VN system

Computational Materials Science 50 (2010) 550–559

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Theoretical investigation of moderate misfit and interface energetics in the Fe/VN system Dan H.R. Fors ⇑, Sven A.E. Johansson, Martin V.G. Petisme, Göran Wahnström Department of Applied Physics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden

a r t i c l e

i n f o

Article history: Received 4 August 2010 Accepted 20 September 2010 Available online 13 October 2010 Keywords: Peierls–Nabarro model Dislocation Misfit Interface energetics Electronic structure Density functional theory

a b s t r a c t In this study an ab initio based approach to determine the effect of moderate misfit on energies and structures for interfaces is presented and applied to the Fe(0 0 1)/VN(0 0 1) system. The interface energetics of the coherent and semicoherent structures of thin VN films in Fe is investigated in order to determine how the misfit is taken up. The coherent interface is directly treated with ab initio calculations, whereas the semicoherent interface energy is accessed by using a Peierls–Nabarro framework, in which ab initio data for chemical interactions across the interface is combined with a continuum description to account for the elastic distortions. The continuum treatment is here extended to thoroughly account for the anisotropy in the materials. Our approach shows that the elastic contribution to the total interface energy dominates for both the coherent and semicoherent structure and must therefore be accurately accounted for in the interface description. Further, the Peierls–Nabarro framework for the semicoherent interface is evaluated by comparing a full scale two-dimensional solution to one-dimensional approximations. We show that the one-dimensional treatment is sufficient in the present case for accurate results, and consequently interactions at dislocation intersections at the interface do not have to be considered. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction The precipitation of secondary phases can have a decisive role for the mechanical properties of a material. Depending on the chemical composition of the precipitates, their coherency with the surrounding lattice, their shape, and their size distribution, the specific impact of the precipitation process ranges from strength enhancement by grain boundary pinning and dispersion strengthening to adverse effects such as the possibility of providing nucleation sites for crack propagation. Consequently, a fundamental understanding of the stability and time evolution of precipitates is important for accurate predictions of the mechanical behavior of the material. In the context of precipitate evolution, the lattice/precipitate interface energetics has a very pronounced role, as it strongly influences the nucleation, growth, and coarsening rates of the precipitates. However, it is difficult to acquire knowledge about interface energetics through experimental measurements. Theoretical tools, in particular ab initio calculations, provide an alternative and powerful approach to access the interface structure and energetics. Ab initio simulations are however restricted to quite small computational unit cells due to large computational costs. For real interfaces any mismatch between the structures of the ⇑ Corresponding author. E-mail address: [email protected] (D.H.R. Fors). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.09.018

two phases will in general give rise to long-ranged stress fields surrounding the interface, and the corresponding computational unit cells will simply become too large for a direct approach with ab initio calculations. The way around this problem is to apply alternative methods that can bridge the different length scales and maintain an accurate description of the chemical interactions across the interface while simultaneously taking into account extended elastic distortions, all at reasonable computational costs. An example of such a theoretical approach was explored in Ref. [1], where the elastic and chemical contributions were accessed from unit cells of moderate sizes. However, the formulation of the interface energy fell short on providing a self-consistent description of the atomic structure and the interactions across the interface. In addition, the accuracy of the invoked approximations used in the technique was hard to estimate without extensive investigation. An alternative theoretical method where the self-consistent description is included has been realized within a Peierls–Nabarro (PN) framework [2–8]. In the model, interface energetics is addressed by combining atomistic ab initio calculations for the chemical interactions with isotropic continuum theory to describe the elasticity of the materials. The framework has been applied to the metal–oxide Al/MgO [8,9], the metal–nitride Fe/VN [10] and the metal–metal NiAl/Mo [11] interfaces and has shown potential for a proper interface description. One of the most significant and widely used group of metallic alloys is steels [12]. They are alloys of the element iron combined

D.H.R. Fors et al. / Computational Materials Science 50 (2010) 550–559

with carbon, nitrogen and usually other elements such as Ti, V, Cr, and Nb. In sufficient concentrations the alloying elements lead to the formation of carbides and nitrides inside grains and at grain boundaries [12]. The interface structure of the precipitates depends on the size and chemical composition, and ranges from very small coherent VN [13] and NbN [14] platelets to larger semicoherent VN [13,15,16], TiC [17] and Nb(C,N) [18] precipitates, where misfit dislocations are present at the interface. The VN precipitates are here of special technological interest, as they tend to form a fine distribution in the Fe lattice, which effectively leads to dispersion strengthening. The Fe/VN interface energetics is thus an important area for investigation. In order to model the Fe/VN interface, detailed knowledge about the geometry must be acquired. The Fe/VN interface orientation relationship is experimentally well established. The VN precipitates nucleate as platelets in a-Fe due to a small (2%) relative misfit parallel to the platelet and an appreciable (44%) misfit perpendicular to the platelet [16]. With respect to the matrix, the orientation relationship for the flat side is well-defined according to the Baker– Nutting relation ð0 0 1Þnacl k ð0 0 1Þbcc , ½1 0 0nacl k ½1 1 0bcc . The misfit is for small precipitates taken up by coherency strains, while for larger particles a semicoherent interface is formed where the defect structure consists of a square dislocation network with Burgers vectors b k ½1 0 0bcc and [0 1 0]bcc [13,15,16]. Previous studies [10,19] on the corresponding Fe/VN interface have shown that the inclusion of the misfit can have a significant impact on the interface energy and must be accounted for in a proper way. In the present paper we extend our previous investigation [10] regarding the effect of moderate misfit on the Fe/VN interface. We here consider the coherent and semicoherent interface energetics of VN films in Fe at different film thicknesses in order to determine how the misfit is taken up. The coherent interface is treated directly with ab initio calculations, while the semicoherent interface energy is accessed by using the PN framework. For the latter we give a thorough account on how to include the anisotropy of the materials in the elastic description. Finally, the ability for the PN framework to describe the semicoherent interface is evaluated by comparing a full scale 2D solution with 1D approximations. The paper is organized as follows: The next section introduces the basic concepts for the coherent and semicoherent interfaces. Section 3 presents the computational techniques for the atomistic and continuum calculations followed by a description of the obtained bulk properties (Section 4). The interface models and corresponding results for the coherent and semicoherent interfaces are summarized in Section 5 and Sections 6–8 respectively. In Section 9 we then discuss and estimate the effect of various approximations and Section 10 summarizes our conclusions. Finally, the appendix presents A: the analytical solution to the semicoherent interface model under the assumption of a sinusoidal restoring force stress at the interface and B: details on how to calculate the energy coefficients used within the anisotropic elasticity theory.

