Evaluating particle-suspension reactor designs for Z-scheme solar water splitting via transport and kinetic modeling

Abstract

Sunlight-driven water splitting to produce hydrogen and oxygen provides a pathway to store available solar energy in the form of stable, energy-dense chemical bonds. Here we investigate a tandem particle-suspension reactor design for solar water splitting comprising micron-scale photocatalyst particles suspended in an aqueous solution with soluble redox shuttles. A porous separator facilitates redox species transport between the hydrogen and oxygen evolution reaction compartments while averting gas crossover. A two-dimensional, transient model of the reactor is presented to illustrate the coupling between light absorption, interfacial electron-transfer kinetics and species transport, and their combined impacts on overall solar-to-hydrogen conversion efficiency. The volumetric reactivity of the suspended semiconductor particles is dictated by combining the (photo)current–voltage behavior of a photodiode with Butler–Volmer electron-transfer kinetics. For the first time, a quantitative approach to determine the impacts of surface-dependent redox shuttle kinetic parameters on reaction selectivity in a Z-scheme system is established. Model results provide insights on the effects of optical, transport and kinetic properties of the semiconductor particles and the redox shuttles on the overall reactor performance. Solar-to-hydrogen reactor efficiencies predicted with BiVO₄ particles for oxygen evolution are at least two times larger than efficiencies achieved with wider band-gap TiO₂ particles due to enhanced visible light absorption; hydrogen evolution with SrTiO₃:Rh particles was considered for both cases. Superior performance is predicted with proton-coupled electron transfer redox shuttles (para-benzoquinone/hydroquinone and iodide/iodate) that absorb little-to-no visible light, while also facilitating operation at near-neutral pH conditions, as compared to the non-proton-coupled triiodide/iodide and iron(III)/iron(II) redox shuttles. For 1 cm tall reaction compartments, diffusive species transport is fast enough to sustain reactor operation at a 1% solar-to-hydrogen conversion efficiency for both para-benzoquinone/hydroquinone and iodate/iodide redox shuttles with less than 2.2 mg L⁻¹ of each of BiVO₄ and SrTiO₃:Rh particles in the solution.

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Broader context

Pairing the most abundant energy resource on our planet, the Sun, with energy storage technologies is essential to overcome the diurnal variations in the availability of sunlight. Taking a leaf from nature's book, artificial photosynthetic devices capture and store sunlight in energy dense fuels by converting water and carbon dioxide to produce hydrogen and other hydrocarbons. For solar hydrogen production, particle-suspension reactor designs comprising suspended photocatalysts in an aqueous electrolyte are projected to be cost-efficient alternatives to the more exhaustively optimized fixed-electrode device architectures. However, there is a lack of a quantitative understanding of the device-scale transport and kinetic processes, and its impacts on the solar-to-hydrogen efficiencies for these designs. An innovative and intrinsically safe reactor design for Z-scheme solar water splitting is proposed. Two reaction compartments with suspended photocatalysts for hydrogen and oxygen production are optically connected in series to more efficiently utilize the incident solar spectrum while facilitating product separation. Soluble redox shuttles relay electrons between these compartments. A computational model is developed to guide device-design and operating conditions to achieve diffusion-sustained, efficient reactor operation. Results reveal the promise of using proton-coupled-electron-transport redox shuttles and the significance of selective surface catalysis to boost efficiency.

Introduction

Stable, scalable and efficient devices for artificial photosynthesis offer the potential to harness and directly store the Earth's most abundant renewable energy resource in the form of energy-dense chemical bonds, such as those in hydrogen and hydrocarbons derived from carbon dioxide.^1–3 These solar fuels are net-carbon-neutral alternatives to fossil fuels (assuming that the carbon is derived from combustion processes or the atmosphere), and enable storage and transport of solar energy for use when and where it is needed. Several solutions are being pursued for solar fuels production including photovoltaic-powered electrochemical devices for water and carbon dioxide splitting,^4–7 solar thermochemical cycles,^8–10 integrated photoelectrochemical (PEC) devices,^11–16 and suspended photocatalyst particles.^17–20 Fixed-electrode, wafer-based PEC devices to split water, that consist of photoactive electrodes immersed in an aqueous electrolyte, have been experimentally demonstrated^12,21–26 and modeled extensively.^27–34 These designs have exhibited solar-to-hydrogen energy-conversion (STH) efficiencies as high as 18%²⁵ at 1 Sun illumination and over 30%²⁶ with concentrated sunlight; however, these demonstrations suffered from stability issues where operational lifetimes were less than one day.¹² Recent technoeconomic analyses^35–37 make evident the need for fixed-electrode PEC devices to be operated at STH efficiencies greater than 25% to reach price parity with hydrogen produced from fossil fuels. However, the electrode-based devices that were stable for at least one day have exhibited STH efficiencies of at best 2% and they utilized designs that were inherently unsafe due to the generation of an explosive mixture of the H₂ and O₂ reaction products.^20,38,39 Reactor designs based on photocatalyst particle suspensions²⁰ coupled with stable, low-cost and efficient material options are conceived to be cost-competitive with hydrogen produced from fossil sources, even at lower STH efficiencies of ∼10%. The projected low levelized cost of hydrogen generated from particle-suspension reactors combined with the lack of in-depth knowledge for performance optimization of these reactors motivates the present study.

Numerous experimental demonstrations pertinent to particle-suspension, Z-scheme reactors with electron transfer mediated by a soluble redox shuttle have been reported,^40–50 and a comprehensive review on this topic has been presented by Fabian and colleagues.²⁰ The predominant focus in the aforementioned studies was the identification of semiconductor materials^40–48 with enhanced activities for implementing the oxygen evolution reaction (OER) and the hydrogen evolution reaction (HER), development of surface treatment techniques and selection of redox shuttles to suppress undesired interfacial electron-transfer reactions,^41,49 and evaluation of the effectiveness of cocatalyst⁵⁰ and coating materials.⁵¹ In prior work, device-scale design and performance dependences on the transport and kinetic processes of the system were not considered. Therefore, from the standpoint of designing particle-suspension reactors, there is a need for a more holistic understanding of the intertwined multiphysics processes of light absorption, species transport in the electrolyte, and heterogeneous surface reactions to maximize the STH efficiency. Moreover, the implications of the kinetic parameters of the redox shuttle on the selectivity towards the desired reactions have not been discussed in the previous studies.

Prior numerical models for solar water and carbon dioxide splitting have mostly been developed to guide the design of the fixed-electrode PEC devices.^27–34 Jaini and Fuller⁵² evaluated mass-transfer correlations for gas-induced convective mixing in a dual-bed, particle-suspension reactor and determined the steady-state redox shuttle concentration profiles assuming spatially homogenous reaction rates. However, the effects of optical absorption by the particles and the influence of interfacial electron-transfer kinetics on the species mass transport were absent from that study. For applications of water treatment with colloidal suspensions of TiO₂, volumetric light absorption and scattering properties of the suspension,^53–55 mass-transport limitations⁵⁶ and radiative transport⁵⁷ have been evaluated using numerical and experimental techniques. However, the target chemical reactions for water treatment are inherently different from water splitting, primarily because the overall reactions are often thermodynamically favorable, and tandem photosystems with soluble redox shuttles have not been considered for these applications.

In this paper, we report on the development of a transient, two-dimensional numerical model of a vertically stacked, Z-scheme particle-suspension reactor for sunlight-driven water splitting. This design (Fig. 1) includes two important advances from prior conceptual designs for dual-bed Z-scheme reactors. The vertical arrangement of the reaction compartments facilitates tandem light absorption, which results in increased theoretical maximum STH efficiencies. By stacking the compartments, this design can afford mass transport over the smallest possible distances, therefore enabling sustained (effectively indefinite) reactor operation in the absence of convection. A notable advantage of the proposed Z-scheme reactor design is the opportunity to evolve H₂ and O₂ in separate physical locations, thus preventing formation of an explosive mixture of gases. The model developed accounts for diurnal variations in the incident solar irradiation and couples the effects of optical absorption in the aqueous suspension, power supply of an ideal photodiode, reversible interfacial reaction kinetics, and species mass transport. By varying reactor height and concentrations of the semiconductor particles and the redox shuttle species the model is used to evaluate transient reactor performance and identifies STH efficiencies that maintain sustained reactor operation.

Fig. 1 (a) Schematic of the vertically stacked particle-suspension reactor design which affords tandem light absorption for Z-scheme solar water splitting, and (b) the desired chemistry and direction of the electron transfer reactions on the surface of the semiconductor particles, assuming acidic conditions. D⁺/D is only used to represent symbolically a redox shuttle and the respective counterions are not shown.

