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Solute effect on basal and prismatic slip systems of Mg
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 445004 (10pp)
doi:10.1088/0953-8984/26/44/445004
Solute effect on basal and prismatic slip systems of Mg Amitava Moitra1 , Seong-Gon Kim2,3 and M F Horstemeyer4,5 1
Thematic Unit of Excellence on Comuputational Materials Science, S.N. Bose National Centre for Basic Sciences, Sector-III, Block-JD, Salt Lake, Kolkata-700098, India 2 Department of Physics and Astronomy, Mississippi State University, Mississippi State, MS 39762, USA 3 Center for Computational Sciences, Mississippi State University, Mississippi State, MS 39762, USA 4 Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS 39762, USA 5 Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS 39762, USA E-mail: [email protected] Received 21 May 2014, revised 19 August 2014 Accepted for publication 3 September 2014 Published 2 October 2014 Abstract
In an effort to design novel magnesium (Mg) alloys with high ductility, we present a first principles data based on the Density Functional Theory (DFT). The DFT was employed to calculate the generalized stacking fault energy curves, which can be used in the generalized Peierls–Nabarro (PN) model to study the energetics of basal slip and prismatic slip in Mg with and without solutes to calculate continuum scale dislocation core widths, stacking fault widths and Peierls stresses. The generalized stacking fault energy curves for pure Mg agreed well with other DFT calculations. Solute effects on these curves were calculated for nine alloying elements, namely Al, Ca, Ce, Gd, Li, Si, Sn, Zn and Zr, which allowed the strength and ductility to be qualitatively estimated based on the basal dislocation properties. Based on our multiscale methodology, a suggestion has been made to improve Mg formability. Keywords: density functional theory, alloy design, generalized stacking fault energy (Some figures may appear in colour only in the online journal)
materials properties or designing novel materials [3, 4]. In the present paper first principles Density Functional Theory (DFT) is used to calculate the generalized stacking fault energies (GSFE). These GSFE are then used to link directly to the generalized Peierls–Nabarro model to study the dislocation properties for Mg. Magnesium alloys have become one of the most promising materials in the automotive, aerospace, biomedical and athletic industries due to its strength to weight ratio, stiffness to weight ratio, biodegradability and abundance [5]. Currently, further use of Mg–alloys in structural applications are restricted due to various metallurgical issues including formability, crash worthiness, strength, corrosion, creep resistance, flammability and fatigue fracture. It is well known for more than a century ago [6–11] that, at sufficiently low temperatures or high strain rates, deformation in double lattice structures, for which hexagonal close-packed
1. Introduction
Alloy design often requires conflicting materials properties to be intelligently optimized for specific usage. Heat treatments, composition control, precipitate strengthening and cold working have a vital role on material’s major functionalities, such as ductility, maximum strength and corrosion resistance to name a few [1]. Due to extraordinary improvements in modern computing facilities, computational alloy design has recently become far more feasible [2]. Nevertheless, the technological challenges are still restrictive either due to spatial (length–scale) or temporal (time–scale) limitations on nanoscale atomistic simulations or the incapability of capturing minute details on continuum scale phase-field or finite-element modeling. A multiscale modeling methodology incorporating these atomistic and continuum scale models (hierarchical or concurrent methods) has a fairly long history in predicting 0953-8984/14/445004+10$33.00
1
© 2014 IOP Publishing Ltd Printed in the UK
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(HCP) structures are an archetypal example is driven by both slip and twinning and their dynamic interactions. Mg is a particular HCP metal where, in contrast to all other HCP metals, twinning is able to consume the entire matrix below 7% of the plastic strain [12, 13]. The associated slip–twin and twin–twin interactions lead to a strong inflection in the stress–strain behavior from convex-up to concave-down. This behavior has been correlated to strong plastic anisotropy and limited ductility, which both stand in the way of a broader industrial applications of Mg [14]. Because dislocations are integral to both slip and twinning (twinning is just a liberation of twin partials), dislocation and twin disconnection core structures must be described at the atomic scale [15]. On one hand, the dislocation core properties dictate dislocation mobility and the ease by which slip and twinning could nucleate. On the other hand, the dislocation core structure is responsible for short range dislocation interactions including slip–slip (latent hardening), slip–twin and twin-twin. Thus, understanding how the crystal structure and the alloying elements influence the dislocation core structure remains a key feature to understand [5] and enhance the mechanical properties of Mg alloys [16]. Numerical modeling for enhancement of such mechanical properties for Mg alloys are underway at several length scales [17–19]. For example, Yoshinaga and Horiuchi [10] investigated from tensile tests at various temperatures that the Li addition on Mg matrix increases the critically resolved shear stress (CRSS) for basal slip systems and the tendency for crossslip to prism planes, thus increasing the strength and ductility, respectively. Out of several Mg alloys, Mg–Al is the most extensively studied system with a focus on improving strength and ductility behavior [20, 21]. One of the possible alternatives to increase ductility in both low and high temperature regimes is to facilitate crossslip of a-type dislocations from basal to prismatic planes. This thermal cross slip can be aided by either lowering the SFE in prism planes or increasing it for basal planes through addition of solutes, hence reducing the anisotropy in slip systems. An ideal combination of both mechanisms remains difficult to achieve. On one hand, an increase of basal SFE can promote bowing and constriction of screw dislocations, so solutes with a positive chemical misfit are needed. Yasi et al [5] developed first principal solute strengthening models that were able to accurately determine dislocation core geometries and compute solute misfits, which pertain to both size misfit and chemical misfit. The solute misfit allows quantification of the volumetric expansion/compression of the lattice and local slip effects in both far-field and core and as such is a good approximation of the solute/dislocation interaction energy. The choice of good strengthening solutes must thus consider a compromise between a large chemical misfit for strengthening capacity and a low chemical misfit for high solubility. On the other hand, decreasing the SFE on prismatic plane depends on the effect of solutes on the energy barrier landscape, impeding formation and movement of double-kinks (high temperatures) and constriction and bowing of screw dislocations (low temperatures). To increase the ductility and also the strength, Ce has been added to Mg [22, 23]. Plastic deformation behaviors
for Mg–Ce–Zn–Zr alloys are studied by Yu et al [22]. Age hardening responses and creep resistances were observed for Mg–Gd and Mg–Gd–Zn systems [24] and attributed to the long period stacking order (LPSO) structure [25]. For biologically friendly alloys, Ca solutes were added to Mg forming Mg2 Ca intermetallics within the alloy [26, 27]. The effects of Si, Sn, Zr and Zn to increase the mechanical strength of Mg was also evaluated by computation and experiment [5, 28, 29, 31, 32]. Although the above mentioned literature have contributed towards the improvement of Mg alloys, there has been a scarcity towards the understanding of how the materials properties could be modified in presence of different solute atoms. This paper focuses on such a particular material property called Formability, which stand in the way of a broader industrial applications of Mg. Defining this Formability parameter, our approach is two folded: one to identify if this parameter can correctly capture the fact that the rare earth solute elements enhance the Mg formability, as found in experimental observation and two to find out additional nonrare-earth solute elements, that can also enhance this material property. Additionally, we estimate the critical shear stress related to solute strengthening for several solute atoms. All of these studies that analyzed different solutes in Mg provided a basis for the current solute (Al, Ca, Ce, Gd, Li, Si, Sn, Zn, Zr) study for alloying effects on the Mg GSFEs in order to provide information for the generalized Peierls–Nabarro dislocation model. Peierls [33], in his hybrid model, first incorporated the dislocation core information into a continuum framework, which was further developed by Nabarro [34] and Eshelby [35] in order to estimate the lattice friction to dislocation motion. Later, Vitek, with a comprehensive approach, introduced the concept of GSFE curve [36, 37] and further discussed the implications in numerous studies in the literature [38, 39]. Let us consider a perfect crystal that is cut across a single plane into two parts that are subjected to a relative displacement f and relaxed. These relaxed lattices have a surplus energy per unit area γ f. As f spans the lattice, γ (f) generates the GSFE curve. The procedure can be performed for basal, prismatic and pyramidal planes for hcp Mg crystal systems [40]. The importance of this curve is that the interfacial restoring force can be deduced from the slope of the γ (f) curve, such that Fb (f) = −∇(γ (f)).
