Membrane-less photoelectrochemical devices for H₂O₂ production: efficiency limit and operational constraint

Abstract

Three types of membrane-less photoelectrochemical (PEC) devices for solar-driven hydrogen peroxide (H₂O₂) production have been proposed, including (a) a H₂O₂ electrolyzer; (b) a photoelectrode-based PEC device; and (c) particulate photocatalysts in a suspension reactor. These devices can produce H₂O₂ from sunlight, water, and oxygen in air as the only inputs. Their respective efficiency limits were calculated by using the measured performance of H₂O₂ evolving electrocatalysts and by assuming idealized 100% selectivity for H₂O₂ accumulation, and by using the energy-conversion performance of light absorbers at their Shockley–Queisser limit. Multi-physics models for the three device configurations were employed for evaluating their electrochemical polarization behaviour as a function of the operating current density and for quantifying the respective contribution from the catalyst overpotential loss, the resistive loss between the cathode and anode, and the Nernstian potential loss due to pH gradients across the electrolyte. For particulate photocatalysts in a suspension, the concept of a non-zero light intensity threshold was for the first time applied to account for net positive H₂O₂ accumulation, which implies that the most efficient photocatalyst suspension should not fully absorb sunlight. The maximum solar-to-H₂O₂ conversion efficiency for a photoelectrode-based device was found to be 20.0%, fundamentally limited by the photocurrent density that is matched by dual absorbers in tandem series under 1-sun illumination. The maximum solar-to-H₂O₂ conversion efficiency for a particulate photocatalyst suspension can achieve 24.5–27.5% with a single-absorber band gap of 1.5–1.7 eV, depending on particle sizes. The modelling outcomes emphasize the importance of water-oxidation selectivity for achieving H₂O₂ accumulation towards molar concentration levels, and should guide the PEC H₂O₂ device implementation as the activity and selectivity of H₂O₂-evolving electro-catalysts continue to improve.

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1. Introduction

Photoelectrochemical (PEC) production of hydrogen peroxide (H₂O₂) is an emerging pathway for converting solar energy to commodity chemicals, which requires only sunlight, oxygen in air, and water as inputs. H₂O₂ is an environmentally friendly commodity chemical with an estimated global gross production of 3 million metric tons per year.¹ It has been widely used in advanced oxidation processes for water and wastewater treatment.^2–5a So far, photoelectrode-based PEC devices have only achieved 0.12–0.66% solar-to-H₂O₂ conversion efficiency, and particulate photocatalysts in a suspension have only achieved 0.15–0.22%.^6–10 There remains a big gap in the demonstrated efficiencies for PEC H₂O₂ production vs. PEC water splitting devices: various models of PEC water splitting devices reported that a solar-to-hydrogen conversion efficiency of 25% is achievable with a tandem light absorber of 1.7/1.1 band gap, coupled with earth-abundant H₂ and O₂ evolving electro-catalysts;^11–13 and to date, the demonstrated solar-to-H₂ conversion device has a record efficiency of 16%.^14,15 However, an analysis for the efficiency limits of PEC H₂O₂ devices and their corresponding operational constraints is lacking. Such an analysis should guide the continuing improvement of new materials or components for PEC H₂O₂ production.

This work is focused on modelling the solar-to-H₂O₂ conversion process in the context of design principles and performance limits under various device configurations. The reported performance of H₂O₂ electro-catalysts will input to the model. Therefore, before constructing PEC H₂O₂ prototypes, one can use this model to understand the operational constraints and to implement sound design choices.

H₂O₂ can be directly produced from H₂O and O₂ through either of the following two routes:

Route 1: O₂ + 2H₂O → 2H₂O₂E_cell = 1.083 V

Cathode: O₂ + 2H⁺ + 2e⁻ → H₂O₂ 0.680 V vs. RHE (I)

Anode: 2H₂O → H₂O₂ + 2H⁺ + 2e⁻ 1.763 V vs. RHE (II)

Route 2: O₂ + H₂O → H₂O₂ + 1/2O₂E_cell = 0.549 V

Cathode: O₂ + 2H⁺ + 2e⁻ → H₂O₂ 0.680 V vs. RHE (I)′

Anode: H₂O → 1/2O₂ + 2H⁺ + 2e⁻ 1.229 V vs. RHE (II)′

For both routes, the cathodic side should perform 2-electron (2-e⁻) reduction of O₂ to H₂O₂, and should be selective over 4-electron O₂ reduction to H₂O. Route 1 is a preferred pathway, not only because it produces twice the amount of cell potentials as Route 2 does which means storing twice amount of energy, but also because H₂O₂ produced by cathodes can become the reactant (reducing reagent) and get consumed in the anodic half-reaction of water oxidation to O₂ in Route 2. To achieve H₂O₂ accumulation to molar quantities in any (photo-)electrochemical pathways, H₂O-to-H₂O₂ oxidative selectivity is crucial and should become a criterion for catalyst benchmarking. But only recently, the literature reported the possibility of selective water oxidation to produce H₂O₂, thereby promising Route 1.^16–18 For the purpose of finding the efficiency limits of PEC H₂O₂ devices, 100% selectivity for water oxidation to H₂O₂ is assumed and should be a key performance target for any H₂O₂-producing (photo)-anodes that are being developed. Facile electro-catalytic kinetics even with a small overpotential has been shown to enable the desired H₂O-to-H₂O₂ selectivity.^19,20 In fact, efficient oxygen-evolution catalysts almost always over-oxidize H₂O₂ into O₂. With the thermodynamic energetics considered, efficient water oxidation usually does not favour H₂O₂ selectivity because the H₂O-to-H₂O₂ pathway in Route 1 exhibits an excess overpotential of 0.534 V (1.763 V vs. RHE to 1.229 V vs. RHE) compared to a non-selective H₂O-to-O₂ pathway. Nonetheless, there are emerging reports about oxidative H₂O₂ selectivity and tuning branching ratios of different water oxidation pathways is a viable route: various H₂O₂ producing (photo-)electrodes, such as g-C₃N₄ materials/Ni mesh composite photoanodes, have been shown to produce H₂O₂ oxidatively but have not yet accumulated H₂O₂ beyond milli-molar concentrations.¹⁸

Route 1 synthesis is chosen in this modelling study also because it can simplify device designs. With Route 1 synthesis, a membrane for product separation is no longer necessary because both reductive and oxidative products are the same, H₂O₂. This membrane-less design is unique to essentially any proposed PEC devices so far because of the disproportionation chemistry of H₂O₂: unlike H₂O₂ cells, a water-splitting cell requires separation of H₂ and O₂ products; and a CO₂ reduction cell requires elimination of cross-over of water-soluble products such as oxygenates to anode compartments.²¹

H₂O₂ is also considered as an alternative liquid fuel with comparable energy density to compressed H₂ gas. Concentrated H₂O₂ (usually 70% in aqueous solution with a stabilizing reagent) has been used as a liquid fuel for propulsion²² and actuation.²³ Its energy density reaches up to 0.75 kW h L⁻¹ which rivals that of compressed H₂ gas (350 bar), useful for stationary energy storage. Chemical energy stored in H₂O₂ can be converted to (i) electricity via a H₂O₂ fuel cell^24–26 or (ii) H₂ gas on-demand by applying a small voltage (∼0.6 V). Up to 0.8 V open-circuit voltage has been achieved by H₂O₂ fuel cells, which rival state-of-the-art H₂ fuel cells.²⁵ Notably, H₂O₂ is one of the very few chemicals that interface applications in both energy and environment with growing interest in both sectors. Such an integration of energy production and water remediation shall be a great example of solar-powered, sustainable, and green chemistry at the nexus of energy and water. Therefore, the PEC H₂O₂ device offers two advantages over existing solar-fuel technologies: (i) not costing additional energy for product separation, provided H₂O₂ (aq.) fuels can accumulate to molar concentration; and (ii) not requiring membrane separators, thus reducing system complexity and cost.

