Stability and critical points

The implementation of critical points in \(\text{FeO}_\text{s}\) follows the algorithm by Michelsen and Mollerup. A necessary condition for stability is the positive-definiteness of the quadratic form (Heidemann and Khalil 1980)

\[\sum_{ij}\left(\frac{\partial^2 A}{\partial N_i\partial N_j}\right)_{T,V}\Delta N_i\Delta N_j\]

The spinodal or limit of stability consists of the points for which the quadratic form is positive semi-definite. Following Michelsen and Mollerup, the matrix \(M\) can be defined as

\[M_{ij}=\sqrt{z_iz_j}\left(\frac{\partial^2\beta A}{\partial N_i\partial N_j}\right)\]

with the molar compositon \(z_i\). Further, the variable \(s\) is introduced that acts on the mole numbers \(N_i\) via

\[N_i=z_i+su_i\sqrt{z_i}\]

with \(u_i\) the elements of the eigenvector of \(M\) corresponding to the smallest eigenvector \(\lambda_1\). Then, the limit of stability can be expressed as

\[c_1=\left.\frac{\partial^2\beta A}{\partial s^2}\right|_{s=0}=\sum_{ij}u_iu_jM_{ij}=\lambda_1=0\]

A critical point is defined as a stable point on the limit of stability. This leads to the second criticality condition

\[c_2=\left.\frac{\partial^3\beta A}{\partial s^3}\right|_{s=0}=0\]

The derivatives of the Helmholtz energy can be calculated efficiently in a single evaluation using generalized hyper-dual numbers. The following methods of State are available to determine spinodal or critical points for different specifications:

	specified	unkonwns	equations
spinodal	\(T,N_i\)	\(\rho\)	\(c_1(T,\rho,N_i)=0\)
critical_point	\(N_i\)	\(T,\rho\)	\(c_1(T,\rho,N_i)=0\) \(c_2(T,\rho,N_i)=0\)
critical_point_binary_t	\(T\)	\(\rho_1,\rho_2\)	\(c_1(T,\rho_1,\rho_2)=0\) \(c_2(T,\rho_1,\rho_2)=0\)
critical_point_binary_p	\(p\)	\(T,\rho_1,\rho_2\)	\(c_1(T,\rho_1,\rho_2)=0\) \(c_2(T,\rho_1,\rho_2)=0\) \(p(T,\rho_1,\rho_2)=p\)