On the Determination of Quantum Efficiencies in Heterogeneous

J. Phys. Chem. B 2000, 104, 10569-10577

10569

On the Determination of Quantum Efficiencies in Heterogeneous Photocatalysis Mathias Corboz,† Ivo Alxneit,*,† Gautier Stoll,‡ and Hans Rudolf Tschudi† Paul Scherrer Institute, 5232 Villigen PSI, Switzerland, and Centre de Physique Theorique, CNRS Luminy, F-13288 Marseilles, Cedex 9 ReceiVed: January 20, 2000; In Final Form: June 23, 2000

A framework to determine the quantum efficiency η of a photoreaction in a porous layer of a photocatalyst is presented. The procedure relies on a model of the photoproduct diffusion in the porous structure of the photocatalyst. The model incorporates a position-dependent source term mirroring the light intensity profile in the layer and an effective diffusion coefficient Deff. It allows for a simultaneous determination of η as well as of Deff. The method is applied to the photosynthesis of CH4 from gaseous H2O and CO2 at the solid/gas interface of a porous layer of TiO2 (Degussa P25). A value of η ) (8.79 ( 0.79) × 10-4 is found for the formation of CH4. The effective diffusion coefficient for the photoproduct CH4 is Deff ) (5.64 ( 2.51) × 10-10 cm2 s-1. This value is much too low to be explained by classical Knudsen diffusion in the layers with an experimentally determined porosity φ ) 0.58. We suggest that the TiO2 layers contain a large number of unconnected pores separated by walls of densely packed microcrystals almost impenetrable for gases such as CH4. Whether processes other than diffusion are also responsible for the small value of Deff is not clear at present.

Introduction Gaining detailed knowledge about the mechanism or the kinetics of photoreactions in a heterogeneous system is extremely complicated. The difficulties originate from the complexity of the interactions of reactants and products with the surface of the photocatalyst and from the effects of photons on these interactions. Serpone et al.1 suggest that the kinetics of processes in heterogeneous photocatalysis should be taken as apparent kinetics. They consider only the formation of the ultimate and stable products to be relevant, arguing that no turnover number nor quantum yield can be defined in heterogeneous photocatalysis since neither the complete reaction chain nor the number and the nature of the catalytic sites nor the number of photons that are absorbed by the photoactive state are usually precisely known. The quantum efficiency η of the reaction must therefore be defined as the number of product molecules divided by the total number of photons absorbed. Because of a subtle interplay between the optical penetration depth and the diffusion length of the free carriers generated, the quantum efficiency of a photochemical reaction at a semiconductor surface can even become wavelength dependent.2,3 In this paper, we propose a new method to determine the quantum efficiency from data of batch experiments in fixedbed photoreactors where gaseous products accumulating above the photocatalyst bed are measured. The method can also be applied to experiments with immersed catalyst beds, as used in water detoxification studies. It yields an effective diffusion coefficient Deff of the photoproduct in addition to the quantum efficiency η of its formation. Many authors have emphasized the essential role of reactant and product transport for heterogeneous reaction in porous media such as a bed of catalyst * To whom correspondence should be addressed. † Paul Scherrer Institute. ‡ Centre de Physique Theorique.

powder4-8 or porous electrodes.9,10 Note, however, that the effective diffusion coefficient does not solely reflect diffusive mass transport within the catalyst bed. Processes such as adsorption and desorption or even reactions among the products or back-reactions influence the value of Deff. The procedure outlined in refs 11 and 12 for the determination of quantum yields in (dilute and stirred) slurries is different from our procedure in several aspects and cannot be used for our system. Two main differences are noteworthy: (a) The total amount of photoproduct formed is not directly accessible in our case because of limitations of diffusive mass transport. (b) The exact light profile inside the layer of photocatalyst has to be known, as it is responsible for the diffusion of the photoproducts out of the layer due to the concentration gradient formed. Experimental Section Experimental Setup. The photochemical experiments were performed in a microphotoreactor (µ-reactor) shown in Figure 1. The µ-reactor and details of the sample preparation are discussed in ref 13. In short, samples were prepared by transferring an aqueous suspension of TiO2 (Degussa P25) onto the sapphire disks forming the windows of the µ-reactor and by subsequent slow evaporation of the solvent. The samples were not subjected to any further pretreatment. With this technique, 100-800 µg of TiO2 were deposited onto the windows as dense, optically homogeneous layers of 11 mm diameter. The two sapphire windows, when mounted, define the µ-reactor volume of 93 mm3. A 200 W Hg/Xe lamp was used as the light source. Infrared radiation was blocked by a cooled water filter. The intense Hg emission line at 365 nm was selected by a narrow band-pass filter. Fluctuations of the light flux were monitored on-line with a Si photodiode. The absolute incident light flux was determined by a calibrated photodetector (IL 17000 Research Radiometer). As reactant and as purge gas, CO2 (Carbagas 5.0) was passed through a bubbler filled with bidistilled H2O (e0.06 µS cm-1),

10.1021/jp000253e CCC: $19.00 © 2000 American Chemical Society Published on Web 10/24/2000

10570 J. Phys. Chem. B, Vol. 104, No. 45, 2000

Corboz et al.

Figure 1. Assembly of the µ-photoreactor: The reactor cavity is formed by two sapphire windows soldered onto metallic flanges (3). The gas flow through the reactor body (1, 2) is accomplished by capillaries connected to a system of fan-shaped channels arc-eroded at the contact sides of the two parts. Metallic seals (4) ensure a high tightness. The volume of the reactor cavity including the channels is 93 mm3.

