Solute–grain boundary interaction and segregation formation in Al: First principles calculations and molecular dynamics modeling

Computational Materials Science 112 (2016) 18–26

Contents lists available at ScienceDirect

Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci

Solute–grain boundary interaction and segregation formation in Al: First principles calculations and molecular dynamics modeling L.E. Karkina a, I.N. Karkin a,b, A.R. Kuznetsov a,b,⇑, I.K. Razumov a,b, P.A. Korzhavyi a,c, Yu.N. Gornostyrev a,b a

Institute of Metal Physics, Ural Branch of RAS, 18 S. Kovalevskaya Street, Yekaterinburg 620990, Russia Institute of Quantum Materials Science, 51-48 Bazhov Street, Yekaterinburg 620075, Russia c Department of Materials Science and Engineering, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden b

a r t i c l e

i n f o

Article history: Received 10 April 2015 Received in revised form 5 October 2015 Accepted 6 October 2015

Keywords: Grain-boundary segregation First principles calculation Molecular dynamics simulation Aluminum alloys

a b s t r a c t The interaction between solute atoms (Mg, Si, Ti) and grain boundaries (GBs) of different types in Al are investigated using two approaches: first principles total energy calculations and large scale atomistic simulations. We have found that both deformation (size effect) and electronic (charge transfer) mechanisms play an important role in solute–GB interaction. The deformation and electronic contributions to GB segregation energy for the considered solutes have been analyzed in dependence on the impurity and the GB type. Mg and Si atoms are calculated to segregate to GBs, while Ti atoms to repel from, GBs in Al. For the case of a symmetric special-type GB the interaction is found to be short-ranged. For a general-type GB the range of GB–solute interaction is found to be considerably longer. A method to estimate the segregation capacity of a GB has been proposed, which takes into account the solute–solute interactions, and shown to be able to correctly describe the GB enrichment in alloying elements. The features of the segregation formation in fine-grained materials produced by severe plastic deformation are discussed. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction Grain boundary (GB) segregation is a very important phenomenon that affects many properties of polycrystals, especially strength and plasticity, recrystallization kinetics, decomposition of solid solution, new phase nucleation [1]. The role of GB segregation becomes especially important in ultrafine grained materials where the fractions of GB and bulk atoms are comparable [2,3]. In this case the formation of segregations on GBs changes significantly the condition of the phase equilibrium [3,4] and thermal stability of ultrafine grained structure [5–9]. Thus, fundamental understanding of the GB segregation is necessary for effective control of structural state and properties of polycrystalline materials. Although the problem of GB segregation has been in the focus of research studies for many years [1], the physical mechanisms that control the interaction between impurities/alloying elements and GBs remain the subject of debates (see, for example, Refs. [10,11]). The investigation of GB chemistry is a challenging task for both experiment [12,13] and theoretical modeling [10,11,14–16]. Over ⇑ Corresponding author at: Institute of Metal Physics, Ural Branch of RAS, 18 S. Kovalevskaya Street, Yekaterinburg 620990, Russia. Tel.: +7 9193748241; fax: +7 (343)3745244. E-mail address: [email protected] (A.R. Kuznetsov). http://dx.doi.org/10.1016/j.commatsci.2015.10.007 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.

the last few years new important information about the GB segregation has been obtained by using 3D atom probe tomography [17–19] which is one of the promising techniques for the quantification of the atomic scale solute distribution in the vicinity of interfaces with a spatial resolution of a few angstroms. The formation of unusually broad regions near GBs (having a half-width of 3– 5 nm) enriched in alloying elements was observed in ultrafine grained alloys produced by severe plastic deformation [20–23]. The formation of such broad segregations may be attributed to structural peculiarities of the GBs after the severe plastic deformation, but the nature of these peculiarities is not exactly known. Here we consider equilibrium segregation (i.e., corresponding to the minimum of Gibbs free energy) of solute atoms to an inhomogeneity in the crystal structure. A thermodynamic description of such equilibrium GB segregation may be obtained using a Langmuir–McLean type of segregation isotherm [1,24] that relates changes in the solute concentration at the GB with changes in the Gibbs free energy, DG ¼ DG0 þ DGex . Here DG0 is the ideal Gibbs free energy and DGex is the excess term that describes the deviation of the real system from the thermodynamic behavior of the ideal solid solution. Segregation energy is defined as the change of internal energy upon moving a solute atom from the bulk to the GB, bulkðGBÞ bulk DEs ¼ EGB is the solution energy of the sol  Esol , where Esol

