-case analysis is organized according to value, which is strongly influenced by the overlap between extinction cross-section and luminescence emission spectra via (see Eqs 6 and 7 and Fig 2).

3.3 Inelastic scattering sensitivity

Inelastic scattering is another conceptual and numerical difficulty when describing photon transport in photosensitized and photocatalytic systems, in addition to multiple scattering. Indeed, wavelengths are redistributed according to the luminescence spectrum at each scattering event. Here we analyze the sensitivity of absorptance and MVRPA to inelastic scattering effects in order to determine if this difficulty must be tackled or can be partially bypassed. For this purpose, we constructed two equivalent models for luminescence radiation that we compared with the reference solution (see Δ(gray) and Δ(elastic) in Table 2). In the first one, gray approximation is used in the luminescence spectral range and in the second, the inelastic collision term in the RTE for luminescence intensity is simply replaced by an elastic collision term. In both cases, the spectral distribution for ballistic photons (zero-th order) is preserved; only luminescence intensity is affected by the approximations.

A gray model for luminescence radiation is constructed based on the distinction between ballistic and scattered intensity (see Eq 18). Gray approximation is used in the RTE for scattered (i.e. luminescence) intensity , which turns the inelastic collision term into an elastic collision term.

The RTE in Eq 19 is integrated over wavelengths λ and then the following gray approximation is used: (30) where and (see Eq 7). Using the normalization and the notation for this gray approximation leads to (31) with the boundary condition (32) Note that in Eq 31 is the solution of Eq 12: ballistic intensity is not affected by the present approximation. Then the contribution of scattered photons to absorptance is obtained by applying the gray approximation Eq 30 to the definition in Eq 24: (33) Finally, our gray model for absorptance is obtained by summing the contributions of ballistic and scattered photons: (34) where is not modified by the approximation (see its definition in Eq 24).

The values presented in Table 2 were obtained with the Monte Carlo algorithm presented in Section 2.5, except that Step 4a.3 was replaced by:

• Step 4a.3:

Whatever the wavelength, the extinction coefficient for scattered photons is set to (see Fig 2).

The relative difference Δ(gray) between and the reference is presented in Table 2. Results do not include , since luminescence radiation does not participate in absorption in this case (see Section 2.3). For other photosensitizers, the fact that Δ(gray) > 0 (always) indicates that gray approximation for luminescence radiation overestimates absorptance (and indeed, in Eq 24 is concave).

Δ(gray) rapidly increases with the scattering optical thickness , because the average number of wavelength redistributions due to inelastic scattering events increases with . Indeed, the analysis in Section 2.3 insists on the significant contribution of scattering orders j = 1, 2, 3 in some tested cases, involving j wavelength redistribution events (see Table 3). For TATA⁺, Δ (gray) ≲ 5 since at least 95% of absorption is due to ballistic radiation that is not affected by inelastic scattering. For Eosin Y, the first and second scattering orders contribute up to 20% and Δ(gray) reaches 11%. For Rhodamine B, we record a significant contribution up to the third scattering order, and Δ(gray) reaches 70%.

Δ(gray) always increases with for a given set of radiative properties, but the magnitude of that increase depends on the extinction cross section and luminescence spectra. For example, if the extinction cross section is gray within the luminescence spectral range (i.e.E_λ does not vary with λ in the spectral range where p^L(λ) ≠ 0), then the gray model and reference model are exactly identical, and therefore . In general, we expect that the higher the spectral variation of E_λ on is, the faster Δ(gray) increases with . We cannot illustrate this behavior here because the typical photosensitizers we selected have comparable spectral variations over the luminescence spectral range (see Fig 2).

To conclude, for the typical cases tested in this paper, wavelength redistribution due to luminescence can hardly be put aside once scattering optical thickness exceeds 0.1. However, the results in Table 3 indicate that the main contribution to luminescence radiation is due to first-order terms in scattering expansion. Therefore we hereafter study a model that preserves luminescence spectral redistribution during the first scattering event only, and neglects the subsequent ones.

The elastic-scattering model for luminescence radiation is simply obtained by replacing the inelastic collision term in the RTE for luminescence intensity (see Eq 19) by an elastic collision term: (35) Luminescence absorptance is defined in Eq 24 with and (36) where is not affected by the approximation.

The values presented in Table 2 were obtained using the Monte Carlo algorithm presented in Section 2.5, except that Step 4a.3 was replaced by:

• Step 4a.3:
  if j = = 1 then
  The wavelength λ is sampled according to the luminescence spectrum p^L(λ) and the extinction coefficient k_λ = CE_λ is interpolated.
  end

The wavelength λ is sampled during the first scattering event only (i.e. for j = 1).

Two spectral integrations are required to solve this model—one over the incident spectrum and one over the luminescence spectrum—whereas only spectral integration over the incident spectrum was considered in the previous gray model. This additional spectral dimension is required to take into account the luminescence spectrum when ballistic photons are scattered and move from population (0) to population (1). The comparison of Δ(elastic) and Δ(gray) in Table 2 indicates that it is a significant improvement in the description of absorption, and yet it corresponds to a considerable conceptual and practical simplification compared to the reference model, which involves an infinite number of spectral integrations (one at each scattering order). This simplified description of inelastic scattering provides results with a relative difference Δ(elastic) below 2% for TATA⁺ and Eosin Y. This is due to the fact that scattering orders j > 1 have little impact with these photosensitizers (see Table 3). For Rhodamine, however, these scattering orders contribute up to 15% of absorptance, in addition to the strong spectral variation in the extinction cross section in the luminescence spectral range (see Fig 2d). In this case, inelastic scattering should be described in all its complexity.

