Ab initio modelling of vacancy–solute dragging in dilute irradiated iron-based alloys

Nuclear Instruments and Methods in Physics Research B 303 (2013) 28–32

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Ab initio modelling of vacancy–solute dragging in dilute irradiated iron-based alloys Luca Messina ⇑, Zhongwen Chang, Pär Olsson Reactor Physics, KTH, Albanova University Centre, 106 91 Stockholm, Sweden

a r t i c l e

i n f o

Article history: Received 13 July 2012 Received in revised form 16 January 2013 Accepted 18 January 2013 Available online 11 February 2013 Keywords: Density functional theory Diffusion RIS

a b s t r a c t The formation of solute–defect nanoclusters in RPV steels is the main cause of radiation induced embrittlement. Solute atoms may diffuse in the alloy by a vacancy drag mechanism, depending on the strength of interaction with point defects. A multifrequency model based on ab initio computed migration barriers was applied in order to investigate the possibility of solute drag in iron-based bcc binary alloys containing Ni, Cr, Cu or Mn, and the obtained solute diffusion coefficients were compared with previous experiments. The results show that Ni is expected to be dragged at temperatures below approximately 900 K, while Cr and Mn are not involved in the dragging mechanism. As for Cu, the results are controversial because the computed migration barriers are strongly affected by the particular choice of the ab initio method. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Ferritic-martensitic steels are widely used as base and weld materials for reactor pressure vessels (RPV) in nuclear power plants. This massive component is well known to suffer radiation induced embrittlement at relatively low temperatures (T  300 °C). The neutron flux generates a vast population of point defects which can alter the local composition of the dilute alloy by interacting with the solute atoms. In particular, the formation of complex solute–defect nanoclusters has been observed in irradiation experiments [1] and in surveillance tests [2] and is considered to be one of the main causes of embrittlement. The full understanding of the clustering process is fundamental for the current plants’ potential lifetime extension and for a better design of Generation-IV reactors. The cluster formation is driven by atom mobilities. Solute–defect interactions play a key-role, as they strongly affect how and at which rate the atoms diffuse throughout the material. This work is focused on solute diffusion in bcc iron-based binary dilute alloys through a monovacancy-mediated mechanism. Some of the most common transition metal solutes known for being involved in the irradiation damage processes in RPV steels (Ni, Cr, Cu, Mn) were considered. For each binary alloy, the Onsager coefficients Lij were computed. Fick’s first law can be rearranged in order to separate the transport properties of the system (included in the Lij matrix) from

⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (L. Messina), [email protected] (Z. Chang), [email protected] (P. Olsson). 0168-583X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nimb.2013.01.049

the driving force inducing the atoms to diffuse (given by the chemical potential gradient) [3]:

Ji ¼ 

N X Lij rlj :

ð1Þ

j¼1

The diagonal coefficients are related to the diffusion of each single species, whereas the off-diagonal terms express the interaction between fluxes of different elements and the combined solute–defect diffusion. The phenomenological coefficients are difficult to measure, but several modelling strategies can be pursued. For instance, random walks of host and solute atoms can be simulated by means of atomistic kinetic Monte Carlo (AKMC) codes and the coefficients derived from the statistical distribution of the random paths of the diffusing atom [4]. As opposed to the stochastic approach, the analytical multifrequency framework developed by Le Claire [5] was chosen for this work. In this model, the transport properties of the system are determined in terms of microscopic jumping frequencies of defects in proximity to a solute atom. More recently, a self-consistent mean field (SCMF) theory was employed by Nastar [6] to extend the multifrequency model to any solute–defect interaction range. A full set of Onsager coefficients provides much information about the vacancy-solute interaction. A parameter of particular interest is the vacancy wind G ¼ LAB =LBB , which establishes whether the solute is dragged along by the vacancy (G < 1) or simply diffuses by vacancy exchange (G > 1). In addition, the solute radiation induced segregation (RIS) at defect sinks can be analyzed. If solute atoms are not dragged by vacancies and diffuse faster than the host atoms, depletion at sinks will occur.

