Diffusion rates of 3d transition metal solutes in nickel by first-principles calculations

Acta Materialia 53 (2005) 2369–2376 www.actamat-journals.com

Diffusion rates of 3d transition metal solutes in nickel by first-principles calculations M. Krcˇmar a, C.L. Fu

a,*

, A. Janotti a, R.C. Reed

b

a

b

Metals and Ceramics Division, Oak Ridge National laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6114, USA Department of Metals and Materials Engineering, The University of British Columbia, 309-6350 Stores Road, Vancouver, Canada V6T 1Z4 Received 5 November 2004; received in revised form 27 January 2005; accepted 30 January 2005 Available online 25 February 2005

Abstract First-principle calculations for the diffusion of 3d transition metal (TM) solutes in nickel demonstrate the existence of a higher diffusion energy barrier for solutes with smaller atomic sizes. The calculations reveal that smaller TM atoms are, actually, among the least compressible due to the formation of incompressible solute-host directional bonds. Magnetism is shown to have a profound effect on the solute diffusion trends across the 3d TM series: the existence of a local minimum in the diffusion energy barrier is accompanied by the occurrence of a maximum in the magnetic moment. The calculated diffusion rates disprove the traditional view that the diffusion of solutes is least rapid when the size misfit with the host is the greatest.
2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Substitutional diffusion; Ab initio calculation; Metal and alloys (nickel)

1. Introduction The behavior of metals and alloys often depends critically upon diffusional rearrangements of atoms on the crystal lattice. This is particularly true for materials like the nickel-base superalloys [1], which are used widely for ultra-high temperature structural applications. At elevated temperatures, a number of diffusional processes can arise in these alloys and thus influence their mechanical performance. For example, creep deformation occurs at a rate dependent upon diffusional rearrangement of atoms at dislocation cores [2]. Oxidation and intermixing with protective coatings are other examples of diffusional phenomena relevant for the operation of these superalloys, since they commonly lead to mechanical degradation. Of course, diffusional phenomena are of the most widespread importance in a range of metallic and ceramic systems [3,4]. *

Corresponding author. Tel.: +1 865 574 5161; fax: +1 865 574 7659. E-mail address: [email protected] (C.L. Fu).

In many systems, the addition of alloying elements has a profound effect on the rate of the diffusion processes. To promote high temperature properties in the nickel-base superalloys, up to 10 or more transition metal elements are typically added as solutes. Generally, the retardation of diffusional processes is important if the very best properties are to be attained. Unfortunately, at the moment, our understanding of the roles played by these alloying elements is far from complete. In particular, the rates of diffusion of transition metals in nickel are not at all well known [5], with no attempts being made so far to correlate the rate of diffusion with the position of the solute within the periodic table. In fact, this appears to be true of many other alloy systems, for example those based upon aluminum, titanium and iron. In the present paper, a first attempt is made to identify systematic trends in the rates of diffusion of the 3d transition metal (TM) solutes in nickel, which is chosen as the solvent in view of its technological importance in the nickel-base superalloys. This work is built upon our recent success in identifying the atomic-scale mechanism

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2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2005.01.044

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governing the diffusion of TM solutes in nickel [6]. Since the effects introduced by each alloying element depend upon the details of their bonding with the host atoms, ab initio methods are used to quantify the effect of solute-vacancy, solvent-vacancy and solute–solvent interactions on the diffusion rates in these systems. Here, we focus our attention on the vacancy-mediated diffusion of 3d solutes with the inclusion of the correlation effect. The correlation effect, which takes into account various site-exchange processes between host atom and vacancy, can have a profound influence on the diffusion rate of solute atoms. We should also point out that the systematic trends for the diffusion of the 3d transition elements are expected to be more complex than those of the non-magnetic 4d and 5d solutes, due to the complexity of magnetic exchange interactions between 3d solutes with the host Ni atoms. The paper is organized as follows. Following the Introduction, Section 2 describes the theoretical model for vacancy-mediated solute diffusion, Section 3 explains our computational approach, and Section 4 presents our calculated results. In Section 5, our predictions on the diffusion rates are compared with the available experimental data.

