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Scripta Materialia 67 (2012) 732–735

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A computational study of impurity diffusivities for 5d transitionmetal solutes in a-Fe

Hong Ding,a,⇑ Shenyan Huang,b Gautam Ghosh,c Peter K. Liawb and Mark Astaa

aDepartment of Materials Science and Engineering, University of California, Berkeley, CA 94720, USAbDepartment of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA

cDepartment of Materials Science and Engineering, The University of Tennesee, Knoxville, TN 37996, USA

Received 13 April 2012; revised 6 June 2012; accepted 7 June 2012Available online 17 June 2012

The impurity diffusivities (D) of the 5d transition metal solutes, Ta–Au, in a-Fe have been computed within a framework com-bining density-functional-theory calculations, harmonic transition-state theory, the Le Claire nine-frequency model and semi-empir-ical corrections for magnetic disorder. The calculated diffusion constants show a trend vs. atomic number featuring minimum valuescorresponding to Re and Os, at the center of the transition metal series. The results for D are correlated with minima and maxima inthe magnitudes of the solute–vacancy binding energy and migration energy, respectively.� 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Impurity diffusion; Ferritic alloys; Density functional theory; Harmonic transition state theory

Due to their relevance for a wide variety of phe-nomena arising in materials processing, extensive effortshave been devoted to the development of databases fordiffusion constants in multicomponent alloys [1–3]. Inthis context, the impurity diffusion constants (D) playan important role, as they set the limiting values of sol-ute mobilities as a function of composition. In caseswhere experimental data is lacking, computationalmethods for estimating D are thus of significant interest,and first-principles calculations based on electronicdensity-functional theory (DFT) have been appliedextensively. For the case of substitutional impuritiesconsidered here, calculations of D have been performedfor Fe [4–7], Ni [8,9], Al [10–14], Mg [6,13,15,16], Mo[13] and Ti [17] alloys.

In addition to providing values that are useful in gen-erating kinetic databases, computational studies alsoprovide insights into the mechanisms underlying varia-tions in D with chemistry. For example, it has long beenassumed that the lowest values of D correspond to thesolutes that are most oversized, as these species are ex-pected to have the largest activation energies; first-prin-ciples calculations (e.g., [8,11,14]) have shown that this“rule-of-thumb” is not generally reliable, and havepointed to the importance of electronic-structure effects,

1359-6462/$ - see front matter � 2012 Acta Materialia Inc. Published by Elhttp://dx.doi.org/10.1016/j.scriptamat.2012.06.010

⇑Corresponding author. Tel.: +1 510 643 9631; e-mail:hongding@berkeley.edu

which underscore the need for DFT-based calculations.In this work we show that both the activation energiesand diffusion prefactors for transition metal solutes ina-Fe show systematic trends vs. bandfilling that cannotbe easily understood based on size effects alone.

We employ the approach described in Ref. [4] for com-puting impurity diffusion coefficients in a-Fe within theframework of Le Claire’s nine-frequency model [18],employing DFT calculations of the relevant hopping fre-quencies for the zero-temperature ferromagnetic state,and a semi-empirical treatment of the effects of magneticdisorder [19,20]. We employ this methodology in the studyof 5d transition metal solutes, Ta–Au, in body-centered cu-bic (bcc) a-Fe. These elements are chosen in part due to thelack of experimental data available for most of these spe-cies (we are aware of experimental data only for W andAu), such that the current results are expected to be usefulin the context of developing kinetic databases. Addition-ally, the work is motivated by the question of whetherany of the 5d solutes are slow diffusers in a-Fe, such thatthey may be useful for developing precipitation-strength-ened alloys [21,22] with reduced coarsening rates. In theremainder of this article we briefly review the computa-tional formalism, and present results along with compari-sons with previous calculations and measurements.