L.E. Karkina et al. / Computational Materials Science 112 (2016) 18–26

alloying element in the bulk(GB) phase. The main contribution due to segregation usually goes into the ideal Gibbs free energy DG0 . Therefore, it is often assumed that segregation of individual atoms is driven by a decrease of the GB surface energy [1,25] and that all entropy contributions (such as anharmonic, and vibrational) can be neglected, except the ideal configurational entropy which is then used for calculating the enrichment of the GB phase by the solute. Although this approach does not take into account the interactions among the segregating atoms, which contribute to DGex , it catches correctly the overall trends in GB segregation phenomenon by introducing just one characteristic parameter DEs. Different contributions to GB segregation energy DEs have been discussed by now and current opinions about their relative importance are divided in two groups. The first group ascribes segregation mainly to the deformation contribution [24,26] or size effect [10,27,28], thereby relating the solute–GB interaction to changes in the GB structure due to the presence of a solute atom. The other group favors the electronic interaction mechanism [26,29–31] associated with a charge transfer or changes in the chemical bonding between the host atoms at the GB in the presence of the solute. To estimate the deformation contribution to GB segregation energy DEs, continual elasticity have been used [1,24,26] for modeling solute–GB interactions. The obtained result is that the elastic interaction is a second order effect due to image forces and DEs  d2 where d is the size misfit between the solute and host atoms. The structure of a real GB may be viewed as consisting of structural elements, or dislocation arrays, which create elastic strain fields that can interact with solute atoms [32,33]; in this case one may expect that DEs  d, i.e. the interaction changes sign depending on the size misfit of the solute. At the same time, the local distortions near the GB core will cause a redistribution of the electron density, which results in the mechanism that usually considered as electronic [10,16,26,30,31]. Moreover, for a symmetric GB having a special misorientation, which does not produce noticeable lattice distortions, the solute–GB interaction is mostly determined by the electronic contribution; this situation is similar to what occurs at a free surface [34]. Thus, the mechanism of solute–GB interaction is rather complex: it may depend on the type of the GB and include the deformation as well as the electronic contributions which are difficult to separate from each other. According to the prevailing point of view, the electronic (charge transfer) contribution is dominant in the majority of cases and is the reason for GB embrittlement. However, as was recently shown, the size effect [10] and changes in the bond order [16] can play a decisive role in the behavior of some solutes at GBs, which includes also the redistribution of the electron density. Therefore, for understanding the solute–GB interactions, a separation of the deformation/structural and electronic/chemical effects is necessary. A consistent model of GB segregation formation requires information about the energy change DEs(R) depending on the position of solute atoms in the vicinity of the GB for different GB types. This information cannot be found in the framework of continuum elasticity, and employment of first principles calculations of total energy or atomistic simulations with suitable interatomic potentials are necessary. Density functional theory (DFT) methods are widely used to study the effects of impurities and alloying elements on the electronic structure and cohesion of GBs [10,11,16,30,31,35–41]. For example, the segregation energy of Mg to different positions near the tilt GB R11(1 1 3) in Al has been computed in [38] using DFT methods. It is shown that DEs varies non-monotonically (and even changes sign) depending on the distance to the GB plane. Such a behavior can originate from the heterogeneity of local distortions as well as from the charge density oscillations which may be significant even near a

19

special-type (ST) GB [26]. It has been found in Ref. [38] that an Mg atom, which preferentially occupies the ‘‘loose” site at the GB, weakens the metallic bonds among the neighboring Al atoms. Strong variation in the DEs value with distance shows that the segregation energy obtained from the experiment should be considered as an effective parameter that characterizes the average enrichment of GB region by the impurity. First principles calculations provide the most reliable information about the GB structure and energetics. However, they require significant computational resources and can be applied to the study of some ST GBs only. Atomistic simulations methods (molecular dynamics (MD), molecular statics (MS) or Monte Carlo (MC)) allow to implement large scale modeling, but are limited by the reliability of the interatomic interaction potentials used. The results of atomistic simulations of impurity segregation at the tilt and twist ST GBs have been reported in a number of papers [42– 50] where important parameters were identified to be responsible for the dependence of segregation on the GB characteristics (misorientation angle, GB energy, and excess volume) or on the characteristics of the impurity atom (solution enthalpy, ionic radius). A correlation has been established between the segregation energy and the GB energy for the ST tilt GB [46]. At the same time, the interaction between solute and GB essentially depends on the fine details of the GB structure. Enrichment of different ST GBs in Al by Mg atoms was investigated in the framework of combined atomistic approach including Monte Carlo simulations [48]. It was shown that local hydrostatic stress in the GB region and size misfit between Mg and Al atoms are the main factors that determine the GB segregation behavior in Al–Mg alloys. However, it is still unclear whether the deformation solute–GB interaction, which dominates in the case of Mg, is also the main contribution to the interaction of other impurities with the GB in Al. According to the results of modeling [48], the thickness of the Mg-enriched layer at the GB is about 1 nm. Thus, consideration of the ST symmetric GB is insufficient for understanding the reasons for the formation of wide segregation profiles, observed in experiments [20–23] after a severe plastic deformation. In this paper, interactions of single solute atoms X with a tilt ST and a general-type (GT) GBs in Al–X (X = Mg, Si, Ti) alloys are studied using two approaches that are effective at different length scales. The first one is based on first-principles DFT calculations of total energy for a crystallite containing a ST GB and a solute atom. The second approach involves MD and MS simulations for a rather large polycrystal containing several GBs of different type and solute atoms, considered at finite temperatures and described using many-body embedded atom method (EAM) interatomic potentials. The results of the calculations reveal the relative roles of the electronic structure features, of the local distortions and size effect in the interaction of substitutional atoms with GBs. In particular, we shown that the width of the segregation layer, enriched by the alloying elements, is about 1 nm for the ST symmetric GB. For asymmetric ST and GT GBs the region of strong solute–GB interaction and the effective width of the enriched layer can reach 2– 3 nm, which is considerably wider than those for a ST symmetric GB. 2. First principles calculations of the interaction of solutes with

R5{2 1 0}[0 0 1] tilt GB in Al The total energy of a crystallite containing a R5{2 1 0}[0 0 1] ST tilt GB and a solute atom (Mg, Si, or Ti) was calculated using DFT approach implemented in VASP (Vienna Ab-initio Simulation Package [51,52]) with pseudopotentials constructed on the basis of projector augmented plane wave (PAW) method [53]. The

