# Phase Diagram App Manual

## Contents

## Introduction

Phase diagrams represent the thermodynamic phase equilibria of multicomponent systems and reveal useful insights into fundamental material aspects regarding the processing and reactions of materials. However, the experimental determination of a phase diagram is an extremely time-consuming process, requiring careful synthesis and characterization of all phases in a chemical system.

Computational tools can accelerate phase diagram construction significantly. By calculating the energies of all known compounds in a given chemical system (e.g., Li-Fe-O), we can determine the phase diagram for that system at 0 K and 0 atm. Furthermore, for systems comprised of predominantly solid phases open with respect to a gaseous element, approximations can be made as to the finite temperature and pressure phase diagrams.

In this manual, we will outline the general usage and thermodynamics methodology of our computational phase diagrams app. The details of the calculations, including computational methodology and accuracy, used in constructing these phase diagrams are available in the Calculations Manual and will not be further covered in this manual.

## Using the Computational Phase Diagram App

To construct the phase diagram of a 2-4 component chemical system (e.g., Li-Fe-O), you only need to select the desired elements from the periodic table and click the "Generate" button.

To generate a grand potential phase diagram, for which one of the components of the phase diagram is considered 'open' to the system, select a projected element from the drop-down list, and input a desired chemical potential either manually or using the temperature slider (for supported elements, O_{2}, N_{2}, H_{2}, F_{2}, Cl_{2} only). When available, the temperature slider has been calibrated using entropy data from NIST.^{[1]}

Once the phase diagram has been generated, there are multiple ways to interact with the data:

- When mousing over nodes in the phase diagram, a pop-up appears that identifies the phase. The pop-up lists both stable and unstable crystal structures computed at that composition. You may click the id link to bring up more details of that compound, such as detailed crystal structure information.
- Checking the 'Show Unstable' box will display the compositions at which unstable compounds have been computed as blue circles.
- You can zoom into a portion of the phase diagram by defining a zoom area: click and hold at the upper-left corner of the desired area, and drag down and right, releasing the mouse at the lower-right corner of the desired area. A pop-up with the zoomed-in portion should appear. To return to the original phase diagram, click anywhere outside the pop-up.

## Interpreting Phase Diagrams

This section provides a short guide on interpreting phase diagrams generated by the PDApp. A more comprehensive discussion and example of phase diagram interpretation can be found in refs ^{[2]} and ^{[3]}.

### Basic Phase Diagram Information

While the PDApp generates several types of phase diagrams, some information is common to all phase diagrams. As an illustrative example, we refer to Figure 1, a ternary Li-Fe-O phase diagram at 0 K and 0 atm calculated using the PDApp.

Several pieces of information can be obtained from Figure 1:

- The red nodes on the phase diagram represent phases that are calculated to be stable under the given conditions.
- Blue circles (not shown) represent phases that are calculated to be unstable under the given conditions.
- An arbitrary composition will thermodynamically decompose to stable phases (red nodes), with the relative proportion of each stable phase determined by the lever rule.

Regarding polymorphic compounds (compounds that change crystal structure with respect to temperature or pressure), only limited information can be obtained because phase diagrams generated by the PDApp do not include an explicit temperature axis. For most solid materials, our experience indicates that temperature and pressure contribute fairly small changes to differences in total energy. Thus, we expect that crystal structures for which the PDApp reports similar formation energies (i.e., within about 50 meV/atom) might be candidates for polymorphism.

The PDApp models only the thermodynamics of a system. Thus, compounds that are metastable (kinetically stabilized) may not show up as stable on the generated phase diagram. We expect that compounds for which the decomposition energy into stable phases is low are potential candidates for metastability; however, compounds with extremely high decomposition energies into stable phases (i.e., over about 200 meV/atom) are less likely to be metastable.

### Compositional Phase Diagrams

In a compositional phase diagram, the system is closed with respect to the environment. We display compositional phase diagrams involving different numbers of components in slightly different ways.

