Corrosion Science 83 (2014) 94–102

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Corrosion Science

journal homepage: www.elsevier .com/locate /corsc i

Vacancy mechanism of oxygen diffusivity in bcc Fe:A first-principles study

http://dx.doi.org/10.1016/j.corsci.2014.02.0090010-938X/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +1 814 863 9957.E-mail address: sus26@psu.edu (S.L. Shang).

S.L. Shang a,b,⇑, H.Z. Fang a,b, J. Wang a,b,c, C.P. Guo a,b,d, Y. Wang a,b, P.D. Jablonski a,e, Y. Du c, Z.K. Liu a,b

a National Energy Technology Laboratory Regional University Alliance, U.S. Department of Energy, Pittsburgh, PA 15236, USAb Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USAc State Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan 410083, Chinad Department of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, Chinae National Energy Technology Laboratory, Department of Energy, Albany, OR 97321, USA

a r t i c l e i n f o

Article history:Received 13 December 2013Accepted 2 February 2014Available online 8 February 2014

Keywords:A. IronB. Modeling studiesC. OxidationC. Internal oxidation

a b s t r a c t

Diffusivity of interstitial oxygen (O) in bcc iron (Fe) with and without the effect of vacancy has beeninvestigated in terms of first-principles calculations within the framework of transition state theory.Examination of migration pathway and phonon results indicates that O in octahedral interstice is alwaysenergetically favorable (minimum energy) with and without vacancy. It is found that vacancy possessesan extremely high affinity for O in bcc Fe, increasing dramatically the energy barrier (�80%) for O migra-tion, and in turn, making the predicted diffusion coefficient of O in bcc Fe in favorable accord withexperiments.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Diffusion is a thermally activated process where a migratingatom passes through an energy barrier moving from a local energyminimum site to an adjacent vacant site. Diffusion determines avast number of materials properties related to the transfer of masssuch as the early stage of oxidation in Ni-based alloys [1], the cor-rosion of Al-based alloys under thin electrolyte layers [2], the mar-ine immersion corrosion of mild steel [3], and the oxidation andcorrosion of iron (Fe) associated with the present work. Despitetechnological importance and considerable experimental studiesperformed for diffusion of oxygen (O) in bcc Fe [4–10], little workregarding its fundamental mechanism has been studied comparedto the interstitial diffusions of nitrogen (N) [11] and especially car-bon (C) [12,13] in iron. Experimentally, diffusion coefficient of O inFe alloys is typically estimated from internal-oxidation [10] owingto a very low solubility of O compared with other interstitial atomsof C and N in Fe and its alloys [8]. The activation energy for diffu-sion of O in bcc Fe is about 1.0 ± 0.1 eV according to experimentsperformed by Takada et al. [8–10], Frank et al. [4,6], and Swisherand Turkdogan [5] (see details in Section 3).

However, first-principles predictions of activation energy for Odiffusion in defect-free bcc Fe are only half and less (<0.54 eV) thanexperimental results regardless of the exchange correlation (X-C)

functional as well as the consideration of charged species (thepresent results, see details in Section 3). The huge difference ofdiffusion activation energy between experiments and theoreticpredictions suggests that some important facts are missed infirst-principles calculations. On the other hand, both experimentsand simulations indicate that the interstitial atoms such as C, H, O,and N have an exceptionally high affinity for vacancy (Va) in iron,and the migrating interstitial atoms are trapped by C–, H–, O–,and N–Va pairs [14–19]. For example, oxygen can serve as aVa-stabilizing agent [20]. Furthermore, Fu et al. [16] showed thatthe O–Va pairs in bcc Fe possess large negative binding energies(�1.45 and �0.6 eV for the first and the second nearest neighbors,respectively, see details and the definition of binding energy in Sec-tion 3), and enables the nucleation of a high density of stablenanoclusters in Fe-based alloys processed by mechanical alloying.

The present work is motivated by the dearth of first-principlesstudy of O diffusion in bcc Fe, and in particular, the huge differenceof activation energy between experiments and predictions regard-ing the diffusion of O in defect-free bcc Fe. Recently it has beendemonstrated that the quantitative diffusion properties as a func-tion of temperature including jump rates and migration barrierscan be predicted in solid phases in terms of first-principles calcu-lations within the framework of transition state theory [21–28].In the present work, diffusion of O in bcc Fe with and withoutthe effect of vacancy is studied through first-principles calculationsbased on the reaction rate theory of Eyring [29] as well as the tran-sition state theory of Vineyard [30]. The climbing image nudged

S.L. Shang et al. / Corrosion Science 83 (2014) 94–102 95

elastic band (CINEB) method [31] is used to search the transitionstate (i.e., the saddle point). The phonon calculations in terms ofthe supercell method [32,33] as well as the Debye model [34] areused to predict thermodynamic properties at finite temperaturesfor various states during diffusion. Details of these computationalmethods are given in Section 2. We find a dramatic effect of va-cancy on the predicted diffusion properties of O in bcc Fe, andthe deeply trapped O by Va is found owing to an increased bendingdeformation resistance of O in terms of differential charge densityand phonon force constants (see Section 3). Finally, in Section 4 theconclusions of the present work are given.

