Modeling of solute hydrogen effect on various planar fault energies

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Modeling of solute hydrogen effect on various planar fault energies Yaxin Zhu a,b, Zhouqi Zheng a, Minsheng Huang a,b, Shuang Liang a, Zhenhuan Li a,b,* a

Department of Mechanics, Huazhong University of Science and Technology, Wuhan, 430074, China Hubei Key Laboratory of Engineering Structural Analysis and Safety Assessment, 1037 Luoyu Road, 430074, Wuhan, China

b

highlights

graphical abstract


The effects of solute hydrogen on various planar fault energies are investigated.
The distribution and migration of solute H atoms play key role in fault energy calculation.
The relationship of fault energy and the H concentration is depicted, quantitatively.
The unstable and stable stacking fault energies can be influenced heavily by the pre-stress.

article info

abstract

Article history:

The strength, plasticity and ductility-brittleness transition of metals are often governed by

Received 28 June 2019

various planar fault energies. Due to interactions between the solute hydrogen (H) and the

Received in revised form

planar faults, the planar fault energies are heavily affected in the H-charged metals. An

4 November 2019

accurate quantitative description of the H-affected planar fault energies is essential for us

Accepted 16 January 2020

to understand correctly the H-induced plasticity mechanism in metals. In this paper, a

Available online xxx

reliable atomistic modeling method is s2uggested to calculate quantitatively the effect of solute H on four frequently-used fault energies. The computed results show that solute H

Keywords:

can increase the unstable stacking fault energy but decrease the other three, and the fault

Stacking-fault energy

energies are linearly related to the equilibrium hydrogen concentration up to 0.020. In

Solute hydrogen

addition, the generalized stacking fault energy curves of the H-charged Ni in the <112> and

Atomistic modeling

<110> directions are also computed to construct the g-surfaces. Finally, the influence of

Hydrogen embrittlement

pre-stress on the unstable and stable stacking fault energies is discussed in detail. These

Plasticity

quantitative results are helpful to understand the dislocation/twin-dominated plastic

* Corresponding author. Department of Mechanics, Huazhong University of Science and Technology, Wuhan, 430074, China. E-mail address: [email protected] (Z. Li). https://doi.org/10.1016/j.ijhydene.2020.01.107 0360-3199/© 2020 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article as: Zhu Y et al., Modeling of solute hydrogen effect on various planar fault energies, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.107

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mechanisms in the H-charged metals and to develop the H-affected discrete dislocation dynamic (DDD) and crystal plasticity (CP) algorithms. © 2020 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.

Introduction Since the middle of 19 century, it has been found that the retention of a small content of hydrogen in metals can degrade heavily their toughness and ductility, causing sudden failure of engineering structures [1]. This is called as hydrogen embrittlement (HE), which is found in almost all traditional and newly developed alloys. Until now, HE has been the focus of academic and industrial circles because of its hazard to the safety of engineering structures [2e6]. HE is extremely complex, which depends not only on the external load environment but also on the complex microstructure inside the material. As the smallest atom in nature, hydrogen can easily enter the material, diffuse rapidly through the lattice under external loads and be trapped by various defects in the material or segregated near them. To thoroughly understand the role of hydrogen in the strength, ductility, damage and fracture of materials, the at-ground level interactions between hydrogen and various defects (such as vacancy, dislocation, grain boundary/phase boundary, micro-crack/void, and so on) in metal should be fully studied. Several mechanisms based on the hydrogen-defect interaction have been put forward after years of research, including the hydrogen enhanced vacancy stabilization mechanism (HEVM) [2,7] based on a detailed analysis of the hydrogen-vacancy interaction, the hydrogen-enhanced localized plasticity (HELP) mechanism [8,9] on the hydrogen-dislocation interaction, the adsorptioninduced dislocation emission (AIDE) mechanism [10e12] on the hydrogen-surface/crack interaction, the hydrogen enhanced decohesion (HEDE) mechanism on the hydrogen-GB interaction [13e15], and so on. As we know, in addition to these defects, the planar faults that accompany the plastic slip are also common defects. By a detailed analysis of the interaction between the solute hydrogen and the planar faults (i.e. stacking fault and twin fault illustrated in Fig. 1), a more comprehensive and thorough understanding of HE can be obtained but less is done in the past years. As is well-known, the stacking fault energy is a key factor that influences heavily the plastic deformation and further the strength, hardening, damage and fatigue of metals [15e19]. In addition, the unstable/stable stacking fault energies and unstable/stable twin fault energies are essential parameters in the classical dislocation nucleation [20,21] and twin nucleation [22,23] theories. The change of stacking fault energy due to hydrogen is usually used to explain H-enhanced dislocation nucleation [24e26], H-facilitated dislocation dissociation [27], H-induced softening [28], H-enhanced slip planarity [29,30], and other experimental phenomena. Quantitative knowledge on how hydrogen affects those planar fault energies is of great significance for accurate description of the HE mechanisms associated with dislocation emission,