Particle-suspension reactor

Our study specifically considers a tandem particle-suspension reactor for solar water splitting that relies on a two-step Z-scheme mechanism⁵⁸ and a reversible redox shuttle dissolved in an aqueous electrolyte (Fig. 1). In a Z-scheme mechanism, the thermodynamic energy requirement for water splitting (WS) is attained using two photosystems (L1 and L2), similar to natural photosynthesis. Absorption of solar photons by a particle results in both an anodic (OER or R_ox) and a cathodic (HER or R_red) half-reaction being driven at distinct semiconductor–electrocatalyst–electrolyte interfaces. Fig. 1 shows OER and R_red in the top compartment, and HER and R_ox in the bottom compartment, but the locations of these reactions is a design variable and the choice of the redox shuttle determines which reactions should optimally be driven in the top and bottom compartments. Electrocatalysts deposited on semiconductor surfaces are commonly referred to as cocatalysts in the pertinent literature,²⁰ and we adopt this terminology in the rest of this paper. As shown in Fig. 1, in the top compartment, light absorption generates holes that oxidize water to O₂ (OER) and electrons that reduce the oxidized state of the redox shuttle, D⁺ → D (R_red). In the bottom compartment, light absorption generates electrons that reduce protons to H₂ (HER), while simultaneously the photo-generated holes regenerate the oxidized state of the redox shuttle, D → D⁺ (R_ox). Cocatalysts deposited on these particles reduce overpotentials for the desired redox reactions and potentially also improve the product selectivity for the desired pair of surface reactions (OER and R_red; HER and R_ox) by influencing the charge-separation and electron transfer mechanisms. Intrinsic separation of product gases makes this design safe, and reduces Faradaic losses and explosion hazards that can occur due to reactions between product gases.

Oxygen evolution reaction (OER) particles	H2O + 2hν + 2D^+ → 2H^+ + ½O2 + 2D (L1)
	H2O + 2h^+ → 2H^+ + ½O2 (OER)	2D^+ + 2e^− → 2D (Rred)
Hydrogen evolution reaction (HER) particles	2H^+ + 2hν + 2D → H2 + 2D (L2)
	2D + 2h^+ → 2D^+ (Rox)	2H^+ + 2e^− → H2 (HER)
Net	H2O + 4hν → H2 + ½O2		(WS)

The ability of a particle-suspension reactor to attain large STH efficiencies and demonstrate sustainable operation relies in part on effective mass transport of the redox shuttle (D⁺/D) and protons, or hydroxides under alkaline conditions, between the physical locations of the OER and the HER particles.²⁰ In the proposed reactor design, the characteristic lengthscale for diffusive mass transport and incident light absorption is the reaction compartment height. To maximize the rate of photon absorption and minimize energy losses associated with large concentration gradients, the reactor compartment height therefore needs to be optimized. The mass-transport lengthscales considered here are on the order of centimeters, in contrast to the lengthscales on the order of meters that were considered for horizontally-adjacent dual-bed designs.^35,36

After surveying the semiconductor materials, dopants, cocatalysts, and redox shuttles reported in the literature,²⁰ we selected semiconductor materials with demonstrated stability as aqueous suspensions, appropriate band energies to drive the desired reactions, and availability of published data on optical properties. Two options were investigated: Option I, which benefited from use of semiconductor particles with complementary bandgaps (E_bg), and Option II, which did not (Table 1). For Option I, the OER material was TiO₂ (E_bg = 3.1 eV,⁵⁹ average value of its two most common crystal structures) and for Option II the OER material was BiVO₄ (E_bg = 2.5 eV),^20,60 both materials with high reported activities for the light-driven OER in Z-scheme designs.^41,49,50,61 The HER material for both these options was SrTiO₃:Rh (rhodium-doped SrTiO₃), with a lowest-energy absorption onset of 2.5 eV, due to excitation from a Rh³⁺ defect band to the SrTiO₃ conduction band,⁶³ and one-of-the-best reported activities for the light-driven HER in Z-scheme designs.^45,48,51Table 1 shows the band gaps of the selected photocatalysts, which have suitable band-edge positions for the valence and conduction bands (denoted by E_vb and E_cb in Table 2) relative to the desired redox reaction potentials. To mediate charge transport across the solutions, various demonstrated redox shuttles were analyzed (I₃⁻/I⁻, IO₃⁻/I^{− 20,41,42,49} and Fe³⁺/Fe^{2+ 48,50,68}) as well as an organic redox couple, p-benzoquinone/hydroquinone (Q/QH₂), which has been used in state-of-the-art aqueous redox flow batteries and can undergo proton-coupled electron transfer redox chemistry.^69–71Table 1 shows the pH conditions required for each redox shuttle to maintain it as the dominant species in solution. Acidic conditions (pH < 4) are required for the I₃⁻/I⁻ and the Fe³⁺/Fe²⁺ redox shuttles; for the Fe³⁺/Fe²⁺ redox shuttle, a one-electron-transfer reaction was analyzed. By coupling the proton-coupled electron transfer reactions for Q/QH₂ with the inherently proton-coupled-electron-transfer OER or HER the overall reactions in each compartment exhibit pH independent thermodynamic potentials,

H₂O + Q → QH₂ + ½O₂ E⁰ = 0.53 V

QH₂ → H₂ + Q E⁰ = 0.7 V

Table 1 Semiconductor materials and redox shuttles for model simulations

	Option I	Option II
Semiconductor materials	OER—TiO2^59,62 E bg = 3.1 eV	OER—BiVO4^20,60 E bg = 2.5 eV
	HER—SrTiO3:Rh^63–65 E bg = 2.5 eV	HER—SrTiO3:Rh^63–65 E bg = 2.5 eV
Redox shuttles (D^+/D)	I3^−/I^− (pH < 4),^41 Fe^3+/Fe^2+ (pH < 2.5),^50 IO3^−/I^− (pH > 7),^41 Q/QH2 (acidic to neutral)^66,67

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Table 2 Baseline values for the modeling parameters

Category	Parameter (unit)	Values
Geometric and porous separator parameters	t sep (mm)	1
	w unit (cm)	1
		0.8
	ε sep	0.26
Semiconductor particles	j rr (A m^−2)	TiO2 and BiVO4	10^−18
		SrTiO3:Rh	10^−15
	E vb (V vs. NHE)^20,59,62,63,65	TiO2	2.7
		BiVO4	2.3
		SrTiO3:Rh	1.5
	E cb (V vs. NHE)^20,59,62,63,65	TiO2	−0.4
		BiVO4	−0.2
		SrTiO3:Rh	−1
	ρ p (g cm^−3)	TiO2	4.23
		BiVO4	6.95
		SrTiO3:Rh	4.81
	γ d	1
	d p (nm)	100
Redox shuttles and electrolytes	D i ^70,78 (10^−9 m^2 s^−1)	I^−, I3^−, IO3^−	2, 1, 1
		Fe^3+, Fe^2+	0.8, 0.8
		Q, QH2	0.6, 0.74
		H^+	9.3
		K^+, Cl^−	2
	Aqueous solubility (M)^79	KI, KIO3	8.92, 0.43
		FeCl3, FeCl2	5.7, 5.2
		Q, QH2	0.1, 0.5
	E ^0 (V vs. NHE)^80	I3^−/I^−; IO3^−/I^−	0.536; 1.085
		Fe^3+/Fe^2+	0.771
		Q/QH2	0.7
Kinetic parameters	j 0,OER (A m^−2)	OER^27,72,73	10^−3
	α a,OER, αc,OER		1, 0.1
	j 0,HER (A m^−2)	HER^27,74,75	10
	α a,HER, αc,HER		2.56, 2.56
	f cat		0.5

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The IO₃⁻/I⁻ redox shuttle also exhibits proton-coupled electron-transfer reactions (six proton, six electron-transfer reactions) for neutral to alkaline pH conditions.⁴¹

The interfacial reaction kinetics for the OER and the HER were modeled by considering the electron-transfer properties of state-of-the-art electrocatalysts, RuO₂ for OER^27,72,73 and Pt for HER,^27,74,75 even though in practice these cocatalysts are not always used.²⁰ The choice of the photocatalyst and the cocatalyst (if present) will impact the overpotentials for each redox reaction and determine the reaction selectivity. Therefore, the impacts of the redox shuttle kinetic parameters (exchange current density, j_0,sh, and charge transfer coefficients, α_a,sh, α_c,sh) were examined in addition to the baseline simulations that assumed ideal selective catalysis with facile redox chemistry.

For each of these cases, the developed model was used to evaluate the effects of the reaction compartment heights (1 to 10 cm), and the concentrations of the semiconductor particles (10⁻⁶ to 1 g L⁻¹) and the redox shuttles (10⁻⁴ to 5 M) on achieving sustained reactor operation, aided by only diffusive species transport, for a baseline STH efficiency of 1%. This modest efficiency target is set to evaluate the theoretical feasibility of the reactor design. Pathways to achieve higher STH efficiencies with the proposed design are also reported.

Theoretical

Modeling domain and assumptions

A transient, two-dimensional, continuum transport model was developed to simulate light absorption and mass transport in the reactor. Fig. 2 depicts the numerical modeling domain with the key governing equations used to model the coupled physical processes and the applied boundary conditions. Table 2 presents the baseline case modeling parameters. The domain comprises the two reaction compartments and a porous separator, and exploits centerline symmetry to reduce the computational expense. The aqueous suspension is assumed to contain a spatially homogeneous and continuous distribution of particles that remain completely soluble. Individual photocatalyst particles are not explicitly modeled and instead are treated as point locations that collectively exhibit diode current–voltage behavior (explained in the subsection on Conversion of light to power). The continuum transport equations formulated are hence applied for an “effective” medium comprising the suspended particles and the aqueous electrolyte. For colloidal suspensions with solid/particle volume fractions less than 0.006, the effective medium and independent light scattering are valid approximations.⁷⁶ These volume fractions translate to particle concentrations less than 20 g L⁻¹, 40 g L⁻¹, and 30 g L⁻¹ based on the bulk densities of TiO₂, BiVO₄, and SrTiO₃:Rh, respectively (listed in Table 2), and we consider concentrations below 1 g L⁻¹ for the model simulations. Therefore, treating the suspension as an effective medium is a reasonable approximation.