2. Computational methods

2.1. Interstitial diffusion coefficient and first-principlesthermodynamics

Diffusion of interstitial solute atoms in solid solutions can bedescribed as random jumps between interstitial positions. Accord-ing to Wert and Zener [35], theoretical diffusion coefficient D of aninterstitial atom in a cubic lattice can be described by,

D ¼ nbd2C ð1Þ

where n is the number of the nearest neighbor (NN) interstitialpositions, 2b the probability when an interstitial atom jumps intoa new interstitial position, and d the projected length onto the dif-fusion direction. For the present octahedral interstitial diffusion inbcc lattice, n = 4, b = 1/6, and d = a/2 with a being the lattice param-eter [35]. Note that these values also hold when a NN vacancy isintroduced to an interstitial atom. Fig. 1 illustrates a 1 � 1 � 1supercell of bcc lattice (atoms located at Wyckoff site 2a of spacegroup Im�3m) together with its octahedral interstitial positions(Wyckoff site 6b) and tetrahedral interstitial positions (Wyckoff site12d). Symbol C in Eq. (1) represents the jump rate [27,28,35],

C ¼ kBTh

ZTS

ZISexp � DE

kBT

� �ð2Þ

where ZTS and ZIS are the partition functions of the transition state(TS) and the ground state of initial state (IS), respectively, kB theBoltzman’s constant, T the temperature, and h the Planck’s constant.DE is the energy difference between the TS (i.e., saddle point) andthe IS, corresponding to the migration energy barrier. At high tem-peratures, the jump rate of Eq. (2) reduces to the result of Vineyard[30],

C ¼Q3N�3

i¼1 mISiQ3N�4

i¼1 mTSi

exp � DEkBT

� �ð3Þ

Fig. 1. 1 � 1 � 1 Supercell of bcc lattice with Fe metals located at Wyckoff site 2a ofspace group Im�3m, together with the octahedral interstitial positions (oct, Wyckoffsite 6b) and tetrahedral interstitial positions (tet, Wyckoff site 12d). As an examplewith one vacancy (Va) at position A, the oct sites B, C, and D are its first, second, andthird nearest neighbors (1NN, 2NN, and 3NN), respectively.

where mi is the real normal modes of vibration, and N the totalvibrating atoms. There are (3N � 3) non-zero optical modes forthe IS and the TS at zone center, and one imaginary optical modeshould be removed for the TS. In the present work, the productsof Pmi at different volumes are combined with the volume vs. tem-perature relationship obtained from the first-principles quasihar-monic approach (see Eq. (6)), therefore, Pmi is expressed as afunction of temperature herein. This volume–temperature transferrefers to the quasistatic approach, which has been used previouslyto predict the temperature dependences of elasticity [36,37], idealshear strength [38], and stacking fault energy [39]. At low temper-atures, the jump rate of Eq. (2) reduces to the reaction rate theoryfound by Eyring [29],

C ¼ kBTh

exp �DEþ DEZPE

kBT

� �ð4Þ

Here, DEZPE is the zero-point vibrational energy difference betweenthe TS and the IS. In practice and also in the present work, DE in Eq.(3) and (DE + DEZPE) in Eq. (4) are replaced by the temperaturedependent Gibbs energy difference DG (or Helmholtz energy differ-ence DF in the case of zero external pressure) between the TS andthe IS estimated using the first-principles quasiharmonic approachof Eq. (6). The application of DG or DF is an efficient and accurateway to predict jump rate at finite temperatures [24,27], even with-out phonon calculations (see below). It is worth mentioning that itis a common practice to represent the diffusion coefficient (e.g., ob-tained by Eq. (1)) according to Arrhenius equation,

D ¼ D0 exp�QkBT

� �ð5Þ

where D0 is a pre-exponential factor, and Q the activation energy(enthalpy) for diffusion.

Within the quasiharmoinc approach, first-principles calcula-tions based on the density functional theory (DFT) yield quantita-tive Helmholtz energy F for a given structure under volume V andtemperature T, which can be approximately separated into[34,40,41],

FðV ; TÞ ¼ EðVÞ þ FvibðV ; TÞ þ FelðV ; TÞ ð6Þ

where E is the static total energy at 0 K without the zero-pointvibrational energy, which is determined herein by fitting the first-principles energy vs. volume data points according to a four-param-eter Birch–Murnaghan equation of state (EOS) with its linear formgiven by [34],

EðVÞ ¼ aþ bV�2=3 þ cV�4=3 þ dV�2 ð7Þ

where a, b, c, and d are fitting parameters. The equilibrium proper-ties estimated from this EOS include the volume (V0), energy (E0),bulk modulus (B0) and its pressure derivative (B00). In the presentwork, six to eight energy-volume data points are used for EOS fit-ting for the structure of interest. The second term Fvib(V,T) andthe third term Fel(V,T) in Eq. (6) are related to contributions at finitetemperatures. Fel is particularly important for metals due to thenon-zero electronic density at the Fermi level, which can be esti-mated from the electronic densities of state at different volumes[34,40,41]. Fvib is the vibrational contribution to F(V,T), which canbe calculated from (i) the Debye model via the equilibrium proper-ties estimated from Eq. (7) for the sake of simplicity and (ii) thephonon density of state via first-principles phonon calculations. Inthe present work, both the Debye–Grüneisen model and the phononmethod are adopted to calculate Fvib with the detailed methodolo-gies given in [34].

Regarding the diffusion of oxygen in vacancy-included bcc Fe,only one major jump of oxygen with the maximum energy barrieris adopted in the present work, this jump determines mainly the

96 S.L. Shang et al. / Corrosion Science 83 (2014) 94–102

jump rate, see explanations by Ling and Scholl [42] regarding theirstudy of carbon diffusion in Pd and Pd-alloys. Unlike the study ofoxygen diffusion in fcc Ni using the combined jump of vacancyand oxygen by Nam et al. [43], we assume that vacancy is fixedin the metal matrix compared with the diffusion of oxygen. It isalso noticed that the proposed model concerning the combinedjump by Nam et al. [43] cannot be used in the present bcc lattice,since we cannot find two adjacent vacancies which are both thenearest neighbors of oxygen in the octahedral interstice of bcc lat-tice (unlike the fcc case proposed by Nam et al. [43]). Besides thepresent results shown in Section 3, our recent study of oxygen dif-fusion in vacancy-containing fcc Ni [44] indicated that the presentmodel in terms of the fixed vacancy and the major jump of oxygenalso could predict satisfactory results in comparison with experi-mental data.