2. Theoretical model Vacancy-mediated solute diffusion in a close-packed face-centered-cubic (fcc) lattice can be analyzed by employing the transition state theory within the LidiardÕs five-frequency exchange model [3,4]. Diffusion coefficient DS of a solute in the vacancy-mediated diffusion process, according to the five-frequency exchange model, can be expressed as DS ¼ fpv a2 CS ;

ð1Þ

where f is the correlation factor, pv is the probability of finding a vacancy adjacent to a solute atom, a is the host lattice parameter and CS is the rate of solute-vacancy exchange. The correlation factor generally depends on various transition rates describing the exchange processes among solute, vacancy and host atoms that result in different host–vacancy–solute configurations, which can enable/disable solute diffusion processes (Fig. 1). Following the LidiardÕs approximation of the five-frequency model, the correlation factor for the fcc lattice can be expressed as [3,4] f ¼

2C1 þ 7C2 ; 2CS þ 2C1 þ 7C2

ð2Þ

where C1 and C2 are the rates of site exchange between vacancy and host atom that result in the rotation and dissociation of a solute–vacancy pair, respectively (cf. Fig. 1). Obviously, if C1
CS or C2
CS, f @ 1 (i.e.,

Ni

Ni

Ni

Ni

Γ1 Ni

Ni

Γ2 Ni

Ni

Γs Ni

S

Ni

Ni

Fig. 1. Schematic illustration of the meanings of the exchange rates CS, C1 and C2, which correspond to the rate of solute–vacancy exchange, and the rates of site exchange between vacancy and host atom that result in the rotation and dissociation of a solute–vacancy pair, respectively. Here the solute atom is denoted by S and the vacancy is represented by circle with dashed line.

the probability of a diffusing atom jumping back to its previous lattice site is negligible small, thus the correlation effect becomes unimportant). On the other hand, if the diffusing atom has very low diffusion activation energy (which satisfies CS
C1 and CS
C2), the correlation factor is quite marked. The probability pv in Eq. (1) at temperature T is given by   EVf pv ¼ C 0 exp  ; ð3Þ kBT where C0 is the lattice coordination number, kB is the VðNiÞ Boltzmann constant, and EVf ¼ Ef þ DEV is the formation energy of vacancy adjacent to the solute atom, i.e., a sum of vacancy formation energy in elemental VðNiÞ nickel Ef and solute–vacancy binding energy DEV. From the transition state theory, CS is given by   Eb CS ¼ m exp  ; ð4Þ kB T where v is the characteristic vibrational frequency for solute migration and Eb is the solute migration barrier. Combining the above results, the diffusion coefficient DS can be written as   Q DS ¼ fD0 exp  ; ð5Þ kBT where D0 is the pre-exponential constant and Q ¼ EVf þ Eb is the diffusion activation energy. In essence, transition state theory describes the diffusion process classically, requiring the solute to overcome the diffusion energy barrier once a vacancy adjacent to the solute lattice site becomes available.

3. Computational approach To calculate diffusion coefficients for the diffusion of 3d solutes in Ni, we carried out ab initio calculations based on the density-functional theory [7] within the local spin density approximation (LDA) [8]. We used the

M. Krcˇmar et al. / Acta Materialia 53 (2005) 2369–2376

ultra-soft pseudo-potential method [9] to solve the local density functional equations. The total energies were calculated with an accuracy of 104 eV. The atoms were considered as fully relaxed once the Hellmann–Feynman ˚ . To forces acting on atoms are smaller than 0.025 eV/A give a better description of the magnetism of the host Ni lattice in the presence of magnetic solutes, the host lattice was represented by a 64 atom fcc super-cell (in contrast to a 32 simple cubic super-cell used previously [6]). A 64 atom fcc super-cell also provides a more realistic description of the atomic relaxation surrounding a migrating solute or solvent atom, especially in the calculation of site-exchange barriers between vacancy and host atom in the presence of a nearby solute atom. ˚ ) was used in The LDA lattice constant of Ni (3.44 A the calculation, which is 2.4% smaller than the experimental value. For the solute diffusion in the fcc lattice, we consider the exchange of a solute atom with the adjacent vacancy along the Æ1 1 0æ direction. Diffusion energy barriers for solutes were calculated by displacing the solutes in the Æ1 1 0æ direction: for each fixed solute atom position, the host atoms positions were fully relaxed; the Eb values were obtained as the total energy differences between solute atom at the saddle point and at the initial lattice VðNiÞ site (Fig. 2). As a benchmark, we calculated Ef and Eb for Ni self-diffusion, obtaining values of 1.706 and 1.152 eV, respectively, which yields the Ni self-diffusion activation energy of 2.858 eV, in excellent agreement with the available experimental results [10]. The characteristic frequencies v were obtained from the calculated vibrational frequencies of solute and its nearest-neighbor host atoms for solute at the saddle point and at the initial lattice site, treating all vibrational degrees of freedom at high temperatures as due to the collection of independent classical oscillators. The characteristic frequencies are found to be similar in values (5–9 THz) for all 3d solutes, yielding the pre-exponential constant D0 of the order of 105 m2 s1 for all solutes.

