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Numerical modeling of nickel-infiltrated gadolinium-doped ceria electrodes reconstructed with focused ion beam tomography
Accepted Manuscript Title: Numerical modeling of nickel-infiltrated gadolinium-doped ceria electrodes reconstructed with focused ion beam tomography Author: Masashi Kishimoto Marina Lomberg Enrique Ruiz-Trejo Nigel P Brandon PII: DOI: Reference:
S0013-4686(15)30954-3 http://dx.doi.org/doi:10.1016/j.electacta.2015.12.044 EA 26193
To appear in:
Electrochimica Acta
Received date: Revised date: Accepted date:
24-4-2015 30-11-2015 4-12-2015
Please cite this article as: Masashi Kishimoto, Marina Lomberg, Enrique Ruiz-Trejo, Nigel P Brandon, Numerical modeling of nickel-infiltrated gadolinium-doped ceria electrodes reconstructed with focused ion beam tomography, Electrochimica Acta http://dx.doi.org/10.1016/j.electacta.2015.12.044 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Numerical modeling of nickel-infiltrated gadolinium-doped ceria electrodes reconstructed with focused ion beam tomography
1,2
1.
Masashi Kishimoto*, 1Marina Lomberg, 1Enrique Ruiz-Trejo, 1Nigel P Brandon Department of Earth Science and Engineering, Imperial College London, London SW7 2AZ, United Kingdom
2.
Department of Aeronautics and Astronautics, Kyoto University, Kyoto 615-8026, Japan
E-mail: [email protected] Tel.: +81-(0)753833652
Graphical abstract FIB Imaging & Reconstruction
Sensitivity analysis Modeling
Ni
Conduction Diffusion Electrochemical reacti reaction
GDC Pore
8µ
m
9.6
9.3
5µ
m
5.63 µm
2 µm
Highlights
A 1D numerical model of Ni-GDC electrodes is developed. Microstructure of a nickel-infiltrated GDC electrode is obtained using FIB-SEM. Overpotential characteristics are successfully reproduced by the developed model. Sensitivity analysis reveals microstructure-performance relationships. The developed model offers guidelines to optimize electrode microstructures.
Abstract A one-dimensional numerical model of a nickel-infiltrated gadolinium-doped ceria (Ni-GDC) electrode has been developed to investigate the effects of electrode microstructure on performance. Electrode microstructural information was obtained with focused ion beam tomography and microstructural parameters were quantified, such as tortuosity factor, surface area and particle/pore sizes. These have been used to estimate the effective transport coefficients and the electrochemical reaction rate in the electrodes. GDC was considered as a mixed ionic and electronic conductor and hence the electrochemical reaction is assumed to occur on the GDC-pore contact surface, i.e. doublephase boundaries (DPBs). Sensitivity analysis was conducted to investigate the effect of electrode microstructure on both transport properties and electrochemical activity, including the effect of DPB density, GDC tortuosity factor and pore size. The developed model offers a basis to understand the microstructure-performance relationships and to further optimize the electrode microstructures.
Key words: solid oxide fuel cells; infiltration; microstructure; FIB-SEM; modeling.
Nomenclature pore size (m) binary molecular diffusivity (m2 s-1) Knudsen diffusivity (m2 s-1) mean diffusivity (m2 s-1) Faraday constant (C mol-1) Gibbs free energy (J mol-1) ,
exchange current per unit DPB area (A m-2) charge-transfer current (A m-3) permeability (m2)
anode thickness (m) molecular weight (kg mol-1) molar flux (mol m-2 s-1) partial pressure (Pa) total pressure (Pa) gas constant (J mol-1 K-1) surface area (m2) temperature (K) volume (m3) volume fraction (-) molar fraction (-) stoichiometric coefficient (-)
Greek symbols Kronecker delta overpotential (V) viscosity (Pa s) chemical potential (J mol-1) electrochemical potential (J mol-1) conductivity (m2 s-1) diffusion volume (m3) tortuosity factor
Subscripts act
activation
CC
current collector
con
concentration
ct
charge-transfer
DPB
double-phase boundary
I
anode-electrolyte interface
t
total
Superscripts bluk
value for bulk material
eff
effective value for porous material
v
volumetric
1. Introduction Doped ceria has been receiving significant attention as an alternative material for solid oxide fuel cells (SOFCs) due to its higher oxide ion conductivity than conventional zirconia-based materials at lower temperatures, making it possible to reduce the operating temperature of SOFCs. Lowering the operating temperature enables reduction of the total cost of SOFC systems by allowing use of inexpensive materials and mitigating thermally induced degradation. One of the important features of doped ceria is its mixed conductivity under the reducing atmospheres [1–4], allowing both electrons and oxide ions to migrate through the material. This broadens the electrochemical reaction site from the triple-phase boundary (TPB) to the ceria-pore double-phase boundary (DPB) [5–8]. It is widely accepted that electrode microstructure plays an important role in determining the performance and durability of SOFCs. Electrodes are required to contain as many reaction sites as possible to promote the electrochemical reaction; at the same time, effective mass and charge transport pathways need to be established to and from the reaction sites. Therefore, the microstructureperformance relationships need to be understood to develop high performance electrodes. Threedimensional (3D) imaging techniques, such as focused ion beam scanning electron microscopy (FIBSEM) [9–15] and X-ray nano CT [16–21], are finding increasing application in the study of complex electrode microstructures. Through the development and application of quantification methodologies and numerical simulation models, new insights have been generated, which offer the prospect of optimizing electrode microstructure to further improve performance and durability. We recently succeeded in imaging the microstructure of nickel-infiltrated gadolinium-doped ceria (GDC) electrodes using FIB-SEM [12], where the infiltrated nickel particles (~100 nm) were precisely imaged. It was revealed that the infiltrated electrodes had one order of magnitude larger TPB density compared with conventional electrodes made by powder mixing and sintering process. To further extend this approach, reliable electrode simulation models are required to investigate the effect of microstructure on the performance. Kishimoto et al. developed 1D and 3D numerical
models based on the finite volume method (FVM) coupled with the sub-grid-scale (SGS) model to analyze the overpotential characteristics and the distribution of chemical species within Ni-YSZ electrodes [11, 22–25]. Shikazono et al. developed 3D models based on the Lattice Boltzmann Method (LBM) and analyzed Ni-YSZ anodes and LSCF cathodes [26–28]. Carraro, Joos and Häffelin et al. developed a 3D model based on the finite element method (FEM) for a MIEC cathode and conducted sensitivity analysis to elucidate the effect of various operating conditions on the electrode performance [29–32]. Shearing and Cai et al. developed a 3D model based on the volume of fluid (VOF) method and discussed the effect of image voxel size and reconstructed volume size on the simulation results [33–35]. Although electrode structures in their model were synthesized by the Monte Carlo method, their findings are particularly important to determine the required volume size for the reconstructed structure to be representative of the whole electrode microstructures. Results from these numerical analyses offer valuable insights to understand the effects of electrode microstructure on electrode performance, which are otherwise very difficult to observe in experiments. For ceria-based materials, Chueh et al. conducted a two-dimensional numerical simulation on a samarium-doped ceria (SDC) electrode to analyze the reaction region in the material, and found that the rate-determining process was electron conduction from the SDC surface to the current collector [36, 37]. They also found that the TPB length is not as influential on electrode performance as in the case of conventional zirconia-based electrodes because the entire SDC-pore contact surface can potentially act as a reaction zone. However, the simulation domain employed in their work was limited to the Ni-SDC-pore contact region; hence it was difficult to assess the effect of microstructure, such as phase volume fraction, DPB density and particle/pore sizes, on performance. In this study, the 3D microstructure of infiltrated Ni-GDC electrodes is obtained using FIBSEM, and microstructural parameters that characterize the porous structures are quantitatively evaluated, such as DPB density, particle/pore sizes and surface area. A 1D numerical model is then developed to analyze the distribution of the charge carriers and gas species within the electrodes as
well as the electrochemical reaction. Sensitivity analysis using the developed model is conducted to investigate the effect of several microstructural parameters on electrode performance, and insights for microstructural optimization are generated.