Fig. 2 Schematic of the two-dimensional unit cell of the reactor (not drawn to scale). It lists the key geometric parameters and the governing equations modeled for evaluating coupled light absorption, reaction kinetics, and species transport in the reactor. The modeling domain comprises one-half of the unit cell and exploits centerline (red dashed line) symmetry; boundary conditions applied on its external boundaries (solid black lines) and the vertical centerline are shown.

A nominal particle diameter, d_p, of 100 nm was used based on a reported powder-to-agglomerate size range of 10–1000 nm⁷⁷ for the OER and the HER semiconductor particles. The particle size does not directly enter the analysis since the model treats the particles as point locations and thus assumes that the rates of charge-carrier transport (diffusion and migration) within the particle are much faster than the reaction rates and device-scale diffusive rates of species transport. This assumption is reasonable because the particle sizes considered here are much smaller relative to the device dimensions (d_p ∼ 100 nm vs. h = 1–10 cm). The choice of the particle size was rather inconsequential because modeling results for reaction rates and STH efficiencies were found to be insensitive to the value of d_p. This insensitivity is a consequence of the model treating the particles as point locations and the particle size only implicitly enters the analysis as a scale factor for the specific surface area, A_0p (as will be shown in eqn (4a) and (6)).

Porous separator

The porous separator dimensions (t_sep, w_sep) and porosity, ε_sep, are selected to restrict diffusive gas crossover losses to less than 2%, a value that results in H₂ and O₂ streams with purity that is large enough to prevent formation of explosive mixtures.²⁸ The conditions for this safe operation were determined by applying a one-dimensional Fickian diffusion model assuming saturated dissolved gas concentrations (1.23 mM for H₂ and 0.78 mM for O₂). For a separator thickness (t_sep) of 1 mm, the following constraint, , arises to minimize gas crossover. More details about this calculation are provided in the ESI.† For the baseline case, the ratio of the separator width to the unit cell width, , is arbitrarily chosen to be 0.8, yielding a required separator porosity, ε_sep < 0.26.

Conversion of light to power

The spatial profile of light intensity in the compartments depends on the effective optical properties (absorption and scattering) and can be approximated by the Beer–Bouguer–Lambert law for solutions with negligible scattering and radiative emission. Negligible radiative emission is a reasonable assumption at low operating temperatures, whereas scattering effects can be neglected only when particle/aggregate sizes are small (typically less than 10 nm) relative to the incident wavelengths of sunlight. For larger particles/aggregates, both absorption and scattering effects have to be taken into consideration to obtain the spatial distribution of photon fluxes,^53,76 and therefore the reaction rates. However, in the absence of literature data to correlate the particle size-optical property dependencies and the added complexities to solve the full radiative transport equation, we applied Beer's law (eqn (1)) to evaluate the spatial profile of photon flux Φ along the height of the reactor,

Φ_y,λ = Φ_AM1.5,λ exp(−κ_eff,λy) (1)

where Φ_AM1.5,λ is the incident AM1.5G⁸¹ photon flux and κ_eff,λ is the effective spectral absorption coefficient of the particle suspension. Applying Beer's law (eqn (1)) represents a worse-case scenario for the spatial gradients of the photon fluxes, and is expected to overpredict light absorption in the suspension. The day/night variation in the incident photon flux was obtained from the TMY3 database⁸² and was scaled to the AM 1.5G spectrum for the peak insolation month of July at Daggitt, which is located in Southern California and was chosen for its prospects of large-scale photovoltaic and concentrated solar power generation. The semiconductor particles and the chemical species in the electrolyte contribute to the effective absorption coefficient,where κ_p,λ is the mass-specific, spectral absorption cross-section of the suspended semiconductor particles, C_p is the semiconductor particle concentration, and [small epsilon, Greek, macron]_i and c_i are the molar absorption coefficient and concentration of the ith species in the electrolyte, including water.⁸³ Fig. S1 (ESI†) depicts the absorption coefficients of the various semiconductor particles^48,77,84 and redox shuttles^85–87 considered. Optical attenuation of incident light by the cocatalysts and the porous separator are neglected. This is a reasonable assumption because shading effects due to the cocatalysts are anticipated to be small in particle-suspension reactor designs. Moreover, cocatalysts are often used at far less than submonolayer coverages and are likely present in discontinuous patches on the particles, and polymer materials suitable for the porous separator exhibit negligibly low absorbance values in the near ultraviolet-visible region of the electromagnetic spectrum (see Fig. S2, ESI†). Values for Φ_AM1.5,λ and κ_eff,λ can also be used to calculate the photon-flux weighted absorption cross-section of the particle, [small kappa, Greek, macron]_p, and the optical thickness, τ_p (eqn (3)). Optical thickness is a critical dimensionless parameter that is linearly dependent on the particle concentration, absorption cross-section of the particles, and optical path length, h. It is a dimensionless indicator of the amount of sunlight absorbed by the semiconductor particles. The photon-flux weighted absorption cross-section of the particle, [small kappa, Greek, macron]_p, is used to determine the particle optical thickness, due to wavelength dependent optical properties.

τ_p = [small kappa, Greek, macron]_pC_ph (3b)

Larger optical thicknesses result in increased sunlight absorption, which can be obtained by increasing the particle concentration and/or absorption properties of the material and/or optical path length. For simplicity, this weighted optical thickness is termed solely the “optical thickness” throughout the text. We chose this value over the photon-flux-weighted, spectrally-averaged percent of light absorbed (absorptance) because optical thickness is linear in both concentration of particles and reactor height, allowing for a more conceptual interpretation of the values. For reference, using 1 g L⁻¹ of the suspended particles and h = 1 cm, the particle optical thicknesses are 54.5, 18.2 and 16.5 for TiO₂, BiVO₄ and SrTiO₃:Rh, respectively.

The ideal photodiode equation has been used in prior modeling of fixed-electrode PEC devices to describe the (photo)current–voltage behavior of rectifying semiconductor–electrolyte junctions.^29,33,88,89 While the photodiode equation may not be appropriate to describe the behavior of individual semiconductor particles whose photon absorption rate is low, we expect that the photodiode equation is appropriate to describe an ensemble of semiconductor particles in a suspension.⁹⁰ In this approach, physical processes within individual particles, such as charge-carrier transport by diffusion and migration, and specific recombination pathways, are not considered, but instead represented by the reverse saturation current density (j_rr in eqn (4a)) for the ensemble of particles. The magnitude of the reverse saturation current density will depend on many factors including the recombination rates and mechanisms that influence charge-carrier lifetimes, optical material properties, dopant densities, surface states, etc. The photodiode equation therefore presents a tractable method to capture the underlying physics of converting the flux of incident photons to an interfacial flux of charge-carriers.

Particle-suspension reactors effectively perform volumetric reactivity as compared to interfacial, areal reactivity in the fixed-electrode, wafer-based PEC geometries. Therefore, the flux of charge carriers transported across the semiconductor–electrolyte interface⁸⁹ was recast as a volumetric rate, r_e/h, to obtain the volumetric version of the ideal photodiode equation,

The first term in eqn (4a), g_ph, is the volumetric absorption rate of the incident photons by the particles in the suspension; only photons with energies greater than the semiconductor absorption onset (oftentimes the bandgap) are considered to be absorbed. The second term is the volumetric recombination rate in the absence of illumination with an exponential dependence on the voltage, V_ph, and influenced by the surface-specific, reverse saturation current density, the diode quality factor, γ_d, the operating temperature, T = 298.15 K, and the specific surface area of the semiconductor particles, , where ρ_p is the bulk density of the semiconductor particle. The reverse saturation current densities modeled for the semiconductor particles influences the open-circuit potentials, with smaller values for j_rr increasing the open-circuit potential and improving the reaction rates by decreasing the recombination rates of the charge-carriers produced. Reverse saturation current densities have not been reported for semiconductors in the particle form factor. Therefore, baseline values (Table 2) were selected from experimentally measured open-circuit potentials for BiVO₄ electrodes affecting OER. For SrTiO₃:Rh particles, the baseline reverse saturation current density was chosen to be several orders of magnitude larger than the detailed balance, Shockley–Queisser limit while also being small enough to yield desired rates of product formation. For comparison, based on the material bandgaps the Shockley–Queisser limits, for TiO₂, BiVO₄, and SrTiO₃:Rh are on the order of 10⁻⁴⁵, 10⁻³⁵ and 10⁻³⁵ A m⁻², respectively at 298.15 K. Furthermore, the sensitivity of model results to the choice of j_rr for the OER and the HER particles was evaluated and the results are shown in Fig. S8 (ESI†). These results indicate that reaction rates expected for reverse saturation current densities dictated by the Shockley–Queisser limit are predicted to be nearly equal to the baseline reaction rates (although the photovoltages are drastically different), suggesting that increasing the sunlight-driven power production of the semiconductor particles, by further decreasing the reverse saturation current densities toward the Shockley–Queisser limits, will not substantially alter the rate of H₂ evolution.