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In the same manner as in the calculation of solute diffusion barrier, the site-exchange barriers between vacancy and host atom in the presence of a nearby solute atom were calculated by moving the host Ni atom along the Æ1 1 0æ direction. However, by symmetry, the diffusing Ni atoms in these exchange processes were allowed to relax in the directions perpendicular to Æ1 1 0æ for each fixed position along the Æ1 1 0æ path.

4. Results and discussion 4.1. Diffusion activation energy In Fig. 3, we present the calculated vacancy formation energies adjacent to the solutes ðEVf Þ, diffusion energy barriers (Eb) and activation energies (Q) for the diffusion of 3d TM solutes in nickel. The trends in the calculated diffusion activation energy using the 64-atom super-cell are very similar to those using the 32-atom super-cell [6]. In terms of the magnitudes of activation energies, these two super-cell approaches differ by about 0.1 eV, which amounts to 5% of the calculated activation energy values. We believe that the numerical results using a 64-atom super-cell are already well-converged with respect to unit cell size. Using the Goldschmidt radii (i.e., metallic radii) as the measure for the atomic size of solute atoms (Fig. 4), we find a seemingly strong correlation between the solute sizes and their diffusion activation energies. The results are, as found previously for the diffusion of 4d and 5d solutes in Ni [6], unexpected and counterintuitive: the larger the solute atom, the lower its diffusion barrier and diffusion activation energy. (Here, we should point out that the atomic radii shown in Fig. 4 are the radii for elemental metals; the magnitude of the atomic radii for 3d solutes in Ni can be somewhat

3.5

Ti

V Cr Mn Fe Co Ni Cu

Energy

Energy (eV)

3.0

Eb

Q

2.5 2.0

E

1.5 1.0

E

V f

b

0.5

Lattice site

Saddle point

Lattice site

<110> Fig. 2. Definition of the diffusion energy barrier, Eb, for solute migration along the Æ1 1 0æ direction in the fcc lattice. The filled circles, unfilled circles and squares are solute atoms, solvent atoms and vacancies, respectively.

0.0

22 23 24 25 26 27 28 29

Atomic number Fig. 3. Variation of the calculated activation energy (Q) with atomic number for the diffusion of the 3d solutes in Ni, and the contributions from the diffusion barrier energy (Eb) and the vacancy formation energy adjacent to a solute ðEVf Þ.

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Goldschmidt atomic radius (A)

1.50 Ti

V

Cr Mn Fe Co Ni Cu

1.45 1.40 1.35 1.30 1.25 1.20

22 23 24 25 26 27 28 29

Atomic number Fig. 4. Variation of the Goldschmidt atomic radius of the elemental 3d transition metals with atomic number.

varied from those shown in Fig. 4 due to the complexity of solute–host magnetic interaction. Nevertheless, the atomic radii shown in Fig. 4 are used as guidance for describing the overall trend in solute size.) We note that the major contribution to the variation of the diffusion activation energies (Q) across the 3d TM series comes from the variation of the diffusion energy barriers (Eb) with atomic number (cf., Fig. 3). By comparison, the vacancy formation energies adjacent to the solute atoms ðEVf Þ depend only weakly on the solute sizes. Nevertheless, the EVf term displays the expected trend: the larger the misfit between solute and host atoms, the larger (i.e., more negative) the solute–vacancy binding energy. But, opposite to the commonly accepted belief, lattice strain effects due to the size of solutes have only a minor effect on the variation of the solute diffusion activation energy across the 3d TM series. 4.2. Correlation factor In order to calculate the correlation factor, it is necessary to know the energy barriers for the site exchange between vacancy and host atoms in the presence of a nearby solute atom for the site-exchange processes C1 and C2 shown in Fig. 1 (the corresponding energy barriers are denoted as Eex(C1) and Eex(C2) hereafter). Since the diffusing Ni atom in the C2 exchange process does not have solute atoms as its nearest-neighbor at the saddle point, the calculated Eex(C2) values are found to be independent of solute atom size and are nearly identical to that of Ni self-diffusion (i.e., 1.15 eV). The site-exchange barrier in the C1 process, by comparison, shows larger variations across the 3d TM row, since the C1 exchange process brings the diffusing Ni atom into direct contact with the nearby solute atom at the saddle point. We find that, as the solute size increases, the Eex(C1) value also increases, since the diffusing Ni atom experiences higher compressive stress from larger solutes than from smaller solutes at the saddle point. For the