að1Þ  að2Þ ; ðað1Þ þ að2Þ Þ 2

f ¼1

551

ð1Þ

where we have used a symmetric definition with respect to the two different phases. We decompose the total interface energy Etot into two terms

Etot ¼ Echem þ Eel ;

ð2Þ

where Echem describes the chemical energy originating from the breaking and creation of bonds in forming the interface and Eel the elastic energy required to create the interface. Moderate misfit (0 < f [ 0.2) can primarily be taken up in two different ways: by coherency strains or by creation of a set of misfit dislocations. A coherent interface can be achieved by compressing (expanding) the slab to match the lattice parameters in the interface plane of the surrounding crystals. The elastic energy cost for these coherency strains will depend on the size of the misfit and increase linearly with slab thickness, Eel / hf2. The chemical interface energy will then correspond to the energy for a perfectly coherent interface Echem = Ecoh (cf. Fig. 1). For thicker films it will be energetically favorable to instead relieve the strain fields by creating a semicoherent interface. This strain relief will occur at some critical thickness hc, where the energy cost to maintain the coherent interface becomes too large compared with the semicoherent interface. In that case the strains are instead taken up periodically by a square network of misfit dis^ directions. The periodicity p of locations arranged along the ^ x and y the misfit dislocations is given by (cf. Fig. 2)

p¼

að1Þ að2Þ : að1Þ  að2Þ

ð3Þ

In the discrete case the periodicity is restricted to p = Pa(1) = (P + 1)a(2), where P is an integer. We will assume p to be given by Eq. (3). The elastic energy is now given by the energy cost to create the network of misfit dislocations. Close to the dislocation cores the atomic structure will be highly distorted which reduces the strength of the bonds across the interface as compared to the coherent case. The chemical energy will thus be larger than in the coherent case, Echem > Ecoh. The optimal atomic configuration and the corresponding energy Esemicoh (cf. Fig. 1) can be obtained by minimizing Etot = Echem + Eel with respect to the displacement fields within a Peierls–Nabarro framework.

2. The interface model In this paper we focus on thin plate-like precipitates which are defined by two interfaces and a thickness h of the slab. The precipitates are assumed to be infinite in the parallel direction and hence, effects such as mismatch at the sides of a real finite precipitate are neglected. We consider the planar interface between two phases, (1) and (2), with the same (cubic) structure but with different lattice spacings, a(1) and a(2). The direction perpendicular to the interface is denoted with ^z and we assume a(1) > a(2). The difference in lattice spacings introduces a misfit f according to

Fig. 1. A schematic illustration of the interface energy as a function of the slab thickness. The circles denote the interface energy in an atomistic treatment, whereas the dashed-dotted (dashed) line represents a continuous model for the interface energy of a coherent (semicoherent) interface. The thickness hc denotes the transition point between the coherent and semicoherent interface.

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ratios m are calculated using the Voigt averages [31], which for cubic crystals give l ¼ 15 c11  15 c12 þ 35 c44 and the Lamé constant k ¼ 15 c11 þ 45 c12  25 c44 . From these constants follow m ¼ 2ðlkþkÞ.

Fig. 2. Schematic figure of the semicoherent structure at the interface along the ^ x direction. The misfit dislocations occur with the periodicity p. Black (gray) circles denote phase 1 (2).

For large misfit the partial lattice matching at the interface will be lost and the interface becomes incoherent with an energy Eincoh (cf. Fig. 1). 3. Computational techniques The atomistic calculations are performed within the framework of density functional theory (DFT) as implemented in the Vienna ab initio simulation package (VASP) [20–22]. Blöchl’s projector-augumented-wave formalism as implemented by Kresse and Joubert [23] is employed for the electron–ion interaction, where the standard Fe, V, and N potentials with eight, five, and five valence electrons, respectively, are used. For Fe it is well-known that spin-polarized generalized gradient approximations for the exchange-correlation functional are needed to give an accurate description of its structural, energetic, and magnetic properties [24]. The exchange-correlation functional is therefore here treated with the spin-polarized generalized gradient approximation according to Perdew and Wang [25]. The k-point integration is performed by using a Monkhorst–Pack grid and the first order Methfessel–Paxton smearing scheme with a smearing width of 0.1 eV. The plane-wave cutoff energy is set to at least 400 eV in all calculations to ensure that the total energies are converged within a few meV/atom. The continuum calculations are performed using the Finite Element Method (FEM) as implemented in the COMSOL Multiphysics simulation environment [26]. We use the Structural Mechanics Module which solves the Navier–Cauchy elastostatic equation

cijkm uk;mj þ fi ¼ 0;

ð4Þ

where cijkm are the elastic constants, uk,mj the derivatives of the displacement fields, and fi the volume forces. (The standard notation with summation over repeated indices is being used.) The Structural Mechanics Module has support for general geometries as well as boundary conditions and it handles anisotropic materials, taking the full elasticity matrices into account. 4. Bulk and elastic properties In the subsequent sections, the elastic constants cijkm are written in a contracted matrix notation [31]. For cubic crystal symmetry we have ciiii = c11, ciijj = c12 and cijij = c44 for i – j. All other constants are zero. The lattice parameters d, the bulk modulus B, and the elastic constants c11, c12 and c44 for Fe and VN are determined by following the procedure of Mehl et al. [33]: (i) an isotropic strain gives B = (c11 + 2c12)/3 from a Birch fit, (ii) a volume conserving orthorhombic strain gives c11  c12 from a polynomial fit of the energy to the strain, and (iii) a volume conserving monoclinic strain gives c44 from a polynomial fit. The isotropic shear moduli l and Poisson

