Sintering Simulation for Porous Material by Integrating Molecular

Article pubs.acs.org/JPCC

Sintering Simulation for Porous Material by Integrating Molecular Dynamics and Master Sintering Curve Kazuhide Nakao,† Takayoshi Ishimoto,‡ and Michihisa Koyama*,‡,§ †

Department of Hydrogen Energy Systems, Graduate School of Engineering, ‡INAMORI Frontier Research Center, and International Institute for Carbon-Neutral Energy Research (I2CNER), Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan §

ABSTRACT: In this study, we propose a combination of molecular dynamics (MD) and the master sintering curve (MSC) approach to analyze the activation energy of sintering for porous materials. MD calculations were performed by using a porous structure with various initial densities, and the change of relative density with simulation time was analyzed. To relate the MD results with long-term sintering behaviors of porous materials, we established a method to obtain the MSC, which is able to determine the activation energy of sintering, on the basis of these MD simulation results. We have successfully obtained sintering behavior and activation energies of sintering depending on temperature range and particle diameter. These activation energies obtained by our approach are in agreement with experimental observations. In addition, the temperature dependence of activation energy of sintering is also in good agreement with that of surface diffusion, which indicates that surface diffusion is the dominant sintering mechanism.

1. INTRODUCTION Sintering is an important industrial process to produce metals, ceramics, and their composite from powder compacting. Understanding the densification of porous materials during sintering is important to design the desired microstructure for many applications. Therefore, prediction of densification and microstructural evolution is important to obtain sintered products that have the desired properties. Monte Carlo (MC) methods1,2 and phase-field models3,4 have been often used to simulate the microstructural evolution during sintering. These methods require some parameters and assumptions for mobility and mechanisms corresponding to actual sintering process. On the other hand, the molecular dynamics (MD) method does not use some empirical macroscopic parameters and assumptions related to sintering mechanisms required for MC or phase-field simulations; however, many studies for sintering simulation using MD have been focused on only two- or threeparticle systems to analyze neck growth behavior and neck growth mechanism.5−14 These simple models are useful for the sintering in the gas phase or on the substrate. However, it is not sufficient to consider parameters related to properties of porous materials, such as relative density, grains, and pores sizes. In addition, it can be questioned whether the sintering mechanism of simple two- or three-particle systems is the same as that of porous material system. Recently, a few studies on sintering of porous materials have been reported by using MD simulation. Xu et al.15 modeled the porous structure of Ni- and yttria-stabilized zirconia (YSZ) cermet and confirmed the framework effect of YSZ on Ni sintering. Cheng and Ngan16 identified the sintering mecha© 2014 American Chemical Society

nism of Cu porous structure due to the variety of plasticity processes. However, these studies are not focused on the parameters related to sintering rate, such as the activation energy. Evaluating the activation energy of sintering is important to know the sinterability of materials. Therefore, we need to develop a methodology to analyze sintering properties based on MD simulation. In order to analyze the densification behavior by using MD simulation, we employ the master sintering curve (MSC) approach17 based on the combined-stage sintering model.18 A MSC is readily constructed by measuring densification with sintering temperature and time. To analyze the sintering phenomena, the sintering process is often divided into three stages: initial, intermediate, and final. In the combined-stage sintering model, these three stages are combined into one model, and densification behavior is modeled from the green body to the dense body. However, MSC strongly depends on the powder and fabrication process, even if the same materials are used. For 8 mol % Y2O3 doped ZrO2, Pouchly et al.19 has reported the activation energies of 750 and 620 kJ mol−1, while Rajeswari et al.20 and Thridandapani et al.21 have reported that of 350 and 200 kJ mol−1, respectively. Although these studies use the same commercially available ceramics powder, the apparent activation energies of sintering obtained by MSC have a large difference between these studies due to the difference in the green body fabrication process. Because MD simulation is able to control the initial structure in the simulation cell and Received: April 8, 2014 Revised: June 25, 2014 Published: July 15, 2014 15766

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with an interval of 100 K, respectively. Temperature and pressure were controlled by a Nose−Hoover thermostat and barostat, respectively. Porous structure was initially modeled by using POCO2 to pack particles in the simulation box, allowing the particle overwrapping25 (Figure 1a). After that, Ni face-

perform under various conditions, such as temperature, pressure, and structure, MD simulation for porous materials sintering and analysis by using MSC could reduce the effect of microstructure to understand the sintering nature. Therefore, in this study we propose an approach combing MD and MSC analysis in order to study the sintering properties. As an example, results for Ni porous structure composed of nanoparticles are compared with experimental results. The sintering mechanism for Ni porous structure is also discussed on the basis of the activation energy of sintering obtained from our approach.