2.2. First-principles and phonon calculations

All DFT-based first-principles calculations in the present studyare performed with the vasp 5.2 code [45,46]. The ion-electroninteraction is described by the projector augmented wave (PAW)method [47] and several exchange-correlation (X-C) functionalsare tested: the generalized gradient approximations (GGA) by Per-dew–Wang (PW91) [48] and by Perdew–Burke–Ernzerhof (PBE)[49], the PBE revised for solids (PBEsol) [50], the Armiento–Matts-son (AM05) [51], as well as the local density approximation (LDA)[52]. Note that the main X-C functional used in the present work isGGA-PBE unless noted otherwise. For Fe, eight electrons (3d62s2)are treated as valence and six for O (2s22p4). As a test, the possiblestrong on-site Coulomb interaction (U) presented in the localized3d electrons of Fe is performed using the GGA + U method accord-ing to the approach of Dudarev et al. [53], which depends on theeffective Coulomb interaction Ueff = U � J, with J being the screenedexchange energy. Here, Ueff = 3 eV is adopted based on the studyfor a-Fe2O3 [54]. Note that the GGA + U method is only used fortest purpose in order to check whether this method could changesignificantly the energy barrier of oxygen diffusion in bcc Fe, andonly one Ueff value (3 eV) is tested for Fe albeit different Ueff valueswere reported for different Fe-containing materials [55]. In addi-tion, calculations by considering the charge of oxygen (O2�) arealso performed with two more charges added in the system usingthe method described by Van de Walle and Neugebauer [56].

During VASP calculations, we employ a 520 eV plane wave en-ergy cutoff together with the 10 � 10 � 10, 6 � 6 � 6, and4 � 4 � 4 k-point meshes for the 2 � 2 � 2, 3 � 3 � 3, and4 � 4 � 4 supercells, respectively. The reciprocal-space energyintegration is performed by the Methfessel–Paxton [57] techniquefor structure relaxations and phonon calculations. For the final cal-culations of total energies and electronic structures, we adopt thetetrahedron method incorporating a Blöchl correction [58]. Theelectronic degrees of freedom are converged to at least 10�6 eVper atom, while the Hellman-Feynman force components are re-laxed to at least 10�3 eV/Å. Due to the ferromagnetic nature ofbcc Fe with its Curie temperature �1044 K [59], all DFT calcula-tions are performed within the spin-polarized approximation.

Saddle point structures and associated minimum energy path-ways are computed with the climbing image nudged elastic band(CINEB) method [31] by using 3 or 7 images and the 3 � 3 � 3supercell, where both the initial structures are fully relaxed beforeand after diffusion. Note that the 7 images are mainly used for testpurpose in order to search for the possible multi-saddle points [24]during the migration of oxygen, and only one saddle point is pre-dicted as shown in Section 3. Phonon calculations are performedusing the supercell method [60] with the force constants predictedby vasp 5.2 code (default displacements of ±0.015 Å are used in thepresent study to perturb the independent atoms) and the phonon

properties calculated by Yphon code [32,33]. It is worth remarkingthat the Yphon code developed recently in our group is a parame-ter-free, mixed-space approach that makes full use of the accurateforce constants from the real-space supercell approach and thelong-range dipole-dipole interactions from the linear response the-ory in the reciprocal space (at the C-point only), and therefore itworks for both polar and nonpolar materials. Here, phonon calcu-lations are performed for the selected 3 � 3 � 3 supercells. Moredetails of phonon calculations using Yphon can be found in[32,33,61–65].

3. Results and discussion

3.1. Site preference and migration energy barrier

Table 1 summarizes the equilibrium properties from first-prin-ciples calculations based on the EOS fitting of Eq. (7) for variousFe–O structures using different X-C functionals and under differentconditions. Note that due to the existence of anti-Invar behavior inbcc Fe [59], these EOS fittings are performed in the low-energylow-spin phase region when possible by examining the changesof magnetic moment and pressures as a function of volume. Con-cerning the relative energy DET–O of O in tetrahedral interstice rel-ative to that in octahedral interstice (see Fig. 1), Table 1 shows thatthe DET–O values predicted by PBE are 0.638, 0.503, and 0.496 eVfor Fe16O (2 � 2 � 2 supercell), Fe54O (3 � 3 � 3 supercell), andFe128O (4 � 4 � 4 supercell), respectively, indicating a convergentDET–O can be achieved using the 3 � 3 � 3 supercell, and thissupercell will be employed mainly in the present work. The totalenergies of O in three positions of bcc Fe (octahedral and tetrahe-dral interstices, and substitutional site) from Fe16O, Fe54O, andFe128O indicate that the octahedral interstice is energetically favor-able while the substitutional site is least favorable. Here the totalenergies of O in substitutional sites are fitted to the same compo-sitions as O in the octahedral and tetrahedral interstices using thecubic Hermite spline method and three energies of Fe15O, Fe53Oand Fe127O (see Table 1). The present results agree well with theprevious findings: C, O, and N bind preferentially at an octahedralinterstitial site of bcc Fe [16].