Interfacial reaction kinetics

On the surface of the semiconductor particles at least one pair of redox half-reactions occurs. One of the factors that influences the reaction rate is the photovoltage (V_ph), which is equal to the sum of the surface overpotentials for the two redox reactions and the species-concentration-dependent thermodynamic equilibrium potential (Nernstian potential), U, for the net reaction on the semiconductor particle surface (L1 or L2)

V_ph = U_L1/L2 + |η_OER/HER| + |η_Rred/Rox| (5)

The baseline case assumed complete selectivity toward the desired reactions, i.e. L1 and L2 (Fig. 1(b)). The flux of electrons/holes across the cocatalyst surface area for the kth reaction (k = OER, HER, R_rd, and R_ox) is

where, f_cat,k is the fractional coverage of the cocatalyst on the semiconductor particles. In the absence of published data for redox-shuttle-specific reaction rate mechanisms and constants, the interfacial electron transfer rates were obtained by applying simplistic Butler–Volmer kinetic formulations,⁹¹where, j_0,k is the exchange current density, α_a,k and α_c,k are the anodic and cathodic charge transfer coefficients, and η_k is the surface overpotential. Typically, j_0,k depends on the electrolyte composition, cocatalyst material, and temperature. For the OER and the HER, the exchange current densities and charge transfer coefficients are based on best-fit parameters obtained from literature data for RuO₂ and Pt, respectively (Table 2). By simultaneously solving eqn (4a) through (7), the variables r_e/h, R_sel,k, V_ph, η_OER/HER and η_Rox/Rred can be evaluated. Based on the assumption of η_Rred ≪ η_OER and η_Rox ≪ η_HER for the facile redox shuttles, the reduction/oxidation overpotentials for the redox shuttle were neglected in eqn (5) and (7) in the baseline case that assumed selective catalysis.

Selective catalysis is crucial to achieve large STH efficiencies in particle-suspension reactors, however, in reality each particle is wetted by electrolyte that contains at least four chemical species that can partake in redox chemistry at the particle surface, and a quanta of separated charge could be lost due to recombination to undesired “back-reactions” with two of the species. Note that only particle-mediated surface reactions were considered in this analysis for non-baseline cases, and the homogeneous chemical reactions of the species in the electrolyte were neglected. For example, on the surface of the HER particles the undesired reactions are the dissolved H₂ oxidation reaction (HOR) by holes in the valence band and/or reduction of the oxidized form of the redox shuttle, D⁺ → D (R_red), by electrons in the conduction band. These reactions occur simultaneously with the desired surface reactions of HER by electrons in the conduction band and oxidation of the reduced form of the redox shuttle, D → D⁺ (R_ox), by holes in the valence band. The overpotentials for these competing reactions on the HER semiconductor particles can be related to the photovoltage in eqn (5) as,

|η_Rred/HOR| = V_ph − |η_Rox/HER| (8)

Butler–Volmer kinetics, as shown in eqn (7), was applied to obtain the rate of competing back-reactions as a function of the respective overpotentials in eqn (8). The effective reaction rate, R_eff, was expressed as the difference of the rate of selective catalysis, R_sel, and the competitive back reactions, R_back. When the back-reaction rates were predicted to be larger than the rates of the desired reactions, particle shunting occurs with unity yield, and the effective reaction rate is zero.

R_eff,j = R_sel,j − R_back,i (9)

For the OER particles j = OER/R_red, i = ORR, R_ox and likewise, for the HER particles, j = HER/R_ox; i = HOR, R_red.

Species transport

For dilute solutions, the Nernst–Planck equation⁷⁸ dictates the migration and diffusion species fluxes (convective fluxes are not considered here as a worse-case stationary system is assumed),

N_i = −z_iu_iFc_i∇ϕ − D_i∇c_i (10)

where, z_i is the charge magnitude, the mobility, u_i, and diffusion coefficient, D_i, for each species are related through the Nernst–Einstein relationship,⁸⁰ϕ is the electrolyte potential and c_i is the species concentration of the ith species. The Nernst–Planck equation was assumed operative in the reactor compartments, as well as in the porous separator where an effective diffusivity was used.

Overall mass conservation,

and solution electroneutrality,was applied. The source/sink term in eqn (11), r_i, is related to the volumetric rate of charge-carrier production from eqn (4a),where, ν_i is the stoichiometric coefficient for the i^th species and n is the number of electrons transferred in the redox reaction considered. In the absence of an external field/current density in the particle-suspension reactor designs,

Combining eqn (10) and (14), electrostatic potential gradient in the bulk solution can be related to the diffusion fluxes of ionic species,

Metrics

The projected areal rate of charge-carrier production, R_a, is obtained by spatially integrating the volumetric rate (eqn (13)) in the respective reaction compartment over the two-dimensional reaction compartment surface area, A, and averaging the integrated value over the reaction compartment height,

This areal reaction rate is reported here instead of a current density, because, unlike fixed-electrode PEC reactors, particle-suspension designs do not result in a net ionic/electronic current that spans the entire length of the reactor necessitating ion migration and the associated ohmic potential drops. The areal H₂ production rate (R_H2,a) is half of the areal rate of charge-carrier production, , based on the reaction stoichiometry (eqn (L1) and (L2)). From the areal H₂ production rate, the peak and day-averaged STH efficiencies are,

where, m in eqn (18) ranges from 2 to 365 days. At the baseline STH efficiency of 1% and at peak incident solar flux, the areal rate of charge-carrier production is 8.43 × 10⁻⁵ mol m⁻² s⁻¹ and the areal H₂ production rate is 4.21 × 10⁻⁵ mol m⁻² s⁻¹.

For the particle-suspension design, potential/energy losses arises due to a change in the concentrations of the reaction species between the two compartments and contributes to spatially variant reaction potentials. This mass-transfer-related potential loss or overpotential, Δϕ_mt, between two locations in the particle-suspension designs, one in the OER compartment and one in the HER compartment, has contributions from pH gradients, Δϕ_pH, and the concentration gradient of the redox shuttle species, Δϕ_D⁺/D,

The species concentrations in eqn (19) are computed as volume-averaged values in the OER and the HER compartments. Ohmic potential losses between the reaction compartments are not present due to the lack of an externally applied field/current density to drive ion transport.

Solution procedure

The coupled governing equations (eqn (1), (2) and (4a)–(13)) were solved on the two-dimensional numerical domain with the appropriate boundary conditions (Fig. 2) using the finite element method on a mapped mesh with a maximum of 10 000 elements. The discretized equations were solved using COMSOL Multiphysics⁹² with quadratic element discretization and a time-dependent direct linear solver (PARADISO). Implicit and adaptive time stepping, with a maximum time-step of 600 s, was employed to attain a relative tolerance of 10⁻⁴ for all the scaled variables. Mesh independence was verified by doubling the mesh size, which yielded a maximum change of 1% in the spatially-averaged species concentrations and the sink-term/source-term computed in eqn (11). Zero current density and zero species flux boundary conditions were applied on the external boundaries and symmetry boundary conditions were imposed at the vertical centerline (Fig. 2). All the simulations were performed on a dual core Intel Xeon X5660 processor (2.8 GHz, 24 GB), and the model developed is robust enough to tackle the various semiconductor particles, redox shuttles and reaction compartment heights investigated in this study. For the largest reaction compartments (10 cm tall) the transient simulations take approximately 8 hours of wall clock time to attain steady-periodic conditions. The procedure adopted for the quasi-steady simulations and determination of initial values of the solution variables for the transient simulations is detailed below.

Model simulations

Quasi-steady simulations were performed to evaluate the impacts of the semiconductor and the electrolyte optical properties on the reactor performance in the absence of any mass transport limitations. These simulations used simplified model conditions of peak incident photon flux (i.e. solar noon) and time-invariant and position-invariant concentrations of the electrolyte species, and only eqn (4a)–(7) are solved along the height of the reactor. The thermodynamic potentials for effecting reactions (L1) and (L2) for the various redox species considered are fixed at the respective standard-state values indicated in Table 2. For all these simulations, only the concentrations of the most strongly sunlight-absorbing redox species (i.e. I₃⁻, Fe³⁺, Q and QH₂) were varied. The concentrations of the other redox species (I⁻, IO₃⁻, and Fe²⁺), which absorb little light at λ > 280 nm, were set to values required to maintain the respective E⁰ values (listed in Table 2) because the areal reaction rates were not affected by changes to the concentrations of these redox species. Moreover, reaction kinetics is not affected by the concentration of these redox species for the baseline cases that assumed selective surface catalysis, because the OER and the HER are rate limiting. For all the quasi-steady simulations, the proton concentration, c_H⁺, was fixed at 1 M. For the Q/QH₂ redox shuttle, equal concentrations of Q and QH₂ were assumed, with a maximum concentration of 0.5 M, which is the aqueous solubility of QH₂.⁷⁰

For the transient simulations, transport, production and consumption of the various species (eqn (8)–(13)) in the reactor were considered in addition to the volumetric charge-carrier generation rate (eqn (4a)–(7)) in the presence of diurnal illumination cycles. The spatial and temporal variations of the electrolyte species concentrations and their influence on light absorption, Nernst reaction potential and reaction kinetics are all inherently accounted for in these simulations. For these simulations, at t = 0 s, the variables r_e/h, R_sel,k, η_OER/HER and ϕ were initialized to 0 and the photovoltage, V_ph, was set equal to the position-dependent open-circuit potential (from eqn (4a)). The concentrations of electrolyte species required to sustain reactor operation at desired STH efficiencies (D⁺, D, H⁺, OH⁻) were evaluated iteratively from the model results. Guess values for these initial concentrations were obtained by solving the transient species transport equations, eqn (10)–(12), with spatially homogeneous reaction rates that were consistent with a 1% STH efficiency at peak solar insolation; the day/night illumination profiles were superimposed on these rates.

