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The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study
Accepted Manuscript The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study Xingming Zhang, Jianfeng Tang, Lei Deng, Gu Zhong, Xunlin Liu, Yifan Li, Huiqiu Deng, Wangyu Hu PII:
S0925-8388(17)30159-7
DOI:
10.1016/j.jallcom.2017.01.135
Reference:
JALCOM 40502
To appear in:
Journal of Alloys and Compounds
Received Date: 17 September 2016 Revised Date:
23 December 2016
Accepted Date: 15 January 2017
Please cite this article as: X. Zhang, J. Tang, L. Deng, G. Zhong, X. Liu, Y. Li, H. Deng, W. Hu, The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study, Journal of Alloys and Compounds (2017), doi: 10.1016/j.jallcom.2017.01.135. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study
Huiqiu Dengd, Wangyu Hub
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Xingming Zhanga, Jianfeng Tanga,b, Lei Denga,* , Gu Zhong c, Xunlin Liua, Yifan Lia,
College of Science, Hunan Agricultural University, Changsha 410128, China
b
College of Materials Science and Engineering, Hunan University, Changsha 410082, China
d
Suzhou Research Institute for Nonferrous Metals, Suzhou 215000, China
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c
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a
Department of Applied Physics, School of Physics and Electronics, Hunan University, Changsha 410082, China
Abstract:
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This study aims to elucidate the effects of interstitial impurities on the mechanical properties of vanadium alloys investigated by means of the first-principles computations. For this purpose, we study the generalized stacking-fault energies and the ductility parameters in different slip system to evaluate the strengthening and ductility potentials of interstitial impurities in V alloys. The results show that the generalized stacking-fault energy values for two slip systems all increase significantly after adding C, O, and N impurities in comparison with pure V. Moreover, the impurities located in the slip plane greatly increase the generalized stacking-fault energy, suggesting that the impurity effect is localized. The calculations reveal a remarkably strong dependence of the USF energies on the impurities concentration. Though evaluating the generalized stacking-fault energies and the ductility parameters, we draw a conclusion that impurities not only strengthen the V alloys but also reduce their plasticity. Keyword: Generalized stacking-fault energy; V alloys; First-principles calculations; Ductility; Interstitial impurity
* Corresponding Author. Email: [email protected] (L. Deng) Tel: +86-0731-84618092; Fax: +86-0731-84618092
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Introduction Vanadium (V) alloys are considered as a candidate for blanket structural
materials for fusion reactor systems due to their low-induced activation characteristics, and remarkable elevated temperature mechanical properties in the fusion
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environment[1-3]. Interstitial impurities (O, N, and C) play key roles in the microstructure, influencing important parameters such as the strength and plasticity. With systematic efforts, studies showed that the mechanical strength of V alloys can be improved by high number density of tiny precipitates dispersed in the matrix [4, 5].
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These nanoscale precipitates in V-4Cr-4Ti alloys are Ti-rich and most likely to be Ti-(O, N, C) with the NaCl structure[6]. In addition to its role in forming precipitates,
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interstitial impurities are considered to effectively harden alloys due to the solid-solution strengthening effects[7, 8]. Even these impurities can also be considered as alloying elements in V, analogous to C in steel or O in the Ti alloys. Though it is known that the alloys strength is strongly related to interstitial impurities, there is no detailed understanding of correlation between the mechanical properties
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and the interstitial impurities in V alloys. Therefore, a deeper insight into their influences on the mechanical properties in V alloys is essential and beneficial to the further design of the new style alloys.
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First-principles calculations based on the density functional theory (DFT) have been the most powerful tool for evaluating the atomic interactions and understanding the basic atomic phenomena involved in the bulk materials. Especially, the nucleation
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tendency of dislocations depends on the generalized stacking-fault (GSF) energies which can be obtained by DFT calculations to evaluate the deformation mode in materials. For instance, Siegel[9] studied the GSF energies in selected Ni alloys to elucidate how alloying alters the mechanical properties. Muzyk et al.[10, 11] calculated the GSF energies in Al and Mg alloys. In addition to the influences of substitutional solute, the interstitial atoms have attracted more attentions. Kwasniak et al.[12] discussed the influence of interstitial atoms on deformation mechanism in Ti. Afterwards, they systematically discussed the solid solution strengthening of a-Ti in 2
ACCEPTED MANUSCRIPT all four active glide modes[13]. Recently, Yu et al.[14] investigated the origin of dramatic oxygen solute strengthening effect in Ti. They found that the distortion of the interstitial sites at the screw dislocation core creates a very strong but short-range repulsion for oxygen, which result in a strong pinning effect on screw dislocations and
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strengthening the a-Ti. In this paper we present a systematic study of the effects of interstitial impurities on the mechanical properties of V alloys. For this purpose we employ the first-principles electronic structure calculations to study the GSF energies and the
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ductility parameters in different slip system to evaluate the strengthening and ductility potentials of interstitial impurities in V alloys; and then the calculated results are
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discussed.