In Table 1 we present the bulk properties as calculated with DFT together with experimental data for the two materials. The lattice parameters in the interface plane for p the ffiffiffi Baker–Nutting orientation VN relationship, aFe = dFe and aVN ¼ d = 2, are also given. The size of the Monkhorst–Pack grid was in each system set to converge the elastic constants to within a few GPa. The VN phase shows good agreement (0.3%) between calculation and experiment for the lattice parameter, while for Fe there is a deviation of 0.04 Å (1.2%). The bulk modulus is in agreement (2%) for Fe, while there is a rather large discrepancy (19%) for VN. Presumably, the underlying reason for the deviation in the latter is the uncertainty in the stoichiometric growth conditions of the VN phase in Ref. [32], because nitrogen vacancies are indisputably of importance for the bond strength. In addition, measurements performed at ambient temperature, in contrast to the DFT calculations which are performed at 0 K, tend to display a softer material due to thermal expansion. Nevertheless, our results for the bulk modulus and elastic constants clearly show that the nitride phase is substantially stiffer than the iron phase. 5. Coherently strained precipitate The coherent interface is created by compressing the VN slab to match the lattice parameter in the interface plane of the surrounding Fe lattice. The strain parallel to the interface plane is given by

VN k ¼

aFe  aVN ; aVN

ð5Þ

and the accompanying induced strain perpendicular to the interface (Poisson effect) is from linear elasticity theory given as

VN ? ¼ 2

cVN 12 VN k : cVN 11

ð6Þ

By using the lattice parameters acquired from the DFT calculations VN we find VN k ¼ 2:94% and ? ¼ 1:66% respectively. The magnitudes of these strains reside within the valid regime of linear elasticity theory ([3%) and therefore justifies the usage of the elastic constants to setup the coherent interface. 5.1. Atomistic model In the DFT calculations, the coherent interface is represented by using a computational unit cell containing a standard slab geometry with nFe Fe layers on top of nVN VN layers. The cell is subject to periodic boundary conditions in all three directions. Each VN layer contains one V atom and one N atom while one Fe layer corresponds to one Fe atom. Table 1 Bulk properties from the DFT calculations and experiments. The lattice parameters are given in Å, the elastic constants in GPa and the magnetic moments M in lB/ (formula unit). The experimental values correspond to room temperature except for the Fe magnetic moment which is given at 0 K.

dFe, dVN aFe, aVN M c11 c12 c44 B

l m

Fe (DFT)

Fe (Exp.)

VN (DFT)

VN (Exp.)

2.831 2.831 2.20 258 133 94 174 81 0.298

2.866 [27] 2.866 2.22 [29] 242 [31] 146.5 [31] 112 [31] 178 86 0.292

4.125 2.917 0.0 633 179 135 330 171 0.279

4.136 [28] 2.925 0.0 [30] 533 [32] 135 [32] 133 [32] 268 159 0.252

D.H.R. Fors et al. / Computational Materials Science 50 (2010) 550–559

At T = 0 K the total interface energy can be evaluated according to

Etot ¼ minzsep

EFe=VN ðzsep Þ  2A

P

i

li N i

;

ð7Þ

where A is the area, Ni the number of atoms for the constituent i and EFe/VN(zsep) is the internal energy for the coherent interface system at the interphase separation zsep. The chemical potentials li are evaluated through separate calculations for the unstrained bulk phases. For each interface geometry the interphase separation zsep is minimized under the constraint that all mutual atomic distances in each respective phase are fixed. The most energetically favorable interface configuration occurs when the Fe atoms are aligned with the N atoms, where a strong covalent N(2p)–Fe(3d) bond can be created together with overlapping metallic V(3d)–Fe(3d) bonds [10,19]. Convergence tests for this geometry with respect to nFe show that nFe = 5 is sufficient to reduce the interface energy error from artificial interface–interface interactions through the Fe phase, originating from the finite size of the computational unit cell, to less than 0.03 J/m2. Based on these convergence results, the interface energy calculations are performed by using nFe = 5 (6) in combination with nVN equal to an odd (even) number of layers. The equal parity for nFe and nVN is here required in order to produce two identical interfaces within the computational unit cell. The corresponding Brillouin zones are sampled with a regular Monkhorst–Pack grid of at least 10 irreducible k-points. In Fig. 3 the interface energy from the above atomistic model (AM) is presented for nVN ranging from one to eight layers. The number of VN layers is translated into a slab thickness h according to

553

thick VN slab. The elastic contribution to the interface energy can within linear elasticity theory be evaluated as

Eel ¼ 

 VN 2 ! c12 V  VN 2 VN k c11 þ cVN  2 12 2A cVN 11 h eel ; 2

ð9Þ

where V = Ah is the volume of the VN phase and eel = 0.614 GPa is the elastic energy per volume unit required to strain the VN slab. The factor 2 in the denominator is included to account for the presence of two interfaces in the computational unit cell. Further, for a VN slab with sufficient thickness no interface–interface interaction occurs through the VN phase. The chemical contribution to the interface energy is therefore independent of h and assumes a constant value Echem ¼ Elim chem . Based on the behavior for the elastic and chemical contributions, a continuous model for the interface energy can now be constructed according to

h Etot ¼ Elim chem þ eel ; 2

ð10Þ

where zsep is the distance between the Fe and the N atom at the interface.

which is asymptotically valid for large h. For thinner VN slabs the chemical bonds across the interface develop a dependence of h and the interface energy will then deviate from the linear model. In Fig. 3 the continuous model is compared to the DFT calculations. For the former we take Elim chem as the average of the chemical energy contributions obtained from the nVN = 7 and nVN = 8 DFT 2 simulations, yielding Elim chem ¼ 0:025 J/m . The continuous model fits well to the DFT interface energies, where any significant deviation occurs only for nVN = 1. The deviation arises due to the inability for one N atom to simultaneously create two full strength N(2p)–Fe(3d) bonds (one bond at each of the two interfaces). For nVN > 1 the DFT interface energies have only small oscillations around the continuous model. The oscillations decay gradually with increasing slab thickness and are expected to disappear completely if additional VN layers were to be used.