20

L.E. Karkina et al. / Computational Materials Science 112 (2016) 18–26

exchange–correlation energy was treated in the generalized gradient approximation (GGA) using the formalism proposed in Refs. [54] and [55]. A 48-atom supercell having the dimensions 8.2a  1.6a  1.0a with periodic boundary conditions was used in the calculations (a is the lattice parameter of Al). The supercell contained two crystallites misoriented (tilted) with respect to each other by 53.1° about the [0 0 1] axis. A shift of the adjacent grains relative to each other along the GB plane can lead to a denser packing of atoms in the GB core and results in lowering of the GB energy [6,31]. Therefore an additional optimization of the GB structure was done, by sliding one part of the crystallite by distance s along the GB. Energetically the most favorable GB structure (Fig. 1a), with the GB energy lowered by about 20%, was achieved after sliding the grains by s  0.2a. We assume that considered solute atoms are substituting Al atoms in a bulk as well as in a GB region. As it is shown in Ref. [56], only light atoms are more stable in the interstitial sites at a GB center; this is even more true for the shifted and denser GB as in our case. An Al atom occupying one of the sites 1–12 near the GB in the crystallite was replaced by a Mg, Si or Ti solute atom, as shown in Fig. 1a, and the total energy of crystallite was calculated with structural relaxation. The segregation energy DEs(R) was determined as the energy difference between the crystallite containing the impurity at a distance R from the nearest GB plane and the crystallite with the same impurity situated on site 12 at the maximum distance from the GB planes. When a solute was placed onto sites 2, 3, or 4, a displacement of the GB plane toward the solute was found to occur during the relaxation in our test calculations. To prevent such displacements, two Al atoms in the GB plane were fixed after sliding and relaxing the structure of the clean GB. The calculated segregation energies DEs(R) of the solute atoms to different positions near the R5{2 1 0}[0 0 1] tilt GB in Al are presented in Fig. 2 as functions of distance R from the GB plane. The figure shows that the interaction between each of the substitutional atoms and the ST GB is short-ranged (concentrated in a narrow layer less 0.5 nm near the GB, positions from 1 to 6) and can vary non-monotonically with the distance, especially in the case of a Ti impurity. Oscillations of the segregation energy DEs(R) are frequently observed and seem to be a typical feature of the interaction of impurities with a GB [26,38,48,57]. The oscillating behavior may be caused by the heterogeneity of local distortions or by Friedel oscillations of the electron density near the GB [26]. That the amplitude of oscillations for a 3d element (Ti) is higher compared to the amplitude observed for the sp elements (Mg, Si) indicates that the electronic contribution to the segregation energy is substantial. Let us compare the segregation energy with the solution energy for impurity atoms X in Al (Table 1), calculated according to the following definition:

Esol ¼ Ecryst  nEAl  EX ;

ð1Þ

where Ecryst is the energy of an AlnX crystallite (or a supercell) containing a solute atom, EAl and EX are the energies of the Al and X elements in their respective ground state structures, and n is the number of Al atoms in the crystallite. In particular, Ti is readily soluble in Al (Esol = 1.03 eV), while Si, on the contrary, has a tendency to precipitate from the Al-based solid solution (Esol = 0.58 eV). It matches with the observed trend of the segregation energy for Ti and Si. However, in the case of Mg, whose segregation behavior is similar to Si (Fig. 2), the calculated Esol value is close to zero, so the argument based on solubility do not work equally well in all the considered cases. We fond that the largest energy gain was achieved for Si, which has one extra p-electron and atomic radius about 10% smaller than that of Al, see Table 1. At the same time, Mg, which has one less pelectron and atomic radius 20% greater than that of Al, interacts with the GB slightly weaker than Si. Finally, for Ti (whose atomic radius is similar to Mg) the segregation formation was calculated to be energetically unfavorable. It means that results of calculation cannot be accounted for by considering only one (e.g., electronic) solute–GB interaction mechanism. In particular, the deformation interaction (due to size mismatch) should also be considered. The atomic radius Rat is not quite reliable parameter to characterize the size of a solute atom in a matrix. To make a more realistic estimation of the size misfit, let us consider the lattice distortions produced by the impurities in the Al crystal lattice. The relative changes in the distance to the nearest (e1) and the next-nearest neighbors (e2) due to the substitution of Al by the solute atoms are given in Table 1. It can be seen that, indeed, a Mg solute in Al behaves as a larger atom, so that the nearest neighbor Al atoms are shifted away from it (e1, e2 > 0). The strain caused by a substitutional Si impurity is very low (Table 1), while a Ti impurity displaces the neighboring Al atoms toward itself, i.e. behaves in the Al lattice as a smaller atom, despite its large Rat. Since a GB is a region with a lower atomic density (or, equivalently, with an excess volume), one can expect, within traditional deformation mechanism, that the movement of a large Mg atom to the GB results in a decrease of energy due to the larger volume available for the impurity at the segregation site. Clearly, the opposite is expected of Ti and Si solutes that are smaller in size than the Al host. Interestingly, the GB segregation of Mg becomes energetically favorable only after the atomic relaxation. The atomic relaxation contribution (defined here as the difference between the relaxed DEs and unrelaxed DEðunrÞ segregation energies, see Table 1) s was calculated to be the largest for Mg, for which it determines the segregation energy value. It means that the atomic displacements and the corresponding redistribution of electron density are decisive factors for the solute–GB interaction in the case of Mg. Following the terminology defined in the Introduction, we will refer to

Fig. 1. Set of solute atom positions (1–12) near the R5{2 1 0}[0 0 1] tilt GB with s = 0.2a (a). Distribution of valence electron density near the GB with segregated Mg (b) and Si (c) atoms. Lighter/darker gray shading in (b) and (c) indicates regions of enriched/depleted electron density.

21

L.E. Karkina et al. / Computational Materials Science 112 (2016) 18–26

0.3

0.3

Segregation energy, eV

(b) 0.4

Segregation energy, eV

(a) 0.4 0.2 Ti

0.1 0

Mg

-0.1 Si

-0.2 -0.3 -0.4

0.2

0 -0.1

0.2

0.4

0.6

0.8

Distance of solute from GB, nm

Mg

-0.2 -0.3 -0.4 -0.5

0

Ti

0.1

0

0.2

0.4

0.6

0.8

Distance of solute from GB, nm

Fig. 2. Segregation energies of solutes to the R5{2 1 0}[0 0 1] tilt GB in Al calculated by ab initio PAW-VASP methods (solid lines) and using MS simulations with EAM potentials (dashed lines on the panel b). The sequence of symbols with increase of distance correspond positions 1, 2, 3, 4, 5, 6, 8, 12 in Fig. 1a.

Table 1 Unrelaxed DEsðunrÞ ð1Þ and relaxed DEs ð1Þ segregation energies of different impurities to position 1 at the R5{2 1 0}[0 0 1] tilt GB in Al. Electronic configuration of free atoms (El. conf.), atomic radius Rat, DFT-calculated solution energy Esol, and relative atomic displacements in the first e1 and second e2 coordination spheres around the impurities in Al.