nucleation and glide, such as the HELP and AIDE mechanisms. Because the planar fault energies are not easy to be obtained directly from experiments, various numerically computational methods, such as DFT and MD/MS simulation, usually are adopted to study the hydrogen effect on those fault energies, especially the generalized stacking fault energy (GSFE). By performing atomic-scale simulations, hydrogen effect on the stacking fault energy has been studied in the past two decades. However, different and even contradictory results were obtained. For example, Lu et al. showed that hydrogen reduced the unstable stacking energy through the ab initio calculation [31], but Apostol and Mishin showed an opposite result in their first-principle based calculation. Wen et al. studied the hydrogen effect on the stable stacking fault energy by measuring the width of the dissociate dislocation [32]. They found that the separation distance between partials increased when placing homogeneous H atoms on the stacking fault plane and then concluded that hydrogen reduced the stable stacking fault energy. As they point out, hydrogen could interact with the partial dislocations, which results in a fuzzy estimation of stacking fault energy. Using the common generalized relax method [33], Song et al. calculated the stacking fault energy, showing that both the unstable and stable stacking fault energies increase substantially with the H concentration [34]. However, Tang and El-Awady found that hydrogen increased the unstable stacking fault energy but decreased the stable stacking fault energy [35]. By looking closely at the binding site of H atoms at the interstitials of the perfect lattice and the stacking fault lattice, they found that H atoms prefer to the octahedral-site (O-site) than the tetrahedral-site (T-site) on the (111) atomic plane, to the T0 site than the O0 -site on the stacking fault plane, however. Accordingly, they proposed that H atom would migrate from the initial stable O-site to the low energy T0 -site with the formation of the stacking fault, which results in the reduction of the stacking fault energy in return. Taketomi et al. showed the stacking fault energy decreased with the increase of H concentration in a  Fe [36], but He et al. showed the stacking fault energy increased by H but decreased by the H filled vacancy in g  Fe [37]. According to some MD results that hydrogen increases the unstable stacking fault energy [34,35], hydrogen is often thought to be an effective inhibitor of dislocation nucleation, which is in opposition to the experiments [24,28,38] but in good agreement with recent observation [39]. Due to the inconsistency between the calculation results of fault energies, it is not surprising that there are different or even contradictory views on the HE mechanism. It is obvious that an accurate calculation of the H-affected fault energies is very important for understanding the HE mechanisms. As mentioned above, the computational method plays a key role in determining quantitatively the fault energies.

Please cite this article as: Zhu Y et al., Modeling of solute hydrogen effect on various planar fault energies, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.107

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Fig. 1 e The illustrations of various faults in FCC metals and their atomic details. However, it should be pointed out, due to the limit domain size in the DFT model, the free surface image effect affects heavily the reliability of the results. In addition, the equivalent hydrogen content is always tremendous (~104 appm) due to the small atomic system in the DFT calculation. Due to these inevitable limitations, the effect of H concentration on the fault energy may not be truly characterized by DFT modeling. Although MD can simulate the atomic details at larger scale, most of them only assign the H atoms within two atomic fault layers and therefore lose sight of the solute H atoms distribution. In fact, the first-principle study has indicated clearly that the stacking fault energy depends on the distance between H atom layer and slip plane [40]. That is, the stacking fault energy is nonlocal, which depends not only on the solute H atoms within the stacking fault plane but also on those solute H atoms adjacent it. However, as far as we know, most of existing atom simulations of the stacking fault energy ignore the influence of hydrogen distribution on the fault energies and therefore needs to be revisited carefully. An accurate quantitative relation between the equilibrium H concentration and the stacking fault energy or other fault energies is very important to clarify the controversy on the HE mechanisms, and is in urgent need for the theoretical or numerical modeling of HE. Motivated by the above background, this paper revisits accurate calculation of the stacking fault energy and other planar fault energies, with special focus on the solute H atoms migration, distribution and the stable occupation during formation of those planar faults. Using the equilibrium solute

segregation model, H atoms around the fault layers are connected with the equilibrium H concentration.

