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Scripta Materialia 60 (2009) 555–558

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Influence of shear strain on the hydrogen trapped in bcc-Fe:A first-principles-based study

Ryosuke Matsumoto,a,b,* Yoshinori Inoue,a Shinya Taketomia,b and Noriyuki Miyazakia,b

aDepartment of Mechanical Engineering and Science, Graduate School of Engineering, Kyoto University, Sakyo-ku,

Kyoto 606-8501, JapanbNational Institute of Advanced Industrial Science and Technology (AIST), Tsukuba Central 2, Tsukuba, Ibaraki 305-8568, Japan

Received 7 August 2008; revised 14 October 2008; accepted 2 December 2008Available online 16 December 2008

To advance our understanding of hydrogen-related fractures of metals and alloys, it is essential to understand the influence oflattice defects and stress on the hydrogen existence state and its concentrations. In this study, we use density functional theory andinteratomic potential to clearly show that shear strain in body-centered cubic Fe yields strong hydrogen trap energy, comparable tothat of volumetric strain.� 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Hydrogen; Iron; Density functional; Simulation; Trap energy

Hydrogen is expected to find increasing use as anenergy source in the near future, due to the escalation ofenvironmental problems with current fuels. Although itis well known that hydrogen weakens the strength ofmost metals and alloys [1–4], the specific mechanisms re-main a subject of debate. The construction of a securehydrogen-energy-based society requires quantitativeprediction of the strength of materials in a hydrogenenvironment.

Extensive research has reported that solute hydrogenatoms and lattice defects have strong interactions, andtrapped hydrogen atoms significantly change the stabil-ity and/or mobility of defects [5–8]. Thus, knowing thehydrogen existence state and its concentrations arounddefects is critical. Sofronis and Birnbaum analyzed thehydrogen distribution around edge dislocations using acontinuum model [9], and explained the promoted dislo-cation activity from the hydrogen shielding effect on theelastic interaction. Here, it is generally believed thathydrogen atoms accumulate at a high hydrostatic stressregion, such as ahead of a mode I crack tip and under anedge-dislocation core. Many previous studies took intoconsideration the gradient of hydrostatic stress in the

1359-6462/$ - see front matter � 2008 Acta Materialia Inc. Published by Eldoi:10.1016/j.scriptamat.2008.12.009

* Corresponding author. Address: Department of Mechanical Engi-neering and Science, Graduate School of Engineering, Kyoto Uni-versity, Sakyo-ku, Kyoto 606-8501, Japan. Tel.: +81 75 753 5894;fax: +81 75 753 5719.; e-mail: matsumoto@solid.me.kyoto-u.ac.jp

hydrogen diffusion equation [9–12]. However, it haslong been known that interstitial atoms in body-cen-tered cubic (bcc) metals and alloys, such as the carbonatom in bcc-Fe, interact strongly with shear deforma-tions [13]. Recent detailed analyses of the distributionsof carbon around dislocations also revealed the impor-tance of shear strain components [14]. Our atomisticstudy of the distribution of hydrogen trap energyaround the edge dislocations in bcc-Fe showed that bothhydrostatic and shear stress are necessary to explain thedistribution [15].

The change in heat of solution caused by elastic strainis equivalent to the trap energy yielded by elastic strainfor solute hydrogen atoms in a crystal lattice. In thispaper, we first present the influence of elastic strain onthe heat of solution of hydrogen incorporated in bcc-Fe using density functional theory (DFT), and comparethe results with those from the embedded atom method(EAM) potential developed by Wen et al. (EAM (Wen)potential) [16]. After verifying the accuracy of the inter-atomic potential, we estimate the trap energy yielded byshear strain for various crystal orientations using thepotential, for sufficiently large calculation models.

We performed DFT calculations within the spin-polarized generalized gradient approximation (GGA)[17] for electron exchange and correlation, using theVienna Ab-initio Simulation Package (VASP) [18–20].The interactions between the ions and electrons are de-scribed by Blochl’s projector augmented wave (PAW)

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556 R. Matsumoto et al. / Scripta Materialia 60 (2009) 555–558

method [21], which has an accuracy similar to an all-electron method within the frozen core approximation.The Monkhorst–Pack scheme [22] is used to define k-points, and the conjugate-gradient method is employedin the relaxation algorithm. We used 30 � 30 � 30 k-points per unit lattice and 425 eV cut-off energy, whichis sufficient for reliable calculations. Zero-point energy(ZPE) corrections, calculated from the Hessian matrix,were considered in all calculations. These calculationconditions provide good agreement with previous re-ports for typical physical properties of Fe–H systems[23,24]. Using these calculation conditions, we obtainedE0 = �16.416 eV as the ground-state energy of a con-ventional bcc-Fe unit lattice containing two atoms,L0 = 0.2832 nm as the relaxed lattice length of bcc-Fe,and EH2

¼ �6:524 eV as the ZPE corrected ground-stateenergy of a hydrogen molecule. Here, the ground-stateenergy is defined as the absolute total energy with re-spect to the origin of the total energy of free ions andelectrons separated by an infinite distance.

