Cellular Automata Modelling of Microstructure ...

Cellular Automata Modelling of Microstructure Evolution of Ni Cermet Anode X. Wang and A. Atkinson Department of Materials, Imperial College, London SW7 2AZ, UK

The degradation of SOFC electrodes is largely determined by their microstructure changes. It is desirable that the microstructure evolution of electrodes in service can be simulated in 3-D for real materials so that the important parameters such as conductivities, TPB (triple phase boundary) density and tortuosity can be predicted. In this contribution a new cellular automaton based method of modelling microstructure evolution is presented. It is shown that the complex topological and morphological changes in a Ni-YSZ anode can be reproduced using this modelling method. The modelling results show that the microstructure evolution is sensitive to the wettability of Ni over YSZ. It is demonstrated that a good wettability is helpful in maintaining a slower coarsening rate and slower degradation rate of TPB density.

Introduction SOFCs offer high chemical to electrical conversion efficiencies due to the absence of the Carnot limitation (1). SOFCs consist of four essential components: the electrolyte, the air electrode, the fuel electrode, and the interconnect. Typically operating at temperatures in the range 700–950°C, the microstructures of many SOFC materials are unstable during operation due to the mobility of some of their constitutive phase(s) (e.g., metals) and long operational lifetimes that are required. Microstructure coarsening (2, 3) and interface degradation (4) were observed in SOFCs, which could lead to severe degradation of cell performance. The microstructure of an electrode is regarded as being almost as important as its composition. This is because fuel cell electrodes typically have a complex micro/nano-structure involving interconnected electronically and ionically conducting phases, gas-phase porosity, and catalytically active surfaces (5). The optimization of durable efficient nickel-cermet anodes in recent decades has relied greatly on empirical improvement of cermet morphology (1). For better understanding of the electrode performance and its time dependence, it is essential to achieve full understanding of microstructure evolution during operation. Porous electrode materials have complicated topological and morphological features in their microstructure that are critical to their properties and functionality. Classical theories and modelling techniques are either unsuitable or computationally too expensive to model real microstructures. Here it is shown that, by using cellular automata and based on a concept of “structural imbalance”, complicated changes of a real cermet anode microstructure can be modelled as the consequence of matter movement guided by interface energy differences.

Methodology Cellular automata (CA) related modelling techniques are powerful methods to describe, understand and simulate the behaviour of complex systems (6). In CA, the space is discretized and made up of cells, each characterized by an internal state property. The system evolves in discrete time steps, like simple automata, according to a set of rules which is a function of the states of the neighboring cells. Materials are composed of atoms or molecules. Any macroscopic behavior of a material would be the natural result of atomic or molecular movements which, in turn must follow certain laws of physics. The key to success of CA modelling is correctly transforming the laws of physics into the interaction between the cells and local transition rules. A concept of structural imbalance has been described to determine the interface energy (7) . The basis of this approach is that any cell is assumed to have a pair-wise interaction energy with other cells in the material which decays with the distance between cells in a similar way to pair-wise interatomic potentials. Thus a cell is in this respect like an atom, but can be much larger, incorporating many atoms having a similar environment. The structural imbalance for an interface cell is determined by integrating the cell interaction energies within a designated range (mask). The interface cell energy for a 3-D case can be calculated using the following formula (7).:

[1]

where Vi=(2i-1)3, rij=i,

is the Kronecker delta: i.e. when the two cells are the same

type (ci=cj), , otherwise . is a mismatch coefficient. The relative structural imbalance with respect to a flat surface is defined as:

[2] where Uf is the structural imbalance of a flat surface cell and for 3-D:

[3] It has been demonstrated in previous work that, with the availability of the interface cell energy ( ), all natural phenomena (in 2D) associated with microstructure changes, such as grain growth, particle shape, coarsening, wetting, grooving can be easily reproduced by the CA model using well-established physical laws as a simple local transition rule (7). Here we will apply the same principle to simulate the 3-D microstructure evolution of a real nickel-cermet anode.