2. THEORETICAL CALCULATIONS 2.1. Master Sintering Curve Based on MD Simulation of Sintering. The MSC, derived from the combined-stage sintering model and relative density for any sintering time and temperature, is plotted by eq 1 Θ(t , T (t )) =

∫0

t

⎛ Q ⎞ 1 ⎟d t exp⎜ − ⎝ RT ⎠ T

Figure 1. Example of the modeled porous structure by using POCO2 (a) and porous structure for MD calculation (b). Modeling conditions of particle size, cell dimension, and target porosity are 5.0 nm, 10.0 nm, and 0.45, respectively.

(1)

where t is the sintering time, T the sintering temperature as a function of sintering time, R the gas constant, and Q the apparent activation energy of sintering. If the MSC description is correct, then the relative density change is uniquely determined by eq 1. Details of the combined-stage sintering model and derivation of MSC from the combined-stage sintering model are described in refs 17 and 18, respectively. Note that the MSC assumes isotropic shrinkage and constant activation energy during sintering. By measuring densification and temperature profiles with different heating rate, MSC is obtained experimentally from the fitting of the apparent activation energy. In this study, results of MD simulation for sintering of porous materials are analyzed on the basis of this concept. However, it is difficult to simulate sintering from the green body to the dense body by using MD calculation because of the difference of time scale; i.e., sintering experiments are performed in the time range from minutes to hours, while MD simulations are for at most on the order of nanoseconds. Therefore, to obtain densification behavior for a long time by using MD calculation, we perform some sintering simulations by using different initial density, and we combine these results to obtain the MSC. To combine simulation results obtained from different initial density, we introduce a new fitting parameter, Δt, into eq 1. This fitting parameter represents for the sintering time from assumed initial density to density of created models. In this study, we perform isothermal sintering simulations to simplify the combination of simulation results. Therefore, eq 1 is rearranged as follows Θ(t , T ) =

⎛ Q ⎞ 1 ⎟(t + Δt ) exp⎜ − ⎝ RT ⎠ T

centered cubic (fcc) unit structure was set in each particle, and the fcc unit structure was randomly rotated. Then, atoms with the interatomic distance in the initial structure lower than 2.0 Å are removed from the simulation box. As a consequence, the surface orientation and contact surface were randomly set (Figure 1b). The initial porosity was set to 0.45, 0.40, 0.35, 0.30, 0.25, and 0.20. In this study, a particle diameter of 5.0, 7.5, and 10.0 nm was packed into the simulation box with dimensions of 10.0, 15.0, and 20.0 nm, respectively. Particle diameter was constant; thus, the particle size distribution was not considered. Three models were prepared for each initial porosity and particle diameter, and the averaged densification profile for three models was used as a densification behavior for each initial porosity and particle diameter. Before running the sintering simulation, all porous structures created here were stabilized through energy minimization at 0 K. To construct the MSC, the relative density was defined as the ratio of density of the simulation box to the density of the bulk Ni fcc crystal at the same temperature.

3. RESULTS AND DISCUSSION 3.1. Sintering Behavior of Ni Porous Structure. To confirm whether Ni porous structure shrinks during MD simulation, we checked the relative density change every step. Figure 2 shows a series of snapshots of crystalline structure during the MD calculation for a particle diameter of 5.0 nm and an initial porosity of 0.45 at 1400 K. Note that the snapshot represents the results of one of the three models, and plots of density change represent the averaged density for three models. Therefore, the relative density for each simulation time in snapshots does not agree with the density change shown in Figure 3. From snapshots in Figure 2, the large pore initially located at the center of the simulation box decreases with simulation time due to the densification. Figure 3 shows the relative density change as a function of time for each temperature. It is found that large densification occurs at the first 100 ps and then gradual densification occurs, regardless of the sintering temperature in Figure 3. This large densification during the first 100 ps was observed for all particle diameters and initial porosities considered in this study. Cheng and Ngan

(2)