Results and discussion

The influence of the optical properties of the semiconductor materials and the redox couples on the reactor performance was first evaluated under quasi-steady conditions and in the absence of any mass transport limitations. Transient reactor performance as well as the impacts of competitive back reactions is then discussed.

Effects of optical properties

Fig. 3 shows the impact of competitive light absorption by the redox shuttle for fixed semiconductor particle concentrations of 1 g L⁻¹, with and without tandem light absorption. For both the tandem (Option I) and the same-bandgap cases (Option II), the areal reaction rate decreased exponentially with redox shuttle concentration (Fig. 3) due to competitive light absorption. Incorporation of triiodide ions led to the lowest predicted reaction rates in the top and bottom compartments, because I₃⁻ exhibits the strongest visible light absorption (Fig. S1, ESI†). The rates were slightly larger for Fe³⁺/Fe²⁺ and even larger for the organic redox couple because these species absorb very little visible light. Option I achieved baseline STH efficiencies in the absence of mass transport limitations when using redox shuttles that contain <2 mM I₃⁻ or <5 mM Fe³⁺, constraints that arise due to competitive absorption of sunlight. It was determined that, for all redox shuttles considered, the rates in the top OER compartment are larger for Option II (Fig. 3(b)) relative to Option I (Fig. 3(a)). This effect is because BiVO₄ absorbs more sunlight than the wide-bandgap TiO₂. However, this improvement in light absorption comes at the expense of the reaction rates in the bottom compartment, which are larger for Option I than Option II owing to improved visible-light transmission to the bottom HER compartment when TiO₂ is used in place of BiVO₄. The consequence is that the reaction rates for Option II in the bottom compartment are negligibly low even for the smallest redox shuttle concentration (10⁻⁴ M) considered (Fig. 3(b)). This result suggests that the BiVO₄ particle concentration and/or pathlength for light absorption have to be further optimized to achieve the baseline STH efficiency of 1% for Option II.

Fig. 3 Areal reaction rates as a function of redox shuttle concentration in the top (OER) and bottom (HER) reaction compartments, for 1 cm tall compartments each containing 1 g L⁻¹ particles with SrTiO₃:Rh in the bottom compartment and the top compartment containing (a) TiO₂ (Option I) and (b) BiVO₄ (Option II). The areal reaction rate corresponding to the baseline STH efficiency of 1% is 8.43 × 10⁻⁵ mol m⁻² s⁻¹ and is indicated by the green dashed and dotted lines. Bottom compartment rates for Option II are so small that they are not visible in (b). I⁻ and IO₃⁻ redox shuttle species are not shown here as they do not absorb sunlight in the wavelengths of interest.

In addition to the type and redox shuttle concentrations, the reaction rates are influenced by the optical thicknesses of the semiconductor particles (eqn (3)). Optical thickness linearly depends on the particle concentration (C_p), absorption cross-section ([small kappa, Greek, macron]_p) of the particles, and reaction compartment height (h). Fig. 4 illustrates the effects of optical thicknesses on the reaction rates in the top and bottom reaction compartments at various concentrations of Q/QH₂ with and without tandem light absorption. The rates in the top compartment are only dependent on the optical thickness of the top compartment, τ_p,top (Fig. 4(a and b)), whereas the rates in the bottom compartment are dependent on the optical thicknesses in both the top and bottom compartments (Fig. 4(c and d)). In both options considered, the reaction rate in the top OER compartment increased asymptotically with optical thickness until a rate of 1.4 × 10⁻⁴ mol m⁻² s⁻¹ for Option I (Fig. 4(a)), and 6.4 × 10⁻⁴ mol m⁻² s⁻¹ for Option II (Fig. 4(b)), values that are dictated primarily by the band gap and spectral absorptivities of the particles. Supported by the data in Fig. 3, reaction rates decrease as the concentrations of the redox shuttle increase due to competitive light absorption. The reaction rates in the bottom reaction compartment are shown in Fig. 4(c and d) for the largest redox shuttle concentration considered, 0.5 M. Results for the other concentrations of Q/QH₂ show qualitatively similar trends and are presented in Fig. S3 (ESI†). In both options, the reaction rates increased as the optical thickness of the SrTiO₃:Rh particles, τ_p,bot, increased, to a maximum value of about 4.5 × 10⁻⁴ mol m⁻² s⁻¹. For Option I (Fig. 4(c)), the rates were largely insensitive to the TiO₂ particle optical thickness owing to its transmission of light beyond 400 nm, which can be absorbed by SrTiO₃:Rh. For Option II, where the particles have similar absorption onsets, a “pseudo-tandem” effect is obtained by modulating the relative optical thicknesses of the top and bottom compartments. If the optical thickness of the top compartment particles is small, τ_top,p < 0.01, reaction rates in the bottom compartment are as large as those obtained for the true tandem condition with Option I. These results clearly indicate that the semiconductor particle optical thicknesses have to be optimized to maximize the reaction rates for the OER and the HER in the two compartments. The optimal values are strongly influenced by not only choice of the semiconductor particles in the two reaction compartments, but also the type and concentration of the redox shuttle species considered.

Fig. 4 Areal reaction rates in the (a and b) top (OER) compartment as a function of quinone redox shuttle concentration and OER particle optical thickness, and in the (c and d) bottom (HER) compartment as a function of OER and HER particle optical thicknesses for (a and c) Option I (TiO₂ (OER) and SrTiO₃:Rh (HER)) (b and d) Option II (BiVO₄ (OER) and SrTiO₃:Rh (HER)) assuming a fixed concentration of 0.5 M for the Q/QH₂ redox shuttle. Particle optical thicknesses linearly depend on the particle concentration, absorption cross-section of the particles, and reaction compartment height.

Sustained reactor operation requires equal rates of production and consumption of the redox shuttle species in the two reaction compartments. In Option I, the maximum attainable STH efficiency with sustained operation is limited to ∼1.7% because the maximum reaction rate for TiO₂ is three times slower than SrTiO₃:Rh, 1.4 × 10⁻⁴ mol m⁻² s⁻¹versus 4.5 × 10⁻⁴ mol m⁻² s⁻¹ (Fig. 4(a and b)). Table 3 presents the particle optical thickness combinations required to attain the specified STH efficiency with sustained reactor operation for each option of the OER particles and each redox shuttle, and assuming rapid mixing that affords fixed redox shuttle concentrations at all spatial locations in the reactor. These results reinforce the negative impacts of competitive light absorption, necessitating rather low concentrations of the I₃⁻/I⁻ and Fe³⁺/Fe²⁺ redox shuttles (<5 mM) to possibly attain the baseline STH efficiency of 1%. These concentrations are prohibitively low, because as discussed next, when mass transport is included in the model, the rate of diffusive species transport is unable to keep up with the rates of redox species consumption/production consistent with a 1% STH efficiency, leading to local depletion of the reactants. For Option II and the quinone redox shuttle, STH efficiencies up to 3% are feasible even with 0.5 M of Q/QH₂, the solubility limit. For lower concentrations of Q/QH₂ (<10⁻³ M), the largest attainable STH efficiency is 3.8% for sustained reactor operation. For the I⁻/IO₃⁻ redox shuttle, which also undergoes proton-coupled electron-transfer, the STH efficiencies are not affected by the redox shuttle concentrations because of their negligible sunlight absorption. However, compared to Q/QH₂, a higher photovoltage (>1.085 V) is required in the HER light absorbers due to the more positive standard-state potential of the I⁻/IO₃⁻ redox shuttle. This large photovoltage requirement can only be obtained when a smaller reverse saturation current density, j_rr, of 10⁻²⁰ A m⁻² is utilized for the SrTiO₃:Rh particles in place of the baseline value of 10⁻¹⁵ A m⁻². In that case, the maximum attainable STH efficiency for the I⁻/IO₃⁻ redox shuttle is 3.8%, the same maximum value obtained for the Q/QH₂ redox shuttle.