2. Types of photoelectrochemical H₂O₂ devices

As shown in Fig. 1, three types of devices can be implemented, including (a) a H₂O₂ electrolyzer; (b) a photoelectrode-based PEC device; and (c) photocatalysts in a particle suspension reactor. The taxonomy of photocatalysts is defined as such: that photocatalysts host co-existing, reductive and oxidative reactions at their electrolyte interfaces.²⁷ The terminology, photocatalysis, in following the convention, refers to light-driven thermodynamically up-hill reactions; although strictly speaking, photocatalysis should only refer to thermodynamically down-hill reactions. PEC, in following the convention, refers to photoelectrode processes exclusively; although by definition, PEC should be inclusive of photocatalyst-based processes. It is noted that under 1-sun solar illumination, the local pH near cathode surfaces becomes basic as protons are consumed during the 2-e⁻ reduction of O₂ to make H₂O₂, and by the same reason the local pH near anodes becomes acidic as protons are liberated via water oxidation to make H₂O₂. This phenomenon motivates the multi-physics modelling of electrochemical polarization losses.

Fig. 1 Schematics of PEC H₂O₂-production devices. The black arrows indicate electrolyte thicknesses: for type (a) H₂O₂ electrolyzers, the thickness ranges from 10 to 100 μm; and for type (b) and (c) photoelectrode devices and type (d) photocatalyst suspension, the thicknesses should be <2 mm to avoid excess sunlight absorption by water.

2.1 H₂O₂ electrolyzers coupled to solar cells

Fig. 1a shows the light-harvesting components, such as a Si photovoltaic module, which provides the electrical energy needed to drive a separately built, H₂O₂-synthesizing electrolyzer. A simple parallel-plate design can readily achieve H₂O₂ electrolysis at high throughput: all the pairs of cathodes and anodes in a H₂O₂ electrolyzer are stacked back-to-back, and a laminar flow plate of μm-length distances is placed in-between the cathodes and anodes. H₂O₂ is produced from both cathodes and anodes, and then collected by a laminar flow electrolyte that travels inside the flow plate parallel with the electrodes and removes the H₂O₂ products. The electrolyzer can operate with neutral pH electrolytes and the cathodes and anodes can be as close as manufacturing allows.^28,29 The H₂O₂ products gradually accumulate in the electrolyte, and eventually get separated and concentrated for delivery.

Different from water-splitting devices, porous gas diffusion electrodes (GDE) as cathodes may be employed for enhancing O₂ transport. Because the O₂ reactant for O₂-reduction half-reactions has limited solubility in water (2.4 × 10⁻⁴ M under standard conditions), the cathodic O₂-reduction current is limited to only −2.5 mA cm⁻².³⁰ GDEs create a triple phase boundary of air, electrodes, and electrolytes, and the electrolyte/catalyst contact in GDEs ensures sufficient supply of O₂ reactants for the cathodic half-reactions. Typically, the photon flux under 1-sun solar illumination produces a current density of 10–20 mA cm⁻² per geometric area. Electrolyzers can operate at 100–2000 mA cm⁻² current densities. Therefore, both PEC and electrolysis devices require the use of GDEs as cathodes.

2.2 Photoelectrode-based PEC H₂O₂ devices

This device scheme, as illustrated in Fig. 1b and c, consists of PEC anode and cathode components which couple the processes of light absorption, charge separation, and catalysis in one device. In this case, direct solar-to-H₂O₂ conversion processes occur at the electrode/electrolyte interface. Fig. 1b schematically illustrates a parallel PEC electrode configuration, which is extended from the electrolyzer's parallel-plate design (Fig. 1a). For example, the upper photoelectrodes facing the sun can be light absorbing photoanodes; and the lower electrodes can be GDE cathodes or photocathodes. A thin electrolyte surrounds the upper photoelectrode and resides above the lower electrode for minimizing solution losses.

It is preferred for a design to utilize a maximum of light harvesting area. Because GDE cathodes can operate in the dark, alternatively, vertical O₂ reduction GDE cathodes can be arranged perpendicular with the upper and lower photoanodes that face illumination for light harvesting (shown in Fig. 1c). The merit of this vertical design is that these GDEs neither obstruct light absorption nor take up the light harvesting area: most reported selective O₂-to-H₂O₂ electro-catalysts are optically opaque and can be readily applied to such GDE configurations by anchoring onto porous carbon supports.^10,31,32

2.3 Particle-based H₂O₂ photocatalysts

In our model, particulate photocatalysts are treated as mono-disperse spheres that are suspended in pure water, as shown in Fig. 1d. Reactants of O₂ will be initially saturated in water, and as the reaction takes place, O₂ will be consumed and replenished via passive exchange with air or active bubbling with air. Both reductive and oxidative sites on individual particles produce the same product, H₂O₂. According to our numerical calculations, O₂ transport can match with the rate of O₂ reduction per particle; and furthermore, particles do not need a supporting electrolyte for lossless ion transport (see Section 3.6). It is also assumed that the reductive and oxidative sites do not further reduce H₂O₂ to water, and do not further oxidize H₂O₂ to O₂, respectively, which is consistent with the required selectivity for Route 1 synthesis, i.e. O₂ reduction to H₂O₂ and water oxidation to H₂O₂. If so, particulate photocatalysts can produce and accumulate, in pure water, a high concentration of H₂O₂ (aq) solution under solar illumination (UV filtered, >400 nm). Through simple filtration, the photocatalysts are easily separated from the as-produced H₂O₂ (aq) concentrate.

2.4 Strategies for H₂O₂ accumulation at molar concentrations

Both thermal instability and photo-decomposition of H₂O₂ are potential concerns for H₂O₂ accumulation at molar concentrations. H₂O₂ accumulation is certainly achievable but needs to be managed carefully.^17,18 A stability benchmark is decomposition by <0.3% over 1000 hours as set by the US Department of Energy.³³ First, H₂O₂ self-decomposes at elevated temperatures: at 65 °C, its concentration reduces by 0.3% over 1000 hours; whereas at 25 °C, the same concentration reduction takes 3 years.³⁴ Therefore, both type (a) electrolyzer and type (b)–(d) PEC devices are recommended to operate at room temperatures with active cooling, but not at elevated temperatures where commercial electrolyzers typically operate for improved catalytic kinetics. Furthermore, metal impurities are known to catalytically decompose H₂O₂via an OH˙ radical mediated mechanism. Metal-free components are thus desirable components for constructing H₂O₂ devices, and OH˙ binding reagents such as Lewis acids²⁵ and sodium stannate³⁴ should help stabilize H₂O₂ molecules for their accumulation.