Figure 3. SEM images of a TiO2 (Degussa P25) layer on sapphire. (Top) Overview (magnification 3000×). (Bottom) Detailed view of the structure of the layer (magnification 300 000×).

Figure 2. Schematic setup of the experiment. The reactor is irradiated while closed.

saturating the gas with H2O. The gas feed was connected to the µ-reactor and to a mass spectrometer (MS) through switching valves, allowing alternation between irradiating the reactor without gas flow and purging the reactor in the dark (see Figure 2). The concentrations of selected gaseous products were analyzed by the MS. The MS was calibrated directly by filling the reactor with known gas mixtures and subsequently measuring the signal intensity with the MS while the reactor was purged. Structural Characterization of Layers. The structure of the TiO2 layers was analyzed in some details by scanning electron microscopy (SEM) (Topcon ATB-60). Fractured sapphire disks covered with different amounts of TiO2 were imaged from the side. An example of such an image is shown in Figure 3. At the bottom of the top image, the sapphire substrate can be seen homogeneously covered with TiO2. Careful inspection of the image reveals that some larger humps are visible in the layer. We suspect that they consist of larger aggregates that survived the sonication treatment applied during the preparation of the TiO2 suspension and that are subsequently incorporated into the layer. The thickness of the TiO2 layers can be determined from these SEM images. Using aliquots of the suspension containing 100 to 800 µg of titania results in layers with a thickness of 0.75 to 6.0 µm. From the density of rutile, the amount of rutile

deposited, and the diameter of the layers, the theoretical thickness of the layers was calculated. Comparing these numbers with the measured ones, the porosity of the layers φ ) 0.58 ( 0.07 was calculated. Images with larger magnification (bottom image of Figure 3) show that the TiO2 layers contain large pores of about 200 nm diameter separated by walls of densely packed TiO2 microcrystals of 20 to 70 nm diameter. Photochemical Experiments. Photoreduction experiments were performed on TiO2 layers with H2O-saturated CO2. The influence of the layer thickness on the product formation was studied for two configurations. In the first configuration (front irradiation), the sapphire window covered with the photocatalyst was mounted such that the TiO2 layer was irradiated on the side facing the cavity of the µ-reactor. In this configuration, the photons are primarily absorbed at the interface between the TiO2 layer and the reactor cavity in which the products are accumulated. In a second configuration (back irradiation), the two windows were exchanged, that is, the TiO2-covered window was mounted on the side facing the light source (see Figure 2). In this configuration, the photons are mainly absorbed at the sapphire/TiO2 interface. Experiments were performed in each configuration with loads of 200, 400, 500, 600, and 800 µg of TiO2, corresponding to layers of 1.5, 3.0, 3.75, 4.5, and 6.0 µm thickness. The samples were irradiated at 300 K with the 365 nm Hg emission line and with a photon flow J0 of 1.3 × 10-4 Einstein m-2 s-1 on a circular surface of 56.7 mm2 for 600 s. The irradiation time of 600 s results as a compromise between the desire to maximize the product concentration inside the reactor in a given cycle and the nonzero leak rate of the reactor. From our earlier experiments performed on this system we know

Determination of Quantum Efficiencies

Figure 4. CH4 yield measured after 600 s of irradiation with the full output of a 200 W Hg/Xe lamp. The solid line represents the back irradiation geometry; the dashed line corresponds to the front irradiation geometry (see text).

that CO, H2, CH4, and C2H6 can be detected as photoproducts14-17 and that the reaction indeed is a photoprocess. We have also excluded that the photoproducts stem from organic contaminants at the photocatalyst surface. In the experiments reported in this paper the detection was limited to methane. The results of these experiments are reported in Figure 4. The curve for front irradiation increases and seems to tend toward a plateau for thick layers. Increasing the thickness of the layer leads to a higher number of photons absorbed, hence to an enhanced photoactivity. The increase, however, is less than linear because the light intensity profile within the layer is, essentially, exponentially decreasing (see eq 8). Saturation of the product evolution rate will occur as soon as the layer becomes totally absorbing since additional photocatalyst remains then unilluminated. The curve for back irradiation has a rather different shape. Starting at 1.2 × 10-3 µmol for a 1.5-µm layer, the curve parallels the one for front irradiation up to d = 3.0 µm. Then the curve starts to fall off to a value of 1.2 × 10-3 µmol for a 6.0-µm layer. Thus the back-irradiation curve exhibits a distinct maximum of 2.2 × 10-3 µmol for layers with d = 3.5 µm. This behavior follows from the observation that the photoactive light is almost completely absorbed in layers thicker than 3.5 µm. For these layers, the same amounts of photoproducts must, therefore, be formed for front and for back irradiation. The much smaller apparent CH4 yield with back irradiation for thicker layers must be attributed to the fact that the reaction products formed near the sapphire window have to diffuse through the TiO2 layer into the reactor cavity. This diffusion process is described in this work by a phenomenological effective diffusion coefficient Deff without any reference to the nature of the different microscopic processes involved. The curves in Figure 4 will be interpreted within this framework. A first approximate value of the effective diffusion coefficient was obtained by exposing a TiO2 sample to CH4 in the dark. Then the µ-reactor was flushed for 5 s every 30 s in the dark with CO2. The amount of CH4 being released from the TiO2 layer was analyzed in the MS. In Figure 5 the cumulative amount of CH4 diffusing out of a 3-µm TiO2 layer is reported. The experimental curve can be interpreted by considering a semi-infinite layer. We further assume that the total amount of CH4 present in the whole layer was initially deposited at t ) 0 in a very narrow region near the window. The amount of CH4

