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Chemical composition–elastic property maps of austenitic stainless steels
Solid State Sciences 5 (2003) 931–936 www.elsevier.com/locate/ssscie
Chemical composition–elastic property maps of austenitic stainless steels B. Johansson a,b,∗ , L. Vitos a,c , P.A. Korzhavyi a a Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, 10044 Stockholm, Sweden b Condensed Matter Theory Group, Physics Department, Uppsala University, 75121 Uppsala, Sweden c Research Institute for Solid State Physics and Optics, PO Box 49, 1525 Budapest, Hungary
Received 10 September 2002; accepted 5 December 2002
Abstract Despite a tremendous development during the last decades, both as regards computer power and methodology, it has remained impossible to treat steel at a fundamental atomic level. However, recently we have shown [L. Vitos, P.A. Korzhavyi, B. Johansson, Phys. Rev. Lett. 88 (2002) 155501] that the most efficient theories of random substitutional alloys combined with advanced numerical techniques have made possible to establish a theoretical insight to the electronic structure of stainless steels. Here a detailed description of the quantummechanical modeling of austenitic stainless steels is presented. We adopt an ab initio electronic structure calculation method based on the coherent potential approximation, implemented within the framework of the exact muffin tin orbitals theory, to map the chemical composition distributions of austenitic stainless steels into the elastic property distributions. The so generated database can be fruitfully used in the design of new class of steel alloys. 2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.
1. Introduction The steels, in general, are far the commercially most important engineering materials: they outperform other materials in mechanical and corrosion resistance properties in relation to their price. The austenitic stainless steels are Fe, Cr and Ni based alloys, with a small amount of (below 0.4%) C and other impurities (such as S, P). The most common austenitic grades belong to the AISI 300 series (see Table 1). They provide high corrosion resistance and unusually fine mechanical properties, and are widely used in both industrial and every day applications. The stainless steels are very complex systems, being normally multiphase and poly-crystalline alloys. The main phase of austenitic stainless steels is the Fe-Cr-Ni austenite. The austenite is a disordered, substitutional solid solution, which has the face centered cubic (fcc) crystallographic structure of the γ -Fe. This structure is mainly stabilized by Ni and by interstitial C plus N, when Ni levels are reduced. At room temperature the austenitic stainless steels
* Corresponding author.
E-mail addresses: [email protected] (B. Johansson), [email protected] (L. Vitos), [email protected] (P.A. Korzhavyi).
are paramagnetic metals with randomly oriented individual Fe magnetic moments [1]. The properties of stainless steels depend on (i) chemical composition (including additional alloying elements) and (ii) manufacturing processes. The representation of structural or thermo-chemical parameters in terms of the chemical composition in the vicinity of some straight grades is an important step in developing new alloy steels. The relative effects of alloying elements can be established either from empirical data or from quantum-mechanical simulations. The empirical approach, however, necessitates an enormous number of measurements performed for alloys with slightly different compositions. Furthermore, in many cases the magnitudes of these effects are comparable with the experimental error bars (e.g., uncertainty in chemical compositions). We adopt the second approach, and perform a series of ab initio calculations to separate the effect of alloying elements on physical and chemical properties and, hence, to map the compositional distribution into the property distribution with high accuracy. The problem addressed in this work is to establish the composition–elastic property relationships of the austenitic stainless steels. We use the exact muffin-tin orbitals (EMTO) method [2–4] to solve the one-electron Kohn–Sham equations of the density functional theory [5,6]. The exchange-correlation term is treated within the generalized gradient approxima-
1293-2558/03/$ – see front matter 2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. doi:10.1016/S1293-2558(03)00118-3
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B. Johansson et al. / Solid State Sciences 5 (2003) 931–936
Table 1 Some typical commercial stainless steel compositions from the AISI (American Iron and Steel Institute) 300 series. Concentrations are given in weight percentages AISI
C
Mn
P
S
Si
Cr
Ni
Mo
Fe
304 310 316 317
0.08 0.25 0.08 0.08
2 2 2 2
0.045 0.045 0.045 0.045
0.03 0.03 0.03 0.03
1.0 1.5 1.0 1.0
18–20 24–26 16–18 18–20
8–10 19–22 10–14 11–15
— — 2–3 3–4
balance balance balance balance
tion [7], which has proved accurate in the case of magnetic 3d metals [8,9]. The problem of chemical and magnetic disorder is described within the coherent potential approximation (CPA) [10–12], and the total energy is obtained using the full charge density (FCD) technique [13]. The paper is divided into three main sections. Section 2 presents the theoretical modeling of alloy steels: this includes a brief overview of the ab initio method (EMTOCPA) for random substitutional alloys, the total energy calculation technique (FCD), and the most important details of the numerical simulations. The semi-empirical correlation between single-crystal and poly-crystal data is given in Section 3. Theoretical results are presented and discussed in Section 4.