Methods
The calculations were performed using the Vienna Ab Initio Simulation Package (VASP) [15, 16] via the projector augmented wave (PAW) method [17, 18] and the
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Perdew-Burke-Ernzerhof function of generalized gradient approximation (GGA-PBE) [19]. The nuclei core with valence electrons on s, p and d orbitals for V and valence electrons on s and p orbitals for impurity atoms are used. To determine the GSF
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energies for (110)<111> slip systems, a 55-atom supercell
(illustrated in Fig. 1(a))
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with 6 (111) layers and 9 host atoms on each layer is employed. A total impurity concentration is 1.85%. The model will ensure that the calculation can remove the interaction of the impurity with its periodic images, which originated from the small model used in the calculation. Similar 6 layers constructions also have successfully been used to study the GSF energies in other systems, such as interstitial N in Fe-N alloys [20] and O in α-Ti [21]. For (112)<111> slip systems, we constructed V supercells (illustrated in Fig. 1(b)) containing 61 atoms with one impurity atom for predicting GSF energies, and the total impurity concentration is 1.67%. Meanwhile, 3
ACCEPTED MANUSCRIPT we consider two octahedral sites (in the slip plane or sub-plane) in each slip system to study the coverage of impurity influence. Similar constructions without defects atoms had been used to study the GSF energies in our previous works [22]. The calculations are obtained with 500 eV plane-wave cutoff. The Monkhorst-Pack mesh of k-points
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was selected to ensure a convergence result: 3×3×1 for the (110)<111> and (112)<111> slip systems. A vacuum width of 10 Å is added to periodical slabs for all configurations. The GSF energies curves are generated with ten equally spaced rigid
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shifts of the two parts of the crystal. For each rigid shift, the atomic positions are relaxed along the Z-axis ([1 1 1] direction) by minimizing the forces to less than 0.02
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eV/Å on each atom. The atoms at the top and bottom layer are fixed during relaxation. The GSF energy γGSF can be formally defined as [23, 24]:
γ GSF =
1 ( ESF − E0 ) S
(1)
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where ESF and E0 are the free energies of the supercell after and before shear deformation, respectively, and S is the area of the faulted region. Based on the PN model, the maximum restoring force τmax of dislocations nucleation can be
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approximated by GSF curve as follow[25]:
τ max = Max ∇ ( γ )
(2)
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Both restoring forces and GSF energies of alloys can be accurately determined by the GSF curves, and their changes are fundamental for understanding the solution strengthening mechanism[13]. In order to evaluate the effects of impurities on crystal plasticity based on the Rice criterion[26], we also determine the ductility parameters (D) with various impurities. The ductility parameters (D) can be formally defined as[26]:
D = γ SF γ US
(3)
where γSF is the surface energy of pure V or V alloys, which is estimated for cleavage between different atomic planes in the slip plane. γUS is the so-called unstable stacking
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Result and Discussion
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3.
The impurities of (C, N and O) are small size atoms, and usually occupy the tetrahedral or octahedral interstices sites in the crystals. The relative stability of an impurity atom between various solution sites has been determined by previous DFT
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studies[28] where the impurity (C, O, and N) prefer the octahedral interstices sites to the tetrahedral interstices ones in the V alloys. Therefore, the initial position for
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impurities is placed in octahedral interstices sites as illustrated in Fig. 1. Fig. 2 presents the GSF energy curves for the (110)<111> and (112)<111> slip systems with C, O, and N impurities in different plane. According to the GSF energy curves, the USF energies γUS and the maximum restoring stress τmax, for Pure V and V alloys (impurity placed in the slip plane) in the different slip systems are listed in Table 1.
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Firstly, from Fig. 2(a) and (c), it can be seen that the GSF energies and maximum restoring stress for two slip systems all increase significantly after adding C, O, and N impurities in comparison with pure V. Moreover, as illustrated in Fig. 2(b) and (d), the
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GSF energies are also determined when impurity is located in sub-plane. Obviously, the impurities located in the slip plane greatly increase the USF energy. Such similar behaviors have been reported in previous studies [13, 20, 29]. Especially, Kwasniak et
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al.[13] studied that the interstitial O significantly improves GSF energies in all slip modes, and the explicit effect is visible only when the O atom is located directly in the glide plane. Therefore, all of this indicates that the impurity effect is localized. In addition, the change in GSF energy with different distances to glide plane are implying segregation tendency [30]. A negative change in SFE implies segregation. Therefore, we conclude that the C, O, and N impurities have no segregation behaviors in the stacking faults since the impurities located in the slip plane present larger GSF energies. 5
ACCEPTED MANUSCRIPT In the classical Peierls-Nabarro (PN) model, Peierls stress σ , the minimum stress required to move a dislocation in one lattice site, can be quantitative predicted by a simple and rigorous analytic formula[31]:
σp =
3 3 a τ max 8 πζ
(4)
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where a is the spacing of atomic planes in the glide plane, τmax the maximum restoring force of dislocations nucleation can be approximated by GSF curve, ζ is the half-width of the dislocation. As listed in Table 1, the maximum restoring stress for
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two slip systems all increase significantly after adding C, O, and N impurities in comparison with pure V. Meanwhile, since the extended dislocation has a reduced SF
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width with increase in USF energy[20], we infer that the doped impurities will effective increase the Peierls stress in V alloy.
As discussing content in the literatures[13, 32], this classic semi-rigid GSF energy calculations approach maybe overestimates the USF energy values, and hide the real slip trace when the considered system includes the interstitial elements (large
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lattice distortion are expected). Therefore, we also used the climbing image-nudged elastic band (CI-NEB) method to determine the GSF energy curve and compare with the semi-rigid approach. From the Fig. 2(e), it should be noted that the differences
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between two approaches can be neglected. Therefore, we can draw a conclusion that the overestimates of unstable stacking fault energy values by the classic "semi -rigid"
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GSFE calculations is absent in our calculations. As mentioned before, the GSF energy values increase significantly after adding
C, O, and N impurities in comparison with pure V for (110)<111> slip system. As an example, the GSF energy from 616.7 to 722.4 mJ/m2 is increased by 17.1% for O impurity, and this effect can be comparable with that of noble metals[22]. As well known, the GSF energy denotes the lowest energy barrier for dislocation nucleation, and is beneficial to the investigation of impurity effects on the mechanical properties of alloys[33]. Note that the dislocation activities of the (110) slip system possibly occur in bcc V and has a dominant function in plastic deformation[22]. Based on this, 6
ACCEPTED MANUSCRIPT we draw a conclusion that impurities can effectively impede the plastic deformation in V alloys by means of suppressing the activation of dislocation. In order to understand the underlying mechanism for impurity effect, partial density of states (DOS) and Bader charge analysis [34] have been performed to study the electron transfers and
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bonding interaction. As shown in Fig. 3, it is clear that there are apparent hybridization interactions between the d states of V atom and the p states of impurity atom. For the stacking fault structure of (110)<111> slip system, the obtained charge for impurity C, O and N are 0.94, 1.17 and 1.31(the order C
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suggesting that the atomic electrons transfer from V host atoms to impurity. Meanwhile, this following order is in consistent with the calculated GSF energies for
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the (110)<111> slip systems, indicating that the electronic effect is the main factor affecting the variations in the GSF energy curves.