(1)
Hence, GSFE calculations have drawn a lot of interest to predict the materials properties [41, 42]. Due to the basal to prismatic slip system anisotropies and their lack of available slip systems, pure Mg has a poor formability. A systematic study on the stacking faults in pure Mg has been performed recently with DFT [43] to show that the unstable stacking fault energy for the first order prismatic plane {1 0 1¯ 0} is one order of magnitude higher than the (0 0 0 1) basal plane. The sources of producing a + c pyramidal dislocations are hard to understand, where nucleation of a + c dislocation is the energetically critical step for plastic deformation of hcp metals such as Mg. Activation of pyramidal a + c dislocation is energetically difficult in general and has not been clearly understood. For example, grain refinement, precipitation 2
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hardening, shear banding, decreased c/a ratio and changed Peierls potentials cannot be the underlying mechanism(s) causing the observed higher activity of the a + c dislocations in MgY alloys [44]. Furthermore, mechanical processes such as, rolling or casting for Mg alloys, involve more a type dislocation and twinning and the improved formability is achieved due to the microstructure involving both grain sizes and the dynamic re-crystallization process. Since the easiest slip system investigated in hcp metals is a basal plane slip system with an a-axis slip direction or a first order prismatic {1 0 1¯ 0} plane slip system, we restrict ourselves to calculate the effect of solute atoms only on basal (0 0 0 1) and {1 0 1¯ 0} prismatic plane GSFE curve. Since basal slip (0 0 0 1) 1 1 2¯ 0 was also found at elevated temperatures, along with (0 0 0 1) 1 0 1¯ 0 dominant slip mechanism at room temperature, we calculate basal plane GSFE on both of these directions. After a brief introduction to the semi discrete Peierls– Nabarro model in section 2, we describe our method and model development in section 3. In section 4, we present our results: we start with the solute effect on Mg GSFE both for basal (section 4.1) and prismatic planes (section 4.2). In section 4.3, we define the parameter Formability and show the ability of this parameter to capture the fact that the rare earth solute elements enhance the Mg formability and the parameter predicts that Ca and Zr solute atoms would also increase Mg formability. In section 5, we discuss how the basal GSFE curves can be correlated with continuum scale dislocation properties (section 5.1) and provide an estimation of precipitate strengthening (section 5.2) related to the solute elements studied in this article. Finally in section 6, we summarize and conclude.
where x is the coordinate of the atomic row and f (x) is the disregistry vector of those atomic row at x connected to the dislocation density by ρ(x) = df (x)/dx
(3)
and K is the materials constant based on the elastic properties, such that for an isotropic solid, µ sin2 θ 2 + cos θ (4) K= 2π 1 − ν where µ and ν are the shear modulus and Poisson’s ratio, respectively. θ is the angle between the burgers vector and the dislocation line, such that: µ for θ =0 Ks = 2π K= (5) µ for θ = π2 . Ke = 2π(1−ν) The dislocation density is such that it satisfies the normalization condition: ∞ ∞ df (x ) ρ(x )dx = = b. (6) dx −∞ −∞ In the PN model a simple sinusoidal form for Fb (f (x)) is assumed and the disregistry vector is reduced to an analytical form: b −1 x b f (x) = tan + (7) π ζ 2 where ζ is the dislocation core half width, given by: ζ =
2. Semi discrete Peierls–Nabarro model
Kb 2Fmax
(8)
where Fmax is the maximum restoring stress.
In the context of multiscale modeling, Peierels–Nabarro model based on lattice friction and dislocation motion, details on the dislocation core that are incorporated essentially into an continuum framework. When the dislocations move along the glide plane, the burgers vector lies on that glide plane. If the burgers vector is along the dislocation line (we consider this as θ = 0), a screw dislocation forms. On the other hand, the burgers vector is along the glide direction (we consider this as θ = 90) for an edge dislocation. For a mixed dislocation the burgers vector will have both the edge and screw components. This glide plane is assumed to contain the dislocation misfits, in the Peierls–Nabarro (PN) formalism. The PN formalism consider two semi-infinite elastic media separated by the glide plane that contains the dislocation. A continuous distribution of dislocations with density ρ(x) can be achieved between the elastic medias. The idea there is to map the deformation of the lattice caused by the dislocations’ displacement field to the adjacent elastic media. Hence, a lattice restoring stress is generated due to the interfacial misfit caused by a dislocation. This restoring force Fb (f (x)) across the glide plane can be evaluated from the PN integrodifferential equation: ∞ 1 df (x ) , dx = Fb (f (x)) (2) K dx −∞ x − x
3. Method and model development
All the energy calculations and geometry optimization in this present work were performed using ab-initio Density Functional Theory, using the projector augmented-wave (PAW) method as implemented in VASP (Vienna ab initio Simulation Package) [45]. For the treatment of electron exchange and correlation, we use the generalized gradient approximation (GGA) using the Perdew–Burke–Ernzerhof scheme [46, 47]. All simulations are carried out using a supercell of 64 atoms corresponding to a double unit cell. The plane-wave cut off energy is set to 300 eV in most of the calculations, 400 eV is used for Mg–Ce calculations. The conjugate-gradient [48] method is used to relax the ions. Geometry relaxations are performed until the energy difference between two successive ionic optimizations is less than 0.001 eV. The Brillouin zone is sampled with a density equivalent to 98 k-points using Monkhorst–Pack scheme [49]. For GSFE curve calculations, an orthonormal supercell was used with 2 × 1 × 12 periodic supercell along [1 2¯ 1 0], [1 0 1¯ 0] and [0 0 0 1] directions, respectively. The supercell contained 24 atomic planes along the c [0 0 0 1] direction. For the prismatic plane a 2 × 12 × 1 supercell was used. 3
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Pure Mg With Al With Ca With Ce With Gd With Li With Si With Sn With Zn With Zr
2
Energy per unit area (mJ/m )
400
300
200
100
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Along [1-210] (Å) Figure 1. Solute effect on Mg on GSFE curve on (0 0 0 1) 1 2¯ 1 0 slip system.
For both of the cases, the simulation cells contained 96 atoms with 12 Åvacuum on top. Initially, a defect-free basal surface supercell was fully relaxed (both along unit cell vectors and internal atomic coordinates). This calculation was then followed up by incorporating a substitutional defect of a solute atom and performing a full internal atomic relaxation. For a large unit cell, the solute atom did not affect the unit cell vectors, but only the atoms adjacent to the solute atom were perturbed. In one hand, the solute concentration gets significantly higher with periodic boundary condition, the DFT cannot handle too many atoms, in the other. Thus a reasonable compromise lead us towards a 4 × 2 × 2 hcp Mg system with 64 atoms, where the formation energies reasonably converged until the atomic forces are less than 0.01 eV A−1 . Cross-section of the supercell was chosen in such a way that when a solute atom replaced one Mg atom on the twelfth atomic plane, above which simple shear takes place, the planar coverage becomes 2.8 × 1014 cm−2 for the basal plane and 3.0 × 1014 cm−2 for the prismatic plane. Such a high value of planar solute concentration reflects only the limitation of DFT based atomistic simulations, at our present computational facility. The convergence tests with respect to the k-points and number of planes confirms that the error bar for the total energy is less than 1 meV/atom.
