Multi-Element Pourbaix Diagram: Species Selection and Mass Balance


I am a graduate student hoping to learn more about the generation of multi-element Pourbaix diagrams. Specifically, I was wondering how the selection of species is chosen for each stability region? Species selection was not clear to me upon looking at the documentation and wiki for Materials Project’s Pourbaix Diagrams App, and when looking up the paper by Thompson et al on Multi-Element Pourbaix diagrams listed as a reference. For an n element diagram, I understand that the species are chosen in combinations of n species such that the change in Gibbs energy is lowest for that combination of n species.

My problem comes from the choice of species combinations that satisfy the mass balance constraint. It seems as though there should always be a general way to write an equation that allows for mass balancing with larger or fractional combinations of elemental species, H+, e- and water species in addition to any products created. How is this mass balance constraint applied?

As an example, reactions that might be written for a multi-element Fe-Ni Pourabix diagram are:
(1) 2Fe + 3H2O = Fe2O3 + 6H+ + 6e-
(2) Ni + H2O = NiO + 2H+ + 2e-
where Fe and Ni are the reference species. When combined:
(3) 2Fe + Ni + 4H2O = NiO + Fe2O3 + 8H+ + 8e-

where equation 3 would allow us to asses the stability of the formation of an NiO + Fe2O3 phase in the multielement diagram. In a single element diagram, reaction 1 would be normalized for by dividing the energy by 2 so that the chemical potential at a specific (pH, potential) point is found per Fe. It seems that equation 3 would then be divided by 2 as well to normalize the equation. I’m sure that I’m not understanding the full implications of mass the mass balance constrain in this case, but am unsure where the issues lie. Thank you in advance for any help!

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Hi Lauren,

Currently, the list of species is generated more or less exhaustively (iteratively through every possible combination) by checking whether or not a valid stoichiometric combination of species exists to satisfy the target composition. The code for this is contained in the pymatgen code in the pymatgen.analysis.pourbaix_diagram code, which you can find at the link below.


Hi Lauren,

I think you are asking two questions here: the first is regarding the mass balance for the available redox reactions (involving H+, e-, and O2), and the second question is regarding the mass balance in a multi-element Pourbaix diagram?

In response to the first question: The Materials Project actually uses a different method to construct Pourbaix Diagrams than is traditionally done. There are two ways to construct a Pourbaix Diagram. In one approach, you can use the Nernst Equation to compute phase boundaries between solid-aqueous stability regions, for example, as discussed below in [1]. Another way is one would make a dense E-pH grid, and evaluate the all available redox reactions at each grid point and find the lowest Gibbs free-energy phase, and then add the boundary lines; for example as done in [2,3].

The Pourbaix code implemented in Pymatgen and used on the Materials Project actually uses a different approach. This is done by constructing ‘Pourbaix free-energy planes’, made by a Legendre Transform of the Gibbs free energy to the E and p H axes, and taking the lower concave envelope of the resulting free-energy planes. This approach was developed in the Ceder Group about 10 years ago, by Denis Kramer and later implemented into the Materials Project by Sai Jayaraman, but the method was actually never published until literally last week [4]. You can find the equation for the Pourbaix Potential on Lines 59-62 of the code in pymatgen, or in the Methods section of [4]. In the Legendre transformed Pourbaix Potential, redox reactions are evaluated implicitly, and so it is unnecessary to formulate any redox balances between the metal and H2O.

In response to your second question, I think you are asking about the normalization of metal species in a multi-element Pourbaix Diagram? This is indeed a very confusing matter, and the documentation on the MP is not very valuable.

First, it is important to consider the boundary conditions of the thermodynamic analysis that is done on a Pourbaix Diagram. In the Legendre transform approach, the thermodynamic boundary conditions are not defined around the full aqueous system including precipitates, but only around the metal-containing solute phases in water (See Figure 1 of the ChemRxiv [5]). The resulting thermodynamic potential corresponds to the free-energy of an aqueous ion precursor or metal oxide precipitate being acted upon by an open external water reservoir at a given p H, redox potential, and dissolved metal ion concentration.

For a multielement system, the Pourbaix diagram has a metal-composition inside the boundary conditions , and the dissolved metal ion concentration outside of the boundary conditions. The Pourbaix diagram shows what would be the stable species inside the boundary conditions.

In the multielemental Pourbaix diagram construction, the user specifies the metal M1:M2 ratio composition inside the boundary conditions. The Pourbaix diagram tells you if that metal composition would be stable, given the external applied E, pH, and [M1], [M2], etc concentrations.

The Pourbaix free-energy plane in a multicomponent Pourbaix diagram actually also has not been published yet, although we are writing up a paper that includes this derivation now and should be submitted in a couple of months. Essentially though, if you imagine Pourbaix planes for each individual phase, the multicomponent Pourbaix plane is a linear interpolation between multiple pure-phase planes. For example, Ψ(AO + 0.5 BO2) = Ψ(AO) + 0.5 Ψ(BO2) / NormalizationConstant. The coefficient weights for the interpolation are the user-specified metal composition inside the boundary conditions.

As Joseph described previously, for each M1:M2 ratio there is an enormous combinatorial problem of what possible phases there are. Joseph and Arunima Singh developed a nice algorithm to construct all the weights for the multielemental Pourbaix planes [3].

Please feel free to ask any other questions you may have. I will try to update the Pourbaix documentation on the MP when I have time.





[5] Figure 1 shows the relevant boundary conditions, although this was removed from [4] by the referees, heh.

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Hi @wenhaosun , I know I’m resurrecting a very old thread but I just wanted to let you know that my group recently found your last post extremely helpful for understanding the compositional constraint argument in the MP Pourbaix diagram code. So thank you for that!

I wanted to ask 2 follow up questions:

The Pourbaix free-energy plane in a multicomponent Pourbaix diagram actually also has not been published yet, although we are writing up a paper that includes this derivation now and should be submitted in a couple of months.

  1. Was this paper ever published, and if so, which one is it? I’d be interested to read.

The resulting thermodynamic potential corresponds to the free-energy of an aqueous ion precursor or metal oxide precipitate

  1. Is this formulation of the potential still valid if the precipitate is not an oxide? For example, if it’s a chloride? Or would that require introducing an additional chemical potential term?