# Modelling the Defect Processes of Materials for Energy Applications
## 2 Methodology  
### Thermodynamic Methodology  
There have been numerous thermodynamic models aiming to model point defects and their processes in solids, but the \(cB\Omega\) model by [PERSON] and [PERSON] is acknowledged as a very efficient and rigorous way to link microscopic and macroscopic properties [36; 37; 45; 46; 47; 48; 49; 50; 51]. The \(cB\Omega\) model links the activation Gibbs free energy of a point defect with macroscopic bulk properties of the solid material:  
\[g^{act}=c^{act}B\Omega \tag{1}\]  
where \(B\) is the isothermal bulk modulus, \(\Omega\) is the mean atomic volume of the host material and \(c^{act}\) is a dimensionless coefficient that is temperature- and pressure-independent. From Equation (1), it can be derived that the Arrhenius behaviour of the diffusion coefficients for a single diffusion mechanism is:  
\[D(T,P)=fga_{0}^{2}ve^{-\frac{c^{act}B\Omega}{\Lambda_{B}T}} \tag{2}\]  
where \(f\) is the correlation factor, \(g\) is a geometrical factor, \(a_{0}\) is the lattice constant, \(v\) is the attempt frequency and \(k_{B}\) is Boltzmann's constant. Additionally, point defect thermodynamic parameters such as the activation entropy \(s^{act}\), activation enthalpy \(h^{act}\) and activation volume \(v^{act}\) can be deduced via the following relations [37]:  
\[s^{act}=-\frac{\partial g^{act}}{\partial T}|_{P}=-c^{act}\Omega\bigg{\{} \frac{\partial B}{\partial T}|_{P}+\beta B\bigg{\}} \tag{3}\]  
\[h^{act}=g^{act}+Ts^{act}=c^{act}\Omega\bigg{\{}B-T\bigg{(}\beta B-\frac{ \partial B}{\partial T}|_{P}\bigg{)}\bigg{\}} \tag{4}\]  
\[v^{act}=\frac{\partial g^{act}}{\partial P}|_{T}=c^{act}\Omega\bigg{\{}\frac{ \partial B}{\partial P}|_{T}-1\bigg{\}} \tag{5}\]  
where \(\beta\) denotes the volumetric coefficient of thermal expansion. These point defect parameters are functions of the elastic and expansion properties of the bulk material. The \(cB\Omega\) model can be employed to self- or hetero-diffusion in the \(A_{1-x}B_{x}\) alloy by considering that the B atoms can be treated as defects [37]. Therefore, the previous relations (Equations (1) and (3)-(5)) can be used to describe the point defect parameters. Henceforth in these expressions, the elastic properties, expansion properties, thermal expansion coefficient and pressure and temperature derivatives will refer to the binary alloy. Therefore, to estimate the defect parameters of \(A_{1-x}B_{x}\), it is imperative to properly determine its bulk properties. The molar volume of the \(A_{1-x}B_{x}\) alloy with respect to the composition and \(T\) (or \(P\)) is expressed via [37]:  
\[V_{A_{1-x}B_{x}}(x,T)=(1-x)V_{A}(T)+xV_{B}(T) \tag{6}\]  
where \(V_{A}\) and \(V_{B}\) are the molar volumes of the two end members and \(x\) is the molar concentration of component B. It should be stressed that this relation assumes that the volume change of the alloy obeys [PERSON]'s law (i.e., the volume change of the alloy due to the replacement of atoms A with atoms B does not depend on the concentration \(x\)). This is a good approximation for a number of systems, but it should be noted that for alloys which strongly deviate from [PERSON]'s law, there should be a third term in Equation (6) representing the deviation from linearity. Compressibility is defined by \(\kappa=1/B=-\left(\frac{1}{V}\right)\left(\partial V/\partial P|_{T}\right)\). The composition and temperature dependence of the isothermal bulk modulus of \(A_{1-x}B_{x}\) alloy can be derived by differentiating its molar volume (Equation (6)) with respect to \(P\)[37]:  
\[B_{A_{1-x}B_{x}}(x,T)=\frac{1+x(\mu-1)}{1+x(\frac{\mu}{\lambda}-1)}B_{A} \tag{7}\]  
where \(\mu=V_{B}/V_{A}=\frac{\Omega_{B}}{\Omega_{A}}\) is the ratio of the molar volumes or the mean atomic volumes of the end members, whereas \(\lambda=B_{B}/B_{A}\) is the ratio of their bulk moduli.