To understand the role of the solute atoms within Mg, we located the solute atoms at the interstitial and substitutional sites as detailed in our earlier report [52]. It was found that all the solute atoms that were considered in this study preferred to stay as a substituting element in bulk Mg. 4.1. GSFE: basal plane
Since the basal plane is the densest atomic plane for HCP Mg crystals, it is the typical slip plane. The basal plane (0 0 0 1) 1 1 2¯ 0 GSFE curve is shown in figure 1, which indicates that there is no local minimum, only a single maximum indicating no dissociation of the dislocation into partials. Also, the maximum value for pure Mg matches exactly with previously calculated results from literature [43]. Figure 1 shows that the substitution of Zr, Gd and Ca increases the maximum GSFE value, whereas Ce, Li, Sn, Al, Zn and Si reduces the maximum GSFE value (It is important to note here that the Zr atom increases the maximum the most and the Si atom reduces the maximum the most.). The GSFE curve indicates the solute strengthening occurs when Zr, Gd and Ca atoms are added and solute softening occurs when Ce, Li, Sn, Al, Zn and Si atoms are added. The Basal plane (0 0 0 1) 1 0 1¯ 0 GSFE curve, as shown in figure 2, reveals that there is one local minimum, indicating dissociation of dislocation into partials. The unstable (γUSF ) and intrinsic stacking (γISF ) fault energies for pure Mg agrees well with previously calculated results [43]. Figure 2 shows that the substitution of Zr, Gd and Ca increase the γUSF energy similar to figure 1, whereas Li, Ce, Zn, Al, Sn and Si reduce the γUSF energy. On the other hand the γISF is increased by Li addition only and all the other solute atoms reduces the γISF from pure Mg case. These results, further elaborated in figure 3, quantitatively agree with the literature value [20, 43, 53–55]. It is remarkable that for the Ce addition the γISF is close to zero, indicating an infinitely large stacking fault width. We will be discussing further about the consequences of the change in GSFE curve later in this paper.
4. Results
The structural optimization was first performed for single crystal Mg. The total energies as a function of atomic volume were then correlated in order to obtain the relaxed lattice parameters and cohesive energy. Our obtained lattice parameter (a = 3.2, c/a = 1.633) and cohesive energy (1.51 eV/atom) agrees well with other theoretical [43, 50] and experimental [51] observations (a = 3.21, c/a = 1.624 and Ec = 1.5 eV/atom) confirming the reliability of the present calculations. 4
2
Energy per unit area (mJ/m )
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Pure Mg With Al With Ca With Ce With Gd With Li With Si With Sn With Zn With Zr
800
600
400
200
0
0
0.5
1
1.5
2
2.5
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3.5
4
4.5
5
5.5
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Along (Å) Figure 2. Solute effect on Mg on GSFE curve on (0 0 0 1) 1 0 1¯ 0 slip system. 120 USF
Basal Slip γ
ISF
Basal Slip γ
80
60
40
γ
USF
and γ
ISF
2
(mJ/m )
100
20
0
Pure Mg With Al With Ca With Ce With Gd With Li With Si With Sn With Zn With Zr
Figure 3. Solute effect on Mg on basal plane γ USF and γ ISF on (0 0 0 1) 1 0 1¯ 0 slip system.
Si and Sn increased the γUSF energy. It should be pointed out here that reducing the γUSF on the prismatic plane by adding Li, Ca, Zr, or Gd solutes actually facilitates the system to nucleate dislocations on the prismatic plane.
4.2. GSFE: prismatic plane
Nonbasal slip systems may be activated by the alloying elements and play a significant role in improving the mechanical properties. Recently, the GSFE for two prismatic planes were carried out by Wen et al [43] in which they found that the γU S for one prismatic plane {1 1 2¯ 0} (second order) is one order of magnitude higher than the other prismatic plane {1 0 1¯ 0} (first order). Accordingly, we studied the solute effects on the GSFE curve on the {1 0 1¯ 0} prismatic plane. For the prismatic plane {1 0 1¯ 0} 1 1 2¯ 0, the unrelaxed GSFE curve is shown in figure 4. Figure 4 shows that there is one local maximum similar to the basal plane 1 1 2¯ 0, indicating that no dissociation of dislocation into partials could occur. The relaxed [57] unstable (γUSF ) stacking fault energy for pure Mg (230 mJ m−2 ) agreed well with previously calculated result. Figure 5 shows that the solute addition of Li, Ca, Zr and Gd decreased the γUSF energy, whereas Zn, Ce, Al,
4.3. Formability
Magnesium alloys have low formability at room temperature making it difficult and expensive to produce complex structural and body parts [58]. Hence, the knowledge and understanding of solute effects on forming technologies and optimized process parameters for magnesium alloys are needed. The low formability of the magnesium alloys at a room temperature is mainly caused by the limited number of slip systems in their hcp lattice structure. Therefore, in order to make use of magnesium sheet metal to produce complex part geometries, it is necessary to activate additional slip systems other than basal slip, such as prismatic slip systems. Activating prismatic slip 5
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Pure Mg With Al With Ca With Ce With Gd With Li With Si With Sn With Zn With Zr
2
Energy per unit area (mJ/m )
400
300
200
100
0
0
0.2
0.4
0.6
0.8
1
Along [1-210] (Å) Figure 4. Solute effect on Mg on prismatic plane GSFE. 400 Prismatic Slip
2
(mJ/m )
300
γ
USF
200
100
0
Pure Mg With Al With Ca With Gd With Li With Ce With Si With Sn With Zn With Zr
Figure 5. Solute effect on Mg on prismatic plane γ USF .
systems essentially demands lowering the unstable stacking fault energy on the GSFE curve on prismatic slip plane. Calculating the basal and prismatic unstable stacking fault energies, we define the formability parameter χ as χ=
B )X (γusf B (γusf )Mg P (γusf )X P (γusf )Mg
Basal Prismatic Table 1. Formability parameter along with γUSF and γUSF for
Mg with different solute atoms.
(9)
B/P
where γusf is the unstable stacking fault energy on related basal (B) or prismatic (P) slip planes. Subscript Mg represents the case for a pure Mg and X for the cases where X solute atom is substituted. χ is defined in such a manner to estimate the effect of solute atoms with reference to the pure Mg case where χ is set to unity. Table 1 shows the unstable stacking fault energies for basal and prismatic planes for the pure Mg and Mg with different solute atoms. χ parameters are also tabulated for those cases
Atom (X)
Basal γusf
Prismatic γusf
Pure Mg With Al With Ca With Ce With Gd With Li With Si With Sn With Zn With Zr
100.59 83.83 99.90 94.61 102.22 99.02 66.60 72.41 85.01 116.71
236.11 266.84 129.51 90.07 55.64 231.81 275.04 250.04 285.21 166.16
B γusf P γusf
0.43 0.31 0.77 1.05 1.84 0.43 0.24 0.29 0.30 0.70
χ 1.00 0.74 1.81 2.47 4.31 1.00 0.57 0.68 0.70 1.65
in table 1, which shows that Gd, Ce, Ca and Zr solute addition increase the formability parameter χ from unity for the pure Mg case, where all other solute atoms reduce the χ . Thus, the present study suppliments a theoretical understanding 6
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Dislocation half-width (Å)
4 Pure Mg With Al With Ca With Ce With Gd With Li With Si With Sn With Zn With Zr
3.5
3
2.5
2
1.5 0
20
40
60 0
80
Angular variation ( ) Figure 6. Solute effect on Mg on dislocation half-width as a function of dislocation angle.
towards the experimentally observed improvement of room temperature formability of Mg when calcium or Rare Earth (RE) elements are added [27, 59].