Simulation model and computational method The model and the computational method for generalized fault energy In order to calculate the generalized fault energies illustrated in Fig. 1, the computational model with three dimensions of pffiffiffi pffiffiffi pffiffiffi 5 6a0 5 2a0 10 3a0 is considered, with the lattice constant a0 being 3.52 and the lattice vectors < 112 > , < 110 > and < 111 > in the X, Y and Z directions, respectively. Periodic boundary conditions are set in the X and Y directions, while the free boundary in the Z direction. A careful check on the domain size shows the present model is large enough that the free surface hardly affects the result. The NieH interaction is depicted by the well established EAM potential which is thought to be accurate for NieH system [41]. The effects of solute hydrogen on four different fault energies are discussed in this paper, i.e., the unstable stacking fault energy gusf , the stable stacking fault energy gssf , the unstable twin fault energy gutf and the stable twin fault energy gstf . As seen from Fig. 1, rigidly shifting one [111] atomic plane by a distance of a60 ½112 (i.e., the Burgers vector bp of a partial dislocation) in the <112> direction with respect to its adjacent plane, the unstable and stable stacking fault energies can be obtained. If one [111] atomic plane adjacent to the stable stacking fault plane is shifted with

Please cite this article as: Zhu Y et al., Modeling of solute hydrogen effect on various planar fault energies, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.107

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another distance of bp , the unstable and stable twin fault energies can be obtained. Alternatively, if the initial [111] atomic plane is shifted continuously, the stacking fault energy gmax can be obtained at a distance of 2bp , as shown in Fig. 1. In the regular calculation of generalized fault energy, the simulation domain is divided into two halves from the middle of the Z direction. Then, the atoms in one half are shifted with respect to the other by a small shear increment in the slip direction, following which the energy minimization is performed with all atoms allowed to move freely only in the direction perpendicular to the slip plane. This is the so-called general relaxation (GR) method. However, when H atoms are introduced in the fault plane, their movement affects strongly the calculation of fault energy. If the H atoms are treated similar to the Ni atoms which can only relax freely in the Z direction, the H atoms are over constrained and lead to an overestimation of the fault energy [34]. In Tang's strategy, the H atoms are allowed to move freely during the stacking fault formation process. In this way, the H atoms can migrate from the initial O-sites to the energy favorable sites at the stable stacking fault plane [35]. Thus, this fault energy calculation is considered more reasonable. The resultant stacking fault energies calculated by these two methods will be compared in this paper.

Reasonable distribution of H atoms by the solute segregation model In previous computations of the solute hydrogen effect on the stacking fault, the H atoms were often assigned at the interstitial sites within the two atomic fault layers (i.e., the 1stnearest interstitial site, as shown in Fig. 1). However, Apostol’s study indicated that those interstitial solute H atoms some distance away from the fault plane also affect markedly the fault energy [40]. To confirm this viewpoint and to decide whether those hydrogen atoms away from fault plane need to be considered in the fault energy calculation, we perform the simulations, with the solute H atoms uniformly distributed in the 2nd- or 3rd-nearest interstitial O-sites (i.e., one or two [111] atomic layers away from the fault plane, as shown in Fig. 1), respectively. The results will be given in the next Section. In order to connect the fault energy with the background equilibrium H concentration, the solute segregation model, which is an effective way to depict the H trapping at the defects [38,42], is adopted to introduce the H atoms around the fault layers. According to the equilibrium segregation model, the probability at which a H atom occupies a potential interstitial site in Ni is given as: .   kB T CH exp  Ebind i .   Ci ¼ kB T 1 þ CH exp  Ebind i

(1)

where CH is the background equilibrium H concentration, kB the Boltzmann constant, T the temperature. kB T is set to be is the binding energy of H atom in an O-site 0:026 eV. Ebind i around the fault layer, which can be obtained as follow:     ¼ EFault  EFault  EH  EBulk  EBulk  EH Ebind i H H and

(2a)

    Ebind ¼ EFault  EFault  EBulk  EBulk i H H

(2b)