Table 3 Optical thickness combinations for sustained reactor operation at specified values of STH efficiency (%) and redox-shuttle concentrations for Option I (TiO₂ (top, OER) and SrTiO₃:Rh (bottom, HER)) and Option II (BiVO₄ (top, OER) and SrTiO₃:Rh (bottom, HER)); U_L1 and U_L2 are Nernstian reaction potentials for the desired reactions in the top and bottom compartments. j_rr,OER = 10⁻¹⁸ A m⁻², j_rr,HER = 10⁻¹⁵ A m⁻² unless otherwise specified; N/A indicates that no optical thickness will yield the indicated STH efficiency (%)

Redox shuttle: I/I3^−; UL1 = −0.694 V and UL2 = −0.536 V vs. NHE
	Option I		Option II
c A/A^− (M)	η STH,peak = 1%		η STH,peak = 1%		η STH,peak = 2%
	τ p,top	τ p,bot	τ p,top	τ p,bot	τ p,top	τ p,bot
10^−4	3.162	0.316	0.178	0.316	0.562	1.778
10^−3	31.62	1.77	0.562	3.16	1	N/A
0.1	N/A	N/A	17.78	N/A	56.23	N/A
1	N/A	N/A	177.82	N/A	562.3	N/A

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Redox shuttle: Fe^2+/Fe^3+; UL1 = −0.459 V and UL2 = −0.771 V vs. NHE
	Option I		Option II
c A/A^− (M)	η STH,peak = 1%		η STH,peak = 1%		η STH,peak = 2%
	τ p,top	τ p,bot	τ p,top	τ p,bot	τ p,top	τ p,bot
10^−4	1	0.316	0.178	0.178	0.316	0.562
10^−3	3.162	1.77	0.178	0.562	0.562	1.778
0.1	316.2	N/A	17.78	N/A	56.23	N/A
1	N/A	N/A	177.82	N/A	562.3	N/A

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Redox shuttle: Q/QH2; UL1 = −0.53 V and UL2 = −0.7 V vs. NHE
	Option I		Option II
c A/A^− (M)	η STH,peak = 1%		η STH,peak = 1%		η STH,peak = 3%		η STH,peak = 3.8%
	τ p,top	τ p,bot	τ p,top	τ p,bot	τ p,top	τ p,bot	τ p,top	τ p,bot
10^−4	0.83	0.186	0.128	0.155	0.47	1	0.77	6.46
10^−3	1	0.186	0.155	0.155	0.57	1.2	0.82	9.4
0.1	28.8	0.225	0.27	0.326	1	5.36	1.75	N/A
0.5	128.27	0.225	0.27	0.393	1	5.36	2.1	N/A

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Redox shuttle: I^−/IO3^−; UL1 = −0.145 V and UL2 = −1.085 V vs. NHE j rr,OER = 10^−18 A m^−2, jrr,HER = 10^−20 A m^−2
	Option I		Option II
	η STH,peak = 1%		η STH,peak = 1%		η STH,peak = 3%		η STH,peak = 3.8%
	τ p,top	τ p,bot	τ p,top	τ p,bot	τ p,top	τ p,bot	τ p,top	τ p,bot
	0.91	0.215	0.142	0.17	0.55	1.53	0.72	7.1

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Effect of species transport

Based on the model predictions that indicate that Option II has larger theoretically achievable efficiencies (Table 3), this option was chosen for further investigation to evaluate the effects of coupled light absorption, species transport, and reaction kinetics. Transient reactor performance is shown in Fig. 5 for the Q/QH₂ redox shuttle, which is projected to attain larger than baseline STH efficiencies for all redox species concentrations considered. The locations of the redox chemistries are opposite of that shown in Fig. 1, with HER/R_ox in the top compartment and OER/R_red in the bottom compartment. The optical properties of Q/QH₂ are less detrimental in this modified configuration, suggesting that this arrangement could be more efficient with this redox shuttle. The data indicate that a steady-periodic state of reactor operation is attained after about 5 days, 125 days, and 500 days for 1 cm, 5 cm, and 10 cm tall reaction compartments, respectively. Fig. 5(b) illustrates that the peak and day-averaged STH efficiencies are ∼1.06%, which remains constant beyond five days of reactor operation. Stable steady-periodic operation is also observed for the 5 cm and 10 cm tall compartments, as shown in Fig. 5(b and c), and for operation at the same baseline STH efficiency of ∼1%. For each of these conditions, the areal reaction rates (not shown) in the two compartments are very similar during steady-periodic reactor operation. The time taken to attain steady-periodic conditions is computed as the time after which the percentage changes in species concentrations (spatially-averaged concentrations in each reaction compartment) are less than 0.1% from the concentrations predicted in the previous illumination cycle and when the STH efficiency remains constant (as shown in Fig. 5(b)). In comparison, an order-of-magnitude estimate for the diffusion timescales, t_diff ∼ h²/D_i, are 2 days, 50 days and 250 days for respective reaction compartment heights of 1 cm, 5 cm and 10 cm, and computed using the diffusivity of para-benzoquinone (slowest diffusing species in the suspension). Model predictions align with the expectation that steady-state conditions will prevail for timescales larger than the diffusion timescales, and in this case, t_steady ∼ 2.5t_diff. The fact that the predicted time to attain the steady-periodic state is approximately proportional to the square of the compartment height (t_steady ∝ h²), is indicative of a diffusion controlled process. Without day-and-night variations in the incident sunlight, local depletion of the redox species (Q) occurs within a day of reactor operation for the 1 cm tall reaction compartments (Fig. S4, ESI†). Therefore, much larger initial redox species concentrations (0.65 M versus 0.3 M for h = 1 cm) are required to attain steady-state conditions for this situation. Irrespective of the pH, the concentration of protons is time-invariant and position-invariant because it was assumed that the Q/QH₂ redox shuttle underwent proton-coupled electron transfer with unity yield in these simulations. Therefore, any pH (pH = 7 in Fig. 5) is sustainable using this reactor design, provided that the redox shuttle undergoes rapid proton-coupled electron transfer. At a fixed STH efficiency, the volumetric reaction rate varies inversely with the reaction compartment height, whereas the rate of diffusive species transport decreases proportional to the square of the height. Therefore, to maintain steady and sustained reactor operation, at roughly 1% STH efficiency, the minimum initial concentrations of the redox shuttle increases nearly linearly with the reaction compartment height (c_steady ∝ h). Slight deviations from linearity occur due to the spatial location and concentration dependence of the reaction rates, and the minor differences in the STH efficiencies obtained with the different heights of the reaction compartments. Overall, these scaling relationships underscore the dominance of diffusion as a mode of species transport in the solution compared to migration. For the neutral Q/QH₂ redox shuttle, results in Fig. 5 demonstrate species mixing entirely controlled by diffusion (in the absence of any forced/natural convection). This result is because with spatially invariant proton concentrations, the net diffusive flux of any charged species is zero, i.e. the numerator in eqn (15) is zero. In the absence of an externally applied field or current density, the driving force for ion migration is essentially zero.

Fig. 5 Transient concentration profiles, (a, c, and d) with Option II for reactor heights and SrTiO₃:Rh (top, HER), BiVO₄ (bottom, OER) concentrations of (a and b) 1 cm and 1.53 × 10⁻³ g L⁻¹, 2.12 × 10⁻³ g L⁻¹, and (b) average and peak values for STH efficiency and dashed lines are provided to guide the eye, (c) 5 cm and 0.32 × 10⁻³ g L⁻¹, 0.43 × 10⁻³ g L⁻¹, and (d) 10 cm and 0.16 × 10⁻³ g L⁻¹ and 0.2 × 10⁻³ g L⁻¹.

For all the reaction compartment heights considered, the potential losses associated with pH gradients are zero with the quinone redox shuttles because the proton concentrations in the solution are spatio-temporally invariant. The mass-transfer overpotential (eqn (19)) due to the redox species concentration gradients attains a maximum of 28 mV, 33 mV, and 35 mV during steady-periodic reactor operation for the 1 cm, 5 cm, and 10 cm tall reactors.

For reaction compartment heights greater than 5 cm the predicted concentration profiles in Fig. 5 suggest that sustained reactor operation, is limited by the aqueous solubility of Q and QH₂. Even for h = 1 cm, the required concentration of Q for sustained reactor operation is larger than its aqueous solubility limit of 0.1 M. While aqueous solubility can be improved through introduction of sulfonate groups on the redox shuttle molecules,⁷⁰ these large groups also reduce species diffusivities and therefore, optimization of these tradeoffs is required. Therefore, one solution to enable sustained reactor operation via diffusive species transport with reasonable concentrations of these redox shuttles, is to reduce the reaction compartment heights to the sub-centimeter length scales with concomitant increase in the semiconductor particle concentration to maintain the same reaction rates.