PEC production and accumulation of H₂O₂ by type (b)–(d) devices is also possible because H₂O₂ can remain stable with the UV portion (λ < 420 nm) of solar spectra filtered, thereby preventing photo-decomposition of H₂O₂.³⁵ Last but not the least, both catalytic and electro-catalytic decomposition of H₂O₂ at the electrodes should be avoided. A platinum-group catalyst, for example, can catalyse H₂O₂'s disproportion reactions with a first-order rate law with respect to its molar concentration. For electro-catalytic decomposition, a non-unity Faradaic efficiency for either the O₂ reduction to H₂O₂ pathway or the water oxidation to H₂O₂ pathway implies the formation of H₂O by H₂O₂ reduction or the formation of O₂ by H₂O₂ oxidation, respectively, just by running non-selective catalysts.

3. Modelling methods

3.1 Performance and modelling of H₂O₂ electrocatalysts

The electro-catalytic behaviour for O₂ reduction to H₂O₂ and H₂O oxidation to H₂O₂ is taken directly from the literature reported, experimentally measured values.^17,31 Although selectivity was not available for the J–E performance of each reported catalyst, 100% selectivity is assigned to all catalytic performances used in efficiency calculations. It is very optimistic to assign both reductive and oxidative catalysts to operate at 100% Faradaic efficiency. It remains appropriate for calculating the upper bound of solar-to-H₂O₂ conversion efficiencies. For cathodic half-reactions, electro-catalysts of nitrogen-doped carbon,³¹ oxidized carbon,³⁰ platinum or palladium–gold composites,³⁷ and graphitic carbon nitrides^8,36 and photo-catalysts of titanium oxides^38–40 and carbon nitrides^9,10 have been surveyed, and the best performing nitrogen-doped carbon electro-catalysts were chosen as modelling inputs.³¹ Because oxygen transport usually affects the current density–potential (J–E) behaviour during O₂ reduction experiments, we used the kinetic limiting current densities (j_kin) extracted from the reported Koutecky–Levich plots. These j_kin–E data reasonably describe the behaviour of GDEs when they are loaded with the respective catalysts which can achieve operating current densities of ca. 20 mA cm⁻² or even above. For anodic half-reactions, catalysts such as bismuth vanadium oxides, manganese oxides, tungsten oxides, and niobates have been evaluated,^{5a,17,41–43} and the anodic J–E behaviour of manganese-alloyed titanium oxides was chosen for modelling.⁵

Butler–Volmer kinetics was used to fit the J–E behaviour of chosen catalysts.^12,44,45 The fitted kinetic parameters were used to describe the reductive and oxidative kinetics of catalyst-loaded cathodes and anodes, respectively, as listed in Table 1. It is noted that the Faradaic efficiency for O₂ reduction to H₂O₂ is often less than unity and that the selectivity of water oxidation catalysis to H₂O₂ is much less explored. Further understanding and improvement of H₂O₂ selectivity should be an area of active investigation.

Table 1 Kinetic parameters for reductive H₂O₂ electro-catalysts (O₂ reduction to H₂O₂) and oxidative H₂O₂ electro-catalysts (H₂O oxidation to H₂O₂), representing the modelled cathodic and anodic behaviour, respectively

Materials	Exchange current density	Transfer coefficient αcathode	Transfer coefficient αanode	Ref.
O 2 -reduction-to-H 2 O 2 catalysts
Nitrogen-doped carbon	0.3078 mA cm^−2	0.3811	0.50	Fellinger^31
Oxidized carbon	0.2497 mA cm^−2	0.3942	0.50	Lu^30
Graphitic carbon nitrides	0.1953 mA cm^−2	0.2105	0.78	Kofuji^36
H 2 O-oxidation-to-H 2 O 2 catalysts
(Ti, Mn)Ox	0.03895 mA cm^−2	0.50	0.35	Siddiqi^5
BiVO4	0.00497 mA cm^−2	0.50	0.2235	Shi^17

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3.2 Optical absorption of light-absorbing photoelectrodes

The photovoltage of light absorbers, or the quasi Fermi level splitting between photo-generated electrons and holes, provides the driving force for solar-to-H₂O₂ conversion and will be calculated by applying the detailed-balance principle at the Shockley–Queisser limit. The same method had been used elsewhere for estimating efficiency limits of PEC water splitting and CO₂ reduction devices.^46,47 Detailed balance refers to the net flux of incoming photons through the thin electrolyte layer and photon dissipation due to radiative recombination in the light absorbers. Mathematically, the net carrier flux is calculated from the gain of generating electron–hole pairs by optical absorption, and the loss of charge carriers due to outgoing radiative recombination and black-body thermal emission. Then, the net outgoing flux of electron–hole pairs determines the H₂O₂ production rates, where the quantum yield of photon-to-charge conversion is assumed to be 100%.

For PEC photoelectrode devices, a dual-absorber configuration should achieve a maximum efficiency during solar-to-H₂O₂ conversion, because (i) dual absorbers provide sufficient up-hill driving force for light-driven H₂O₂ production; (ii) the matching photocurrent flux of more than two absorbers decreases and reduces H₂O₂ production rates; and (iii) dual absorbers offer a two-electrode design necessary for PEC H₂O₂ devices which is not possible with single absorbers. The thickness of electrolytes above the tandem light absorbers was chosen to be 2 mm, which is thick enough to ensure effective ion transport thereby minimizing solution losses (see Fig. 6a), but not too thick to absorb the infrared photons of 1.0–1.3 eV energy thereby limiting the bottom absorber's photocurrent.⁴⁸

During operation, the respective photocurrent densities, j_top(V₁) and j_bottom(V₂), of the top and bottom absorbers are equal, j_ph(V_ph) = j_top(V₁) = j_bottom(V₂). The tandem absorber photovoltage at j_ph is V_ph = V₁ + V₂. To make realistic modelling of dual absorbers, an effective series resistance, R_s, was introduced to adjust the j_ph–V_ph behavior comparable to the reported record efficiency solar cells. Typically, calculations at the Shockley–Queisser limit usually give a fill-factor of ca. 0.90, which will be adjusted to 0.85 to match with record efficiency solar cells.⁴⁹

3.3 Electrolyte modelling

A buffer electrolyte was used for both H₂O₂ electrolyzers and PEC photoelectrode devices but not for photocatalysts, which are suspended in pure water. The borate buffer was chosen for modelling with their literature parameters listed in Table 2.^21,44 Both electrolytes of phosphates (pKa = 7.21) and borates (pKa = 9.23) buffered at near neutral pH were considered and both resulted in similar behaviour of ion transport and pH gradient in solution modelling (see Section 3.4). To illustrate the benefits of the high ionic strength of buffered electrolytes, a weak electrolyte of pH = 2 sulfuric acid was also used for comparison.