El. conf. DEs ð1Þ; eV

DEðunrÞ ð1Þ; eV s Rat, pm Esol, eV e1 e2

Mg

Al

Si

Ti

3s2 0.26 0.09

3s2 3p1 – –

3s2 3p2 0.32 0.25

3d2 4s2 0.33 0.62

150 0.03 1.4% 0.3%

125 – – –

110 0.58 0.5% 0.1%

140 1.03 1.2% 0.1%

this segregation mechanism as a ‘‘structural” or ‘‘deformation” mechanism. The atomic relaxation contributions for Ti and Mg solutes are similar to each other, while that for Si is remarkably smaller, so the electronic interaction mechanism should play the dominant role in the last case. The distribution of valence electron density (shown in Fig. 1c) indicates that the Si impurity, which has one extra electron in comparison with Al, forms strong covalent Si–Al bonds with its neighbors at the GB. Such an increase of the electron density is not observed in the case of Mg which has one electron less than Al; in this case a reduction of the electron density is observed near the segregated Mg atoms (Fig. 1b), indicating weakening of the chemical bond. The results show that both factors, local distortions and changes in the chemical bonding, can contribute to the solute–GB interactions. For the solutes that are acceptors of electrons from the matrix and have a bigger atomic size, one can expect that the deformation contribution will be dominant. At the same time, for solutes that are electron donors the chemical contribution to solute–GB interaction is more important. A special case is the ‘‘anti-segregation” behavior of a Ti solute. The energy of lattice relaxations around Ti at the GB is rather large and close to that for Mg (Table 1). Nevertheless, the segregation energy DEs is found to be positive, which points to the decisive role of the chemical contribution to solute–GB interactions in this case. By analyzing the charge distribution we found that segregation of Ti atoms to the GB depletes the electron density just as in the case of Mg. Thus, the substitution of Al by Ti weakens the chemical bonding at the GB and increases the GB energy. Because Ti has a large negative solution energy (Table 1) due to the formation of strong covalent-like Ti–Al bonds [58] in the fcc lattice of Al, we conclude that the main contribution to Ti–GB interaction is due

to breaking the favorable nearest-neighbor coordination when the Ti impurity is moved to the GB core. A similar mechanism was discussed recently in Ref. [16] in terms of a bond order parameter; both structural and electronic contributions to solute–GB interactions are important in this case as was shown in Refs. [10,16].

3. Simulations of interaction of Ti and Mg solutes with GBs in a polycrystal As discussed in the previous section, the ST GBs are predominant in the structure of well-annealed materials. In fine-grained polycrystals, GT GBs are more frequent, and the fraction of these GBs is maximal in materials subjected to severe plastic deformation. It is impractical to consider such GBs in the framework of ab initio approaches because their modeling requires very large supercells. In this section MD method is used for the calculation of the interaction of Mg and Ti atoms with GT GBs as well as with ST asymmetric GBs. To describe the interatomic interactions we use many-body potentials that have been developed in the scheme of embedded atom method (EAM) for Al–Ti [59] and Al–Mg [60]. For quantitative comparison of the two methods used, we have made EAM simulations for the symmetric R5{2 1 0}[0 0 1] ST tilt GB by taking the supercell used for ab initio calculations described in Section 2 and increasing it 3, 2 and 4 times in the x, y and z directions, respectively. The EAM relaxation of atomic position in the supercell was performed by using MD simulations with cooling to 0 K; this method provides the same results as molecular statics (MS) calculation. The solute atoms were considered in the same substitutional positions as shown in Fig. 1a. The obtained segregation energies are compared with the corresponding results of ab initio calculations in Fig 2b. As it is seen from Fig 2b, the segregation energies obtained by using EAM and ab initio calculations are qualitatively similar. In case of Ti there is rather good agreement between EAM results and ab initio calculations for position n = 1, 2, and for n > 2 discrepancy does not exceed 0.2 eV. At the same time, for Mg the EAM simulations yield stronger segregation energy by about 0.2 eV for n = 1, 2, 3, 4. One of possible reasons for such disagreement may be connected with redistribution of electron density in the vicinity of the GB, and this effect is not completely reproduced by the EAM interatomic interaction model. Thus, it is expected from comparison of the solid and dashed curves in Fig 2b that the EAM simulation should reproduce correctly the trends in solute–GB interaction but can overestimate the segregation tendency in case of Mg.

22

L.E. Karkina et al. / Computational Materials Science 112 (2016) 18–26

Fig. 3. Projection on the (0 0 1) plane of the crystallite containing two R5 GBs, two low-angle GBs and two high-angle GT GBs (Tan = 500 K). The rectangular region selected by a thin solid line (in the center of the figure) is enlarged in the insert to show the atomic structure of the R5 GB consisting of units B and C [61]. Filled and open circles in the insert show the atoms in the consecutive (0 0 1) atomic planes.

3.1. GBs structure simulation in Al To investigate the interaction of solute atoms with both ST and GT GBs, MD-EAM simulation was performed for a sufficiently large crystallite which contained several GBs. A projection of the crystallite onto the (0 0 1) plane is shown in Fig. 3. As the first step a R5 [0 0 1]{7 1 0}/{1 1 0} ST asymmetric tilt GB was created in the center of the crystallite according to the scheme of coincidence site lattice (CSL) [61]. As described in the previous Section, an additional energy minimization was carried out for this GB by a small relative shift of the conjugated grains along the GB plane. The insert in Fig. 3 shows the crystal geometry of the R5 GB, where the main structural units describing this GB [61] are marked as C and B. The size of the crystallite in the direction perpendicular to the R5 GB plane was chosen in such a way that periodic boundary conditions lead to the appearance of the second (fictitious) R5 GB of the opposite sign. The two grains, divided by the R5 GB, were  5 0}, which placed inside the third grain bounded by the planes {3 form angles close to 120° with the R5 GB. Thereby, two triple junctions were formed in the crystallite, which connected GBs of three types: asymmetric tilt ST GB R5, GT GB with a misorientation angle of 45°, and a low-angle (LA) GB with a misorientation angle of 8.13° (Fig. 3). Such a structure allows MD simulations with periodic boundary conditions along all the three crystallographic directions. The supercell dimensions were 20 nm in the [1 0 0] and [0 1 0] directions and 10 nm in the [0 0 1] direction (which coincided with the tilt axis for all the GBs); the supercell contained about 250,000 atoms. During the supercell construction, the upper and lower parts of the matrix were moved relative to each other along the [0 1 0], to achieve the densest possible arrangement of the atoms along the