Profiles along the reactor height are informative when visualizing the conditions present under steady-periodic operation. Therefore, depicted in Fig. 6 are the following parameter profiles at solar noon for h = 1 cm: redox shuttle concentrations, volumetric reaction rates, photovoltages, and reaction overpotentials. The concentration profiles (Fig. 6(a)) are reflected about the center of the reactor (x = 0, y = 0) with equal rates of production and consumption of the redox shuttle in the two compartments during steady-periodic operation. The spatial profiles of the volumetric reaction rates, photovoltage, and overpotentials (Fig. 6(b–d)) follow the trend of the absorbed photon flux and therefore, decrease along the reactor height. The relatively more linear spatial profiles of the volumetric reaction rate in the bottom compartment in comparison to the top compartment are due to the differences in the optical properties of the semiconductor materials and the increased light attenuation by the electrolyte along the reactor height. The average photovoltage produced by the photocatalysts that drive R_ox/HER is 0.734 V, while the photovoltage is ∼100 mV larger at 0.824 V for the photocatalyst particles that implement OER/R_red, which is in part required to supply the larger overpotentials for the OER (Fig. 6(d)).

Fig. 6 Spatial profiles along the reactor centerline at x = 0 (see Fig. 2) during steady-periodic operation at solar noon with 1.53 × 10⁻³ g L⁻¹ of SrTiO₃:Rh (top, HER) and 2.12 × 10⁻³ g L⁻¹ of BiVO₄ (bottom, OER) particles: (a) redox shuttle concentrations, (b) volumetric rates of charge-carrier production, r_e/h, (c) photovoltage, V_ph, and (d) reaction overpotentials, η_OER/HER.

In addition to the Q/QH₂ redox shuttle, transient reactor performance was also evaluated for the IO₃⁻/I⁻ redox shuttle (Fig. S5, ESI†), which is also predicted to achieve baseline and up to 3.8% STH efficiency like the quinone redox shuttles (Table 3). With exceptions of the requirements of larger photovoltages from the HER semiconductor particles than with the Q/QH₂ redox shuttle and the need for more alkaline conditions to favor conversion of I⁻ to IO₃⁻ instead of strongly-visible-light-absorbing I₃⁻, the results using this redox shuttle are promising. Sustained reactor operation at a 1% STH efficiency is achieved with redox shuttle concentrations well below their solubility limits even for 10 cm tall reactor compartments (see Fig. S5, ESI†), and Fig. S6 (ESI†) shows that diffusion-driven species transport is able to sustain reactor operation at 3.8% STH efficiency with 1 cm tall reaction compartments for the IO₃⁻/I⁻ redox shuttle. The mass-transfer overpotential due to the redox species concentration gradients has a maximum of 11 mV for the 10 cm tall reaction compartments during steady-periodic operation.

For the strongly-visible-light-absorbing I₃⁻/I⁻ and Fe³⁺/Fe²⁺ redox shuttles, because concentrations less than 5 mM are required to minimize the impacts of competitive light absorption, diffusion distances less than 0.01 cm are needed to sustain the reaction rates corresponding to the baseline STH efficiency of 1%. However, this short distance implies that impractically large semiconductor particle concentrations, i.e. >10 g L⁻¹, are required to achieve the required optical thicknesses reported in Table 3. Moreover, reactor heights on the sub-mm lengthscale will be extremely challenging to fabricate and to prevent bubble occlusion. For taller reaction compartments, h > 0.01 cm, the diffusion-sustained STH efficiencies are restricted to less than 0.2% due to competitive light absorption by I₃⁻ and Fe³⁺ redox species, as suggested by Fig. S7 (ESI†). Because these redox shuttles are not modeled to undergo proton-coupled electron-transfer, spatial gradients in pH and the redox shuttle species concentration gradients result in mass-transfer-related overpotentials. However, at low STH efficiencies of 0.2%, these overpotentials are less than 1 mV. These results indicate that reactor heights on the lengthscales of centimeters are best to achieve the 1% baseline STH efficiency for the I₃⁻/I⁻ and Fe³⁺/Fe²⁺, provided that some form of active species mixing is possible.

Effects of competitive surface reactions

The above analysis assumed perfectly selective reactions without considering the overpotentials for the undesired back reactions. Fig. 7 shows the impacts of the redox shuttle kinetic parameters for Q/QH₂ on the competing back reactions of redox chemistry with the redox shuttle species. The effective reaction rates (R_eff, eqn (9)) for HER/R_ox and OER/R_red are normalized with the baseline rates obtained in Fig. 6, which assumed selective surface catalysis. The overpotential for oxidation of liberated H₂ by holes in the valence band for the HER particles is large (>1.23 V), and the models predict very rapid hydrogen oxidation, so much so that in the absence of active removal of H₂ or asymmetric charge transfer coefficients for the HER there is no net desired H₂ formation (data not shown). For all exchange current densities investigated for the redox shuttle, symmetric charge-transfer coefficients (α_a,sh = α_c,sh = 0.5) resulted in net rates that were zero. Fig. 7(a) suggests that for asymmetric charge-transfer coefficients there is an optimum redox shuttle exchange current density to maximize the resultant surface reaction rate in the presence of back reactions of the redox shuttle, in this case, the reduction of Q. The contending effects are that an increase in the exchange current density reduces the overpotential for the oxidation (QH₂ to Q) and thereby, intensifies the driving force for the reduction reaction, which acts as a recombination pathway shunt. For smaller exchange current densities, the overpotential increases for the oxidation reaction, and shifts the photovoltage closer to the open-circuit potential of the photodiode, which diminishes the reaction rate. With increase in the anodic charge transfer coefficient, α_a,sh, the oxidation of QH₂ is favored more than the reduction of Q, and therefore the resultant reaction rates increase. The highest predicted rate for the case of α_a,sh = 0.9 with the selected concentration ratio is at least 15% smaller than the baseline reaction rate without competing surface reactions.

Fig. 7 Effective redox shuttle reaction rates, R_eff, normalized to the baseline reaction rate, R_sel,k, that assumed selective catalysis, as a function of kinetic parameters (j_0,sh, α_a,sh and α_c,sh = 1 − α_a,sh) for h = 1 cm with the Q/QH₂ redox shuttle at concentration ratios that are consistent with the steady-periodic results from Fig. 5 (i.e. equals 2.56 and 0.44 in the HER (top) and the OER (bottom) compartments respectively) for (a) SrTiO₃:Rh (top, HER) and (b) BiVO₄ (bottom, OER) particles.

Oxygen reduction using the OER particles is much more challenging than H₂ oxidation using the HER particles owing to the relatively low cathodic transfer coefficient to effect the oxygen reduction reaction (α_c,OER = 0.1 compared to α_a,OER = 1^27,72,73). The resultant reaction rates change by less than 3% when back reactions with dissolved O₂ are taken into account, and therefore, those results are indistinguishable from the results shown in Fig. 7(b). Similar to the HER particles, symmetric charge transfer coefficients result in no net desired reactions and an optimum exchange current density arises due to the tradeoffs in decreasing the overpotential of the desired reaction (R_red in this case) for larger values of j_0,sh and operating near the open-circuit potential of the modeled photodiode when the exchange current density is low. The optimum exchange current densities are in general lower compared to those of the HER particles for all cases considered, and the highest reaction rate for the case of α_c,sh = 0.9 is only 2% lower than the baseline reaction rate. These contrasting effects are due to the relatively larger open-circuit photovoltage modeled for the BiVO₄ particles than in the SrTiO₃:Rh particles of 1.06 V versus 0.88 V, respectively. Fig. 7 is suggestive that redox shuttles with asymmetric charge-transfer coefficients and small exchange current densities (typically <1 A m⁻²) are likely to result in larger STH efficiencies.

Fig. 8 shows the variations in the effective OER/R_red and HER/R_ox reaction rates normalized to the baseline reaction rates as the ratio of the concentrations of the oxidized and reduced forms of the redox shuttle, , changes from 0.1 to 10. The concentration ratio affects the driving force for the redox shuttle reactions via the Nernst equation such that larger driving forces result in faster reactions. The optimum exchange current density decreases as the concentration ratio is modified such that it favors the desired reactions by reducing the reaction overpotentials (decreased/increased redox shuttle concentration ratio for the HER/OER particles, respectively).

Fig. 8 Effective redox shuttle reaction rates normalized to the baseline reaction rate, R_sel,k, that assumed selective catalysis as a function of the exchange current density (j_0,sh) and the concentration ratio of the Q/QH₂ redox shuttle (c_Q/c_QH2) for h = 1 cm with (a) SrTiO₃:Rh (top, HER), α_a,sh, α_c,sh = 0.9, 0.1, and (b) BiVO₄ (bottom, OER), α_a,sh, α_c,sh = 0.1, 0.9 particles. The solid red line corresponds to values of consistent with steady-periodic results in Fig. 5 with (a) and (b) .

