Editing Solid oxide fuel cell (section)

===Activation polarization===
The activation polarization is the result of the kinetics involved with the electrochemical reactions. Each reaction has a certain activation barrier that must be overcome in order to proceed and this barrier leads to the polarization. The activation barrier is the result of many complex electrochemical reaction steps where typically the rate limiting step is responsible for the polarization. The polarization equation shown below is found by solving the [[Butler–Volmer equation]] in the high current density regime (where the cell typically operates), and can be used to estimate the activation polarization:

:<math> {\eta}_{act} = \frac {RT} {{\beta}zF} \times ln \left(\frac {i} {{i}_0} \right) </math>

where:
*<math>R</math> = gas constant
*<math>{T}_0</math> = operating temperature
*<math>{\beta}</math> = electron transfer coefficient
*<math>z</math> = electrons associated with the electrochemical reaction
*<math>F</math> = Faraday's constant
*<math>i</math> = operating current
*<math>i_0</math> = exchange current density

The polarization can be modified by microstructural optimization. The Triple Phase Boundary (TPB) length, which is the length where porous, ionic and electronically conducting pathways all meet, directly relates to the electrochemically active length in the cell. The larger the length, the more reactions can occur and thus the less the activation polarization. Optimization of TPB length can be done by processing conditions to affect microstructure or by materials selection to use a mixed ionic/electronic conductor to further increase TPB length.

==Mechanical Properties==
Current SOFC research focuses heavily on optimizing cell performance while maintaining acceptable mechanical properties because optimized performance often compromises mechanical properties. Nevertheless, mechanical failure represents a significant problem to SOFC operation. The presence of various kinds of load and [[Thermal stress]] during operation requires high mechanical strength. Additional stresses associated with changes in gas atmosphere, leading to reduction or oxidation also cannot be avoided in prolonged operation.<ref name="Progress in Material Selection">{{cite journal |last1=Mahato |first1=N |last2=Banerjee |first2=A |last3=Gupta |first3=A |last4=Omar |first4=S |last5=Balani |first5=K |title=Progress in material selection for solid oxide fuel cell technology: A review |journal=Progress in Materials Science |date=2015-07-01 |volume=72 |pages=141–337 |doi=10.1016/j.pmatsci.2015.01.001}}</ref> When electrode layers delaminate or crack, conduction pathways are lost, leading to a redistribution of current density and local changes in temperature. These local temperature deviations, in turn, lead to increased thermal strains, which propagate cracks and [[Delamination]]. Additionally, when electrolytes crack, separation of fuel and air is no longer guaranteed, which further endangers the continuous operation of the cell.<ref>{{cite journal |last1=Nakajo |first1=Arata |last2=Kuebler |first2=Jakob |last3=Faes |first3=Antonin |last4=Vogt |first4=Ulrich |last5=Schindler |first5=Hansjürgen |last6=Chiang |first6=Lieh-Kwang |last7=Modena |first7=Stefano |last8=Van Herle |first8=Jan |title=Compilation of mechanical properties for the structural analysis of solid oxide fuel cell stacks. Part I. Constitutive materials of anode-supported cells. |journal=Ceramics International |date=2012-01-25 |volume=38 |pages=3907–3927 |doi=10.1016/j.ceramint.2012.01.043}}</ref>

Since SOFCs require materials with high oxygen conductivity, thermal stresses provide a significant problem. The [[Coefficient of thermal expansion]] in mixed ionic-electronic perovskites can be directly related to oxygen vacancy concentration, which is also related to ionic conductivity.<ref>{{cite journal |last1=Ullmann |first1=H. |last2=Trofimenko |first2=N. |last3=Tietz |first3=F. |last4=Stöver |first4=D. |last5=Ahmad-Khanlou |first5=A. |title=Correlation between thermal expansion and oxide ion transport in mixed conducting perovskite-type oxides for SOFC cathodes |journal=Solid State Ionics |date=1 December 2000 |volume=138 |issue=1–2 |pages=79–90 |doi=10.1016/S0167-2738(00)00770-0}}</ref> Thus, thermal stresses increase in direct correlation with improved cell performance. Additionally, however, the temperature dependence of oxygen vacancy concentration means that the CTE is not a linear property, which further complicates measurements and predictions.

