Calculating Phase Diagrams

Why do I love phase diagrams so much? I've always been fascinated by these relatively cursory-looking plots that show where the phases of matter at different thermodynamic conditions are stable. I remember my excitement in the intro lecture and lab, MATE 25 at SJSU, where in the lab we constructed points on the phase diagram of a lead alloy system. I thought this was the coolest thing that we could build these maps and then use them later to determine what phase a material would be in a given temperature and composition. At the time I had no thermodynamics course work so I didn't realize the underlying driving force of this phenomenon nor did I realize you can calculate these phase diagrams using the CALPHAD method. When I got to grad school I took the required thermodynamics course and then I was even more blown away at how powerful this framework was. I was particularly lucky because the course was taught from a "grassroots" approach where everything was built from the ground up given a set of postulates (you can see the book by H. Callen to get the gist). 

So what does a phase diagram look like and how does one use it? Here I'm going to leverage the excellent Python library pycalphad [1]. Which lets you construct phase diagrams using thermochemical databases, if available. Let's take an example of Cu-Ni system, you can work through the CALPHAD calculation with pycalphad in this google colab notebook. Here is the  binary phase diagram predicted for Cu-Ni:

Phase diagram predicted using cost507.tdb and pycalphad.

How do you read this? Well the blue and yellow points indicate the equilibrium phase boundary. The regions between phase boundaries indicates what phases are in eqiulibrium and how much (see my old notes on tie-lines). So how does the prediction look? Not very good if you consider what the textbook phase diagram looks like:

Textbook phase diagram for Cu-Ni from adapted by ref. [1] 

As we see the predicted phase diagram isn't even close to that shown in the textbook version. This is a direct consequence of the thermodynamic database. However, if we use the same thermodynamic database and look at another system like Al-Zn its much better:

Phase diagram prediction using cost507.tdb. Not bad!

This is much better when you compare it to the phase diagram reported in ref [2]. I just think the CALPHAD approach is so cool in that you just need thermodynamic descriptions of various phases of a material system to make predictions about stability regions. To make a CALPHAD calculation work you would need the following:

1. Thermodynamic data/descriptions of individual phases such as enthalpy, entropy, and Gibbs free energy.
2. The phases that could exist and their structure.
3. Well-defined reference states (e.g.pure metal) allow for consistent and accurate calculations.
4. Interaction model/parameters to describe mixing behavior of components/species.

The CALPHAD framework then enables the building of a model with these inputs to predict the phase equilibria and diagrams. You can also calculate other thermodynamic properties like heat capacity or even more useful is that the free energy models can be used within the context of phase-field simulations to evolve microstructures.

References

[1] R. Otis, Z.-K. Liu, pycalphad: CALPHAD-based Computational Thermodynamics in Python, JORS. 5 (2017) 1. https://doi.org/10.5334/jors.140.

[2] https://sv.rkriz.net/classes/MSE2094_NoteBook/96ClassProj/examples/cu-ni.html, reproduced from Callister, William D., Materials Science and Engineering: An Introduction. United States, Wiley.

[3] A. Pola, M. Tocci, F.E. Goodwin, Review of Microstructures and Properties of Zinc Alloys, Metals. 10 (2020) 253. https://doi.org/10.3390/met10020253.

https://doi.org/10.5281/zenodo.7901067

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