L. Messina et al. / Nuclear Instruments and Methods in Physics Research B 303 (2013) 28–32

Several irradiation experiments, Monte Carlo simulations and ab initio studies have shown that most of the mentioned impurities are expected to strongly interact with vacancies. For instance, the formation of solute–vacancy clusters in FeCu and FeNi irradiated alloys was observed. In a recent work, Choudhury et al. [7] applied the multifrequency framework in order to compute the Onsager matrix for FeNi and FeCr binary alloys, drawing the conclusion that neither Ni nor Cr are expected to be involved in any vacancy dragging mechanism. Whereas this deduction is confirmed by several works for FeCr alloys, it is controverted by most experimental evidences when the FeNi alloys are concerned [8,9]. Given the strong second nearest neighbour (nn) interaction between Ni solute atoms and monovacancies, a drag mechanism for Ni is very likely to occur. Therefore, in this work the vacancy wind computation was repeated for Ni and Cr impurities and extended to Cu and Mn. A more accurate model for the Onsager matrix computation was used, as is explained in the following section. 2. Analytical model The multifrequency model developed by Le Claire [5] was used to determine the Onsager matrices for the binary alloys here studied. The solute concentration is considered to be sufficiently low so that the probability for two solute atoms to be close to each other is negligible (dilute alloy limit). In body-centered cubic (bcc) alloys, the solute–vacancy interaction is usually strong up to the 2nn, because the distance between 1nn and 2nn is small compared to face-centered cubic (fcc) structures. In the multifrequency framework, the analytical expressions of the Onsager and diffusion coefficients depend on the cut-off distance assumed for the solute– vacancy interaction. Choudhury et al. [7] chose a model that limits the solute–vacancy interaction to the 1nn. This kind of model is not applicable to the vacancy drag prediction in bcc lattices. As a matter of fact, for geometrical reasons the 2nn configuration is part of the dragging mechanism and the 2nn vacancy–solute interaction is hence a necessary condition for this phenomenon to occur. For this reason it was decided to make use of a more accurate (but still approximated) set of analytical equations derived by Serruys et al. [10], where the 2nn interaction is partially taken into account. The 9-frequency model owes its name from the fact that 9 different jump frequencies must be distinguished from the background jump rate, since the presence of a solute atom alters the hopping frequencies of the host atoms around it. These configurations and their nomenclature are shown in Fig. 1. Solute dragging arises when the solute–vacancy couple does not dissociate during the diffusion mechanism, which is the case if the dissociative

jumps occur at a lower rate than the associative jumps and the jump types actually involved in the dragging path (x2 ; x3 and x4 ). For each of the 10 jump types, the jumping rate has an Arrhenius-like dependence on temperature:



xi ¼ mi exp 

 Hm i : kB T

ð2Þ

The coefficients mi are the attempt frequencies (which contain the migration entropy term) and Hm i the migration barriers that were computed by ab initio calculations. The latter are defined as the difference between the energy at the transition saddle point and the initial state energy. Attempt frequencies might also be computed by means of DFT calculations, by determining the vibrational eigenfrequencies of the lattice and applying transition state theory [11]. However, in this work it was chosen to make use of an accurate calculation performed by Chang et al. [12] for the pure Fe system. The analytical expressions of the Lij were derived by Serruys et al. [10]. The chosen approximated model is based on the assumption that the 2nn interaction is weak (although it is often not the case for the solutes included in this paper), which is equivalent to assuming that x40 ¼ x400 ¼ x6 ¼ x0 . The consequences of the weak-interaction assumption are discussed in the following sections. While the analytical expressions for fcc structures can be easily written down and found in several occasions in literature (for dilute alloys), it is not as easy to develop an exact analytical model for bcc crystals, because of the greater number of frequencies involved. Serruys et al. [10] and Allnatt et al. [4] provided two different ways for an exact derivation of the coefficients, although in both cases a numerical solution is required. Nastar’s SCMF-based model [6] allows for the removal of the interaction cut-off distance approximation, but it requires at any rate a numerical solution. As a first step towards a more accurate determination of the transport properties, it was chosen to rely on Serruys’s approximated model that provides analytical expressions of the Lij . The vacancy equilibrium concentration C eq v , which appears in the Onsager coefficient expressions, can be shown with statistical mechanics arguments to be