 EFault  EH is the energy of the relaxed In Eq. (2a), EFault H system with a H atom sited at the possible O-site i around the  EBulk  EH is the energy of the relaxed fault layer and EBulk H system with a H atom sited at one interstitial O-site far from the fault layer. Thus, the difference between them is the binding energy of solute H atom around the fault layer. The  EBulk in Eq. (2b) is  2:180eV. Using calculated value of EBulk H Eq. (2b), the binding energies of H atom at different-nearest Osites with respect to the fault plane are calculated and the results will be given below. Through employing the calculated binding energy, the occupation probabilities Ci at different equilibrium H concentrations CH are obtained by Eq. (1). Then, the total number of H atoms can be calculated by N ¼ N1 þ 2N2 þ 2N3 þ ,,,, where N1 ¼ C1i ,NOsite is the number of H atoms at the 1stnearest O-sites, N2 ¼ C2i ,NOsite the number of H atoms at the 2nd-nearest O-sites, and so on. NOsite ¼ 200 is the total number of O-sites within two adjacent (111) planes in the j present simulation domain. Ci is the occupation probability at the jth-nearest O-sites with respect to the fault layer, respectively. And then, those H atoms with their number of N1 are uniformly distributed at the 1st-nearest O-sites, and Nj (j > 2) distributed at the jth-nearest O-sites on both sides of the fault layers. In this way, a reliable computational model can be easily constructed, by which the effect of solute hydrogen with different equilibrium concentrations on the planar fault energies can be calculated more accurately.

Computational results The effects of H migration and H distribution on the fault energy By employing the NieH EAM potential, the fault energy vs. slip displacement curves of the pure Ni and the impure Ni with the solute H atoms embedded at the 1st-nearest O-sites are computationally obtained and plotted in Fig. 2, respectively. The areal density of H atom is set as 4.67 atom/nm2 for comparison. It is seen from Fig. 2, if the general relaxation (GR) method is employed, the computational unstable and stable stacking fault energies of impure Ni are both greatly enhanced, compared with the results of pure Ni. By examining the atomic structure obtained by the GR method as shown in the upper right insert in Fig. 2, it is found that the interstitial site of H atom transits from the initial O-site to the tetrahedral site (T-site) where the H atom is energy unfavorable. Because the lateral movement of H atoms is forbidden, the computed fault energy is over-estimated. If we employ the Tang's strategy which allows the H atoms to move freely from the O-sites to the energy favorite site within the two stacking fault layers, the computed unstable stacking fault energy g1usf is enhanced but the stacking fault energy g1ssf is lowered, compared with the results of pure Ni. Obviously, compared with the GR method, Tang's strategy is more appropriate. It is to be mentioned that the energy stable interstitial site within the stacking fault layers is assigned to the T0 -site in Tang's paper

Please cite this article as: Zhu Y et al., Modeling of solute hydrogen effect on various planar fault energies, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.107

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Fig. 2 e The fault energy vs. slip displacement curves obtained by the general relaxation method and Tang's strategy. The slip displacement is normalized by bp , i.e. a0 =6½112. [35], but it is also an octahedral site, as shown by the lower insert in Fig. 2. To eliminate ambiguity, we call it the O0 -site below. When calculating the generalized fault energies of impure Ni with solute H atoms, a very important problem is whether the solute H atoms at interstitial O-sites away from the fault layer affect the calculation results. In order to verify this, the H atoms are assigned uniformly to the 2nd- or 3rd-nearest interstitial O-sites near the fault plane, as mentioned in Section Reasonable distribution of H atoms by the solute

segregation model and as seen by the inserts in Fig. 3(a), (b). The resultant fault energy vs. slip displacement curves are plotted in Fig. 3(a), (b), respectively, where six areal densities of H atom are considered for comparison. As seen from Fig. 3(a), when the H atoms are assigned to the 2nd-nearest Osites, the computed fault energies are obviously reduced compared with that of pure Ni. With the increase of H atomic areal density, the unstable stacking fault energy gusf is decreased, opposing to all of the results in Fig. 2 where H atoms stay at the 1st-nearest O-sites. In addition, the stable

Fig. 3 e The H-affected fault energy vs. slip displacement curves for the solute H atoms at the (a) 2nd-nearest O-sites and (b) 3rd-nearest O-sites away from the fault plane. The slip displacement is normalized by bp . Please cite this article as: Zhu Y et al., Modeling of solute hydrogen effect on various planar fault energies, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.107

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stacking fault energy gssf , the unstable twin fault energy gutf and the stable twin fault energygstf are also significantly decreased with the increasing H atomic areal density. This clearly shows that H atoms at the 2nd-nearest O-sites should not be removed artificially when calculating the generalized fault energies. Moreover, as shown in Fig. 3(b), when the H atoms are assigned to those 3rd-nearest O-sites, their influences on the fault energy becomes almost negligible, unless the H atomic areal density gets very high. It seems that those H atoms at the 3rd-nearest O-sites are insignificant for the fault energy calculation to some extent. The above results convince us that when calculating the generalized fault energies of impure Ni with solute H atoms, the migration of H atoms should be considered carefully, and the H atomic distribution should be assigned properly. Otherwise, the computed fault energies may be inaccurate or even wrong.