As for (112)<111> slip system, and shown in Fig. 2(b), it should be emphasized that the energy peak is not located in the center of the GSF energy curve, indicating the non-generic strengthening effect stemming from the interaction of impurities and
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stacking faults. Through the structural analysis, we found that the new phenomenon is ascribed to the structure transformation which the initial octahedral site for impurity occupied is transformed to the hexahedral site after shear deformation by (112)<111>
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type mode. With O impurity, for example, the corresponding transformation process is illustrated in Fig. (4). Through the shear deformation, one V-O bond (initial V-O distance = 2.08 ) is interrupted and the host V atom away from the impurity. The
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other V atom (initial V-O distance = 1.88Å) still remains the nearest-neighbor distance to impurity atom after the shear deformation. To elucidate the V-impurity bonding in the stacking fault structure, we illustrate the partial density of states (DOS) of the impurity atom and nearest-neighbor V atom (shown in Fig. 5). It is clear that there are apparent hybridization interactions between the d states of V atom and the p states of impurity atom. Since hexahedral sites exist widely in the close-packed crystals, but not in the body-centered cubic (BCC) ones. Meanwhile, note that the BCC (110) plane transforms into the HCP (0001) plane mainly through the (112)<111> shear in the 7
ACCEPTED MANUSCRIPT BCC-to-HCP phase transition according the Burgers mechanisms[35]. In addition, recently, Henning et al.[36] obtained a conclusion that the impurities (C, O, and N) block the α to ω martensitic transformation in titanium due to the initial octahedral site placed with impurity transform to tetra- or hexahedral ones. Therefore, from these
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aspects, we infer that the impediment effect on the martensitic transformation stemming from the impurities reconstructions also exist in the V alloys.
To obtain the further information of the impurity influence, we consider the different concentrations of impurity (C, O, and N) doped in the supercells. Due to the
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pronounced repulsion in the impurity pair[37], the distance between two impurities in the supercell is not less than one lattice constant of host V bulk (approximately 3 ),
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and assuming that the distribution of impurities are homogeneous in the V alloy. Fig. 6 presents the dependence of the USF energies in the V alloy as functions of the impurity concentration in the (110)<111> and (112)<111> slip systems. The calculations reveal a remarkably strong dependence of the GSF energies on the impurities concentration, changing by more than 60 mJ/m2 for (110)<111> slip system
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if the impurity content is increased by 1% of atomic number, and that for (112)<111> slip system, even increased more than 170 mJ/m2. To specially mention, the strong effect of impurity concentration on the GSF energy is also found in the Fe-C
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alloy[29].
Based on Rice analysis of the competition between dislocation emission from a crack tip and crack cleavage [26], we calculate the ductility parameter (D) to examine
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how impurities effect on the ductility of V. According to this analysis, a metal will be ductile one if the ductility parameter satisfies D>3.5. Corresponding surface energies γSF, and ductility parameters D for pure V and V alloys in the different slip systems are
also listed in Table 1. As for pure V, the calculated surface energies for (111) and (112) plane are 2428.0 and 2703.3 mJ/m2 respectively, and yields corresponding ductility parameters D is 3.93 and 3.76, which is consistent with the reported results[26, 38]. Meanwhile, the D value is larger than 3.5 indicating that pure V is an intrinsically ductile metal, and therefore the metal tend to fail by dislocation-mediated slip rather 8
ACCEPTED MANUSCRIPT than by cleavage fracture. By contrast, with the addition of C, O, and N impurities resulting in the D values lower than 3.5, the fracture models of V alloys maybe take a turn from ductility to brittle collapse. In other words, it demonstrates that impurities can reduce the plasticity in V alloys. These results are in accordance with the
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experimental observations[39, 40]. It is also noteworthy that the decreases of the ductility parameters D by adding impurities are mainly due to the increase of USF energies which plays a major role.
Summary
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4.
In summary, first-principles calculations are used to elucidate the effect of
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solutes on the mechanical properties of V alloys. The results show that the GSF energy values for two slip systems all increase significantly after adding C, O, and N impurities in comparison with pure V. Moreover, the impurities located in the slip plane greatly increase the generalized stacking-fault energy, suggesting that the impurity effect is localized. As for (112)<111> slip system, impurities (C, O, and N)
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can block the α to ω martensitic transformation in vanadium. Considering the concentration effect, the calculations reveal a remarkably strong dependence of the USF energies on the impurities concentration, changing by more than 60 mJ/m2 for (110)<111> slip system if the impurity content is increased by 1% of atomic number,
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and that for (112)<111> slip system, even increased more than 170 mJ/m2. Though evaluating the ductility parameters, we draw a conclusion that pure V is an
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intrinsically ductile metal, and therefore the metal tend to fail by dislocation-mediated slip rather than by cleavage fracture. By contrast, with the addition of C, O, and N impurities, the fracture models of V alloys maybe take a turn from ductility to brittle collapse. We conclude that impurities not only strengthen the V alloys but also reduce their plasticity. Our new prediction provides a theoretical basis to design and fabricate novel V alloys with excellent mechanical properties.
