I have seen examples for binary structures; for each ratio of A and B, the energy above hull is simply calculated by comparison with structures of the same composition, and the structure with the lowest energy above hull (i.e., E = 0) is set as the standard.
However, with ternary (3 atomic species) and quarternary (4 species) structures, how are they calculated? Since there are usually much less reference structures on Materials Project, so I am assuming that a different method is used.
If anyone can refer me to a documentation or paper that explains how they are calculated and/or what equations are used for the calculation, that would be most helpful. Thank you!
Check out @CJBartel’s paper, which covers the theory behind metrics of thermodynamic stability: Review of computational approaches to predict the thermodynamic stability of inorganic solids | Journal of Materials Science
The convex hull algorithm calculates the minimum energy “envelope” in any number of dimensions in energy-composition space (E vs N-1 elements). The energy above hull is measured as the distance to that lower envelope, regardless of how many species are in your phase diagram.
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Hello Matt,
Thank you for the paper recommendation and the link you provided in the other post! I have tried reading the paper, but I still have questions because there are no concrete equations other than those for the formation energy (Ef) and decomposition energy (Ed).
From my understanding, Ef is the energy required to create a composition from its elemental constituents (hence atoms, and not molecules? If so, I suppose that definition-wise Ef is somewhat equivalent to cohesive energy), and Ed is the energy required/released to form other phases (instead of atoms, it can be, for example, the energy required/released for a reaction ABO3 → AO2 + BO or a reaction A2B2O7 → A2BO5 + BO2).
So it can be said that the more accurate measure of whether a material is more stable compared to other phases is the decomposition energy Ed.
Now, a part in the paper I am confused about is when it mentions Ehull. The paper mentioned that the value of delta being Ehull = 0 or > 0 only determines whether a material is stable or not, while Ed measures the magnitude of the stability/instability. However it mentions not how Ehull is calculated, unlike Ed and Ef.
My first question, is Ehull what is known as Energy above hull in the MaterialsProject database?
Second, if so, how is it calculated if there are no structures with the same composition to compare to?
Perhaps I am missing something from the paper?
I apologize if the question is still somewhat vague, I don’t think I have yet to have a good grasp on what I’m missing.
Thank you for your time!
Totally understand where you’re coming from — it’s a tricky concept.
It sounds like the aspect you are stuck on is understanding what the “convex hull” really is. It’s a geometrical construction and the solution to essentially a big system of equations. The convex hull can be thought of like a surface, and it gives the lowest energy phase(s) for every composition in a chemical space. If a phase is above the hull, it will decompose into the phases directly below it. Sometimes that is 1 phase decomposing into 1 directly below it. But sometimes it’s 1 phase decomposing into several, because there are lines (2d) or triangles (3d) or tetrahedra (4d) where multiple phases are in thermodynamic equilibrium at that composition. The energy above hull is probably most easily thought of as the vertical distance (an energy) to this surface. It could be written as an equation I suppose, but the terms would differ based on which phases were in equilibrium at that point. And yes, if you see Ehull on Materials Project, that is just the symbol for this same energy above hull we are discussing. I would try to fully understand Ehull first before jumping into E_d, but as a rough approximation, E_d is the energy “below” the hull. More accurately said, it’s the upwards energy distance to the new convex hull that is formed when that phase is removed.
As a general tip, it is helpful to try to think geometrically and visually when working with thermodynamics (particularly phase equilibria). Hope this helps.
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Thank you Matt! Your explanation helped a lot towards my conceptual understanding of what a convex hull is, though I still have some doubts and questions.
Please correct me if I’m wrong, but from what I gathered from your explanation, it seems that if our final goal is to generate a convex hull and the energy above hull, we do not necessarily “need to know” the equation or how it is calculated, just that it is the vertical distance from a given point to the convex hull.
Two questions:
- If I am to generate a convex hull from the PyMatGen module using the free energies from my own DFT calculations, how can I ensure proper energies above hull? In other words, how many other reference structures would I need to generate it? What other considerations must I take (other than ensuring the same reaction conditions and input parameters (e.g., functionals, potentials, convergence algorithms, etc.)?