The reasonable distribution of solute H atoms and its effect on the fault energy The binding energies of solute H atoms at different-nearest Osites with respect to the fault plane are computed and listed in Table 1. From Table 1, it can be seen that the binding energy of the solute H at the 1st-nearest interstitial O-site is the largest, about 0.08 eV. Therefore, the solute H atoms tend to segregate to the 1st-nearest O-sites within the two atomic fault layers. The binding energy at the 2nd-nearest O-site is 0.041 eV, implying the 2nd-nearest O-sites are also the favorite sites for solute H atoms. Nevertheless, the binding energies of solute H atoms at the 3rd- and 4th-nearest O-sites become positive, so the solute H is less prone to segregate to these sites due to the increasing energy state. From the point view of binding energy, when calculating the fault energy, the solute H atoms at the 1st- and 2nd-nearest O-sites should be taken into account, but those at the 3rd- and 4th-nearest Osites can be ignored. Using the solute segregation model and the binding energies listed in Table 1, the probabilities C1i of the solute H atom segregating at the 1st-nearest O-site and C2i at the 2ndnearest O-site can be calculated, with different equilibrium H concentrations CH considered. From the calculated segregation probabilities C1i and C2i , the number of H atoms at those Osites (N1 and N2 ) can be easily obtained, which are all listed in Table 2. Then, those solute H atoms are uniformly assigned to the 1st-nearest O-sites within the two atomic fault layers and the 2nd-nearest O-sites on both sides of the fault layer. By employing Tang's strategy stated in Section The model and the computational method for generalized fault energy, the

Table 2 e The segregation probabilities C1i , C2i and the number of H atoms N1 , N2 at the 1st- and 2nd-nearest Osites under different equilibrium H concentrations CH , respectively. CH 0.001 0.005 0.010 0.020 0.030 0.050

C1i

C2i

N1

N2

0.0217 0.0978 0.1783 0.3026 0.3942 0.5203

0.0048 0.0236 0.0462 0.0883 0.1268 0.1948

5 20 36 61 79 104

1 5 9 18 25 39

fault energy vs. slip displacement curves for different equilibrium H concentrations CH are obtained and plotted in Fig. 4. It can be seen from Fig. 4, the unstable stacking fault energy gusf increases with increasing equilibrium H concentration, which indicates the dislocation nucleation becomes difficult in the hydrogen charged Ni system [35]. This seems to contradict some experimental observation that hydrogen induces softening [24,28] but agree with the recent experimental observation that hydrogen suppresses dislocation nucleation in the Ag nano-wires [39]. In fact, hydrogen-induced softening may be governed by other mechanisms rather than just dislocation nucleation. In contrast, the stable stacking fault energy gssf , the unstable twin fault energy gutf and the stable stacking fault energy gstf all decrease with increasing equilibrium H concentration. As we know, when the values of gssf = gusf and gutf =gusf all approach 1, the perfect dislocation is easy to nucleate; otherwise, the partial dislocation or twin prefers to nucleate [43]. According to this, with increasing hydrogen concentration in Ni, the partial dislocation or twin nucleation mechanism will be dominant due to increasing gusf but decreasing gssf and gutf . It is interesting that the stable stacking fault energy gssf is negative if the equilibrium H

Table 1 e The binding energy of solute H atom at different O-sites. Interstitial site 1st-nearest O-site 2nd-nearest O-site 3rd-nearest O-site 4th-nearest O-site

EFault  EFault (eV) H

Ebind (eV) i

2.260 2.221 2.176 2.167

0.080 0.041 0.004 0.013

Fig. 4 e The fault energy vs. slip displacement curves of Ni with the solute H atoms distribution assigned by the solute segregation model. The slip displacement is normalized by bp .