been a long time investigation. σp in this section has been calculated following the expression of Jo´os and Duesbery [56] as the following: 2π Kb 2π ζ σp = (10) exp − a a
5. Discussions
where a is the atomic separation perpendicular to the dislocation line. Figure 7 shows the angular dependency of the Peierls stresses. Figure 7 shows that the Peierls stresses are several magnitude greater than the reported observation [7]. This major variation is due to the fact that equation (10) cannot treat the dissociated dislocation as it is based on the sinusoidal approximation of the restoring stress Fmax [15]. Nevertheless, an overall idea can be deduced that the Peierls stress is reduced by solute addition of Ce, Li, Al, Zn, Sn and Si atoms, whereas it gets increased by Gd and Zr solute atoms. If the Jo´os and Duesbery [56] model exponential factor is multiplied by two, then the Peierls stress for edge and screw dislocations decreases to reported values [7] of 0.8 MPa and 10.1 MPa, respectively. We next show the effect of solute atoms on the stacking fault width on Mg. A stacking fault is the region bounded by the two partials: leading and trailing. Continuum elastic theory predicts the stacking fault width or the partial separation distance from the following equation:
5.1. Dislocation properties
The Dislocation half-width (ζ ) can be determined by calculating the disregistry vector f as a function of the angle θ between the burgers vector b and the dislocation line. The dislocation half width is defined in this manner as the atomic distance over which the component of this disregistry vector changes from b4 to 3b . A recent study [15] shows that the 4 dislocation half width calculated in this manner is very similar if one calculates the same from equation (8) (see figure 8 of the reference), where a sinusoidal form of dislocation stress is assumed. The maximum restoring stress Fmax is obtained from the second maximum of the basal GSFE curve along 1 0 1¯ 0. Figure 3 shows the variation of dislocation half-width (ζ ) with respect to the angular orientation of the dislocation from 0◦ for a screw dislocation up to 90◦ for an edge dislocation assuming µ = 17 GPa, ν = 0.29 and b = |b| = 1.842 Å. The ζ increases monotonically with the dislocation angle for pure Mg and Mg with any solute atoms. This is due to the inherent assumption of the sinusoidal form of dislocation stress. Figure 6 also shows that ζ does not alter much from pure Mg when Ca was added but slightly increased when Li and Ce were added and slightly decreased when Gd was added. Other than Gd and Zr, all the other solute elements (Si, Sn, Al and Zn) increased the ζ . The maximum decrement occurred from the pure Mg case when Zr was added due to the fact that the Fmax became a maximum when Zr was added to Mg. Similarly, since Fmax is the minimum with Si addition, ζ is the maximum for Mg with Si addition. The correlation between dislocation mobility through Peierls stress (σp ) and macroscopic materials behavior has
d=
µ bp2 8π γISF
×
2+ν 1−ν
(11)
where bp is the partial burgers vector and γISF is the intrinsic stacking fault energy. Figure 8 shows the variation of stacking fault width as predicted by the equation (11). Figure 8 clearly shows that the stacking fault width does not change much for solute additions when compared to the pure Mg case, except for the Ce addition. This is due to the fact that the γISF , which comes as a denominator to the equation (11), reduces to a minimum for Ce addition in Mg. 7
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Pure Mg With Al With Ca With Ce With Gd With Li With Si With Sn With Zn With Zr
Peierls Stress (GPa)
1.2
1
0.8
0.6
0.4
0.2
0
0
20
40
60
80
0
Angle ( ) Figure 7. Solute effect on Mg on Peierls stress as a function of dislocation angle.
Figure 8. Solute effect on Mg on partial–separation distance.
of the elastic modulus mismatch between the precipitate and the host hcp Mg. If the precipitate elastic modulus is less than or comparable to the Mg host, dislocations can easily cut through the precipitate and be restructured to its original form. If the precipitate is much stronger having large elastic modulus compared to the Mg host, the local matrix becomes impenetrable obstacle to the dislocation motion. Friedel [60] developed a statistical model to calibrate the critical shear stress with randomly distributed precipitates introducing effective spacing between precipitates, volume fraction of the precipitates, Burgers vectors and self forces on the dislocations. Nembach [61] found that the critical shear stress (CSS) for dislocations to penetrate the precipitate is proportional to the µ3/2 and rp0.22 , where µ is modulus difference between precipitate and host matrix and rp is the radius of the spherical precipitate. Recently Takahasi and Ghoniem [1] in their more general and complex model, showed that the CSS was proportional to the µ, such that the CSS
Out of the several solute elements that we studied in this paper, Ce has a unique feature in its electronic state as: [Kr 4d1 0] 5s2 5p6 4f 1 4d1 6s2 . The valence electrons of Ce are from all the orbitals: s, p, d and f; that makes Ce atom with large energy band. We believe that these valence electrons from all the orbitals play a crucial role in Mg matrix, specially when the matrix itself is in a faulted structure. However, further investigation on this field is much needed where one can use hybridization index or advanced electronic structure theory to elucidate this behavior and consequently shed light on ductility enhancement of Mg alloys. 5.2. Precipitation hardening
Precipitation strengthening is dependent on several factors, such as precipitate geometry, spatial arrangement and the relative magnitude of the elastic modulus compared to the host matrix. It is in the scope of our study to estimate the effect 8
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atoms reduces the γISF from pure Mg case. (4) Ce addition leads to the fact that the ISF is close to zero, indicating an infinitely large stacking fault width on basal plane. (5) For basal plane, dislocation half-width (ζ ) does not alter much from pure Mg when Ca was added but slightly increased when Li and Ce were added and slightly decreased when Gd was added. Other than Gd and Zr, all the other solute elements (Si, Sn, Al and Zn) increased the ζ . The maximum decrement occurred from the pure Mg case when Zr was added. (6) Basal plane Peierls stress is reduced by solute addition of Ce, Li, Al, Zn, Sn and Si atoms, whereas it gets increased by Gd and Zr solute atoms. (7) Ce addition increases the stacking fault width enormously as compared to any other solute elements in Mg. (8) For the prismatic plane (1 0 1¯ 0) 1 1 2¯ 0, the solute addition of Li, Ca, Zr and Gd decreased the USF energy, whereas Zn, Ce, Al, Si and Sn increased the γUSF energy. (9) Comparing the basal and prismatic plane GSFE, a parameter to quantify formability is defined and validated that rare earth elements Ce and Gd would increase Mg formability and predict that nonrare-earth elements like Ca and Zr solutes would increase the same. (10) It is noted that Mg2 Si precipitate would enhance the precipitate hardening the most, compared to any other precipitates formed by the elements that are studied in this article.
Table 2. Relative shear moduli for precipitates, formed by solute atoms considered in this study.
Solute atom (X)
Precipitate phase
µppt (GPa)
µ (GPa)
Al Si Sn Ca Ce Gd Li Zn
Mg17 Al12 Mg2 Si Mg2 Sn Mg2 Ca MgCe MgGd MgLi (poly) MgZn2
22 [9] 47.6 [30] 34.2 [30] 21.3 [30] 25.39 [23] 29.06 [23] 24 [31] 33.7 [32]
5 30.6 17.2 4.3 8.39 12.06 7.0 16.7
was in the following form, β1
τ=
µb2 α1 rp bβ1 +1 L
(12)
where b is the burgers vector, α1 and β1 are the material parameters and L is the spacing between the precipitates. Consider a situation where the solute atoms are added beyond their solubility, such that they form specific precipitates at ambient conditions. Here we evaluated this situation by comparing the elastic constants for all solute elements, except Zr (since there is not a particular phase of Zr with Mg). From the table 2 one can observe that the µ is quite high for Mg2 Si, compared to any other intermetalics studied in this article. Accordingly, for a situation where Si is excessively added to the Mg host and Mg2 Si precipitate forms dispersively, the CSS for the dislocations to penetrate the precipitate is the highest compared to the other precipitates studied in this article.
Acknowledgments
This work is in part supported by both the Center for Advanced Vehicular Systems (CAVS) at the Mississippi State University, USA and Thematic Unit of Excellence on Computational Materials Science at S N Bose National Centre for Basic Sciences at Kolkata, India. Computer time allocation has been provided by both the High Performance Computing Collaboratory (HPC2 ) at Mississippi State University and the computational resources at IUAC, New Delhi, India.