#### Binary Compositional Phase Diagrams

Figure 2 is an example of a calculated binary Fe-P phase diagram at 0K and 0 atm using the PDApp. Binary phase diagrams show the complete convex hull for the system, where the y-axis is the formation energy per atom and the x-axis is the composition (e.g., in the Fe-P phase diagram, the x-axis is the fraction of P).

The black lines show the convex hull construction, which connects stable phases. Unstable phases will always appear above the convex hull line; one measure of the thermodynamic stability of an arbitrary compound is its distance from the convex hull line, which predicts the decomposition energy of that phase into the most stable phases.

#### Ternary Compositional Phase Diagrams

Figure 1 is an example of a calculated ternary Li-Fe-O phase diagram at 0 K and 0 atm using the PDApp. The presentation of ternary phase diagrams differs from that of binary phase diagrams; the energy axis is removed so that the entire compositional space can be represented. (The energy axis, not shown, can be thought of as coming 'out of the page').

The black lines in the ternary phase diagrams are projections of the convex hull construction into compositional space. The lines form Gibbs triangles, which can be used to find stable phases at an arbitrary composition. At any point in the phase diagram other than the stable nodes, the equilibrium phases are given by vertices of the triangle bounding that composition. For example, the equilibrium phases for a composition with Li:Fe:O ratio of 1:1:1 (i.e., in the center of the phase diagram), is predicted to be Li_{5}FeO_{4}, LiFeO_{2} and Fe.

#### Quaternary Compositional Phase Diagrams

Quaternary phase diagrams are presented in the same general way as ternary phase diagrams, but an additional axis is needed to represent the fourth composition. Therefore, quaternary phase diagrams are shown in three dimensions rather than as a 2D plot.

The lines in quaternary phase diagrams define polyhedra with 4 vertices each rather than triangles. At any point in the phase diagram other than the stable nodes, the equilibrium phases are given by the vertices of the polyhedron bounding that composition.

### Grand Potential Phase Diagrams

Grand potential phase diagrams are phase diagrams representing phase equilibria that are open to a particular element. In these phase diagrams, the chemical potential of an element is an external input variable that the user must specify. In some cases (e.g., a gaseous element such as oxygen), the chemical potential can be related to temperature and partial pressure via thermodynamic relations and known experimental data[5].

A grand potential phase diagram involving *n* components is presented in the same way as a compositional phase diagram involving *n-1* components.

Figure 3 shows the calculated grand potential phase diagram for the Li-Fe-O system at an oxygen chemical potential consistent with 700 K and 0 atm. The phase diagram provides information on the equilibrium phases under these conditions for any Li:Fe composition (note that the O fraction is controlled by interactions with the environment). For example, we may observe that a compositon with a relatively high Fe ratio (near composition = 1) will comprise Fe_{2}O_{3} and LiFeO_{2}. In addition, by examining the changes in the equilibrium phases as the chemical potential changes, we can obtain insights into how the phase equilibria are modified with changes in the chemical potential. A detailed discussion of this application can be found in refs ^{[2]} and ^{[3]}.

## Accuracy of Calculated Phase Diagrams

Figure 4 shows the experimentally determined phase diagram for the Li-Fe-O system at 673K.^{[4]} Even though the experimental phase diagram is at a much higher temperature, we can see that our calculated phase diagram reproduces the features very well. The phases Li_{2}O, Fe_{3}O_{4}, Fe_{2}O_{3}, LiFeO_{2}, Li_{5}FeO_{4} are stable in both the experimental diagram and our calculated phase diagram. The only additional phase in our calculated diagram is FeO, which is well known to be difficult to obtain in stoichiometric proportion under normal conditions. Our calculations correctly reproduce the FeO formation enthalpy to within experimental accuracy, and our calculated phase diagram is fullly consistent with known experimental thermochemical data along the Fe-O line at 1 atm and 298K.