There are two types of interstices in a bcc lattice: tet-rahedral (t)-sites and octahedral (o)-sites. It is wellknown that, in bcc-Fe, hydrogen prefers to occupy a t-site rather than an o-site, whereas other interstitialatoms prefer to occupy an o-site rather than a t-site.However, it has been reported that o-site occupation be-comes more stable under monoaxial tensile conditionsfor the hydrogen in bcc metals [25]; thus, o-site occupa-tion possibly influences the hydrogen distributionaround the stress singular points.

We introduced a hydrogen atom into a t- or an o-sitein a strained bcc-Fe lattice consisting of N = 16 Featoms (2 � 2 � 2 lattices), and calculated the ground-state energy EFeN Hðeij; nÞ. Here, n can be t- or o-site,and index i and j can be x, y or z. According to conven-tional notation, we describe exx, eyy and ezz as ex, ey andez, respectively. We define the strain by Cauchy’s infini-tesimal strain tensor using the lattice size of bcc-Fe (L0).Both occupation sites have an anisotropic geometry, asillustrated in Figure 1a and b, i.e. the occupation sitesshown in Figure 1a (t-site) and Figure 1b (o-site) aresymmetrical only in the y- and z-axes. We call thesethe tx- and ox-sites, respectively. According to the sym-metrical property of the occupation sites, here we applytwo different tensile strains (ex and ey (= ez)) and two dif-ferent shear strains (exy and eyz (= ezx)) for both tx- andox-site occupations. The relationship for a differentoccupation site (ty-, tz-, oy- or oz-site) is easily obtained

Figure 1. Bcc-Fe lattices consist of N = 16 Fe atoms (2 � 2 � 2lattices) and one hydrogen atom: (a) and (b) show the tx- and ox-siteoccupations, respectively.

by changing indexes. The strain effects are estimated inthe range of jeijj 6 0:05, which is almost equal to theelastic strain around the dislocation core (r > 0.5 nm).All atomic positions are relaxed under fixed lattice con-ditions. The heat of solution for incorporation of hydro-gen into the strained lattice, EHOS(eij,n), is estimated by:

EHOSðeij; nÞ ¼ EFeN Hðeij; nÞ � N � 1

2E0ðeijÞ þ

1

2EH2

� �;

ð1Þwhere E0(eij) is the ground-state energy of the bcc-Feunit lattice under strain eij. EHOS(eij, n) includes the influ-ences of the nonlinear elasticity and the chemicalinteractions.

The heats of solution under tensile and shear strainare shown in Figure 2a and b, respectively. It is foundthat changes in the heats of solution notably differaccording to the strain directions for the o-site occupa-tion (Fig. 2a). This result corresponds to the large tetrag-onal strain caused by an o-site occupation. The heat ofsolution for incorporation of hydrogen into the o-site be-comes almost the same as that into the t-site forex > 0.04. The magnitudes of the changes of heat of solu-tion from tensile strain (Fig. 2a) are much larger thanthose from shear strain (Fig. 2b) in this reference crystal,which is oriented in the [100], [010] and [001] directionsto the x-, y- and z-axes, respectively. However, since theheat of solution for incorporation of hydrogen into theo-site is significantly influenced by the direction of tensiledeformation, strong shear strain effects can appear fordifferent crystal orientations, as shown later.

Although DFT calculations can give quantitative pre-dictions for given atomic structures, the model size issmall because of the calculation costs, and therefore

igure 2. Strain effects on the heat of solution of incorporation ofydrogen into bcc-Fe: (a) and (b) show the tensile and shear strainffects estimated by DFT calculations, respectively. (c) and (d) showhe heats of solution into tensile strained bcc-Fe estimated by the EAMWen) potential for 2 � 2 � 2 and 10 � 10 � 10 lattices, respectively.