The temperature dependence of D is expressed in thenormal Arrhenius form:

D ¼ D0 expð�Q=kBT Þ ð1Þ

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Figure 1. The diffusion coefficients of W and Au calculated in thiswork in comparison with available experimental data.

H. Ding et al. / Scripta Materialia 67 (2012) 732–735 733

where D0 is the pre-exponential factor and the Q is acti-vation energy. Assuming a monovacancy mechanism,the pre-exponential factor can be expressed in terms ofthe lattice constant (a), the correlation factor (f2), the at-tempt frequency for the hop of a solute atom to a near-est-neighbor vacancy (m*), the entropy of vacancyformation in bcc Fe DSf

v

� �and the entropy of vacancy

binding to a nearest-neighbor solute (DSb) as follows:

D0 ¼ a2f2m� exp DSf

v þ DSb

� �=kB

� �ð2Þ

In Eq. (2), m* can be described within harmonic transi-tion-state theory [23] as the quotient of product of thevibrational frequencies corresponding to a configurationwith a nearest-neighbor solute–vacancy pair mvac;sol

i

� �

and the product of frequencies for a saddle-point config-uration for the exchange of a solute and vacancymsad

i

� �: m� ¼

Qmvac;sol

i =Q

msadi , where the product in the

denominator excludes the imaginary frequency corre-sponding to motion along the diffusion path. For impu-rity diffusion, the activation energy is the sum of thevacancy formation energy in pure a-Fe DH f

v

� �, the near-

est-neighbor binding energy between a solute andvacancy (DHb), and the migration energy for solute–va-cancy exchange (DHm):

Q ¼ DH fv þ DH b þ DH m: ð3Þ

In Eq. (2) the correlation factor (f2) is expressed in themodel of Le Claire in terms of nine distinct jump fre-quencies, involving hops of the vacancy between siteswithin the second neighbor of the solute, as well as thatin the bcc lattice far from the solute. Formulas for f2 canbe found in Ref. [18]. In general, f2 can display an appre-ciable temperature dependence such that D0 as definedabove cannot be taken as independent of T.

As discussed previously [24–26], for a-Fe the slope ofthe self-diffusion and impurity-diffusion coefficients onan Arrhenius plot typically show a pronounced curva-ture below the Curie temperature. This effect is generallyattributed to the dependence of Q on the state of mag-netic order in the matrix. Following Girifalco [19], weexpress the dependence of Q on magnetic disorderthrough the formula

QF ðT Þ ¼ QP ½1þ aSðT Þ2� ð4Þ

where a depends on the type of solute atom,S(T) = M(T)/M(T = 0 K) is the ratio of the magnetiza-tion at temperature T to that at zero temperature, andQF(T) and QP denote values of the activation energyin the ferromagnetic and paramagnetic states, respec-tively. The DFT calculations give direct access toQF(T = 0 K), the magnitude of the activation energy inthe fully magnetically ordered ferromagnetic state atzero temperature. Given the value of a, derived throughthe procedure described below, we can extract a valuefor QP through the relation QP = QF(T = 0 K)/(1 + a).We can then compute QF(T) from Eq. (4) using experi-mentally measured values of S(T) [27,28].

For several solutes, the value of a has been reportedfrom fits of experimental data over temperature rangesspanning the paramagnetic and ferromagnetic states[20]. Based on these values, it has been suggested [20]that there is a linear correlation between the magnitudes

of a and DM12, the sum of the change in the local mag-netic moments induced on the Fe atoms in the first- andsecond-neighbor shells of an impurity atom:

DM12 ¼X8

i¼1

DM1sti þ

X6

i¼1

DM2ndi ð5Þ

In the current work we make use of this empirical rela-tion employing computed values of DM12 (see Supple-mentary materials for further detail).