Please cite this article as: Zhu Y et al., Modeling of solute hydrogen effect on various planar fault energies, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.107

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concentration CH is 0.03 or higher. This implies that the HCP phase is energetically more favorable than the FCC phase at high H concentration [44]. In order to quantify the relationships between the generalized fault energies and the equilibrium H concentration CH , the computationally obtained fault energies gusf , gssf , gutf and gstf vs. CH are plotted in Fig. 5(a)e(d). It can be seen clearly that when the equilibrium H concentration CH is less than or equal to 0.020, the fault energies gusf , gssf , gutf and gstf are almost

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linearly dependent on CH , similar to Wen’s result [32] and can all be formulated as g ¼ a þ b*CH , with g being the fault energy and CH the equilibrium H concentration. To be pointed out, those fault energies derive from the linear relationship when CH is greater than 0.02. This mainly comes from very high local H concentration in the 1st- and 2nd-nearest O-sites (as seen from Table 2) which leads to the formation of local hydrides or very strong HeH interaction. By fitting the computationally obtained the fault energies with CH being smaller than 0.020 in

Fig. 5 e The fault energy vs. the H concentration curves. Please cite this article as: Zhu Y et al., Modeling of solute hydrogen effect on various planar fault energies, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.107

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Table 3 e The fitting coefficients a and b of the various fault energies.

gusf gssf gutf gstf gmax112 gmax110

a

b

215.0 83.1 250.6 97.8 986.8 634.7

2524.4 4204.1 2870.0 2733.9 17130.5 8500.3

Fig. 5, all coefficients of four fault energies are easily obtained and listed in Table 3. With these quantitative relations between the various fault energies and the equilibrium H concentration, we can study quantitatively the short-range effects of hydrogen on the dislocation dominated plasticity, including the H-affected dislocation emission/nucleation from crack-tip/interface/surface and various sources, dislocation glide resistance, dislocation dissociation, and so on. Moreover, these quantitative relationships at the atomic scale also can be used in various the up-scaled modelings of Haffected plasticity, such as discrete dislocation dynamic modeling (DDD) at the meso-scale and crystal plastic (CP) modeling at the continuum scale [45e47].

Discussion The GSFE curves in the < 112 > and < 110 > directions The GSFE surface, i.e. the g-surface, is an essential feature, which plays a key role in predicting brittleeductile transition [48] and the misfit energy calculation [44,49]. Specially, in the theoretical study of the hydrogen-induced material property degradation, the H-affected g-surface is a prerequisite. To

construct the g-surface, the GSFEs in the < 112 > and < 110 > directions are required at first. The GSFE vs. slip displacement curves in the <112> direction at different equilibrium H concentrations are plotted in Fig. 6(a). In the first stage of bp < 1 where H atom migrates from the initial O-site to the new O0 -site, as shown in Fig. 6(b), (c), the computed curves are the same as that in Fig. 4. When the slip displacement in the <112> direction is 2bp , the fault energy reaches the maximum value gmax112 , as shown by the fault energy curve at H concentration of 0.000. If H atoms are introduced, their migration path will affect heavily the fault energy. When the slip displacement proceeds from bp to 2bp , if the migration path of H atom is not assigned artificially, the H atom can spontaneously find a low energy site, which is shown in Fig. 6(d). A careful polyhedral structure check on the atomic details in Fig. 6(d) shows that the H atom is in the interstitial Triangular Prism-site (TP-site), as shown by the insert in Fig. 6(a). During this spontaneous migration, the H atom jumps from the O0 -site to the TP-site at one moment, which induces a sudden drop in the computed fault energy vs. slip displacement curve, as observed by Tang et al. [35]. This sudden drop makes it difficult to accurately construct the g-surface. To conquer this difficulty, those H atoms are artificially forced to migrate proportionally from O0 -site to the TPsite, similar to Tang's treatment [35]. After that, as the slip displacement proceeding from 2bp to 3bp , all of H atoms are allowed to migrate spontaneously. During the last stage of bp from 2 to 3, those H atoms seem to stay nearly statically, but their interstitial sites regain from the TP-sites to the O-sites, as shown in Fig. 6(e). In this way, the GSFE vs. slip displacement curves in the <112> direction at different equilibrium H concentrations are all calculated and shown in Fig. 6(a). It can be seen that the fault energy curves are smooth without sudden drop, but the flag values of the fault energy are not changed. This treatment enables to describe the H-affected g-surface

Fig. 6 e (a) The generalized stacking fault energy vs. slip displacement curves in the <112> direction, and (bee) the migration paths of solute H atoms. The slip displacement is normalized by bp . Please cite this article as: Zhu Y et al., Modeling of solute hydrogen effect on various planar fault energies, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.107