Fig. 2 illustrates the CINEB predicted minimum energy path-ways for O migration from one octahedral interstitial site to itsnearest octahedral interstitial site (such as from B to C or from Cto D as shown in Fig. 1) based on the X-C functional of PBE. Forthe prefect bcc Fe without vacancy, Fig. 2 shows that the transitionstate (saddle point) of O migration locates at the middle of the CI-NEB curve, i.e., the tetrahedral interstice. For the vacancy-contain-ing bcc Fe, the CINEB results of the 1NN to the 2NN case alsoindicate that oxygen in octahedral interstice adjacent to the near-est vacancy possesses the lower energy with respect to the oxygenin tetrahedral interstice adjacent to the nearest vacancy. Our pho-non results for both the transition state from the CINEB predictionand the structure of O in tetrahedral interstice give almost identi-cal results and especially both structures possess only one imagi-nary phonon mode. As examples, Fig. 3 shows the phonondensities of state of bcc Fe and the structures of O in octahedraland tetrahedral interstices of bcc Fe predicted at their theoreticalequilibrium volumes (see Table 1) in terms of the 3 � 3 � 3 super-cells and the PBE X-C functional. Experimental phonon density ofstate for bcc Fe at room temperature based on the Born-von Kar-man force constants [66] is also included for comparison, agreeingreasonably well with the present prediction. For the structure of Oin tetrahedral interstice, an imaginary phonon mode appearsaround �4.2 THz in Fig. 3, indicating that it is an unstable structureof transition state during O migration as mentioned above. In com-parison, the structure of O in octahedral interstice of bcc Fe is a

Table 1Equilibrium properties of bcc Fe due to the effects of O and vacancy (Va) predicted using the EOS of Eq. (7) and different X-C functionals (PBE, PBE + U, PW91, PBEsol, AM05, andLDA), where the atom O is located at the octahedral (oct) and tetrahedral (tet) interstitial sites as well as the substitutional (sub) site of bcc lattice, and the Fe16 (or Fe15), Fe54 (orFe53), and Fe128 (or Fe127) indicate the 2 � 2 � 2, 3 � 3 � 3, and 4 � 4 � 4 supercells of bcc Fe. The properties include equilibrium volume V0 (Å3/atom), total energy (eV/supercell),bulk modulus B0 (GPa), the first derivative of bulk modulus with respect to pressure (B00), and the magnetic moment MM (lB/atom). DE0 (eV) represents the relative energy of O inoct site with respect to the corresponding O in tet site (DET–O), or the energy difference between the Fe53OVa structures with the O–Va pair being the first nearest neighbor (1NN),the second nearest neighbor (2NN), and the third nearest neighbor (3NN). The ‘‘charged’’ indicates the charged species of O2� is considered. The available experimental data of bccFe are also included for comparison.

Structure Note V0 Total energy DE0 B0 B00 MM

Fe Expt. 11.783a 173b 5.0c 2.22d

Fe54 PBE 11.361 �448.726 186.1 5.32 2.28Fe53Va PBE 11.537 �438.249 174.3 5.37 2.23Fe16O (oct) PBE 11.571 �138.709 0 170.1 3.99 2.32Fe16O (tet) PBE 11.553 �138.071 0.638 156.3 5.16 2.20Fe15O (sub) PBE 11.504 �128.276 145.4 3.12 2.29Fe16O (sub)e PBE �130.3Fe54O (oct) PBE 11.424 �454.175 0 171.1 5.49 2.21Fe54O (tet) PBE 11.409 �453.672 0.503 171.2 5.34 2.17Fe53O (sub) PBE 11.401 �444.202 170.4 5.34 2.22Fe54O (sub)e PBE �450.1Fe128O (oct) PBE 11.399 �1069.043 0 180.8 5.07 2.20Fe128O (tet) PBE 11.387 �1068.547 0.496 173.8 6.03 2.18Fe127O (sub) PBE 11.398 �1059.576 170.8 6.13 2.19Fe128O (sub)e PBE �1063.6Fe53OVa (1NN)f PBE 11.430 �445.229 0g 166.3 6.47 2.15Fe53OVa (2NN)f PBE 11.556 �444.351 0.878g 174.5 7.00 2.19Fe53OVa (3NN)f PBE 11.583 �443.782 1.447g 160.3 5.95 2.21Fe53OVa (TS)h PBE 11.506 �444.337 0.892g 165.1 6.33 2.18Fe54O (oct) charged PBE 11.802 �442.972 0 154.1 5.66 2.10Fe54O (tet) charged PBE 11.786 �442.434 0.538 148.4 6.63 2.26Fe54O (oct) PBE + U 12.657 �330.615 0 111.5 5.24 2.69Fe54O (tet) PBE + U 12.740 �330.556 0.059 109.6 5.14 2.70Fe54O (oct) PW91 11.326 �446.208 0 181.0 5.19 2.17Fe54O (tet) PW91 11.309 �445.806 0.402 185.8 4.71 2.13Fe54O (oct) PBEsol 10.856 �475.917 0 206.7 5.30 2.13Fe54O (tet) PBEsol 10.841 �475.503 0.414 207.5 5.24 2.09Fe54O (oct) AM05 10.833 �470.474 0 203.0 5.42 2.15Fe54O (tet) AM05 10.814 �470.053 0.421 203.9 5.58 2.11Fe54O (oct) LDA 10.334 �504.398 0 239.1 4.49 1.92Fe54O (tet) LDA 10.326 �504.214 0.184 241.6 4.78 1.88

a Measured value at room temperature [71].b Extrapolated value to 0 K [72].c Ref. [73].d Ref. [74].e Fitted results using the cubic Hermite spline method and the energies of Fe15O (sub), Fe53O (sub) and Fe127O (sub).f O in octahedral interstitial site.g Relative energy with respect to the structure Fe53OVa (1NN).h Transition state (TS) predicted by the CINEB calculations, see Fig. 2, with the initial structure for CINEB calculations being Fe53OVa (1NN) as shown in this table. In

addition, the cell shape and atomic positions of TS are fixed during first-principles calculations for different volumes.

