Thermodynamic assessment of Gd-doped CeO₂ for microwave-assisted thermochemical reduction

Abstract

Microwave-assisted hydrogen production is a promising technology with the capability to decompose H₂O into H₂ economically. The potential of this technology depends on the parameter f_r that measures the fraction of microwave energy directly contributing to the reduction reaction of metal oxides by extracting the lattice oxygen. We quantitatively examine f_r for Gd-doped ceria (CeO₂), a well-known benchmark material, using the Van't Hoff method. Our study reveals that approximately more than 1/2 of the reduction enthalpy is attributed to microwave energy, suggesting that f_r exceeds 0.5. Simultaneously, we introduce a defect equilibria model to identify the contribution of f_r and derive equilibrium constants for isolated defects and associated defects at T = 450–600 °C and P(O₂) = 2 × 10⁻⁴ to 2.1 × 10⁻¹ atm. The results advocate that microwave energy significantly contributes to defect formation under alleviated conditions (lower T and higher P(O₂)) with a shorter timescale compared to conventional thermal reduction. Our study reaffirms the importance of f_r in microwave-assisted reduction and provides a new thermodynamic insight into the interaction between defects and microwave fields in doped ceria.

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1. Introduction

Considerations of the sustainability of the global energy system, beyond the traditional fossil fuel-based system, are intensifying. This drives the development of carbon-free or carbon-neutral energy grids integrated with renewable energy such as wind and solar.^1,2 The proposed grid system, however, inherently experiences temporal and spatial imbalances in energy demand and supply. Various technologies have been developed to address the issues, such as thermal energy storage, batteries, and hydrogen energy.^3–6 Among these, hydrogen energy is of critical importance because of its role not only as an energy carrier but also as a precursor in large-scale industrial processes such as ammonia synthesis, Fischer–Tropsch synthesis, and hydrogen-based iron ore reduction.^7–9

Meeting the demand for carbon-free hydrogen in various sectors requires economically viable technologies for splitting H₂O into H₂. While photochemical, electrochemical, and thermochemical pathways are being explored,^8,10–13 a microwave-assisted route has recently garnered attention, as it can significantly lower the required reduction temperature for thermochemical reactions to below 600 °C, while enabling contactless or electrodeless operation.^14–16 The microwave route consists of a reduction of metal oxides followed by a subsequent water splitting step, similar to a conventional thermochemical two-step reaction. In the reduction step (of metal oxides), microwave-assisted reduction creates oxygen vacancies by releasing lattice oxygen from metal oxides at temperatures below 600 °C (eqn (1)). Subsequently, in the water splitting step (with the microwaves turned off), the oxygen vacancies are re-filled by oxygen originating from dissociating the bonds of H₂O, generating H₂ with the remaining thermal energy (eqn (2)).

MO_x−δ + δH₂O → MO_x + δH₂ (2)

The microwave-assisted reduction has been preliminarily investigated for materials with various crystal structures such as CaTiO₃, ZrO₂, and CeO₂.^14,15,17 Among these, CeO₂ doped with trivalent cations (Gd³⁺, La³⁺, Y³⁺, Yb³⁺, Er³⁺, and Nd³⁺) has been considered desirable for hydrogen production due to the thermodynamic favorability to split water using reduced oxides, as well as the optimization potential provided by the dopants.^14,18,19 Notably, Gd_0.2Ce_0.8O_1.9−δ is one of the benchmark materials, potentially enabling an energy-efficient process with the energy required to produce a unit mass of hydrogen, i.e., a specific energy of 65–80 kW h kg⁻¹ H₂.¹⁵

Eqn (3) shows that the microwave energy (E_MW) is used to induce changes in the enthalpy and entropy of the material during the reduction (ΔH_O and ΔS_O, respectively), after taking into account the fraction of non-absorbed microwave energy (η_MW) and the fraction of energy involved in the reduction relative to the total absorbed energy (f_r).¹⁵ Note that f_r represents the intrinsic interaction between the microwave fields and the material, while η_MW depends on the details of the microwave reactor system such as geometry, wall resistivity, and cable feedthrough. In thermal reduction, all thermal energy input is used to meet the enthalpy of reduction. In contrast, in microwave-assisted reduction, the fraction f_r of microwave energy directly contributes to the reduction process, and the remaining part (1 − f_r) satisfies the heat demand (T·ΔS_O) (Fig. 1). The change in Gibbs free energy ΔG_O is expressed as ΔH_O − T·ΔS_O.