6. Conclusions
We calculated the GSFE curves for pure Mg and Mg with several solute elements using Density Functional Theory as implemented in VASP. The calculated GSFE results are used in the Peierls–Nabarro model to extract the basal plane dislocation properties for Mg alloy design. Nine solute elements are substituted in Mg in order to observe their effect on the basal-plane dislocation-width, stacking fault width and Peierls stress. Solute effects for those nine elements are also examined for the prismatic plane GSFE. In order to compare the basal and prismatic plane GSFE, a parameter to quantify the formability is defined and estimated for these Mg alloys. Finally, solute strengthening has been estimated with the literature data for the shear modulus for different Mgprecipitates. This study provides a basis for materials design of magnesium, particularly when solute elements become very critical. Our major findings of this work are summarized below: (1) The basal plane (0 0 0 1) 1 1 2¯ 0 GSFE curve shows Zr, Gd and Ca increases the maximum GSFE value, whereas Ce, Li, Sn, Al, Zn and Si reduces the maximum GSFE value. (2) The Basal plane (0 0 0 1) 1 0 1¯ 0 GSFE curve shows that the substitution of Zr, Gd and Ca increase the USF energy, whereas Li, Ce, Zn, Al, Sn and Si reduce the γUSF energy. (3) The Basal plane (0 0 0 1) 1 0 1¯ 0 GSFE curve shows that the ISF is increased by Li addition only and all the other solute
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Solute effect on basal and prismatic slip systems of Mg
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 445004 (10pp)
doi:10.1088/0953-8984/26/44/445004
Solute effect on basal and prismatic slip systems of Mg Amitava Moitra1 , Seong-Gon Kim2,3 and M F Horstemeyer4,5 1
Thematic Unit of Excellence on Comuputational Materials Science, S.N. Bose National Centre for Basic Sciences, Sector-III, Block-JD, Salt Lake, Kolkata-700098, India 2 Department of Physics and Astronomy, Mississippi State University, Mississippi State, MS 39762, USA 3 Center for Computational Sciences, Mississippi State University, Mississippi State, MS 39762, USA 4 Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS 39762, USA 5 Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS 39762, USA E-mail: [email protected] Received 21 May 2014, revised 19 August 2014 Accepted for publication 3 September 2014 Published 2 October 2014 Abstract
In an effort to design novel magnesium (Mg) alloys with high ductility, we present a first principles data based on the Density Functional Theory (DFT). The DFT was employed to calculate the generalized stacking fault energy curves, which can be used in the generalized Peierls–Nabarro (PN) model to study the energetics of basal slip and prismatic slip in Mg with and without solutes to calculate continuum scale dislocation core widths, stacking fault widths and Peierls stresses. The generalized stacking fault energy curves for pure Mg agreed well with other DFT calculations. Solute effects on these curves were calculated for nine alloying elements, namely Al, Ca, Ce, Gd, Li, Si, Sn, Zn and Zr, which allowed the strength and ductility to be qualitatively estimated based on the basal dislocation properties. Based on our multiscale methodology, a suggestion has been made to improve Mg formability. Keywords: density functional theory, alloy design, generalized stacking fault energy (Some figures may appear in colour only in the online journal)
materials properties or designing novel materials [3, 4]. In the present paper first principles Density Functional Theory (DFT) is used to calculate the generalized stacking fault energies (GSFE). These GSFE are then used to link directly to the generalized Peierls–Nabarro model to study the dislocation properties for Mg. Magnesium alloys have become one of the most promising materials in the automotive, aerospace, biomedical and athletic industries due to its strength to weight ratio, stiffness to weight ratio, biodegradability and abundance [5]. Currently, further use of Mg–alloys in structural applications are restricted due to various metallurgical issues including formability, crash worthiness, strength, corrosion, creep resistance, flammability and fatigue fracture. It is well known for more than a century ago [6–11] that, at sufficiently low temperatures or high strain rates, deformation in double lattice structures, for which hexagonal close-packed
1. Introduction
Alloy design often requires conflicting materials properties to be intelligently optimized for specific usage. Heat treatments, composition control, precipitate strengthening and cold working have a vital role on material’s major functionalities, such as ductility, maximum strength and corrosion resistance to name a few [1]. Due to extraordinary improvements in modern computing facilities, computational alloy design has recently become far more feasible [2]. Nevertheless, the technological challenges are still restrictive either due to spatial (length–scale) or temporal (time–scale) limitations on nanoscale atomistic simulations or the incapability of capturing minute details on continuum scale phase-field or finite-element modeling. A multiscale modeling methodology incorporating these atomistic and continuum scale models (hierarchical or concurrent methods) has a fairly long history in predicting 0953-8984/14/445004+10$33.00
1
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(HCP) structures are an archetypal example is driven by both slip and twinning and their dynamic interactions. Mg is a particular HCP metal where, in contrast to all other HCP metals, twinning is able to consume the entire matrix below 7% of the plastic strain [12, 13]. The associated slip–twin and twin–twin interactions lead to a strong inflection in the stress–strain behavior from convex-up to concave-down. This behavior has been correlated to strong plastic anisotropy and limited ductility, which both stand in the way of a broader industrial applications of Mg [14]. Because dislocations are integral to both slip and twinning (twinning is just a liberation of twin partials), dislocation and twin disconnection core structures must be described at the atomic scale [15]. On one hand, the dislocation core properties dictate dislocation mobility and the ease by which slip and twinning could nucleate. On the other hand, the dislocation core structure is responsible for short range dislocation interactions including slip–slip (latent hardening), slip–twin and twin-twin. Thus, understanding how the crystal structure and the alloying elements influence the dislocation core structure remains a key feature to understand [5] and enhance the mechanical properties of Mg alloys [16]. Numerical modeling for enhancement of such mechanical properties for Mg alloys are underway at several length scales [17–19]. For example, Yoshinaga and Horiuchi [10] investigated from tensile tests at various temperatures that the Li addition on Mg matrix increases the critically resolved shear stress (CRSS) for basal slip systems and the tendency for crossslip to prism planes, thus increasing the strength and ductility, respectively. Out of several Mg alloys, Mg–Al is the most extensively studied system with a focus on improving strength and ductility behavior [20, 21]. One of the possible alternatives to increase ductility in both low and high temperature regimes is to facilitate crossslip of a-type dislocations from basal to prismatic planes. This thermal cross slip can be aided by either lowering the SFE in prism planes or increasing it for basal planes through addition of solutes, hence reducing the anisotropy in slip systems. An ideal combination of both mechanisms remains difficult to achieve. On one hand, an increase of basal SFE can promote bowing and constriction of screw dislocations, so solutes with a positive chemical misfit are needed. Yasi et al [5] developed first principal solute strengthening models that were able to accurately determine dislocation core geometries and compute solute misfits, which pertain to both size misfit and chemical misfit. The solute misfit allows quantification of the volumetric expansion/compression of the lattice and local slip effects in both far-field and core and as such is a good approximation of the solute/dislocation interaction energy. The choice of good strengthening solutes must thus consider a compromise between a large chemical misfit for strengthening capacity and a low chemical misfit for high solubility. On the other hand, decreasing the SFE on prismatic plane depends on the effect of solutes on the energy barrier landscape, impeding formation and movement of double-kinks (high temperatures) and constriction and bowing of screw dislocations (low temperatures). To increase the ductility and also the strength, Ce has been added to Mg [22, 23]. Plastic deformation behaviors
for Mg–Ce–Zn–Zr alloys are studied by Yu et al [22]. Age hardening responses and creep resistances were observed for Mg–Gd and Mg–Gd–Zn systems [24] and attributed to the long period stacking order (LPSO) structure [25]. For biologically friendly alloys, Ca solutes were added to Mg forming Mg2 Ca intermetallics within the alloy [26, 27]. The effects of Si, Sn, Zr and Zn to increase the mechanical strength of Mg was also evaluated by computation and experiment [5, 28, 29, 31, 32]. Although the above mentioned literature have contributed towards the improvement of Mg alloys, there has been a scarcity towards the understanding of how the materials properties could be modified in presence of different solute atoms. This paper focuses on such a particular material property called Formability, which stand in the way of a broader industrial applications of Mg. Defining this Formability parameter, our approach is two folded: one to identify if this parameter can correctly capture the fact that the rare earth solute elements enhance the Mg formability, as found in experimental observation and two to find out additional nonrare-earth solute elements, that can also enhance this material property. Additionally, we estimate the critical shear stress related to solute strengthening for several solute atoms. All of these studies that analyzed different solutes in Mg provided a basis for the current solute (Al, Ca, Ce, Gd, Li, Si, Sn, Zn, Zr) study for alloying effects on the Mg GSFEs in order to provide information for the generalized Peierls–Nabarro dislocation model. Peierls [33], in his hybrid model, first incorporated the dislocation core information into a continuum framework, which was further developed by Nabarro [34] and Eshelby [35] in order to estimate the lattice friction to dislocation motion. Later, Vitek, with a comprehensive approach, introduced the concept of GSFE curve [36, 37] and further discussed the implications in numerous studies in the literature [38, 39]. Let us consider a perfect crystal that is cut across a single plane into two parts that are subjected to a relative displacement f and relaxed. These relaxed lattices have a surplus energy per unit area γ f. As f spans the lattice, γ (f) generates the GSFE curve. The procedure can be performed for basal, prismatic and pyramidal planes for hcp Mg crystal systems [40]. The importance of this curve is that the interfacial restoring force can be deduced from the slope of the γ (f) curve, such that Fb (f) = −∇(γ (f)).