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quantitatively. The calculated maximum fault energies gmax112 decrease with increasing H concentration. Interestingly, in the cases with the equilibrium H concentration less than 0.020, the fault energy gmax112 at the slip displacement of 2bp is the highest energy points on the fault energy curve. However, in the cases with the equilibrium H concentration larger than 0.020, the fault energy at the slip displacement of 2bp is a local minimum points as seen from the fault energy curves with solid rhombus and hollow nablas in Fig. 6(a). This unexpected local minimum might come from the local high H concentration around the fault layers. The fault energy gmax112 vs. H concentration CH curve is plotted in Fig. 5(e), which shows clearly that gmax112 decreases almost linearly with the increase of CH , as long as CH is less than or equal to 0.020. The linear fitting coefficients are listed in Table 3. By alternatively shifting one [111] atomic layer a distance of a Burger vector b (a20 ½110) in the <110> direction with respect to its adjacent layer, other important GSFE vs. slip displacement curves can be obtained, which are plotted in Fig. 7(a). Similar to the fault energy calculation in the <112> direction, the migration path of H atoms also should be treated carefully in these cases. For the case of CH ¼ 0.000, the fault energy reaches its peak value at the slip displacement of 0:5b, as seen from Fig. 7(a). For the cases of CH > 0, if the migration path of H atoms is not assigned artificially, the H atoms can spontaneously find a low energy site, i.e. those H atoms migrate from the initial O-site to the new site, as shown from Figs. 7(b) to (c). A careful check on the polyhedral structure shows that this new site is also an Octahedron site. Considering that this Octahedron site is slightly different from the strict O0 -site in Fig. 6(c), it is called the O00 site instead. The calculation of the binding energy shows that the interstitial energy of H atom at the O00 -site is 0.047 eV, which is less than that at the O0 -site (0.072 eV), implying the O00 -site is the actual energy more stable than the O0 -site. In order to avoid the sudden jump in the fault

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energy curve mentioned above, we force the H atom to migrate proportionally from the initial O-site to the O00 -site with the displacement from 0 to 0:5b, and then from the O00 site to the nearest O-site with the displacement from 0:5b to b, as shown from Fig. 7(b) to (d). By this treatment, the GSFE vs. slip displacement curves in the <110> direction at different equilibrium H concentrations are computationally obtained and plotted in Fig. 7(a). From those curves, it can be seen clearly that the gmax110 also decreases with increasing H concentration and the local minimum values are also found when CH is larger than 0.020. The computationally obtained fault energy gmax110 vs. H concentration CH curve is plotted in Fig. 5(f), which shows the maximum fault energy gmax110 also decreases linearly with the increase of CH from 0 to 0.020, similar to the result shown in Fig. 5(e). The linear fitting coefficients are also listed in Table 3. With the above fault energy curves in the <110> and <112> directions, the g-surface can be constructed by choosing a suitable interpolation function. Considering the angular symmetry of the FCC lattice in the {111} slip planes, a function with the Fourier series approximation [49] is usually taken to depict the target g-surface by fitting those feature points on the fault energy curves. Due to the paper length, the fitting process will not be given in detail here.

The effect of the pre-stress on the stacking fault energy The GSFE obtained at zero pre-stress is often used to capture the dislocation nucleation. However, as we know, the dislocation nucleation is a complex process, depending closely on the stress state [50,51], temperature [52,53], and other physical or chemical conditions [54]. How the pre-stress cooperating with solute hydrogen affects the stacking fault energy is seldom discussed to our knowledge. In this part, the effects of pre-tensile or pre-compressive stress on the stacking fault energy are studied.

Fig. 7 e (a) The generalized stacking fault energy vs. slip displacement curves in the <110> direction, and (bed) the ‾ migration paths of solute H atom. The slip displacement is normalized by b, i.e. a0 =2½110. Please cite this article as: Zhu Y et al., Modeling of solute hydrogen effect on various planar fault energies, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.107

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Fig. 8 e (a) The unstable stacking fault energy and (b) the stable stacking fault energy vs. the pre-stress curves of Ni with different hydrogen concentrations.