In general, we can expect that compositional phase diagrams comprising of predominantly solid phases to be reproduced fairly well by our calculations. However, it should be noted that there are inherent limitations in accuracy in the DFT calculated energies. Furthermore, our calculated phase diagrams are at 0K and 0atm, and differences with non-zero temperature phase diagrams are to be expected.

For grand potential phase diagrams, further approximations are made as to the entropic contributions.^{[2]}^{[3]} They are therefore expected to be less accurate, but nonetheless provide useful insights on general trends.

## Thermodynamics Methodology

In the remainder of this section, we will use an Li-Fe-O system to illustrate the methodology without loss of generality.

### Compositional Phase Diagrams

To construct a phase diagram, one would need to compare the relative thermodynamic stability of phases belonging to the system using an appropriate free energy model. For an isothermal, isobaric, closed system, the relevant thermodynamic potential is the Gibbs free energy, <math>G</math>, which can be expressed as a Legendre transform of the enthalpy, <math>H</math>, and internal energy, <math>E</math>, as follows:

<math>
\begin{align} G(T, P, N_{Li}, N_{Fe}, N_{O}) & = H(T, P, N_{Li}, N_{Fe}, N_{O}) - TS(T, P, N_{Li}, N_{Fe}, N_{O}) )\\
&= E(T, P, N_{Li}, N_{Fe}, N_{O}) + PV(T, P, N_{Li}, N_{Fe}, N_{O}) - TS(T, P, N_{Li}, N_{Fe}, N_{O})
\end{align}
</math>

where <math>T</math> is the temperature of the system, <math>S</math> is the entropy
of the system, <math>P</math> is the pressure of the system, <math>V</math> is the volume
of the system, and <math>N_i</math> is the number of atoms of species i in the system.

For systems comprising primarily of condensed phases, the <math>PV</math> term can be neglected and at 0K, the expression for <math>G</math> simplifies to just <math>E</math>. Normalizing <math>E</math> with respect to the total number of particles in the system, we obtain <math>\bar{E}(0,P,x_{Li}, x_{Fe}, x_O)</math>, where <math>x_i = \frac{N_i}{N_{Li} + N_{Fe} + N_O}</math>. By taking the convex hull^{[5]} of <math>\bar{E}</math> for all phases belonging to the <math>M</math>-component system and projecting the stable nodes into the <math>(M - 1)</math>-dimension composition space, one can obtain the 0 K phase diagram for the closed system at constant pressure. The convex hull of a set of points is the smallest convex set containing the points. For instance, to construct a 0 K, closed Li-Fe-O system phase diagram, the convex hull is taken on the set of points in <math>(\bar{E}, x_{Li}, x_{Fe})</math> space with <math>x_O</math> being related to the other composition variables by <math>x_O = 1 - x_{Li} - x_{Fe}</math>.

### Grand Potential Phase Diagrams

In many scientific applications, the phase equilibria of interest is not that of a closed system, but rather one which is open to a particular element. For instance, synthesis might be carried out in flowing Argon gas at a particular temperature, which would mean the system is open to gaseous elements such as oxygen. Other experiments may be carried out in air, which essentially serves as an infinite reservoir of atmospheric elements such as oxygen, nitrogen and others.

In environments that are open to a particular element (say oxygen), the relevant control variable is the chemical potential of that element (<math>\mu_{O}</math>). The relevant thermodynamic potential to study phase equilibria in an open system is the grand potential, which is defined as the following for an Li-Fe-O system that is open with respect to oxygen:

<math>
\begin{align}
\phi(T, P, N_{Li}, N_{Fe}, \mu_{O}) & = G(T, P, N_{Li}, N_{Fe}, \mu_{O}) - \mu_O N_O(T, P, N_{Li}, N_{Fe}, \mu_{O})\\
& = E(T, P, N_{Li}, N_{Fe}, \mu_{O}) - TS(T, P, N_{Li}, N_{Fe}, \mu_{O}) - \mu_O N_O(T, P, N_{Li}, N_{Fe}, \mu_{O})
\end{align}
</math>

where the <math>PV</math> term is again ignored.