5.2. Continuous model

6. Semicoherent interface – 2D numerical treatment

In addition to the DFT calculations, a simple continuous model (CM) can be constructed for the interface energy by considering a

The semicoherent interface is treated within an extended Peierls–Nabarro framework, where the interface energy is decomposed into a chemical and an elastic term (cf. Eq. (2)). The chemical interactions across the interface are approximated by an effective two-dimensional potential energy surface, referred to as the c-surface, evaluated through DFT calculations. The elastic distortions in the materials are accounted for by using a continuum description where the boundary conditions are derived using the c-surface.

VN 

h¼

d

 1 þ VN ? ðnVN  1Þ þ zsep ; 2

ð8Þ

6.1. The interface and the c-surface

Fig. 3. The total interface energy for the coherent ( ) and the semicoherent (dashed line) systems as a function of the slab thickness of the VN phase. The dashed-dotted line corresponds to the continuous model for the coherent interface.

The c-surface is determined through evaluation of the interface energy within a coherent interface approximation for different relative translations s of the two phases. The coherent interface sys^ tem is constructed by straining the two phases in both the ^ x and y directions into a common lattice parameter a in the interface plane. The parameter a is determined from linear elasticity theory and is chosen to minimize the sum of the elastic energy per volume of the two strained bulk systems. The expressions for the strains in each bulk system will depend on their respective symmetry and orientation with respect to the interface. For two phases, (1) and (2), with cubic structure and subjected to the Baker–Nutting orientation relationship, it is straightforward to derive that the total elastic energy per volume of the two strained bulk phases is minimized when the parallel strain in phase (1) is

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D.H.R. Fors et al. / Computational Materials Science 50 (2010) 550–559

ð1Þ k ¼

  að2Þ  að1Þ ð1Þ  ð2Þ ð2Þ ð2Þ ð2Þ c11 c11  c12 c11 þ 2c12 ð1Þ n a   ð1Þ

ð2Þ

ð2Þ

ð2Þ

B

ð11Þ

The derivation here assumes equal heights of the two unstrained bulk phases. Accordingly, the common lattice parameter is given   ð1Þ as a ¼ 1 þ k að1Þ and the associated Poisson effect is given by the relation in Eq. (6). The strains in phase (2) follow by replacing (1) ? (2), (2) ? (1). The above approach is expected to give the best description of the chemical environment at the interface, especially in the near coherent regions where the two phases will be close to satisfying the assumed strain condition (cf. Fig. 2). The computational unit cell for the coherent Fe/VN interface is set up by using a standard slab geometry as discussed in Section 5. The corresponding common lattice parameter and strains are given in Table 2. The former is, as expected from the elastic constants, closer to the more rigid VN phase and the largest strains thus occur in the Fe phase. Convergence tests with respect to the number of Fe and VN layers yield a relative error in the interface energy identical to the tests performed in Section 5 (0.03 J/m2). Based on these tests we therefore use a setup with nFe = 5 and nVN = 5 for the interface energy calculations. The c-surface is constructed by using six relative translation vectors s (A–F), cf. Fig. 4. For each relative translation the interface energy is in accordance with Eq. (7) evaluated as

EFe=VN ðs; zsep Þ  2A

C

E

D

ð2Þ

c11 c11  c12 c11 þ 2c12   o1 ð2Þ ð1Þ ð1Þ ð1Þ ð1Þ : þc11 c11  c12 c11 þ 2c12

cðsÞ ¼ minzsep

F

P

i

li N i

ð12Þ

:

However, the chemical potentials are here obtained from separate calculations for the strained bulk phases (instead of the unstrained bulk phases used in Section 5) and hence only the chemical contribution to the interface energy is included. A continuous potential energy surface c(x, y) is then created through interpolation of the six acquired minima by using a superposition of cosine functions, cos 2ap ðmx þ nyÞ (m; n 2 Z), obeying the symmetry of the interface. The results for the sample points (A–F) of the c-surface can be found in Table 3 and the corresponding interpolation is plotted in Fig. 4. We define the energy Ecoh to be the minimum of the c-surface, corresponding to the interface energy of a fully coherent interface (f = 0). We also define Eincoh to be the mean of the c-surface, which is a measure of the interface energy for an incoherent (in both ^x and ^ directions) structure. The interface energy of the semicoherent y interface will be bounded by Ecoh and Eincoh. The calculated local magnetic moments at the A, B, and C sites at the c-surface are presented in Table 4, where the Wigner–Seitz radii used for the site projections were set to rFe WS ¼ 1:30 Å, rVWS ¼ 1:32 Å and r NWS ¼ 0:74 Å. At site B, the magnetic moments in the first Fe layer are found to be depleted by around two tenths of lB, whereas the polarization in the second Fe layer is enhanced

Table 2 Properties of the semicoherent interface together with the strains used to obtain the c-surface. Corresponding data using experimental values for the lattice parameters and elastic constants are also given. Fe/VN (DFT)

Fe/VN (Exp.)

f (%) p (Å) a (Å) Fe k (%)

3.0 96.5 2.894 2.23

2.0 141.4 2.909 1.51

Fe ? (%) VN (%) k VN ? (%)

2.29

1.83

0.77

0.52

0.44

0.26

A

Fig. 4. The c-surface of the Fe/VN interface. Consecutive contour lines differ by 0.1 J/m2. The maximum of the c-surface lies in the middle (corresponding to Fe over V) and the minimum lies in the corner (corresponding to Fe over N).

Table 3 Values of the c-surface at the six sample points together with the mean value Eincoh. All values are in J/m2. Interface

A

B, Ecoh

C

D

E

F

Eincoh

Fe/VN

2.80

0.06

1.98

2.50

1.87

0.88

1.73

Table 4 Calculated local magnetic moments at the Fe/VN interface for the first two Fe and VN layers, together with the corresponding bulk values. Layer \Site

A (lB)

B (lB)

C (lB)

Fe(1) Fe(2) Fe(Bulk) V(1) V(2) V(Bulk) N(1) N(2) N(Bulk)

2.69 2.30 2.24 0.68 0.28 0.00 0.05 0.00 0.00

2.03 2.44 2.24 0.48 0.49 0.00 0.00 0.02 0.00

2.32 2.38 2.24 0.64 0.28 0.00 0.04 0.01 0.00

by the same amount. In contrast, the polarization of the first two Fe layers at site A and C are simultaneously increased. Furthermore, we find that the presence of the interface induces an antiferromagnetic spin structure of the V atoms throughout the VN slab at all sites, together with a very weak polarization of the N atoms. The magnitudes of the magnetic moments are consistent with previous investigations of the Fe(0 0 1)/VN(0 0 1) interface [19], the Fe(0 0 1)/ V(0 0 1) interface [34], and chemisorption of N on the Fe(0 0 1) surface [35]. However, it should be noted that for the Fe(0 0 1)/V(0 0 1) interface the induced magnetic moments on the V atoms decay rapidly to zero away from the interface instead of adopting an antiferromagnetic spin structure [34].