Just as thermal stresses increase as cell performance improves through improved ionic conductivity, the fracture toughness of the material also decreases as cell performance increases. This is because, to increase reaction sites, porous ceramics are preferable. However, as shown in the equation below, fracture toughness decreases as porosity increases.<ref>{{cite journal |last1=Radovic |first1=M. |last2=Lara-Curzio |first2=E. |title=Mechanical properties of tape cast nickel-based anode materials for solid oxide fuel cells before and after reduction in hydrogen |journal=Acta Materialia |date=December 2004 |volume=52 |issue=20 |pages=5747–5756 |doi=10.1016/j.actamat.2004.08.023|bibcode=2004AcMat..52.5747R }}</ref>

<math> K_{IC} = K_{IC,0}\exp{(-b_{k}p')} </math>

Where:

<math>K_{IC}</math> = fracture toughness

<math>K_{IC,0}</math> = fracture toughness of the non-porous structure

<math>b_k</math> = experimentally determined constant

<math>p' </math> = porosity

Thus, porosity must be carefully engineered to maximize reaction kinetics while maintaining an acceptable fracture toughness. Since fracture toughness represents the ability of pre-existing cracks or pores to propagate, a potentially more useful metric is the failure stress of a material, as this depends on sample dimensions instead of crack diameter. Failure stresses in SOFCs can also be evaluated using a ring-on ring biaxial stress test. This type of test is generally preferred, as sample edge quality does not significantly impact measurements. The determination of the sample's failure stress is shown in the equation below.<ref>{{cite web |last1=ASTM |title=Standard Test Method for Monotonic Equibiaxial Flexural Strength of Advanced Ceramics at Ambient Temperature, ASTM Standard C1499-04 |url=https://www.astm.org/c1499-19.html}}</ref>

<math> \sigma_{cr}= \frac{3F_{cr}}{2\pi h_{s}^{2}}+
\Biggl((1-\nu)\frac{D_{sup}^{2}-D_{load}^{2}}{2D_{s}^{2}}+(1+\nu)\ln\left ( \frac{D_{sup}}{D_{load}} \right )\Biggr) </math>

Where:

<math> \sigma_{cr} </math> = failure stress of the small deformation

<math> F_{cr} </math> = critical applied force

<math> h_s </math> = height of the sample

<math> \nu </math> = Poisson's ratio

<math> D </math> = diameter (sup = support ring, load = loading ring, s = sample)

However, this equation is not valid for deflections exceeding 1/2h,<ref>{{Cite journal |last1=Kao |first1=Robert |last2=Perrone |first2=Nicholas |last3=Capps |first3=Webster |date=1971 |title=Large-Deflection Solution of the Coaxial-Ring-Circular-Glass-Plate Flexure Problem |url=https://onlinelibrary.wiley.com/doi/10.1111/j.1151-2916.1971.tb12209.x |journal=Journal of the American Ceramic Society |language=en |volume=54 |issue=11 |pages=566–571 |doi=10.1111/j.1151-2916.1971.tb12209.x |issn=0002-7820|url-access=subscription }}</ref> making it less applicable for thin samples, which are of great interest in SOFCs. Therefore, while this method does not require knowledge of crack or pore size, it must be used with great caution and is more applicable to support layers in SOFCs than active layers. In addition to failure stresses and fracture toughness, modern fuel cell designs that favor mixed ionic electronic conductors (MIECs), [[Creep (deformation)]] pose another great problem, as MIEC electrodes often operate at temperatures exceeding half of the melting temperature. As a result, diffusion creep must also be considered.<ref>{{Cite journal |last1=Nakajo |first1=Arata |last2=Kuebler |first2=Jakob |last3=Faes |first3=Antonin |last4=Vogt |first4=Ulrich F. |last5=Schindler |first5=Hans Jürgen |last6=Chiang |first6=Lieh-Kwang |last7=Modena |first7=Stefano |last8=Van herle |first8=Jan |last9=Hocker |first9=Thomas |date=2012-01-25 |title=Compilation of mechanical properties for the structural analysis of solid oxide fuel cell stacks. Constitutive materials of anode-supported cells |url=https://linkinghub.elsevier.com/retrieve/pii/S0272884212000466 |journal=Ceramics International |language=en |volume=38 |issue=5 |pages=3907–3927 |doi=10.1016/j.ceramint.2012.01.043|url-access=subscription }}</ref>

<math> \dot{\epsilon}_{eq}^{creep}=\frac{\tilde{k}_0D}{T}\frac{\sigma_{eq}^{m}}{d_{grain}^{n}} </math>

Where:

<math> \dot{\epsilon}_{eq}^{creep} </math> = equivalent creep strain

<math> D </math> = [[Diffusion]] coefficient

<math> T </math> = temperature

<math> \tilde{k}_0 </math> = kinetic constant

<math> \sigma_{eq} </math> = equivalent stress (e.g. von Mises)

<math> m </math> = creep stress exponential factor

<math> n </math> = particle size exponent (2 for [[Nabarro–Herring creep]], 3 for [[Coble creep]])

To properly model creep strain rates, knowledge of [[Microstructure]] is therefore of significant importance. Due to the difficulty in mechanically testing SOFCs at high temperatures, and due to the microstructural evolution of SOFCs over the lifetime of operation resulting from [[Grain growth]] and coarsening, the actual creep behavior of SOFCs is currently not completely understood