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Density Functional Theory Study for Ni Diffusion on Ni(111) Surface under Solid Oxide Fuel Cell Operating Condition Kazuhide Nakao, Takayoshi Ishimoto, and Michihisa Koyama J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b03440 • Publication Date (Web): 11 Jul 2016 Downloaded from http://pubs.acs.org on July 15, 2016

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The Journal of Physical Chemistry

Density Functional Theory Study for Ni Diffusion on Ni(111) Surface under Solid Oxide Fuel Cell Operating Condition Kazuhide Nakao†, Takayoshi Ishimoto §, and Michihisa Koyama*,†,§, ǁ †

Department of Hydrogen Energy Systems, Graduate School of Engineering, Kyushu University,

744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan §

INAMORI Frontier Research Center, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka

819-0395, Japan ǁ

International Institute for Carbon-Neutral Energy Research (I2CNER), Kyushu University, 744

Motooka, Nishi-ku, Fukuoka 819-0395, Japan

ACS Paragon Plus Environment

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ABSTRACT Understanding the sintering mechanism of Ni in solid oxide fuel cell (SOFC) anode is one of the important issues to discuss the long-term durability of SOFC performance. The sintering behavior of Ni is affected not only by the operating temperature but also by the gas composition in the anode chamber. We analyzed the surface diffusion of Ni adatom and Ni complexes on Ni(111) surface under the operating temperature and gas compositions in anode by using density functional theory. The Ni adatom, Ni-H and Ni-S complexes, which can be formed by H2 and H2S in anode gas, were considered as diffusion species on Ni surface. It is theoretically confirmed that the formation of NiS complex influences the sintering behavior of Ni depending on the temperature and the impurity H2S concentration in the fuel. Our calculated results are compared with the experimental observation on Ni sintering under various temperatures and H2S concentrations to find a good agreement. We clearly showed the important effect of gas composition in anode on the sintering properties of Ni in SOFC anode.

Table of Contents


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1. INTRODUCTION Solid oxide fuel cell (SOFC) is one of the promising technologies to reduce the CO2 emission associated with the use of fossil fuels.1 SOFC is expected to be used for a wide range of purposes, from residential use to large scale power generation. While some systems based on SOFC is commercialized, realizing the high durability and reliability with a low production cost is still a remaining challenge. For this purpose, the suppression of SOFC performance degradation is one of the important issues. 2-8 SOFC consists of the anode, cathode, and electrolyte as it main componets. As an anode material, various elements have been investigated in its early phase of evelopment.9,10 Nowadays, the Ni metal is exclusively used as an anode material in the cells of the leading SOFC stack developers.11 While Ni shows an excellent anodic properties, there still remains some problems in Ni as an SOFC anode material. One of the most serious problems is the microstructural change of the anode as results of the sintering of Ni, which occurs during the long-term operation at high temperature.12-16 For example, Simwonis et al. observed the particle size growth of Ni with increasing operation time under 1273 K.12 Tanasini et al. found the relation between decrease of the cell performance and the sintering of Ni by the impedance measurement at 1123 K.15 These results indicate the decrease in electronic conductivity and triple phase boundary (TPB) length where Ni, electrolyte, and gas phases meet, as results of the sintering of Ni. The sintering of Ni is influenced not only by the operating temperature but also by the gas composition in anode.17,18 Matsui et al. studied the effect of water partial pressure on SOFC degradation at 1273 K.17 They reported that the increase of water partial pressure leads to the increase of degradation degree due to sintering of Ni. Lee et al. measured the ohmic and


