APS/123-QED

Non-empirical prediction of impurity segregation in α-Fe from firstprinciples
T. Tsuru,1, ∗ C. Suzuki,1 Y. Kaji,1 and T. Tsukada1
1

Nuclear Science and Engineering Directorate, Japan Atomic Energy Agency,
2-4 Shirakata-Shirane, Tokai-mura, Ibaraki, Japan

Abstract
The segregation and clustering of impurities in α-Fe were investigated by first principle density functional
theory calculations. The segregation tendencies of various elements observed in reactor pressure vessels
were considered and the interaction characteristics between Fe and each impurity element were estimatedby mean field approximation. Stable N-atom impurity clusters were subsequently chosen to evaluate thechanges in free energy for clustering. These calculations show that Cu and Mn impurities embedded in α-Fe
are more stable when they are in the segregated state. Conversely, Nb and Ta are stable in the separately
solute state. The present estimates provide reliable suggestions for the segregation characteristics, and thetendencies are in good agreement with the recent atom probe observations. We suggest that the segregation
tendency is derived from the d-orbital interaction and that the solubility limit is not necessarily correlatedwith the tendency of clustering formation.
PACS numbers: Valid PACS appear here

1

Irradiation-induced embrittlement of low-alloy reactor pressure vessels (RPVs) is a central issue in the aging of nuclear power plants. The structural factors that contribute to hardening andembrittlement have been extensively studied using test materials1–5 . The mechanical properties of
RPVs change due to aging and neutron irradiation, as neutron irradiation enhances the nucleationof lattice defects such as the highly concentrated vacancy formations and anomalous interstitialatoms. In general, most of the impurity atoms dominantly diﬀuse via vacancy diﬀusion, and thehigh-density vacancy condition under irradiation promotes that diﬀusion. The recently available
three-dimensional atom probe (3DAP) allows the formation of the impurity clusters in RPVs tobe directly observed under neutron and ion irradiation6,7 and under deformation8 . RPV steels are
generally made from low-alloy ferritic steels containing traces of impurities, and among themCu impurities have the pronounced tendency to segregate from the matrix as confirmed using
3DAP. This segregation interferes with plastic deformation and results in the embrittlement of thematerial9,10 .
In addition to the research with test materials, atomistic simulations have been studied to investigate the singular state atomic structure under irradiation. In previous works, point defects
in Fe and dilute Fe-Cu alloys and dumbbells11 , and the stability and mobility of self-interstitialatoms in α-Fe were investigated using first principle calculations12 . The diﬀusion mechanism
and temperature-dependent precipitation kinetics were integrated into a Monte Carlo simulation
with the embedded atom method13 . A great number of molecular dynamics simulations of theinteraction between Cu precipitation and edge/screw dislocations in α-Fe have been performed,
and the dependence of the yield stress on the radius of precipitation and temperature has been
investigated14–18 . However, few satisfactory investigations of the other impurity elements in α-Fe
exist, although 3DAP observations have indicated the possibility of the occurrence Mn and Ni
impurity clusters as well. Therefore, a systematic investigation of the various impurity elements
was considered necessary.
Although unusual defect structures in a material such as cascade damage arise under irradiation, basic information on the equilibrium state can suﬃciently illuminate the segregation problem.
Some excellent ways of using the vibration entropy between two phases and cluster expansions19,20
in conjunction with first principle analyses have been proposed and yielded acceptable results21,22 .
These approaches are useful with specific materials and phases; however, from the perspective of
the cost of calculation, it is not worth considering the various impurity elements and the corresponding possible number of configurations. In this study, the aim was to consider the various
2

types of impurities in α-Fe and investigate the tendency prediction of segregation and clustering. First principle density functional theory (DFT) in conjunction with the Bragg-Williams (BW)approximation23,24 was eﬃciently applied without considering actual cluster configurations. Subsequently, the change in enthalpy for clustering of the several impurity atom clusters with typicallystable cluster shapes in α-Fe was evaluated, where a characteristic association between electronic
states and segregation tendency was discussed.
A binary mixture was first considered to understand the segregation characteristics of an Febased binary system. Binary mixtures can be regarded simply as a regular solution model according to the BW approximation, and the change in free energy by mixture is given by the equation∆Gmix = ∆Hmix − T ∆S mix , where the entropy term can be evaluated using Boltzmann statistics,
provided the contribution of vibration is ignored. As the enthalpy term contributes mainly to thetotal free energy, the change in enthalpy is evaluated nonempirically in this study by combining
the classical analysis and first principle DFT calculations. Assuming a regular solution model, thechange in enthalpy is obtained from the diﬀerence in energies between the segregation limit andsolid solution state using the following equation:(
EFe−Fe + EM−M )∆Hmix = NA z EFe−M −xFe xM = W xFe xM .
2

(1)