2. Experimental In this study, the electrochemical performance of an infiltrated Ni-GDC electrode was characterized using an electrolyte-supported button cell, whose performance was used to validate the proposed simulation model. First, a GDC scaffold (Ce0.9Gd0.1O1.95) was fabricated on an 8YSZ electrolyte disk (Fuel Cell Materials) by screen printing an ink and sintering at 1350 °C for 2 h. Second, the cathode was screen printed on the other side of the electrolyte using an LSCF-GDC ink ((La0.60Sr0.40)0.995(Co0.20Fe0.80)O3-x:Ce0.9Gd0.1O1.95 = 50:50 wt.%, Fuel Cell Materials) and sintering at 1100 °C for 2 h. Finally, nickel oxide nanoparticles were infiltrated into the scaffold by introducing Ni(NO3)2 solution from the top of the scaffold followed by decomposition in air at 500 °C for 30 min. This infiltration process was repeated ten times, followed by a reduction process under a hydrogennitrogen gas mixture (760 °C, H2:N2 = 50:50, 1 h). More details can be found elsewhere [38]. Electrochemical characterization of the electrodes was performed under a three-electrode configuration. Measurement of current-voltage characteristics and electrochemical impedance spectra was performed using an Autolab PGSTAT302 (Metrohm) over a range of temperatures (600-700 °C) under
a
hydrogen-nitrogen
gas
mixture humidified with water
at
room
temperature
(H2:H2O:N2 = 48.5:3.0:48.5). The efficacy of our three-electrode measurement was demonstrated by Lomberg et al. [38]
3. 3D Imaging and Reconstruction
In our previous report, we successfully imaged nickel-infiltrated GDC electrodes with 5 nm resolution and then reconstructed and quantified the electrode microstructures [12]. The process for the SEM sample preparation and 3D imaging is briefly summarized below. After the electrochemical measurements, the anode samples were impregnated with epoxy resin (Specifix20, Struers) under vacuum conditions in order to minimize the undesirable damage on the specimen and to avoid accumulation of gallium ions in the pores during FIB milling, as well as to help distinguish the pores from the solid phases in SEM imaging. The cured samples were cut and mechanically polished, mounted onto a specimen stub with silver paste, and sputtered with gold to ensure sufficient electron conductivity. The 3D microstructure of the anodes was imaged with an Auriga (Zeiss) FIB-SEM system equipped with a backscattered electron detector, which enables to clearly distinguish the two solid phases in the SEM images. Regions were selected for microstructural analysis and went through the following image processing using Avizo (FEI): (i) alignment of the stack images using the least square method, (ii) noise reduction using the edge preserving smoothing filter (a combination of the Canny filter and the Gaussian smoothing filter) and (iii) segmentation using a 2D histogram segmentation algorithm based on the water shed algorithm, followed by manual correction. Then, the 3D porous microstructure was virtually reconstructed, from which various microstructural parameters, such as phase volume fractions, surface area, particle/pore sizes, phase connectivity and tortuosity factor, were quantified. Volume fractions were evaluated by counting the number of voxels corresponding to each phase, followed by division by the total number of voxels in the reconstructed volume [11]. Surface area was measured by the marching cube algorithm and then used to obtain the GDC-Pore double-phase boundary
as follows:
where
is the total surface area of phase and
(1)
is the entire reconstructed volume. Particle/pore
sizes were measured using the 3D mean intercept method, the 2D version of which was originally proposed by Simwonis et al. [39]. This 3D mean intercept method is also a simplified approach of the
ray-shooting method [40]. Tortuosity factors were measured by the diffusion-based method [23]. Connectivity was judged based on the 6-neighboring voxel connection using an in-house image processing program.
4. Numerical Modeling A 1D numerical simulation model was developed to predict the electrochemical performance of the Ni-GDC electrodes. Conservation of electrons, ions and gas species were considered, and were coupled by the electrochemical oxidation of hydrogen. Since GDC is a mixed ionic electronic conductor (MIEC) in the fuel environment, the electrochemical reaction was assumed to occur at the contact surface between GDC and pore, i.e. double-phase boundaries (DPBs). This assumption, considered reasonable according to the experimental works in literature [5–8], is one of the distinguishing features of Ni-GDC anodes. Microstructural parameters obtained from the real porous microstructure were used to evaluate the effective transport rates and the electrochemical activity within the electrodes. Conservation of electrons and oxide ions was described as follows using the electrochemical potential of the charge carriers
and
where
: ⋅
(2)
⋅
(3)
are the effective electronic and ionic conductivities, and
is the volumetric
density of the charge-transfer current. The dusty-gas model [41,42] was used to solve multi-component gas diffusion in the porous anodes. ⋅
(4)
∑
,
is the stoichiometric coefficient ( 1 ,
where
respectively),
,
respectively.
is the total pressure, while
,
1
and
and
,
(5)
1 and 0 for hydrogen, steam and nitrogen,
are the molar fraction, partial pressure and molar flux of gas species , and
are the mixture viscosity and permeability, and
are the effective Knudsen and molecular diffusion coefficients, respectively.
The effective conductivities and diffusivities in eq. 2, 3 and 5 were estimated by modifying bulk properties with the quantified volume fractions
and tortuosity factors
:
(6)
Bulk conductivities and diffusivities were evaluated as follows [3,43–45]. 3.27 .
10
1065.3 .
exp .
(7)
exp
.
∗
.
(8) (9)
/
.
. /
/
,
where
and Σ
are the molecular mass and the diffusion volume of gas species i, respectively.
(10)
(11)
The local oxygen partial pressure (or activity) in the solid GDC phase used in eq. 8 is obtained by assuming the following local equilibrium [26], as there is no electrochemical reaction taking place inside the solid phase: 2
O
1 2
ln
∗ O2
(12)
The electrochemical oxidation of hydrogen was assumed to occur at the DPB [5–8], and described by the Butler-Volmer equation, where the exchange current density is considered as a function of the DPB density
. exp
,
exp
(13)
The exchange current density per unit DPB surface area
,
, or the surface exchange
coefficient, was assumed to be a function of the local oxygen partial pressure [5, 8, 37]. ,
5.9
10
.
exp
.