Fig. 1 Sankey diagram of the reduction process comparing thermal reduction and microwave-assisted reduction. Reproduced from ref. 15.

Domínguez-Saldaña et al. explored the interaction between microwaves and materials in-depth,¹⁴ suggesting that microwaves applied to the materials cause a dielectric breakdown, resulting in electron excitation from the valence band to the conduction band. For instance, in cerium oxide systems, reduction occurs as two electrons are transferred from the valence band, mainly filled by oxygen 2p states, to the empty 4f states of the cerium. During this process, part of the electron transport is converted into Joule heating, contributing to the increase of material temperature.²⁰ The microwave-assisted reduction mechanism is known to not only induce the reduction of materials but also further facilitate the formation of charge carriers such as polarons.^21,22 The generation of polarons by microwaves was supported by a significant increase in electronic conductivity in a microwave environment compared to conventional heating.^15,17

Building on the insights gained into the possible mechanism for microwave-assisted reduction, the next important step is to obtain quantitative estimations of the parameter f_r, that is, the direct contribution of microwaves to the reduction process for a practical range of operating temperature and oxygen partial pressure. For instance, Serra et al. predicted that an increase in f_r would lead to a reduction in the energy cost for hydrogen production using microwave technology.¹⁵ In addition, there also exists a need for an in-depth investigation on the specific types of charge carriers induced by microwaves, considering the fact that various carriers such as isolated and associated defects can exist in cerium oxide systems. The defects can influence polaron hopping and thus affect surface charge transport as well as oxygen ion transport, suggesting that the quantitative analysis of the defect species is imperative.^23,24

In the present study, we have estimated ΔH_O and ΔS_O of microwave-assisted reduction using the Van't Hoff method. ΔH_O of Gd_0.2Ce_0.8O_1.9−δ decreases according to our estimates, indicating that part of the reduction enthalpy comes from the direct work of microwaves (f_r). To identify where this microwave-to-reduction direct work contributes, a defect equilibria model of ceria is introduced, deriving equilibrium constants for isolated and associated defects in the temperature range of 450–600 °C and the oxygen partial pressure range of 2 × 10⁻⁴ to 2.1 × 10⁻¹ atm. Considering that conventional thermochemical reduction requires a temperature of 1500 °C to achieve a similar order of magnitude of the equilibrium constants,²⁵ the defect equilibria model suggests that the microwave work is used to form the defect species.

2. Theoretical method

2.1 Van't Hoff method

The thermodynamic properties of a chemical reaction, such as changes in enthalpy and entropy, can be estimated using the Van't Hoff method. Specifically, for the reduction reaction of an oxide where oxygen evolution results in the formation of oxygen non-stoichiometry (δ) within the lattice, the reduction enthalpy and entropy can be estimated from the slope and y-intercept of the ln(P(O₂)) versus 1/T plot at a given δ, as described by eqn (4) (ref. 26–29)where P(O₂) is the oxygen partial pressure and P₀ corresponds to the standard pressure of 1 atm. ΔH_O(δ) and ΔS_O(δ) are the functions of the enthalpy and entropy changes in the reduction reaction at specific δ, respectively. T is the temperature and R is the molar gas constant.

2.2 Defect model

The oxygen evolution reaction in Gd_0.2Ce_0.8O_1.9−δ can be modeled using the defect equilibria of point defects and defect association formation reactions, as described by Bishop et al., using the Kröger–Vink notation in eqn (5)–(8).^25,30–32 For the point defect formation reaction (eqn (5)), O²⁻ anions at the oxygen lattice site, , transform into double positively charged oxygen vacancies, , due to the oxygen evolution. The remaining two electrons are transferred to Ce⁴⁺ cations at the cerium lattice site, , forming single negatively charged point defects, . The equilibrium constant K₁ for the reaction is defined by the concentration ratio of the products to the reactants, as shown in eqn (6).

For the defect association formation reaction, while various defect associations can be potentially formed, among these, the trimer is generally considered one of the most significant associations based on the previous literature.^32–35 The equilibrium equation and the equilibrium constant K₂ can be defined as eqn (7) and (8), respectively.

Each defect equilibrium should satisfy the conservation eqn (9)–(11), which represent anion site conservation, cation site conservation, and electroneutrality conservation, respectively, where is the Gd³⁺ cation substituted into the Ce lattice site that is constant at 0.2 in this work.