Here NA is the Avogadro constant, z is the coordination number, EFe−Fe , EM−M , and EFe−M are thebinding energies between suﬃxes, and xFe and xM are the mixing ratios. The pressure term can beignored when considering the aggregation system. W, the interaction parameter, is the principal
factor in determining the stability of the impurities. Summing the binding energy between nearestneighbor atoms equals the total energy of the system with the BW approximation; we assume thatthe total energy calculated in the DFT simulation can be distributed equally to each bond resulting
in the total energy being obtained at a desirable accuracy.
The further assumption was made in the present approach that dissimilar atoms are replacedat the substitutional position. The binding energies of EFe−Fe and EM−M were evaluated using
DFT calculations prior to that of EFe−M . DFT calculations were then performed using the Vienna
Ab initio Simulation Package (VASP)25,26 with the Perdew-Burke-Ernzerhof generalized gradientapproximation exchange-correlation density functional from27 and the projector augmented-wave
method. The cutoﬀ energy for the plane wave basis set was 500 eV. For the calculations of EFe−Feand EM−M , 17 × 17 × 17 Brillouin zone k-point sampling was selected using the Monkhorst-Packalgorithm28 . The lattice constant of α-Fe was found to be 2.8335 Å and the total energy is directly3correlated with EFe−Fe . The binding energies between impurities were obtained similarly from thebcc phase at the same lattice constant as the stable α-Fe matrix. In this study, various kinds oftransition metals of V, Cr, Mn, Co, Ni, Cu, Ni, Cu, Zn, Nb, Pd, Ag, Ta, and some representative
elements were taken to be the impurities in α-Fe, because most of them are well-known impurities
in reactor pressure vessels. Subsequently, the binding energies between α-Fe and the impurity
elements were investigated. The simulation cell was a 3 × 3 × 3 bcc lattice that included fifty-three
Fe atoms and an impurity atom. Moreover, 7 × 7 × 7 Brillouin zone k-point sampling was used,
and the fully relaxed configurations were obtained by the conjugate gradient (CG) method with aconvergence condition of 0.02 eV/Å. The interaction energy between α-Fe and impurity EFe−M isderived from both the total energy of the intended system E tot and the binding energy of matrix
atoms EFe−Fe and is expressed by the following equation:
EFe−M = {E tot − z(N/2 − 1)EFe−Fe }/z.

(2)Here N is the total number of atoms in the simulation cell. The definitional Eq. (2) was used toevaluate the interaction parameter. Although the regular solution approximation includes essentially inevitable error, the long-distance contribution to the nearest neighbor can be summarizedusing EFe−M , and thus the eﬀect of the impurity element is suﬃciently considered.
As impurities are first segregated at the lattice site of α-Fe, the interaction parameter is usefulin predicting the early-stage nucleation process. The equivalent binding energies of Fe and theimpurities and resulting interaction parameters were calculated using DFT and are summarizedin Table I, where the total number of atoms is fifty-four and the coordination number is eight inconsideration of the bcc phase. It is confirmed from Table I that the interaction parameters of
Cu and Mn are positive. The results imply that these impurities tend to segregate from the Fematrix. Conversely, some transition metals such as Nb, Mo, and Ta and all representative elements
are negative values with large absolute values, thus predicting that these impurities are diﬃcult tosegregate.
To validate the above estimation, the change in free energy was considered in the case that
randomly distributed impurities segregated and made specific clusters. Classical nucleation theory
was introduced to evaluate the free energy for clustering. According to nucleation theory, changes
in free energy are expressed by the enthalpy that includes the volumetric and interfacial terms andthe entropy term given by the following equation: ∆G = (∆HV + HS ) − T ∆S . In reality, however,
it is simply impractical to evaluate both the volumetric and interfacial terms separately because
4TABLE I: Equivalent binding energies of Fe, impurities, and Fe-impurity calculated at bcc phase at thelattice constant of most stable α-Fe (a0 = 2.8335 Å) and interaction parameters of the Fe-M binary system.
Binding energy in bcc phase of FeInteraction parameter

Mater.

EFe−Fe , EM−M (eV)EFe−M (eV)

W/Nz (eV)

Fe

−2.08

-

-

V

−2.21

−2.26

−0.12

Cr

−2.38

−2.25

−0.017

Mn

−2.22

−2.13

0.015

Co

−1.75

−1.94

−0.030

Ni

−1.34

−1.72

−0.014

Cu

−0.92

−1.41

0.084

Zn

−0.19

−1.18

−0.023

Nb

−1.76

−2.31

−0.39

Mo

−2.32

−2.40

−0.20

Pd

−1.00

−1.63

−0.093

Ag

−0.20

−1.15

−0.013

Ta

−2.12

−2.56

−0.46

Al

−0.72

−1.60

−0.20

Si

−1.14

−1.87

−0.26

P

−1.11

−1.84

−0.25

S

−0.62

−1.53

−0.18

various types of cluster shapes need to be considered. Additionally, the interfacial energies of avariety of planes cannot be calculated. If the generality of the analytical theory is sacrificed, thechange in enthalpy for clustering can be replaced by the sum of both volumetric and interfacial
contributions on the assumption that the cluster shape is preliminarily presented and expressed by
the following equation: ∆HV + HS ⇒ ∆HCNC , where Nc is the number of impurity atoms of theintended specific cluster. In this study, impurity clusters including two-six impurity atoms wereconsidered, where comparatively stable shapes in the bcc phase were selected. DFT calculations
were performed to evaluate the heats of solution of impurity clusters under the same conditions as5the above calculation. Atomic relaxation was implemented using the CG method with the abovementioned convergence condition. Thus, the heats of solution of both a single impurity atom
and N-atom cluster were calculated. Therefore, the assumption can be made that the change inenthalpy for clustering can be approximated using the heats of solution of both the single impurityNCatom EHoS and the intended N-atom cluster EHoS
using the following equation:NC− NC EHoS .
∆HCNC ≃ EHoS