! Sfv DHfv ; C v ¼ exp  kB kB T eq

ð3Þ

where Hfv and Sfv are the vacancy formation enthalpy and entropy. The former one can be obtained as Hfv ¼ EðN  1Þ  ðN  1Þ=N  EðNÞ [13], where E(N) is the energy of the perfect bulk lattice and E(N  1) the energy of the system with one vacancy. The formation entropy is a function of the lattice vibrational eigenfrequencies obtainable by phonon spectrum calculations [13]. Also in this case the results provided by Chang et al. [12] were used. The drag mechanism is correlated to all jump frequencies defined in the 9-frequency model; however, some conclusions can be drawn just by analyzing the solute–vacancy binding energy, which is a function of the solute–vacancy distance. The binding energy at the x-nn position is defined as N1 N1 N HbV;xnn ¼ EN2 1V;1sol  E1V  E1sol þ E ;

Fig. 1. Hopping frequency nomenclature in a bcc lattice, according to the multifrequency framework defined by Le Claire [5], where the solute–vacancy interaction range is limited to the 2nn distance. x0 identifies any jump type not affected by the solute presence. The jump rates in parentheses are those that were set equal to x0 in the approximated model described in Section 2 [10].

29

ð4Þ

where EN2 1V;1sol is the energy of a N-atom supercell containing one solN1 ute atom and one vacancy at the x-nn site, EN1 1V and E1sol refer to the N same supercell with a vacancy or a solute, and E is the energy of the reference pure iron supercell. With this sign convention, a negative binding energy stands for attractive interaction. The benchmarking of the model must be performed by comparing the ab initio computed diffusion coefficients to experimental measurements. The self-diffusion coefficient is given by [5]

L. Messina et al. / Nuclear Instruments and Methods in Physics Research B 303 (2013) 28–32

ð5Þ

where a is the lattice parameter and f0 ¼ 0:7272[14] is the correlation factor for self-diffusion in bcc lattices. The effect of a small solute concentration C B on self-diffusion is accounted for by the solute enhancement factor b, which depends in an intricate manner on all jump frequencies [10]. The solute diffusion coefficient is instead expressed as [5]

Dsol ¼ a2 f2 x2

x40 eq C ; x30 v

ð6Þ

where the solute correlation factor f2 is a function of all 9 frequencies defined in our model [14]. Since diffusion experiments are necessarily performed at relatively high temperatures, the effect of the order–disorder magnetic transition in iron on the diffusion activation energy must be taken into account. The model developed by Chang et al. [12] was adopted. The activation energy Q ¼ Hf þ Hm is affected by the magnetic transformation according to

DQ ðTÞ ¼ DQ F0  aH;

ð7Þ

where H is a normalized coefficient depending on the magnetic excess enthalpy in pure iron DHmag as HðTÞ ¼ 1  DHmag ðTÞ=DHmag ð0Þ. The latter quantity was measured experimentally by Jönsson [15]. In pure ferromagnetic state H ¼ 0 and in pure paramagnetic state H ¼ 1. From this definition, the proportionality factor a is determined as the difference between activation energy in ferromagnetic state at T ¼ 0 K and the activation energy in a completely random alloy: a ¼ Q F0  Q P [15,12]. The value of Q P can be either measured in experiments or computed ab initio. 3. Ab initio methodology All bulk properties, solute–vacancy binding energies, migration barriers and attempt frequencies were computed in the framework of Density Functional Theory (DFT) by using the Vienna ab initio simulation package (VASP) [16–18]. The computations were performed with the pseudopotentials included in the VASP library and developed within the projector augmented wave (PAW) method [19,20]. For the exchange–correlation functional the Perdew– Burke–Ernzerhof (PBE) [21] parametrization of the generalized gradient approximation (GGA) was used. The spin interpolation of the correlation potential was accomplished with the Vosko– Wilk–Nusair (VWN) algorithm [22], while the Monkhorst–Pack scheme was used in order to sample the Brillouin zone. All calculations were spin-polarized. Standard potentials were chosen for Fe, Ni and Cu, whereas the semi-core potential was used for Cr. Concerning Mn, both potentials were tested and no substantial differences were found (the largest difference in solute–vacancy interaction energy was 3 meV/atom). The rest of the computations were thus run with the Mn standard potential. Periodic boundary conditions were applied to a 128-atom bcc supercell. Defect calculations were carried out restraining cell shape and volume, but allowing for atomic relaxations. Following convergence tests of a previous work [23], a 3  3  3 k-point mesh and a plane wave cutoff energy of 300 eV were selected. The migration barrier calculations were carried out by means of the nudged elastic band (NEB) method [24,25], with 3 images and the application of the climbing image algorithm. A refinement of the charge density mixing algorithm was necessary in order to improve the convergence of the calculations for the FeMn alloy. The simulations were in fact trapped into local minima characterized by incorrect local magnetic states, as previously reported [26]. The application of a linear mixing in the starting guess of the charge dielectric function (in place of the default Kerker