2. Theoretical modeling of alloy steels 2.1. Computational tool: the EMTO-CPA method
viation between the exact and overlapping potentials and (b) the errors coming from the overlap between spheres. The EMTO’s are defined for each lattice site and for each angular momentum quantum number L ≡ (l, m) with l lmax (usually lmax = 3). They are constructed from the screened spherical waves, which are solutions of the wave equation with boundary conditions given in conjunction with non overlapping hard spheres [2]. Inside the potential spheres the low l (l lmax ) projections of the orbitals onto the spherical harmonics YL (ˆr ) are the partial waves [2]. The matching between the screened spherical waves and the partial waves is realized by additional free-electron solutions [2]. The EMTO or the so-called kink-cancellation equations are solved using the Green function formalism [3,4]. In the case of substitutionally disordered alloys the average alloy density of states for the energy z is determined from the average Green function [12] ˙ − = g(z) ci g i (z)D˙ i (z) + Gi0 (z) , G(z) ˜ S(z) (1) i
The original formulation of the exact muffin-tin orbitals theory may be found in Refs. [2,14–16]. A self-consistent implementation of this theory, within the spherical cell approximation, is presented in Ref. [3], while a comprehensive description of the total energy calculation method, based on the EMTO and the full charge density technique [13] is given in Ref. [4]. The application of the coherent potential approximation (CPA) [10,11] to the compositional and/or magnetic disorder, formulated in the framework of the EMTO theory, is demonstrated in Refs. [12,17,18]. Here we overview the most relevant features of the EMTO-CPA method and outline some important numerical details. The EMTO theory formulates an efficient and accurate method of solving the density functional problem. For simplicity, it may be considered as an improved Korringa– Kohn–Rostoker method [19], where the exact Kohn–Sham potential is represented by large overlapping potential spheres. Inside these spheres the potential is spherically symmetric, and constant between the spheres. However, within the EMTO theory, in contrast to the usual muffin-tin based methods, the one-electron states are determined exactly for an optimized overlapping muffin-tin potential. This potential is chosen as the best possible spherical approximation to the exact potential [4,14]: the radii of the potential spheres, the spherical potential waves, and the constant value from the interstitial, are calculated by minimizing (a) the de-
ci
where is the concentration of the alloy component i. For the sake of simplicity the site and angular momentum ˙ indices in Eq. (1) have been omitted. S(z) and D˙ i (z) denote the energy derivative of the slope matrix, and the logarithmic derivative function of the alloy component i, respectively. Gi0 (z) is the onsite component of the EMTO Green function. For definitions see Refs. [2,12,17]. The coherent Green function, g(z), ˜ and the Green functions for alloy components, g i (z), are determined from the selfconsistent solution of the CPA equations [12]. In Eq. (1) the first term corresponds approximately to the average density of states from the interstitial, while the second term gives the contribution from the spheres centered on each alloy component. Note that the first term has a multicenter form [17], and is not projected onto the alloy components. Within the single-site approximation to the impurity problem, the EMTO-CPA Green function (1) leads to the exact density of states for the optimized overlapping potential. The complete non-spherically symmetric charge density of each alloy component is represented in one-center form around the lattice sites, i.e., niL (r)YL (ˆr ). ni (r) = (2) L
The above one-center expansion can formally be obtained from the real-space expression of the average Green func-
B. Johansson et al. / Solid State Sciences 5 (2003) 931–936
r, r) = i ci Gi (z, r, r). To this end one needs to tion, G(z, transform the interstitial term from (1) to one-center form. However, due to this transformation the angular momentum expansion of Gi (z, r, r), and thus of ni (r), will include the high l terms as well. In practice these terms are truncated at h = 8–12. lmax The total energy of the random alloy is calculated as [12] 1 Etot = ci voi (r) dr + Finter [ni ] z G(z) dz − 2πi i i i ci Fintra [ni ] + Exc [ni ] + i
−
i
ci
αc i i i 2 Q − cQ , w
(3)
i
where vol (r) is the overlapping potential for the alloy component i, and Finter is the average Madelung energy. i i are the electrostatic and exchange-correlation Fintra and Exc energies of the alloy component i due to the charges from the Wigner–Seitz cell. The last term in (3) is the correction to the electrostatic energy calculated within the screened impurity model [20], αc ≈ 0.6, w is the average atomic radius, and Qi denotes the total number of electrons inside the cell for the alloy component i. The individual energy functionals are evaluated using the FCD technique [13]. The accuracy of the EMTO-CPA method has been demonstrated for the ground state properties of metals, semiconductors and oxides [8,21–23], ordered [3] and random alloys [12,18,24,25]. 2.2. Details of the numerical calculations Using the EMTO-CPA method we have calculated the three cubic elastic constants (c11 , c12 and c44 ) of the Fe100−(c+n) Crc Nin alloys as a function of the chemical composition, for the range 13.5 c 25.5 and 8 n 24. This concentration interval includes the basic compositions of the well-known commercial grades from the AISI 300 series. In the present application the one-electron equations were solved within the scalar-relativistic and frozen-core approximations. The Green function was calculated for 16 complex energy points distributed exponentially on a semicircular contour. In the EMTO basis set we included s, p, d and f orbitals, and in the one-center expansion of the full h = 10. The FCD total energy charge density (2) we used lmax was evaluated by the shape function technique [13]. The m = 8. conventional Madelung energy was calculated for lmax At each concentration the theoretical equilibrium volume and the bulk modulus were determined from an exponential Morse type function [26] fitted to the ab initio total energies of fcc structures for nine different atomic volumes. In order to obtain the two cubic shear moduli c = (c11 − c12 )/2 and c44 we used volume conserving tetragonal and orthorhombic
deformations, i.e., 0 1 + εt 0 1 + εt 0 0 1 + εo 0 0 1 − εo 0 0
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0 0
1 (1+εt )2
and
0 0 ,
1 1−εo2
(4)
respectively. We calculated the total energies, E(εt ) and E(εo ), for five tetragonal, εt = −0.02, −0.01, . . ., 0.02, and six orthorhombic, εo = 0.00, 0.01, . . ., 0.05, distortions. Finally, to obtain the accuracy needed for the calculation of elastic constants in the irreducible wedge of the Brillouin zones we used 2000 k-points. The room temperature magnetic structure of Fe-Cr-Ni ternary alloy was described by the so-called disordered local moment (DLM) model [27], which is known as a useful approximation for the paramagnetic state well above the magnetic transition temperature (approximative 100 K for austenitic stainless steels [1]). We used a completely random mixture of the two magnetic states of Fe, and treated the Fe100−(c+n) Crc Nin ternary system as an ↑ ↓ Fe50−(c+n)/2 Fe50−(c+n)/2 Crc Nin quaternary alloy, accounting by this for the possibility of local antiparallel spin alignments with zero net magnetic moment.