To assess the accuracy of the computational proce-dure, we compare our results with experimental datafor the only two 5d solute species in a-Fe for which mea-sured data is available in the literature (to the best of ourknowledge). The comparison shown in Figure 1 is forW, which we compare to the two most recently pub-lished measurements [29,30], and for Au, which we com-pare to the relatively older results of Borg et al. [31]. ForW, our calculated results for temperatures near 1000 Kshow good agreement with the measurements of Takem-oto et al. [29], while they are larger than the more re-cently published measured values from Perez et al. [30](by a factor of 3–4). A comparison of our calculationsand measurements for Au shows that the calculated val-ues are larger by a factor of up to 6. We note that thislevel of discrepancy is greater than that found hereand in Ref. [4] for W and Mo impurities, and for self-dif-fusion in Fe, when the comparisons are based on themost recently published measured values. Whether thelarger discrepancies found for Au are due to limitationsinherent in the older measurements or due to larger er-rors inherent in the approximations underlying the cal-culations for Au warrants further investigation.Specifically, measurements that make use of the mostmodern techniques would be of interest, as would moredetailed calculations employing alternate exchange-cor-relation potentials. In what follows, we will focus mainlyon the trends in the calculated results, as they are ex-pected to be more accurate than the absolute valuesfor a given solute species.

The calculated results are summarized in Table 1 andFigures 2 and 3. We consider first the species-dependentcontributions to Q in Eq. (1): DHb and DHm. Both quan-

Figure 2. Dependence of the solute–vacancy binding and migrationenergies on atomic radius (i.e. Goldschmidt radii) for 5d transitionelement solutes.

Table 1. Solute–vacancy binding enthalpy DHb, solute migration enthalpy DHm and the ferromagnetic activation energy QF at 0 K in a-Fe (in unitsof eV), are in the first three rows, respectively (QP is derived as QP = QF/(1 + a)); the correlation factors, f2, and pre-exponential factors, D0

(mm2 s�1), obtained by Le Claire’s model at 1000 K, are listed in the last two rows.

Ta W Re Os Ir Pt Au

DHb �0.33 �0.17 �0.13 �0.13 �0.16 �0.28 �0.32DHm 0.43 0.71 0.86 0.90 0.88 0.85 0.76QF 2.30 2.74 2.93 2.97 2.92 2.77 2.64a 0.06 0.09 0.08 0.12 0.18 0.24 0.25Qp 2.17 2.52 2.71 2.65 2.48 2.23 2.11f2 0.03 0.73 0.93 0.97 0.96 0.90 0.57D0 4.04 138 298 310 118 67 21

Figure 3. Calculated temperature dependence of diffusion coefficientsfor 5d transition metal solutes in a-Fe. The dashed line corresponds toour previous calculation of the self-diffusion coefficient in a-Fe.

734 H. Ding et al. / Scripta Materialia 67 (2012) 732–735

tities are plotted in Figure 2 as a function of the atomicnumber of the 5d solute species. In the conventionadopted here, negative values of DHb correspond to anattractive interaction between vacancy and solute, andthe more negative the value the larger the effect onreducing the magnitude of Q. It can be seen in Fig-ure 2(a) that the strongest binding is obtained for solutesat the beginning and end of the transition metal series,with Re and Os, which are in the middle of the series,showing the weakest binding. This trend is consistentwith recent computational results obtained for DHb vs.atomic number for 4d solute elements in a-Fe [32]. Alsoshown in Figure 2(a) is the atomic radius across the ser-ies [33]. A comparison of these values with DHb shows ageneral trend that the largest magnitudes for the bindingenergies are obtained for the largest solute atoms. Thissame trend was obtained for transition metal solute ele-ments in Ni [8], and collectively the results support asimple picture that large solute atoms prefer to be nearvacancies to minimize elastic energy. However, the cor-relation between size and DHb in Figure 2(a) is not per-fect, as Ta is predicted to have the largest binding energydespite the fact that its atomic radius is slightly smallerthan that of Au, and both Os and Re have very similarvalues of DHb despite the smaller size of the former.