model [27,28]) was successful in compelling the system into the correct global minimum. 4. Results and discussion The relaxation of the pure iron supercell gave an equilibrium lattice parameter of a ¼ 2:831 Å. The vacancy formation and migration energies were determined to be DHfv ¼ 2:18 eV and DHmig ¼ 0:70 eV, which are in well accordance with previous com0 putations [23,29]. The total activation energy in ferromagnetic state is therefore Q Fv ¼ Hfv þ Hm v ¼ 2:88 eV and is close to the experimental value of about 2.95 eV [29]. The vacancy formation entropy Sfv ¼ 2:43 kB and the attempt frequency for the background jump m0 ¼ 53:2 THz were provided by Chang et al. [12]. This attempt frequency is assumed to be unaffected by the solute atoms. The activation energy in the paramagnetic state was taken from experimental data [15] to be Q P ¼ 2:26 eV, which gave a parameter a ¼ Q F  Q P ¼ 0:62 eV for pure iron. The presence of solute atoms in the limit of a dilute alloy is assumed to have a negligible effect on the activation energy in the paramagnetic state, hence the factor a is taken as constant for all considered impurities. Solute–vacancy binding energies where obtained by relaxing different supercell configurations and applying the definition provided in Eq. 4. The results are reported in Fig. 2, which shows also a comparison with previous ab initio calculations. The binding energies computed in the present work match in most cases very well with the previous calculations. Most of the solutes show a strong binding up to the 2nn, as is expected in bcc lattice. Ni has a strong 2nn interaction, which is the reason why one would expect to obtain Ni dragging by vacancies. On the contrary, Cr is almost completely transparent to vacancies. A slight 5nn interaction is present for Ni, Cu and Mn, which is due to the particular symmetry of the bcc structure. Nevertheless, the approximated model used to compute the Onsager coefficients in this work does not take into account interaction beyond the 2nn distance, which may lead to an error in the model, especially for Cu. Some appreciable deviations are observed in comparison with the results obtained by using the ultrasoft pseudopotentials (USPP) [30]. The difference is especially remarkable in the vacancy-copper case: according to USPP, the 1nn interaction in Cu is smaller than the 2nn interaction, which would make Cu behave more similarly to Ni, while according to our calculations the 1nn V–Cu interaction is considerably stronger. This may have strong effects on the dragging phenomenon for Cu, as discussed below. The migration barriers computed ab initio are listed in Table 1 and compared with previous computations [7,32,33]. In general, one can observe a good agreement between the current work’s results and the comparison values. The small discrepancies for Ni and Cr are due to slight differences in the parameter settings of the DFT

0.1 0 −0.1 −0.2 −0.3

−0.1 −0.2 −0.3

eV

DFe ðC B Þ ¼ a2 f0 x0 C eq v  ð1 þ bC B Þ;

Ni

0 −0.1 −0.2 −0.3 Cu −0.4

1

2

3

4

−0.1 −0.2 −0.3 1

2 3 4 NN distance

5

This work [7] [24] [28] [29]

Cr

5

eV

30

1

2

3

4

5

1

2 3 4 NN distance

5

Mn

Fig. 2. Ab-initio computed solute–vacancy binding energies as function of the impurity-defect distance, for the FeNi, FeCr, FeCu and FeMn binary alloys, and comparison with previous calculations [7,26,30,31].