J. Phys. Chem. B, Vol. 104, No. 45, 2000 10571

Figure 5. Cumulative amount of CH4 diffusing out of the TiO2 layer. Each point corresponds to an accumulation time of 25 s during which the product diffused out of the layer.

found after a certain time t in the layer between d ) x and d ) ∞ approximates the amount of CH4 in the cavity of the µ-reactor. This quantity is given by

1 - erf

(x ) ( x

2 Defft

∝ exp -

x2 4Defft

)

(1)

in a rough but sufficient approximation (see ref 18, formula 7.1.25). A value of Deff ) 2.5 × 10-9 cm2 s-1 follows from a linear fit. This value has to be compared with the diffusion coefficient of CH4 in CO2 being 8 orders of magnitude higher (D0 ) 0.2 cm2 s-1).19 Modeling and Discussion A more detailed interpretation of the results presented has to take into account that photoreaction occurs in the whole finite layer and depends on position due to the spatial variation of the light intensity in the layer. One needs, therefore, a model for the light intensity profile in the layer and one has to solve the diffusion problem with the corresponding source term and appropriate boundary conditions. Modeling of Light Intensity in the Layer. Radiation in the layer is modeled and analyzed in the framework of the Kubelka-Munk theory.20 The simplified description of the radiation field considers only two total photon flows J+, J- in a positive and a negative space direction. The interaction of light with matter is described with two phenomenological constants, the absorption coefficient K and the scattering coefficient S, in the governing differential equations

d J ) -(K + S) J+ + S Jdx + d J ) -S J+ + (K + S) Jdx -

(2)

for the photon flows. It follows from these equations that the quantity K(J+ + J-) is the absorbed number of photons per unit time and unit volume. The general solution of the differential equations is given by


[

J+(x) ) cosh(Lx) -

Corboz et al.

K+S sinh(Lx) J+(0) + L S sinh(Lx) J-(0) (3) L

J-(d) ) Fb J+(d)

]

With the integrating sphere, the quotient

Reff ≡

S J-(x) ) - sinh(Lx) J+(0) + L

[cosh(Lx) + K +L S sinh(Lx)] J (0) (4) -

(5)

The quantity 1/L can be interpreted as the optical penetration depth of the layer. The final expression for the radiation field in a layer of thickness d follows from these equations and from the boundary conditions

J+(0) ) (1 - F) J0 + F J-(0)

(6)

J-(d) ) F J+(d)

(7)

where the quantity F denotes the reflectance of the sapphire windows and J0 is the incident photon flow. Absorption and scattering in the windows and in the gas volume of the reactor are neglected. With these assumptions, the following expression for the quantity E ) J+ + J- can be derived:

E(x) ) J0 (1 - F2) L cosh(Ld - Lx) + (1 - F)2(K + 2S) sinh(Ld - Lx) (1 - F2) L cosh(Ld) + [K + S(1 - F)2] sinh(Ld) (8) The coefficients K and S were determined from total (diffuse + direct) transmission and total (diffuse + specular) reflectance spectra of the various TiO2-covered sapphire windows. These spectra were measured between 320 and 620 nm with a UV/ Vis spectrometer (Perkin-Elmer, Lambda 19) equipped with a 60-mm integrating sphere. A measured value of F ) 0.15 was used. For both types of measurement, the sapphire windows were mounted on the integrating sphere with the TiO2 layer on the outside. To measure transmission spectra the sample was mounted at the entrance of the integrating sphere. This implies that the radiation field for the transmittance measurements is characterized by the boundary conditions

J+(0) ) J0

(9)

J-(d) ) rJ+(d)

(10)

where the constant r includes the reflection at the sapphire window and the response of the integrating sphere. The transmission T is, therefore, given by the expression

J0

A 1 ≡ ) cosh(Ld) + sinh(Ld) L J+(d) T

(1 - F)2 1 +F) ) Reff - F R

x

1+

L2 L + cosh(Ld) - Fb S2 S

(11)

(12)

(14)

[x

1 - Fb

]

(15)

L2 L 1 + 2 - coth(Ld) S S

The three unknown parameters L, S, and A are determined by a nonlinear least-squares fit (Levenberg-Marquardt21) of measured values of 1/T and 1/R for different layer thicknesses d. The absorption coefficient K follows then from the identity

K)

L2

xL2 + S2 + S

(16)

All these parameters are of course wavelength dependent. The wavelength dependence is indicated only where necessary to keep the nomenclature simple. From the analysis of the measured data the values L(365 nm) ) 0.77 ( 0.12 µm-1 and S(365 nm) ) 0.68 ( 0.07 µm-1 are extracted corresponding to K(365 nm) ) 0.34 ( 0.11 µm-1. Modeling of Diffusion Inside A Porous Photocatalyst. The modeling of the material flows in the porous layer is much simplified by the fact that only the flow of products must be considered. Because of the very small quantum efficiency of the reaction under consideration (see below), the reactants in the layer are not so severely depleted during the irradiation time of 600 s that their surface coverage on the photocatalyst is expected to be altered significantly. We further assume that the product flux is proportional to the concentration gradient

j(x) ) -Deff

∂ c(x,t) ∂x

(17)

The microscopic processes involved will not be specified and the effective diffusion coefficient Deff is a phenomenological parameter determined by measurement. Mass conservation yields the diffusion equation

∂2 ∂ c(x,t) ) Deff 2 c(x,t) + q(x) ∂t ∂x

(18)

where the source term q(x) is given by

q(x) ) ηKE(x)

with A ) K + (1 - r)S. For the reflectance measurements a black background (Fb ) 0.08) was mounted behind the samples to minimize back reflection. The reflectance measurements are then characterized by the boundary conditions