3. Elastic constants of alloy steels The main difference between a single crystal alloy considered in ordinary first-principles simulations and a real material is the inherent disorder. The most common form of disorder is the breakdown of the long range order of the crystal lattice sites. Most of the real solid materials have a hierarchy of structures beginning with atoms and ascends through various nano or micro-meter level crystalline grains. The misoriented single crystals are separated by stacking faults, interphase boundaries, etc. The only way to establish first-principles parameters of these poly-crystalline systems is to first derive data of microscopic nature and then to transform these data to macroscopic quantities by suitable averaging methods based on statistical mechanics. In the present work we adopted the Hershey’s averaging method [28], which turned out to give the most accurate correlation between single- and poly-crystalline data [29]. According to this approach the isotropic shear modulus is given by the cubic equation G3 + αG2 + βG + γ = 0,
(5)
where α = (5c11 + 4c12 )/8, β = −c44 (7c11 − 4c12 )/8 and γ = −c44 (c11 − c12 )(c11 + 2c12 )/8. The isotropic bulk modulus is equivalent with the cubic bulk modulus, i.e. B = (c11 + 2c12)/3. There are empirical correlations between the above engineering or poly-crystalline elastic moduli and technologically important properties such as strength, hardness and
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wear. Particularly, the shear modulus encompasses a large set of materials as a general indicator of mechanical hardness [30,31]. Recently, Clerc and Ledbetter [31] have shown that the mechanical hardness (H ) of a ternary metallic alloy, e.g. Fe100−(c+n) Crc Nin , to a good approximation can be written as H (c, n) = constG(c, n)h(c, n)[1 + D(c, n)],
(6)
where h is the so-called chemical hardness and D accounts for the interaction between dislocations and the alloying elements. h(c, n) is a weakly decreasing function of c and n, and at small concentrations D(c, n) was proposed to be proportional with the linear concentrations of solute atoms [31]. In the case of annealed austenite (here we consider neither the solid-solution strengthening nor workhardening) for relatively high Cr and Ni concentrations the h(c, n)D(c, n) function may be approximated by a constant, and therefore the concentration dependence of H (c, n) will mainly be determined by G(c, n). The bulk modulus is proportional to the cohesive energy [32], and represents, on the average, the opposition of the material to bond rupture [33]. Since the shear modulus is associated with the resistance to plastic deformation, the ratio B/G may be considered as a measure of the ductility/brittleness performance of solids. Ductility (ability to change shape without fracture) is, therefore, characterized by high B/G ratio ( 1.75), while low B/G is representative of brittleness (fracture without appreciable plastic deformation) [33]. Finally, in the case of austenitic stainless steel, both the bulk modulus and the B/G ratio show correlations with the corrosion resistance, in particular, with the resistance against stress-corrosion cracking [18].
4. Results and discussion First we demonstrate the accuracy of the EMTO-CPA method in the case of stainless steels by comparing the present theoretical results with the available experimental data. The theoretical and experimental elastic moduli of some selected alloys are shown in Fig. 1. The average deviation between these theoretical and experimental shear and bulk moduli are 1.0% and 5.1%, respectively. Note that such small differences are typical for what are obtained for the elemental 3d transition metals using the most accurate density functional methods [8,9,34]. In Fig. 2 the effect of alloying with Mo on the shear modulus of grade 316 is shown. Both theory and experiment predict a substantial decrease in G. It is interesting to compare the effect of Mo addition on the shear modulus with the composition dependence of the stability of the austenitic (fcc) phase relative to the ferrite phase, which has the body centered cubic (bcc) crystallographic structure. We recall that in the case of transition metals the cubic elastic constant associated with the tetragonal distortion of the lattice, shows a proportionality to the energy difference between the bcc
Fig. 1. Comparison between the theoretical [18] and experimental [35–37] shear and bulk moduli of the alloy steels (for composition see Table 1).