LA and GT GBs corresponding to the minimum of energy of the polycrystal. During the rigid shifts some of the atoms, belonging to neighboring grains and located at a distance less than half the radius of the first coordination shell, were removed. To reach the equilibrium structure of the GBs, the crystallite was treated by MD annealing, including a long exposure to temperature Tan = 500 K (0.5 of the melting point Tm) or at Tan = 800 K (0.8Tm), and then gradually cooled to T = 0 K with a 100 K step. The total duration of the treatment ranged from 75 to 225 ps. The MD simulations were performed with a MD time step of 2 fs. Through the annealing a relaxation of the GB atomic structure occurred, to form a more uniform distribution of the atomic density and to reduce local internal stresses. The central part of the R5 ST GB remains narrow after annealing at 500 K (Fig. 3) and can be described in terms of two structural units; significant structural changes occur only the regions near the triple junctions. After a prolonged annealing at 800 K, the triple junction migration was observed, which occurred by correlated displacements of atoms according to the mechanism discussed in [62]. In this case the regular structure of the ST GB collapsed; it was no longer possible to distinguish the ordered sequence of structural units. The reconstruction of the R5 GB during the relaxation process started from the triple junction and was initiated by the processes that occurred at the GT GB. For ST GBs a structural collapse can be expected in nanocrystalline materials having a high density of triple junctions. 3.2. Interaction of Ti and Mg atoms with the R5 asymmetric and GT GBs The segregation energies DEs(R) of Ti and Mg atoms were calculated using MS-EAM simulation for the asymmetric R5 and GT GBs

L.E. Karkina et al. / Computational Materials Science 112 (2016) 18–26

(a)

(b)

(c)

(d)

23

Fig. 4. Variation of the segregation energy DEs(R) for Ti (a, b) and Mg (c, d) atoms in dependence on the distance from the R5 (a, c) and GT (b, d) GBs. Curves 1 and 2 correspond to the annealing temperatures 800 K and 500 K, respectively. Dashed lines pick out energy interval corresponding to ±300 K.

by placing the solute atom in different substitutional positions near the GBs and subsequently relaxing the atomic structure. The calculated segregation energy as a function of distance to the GB plane along two chosen directions X1 and X2 (indicated in Fig. 3) perpendicular to the GBs plane is shown in Fig. 4. The coordinate X = 0 in the plot corresponds to a point located at a distance of about 1.5 nm from the GB, where the segregation energy is close to zero. As Fig. 4 shows, the solute interaction energy with both R5 and GT GBs is negative (attraction) for Mg and positive (repulsion) for Ti. These findings made in MS-EAM simulations are consistent with the results of ab initio calculations (Section 2). The agreement is readily expected in the case of Mg for which the size effect is important, so the movement of the solute into the GB region with a reduced atomic density is accompanied by an energy gain. However, the agreement is not self-evident in case of Ti whose antisegregation behavior is due to both structural and electronic contributions. The fact that MS simulations of Ti segregation reproduce the ab initio results indicates that the used many-body EAM potentials successfully captures the electronic structure effects related to the changes in atomic coordination. These changes include the variations in bond lengths and bond angles, as well as the decrease in the coordination number. The maximum value of Ti interaction energy with the R5 GB was found to be close to the results of ab initio calculations. At the same time, in the case of Mg EAM simulation gave significantly higher segregation energy for the R5 GB in comparison with ab initio results (see Figs. 2b and 4c). Another possible reason for this discrepancy could be that the EAM scheme underestimates the energy changes due to the electron density redistribution in the GB core region when an Al atom is substituted by Mg (Fig. 1b).

As can be seen in Fig. 4, the width LGB of the region where segregation energy exceeds the thermal fluctuations is about 1 nm and 1.5 nm for Mg and Ti, respectively; the GB is about twice as wide as the symmetric ST GB in Fig. 2. The increase in the annealing temperature leads to a broadening of the highly distorted region near the GB and to an increase in LGB value, particularly pronounced in the case of Ti (Fig. 4a and b). Thus, an increase in the range of the interaction and, consequently, an increase in the GBs segregation capacity may be the results of a loss of the GBs structure regularity due to thermal disorder when T approaches Tm. Fig. 5a and b shows the space distribution of the segregation energy for Mg and Ti in the vicinity of the R5 GB as contour plots (the dark- and light-gray shadings indicate, respectively, the attraction and repulsion regions). One can see that the segregation energy depends substantially on the position of the impurity showing regions of strong and weak interactions near the GB (their positions correlate with the distribution of strain in the GB as well as with the location of the structural elements, see Fig. 5c). A similar, but less regular, distribution of segregation energies has been found near the GT GB. Thus, structural features of the GB play a decisive role in the interaction of the considered substitutional impurities with asymmetric ST or GT GBs in polycrystals of Al. It is known that a high-angle GB may be viewed as composed of GB dislocations and partial disclination dipoles [63,64], forming GB structural elements and being the sources of internal stresses. The density of such defects at GBs should be particularly large in the fine-grained materials produced by severe plastic deformation [65]. For such non-equilibrium GBs more intensive interaction with the impurities and, correspondingly, a higher GB segregation capacity can be expected.

