Theory-based design of sintered granular composites triples three-phase boundary in fuel cells

Solid-oxide fuel cells produce electric current from energy released by a spontaneous electrochemical reaction. The efficiency of these devices depends crucially on the microstructure of their electrodes and in particular on the three-phase boundary (TPB) length, along which the energy-producing reaction occurs. We present a systematic maximization of the TPB length as a function of four readily controllable microstructural parameters, for any given mean hydraulic radius, which is a conventional measure of the permeability to gas flow. We identify the maximizing parameters and show that the TPB length can be increased by a factor of over 300% compared to current common practices. We support this result by calculating the TPB of several numerically simulated structures. We also compare four models for a single intergranular contact in the sintered electrode and show that the model commonly used in the literature is oversimplified and unphysical. We then propose two alternatives.

Figure 1

Shown on top is a simulated composite electrode, made of a sintered powder of ion-conducting (blue) grains of radius $r_1$ and electron-conducting (red) grains of radius $r_2=r_1/P$ with $P=2$. The inset shows three interphase contact disks and their respective TPBs as dark circular lines. The bottom shows our predicted improvement factor for the TPB density (blue line) and conductivity (green line) as functions of $P$, compared to the common design of $P=1$, for the same mean hydraulic radius. The red closed circles are respective maximal improvements, obtained for $P_{\max }$. The blue open circles are measured TPB improvements for simulated systems. These match our prediction to an accuracy of 6%.

Figure 2

Volume fraction ${\psi }_1$ as a function of the two grain radii $r_1\ge r_0\ge r_2$. In black are equal ${\psi }_1$ contour lines and the top thick line is the maximum-size ratio limit $r_1/r_2=P_{\max }$. Note that the ordinate is labeled in decreasing values of $r_2$.

Figure 3

Four types of geometries to model intergranular contacts, and their respective radii of contact disks $r_c$: (a) the H model, based on a Hertzian contact force model; (b) the C model, taking into consideration a curved interface due to sintering; (c) the geometric I model, based on an inflation of grains due to sintering; and (d) the $\theta$ model, assuming a constant contact angle of the smaller grain. The thin line in (d) represents an even larger $r_1$, demonstrating that this model is unrealistic.

Figure 4

Values of the contact disk radius $r_c$ given by the four contact models. The less realistic $r_{c,\min }$ and $r_{c,\max }$ bound, respectively, from below and from above, the values of the more realistic $r_{c,\mathrm{H}}$ and $r_{c,\mathrm{I}}$. Although $r_1\ge r_2$ in our analysis, here we show also $r_1<r_2$ for clarity.

Figure 5

Single TPB length $l_{\mathrm{TPB}}$ as a function of the two grain radii $r_1$ and $r_2$ for (a) the H model, with $C=0.518$; (b) the I model, with $a=1.032$; (c) the $\theta$ model, with the smaller radius $\theta ={15}^{\circ }$; and (d) the $\theta$ model, with the larger radius $\theta ={15}^{\circ }$. The axes, which have been removed for clarity, are the same as in Fig. 2. The value ranges of $l_{\mathrm{TPB}}$ are (a) $\left[0.14\text{–}0.7\right]$, (b) $\left[0.19\text{–}0.77\right]$, (c) $\left[0.08\text{–}0.49\right]$ and (d) $\left[0.49\text{–}1.22\right]\mu \mathrm{m}$; $l_{\mathrm{TPB}}(r_0,r_0)$ (bottom left corner) is equal for all models ($0.488\mu \mathrm{m}$).

Figure 6

Number of 1-2 contacts per $1\mu {\mathrm{m}}^3$, $c_{1-2}$, as a function of the two grain radii $r_1$ and $r_2$. The values of ${\psi }_1$ are obtained using the same fixed value of $R_h$ as in Fig. 2.

Figure 7

Total active TPB length per $\mu {\mathrm{m}}^3$, $L_{\mathrm{TPB}}$, as a function of the two grain radii $r_1$ and $r_2$ for the four contact models and for the same value of $R_h=54\mathrm{nm}$. The single-contact TPB length is calculated for (a) the H model, with $C=0.518$; (b) the I model, with $a=1.032$; (c) the $\theta$ model, with the smaller radius $\theta ={15}^{\circ }$; and (d) the $\theta$ model, with the larger radius $\theta ={15}^{\circ }$. The axes, which have been removed for clarity, are the same as in Figs. 2 and 6. The highest values of $L_{\mathrm{TPB}}$ are obtained at $(r_1,r_2)=(0.500\mu \mathrm{m},0.077\mu \mathrm{m})$ and their values are (a) 8.95, (b) 13.14, (c) 5.17, and (d) $33.40\mu {\mathrm{m}}^{-2}$. The value of $L_{\mathrm{TPB}}(r_0,r_0)$ (bottom left corner) is equal for all contact models ($\simeq 3.53\mu {\mathrm{m}}^{-2}$). The black dots are at $(r_1^{\left(\max \right)},r_2^{\left(\max \right)})=(0.543\mu \mathrm{m},0.084\mu \mathrm{m})$, which is the best grain size pair that can be compared fairly to $(r_0,r_0)$, and they correspond to (a) $L_{\mathrm{TPB}}=7.60\mu {\mathrm{m}}^{-2}$, (b) $L_{\mathrm{TPB}}=11.15\mu {\mathrm{m}}^{-2}$, (c) $L_{\mathrm{TPB}}=4.39\mu {\mathrm{m}}^{-2}$, and (d) $L_{\mathrm{TPB}}=28.36\mu {\mathrm{m}}^{-2}$.

Figure 8

Comparison of a monodisperse system with grain radius $r_0=0.3\mu \mathrm{m}$ (dashed lines) and a bidisperse system with grain radii $0.543$ and $0.084\mu \mathrm{m}$ (solid lines), for the I model. The value of $L_{\mathrm{TPB}}$ is shown by thick blue lines and $R_h$ by thin green lines. The maximum value of $L_{\mathrm{TPB}}$ for the bidisperse mixture is 3.16 times as high. At the peak, $R_h$ is the same for both mixtures (black circles). The inset shows the effective conductivity of type-1 grains for the two mixtures (thin green lines). At the peak, the effective conductivity of the bidisperse mixture is twice as high (black circles).

Figure 9

Relative improvement $L_{\mathrm{TPB}}^{\left(\max \right)}/L_{\mathrm{TPB}}^{\left(0\right)}$ vs grain size ratio $P=r_1/r_2$ for the H model (blue circles) and I model (green triangles). The vertical thick red line marks the maximum possible ratio $P_{\max }$. The inset shows the values of $r_1/r_0$ (red squares) and ${\psi }_1$ (magenta crosses), for which $L_{\mathrm{TPB}}$ is maximal, vs $P$. The black lines are cubic polynomial fits.