J+(0) ) (1 - F) J0 + F J-(0)

(1 - F) J-(0) + F J0 J0

is measured. It follows from the boundary conditions that the reflectance R ) J-(0)/J+(0) of the layer is given by

with

L ) xK2 + 2KS

(13)

(19)

η denotes the quantum efficiency and E(x) is given by eq 8. Equation 19 implies that η is constant and independent of the light intensity E, which is not necessarily the case. Equation 19 states furthermore that the source strength does not depend on the concentration of the reactant, that is, that the effect of reactant consumption can be neglected. This point will be discussed at the end of this section. The differential equation 18 has to be integrated with the initial condition



c(x,0) ) 0

(20)

and the boundary conditions

∂ c(0,t) ) 0 ∂x

(21)

(no product flow through the sapphire window) and

c(d,t) ) -

Deff w

∂c (d,t′) ∫0t dt′∂x

(22)

(products leaving the porous layer accumulate in the gas volume of the µ-reactor and the diffusion coefficient D0 in the gas is so high that there is always a spatially constant product concentration). The parameter w is the ratio of the gas volume to the product of the illuminated cross-section of the porous layer with its porosity (w ) 2.83 mm . d e 6 µm). The measured quantity is the photoproduct concentration in the gas volume cV(t):

cV(t) ≡ c(d,t) ) ηJ0 f(t, d, w, K, S, Deff, F)

(23)

Figure 6. Simulated gas concentration profiles obtained after 600 s of back irradiation. Deff is set equal to 5 × 10-10 cm2 s-1 and η ) 10-4. The lines represent the concentration profile inside the layer (d ) 1.5, 3.0, 3.75, 4.5, and 6.0 µm) and the circles represent the gas concentration in the reactor void volume. On one profile (d ) 3.75 µm) the analytical solution is indicated at certain points with diamonds.

The factorization follows from the homogeneity of the boundary and initial conditions. All arguments, except η and Deff, are external experimental parameters (t, d, w, J0) or known from other experiments (K, S, F). The remaining parameters η and Deff shall now be determined by a fit from measured values of cV(t). The analytical solution of the stated problem is given in Appendix A. The resulting expression for the observable concentration cV(t) reveals the explicit dependence of this quantity on J0, η, Deff, and t

cV(t) )

t

∫ w+d 0

d

dx q(x) +

1 Deff

∞

(b0 +

bnγn e-D ∑ n)1

2

ν eff n˜

t

) (24)

the coefficients bn having the form bn ) ηJ0βn(d, w, K, S, F). For times t . 1/(Deffν21) the system reaches a steady state and cV(t) becomes a simple linear function in t. The steady state is reached after a time delay proportional to 1/Deff since b0 < 0. An analysis of the curve cV(t) for a single layer is not suited to determine the parameters Deff and η. For physical reasons, this curve increases monotonically in time for all values of η and Deff. Therefore, the curve does not discriminate well between different values of these parameters. More significant information can be expected if the layer thickness is varied at a constant accumulation time as indicated by Figure 4. To deduce the dependence of cV on the layer thickness d for a fixed accumulation time t implies rather lengthy calculations. We decided, therefore, to perform the analysis by numerical integration on the basis of a discretization approach as described in Appendix B. This method can also be more easily adapted to solve slightly modified problems (e.g., if reactant diffusion has to be incorporated), in contrast to the analytical solution. A parameter study was performed with this numerical solution where the following parameter values were kept constant: the -9 Einstein s-1, F ) 0.15, w total photon flow Jtot 0 ) 8.7 × 10 -1 ) 2.83 mm, K ) 0.34 µm , and S ) 0.68 µm-1 for monochromatic irradiation at 365 nm. In Figure 6, concentration profiles for CH4 in the TiO2 layer obtained after 600 s of irradiation from the sapphire side are shown. The thickness of the layer is varied from 1.5 µm to 6.0 µm. Deff is set equal to 5 × 10-10 cm2 s-1 and a value of η ) 10-4 is used.16 The

Figure 7. Amount of photoproduct diffused into the reactor cavity after 600 s of back irradiation in function of Deff for d ) 1.5, 3.75, and 6.0 µm. For a discussion of regions I, II, and III see text.

resulting profiles have a simple shape. They are horizontal at the sapphire window and turn then gradually to a constant negative slope near the interface between the layer and the reactor cavity. The product concentration in the reactor cavity is denoted by empty circles representing the measured quantity cV. Inside the layer, the local as well as the total amount of product rises monotonically with increasing thickness of the layer. In Figure 7, the dependence of cV on the diffusion coefficient Deff for three different layers (d ) 1.5, 3.75, 6.0 µm) after 600 s of back irradiation is reported. The sigmoid form of the curves corresponds to the analytical solution (eq 24). Three different regimes (I, II, and III) of Deff values may be discerned. Each regime is characterized by different shapes of the curves cV(d) as shown in Figure 8. For small values of Deff (regime I), the amount of detectable photoproduct decreases monotonically with increasing thickness of the layer because the system is then diffusion-limited and the detectable products stem from a thin layer just below the interface of the photocatalyst layer and the gas volume. The decrease of cV(d) represents the decrease of q(d) ) ηKE(d). An opposite behavior can be found for regime


Figure 8. Amount of photoproduct diffused into the reactor cavity after 600 s of back irradiation in function of the thickness d of the TiO2 layer and for values of Deff characteristic for the regimes I to III in Figure 7 (Deff ) 5 × 10-11, 5 × 10-10, 5 × 10-9 cm2 s-1).