Various experimental evidence implies that surface evaporation-deposition enhanced by the presence of water vapour is the dominant mechanism that leads to microstructure changes in nickel cermet anodes (eg., Ref (2)). In the CA model, evaporation-deposition is described by removing matter from an occupied surface cell and depositing it at a vacant cell elsewhere on the surface. Thus each time step in the CA contains two sub-steps: one for detachment, and another for attachment. Here it is also assumed that detachment / attachment is the rate controlling step and that the partial pressure of nickel-containing vapour species is constant. Surface cells can be either occupied cells or vacant cells. At each detachment step, all the surface occupied cells are allowed a chance of leaving their original positions with probabilities proportional to their energy. A maximum possible cell energy is used as a reference, and assumed to have a probability of 1 to change its state. For a specific surface cell, the probability to change its state is its energy level divided by the maximum possible energy, with the cell energy level calculated by Eqs.1-3. A final decision on whether or not the cell will detach is made by a Monte Carlo step. At each attachment step, the vacant surface cells with higher energies are filled preferentially with the detached material (7). Although in reality, nickel could be lost from the system through the vapour phase, in the following modelling a conservation of nickel mass is assumed, which means the number of cells detached in the detachment step is equal to the number of cells attached in attachment step. In the meantime the zirconia phase is assumed to be immobile. To demonstrate how this CA model works on a simple microstructure, Fig.1a shows two nickel cubes sitting on a flat square YSZ plate. YSZ is assumed immobile at operating temperatures (800-950ºC), while Ni is mobile via evaporation-deposition. Fig. 1b gives the CA simulated microstructure after 200 time steps assuming the mismatch coefficient Cmis=0.8, while Fig. 1c gives the CA simulated microstructure after 200 time steps for the case of Cmis=0.2. Here Cmis represents the non-compatibility between YSZ and nickel. In the CA model, the influence of Cmis on microstructure is due to its influence on the cell energy (via Eq.1). According to (7), Cmis = 0.8 corresponds to a wetting angle of 125º and 0.2 corresponds to a wetting angle of 40º. From Fig.1b, it can be seen that, due to a relatively poor wettability of the Ni on YSZ, the original two cubic Ni particles have merged into one large sphere-like particle after 200 time steps. The particle is not exactly spherical because particle growth has reached the top of the modeled space (due to the periodic boundary condition, touching at the top means touching the bottom of the repeated YSZ plate). In contrast, the relatively good wettability (in Fig.1c) has led to a very different microstructure from Fig.1b. Therefore, the wettability of Ni on YSZ can be expected to play a very important role in determining the topological and morphological features in real anode materials.

a

The flat top is due to contact with YSZ above (periodic boundary)

b

C

Figure 1 a) 3D representation of 2 cubic nickel particles sitting on a square flat YSZ plate. b) The CA modelled microstructure after 200 time steps assuming Cmis =0.8. c) The evolved microstructure after 200 time steps assuming Cmis =0.2 Microstructure Evolution of Nickel-Cermet Anode 3-D Microstructure Reconstruction To do simulation on the real microstructure of Ni cermet anode, it is necessary to obtain the initial 3-D microstructure in digital form. For the current work, the 3-D microstructure was a sampled volume from a real anode material (as reduced anode functional layer on an anode substrate, provided by Forschungszentrum Juelich GmbH). The 3-D microstructure reconstruction was based on 2-D images obtained from slice and view experiments using dual beam focused ion beam-scanning electron microscopy (FIB-

SEM, Zeiss, Auriga). Good contrast between Ni and YSZ was achieved by using an inlens detector with an acceleration voltage of 1kV. Fig.1 shows a typical 2D image.

Functional layer

Figure 2. A typical 2D image of a Ni/YSZ anode obtained from FIB slice and view A series of such cross-sectional images were collected, cropped (to choose the area of interest), segmented (to distinguish different phases) and stacked to form the 3D microstructure in digital form. Figs. 3a and b show the reconstructed 3-D microstructure of the anode functional layer. In Fig.3a, the black phase is nickel and the white phase is YSZ. Whereas the nickel phase is white in Fig. 3b for a better visual effect. The data set size is 200 x 200 x 100 cells and the corresponding actual volume of this sampled section is 3 x 3 x 1.6 (µm). The 3-D microstructure is clearly complicated. The topology can be observed more readily from the single phase microstructure, such as Fig.3b which is a Ni-only representation after digital removal of the YSZ phase. From Fig.3b it can be seen that almost all the nickel particles are inter-linked in some way. Microstructure evolution Simulated by the CA Model Figs. 3c and d represent the simulated microstructure after 200 time steps by assuming the mismatch coefficient Cmis=0.8, whereas Figs. 3e and 3f are the simulated microstructure after 200 time steps by assuming Cmis=0.2. Upon comparing the evolved microstructure in Figs.3c and 3d (Cmis=0.8) with the original microstructure (Figs 3a and 3b) it can be seen that some of the original ligaments between neighbouring nickel particles have been broken. Thus poor wettability of Ni on YSZ leads to reduced percolation of the Ni. Conversely, good compatibility (Cmis=0.2) encourages the spreading of the nickel phase, leading to more local linkages between neighbouring particles as shown in Figs. 3e and 3f.

