feos-org · prehner · Oct 24, 2024 · Dec 26, 2022 · Dec 27, 2022 · Jan 2, 2023
diff --git a/docs/theory/eos/critical_points.md b/docs/theory/eos/critical_points.md
@@ -0,0 +1,30 @@
+# Stability and critical points
+
+The implementation of critical points in $\text{FeO}_\text{s}$ follows the algorithm by [Michelsen and Mollerup](https://tie-tech.com/new-book-release/). A necessary condition for stability is the positive-definiteness of the quadratic form ([Heidemann and Khalil 1980](https://doi.org/10.1002/aic.690260510))
+
+$$\sum_{ij}\left(\frac{\partial^2 A}{\partial N_i\partial N_j}\right)_{T,V}\Delta N_i\Delta N_j$$
+
+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](https://doi.org/10.3389/fceng.2021.758090). 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`|$N_i$|$T,\rho$|$c_1(T,\rho,N_i)=0$<br/>$c_2(T,\rho,N_i)=0$|
+|`critical_point_binary_t`|$T$|$\rho_1,\rho_2$|$c_1(T,\rho_1,\rho_2)=0$<br/>$c_2(T,\rho_1,\rho_2)=0$|
+|`critical_point_binary_p`|$p$|$T,\rho_1,\rho_2$|$c_1(T,\rho_1,\rho_2)=0$<br/>$c_2(T,\rho_1,\rho_2)=0$<br/>$p(T,\rho_1,\rho_2)=p$|
diff --git a/docs/theory/eos/index.md b/docs/theory/eos/index.md
@@ -6,6 +6,7 @@ This section explains the thermodynamic principles and algorithms used for equat
    :maxdepth: 1
 
    properties
+   critical_points
 ```
 
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