S.L. Shang et al. / Corrosion Science 83 (2014) 94–102 97

stable one without imaginary phonon modes. Therefore the rela-tive energy DET–O between O in tetrahedral and octahedral inter-stices is the O migration barrier in defect-free bcc Fe.

Table 2 shows that the O migration barrier of 0.526 eV from CI-NEB is slightly larger than the DET-O of 0.503 eV from the 3 � 3 � 3supercell, due to the constraints imposed on the CINEB calculationssuch as the fixed cell shape. Using different X-C functionals ofPW91, PBEsol, AM05, and LDA, as well as different treatments ofFe (the GGA + U method) and O (charged species), Table 1 showsthat the predicted DET–O (i.e., migration barriers) based on the3 � 3 � 3 supercells are 0.402, 0.414, 0.421, 0.184, 0.059, and0.538 eV, respectively. These migration barriers (viz., the rough dif-fusion activation energies) are smaller than half of the experimen-tal activation energies of 1.0 ± 0.1 eV (0.89–0.96 eV by Takada et al.[8–10], 0.98 ± 0.1 eV by Frank et al. [4,6], 1.01 eV by Swisher andTurkdogan [5], and 1.73 eV by Barlow et al. [7], see details in Ta-ble 2. Here, the larger activation energy of 1.73 eV may be for thepenetration of internal oxidation front [8]), indicating the failureof DFT prediciton of migration energy barrier using the prefect, de-fect-free bcc Fe.

Since vacancy has exceptionally high affinity for interstitialatoms (C, H, O, and N, etc.) in iron according to experiments andsimulations [14–19], we introduce one vacancy in bcc Fe, e.g., the

site A in Fig. 1 and put one O atom at the octahedral interstitialsites of B, C, and D, which are the first nearest neighbor (1NN),the second nearest neighbor (2NN), and the third nearest neighbor(3NN), respectively, with respect to vacancy located at site A. Here,the O–Va binding energy for the present 3 � 3 � 3 supercell of bccFe is defined as,

DEb ¼ ½EðFe53OVaÞ þ EðFe54Þ� � ½EðFe53VaÞ þ EðFe54OÞ� ð8Þ

where E(Fe53OVa) is the energy for O at octahedral interstice andclose to Va, E(Fe54) the energy of defect-free bcc Fe, E(Fe53Va) theenergy of bcc Fe with one vacancy, and E(Fe54O) the energy of bccFe with O in octahedral interstice. Note that the binding energy de-fined here is between the bound and unbound states and thus neg-ative, the same as the one used in the thermodynamic community[67], but other researchers also use the positive value to definethe binding energy [28,68]. In addition, we use Fe54O instead ofthe isolated O atom as reference state in order to avoid the lessaccurate first-principles results for isolated atoms. In fact, the bind-ing energy defined in Eq. (8) gives the energy difference betweenthe bound O–Va pair and the fully separate O–Va pair, and the en-ergy of [E(Fe53Va) � (53/54)E(Fe54)] is the vacancy formation en-ergy. Table 1 shows that the predicted O–Va binding energies are�1.53, �0.65, and �0.08 eV, respectively, for the 1NN, 2NN, and

1.0

0.8

0.6

0.4

0.2

0.0

Ener

gy d

iffer

ence

(eV)

1.00.80.60.40.20.0

Reaction coordinate

Fe53OVa (1NN to 2NN) Fe53OVa (2NN to 3NN) Fe54O (no Va)

TS (tet interstice)

TS

TS

Fig. 2. Minimum energy pathways of O diffusion in bcc Fe with Va (Fe53OVa) andwithout Va (Fe54O) predicted in terms of the PBE X-C functional and the CINEBmethod. For Fe54O, O diffuses from one oct site to its nearest oct site (e.g., B to C or Cto D in Fig. 1). For Fe53OVa with Va at e.g. A (see Fig. 1), O diffuses from 1NN to 2NN(B to C in Fig. 1) and from 2NN to 3NN (C to D in Fig. 1). Without Va, the tetinterstice is the transition state (TS), and the migration energy barriers for theseCINEB calculations are reported in Table 2. Note that the reaction coordinaterepresents the migration pathway of oxygen diffusion.

Phon

on D

OS

2520151050-5Frequency (THz)

Oxygen at oct site Oxygen at tet site bcc Fe bcc Fe (expt.)

Fig. 3. Phonon densities of state (DOS) for bcc Fe and the structures of O atoctahedral and tetrahedral interstices of bcc Fe predicted at their theoreticalequilibrium volumes (see Table 1) in terms of the 3 � 3 � 3 supercells and the PBEX-C functional. The experimental phonon DOS for bcc Fe at room temperature basedon the Born–von Karman force constants [66] is also plotted for comparison.

Table 2Predicted properties using the X-C functional of GGA-PBE and the measured results ofO diffusion in bcc Fe, including the migration energy barriers DEm (eV) at 0 K, the pre-exponential factor D0 (m2/s), and the activation energy Q (eV/atom) of Arrheniusequation (see Eq. (5)).