Table 2 Baseline values for the modelling parameters

Category	Parameter (unit)	Value
Electrolyte property	H^+ diffusion coefficient	9.31 × 10^−5 cm^2 s^−1
	OH^− diffusion coefficient	5.27 × 10^−5 cm^2 s^−1
	H2BO3^− diffusion coefficient	0.879 × 10^−5 cm^2 s^−1
	H3BO3 diffusion coefficient	1.33 × 10^−5 cm^2 s^−1
	K^+ diffusion coefficient	1.96 × 10^−5 cm^2 s^−1
	H^+ charge number	1
	OH^− charge number	−1
	H2BO3^− charge number	−1
	K^+ charge number	1
	Electrolyte conductivity, κ	0.35 S m^−1
Reaction kinetics
	Forward rate constant, k1f	1 × 10^7 s^−1
	Backward rate constant, k1b	1.74 × 10^16 M^−1 s^−1
	Forward rate constant, k2f	1 × 10^8 s^−1
	Backward rate constant, k2b	1.74 × 10^19 M^−1 s^−1
Semiconductor parameters	Optical absorption coefficient (typical GaAs), α	5 × 10^3 cm^−1
	Particle concentration, cpar	1.0 g L^−1
	Particle density, ρpar	2.3 g cm^−3
Constants	Equilibrium cell potential, E0	1.083 V
	Reaction temperature, T	293 K
	Faraday's constant, F	96 485C mol^−1
	Unit charge of electrons, qe	1.6 × 10^−19 C
	Ideal gas constant, R	8.314 J mol^−1 K^−1

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In a 1-D model for parallel-plate H₂O₂ electrolyzers, the local current density is identical across the electrode surface because of the translational symmetry. In a 2-D model of PEC devices, the local current densities and the electrolyte potentials vary spatially across the electrode surface. The potential of the electrode is set constant so that the operating voltage of PEC devices is quoted as the anode–cathode potential difference.

3.4 Electrochemical load behaviour at the device level

In the COMSOL Multi-physics model, the boundary conditions were set for identical total currents, I, while sweeping the total current and monitoring the anode–cathode potential difference, V. The total current is equal to the integration of local current densities over the electrode surface. The electrochemical load behaviour is represented by the modelled I–V performance under a specific device configuration, and will be used in conjunction with the light absorber optoelectronic modelling (Section 3.2) to determine the device operation point and solar-to-H₂O₂ conversion efficiency (see Section 3.5).

For both electrolyzers and PEC devices, the potential difference, V, or the operating voltage, consists of the thermodynamic cell potential (E₀), kinetic overpotentials for reduction and oxidation half-reactions (η_C and η_A, respectively), ion transport losses in electrolytes (Δϕ_transport), and pH gradient losses (Δϕ_pH).

V = E₀ + |η_A| + |η_C| + Δϕ_transport + Δϕ_pH, (1)

where η_C is the cathodic overpotential (η_C < 0), and η_A is the anodic overpotential (η_A > 0). The ion transport losses, Δϕ_transport, are due to the finite mobility of ions during transport,^44,50,51 which is contributed by two terms: (i) the ohmic resistive loss defined as in the first term; and (ii) the concentration gradient or electrodialysis loss defined as in the second term:where κ is the electrolyte conductivity, J is the current density, is the position vector of the 2-D and 3-D device models, F is Faraday's constant, D_i is the diffusion coefficient for the ith species in the electrolyte, and z_i is the charge number and c_i is the molar concentration for the ith species (see Table 2 for modelling parameters). The major contribution to the ion transport losses is the ohmic resistance. Electrodialysis loss is negligible for all the membrane-less devices that operate with the current density and geometric parameters investigated herein.

The loss due to pH gradient, Δϕ_pH, was obtained by the Nernstian term:

where R is the ideal gas constant, and T is the absolute temperature taken as T = 298.15 K. During operation, pH_cathode > pH_anode because locally cathodes consume H⁺ becoming more basic and anodes release H⁺ becoming more acidic. The total polarization loss in electrolytes is a sum of the electrode losses (catalyst overpotentials) and the electrolyte losses (consisting of ion transport and pH gradient losses).

3.5 Efficiency and operation point of PEC devices

The operation point of PEC photoelectrode H₂O₂ devices determines the device efficiency.^46,52 Under steady-state illumination, the device will settle at a single operation point where the dual-absorber photovoltage should match with the operating voltage for electrochemical catalysis under the same operating photocurrent density, i.e. V_ph(j_op) = V(j_op). Generally, one combination of top and bottom absorbers, one set of electrocatalysts, and one device configuration determine one efficiency number. As the absorber performance reaches its Shockley–Queisser limit and as the catalyst selectivity approaches 100%, this calculation represents the upper limit of solar-to-H₂O₂ conversion efficiencies. The operation point at any time is given by intersecting the electrochemical load curve (I–V behaviour) and the light-absorber drive curve (j_ph–V_ph behaviour) as illustrated in Fig. 2. The solar-to-H₂O₂ efficiency limit is defined as:where j_op is the operating photocurrent density (mA cm⁻²) at the intersect, and P_in is the total incident solar irradiance (mW cm⁻²).

Fig. 2 Illustration of operation points for PEC H₂O₂ devices operating at the optimal efficiency point as shown in Fig. 6d. The specific calculation presented involves the photovoltaic performance of tandem dual absorbers of 1.75 eV/1.15 eV band gaps adjusted to a realistic fill factor of 0.85 (red curve), and electrochemical load (black curve) of the lower photoanode + vertical GDE cathode device configuration.

Photovoltaic-coupled electrolyzers can use the same intersection method to obtain operation points. In this case, the solar-to-H₂O₂ conversion efficiency is the product of the solar-cell efficiency at the maximum power point, the electrolyzer efficiency determined by the operating currents voltages of the electrolyzer, and power transmission efficiency from the solar cell to the electrolyzer. The areal ratio of PV cells and electrolyzer electrodes strongly affects the operation point: for example, a small area PV cell driving a large surface area electrode will skew the conversion efficiency to a high efficiency number. Practical engineering and economic considerations will settle the combined PV–H₂O₂ electrolyzer efficiency on a reasonable number, which is not the subject of this study.

3.6 Detailed-balance modelling of particulate absorbers

For particles made of one type of absorber, the detailed balance formalism remains applicable. Individual particles are treated as geometric points despite their realistic shapes are often aggregated spheres. For each particle, the net current flux, J_ph, is proportional to the rate of H₂O₂ production, ṅ_H2O2:

2Fṅ_H2O2 = J_ph/q = G_y + ṙ_th − ṙ_rad(V_ph), (5)

where G_y is the total optical generation rate within the volume of the particle that is located at a distance of y from the top surface of the particle suspension; ṙth is the rate of thermal radiation which is negligibly small at room temperature for the absorbers modelled herein; and ṙ_rad is the rate of radiative emission at a photovoltage of V_ph, equal to quasi-Fermi level splitting of electrons and holes. The pre-factor of 2F indicates that transfer of two electrons produces one H₂O₂ molecule, where F is Faraday's constant.

