# Decide the position of oxygen vacancy via the configuration with the lowest energy?

I am trying to determine the position of oxygen vacancy in $$\ce{TiO2}$$, is it reasonable to decide it by choosing the configuration with the lowest energy?

It doesn't feel quite right to always choose the configuration with the lowest energy, because it means this configuration is quite stable, if I wanna use it as a catalyst, I might want it to be active.

What is the rule for deciding the position of defects?

According to the minimum energy criterion, the oxygen-3 should be lost and form a vacancy, but it doesn't fit the EXAFS experiment, because when oxygen-3 is lost, there would be two nearest Ti atoms around the Cu atom, but the experiment results show there is only one nearest Ti atom.

• Is the preparation method of this particular TiO2 catalyst known? Choosing the lowest energy configuration gives the thermodynamically favored configuration, but if the catalyst preparation procedure is such that the locations of the vacancies are kinetically controlled, then the vacancies can indeed form at high-energy sites, but in that case you will have to somehow simulate the catalyst preparation procedure in order to determine the locations of the vacancies. Apr 25 at 20:28
• Could you give some examples in which the defect forms at high-energy sites?
– Jack
Apr 25 at 22:35
• This is a question that generally needs some critical thinking as well as digging into literature. If your slab model is too small, nothing guarantees that the calculated lowest energy vacancy position will be the same as the one in reality. If there is no huge energy difference between different positions, they may be present in your sample. If you surface has water etc coverage, vacancy stability may be different than your calculated one if you miss those details. If the difference in activity is large, you may have to study one of the less stable vacancy position... etc etc
– Greg
Apr 26 at 2:05
• if I choose to use the minimum energy criterion, should I use Gibbs free energy or internal energy? I mean the difference between these two parameters is caused by vibrations, is the vibration important when choosing the stable configuration?
– Jack
Apr 26 at 4:53
• @Jack For example if the vacancies are introduced by a plasma at room temperature, then it is well possible that all possible vacancies form with roughly equal probability, because they are produced in a high energy process and are rapidly quenched by cooling to room temperature. Apr 26 at 6:53

# Introduction

TiO$$_2$$ has an equilibrium concentration $$c_{eq}$$ of all symmetry-unique oxygen vacancies, which is given by

$$c_{eq} = \frac{N_{O_i}}{N_O} = \exp \left( -\frac{\Delta G_{vac}}{k_B T} \right)$$

where $$N_{O_i}$$ is the number of oxygen ($$O$$) of symmetry type $$i$$ in a sample or simulation cell of TiO$$_2$$, $$N_O$$ is the total number of oxygen in the same sample or simulation cell, $$k_B$$ is the Boltzmann constant, and $$T$$ is the temperature.

Note that this equation assumes that the oxygen vacancies do not interact, which allows the use of the mixing entropy of an ideal solution. This equation also assumes the formation of a neutral oxygen vacancy, i.e., O$$^{2-}$$ donates two electrons to TiO$$_2$$, which usually reduces two neighboring Ti$$^{4+}$$ to Ti$$^{3+}$$, as it leaves to form half a mole of O$$_2$$ gas. Since TiO$$_2$$ has a bandgap, it also can form charged vacancies (you can find an excellent and very recent summary of this here, https://doi.org/10.1038/s41578-022-00433-0).

$$\Delta G_{vac}$$ is the Gibbs free energy ($$G$$) of oxygen vacancy ($$vac$$) formation, which contains the O$$_2$$ gas pressure dependence and the effect of changes in the vibrational entropy.

Question 1: is it reasonable to decide it by choosing the configuration with the lowest energy?"

If one oxygen vacancy configuration has a much lower formation energy than the others, probably.

Question 2: What is the rule for deciding the position of defects?

All defect positions (i.e., symmetry-unique sites) have an equilibrium concentration...some larger than others...some experimentally detectable, others not. The "rules" deciding their concentrations are the first and second laws of thermodynamics, which give the equation for $$c_{eq}$$ above.

it doesn't feel quite right to always choose the configuration with the lowest energy, because it means this configuration is quite stable, if I wanna use it as a catalyst, I might want it to be active.

Unfortunately, we don't get to choose the lowest energy – or, better yet, highest concentration – defect configuration. This is controlled by crystal bond strengths (O$$^{2-}$$–Ti$$^{4+}$$), cation reducibility (Ti$$^{4+} \rightarrow$$ Ti$$^{3+}$$), the bandgap, polymorph stability (Rutile vs. Anatase vs. Brookite), etc. (Shameless plug: To learn more about the factors controlling neutral oxygen vacancy formation in metal oxides, check out https://doi.org/10.1021/jacs.1c05570)

I agree that sometimes a material isn't active enough (i.e., it doesn't form oxygen vacancies easily enough) to be a catalyst. First, you need to identify what is active enough, i.e., an acceptable range for the oxygen vacancy formation energy for your catalysis application; this is a prerequisite for catalyst design! Then, you can search for either (1) bulk materials with good oxygen vacancy formation energies or (2) dopants that tune a material's oxygen vacancy formation energies to within the acceptable range.

• Great answer! Just a small stylistic note: you can nicely format chemical names using the mhchem package, e.g. $\ce{TiO2}$ $\ce{TiO2}$