(1)
Hence, GSFE calculations have drawn a lot of interest to predict the materials properties [41, 42]. Due to the basal to prismatic slip system anisotropies and their lack of available slip systems, pure Mg has a poor formability. A systematic study on the stacking faults in pure Mg has been performed recently with DFT [43] to show that the unstable stacking fault energy for the first order prismatic plane {1 0 1¯ 0} is one order of magnitude higher than the (0 0 0 1) basal plane. The sources of producing a + c pyramidal dislocations are hard to understand, where nucleation of a + c dislocation is the energetically critical step for plastic deformation of hcp metals such as Mg. Activation of pyramidal a + c dislocation is energetically difficult in general and has not been clearly understood. For example, grain refinement, precipitation 2
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hardening, shear banding, decreased c/a ratio and changed Peierls potentials cannot be the underlying mechanism(s) causing the observed higher activity of the a + c dislocations in MgY alloys [44]. Furthermore, mechanical processes such as, rolling or casting for Mg alloys, involve more a type dislocation and twinning and the improved formability is achieved due to the microstructure involving both grain sizes and the dynamic re-crystallization process. Since the easiest slip system investigated in hcp metals is a basal plane slip system with an a-axis slip direction or a first order prismatic {1 0 1¯ 0} plane slip system, we restrict ourselves to calculate the effect of solute atoms only on basal (0 0 0 1) and {1 0 1¯ 0} prismatic plane GSFE curve. Since basal slip (0 0 0 1) 1 1 2¯ 0 was also found at elevated temperatures, along with (0 0 0 1) 1 0 1¯ 0 dominant slip mechanism at room temperature, we calculate basal plane GSFE on both of these directions. After a brief introduction to the semi discrete Peierls– Nabarro model in section 2, we describe our method and model development in section 3. In section 4, we present our results: we start with the solute effect on Mg GSFE both for basal (section 4.1) and prismatic planes (section 4.2). In section 4.3, we define the parameter Formability and show the ability of this parameter to capture the fact that the rare earth solute elements enhance the Mg formability and the parameter predicts that Ca and Zr solute atoms would also increase Mg formability. In section 5, we discuss how the basal GSFE curves can be correlated with continuum scale dislocation properties (section 5.1) and provide an estimation of precipitate strengthening (section 5.2) related to the solute elements studied in this article. Finally in section 6, we summarize and conclude.
where x is the coordinate of the atomic row and f (x) is the disregistry vector of those atomic row at x connected to the dislocation density by ρ(x) = df (x)/dx
(3)
and K is the materials constant based on the elastic properties, such that for an isotropic solid, µ sin2 θ 2 + cos θ (4) K= 2π 1 − ν where µ and ν are the shear modulus and Poisson’s ratio, respectively. θ is the angle between the burgers vector and the dislocation line, such that: µ for θ =0 Ks = 2π K= (5) µ for θ = π2 . Ke = 2π(1−ν) The dislocation density is such that it satisfies the normalization condition: ∞ ∞ df (x ) ρ(x )dx = = b. (6) dx −∞ −∞ In the PN model a simple sinusoidal form for Fb (f (x)) is assumed and the disregistry vector is reduced to an analytical form: b −1 x b f (x) = tan + (7) π ζ 2 where ζ is the dislocation core half width, given by: ζ =
2. Semi discrete Peierls–Nabarro model
Kb 2Fmax
(8)
where Fmax is the maximum restoring stress.
In the context of multiscale modeling, Peierels–Nabarro model based on lattice friction and dislocation motion, details on the dislocation core that are incorporated essentially into an continuum framework. When the dislocations move along the glide plane, the burgers vector lies on that glide plane. If the burgers vector is along the dislocation line (we consider this as θ = 0), a screw dislocation forms. On the other hand, the burgers vector is along the glide direction (we consider this as θ = 90) for an edge dislocation. For a mixed dislocation the burgers vector will have both the edge and screw components. This glide plane is assumed to contain the dislocation misfits, in the Peierls–Nabarro (PN) formalism. The PN formalism consider two semi-infinite elastic media separated by the glide plane that contains the dislocation. A continuous distribution of dislocations with density ρ(x) can be achieved between the elastic medias. The idea there is to map the deformation of the lattice caused by the dislocations’ displacement field to the adjacent elastic media. Hence, a lattice restoring stress is generated due to the interfacial misfit caused by a dislocation. This restoring force Fb (f (x)) across the glide plane can be evaluated from the PN integrodifferential equation: ∞ 1 df (x ) , dx = Fb (f (x)) (2) K dx −∞ x − x
3. Method and model development
All the energy calculations and geometry optimization in this present work were performed using ab-initio Density Functional Theory, using the projector augmented-wave (PAW) method as implemented in VASP (Vienna ab initio Simulation Package) [45]. For the treatment of electron exchange and correlation, we use the generalized gradient approximation (GGA) using the Perdew–Burke–Ernzerhof scheme [46, 47]. All simulations are carried out using a supercell of 64 atoms corresponding to a double unit cell. The plane-wave cut off energy is set to 300 eV in most of the calculations, 400 eV is used for Mg–Ce calculations. The conjugate-gradient [48] method is used to relax the ions. Geometry relaxations are performed until the energy difference between two successive ionic optimizations is less than 0.001 eV. The Brillouin zone is sampled with a density equivalent to 98 k-points using Monkhorst–Pack scheme [49]. For GSFE curve calculations, an orthonormal supercell was used with 2 × 1 × 12 periodic supercell along [1 2¯ 1 0], [1 0 1¯ 0] and [0 0 0 1] directions, respectively. The supercell contained 24 atomic planes along the c [0 0 0 1] direction. For the prismatic plane a 2 × 12 × 1 supercell was used. 3
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Pure Mg With Al With Ca With Ce With Gd With Li With Si With Sn With Zn With Zr
2
Energy per unit area (mJ/m )
400
300
200
100
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Along [1-210] (Å) Figure 1. Solute effect on Mg on GSFE curve on (0 0 0 1) 1 2¯ 1 0 slip system.
For both of the cases, the simulation cells contained 96 atoms with 12 Åvacuum on top. Initially, a defect-free basal surface supercell was fully relaxed (both along unit cell vectors and internal atomic coordinates). This calculation was then followed up by incorporating a substitutional defect of a solute atom and performing a full internal atomic relaxation. For a large unit cell, the solute atom did not affect the unit cell vectors, but only the atoms adjacent to the solute atom were perturbed. In one hand, the solute concentration gets significantly higher with periodic boundary condition, the DFT cannot handle too many atoms, in the other. Thus a reasonable compromise lead us towards a 4 × 2 × 2 hcp Mg system with 64 atoms, where the formation energies reasonably converged until the atomic forces are less than 0.01 eV A−1 . Cross-section of the supercell was chosen in such a way that when a solute atom replaced one Mg atom on the twelfth atomic plane, above which simple shear takes place, the planar coverage becomes 2.8 × 1014 cm−2 for the basal plane and 3.0 × 1014 cm−2 for the prismatic plane. Such a high value of planar solute concentration reflects only the limitation of DFT based atomistic simulations, at our present computational facility. The convergence tests with respect to the k-points and number of planes confirms that the error bar for the total energy is less than 1 meV/atom.