F

het(

Figure 3. Hydrogen trap energy yielded by pure shear ex0y0 in thecoordinate system, where the [111], ½�1�12� and ½1�10� direction aredefined as the x0, y0 and z0-axes, respectively. (a) The schematicdiagram of the crystal structure and hydrogen occupation sites. (b) Therelationship between shear strain and hydrogen trap energy. Thesymbol j is the maximum trap energy for ex0y0 ¼ 0:05, and h is that forex0y0 ¼ �0:05. These points correspond to those shown in Figure 4c.

R. Matsumoto et al. / Scripta Materialia 60 (2009) 555–558 557

the results suffer from model size effects. Here we need totreat a deformed lattice, which has a low symmetricalproperty for a k-point configuration; thus, using a largermodel is computationally too expensive. We considerusing an interatomic potential that can overcome thesize limitation.

Figure 2c shows the effect of tensile strain on the heatof solution estimated by the EAM (Wen) potential. Weperformed relaxation of the atomic positions and usedthe N = 16 (2 � 2 � 2 lattices) model, which is the sameas that used for DFT calculations. Comparing Figure 2aand c shows that, in spite of the high hydrogen concen-tration, the EAM (Wen) potential represents thechanges in the heats of solution under elastic strain withgood accuracy especially for the o-site occupations. It isalso found in the EAM (Wen) potential calculations, aswell as from first-principles calculations, that hydrogenatoms occupy t-sites under small deformations. The re-sults under shear deformation are omitted because therange of the change is small, as for the DFT calcula-tions. We consider that the difference in the heat of solu-tion for incorporation of hydrogen into a t-site in anundeformed lattice between this calculation (0.33 eV)and the reference value (0.30 eV [26]) is mainly due tothe elastic energy attributed to lattice dilatation.

We estimated the cell size influence by changing thenumber of lattices. The heat of solution for t-site occu-pation becomes almost constant for a unit cell size of3 � 3 � 3 lattice; however, for o-sites, we need a4 � 4 � 4 or even larger unit cell. We confirmed that a10 � 10 � 10 lattice model (N = 2000) is sufficientlylarge for both occupation sites. The strain effects onthe heat of solution for the 10 � 10 � 10 lattice modelare shown in Figure 2d. The heats of solution for incor-poration of hydrogen into the undeformed lattice areEHOS(0, t-site) = 0.29 eV and EHOS(0, o-site) = 0.32 eVfor t- and o-site occupation, respectively, and these val-ues are consistent with the previous large-scale DFT cal-culations [23]. The potential can reproduce elasticconstants accurately; therefore, we consider that theEAM (Wen) potential can predict the effect of strainon the heat of solution for various hydrogen concentra-tions, with at least N P 16 lattice models. Thus, the rela-tionship shown in Figure 2d has quantitatively sufficientaccuracy, without being affected by cell size limitationsand high hydrogen concentration. It was found thattx-site occupation becomes unstable for ey > 0.01 orey < �0.015 (ez > 0.01 or ez < �0.015), and hydrogenmoves toward a neighboring oy-site (oz-site); thus welimited the curve for the small strain.

The contribution to the trap energy of strain eij is ex-pressed as:

DEtrapðeij; nÞ ¼ � EHOSðeij; nÞ � EHOSð0; nÞ� �

: ð2Þ

The strain effect on the hydrogen trap energy can beapproximated using the amount of lattice dilatationcaused by hydrogen. However, for hydrogen, the stableoccupation site changes depending on the strain, and insome cases, different kinds of occupation sites combineinto one or certain occupation sites disappear (Fig. 2dand Refs. [15,25]). Thus, the actual estimation of therelation is important.

We estimate the trap energy yielded by elastic strain e0

at the occupation site positioned at x0 in a different crys-tal orientation as follows: (1) using the rotation tensor,the strain tensor e0 and the coordinates of the hydrogenoccupation site x0 are transformed to the reference coor-dinate system, where DEtrap was preliminarily estimatedas a function of strain component eij and occupation siten; (2) the occupation site is classified as tx-, ty- or tz-site(or ox-, oy- or oz-site); (3) the trap energy yielded by eachstrain component is obtained using DEtrap(eij,n); (4) theabsolute contribution to trap energy from all straincomponents DEtrap_total is estimated as the linear sumof the trap energy yielded by each strain component;and (5) finally, the trap energy Etrap is estimated by con-sidering the difference in the heat of solution between t-and o-sites at an undeformed state (e.g. for o-site occu-pations in bcc-Fe; Etrap = DEtrap_total + (HHOS(0, t-si-te) � HHOS(0, o-site)).