For simplicity, we use an average volumetric optical generation rate by assigning a particle as a geometric point, the centre point of a particle. This assumption is reasonable for efficiency bound calculations because it captures the physics of position dependent light intensity as a function of the particle-centre depth, y. The volumetric optical generation rate, g_y, taking units of # cm⁻³, is the photons absorbed per unit volume at the depth, y, in units of cm, following the Beer–Lambert Law of light absorption:

, where q_e is the unit charge of electrons, and σ(E) is the effective optical cross-section according to the Mie theory, the mathematical description of which can be found in Bohen and Huffman.⁵³ The photon flux, ϕ_y,E, in units of # (eV s)⁻¹ cm⁻² at the depth of particle suspension, y, is described as a diluted effective medium:

ϕ_y,E = ϕ_AM1.5,ET_(y,E)exp[−f_parα(E)_y] (7)

, where ϕ_AM1.5,E is the incident AM 1.5G photon flux taken from the NREL database;¹⁷T_(y,E) is the wavelength-dependent optical transmission of water at depth y; and f_par is the volume fraction of particle in suspension. f_par is a dimensionless number, and can be calculated from c_par and ρ_par, following f_par = c_par/(ρ_par1L)α(E) should be a function of photon energy.

For baseline values of modelling, α(E) is taken as a constant, α, of 5 × 10³ cm⁻¹ because typical direct band gap absorbers like GaAs have a value of 10⁴ cm⁻¹ for absorbed photons and typical indirect band gap absorbers like Si have a value of 10²–10³ cm⁻¹. To simplify the calculation of particle efficiency limits and to account for realistic water absorption, T_(y,E) is taken as depth-independent values at y = 2 mm.¹¹ Using Si's materials property, the bulk density of semiconductor particles, ρ_par, is 2.3 g cm⁻³; and the particle concentration, c_par, is 10.5 g L⁻¹, which is a reasonable value for particle loading in a suspension.

The radiative emission rate of ṙ_rad(V_ph) adopts the formalism by Henry⁵⁴ and its calculation may incorporate the term for the Mie resonance according to the reciprocity of optics.

, where A_par is the particle cross-sectional area; σ(E) is the effective optical cross section; b(E, T) is Plank's law of black body radiation for the photon energy of E at a temperature of T; and ρ is the product of the density of optical modes per unit volume and unit solid angle. ṙth is taken as ṙ_rad(V_ph). We note that for particles, the angular integral ∫dΩ = 4π.

The total optical generation rate of individual particles, G_y, can be approximated by multiplying the particle volume, V_par, and the position-dependent, average volumetric optical generation rate, i.e. G_y = g_yV_par.⁵⁵ In this study, we do not consider the size-dependent effects of light scattering and Mie resonance at individual particles, i.e. σ(E) ≈ 1. The enhancement due to the Mie resonance is at most 4n² where n is the refractive index of particle absorbers.⁵³

Notably, the rate of particulate photocatalysis depends on particle sizes, because optical absorption scales with particle volume and radiative emission rate scales with surface area. At each particle, the net outgoing current of electron/hole pairs for photocatalytic H₂O₂ production is

where V_par and r_par are the particle volume and radius, respectively; and j₀ is the dark current due to radiative emission, which follows the detailed-balance calculations for planar light absorbers. Light absorption and radiative photon emission correlate with the geometric parameters of particle volume and particle surface area, respectively. To calculate j₀, we treat the top interface as a semiconductor–water interface and the bottom interface as being perfectly reflective, and assume that radiative recombination is the only carrier loss mechanism. In a realistic assessment, the non-radiative Shockley-Read-Hall recombination at the particle surface is almost always a determining factor for photocatalyst efficiency, but is neglected here for calculations of the efficiency upper bound.

3.7 Efficiency of particulate photocatalyst suspension

We will first analyse the operation points of individual particles and then sum up H₂O₂ production rates of all the particles in the reactor. For particulate photocatalysts of <1 μm diameter, the equivalent photocurrent densities at particle surfaces are typically two orders of magnitude smaller than those of planar photoelectrodes. Thus, the total polarization losses including the overpotential losses and solution transport losses (see Section 4.3) at individual particles are negligible under 1-sun illumination.

Another distinction with photoelectrodes is that each individual particle is simultaneously in contact with both O₂/H₂O₂ and H₂O₂/H₂O redox levels in the same electrolyte. It requires an additional consideration of a light intensity threshold, at which a minimal photovoltage is required for the net accumulation of H₂O₂ products in the electrolyte surrounding the particles. According to Shreve and Lewis, a non-zero quasi Fermi-level splitting at photoelectrode/electrolyte interfaces is a necessary condition for net accumulation of chemical products.⁵⁶ In this case, the threshold photovoltage is the cell potential, i.e. 1.083 V, to achieve the microscopic reversibility of forward H₂O₂ reactions (photo-electrolysis) and H₂O₂ recombination back-reactions (fuel–cell reactions). With particles in complete darkness, H₂O₂ can disproportionate back to oxygen and water via cathodic and anodic backward half-reactions, which is identical to running a H₂O₂ fuel cell.^24–26 The particles recombine their excess electrons and holes to emit light or to generate heat, thereby catalysing H₂O₂'s disproportionation. In this case, one does need to separate solid particles from H₂O₂ containing water when there is no illumination.

Once the photovoltage threshold is met, the H₂O₂ production rates by individual particles (typically μm size, under 1-sun illumination) are essentially determined by the outgoing currents, j_ph because of particles' negligible polarization losses. The operating point of each particle, therefore, is at j_ph = j_ph(V_ph = 1.083 V).

The bottom of a photocatalytic reactor should not be completely dark: the light intensity at the reactor bottom is just enough to produce 1.083 V. In other words, a well-designed particle reactor should not fully absorb the incident solar illumination, as illustrated in Fig. 3. As the intensity of light impinging onto particles attenuates from top to bottom of the particle reactor, the particle photovoltage, or quasi Fermi level splitting, decreases down to the threshold voltage of 1.083 V.

Fig. 3 Particle-based photocatalysts producing H₂O₂ under depth dependent light intensity. (a) Non-zero light intensity exiting bottom of a photocatalyst suspension. (b) H₂O₂-producing photocatalysts and their quasi Fermi level positions of light-excited electrons and holes, for those at the top and bottom of the particle reactor. (c) Schematic light intensity profile vs. depth of particle photocatalysts, where the minimal light intensity at the reactor bottom ensures a threshold photovoltage of 1.083 V.

For individual particles under various illumination intensities, their j_ph (j_ph > 0) is taken at V_ph = 1.083 V using eqn (9), because the outgoing current flux is equivalent to 10⁻² mA or less, which is so small that V_ph is poised at ca. 1.083 V. Specifically, particles at the reactor bottom give a net zero H₂O₂ production rate, mathematically expressed as,

j_ph(V_ph = 1.083 V) = 0 (10)

The minimal intensity at the reactor bottom, g_y,min, is obtained by solving the above equation. The depth to achieve the minimal intensity is defined as y_min, which is the maximum height of the reactor. Combining eqn (6) and (7), y_min can be solved. y_min is a function of the critical variables of particle radius r_par, and band gap, E_g, according to eqn (6)–(9).