To understand the role of the solute atoms within Mg, we located the solute atoms at the interstitial and substitutional sites as detailed in our earlier report [52]. It was found that all the solute atoms that were considered in this study preferred to stay as a substituting element in bulk Mg. 4.1. GSFE: basal plane
Since the basal plane is the densest atomic plane for HCP Mg crystals, it is the typical slip plane. The basal plane (0 0 0 1) 1 1 2¯ 0 GSFE curve is shown in figure 1, which indicates that there is no local minimum, only a single maximum indicating no dissociation of the dislocation into partials. Also, the maximum value for pure Mg matches exactly with previously calculated results from literature [43]. Figure 1 shows that the substitution of Zr, Gd and Ca increases the maximum GSFE value, whereas Ce, Li, Sn, Al, Zn and Si reduces the maximum GSFE value (It is important to note here that the Zr atom increases the maximum the most and the Si atom reduces the maximum the most.). The GSFE curve indicates the solute strengthening occurs when Zr, Gd and Ca atoms are added and solute softening occurs when Ce, Li, Sn, Al, Zn and Si atoms are added. The Basal plane (0 0 0 1) 1 0 1¯ 0 GSFE curve, as shown in figure 2, reveals that there is one local minimum, indicating dissociation of dislocation into partials. The unstable (γUSF ) and intrinsic stacking (γISF ) fault energies for pure Mg agrees well with previously calculated results [43]. Figure 2 shows that the substitution of Zr, Gd and Ca increase the γUSF energy similar to figure 1, whereas Li, Ce, Zn, Al, Sn and Si reduce the γUSF energy. On the other hand the γISF is increased by Li addition only and all the other solute atoms reduces the γISF from pure Mg case. These results, further elaborated in figure 3, quantitatively agree with the literature value [20, 43, 53–55]. It is remarkable that for the Ce addition the γISF is close to zero, indicating an infinitely large stacking fault width. We will be discussing further about the consequences of the change in GSFE curve later in this paper.
4. Results
The structural optimization was first performed for single crystal Mg. The total energies as a function of atomic volume were then correlated in order to obtain the relaxed lattice parameters and cohesive energy. Our obtained lattice parameter (a = 3.2, c/a = 1.633) and cohesive energy (1.51 eV/atom) agrees well with other theoretical [43, 50] and experimental [51] observations (a = 3.21, c/a = 1.624 and Ec = 1.5 eV/atom) confirming the reliability of the present calculations. 4
2
Energy per unit area (mJ/m )
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Pure Mg With Al With Ca With Ce With Gd With Li With Si With Sn With Zn With Zr
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600
400
200
0
0
0.5
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2
2.5
3
3.5
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4.5
5
5.5
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Along (Å) Figure 2. Solute effect on Mg on GSFE curve on (0 0 0 1) 1 0 1¯ 0 slip system. 120 USF
Basal Slip γ
ISF
Basal Slip γ
80
60
40
γ
USF
and γ
ISF
2
(mJ/m )
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0
Pure Mg With Al With Ca With Ce With Gd With Li With Si With Sn With Zn With Zr
Figure 3. Solute effect on Mg on basal plane γ USF and γ ISF on (0 0 0 1) 1 0 1¯ 0 slip system.
Si and Sn increased the γUSF energy. It should be pointed out here that reducing the γUSF on the prismatic plane by adding Li, Ca, Zr, or Gd solutes actually facilitates the system to nucleate dislocations on the prismatic plane.
4.2. GSFE: prismatic plane
Nonbasal slip systems may be activated by the alloying elements and play a significant role in improving the mechanical properties. Recently, the GSFE for two prismatic planes were carried out by Wen et al [43] in which they found that the γU S for one prismatic plane {1 1 2¯ 0} (second order) is one order of magnitude higher than the other prismatic plane {1 0 1¯ 0} (first order). Accordingly, we studied the solute effects on the GSFE curve on the {1 0 1¯ 0} prismatic plane. For the prismatic plane {1 0 1¯ 0} 1 1 2¯ 0, the unrelaxed GSFE curve is shown in figure 4. Figure 4 shows that there is one local maximum similar to the basal plane 1 1 2¯ 0, indicating that no dissociation of dislocation into partials could occur. The relaxed [57] unstable (γUSF ) stacking fault energy for pure Mg (230 mJ m−2 ) agreed well with previously calculated result. Figure 5 shows that the solute addition of Li, Ca, Zr and Gd decreased the γUSF energy, whereas Zn, Ce, Al,
4.3. Formability
Magnesium alloys have low formability at room temperature making it difficult and expensive to produce complex structural and body parts [58]. Hence, the knowledge and understanding of solute effects on forming technologies and optimized process parameters for magnesium alloys are needed. The low formability of the magnesium alloys at a room temperature is mainly caused by the limited number of slip systems in their hcp lattice structure. Therefore, in order to make use of magnesium sheet metal to produce complex part geometries, it is necessary to activate additional slip systems other than basal slip, such as prismatic slip systems. Activating prismatic slip 5
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Pure Mg With Al With Ca With Ce With Gd With Li With Si With Sn With Zn With Zr
2
Energy per unit area (mJ/m )
400
300
200
100
0
0
0.2
0.4
0.6
0.8
1
Along [1-210] (Å) Figure 4. Solute effect on Mg on prismatic plane GSFE. 400 Prismatic Slip
2
(mJ/m )
300
γ
USF
200
100
0
Pure Mg With Al With Ca With Gd With Li With Ce With Si With Sn With Zn With Zr
Figure 5. Solute effect on Mg on prismatic plane γ USF .
systems essentially demands lowering the unstable stacking fault energy on the GSFE curve on prismatic slip plane. Calculating the basal and prismatic unstable stacking fault energies, we define the formability parameter χ as χ=
B )X (γusf B (γusf )Mg P (γusf )X P (γusf )Mg
Basal Prismatic Table 1. Formability parameter along with γUSF and γUSF for
Mg with different solute atoms.
(9)
B/P
where γusf is the unstable stacking fault energy on related basal (B) or prismatic (P) slip planes. Subscript Mg represents the case for a pure Mg and X for the cases where X solute atom is substituted. χ is defined in such a manner to estimate the effect of solute atoms with reference to the pure Mg case where χ is set to unity. Table 1 shows the unstable stacking fault energies for basal and prismatic planes for the pure Mg and Mg with different solute atoms. χ parameters are also tabulated for those cases
Atom (X)
Basal γusf
Prismatic γusf
Pure Mg With Al With Ca With Ce With Gd With Li With Si With Sn With Zn With Zr
100.59 83.83 99.90 94.61 102.22 99.02 66.60 72.41 85.01 116.71
236.11 266.84 129.51 90.07 55.64 231.81 275.04 250.04 285.21 166.16
B γusf P γusf
0.43 0.31 0.77 1.05 1.84 0.43 0.24 0.29 0.30 0.70
χ 1.00 0.74 1.81 2.47 4.31 1.00 0.57 0.68 0.70 1.65
in table 1, which shows that Gd, Ce, Ca and Zr solute addition increase the formability parameter χ from unity for the pure Mg case, where all other solute atoms reduce the χ . Thus, the present study suppliments a theoretical understanding 6
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Dislocation half-width (Å)
4 Pure Mg With Al With Ca With Ce With Gd With Li With Si With Sn With Zn With Zr
3.5
3
2.5
2
1.5 0
20
40
60 0
80
Angular variation ( ) Figure 6. Solute effect on Mg on dislocation half-width as a function of dislocation angle.
towards the experimentally observed improvement of room temperature formability of Mg when calcium or Rare Earth (RE) elements are added [27, 59].