III. For large values of Deff, photoproducts easily diffuse out of the layer and cV(d) being roughly proportional to the total amount of absorbed light increases with increasing thickness until the layer becomes totally absorbing. In regime II, intermediate values of Deff lead to a maximum in the curves cV(d). The effects of diffusion are less important for small values of d and we expect there a behavior as in region III, whereas diffusion becomes limiting for thick layers and we expect there a behavior as in region I. The exact shape of the curve is the result of a subtle interplay of diffusion and optical characteristics of the layer. The position of the maximum of the curve cV(d) is a sensitive indicator for the value of Deff because the quantum efficiency η enters only as a multiplicative factor and does not influence the relative shape of the curve. Strictly, this is only the case if η is constant and independent of the photon flow. In the more general case when η depends on the photon flow, η can affect the form of the curve. Sensitivity studies with respect to variations of the optical parameters K and S revealed that variations of these parameters up to 20% had only marginal effects on the position of the maximum. Separation of the parameters Deff and η is much more difficult in situations belonging to regime I or to regime III. In Figure 9, the experimental data reported in Figure 4 are reported with a single set of parameter values. For thν, J0, K(365), and S(365), the measured values, given above, were used. The values Deff ) (5.64 ( 2.51) × 10-10 cm2 s-1 and η ) (8.79 ( 0.79) × 10-4 were determined by a nonlinear least-squares fit from the measured CH4 yields for the different layer thicknesses. We judge the fit to be quite satisfactory, taking into account the uncertainties of the measured layer thicknesses and the porosity, respectively. Until now the whole treatment of the problem relied on the validity of eq 19, which states that the source is independent of the concentration of the reactant. In the simulations it is thus assumed that the concentration of the reactant is constant throughout the whole layer and remains so during the reaction. Thus reactants are not consumed during the reaction or are replaced instantaneously. This assumption needs some justification, especially in view that Deff of CO2 may be as small as Deff of CH4. In Figure 10 the influence of reactant depletion and diffusion on the concentration profiles inside the layer is reported. Equations 18-22 are here completed with a corre-

Corboz et al.

Figure 9. Fit of the experimental data of Figure 4. For both curves Deff ) 5.64 × 10-10 cm2 s-1 and η ) 8.79 × 10-4.

Figure 10. Influence of the consumption of reactants on the product concentration profiles. Solid lines: reactant; dashed lines: product. Thick lines include product consumption, thin lines do not. Curves are for back-irradiation geometry and Deff ) 5.64 × 10-10 cm2 s-1, η ) 8.79 × 10-4, d ) 3.75 µm, t ) 600 s.

sponding set of equations for the reactant. The rate of consumption of the reactant qR is set equal to

qR(x) ) -nηKE(x)

(25)

where n ) 46/4 ) 11.5 follows from the stoichiometry of eq 28. For the numerical simulation, t ) 600 s, η ) 8.79 × 10-4, and Deff ) DCH4 ) DCO2 ) 5.64 × 10-10 cm2 s-1 were used. In Figure 10 one immediately sees that the profiles for the reactant look very different in the two cases. In the region of large E(x) the reactant is strongly depleted. The concentration profile for the product also changes, but less drastically. The difference at the sapphire window is about 33%. cV(t), the concentration of the product in the reactor cavity, the ultimately measurable quantity, changes even less (about 18%). These numbers have to be put into perspective; as in a heterogeneous photoreaction, the first step is the adsorption of the reactants at the surface of the photocatalyst. Generally the associated adsorption/desorption equilibrium can be assumed to be shifted toward adsorption at realistic temperatures. As a result of this, changes in the local gas-phase concentration of the reactants only translate into small variations of the surface coverage on the photocatalyst. Only if the gas phase becomes severely depleted of reactants does their surface coverage change



strongly.22 Thus the results of the simulation above represent a worst-case scenario as the local gas-phase concentration of the reactant is used instead of the surface coverage. After which irradiation time the surface coverage actually starts to decrease strongly depends on the Deff of the reactant, as its actual value defines how fast reactants are resupplied. Finally, we shortly discuss the implications due to the negligence of back- or secondary reactions of the product. Generally, such adverse reactions cannot be excluded a priori. We are well aware that also for the reaction studied in this paper at least the photooxidation of methane does occur. In the case of a photochemical decomposition reaction of the product occurring with a quantum efficiency η′, eq 19 has to be replaced by

q′(x) ) ηKE(x) - η′KE(x)

(26)

In this case the analysis will determine only the difference in the quantum efficiencies between the source and the sink. In the case of a (thermal) monomolecular decay of the product, eq 19 has to be completed by a sink term or the form

-kbc(x,t)

(27)

An analytical solution can be found for this more complicated system much along the lines indicated in Appendix A. The structure of the solution is similar to the one given by eq 24. Because in the beginning c(x,0) ) 0 the solution is close to the previous one at the beginning of the time evolution. The main difference is that cV(t) will reach a steady-state value after some time instead of increasing linearly in time. We stress the fact that the numerical method described in Appendix B can easily be adapted to include the monomolecular decay of the product. Our experimental results, however, do not allow us to determine the additional constant kb since the parameters Deff and kb are statistically highly correlated as long as the steady state has not been reached. Discussion. Comparing the value of Deff in the TiO2 layer with the value of the coefficient for free diffusion D0, one notes the tremendous decrease of about 9 orders of magnitude. This low value must be a consequence of the structure of the TiO2 layer. In an extensive literature study, several models applicable to the TiO2 layers were considered: binary representations,9,23 flooded agglomerates,10 hard spheres,24,25 overlapping spheres,26-28 compressed or sintered spheres.5,29 The main common model parameters are the porosity (void fraction) and the tortuosity (connectivity of the individual pores) of the layer. For all these models, a Knudsen type of diffusion predicts values of Deff g 10-4 cm2 s-1 for layers of microcrystals of 20 to 70 nm in diameter, of a porosity of φ ) 0.58, and with tortuosity values in the range of 1 to 10.9,23,25,26,28,30,31 The experimental value Deff ) 5.64 × 10-10 cm2 s-1 can be obtained from these models only by an extremely high tortuosity of about 108. The SEM pictures shown in Figure 3, suggest that the TiO2 layers consist of rather large pores separated by walls of densely packed TiO2 microcrystals. We conjecture that a large number of these pores are disconnected from each other and that the densely packed walls are almost impenetrable to gases. This structure could account for the large tortuosity necessary to explain the small experimental value of Deff. Comparable systems might be zeolites. Mueller and Calzaferri applied in situ transmission Fourier transform infrared spectroscopy to find Deff ≈ 10-13 cm2 s-1 for the sorption of Mo(CO)6 in activated NaY zeolites.32 Two other results should be mentioned as they point out that porous systems with comparable porosity (φ )