Fig. 2. The effect of Mo addition on the shear modulus of austenitic stainless steel 316 (for composition see Table 1). The theoretical results are compared with the experimental data from Ref. [37].
and fcc structures [34]. In the case of alloy 316 we have found that Mo strongly stabilizes the ferrite phase (∼ 3 meV per at.%), which correlates very well with the trend of the shear modulus. From Figs. 1 and 2 one can conclude that quantum mechanics formulated via density functional theory and expressed by the EMTO-CPA method reproduces accurately the observed trends, including the effect of additional Mo, of the elastic moduli of stainless steels. Therefore, theoretical results calculated using the above computational tool can be used for prediction of new data. Recently, with this object, a series of composition–elastic moduli maps of austenitic stainless steels have been generated [18]. In Figs. 3 and 4 the chemical composition distribution of the bulk and shear moduli are shown. The bulk modulus from Fig. 3 varies between a minimum value of 161 GPa, corresponding to Fe13.5Cr12Ni (i.e., c = 13.5 and n = 12), and a maximum value of 178 GPa, belonging to Fe25Cr24Ni. It follows from figure that both Cr and Ni enhance the bulk modulus of alloy steel. In terms of the shear modulus, Fig. 4, three families of alloys can be distinguished. Alloys with large shear modulus correspond
B. Johansson et al. / Solid State Sciences 5 (2003) 931–936
Fig. 3. The calculated bulk modulus of austenitic stainless steels as a function of Cr and Ni contents (balance Fe).
Fig. 4. The calculated shear modulus of austenitic stainless steels as a function of Cr and Ni contents (balance Fe).
to low and intermediate Cr (< 20%) and low Ni (< 15%) concentrations. Within this group of alloys G decreases monotonically with both Cr and Ni from a pronounced maximum about 81 GPa (near Fe14Cr8Ni) to approximately 77 GPa. The high Cr content alloys define the second family of austenites possessing the lowest shear moduli ( 75 GPa) with a minimum around the composition Fe25Cr20Ni. The third family of austenites, with intermediate G values, is located at moderate Cr (< 20%) and high Ni (> 15%) concentrations, where G shows no significant chemical composition dependence. It is interesting to compare the effect of alloying elements on G with the composition dependence of the stability of the austenitic (fcc) phase relative to the ferrite (bcc) phase. The cubic elastic constant c = (c11 − c12 )/2, associate with the tetragonal distortion of the fcc lattice, in the case of transition metals shows strong proportionality with the energy difference between the bcc and fcc structures [34]. In the case of austenitic stainless steels we have found that Ni always stabilizes the austenitic phase (with ∼ 0.10 eV per at.%), even so, except for large concentrations, it decreases the cubic and thus the poly-crystalline elastic moduli. On the other hand, Cr has a strong ferrite stabilizer effect (approximately −0.13 eV per at.%), which correlates reasonably well with the trend of the shear modulus. Using the maps from Figs. 3 and 4 one can predict new alloy compositions, which have improved and controllable properties relative to the commercial grades. Here we consider the following two prospects: (a) low Cr and Ni laden austenite with approximate composition of Fe13Cr8Ni, and
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(b) high Ni content austenite with composition around Fe18Cr24Ni. Hard stainless steels can be obtained from low Cr and Ni alloys (Fig. 4). However, these alloys have the lowest B (Fig. 3) and B/G ratio (∼ 2.0), which reflects a decreased cohesion and an increased brittleness relative to the commercial grades 304 and 316 (see Table 1). These shortcomings can be retrenched by alloying the basic composition with a small amount of Mo, for example. In particular, we have found that a 4% increase of the bulk modulus of Fe13.5Cr8Ni can already be achieved by substituting 2% Fe by Mo. A similar steel composition (Fe13Cr8Ni2Mo1Al) with high strength and toughness, has recently been developed experimentally by the British company Allvac. Austenitic stainless steels with compositions near 18% Cr and 24% Ni possess intermediate hardness (Fig. 4). But these alloys are more ductile compared to the ordinary low Ni austenites (e.g., 304 and 316), and, due to the relatively large bulk modulus, they should exhibit excellent resistance against localized damage. Additionally, the relative large Ni content, that renders the Fe-Cr alloys austenitic, allows for a reduction of the amount of C, providing by this extra corrosion resistance (note: at high temperatures the C combines with Cr at the grain boundaries and thereby it worsens the stainless properties).
5. Conclusions We have shown that efficient quantum level theories of random substitutional alloys combined with the most sophisticated numerical techniques allows one to establish a theoretical insight to the electronic structure of austenitic stainless steels. Using the EMTO-CPA ab initio computational tool, based on the density functional theory, we have calculated the elastic moduli of a series of Fe-Cr-Ni alloys of industrial relevance. From the chemical composition– elastic moduli maps we predict new outstanding class of alloy steels. The present achievements point to new perspectives concerning theoretical modeling of real materials.
Acknowledgements The Swedish Natural Science Research Council, the Swedish Foundation for Strategic Research and Royal Swedish Academy of Sciences are acknowledged for financial support. Part of this work was supported by the research project OTKA T035043 of the Hungarian Scientific Research Fund and by the Hungarian Academy of Science. The calculations were performed at the Swedish National Supercomputer Center, Linköping and Hungarian National Supercomputer Center, Budapest.