24

L.E. Karkina et al. / Computational Materials Science 112 (2016) 18–26

of V indicates that the solute atoms repel each other, thereby making the alloy stable against the solute precipitation. Note that Eq. (2) is obtained under the assumption that the grain size is so large that the bulk concentration C inside the grains is not affected by segregation. For simplicity, we assume here that the values of Vn are the same in the GB region as in the bulk. The effective interaction energies of Mg solutes in the Al matrix were determined using ab initio calculations by PAW-VASP method for a 32 atom supercell. The calculated Vn values for the first three coordination shells are positive and equal, respectively, to 58, 19, and 11 meV, which gives a mixing energy of V = 0.71 eV. The concentration of Mg at the center of the asymmetric R5 GB is plotted in Fig. 7a as a function of temperature. The maximum value of CGB at the center of the GB calculated using Eq. (2) with the obtained parameter V was found to be about 25 at.% for the bulk Mg concentration of 1 at.% (Fig. 7a). If we neglect the interaction between Mg atoms, the calculations give very high enrichment of GB in Mg; for temperatures T < 600 K, the concentration of Mg at the center of the GB is close to 100%, in disagreement with the experiment [20,21,23]. Thus, the repulsive interactions among the Mg atoms result in a considerable decrease in the concentration of alloying element at the GB and should be taken into account for a correct description of segregation. The distribution of Mg near the asymmetric R5 GB at 500 K calculated using Eq. (2) is shown in Fig. 7b. One can see that the width of the enriched region near the GB is about 3 nm, which is similar to the range of the solute– GB interaction (Fig. 6). The calculated enrichment of the R5 GB in Al by Mg (Fig. 7b) is found to be close to experiment. As observed in Refs. [23,66], the maximal concentration of Mg at some GBs reached 20–25 at.% after annealing of a Al–5.7%Mg alloy subjected to severe plastic deformation.

4. Segregation of Mg to GBs in Al The strong oscillations of the segregation energy DEs(R) (Fig. 5) can cause inhomogeneity of the alloying element distribution along the GB. To characterize the capacity of the GB as a whole, we have averaged the values DEs(R) along the GB plane at fixed distance X from the GB. The so obtained mean segregation energy hDEs i shows a much regular behavior as a function of X (Fig. 6) in comparison with that of DEs(R) in Fig. 5. Note that the magnitude of hDEs i variation is remarkably smaller than that of the corresponding dependence shown in Fig. 4, and the energy gain given by hDEs i for the positions in the GB core is comparable with that obtained in ab initio calculations (Section 2). At the same time, the region of strong solute–GB interaction for the asymmetric R5 GB is much wider in comparison with the corresponding region near the R5 ST GB. Let us now estimate the enrichment of the GB in Mg by using obtained mean segregation energy hDEs i profile presented in Fig. 6. In the framework of the regular solid solution model, GB segregation can be described by the Fowler equation [1]


 C GB C DEs þ VðC  C GB Þ exp ; ¼ kT 1  C GB 1  C

ð2Þ

P where V ¼ n Z n V n is the mixing energy, Vn is the effective interaction energy between the solute atoms, Zn is the coordination number for nth coordination shell. Eq. (2) is similar to the Fowler– Guggenheim adsorption isotherm. In the framework of the model of central and pairwise interatomic interactions the value Vn can be presented as V n ¼ ðV nAA þ V nBB  2V nAB Þ=2, where V nXY is the interaction potential between chemical species X and Y. A positive value

(a)

(b)

(c)

Fig. 5. Contour maps of the segregation energy DEs(R) of Mg (a) and Ti (b) atoms in the vicinity of the structural units B and C (indicated by solid lines) of the asymmetric R5 P12 1 0 0 0 GB and corresponding distribution of strain dU i ¼ 12 j¼1 ðr ij  r ij Þ=r ij (r ij is the equilibrium distance between neighboring atoms i and j, r ij is the distance between a pair of atoms near the GB) (c).

25

L.E. Karkina et al. / Computational Materials Science 112 (2016) 18–26

<ΔEs>, eV

0

-0.1

-0.2

-0.3

-0.4 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

X, nm Fig. 6. Mean segregation energy hDEs i of Mg atoms to the asymmetric R5 GB, as a function the solute–GB distance in the direction perpendicular to the GB plane.

5. Discussion The results of ab initio calculations and EAM simulations show the role of different factors contributing to solute–GB interaction, such as the GB’s structure features, local distortions due to size mismatch of solute atoms with the matrix, and electronic structure reconstructions. These factors affect the magnitude of the segregation energy DEs and can be divided in two groups according to the mechanism of their influence on solute–GB interactions. The first group involves mechanisms acting via deformation (or structural distortions) and the second one is related to the electronic (or chemical) interaction mechanisms. In fact, the two different types of solute–GB interactions mechanisms are closely interrelated and, therefore, are difficult to separate from each other. We fond that the energy gain due to atomic relaxation at GB determines the value of the segregation energy for Mg, which behaves as a large atom in the Al lattice, and therefore the size effect in solute–GB interaction plays the decisive role in this case. A Si atom gives an example of the opposite situation: lattice distortions and the energy of relaxation caused by a Si atom segregating to a GB in Al are very small, so the electronic interaction mechanism plays the dominant role in this case. Because Si has an extra valence electron compared to Al, it forms strong covalent bonds with Al when placed on a GB site having a reduced atomic coordi-

(b) 0.4

1

2 0.8

Mg concentration, at%

Mg concentration at center GB, at%

(a)

nation. As a result, the segregation energy of Si to GBs in Al is mainly determined by the charge transfer and electron density redistribution. Thus, for sp solutes that are acceptors of electrons from the matrix, and have a bigger atomic size, one can expect the deformation contribution to be dominant. At the same time, for solutes that are electron donors the chemical contribution to solute–GB interaction is more important. Anti-segregation behavior of Ti solute is an example when both structural and chemical mechanisms make similarly important contributions to the solute–GB interaction. In this case the segregation energy DEs is high and positive without lattice relaxation. At the same time Ti atom has large negative solution energy in fcc Al due to the hybridization between Al 3p–Ti 3d electron states and the formation of strong covalent-like Ti–Al bonds. It is means that main contribution to Ti–GB interaction is due to breaking the favorable nearest-neighbor coordination when the Ti atom is moved to the GB core. This conclusion is supported by the calculated depletion of charge density at the GB near the Ti atoms, which was found to be qualitatively similar to the electron density depletion obtained for Mg. A similar behavior is expected for segregation of the other 3d transition elements (Ti to Ni) to GBs in Al. The cases of Cu or Zn impurities, whose 3d shells are filled and the strain caused by them in the Al lattice is small, require special consideration. It should be noted that there is a certain similarity between segregation of solutes to a GB and to a free surface. The latter phenomenon is much better understood, and, within its simplified model one expects that an alloy component which has a lower surface energy than the host will segregate toward the surface. Because this model ignores other significant contributions to the GB segregation energy [67], we find it surprising that our calculations predict the same segregation behavior of Mg and Ti solutes at GBs as their expected behavior at a free surface of Al (the surface energy is lower for Mg and higher for Ti in comparison with Al). As essential structural parameters, namely the coordination numbers, bond lengths, and bond angles at free surfaces are very different from those at typical GBs, so we believe that the mechanisms making the dominant contributions to the segregation energies of Mg and Ti to free surfaces of Al are different from the mechanisms of the solute–GB interactions. The interaction of solute atoms with symmetric ST GBs is calculated to be very short-ranged and to decay at a distance of the order of one lattice parameter (see Fig. 2). Such ordered GBs are typical of well-annealed materials with large grains. However, in