Fig. 8(a), (b) plot the unstable and stable stacking fault energy vs. the pre-stress, respectively. From them, it is seen clearly both the unstable stacking fault energy gusf and the stable stacking fault energy gssf decrease with increasing prestress for the pure Ni, showing good agreement with to the former result [51]. The pre-tensile stress plays the same role on the fault energies in the H-charged Ni, as seen from curves with the H concentration CH of 0.001e0.020. In addition, the unstable/stable stacking fault energy is linearly related to the pre-stress when the H concentration is lower than 0.02. In this condition, the fault energy formula proposed above can be further extended as g ¼ a þ b*CH þ d*Pstr with Pstr being the pre-stress. In Fig. 8(a), the slopes of the fitted lines vary with the H concentration, indicating the coefficient d correlates with the H concentration CH , i.e., d ¼ dðCH Þ. By fitting the slope of those lines in Fig. 8(a), the coefficient d can be expressed as d ¼  11:0  20*CH , as seen in the insert of Fig. 8(a). Different from these, the slopes of the fitted lines in Fig. 8(b) are about 11.4, independent of the H concentration. By carefully comparing all curves in Fig. 8(a), it can be found that the unstable stacking fault energy gusf , which primarily dominates the dislocation nucleation, increases with increasing equilibrium H concentration for the zero pre-stress case; however, it decreases with increasing pre-stress for the H-charged and H-free cases. As a consequence, the unstable stacking fault energy gusf of the H-charged Ni at the pre-tensile stress may be smaller than that of the pure Ni at the zero pretensile stress. This means even in the H-charged Ni, the dislocation nucleation still can be promoted rather than inhibited by the solute H at the localized site whose stress state is tensile and normal to the slip plane, such as at the opening crack tip, the stretched grain boundary (GB) or other. On the contrary, the dislocation nucleation at some sites with a compressive pre-stress normal to the slip plane becomes more difficult in the H-charged Ni than in the pure Ni. In our opinion, whether hydrogen hinders or facilitates dislocation nucleation depends not only on hydrogen concentration but also on the stress state, which seems to shed some light on

different H-induced dislocation nucleation behaviors observed in different experiments [24,28] and simulations [27,55]. Interestingly, the stable stacking fault energies of the charged-H Ni with H concentration of 0.010 and 0.020 are all positive at zero pre-stress in Fig. 8(b), but they become negative at the pre-tensile stress higher than 4.0 GPa and 2.0 GPa. This means that the HCP phase which is energetically stable than the FCC phase may form at the opening crack tip where the tensile stress usually is very high.

Main conclusions and remarks In summary, the H migration during the formation of planar fault and the H distribution based on the solute segregation model are considered to calculate the H-affected planar fault energies, accurately. The influence of solute H atoms on four main planar fault energies in FCC Ni, i.e. the unstable stacking fault energy gusf , the stable stacking fault energy gssf , the unstable twin fault energy gutf and the stable twin fault energy gstf , are quantitatively depicted. Moreover, the g-surfaces of the H-charged Ni with different H concentrations are constructed, and the influences of pre-stress on the generalized stacking fault energy are also discussed. The main results include: 1. The computationally obtained fault energies depend not only on the distribution of solute H atoms but also their migration path during fault formation. Therefore, a random distribution of solute H atoms in Ni or an inappropriate treatment to the migration path of solute H atoms may lead to inaccurate or unusable results. 2. The solute H can increase the unstable stacking fault energy gusf but decrease the stable stacking fault energy gssf , the unstable twin fault energy gutf and the stable twin fault energy gstf . Therefore, the solute H can influence heavily the dislocation-dominated plasticity mechanisms at the atomic scale.

Please cite this article as: Zhu Y et al., Modeling of solute hydrogen effect on various planar fault energies, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.107

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3. Both the unstable and stable stacking fault energies in the pure Ni and the H-charged Ni can be influenced heavily by the pre-stress. Whether the solute hydrogen hinders or facilitates dislocation nucleation depends on the pre-stress normal to the slip plane besides the hydrogen concentration. Although this is a pure modeling work without any experimental results, our calculated results are partially validated by indirect comparison with the experimental results mentioned above and direct comparison with some of the calculated results by Tang and El-Awady (2012). In addition, in comparison with others in the literature, the present modeling gave some reliable quantitative relationships of H concentration with various fault energies, which not only can shed light on dislocation-dominated plasticity in the Hcharged Ni but also can be used to develop the up-scaled modeling methods, such as discrete dislocation dynamic (DDD) at the meso-scale and crystal plasticity (CP) at the continuous scale.

Acknowledgement The work was supported by the National Nature Science Foundation of China (No. 11632007 and 11802099) and by Science Challenge Project (No. TZ2018001) and the Fundamental Research Funds for the Central Universities (No. 2019kfyXJJS144).

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Please cite this article as: Zhu Y et al., Modeling of solute hydrogen effect on various planar fault energies, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.107