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polarization resistances under two humidified conditions (10 % and 40 % H2O) at three different temperatures (1273, 1373, and 1473 K).18 They reported the large degradation associated with the microstructural change under 40 % humidified condition above 1373 K. Also, it is well known that adsorbed sulfur on Ni surface hinders the electrochemical reactions. Although the sulfur poisoning is recovered by the removal of the adsorbed sulfur, the irreversible degradations due to microstructural changes by the existence of sulfur species in the gas phase were also reported.19-21 Lohsoontorn et al. found the irreversible degradation of Ni by using the polarization resistance measurement for Ni-CGO anode under 97 % H2/3 % H2O with 1 and 3 ppm H2S at 867 K.19 This irreversible degradation is also affected by temperature. Li et al. measured the cell performance using Ni-YSZ anode at 1023 and 1073 K with 0.2 % H2S.20 They showed that the cell voltage recovery from the sulfur poisoning depends on the temperature. They also pointed out that the lower temperature leads to the large irreversible degradation. These results clearly indicate that the sintering of Ni is affected by the anode gas compositions and the temperature. Although the effects of H2S on the sintering of Ni have been reported, the promotion mechanism of sintering of Ni by H2S is unclear. Generally, the diffusion mechanism on sintering of Ni is classified into the surface, grain boundary, and volume diffusions. In this study, we focused on the surface diffusion as sintering mechanism of Ni referring the experimental and theoretical studies,22-27 which report the promotion of surface diffusion due to the formation of complex composed of surface adatom and adsorbate. To predict the sintering behavior of Ni under SOFC operating condition, it is important to understand the essential effect of gas compositions on Ni diffusion. The purpose of this study is to evaluate the effect of gas compositions on the surface diffusion of Ni by using density functional theory (DFT) method. First, we analyzed the trend of


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Ni diffusion under vacuum condition as a reference of this study. The H2 gas, which is a fuel for SOFC, was used to understand the effect of gas composition. We also analyzed the Ni diffusion with H2S, which is an example of impurity in fuel, and compared with the experimental data.

2. THEORETICAL CALCULATION 2.1. Calculation of Surface Diffusion Coefficient The surface diffusion coefficient is represented by the concentration and the mobility of diffusion species28

=

,

(1)

where is the concentration of diffusion species on substrate, the concentration of substrate atoms, and the diffusion coefficient of single diffusion species. Both of and depend on temperature and are represented by Arrhenius form = exp (−

∆

= exp (−

!"#

),

(2)

),

(3)

where $ is the gas constant, % the absolute temperature, ∆&'()* the Gibbs formation energy of diffusion species, and and + ,'' the pre-exponential factor and the activation energy for single diffusion species, respectively. Thus, we estimated the ∆&'()* , , and + ,'' for surface diffusion coefficient based on the DFT calculations. In this study, we assumed that the Ni adatom comes from kink site to form complex with surface adsorbates. This assumption implies that the effect of surface defect on surface diffusion is included as the adatom formation in this calculation. Then, the change of Gibbs formation energy of diffusion species is represented by


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∆&'()* = & ,'' − +- − ./ − .-0 ,

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(4)

where & ,'' is the Gibbs free energy of the Ni slab model with diffusion species, +- the total energy of the clean slab, ./ and .-0 the chemical potential of the gas species and the atom in the bulk system, respectively. To consider the temperature effect on the Gibbs free energy, the enthalpy and entropy terms were calculated on the basis of vibrational frequency analysis. We note that the zero point vibrational energy was also considered. In this analysis, we assumed that the vibrational frequencies of bulk atoms do not change with and without adsorbates. Therefore, the zero point vibrational energy, enthalpy, and entropy of the slab system were not considered due to the cancellation of these values with and without the diffusion species on the surface. The enthalpy and entropy for gas phase were calculated referring the database values.29,30 According to the random-walk theory,31 pre-exponential factor of diffusion for single diffusion species is given by =

12 3 4 - 5 6

,

(5)

where 78 is the number of equivalent diffusion pathways, 9 the pre-exponential factor of attempt frequency for diffusion, : the diffusion distance, and ; the dimension of diffusion (in the case of surface diffusion, ; equals to 2). In Eq. (5), 9 is calculated from vibrational frequencies at equilibrium and transition state structures32 =>C@ B 9 = ∏=> ,?@ 9, D∏A?@ 9A ,

(6)

where 9, and 9AB are vibrational frequency at equilibrium and transition state structures, respectively. To obtain the pre-exponential factor of the attempt frequency and the activation energy for diffusion, we performed the transition state calculations by using the climbing image nudged


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elastic band (CI-NEB) method.33,34 We assumed the hopping mechanism as a diffusion mechanism in this study while there are some diffusion mechanisms, such as hopping, exchange, vacancy, and so on.35

2.2. DFT Calculation All DFT calculations were performed under spin-polarized conditions by using Vienna ab-initio simulation package (VASP).36-39 Generalized gradient approximation (GGA)-Perdew-BurkeErnzerhof (PBE) functional was used for exchange-correlation functional,40,41 and core electrons were treated by using projector-augmented wave (PAW) method.42,43 Plane wave energy cutoff was set to 400 eV and k-points were sampled by using Monkhorst-Pack method.44 To construct the Ni slab model, the Ni face-centered cubic (fcc) bulk structure was optimized by 14×14×14 k-points. Three layers Ni(111) slab with 10 Å vacuum region was modeled based on the optimized lattice constant and bottom layer of this model was fixed. Ni(111) slab model was prepared by using 3 × 3 and 6 × 6 supercells and k-points were set to 4 × 4 × 1 and 2 × 2 × 1, respectively, to keep the same k-points density in reciprocal space.