Acknowledgements 9
ACCEPTED MANUSCRIPT This work is financially supported by the National Nature Science Foundation of China (51501063, 51301066, and 51371080), the Natural Science Foundation of Hunan Province (No. 14JJ2080), and the Talents Foundation of Hunan Agricultural University (No. 14YJ04). The authors are thankful for the computational resource
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provided by the National Supercomputer Center in Changsha and Tianjin, China.
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ACCEPTED MANUSCRIPT Table 1 Calculated USF energies γUS, maximum restoring stress τmax, surface energies γSF, and ductility parameters D for Pure V and V alloys in the different slip systems. The forces and energies are given in GPa and mJ/m2, respectively. γUS
τmax
Pure V
616.7
7.11
(112)<111>
V+C V+O V+N Pure V V+C V+O V+N
706.3 722.4 727.3 718.5 1054.1 980.5 1066.3
8.29 8.41 8.49 9.09 13.77 12.43 13.71
Rice [26].
b
Lee [38]
3.93, 4.0a 3.33 3.21 3.22 3.76 2.63 2.62 2.61
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a
2428.0, 2280 2636b 2353.3 2339.3 2329.6 2703.3 2777.0 2564.0 2782.3
a
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(110)<111>
D
γSF
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System
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Slip mode
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ACCEPTED MANUSCRIPT Figure Captions Fig. 1. Configurations used to calculate the GSF energies for the (a) (110)<111> and (b) (112)<111> slip systems. The dashed line indicates the slip plane. Normal spheres represent host V atoms, and relatively small ones represent interstitial impurities (C, sub-plane where the impurity is placed.
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O, or N). The value i = 1or 2 represents the octahedral site in the slip plane or
Fig. 2. The GSF energy curves for the (110)<111> and (112)<111> slip systems when the C, O, and N impurities are placed in (a), (c) the slip plane or (b), (d) the sub-plane.
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the GSF energy curve is presented in (e).
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The comparison of the classic semi-rigid method and CI-NEB method for calculating
Fig. 3. Partial DOS of the impurity atom and nearest-neighbor V atom in the stacking fault structure of (110)<111> slip system: (a)C, (b)O, and (c)N. The black solid line corresponds to the d-projected DOS of the V atom; the red dashed-dotted line represents the p-projected DOS of the impurity atom.
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Fig. 4. The transformation process of impurities from initial octahedral site to the hexahedral one after shear deformation by (112)<111> type glide mode. Normal spheres represent host V atoms, and relatively small ones represent interstitial
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impurities.
Fig. 5. Partial DOS of the impurity atom and nearest-neighbor V atom in the stacking fault structure of (112)<111> slip system: (a)C, (b)O, and (c)N. The black solid line
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corresponds to the d-projected DOS of the V atom; the red dashed-dotted line represents the p-projected DOS of the impurity atom. Fig. 6. Dependence of the USF energies in the V alloy as functions of the impurity concentration in the (a) (110)<111> and (b) (112)<111> slip systems.
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The influence of interstitial impurities on GSF energy is systematically studied. Interstitial impurities block the α to ω martensitic transformation in V alloys. The impurities effects on the ductility of V alloys are evaluated. The dependence of the USF energies on the impurities concentration is determined. Impurities not only strengthen the V alloys but also reduce their plasticity.
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1. 2. 3. 4. 5.
S0925-8388(17)30159-7
DOI:
10.1016/j.jallcom.2017.01.135
Reference:
JALCOM 40502
To appear in:
Journal of Alloys and Compounds
Received Date: 17 September 2016 Revised Date:
23 December 2016
Accepted Date: 15 January 2017
Please cite this article as: X. Zhang, J. Tang, L. Deng, G. Zhong, X. Liu, Y. Li, H. Deng, W. Hu, The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study, Journal of Alloys and Compounds (2017), doi: 10.1016/j.jallcom.2017.01.135. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study
Huiqiu Dengd, Wangyu Hub
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Xingming Zhanga, Jianfeng Tanga,b, Lei Denga,* , Gu Zhong c, Xunlin Liua, Yifan Lia,
College of Science, Hunan Agricultural University, Changsha 410128, China
b
College of Materials Science and Engineering, Hunan University, Changsha 410082, China
d
Suzhou Research Institute for Nonferrous Metals, Suzhou 215000, China
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c
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Department of Applied Physics, School of Physics and Electronics, Hunan University, Changsha 410082, China
Abstract:
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This study aims to elucidate the effects of interstitial impurities on the mechanical properties of vanadium alloys investigated by means of the first-principles computations. For this purpose, we study the generalized stacking-fault energies and the ductility parameters in different slip system to evaluate the strengthening and ductility potentials of interstitial impurities in V alloys. The results show that the generalized stacking-fault energy values for two slip systems all increase significantly after adding C, O, and N impurities in comparison with pure V. Moreover, the impurities located in the slip plane greatly increase the generalized stacking-fault energy, suggesting that the impurity effect is localized. The calculations reveal a remarkably strong dependence of the USF energies on the impurities concentration. Though evaluating the generalized stacking-fault energies and the ductility parameters, we draw a conclusion that impurities not only strengthen the V alloys but also reduce their plasticity. Keyword: Generalized stacking-fault energy; V alloys; First-principles calculations; Ductility; Interstitial impurity
* Corresponding Author. Email: [email protected] (L. Deng) Tel: +86-0731-84618092; Fax: +86-0731-84618092
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Introduction Vanadium (V) alloys are considered as a candidate for blanket structural
materials for fusion reactor systems due to their low-induced activation characteristics, and remarkable elevated temperature mechanical properties in the fusion
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environment[1-3]. Interstitial impurities (O, N, and C) play key roles in the microstructure, influencing important parameters such as the strength and plasticity. With systematic efforts, studies showed that the mechanical strength of V alloys can be improved by high number density of tiny precipitates dispersed in the matrix [4, 5].