System DEm D0 Q

Fe54O (without Va)a 0.526b 3.75 � 10�7(e) 0.49e

0.503c 5.72 � 10�7(f) 0.56f

2.86 � 10�7(g) 0.54g

Fe53OVa (1NN to 2NN)a 0.897b 0.63 � 10�7(g) 0.80g

0.892d

Fe53OVa (2NN to 3NN)a 0.772b

Expt: O in dilute Al–Feh1:79þ1:76

�0:89

� �� 10�7 0.89 ± 0.06

Expt: O in dilute Si–Feh2:91þ3:40

�1:57

� �� 10�7 0.93 ± 0.07

Expt: O in dilute Ti–Feh3:78þ3:55

�1:83

� �� 10�7 0.95 ± 0.06

Expt: O in dilute Si–Fei (400 ± 100) � 10�7 1.73 ± 0.01Expt: O in dilute Al–Fej 37.2 � 10�7 1.01Expt: estimated valuesk 0.98 ± 0.1

a Oxygen in octahedral (oct) interstitial site and it diffuses from one oct site to thenearest oct site. Here, 1NN, 2NN, and 3NN indicate the first, the second, and thethird nearest neighbor of O with respect to vacancy (see also Table 1).

b Results from CINEB calculations.c Energy difference from EOS fittings (see Eq. (7)) between O in tet and oct sites

(see also Table 1).d Energy difference from EOS fittings (see Eq. (7)) between the initial state and

the transition state.e Jump rate obtained from the Vineyard method (i.e., Eq. (3)) and the vibrational

contribution to Gibbs energy from phonon calculations.f Jump rate obtained from the Eyring method (i.e., Eq. (4)) and the vibrational

contribution to Gibbs energy from phonon calculations.g Jump rate obtained from the Eyring method (i.e., Eq. (4)) and the vibrational

contribution to Gibbs energy from Debye model.h From the internal oxidation measurements [8–10], and the results were

extrapolated to matrix metal of bcc Fe.i Exclusive internal oxidation based on dilute 0.072 wt.% Si–Fe [7], and the

results were extrapolated to matrix metal of bcc Fe.j Combining the permeability and solubility data determined on dilute 0.1 wt.%

Al–Fe [5], and the results were extrapolated to matrix metal of bcc Fe.k Determined from data on the reduction of FeO with H and the internal oxida-

tion of Si in bcc Fe [4].

-1.6

-1.2

-0.8

-0.4

0.0

O-V

a bi

ndin

g en

ergy

(eV)

NN between O and Va

1NN

2NN

3NNfully separate O-Va

0.88

eV

0.57

eV

0.08

eV

Fe53Va Fe54OFe53OVa Fe54_

Energy 1 Energy 2

Fig. 4. Predicted O–Va binding energy in bcc Fe using the 3 � 3 � 3 supercell andthe PBE X-C functional with the O–Va pair being the first nearest neighbor (1NN,e.g., A–B in Fig. 1), the second nearest neighbor (2NN, e.g., A–C in Fig. 1), the thirdnearest neighbor (3NN, e.g., A–D in Fig. 1), and fully separate with infinite distanceof O–Va pair. The binding energy is defined in Eq. (8) and shown in the figure. Thepresent binding energies for the 1NN and the 2NN O–Va pairs are �1.53 and�0.65 eV, respectively, agreeing well with Fu et al.’s predictions [16] of �1.45 and�0.60 eV (a different definition of binding energy and first-principles methodologyof USPP-PW91 were used by Fu et al. [16]).

98 S.L. Shang et al. / Corrosion Science 83 (2014) 94–102

3NN pairs, agreeing well with the predictions of �1.45 eV (for 1NN)and �0.60 eV (for 2NN) by Fu et al. [16] using the PW91 X-C func-tional and the isolated O atom as one of the reference states. Thepresent binding energies are also plotted in Fig. 4 for easy view. Itis seen that the 1NN and even the 2NN O–Va pairs have quite largenegative binding energies, but after the 3NN, the absolute bindingenergies are rather small (<0.08 eV). The quick decrease of the abso-lute O–Va binding energy with increasing the O–Va distance indi-cates further the 3 � 3 � 3 supercell is large enough to mimic thedilute environment of vacancy in bcc Fe, since the 3NN O–Va paircan even be built within the 1 � 1 � 1 supercell as shown inFig. 1. The O–Va binding energies also indicate that the diffusionbehavior of oxygen in the vacancy-containing bcc Fe is similar tothat in the vacancy-free bcc Fe when the O–Va distance is largerthan the 3NN. Additionally, the predicted vacancy formation energyfrom PBE and 3 � 3 � 3 supercell is 2.17 eV, which is lower than theexperimental 1.38–1.84 eV [69,70], but agrees with other first-prin-ciples predictions (>2.0 eV) [18,69].

By including one vacancy, e.g., site A in Fig. 1, the migrations ofO from its 1NN position (site B) to its 2NN position (site C) andfrom its 2NN position (site C) to its 3NN position (site D) are calcu-lated. Fig. 2 shows that the CINEB predicted migration energy in-creases dramatically due to the existence of vacancy, e.g., from

-11.0

-10.5

-10.0

-9.5

-9.0

-8.5

Log 1

0[D (m

2 /s)]

10.09.59.08.58.0

1/T (104/K)

Expt. Without Va (Phonon): Vineyard Without Va (Phonon): Eyring Without Va (Debye): Eyring With Va (Debye): Eyring

1200 1100 1050 10001150Temperature (K)

1250

Fig. 5. Arrhenius plots of O diffusion in bcc Fe for the present predictions (lines) andexperimental data from internal oxidation measurements by Takada et al. (sym-bols) [8–10]. The jump rates are calculated using both the methods by Vineyard (Eq.(3)) and by Eyring (Eq. (4)), and the vibrational contributions to Helmholtz energy(Eq. (6)) are predicted by both the phonon method and the Debye model. Detaileddiffusion properties of O in bcc Fe are reported in Table 2.

S.L. Shang et al. / Corrosion Science 83 (2014) 94–102 99

0.526 eV without vacancy to 0.897 eV (migration from 1NN to 2NNpositions) and 0.772 eV (migration from 2NN to 3NN positions),see also Table 2. The O migration energy in bcc Fe due to the exis-tence of vacancy agrees roughly the experimental activation en-ergy of 1.0 ± 0.1 eV (see Table 2), implying the vacancy-limited Omigration in bcc Fe.

