Insight Into Point Defects and Impurities in Titanium from First Principles

Insight into Point Defects and Impurities in Titanium from First Principles

Sanjeev K. Nayak,a Cain J. Hung, aVinit Sharma, a,[†]S. Pamir Alpay, a,bAvinash M. Dongare, a

William J. Brindley,c and Rainer J. Hebert a,d

a Department of Materials Science and Engineering and Institute of Materials Science,

University of Connecticut, Storrs, CT 06269, USA

b Department of Physics, University of Connecticut, Storrs, CT 06269, USA

c Pratt & Whitney, East Hartford, CT 06108, USA

d Additive Manufacturing Innovation Center, University of Connecticut, Storrs, CT 06269, USA

Supplementary Methods

Formation energy obtained from the first-principles total energies are dependent on the values of chemical potentials. Determining the precise value of chemical potential of a component needs thorough knowledge of the reaction mechanism and the associated chemical environment. While the precise value of chemical potential can be difficult to determine, one can define the allowed range of chemical potential by defining its bounds from thermodynamic considerations 1,2,3,4,5. In our work, the upper bound of chemical potential is taken from the elemental sources and the allowed range is calculated from the formation enthalpy of the corresponding oxide compound. For example, consider the compound Al2O3 for which the formation enthalpy satisfies the equation:

(Al2O3) = .(A1)

The upper bound of is chosen for pure aluminum (from the total energy of fcc Al). The chemical potential is bounded by (Al2O3) of Eq. (A1), thus the lower bound of can be computed from and , accordingly. For the thermodynamic stability of Al2O3, a large value of automatically means smaller value of and vice versa, coupled via Eq. (A1).

There exists a second method for the choice of chemical potential 6,7, which is more applicable for an ideal gas element as one of the components. The chemical potential for an ideal gas as function of temperature (T) has been determined experimentally and documented in standard data tables. The pressure dependence of the chemical potential can be computed from the thermodynamic relation:

(T, p) = (T, p0) + (1/2) kBT ln(p/p0),(A2) where, (T, p) and (T, p0) are the chemical potentials at (T, p) and (T, p0), respectively. p0 is the standard pressure of 1 atmosphere and kB is the Boltzmann’s constant. One advantage of this approach is that one can map the data to partial pressure of respective gases. However, the limitation in this analysis can easily be seen in systems that do not have gaseous element components. There are other approaches proposed for an improved description of formation enthalpies and chemical potential of alloys 8,9,10 and liquids 11, which has not been pursued in this work.

We have combined the above two approaches to bring forth two sets of chemical potentials. The elemental chemical potentials provide the upper bound for value of () for the impurity elements, while the oxide compounds provide the choice of () for the other extremum. The oxide components can also be used to analyze the pressure dependence of formation energies. Thus, we have, for example, the chemical potentials of Al as,

(Al fcc) and(A3)

(Al2O3)/2,(A4)

where (O2)/2. Likewise, the chemical potentials for oxygen is taken as,

(O2)/2, and (A5)

(TiO2), (A6)

where is derived from total energy of hcpTi.

We use a similar methodology for the chemical potentials for other impurities. The compounds chosen for deriving : H2 molecule, bcc Li, hcp Be, rhombohedral B, hcp C (graphite), N2 molecule, O2 molecule, F2 molecule, bcc Na, hcp Mg, fcc Al, cubic Si (diamond structure), orthorhombic P, orthorhombic S, Cl2 molecule, bcc K, fcc Ca, bcc V, fcc Ni,fcc Cu, hcpZr, fcc Mo, fcc Ag, bcc Ta, bcc W, fcc Au, fccPb.

Theis derived from the total energies of following compounds: H2O molecule, LiO2, BeO, B2O3, CO2 molecule, N2O3 molecule, TiO2 (for ), FO2 molecule, NaOH molecule, MgO, Al2O3, SiO2, P2O5 molecule, SO2 molecule, Cl2O3 molecule, K2O, CaO, V2O5, NiO, CuOZrO2, MoO2, AgO, Ta2O5, WO3, Au2O3, PbO. The crystal structure of solids are taken from the American Mineralogist Crystal Structure Database 12 wherever possible. The chemical potential of Ti, per atom, is derived from the hcpTi metal in order to be consistent with the host metal. The structuresof reference compounds are geometrically optimized before the total energies are taken into account.

Supplementary References

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Nayaket al., Point Defects in Ti, page 1

[†]Current address: Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831