According to eqn (9), j_ph varies with the optical generate rate, g_y, radiative emission dark currents, j₀, and particle sizes, r_par. To calculate the efficiency of a particle suspension, we define a volumetric H₂O₂ production rate, in units of # cm⁻³, which is expressed by normalizing with the particle volume because each particle is assigned as a geometric point:

and j_ph = q_eṙ_y. We then integrate the j_ph for individual particles across the whole particle reactor to obtain the areal current density for particles, j_op. It is calculated by integrating the depth-dependent volumetric rate in eqn (11):j_op will vary with a range of r_par and E_g values, and j_op takes units of mA cm⁻². Finally, the solar-to-H₂O₂ efficiency of a particle suspension is calculated by following eqn (4) and by taking the integrated j_op as inputs. The efficiency limit of a particle suspension will be plotted as a function of r_par and E_g.

4. Results and discussion

4.1 Parallel-plate H₂O₂ electrolysis

A parallel-plate H₂O₂ electrolyzer employing the state-of-the-art, reductive and oxidative H₂O₂ evolving electrocatalysts (their kinetic parameters are listed in Table 1) was modelled in 1-D. The gas-diffusion cathode of nitrogen-doped carbon³¹ and the water oxidation anodes of manganese-alloyed titanium oxides^5a simultaneously produce H₂O₂ across a thin-layer electrolyte of a thickness of w (μm). The anode is assumed to selectively produce H₂O₂ from water at 100% Faradaic efficiency.^5b We use this simple 1-D model to illustrate the electro-catalytic production of H₂O₂ in a membrane-less electrolyzer and to quantify the various polarization losses during its operation. Despite a laminar flow of thin electrolytes, they are modelled as stagnant layers because the ion transport across the thin electrolyte should not be affected by the traversing flow field.

Fig. 4 shows the current density–voltage (J–V) behaviour and polarization-loss breakdown of parallel-plate H₂O₂ electrolyzers that operate in a borate buffer with an initial pH = 9.2. A current density of 20 mA cm⁻² is obtained at an applied two-electrode potential of 1.85 V for electrode distances of less than 50 μm. With the cathode–anode distance of w < 50 μm, the sum of the ion transport loss (Fig. 4d) and pH gradient loss (Fig. 4b) consist of less than 10% of overall polarization losses. Fig. 4c exhibits the identical current density–overpotential behaviour of anodes and cathodes irrespective of various electrode distances, which validates our 1-D model.

Fig. 4 Current density versus cell potential (J–V) behaviour and polarization losses for parallel-plate, membrane-less H₂O₂ electrolyzers containing a 1 M borate electrolyte buffered at pH = 9.2. The cathode–anode distance varied from 10–200 μm. (a) J–V behaviour of the electrolysis process in a stagnant electrolyte. (b)–(d) Analysis of polarization losses as a function of the operating current density, due to (b) pH difference, (c) anodic (left axis) and cathodic (right axis) overpotentials, and (d) ion transport in electrolytes.

We found that the catalyst overpotentials contribute to the major polarization losses for any reasonable electrode distances of w < 100 μm. The loss of (1.850–1.083) V = 0.767 V at 20 mA cm⁻² mostly consists of overpotential losses, i.e. 0.450 V for anodic and 0.277 V for cathodic overpotentials, respectively. At a shorter distance of w < 50 μm, both the ion transport loss and pH gradient become negligible, and appears to be in a linear relationship with the electrolyzer's operating currents. For proper operation of the H₂O₂ electrolyzer, a cathode–anode distance should be 100 μm or less and this operational constraint is consistent with typical designs of membrane-based water-splitting electrolyzers.²⁹

To illustrate the necessity of buffered electrolytes, we further assessed the polarization losses during the H₂O₂ electrolysis in a weak acid with a low ionic strength (weak base is known to gradually decompose H₂O₂ and was not chosen³⁴). Fig. 5 shows the J–V and loss behaviour of the same set of electrolyzer configurations as in Fig. 4, but in a diluted sulfuric acid of pH = 2. As shown in Fig. 5b, the loss due to pH gradient becomes significant at a j_op of above 20 A cm⁻² for intermediate electrode distances of w > 70 μm. Without buffer, the concentration polarization of local H⁺ or OH⁻ ions at high current densities can dominate over the overpotential losses and become the major contributor to the overall polarization losses. Beyond this point, the loss due to pH gradient could increase so dramatically that no further increments in electrolysis currents can be made even with operating voltages, V, continuously increased. Fig. 5c shows that the catalyst overpotential losses at various operating current densities are identical with the case of buffered electrolytes (Fig. 4c) which further validated our model. The ion transport losses in a weak acid (Fig. 5d) are comparable with those calculated for a buffered electrolyte (Fig. 4d).

Fig. 5 Current density versus cell potential (J–V) behaviour and polarization losses for parallel-plate, membrane-less H₂O₂ electrolyzers containing a diluted sulfuric acid electrolyte of pH = 2. The cathode–anode distance varied from 10–200 μm. (a) J–V behaviour of the electrolysis process in a stagnant electrolyte. (b)–(d) Analysis of polarization losses as a function of the operating current density, due to (b) pH difference, (c) anodic (left axis) and cathodic (right axis) overpotentials, and (d) ion transport in electrolytes.

With no buffer or with weak ionic strength, losses due to pH gradient can become significant at high current densities (>20 mA cm⁻²). This phenomenon of concentration polarization is very sensitive to the electrode distance. With weak electrolytes, the cathode–anode distance becomes a major operational constraint for membrane-less electrolysis. This is because H⁺ and OH⁻ ion mobility is finite and they do not effectively transport across the thin-layer electrolyte by drift under a small electrical field and/or across a large distance. When j_op increases to the magnitude of 1 A cm⁻², which is the typical operating current density of a commercial water electrolysis unit, both losses of ion transport resistance and pH difference can become significant. With weak electrolytes, the cathode–anode distance needs to be small. In a pH = 2 acid, it is still possible to operate efficiently, but the electrode distance needs to be <50 μm for a j_op of ca. 20 mA cm⁻². For a j_op of 1 A cm⁻², the electrode distance needs to be <2 μm.

4.2 Photoelectrode-based H₂O₂ devices under illumination

The design principles and operational constraints that have been studied for H₂O₂ electrolysis are expandable to PEC H₂O₂ production with the photoelectrode-based devices and photocatalyst-based suspension. For photoelectrodes, we will apply the same Butler–Volmer kinetics for the cathodic and anodic catalysis using a 2-D device model, and additionally couple the light-driven electro-catalysis with light absorption and charge separation using the operation point analysis outlined in Section 3.5.

While the 1-D electrolyzer model shows identical current densities across the electrode surface due to a translational symmetry, the 2-D PEC device model only exhibits a mirror symmetry.