majority of 3d solutes (with small size misfit with Ni), Eex(C1) is within 0.1 eV of that of Ni self-diffusion. But, for solutes with the largest size misfit with Ni (i.e., Ti and Mn), Eex(C1) exceeds that of Ni self-diffusion by 0.25 and 0.35 eV for Ti and Mn, respectively. Taking the Ni self-diffusion barrier as the reference, we note that, as the size misfit between solute and Ni increases, the solute diffusion barrier decreases but the site-exchange barrier between Ni and vacancy (i.e., Eex(C1)) increases. For the cases of Ti and Mn solutes, the solute diffusion barrier (i.e., the CS process in Fig. 1) becomes considerably lower than the site-exchange barriers between Ni and vacancy in the presence of a solute atom (i.e., the C1 and C2 processes in Fig. 1). Under this circumstance, since the vacancy left behind by the diffusing solute atom is more likely to be re-occupied by the ‘‘jumping back’’ solute atom than by the nearby host Ni atoms, the diffusion process of solute atoms is considerably slowed down by the presence of un-dissociated solute–vacancy pair. Thus, for solutes with low diffusion barrier, the correlation factor is indispensable in the calculation of the diffusion rate. To evaluate the correlation factor, the characteristic frequency values for all exchange processes in Eq. (2) are assumed to be equal. We believe that this approximation is valid, since the characteristic frequencies for 3d solute diffusion are found to be similar in magnitude across the TM row. The calculated correlation factor values for 3d solutes range from 103 to 1 at 1300 C; this will be discussed further below. 4.3. Electronic and magnetic structures of 3d solutes in Ni To understand the origin of lower diffusion barriers for larger solutes, one has to examine the bonding characteristics of the solute atoms in the host lattice. In our previous study of the diffusion of 4d and 5d TM solutes in Ni [6], we have shown that the electronic structure of the solutes is a dominant factor in determining the variation of the diffusion barriers and diffusion activation energies. Since solute atoms are considerably compressed near the saddle point, the diffusion energy barrier should reflect the degree of bonding directionality of the solutes. Smaller TM atoms are among the least compressible of all the transition metals due to the formation of solute–host directional bonds, which do not favor solute–vacancy exchange. We find that the compressibility of elemental transition metals provides a comprehensive physical basis to describe the trend in the diffusion energy barrier of TM solutes. Since the magnitude of the compressibility manifests the size of atomic radius within a TM series, it is possible to establish a correlation between atomic radius and diffusion activation energy across a TM row. In the following, we will show that the same physical principle can also be applied to describe the diffusion of 3d solutes in Ni.

M. Krcˇmar et al. / Acta Materialia 53 (2005) 2369–2376

Majority-spin

3.0

Cr solute 2.5

Density-of-states

Density-of-states

EF

2.0 1.5 1.0

2.0

1.0 0.5

0.0 3.0

0.0 3.0

Mn solute

Mn solute 2.5

Density-of-states

Density-of-states

EF

1.5

0.5

2.5 2.0 1.5

EF

1.0

2.0

EF

1.5 1.0

0.5

0.5

0.0 3.0

0.0 3.0

Fe solute

Fe solute 2.5

Density-of-states

2.5

Density-of-states

Minority-spin

3.0

Cr solute 2.5

2.0 1.5

EF

1.0

2.0 1.5

E

F

1.0 0.5

0.5 0.0 -10

2373

-8

-6

-4

-2

0

2

Energy (eV)

4

0.0 -10

-8

-6

-4

-2

0

2

4

Energy (eV)

Fig. 5. The local density-of-states (in units of states/eV) of Cr, Mn and Fe solutes in Ni. Note that there is a change in the solute–host magnetic coupling from Cr to Mn. Left panels corresponds to the majority-spin channels (spin-up for Mn and Fe, and spin-down for Cr), whereas right panels corresponds to the minority-spin channels (spin-down for Mn and Fe, and spin-up for Cr).