The trimer can be expressed as a function of by dividing eqn (8) by eqn (6), as shown in eqn (12). By substituting the conservation eqn (9)–(11) into the equilibrium constant eqn (6) and (8), both K₁ and K₂ can also be expressed as a function of , as shown in eqn (13) and (14), respectively.

By inputting and numerically, K₁ and K₂ can be determined for the unique P(O₂). This allows for the calculation of δ in Gd_0.2Ce_0.8O_1.9−δ, considering both the intrinsic point defects and the defect association , as defined in eqn (15). By minimizing the difference between the simulated δ and the experimentally derived δ, the values of K₁ and K₂ can be determined through numerical iteration.^25,30,33

3. Experimental method

3.1 Microwave reaction and analysis system

Gd_0.2Ce_0.8O_1.9−δ was purchased as a commercial product from Sigma-Aldrich and was calcined at 1300 °C for 1 hour to obtain the as-prepared sample. Thermodynamic assessment of the as-prepared Gd_0.2Ce_0.8O_1.9−δ under a microwave environment was conducted using a custom-designed microwave cavity resonator setup (Fig. 2). Among the various types of microwave resonators such as rectangular, cylindrical, and spherical, we selected a cylindrical type for ease of manufacture. This resonator theoretically supports infinite resonance modes, with the corresponding electromagnetic field distribution inside the resonator dependent on the specific modes. We utilized the TE₁₁₁ mode for microwave irradiation of the sample, as it creates a uniform and strong electric field distribution at the center of the cylindrical cavity.^15,36 The cavity dimensions were optimized using Ansys HFSS (High-Frequency Structure Simulator) to ensure the resonance of the TE₁₁₁ mode at approximately 2.45 GHz, which lies within the ISM (Industry-Science-Medical) band, thereby ensuring cost-effectiveness.

Fig. 2 Schematic diagram of the overall microwave irradiation system.

The cylindrical cavity had an inner radius of 50 mm and an inner height of 85 mm, with two monitoring holes (3.5 mm radius) positioned at the center of the cavity side to measure the sample temperature. A quartz tube with an inner diameter of 10 mm and an outer diameter of 12.6 mm featured a quartz protrusion on the inner wall to secure a quartz frit of 10 mm diameter. The Gd_0.2Ce_0.8O_1.9−δ sample was placed on top of the quartz frit at the center of the cavity. An easily tunable coaxial probe feed was coupled to the cavity.

Two pyrometers (3MH1-CF-Video-CB3C & 3ML-CF3-CB3, Optris, Germany) measure wavelengths of 2.3 μm, which penetrate quartz, allowing direct measurement of the sample surface temperature. The emissivity of the sample was corrected using a programmable digital furnace and a K-type thermocouple. The microwave irradiation sources consist of a signal generator (83640B, Hewlett Packard, USA) and a solid-state amplifier (SSPA, KRF, Republic of Korea) with a bandwidth of 2.4 GHz ± 100 MHz and a typical gain of 57 dB. A circulator (UIYCC4546A2400T2600NF, UIY, China) protected the amplifier by redirecting power reflected from the cavity to a terminal load. Forward and reflected power measurements were conducted using a dual-directional coupler (DC0031, SUNGSAN E&C Co., Ltd, Republic of Korea), as it is important to calculate the accurate microwave power delivered to the cavity and sample. A spectrum analyzer (E4404B, Agilent Technologies, USA) was installed behind two attenuators (UIYCA50AN6A30, UIY, China) to measure the microwave power.

The gas distribution into the reactor system was precisely controlled using mass flow controllers (VIC-D200 Series, MKP Co., Ltd, Republic of Korea). Ultra-high purity grade N₂ and high purity grade N₂ + O₂ mixture (79% N₂ and 21% O₂) were supplied through individual mass flow controllers, with proper gas correction factors considered to achieve the desired flow rates. This setup allowed for P(O₂) ranging from 2 × 10⁻⁴ to 2.1 × 10⁻¹ atm. The variation of P(O₂) during microwave reaction was analyzed in real-time using a quadrupole mass spectrometer (HPR-20, Hiden Analytical, UK) equipped with a secondary electron multiplier detector, chosen for its high sensitivity at low P(O₂) and fast response time.