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Chemical composition–elastic property maps of austenitic stainless steels B. Johansson a,b,∗ , L. Vitos a,c , P.A. Korzhavyi a a Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, 10044 Stockholm, Sweden b Condensed Matter Theory Group, Physics Department, Uppsala University, 75121 Uppsala, Sweden c Research Institute for Solid State Physics and Optics, PO Box 49, 1525 Budapest, Hungary
Received 10 September 2002; accepted 5 December 2002
Abstract Despite a tremendous development during the last decades, both as regards computer power and methodology, it has remained impossible to treat steel at a fundamental atomic level. However, recently we have shown [L. Vitos, P.A. Korzhavyi, B. Johansson, Phys. Rev. Lett. 88 (2002) 155501] that the most efficient theories of random substitutional alloys combined with advanced numerical techniques have made possible to establish a theoretical insight to the electronic structure of stainless steels. Here a detailed description of the quantummechanical modeling of austenitic stainless steels is presented. We adopt an ab initio electronic structure calculation method based on the coherent potential approximation, implemented within the framework of the exact muffin tin orbitals theory, to map the chemical composition distributions of austenitic stainless steels into the elastic property distributions. The so generated database can be fruitfully used in the design of new class of steel alloys. 2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.
1. Introduction The steels, in general, are far the commercially most important engineering materials: they outperform other materials in mechanical and corrosion resistance properties in relation to their price. The austenitic stainless steels are Fe, Cr and Ni based alloys, with a small amount of (below 0.4%) C and other impurities (such as S, P). The most common austenitic grades belong to the AISI 300 series (see Table 1). They provide high corrosion resistance and unusually fine mechanical properties, and are widely used in both industrial and every day applications. The stainless steels are very complex systems, being normally multiphase and poly-crystalline alloys. The main phase of austenitic stainless steels is the Fe-Cr-Ni austenite. The austenite is a disordered, substitutional solid solution, which has the face centered cubic (fcc) crystallographic structure of the γ -Fe. This structure is mainly stabilized by Ni and by interstitial C plus N, when Ni levels are reduced. At room temperature the austenitic stainless steels
* Corresponding author.
E-mail addresses: [email protected] (B. Johansson), [email protected] (L. Vitos), [email protected] (P.A. Korzhavyi).
are paramagnetic metals with randomly oriented individual Fe magnetic moments [1]. The properties of stainless steels depend on (i) chemical composition (including additional alloying elements) and (ii) manufacturing processes. The representation of structural or thermo-chemical parameters in terms of the chemical composition in the vicinity of some straight grades is an important step in developing new alloy steels. The relative effects of alloying elements can be established either from empirical data or from quantum-mechanical simulations. The empirical approach, however, necessitates an enormous number of measurements performed for alloys with slightly different compositions. Furthermore, in many cases the magnitudes of these effects are comparable with the experimental error bars (e.g., uncertainty in chemical compositions). We adopt the second approach, and perform a series of ab initio calculations to separate the effect of alloying elements on physical and chemical properties and, hence, to map the compositional distribution into the property distribution with high accuracy. The problem addressed in this work is to establish the composition–elastic property relationships of the austenitic stainless steels. We use the exact muffin-tin orbitals (EMTO) method [2–4] to solve the one-electron Kohn–Sham equations of the density functional theory [5,6]. The exchange-correlation term is treated within the generalized gradient approxima-
1293-2558/03/$ – see front matter 2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. doi:10.1016/S1293-2558(03)00118-3
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Table 1 Some typical commercial stainless steel compositions from the AISI (American Iron and Steel Institute) 300 series. Concentrations are given in weight percentages AISI
C
Mn
P
S
Si
Cr
Ni
Mo
Fe
304 310 316 317
0.08 0.25 0.08 0.08
2 2 2 2
0.045 0.045 0.045 0.045
0.03 0.03 0.03 0.03
1.0 1.5 1.0 1.0
18–20 24–26 16–18 18–20
8–10 19–22 10–14 11–15
— — 2–3 3–4
balance balance balance balance
tion [7], which has proved accurate in the case of magnetic 3d metals [8,9]. The problem of chemical and magnetic disorder is described within the coherent potential approximation (CPA) [10–12], and the total energy is obtained using the full charge density (FCD) technique [13]. The paper is divided into three main sections. Section 2 presents the theoretical modeling of alloy steels: this includes a brief overview of the ab initio method (EMTOCPA) for random substitutional alloys, the total energy calculation technique (FCD), and the most important details of the numerical simulations. The semi-empirical correlation between single-crystal and poly-crystal data is given in Section 3. Theoretical results are presented and discussed in Section 4.