(14)
where the pre-exponent constant and the activation energy terms were determined so as to match the numerically predicted overpotential characteristics to the experimental results. It should be noted that the
.
dependence was originally obtained from a samarium-doped ceria (SDC) electrode; this is
a temporary treatment due to the lack of experimental data related to hydrogen oxidation kinetics on GDC surfaces. The activation energy was similar to the value reported by Lai and Haile for SDC (2.67 eV = 2.57
10 J mol-1 K-1) [5]. The oxygen partial pressure used in eq. (14) is estimated from
the gas-phase equilibrium and different from that obtained from eq. (12). Local activation overpotential
and concentration overpotential
were expressed as
follows: 2 ln
Δ
ln
(15) (16)
where
is the standard Gibbs free energy associated with the hydrogen oxidation reaction, and is the gas partial pressure on the anode surface. The total anode overpotential can be expressed
as follows: 2 where
,
,
Δ
ln
(17)
denotes the current collector side, and the anode-electrolyte interface side.
5. Results and Discussion 5.1 Microstructure Fig. 1 shows one of the cross-sectional SEM images, its histogram and the segmented image. By using lower acceleration voltage of the electron beam (1.5 keV), the particles were imaged with 12.5 nm3 voxel size. As can be seen from the histogram, the three phases were clearly distinguished using the back-scattering electron detector. This clear contrast enabled precise image segmentation using the 2D histogram segmentation algorithms. In the segmented images, the white, gray and black regions correspond to GDC, Ni and pore phase, respectively. Fig. 2 shows the reconstructed structures of the infiltrated Ni-GDC anode. The reconstructed size was 9.38 x 5.63 x 9.65 m3. Table 1 shows the quantified microstructural parameters. In order for the reconstructed volume to be representative of the whole anode structure, the structure needs to contain a certain number of particles/pores in each direction. Given the largest characteristic length scale of the anode structure is found in the GDC phase (1.23 μm), the reconstructed structure has 7.63, 4.58 and 7.85 particles in each direction, which is considered to be large enough based on the analysis by Cai et al [34]. Also, for the precise quantification of the microstructural parameters, characteristic features of the anode need to be represented by sufficient number of image elements (voxels). Given the smallest characteristic length scale is found in the pore phase (0.341 μm), the particle and pores are resolved by at least 27 voxels in each direction. This is also sufficient to accurately quantify the microstructural parameters [34].
The nickel particle size found in the infiltrated electrode was much smaller than that in the conventional electrodes [23], which is one of the advantages of the infiltration technique. However, due to the lower volume fraction of the phase, there was limited percolation in the nickel phase. In this case the nickel tortuosity factor becomes indefinite, and hence the effective electronic conductivity (eq. 6) is determined by that of the GDC phase. Note that the electronic conductivity of the GDC phase is still one or two orders of magnitude larger than the ionic conductivity of the phase. Therefore the distribution of the electrochemical reaction is likely to be determined by the ionic conduction [25], and the effect of the electronic conduction is limited. Although not all of the nickel particles are percolated to form electronic conduction pathways, they still serve as a catalyst for the dissociative adsorption of hydrogen molecules. In the anode used in this study, the nickel particles are homogeneously distributed within the GDC scaffold. Therefore the supply of the adsorbed hydrogen atoms to the reaction sites is considered to be sufficient, and less likely to be the rate-determining process.
5.2 Simulation Table 2 summarizes the default computational parameters. The microstructural parameters used in eq. 6 and 13 are taken from Table 1. Fig. 3 shows the comparison of the anode overpotential between the experiment and the simulation under three different temperature conditions with the anode fuel composition of H2:H2O:N2 = 48.5:3.0:48.5. It was found that the numerical results have a reasonable agreement with the experimental results at 700 °C and 650 °C, whereas the curvature in the overpotential characteristics is not fully reproduced, especially at increasing current density at 600 °C. There are several reasons that can explain the discrepancies: (i) the electrochemical reaction model described by the Butler-Volmer equation (eq. 13 and 14) oversimplifies the complex reaction kinetics around the reaction sites. Similar discrepancies have also been found in the case of Ni-YSZ electrode simulation [11]. (ii) The assumption that the entire DPB is active for the reaction may not be always accurate; instead, the reaction may be localized around the TPB lines, i.e. an “extended TPB area”.
This may become significant at lower temperature due to the smaller electronic conductivity in the GDC phase. However its width from the TPB line is currently not clear. (iii) The contribution of the isolated nickel phase was not taken into account in the 1D model with the homogenization approach. However such a phase might alter the migration pathways of the electrons and affect the reaction distribution within the electrode. In order to improve the accuracy of the model, the kinetics of adsorption and desorption of gas species on nickel and GDC surfaces can be included; examples of such approach are found in literature [46–48]. However, this will introduce a number of unknown parameters, such as the total adsorption site density and equilibrium coverage, and may increase uncertainty of the model. The model presented in this study is intended to simulate electrochemical behavior of Ni-GDC electrodes with a minimum number of fitting parameters. The electrode performance is significantly influenced by the electrode microstructure through species transport and electrochemical reaction; therefore optimization of the microstructure is required to improve electrode performance and durability. However, it is difficult to enhance both the transport and electrochemical properties of the electrodes at the same time because these are inherently coupled and sometimes in conflict with each other. For example, reducing the particle size helps to increase the number of reaction sites, i.e., the DPB density, however it also reduces the pore size, increasing the gas diffusion resistance. Therefore the sensitivity of the electrode performance to the microstructural parameters needs to be understood in order to determine which parameter needs to be prioritized for microstructural optimization. In this study the electrode area-specific resistance (ASR) was chosen as a performance indicator and its sensitivity to the electrode microstructure, such as DPB density, GDC tortuosity factor and mean pore size, was investigated to provide insights into possible strategies to improve electrodes. Fig. 4 shows the sensitivity of the area-specific resistance to the DPB density at different current densities. The DPB density was varied from 10-2 to 102 μm2 μm-3, and the current density from