There are two standout differences between 3d and 4d/5d solutes in Ni: (1) the size misfit with the host is less pronounced for the 3d solutes and (2) the effect of magnetism, which is intrinsic to the 3d TM elements, adds complexity to the electronic structure already discussed for the 4d and 5d solutes. Thus the system under study is no longer restricted to the diffusion of larger solute atoms in a smaller lattice. On the other hand, the effect of magnetism changes the characteristic of chemical bonds, which should also be reflected in the calculated barrier height of the 3d solutes. We find that in the presence of host–solute magnetic interactions, the diffusion activation energy across the 3d TM series exhibits a local minimum corresponding to the Mn solute. Such minimum is absent in the cases of non-magnetic 4d and 5d solutes. Most significantly, the activation energy minimum is accompanied by the

occurrence of a maximum in the magnetic moment among the 3d solutes. Calculated magnetic moments of 3d solutes in Ni, evaluated within their Goldschmidt atomic radii, are as follows: 0.2lB (Ti), 0.4lB (V), 1.2lB (Cr), 3.1lB (Mn), 2.7lB (Fe), 1.7lB (Co) and 0.0lB (Cu). Here, a positive (negative) sign indicates that the solute magnetic moment is parallel (anti-parallel) to the host atoms magnetic moments (0.6lB for bulk Ni). These results clearly confirm that magnetism has a profound effect on the magnitudes of the diffusion energy barrier of 3d solutes in Ni. To verify the accuracy of pseudopotential approach for these magnetic systems, we have carried out independent calculations employing the full-potential linearized augmented plane wave (FLAPW) method [11] to examine the magnetic moments of 3d solutes in Ni. The calculated magnetic moments of Mn and Fe solutes by

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the FLAPW method (3.2lB and 2.8lB, respectively) agree well with those obtained from the pseudopotential method. However, a larger discrepancy is found in the calculated magnetic moment of Cr (1.7lB by FLAPW and 1.2lB by pseudopotential). Nevertheless, the discrepancy in the calculated magnetic moments between different methods does not affect the overall trend in the diffusion activation energy, since it is the change in the solute–host magnetic coupling from Cr to Mn (which both methods agree) that is more important in causing a local minimum in the diffusion barrier across the 3d TM series. In the presence of magnetic exchange interaction, the d-bands of 3d solutes are split into majority-spin and minority-spin channels. A change in the solute–host magnetic coupling from Cr solute to Mn solute indicates an abrupt change in their d-band bonding characteristics. To understand the origin of low diffusion barrier for Mn solute, we present in Fig. 5 the local densityof-states (DOS) profiles of Cr, Mn and Fe solutes in Ni (with the majority-spin of Ni denoted as spin-up); the left panels correspond to the majority-spin channels (spin-up for Mn and Fe, and spin-down for Cr), whereas the right panels correspond to the minority-spin channels (spin-down for Mn and Fe, and spin-up for Cr). For Cr solute, the majority-spin states are more than half filled and the minority-spin states are mostly empty. Occupancy of the d-band in the majority-spin channel brings in the directional Æ1 1 0æ bonding of the Cr solute with its nearest neighbor host atoms, resulting in a significant diffusion barrier for this solute. For Mn solute, having the largest magnetic moment of 3.1lB, the majority-spin d-states are nearly fully occupied and do not participate actively in solute–host chemical bonding; on the other hand, the minority-spin d-states are less than half-filled and are responsible for the Mn–host chemical interactions. However, these occupied minority-spin states lack the appreciable Æ1 1 0æ directional bonding components, thus the Mn solutes are easy to compress and their diffusion energy barrier is low. For the Fe solute, a decrease in the magnetic moments from that of Mn (to 2.7lB) implies an increase in the occupation of the minority-spin d-states from that of Mn; as a result, the minority-spin d-states with the directional bonding components becomes progressively occupied from Mn to Fe. This can be seen in Fig. 6 where we plot charge density difference in the (0 0 1) plane of the fcc lattice between Fe and Mn solutes in Ni (where Ni atoms are in the un-relaxed positions in these plots), both for the majority-spin (Fig. 6(a)) and minority-spin (Fig. 6(b)) channels. In the majority-spin channel, we do not see any appreciable change in the charge polarization near the solute atom from Mn to Fe; by contrast, in the minority-spin channel, it is evident that the excess electron contributes to the charge polarization in the Æ1 1 0æ directions near the solute atom, indicating an in-