6.2. Elastic contribution Within the Peierls–Nabarro framework we write the total interface energy as ð1Þ

ð2Þ

Etot ¼ Echem ½U þ Eel ½uð1Þ  þ Eel ½uð2Þ ;

ð13Þ

where u(a)(x, y, z) (a = 1,2) is the displacement field in phase (1) and (2), respectively, and U(x, y) is the disregistry at the interface, a

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two-dimensional vector. In our case the two components of the disregistry can be expressed as

Eq. (14). After each iteration step n the disregistry is updated according to

a a ð2Þ þ x þ uð1Þ x ðx; y; z ¼ 0Þ  ux ðx; y; z ¼ 0Þ; 2 p a a ð2Þ U y ðx; yÞ ¼ þ y þ uð1Þ y ðx; y; z ¼ 0Þ  uy ðx; y; z ¼ 0Þ: 2 p

U nþ1 ¼ ð1  ÞU n þ U sol;n ;

U x ðx; yÞ ¼

ð14Þ

In order to treat a periodic square network of misfit dislocations, we set up the interface system as two blocks, stacked on top of each other, with the dimensions p  p  h(a)(a = 1, 2). On the sides of the blocks we impose Dirichlet boundary conditions for the displacement field in the perpendicular direction

ux ðp=2; y; zÞ ¼ ux ðp=2; y; zÞ ¼ 0; uy ðx; p=2; zÞ ¼ uy ðx; p=2; zÞ ¼ 0;

ð15Þ

and periodic boundary conditions in the two other directions

uy ðp=2; y; zÞ ¼ uy ðp=2; y; zÞ; uz ðp=2; y; zÞ ¼ uz ðp=2; y; zÞ; ux ðx; p=2; zÞ ¼ ux ðx; p=2; zÞ;

ð16Þ

uz ðx; p=2; zÞ ¼ uz ðx; p=2; zÞ: On the top and bottom of the blocks we impose periodic boundary conditions for the displacement field in the perpendicular direction ð1Þ

ð1Þ uð1Þ z ðx; y; 0Þ ¼ uz ðx; y; h Þ;

ð17Þ

and correspondingly for block No. (2). In the parallel directions we use a force obtained from the c-surface

@ cðU x ðx; yÞ; U y ðx; yÞÞ; @U x @ cðU x ðx; yÞ; U y ðx; yÞÞ; F y ðx; y; 0Þ ¼  @U y

F x ðx; y; 0Þ ¼ 

ð18Þ

as the boundary condition. The force is periodically repeated on the top and bottom of the blocks according to Fig. 5. The force depends on the disregistry (and, hence, the displacements in the parallel direction) and therefore the problem has to be solved in a self-consistent manner. The computations are initiated by assuming a reasonable initial form for the disregistry U0. The elastostatic equation (Eq. (4)) is solved for the two blocks using the COMSOL package and a new disregistry Usol,0 is obtained from the displacement field through

ð19Þ

and the solution is iterated until the total energy Etot = Eel + Echem has converged. The elastic energy Eel is during the iteration scheme obtained through integration over the resulting strain fields and the chemical energy is given by

Echem ½U x ðx; yÞ; U y ðx; yÞ ¼

1 p2

Z

p=2

p=2

Z

p=2

cðU x ðx; yÞ; U y ðx; yÞÞdx dy:

p=2

ð20Þ We have performed a full scale computation using two blocks stacked on top of each other with the size hFe = hVN = 2p. The size 2p has been confirmed to be sufficiently large to represent an interface between two semi-infinite crystals by comparing the numerical result in an one-dimensional case to the known analytical solution. The interface-interface interaction through the elastic fields is thus eliminated. The c-surface presented in Section 6.1 was used to describe the force at the interface. In order to obtain a self-consistent solution we had to iterate about 100 times where the parameter  was linearly varied between 2% and 5%. In Table 5 we give the values for the final converged chemical and elastic energies (denoted ‘‘2D-numerical”). The width of the dislocations varies from 3.33 Å far away from the dislocation intersections to 4.88 Å at the intersections. In Fig. 6 we show the sum of the elastic energy distributions for the two blocks as function of x and y and integrated over the z direction. The two-dimensional network of dislocations is clearly visible. 7. Semicoherent interface – 1D numerical treatment If the spacing between the dislocations is sufficiently large, the two-dimensional dislocation network can be approximated with two independent periodic arrays of edge dislocations. The problem is then reduced from 2D to 1D, and the interface system can be modeled by two ‘‘blocks” with dimension p  h(a)(a = 1, 2). In order to perform the calculations an 1D c-surface has to be constructed. This construction is carried out by choosing the cut in the 2D c-surface that gives the lowest mean energy when integrated along the corresponding line, i.e. the cut between the coherent B sites (the line BFC in Fig. 4). The resulting 1D c-surface is shown in Fig. 8. The problem can now be solved in the same iterative manner as described in Section 6, where the computational time is substantially reduced due to the reduction to a 1D problem. The elastic energy is obtained through integration of the displacement fields in the two ‘‘blocks” and the chemical energy is evaluated from Eq. (A.3). Finally, the interface energy for a square network of misfit dislocations is then approximated by scaling the obtained interface energy, for the periodic array of edge dislocations, according to Eq. (A.11). We have performed a calculation using two ‘‘blocks” with the size hFe = hVN = 2p. The obtained interface energy and dislocation width are presented in Table 5 (denoted ‘‘1D-numerical”). In addition, we have also generated the elastic energy in the two blocks as function of the lateral coordinate and superimposed two such periodic arrays. The corresponding elastic energy distribution is

Table 5 Results of the calculations of elastic, chemical, total energies, and dislocation widths of the interfaces. All energies in J/m2.