10 to 300 mA cm-2 at 700 °C. The ASR values are shown in the log scale. Overall, the area-specific resistance decreases as the DPB density increases. Also, the sensitivity to the DPB density is significantly higher in the region where the DPB density is smaller than ca. 1 μm2 μm-3. The point at which the area-specific resistance becomes almost stable is not significantly affected by the current density. Increasing the current density decreases the area-specific resistance in the higher current density region when the DPB density is lower than ca. 0.1 μm2 μm-3. This may be due to the nonlinearity of the electrochemical reaction kinetics employed in this model. Considering the fact that the DPB density quantified from the reconstructed anode is 2.13 μm2 μm-3, the electrode performance can still be improved if the DPB density is increased; a 10% increase in the DPB density would result in 10% decrease of the area-specific resistance. Fig. 5 shows the sensitivity of the area-specific resistance to the GDC tortuosity factor at different current densities. The GDC tortuosity factor was varied from 1 to 10, and current density from 10 to 300 mA cm-2 at 700 °C. Regardless of the current density, the area-specific resistance of the anode decreases as the tortuosity factor decreases, which is not surprising as the lower tortuosity factor decreases the Ohmic resistance associated with both electron and ion conduction. Gas diffusion in the porous material with submicron scale pores is influenced by the Knudsen diffusion because the characteristic length scale of the pores is comparable to or smaller than the mean free path length of the gas molecules. According to eq. 11, the Knudsen diffusivity linearly depends on the mean pore size and becomes significantly smaller than the molecular diffusivity (eq. 10) when the pore size is small. The sensitivity analysis on the DPB density indicated that the smaller characteristic length scale of the porous electrode may help enhance the electrochemical reaction rate by increasing the reaction site density. However, the smaller characteristic length scale is likely to inhibit gas diffusion through the electrode. Prior to analysis of the area-specific resistance, the dependency of the gas diffusivities on the mean pore size is investigated. For this purpose “mean gas diffusivity” in a gas mixture is defined as
follows. First, the DGM model (eq. 5) is analytically transformed to an explicit form of the molar flux as follows: ∑
(18)
,
where the matrix
(with components of
) is defined as follows with gas diffusivities and molar
fractions: ∑ ,
where
,
is the Kronecker delta. The mean gas diffusivity
of the matrix
1
(19)
is then defined using the Frobenius norm
: ‖ ‖
(20) ∑ ∑
The dependency of the mean gas diffusivity on the mean pore size is shown in Fig. 6, as well as that of the Knudsen diffusivities and the molecular diffusivities. Overall, there are two distinct regions found in the graph. In the region where the pore size is larger, the mean gas diffusivity is almost constant and comparable to the molecular diffusivities. This is because the molecular diffusion is dominant in such a scale. On the other hand, it asymptotically approaches the Knudsen diffusivities as the pore size becomes smaller, which indicates that Knudsen diffusion dominates mass transfer in the region. The intermediate region appears around a mean pore size of 1 μm. Fig. 7 shows the effect of the mean pore size on the anode area-specific resistance at the same range of pore sizes. The area specific resistance of the electrode reflects the behavior of the mean gas diffusivity. The current density was varied from 10 to 300 mA cm-2 at 700 °C. In the moleculardiffusion-dominant region, the area-specific resistance is not considerably influenced by the mean pore size. On the other hand in the Knudsen-diffusion-dominant region the value gradually increases due to the decrease in the mean diffusivity. The effect of Knudsen diffusion begins to appear at the mean pore size range of 1 μm when the current density is higher, but around 0.1 μm when the current density is
lower. This is because the higher current density requires faster fuel diffusion through the porous electrode and hence the electrode performance is likely to be more sensitive to the difference in gas diffusivity. Therefore it is recommended for SOFC anodes to have pores with the size of at least 1 μm particularly when SOFC systems are operated at high fuel consumption rates; otherwise gas transport in the pores are governed by the Knudsen diffusion and the electrode performance significantly decreases. The variation in the area-specific resistance in the sensitivity analysis was largest when the DPB density was varied, spanning two orders of magnitude, whereas values remained in the same order of magnitude when the GDC tortuosity factor and the mean pore size were varied. This indicates that the electrochemical reaction on the DPB is rate-limiting in these Ni-GDC anodes. Similar trends were also observed experimentally by Nakamura et al [7]. Therefore, increasing the DPB density in Ni-GDC electrodes is confirmed to be the most effective approach to improve electrode performance, when compared to improving species transport through the electrode. Decreasing the primary particle size of the GDC phase is one of the ways to achieve this. However, smaller particle sizes inevitably decrease the pore size, which inhibits gas diffusion through the electrodes. Therefore adding pore former, such as carbon black or other organic nano particles, would be effective to keep the pore size no less than few hundred nanometers to avoid Knudsen diffusion in the gas diffusion process. Note that the smaller GDC particle size does not increase the tortuosity factor of the phase because theoretically the geometrically analogous structures have the same tortuosity factors. Although adding pore former changes the geometry of the electrode and hence may increase the GDC tortuosity factor, the amount of the increase needs to be limited to avoid an increase in the Ohmic loss associated with the ion conduction. These insights and directions obtained in this study are expected to be useful in the future attempt to further optimize the electrode microstructures.
6. Conclusions
A one-dimensional numerical model of Ni-GDC electrodes has been developed to analyze the effect of the electrode microstructure on performance. The electrode microstructure was obtained with FIB-SEM tomography and the microstructural parameters were quantified and applied to a numerical model to evaluate the transport and electrochemical properties. GDC was considered as a mixed conductor and hence the electrochemical reaction was assumed to occur on the entire GDC-pore contact area (DPB). The numerical model successfully reproduced the overpotential characteristics measured at 700 °C and 650 °C, though the curvature of the graph was not as well reproduced at 600 °C at high current density (>150 mA cm-2). The contribution of the nickel phase to help local electron conduction may need to be taken into account to improve the accuracy of the model. A sensitivity analysis was also conducted at 700 °C to investigate the effect of electrode microstructure on electrode performance. This revealed that the electrochemical reaction on the DPB is the rate-determining process within the electrodes; therefore increasing the DPB density is recommended as the most effective route to improving performance of ceria-based electrodes, rather than improving species transport rate. The model developed in this study offers a basis to understand the microstructureperformance relationships in Ni-GDC electrodes, which is useful to further optimize their porous microstructures.
Acknowledgments The authors would like to acknowledge the financial support from Japan Society for the Promotion of Science (JSPS), Alan Howard scholarship and Engineering and Physical Sciences Research Council (EPSRC) grant number EP/I037016/1.
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Figure Captions
5.63 µm
Segmentation Histogram GDC
Ni
GDC Pore
9.38 µm
Pore Ni
2 µm
Fig. 1
FIB imaging of a Ni-GDC electrode and the image segmentation.
Fig. 2
Reconstructed 3D microstructure of the three phases.
Fig. 3
Comparison of the anode polarization overpotential between simulation and experiment.
Fig. 4
Effect of DPB density on the anode area-specific resistance at 700 °C.
Fig. 5
Effect of GDC tortuosity factor on the anode area-specific resistance at 700 °C.
Fig. 6
Effect of the mean pore size on the diffusion coefficients at 700 °C.
Fig. 7
Effect of the mean pore size on the anode area-specific resistance at 700 °C.
Table Captions Table 1
Microstructural parameters.