2. Theoretical modeling of alloy steels 2.1. Computational tool: the EMTO-CPA method
viation between the exact and overlapping potentials and (b) the errors coming from the overlap between spheres. The EMTO’s are defined for each lattice site and for each angular momentum quantum number L ≡ (l, m) with l lmax (usually lmax = 3). They are constructed from the screened spherical waves, which are solutions of the wave equation with boundary conditions given in conjunction with non overlapping hard spheres [2]. Inside the potential spheres the low l (l lmax ) projections of the orbitals onto the spherical harmonics YL (ˆr ) are the partial waves [2]. The matching between the screened spherical waves and the partial waves is realized by additional free-electron solutions [2]. The EMTO or the so-called kink-cancellation equations are solved using the Green function formalism [3,4]. In the case of substitutionally disordered alloys the average alloy density of states for the energy z is determined from the average Green function [12] ˙ − = g(z) ci g i (z)D˙ i (z) + Gi0 (z) , G(z) ˜ S(z) (1) i
The original formulation of the exact muffin-tin orbitals theory may be found in Refs. [2,14–16]. A self-consistent implementation of this theory, within the spherical cell approximation, is presented in Ref. [3], while a comprehensive description of the total energy calculation method, based on the EMTO and the full charge density technique [13] is given in Ref. [4]. The application of the coherent potential approximation (CPA) [10,11] to the compositional and/or magnetic disorder, formulated in the framework of the EMTO theory, is demonstrated in Refs. [12,17,18]. Here we overview the most relevant features of the EMTO-CPA method and outline some important numerical details. The EMTO theory formulates an efficient and accurate method of solving the density functional problem. For simplicity, it may be considered as an improved Korringa– Kohn–Rostoker method [19], where the exact Kohn–Sham potential is represented by large overlapping potential spheres. Inside these spheres the potential is spherically symmetric, and constant between the spheres. However, within the EMTO theory, in contrast to the usual muffin-tin based methods, the one-electron states are determined exactly for an optimized overlapping muffin-tin potential. This potential is chosen as the best possible spherical approximation to the exact potential [4,14]: the radii of the potential spheres, the spherical potential waves, and the constant value from the interstitial, are calculated by minimizing (a) the de-
ci
where is the concentration of the alloy component i. For the sake of simplicity the site and angular momentum ˙ indices in Eq. (1) have been omitted. S(z) and D˙ i (z) denote the energy derivative of the slope matrix, and the logarithmic derivative function of the alloy component i, respectively. Gi0 (z) is the onsite component of the EMTO Green function. For definitions see Refs. [2,12,17]. The coherent Green function, g(z), ˜ and the Green functions for alloy components, g i (z), are determined from the selfconsistent solution of the CPA equations [12]. In Eq. (1) the first term corresponds approximately to the average density of states from the interstitial, while the second term gives the contribution from the spheres centered on each alloy component. Note that the first term has a multicenter form [17], and is not projected onto the alloy components. Within the single-site approximation to the impurity problem, the EMTO-CPA Green function (1) leads to the exact density of states for the optimized overlapping potential. The complete non-spherically symmetric charge density of each alloy component is represented in one-center form around the lattice sites, i.e., niL (r)YL (ˆr ). ni (r) = (2) L
The above one-center expansion can formally be obtained from the real-space expression of the average Green func-
B. Johansson et al. / Solid State Sciences 5 (2003) 931–936
r, r) = i ci Gi (z, r, r). To this end one needs to tion, G(z, transform the interstitial term from (1) to one-center form. However, due to this transformation the angular momentum expansion of Gi (z, r, r), and thus of ni (r), will include the high l terms as well. In practice these terms are truncated at h = 8–12. lmax The total energy of the random alloy is calculated as [12] 1 Etot = ci voi (r) dr + Finter [ni ] z G(z) dz − 2πi i i i ci Fintra [ni ] + Exc [ni ] + i
−
i
ci
αc i i i 2 Q − cQ , w
(3)
i
where vol (r) is the overlapping potential for the alloy component i, and Finter is the average Madelung energy. i i are the electrostatic and exchange-correlation Fintra and Exc energies of the alloy component i due to the charges from the Wigner–Seitz cell. The last term in (3) is the correction to the electrostatic energy calculated within the screened impurity model [20], αc ≈ 0.6, w is the average atomic radius, and Qi denotes the total number of electrons inside the cell for the alloy component i. The individual energy functionals are evaluated using the FCD technique [13]. The accuracy of the EMTO-CPA method has been demonstrated for the ground state properties of metals, semiconductors and oxides [8,21–23], ordered [3] and random alloys [12,18,24,25]. 2.2. Details of the numerical calculations Using the EMTO-CPA method we have calculated the three cubic elastic constants (c11 , c12 and c44 ) of the Fe100−(c+n) Crc Nin alloys as a function of the chemical composition, for the range 13.5 c 25.5 and 8 n 24. This concentration interval includes the basic compositions of the well-known commercial grades from the AISI 300 series. In the present application the one-electron equations were solved within the scalar-relativistic and frozen-core approximations. The Green function was calculated for 16 complex energy points distributed exponentially on a semicircular contour. In the EMTO basis set we included s, p, d and f orbitals, and in the one-center expansion of the full h = 10. The FCD total energy charge density (2) we used lmax was evaluated by the shape function technique [13]. The m = 8. conventional Madelung energy was calculated for lmax At each concentration the theoretical equilibrium volume and the bulk modulus were determined from an exponential Morse type function [26] fitted to the ab initio total energies of fcc structures for nine different atomic volumes. In order to obtain the two cubic shear moduli c = (c11 − c12 )/2 and c44 we used volume conserving tetragonal and orthorhombic
deformations, i.e., 0 1 + εt 0 1 + εt 0 0 1 + εo 0 0 1 − εo 0 0
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0 0
1 (1+εt )2
and
0 0 ,
1 1−εo2
(4)
respectively. We calculated the total energies, E(εt ) and E(εo ), for five tetragonal, εt = −0.02, −0.01, . . ., 0.02, and six orthorhombic, εo = 0.00, 0.01, . . ., 0.05, distortions. Finally, to obtain the accuracy needed for the calculation of elastic constants in the irreducible wedge of the Brillouin zones we used 2000 k-points. The room temperature magnetic structure of Fe-Cr-Ni ternary alloy was described by the so-called disordered local moment (DLM) model [27], which is known as a useful approximation for the paramagnetic state well above the magnetic transition temperature (approximative 100 K for austenitic stainless steels [1]). We used a completely random mixture of the two magnetic states of Fe, and treated the Fe100−(c+n) Crc Nin ternary system as an ↑ ↓ Fe50−(c+n)/2 Fe50−(c+n)/2 Crc Nin quaternary alloy, accounting by this for the possibility of local antiparallel spin alignments with zero net magnetic moment.