Further important information can be obtained from quantitative analysis of the 3-D microstructures. Automated image analysis was applied to determine the volume percentage of different phases and surface /interface areas in the microstructures. The results are shown in Table 1. As expected the volume percentage of different phases does not change with time, whereas the interface areas do change with time and also depend on the wettability of Ni on YSZ. The interface area between Ni and pore (SANi, or nickel surface area) decreased nearly 18% after 200 time steps, almost independent of the wettability, whereas the interface area between YSZ and pore (SAZr, or YSZ surface area) can either increase or decrease, depending on wettability. Good wettability decreases the YSZ surface area, while poor wettability increases it. This is consistent with the simple case shown in Figs.1b and 1c. Similarly, the changes of interface area (Aint in Table 1) between Ni and YSZ also depend on wettability: good wettability increases the contact area between the two solid phases while poor wettability decreases it. This is also consistent with the simple case represented in Figs. 1b and 1c. Based on the interface areas, the equivalent particle sizes can be calculated:

Deq=6V/A

[4]

where V is the volume of a specific phase, A is the total external area (for example, for Ni phase A= SANi+Aint). Table 2 lists the equivalent particle size for the three different phases. Since YSZ is set as immobile in CA model, its particle size is unchanged (as expected). However, poor wettability leads to a coarsening of the Ni phase and a smaller average pore size, whereas good wettability leads to a smaller Ni particle size and larger pore size. TPB length per unit volume is directly related to the electrochemical performance of the anode. The TPB densities for the different microstructures are given in Table 3. The poor wettability (Cmis=0.8) has led to decreases of both percolated TPB density (from 5.16 to 4.06 µm/µm3) and total TPB density (from 6.14 to 5.09 µm/µm3). In contrast, a good wettability (Cmis=0.2) has led to a big increase of the total TPB density (6.14 to 10.6 µm/µm3), but a decrease of the percolated TPB density (from 5.16 to 4.74 µm/µm3). The spatial distribution of TPB lines in the sampled volume is shown in Figs.4a-c for the initial microstructure, the evolved microstructure after 200 time steps with a poor wettability and that with a good wettability, respectively. It is noticeable that the density of TPB lines in Fig.4b is reduced in comparison with Fig.4a, while it is remarkable that the TPB density is significantly increased in Fig.4c. However, a large fraction of the TPB lines shown in Fig.4c are non-percolated (as summarized in Table 3).

a

b

c

d

e f Figure 3 a) Reconstructed 3D microstructure of a Ni cermet anode; b) Ni-only microstructure corresponding to Fig.3a. c) The simulated 3D microstructure after 200 time steps assuming Cmis=0.8; d) Ni-only microstructure corresponding to Fig.3c. e) The simulated 3D microstructure after 200 time steps assuming Cmis =0.2; f) Ni-only microstructure corresponding to Fig.3e. Note: the nickel phase is black and YSZ is white in a, c and e, but nickel phase in b, d and f is white. TABLE I. Quantitative Information of Different Microstructures Microstructure VNi (%) VZr (%) Vpore (%) SANi (µm2) As-reduced After 200 time steps (Cmis=0.8) After 200 time steps (Cmis=0.2)

steps

Aint (µm2)

35 35

43.4 43.4

21.6 21.6

9.4 7.8

16.8 20.5

20.6 16.9

35

43.4

21.6

7.7

14.7

22.7

TABLE II. Equivalent Particle Size for Different Phases Microstructure Deq of Ni (µm) Deq of YSZ (µm) As-reduced 1.01 1.0 200 time (Cmis=0.8) 200 time steps (Cmis=0.2)

SAZr (µm2)

Deq of Pore (µm) 0.72

1.23

1.0

0.66

0.99

1.0

0.84

TABLE III. Microstructure As-reduced 200 time (Cmis=0.8) 200 time (Cmis=0.2)

Percolated TPB density (µm/µm3)

Total TPB density (µm/µm3)

steps

5.16 4.06

6.14 5.09

steps

4.74

10.6

Summary We have demonstrated that the new CA approach of simulating microstructure evolution can be applied to model the complicated topological and morphological changes in the real microstructure of a porous nickel cermet anode. It has been demonstrated that at least in the initial stage the microstructure changes are very sensitive to the wettability of Ni on YSZ. A better wettability of Ni on YSZ would lead to slower, or even no, coarsening of Ni the phase and a slower decrease of the percolated TPB length density. Poor wettability leads to rapid coarsening and loss of percolation.

a

b

c Fig.4 a) Total TPB lines corresponding to the microstructure shown by Figs. 3a and 3b; b) total TPB lines corresponding to the microstructure given by Figs. 3c and 3d; c) total TPB lines corresponding to the microstructure given by Figs. 3e and 3f.

Acknowledgements This work was carried out as part of the European Union’s Seventh Framework Programme (FP7/20017-1013) for the Fuel Cells and Hydrogen Joint Undertaking ( under grant agreement 256885, Solid Oxide Fuel Cells – Integrating Degradation Effects into Lifetime Prediction Models) and the UK Supergen consortium project on “Fuel Cells: Powering a Greener Future” (EPSRC Grant EP/G030995/1).

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