3.2 Redox cycle protocol

To estimate δ of Gd_0.2Ce_0.8O_1.9−δ at the given temperature and P(O₂) under a microwave environment, 2 g of sample was loaded into the cavity system, and then, redox cycles consisting of reduction and re-oxidation steps were conducted. For the reduction step, Gd_0.2Ce_0.8O_1.9−δ lost the lattice oxygen as the temperature was increased to the target level through microwave irradiation. For the re-oxidation step, subsequently, when the target temperature was achieved and P(O₂) returned to the initial level, the microwave was turned off, resulting in the re-oxidation of Gd_0.2Ce_0.8O_1.9−δ. The oxygen gain and loss were converted into δ at the given temperature and P(O₂) based on the molar mass of Gd_0.2Ce_0.8O_1.9−δ.

Fig. 3a shows the simulated normalized electric field distribution of the TE₁₁₁ mode with the sample loaded, along with the average temperature recorded by the IR pyrometer under actual experimental conditions. The calculated electric field closely followed the field distribution trend reported in the literature.¹⁵ It is worth noting that the increased brightness of the IR image with increasing temperature in the radial direction strongly suggests that heating initiates from the bulk of the sample.^37,38

Fig. 3 (a) A quarter cross-section diagram of the cylindrical cavity system and captured visible images of the sample during microwave irradiation under a N₂ environment. The red circle in the image indicates the center of the sample. (b) Procedure of microwave irradiation for the redox reaction.

Fig. 3b shows the microwave control protocol during the reduction step to precisely achieve the target temperature, considering the optimization of microwave power delivered to the cavity. The relationship between power transfer and tuning in the microwave frequency range is based on transmission line theory,³⁹ and a detailed description of power transfer is given in the ESI.† Initially, the single frequency (f_d) of the signal generator was set equal to the resonant frequency (f₀) of the sample-loaded cavity. By applying microwave power to the cavity, the resonant frequency decreased as the temperature of the sample increased . Since resulted in a large amount of reflected power that was not absorbed by the sample, f_d was controlled iteratively to minimize the reflected power . The iteration was finished when the sample reached the target temperature, at which the power delivered to the sample-loaded cavity was calculated by subtracting the reflected power from the forward power.

3.3 Sample characterization method

3.3.1 SEM. The surface microstructure of the as-synthesized and redox-cycled samples was characterized using field emission scanning electron microscopy (FE-SEM, SU6600, Hitachi, Ltd, Japan) operating at an accelerated voltage of 5 kV and a beam current of 32 μA. The working distance was consistently maintained between 8.1 and 8.3 mm. The samples were uniformly dispersed on carbon tape and then coated with platinum for 60 seconds at a discharging current of 30 mA using vacuum coating systems (108 Auto Sputter Coater, Cressington Scientific Instruments, UK) to minimize charging effects.

3.3.2 X-ray diffraction. The crystal structure of the materials was investigated using an X-ray diffraction (XRD) system (D/MAX-2500, Rigaku, USA) equipped with a copper source. Data acquisition was conducted over a 2θ range from 20° to 80°, with a step size of 0.02°, at ambient temperature. The operational parameters for the source were maintained at 40 kV and 200 mA. To accurately identify the peak position of Gd_0.2Ce_0.8O_1.9−δ, a reference material, Al₂O₃ was added during the measurement. Each representative peak was characterized using JADE software (Materials Data, Inc. (MDI)), leading to the identification of peaks for Gd_0.2Ce_0.8O_1.9−δ (Fm3̄m) and Al₂O₃ (R3̄c).

3.3.3 Raman spectroscopy. The bonding characteristics of cerium and oxygen were investigated through a confocal Raman spectrometer (HEDA, WEVE, Republic of Korea) at ambient temperature. The excitation laser for Raman scattering had a wavelength of 785 nm with an objective lens of 20× magnification. The beam diameter and the power of the laser were approximately 40 μm and 0.5 mW, respectively. Raman shift (cm⁻¹) was accurately calibrated using cyclohexane.

4. Results and discussion

4.1 Oxygen non-stoichiometry estimation from redox cycles

The re-oxidation behavior of Gd_0.2Ce_0.8O_1.9−δ was investigated under a given P(O₂) ranging from 6.7 × 10⁻³ to 2.1 × 10⁻¹ atm to confirm whether the oxide is sufficiently re-oxidized and whether the reaction is repeatable over multiple cycles. As shown in Fig. 4a, the reduction reaction was consistently conducted at T = 480 °C and P(O₂) = 2 × 10⁻⁴ with consistent microwave power, followed by re-oxidation reactions in the given P(O₂) ranges. Fig. 4b shows the amount of oxygen released during the reduction reaction of each cycle, which was observed to be independent of P(O₂) of the preceding re-oxidation reaction. This indicates that the P(O₂) range is sufficient to fully oxidize Gd_0.2Ce_0.8O_1.9−δ while confirming the repeatability of the reactions over multiple cycles, as reported in previous work.¹⁷