GDC
Pore
11.4
73.3
15.3
Volume fraction
[%]
Specific surface area
[μm μm ]
1.75
2.98
3.02
Particle/pore size
[μm]
0.349
1.23
0.341
Connectivity
[%]
0.00
99.9
98.2
N/A
1.21
6.25
2
-3
Tortuosity factor
Ni
2
-3
2.13
-3
8.27
DPB density
[μm μm ]
TPB density
[μm μm ]
Table 2
Default computational parameters. Parameter
Total Pressure
1.013
10
Unit [Pa]
600, 650, 700
[°C]
H2 Partial Pressure
0.485
[Pa]
H2O Partial Pressure
0.03
[Pa]
N2 Partial Pressure
0.485
[Pa]
Temperature
Anode Thickness
Value
12.5
[μm]
S0013-4686(15)30954-3 http://dx.doi.org/doi:10.1016/j.electacta.2015.12.044 EA 26193
To appear in:
Electrochimica Acta
Received date: Revised date: Accepted date:
24-4-2015 30-11-2015 4-12-2015
Please cite this article as: Masashi Kishimoto, Marina Lomberg, Enrique Ruiz-Trejo, Nigel P Brandon, Numerical modeling of nickel-infiltrated gadolinium-doped ceria electrodes reconstructed with focused ion beam tomography, Electrochimica Acta http://dx.doi.org/10.1016/j.electacta.2015.12.044 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Numerical modeling of nickel-infiltrated gadolinium-doped ceria electrodes reconstructed with focused ion beam tomography
1,2
1.
Masashi Kishimoto*, 1Marina Lomberg, 1Enrique Ruiz-Trejo, 1Nigel P Brandon Department of Earth Science and Engineering, Imperial College London, London SW7 2AZ, United Kingdom
2.
Department of Aeronautics and Astronautics, Kyoto University, Kyoto 615-8026, Japan
E-mail: [email protected] Tel.: +81-(0)753833652
Graphical abstract FIB Imaging & Reconstruction
Sensitivity analysis Modeling
Ni
Conduction Diffusion Electrochemical reacti reaction
GDC Pore
8µ
m
9.6
9.3
5µ
m
5.63 µm
2 µm
Highlights
A 1D numerical model of Ni-GDC electrodes is developed. Microstructure of a nickel-infiltrated GDC electrode is obtained using FIB-SEM. Overpotential characteristics are successfully reproduced by the developed model. Sensitivity analysis reveals microstructure-performance relationships. The developed model offers guidelines to optimize electrode microstructures.
Abstract A one-dimensional numerical model of a nickel-infiltrated gadolinium-doped ceria (Ni-GDC) electrode has been developed to investigate the effects of electrode microstructure on performance. Electrode microstructural information was obtained with focused ion beam tomography and microstructural parameters were quantified, such as tortuosity factor, surface area and particle/pore sizes. These have been used to estimate the effective transport coefficients and the electrochemical reaction rate in the electrodes. GDC was considered as a mixed ionic and electronic conductor and hence the electrochemical reaction is assumed to occur on the GDC-pore contact surface, i.e. doublephase boundaries (DPBs). Sensitivity analysis was conducted to investigate the effect of electrode microstructure on both transport properties and electrochemical activity, including the effect of DPB density, GDC tortuosity factor and pore size. The developed model offers a basis to understand the microstructure-performance relationships and to further optimize the electrode microstructures.
Key words: solid oxide fuel cells; infiltration; microstructure; FIB-SEM; modeling.
Nomenclature pore size (m) binary molecular diffusivity (m2 s-1) Knudsen diffusivity (m2 s-1) mean diffusivity (m2 s-1) Faraday constant (C mol-1) Gibbs free energy (J mol-1) ,
exchange current per unit DPB area (A m-2) charge-transfer current (A m-3) permeability (m2)
anode thickness (m) molecular weight (kg mol-1) molar flux (mol m-2 s-1) partial pressure (Pa) total pressure (Pa) gas constant (J mol-1 K-1) surface area (m2) temperature (K) volume (m3) volume fraction (-) molar fraction (-) stoichiometric coefficient (-)
Greek symbols Kronecker delta overpotential (V) viscosity (Pa s) chemical potential (J mol-1) electrochemical potential (J mol-1) conductivity (m2 s-1) diffusion volume (m3) tortuosity factor
Subscripts act
activation
CC
current collector
con
concentration
ct
charge-transfer
DPB
double-phase boundary
I
anode-electrolyte interface
t
total
Superscripts bluk
value for bulk material
eff
effective value for porous material
v
volumetric
1. Introduction Doped ceria has been receiving significant attention as an alternative material for solid oxide fuel cells (SOFCs) due to its higher oxide ion conductivity than conventional zirconia-based materials at lower temperatures, making it possible to reduce the operating temperature of SOFCs. Lowering the operating temperature enables reduction of the total cost of SOFC systems by allowing use of inexpensive materials and mitigating thermally induced degradation. One of the important features of doped ceria is its mixed conductivity under the reducing atmospheres [1–4], allowing both electrons and oxide ions to migrate through the material. This broadens the electrochemical reaction site from the triple-phase boundary (TPB) to the ceria-pore double-phase boundary (DPB) [5–8]. It is widely accepted that electrode microstructure plays an important role in determining the performance and durability of SOFCs. Electrodes are required to contain as many reaction sites as possible to promote the electrochemical reaction; at the same time, effective mass and charge transport pathways need to be established to and from the reaction sites. Therefore, the microstructureperformance relationships need to be understood to develop high performance electrodes. Threedimensional (3D) imaging techniques, such as focused ion beam scanning electron microscopy (FIBSEM) [9–15] and X-ray nano CT [16–21], are finding increasing application in the study of complex electrode microstructures. Through the development and application of quantification methodologies and numerical simulation models, new insights have been generated, which offer the prospect of optimizing electrode microstructure to further improve performance and durability. We recently succeeded in imaging the microstructure of nickel-infiltrated gadolinium-doped ceria (GDC) electrodes using FIB-SEM [12], where the infiltrated nickel particles (~100 nm) were precisely imaged. It was revealed that the infiltrated electrodes had one order of magnitude larger TPB density compared with conventional electrodes made by powder mixing and sintering process. To further extend this approach, reliable electrode simulation models are required to investigate the effect of microstructure on the performance. Kishimoto et al. developed 1D and 3D numerical
models based on the finite volume method (FVM) coupled with the sub-grid-scale (SGS) model to analyze the overpotential characteristics and the distribution of chemical species within Ni-YSZ electrodes [11, 22–25]. Shikazono et al. developed 3D models based on the Lattice Boltzmann Method (LBM) and analyzed Ni-YSZ anodes and LSCF cathodes [26–28]. Carraro, Joos and Häffelin et al. developed a 3D model based on the finite element method (FEM) for a MIEC cathode and conducted sensitivity analysis to elucidate the effect of various operating conditions on the electrode performance [29–32]. Shearing and Cai et al. developed a 3D model based on the volume of fluid (VOF) method and discussed the effect of image voxel size and reconstructed volume size on the simulation results [33–35]. Although electrode structures in their model were synthesized by the Monte Carlo method, their findings are particularly important to determine the required volume size for the reconstructed structure to be representative of the whole electrode microstructures. Results from these numerical analyses offer valuable insights to understand the effects of electrode microstructure on electrode performance, which are otherwise very difficult to observe in experiments. For ceria-based materials, Chueh et al. conducted a two-dimensional numerical simulation on a samarium-doped ceria (SDC) electrode to analyze the reaction region in the material, and found that the rate-determining process was electron conduction from the SDC surface to the current collector [36, 37]. They also found that the TPB length is not as influential on electrode performance as in the case of conventional zirconia-based electrodes because the entire SDC-pore contact surface can potentially act as a reaction zone. However, the simulation domain employed in their work was limited to the Ni-SDC-pore contact region; hence it was difficult to assess the effect of microstructure, such as phase volume fraction, DPB density and particle/pore sizes, on performance. In this study, the 3D microstructure of infiltrated Ni-GDC electrodes is obtained using FIBSEM, and microstructural parameters that characterize the porous structures are quantitatively evaluated, such as DPB density, particle/pore sizes and surface area. A 1D numerical model is then developed to analyze the distribution of the charge carriers and gas species within the electrodes as
well as the electrochemical reaction. Sensitivity analysis using the developed model is conducted to investigate the effect of several microstructural parameters on electrode performance, and insights for microstructural optimization are generated.