(3)

Here NC is the number of impurities. Assuming that the heats of solution in an infinite matrix are
calculated, Eq. (3) becomes a true equality. Thus, the changes in enthalpy for clustering can beevaluated. The atomic configurations of selected N-atom clusters and their changes in enthalpy
are shown in Fig. 1 (a) and (b), respectively. It is found from this figure that the ∆HCNC for clusters
composed of Cu, Ni, Mn, Zn, and Ag are negative, which shows that these reactions are exoergic,
and thereby reveals that all these impurities prefer to segregate from the matrix to make clusters.
Conversely, the changes in enthalpy of the other impurity elements are positive, which meansthat these elements resist segregation. This suggestion is close to previously obtained tendencies
based on the BW approximation. Clusters composed of Cu, Mn, and Ni were observable using
the recently available 3DAP7 , although Mn and Ni are known as comparatively soluble elements.
Contrariwise, Nb and Ta clusters have not been detected. Our predictions agree rather well withthe experimental observation, and therefore it is particularly worth noting that the solubility limit
estimated from the phase diagram of Fe-based binary alloys29 does not necessarily correspond to
the segregation characteristics.
The clustering of impurities in α-Fe is determined by the binding characteristics; that is, theelectronic interaction between the d-electron of α-Fe and the outer electron of the impurity element. The local density of states (DOS) of Cu and Nb impurities embedded in α-Fe are shownin Fig. 2, which are taken as the typical cases of segregation and antisegregation elements. Herethe local DOS patterns of the nearest neighbor Fe atom to the impurities and those of singlecomponent Cu and Nb at bcc phase are drawn for comparison. It is found in Fig. 2 (a) that theoccupied or closed d-shell structure of Cu produces a weak interaction with the matrix element.
On the other hand, in Fig. 2 (b), the local DOS of Nb changes shape dramatically when it is embedded in α-Fe. A strong adhesive interaction is observed in the range of −4.0 to −2.0 eV ofup-spin state due to the unoccupied d-electron of Nb. Finally, a sequence of electronic state calculations presents the reliable prediction for segregation and clustering and supply a quantitative
6

(a)2 atom
4 atom

3 atom

5 atom
6 atom(b)
4.0

2 atom
5 atom

3 atom
6 atom

4 atomChange in enthalpy (eV)

3.0
2.0
1.0
Ni Cu Zn

Mn
0.0

V

Cr

Ag
Nb Mo PdCo

Ta Al Si

-1.0
-2.0

FIG. 1: (Color online) (a) Selected cluster shapes of two-six atom clusters. (b) Enthalpy changes from theisolated impurity to the segregated N-atom clusters evaluated by Eq. (3).

underpinning.
In summary, the segregation and clustering of various types of impurities in α-Fe were predictedusing DFT calculations in conjunction with analytical models. Interaction characteristics based on
the BW approximation were evaluated using equivalent binding energies between α-Fe and theimpurities. The interaction parameters of Cu and Mn are extremely large positive values, whichindicate the tendency to segregate from the α-Fe matrix when compared with other impurities.
Several clusters including two-six impurity atoms were then selected, and the enthalpy changes forclustering were evaluated, which is represented by the heats of solution of both a single impurity
and a N-atom cluster. Enthalpy changes of clustering composed of Cu, Ni, Mn, Zn, and Ag
are negative, which reveal that they prefer to make clusters. These tendencies agree well with7

(a)Local density of states (1/eV)

3α−Fe
singlecomponent Cu
Cu in α−Fe

2

1 up
0
−1 down
−2

−3
−10

−5

0

5

10

Energy (eV)(b)

Local density of states (1/eV)

3α−Fe
singlecomponent Nb
Nb in α−Fe

2

1 up
0
−1 down
−2

−3
−10

−5

0

5

10

Energy (eV)FIG. 2: (Color online) Local DOS of the typical segregated and nonsegregated impurities of (a) Cu and (b)Nb embedded in α-Fe, where the local DOS of the nearest neighbor Fe atom and those of single-component
Cu and Nb at bcc phase are drawn.

the 3DAP observation. These segregation tendencies are derived from the d-orbital interaction
with the Fe matrix, and therefore our results indicate that the solubility limit in the matrix cannot
necessarily be correlated with clustering. The physically meaningful predictions for segregation
characteristics can be estimated.
The author T.T. acknowledges the financial support from the Japan Society for the Promotion of
Science (JSPS), Grant-in-Aid for Young Scientists (B, 21760090). Technical support for parallel
computing from the Center for Computational Science and e-Systems at the Japan Atomic Energy

8Agency is appreciated.

∗tsuru.tomohito@jaea.go.jp

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