been a long time investigation. σp in this section has been calculated following the expression of Jo´os and Duesbery [56] as the following: 2π Kb 2π ζ σp = (10) exp − a a
5. Discussions
where a is the atomic separation perpendicular to the dislocation line. Figure 7 shows the angular dependency of the Peierls stresses. Figure 7 shows that the Peierls stresses are several magnitude greater than the reported observation [7]. This major variation is due to the fact that equation (10) cannot treat the dissociated dislocation as it is based on the sinusoidal approximation of the restoring stress Fmax [15]. Nevertheless, an overall idea can be deduced that the Peierls stress is reduced by solute addition of Ce, Li, Al, Zn, Sn and Si atoms, whereas it gets increased by Gd and Zr solute atoms. If the Jo´os and Duesbery [56] model exponential factor is multiplied by two, then the Peierls stress for edge and screw dislocations decreases to reported values [7] of 0.8 MPa and 10.1 MPa, respectively. We next show the effect of solute atoms on the stacking fault width on Mg. A stacking fault is the region bounded by the two partials: leading and trailing. Continuum elastic theory predicts the stacking fault width or the partial separation distance from the following equation:
5.1. Dislocation properties
The Dislocation half-width (ζ ) can be determined by calculating the disregistry vector f as a function of the angle θ between the burgers vector b and the dislocation line. The dislocation half width is defined in this manner as the atomic distance over which the component of this disregistry vector changes from b4 to 3b . A recent study [15] shows that the 4 dislocation half width calculated in this manner is very similar if one calculates the same from equation (8) (see figure 8 of the reference), where a sinusoidal form of dislocation stress is assumed. The maximum restoring stress Fmax is obtained from the second maximum of the basal GSFE curve along 1 0 1¯ 0. Figure 3 shows the variation of dislocation half-width (ζ ) with respect to the angular orientation of the dislocation from 0◦ for a screw dislocation up to 90◦ for an edge dislocation assuming µ = 17 GPa, ν = 0.29 and b = |b| = 1.842 Å. The ζ increases monotonically with the dislocation angle for pure Mg and Mg with any solute atoms. This is due to the inherent assumption of the sinusoidal form of dislocation stress. Figure 6 also shows that ζ does not alter much from pure Mg when Ca was added but slightly increased when Li and Ce were added and slightly decreased when Gd was added. Other than Gd and Zr, all the other solute elements (Si, Sn, Al and Zn) increased the ζ . The maximum decrement occurred from the pure Mg case when Zr was added due to the fact that the Fmax became a maximum when Zr was added to Mg. Similarly, since Fmax is the minimum with Si addition, ζ is the maximum for Mg with Si addition. The correlation between dislocation mobility through Peierls stress (σp ) and macroscopic materials behavior has
d=
µ bp2 8π γISF
×
2+ν 1−ν
(11)
where bp is the partial burgers vector and γISF is the intrinsic stacking fault energy. Figure 8 shows the variation of stacking fault width as predicted by the equation (11). Figure 8 clearly shows that the stacking fault width does not change much for solute additions when compared to the pure Mg case, except for the Ce addition. This is due to the fact that the γISF , which comes as a denominator to the equation (11), reduces to a minimum for Ce addition in Mg. 7
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Pure Mg With Al With Ca With Ce With Gd With Li With Si With Sn With Zn With Zr
Peierls Stress (GPa)
1.2
1
0.8
0.6
0.4
0.2
0
0
20
40
60
80
0
Angle ( ) Figure 7. Solute effect on Mg on Peierls stress as a function of dislocation angle.
Figure 8. Solute effect on Mg on partial–separation distance.
of the elastic modulus mismatch between the precipitate and the host hcp Mg. If the precipitate elastic modulus is less than or comparable to the Mg host, dislocations can easily cut through the precipitate and be restructured to its original form. If the precipitate is much stronger having large elastic modulus compared to the Mg host, the local matrix becomes impenetrable obstacle to the dislocation motion. Friedel [60] developed a statistical model to calibrate the critical shear stress with randomly distributed precipitates introducing effective spacing between precipitates, volume fraction of the precipitates, Burgers vectors and self forces on the dislocations. Nembach [61] found that the critical shear stress (CSS) for dislocations to penetrate the precipitate is proportional to the µ3/2 and rp0.22 , where µ is modulus difference between precipitate and host matrix and rp is the radius of the spherical precipitate. Recently Takahasi and Ghoniem [1] in their more general and complex model, showed that the CSS was proportional to the µ, such that the CSS
Out of the several solute elements that we studied in this paper, Ce has a unique feature in its electronic state as: [Kr 4d1 0] 5s2 5p6 4f 1 4d1 6s2 . The valence electrons of Ce are from all the orbitals: s, p, d and f; that makes Ce atom with large energy band. We believe that these valence electrons from all the orbitals play a crucial role in Mg matrix, specially when the matrix itself is in a faulted structure. However, further investigation on this field is much needed where one can use hybridization index or advanced electronic structure theory to elucidate this behavior and consequently shed light on ductility enhancement of Mg alloys. 5.2. Precipitation hardening
Precipitation strengthening is dependent on several factors, such as precipitate geometry, spatial arrangement and the relative magnitude of the elastic modulus compared to the host matrix. It is in the scope of our study to estimate the effect 8
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atoms reduces the γISF from pure Mg case. (4) Ce addition leads to the fact that the ISF is close to zero, indicating an infinitely large stacking fault width on basal plane. (5) For basal plane, dislocation half-width (ζ ) does not alter much from pure Mg when Ca was added but slightly increased when Li and Ce were added and slightly decreased when Gd was added. Other than Gd and Zr, all the other solute elements (Si, Sn, Al and Zn) increased the ζ . The maximum decrement occurred from the pure Mg case when Zr was added. (6) Basal plane Peierls stress is reduced by solute addition of Ce, Li, Al, Zn, Sn and Si atoms, whereas it gets increased by Gd and Zr solute atoms. (7) Ce addition increases the stacking fault width enormously as compared to any other solute elements in Mg. (8) For the prismatic plane (1 0 1¯ 0) 1 1 2¯ 0, the solute addition of Li, Ca, Zr and Gd decreased the USF energy, whereas Zn, Ce, Al, Si and Sn increased the γUSF energy. (9) Comparing the basal and prismatic plane GSFE, a parameter to quantify formability is defined and validated that rare earth elements Ce and Gd would increase Mg formability and predict that nonrare-earth elements like Ca and Zr solutes would increase the same. (10) It is noted that Mg2 Si precipitate would enhance the precipitate hardening the most, compared to any other precipitates formed by the elements that are studied in this article.
Table 2. Relative shear moduli for precipitates, formed by solute atoms considered in this study.
Solute atom (X)
Precipitate phase
µppt (GPa)
µ (GPa)
Al Si Sn Ca Ce Gd Li Zn
Mg17 Al12 Mg2 Si Mg2 Sn Mg2 Ca MgCe MgGd MgLi (poly) MgZn2
22 [9] 47.6 [30] 34.2 [30] 21.3 [30] 25.39 [23] 29.06 [23] 24 [31] 33.7 [32]
5 30.6 17.2 4.3 8.39 12.06 7.0 16.7
was in the following form, β1
τ=
µb2 α1 rp bβ1 +1 L
(12)
where b is the burgers vector, α1 and β1 are the material parameters and L is the spacing between the precipitates. Consider a situation where the solute atoms are added beyond their solubility, such that they form specific precipitates at ambient conditions. Here we evaluated this situation by comparing the elastic constants for all solute elements, except Zr (since there is not a particular phase of Zr with Mg). From the table 2 one can observe that the µ is quite high for Mg2 Si, compared to any other intermetalics studied in this article. Accordingly, for a situation where Si is excessively added to the Mg host and Mg2 Si precipitate forms dispersively, the CSS for the dislocations to penetrate the precipitate is the highest compared to the other precipitates studied in this article.
Acknowledgments
This work is in part supported by both the Center for Advanced Vehicular Systems (CAVS) at the Mississippi State University, USA and Thematic Unit of Excellence on Computational Materials Science at S N Bose National Centre for Basic Sciences at Kolkata, India. Computer time allocation has been provided by both the High Performance Computing Collaboratory (HPC2 ) at Mississippi State University and the computational resources at IUAC, New Delhi, India.
6. Conclusions
We calculated the GSFE curves for pure Mg and Mg with several solute elements using Density Functional Theory as implemented in VASP. The calculated GSFE results are used in the Peierls–Nabarro model to extract the basal plane dislocation properties for Mg alloy design. Nine solute elements are substituted in Mg in order to observe their effect on the basal-plane dislocation-width, stacking fault width and Peierls stress. Solute effects for those nine elements are also examined for the prismatic plane GSFE. In order to compare the basal and prismatic plane GSFE, a parameter to quantify the formability is defined and estimated for these Mg alloys. Finally, solute strengthening has been estimated with the literature data for the shear modulus for different Mgprecipitates. This study provides a basis for materials design of magnesium, particularly when solute elements become very critical. Our major findings of this work are summarized below: (1) The basal plane (0 0 0 1) 1 1 2¯ 0 GSFE curve shows Zr, Gd and Ca increases the maximum GSFE value, whereas Ce, Li, Sn, Al, Zn and Si reduces the maximum GSFE value. (2) The Basal plane (0 0 0 1) 1 0 1¯ 0 GSFE curve shows that the substitution of Zr, Gd and Ca increase the USF energy, whereas Li, Ce, Zn, Al, Sn and Si reduce the γUSF energy. (3) The Basal plane (0 0 0 1) 1 0 1¯ 0 GSFE curve shows that the ISF is increased by Li addition only and all the other solute
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