4.1 Zero-order scattering approximation

In Section 3.2 we observed that the main contribution to absorption is due to ballistic radiation. The expansion in successive orders of scattering is therefore truncated here at order zero: . In comparison with the reference solution in Eq 25, the absorption of luminescence radiation is not taken into account, leading to quite a simple analytic expression: the solution of the RTE Eq 12 for ballistic intensity is (41) (42) and substituting this expression into Eq 24 leads to (43) Let us emphasize that is very easily obtained by multiplying , which is routinely calculated or measured in photochemistry practice, by (1 − Φ).

The solution for is the same as for , but a crucial difference is that the luminescence quantum yield Φ is here taken into account in the expression of and . When luminescence is neglected, every photon that interacts with the photosensitizer is counted as being absorbed, leading to an overestimation of : Δ(Φ = 0) > 0 in Table 2. Here, in contrast, luminescence scattering is modeled but only ballistic photon absorption is accounted for, leading to the prefactor (1 − Φ) in Eq 43. Since the absorption of luminescence radiation is ignored, the zero-order scattering approximation underestimates absorptance: Δ(j = 0) < 0 in Table 2. The impact of truncating the expansion in successive orders of scattering has already been discussed in Sections 2.3 and 3.2, and indeed, Δ(j = 0) in Table 2 is by definition equal to −P(j > 0) in Table 3. As seen in previous discussions, the error associated with zero-order scattering approximation is governed by the scattering optical thickness : the higher is, the higher the error Δ(j = 0).

Of course, when , the zero-order scattering approximation gives exact results: , and . This situation is encountered in two cases. First, for photosensitizers that do not emit luminescence radiation (i.e. when Φ = 0). In this case , and . Second, and more importantly, for photosensitizers with extinction cross-section and luminescence emission spectra that do not overlap, as illustrated with in Section 3.1 (see Case 1). In this case, the error obtained with a model neglecting luminescence is . In Table 2, indeed, Δ(Φ = 0) = 6.6% for with luminescence quantum yield Φ = 0.062. Note that a photosensitizer with a higher Φ could lead to a significantly greater error: e.g. Δ(Φ = 0) = 25% for Φ = 0.2. In these situations where , the very simple solution provided in Eq 43 should be used.

In Table 2, zero-order scattering approximation is also tested for three photosensitizers with : TATA⁺, Eosin Y and Rhodamine B. As discussed in Section 3.1, the luminescence effect is significant in these cases and should not be neglected (Δ(Φ = 0) = 10% to 170%):

• For TATA⁺, the zero-order scattering approximation provides accurate results with error Δ(j = 0) ≤ 5% in every tested situation. Indeed, Table 3 indicates that at least 95% of absorption is due to ballistic photons. The relative error is 2 to 20 times lower than the error Δ(Φ = 0) obtained when ignoring luminescence.
• For Eosin Y and Rhodamine B, even if the zero-order scattering approximation is not sufficient to describe photon transport with accuracy below 5%, we nevertheless record a significant improvement in relative error compared to Δ(Φ = 0).

Overall, we record the following accuracy levels for the zero-order scattering approximation as a function of scattering optical thickness :

• Δ(j = 0) < 5% for < 0.05,
• Δ(j = 0) < 10% for 0.05 < 0.3,
• Δ(j = 0) < 30% for 0.3 < < 1.

In conclusion, zero-order scattering is a relevant approximation that provides accurate results for optical thicknesses and for TATA⁺ in general; it includes half the cases investigated in this paper. In addition, this approximation is exceedingly simple because it only requires multiplying , which is routinely calculated or measured in photochemistry practice, by (1 − Φ).

4.2 P1 approximations for luminescence radiation

For optical thicknesses , the absorption of luminescence radiation must be described to improve accuracy compared to zero-order approximation. Here, the luminescence radiation in Eq 25 is described by diffusion equations derived thanks to P1 approximation [35, 50]. But a simple analytical solution is only accessible for elastic scattering processes, and therefore P1 approximation is applied to the gray and elastic models presented in Section 3.3.

4.3 Single-scattering approximation

The approximation developed in the previous section fails to describe absorptance for Rhodamine B photosensitizer, with errors higher than the contribution of scattering orders j > 1 (see Table 3). Therefore, the description of luminescence radiation thanks to first-order scattering expansion is investigated here.

5.1 Materials and methods

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 versus MVRPA is a straight line, the slope of which being the overall quantum yield φ^(•) (see Eq 2): (63) From the experimental hydrogen production rate , the overall quantum yield is identified by Eq 63 according to several methods used to obtain MVRPA, which are:

5.2.2 Calculation of MVRPA from models.

MVRPA is calculated with Eq 23, from the following absorptance models that have been presented in previous sections:

• Neglecting fluorescence (see Section 4): notation (•) ≡ (Φ = 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

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

• k_λ → k_λ + k_cat,λ with the extinction coefficient of the catalyst k_cat,λ = C_cat E_cat,λ
• 

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 Carlo algorithm) are multiplied by the probability f_λ that a photon absorbed is absorbed by Eosin Y molecules (rather than by the catalyst): (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.

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Table 4. Overall quantum yields of the homogeneous photoreactive system at different catalyst (C_cat) and Eosin Y (C) concentrations in mol.m⁻³.

Results obtained from Eq 63, using several methods to estimate : φ^(exp,0) uses measurements on ballistic radiation (fluorescence is neglected), φ uses the reference Monte Carlo calculation, φ^(Φ=0) uses model results when fluorescence is neglected, 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 and are calculated from Eqs 4 and 6, with for the LED emission spectrum.

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

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 (). Therefore, only 1% of the incident radiation is reflected back into the medium.

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 Et₃N, H₂O, 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.