Fig. 5. A x–z plane view of the computational unit cell used in the FEM calculations.

Method

Eel

Echem

Etot

Width f (Å)

2D-numerical 1D-numerical 1D-analytical

0.338 0.336 0.349

0.140 0.131 0.133

0.478 0.467 0.482

3.33 3.10 3.07

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shown in Fig. 6. The full 2D c-surface is found to have an impact on the displacements mainly in the vicinity of the dislocation intersections, where the interaction between the dislocations gives rise to a noticeable larger dislocation width, see Fig. 6. Yet, the elastic energy still remains essentially identical to the 1D treatment with only minor redistribution between the two phases, see Fig. 7. Therefore, going from a 2D to a 1D c-surface has limited effect on the total interface energy, see Table 5, and the computationally much less expensive 1D case can be used. 7.1. Finite thickness

Fig. 6. The sum of the elastic energy distributions for the two blocks in the FEM calculations as function of x and y and integrated over the z direction. The top (lower) panel shows the result from the 2D (1D) numerical treatment.

Up until now only the semi-infinite Fe/VN interface has been considered. However, for a finite VN slab thickness the displacements fields from the misfit dislocations at the respective interface will start to interact with each other and consequently the interface energy can change. In order to investigate the applicability of the solution in the semi-infinite case to a finite VN slab, we set hFe = 2p and evaluate the interface energy with the 1D numerical approach for different values of hVN. The c-surface is kept fixed to the semi-infinite case, which by judging from Fig. 3 will be a valid approximation down to hVN = 10 Å. The difference in interface energy with respect to the semi-infinite case is visualized in Fig. 9. The overall change is found to be very small, being of the order 0.02 J/m2 at maximum, and we thus conclude that the semiinfinite solution constitutes a very good approximation even for a finite VN slab thickness. 8. Semicoherent interface – 1D analytical treatment

Fig. 7. The elastic energy integrated from a distance 0 to h from the interface. The solid (dashed) lines show the results from the 2D (1D) numerical treatment.

Fig. 8. The 1D c-surface from the 2D interpolation (solid line) and from the 1D sinusoidal interpolation (dashed line).

If the 1D c-surface is assumed to have a sinusoidal variation the problem can be solved analytically, see Appendix A. A comparison between the 1D c-surface and a cosine c-surface obtained by using the values of the c-surface at position B and C, shows that the sinusoidal form in the Fe/VN case constitutes a good approximation for the 1D c-surface, see Fig. 8. The semicoherent interface energy for the semi-infinite case is obtained through evaluation of the expres¼ 148 GPa, and subsesions Eqs. (A.7) and (A.8), with K e ¼ K aniso e quently scaled to 2D according to Eq. (A.11). The result is presented in Table 5 (denoted ‘‘1D-analytical”). By comparing the elastic and chemical contributions to the interface energy from the 1D analytical treatment with the 1D numerical approach, we conclude that the effect of going from a simple sinusoidal force

Fig. 9. The difference in the total interface energy (solid line) and the elastic energy contribution (dashed line) with respect to 1D numerical results for the semi-infinite case.

D.H.R. Fors et al. / Computational Materials Science 50 (2010) 550–559

to a more accurate force is small. Therefore, for the current Fe/VN interface it is sufficient to use the 1D analytical treatment in order to obtain accurate results. 8.1. Effect of anisotropy In order to investigate the effect from anisotropy we evaluate the isotropic energy coefficient according to Eq. (A.2), which yields K iso e ¼ 157 GPa. The corresponding interface energy is found to be Etot = 0.504 J/m2. The difference in the interface energy between the isotropic and anisotropic treatment is thus small and we therefore deduce, that for the current Fe/VN interface, the anisotropy has a minor influence and can be disregarded. However, for other orientation relationships and/or a different set of elastic constants the effect can be larger and the influence from anisotropy should be confirmed in each case. 9. Discussion It should be noted that our results in this study are based on DFT calculations. This method raises a question about accuracy due to the choice of an exchange-correlation functional. In particular, any error in the lattice parameters of the two materials will influence our results since the value of the interface energy is sensitive to the misfit. For the current Fe/VN interface the largest discrepancy arises between the calculated and the measured lattice parameter of Fe. In order to estimate the corresponding effect on the interface energy we use the 1D analytical approach and evaluate the interface energy for different aFe values while keeping aVN fixed at its calculated value, see Fig. 10. We find that the use of the experimental lattice parameter of Fe (aFe = 2.866 Å) in the model causes a large correction to the DFT results, where the Fe/VN interface energy is reduced by one third down to 0.32 J/m2. The misfit is therefore an important parameter that must be properly accounted for in order to obtain an accurate interface description. Furthermore, the interface energy has consistently been evaluated with a stoichiometric VN phase and with no atomic relaxation occurring in the two materials. In reality stoichiometric nacl-structured bulk VN is known to have dynamical instabilities at the X and K high symmetry points in the phonon spectrum [36,37], which consequently distort the nacl structure. Allowing for atomic relaxation in our Fe/VN interface system induces the same distortions in

Fig. 10. The interface energy as a function of misfit evaluated by the 1D analytical method. The circle (star) denotes the interface energy for a misfit resulting from the calculated (experimental) lattice parameter of Fe. In both cases, the calculated lattice parameter of the VN phase is used.