3. RESULTS AND DISCUSSION 3.1. Diffusion Coefficient of Surface Ni Species on Ni(111) Surface Before the evaluation of the effect of gas composition on surface diffusion on Ni, we calculated the surface diffusion coefficient of Ni adatom and Ni clusters. This calculation corresponds to the surface diffusion of Ni under vacuum condition. We considered three kinds of Ni species (Ni adatom, Ni2 cluster, and Ni3 cluster) on Ni(111) surface. Figure 1 shows the most stable structures for (a) Ni adatom, (b) Ni2 cluster, and (c) Ni3 cluster on Ni(111) surface.


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Figure 1. The most stable structures for (a) Ni adatom, (b) Ni2 cluster, and (c) Ni3 cluster on Ni(111) surface.

The most stable site for Ni adatom and Ni2 cluster on Ni(111) surface was 3-fold fcc hollow one. Contrary, the most stable site for Ni3 surface cluster on Ni(111) surface was 3-fold hcp hollow one. The formation energy of these structures at 0 K calculated by Eq. (4) is summarized in Table 1.

Table 1. Formation Free Energy of Ni Adatom, Ni2 Cluster, and Ni3 Cluster on Ni(111) Surface at 0 K on Fcc and Hcp 3-Fold Sites. Formation Free energy (eV) fcc 3-fold site

hcp 3-fold site

Ni adatom

1.045

1.047

Ni2 cluster

1.741

1.831

Ni3 cluster

2.233

2.086


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To consider the SOFC operation temperature, the temperature dependency of formation free energy of Ni species was analyzed as shown in Figure 2. The zero point vibrational energy, enthalpy, and entropy at 1073 K per Ni atom are listed in Table 2.

Figure 2. Temperature dependency of the formation free energy of Ni species on Ni(111) surface.

Table 2. Zero Point Vibrational Energy, Enthalpy, and Entropy at 1073 K Per Ni Atom for Ni Adtom, Ni2 Cluster, and Ni3 Cluster on Ni(111) Surface. Zero point vibrational energy (eV)

Enthalpy (eV)

Entropy (×10-3 eVK-1)

Ni adatom

0.051

0.558

0.517

Ni2 cluster

0.027

0.417

0.701

Ni3 cluster

0.029

0.370

0.678


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The formation free energy of Ni species became smaller with increasing the temperature. This trend was enhanced with increasing the number of Ni atoms. This result indicates that the most stable or most abundant Ni species on Ni(111) surface depends on temperature. To obtain the activation energy for surface diffusion of Ni, we calculated the transition state between the most stable and the next stable sites by using CI-NEB method. The transition state structure was confirmed by the single imaginary vibrational frequency of the obtained structure. Figure 3 shows the minimum energy path of surface diffusion of Ni adatom, Ni2 cluster, and Ni3 cluster on Ni(111) surface.


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Figure 3. Minimum energy path of surface diffusion of (a) Ni adatom, (b) Ni2 cluster, and (c) Ni3 cluster on Ni(111) surface. Top view of optimized structures of initial, transition, and final states are shown.