3. Elastic constants of alloy steels The main difference between a single crystal alloy considered in ordinary first-principles simulations and a real material is the inherent disorder. The most common form of disorder is the breakdown of the long range order of the crystal lattice sites. Most of the real solid materials have a hierarchy of structures beginning with atoms and ascends through various nano or micro-meter level crystalline grains. The misoriented single crystals are separated by stacking faults, interphase boundaries, etc. The only way to establish first-principles parameters of these poly-crystalline systems is to first derive data of microscopic nature and then to transform these data to macroscopic quantities by suitable averaging methods based on statistical mechanics. In the present work we adopted the Hershey’s averaging method [28], which turned out to give the most accurate correlation between single- and poly-crystalline data [29]. According to this approach the isotropic shear modulus is given by the cubic equation G3 + αG2 + βG + γ = 0,
(5)
where α = (5c11 + 4c12 )/8, β = −c44 (7c11 − 4c12 )/8 and γ = −c44 (c11 − c12 )(c11 + 2c12 )/8. The isotropic bulk modulus is equivalent with the cubic bulk modulus, i.e. B = (c11 + 2c12)/3. There are empirical correlations between the above engineering or poly-crystalline elastic moduli and technologically important properties such as strength, hardness and
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wear. Particularly, the shear modulus encompasses a large set of materials as a general indicator of mechanical hardness [30,31]. Recently, Clerc and Ledbetter [31] have shown that the mechanical hardness (H ) of a ternary metallic alloy, e.g. Fe100−(c+n) Crc Nin , to a good approximation can be written as H (c, n) = constG(c, n)h(c, n)[1 + D(c, n)],
(6)
where h is the so-called chemical hardness and D accounts for the interaction between dislocations and the alloying elements. h(c, n) is a weakly decreasing function of c and n, and at small concentrations D(c, n) was proposed to be proportional with the linear concentrations of solute atoms [31]. In the case of annealed austenite (here we consider neither the solid-solution strengthening nor workhardening) for relatively high Cr and Ni concentrations the h(c, n)D(c, n) function may be approximated by a constant, and therefore the concentration dependence of H (c, n) will mainly be determined by G(c, n). The bulk modulus is proportional to the cohesive energy [32], and represents, on the average, the opposition of the material to bond rupture [33]. Since the shear modulus is associated with the resistance to plastic deformation, the ratio B/G may be considered as a measure of the ductility/brittleness performance of solids. Ductility (ability to change shape without fracture) is, therefore, characterized by high B/G ratio ( 1.75), while low B/G is representative of brittleness (fracture without appreciable plastic deformation) [33]. Finally, in the case of austenitic stainless steel, both the bulk modulus and the B/G ratio show correlations with the corrosion resistance, in particular, with the resistance against stress-corrosion cracking [18].
4. Results and discussion First we demonstrate the accuracy of the EMTO-CPA method in the case of stainless steels by comparing the present theoretical results with the available experimental data. The theoretical and experimental elastic moduli of some selected alloys are shown in Fig. 1. The average deviation between these theoretical and experimental shear and bulk moduli are 1.0% and 5.1%, respectively. Note that such small differences are typical for what are obtained for the elemental 3d transition metals using the most accurate density functional methods [8,9,34]. In Fig. 2 the effect of alloying with Mo on the shear modulus of grade 316 is shown. Both theory and experiment predict a substantial decrease in G. It is interesting to compare the effect of Mo addition on the shear modulus with the composition dependence of the stability of the austenitic (fcc) phase relative to the ferrite phase, which has the body centered cubic (bcc) crystallographic structure. We recall that in the case of transition metals the cubic elastic constant associated with the tetragonal distortion of the lattice, shows a proportionality to the energy difference between the bcc
Fig. 1. Comparison between the theoretical [18] and experimental [35–37] shear and bulk moduli of the alloy steels (for composition see Table 1).