Similar to the results for DHb, the migration energyshows a clear trend vs. the atomic number of the solutespecies. Specifically, DHm shows a maximum for Os, inthe middle of the transition metal series, with decreasingvalues obtained with increasing or decreasing atomicnumber. This trend is somewhat counter-intuitive, be-cause the largest values of DHm are obtained for the sol-ute species with the smallest atomic radius. However, asimilar trend has been obtained in previous calculations,e.g. for Ni-based alloys [8].

In Table 1, the calculated results for D0 at the repre-sentative temperature of 1000 K are seen to displaystrong variations between the solute species. Specifically,the values are computed to vary by approximately twoorders of magnitude. While the values for W, Re, Os,Ir and Pt are in the range of values reported for self-dif-fusion in a-Fe and for impurity-diffusion coefficientsavailable for 4d and 5d impurities (see Ref. [4] and refer-ences therein), the values for Au and especially Ta areanomalously low. For Ta, the low value of D0 originatesfrom a small value for the correlation factor, f2 = 0.03,reflecting the strong solute–vacancy binding for this ele-ment, which gives rise to a large fraction of solute–va-cancy hops which are followed by hops back to the

H. Ding et al. / Scripta Materialia 67 (2012) 732–735 735

original solute position (see Supplementary Materialsfor further details). Similarly low values of f2 have beenreported for Y in Fe [6] and Mg in Al [14]. For Au, thelow value of D0 is due in part to a large negative valuefor the vacancy solute binding entropy.

The temperature dependencies of the impurity diffu-sion coefficients in a-Fe calculated in the present workare plotted in Figure 3, along with the calculated resultsfor the self-diffusion in a-Fe obtained in Ref. [4] usingthe same computational approach. A comparison ofthe results in the paramagnetic state (from 1043 to1183 K) shows that the values of D can be divided intothree groups. The first are the solutes which display Dvalues higher than that for self-diffusion in Fe: Pt andAu. The second group is composed of solute atoms withD close to that for self-diffusion: W, Ir and Ta (the com-parable values of D for Ta and W obtained here is incontrast to the values for interdiffusion measurementsreported in Ref. [34] for more concentrated alloys, whereTa shows much larger values). The third group, contain-ing Re and Os, are the “slow diffusers”, which show val-ues of D lower than that for self-diffusion. From theanalysis of the trends discussed above, the relativelylow values of D for these solute species originates fromthe larger values of DHm and the smaller magnitudes forDHb. Of all the solute species considered in this work,Re has the lowest calculated impurity diffusion coeffi-cient, with a value of D that is approximately an orderof magnitude lower than that for self-dffusion at temper-atures near 1000 K. This solute species may thus be ofinterest as potential additions to precipitation-strength-ened ferritic alloys for slowing high-temperature coars-ening rates.

In summary, we have presented first-principles calcu-lations of impurity diffusion coefficients, and associatedsolute–vacancy binding energies, migration energies andcorrelation factors for 5d transition metal solutes in a-Fe. The solute–vacancy binding and migration energiesshow clear trends vs. atomic number, with maximumvalues for the elements Re and Os, in the centre of thetransition metal series. This trend is reflected in the pre-dicted result that Re and Os feature the slowest impuritydiffusion coefficients amongst the elements consid-ered.The slower diffusion rates of the solute species orig-inate from the significantly larger values for both Q andD0 relative to other solutes in the 5d series.

This work was supported by the US Departmentof Energy (DOE), Office of Fossil Energy, under GrantNo. DE-FG000568. The work made use of computa-tional resources provided by (i) the National Energy Re-search Scientific Computing Center, supported by theOffice of Science of the US DOE under Contract No.DE-AC02-05CH11231 and (ii) an allocation providedby the National Science Foundation, on Kraken at theNational Institute for Computational Sciences.

Supplementary data associated with this article canbe found, in the online version, at http://dx.doi.org/10.1016/j.scriptamat.2012.06.010.

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