The activation energies for surface diffusion of Ni adatom, Ni2 cluster, and Ni3 cluster were 0.069, 0.066, and 0.371 eV, respectively. Pre-exponential factors of Ni adatom, Ni2 cluster, and Ni3 cluster were 1.98×10-7, 7.05×10-8, and 2.39×10-7 m2s-1, respectively. We calculated the surface diffusion coefficient of Ni adatom, Ni2 cluster, and Ni3 cluster by using the formation energy, activation energy, and pre-exponential factor. However, it is difficult to use the formation energy directly for Ni2 and Ni3 clusters, because our assumption is formation at kink site or detachment from kink site. Therefore, we assumed the coalescence of two and three Ni adatoms to form Ni2 and Ni3 clusters. Figure 4 shows the Arrhenius plot of the surface diffusion coefficient of Ni species on Ni(111) surface with experimental values measured under vacuum condition.45 Table 3 summarizes the calculated apparent activation energy and pre-exponential factor for Ni surface diffusion with experimental values.45 Here, the apparent activation energy in this study means the activation energy of surface diffusion considering the concentration of diffusion species. This apparent activation energy is obtained from the slope of Arrhenious plot considering the concentration of diffusion species. The surface diffusion coefficient of Ni calculated by DFT was described by the summation of diffusion coefficients for Ni adatom, and Ni2 cluster, and Ni3 cluster.


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Figure 4. Arrhenius plot for the calculated surface diffusion coefficient of Ni species on Ni(111) surface and the experimental values measured in vacuum condition.45 Solid and dashed lines mean the calculated and experimental results, respectively.

Table 3. Calculated Apparent Activation Energies and Pre-exponential Factors for Surface Diffusion of Ni with Experimental Values.45 Apparent activation energy (eV)

Pre-exponential factor (m2s-1)

This work

1.276

2.45×10-6

Expt. (high purity Ni)

0.788

8.0×10-8

Expt. (low purity Ni)

0.919

1.8×10-7

The calculated surface diffusion coefficient of Ni is in a reasonable agreement with experimental values. The contribution of each Ni adatom, Ni2 cluster, and Ni3 cluster to the overall diffusion coefficient at 1073 K was 0.979, 0.021, and 0.000, respectively. This result indicates that the Ni adatom is a dominant species for surface diffusion on Ni(111).


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3.2. Effect of Hydrogen on Surface Diffusion of Ni We evaluated the effect of hydrogen on the surface diffusion of Ni. Six compositions of Ni-H complexes, NiH, Ni2H, Ni2H2, Ni3H, Ni3H2, and Ni3H3, were considered in this study. Figure 5 shows the most stable structure for each Ni-H complex. Table 4 summarizes the formation energy for each Ni-H complex at 0 K.

Figure 5. The most stable structures for (a) NiH, (b) Ni2H, (c) Ni2H2, (d) Ni3H, (e) Ni3H2, and (f) Ni3H3 complexes on Ni(111) surface.

Table 4. Formation Energy of Ni-H Complexes at 0 K. Ni-H Complex

Formation energy (eV)

NiH

0.653

Ni2H

1.253

Ni2H2

1.146

Ni3H

1.720


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Ni3H2

1.559

Ni3H3

1.272

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It was found that the NiH complex is the most stable although the positive formation energy was obtained for all the Ni-H complexes investigated. Therefore, we focused on the NiH complex to evaluate the effect of hydrogen on surface diffusion of Ni. From vibrational frequency analysis, the zero point vibrational energy, enthalpy, and entropy of NiH complex at 1073 K were 0.17 eV, 0.858 eV, and 0.983×10-3 eVK-1, respectively. To obtain the diffusion coefficient mediated by NiH complex, we calculated the minimum energy path of diffusion of NiH complex on Ni(111) as illustrated in Figure 6.

Figure 6. Minimum energy path of surface diffusion of NiH complex on Ni(111) surface. Top view of optimized structures of initial, transition, and final states are shown.

The activation energy and the pre-exponential factor of NiH diffusion were obtained by 0.064 eV and 2.01×10-8 m2s-1, respectively. We calculated the surface diffusion coefficient of NiH


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complex based on our calculated results. Figure 7 shows the Arrhenius plot of surface diffusion coefficient of NiH complex under the different hydrogen partial pressure.

Figure 7. Arrhenius plot of surface diffusion coefficient of NiH complex with different hydrogen partial pressure.

It was found that the hydrogen has no acceleration effect on the surface diffusion on Ni(111). Experimentally, the surface diffusion coefficient of Ni on Ni surface measured under hydrogen atmosphere showed the acceleration of the surface diffusion of Ni by hydrogen.46-48 In literature, the acceleration effect by hydrogen was reported for Pt(110) surface.22 The result of Pt surface shows the possibility that the surface orientation influences the surface diffusion. The diffusion analysis of Ni not only (111) but also other surface orientation, such as (110) and (100), might be necessary in the future work to understand the origin of acceleration effect by hydrogen on the Ni sintering.