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These nanoscale precipitates in V-4Cr-4Ti alloys are Ti-rich and most likely to be Ti-(O, N, C) with the NaCl structure[6]. In addition to its role in forming precipitates,
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interstitial impurities are considered to effectively harden alloys due to the solid-solution strengthening effects[7, 8]. Even these impurities can also be considered as alloying elements in V, analogous to C in steel or O in the Ti alloys. Though it is known that the alloys strength is strongly related to interstitial impurities, there is no detailed understanding of correlation between the mechanical properties
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and the interstitial impurities in V alloys. Therefore, a deeper insight into their influences on the mechanical properties in V alloys is essential and beneficial to the further design of the new style alloys.
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First-principles calculations based on the density functional theory (DFT) have been the most powerful tool for evaluating the atomic interactions and understanding the basic atomic phenomena involved in the bulk materials. Especially, the nucleation
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tendency of dislocations depends on the generalized stacking-fault (GSF) energies which can be obtained by DFT calculations to evaluate the deformation mode in materials. For instance, Siegel[9] studied the GSF energies in selected Ni alloys to elucidate how alloying alters the mechanical properties. Muzyk et al.[10, 11] calculated the GSF energies in Al and Mg alloys. In addition to the influences of substitutional solute, the interstitial atoms have attracted more attentions. Kwasniak et al.[12] discussed the influence of interstitial atoms on deformation mechanism in Ti. Afterwards, they systematically discussed the solid solution strengthening of a-Ti in 2
ACCEPTED MANUSCRIPT all four active glide modes[13]. Recently, Yu et al.[14] investigated the origin of dramatic oxygen solute strengthening effect in Ti. They found that the distortion of the interstitial sites at the screw dislocation core creates a very strong but short-range repulsion for oxygen, which result in a strong pinning effect on screw dislocations and
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strengthening the a-Ti. In this paper we present a systematic study of the effects of interstitial impurities on the mechanical properties of V alloys. For this purpose we employ the first-principles electronic structure calculations to study the GSF energies and the
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ductility parameters in different slip system to evaluate the strengthening and ductility potentials of interstitial impurities in V alloys; and then the calculated results are
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discussed.
Methods
The calculations were performed using the Vienna Ab Initio Simulation Package (VASP) [15, 16] via the projector augmented wave (PAW) method [17, 18] and the
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Perdew-Burke-Ernzerhof function of generalized gradient approximation (GGA-PBE) [19]. The nuclei core with valence electrons on s, p and d orbitals for V and valence electrons on s and p orbitals for impurity atoms are used. To determine the GSF
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energies for (110)<111> slip systems, a 55-atom supercell
(illustrated in Fig. 1(a))
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with 6 (111) layers and 9 host atoms on each layer is employed. A total impurity concentration is 1.85%. The model will ensure that the calculation can remove the interaction of the impurity with its periodic images, which originated from the small model used in the calculation. Similar 6 layers constructions also have successfully been used to study the GSF energies in other systems, such as interstitial N in Fe-N alloys [20] and O in α-Ti [21]. For (112)<111> slip systems, we constructed V supercells (illustrated in Fig. 1(b)) containing 61 atoms with one impurity atom for predicting GSF energies, and the total impurity concentration is 1.67%. Meanwhile, 3
ACCEPTED MANUSCRIPT we consider two octahedral sites (in the slip plane or sub-plane) in each slip system to study the coverage of impurity influence. Similar constructions without defects atoms had been used to study the GSF energies in our previous works [22]. The calculations are obtained with 500 eV plane-wave cutoff. The Monkhorst-Pack mesh of k-points
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was selected to ensure a convergence result: 3×3×1 for the (110)<111> and (112)<111> slip systems. A vacuum width of 10 Å is added to periodical slabs for all configurations. The GSF energies curves are generated with ten equally spaced rigid
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shifts of the two parts of the crystal. For each rigid shift, the atomic positions are relaxed along the Z-axis ([1 1 1] direction) by minimizing the forces to less than 0.02
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eV/Å on each atom. The atoms at the top and bottom layer are fixed during relaxation. The GSF energy γGSF can be formally defined as [23, 24]:
γ GSF =
1 ( ESF − E0 ) S
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where ESF and E0 are the free energies of the supercell after and before shear deformation, respectively, and S is the area of the faulted region. Based on the PN model, the maximum restoring force τmax of dislocations nucleation can be
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approximated by GSF curve as follow[25]:
τ max = Max ∇ ( γ )
(2)
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Both restoring forces and GSF energies of alloys can be accurately determined by the GSF curves, and their changes are fundamental for understanding the solution strengthening mechanism[13]. In order to evaluate the effects of impurities on crystal plasticity based on the Rice criterion[26], we also determine the ductility parameters (D) with various impurities. The ductility parameters (D) can be formally defined as[26]:
D = γ SF γ US
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where γSF is the surface energy of pure V or V alloys, which is estimated for cleavage between different atomic planes in the slip plane. γUS is the so-called unstable stacking
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Result and Discussion
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3.