31

L. Messina et al. / Nuclear Instruments and Methods in Physics Research B 303 (2013) 28–32 Table 1 Ab-initio computed migration barriers (in eV) for the FeNi, FeCr, FeCu and FeMn binary alloys, compared with previous calculations [7,32,33]. In reference [32], the USPP method in place of the PAW scheme was used.

DHmig 0

DHmig 2

DHmig 3

DHmig 4

DHmig 30

DHmig 300

DHmig 5

Ni Ni [7]

0.70 0.67

0.63 0.68

0.59 0.55

0.69 0.69

0.72 0.70

0.66 0.62

0.80

Cr Cr [7] Cr [33]

0.70 0.67 0.66

0.55 0.58 0.52

0.70 0.69 0.66

0.66 0.65 0.62

0.69 0.67 0.65

0.67 0.64 0.60

0.72 0.74

Cu Cu [32]

0.70 0.65

0.51 0.55

0.73 0.56

0.64 0.60

0.74 0.65

0.67 0.59

0.75 0.70

Mn

0.70

0.42

0.66

0.61

0.70

0.66

0.76

1 T

OP

Vacancy wind G

0.5

0

−0.5

−1

Ni (2nn) Cr (2nn) Cu (2nn) Mn (2nn) Cu [30] Ni (1nn)

−1.5

300

600

900 1200 Temperature [K]

1500

1800

Fig. 3. Vacancy wind for the FeNi, FeCr, FeCu and FeMn dilute binary alloys. Vacancy drag occurs at temperatures where G < 1. (2nn) stands for curves determined with the 2nn approximation model, while (1nn) marks those obtained when only the 1nn solute–vacancy interaction is taken into account. The light gray dashed curve shows the vacancy wind obtained with the 2nn model applied on the USSP-computed migration barriers [30,32]. T OP ¼ 573 K is the RPV operational temperature [2].

DFe/X [m2/s]

10 10 10 10

−10

−10

[m2/s] Fe/X

D

10 10 10

10

Ni −13

10

−16

−19

−16

10 Self−diffusion Solute diffusion [32] [33]

9

−19

10 10

11Cu

7

9

10

11 Mn

−13

10

−16

−22

Self−diffusion Solute diffusion [34] [35]

8

−13

−19

Cr

−13

8 10

calculations. Specifically, different PAW potentials for Fe and Ni and a different cell size (128 atoms instead of 54) were used. A remarkable difference for all jump types is instead observable for the Cu hopping rates computed with the USPP method [30,32]. After having determined the diagonal and off-diagonal coefficients for all alloys, the vacancy wind parameter for the different solutes was computed. Its behaviour as function of temperature is shown in Fig. 3. In the same figure the results of the 1nn multifrequency model for Ni are represented, as well as the curve yielded by the 2nn model with the use of the USPP migration barriers in the FeCu alloy. Dragging occurs when G < 1. Among all considered solutes, only Ni impurities follow the vacancies in their diffusion, when the temperature is below 900 K. The comparison with the 1nn model shows clearly that the latter model does not detect Ni dragging at any temperature. Cr is not dragged, which is reasonable given its transparency to vacancies; as opposed to that, the results for Cu and Mn are more unexpected. The V–Mn interaction is not negligible, although it is weaker than other solutes. The approximations of the used model might have a strong effect in this case, hence the result must be verified with a more accurate model that includes more hopping frequencies. In the case of Cu solute atoms, the alternative set of jumping frequencies listed in [32] shows that Cu dragging by vacancies is possible. In spite of its deficiencies, the USPP method appears to describe better the Cu behaviour which is observed in experiments [1]. A deeper analysis of the FeCu alloy using different ab initio methods might be necessary. The self- and solute diffusion coefficients were computed through Eqs. 5 and 6, by making use of the ab initio determined hopping frequencies. Fig. 4 shows the coefficient behaviour as function of temperature for the four different species. The self-diffusion coefficients include the effect of the solute enhancement factor b. For Ni and Cr the agreement with the experimental values is within one order of magnitude, while the discrepancies in the case of Cu and Mn are more considerable, although they are at any rate smaller than two orders of magnitude. In all cases the solute is a faster diffuser than the solvent atoms, which means that Cr and Mn are expected to deplete at grain boundaries. Nothing conclusive can be said about the Ni RIS behaviour, as vacancy drag is expected to be present, as well as for Cu.