Concerning the concentrations of vacancy and interstitial O inbcc Fe, the vacancy concentration (i.e., mole fraction) can be esti-mated using a partition function,

CVa ¼ exp �1:61� 0:23kBT

� �ð9Þ

Here the value of 1.61 ± 0.23 eV is estimated according to the exper-imental vacancy formation energy of bcc Fe (1.38–1.84 eV) [69,70].Note that the energy unit of eV is adopted in the present work un-less noted otherwise. The O concentration in bcc Fe in thermody-namic equilibrium with solid FeO can be estimated by [10],

CO ¼ 0:381 exp �1:08kBT

� �ð10Þ

Or [6],

CO ¼ 1:5 exp �1:33� 0:15kBT

� �ð11Þ

For example of 1100 K and based on the above equations,CVa = (0.04–4.8) � 10�7 is roughly 10 times smaller than CO =(0.2–5.9) � 10�6. However, each Va may attract six O in its 1NNoctahedral interstitial sites and twelve O in its 2NN octahedralinterstitial sites, which could make the concentrations of vacancyand O in bcc Fe comparable. In addition, Fu et al. [16] also claimedthat the form of O-enriched nanoclusters in Fe-based alloys iscaused by the enriched vacancies processed by mechanical alloying.

Regarding other equilibrium properties from EOS fittings, Ta-ble 1 shows that the predicted equilibrium volume V0 at 0 K forbcc Fe is slightly lower than the experimental V0 at room temper-ature (11.36 vs. 11.78 Å/atom) [71], while the predicted bulk mod-ulus B0 is larger than the measured B0 (186 vs. 173 GPa) [72]. Withintroducing one vacancy, both the predicted V0 and especially B0

are closer to experiments, indicating again the vacancy mechanismin bcc Fe. With O in bcc Fe, Table 1 shows that the predicted V0 of Oin octahedral site is larger than that in tetrahedral site, and corre-spondingly the predicted B0 shows an inverse trend for all the cal-culations using different X-C functionals. With increasing Oconcentration in bcc Fe (from Fe128O, Fe54O to Fe16O), the predictedV0 increases, while the predicted B0 decreases roughly. With intro-ducing Va, it is found that the farther the distance between O andVa, the larger the predicted V0 and the smaller the predicted B0.Regarding the predicted pressure derivative of bulk modulus B00,they are all around the experimental value of 5 [73], except forthe structures with O–Va inside. The predicted magnetic momentsare all around the measured 2.2 lB/atom for bcc Fe [74], except forthe results from LDA (see details in Table 1).

3.2. Diffusion coefficients of O in bcc Fe

Using Eq. (1) and the methodologies described in Section 2, dif-fusion coefficients of O in bcc Fe with and without the effect of va-cancy can be predicted and then fitted using Arrhenius equation ofEq. (5). Fig. 5 illustrates the Arrhenius plots for the present predic-tions compared with experimental data from the internal oxida-tion measurements by Takada et al. [8–10]. For the case ofdefect-free bcc Fe, three methods are adopted to predict diffusioncoefficients: (i) jump rate from the Vineyard method of Eq. (3) withthe volume-temperature relation from first-principles phonon cal-culations (Vineyard + phonon); (ii) jump rate from the Eyring

method of Eq. (4) with Gibbs energy (Helmholtz energy) at zeroexternal pressure from first-principles phonon calculations(Eyring + phonon); and (iii) jump rate from the Eyring method ofEq. (4) with Gibbs energy (Helmholtz energy) at zero external pres-sure from first-principles Debye model (Eyring + Debye). In addi-tion, the accurate migration barrier of 0.503 eV (i.e., the energydifference DET–O as shown in Tables 1 and 2) is used. The predicteddiffusion coefficients of O in defect-free bcc Fe from these threemethods are roughly comparable with each other, and all these dif-fusion coefficients are much higher than experimental results (seeFig. 5). By considering the nearest Va of O, the predicted diffusioncoefficients of O in bcc Fe using the (Eyring + Debye) method agreereasonably well with experiments as shown in Fig. 5, indicatingagain the vacancy-limited O diffusion.

The predicted pre-exponential factor D0 and activation energy Qaccording to the Arrhenius equation of Eq. (5) are listed in Table 2.Calculated D0 is about (4.3 ± 1.4) � 10�7 m2/s for O diffusion in de-fect-free bcc Fe, and (0.6 ± 1.4) � 10�7 m2/s for O diffusion in Va-included bcc Fe (here the same error bar is adopted based on thedefect-free case, also for Q), agreeing reasonably well with the scat-tered D0 from experiments (see Table 2). Note that the error bar‘‘±1.4’’ is estimated based on the predicted results from differentdiffusion models as shown in Eqs. (3) and (4) (see results in Ta-ble 2), and this large error bar indicates also the sensitivity offirst-principles diffusion properties from different models. Calcu-lated Q is about 0.53 ± 0.04 eV for O diffusion in defect-free bccFe and 0.80 ± 0.04 eV for O diffusion in Va-included bcc Fe, wherethe latter case agrees reasonably well with experimental Q of1.0 ± 0.1 eV (from 0.89 ± 0.06 to 1.01, and up to the unreasonable1.73 ± 0.01 eV, see Table 2). Although it is not always correct, theagreement with experiment is an extremely important criterionto verify a model. In this regard, the present vacancy mechanismis a correct model to study the oxygen diffusivity in bcc Fe, andthe mono-vacancy mechanism is predominated. It is worth men-tioning that the presence of multiple vacancies may improve fur-ther the predicted diffusion properties but it also makes theprediction more complex and these predictions are beyond thescope of the present work.