Normalizing with respect to Li-Fe composition and dropping the explicit expression of the functional dependence of <math>E</math>, <math>S</math>, and <math>N_O</math> on the right-hand side henceforth for brevity, we can obtain

<math>
\bar{\phi}(T, P, x_{Li}, x_{Fe}, \mu_O) = \frac{E - TS - \mu_O N_O}{N_{Li} + N_{Fe}}
</math>

where <math>x_i = \frac{N_i}{N_{Li} + N_{Fe}}</math> is the fraction of component <math>i</math> in Li-Fe composition space.

To formally introduce temperature into ab initio phase stability calculations, one would usually need to take into account all the relevant excitations (e.g., vibrational, configurational, and electronic) that contribute to entropy. However, for systems where phase equilibria changes takes place primarily between solid phases with absorption or loss of as gas, a few simplifying assumptions can be made that allow us to obtain a useful approximate phase diagram with less effort. In such reactions, the reaction entropy is dominated by the entropy of the gas, and the effect of temperature is mostly captured by changes in the chemical potential of the gas. Since the <math>TS</math> term is the entropy contribution of the condensed system, it can be neglected compared to the entropy effect of the gas, and the expression for <math>\bar{\phi}</math> simplifies to

<math> \bar{\phi} = \frac{E - \mu_O N_O}{N_{Li} + N_{Fe}} </math>

Using the above assumption, the effect of temperature and partial pressure can be fully captured in a single chemical potential variable, with a more negative value corresponding to higher temperatures or lower partial pressures. The chemical potential is then treated as an external variable to obtain a grand potential phase diagram under a particular condition by taking the convex hull of <math>\bar{\phi}</math> for all phases and projecting the stable nodes into the Li-Fe composition space.

## Other Resources

- Matlab package developed by this author (S.P. Ong) to generate publication-quality figures from a csv data file.

## Citation

To cite the Computational Phase Diagram App, please reference the following works:

- S. P. Ong, L. Wang, B. Kang, G. Ceder., The Li-Fe-P-O
_{2}Phase Diagram from First Principles Calculations, Chemistry of Materials, vol. 20, Mar. 2008, pp. 1798-1807. - S.P. Ong, A. Jain, G. Hautier, B. Kang, and G. Ceder, Thermal stabilities of delithiated olivine MPO
_{4}(M=Fe, Mn) cathodes investigated using first principles calculations, Electrochemistry Communications, vol. 12, 2010, pp. 427-430.

## References

- ↑ NIST Chemistry WebBook, NIST Standard Reference Database Number 69, Eds. P.J. Linstrom and W.G. Mallard.
- ↑
^{2.0}^{2.1}^{2.2}S. P. Ong, L. Wang, B. Kang, G. Ceder., The Li-Fe-P-O_{2}Phase Diagram from First Principles Calculations, Chemistry of Materials, vol. 20, Mar. 2008, pp. 1798-1807. - ↑
^{3.0}^{3.1}^{3.2}S.P. Ong, A. Jain, G. Hautier, B. Kang, and G. Ceder, Thermal stabilities of delithiated olivine MPO_{4}(M=Fe, Mn) cathodes investigated using first principles calculations, Electrochemistry Communications, vol. 12, 2010, pp. 427-430. - ↑ V. Raghavan, Fe-Li-O Phase Diagram, ASM Alloy Phase Diagrams Center, P. Villars, editor-in-chief; H. Okamoto and K. Cenzual, section editors; http://www1.asminternational.org/AsmEnterprise/APD, ASM International, Materials Park, OH, 2006.
- ↑ C. B. Barber, D. P. Dobkin, & H. Huhdanpaa, 1996. The quickhull algorithm for convex hulls. ACM Transactions on Mathematical Software (TOMS), 22(4), p.469.

## Authors

- Shyue Ping Ong
- Anubhav Jain