**2. Image-based microstructural finite element models**

The first step in this work was to digitally reconstruct a three-dimensional (3-D) anode microstructure from two-dimensional (2-D) images of anode cross-sections obtained using focused ion beam-scanning electron microscopy (FIB-SEM). The 2-D images of the anode and cathode microstructures were obtained from Dr. Scott Barnett's research group at Northwestern University (Wilson et al., 2006, 2009). The initial 3-D reconstruction was achieved using IMOD (Kramer et al., 1996), a free collection of image processing programs developed by scientists at the Boulder Laboratory for 3-D Electron Microscopy of Cells. IMOD is capable of creating a stack of 2-D images, interpolating the gaps between consecutive images, and creating and displaying the 3-D model. A few representative 2-D images and the 3-D reconstruction of the anode microstructure are shown in Figure 1. The in-plane dimensions of the reconstructed anode are 5 μm x 6 μm, while the thickness is 3.54 μm.

Fig. 1. Two-dimensional SEM images of anode (left) and cathode (center) cross-sections (Wilson et al., 2006, 2009) and a reconstructed three-dimensional anode (right)

In the SEM images of the anode, white (pixel value 255) corresponds to nickel, gray (pixel value 127) to YSZ, and black (pixel value 0) to the pores. In the images of the cathode, white (pixel value 255) represents LSM, gray (pixel value 127) represents YSZ, and black (pixel value 0) represents the pores. The reconstructed 3-D model was used as a check on the geometry of the 3-D FE model.

Continuum Mechanics of Solid Oxide Fuel Cells

Case 1: Temperatureindependent

Case 2: Temperaturedependent CTEs

Case 3: Elasticplastic behavior of Ni

Case 1: Temperatureindependent

Case 2: Temperaturedependent

**3.2 Material properties** 

Using Three-Dimensional Reconstructed Microstructures 77

The FE analyses of the anode and cathode models were divided into different categories as explained in Tables 1 and 2. In each case, the FE model was subjected to fixed boundary conditions (i.e. all nodes on each of the six faces were allowed neither to translate nor to rotate). The behavior of the model with increasing temperature loads was investigated by subjecting the model to eight different spatially uniform predefined temperature fields of magnitude 120⁰C, 220⁰C, 320⁰C, ..., 820⁰C. In each analysis, the initial temperature was specified as 20⁰C (room temperature), so that the model was subjected to eight different

dependence (Ni)

Temperaturedependence (LSM)

(GPa) Poisson's ratio CTE (10-6 <sup>⁰</sup>C-1)

None None

CTE CTE

CTE Young's

None None

Young's modulus Young's

Temperaturedependence (YSZ)

modulus, CTE

Temperaturedependence (YSZ)

modulus, CTE

magnitudes of temperature change (ΔT = 100⁰C, 200⁰C, 300⁰C, ..., 800⁰C)..

Case Ni YSZ Temperature-

Linear elastic

Linear elastic

Linear elastic

Linear elastic

Linear elastic

Table 3 lists the room temperature material properties used for nickel, YSZ and LSM

Nickel 207 0.31 12.50 YSZ 205 0.30 10.40 LSM 40 0.25 11.40

Linear elastic

Linear elastic

Elasticplastic

Table 1. Metrics for finite element analyses of anode

Case LSM YSZ

Linear elastic

Linear elastic

Table 2. Metrics for finite element analyses of cathode

(Johnson & Qu, 2008; Anandakumar et al., 2010).

Material Young's modulus

Table 3. Room temperature material properties used in FE analyses

The next step involved the creation of a single 2-D FE model from a single 2-D SEM image. FE modeling was carried out using the commercial FE software ABAQUS v6.9 (Dassault Systems Simulia Corp., Providence, Rhode Island, USA). This was done by writing a MATLAB ® program (R2010a, The MathWorks, Inc., Natick, Massachusetts, USA) to recreate the geometry of the image using 2-D finite elements (4-node quadrilateral elements) and write the geometry data to an ABAQUS input file. Exactly one element was assigned to each pixel in the image, and the element was assigned to the appropriate element set (nickel or YSZ) based on the pixel value. Information concerning the material properties, boundary conditions, initial temperature, temperature field, and required outputs (e.g. principal stresses) was also specified in the input file. The input file was then run using ABAQUS to generate the 2-D FE model as shown in Figure 2.

Fig. 2. Two-dimensional FE model of a single cross-section of the SOFC anode

The 3-D FE anode and cathode models were created by making a stack of all the 2-D images and introducing a "buffer" plane between each pair of consecutive images. This was necessary and useful to ensure a simple step variation in material properties between corresponding regions in two consecutive images. Then the gaps between consecutive images were interpolated by assigning one three-dimensional 8-node brick element to each volumetric pixel (or voxel). Thus, the 3-D geometries of the anode and cathode microstructures were recreated in the 3-D FE models of the anode and cathode. Various free-body cuts of the 3-D FE anode model are shown in Figure 3.

Fig. 3. Free-body cuts of the three-dimensional FE model of the SOFC anode