0.6) can have high tortuosity because of their low amount of connected pores. Bentz and Garbozczi have shown that dry cement paste exhibits a drastically decreasing porosity as a function of an increasing degree of hydration.33 They suggest that the microcrystals actively aggregate and form unconnected pores. Schwartz developed a new model of “consolidated grain packing” that simulates porous media with a high amount of unconnected pores.34 This model might be a realistic description of our samples. The quantum efficiency η of a porous layer of a microcrystalline semiconductor follows from the least-squares fit with a good precision. The experimental value η ) (8.79 ( 0.79) × 10-4 for the formation of CH4 may appear rather small but can be explained by the photophysical properties of Degussa P25 and the peculiar stoichiometry of the photoreaction (eq 28). In a previous study on the photophysical behavior of monodisperse TiO2 particles of different sizes,35 the authors found that the relaxation dynamics strongly depend on the particle size. For particles up to 30 nm, comparable to Degussa P25, more than 90% of the photogenerated electron/hole pairs have recombined in the bulk after 10 ns. Therefore only about 10% of the photoproduced electron/hole pairs are trapped at the surface of the TiO2 crystallite where they subsequently can react with absorbed molecules. This implies that the quantum efficiency for a photocatalytic process in such a system must be 10% or less. The reaction in our case consists of a complex sequence of individual steps starting with the absorption of a photon. The first chemical step then is the transfer of an electron from TiO2 onto adsorbed CO2, forming the CO2- radical. Until the final products, one of which is CH4, are formed and released, several intermediate steps are necessary. They are either associated with photochemical or thermal reactions. In our previous experiments the following average stoichiometry of the overall reaction has been established13-16

46 CO2 + 13 H2O f 4 CH4 + 2 H2 + C2H6 + 40 CO (28) In this reaction 130 e- are consumed to form the various photoproducts. However, photoproducts from the reaction of the associated holes such as, for example, dioxygen could not be identified until now. Thus eq 28 should be used with care. Taking this stoichiometry and the fact that ultimately eight electrons are needed to reduce CO2 to CH4,16 the expected maximum quantum efficiency for the formation of CH4 is about 3 × 10-3. Comparing this number with the value of η ) 8.79 × 10-4 from the present work, we conclude that TiO2 photoreduces carbon dioxide to methane with an efficiency of about 29% relative to the theoretical maximum efficiency. Conclusion Irradiating thin TiO2 layers acting as photocatalysts from the front and from the back leads to a completely different dependence of the photoproduct concentration above the layer on the thickness of the layer. This behavior can be quantitatively modeled by taking into account the locally varying photoproduction and the diffusion of the photoproducts out of the layer. The corresponding effective diffusion coefficient Deff and the quantum efficiency of the reaction η can be determined from this model if the experimental conditions are suitably chosen. It should be noted that the value of Deff is an intrinsic property of the layer and thus depends very much on the exact structure of the layer. Different methods of preparation used may well lead to different values of Deff. A surprisingly small value of Deff ) (5.64 ( 2.51) × 10-10 cm2 s-1 was experimentally found. We explain this low value

10576 J. Phys. Chem. B, Vol. 104, No. 45, 2000 by a high tortuosity of our samples produced by slow sedimentation of aqueous suspensions. We stress the fact that Deff has to be considered as an effective diffusion coefficient eventually comprising processes such as product adsorption and desorption. To clarify the microscopic nature of the processes covered by Deff, the layer structure must be characterized in more detail in a way that porosity and tortuosity can be determined independently. Information to clarify this point might result from a study on the temperature dependence of the reaction. The low value of the quantum efficiency η ) (8.79 ( 0.79) × 10-4 experimentally found conforms to independent findings on the photophysical behavior of Degussa P25 and to the experimental stoichiometry of the reaction. From the analysis presented in this paper it is evident that front irradiation of the layer of photocatalyst is to be preferred to minimize the influence of diffusive mass transport inside the layer. Nevertheless, some ingenious designs reported in the literature rely on the back irradiation geometry. Examples are TiO2-coated waveguides37 or some annular flow like reactors.38,39 In these cases the optical characteristics of the photocatalyst layer have to be determined and the thickness of the layer has to be chosen carefully. We have indicated that the effect of reactant depletion does not play a significant role as long as the local reactant concentration in the pores of the layers does not become extremely small. Only then does the surface coverage of the reactants on the photocatalyst surface start to decrease significantly. We have also shown that certain types of back- or secondary reactions can be included into the analysis but that the present experimental results do not allow us to determine the additional phenomenological parameters. We think that this can be justified, as we are not talking about quantum yields but about quantum efficiencies. Thus the parameters η and Deff are intrinsic to layers of a common microstructure (characterized by φ, K, and S) and the photoreaction under consideration. They are generic in the sense that they are independent of the thickness and the geometry of the layer and of the experimental setup used for their determination. The procedure described in this study is rather general and may also be applied with minor modifications to traditional thermal catalysis if a temperature gradient through a catalyst bed is considered. In this case, the spatially varying temperature plays the role of the spatially varying light intensity, and the local product formation will be described by some Arrhenius expression. We finally remark that the photocatalytic layers used in our experiments are neither compressed nor specially treated otherwise, but are uniquely the result of a slow sedimentation process in water. This method of layer formation is commonly used. The unexpectedly low value of the gas diffusion coefficient may apply quite generally to layers of this type.