−16

10 Self−diffusion Solute diffusion [36] [37]

8

9 10 104/T [K−1]

−19

10

−22

11

12

10

7

Self−diffusion Solute diffusion [35]

8

9 10 104/T [K−1]

11

12

Fig. 4. Ab-initio computed self- (dashed line) and solute (solid line) diffusion coefficients for the FeNi, FeCr, FeCu and FeMn dilute binary alloys, compared with experimental measurements [34–39]. The dash-dotted lines mark the Curie temperature T C ¼ 1043 K in pure Fe. The experiment data concern exclusively the solute diffusion. The computed self-diffusion coefficients take into account the solute enhancement factor. It is assumed a solute concentration of 1 at.%.

32

L. Messina et al. / Nuclear Instruments and Methods in Physics Research B 303 (2013) 28–32

5. Conclusions The choice of the range for the solute–vacancy interaction to be considered in the multifrequency framework greatly affects the model outcome. It has been shown that any meaningful treatment of the dragging mechanism in bcc structures requires to consider at least the 2nn solute–vacancy interactions. The difference between 1nn and 2nn models is conceptually remarkable and leads to contradictory results. The here presented ab initio results are in good agreement with experiments [29] and previous calculations [26,32,7]. The approximated model for the Onsager coefficient and vacancy wind computation was benchmarked by comparing the obtained solute diffusion coefficients with measurements. The change of diffusion activation energy due to magnetic disordering in the alloy was included in the model. The computed values were in most cases in the same order of magnitude as the experiments. The following conclusions can be hence drawn for each of the considered solutes. 1. Nickel has a strong 2nn interaction with vacancies and is expected to be dragged at temperatures lower than 900 K. If the 2nn interaction is neglected, no dragging is obtained at any temperature. 2. Chromium is the only impurity in this work that does not interact at all with vacancies. Since Cr diffuses faster than Fe, it is expected to deplete at grain boundaries, as far as the interaction with vacancies is concerned. As many previous works confirm [23,7], this species is more likely to participate in an interstial-mediated diffusion mechanism, which may lead to an opposite RIS behaviour. 3. Copper interacts with vacancies at the 1nn site stronger than at the 2nn, and the model does not show dragging, which is not straight-forward to concile with the analysis of irradiation experiments [1]. Nevertheless, a set of migration barriers computed with the USPP method led to the opposite conclusion. The reason for this discrepancy has to be further studied. 4. The FeMn alloy yielded similar results as in the FeCu case. The weaker V–Mn 2nn interaction is not strong enough for the Mn impurity to be dragged along. The diffusion of Mn in bcc iron does probably occur through interaction with self-interstitials. The sensitivity of the model to the cut-off interaction distance obliges to countercheck the conclusions of the current work by applying other methods mentioned above (SCMF theory [6] and Monte Carlo [4]). A further step will be the extension of the model and the ab initio calculations to the interaction between solutes and self-interstitials. Acknowledgments This work was carried out with the financial support from Vattenfall and the Göran Gustafsson Foundation. The high performance computations were performed on the resources provided by the Swedish National Infrastructure for Computing (SNIC) via PDC and by PRACE on Monte Rosa, CSCS based in Switzerland. The authors acknowledge L. Malerba, F. Soisson and M. Nastar for the fruitful conversations and A. Claisse for the graphics support. References [1] Y. Nagai, Z. Tang, M. Hassegawa, T. Kanai, M. Saneyasu, Irradiation-induced Cu aggregations in Fe: an origin of embrittlement of reactor pressure vessel steels, Phys. Rev. B 63 (2001) 134110. [2] P. Efsing, C. Jansson, T. Mager, G. Embring, Analysis of the ductile-to-brittle transition temperature shift in a commercial power plant with high nickel containing weld material, J. ASTM Int. 4 (7) (2007).

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