crease in the bonding directionality between solute and its nearest-neighbor Ni atoms from Mn to Fe. Due to the increase in Æ1 1 0æ bonding components, Fe solute becomes relatively harder to compress than Mn solute; thus the diffusion energy barrier increases from Mn to Fe. 4.4. Diffusion rates of 3d transition metal solutes in Ni In order to address which 3d solutes diffuse faster in the Ni–host lattice, solute diffusivities have to be calculated. If a solute has low diffusion barrier, the solute–vacancy exchange probability increases; however, whether this exchange results in the solute migration away from or back to its original lattice site depends on the correlation effects. We find that solutes with the largest size

(a)

Ni S

(b) 0.020 0.018 0.016 0.014 0.012

Ni

0.010 0.008 0.006 0.004

S

0.002 0 -0.002 -0.004 -0.006 -0.008 -0.010

Fig. 6. The electronic charge density difference (in units of e/(a.u.)3) in the (0 0 1) plane of the fcc lattice between Fe and Mn solutes in Ni (where Ni atoms are in the un-relaxed positions in these plots) for (a) majority-spin and (b) minority-spin channels. The solid (dashed) lines denote contours of increased (decreased) electronic density from Mn to Fe. The position of the solute is denoted by S.

Diffusion co-efficient (m2s-1)

Diffusion co-efficient (m2s-1)

M. Krcˇmar et al. / Acta Materialia 53 (2005) 2369–2376

10 -11

10

-13

10

-15

Ti

V Cr Mn Fe Co Ni Cu

2375

(a) o

1300 C o 1200 C o 1100 C o 1000 C o

900 C 10 -17

10

-19

10

-11

(b)

10 -13 1300oC o 1200 C 1100oC 1000oC

10 -15

Fig. 8. Variation of measured inter-diffusion coefficients with nickel for various substitutional 3d transition metal solutes. See text for details.

o

10

900 C

-17

10 -19

22 23 24 25 26 27 28 29

Atomic number Fig. 7. Predicted variation of the diffusion coefficient of the 3d elements in Ni: (a) without the correlation effect account for, and (b) with the correlation effect included. The prediction in (b) should be compared with the experimental data in Fig. 8.

misfit with the host atoms not only have the lowest activation energies but also have the most prominent correlation effects. The correlation effects effectively decrease the diffusivities of solutes with low diffusion activation energy. More specifically, for the 3d solutes in Ni, we find that f  103–102 for Ti and Mn, f  101 for V, Fe, and Cu, and f  1 for Cr and Co in the temperature range plotted in Fig. 7. The calculated diffusion coefficients of the 3d solutes and their variation with atomic number are given in Fig. 7(a) with the correlation factor ignored (i.e., by setting f = 1) and in Fig. 7(b) with the correlation factor included. We note that, the larger the solute is, the faster is its diffusion rate. For solute atoms with smaller atom size and higher diffusion activation energy, the correlation effect has little influence on the predicted values of the diffusion coefficients. But for solute atoms with larger atom size and lower diffusion activation energy, the correlation effect is quite marked. With the correlation factor included in the calculation, the predicted diffusion coefficients are in reasonable agreement with the experimental data given in Fig. 8. We emphasize that, as shown in Fig. 7, Ti and Mn have the fastest diffusion rate in Ni, despite the fact that they have the largest atomic radii among 3d solute elements (cf., Fig. 4). It is worthy to point out that, experimentally, the conventional approach to obtained the diffusion activa-

tion energy of a solute is through the measured diffusion coefficients by using the relationship given in Eq. (5), but neglecting the correlation factor and its temperature dependence (i.e., the correlation factor f is set to be unity). Here, we have demonstrated that the correlation factor should be included in the calculation of diffusion rate. Our results indicate that neglecting of correlation factor in experimental analyses can lead to an overestimate of the diffusion activation energy of solutes, particularly for solutes with diffusion energy barrier much lower than that of the self-diffusion of host atoms.