Fig. 4 (a) Reaction scheme showing microwave-assisted reduction at P(O₂) = 2 × 10⁻⁴ atm and T = 480 °C and re-oxidation at various P(O₂) ranging 6.7 × 10⁻³ to 2.1 × 10⁻¹ atm. Each reaction step is highlighted in red and blue, respectively. (b) Oxygen evolution during the reduction step at P(O₂) = 2 × 10⁻⁴ atm and T = 480 °C, which was subsequently conducted after the re-oxidation step with various P(O₂).

Fig. 5a shows the representative profile of oxygen evolution and consumption during the reduction and re-oxidation steps, respectively, at T = 520.3 °C and P(O₂) = 6.7 × 10⁻³ atm. The redox cycle was completed within 15 minutes, demonstrating notably rapid kinetics considering that the timescale of a conventional thermochemical cycle is on the order of 1–2 hours. It could possibly be attributed to the defect species, which will be discussed in a later section, influencing polaron hopping in the microwave environment. The non-monotonic profile of oxygen during the reduction step is due to the control protocol of the frequency f_d and the power of the microwave system. Considering the non-monotonic characteristics, we consistently used the amount of oxygen consumption rather than that of oxygen evolution for δ estimation. Meanwhile, the temperature of the sample depends linearly on the applied microwave power (Fig. 5b), and the linear dependency is consistent across all given P(O₂) ranges. It suggests that the electromagnetic field of the microwave is uniformly distributed within the sample and that the formation of undesirable hot spots is minimized, indicating the occurrence of homogeneous chemical reactions.⁴⁰

Fig. 5 (a) Microwave-assisted reduction at P(O₂) = 6.7 × 10⁻³ atm and T = 520.3 °C with the microwave power on, followed by re-oxidation at the same P(O₂) with the microwave power off. (b) Temperature of Gd_0.2Ce_0.8O_1.9−δ as a function of microwave power. (c) Oxygen non-stoichiometry (δ) in Gd_0.2Ce_0.8O_1.9−δ depending on T and P(O₂).

Fig. 5c shows δ as a function of T and P(O₂) under microwave irradiation, wherein the amount of oxygen consumption was converted to δ on the corresponding y-axis scale. The increase in δ was achieved not only by increasing T but also by decreasing P(O₂), consistent with previous literature.¹⁷δ formation at relatively high P(O₂) from 2 × 10⁻⁴ atm to 2.1 × 10⁻¹ suggests that the large energy penalties required for oxygen pumping or inert gas sweeping could be alleviated by microwave technology. To investigate the relation between δ, T, and P(O₂) in-depth, the experimental data points were fitted using linear regression, and the error bars were defined by multiplying the residual standard deviation of the data points from the regression line by 1.645, representing a 90% confidence interval.

In addition, to accurately convert the oxygen consumption to δ over multiple cycles, full re-oxidation should be ensured, which was confirmed through the investigation into the crystal structure and bonding characteristics (Fig. 6). Fig. 6a shows the X-ray diffraction patterns (XRD) of Gd_0.2Ce_0.8O_1.9−δ after being re-oxidized at each P(O₂) compared to the as-prepared state. The shifts of the XRD pattern were accurately calibrated using the reference material Al₂O₃. The analysis revealed that the crystal structure of the samples remained stable as well as that no shifts were observed, indicating that δ returned to the as-prepared state at all P(O₂) values (Fig. 6b and c). This was also consistently observed in Raman spectra analysis. As shown in Fig. 6d, the main peak F_2g, corresponding to the triply degenerate Raman active mode of cerium-based oxide, was detected near 463.7 cm⁻¹.⁴¹ It is a sensitive indicator of change in the oxide bond length and formation of oxygen vacancies.^30,42,43 The intensity decreased after the reduction but returned to the as-prepared state after the re-oxidation. The shoulder at ∼480 cm⁻¹, as well as the peaks near 250 cm⁻¹ and 556 cm⁻¹, corresponds to Gd doping effects not detected in pure CeO₂.^30,44 As the bonding characteristics are also affected by vacancy formation,⁴⁴ changes in the intensities were observed in the reduced Gd_0.2Ce_0.8O_1.9−δ (Fig. 6e). All the samples showed similar surface morphology, with an agglomerated grain diameter of around 500 nm compared to 300 nm in the as-prepared sample (Fig. 6f–k).

