# Chemical Potential Diagrams (CPDs)

## Introduction

The chemical potential diagram is the mathematical dual to the compositional phase diagram. To create the diagram, convex minimization is performed in energy (E) vs. chemical potential (μ) space by taking the lower convex envelope of hyperplanes. Accordingly, “points” on the compositional phase diagram become N-dimensional convex polytopes (domains) in chemical potential space.

For more information on this specific implementation of the algorithm, please cite/reference the paper below:

## Methodology

### Constructing hyperplanes

### Calculating lower convex envelope (halfspace intersection)

![Figure by Matthew McDermott.](/files/mxIZ4jirIvLKM2cgEzWJ)

### Visualizing the chemical potential diagram

![Two dimensional (2-D) chemical potential diagram for the V-S chemical system. Energies are DFT-calculated energies directly acquired from MP database.](/files/3DCUlLPPj5divYvICf7f)

![Three dimensional (3-D) chemical potential diagram for the V-S-O chemical system. Energies are DFT-calculated energies directly acquired from MP database.](/files/X79vd2rTCIdkJXwAXpgY)

### Relationship to predominance diagrams

![Relationship between 3-D chemical potential diagram and predominance diagrams, which are 2-D views of the full three-dimensional chemical potential diagram surface. Figure by Matthew McDermott.](/files/14nUSMj6z6nuDVarD09w)

## Citations

{% hint style="info" %}

#### **Methodology**

Todd, P. K., McDermott, M. J., Rom, C. L., Corrao, A. A., Denney, J. J., Dwaraknath, S. S., Khalifah, P. G., Persson, K. A., & Neilson, J. R. (2021). Selectivity in Yttrium Manganese Oxide Synthesis via Local Chemical Potentials in Hyperdimensional Phase Space. Journal of the American Chemical Society, 143(37), 15185-15194. <https://doi.org/10.1021/jacs.1c06229>
{% endhint %}

## References

\[1] Yokokawa, H. “Generalized chemical potential diagram and its applications to chemical reactions at interfaces between dissimilar materials.” JPE 20, 258 (1999). <https://doi.org/10.1361/105497199770335794>

\[1] Todd, P. K., McDermott, M. J., Rom, C. L., Corrao, A. A., Denney, J. J., Dwaraknath, S. S., Khalifah, P. G., Persson, K. A., & Neilson, J. R. (2021). Selectivity in Yttrium Manganese Oxide Synthesis via Local Chemical Potentials in Hyperdimensional Phase Space. Journal of the American Chemical Society, 143(37), 15185-15194. <https://doi.org/10.1021/jacs.1c06229>


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://docs.materialsproject.org/methodology/materials-methodology/thermodynamic-stability/chemical-potential-diagrams-cpds.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.