By using eq 2, we combine calculation results of different initial densities and finally obtain the MSC from green density to sintered density by using MD. 2.2. Calculation Condition. In this study, we used pure nickel porous structure models in MD simulation as a simple material system. All MD calculations were performed by using LAMMPS,22,23 and the embedded atom method (EAM) developed by Mishin et al.24 was used for Ni−Ni interaction. For MD simulation, a time step of 2.0 fs, total steps of 500 000, and the NPT ensemble were employed. Pressure and temperature were set to 1.0 atm and from 600 to 1400 K 15767

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and 1000 ps as a result of densification during MD calculation to avoid the influence of the structure stabilization. Then, a simulation time of 200 ps is set to 0 in eq 2 and Δt is fitted as a sintering time from the initial density of each model to the simulated density at 200 ps. Figure 4a shows a plot of the

Figure 2. Snapshot of microstructural evolution during MD simulation at 1400 K (particle diameter of 5.0 nm and initial porosity of 0.45). Figure 4. Plot of relative density calculated by MD for a particle diameter of 5.0 nm and an initial porosity of 0.45 with eq 2. (a) Result for the tentative value of activation energy. (b) Fitted result by using data for the temperature range from 600 to 1400 K. (c) Fitted result for the temperature range from 600 to 900 K. (d) Fitted result for the temperature range from 1000 to 1400 K.

sampled relative density for Figure 2 by assigning tentative values (Δt = 0 and Q = 200 kJ mol−1) to eq 2. After the fitting, it is found that a series of plots for each temperature lies almost on a single curve (Figure 4b). However, by dividing plot data for low- and high-temperature regions (600−900 and 1000− 1400 K), the fitting error described by eq 3 for two temperature regions is lower than that for the whole temperature region (3.36 × 10−2). Error for low- and high-temperature regions are 0.18 × 10−3 (Figure 4c) and 2.22 × 10−3 (Figure 4d). Different activation energies are calculated for different temperature regions. The reason for these two different temperature regions is discussed in the next section. For other particle diameters and initial porosities, it is possible to use two temperature regions. Therefore, we analyze calculated results by using two temperature regions in the subsequent section. Figure 5 shows plots of the activation energy obtained from the MSC for the different initial particle sizes and densities. The initial density is defined as unity minus the initial porosity. It is found that the activation energy at both high- and low-temperature regions fluctuated for each initial density and particle diameter. It is indicated that the microstructure affects the activation energy of sintering. However, we do not discuss the detailed effects of microstructure on the activation energy of sintering in this paper. In order to calculate the finally obtained apparent activation energy of sintering in the below section, we assumed that the activation energy is constant for the same particle diameter systems based on the assumption of the MSC. We used the activation energy for each particle diameter averaged by the whole initial density as an initial value for the combination of simulation results. Fitting parameters a and b were also obtained from each particle diameter and initial density, and averaged values are used for initial values. 3.4. MSC Construction for Overall Density Region from the Results of MD Simulation. By using eq 2 and overall simulation results, we constructed a MSC through

Figure 3. Plot of relative density change with simulation time (particle diameter of 5.0 nm and initial porosity of 0.45).

also have reported the same behavior for Cu porous structure by MD simulation.16 We used the densification profiles after 200 ps to construct the MSC in this study, because the drastic changes of density at the beginning of simulation possibly include the effect of stabilization. 3.2. MSC Construction from MD Simulation. At first, we constructed the MSC for each particle diameter and initial porosity. To obtain the MSC based on eq 2, we minimized the error between the densification profile for each temperature and sigmoid type function, because it is reported that the MSC is well-represented by a sigmoid function,26 ⎡ ⎛ 1 − ρ0 ⎢ ⎜ error = ∑ ⎢ρcalc (Ti , t ) − ⎜ρ0 + ln θ − a ⎜ 1 + exp − b i ⎢ ⎝ ⎣

(

)

⎞⎤ ⎟⎥ ⎟⎟⎥ ⎠⎥⎦

2

(3)

where ρcalc is the relative density obtained from MD calculation given as a function of temperature (T ) and simulation time (t), ρ0 is the initial density, a and b are fitting parameters, and θ at T and t is determined by using eq 2. Note that the initial densities used in sections 3.3 and 3.4 are defined by the average density of each initial density model and by the initial porosity of 0.45, respectively. 3.3. Effect of Temperature, Particle Diameter, and Initial Porosity for MSC Construction. We used the averaged relative density at 200, 300, 400, 500, 600, 700, 800, 15768