1 0.6

0.4

4 0.2

0.3

2

0.2

1

0.1

3 0 400

500

600

700

T, K

800

900

0

-1.5

-1

-0.5

0

0.5

1

1.5

2

distance from GB, nm

Fig. 7. Concentration of Mg at the center of the asymmetric R5 GB as a function of temperature (a), calculated using Eq. (1) (curves 1, 2) and Eq. (2) (curves 3, 4) for Al–Mg alloys with the bulk Mg concentration of 1 at.% (curves 1, 3) and 5 at.% (curves 2, 4). Distribution of Mg near the R5 GB at T = 500 K (b) determined by Eq. (2) for the bulk Mg concentration of 1 at.% (curve 1) and 5 at.% (curve 2).

26

L.E. Karkina et al. / Computational Materials Science 112 (2016) 18–26

fine-grained materials the GB structure is considerably more disordered, it contains a high density of structural defects creating sizeable local stresses. Since a GB is a region with reduced atomic density (possessing excess volume), for most lattice sites in the GB core the nearest-neighbor coordination number is reduced and the bond lengths are increased (see Fig. 5c). As a result, the substitution of an Al atom by Ti at the GB leads to an increase in energy, while the substitution of Al by Mg is energetically favorable. Thus, for asymmetric ST and GT GBs the region of strong solute–GB interactions is considerably wider than for ST symmetric GBs; the interaction width LGB can reach 2 nm. For the nonequilibrium GBs that are formed under severe plastic deformation, and characterized by strong disorder, much higher values of LGB can be expected. The observed unusually wide GB segregation profiles in Al after severe plastic deformation [20–23] may indicate the formation of broad regions of structural disorder at the GBs in this material. The calculated segregation energies give the basis for qualitative prediction of the GB enrichment by the solute elements. However, for quantitative estimates of the segregation effect the interactions among the segregated solute atoms must be taken into account. In the considered case of Mg dissolved in Al, the predicted GB enrichment was found to be close to experimental data [23,66]. One possible reason for the remaining differences between theory and experiment may be attributed to the competition between Mg and other alloying elements segregating to the same GBs. Also, in the nano-crystalline alloys studied in Refs. [23,66] a significant fraction of atoms is situated at GBs and, therefore, the solute concentration inside the grains should be affected by the GB segregation. This feature has not been taken into account in the present calculation and requires further consideration. Acknowledgements The authors acknowledge financial support from the Russian Science Foundation (Grant 14-12-00673). The results of the work have been obtained using the computational resources of MCC National Research Center ‘‘Kurchatov Institute” (http://computing.kiae.ru) and using Uran supercomputer of IMM Ural Branch of the Russian Academy of Sciences. Fruitful discussions with Prof. R.Z. Valiev are acknowledged. References [1] P. Lejcˇek, Grain Boundary Segregation in Metals, Springer Series in Materials Science, New York, 2010. [2] X. Sauvage, G. Wilde, S.V. Divinski, Z. Horita, R.Z. Valiev, Mater. Sci. Eng. A 540 (2012) 1–12. [3] A.Y. Yermakov, Mater. Sci. Forum 455 (1995) 179–181. [4] P.P. Chatterjee, S.K. Pabi, I. Manna, J. Appl. Phys. 86 (1999) 5912–5914. [5] J. Weissmuller, Nanostruct. Mater. 3 (1993) 261–272. [6] R. Kirchheim, Acta Mater. 50 (2002) 413–419. [7] H.Q. Li, F. Ebrahimi, Acta Mater. 51 (2003) 3905–3913. [8] J.R. Trelewicz, C.A. Schuh, Phys. Rev. B 79 (2009) 094112. [9] T. Chookajorn, H.A. Murdoch, C.A. Schuh, Science 337 (2012) 951–954. [10] A.Y. Lozovoi, A.T. Paxton, M.W. Finnis, Phys. Rev. B 74 (2006) 155416. [11] E. Wachowicz, T. Ossowski, A. Kiejna, Phys. Rev. B 81 (2010) 094104. [12] B.W. Krakauer, D.N. Seidman, Acta Mater. 46 (1998) 6145–6161. [13] A.A. Salem, T.G. Langdon, T.R. McNelley, S.R. Kalidindi, S.L. Semiatin, Metall. Mater. Trans. A 37 (2006) 2879–2891. [14] J.S. Braithwaite, P. Rez, Acta Mater. 53 (2005) 2715–2726.