Comparison with fixed-electrode PEC designs

An important aspect of the particle-suspension reactor is that local currents between sites of anodic (holes) and cathodic (electrons) reactions are very small, even for efficient designs that generate O₂ and H₂ at a rate consistent with the benchmark current density for an efficient solar water splitting device of 20 mA cm⁻². This is because these reaction rates are distributed over the height of the compartment such that on average the rates are consistent with a current density of 20/n mA cm⁻², for a reactor with n particles in the optical pathlength to attain the benchmark current density. For the particles modeled in this study the current densities were at most 20 μA cm⁻² particle⁻¹, which results in a reaction rate expressed as an effective volumetric current density of 250 attoA per particle. These calculations assumed 1 cm tall reaction compartments, a maximum particle concentration of 5 g L⁻¹ and d_p = 100 nm. One benefit of the small local currents present in particle-suspension reactors is that it results in negligibly low ohmic potential losses in the solution. For a single semiconductor particle with radius of 100 nm to at most several microns, the ohmic potential drop due to migration of ions between the sites of anodic and cathodic reactions is expected to be negligibly small (<1 mV), even when using water with little-to-no supporting electrolyte and redox shuttles.²⁰ However, ionic migration over the small lengthscales of individual particles makes separation of the anodic and cathodic reactions on their surfaces nearly impossible, meaning that incorporation of a membrane between those sites becomes challenging. Therefore, highly selective particle reactivity, as shown in Fig. 7, is required to achieve large efficiencies. This is in contrast to the fixed-electrode PEC designs where membrane separators are common and the device dimensions, including species transport lengthscales and operating conditions, must be carefully optimized to minimize the ohmic potential drop and the overall solution transport losses, often to a target value of less than 100 mV.^28,31

Another benefit of the small local currents present in efficient particle-suspension reactors is that near-neutral pH conditions can be used, yet, little-to-no variation in the local pH is generated during operation on the surface of an individual particle. Changes in local pH are detrimental to device efficiency because they result in potential losses, as described in eqn (19). This is in stark contrast to the fixed-electrode designs that are incorporated with membranes, where local pH gradients are much more likely to be generated, especially at near-neutral pH conditions. Even for pH 1 electrolytes the pH gradient potential loss is roughly 70 mV for a fixed-electrode PEC device operating at a 10% STH efficiency.³¹

Even in the absence of a local pH gradient on the particle surface, there are potential losses due to pH gradients between the OER compartment and the HER compartment in the particle-suspension reactor designs. These losses are anticipated to be similar in magnitude to those observed in fixed-electrode designs. An approximately 60 mV pH gradient loss was evaluated for a hypothetical simulation case operating at 10% STH efficiency and using the I₃⁻/I⁻ redox shuttle. The pH losses are however nearly zero for redox shuttles that undergo rapid proton-coupled electron-transfer. Also, in the particle-suspension reactor designs, additional potential losses occur due to redox shuttle concentration gradients, the magnitude of which depends greatly on the diffusion lengthscales in the absence of active mixing. Shorter compartment heights facilitate rapid mass transport and decrease this loss, but require a concomitant increase in the concentrations of semiconductor particles to maintain the same performance. These losses are not present in fixed-electrode PEC designs.

A related aspect to the small local currents is that particle-suspension reactors facilitate volumetric, three-dimensional reactivity in comparison to the fixed-electrode, wafer-based PEC device designs that drive areal, interfacial activity. In the suspension reactors, the overall reaction rates are distributed throughout the active semiconductor particle surface sites, behavior that is similar to mesoporous electrodes^93,94 and microwire/nanowire array designs.^95–97 This redistribution of the charge carriers and reaction sites reduces the charge-carrier flux to the surface and lowers the reaction overpotentials at any point location. Similar to other micro-structured and nano-structured PEC designs, the high specific-surface areas of the suspended semiconductor particles can enhance the redox-active surface area per unit sunlight collection area compared to the fixed electrode devices and increase efficiency and scalability, assuming that recombination rates are not limited by surface processes. If surface or near-surface sites facilitate recombination, then the benefit of low electrocatalytic rates is at least in part negated by lower observed photovoltages.⁹⁸ Finally, particle-suspension reactors can tolerate particle failure (e.g., due to corrosion), because it may not impact other particles to cause a catastrophic failure in the entire device. This is not the case in many fixed-electrode designs where impurities or surface oxides introduce detrimental recombination centers, even at very low concentrations, which then affect the efficacy of charge carriers over a much larger area than just the size (cross-section) of the recombination center.

Conclusion

For the first time, the combined effects of device design, and choice and concentrations of semiconductor particles and redox shuttles were numerically evaluated to understand their effects on the overall performance of a vertically stacked, particle-suspension reactor with soluble redox shuttles implementing Z-scheme solar water splitting. The feature that distinguishes the proposed reactor design from prior work is that the compartments effecting the OER and the HER were optically in series, which facilitated tandem light absorption and larger possible STH efficiencies. A transient, two-dimensional reactor model was developed to identify the reaction compartment heights and the operating conditions to achieve sustained reactor operation at 1% STH, facilitated by only diffusion. The volumetric rates of H₂ and O₂ generation in the reaction compartments were evaluated by coupling the photodiode (photo)current–voltage relationship with Butler–Volmer electron transfer kinetics, and taking into account volumetric light absorption and recombination effects in the suspension. The two materials options that were considered for the OER semiconductor particles were TiO₂ and BiVO₄, while SrTiO₃:Rh was the HER semiconductor particle for all cases. The redox shuttles investigated included I₃⁻/I⁻, Fe³⁺/Fe²⁺, IO₃⁻/I⁻, and Q/QH₂, with the last two modeled as proton-coupled electron transfer redox reactions.

The optical thickness of the semiconductor particles, which is proportional to particle concentration and absorption cross-section, and reaction compartment height, emerges as an important dimensionless parameter, and was optimized specific to the redox shuttle considered to attain the desired STH efficiency. Results illustrated that amongst the redox shuttles considered, Q/QH₂ and IO₃⁻/I⁻ are more promising than I₃⁻/I⁻ and Fe³⁺/Fe²⁺ owing to substantial visible light absorption by I₃⁻ and Fe³⁺ redox species. Competitive light absorption by the I₃⁻ and Fe³⁺ redox species restricts their concentrations to 5 mM, to attain 1% STH efficiency, requiring small reaction compartment heights and subsequently unrealistically high particle concentrations. OER semiconducting particles consisting of BiVO₄ outperformed those consisting of TiO₂, and were projected to achieve STH efficiencies of 3.8% and 1.7% respectively with the Q/QH₂ and IO₃⁻/I⁻ redox shuttles. Species transport is dominated by diffusion over migration for all the reaction compartment heights considered, and purely diffusion driven for the neutral, proton-coupled-electron-transfer quinone redox shuttles. The baseline STH efficiency and steady-periodic reactor operation was attained within the first five days of operation for 1 cm tall reaction compartments with 2.12/1.53 mg L⁻¹ of BiVO₄/SrTiO₃:Rh particles for the Q/QH₂ redox shuttles. Because of the drawback of a low aqueous solubility for p-benzoquinone, the quinone redox shuttle is not anticipated to sustain larger STH efficiencies. On the other hand, the IO₃⁻/I⁻ redox shuttles achieved sustainable baseline STH efficiency while maintaining concentrations below their solubility limits even for 10 cm tall reaction compartments. These model predictions assumed that the HER particles generated photovoltages larger than 1.04 V, which poses both challenges and great opportunities for continued research into these HER semiconducting materials. For both these proton-coupled electron transfer redox shuttles potential losses due to pH gradients are absent, and the species concentration gradients yield a maximum mass-transfer overpotential of 35 mV and 11 mV with the Q/QH₂ and IO₃⁻/I⁻ redox shuttles for 10 cm tall reaction compartments.

In the particle-suspension reactor designs, a fixed areal reaction rate or current density of H₂ generation is divided into much smaller localized anodic and cathodic reaction rates (current densities) as compared to the fixed-electrode counterparts, which results in negligible ohmic/migration potential losses. However, this attribute also poses challenges for placement of membranes between the reaction sites, making selective surface catalysis a crucial requirement for improving the efficiency of these reactor designs. Facile/reversible redox shuttles with symmetric charge transfer coefficients and large exchange current densities lead to large rates for shunting reactions. Asymmetric charge transfer coefficients and optimal exchange current densities for the redox shuttle are required to suppress these undesired back reactions.

In summary, the proposed reactor design is projected to sustain at least a 1% STH efficiency with BiVO₄ and SrTiO₃:Rh photocatalyst particles, aided by diffusive transport of Q/QH₂ and IO₃⁻/I⁻ redox shuttles; the latter redox shuttle exhibits promise of achieving STH efficiencies up to 3.8%. While the model was applied to consider specific photocatalyst materials and redox shuttles, the model formulation is general and can be widely applied for any material and redox shuttle of interest. The effects of particle size and detailed particle-scale photophysics, and natural convection induced species mixing on the overall reactor performance were not considered in this study, but we anticipate that they will be influential parameters to be considered in future work.

Conflicts of interest

There are no conflicts of interest to declare for the work presented in this manuscript.

Acknowledgements

This work was supported by the U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, Fuel Cell Technologies Incubator Program under Award No. DE-EE0006963 and Lawrence Berkeley National Laboratory under Contract No. DE-AC02-05CH11231. We cordially thank Dr Chengxiang Xiang at the Joint Center for Artificial Photosynthesis at the California Institute of Technology for providing critical insights and suggestions towards the model development. The authors also acknowledge inputs from colleagues at University of California, Irvine—Dr Houman Yaghoubi for providing current–voltage data for BiVO₄ electrodes, and Mr William Gaieck and Mr Kevin Tkacz for obtaining spectroscopic measurements on the redox shuttles and the porous separators.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7ee01360d

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