Fig. 6a shows the pH distribution under a steady-state, average operating current density of 20 mA cm⁻², which is consistent with the expected mirror symmetric distribution of local current densities and electrolyte potentials. The widths of top and bottom photoanodes are 1 mm and 0.4 mm, respectively. The dual absorber tandem by adjusting their band gaps can provide an upper limit of matching photocurrent densities at ca. 20 mA cm⁻² under 1-sun illumination. The pH gradient between photoanodes and vertical GDE cathodes under this limiting photocurrent density is less than 1 pH unit with buffered electrolytes.

To survey the design space, two geometric parameters including the upper anode width and the width ratios of lower/upper anodes were varied in the 2-D PEC model. Their effects on the light-driven electrolysis performance and associated solution losses were evaluated. The I–V behaviour of upper photoanode/vertical cathode and lower photoanode/vertical cathode pairs is plotted in Fig. 6b as a function of average operating current densities. The solar-to-H₂O₂ conversion efficiency can be improved with less polarization losses, because the electro-catalytic component of the photoelectrodes is coupled to the dual absorber tandem (see Section 3.5). An upper anode width of 1 mm and a width ratio of 0.4 were chosen as the optimized configuration, because the calculated total polarization losses in this case had already matched with the Shockley–Queisser limit of tandem absorbers, i.e. a photovoltage of 1.8 V at a photocurrent density of 20 mA cm⁻². Further reducing width ratios to less than 0.2 only marginally improves the cell J–V performance but with increased manufacturing complexity. In principle, the upper anode width and lower anode width can be 10s of μm as long as manufacturing allows⁵⁷ but its complexity and cost render this geometry impractical.

There will be an optimal combination of band gaps for each parameter set of catalyst performance and device geometry. Fig. 6c and d show solar-to-H₂O₂ iso-efficiency contour plots for the upper photoanode + vertical GDE cathode and for the lower photoanode + vertical GDE cathode configurations, respectively. The two-electrode I–V curves at a width ratio of 0.4 were quoted from Fig. 6b and used for generating both iso-efficiency plots. The upper photoanode component can achieve an optimal solar-to-H₂O₂ efficiency of 19.3% at a band gap combination of 1.85 eV/1.25 eV for the upper photoanode tandem absorber. The lower photoanode can achieve an optimal solar-to-H₂O₂ efficiency of 21.7% at a band gap combination of 1.75 eV/1.15 eV for the lower photoanode tandem absorber. Photoelectrode-based PEC H₂O₂ devices can reach an overall solar-to-H₂O₂ efficiency of 20.0%.† Because the maximum photocurrent density at the Shockley–Queisser limit is ca. 20 mA cm⁻² at 1-sun, 20.0% is near the solar-to-H₂O₂ efficiency upper limit.

The iso-efficiency plots exhibit three characteristic regions of efficiency contour lines: (i) the upper region: top-absorber band gaps are too large to limit photocurrents in series; (ii) the lower left region: band gaps of both top and bottom absorbers are so small that the total output photovoltage is not sufficient to drive H₂O₂ electrolysis; and (iii) the central high-efficiency plateau: the device at the given parameter set achieves optimal conversion efficiencies at an operating point of sufficient photovoltage matching with the modelled electrochemical polarization behaviour.

The geometric parameter space can be further explored under the trade-off between the catalyst overpotential losses and the solution transport losses. Improving the catalyst performance will relax many operational constraints such as the PEC electrode width (wider than 1 mm), GDE cathode height (lower anode/cathode aspect-ratio), and optical concentration (brighter than 1-sun). Mild optical concentration (10×) can effectively utilize the real-estates of electrode area without complex optics. We had fixed the height of the vertical GDE cathode to be 4 times the width of top photoanodes: in principle, an increase in the height length of GDE cathodes can further relieve the turn-over-frequency limitation of current O₂-reduction electro-catalysts; however too tall of vertical cathodes may easily shadow the bottom photoelectrode making manufacturing impractical. Other configurations may help reduce solution losses, such as an angled arrangement of photoanodes and GDE cathodes but at a cost of increased light absorber area.

4.3 Efficiency limits of photocatalytic H₂O₂ production

A particle at mesoscopic sizes (<10 μm diameter) effectively reduces both catalyst overpotential and solution transport losses by shortening distances between cathodic and anodic catalytic sites. First, a 3-D multi-physics model was employed to verify the nuance variation in the pH gradient of electrolytes surrounding a 1 μm diameter, strongly absorbing particle (e.g. GaAs) under 1-sun illumination. As shown in Fig. 7a, the local pH difference between any cathodic and anodic sites is less than 10⁻³ pH units, negligible as compared to the electrolysis and photoelectrode devices (see Sections 4.1 and 4.2). Similar analysis has been applied to O₂ transport, indicating that the solubility of O₂ (2.4 × 10⁻⁴ M, aerated with 1 atm and 25 °C air under standard conditions) does not limit the photocatalytic rates either.

Fig. 6 Component analysis and efficiency limits of photoelectrode-based PEC H₂O₂ devices under steady-state operation in a stagnant, 1 M borate electrolyte buffered at pH = 9.2. (a) pH distribution in the electrolyte for the device designed with a top photoanode width of 1 mm and a lower photoanode width of 0.4 mm. (b) Two-electrode cell J–V behaviour of H₂O₂ catalysis components with varying width ratios. The J–V curves were calculated while both upper and lower photoanodes are simultaneously operating. Iso-efficiency plots of solar-to-H₂O₂ efficiencies as a function of band gaps of top and bottom absorbers that operate at the Shockley–Queisser limit, for (c) an upper photoanode + vertical GDE cathode device configuration, and for (d) a lower photoanode + vertical GDE cathode device configuration, both at a width ratio of 0.4. Red points indicate the maximum efficiency points in the plots.

Fig. 7 Component analysis and efficiency limits of particle-based H₂O₂ photocatalysts suspended in pure water (pH = 7). (a) 3D plot of local pH at individual particles with exaggerated colour legend spanning ΔpH < 0.001 (pH units). (b) Iso-efficiency plot as a function of particle band gaps and particle diameters, showing the maximum solar-to-H₂O₂ efficiency of particle systems that operate at the Shockley–Queisser limit. (c) Efficiency limit of a suspension of 1 μm diameter spherical photocatalysts as a function of their band gaps, with a maximum efficiency of 27.5% at a band gap of 1.55 eV.

Fig. 7a shows the upper bound of pH difference, at the maximum optical generation rate of particles by using the most absorbing materials and most efficient band gaps: optical absorption coefficient is taken as 10⁵ cm⁻¹, which is GaAs's absorption coefficient at 600 nm (much greater than α = 5 × 10³ cm⁻¹ used for particle efficiency calculations); particles are assumed to be perfect spheres of 1 μm diameter; and the band gap of particles is taken as 1.55 eV, which will be presented below as the optimal band gap for solar-to-H₂O₂ conversion. The estimated upper bound of optical generation rates for individual particles has an upper bound of 1.69 × 10²² cm⁻³ s⁻¹, because as light penetrates deeper through the particle suspension in the reactor, the light intensity is effectively decreased, and so are optical generation rates. Even when the generation rate increases by two orders of magnitude, e.g. by considering light trapping under Mie resonance and under 10× mild concentration, the local polarization losses at individual particles are still negligible. As particle diameters become comparable with the wavelength of incident visible light, the size-dependent Mie resonance can enhance both optical absorption and photon emission. This effect will be subject to future analysis and modelling.