[15] S. Zhang, O.Y. Kontsevoi, A.J. Freeman, G.B. Olson, Phys. Rev. B 82 (2010) 224107. [16] D. Scheiber, V.I. Razumovskiy, P. Puschnig, R. Pippanc, L. Romaner, Acta Mater. 88 (2015) 180–189. [17] M.K. Miller, A. Cerezo, M.G. Hetherlington, G.D.W. Smith, Atom Probe Field Ion Microscopy, Oxford Science, Oxford, 1996. [18] A. Cerezo, P.H. Clifton, S. Lozano-Perez, P. Panayi, G. Sha, G.D.W. Smith, Microsc. Microanal. 13 (2007) 408–417. [19] D. Blavette, E. Cadel, C. Pereige, B. Deconihout, P. Caron, Microsc. Microanal. 13 (2007) 464–483. [20] G. Nurislamova, X. Sauvage, M. Murashkin, R. Islamgaliev, R. Valiev, Philos. Mag. Lett. 88 (2008) 459–466. [21] G. Sha, R.K.W. Marceau, X. Gao, B.C. Muddle, S.P. Ringer, Acta Mater. 59 (2011) 1659–1670. [22] G. Sha, L. Yao, X. Liao, S.P. Ringer, Z.C. Duan, T.G. Langdon, Ultramicroscopy 111 (2011) 500–505. [23] X. Savage, A. Ganeev, Y. Ivanisenko, N. Enikeev, M. Murashkin, R. Valiev, Adv. Eng. Mater. 14 (2012) 968. [24] D. McLean, Grain Boundaries in Metals, Oxford University Press, London, 1957. [25] The Collected Works of J.W. Gibbs, vol. 1, Thermodynamics, Green and Co., New York, London, Toronto, 1928. [26] P. Guyot, J.-P. Simon, J. Phys. Colloq. 36 (C4) (1975) 141–149. [27] M.P. Seah, Acta Metall. 28 (1980) 955. [28] A.P. Sutton, V. Vitek, Acta Metall. 30 (1982) 2011. [29] R. Messmer, C.L. Briant, Acta Metall. 30 (1982) 457. [30] R. Wu, A.J. Freeman, G.B. Olson, Science 265 (1994) 376–380. [31] W.T. Geng, A.J. Freeman, R. Wu, G.B. Olson, Phys. Rev. B 62 (2000) 6208–6215. [32] J.C.M. Li, Surf. Sci. 31 (1972) 12–26. [33] A.A. Nazarov, O.A. Shenderova, D.W. Brenner, Mater. Sci. Eng. A 281 (2000) 148–155. [34] G. Treglia, B. Legrand, F. Ducastelle, A. Saul, C. Gallis, I. Meunier, C. Mottet, A. Senhaji, Comput. Mater. Sci. 15 (1999) 196–235. [35] R. Schweinfest, A.T. Paxton, M.W. Finnis, Nature 432 (2004) 1008–1011. [36] M. Yamaguchi, M. Shiga, H. Kaburaki, Science 21 (2005) 393–397. [37] L. Zhang, X. Shu, S. Jin, Y. Zhang, G.H. Lu, J. Phys.: Condens. Matter 22 (2010) 375401. [38] X. Liu, X. Wang, J. Wang, H.J. Zhang, J. Phys.: Condens. Matter 17 (2005) 4301– 4308. [39] M. Christensen, T.M. Angeliu, J.D. Ballard, J. Vollmer, R. Najafabadi, E. Wimmer, J. Nucl. Mater. 404 (2010) 121–127. [40] M. Yuasa, M.J. Mabuchi, J. Phys.: Condens. Matter 22 (2010) 505705. [41] E. Wachowicz, A. Kiejna, Comput. Mater. Sci. 43 (2008) 736–743. [42] A.P. Sutton, V. Vitek, Acta Metall. 30 (1982) 2011–2033. [43] S.M. Foiles, Phys. Rev. B 32 (1985) 7685–7693. [44] A. Seki, D.N. Seidman, Y. Oh, S.M. Foiles, Acta Metall. Mater. 39 (1991) 3179– 3185. [45] D. Udler, D.N. Seidman, Interface Sci. 3 (1995) 41–73. [46] J. Rittner, D.N. Seidman, Acta Mater. 45 (1997) 3191–3202. [47] J. Creuze, F. Berthier, R. Tetot, B. Legrand, Phys. Rev. Lett. 86 (2001) 5737. [48] X.-Y. Liu, J.B. Adams, Acta Mater. 46 (1998) 3467–3476. [49] N.R. Rhodes, M.A. Tschopp, K.N. Solanki, Modell. Simul. Mater. Sci. Eng. 21 (2013) 035009. [50] P.C. Millett, R.P. Selvam, A. Saxena, Acta Mater. 54 (2006) 297–303. [51] G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169–11186. [52] G. Kresse, J. Hafner, J. Phys.: Condens. Matter 6 (1994) 8245–8258. [53] G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758–1775. [54] J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Phys. Rev. B 46 (1992) 6671–6687. [55] S.H. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 58 (1980) 1200–1211. [56] M. Yamaguchi, M. Shiga, H. Kaburaki, J. Phys.: Condens. Matter 16 (2004) 3933–3956. [57] H.B. Lee, F.B. Prinz, W. Cai, Acta Mater. 61 (2013) 3872–3887. [58] S. Znamy, D. Nguyen-Manhz, D.G. Pettifor, V. Vitek, Phil. Mag. 83 (2003) 415– 438. [59] R.R. Zope, Y. Mishin, Phys. Rev. B 68 (2003) 024102. [60] M.I. Mendelev, M. Asta, M.J. Rahman, J.J. Hoyt, Phil. Mag. 89 (2009) 3269–3285. [61] M.A. Tschopp, D.L. McDowell, Phil. Mag. 87 (2007) 3871–3892. [62] Z.T. Trautt, Y. Mishin, Acta Mater. 60 (2012) 2407–2424. [63] H. Gleiter, B. Chalmers, High-angle Grain Boundaries, Prog. Mater. Sci., vol. 16, Pergamon Press, Oxford, New York, Toronto, Sydney, Braunschweig, 1972. [64] K.K. Shin, J.C.M. Li, Surf. Sci. 50 (1975) 109–124. [65] R.Z. Valiev, T.G. Langdon, Prog. Mater Sci. 51 (2006) 881–981. [66] X. Savage, N. Enikeev, R. Valiev, Y. Nasedkin, M. Murashkin, Acta Mater. 72 (2014) 125–136. [67] A.V. Ruban, H.L. Skriver, J.K. Norskov, Phys. Rev. B 59 (1999) 15990–16000.