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Figure 6. Errors for MSC fitting at each particle diameter and highand low-temperature regions: (a) 5.0 nm and low-temperature region, (b) 5.0 nm and high-temperature region, (c) 7.5 nm and lowtemperature region, (d) 7.5 nm and high-temperature region, (e) 10.0 nm and low-temperature region, and (f) 10.0 nm and hightemperature region.

behavior is smaller with increasing particle diameter. Therefore, this effect of microstructure may contribute to the large fluctuation with decreasing particle diameter. Nevertheless, all activation energy obtained from these MSCs is similar to the average activation energy for every initial density. It is thus indicated that the sintering mechanism is the same during sintering and the change of particle diameter is negligible, because the particle diameter has an effect on the activation energy of sintering and sintering rate. 3.5. Comparison with the One-Particle Model. Assuming that atomic diffusion is the dominant mechanism for densification, the activation energy of sintering corresponds to that of diffusion for the dominant diffusion mechanism. According to the reported activation energy for each Ni diffusion mechanism, such as lattice, grain boundary, and surface diffusion,27−36 the activation energy of sintering corresponds to the surface diffusion. To evaluate the activation energy of surface diffusion for each particle diameter, we calculated the surface diffusion coefficient by using the oneparticle model. One particle is prepared by using the same method as in section 2 and placed in a large simulation box. The surface diffusion coefficient was calculated from the mean square displacement of surface atoms. Here, surface atom is defined as atoms initially located 2.0 Å from the surface. The Arrhenius plot of the surface diffusion coefficient for each particle diameter is shown in Figure 8. The activation energy of the surface diffusion is calculated from the slope of the Arrhenius plot, and results are summarized in Table 1. These activation energies are in good agreement with experimental values.33−36 From Figure 8 and Table 1, it is found that the activation energy changed around 900 K. This result indicates that the activation energy of surface diffusion shows the same trend as that of sintering for porous structure obtained from MSC construction. In addition, these values are also similar to the activation energy of sintering, especially for the high-temper-

Figure 5. Relationship between the activation energy and initial density for each particle diameter: (a) 5.0 nm, (b) 7.5 nm, and (c) 10.0 nm.

averaged values for fitting parameters obtained from each initial density. Errors expressed in eq 3 and results of the MSC construction for each particle diameter are shown in Figures 6a−f and 7a−c, respectively. The activation energy of sintering, which was determined as the activation energy minimizing the errors in eq 3, is listed in Table 1. In Figure 7, the dots and curve represent simulation results and the fitted MSC, respectively. From Figure 6, the change of errors depending on activation energy is not so much, especially for the hightemperature region. This indicates that the activation energy of sintering listed in Table 1 includes some errors, and the activation energy around values obtained in this study could also obtain almost the same MSCs. From Figure 7, it is found that almost all of calculation results lie on the fitted MSCs. However, fluctuation becomes larger with decreasing particle diameter. Because density change became larger with decreasing particle diameter at the same simulation time, it is assumed that the effect of microstructure on densification 15769

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Figure 8. Arrhenius plot of Ni surface self-diffusion coefficient for each particle diameter.

Moon et al. reported an activation energy for nickel nanoparticle sintering in the gas phase (60 ± 10 and 63 kJ/ mol).37,38 They suggested that the dominant sintering mechanism is surface diffusion and particle migration, respectively. In addition, Panigrahi reported the activation energy of 66.2 ± 3 kJ/mol by using nickel nanoparticle compaction and suggested that surface diffusion is the dominant sintering mechanism.39 In the case of compact nickel powder composed of micrometer powders, Blaine et al. reported the activation energy of 108 kJ/mol by using MSC analysis.26 This value is in good agreement with the activation energy of the grain boundary diffusion,40 which indicates that sintering of nickel compacts composed of micrometer scale powders is dominated by the grain boundary diffusion. Activation energies obtained in this study at high temperature are relatively higher than reported values for a nanoparticles system. This might be caused by the difference between particle shape and surface morphology. Although porous models in this study are composed of identical spherical particles without particle size distribution and surface defects, for practical conditions there is a particle size distribution, and surface defects and particle shape are also inhomogeneous. On the other hand, activation energies in this study are similar to a large-particle system. This might be caused by similar activation energies between surface and grain boundary diffusion of Ni. As mentioned in above sections, however, the activation energy and temperature dependency of sintering and surface diffusion calculated in this study are in good agreement with each other. If the activation energy of sintering obtained from constructed MSC corresponds to the activation energy of the dominant diffusion mechanism, we could suggest that the surface diffusion is the dominant sintering mechanism for a porous structure composed of Ni nanoparticles. It is generally known that the surface diffusion mainly contributes to grain growth and has little effect on densification. According to the sintering model, by coupling surface and grain boundary diffusion, as developed by Wakai and Brakke,41 however, surface diffusion could also contribute to shrinkage of particles. If the product of surface diffusion coefficient and surface diffusion area is large enough compared with the product of grain boundary diffusion coefficient and grain boundary diffusion area, shrinkage of particles is dominated by the surface diffusion. Our models are composed of Ni nanoparticles, and the surface area may be large enough to be dominated by surface diffusion. Therefore, it is reasonable to suggest that densification for our model is dominated by surface diffusion.