- Would you happen to have reference materials that goes more in-depth into Ehull? The Bartel paper is quite exhaustive with regards to other definitions and the convex hull, but has only brief mentions of Ehull.
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Yes, you need to ensure that you are plotting energies that are calculated with the same approach/conditions. First, they need to be the same thermodynamic potential (energy vs. enthalpy vs. free energy). With DFT we usually assume energy is approximately equal to enthalpy due to how relatively small differences in the PV term are between structures. As you mentioned, you also need to ensure the same DFT settings. If you are applying any energy corrections (which MP does extensively), you need to apply the correction scheme uniformly to all entries. Your question regarding how many reference structures you need has no answer – you will never be able to know if you have enough structures, because there could hypothetically always be a phase missing from your phase diagram that would destabilize (or further destabilize) the phase of interest you’re calculating Ehull for. Therefore, the general approach is to include as many phases as you can when constructing the convex hull. Because the MP database is originally based off the ICSD (experimental) structure database, we have some certainty that structures we see in nature, which are generally on or close to the hull (within the amorphous limit), are being represented, and thus our Ehull values are reasonable reflections of metastability.
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I am not aware of any papers that are specifically about Ehull, although reading papers about metastability (like the one I linked in the first paragraph) may be useful. I think even better than reading is just to do it yourself. Try making your own compositional phase diagrams with pymatgen and you’ll develop an intuition for these concepts.
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Dear Matt,
Apologies for the late reply. Thank you again for your answers.
At the moment, I do not know if I have sufficient know-how to craft the convex hull with 4 materials, which is why I am opting to do decomposition energy (Ed) calculation.
I have the following reaction equation:
A2B2O7 + 2NH3 → 2ABO2N + 3H2O
Where the synthesis target is ABO2N, and A2B2O7 is a commonly used composition as the precursor structure.
If I calculated the DFT energies of all the structures, and calculated them by using the following equation and normalizing per atom:
Ed(ABO2N) = E(ABO2N)/n(ABO2N) - E(A2B2O7)/n(A2B2O7) - E(NH3)/n(NH3) + E(H2O)/n(H2O)
And I obtain a negative Ed value, can I conclusively say that ABO2N is thus synthesizable?
If not, what other studies should I do in order to have conclusive results?
Thank you for your time and attention!
Hi @rputra17 ,
No, if you obtain a negative Ed(ABO2N) using the equation you wrote, you cannot say that ABO2N is synthesizable. This is because A2B2O7, NH3, and H2O are not necessarily the decomposition products of ABO2N.
If we take BaTaNO2 as an example of an oxynitride with formula ABO2N, you’ll see that its calculated to be 32 meV/atom above the hull on the Materials Project. Its decomposition products are a mixture of its nearest stable neighbors: 2⁄3 Ba₄Ta₂O₉ + 7⁄45 Ba(TaN₂)₂ + 8⁄45 Ta₃N₅.
Unless you build the full convex hull for the A-B-N-O system, you cannot ensure that your ABO2N is stable with respect to other competing phases. Furthermore, even if you calculate that the energy above hull is >0, the phase may be metastable and still synthesizable. BaTaNO2 is an example of this. Note that getting an Ehull > 0 doesn’t necessarily mean it’s not actually stable either; it could be a small DFT error, perhaps related to unaccounted for disorder / magnetic orderings / etc.
Also, note that writing a synthesis reaction (e.g., A2B2O7 + 2NH3 → 2ABO2N + 3H2O) and finding a negative reaction energy does not mean that the phase is synthesizable either. However, it does indicate that a spontaneous reaction is likely to occur (it just may not form the product phases you wrote on the right side of the reaction).
My recommendation is to calculate the energy of your ABO2N phase with Materials Project settings (see the VASP input sets in pymatgen or automated workflows in atomate2), apply MP corrections, and add your custom entry to the A-B-N-O phase diagram created with MP entries. Then, calculate the Ehull to get a first guess at synthesizability. If you don’t have access to VASP, consider running a structural relaxation and energy calculation with an MLIP trained on Materials Project data (such as CHGNET) as an approximation of its energy. Hope this helps.