Fig. 6 Spectroscopic characterization using XRD and Raman spectroscopy. (a) Crystal structure of as-prepared Gd_0.2Ce_0.8O_1.9−δ and samples oxidized subsequently after the reduction at given P(O₂). The star marks (*) indicate the peaks of the reference material aluminum oxide, which was added during the XRD measurement for accurate peak shift calibration. (b) Representative XRD peak of the reference aluminum oxide and (c) XRD peak of Gd_0.2Ce_0.8O_1.9−δ. Raman spectra for all samples from the experiment conducted in Fig. 5c (d) and (e) comparison of selected sample sets. The reduced sample was obtained by quenching in UHP-grade N₂ after the reduction reaction at P(O₂) = 2 × 10⁻⁴ atm and T = 485.6 °C. (f–k) SEM images of as-prepared, 2 × 10⁻⁴ atm, 6.7 × 10⁻³ atm, 1.12 × 10⁻² atm, 1.02 × 10⁻¹ atm, and 2.1 × 10⁻¹ atm samples, respectively.

4.2 Thermodynamic properties under a microwave environment

The relationship between δ, T, and P(O₂) at equilibrium during the microwave-assisted reduction reaction was presented using the Arrhenius relation (Fig. 7). The data points were fitted using linear regression, and the statistical results, such as R-square, slope, y-intercept, and standard error, are summarized in Table 1. It is important to note that, due to the limited number of data points and the observed systematic deviations from linear behavior, the assumption of linearity should be approached with caution. Specifically, the datasets generally exhibited a decreasing slope with increasing 1/T in Fig. 7. While various models can be used to approach cases exhibiting experimental non-linearity,^45,46 we have opted to use a simplified linear Van't Hoff model in the context of this nascent stage of research. The decision was based on the straightforward nature of the approach as well as the fact that the coefficients of determination (R²) were generally above 0.89 in the range of 0.0045 < δ < 0.0145, suggesting some degree of linearity of the Arrhenius relation for the given δ, though not without significant deviations. Future studies may explore the use of non-linear models, which could offer further insights.^45,46

Fig. 7 Determination of thermodynamic properties using the Van't Hoff method on the data extracted from the linear regression of the experimental results.

Table 1 Results of linear regression for the relationship between δ, T, and P(O₂) in the microwave-assisted reduction reaction. SE(m) and SE(y₀) represent the standard error of the slope and y-intercept, respectively. Those errors were calculated using SE(m) = (∑(y_i − ŷ)²/[(n − 2)∑(x_i − x̄)²])^0.5 and SE(y₀) = SE(m)·(∑x_i2/n)^0.5, where y_i is the observed value, ŷ is the predicted value, x_i is the variable value, x̄ is the mean of x_i, and n is the number of data points

δ	Slope (m)	SE(m)	y-intercept (y0)	SE(y0)	R ^2
0.0045	−22.01	5.60	24.44	7.53	0.89
0.0065	−19.24	4.07	19.19	5.15	0.92
0.0085	−17.56	3.22	15.84	3.86	0.94
0.0105	−21.32	4.09	19.41	4.88	0.97
0.0125	−19.56	3.67	16.27	4.19	0.97
0.0145	−18.35	3.37	14.00	3.68	0.97

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The Van't Hoff method was applied to derive the fundamental thermodynamic properties, ΔH_O and ΔS_O, as discussed earlier in eqn (4). From the slope and the y-intercept of the Arrhenius relation fitting line, ΔH_O and ΔS_O were estimated, respectively, with their corresponding standard errors. ΔH_O ranged from 73.0 ± 13.4 kJ mol⁻¹ to 91.5 ± 23.3, and ΔS_O ranged from 58.2 ± 15.3 J (mol⁻¹ K⁻¹) to 101.6 ± 31.3 J (mol⁻¹ K⁻¹) in microwave-assisted reduction. Although the wide error ranges make it difficult to rigorously discuss the trends of the thermodynamic parameters depending on δ in the microwave environment, it is noteworthy that ΔH_O and ΔS_O are significantly reduced compared to the conventional thermal reduction of Gd_0.2Ce_0.8O_1.9−δ, being approximately 1/4 and 1/2, respectively, of those of the thermal reduction, which are about 385 kJ mol⁻¹ and 200 J mol⁻¹ K⁻¹, respectively (Table 2).^47,48 This observation is consistent with other endothermic chemical reactions, where both ΔH_O and ΔS_O are reduced when using microwave energy.⁴⁹ Further analysis was performed by calculating the steepest and shallowest slopes between the consecutive data points. This approach captures variability in the data that may not be fully addressed by linear regression alone. For example, with a small number of data points, the steepest and shallowest slopes can be qualitatively interpreted as the most conservative and lenient scenarios, respectively. Interestingly, even in the most conservative scenario, ΔH_O was found to be approximately 1/2 of that of conventional thermal reduction (Table S1†). This propounds that while thermal reduction requires significant T and ΔS_O to compensate for endothermic ΔH_O satisfying ΔG_O < 0, the microwave-assisted reduction process can contribute more than 3/4 or even conservatively 1/2 of ΔH_O in the form of electrical work, approximately, suggesting that the microwave-to-reduction effectiveness f_r ≥ 0.75 or conservatively > 0.5. This suggests that microwave technology, by compensating for a portion of ΔH_O, could offer a variety of research topics.