5. Comparison with experiments Here, we compare the theoretical predictions of the diffusion rates of 3d TM solute in Ni with available experimental data collated from the literature; this mostly pertains to the rates of inter-diffusion determined by the analysis of diffusion profiles set-up in diffusion couples. In most cases, the couples consisted of pieces of nickel diffusion-bonded to nickel alloyed with a small concentration of the 3d transition metal: the diffusion coefficients obtained by the Boltzmann–Matano analysis or similar approaches therefore correspond to inter-diffusion coefficients rather than true impurity coefficients in the limit of an infinitely dilute solution in nickel (which is more pertinent to the analysis carried out here). The data used for the comparison are from the papers by Komai et al. [12] for Ti, Davin et al. [13] for V, Jung et al. [14] for Cr, Swalin and Martin [15] for Mn, Ustad and Sorum [16] for Fe and Co, Jonsson [10] for the self-diffusion of Ni and Anand et al. [17] for Cu. In these papers, the inter-diffusion coefficients were not always measured or quoted specifically at the temperatures of 900, 1000, 1100, 1200 and 1300 C, which are of interest here; in these instances, we have estimated the values by extrapolation through the data given.

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The experimental data in Fig. 8 are consistent with many aspects of the predictions. In particular, the experimental data indicate that Ti, Mn and Cu are the elements which diffuse at the fastest rates, as was predicted by our analyses, with the diffusion rates for V and Cr (which lie between Ti and Mn) and Fe, Co and Ni (which lie between Mn and Cu) falling below these values for Ti, Mn and Cu. These broad trends are consistent with the calculated results given in Fig. 7(b). It should be pointed out that, however, the predicted variation across the 3d series in the values of the diffusion coefficient, at about two orders of magnitude for each of the temperatures quoted, is about one order of magnitude greater than as observed in the experiments. Nevertheless, one should realize too that much of the experimental data used here for comparison is not only rather old, but also collected from seven different sources. It therefore seems timely to repeat the measurements in the light of the predictions made here, in the manner reported for the 4d and 5d elements [5]. This work is ongoing.

ute–host misfit strain in determining the trend of the diffusion activation energy with atomic number. We find that it is important to include the correlation factor in the calculation of solute diffusion rate. For solute atoms with smaller atom size and higher diffusion activation energy, the correlation effect has little influence on the magnitude of the diffusion coefficients. On the other hand, for solute atoms with larger atom size and lower diffusion activation energy, the correlation effect is quite marked. We suggest that extra caution should be paid in interpreting the diffusion activation energy from the experimentally measured diffusion coefficients.

Acknowledgments This work was sponsored by the Division of Materials Sciences and Engineering (C.L.F., M.K., and A.J.), the US Department of Energy, under contract DEAC05-00OR-22725 with UT-Battelle, LLC. R.C.R. acknowledges funding from the Natural Sciences and Engineering Research Council of Canada (NSERC).

6. Conclusion References First-principles quantum-mechanical electronic structure methods have been used to study the diffusion rates of 3d TM solutes in nickel. Our calculations disprove the traditional view that the diffusion of solutes is least rapid when the size misfit with the host is the greatest [3]. The variation of the diffusion rate with atomic number is largely due to differences in the diffusion barrier energy of solutes. The size effect (i.e., the size misfit between solute and host atom) is not so important; instead it is the electronic bonding characteristics of solute atoms which causes these phenomena. The development of bonding directionality for solutes with smaller atomic radii leads to incompressible solute–host bonds (and thus higher diffusion energy barriers), which hinder solute diffusion. Furthermore, we find that magnetism leads to a profound effect on the solute diffusion trends across the 3d TM series: the existence of a local minimum in the diffusion energy barrier is accompanied by the occurrence of a maximum in the magnetic moment. Atom with the largest magnetic moment (i.e., Mn) also has the largest atomic size among magnetic solutes, and is softer and easier to compress. The compressibility of the solutes overwhelmingly dominates over any differences in the solute-induced vacancies and any influence of the sol-

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