Fig. 2. The effect of Mo addition on the shear modulus of austenitic stainless steel 316 (for composition see Table 1). The theoretical results are compared with the experimental data from Ref. [37].
and fcc structures [34]. In the case of alloy 316 we have found that Mo strongly stabilizes the ferrite phase (∼ 3 meV per at.%), which correlates very well with the trend of the shear modulus. From Figs. 1 and 2 one can conclude that quantum mechanics formulated via density functional theory and expressed by the EMTO-CPA method reproduces accurately the observed trends, including the effect of additional Mo, of the elastic moduli of stainless steels. Therefore, theoretical results calculated using the above computational tool can be used for prediction of new data. Recently, with this object, a series of composition–elastic moduli maps of austenitic stainless steels have been generated [18]. In Figs. 3 and 4 the chemical composition distribution of the bulk and shear moduli are shown. The bulk modulus from Fig. 3 varies between a minimum value of 161 GPa, corresponding to Fe13.5Cr12Ni (i.e., c = 13.5 and n = 12), and a maximum value of 178 GPa, belonging to Fe25Cr24Ni. It follows from figure that both Cr and Ni enhance the bulk modulus of alloy steel. In terms of the shear modulus, Fig. 4, three families of alloys can be distinguished. Alloys with large shear modulus correspond
B. Johansson et al. / Solid State Sciences 5 (2003) 931–936
Fig. 3. The calculated bulk modulus of austenitic stainless steels as a function of Cr and Ni contents (balance Fe).
Fig. 4. The calculated shear modulus of austenitic stainless steels as a function of Cr and Ni contents (balance Fe).
to low and intermediate Cr (< 20%) and low Ni (< 15%) concentrations. Within this group of alloys G decreases monotonically with both Cr and Ni from a pronounced maximum about 81 GPa (near Fe14Cr8Ni) to approximately 77 GPa. The high Cr content alloys define the second family of austenites possessing the lowest shear moduli ( 75 GPa) with a minimum around the composition Fe25Cr20Ni. The third family of austenites, with intermediate G values, is located at moderate Cr (< 20%) and high Ni (> 15%) concentrations, where G shows no significant chemical composition dependence. It is interesting to compare the effect of alloying elements on G with the composition dependence of the stability of the austenitic (fcc) phase relative to the ferrite (bcc) phase. The cubic elastic constant c = (c11 − c12 )/2, associate with the tetragonal distortion of the fcc lattice, in the case of transition metals shows strong proportionality with the energy difference between the bcc and fcc structures [34]. In the case of austenitic stainless steels we have found that Ni always stabilizes the austenitic phase (with ∼ 0.10 eV per at.%), even so, except for large concentrations, it decreases the cubic and thus the poly-crystalline elastic moduli. On the other hand, Cr has a strong ferrite stabilizer effect (approximately −0.13 eV per at.%), which correlates reasonably well with the trend of the shear modulus. Using the maps from Figs. 3 and 4 one can predict new alloy compositions, which have improved and controllable properties relative to the commercial grades. Here we consider the following two prospects: (a) low Cr and Ni laden austenite with approximate composition of Fe13Cr8Ni, and
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(b) high Ni content austenite with composition around Fe18Cr24Ni. Hard stainless steels can be obtained from low Cr and Ni alloys (Fig. 4). However, these alloys have the lowest B (Fig. 3) and B/G ratio (∼ 2.0), which reflects a decreased cohesion and an increased brittleness relative to the commercial grades 304 and 316 (see Table 1). These shortcomings can be retrenched by alloying the basic composition with a small amount of Mo, for example. In particular, we have found that a 4% increase of the bulk modulus of Fe13.5Cr8Ni can already be achieved by substituting 2% Fe by Mo. A similar steel composition (Fe13Cr8Ni2Mo1Al) with high strength and toughness, has recently been developed experimentally by the British company Allvac. Austenitic stainless steels with compositions near 18% Cr and 24% Ni possess intermediate hardness (Fig. 4). But these alloys are more ductile compared to the ordinary low Ni austenites (e.g., 304 and 316), and, due to the relatively large bulk modulus, they should exhibit excellent resistance against localized damage. Additionally, the relative large Ni content, that renders the Fe-Cr alloys austenitic, allows for a reduction of the amount of C, providing by this extra corrosion resistance (note: at high temperatures the C combines with Cr at the grain boundaries and thereby it worsens the stainless properties).
5. Conclusions We have shown that efficient quantum level theories of random substitutional alloys combined with the most sophisticated numerical techniques allows one to establish a theoretical insight to the electronic structure of austenitic stainless steels. Using the EMTO-CPA ab initio computational tool, based on the density functional theory, we have calculated the elastic moduli of a series of Fe-Cr-Ni alloys of industrial relevance. From the chemical composition– elastic moduli maps we predict new outstanding class of alloy steels. The present achievements point to new perspectives concerning theoretical modeling of real materials.
Acknowledgements The Swedish Natural Science Research Council, the Swedish Foundation for Strategic Research and Royal Swedish Academy of Sciences are acknowledged for financial support. Part of this work was supported by the research project OTKA T035043 of the Hungarian Scientific Research Fund and by the Hungarian Academy of Science. The calculations were performed at the Swedish National Supercomputer Center, Linköping and Hungarian National Supercomputer Center, Budapest.
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