Table 2 Comparison of the reduction enthalpy and entropy values of Gd_0.2Ce_0.8O_1.9−δ during thermal reduction and microwave-assisted reduction for the corresponding δ. The data for thermal reduction are sourced from the literature,^47,48 while the data for microwave-assisted reduction are estimated using the Van't Hoff method

δ	Thermal reduction		Microwave-assisted reduction
	ΔHO (kJ mol^−1)	ΔSO (J mol^−1 K^−1)	ΔHO (kJ mol^−1)	ΔSO (J mol^−1 K^−1)
0.0045	385.0	206.0	91.5 ± 23.3	101.6 ± 31.3
0.0065	385.0	200.9	80.0 ± 16.9	79.8 ± 21.4
0.0085	385.0	196.7	73.0 ± 13.4	65.8 ± 16.0
0.0105	385.0	192.8	88.6 ± 17.0	80.7 ± 20.3
0.0125	385.0	189.0	81.3 ± 15.2	67.6 ± 17.4
0.0145	385.0	185.3	76.3 ± 14.0	58.2 ± 15.3

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Meanwhile, the potential of microwave technology should be assessed cautiously alongside our findings. For instance, establishing a comprehensive capital expenditure (CAPEX) analysis—including the cost of redox materials, which was beyond the scope of the previous study—will be essential to define a realistic benchmark f_r.¹⁵ Such an analysis could provide a more accurate basis for evaluating the economic feasibility of microwave-assisted hydrogen production. Furthermore, as noted by recent analyses, the small oxygen storage per cycle (δ) of ceria-based oxides limits energy production per redox cycle, posing scalability challenges due to low power density.⁵⁰ These material challenges indicate the need to explore alternative materials, such as other doped ceria, perovskites, and ferrites, which may offer improved power density as well as reaction efficiency. This potential for material diversification opens up broader research opportunities for microwave technology.

4.3 Defect equilibria under a microwave environment

As described in eqn (5)–(8), the defect equilibria model was applied to investigate in-depth the defect formation mechanism influenced by the electric work of microwaves. According to the well-established models, ceria-based materials are not adequately described by a single formulation of defects, which is widely known as oxygen vacancy formation accompanied by oxygen evolution; instead, there is evidence that a comprehensive model better describes the experimental data, wherein the model accounts for intrinsic point defects being dominant at low oxygen vacancy concentration and defect association being dominant at high oxygen vacancy concentration.^31–33,35

Fig. 8a shows the dependence of log(δ) on log(P(O₂)) at the corresponding T, extracted from the linear regression fitting line of the experimental data. The errors of the extracted data points were defined by the error propagation of the confidence intervals of the linear regression lines shown in Fig. 5c. The dependence of log(δ) on log(P(O₂)) was numerically fitted using the equilibria model for T = 450–650 °C, which was consolidated through conservation equations as described in eqn (12)–(15). The solid lines in Fig. 8a show the numerical results at the given T. Based on previous studies, d log(δ)/d log(P(O₂)) are −1/4 and −1/2 for isolated defects and defect associations, respectively.^30,32,33 The slopes of our results consistently ranged between −1/4 and −1/3, suggesting not only the presence of point defects but also the formation of defect associations, with the coefficients of determination ranging from 0.85 ≤ R² ≤ 0.91.

