Chemical potentials#

Chemical potentials determine the formation energy of point defects, as shown in the Formation energies tutorial:

\[\Delta E_f = E_D - E_B + q(\epsilon_{VBM} + \epsilon_F) - \sum_i \Delta n_i \; \mu_i + E_{corr}\]

where \(\mu_i\) is the chemical potential of particle \(i\). When we compute formation energies, we make a choice of which system (reservoir) exchanges the particles with our target phase. We therefore define a set of chemical potentials, which in defermi can be passed as dictionaries or Chempots objects, which behave like dictionaries. Chempots objects provide additional methods to convert values from absolute to referenced and to convert dictionaries from and to the pymatgen format ({Element object: value}).

[21]:

from defermi.chempots.core import Chempots

chempots_dict = {'Sr':-3,'O':-7} # absolute chemical potentials in eV

chempots = Chempots(chempots_dict)
print(chempots['Sr']) # subscriptable like a dict
print(chempots.keys())

chempots.to_pmg_elements()

-3
dict_keys(['Sr', 'O'])

[21]:

{Element Sr: -3, Element O: -7}

It is often convenient to express the chemical potentials relative to the elemental phase:

\[\Delta \mu_i = \mu_i - \mu^{\mathrm{ref}}_i\]

Use the get_absolute and get_referenced functions to change from one convention to the other.

[22]:

mu_refs = {'Sr':-2,'O':-5}
chempots.get_referenced(mu_refs)

[22]:

{'Sr': -1, 'O': -2}

A typical use case is to show the formation energies for different sets of chemical potentials (see the Formation energies tutorial). We can more easily manipulate a dictionary of sets of chemical potentials with the Reservoirs class. Provide the elemental chemical potentials to be used as reference with the mu_refs argument. Use the are_chempots_delta argument to declare whether you are providing relative (delta) or absolute values.

[23]:

from defermi.chempots.reservoirs import Reservoirs

res_dict = {
        'A':Chempots({'Sr':-2,'O':-8}),
        'B':Chempots({'Sr':-3,'O':-6.5}),
        'C':Chempots({'Sr':-4,'O':-5})
        }

res = Reservoirs(res_dict=res_dict,mu_refs=mu_refs,are_chempots_delta=False)
res

[23]:

	Sr	O
A	-2.0	-8.0
B	-3.0	-6.5
C	-4.0	-5.0

Move from absolute to referenced values easily:

[24]:

res.set_to_referenced()
print(res.are_chempots_delta)
res

True

[24]:

	Sr	O
A	0.0	-3.0
B	-1.0	-1.5
C	-2.0	0.0

Filter reservoirs for specific elements
create latex tables

[25]:

res.filter_reservoirs(elements=['O'])

[25]:

	O
A	-3.0
B	-1.5
C	0.0

[26]:

print(res.get_latex_table())

\begin{tabular}{lrr}
\toprule
 & $\Delta \mu_{\text{Sr}}$ & $\Delta \mu_{\text{O}}$ \\
\midrule
A & 0.000000 & -3.000000 \\
B & -1.000000 & -1.500000 \\
C & -2.000000 & 0.000000 \\
\bottomrule
\end{tabular}

Phase diagram#

Reservoirs are often chosen from the stability region of the target phase, obtained from the phase diagram. We can store the phase diagram in the Reservoirs object. Phase diagrams computed with DFT can be also pulled from the Materials Project. defermi provides wrapper functions for the MP API in the tools.materials_project module. You

[27]:

from defermi.tools.materials_project import MPDatabase
from pymatgen.core.composition import Composition

phase_diagram = MPDatabase().get_phase_diagram_from_chemsys('Ba-Ti-O') #if not stored in your .pmgrc file, provide the MP API_KEY
phase_diagram

[27]:

Ba-Ti-O phase diagram
21 stable phases:
TiO, BaO2, Ti3O5, BaO10, Ti2O, Ba, O2, BaO, Ti3O, Ba2Ti13O22, Ba4Ti13O30, Ba2TiO4, Ti2O3, BaTiO3, Ba3Ti20O40, Ti6O, TiO2, BaTi2O5, Ti, BaTi8O16, Ba2Ti6O13

[28]:

boundaries = phase_diagram.get_all_chempots(Composition('BaTiO3'))
res_dict = {key: Chempots.from_pmg_elements(value) for key,value in boundaries.items()}
res = Reservoirs(res_dict,phase_diagram=phase_diagram,are_chempots_delta=False)
res

[28]:

	Ba	Ti	O
BaTiO3-BaTi2O5-Ba2Ti6O13	-3.755598	-9.033464	-9.776825
BaTiO3-TiO-Ba2TiO4	-2.938610	-8.398586	-10.260780
BaTiO3-BaO2-Ba2TiO4	-8.073304	-18.667975	-5.126085
BaTiO3-BaO10-BaO2	-8.402881	-18.832763	-4.961297
BaTiO3-BaO10-O2	-8.536242	-18.739409	-4.947961
BaTiO3-BaTi2O5-O2	-8.584461	-18.691191	-4.947961
Ba2Ti13O22-BaTiO3-TiO	-3.304013	-8.581288	-10.078078
Ba2Ti13O22-BaTiO3-Ba2Ti6O13	-3.696412	-8.950604	-9.824173

We can use the functions in the chempots.generator module to speed-up with workflow.

[29]:

from defermi.chempots.generator import generate_chempots_from_mp

res = generate_chempots_from_mp(composition='BaTiO3')
res

Pulling chemical potentials from Materials Project database...
Chemical potentials:
                 Ba         Ti          O
O-poor   -2.938610  -8.398586 -10.260780
O-middle -5.737426 -13.568998  -7.604370
O-rich   -8.536242 -18.739409  -4.947961

[29]:

	Ba	Ti	O
O-poor	-2.938610	-8.398586	-10.260780
O-middle	-5.737426	-13.568998	-7.604370
O-rich	-8.536242	-18.739409	-4.947961

Chemical potentials

Contents

Chemical potentials#

Phase diagram#