The impurities of (C, N and O) are small size atoms, and usually occupy the tetrahedral or octahedral interstices sites in the crystals. The relative stability of an impurity atom between various solution sites has been determined by previous DFT
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studies[28] where the impurity (C, O, and N) prefer the octahedral interstices sites to the tetrahedral interstices ones in the V alloys. Therefore, the initial position for
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impurities is placed in octahedral interstices sites as illustrated in Fig. 1. Fig. 2 presents the GSF energy curves for the (110)<111> and (112)<111> slip systems with C, O, and N impurities in different plane. According to the GSF energy curves, the USF energies γUS and the maximum restoring stress τmax, for Pure V and V alloys (impurity placed in the slip plane) in the different slip systems are listed in Table 1.
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Firstly, from Fig. 2(a) and (c), it can be seen that the GSF energies and maximum restoring stress for two slip systems all increase significantly after adding C, O, and N impurities in comparison with pure V. Moreover, as illustrated in Fig. 2(b) and (d), the
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GSF energies are also determined when impurity is located in sub-plane. Obviously, the impurities located in the slip plane greatly increase the USF energy. Such similar behaviors have been reported in previous studies [13, 20, 29]. Especially, Kwasniak et
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al.[13] studied that the interstitial O significantly improves GSF energies in all slip modes, and the explicit effect is visible only when the O atom is located directly in the glide plane. Therefore, all of this indicates that the impurity effect is localized. In addition, the change in GSF energy with different distances to glide plane are implying segregation tendency [30]. A negative change in SFE implies segregation. Therefore, we conclude that the C, O, and N impurities have no segregation behaviors in the stacking faults since the impurities located in the slip plane present larger GSF energies. 5
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σp =
3 3 a τ max 8 πζ
(4)
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where a is the spacing of atomic planes in the glide plane, τmax the maximum restoring force of dislocations nucleation can be approximated by GSF curve, ζ is the half-width of the dislocation. As listed in Table 1, the maximum restoring stress for
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two slip systems all increase significantly after adding C, O, and N impurities in comparison with pure V. Meanwhile, since the extended dislocation has a reduced SF
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width with increase in USF energy[20], we infer that the doped impurities will effective increase the Peierls stress in V alloy.
As discussing content in the literatures[13, 32], this classic semi-rigid GSF energy calculations approach maybe overestimates the USF energy values, and hide the real slip trace when the considered system includes the interstitial elements (large
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lattice distortion are expected). Therefore, we also used the climbing image-nudged elastic band (CI-NEB) method to determine the GSF energy curve and compare with the semi-rigid approach. From the Fig. 2(e), it should be noted that the differences
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between two approaches can be neglected. Therefore, we can draw a conclusion that the overestimates of unstable stacking fault energy values by the classic "semi -rigid"
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GSFE calculations is absent in our calculations. As mentioned before, the GSF energy values increase significantly after adding
C, O, and N impurities in comparison with pure V for (110)<111> slip system. As an example, the GSF energy from 616.7 to 722.4 mJ/m2 is increased by 17.1% for O impurity, and this effect can be comparable with that of noble metals[22]. As well known, the GSF energy denotes the lowest energy barrier for dislocation nucleation, and is beneficial to the investigation of impurity effects on the mechanical properties of alloys[33]. Note that the dislocation activities of the (110) slip system possibly occur in bcc V and has a dominant function in plastic deformation[22]. Based on this, 6
ACCEPTED MANUSCRIPT we draw a conclusion that impurities can effectively impede the plastic deformation in V alloys by means of suppressing the activation of dislocation. In order to understand the underlying mechanism for impurity effect, partial density of states (DOS) and Bader charge analysis [34] have been performed to study the electron transfers and
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bonding interaction. As shown in Fig. 3, it is clear that there are apparent hybridization interactions between the d states of V atom and the p states of impurity atom. For the stacking fault structure of (110)<111> slip system, the obtained charge for impurity C, O and N are 0.94, 1.17 and 1.31(the order C
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suggesting that the atomic electrons transfer from V host atoms to impurity. Meanwhile, this following order is in consistent with the calculated GSF energies for
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the (110)<111> slip systems, indicating that the electronic effect is the main factor affecting the variations in the GSF energy curves.
As for (112)<111> slip system, and shown in Fig. 2(b), it should be emphasized that the energy peak is not located in the center of the GSF energy curve, indicating the non-generic strengthening effect stemming from the interaction of impurities and
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stacking faults. Through the structural analysis, we found that the new phenomenon is ascribed to the structure transformation which the initial octahedral site for impurity occupied is transformed to the hexahedral site after shear deformation by (112)<111>
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type mode. With O impurity, for example, the corresponding transformation process is illustrated in Fig. (4). Through the shear deformation, one V-O bond (initial V-O distance = 2.08 ) is interrupted and the host V atom away from the impurity. The
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other V atom (initial V-O distance = 1.88Å) still remains the nearest-neighbor distance to impurity atom after the shear deformation. To elucidate the V-impurity bonding in the stacking fault structure, we illustrate the partial density of states (DOS) of the impurity atom and nearest-neighbor V atom (shown in Fig. 5). It is clear that there are apparent hybridization interactions between the d states of V atom and the p states of impurity atom. Since hexahedral sites exist widely in the close-packed crystals, but not in the body-centered cubic (BCC) ones. Meanwhile, note that the BCC (110) plane transforms into the HCP (0001) plane mainly through the (112)<111> shear in the 7
ACCEPTED MANUSCRIPT BCC-to-HCP phase transition according the Burgers mechanisms[35]. In addition, recently, Henning et al.[36] obtained a conclusion that the impurities (C, O, and N) block the α to ω martensitic transformation in titanium due to the initial octahedral site placed with impurity transform to tetra- or hexahedral ones. Therefore, from these
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aspects, we infer that the impediment effect on the martensitic transformation stemming from the impurities reconstructions also exist in the V alloys.