557

the VN slab as in the corresponding bulk system. The structural relaxation modifies the c-surface slightly, where the coherent value is decreased to Ecoh = 0.10 J/m2. The relaxation has only a minor effect on the magnetic moments as compared to the unrelaxed interface, where the changes are found to be less than 0.05lB/atom. Further, the distorted bulk VN phase leads to a reduction of the unstrained lattice parameter aVN by 1%, which consequently decreases the elastic contribution to the interface energy. The atomic relaxation thus affects both the coherent and the semicoherent interface energy; however, the same qualitative behavior as in Fig. 3 will still be obtained. Finally, it should be noted that the cubic nacl-structured VN phase is from experimental investigations [38,39] and theoretical studies [40] known to be stabilized by nitrogen vacancies (VN1x, x [ 0.2). This effect gives further support to our decision to not consider the atomic relaxation of a stoichiometric VN phase in more detail. The influence of nitrogen vacancies on the coherent Fe/VN interface energy has been investigated in Ref. [19]. The results show that the presence of vacancies in the VN phase will increase the chemical contribution to the interface energy by approximately 0.2 J/m2, although this high increase was expected to be compensated by a reduction in the elastic energy due to a smaller VN lattice parameter. The results in Ref. [19] alone do not allow us to estimate the influence on the elastic contribution to the coherent and semicoherent interface energies when vacancies are present. For this purpose a more detailed treatment including vacancies and structural relaxation must be performed. However, such a treatment is beyond the scope of the present work and has been postponed to future studies.

10. Conclusions In this work we develop a general approach to determine effects of moderate misfit on interface energetics. The approach is applied to the Fe/VN interface, motivated by the dispersion strengthening of VN precipitates in steel alloys. We consider the coherent and semicoherent interface structures of VN films in Fe by evaluating the interface energy for different film thicknesses. The films are considered to have an infinite two-dimensional extension and the impacts from the rim of the precipitate are thus omitted. The coherent interface structure is treated directly by using ab initio DFT calculations. We show that the chemical contribution to the interface energy is very small (25 mJ/m2), which implies a near negligible cost for the breaking and creation of bonds in the interface formation. Further, the elastic contribution, associated with the coherency strains, is found to increase linearly as a function of the film thickness with a small slope corresponding to 63 mJ/m2 per atomic layer. The low elastic cost arises as a direct consequence of the small misfit for the current interface. Our results for the coherent interface energy show that very thin coherent VN films (2–3 atomic layers) are stabilized in Fe, which is consistent with the experimental observations in Ref. [13]. For thicker films the elastic energy cost can be relieved by forming a semicoherent interface, where the strains are periodically taken up in a square network of misfit dislocations. The corresponding interface energy is here accessed by using an extended Peierls– Nabarro framework, in which DFT data for the chemical interactions across the interface are combined with a continuum description to account for the elastic distortions in the materials. The interface energy is found to be 0.48 J/m2, where the elastic contribution dominates and accounts for two-thirds of the formation cost. Further, by comparison of the coherent and semicoherent interface energy, the transition point between the two structures is found to occur when the thickness of the VN slab is hc = 16.5 Å or equivalently eight VN layers, cf. Fig. 3. This thickness is in line with the experimental

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results in Ref. [16], where a VN precipitate observed to have a misfit dislocation structure parallel to the platelet is approximately 15 Å thick. Furthermore, we compare a full scale two-dimensional solution of the Peierls–Nabarro model with a one-dimensional treatment. The comparison revealed only a minor difference in the interface energy (<0.01 J/m2) between the two cases, which shows that the interactions at dislocation intersections do not have to be considered. We can therefore conclude that the semicoherent interface energy can be evaluated by using the simplified one-dimensional Peierls–Nabarro model, where only knowledge about the elastic coefficients of the two materials and an one-dimensional potential energy surface to describe the chemical interactions across the interface are required as input parameters. In this study we also extend the Peierls–Nabarro model to account for the anisotropy of the materials. For the present Fe/VN interface system our investigations show that the effect of anisotropy is quite small as compared to the isotropic case, and can thus be disregarded. In summary, very thin VN films can be studied directly with DFT calculations. The results provide important information for the formation of thin precipitates in Fe. For thicker films the effect of the misfit dislocations can be treated with a simplified one-dimensional Peierls–Nabarro model. For both the coherent and semicoherent structures the interface energy is dominated by the elastic contribution and an accurate account for the elastic distortions is thus essential for a proper interface description.

Acknowledgments This work was supported by the Swedish Foundation for Strategic Research (SSF). The allocations of computer resources via C3SE and the Swedish National Infrastructure for Computing (SNIC) are gratefully acknowledged. The authors wish to thank Mikael Christensen for valuable comments.

ing Ke). For two isotropic media with shear moduli l(1) and l(2), and Poisson’s ratios m(1) and m(2), one can derive [42] (see Appendix B) ð1Þ ð2Þ K iso e ¼ 2l l  1 1 : þ lð1Þ þ lð2Þ ð3  4mð1Þ Þ lð2Þ þ lð1Þ ð3  4mð2Þ Þ

For two anisotropic media an explicit form for Ke expressed in the elastic constants of the two phases is generally not available. It can however be obtained numerically with knowledge of elastic constants and orientation of the crystals (see Appendix B). The chemical energy per unit area is given by an integral over the c-surface according to

Eedge chem ½UðxÞ ¼

Ke 4p p

Z

p=2

p=2

Z

p=2

p=2

Z

p=2

cðUðxÞÞdx;

ðA:3Þ

p=2

a a þ x þ uðxÞ; 2 p

UðxÞ ¼

ðA:4Þ

with the boundary conditions U(p/2) = 0 and U(p/2) = a. Finally, the total interface energy is obtained by minimizing edge edge Eedge tot ½uðxÞ ¼ Eel ½uðxÞ þ Echem ½uðxÞ;

ðA:5Þ

with respect to u(x). By taking the functional derivative of Eq. (A.5) with respect to u(x), the usual PN integro-differential equation is obtained. Generally, the functional (A.5) would have to be minimized numerically. However, if the c-surface has a sinusoidal variation, analytical solutions exist [6]. By setting

cðxÞ ¼

cmin þ cmax 2

þ

cmin  cmax 2

cos

2p x; a

ðA:6Þ

one finds

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi K e a2 ln 2b 1 þ b2  2b2 ; 4p p

