Theory guide: critical points

feos-org · prehner · Oct 24, 2024 · Dec 26, 2022 · Dec 27, 2022 · Jan 2, 2023
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diff --git a/docs/theory/eos/critical_points.md b/docs/theory/eos/critical_points.md
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+# Spinodal and critical points
+
+The stability of an isothermal system is determined by the inequality[1]
+
+$$A-A_0-\sum_i\mu_{i0}\Delta N_i>0$$
+
+where $A_0$ and $\mu_{i0}$ are the Helmholtz energy and chemical potentials of the state which stability is being assessed and $A$ is the Helmholtz energy of any state that is obtained from state 0 by adding the amounts $\Delta N_i$. The Helmholtz energy can be expanded around the test point, giving
+
+$$\sum_{ij}\left(\frac{\partial^2 A}{\partial N_i\partial N_j}\right)\Delta N_i\Delta N_j+\sum_{ijk}\left(\frac{\partial^3 A}{\partial N_i\partial N_j\partial N_k}\right)\Delta N_i\Delta N_j\Delta N_k+\mathcal{O}\left(\Delta N^4\right)>0$$(eqn:stability)
+
+The inequality must be fulfilled for any $\Delta N_i$. Therefore, a necessary criterion for stability is that the quadratic form (the first term) in eq. {eq}`eqn:stability` is [positive definite](https://en.wikipedia.org/wiki/Definite_quadratic_form). The **spinodal** or limit of stability consists of the points for which the quadratic form is positive semi-definite. The stability is then determined by the higher-order terms in eq. {eq}`eqn:stability`. Mathematically there exists a $\Delta N_i$ that fulfills
+
+$$\sum_i\left(\frac{\partial^2 A}{\partial N_i\partial N_j}\right)\Delta N_i=0$$
+
+A **critical point** is defined as a stable point on the limit of stability.[2] For that to be the case, the quadratic and cubic terms in eq. {eq}`eqn:stability` need to vanish[1] which leads to the second criticality condition
+
+$$\sum_{ijk}\left(\frac{\partial^3 A}{\partial N_i\partial N_j\partial N_k}\right)\Delta N_i\Delta N_j\Delta N_k=0$$
+
+The criticality conditions can be reformulated by defining the matrix $M$ with[3]
+
+$$M_{ij}=\sqrt{x_ix_j}\left(\frac{\partial^2\beta A}{\partial N_i\partial N_j}\right)$$
+
+and $u$ as the eigenvector corresponding to the smallest eigenvalue $\lambda_1$ of $M$. $M$ and in conclusion the quadratic form in eq. {eq}`eqn:stability` is positive semi-definite if and only if its smallest eigenvalue is 0. Therefore the first criticality condition simplifies to
+
+$$\lambda_1=0$$
+
+For the second criticality condition $\Delta N_i$ a step $s$ is defined that acts on the mole numbers $N_i$ as
+
+$$N_i=z_i+su_i\sqrt{z_i}$$
+
+Then the derivative $\left.\frac{\partial^3 A}{\partial s^3}\right|_{s=0}$ can be rewritten as
+
+$$\left.\frac{\partial^3 A}{\partial s^3}\right|_{s=0}=\sum_{ijk}\left(\frac{\partial^3 A}{\partial N_i\partial N_j\partial N_k}\right)u_iu_ju_k\sqrt{z_iz_jz_k}$$
+
+$$C=\sum_{ij}\left(\sum_k\left(\frac{\partial^3 A}{\partial N_i\partial N_j\partial N_k}\right)\Delta N_k\right)\Delta N_i\Delta N_j$$
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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