Figure 7. Constructed MSC for the overall density range at each particle diameter: (a) 5.0 nm, (b) 7.5 nm, and (c) 10.0 nm.

Table 1. Activation Energy of Sintering (Qsinter) and Surface Diffusion (Qdiff) for Each Particle Diameter particle diameter (nm)

temp (K)

Qsinter (kJ/mol)

Qdiff (kJ/mol)

5.0 5.0 7.5 7.5 10.0 10.0

600−900 1000−1400 600−900 1000−1400 600−900 1000−1400

16.6 107.3 11.3 87.7 8.7 74.9

18.1 89.3 27.6 96.8 70.9 102.6

ature region. Therefore, it is suggested that the dominant sintering mechanism for Ni porous structure composed of nanoparticles is surface diffusion. From the trajectories of MD simulation of the one-particle model, it is found that mobile Ni adatom is only formed from the relatively unstable (110) facet and edge regions at low temperature, while it is formed from more stable (111) and (100) facets at high temperature. This difference of mobile nickel adatom formation mechanism could lead to the change of activation energy of surface diffusion by temperature. 3.6. Sintering Mechanism for Ni Porous Structure. The activation energy obtained in this study is comparable with available experimental nickel sintering data. Tsyganov et al. and 15770

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(13) Nguyen, N. H.; Henning, R.; Wen, J. Z. J. Molecular Dynamics Simulation of Iron Nanoparticle Sintering during Flame Synthesis. J. Nanopart. Res. 2011, 13, 803−815. (14) Hussain, F.; Hayat, S. S.; Imran, M.; Ahmad, S. A.; Bouafia, F. Sintering and Deposition of Nanoparticles on Surface of Metals: A Molecular Dynamics Approach. Comput. Mater. Sci. 2012, 65, 264− 268. (15) Xu, J.; Sakanoi, R.; Higuchi, Y.; Ozawa, N.; Sato, K.; Hashida, T.; Kubo, M. Molecular Dynamics Simulation of Ni Nanoparticles Sintering Process in Ni/YSZ Multi-Nanoparticle System. J. Phys. Chem. C 2013, 117, 9663−9672. (16) Cheng, B.; Ngan, A. H. W. The Sintering and Densification Behavior of Many Copper Nanoparticles: A Molecular Dynamics Study. Comput. Mater. Sci. 2013, 74, 1−11. (17) Su, H.; Johnson, D. L. Master Sintering Curve: A Practical Approach to Sintering. J. Am. Ceram. Soc. 1996, 79, 3211−3217. (18) Hansen, J. D.; Rusin, R. P.; Teng, M.-H.; Johnson, D. L. Combined-Stage Sintering Model. J. Am. Ceram. Soc. 1992, 75, 1129− 1135. (19) Pouchly, V.; Maca, K.; Shen, Z. Two-Stage Master Sintering Curve Applied to Two-Step Sintering of Oxide Ceramics. J. Eur. Ceram. Soc. 2013, 33, 2275−2283. (20) Rajeswari, K.; Padhi, S.; Reddy, A. R. S.; Johnson, R.; Das, D. Studies on Sintering Kinetics and Correlation with the Sinterability of 8Y Zirconia Ceramics Based on the Dilatometric Shrinkage Curves. Ceram. Int. 2013, 39, 4985−4990. (21) Thridandapani, R. R.; Folz, D. C.; Clark, D. E. Development of a Microwave Dilatometer for Generating Master Sintering Curves. Meas. Sci. Technol. 