c(x,t) )

(

x2 - d2 2Deff

1 Deff 1 Deff

The analytical solution of the differential equation 18 with the boundary conditions (eq 21) and (eq 22) and the initial condition (eq 20) can be found as a superposition of a special solution of the inhomogeneous differential equation satisfying the boundary conditions but not the initial condition and the general solution of the homogeneous differential equation matching the boundary conditions

)

1

d dx q(x) + ∫ 0 w+d

∫xd dx′ ∫0dx′ dx′′ q(x′′) + ∞

∑bn[e-D n)1

2 effνnt

cos (νnx) - cos (νnd)] (29)

where the νn are the positive solutions of the equation

tan(νd) ) -νw

(30)

The functions exp(-Deffν2nt) cos (νnx) - cos (νnd) are the only solutions of the homogeneous differential equation with exponential time dependence satisfying the boundary conditions. Since w . d in our case, the quantity νnd should be written as

ν nd )

2n - 1 π + µn, n ) 1, 2,... 2

(31)

where µn is the solution of the equation

tan µn )

d 1 w 1 n - π + µn 2

(

(32)

)

with 0 < µn < π/2. It is, therefore,

νn )

1 2n - 1 π d 2 1+O + 2 d (2n - 1)π w w

[

( )]

(33)

The coefficients bn follow now from the initial condition c(x,0) ) 0. They must be chosen such that ∞

-

x2 - d 2

∑bnφn(x) ) 2(w + d) ∫0 n)1

d

dx q(x) +

∫xd dx′ ∫0x′ dx′′ q(x′′) ≡ F(x)

(34)

where

φn(x) ) cos(νnx) - cos(νnd)

(35)

The determination of the coefficient bn is astonishingly easy. Note that

∫0d dx cos(νmx)φn(x) ) 21(d + wγ2m)δmn,

n, m ) 1, 2, ... (36)

where

Acknowledgment. We thank Prof. G. Calzaferri for his continuing support and interest in our work. This work has been supported by the Swiss Federal Office of Energy. Appendix A

+t

Corboz et al.

γm ) cos(νmd) )

(-1)m

x1 + ν2mw2

(37)

as a consequence of eq 30 defining the numbers νm. Inserting now the series expansion eq 34 of the function F(x) into the integral ∫d0dx cos(νmx)F(x) one readily obtains

bm )

-2 d + wγ2m

∫0d dx cos(νmx)F(x), m ) 1, 2, ...

(38)

A similar evaluation of the integral ∫d0 dxφm(x)F(x) yields the additional equation

Determination of Quantum Efficiencies ∞

b0 ≡ -

-1


∑bnγn ) d + w ∫0

d

dx F(x)

(39)

n)1

where eq 30 has been used again. The observable quantity cV(d,t) is given by

cV(d,t) )

t

1

d dx q(x) + ∫ 0 w+d D

∞

(b0 +

eff

∑bnγne-D n)1

ν2t eff n

) (24)

The dependence on t and Deff is here explicitly given [bn ) ηJ0βn(d, w, K, S, F)], but the dependence on d is rather complicated. Appendix B The numerical solution of the differential eq 18 is performed by discretization. The whole spatial interval [0, d+w] is divided into N cells, each cell being characterized by its spatial extension δn and spatially constant parameters cn, Dn, and qn, the mean values of the concentration, the diffusion coefficient, and the source strength in the cell, respectively. The cells 1, ..., N - 1 contain the porous solid, the cell N the entire gas volume. One may choose δ1 ) δ2 ) ... ) δN-1) δ ) d/(N - 1), δN ) w and D1, D2, ..., DN-1) Deff, DN ) D0 f ∞. There are diffusion currents between adjacent cells, the current jn flowing from cell n to cell n + 1. Equation 17 is replaced by a system of difference equations

jn ) -

2Dn / 2Dn+1 (c n - cn) ) (c - c *n), δn δn+1 n+1 n ) 1, 2, ..., N - 1 (41)

The auxiliary quantity c/n, the “concentration at the boundary n”, is eliminated from these equations yielding

2DnDn+1 (c - cn), n ) 1, ..., N - 1 jn ) δn+1Dn + δnDn+1 n+1

(42)

With the additional conditions

j0 ) jN ) 0

(43)

(no diffusion currents through the sapphire windows), mass conservation for the nth cell yields

dcn 1 ) (jn-1 - jn) + qn, n ) 1, ..., N dt δn

(44)

where qN ) 0. Inserting the currents jn into eq 44 yields a linear differential equation for the vector b c ≡ (c1, c2, ... cN)

dc b ) Hc b+b q dt

(45)

with a tridiagonal matrix H. The formal solution of this differential equation is given by

b c (dt + t0) ) b c (t0) + P(dt)[Hc b(t0) + b q]

(46)

where the matrix P(dt) equals -1

P(dt) ≡ H (e

∞

Hdt

- 1) ) dt

∑

(H dt)n

n)0(n

+ 1)!