To obtain the further information of the impurity influence, we consider the different concentrations of impurity (C, O, and N) doped in the supercells. Due to the
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pronounced repulsion in the impurity pair[37], the distance between two impurities in the supercell is not less than one lattice constant of host V bulk (approximately 3 ),
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and assuming that the distribution of impurities are homogeneous in the V alloy. Fig. 6 presents the dependence of the USF energies in the V alloy as functions of the impurity concentration in the (110)<111> and (112)<111> slip systems. The calculations reveal a remarkably strong dependence of the GSF energies on the impurities concentration, changing by more than 60 mJ/m2 for (110)<111> slip system
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if the impurity content is increased by 1% of atomic number, and that for (112)<111> slip system, even increased more than 170 mJ/m2. To specially mention, the strong effect of impurity concentration on the GSF energy is also found in the Fe-C
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alloy[29].
Based on Rice analysis of the competition between dislocation emission from a crack tip and crack cleavage [26], we calculate the ductility parameter (D) to examine
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how impurities effect on the ductility of V. According to this analysis, a metal will be ductile one if the ductility parameter satisfies D>3.5. Corresponding surface energies γSF, and ductility parameters D for pure V and V alloys in the different slip systems are
also listed in Table 1. As for pure V, the calculated surface energies for (111) and (112) plane are 2428.0 and 2703.3 mJ/m2 respectively, and yields corresponding ductility parameters D is 3.93 and 3.76, which is consistent with the reported results[26, 38]. Meanwhile, the D value is larger than 3.5 indicating that pure V is an intrinsically ductile metal, and therefore the metal tend to fail by dislocation-mediated slip rather 8
ACCEPTED MANUSCRIPT than by cleavage fracture. By contrast, with the addition of C, O, and N impurities resulting in the D values lower than 3.5, the fracture models of V alloys maybe take a turn from ductility to brittle collapse. In other words, it demonstrates that impurities can reduce the plasticity in V alloys. These results are in accordance with the
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experimental observations[39, 40]. It is also noteworthy that the decreases of the ductility parameters D by adding impurities are mainly due to the increase of USF energies which plays a major role.
Summary
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4.
In summary, first-principles calculations are used to elucidate the effect of
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solutes on the mechanical properties of V alloys. The results show that the GSF energy values for two slip systems all increase significantly after adding C, O, and N impurities in comparison with pure V. Moreover, the impurities located in the slip plane greatly increase the generalized stacking-fault energy, suggesting that the impurity effect is localized. As for (112)<111> slip system, impurities (C, O, and N)
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can block the α to ω martensitic transformation in vanadium. Considering the concentration effect, the calculations reveal a remarkably strong dependence of the USF energies on the impurities concentration, changing by more than 60 mJ/m2 for (110)<111> slip system if the impurity content is increased by 1% of atomic number,
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and that for (112)<111> slip system, even increased more than 170 mJ/m2. Though evaluating the ductility parameters, we draw a conclusion that pure V is an
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intrinsically ductile metal, and therefore the metal tend to fail by dislocation-mediated slip rather than by cleavage fracture. By contrast, with the addition of C, O, and N impurities, the fracture models of V alloys maybe take a turn from ductility to brittle collapse. We conclude that impurities not only strengthen the V alloys but also reduce their plasticity. Our new prediction provides a theoretical basis to design and fabricate novel V alloys with excellent mechanical properties.
Acknowledgements 9
ACCEPTED MANUSCRIPT This work is financially supported by the National Nature Science Foundation of China (51501063, 51301066, and 51371080), the Natural Science Foundation of Hunan Province (No. 14JJ2080), and the Talents Foundation of Hunan Agricultural University (No. 14YJ04). The authors are thankful for the computational resource
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provided by the National Supercomputer Center in Changsha and Tianjin, China.
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faults in γ
[31] B. Joós, M.S. Duesbery, The Peierls Stress of Dislocations: An Analytic Formula, Physical Review Letters, 78 (1997) 266-269.
[32] P. Kwaśniak, P. Śpiewak, H. Garbacz, K.J. Kurzydłowski, Plasticity of hexagonal systems: Split slip modes and inverse Peierls relation in Ti, Physical Review B, 89 (2014) 144105. [33] J.A. Yan, C.Y. Wang, S.Y. Wang, Generalized-stacking-fault energy and dislocation properties in bcc Fe: A first-principles study, Physical Review B, 70 (2004) 174105. [34] W. Tang, E. Sanville, G. Henkelman, A grid-based Bader analysis algorithm without lattice bias, Journal of Physics: Condensed Matter, 21 (2009) 084204. [35] W.G. Burgers, On the process of transition of the cubic-body-centered modification into the hexagonal-close-packed modification of zirconium, Physica, 1 (1934) 561-586. [36] R.G. Hennig, D.R. Trinkle, J. Bouchet, S.G. Srinivasan, R.C. Albers, J.W. Wilkins, Impurities block the α to ω martensitic transformation in titanium, Nat Mater, 4 (2005) 129-133.