Because both the overpotential losses and solution transport losses are negligible for particles, the detailed-balance principle is sufficient for modelling the efficiency limits of photocatalytic H₂O₂ production. During particle operation, the splitting between electron and hole quasi-Fermi levels at individual particles is very close to the thermodynamic cell potential. Under the principle of detailed balance, the operation points of individual particles are taken as j_ph at which V_ph = 1.083 V (see Sections 3.6 and 3.7). According to eqn (9) and (10), the particle size and the particle band gap determine the minimal light intensity at the reactor bottom, for the particles at which the detailed balance of light absorption and emission gives a net zero photocatalytic rate. We note that for realistic photocatalysts, the Shockley-Read-Hall surface recombination is usually dominant due to the slow catalytic kinetics, but it is not considered here for estimating the efficiency limits of particle-based solar-to-H₂O₂ conversion.

Fig. 7b shows the limiting particle efficiency as a function of band gaps and particle diameter, which is expected from the mathematics. With the same band gap, as particles become smaller, the efficiency limit reduces monotonically. Fig. 7c presents a trade-off between the efficiency and band gaps when the particle diameter is fixed to 1 μm. A maximum efficiency of 27.5% is achievable at a band gap of 1.55 eV, above which the reduced photon absorption limits the efficiency and below which the reduced photovoltage results in insufficient driving force for H₂O₂ photocatalysis. As expected, the iso-efficiency contour plots exhibit a “nose” shape: with particle diameters >100 nm, it's possible to achieve >25% solar-to-H₂O₂ efficiency. When particle diameters are reduced to 50 nm, the maximum efficiency reduces to 24.5% at a different band gap of 1.65 eV.

4.4 Device applications at the energy–water nexus

H₂O₂ produced from renewable energy sources, e.g. solar, wind, and nuclear, can have an immediate impact on energy and environment at once: H₂O₂ can be converted to electricity in a fuel cell, and can be used for water treatment at the same location. PEC and photocatalytic devices enable a distributed approach for chemical and energy production, irrespective of the local infrastructure. If H₂O₂ can be accumulated to molar concentration levels, it can be considered as an energy-storage medium in a flow battery: electrolysis stores electrical energy in the chemical energy of H₂O₂, and a separate fuel cell will convert the stored chemical energy back to electricity. Under an idealized condition of 100% selectivity, one would not need to run a flow system to constantly remove H₂O₂ products but rather let H₂O₂ molecules accumulate.

In the long term, photocatalysts offer the biggest potential for cost effective solar-to-H₂O₂ conversion. A simple filtration can separate photocatalysts from H₂O₂-containing water. At the Shockley–Queisser limit where radiative recombination is the only loss mechanism, model spherical particles of 50 nm to 1 μm diameter can achieve a maximum conversion efficiency of 24.5–27.5% with a single band gap ranging from 1.5 to 1.7 eV. Numerous efficient semiconductor absorbers of this band gap range are readily available. However, challenges of poor charge separation, rapid surface recombination, and instability against photo-corrosion remain to be addressed. These aspects of challenges are common not just for H₂O₂ photocatalysis but also for particle-based water splitting.^58–60

Integrated, photoelectrode-based PEC devices can find their niche applications in portable H₂O₂ production and water treatment, by flowing water across the active surfaces of these flat-panel devices under sunlight. So far these PEC devices hold the record efficiency because photoelectrodes employ semiconductor/liquid junctions or heterojunction interfaces to separate and collect light-excited charge effectively.⁷ Surface coatings over photoelectrodes have shown multi-functionalities of protecting light absorbers against photo-corrosion and facilitating effective charge separation.^5a,61,62 Hybrids of photovoltaics or photoelectrodes in conjunction with parallel-plate electrolyzers can also have immediate impacts: one can directly stack a photovoltaic cell or a photoanode over a thin electrolyte channel with a parallel counter electrode just like the H₂O₂ electrolyzer (see Fig. 1a). Our break-down analysis of electrolysis polarization losses is equally applicable to such hybrid devices.

5. Conclusions

A modelling approach was primarily used to evaluate the efficiency limits and operational constraints of three types of H₂O₂-evolving photoelectrochemical (PEC) devices using the measured and reported performance of H₂O₂-evolving catalyst materials. Both cathodic O₂ reduction and anodic water oxidation should be close-to-100% selective towards H₂O₂ production for H₂O₂ accumulation at molar concentrations, and if so, efficient solar-to-H₂O₂ conversion can be realized taking advantage of membrane-less configurations. The membrane-free design can afford operational conditions where catalyst overpotentials are the major loss for H₂O₂ electrolysis, and where total polarization losses in photoelectrode PEC devices should not limit the maximum photocurrent density of ca. 20 mA cm⁻² that is offered by a current-matching, tandem light absorber under 1-sun illumination. Manageable solution losses (<100 mV at 20 mA cm⁻²) can be achieved both in strong borate buffer (1 M, pH = 9.2) and in a diluted acid (pH = 2) but with thin-layer electrolytes across the cathode and anode (distance <50 μm). For particulate H₂O₂ photocatalysts, both the overpotential losses and solution transport losses are negligible. Therefore, the detailed-balance principle is sufficient for modelling their solar-to-H₂O₂ conversion efficiency as a function of absorber band gaps and particle sizes. A non-zero light intensity threshold was for the first time applied to account for the net accumulation of cathodic and anodic co-evolved H₂O₂, which means that the most efficient particle suspension should not be fully opaque.

The efficiency limits of photoelectrode PEC devices are found to be ca. 20.0% with an upper anode width of 1 mm, a lower anode width of 0.4 mm, and the upper and lower absorber band gap pairs of 1.85 eV/1.25 eV and 1.75 eV/1.15 eV, respectively. A single band gap for model spherical absorbers can achieve a solar-to-H₂O₂ conversion efficiency of 27.5% in their suspension, and particle sizes of 50 nm–1 μm diameter can achieve a maximum conversion efficiency of 24.5–27.6% with a band gap of 1.5–1.7 eV. It is concluded that within the boundary of reported, realistic parameters of light absorbers, electro-catalysts, and electrolyte properties, H₂O₂ (aq.) can be considered as a promising energy carrier for long-term storage of renewable energy and for water remediation. The model will be useful for guiding the development of sunlight-driven H₂O₂-production devices, and for improving and benchmarking the materials and components for efficient H₂O₂ conversion.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The author thanks William Rosenbluth and Brenna Hodges for participating in 1-D model validation, and would like to acknowledge Prof. Jaehong Kim, Prof. Owen Miller, Xi Chen, Jiahui Li, and Qianhong Zhu for very helpful discussions. The author also thanks the start-up support from Yale Energy Sciences Institute.

Notes and references

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Footnote

† This value is calculated by (i) following the optimal combinations for both upper and lower tandem absorbers, namely 19.3% for the upper photoanode/vertical cathode pair and 21.7% for the lower photoanode/vertical cathode pair; and (ii) taking the areal average following the width ratio of 0.4.

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