2011, 22, 105706. (22) http://lammps.sandia.gov. (23) Plimpton, S. Fast Parallel Algorithms for Short-Range Molecular Dynamics. J. Comput. Phys. 1995, 117, 1−19. (24) Mishin, Y.; Farkas, D.; Mehl, M. J.; Papaconstantopoulos, D. A. Interatomic Potentials for Monoatomic Metals from Experimental Data and Ab Initio Calculations. Phys. Rev. B 1999, 59, 3393−3407. (25) Koyama, M.; Ogiya, K.; Hattori, T.; Fukunaga, H.; Suzuki, A.; Sahnoun, R.; Tsuboi, H.; Hatakeyama, N.; Endou, A.; Takaba, H.; Kubo, M.; Del Carpio, C. A.; Miyamato, A. Development of ThreeDimensional Porous Structure Simulator POCO2 for Simulations of Irregular Porous Materials. J. Comput. Chem. Jpn. 2008, 7, 55−62. (26) Blaine, D. C.; Park, S. J.; Suri, P.; German, R. M. Application of Work-of-Sintering Concepts in Powder Metals. Metall. Mater. Trans. A 2006, 37, 2827−2835. (27) Wazzan, A. R. Lattice and Grain Boundary Self-Diffusion in Nickel. J. Appl. Phys. 1965, 36, 3596−3599. (28) Maier, K.; Mehrer, H.; Lessmann, E.; Schle, W. Self-Diffusion in Nickel at Low Temperatures. Phys. Status Solidi B 1976, 78, 689−688. (29) Bokstein, B. S.; Bröse, H. D.; Trusov, L. I.; Khvostantseva, T. P. Diffusion in Nanocrystalline Nickel. Nanostruct. Mater. 1995, 6, 873− 876. (30) Zhilyaev, A. P.; Nurislamova, G. V.; Suriñach, S.; Baró, M. D.; Langdon, T. G. Calorimetric Measurements of Grain Growth in Ultrafine-Grained Nickel. Mater. Phys. Mech. 2002, 5, 23−30. (31) Spingarn, J. R.; Jacobson, B. E.; Nix, W. D. High Ductilities in Physically Vapor-Deposited Nickel. Thin Solid Films 1977, 45, 507− 515. (32) Karashima, S.; Oikawa, H.; Motomiya, T. Steady-State Creep Characteristics of Polycrystalline Nickel in the Temperature Range 500° to 1000°C. Trans. Jpn. Inst. Met. 1969, 10, 205−209. (33) Melmed, A. J. Surface Self-Diffusion of Nickel and Platinum. J. Appl. Phys. 1961, 38, 1885−1892. (34) Bonzel, H. P.; Latta, E. E. Surface Self-Diffusion on Ni(110): Temperature Dependence and Directional Anisotropy. Surf. Sci. 1978, 76, 275−295. (35) Blakely, J.; Mykura, H. Surface Self Diffusion Measurements on Nickel by the Mass Transfer Method. Acta Metall. 1961, 9, 23−31. (36) Bonzel, H.; Gjostein, N. Diffraction Theory of Sinusoidal Gratings and Application to In Situ Surface Self Diffusion Measurements. J. Appl. Phys. 1968, 39, 3480−3491.

4. CONCLUSIONS In this study, we applied the MSC approach to analyze the sintering behavior for porous material calculated by MD simulation. By using the MSC approach, we obtain the activation energy of sintering for porous structure composed of Ni nanoparticles from MD simulation. In addition, we succeeded in obtaining the densification behavior for long time scale by combining simulation results of different initial densities. The activation energy of sintering was constant, which indicated that the mean particle diameter is constant from the initial to the intermediate sintering stage. Obtained activation energies of sintering depend on the temperature range and particle diameter. These values are in good agreement with the activation energy for surface diffusion, and we concluded that sintering for porous structure composed of Ni nanoparticles is dominated by the surface diffusion.

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AUTHOR INFORMATION

Corresponding Author

*Tel/Fax: +81-92-802-6968. E-mail: [email protected]. jp. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI, Grant-in-Aid for JSPS Fellows (127877). Activities of INAMORI Frontier Research Center is supported by KYOCERA Corp.

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REFERENCES

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