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ACCEPTED MANUSCRIPT [37] R. Li, P. Zhang, X. Li, C. Zhang, J. Zhao, First-principles study of the behavior of O, N and C impurities in vanadium solids, Journal of Nuclear Materials, 435 (2013) 71-76. [38] B.-J. Lee, M.I. Baskes, H. Kim, Y. Koo Cho, Second nearest-neighbor modified embedded atom method potentials for bcc transition metals, Physical Review B, 64 (2001) 184102. [39] R.J. Kurtz, Effect of oxygen on the crack growth behavior of V–4Cr–4Ti at 600°C, Journal of Nuclear Materials, 283–287, Part 2 (2000) 822-826. [40] J.R. DiStefano, J.H. DeVan, Reactions of oxygen with V-Cr-Ti alloys, Journal of Nuclear Materials,
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249 (1997) 150-158.
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ACCEPTED MANUSCRIPT Table 1 Calculated USF energies γUS, maximum restoring stress τmax, surface energies γSF, and ductility parameters D for Pure V and V alloys in the different slip systems. The forces and energies are given in GPa and mJ/m2, respectively. γUS
τmax
Pure V
616.7
7.11
(112)<111>
V+C V+O V+N Pure V V+C V+O V+N
706.3 722.4 727.3 718.5 1054.1 980.5 1066.3
8.29 8.41 8.49 9.09 13.77 12.43 13.71
Rice [26].
b
Lee [38]
3.93, 4.0a 3.33 3.21 3.22 3.76 2.63 2.62 2.61
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a
2428.0, 2280 2636b 2353.3 2339.3 2329.6 2703.3 2777.0 2564.0 2782.3
a
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(110)<111>
D
γSF
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System
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Slip mode
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ACCEPTED MANUSCRIPT Figure Captions Fig. 1. Configurations used to calculate the GSF energies for the (a) (110)<111> and (b) (112)<111> slip systems. The dashed line indicates the slip plane. Normal spheres represent host V atoms, and relatively small ones represent interstitial impurities (C, sub-plane where the impurity is placed.
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O, or N). The value i = 1or 2 represents the octahedral site in the slip plane or
Fig. 2. The GSF energy curves for the (110)<111> and (112)<111> slip systems when the C, O, and N impurities are placed in (a), (c) the slip plane or (b), (d) the sub-plane.
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the GSF energy curve is presented in (e).
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The comparison of the classic semi-rigid method and CI-NEB method for calculating
Fig. 3. Partial DOS of the impurity atom and nearest-neighbor V atom in the stacking fault structure of (110)<111> slip system: (a)C, (b)O, and (c)N. The black solid line corresponds to the d-projected DOS of the V atom; the red dashed-dotted line represents the p-projected DOS of the impurity atom.
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Fig. 4. The transformation process of impurities from initial octahedral site to the hexahedral one after shear deformation by (112)<111> type glide mode. Normal spheres represent host V atoms, and relatively small ones represent interstitial
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impurities.
Fig. 5. Partial DOS of the impurity atom and nearest-neighbor V atom in the stacking fault structure of (112)<111> slip system: (a)C, (b)O, and (c)N. The black solid line
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corresponds to the d-projected DOS of the V atom; the red dashed-dotted line represents the p-projected DOS of the impurity atom. Fig. 6. Dependence of the USF energies in the V alloy as functions of the impurity concentration in the (a) (110)<111> and (b) (112)<111> slip systems.
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(a)
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(b)
1
1
2
2
Z [110]
Z [112]
X [111]
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X [111]
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Fig. 1 (a)-(b)
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C N O Pure V
600
400
200
0 2
4
6
8
10
Dispalcement along <1-11> (c)
2
(112) Slip-plane
C N O Pure V
600
300
0 2
4
6
8
10
600
400
200
0 0
4
6
8
10
1200
(d)
(112) Sub-plane 900
C N O Pure V
600
300
0
0
2
4
6
8
10
Dispalcement along <1-11>
(e)
1200
N-rigid Pure-rigid N-CINEB Pure-CINEB
900
(112) Slip-plane
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2
2
Dispalcement along <1-11>
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Dispalcement along <1-11>
Enegy relative to unfaulted geometry (mJ/m )
C N O Pure V
(110) Sub-plane
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900
0
Enegy relative to unfaulted geometry (mJ/m )
1200
800
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(110) Slip plane
0
(b)
2
(a)
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Enegy relative to unfaulted geometry (mJ/m )
800
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2
Enegy relative to unfaulted geometry (mJ/m )
2
Enegy relative to unfaulted geometry (mJ/m )
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600
300
0 0
2
4
6
8
10
Dispalcement along <1-11>
Fig. 2 (a)-(e)
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V-d C-p
4 3 2
0 4
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(a)
V-d O-p
3 2 1
(b)
0 4
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-1
Partial DOS (states eV )
1
V-d N-p
2 1
(c)
0 -12
-9
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3
-6
-3
0
3
E-EF (eV)
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Fig. 3 (a)-(c)
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2.08 Å 1.85 Å
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Fig. 4
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1.88 Å
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1.5 V-d C-p
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1.0
0.5
0.0 1.5
V-d O-p
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1.0
0.5
(b) 0.0 1.5 V-d N-p
1.0
0.5
(c) 0.0 -12
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Partial DOS (states eV )
(a)
-6
-3
0
3
E-EF (eV)
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Fig. 5 (a)-(c)
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a) 900 C N O
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1
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USF energies for (110)<111> slip systems
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2
3
4
1400
C N O
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1200
1000
800
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USF energies for (112)<111> slip systems
b)
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Impurity concentration in supercell (%)
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600
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Impurity concentration in supercell (%)
Fig. 6 (a)-(b)
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The influence of interstitial impurities on GSF energy is systematically studied. Interstitial impurities block the α to ω martensitic transformation in V alloys. The impurities effects on the ductility of V alloys are evaluated. The dependence of the USF energies on the impurities concentration is determined. Impurities not only strengthen the V alloys but also reduce their plasticity.
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1. 2. 3. 4. 5.