2. Experimental In this study, the electrochemical performance of an infiltrated Ni-GDC electrode was characterized using an electrolyte-supported button cell, whose performance was used to validate the proposed simulation model. First, a GDC scaffold (Ce0.9Gd0.1O1.95) was fabricated on an 8YSZ electrolyte disk (Fuel Cell Materials) by screen printing an ink and sintering at 1350 °C for 2 h. Second, the cathode was screen printed on the other side of the electrolyte using an LSCF-GDC ink ((La0.60Sr0.40)0.995(Co0.20Fe0.80)O3-x:Ce0.9Gd0.1O1.95 = 50:50 wt.%, Fuel Cell Materials) and sintering at 1100 °C for 2 h. Finally, nickel oxide nanoparticles were infiltrated into the scaffold by introducing Ni(NO3)2 solution from the top of the scaffold followed by decomposition in air at 500 °C for 30 min. This infiltration process was repeated ten times, followed by a reduction process under a hydrogennitrogen gas mixture (760 °C, H2:N2 = 50:50, 1 h). More details can be found elsewhere [38]. Electrochemical characterization of the electrodes was performed under a three-electrode configuration. Measurement of current-voltage characteristics and electrochemical impedance spectra was performed using an Autolab PGSTAT302 (Metrohm) over a range of temperatures (600-700 °C) under
a
hydrogen-nitrogen
gas
mixture humidified with water
at
room
temperature
(H2:H2O:N2 = 48.5:3.0:48.5). The efficacy of our three-electrode measurement was demonstrated by Lomberg et al. [38]
3. 3D Imaging and Reconstruction
In our previous report, we successfully imaged nickel-infiltrated GDC electrodes with 5 nm resolution and then reconstructed and quantified the electrode microstructures [12]. The process for the SEM sample preparation and 3D imaging is briefly summarized below. After the electrochemical measurements, the anode samples were impregnated with epoxy resin (Specifix20, Struers) under vacuum conditions in order to minimize the undesirable damage on the specimen and to avoid accumulation of gallium ions in the pores during FIB milling, as well as to help distinguish the pores from the solid phases in SEM imaging. The cured samples were cut and mechanically polished, mounted onto a specimen stub with silver paste, and sputtered with gold to ensure sufficient electron conductivity. The 3D microstructure of the anodes was imaged with an Auriga (Zeiss) FIB-SEM system equipped with a backscattered electron detector, which enables to clearly distinguish the two solid phases in the SEM images. Regions were selected for microstructural analysis and went through the following image processing using Avizo (FEI): (i) alignment of the stack images using the least square method, (ii) noise reduction using the edge preserving smoothing filter (a combination of the Canny filter and the Gaussian smoothing filter) and (iii) segmentation using a 2D histogram segmentation algorithm based on the water shed algorithm, followed by manual correction. Then, the 3D porous microstructure was virtually reconstructed, from which various microstructural parameters, such as phase volume fractions, surface area, particle/pore sizes, phase connectivity and tortuosity factor, were quantified. Volume fractions were evaluated by counting the number of voxels corresponding to each phase, followed by division by the total number of voxels in the reconstructed volume [11]. Surface area was measured by the marching cube algorithm and then used to obtain the GDC-Pore double-phase boundary
as follows:
where
is the total surface area of phase and
(1)
is the entire reconstructed volume. Particle/pore
sizes were measured using the 3D mean intercept method, the 2D version of which was originally proposed by Simwonis et al. [39]. This 3D mean intercept method is also a simplified approach of the
ray-shooting method [40]. Tortuosity factors were measured by the diffusion-based method [23]. Connectivity was judged based on the 6-neighboring voxel connection using an in-house image processing program.
4. Numerical Modeling A 1D numerical simulation model was developed to predict the electrochemical performance of the Ni-GDC electrodes. Conservation of electrons, ions and gas species were considered, and were coupled by the electrochemical oxidation of hydrogen. Since GDC is a mixed ionic electronic conductor (MIEC) in the fuel environment, the electrochemical reaction was assumed to occur at the contact surface between GDC and pore, i.e. double-phase boundaries (DPBs). This assumption, considered reasonable according to the experimental works in literature [5–8], is one of the distinguishing features of Ni-GDC anodes. Microstructural parameters obtained from the real porous microstructure were used to evaluate the effective transport rates and the electrochemical activity within the electrodes. Conservation of electrons and oxide ions was described as follows using the electrochemical potential of the charge carriers
and
where
: ⋅
(2)
⋅
(3)
are the effective electronic and ionic conductivities, and
is the volumetric
density of the charge-transfer current. The dusty-gas model [41,42] was used to solve multi-component gas diffusion in the porous anodes. ⋅
(4)
∑
,
is the stoichiometric coefficient ( 1 ,
where
respectively),
,
respectively.
is the total pressure, while
,
1
and
and
,
(5)
1 and 0 for hydrogen, steam and nitrogen,
are the molar fraction, partial pressure and molar flux of gas species , and
are the mixture viscosity and permeability, and
are the effective Knudsen and molecular diffusion coefficients, respectively.
The effective conductivities and diffusivities in eq. 2, 3 and 5 were estimated by modifying bulk properties with the quantified volume fractions
and tortuosity factors
:
(6)
Bulk conductivities and diffusivities were evaluated as follows [3,43–45]. 3.27 .
10
1065.3 .
exp .
(7)
exp
.
∗
.
(8) (9)
/
.
. /
/
,
where
and Σ
are the molecular mass and the diffusion volume of gas species i, respectively.
(10)
(11)
The local oxygen partial pressure (or activity) in the solid GDC phase used in eq. 8 is obtained by assuming the following local equilibrium [26], as there is no electrochemical reaction taking place inside the solid phase: 2
O
1 2
ln
∗ O2
(12)
The electrochemical oxidation of hydrogen was assumed to occur at the DPB [5–8], and described by the Butler-Volmer equation, where the exchange current density is considered as a function of the DPB density
. exp
,
exp
(13)
The exchange current density per unit DPB surface area
,
, or the surface exchange
coefficient, was assumed to be a function of the local oxygen partial pressure [5, 8, 37]. ,
5.9
10
.
exp
.