(47)

c(kdt) and b c0 ) B 0, eq 46 defines a recursion Setting b ck ) b with the constant matrix P(dt). It is recommended to use this matrix in the form -1 1 1 P(dt) ) dt 1 - (H dt) + (H dt)2 + δ(dt5) (48) 2 12

[

)

to improve numerical stability (see ref 36, section 15-6). The recursion amounts then to a repeated solution of a system of linear equations with always the same coefficient matrix but varying right-hand sides. This task can be achieved advantageously with a LU decomposition of the matrix (see ref 36, section 12-3), which has to be performed only once, and the consecutive solutions of the upper and lower triangular sets of equations by forward and back substitution. References and Notes (1) Serpone, N.; Pelizzetti, E.; Hidaka, H. Proceedings IPS-9: Beijing, 1992; pp 33-73. (2) Emeline, A. V.; Ryabchuk, V. K.; Serpone, N. J. Phys. Chem. B 1999, 103, 1316. (3) Emeline, A. V.; Lobyntseva, E. V.; Ryabchuk, V. K.; Serpone, N. J. Phys. Chem. B 1999, 103, 1325. (4) Pollard, W. G.; Present, R. D. Phys. ReV. B 1948, 73, 763. (5) Reyes, S. C.; Iglesia, E. J. Catal. 1991, 129, 457. (6) Strieder, W. J. Chem. Phys. 1969, 51, 566. (7) Strieder, W.; Aris, R. J. Inst. Math. Its Appl. 1971, 8, 328. (8) Garza-Lopez, R. A.; Kozak, J. J. J. Phys. Chem. B 1999, 103, 9200. (9) Deng, H.; Zhou, M.; Abeles, B. Solid State Ionics 1995, 80, 213. (10) Giner, J.; Hunter, C. J. Electrochem. Soc. 1969, 116, 1124. (11) Serpone, N.; Salinaro, A. Pure Appl. Chem. 1999, 71, 303. (12) Salinaro, A.; Emeline, A. V.; Zhao, J.; Hidaka, H.; Ryabchuk, V. K.; Serpone, N. Pure Appl. Chem. 1999, 71, 321. (13) Saladin, F.; Meier, A.; Kamber, I. ReV. Sci. Instrum. 1996, 67, 2406. (14) Saladin, F.; Forss, L.; Kamber, I. J. Chem. Soc., Chem. Commun. 1995, 533. (15) Saladin, F.; Alxneit, I. J. Chem. Soc., Faraday Trans. 1997, 93, 4159. (16) Saladin, F. Versuche zur direkten Umwandlung konzentrierter Solarstrahlung in chemische EnergiesPhotosynthetische CO2 Reduktion und H2O Oxidation an der Gas/Fest-Phasengrenze, Dissertation, Universita¨t Bern, 1997. (17) Alxneit, I.; Corboz, M. J. Phys. IV 1999, 9, 295. (18) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover Publications: New York, 1972. (19) Landolt-Bo¨ rnstein, 6th ed.; Springer: Berlin, 1985; Vol II/5a. (20) Kortu¨m, G. Reflexionsspektroskopie; Springer: Berlin, 1969. (21) Marquardt, D. W. J. Soc. Ind. Appl. Math. 1963, 11, 431. (22) Very preliminary experiments show that changing the reactant from 100% CO2 to 25% CO2 in N2 does not change the CH4 evolution rate. (23) Burganos, V. N. J. Chem. Phys. 1998, 109, 6772. (24) Evans, J. W.; Abbasi, M. H.; Sarin, A. J. Chem. Phys. 1980, 72, 2967. (25) Levitz, P. J. Phys. Chem. 1993, 97, 3813. (26) Ho, F. G.; Strieder, W. Chem. Eng. Sci. 1981, 36, 253. (27) Ho, F. G.; Strieder, W. J. Chem. Phys. 1982, 76, 673. (28) Akanni, K. A.; Evans, J. W.; Abramson, I. S. Chem. Eng. Sci. 1987, 42, 1945. (29) Reyes, S. C.; Iglesia, E.; Chiew, Y. C. Mater. Res. Soc. Symp. Proc. 1990, 195, 553. (30) Chen, L. L.; Katz, D. L.; Tek, M. R. AIChE J. 1977, 23, 336. (31) Wo¨hr, M.; Bolwin, K.; Schnurnberger, W.; Fischer, M.; Neubrand, W.; Eigenberger, G. Int. J. Hydrogen Energy 1998, 23, 213. (32) Mu¨ller, B. R.; Calzaferri, G. Microporous Mesoporous Mater. 1998, 21, 59. (33) Bentz, D. P.; Garboczi, E. J. Mater. Res. Soc. Symp. 1990, 195, 523. (34) Schwartz, L. M. Mater. Res. Soc. Symp. 1990, 195, 537. (35) Serpone, N.; Lawless, D.; Khairutdinov, R.; Pelizzetti, E. J. Phys. Chem. 1995, 99, 16655. (36) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes in Pascal; Cambridge University Press: Cambridge, 1994. (37) Miller, L. W.; Tejedor-Tejedor, M. I.; Anderson, M. A. EnViron. Sci. Technol. 1999, 33, 2070. (38) Ray, A. K. Chem. Eng. Sci. 1999, 54, 3113. (39) Shiraishi, F.; Toyoda, K.; Fukinbara, S.; Obuchi, E.; Nakano, K. Chem. Eng. Sci. 1999, 54, 1547.

On the Determination of Quantum Efficiencies in Heterogeneous

Recommend Documents