Fig. 8 (a) Dependence of log(δ) on the log(P(O₂)). Points represent interpolated data extracted from linear regression of the experimental data. Lines represent the theoretical values from the defect model. (b) Total log(δ) calculated from the defect model, along with contributions from isolated () and associated defects (), represented by dotted (•••) and dashed-single dotted (–•–) lines, respectively. (c) Equilibrium constant K_i for the isolated defect formation (i = 1) and the associated defect formation reactions (i = 2) at the given reduction temperature. (d) Dependence of log(δ) on log(P(O₂)) for conventional thermal reduction of CeO₂ along with the corresponding fitting lines.²⁵

The total δ obtained from the defect equilibria was deconvoluted into contributions from intrinsic point defects and defect association (Fig. 8b). As P(O₂) decreased, the contribution of defect association to the total δ gradually increased, while an increase in T simultaneously enhanced the contributions of both defects. Specifically, the natural logarithms of equilibrium constants ln(K₁) and ln(K₂) for each defect increased from −16.3 to −14.2 and from −10.8 to −9.6, respectively (Fig. 8c). Meanwhile, when the sourced data from the conventional thermal reduction of CeO₂ were applied to the same defect equilibria model in this work, ln(K₁) and ln(K₂) at 1500 °C were found to be −14 and −10.3, respectively, following the fitting lines of the two defect contributions shown in Fig. 8d.²⁵ Comparing the equilibrium constants of microwave-assisted reduction with those of conventional thermal reduction, it is evident that similar degrees of equilibrium constants were achieved at much lower operating temperatures in microwave-assisted reduction. It is attributed to the electrical work of the microwave in the reduction reaction, consistent with the mitigated ΔH_O in Van't Hoff analysis. The formation of the defect species can possibly influence polaron hopping and reaction kinetics as well,^23,24 considering that the redox cycle was completed within 15 minutes in the microwave environment, as shown earlier in Fig. 5a. For future work, investigating the interactions between microwaves and defect species is expected to provide important insights into the relevant field.

5. Conclusions

In summary, we investigated the enthalpy and entropy changes of Gd_0.2Ce_0.8O_1.9−δ in microwave-assisted reduction by applying the Van't Hoff method, with a particular focus on the parameter f_r. Microwaves can facilitate the release of oxygen at lower temperatures (450–600 °C) and higher oxygen partial pressures (2 × 10⁻⁴ to 2.1 × 10⁻¹ atm) compared to conventional thermal reduction. ΔH_O lowered to 1/2 or more, with the remaining part directly satisfied by microwave energy. This suggests that f_r could conservatively be greater than 0.5, thereby providing a rationale for further investigation into the potential of microwave technology.

To elucidate the contribution of f_r to defect formation, we applied the defect equilibria model for both isolated and associated defects. The slopes d log(δ)/d log(P(O₂)) ranged between −1/4 and −1/3, indicating the presence of both point defects and defect associations. As P(O₂) decreased, defect associations contributed more significantly to the total δ, while higher T increased the contributions of both defects. The equilibrium constants for the defect species formation were comparable to those for the thermal reduction reaction at 1500 °C. It was worth noting that microwave energy significantly contributed to the formation of abundant defects within a short time, even at much lower T and higher P(O₂).

From a fundamental point of view, our thermodynamic interpretation of microwave-assisted reduction is expected to reaffirm the importance of f_r and provide useful insights for future studies on the interaction between microwaves and defect species. Furthermore, from a practical point of view, the possibility of the reduction at high oxygen partial pressure and the short time scale of the reaction are expected to be attractive process conditions of microwave technology.

Data availability

The data supporting this article have been included as part of the ESI.†

Author contributions

D. Lee: conceptualization, data curation, formal analysis, investigation, methodology, validation, visualization, software, writing – original draft, and writing – review & editing. J. Yoo: conceptualization, data curation, formal analysis, investigation, methodology, validation, visualization, software, writing – original draft, and writing – review & editing. G. S. Yun: conceptualization, funding acquisition, investigation, methodology, project administration, resources, supervision, and writing – review & editing. H. Jin: conceptualization, funding acquisition, investigation, methodology, project administration, resources, supervision, and writing – review & editing.

Conflicts of interest

There are no conflicts of interest to declare.

Acknowledgements

This research was supported by the Circle Foundation of Korea (2023 TCF Innovative Science Project-02) and the National Research Foundation of Korea (BK21 PLUS program; grant No. RS-2024-00345022). This work was also supported by the Basic Science Research Institute (BSRI) (grant No. 2021R1A6A1A10042944).

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Footnotes

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4ta05804f

‡ D. Lee and J. Yoo contributed equally to this work.

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