3.3. Effect of Sulfur on Surface Diffusion of Ni


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To evaluate the effect of sulfur on the surface diffusion of Ni, we considered seven Ni-S complexes, NiS, Ni2S, Ni2S2, Ni3S, Ni3S2, and Ni3S3. Figure 8 shows the most stable structures for each Ni-S complex. Table 5 summarizes the formation energy for each complex at 0 K.

Figure 8. The most stable structures for (a) NiS, (b) Ni2S, (c) Ni2S2, (d) Ni2S3, (e) Ni3S, (f) Ni3S2, and (g) Ni3S3 complexes on Ni(111) surface.

Table 5. Formation Energy of Ni-S Complexes at 0 K. Ni-S complex

Formation energy (eV)

NiS

-1.048

Ni2S

0.242

Ni2S2

0.171

Ni2S3

-4.612

Ni3S

0.004

Ni3S2

-2.032


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Ni3S3

-4.994

It was found that NiS, Ni2S3, Ni3S2, and Ni3S3 are stable. As mentioned in 3.1, these formation energies are defined as the formation of Ni adatom from the kink site. Therefore, we assumed that larger and stable clusters, Ni2S3, Ni3S2, and Ni3S3 are not directly formed at kink site but are formed as results of coalescence between atoms or smaller surface species. Thus, we focused on the formation and diffusion of NiS complex as a first step in the subsequent analysis in this study. We calculated the zero point vibrational energy, enthalpy, and entropy of NiS complex at 1073 K based on the vibrational frequency analysis. The results were 0.073 eV, 0.836 eV, and 1.264×103

eVK-1, respectively. Based on these results, the temperature dependency of NiS complex

coverage on Ni(111) surface with different H2S concentration is calculated as shown in Figure 9.

Figure 9. Temperature dependency of NiS complex coverage on Ni(111) surface with different H2S concentration.

It was found that the coverage of NiS complex increases with increasing the H2S concentration and decreasing the temperature. The reason is that the formation of NiS complex becomes stable


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with decreasing temperature. Note that the coverage of NiS complex in the lower temperature region and the higher H2S concentration is not accurate because the low surface coverage was assumed in Eq. (2). However, the coverage of NiS complex under a typical SOFC operating condition, such as above 900 K and below several ppm H2S, is sufficiently low. Thus it is worthwhile to discuss the effect of NiS complex formation on surface diffusion of Ni focusing on the low coverage region. To obtain the surface diffusion coefficient of NiS complex, we calculated the activation energy and pre-exponential factor for diffusion of NiS complex. Figure 10 shows the minimum energy path of surface diffusion of NiS and Ni3S3 complexes on Ni(111) surface.

Figure 10. Minimum energy path of surface diffusion of (a) NiS complex and (b) Ni3S3 complex on Ni(111) surface. Top view of optimized structures of initial, transition, and final states are shown.


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The activation energy and pre-exponential factor for diffusion of NiS complex were 0.299 eV and 5.93×10-8 m2s-1, respectively. Calculated surface diffusion coefficient of NiS complex is shown in Figure 11. The surface diffusion of Ni adatom on Ni surface calculated in 3.1, which is a model of vacuum condition, is also shown in Figure 11.

Figure 11. Arrhenius plot of surface diffusion coefficient of NiS complex with different H2S concentration. Filled and open symbols are reversible and irreversible degradation conditions in experiments, respectively.19,49-53

It was found that the surface diffusion of NiS complex becomes faster with increasing H2S concentration and decreasing temperature. The opposite effect of H2 and H2S on the surface diffusion was observed as results of the different concentration of surface species. The formation energy of NiS is smaller than that of Ni adatom, thus, the NiS concentration on the surface becomes larger than that of Ni adatom. The opposite case is applicable to NiH from our calculation. Thus, the NiS complex becomes a main diffusion species on the surface under lower temperature and higher H2S concentration condition, while Ni atom is the dominant species of