ðA:7Þ

and

For a periodic array of misfit dislocations at a semicoherent interface, an analytical result can be derived assuming a sinusoidal c-surface. We consider an interface between two semi-infinite crystals, (1) and (2), where the elastic response of the two crystals is assumed to be described by linear elasticity theory with two different sets of elastic constants. The misfit dislocations are assumed ^ directo be of pure edge type with the dislocation line along the y tion. For an edge dislocation the displacements of the elastic media just above or below the interface, u(1)(x) and u(2)(x), will lie along the ^ x direction. The relative displacement parallel to the interface is denoted with u(x) = u(1)(x)  u(2)(x), while the relative displacement perpendicular to the interface is set to zero. The displacements are all functions of the periodicity p, and we impose the boundary condition u(p/2) = u(p/2) = 0. Assuming continuity of the stress field across the interface, the ¼ Eedge elastic energy per interface area is a functional Eedge el el ½uðxÞ of the relative displacement u(x), despite the difference in elastic constants. For a periodic array of edge dislocations we have

Eedge el ½uðxÞ ¼ 

1 p

where the disregistry U(x) across the interface is related to the relative displacement u(x) by

Eedge ¼ el Appendix A. Analytical solution

ðA:2Þ

   duðx0 Þ duðxÞ 0  p ln sin ðx  x0 Þ dx dx; dx0 dx p ðA:1Þ

where Ke is the energy coefficient [31] for an interfacial edge dislocation (see Appendix B and Ref. [41] for a description on determin-

Eedge chem ¼

cmin þ cmax 2

þ

cmin  cmax 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ b2  b ;

ðA:8Þ

where

b¼

K e a2 : 2ppðcmax  cmin Þ

ðA:9Þ

By defining the half width of the misfit dislocation core f from U(b1) = a/4, U(b2) = 3a/4, f = b2  b1, one also finds

f¼

2p

p

0qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ b2  1 A: arctan @ b

ðA:10Þ

Up to now, the description of our method has only dealt with a periodic 1D array of edge dislocations. For the system exemplified in this paper, misfit exists in both directions of the interface and is taken up by a square dislocation network. To account for this square network, we transfer the 1D solution to 2D by (i) multiplying the elastic energy by 2 and hence assuming that perpendicular dislocations do not interact, and (ii) multiplying the deviation of the chemical energy from its minimum value Ecoh (the minimum of the c-surface) by 2. Accordingly, the final result for the 2D semicoherent interface energy is given by

  Etot ¼ 2Eedge þ 2 Eedge el chem  Ecoh þ Ecoh :

ðA:11Þ

D.H.R. Fors et al. / Computational Materials Science 50 (2010) 550–559

and

Appendix B. Energy factor Ke This appendix is included to give a brief description on how to calculate the energy factor Ke appearing in Eq. (A.1). Throughout the appendix, tensor notation is used where double indices imply summation. The elastic energy per length W/L of a straight dislocation has the functional form

W R K ij bi bj R ln ; ¼ E ln ¼ L r0 r0 4p

ðB:1Þ

where R (r0) is an outer (inner) cutoff radius. E is called the prelogarithmic energy factor and Kij is called the energy coefficient [31]. In an isotropic medium, an edge dislocation has Ke = l/(1  m) and a screw dislocation Ks = l. Barnett and Lothe [41] have derived K for a dislocation at the interface of two dissimilar linear elastic anisotropic media. In terms of the matrices B and S associated with the integral formalism, it takes the form

K ¼ 8pBð2ÞN 1 Bð1Þ;

ðB:2Þ

where

N ¼ ðBð1Þ þ Bð2ÞÞ þ ðST ð1ÞBð2Þ þ Bð1ÞSð2ÞÞ  ðBð1Þ þ Bð2ÞÞ1 ðST ð1ÞBð2Þ þ Bð1ÞSð2ÞÞ:

ðB:3Þ

Here, B(1) (B(2)) and S(1) (S(2)) are the B and S matrix of phase 1 (2). For an interface between two cubic materials, with the normal ^z and ^, the energy coefficient is given as the dislocation line parallel to y K = diag(Ke, Ks, Kv). A good account of how to calculate B and S is given in Ref. [43]. The shortest route is to follow the integral formalism as outlined ^ and n ^ be here. Let ^t be the direction of the dislocation line, m two mutually orthogonal vectors perpendicular to ^t and let x be ^ and a fixed vector in the mn-plane. The matrithe angle between m ^ and n ^ ces B and S are then given as integrals of x according to (m rotating positively around ^t)

Bij ¼

1 8p2

Z 2p h i ðmmÞij  ðmnÞik ðnnÞ1 kl ðnmÞlj dx;

ðB:4Þ

0

and

Sij ¼ 

1 2p

Z 2p 0

ðnnÞ1 ik ðnmÞkj dx;

ðB:5Þ

where (ab)jk = aicijkmbm. Caution must be taken so that B(1) and B(2) are given in the same basis appropriate for the interface system, since the elastic constants cijkm are most often given in a basis of the crystal axes. Therefore, one must transform cijkm to elastic constants c0ijkm in the basis used for the interface at hand, through c0ijkm = TipTjq cpqrsTkrTms, where Tij is a transformation tensor. An alternative is to use the unprimed cijkm in the calculation of B and S, still ^ n ^ ? ^t, and to transform the resulting B and S to the basis provided m; of the interface. For the isotropic case, it is possible to calculate B and S starting from cijkm = kdijdkm + l (dikdjm + dimdjk), where k = 2 ml/(1  2m). One finds

 kþl ðmmÞjk ¼ l djk þ mj mk ; l  1 k þl ðmmÞ1 djk  mj mk ; jk ¼ l k þ 2l

559

ðB:6Þ ðB:7Þ

ðmnÞjk ¼ kmj nk þ lmk nj :

ðB:8Þ

^ perpendicular to ^t, one has By fixing two orthogonal vectors ^x and y mi = xicosx + yisinx, ni = xisinx + yicosx. With ^x ¼ ð1; 0; 0Þ and ^ ¼ ð0; 1; 0Þ, a straightforward integration yields B11 ¼ B22 ¼ 41p 1l m, y 12m B33 ¼ 41p l and S12 ¼ S21 ¼  2ð1 mÞ (all other elements of B and S are zero). Using these expressions for B and S, it is a matter of algebra using Eqs. (B.2) and (B.3) to derive the energy coefficient of Eq. ð1Þ ð2Þ (A.2). As an additional result, one also finds that K iso s ¼ 2l l = iso for vertical disðlð1Þ þ lð2Þ Þ for a screw dislocation and K iso v ¼ Ke placements at the interface.

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