Fig 1.
(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 Eλ and luminescence spectra pL(λ) of the photosensitizer.
Fig 2.
Photosensitizer radiative properties.
In all plots, plain lines stand for UV-visible molar extinction cross-section spectrum Eλ and dashed lines for luminescence emission spectrum pL(λ). Luminescent quantum yield Φ is provided in a box. (a) Ru[Bpy]3Cl2 (40 μM) in aqueous buffer (pH 6.8) solution [48]. (b) TATA(Cl) (0.5 mM) UV-visible molar extinction cross-section, TATA(Cl) (10 μM) fluorescence emission spectrum and quantum yield under argon in acetate buffer (1 M) at pH 4.5 [25] (c) Eosin Y (24 μM) in a mixture of triethylamine (10% vol.) in water at pH 10.5 [18, 21] (d) Rhodamine B in ethanol [26, 49]. Extinction cross-sections are obtained assuming that absorbance Aλ measured by the authors cited above using spectrophotometry experiments can be interpreted as Aλ = 1 − exp(−Eλ CL). Overlapping between molar extinction cross-section and luminescence emission spectrum is represented for each photosensitizer (except Ru[Bpy]3Cl2 for which no overlapping is observed) by 2 parallel lines linked by a double arrow.
Fig 3.
Solar AM1.5 incident spectrum in moles of photons.
Probability density function pi(λ) between nm and
nm corresponding to an incident photon flux q0 = 1743 μmolhν.m−2.s−1 (for TATA+, we work between
nm and
nm and therefore q0 = 1721 μmolhν.m−2.s−1). Note that pi(λ) = 0 for
and spectral integration over incident radiation therefore gives
.
Table 1.
The table summarizes mean molar extinction cross-sections for each photosensitizer.
is weighted by the incident AM1.5 source in the spectral range [280 nm, 650 nm] for
, Eosin Y and Rhodamine B, and in the spectral range [335 nm, 650 nm] for TATA+ due to extinction coefficient data availability (see Eq 5 and Fig 3).
is weighted by the luminescence emission spectra (see Eq 7). Note that pL(λ) = 0 for
, with
nm and
nm for
,
nm and
nm for TATA+,
nm and
nm for Eosin Y and
nm and
nm for Rhodamine B. Thus spectral integration over luminescence spectrum gives
.
for
because no overlapping is observed between extinction cross section and luminescence spectra (see Fig 2).
Table 2.
Results for , TATA+, Eosin Y and Rhodamine B photosensitizers (radiative properties are in Fig 2) for the incident solar spectrum in Fig 3, at selected optical thicknesses
and
(see Eqs 4–7 and Table 1): Reference absorptance
, reference MVRPA
(slab thickness L = 1 cm; incident flux density q0 = 1743 μmolhν.m−2.s−1 and 1721 μmolhν.m−2.s−1 for TATA+), relative difference Δ(X) between these values and those obtained with a model denoted X (see Eq 29).
X ≡ (Φ = 0) when luminescence is neglected, X ≡ (j = 0) for zero-order scattering expansion () as presented in Section 4.1, X ≡ (j ≤ 1) for first-order scattering expansion (
) as presented in Section 4.3, X ≡ (elastic) and X ≡ (gray) for elastic and gray models presented in Section 3.3,
and
for elastic and gray P1 approximations presented in Section 4.2 and finally,
for the gray single-scattering approximation presented in Section 4.3. Results for
do not include all X models, since zero-order expansion already provides a reference solution. The relative uncertainty of the Monte Carlo absorptance estimation is always lower than 1%; the precision indicated accounts for case-by-case absolute uncertainty.
Table 3.
Successive orders of scattering for , TATA+, Eosin Y and Rhodamine B photosynthetizers (radiative properties are presented in Fig 2) at selected optical thicknesses
and
(see Eqs 4–7 and Fig 2): distribution
of scattering orders (or the probability that an absorbed photon is absorbed after j scattering events) and cumulative tail distribution
are indicated with a gray background (see Eq 26).
P(j>q) is the relative error on both absorptance and MVRPA
when the expansion in successive scattering orders is truncated at the q-th order. The weight of luminescence radiation is P(j>0) = P(S).
Fig 4.
Hydrogen production with the fluorescent photocatalytic system in homogeneous phase [21, 51, 52].
Fig 5.
(a) Photocatalytic system radiative properties—left: Eosin Y (24 μM) UV-visible molar extinction cross-section Eλ, fluorescence emission spectrum pL(λ) and quantum yield Φ = 0.35 in a mixture of triethylamine (10% vol.) in water at pH 10.5 [18, 21]; right: Catalyst (1.66 mM) molar absorption cross section in methanol. UV-visible spectra were obtain using a Shimadzu UV-visible UV-160 A spectrophotometer and emission spectra from a Shimadzu RF-1501 spectrofluorimeter [18]—(b) Example of experimental pressure time course presenting transient and linear regimes with the slope in the linear regime—(c) wavelength probability density function incident source i.e. LED panel emission spectrum pi(λ)—(d) photograph of the photoreactor containing the fluorescent photocatalytic system illuminated by the blue LED panel—(e) Example of the wavelength probability density function spectrum pr(λ) of photons measured at the rear of the reactor composed of transmitted LED panel photons and fluorescent photons.
Table 4.
Overall quantum yields of the homogeneous photoreactive system at different catalyst (Ccat) 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.