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surface diffusion under the higher temperature and lower H2S concentration. As mentioned above, the Ni3S3 complex was the most stable complex at 0 K. To estimate the effect of larger complex formation, we simply used three assumptions; 1) all of NiS complex transform to Ni3S3 complex, 2) the coverage of Ni3S3 complex is equal to that of NiS complex, and 3) the change of diffusion coefficient can be described by ratio of diffusion coefficient of single Ni3S3 and NiS complex. Obtained activation energy and pre-exponential factor for diffusion of Ni3S3 complex were 0.610 eV and 1.11×10-7 m2s-1, respectively. The ratio of diffusion coefficient of single Ni3S3 and NiS complexes was about 2. This indicates that the surface diffusion coefficient of Ni3S3 complex at the same temperature and H2S concentration is twice larger than that of NiS. Although this assumption is simple for the estimation of diffusion of larger Ni-S complex, we clearly showed the importance of the treatment of the real coverage conditions by all Ni-S complexes considering with all element reaction for complex formation and diffusion by using diffusion-reaction equation.

3.4. Comparison with Experimental Results In above section, we theoretically found that the H2S contamination affects to the surface diffusion of Ni by forming the Ni-S complex on the Ni surface. To confirm the reliability of our results, we compared our results with the experimental results about sintering of Ni.19,49-53 Matsuzaki and Yasuda measured the electrochemical performance of Ni-YSZ anode with 0.05, 0.5, and 2.0 ppm H2S at 1073, 1173, and 1273 K, respectively.49 They observed the performance degradation due to the sulfur poisoning and the performance recovery after the removal of adsorbed sulfur. This result corresponds to the reversible degradation condition. On the other hand, Ivey et al. reported the promotion of Ni agglomeration in Ni-YSZ thin film with


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1.0 ppm H2S at 973 K.50 Zha et al. also observed the sulfur poisoning degradation from the measurement of the electrochemical performance of Ni-YSZ anode with 2.0 ppm H2S at 1073 K.51 Their anode showed the Ni sintering after H2S removal. The calculated results can be classified into two regions, i.e. the region where the diffusion coefficient of is larger than that of Ni and vice versa. The reversible degradation data are plotted in the former region while the irreversible degradation data are plotted in the latter. Clearly, our calculation results shown in Figure 11 are in good agreement with the literature in terms of the reversible and irreversible degradation conditions. Contrary, the formation of Ni sulfide layer has been reported under the lower temperature or higher H2S concentration condition. Harris et al. observed the Ni sulfide layer formation during the cooling down of the Ni-YSZ anode from 1073 K to room temperature with 100 ppm H2S.52 Deleebeeck et al. also observed the Ni sulfide layer under 5 ppm H2S at 873 to 773 K condition.53 Although these results are also categorized as irreversible degradation condition, the Ni sulfide layer formation is not explained by our present calculation results because we only assumed the Ni-S complex formation on Ni(111) surface. From the comparison with experimental results, we clearly demonstrated that our calculation results are effective to understand the Ni sintering degradation under the different temperature and H2S concentration conditions. By performing similar calculations for various gas compositions, we will be able to predict the sintering behavior of Ni under complicated anode gas compositions.

4. CONCLUSIONS The surface diffusion coefficient of Ni on Ni(111) surface considering with H2 and H2S contamination is calculated by using DFT. Calculated diffusion coefficient of Ni adatom, Ni2


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cluster, and Ni3 clusters on Ni surface were in agreement with experimental values under the vacuum condition. Main diffusion species on Ni(111) was Ni adatom under the H2 gas. To consider the H2S as an anode gas composition, we assumed the formation of NiS complex on Ni surface. Our calculation suggests that the diffusion coefficient of NiS complex increases with decreasing the temperature and increasing the H2S concentration. The experimental results for the temperature and H2S concentration dependency on sintering of Ni were well explained by our calculation results. Therefore, our DFT calculation is effective to predict the sintering behavior of Ni under different temperature and H2S concentration conditions.

AUTHOR INFORMATION Corresponding Author *Tel/Fax +81-92-802-6968 E-mail: [email protected]

ACKNOWLEDGMENT This work was supported by CREST, JST and JSPS KAKENHI, Grant-in-Aid for JSPS Fellows, 127877. Activities of INAMORI Frontier Research Center is supported by KYOCERA Corporation. The part of the computation was carried out using the computer facilities at Research Institute for Information Technology, Kyushu University.

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Supporting Information. The temperature dependency of enthalpy and entropy terms for Ni adatom, Ni2 cluster, and Ni3 cluster are shown in supporting information. This material is available free of charge via the Internet at http://pubs.acs.org.”

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