In an effort to probe the effect of Va, Fig. 6 plots the differentialcharge densities of the (001) slices of Fe54O and Fe53OVa with O inthe octahedral interstitial site. Here the differential charge density

14

13

12

11

10

9

Stre

tchi

ng F

C (e

V/Å2 )

Fe54O (without Va) Fe53O (with Va)

-0.4

0.0

0.4

endi

ng F

C (e

V/Å2 )

V0 of Fe54O V0 of Fe53O

100 S.L. Shang et al. / Corrosion Science 83 (2014) 94–102

is defined as the difference between the charge density of theinterstitial-containing Fe system and the superposition densitiesof the corresponding free atoms. An asymmetrical feature of thedifferential charge density is shown for O in Fe54O along e.g. theFe–O–Fe direction, while a symmetrical feature is shown for O inFe54OVa along e.g. the Fe–O–Va direction. By examining the majorstretching and bending force constants of Fe–O pairs in Fe54O andFe53OVa obtained from phonon calculations (see Fig. 7), it is foundthat all the stretching force constants are positive for both struc-tures, while the major bending force constants are negative forFe54O but positive for Fe53OVa caused by the asymmetrical andsymmetrical differential charge densities, respectively. It shouldbe remarked that force constants quantify the extent of interactionor bonding between the atoms. A positive force constant suggestsbonding, while a negative force constant suggests that the twoatoms in question would prefer to move apart [61,62,75]. The

Fig. 6. Differential charge densities of the (001) slices of Fe54O (a) and Fe53OVa (b)with O in the octahedral interstitial site and the reference charge densities being thesuperposition densities of the corresponding free atoms. Here, the inversed greyscale is used with the dark region indicating the gain of electrons (of O) and thewhite region indicating the loss of electrons (of Fe). An asymmetrical feature aroundO is shown for Fe54O along the horizontal line of Fe–O–Fe, while a symmetrical oneis shown for O in Fe54OVa due to the existence of Va along the horizontal line of Fe–O–Va.

Phon

on D

OS

2520151050

Frequency (THz)

Fe54O (without Va) Fe53O (with Va)

-0.8B

1.861.841.821.80

Bond length between Fe-O (Å)

Fig. 7. Stretching and bending force constants (FCs) of the nearest Fe–O pairs inFe54O and Fe53OVa together with their phonon densities of state (DOS) at V0. Here,V0 is the equilibrium volume (see Table 1).

major stretching and bending force constants of Fe–O pairs areall positive in Fe53OVa, while the negative bending force constantsexists in Fe54O, indicating again the more stability of O in Va-con-taining bcc Fe with an increased resistance of O against bendingdeformation. Since the largest stretching force constants of Fe–Opairs in Fe54O at its equilibrium volume are larger than those inFe53OVa, the highest phonon frequency in Fe54O is thus higher thanthat in Fe53OVa, see Fig. 7 the phonon densities of state for bothstructures. At the lower phonon frequency region (<10 THz), thephonon densities of state show similar behaviors for both Fe54Oand Fe53OVa.

4. Summary

Within the framework of first-principles transition state theory,diffusion coefficients and associated activation energy and pre-exponential factor in Arrhenius equation have been computed forinterstitial O in bcc Fe with and without lattice vacancy. It is foundthat O in octahedral interstitial site is always energetically favor-able (energy minimum) with and without the effect of Va in bccFe according to the migration pathway predicted by the climbingimage nudged elastic band method. The present work shows thatvacancy possesses a high affinity for O atoms in bcc Fe, e.g., thebinding energies of �1.53 and �0.60 eV for the first and the secondnearest O–Va pairs. As a consequence, vacancy increases dramati-cally the migration energy barrier (from �0.5 eV up to 0.9 eV basedon PBE) for O diffusion in bcc Fe, making the predicted diffusioncoefficient of O in bcc Fe in favorable accord with experiments.

S.L. Shang et al. / Corrosion Science 83 (2014) 94–102 101

The stability of O in bcc Fe with Va is traceable from the increasedresistance of O against the bending deformation according to thedifferential charge density as well as the phonon force constants.The present work suggests that the stabilities and diffusion coeffi-cients of interstitial atoms in metals can be elucidated satisfacto-rily by the vacancy-limited mechanism.

Acknowledgements

This work was funded by the Cross-Cutting Technologies Pro-gram at the National Energy Technology Laboratory (NETL), man-aged by Susan Maley (Technology Manager) and Charles Miller(Technical Monitor). The Research was executed through NETL Of-fice of Research and Development’s Innovative Process Technolo-gies (IPT) Field Work Proposal. This work was financiallysupported at The Pennsylvania State University by NETL throughthe RES Contract No. DE-FE00400, and also by the U.S. Natural Sci-ence Foundation (NSF) through Grant Nos. CHE-1230924 andDMR-1310289. First-principles calculations were carried out par-tially on the LION clusters supported by the Materials SimulationCenter and the Research Computing and Cyber infrastructure unitat the Pennsylvania State University, and partially on the resourcesof NERSC supported by the Office of Science of the US DOE undercontract No. DE-AC02-05CH11231. ZKL and YD would like to thankthe support from National Natural Science Foundation of China(NSFC) with Grant No. 51028101. This report was prepared as anaccount of work sponsored by an agency of the United States Gov-ernment. Neither the United States Government nor any agencythereof, nor any of their employees, makes any warranty, expressor implied, or assumes any legal liability or responsibility for theaccuracy, completeness, or usefulness of any information, appara-tus, product, or process disclosed, or represents that its use wouldnot infringe privately owned rights. Reference herein to any spe-cific commercial product, process, or service by trade name, trade-mark, manufacturer, or otherwise does not necessarily constituteor imply its endorsement, recommendation, or favoring by the Uni-ted States Government or any agency thereof. The views and opin-ions of authors expressed herein do not necessarily state or reflectthose of the United States Government or any agency thereof.

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