(14)
where the pre-exponent constant and the activation energy terms were determined so as to match the numerically predicted overpotential characteristics to the experimental results. It should be noted that the
.
dependence was originally obtained from a samarium-doped ceria (SDC) electrode; this is
a temporary treatment due to the lack of experimental data related to hydrogen oxidation kinetics on GDC surfaces. The activation energy was similar to the value reported by Lai and Haile for SDC (2.67 eV = 2.57
10 J mol-1 K-1) [5]. The oxygen partial pressure used in eq. (14) is estimated from
the gas-phase equilibrium and different from that obtained from eq. (12). Local activation overpotential
and concentration overpotential
were expressed as
follows: 2 ln
Δ
ln
(15) (16)
where
is the standard Gibbs free energy associated with the hydrogen oxidation reaction, and is the gas partial pressure on the anode surface. The total anode overpotential can be expressed
as follows: 2 where
,
,
Δ
ln
(17)
denotes the current collector side, and the anode-electrolyte interface side.
5. Results and Discussion 5.1 Microstructure Fig. 1 shows one of the cross-sectional SEM images, its histogram and the segmented image. By using lower acceleration voltage of the electron beam (1.5 keV), the particles were imaged with 12.5 nm3 voxel size. As can be seen from the histogram, the three phases were clearly distinguished using the back-scattering electron detector. This clear contrast enabled precise image segmentation using the 2D histogram segmentation algorithms. In the segmented images, the white, gray and black regions correspond to GDC, Ni and pore phase, respectively. Fig. 2 shows the reconstructed structures of the infiltrated Ni-GDC anode. The reconstructed size was 9.38 x 5.63 x 9.65 m3. Table 1 shows the quantified microstructural parameters. In order for the reconstructed volume to be representative of the whole anode structure, the structure needs to contain a certain number of particles/pores in each direction. Given the largest characteristic length scale of the anode structure is found in the GDC phase (1.23 μm), the reconstructed structure has 7.63, 4.58 and 7.85 particles in each direction, which is considered to be large enough based on the analysis by Cai et al [34]. Also, for the precise quantification of the microstructural parameters, characteristic features of the anode need to be represented by sufficient number of image elements (voxels). Given the smallest characteristic length scale is found in the pore phase (0.341 μm), the particle and pores are resolved by at least 27 voxels in each direction. This is also sufficient to accurately quantify the microstructural parameters [34].
The nickel particle size found in the infiltrated electrode was much smaller than that in the conventional electrodes [23], which is one of the advantages of the infiltration technique. However, due to the lower volume fraction of the phase, there was limited percolation in the nickel phase. In this case the nickel tortuosity factor becomes indefinite, and hence the effective electronic conductivity (eq. 6) is determined by that of the GDC phase. Note that the electronic conductivity of the GDC phase is still one or two orders of magnitude larger than the ionic conductivity of the phase. Therefore the distribution of the electrochemical reaction is likely to be determined by the ionic conduction [25], and the effect of the electronic conduction is limited. Although not all of the nickel particles are percolated to form electronic conduction pathways, they still serve as a catalyst for the dissociative adsorption of hydrogen molecules. In the anode used in this study, the nickel particles are homogeneously distributed within the GDC scaffold. Therefore the supply of the adsorbed hydrogen atoms to the reaction sites is considered to be sufficient, and less likely to be the rate-determining process.
5.2 Simulation Table 2 summarizes the default computational parameters. The microstructural parameters used in eq. 6 and 13 are taken from Table 1. Fig. 3 shows the comparison of the anode overpotential between the experiment and the simulation under three different temperature conditions with the anode fuel composition of H2:H2O:N2 = 48.5:3.0:48.5. It was found that the numerical results have a reasonable agreement with the experimental results at 700 °C and 650 °C, whereas the curvature in the overpotential characteristics is not fully reproduced, especially at increasing current density at 600 °C. There are several reasons that can explain the discrepancies: (i) the electrochemical reaction model described by the Butler-Volmer equation (eq. 13 and 14) oversimplifies the complex reaction kinetics around the reaction sites. Similar discrepancies have also been found in the case of Ni-YSZ electrode simulation [11]. (ii) The assumption that the entire DPB is active for the reaction may not be always accurate; instead, the reaction may be localized around the TPB lines, i.e. an “extended TPB area”.
This may become significant at lower temperature due to the smaller electronic conductivity in the GDC phase. However its width from the TPB line is currently not clear. (iii) The contribution of the isolated nickel phase was not taken into account in the 1D model with the homogenization approach. However such a phase might alter the migration pathways of the electrons and affect the reaction distribution within the electrode. In order to improve the accuracy of the model, the kinetics of adsorption and desorption of gas species on nickel and GDC surfaces can be included; examples of such approach are found in literature [46–48]. However, this will introduce a number of unknown parameters, such as the total adsorption site density and equilibrium coverage, and may increase uncertainty of the model. The model presented in this study is intended to simulate electrochemical behavior of Ni-GDC electrodes with a minimum number of fitting parameters. The electrode performance is significantly influenced by the electrode microstructure through species transport and electrochemical reaction; therefore optimization of the microstructure is required to improve electrode performance and durability. However, it is difficult to enhance both the transport and electrochemical properties of the electrodes at the same time because these are inherently coupled and sometimes in conflict with each other. For example, reducing the particle size helps to increase the number of reaction sites, i.e., the DPB density, however it also reduces the pore size, increasing the gas diffusion resistance. Therefore the sensitivity of the electrode performance to the microstructural parameters needs to be understood in order to determine which parameter needs to be prioritized for microstructural optimization. In this study the electrode area-specific resistance (ASR) was chosen as a performance indicator and its sensitivity to the electrode microstructure, such as DPB density, GDC tortuosity factor and mean pore size, was investigated to provide insights into possible strategies to improve electrodes. Fig. 4 shows the sensitivity of the area-specific resistance to the DPB density at different current densities. The DPB density was varied from 10-2 to 102 μm2 μm-3, and the current density from