Figure 3b shows the trap energy yielded by pure shearex0y0 in the coordinate system, in which the [111],½�1�12� and ½1�10� directions are defined as the x0-, y0-and z0-axes, respectively (Fig. 3a). Since tx- and ty-siteoccupations become unstable under the severe strainas mentioned in Figure 2d, the curve is limited for smallstrain in the figure. It is found that the trap energy isstrongly dependent on the shear strain, and the sensitiv-ity is nearly equal to that of the tensile strain in the ref-erence crystal orientation (Fig. 2d). A small amount ofshear strain ðex0y0 � 0:01Þ changes the most stable occu-pation site from tx-site (or ty-site) to an oz-site. Further-more, although the energy difference among occupationsites is only 0.03 eV for an undeformed lattice, the differ-ence becomes very large under strained condition (e.g.�0.42 eV when ex0y0 ¼ �0:05 and �0.1 eV whenex0y0 ¼ �0:01Þ. Thus, the elastic strain makes the poten-tial surface of hydrogen diffusion uneven, and the diffu-sion coefficient and diffusion pass are significantlyaltered by strain. We confirmed that the relationship ob-tained by the DFT calculations (Fig. 2a and b) also givessimilar curves.

The hydrogen trap energies yielded by a constantshear strain are calculated for various crystal orienta-tions. The coordinate system shown in Figure 4a, where

Figure 4. Hydrogen trap energy yielded by shear strain for various crystal orientations. (a) The schematic diagram of crystal orientation. (b) Thedependence of the maximum trap energy from shear strain on the rotation angles h and /. (c) The trap energy in the cross-section of h = 0.

558 R. Matsumoto et al. / Scripta Materialia 60 (2009) 555–558

the ½1�1�1�, [110] and ½1�12� directions are defined as thex0-, y0- and z0-axes, respectively, is rotated around they 0-axis by h (transformed from the x0y0z0 to the x00y00z00

coordinate system), and then rotated around the x00-axisby / (transformed from the x00y 00z00 to the x000y000z000 coordi-nate system). Finally, the pure shear strain ex000y000 ¼ 0:05is applied, and the maximum value of hydrogen trap en-ergy among the tx-, ty-, tz-, ox-, oy- and oz-sites is esti-mated for various rotation angles. Figure 4b shows thedependence of the maximum trap energy from shearstrain on the rotation angles h and /, and Figure 4cshows the trap energy in the cross-section of h = 0,which includes slip planes of edge dislocations in bcc-Fe, i.e. {110}, {112} and {123} planes. In the figures,the trap energy arising from volumetric strainev = 0.05 is shown by blue lines (0.154 eV), and the max-imum trap energies indicated by (j) and (h) in Figure3b1 are also shown. It is found that the maximum trapenergy yielded from shear strain exceeds the valueyielded by the same magnitude of volumetric strain ina wide range of crystal orientations. Therefore, for thosecrystal orientations, the hydrogen distributions aroundstress singular points are more influenced by shear strainthan volumetric strain.

Although the shear strain effect can be stronger thanvolumetric strain, the value (maximum of �0.2 eV inFig. 4b) may not be sufficiently large to trap hydrogenat room temperature under a practical hydrogen con-centration, because the hydrogen concentration in lat-tice sites is very low in bcc-Fe. However, remarkableinfluences appear when various strain contributionsoverlap. Moreover, the shear strain effects observed inthis study should appear in bcc metals other than Fe,and even in other crystal structures, which have stronganisotropy of occupation sites, and the influence be-comes significant for metals that absorb a large amountof hydrogen.

In this study, we first estimated the strain effects onthe heat of solution for incorporation of hydrogen intobcc-Fe by DFT calculations, and compared the resultswith those from EAM (Wen) potential. After verifyingthe accuracy of the interatomic potential, we estimatedthe hydrogen trap energy yielded by shear strain for var-ious crystal orientations, using calculation models of

1 For interpretation of the references to colour in this figure, the readeris referred to the web version of this paper.

sufficient size, and revealed the significant influence ofshear strain. This result indicates that hydrogen distri-bution and diffusion behaviour are also influenced byshear strain.

Part of this research was supported by the‘‘Fundamental Research Project on Advanced Hydro-gen Science” funded by the New Energy and IndustrialTechnology Development Organization (NEDO).

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