10 to 300 mA cm-2 at 700 °C. The ASR values are shown in the log scale. Overall, the area-specific resistance decreases as the DPB density increases. Also, the sensitivity to the DPB density is significantly higher in the region where the DPB density is smaller than ca. 1 μm2 μm-3. The point at which the area-specific resistance becomes almost stable is not significantly affected by the current density. Increasing the current density decreases the area-specific resistance in the higher current density region when the DPB density is lower than ca. 0.1 μm2 μm-3. This may be due to the nonlinearity of the electrochemical reaction kinetics employed in this model. Considering the fact that the DPB density quantified from the reconstructed anode is 2.13 μm2 μm-3, the electrode performance can still be improved if the DPB density is increased; a 10% increase in the DPB density would result in 10% decrease of the area-specific resistance. Fig. 5 shows the sensitivity of the area-specific resistance to the GDC tortuosity factor at different current densities. The GDC tortuosity factor was varied from 1 to 10, and current density from 10 to 300 mA cm-2 at 700 °C. Regardless of the current density, the area-specific resistance of the anode decreases as the tortuosity factor decreases, which is not surprising as the lower tortuosity factor decreases the Ohmic resistance associated with both electron and ion conduction. Gas diffusion in the porous material with submicron scale pores is influenced by the Knudsen diffusion because the characteristic length scale of the pores is comparable to or smaller than the mean free path length of the gas molecules. According to eq. 11, the Knudsen diffusivity linearly depends on the mean pore size and becomes significantly smaller than the molecular diffusivity (eq. 10) when the pore size is small. The sensitivity analysis on the DPB density indicated that the smaller characteristic length scale of the porous electrode may help enhance the electrochemical reaction rate by increasing the reaction site density. However, the smaller characteristic length scale is likely to inhibit gas diffusion through the electrode. Prior to analysis of the area-specific resistance, the dependency of the gas diffusivities on the mean pore size is investigated. For this purpose “mean gas diffusivity” in a gas mixture is defined as
follows. First, the DGM model (eq. 5) is analytically transformed to an explicit form of the molar flux as follows: ∑
(18)
,
where the matrix
(with components of
) is defined as follows with gas diffusivities and molar
fractions: ∑ ,
where
,
is the Kronecker delta. The mean gas diffusivity
of the matrix
1
(19)
is then defined using the Frobenius norm
: ‖ ‖
(20) ∑ ∑
The dependency of the mean gas diffusivity on the mean pore size is shown in Fig. 6, as well as that of the Knudsen diffusivities and the molecular diffusivities. Overall, there are two distinct regions found in the graph. In the region where the pore size is larger, the mean gas diffusivity is almost constant and comparable to the molecular diffusivities. This is because the molecular diffusion is dominant in such a scale. On the other hand, it asymptotically approaches the Knudsen diffusivities as the pore size becomes smaller, which indicates that Knudsen diffusion dominates mass transfer in the region. The intermediate region appears around a mean pore size of 1 μm. Fig. 7 shows the effect of the mean pore size on the anode area-specific resistance at the same range of pore sizes. The area specific resistance of the electrode reflects the behavior of the mean gas diffusivity. The current density was varied from 10 to 300 mA cm-2 at 700 °C. In the moleculardiffusion-dominant region, the area-specific resistance is not considerably influenced by the mean pore size. On the other hand in the Knudsen-diffusion-dominant region the value gradually increases due to the decrease in the mean diffusivity. The effect of Knudsen diffusion begins to appear at the mean pore size range of 1 μm when the current density is higher, but around 0.1 μm when the current density is
lower. This is because the higher current density requires faster fuel diffusion through the porous electrode and hence the electrode performance is likely to be more sensitive to the difference in gas diffusivity. Therefore it is recommended for SOFC anodes to have pores with the size of at least 1 μm particularly when SOFC systems are operated at high fuel consumption rates; otherwise gas transport in the pores are governed by the Knudsen diffusion and the electrode performance significantly decreases. The variation in the area-specific resistance in the sensitivity analysis was largest when the DPB density was varied, spanning two orders of magnitude, whereas values remained in the same order of magnitude when the GDC tortuosity factor and the mean pore size were varied. This indicates that the electrochemical reaction on the DPB is rate-limiting in these Ni-GDC anodes. Similar trends were also observed experimentally by Nakamura et al [7]. Therefore, increasing the DPB density in Ni-GDC electrodes is confirmed to be the most effective approach to improve electrode performance, when compared to improving species transport through the electrode. Decreasing the primary particle size of the GDC phase is one of the ways to achieve this. However, smaller particle sizes inevitably decrease the pore size, which inhibits gas diffusion through the electrodes. Therefore adding pore former, such as carbon black or other organic nano particles, would be effective to keep the pore size no less than few hundred nanometers to avoid Knudsen diffusion in the gas diffusion process. Note that the smaller GDC particle size does not increase the tortuosity factor of the phase because theoretically the geometrically analogous structures have the same tortuosity factors. Although adding pore former changes the geometry of the electrode and hence may increase the GDC tortuosity factor, the amount of the increase needs to be limited to avoid an increase in the Ohmic loss associated with the ion conduction. These insights and directions obtained in this study are expected to be useful in the future attempt to further optimize the electrode microstructures.
6. Conclusions
A one-dimensional numerical model of Ni-GDC electrodes has been developed to analyze the effect of the electrode microstructure on performance. The electrode microstructure was obtained with FIB-SEM tomography and the microstructural parameters were quantified and applied to a numerical model to evaluate the transport and electrochemical properties. GDC was considered as a mixed conductor and hence the electrochemical reaction was assumed to occur on the entire GDC-pore contact area (DPB). The numerical model successfully reproduced the overpotential characteristics measured at 700 °C and 650 °C, though the curvature of the graph was not as well reproduced at 600 °C at high current density (>150 mA cm-2). The contribution of the nickel phase to help local electron conduction may need to be taken into account to improve the accuracy of the model. A sensitivity analysis was also conducted at 700 °C to investigate the effect of electrode microstructure on electrode performance. This revealed that the electrochemical reaction on the DPB is the rate-determining process within the electrodes; therefore increasing the DPB density is recommended as the most effective route to improving performance of ceria-based electrodes, rather than improving species transport rate. The model developed in this study offers a basis to understand the microstructureperformance relationships in Ni-GDC electrodes, which is useful to further optimize their porous microstructures.
Acknowledgments The authors would like to acknowledge the financial support from Japan Society for the Promotion of Science (JSPS), Alan Howard scholarship and Engineering and Physical Sciences Research Council (EPSRC) grant number EP/I037016/1.
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Figure Captions
5.63 µm
Segmentation Histogram GDC
Ni
GDC Pore
9.38 µm
Pore Ni
2 µm
Fig. 1
FIB imaging of a Ni-GDC electrode and the image segmentation.
Fig. 2
Reconstructed 3D microstructure of the three phases.
Fig. 3
Comparison of the anode polarization overpotential between simulation and experiment.
Fig. 4
Effect of DPB density on the anode area-specific resistance at 700 °C.
Fig. 5
Effect of GDC tortuosity factor on the anode area-specific resistance at 700 °C.
Fig. 6
Effect of the mean pore size on the diffusion coefficients at 700 °C.
Fig. 7
Effect of the mean pore size on the anode area-specific resistance at 700 °C.
Table Captions Table 1
Microstructural parameters.
GDC
Pore
11.4
73.3
15.3
Volume fraction
[%]
Specific surface area
[μm μm ]
1.75
2.98
3.02
Particle/pore size
[μm]
0.349
1.23
0.341
Connectivity
[%]
0.00
99.9
98.2
N/A
1.21
6.25
2
-3
Tortuosity factor
Ni
2
-3
2.13
-3
8.27
DPB density
[μm μm ]
TPB density
[μm μm ]
Table 2
Default computational parameters. Parameter
Total Pressure
1.013
10
Unit [Pa]
600, 650, 700
[°C]
H2 Partial Pressure
0.485
[Pa]
H2O Partial Pressure
0.03
[Pa]
N2 Partial Pressure
0.485
[Pa]
Temperature
Anode Thickness
Value
12.5
[μm]