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Atomistic simulation of hydrogen dynamics near dislocations in vanadium hydrides
Journal of Alloys and Compounds xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jalcom
Atomistic simulation of hydrogen dynamics near dislocations in vanadium hydrides Hiroshi Ogawa ⇑ National Institute of Advanced Industrial Science and Technology (NRI-AIST), 1-1-1, Umezono, Tsukuba, Ibaraki 305-8568, Japan
a r t i c l e
i n f o
Article history: Available online xxxx Keywords: Dislocation Hydrogen Vanadium hydride Diffusion Molecular dynamics
a b s t r a c t Kinetics of interstitial hydrogen atoms near dislocation cores were analyzed by atomistic simulation. Classical molecular dynamics method was applied to model structures of edge and screw dislocations in a-phase vanadium hydride. Simulation showed that hydrogen atoms aggregate near dislocation cores. The spatial distribution of hydrogen has a planner shape at edge dislocation due to dislocation partialization, and a cylindrical shape at screw dislocation. Simulated self-diffusion coefficients of hydrogen atoms in dislocation models were a half- to one-order lower than that of dislocation-free model. Arrhenius plot of self-diffusivity showed slightly different activation energies for edge and screw dislocations. Directional dependency of hydrogen diffusion near dislocation showed high and low diffusivity along edge and screw dislocation lines, respectively, hence so called ‘pipe diffusion’ possibly occur at edge dislocation but does not at screw dislocation. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction Lattice defects are known to play important roles on materials property of metals. They are inevitably generated during hydrogen adsorption and desorption processes, and possibly affect hydrogen storage property. Hydrogen atoms firstly adsorb on metallic surface (planner defect) and migrate via subsurface layer into bulk region by accompanying dislocation (line defect) or point defects at the phase boundary. Some of these defects works as a hydrogen trap and possibly affect hydrogen diffusivity. Dynamical feature of hydrogen atoms in lattice defects is one of the important issues to understand hydrogen storage processes. Relation between lattice defects and hydrogen storage property has been extensively studied. Pundt and Kirchheim [1] reviewed such studies and showed that hydrogen solubility and other properties of thin films or nanoparticles are strongly affected by lattice defects. However, effects of lattice defects, especially dislocation, in bulk region to hydrogen storage property are not fully understood yet. Relation between hydrogen and dislocation has also been studied in aspect of hydrogen embrittlement. It is known that segregation of hydrogen atoms at defects causes degradation of structural materials. One of the interesting topics in this field is so called ‘pipe diffusion’ [2,3] which denotes high diffusivity of atoms along dislocation line. Experimental data of hydrogen diffusivity, however, exhibit large scattering due to incompleteness of the samples such ⇑ Tel.: +81 29 861 9212. E-mail address: [email protected]
as surface [4]. Hence it is difficult to clarify the pure effect of dislocation among several types of lattice defects. Atomistic simulation is useful to understand the individual effect of each defect structure. In the previous study [5], the present author simulated point, line, and planner defects in vanadium hydride by using density functional theory and classical molecular dynamics (MD). Kim et al. [6] recently observed the structure variation of vanadium hydride during hydrogen sorption cycles by X-ray total scattering. They suggested by comparing the observed and simulated values that damping of pair correlation in middle and long ranges after repeated hydrogen sorption could be explained by dislocation pileup. In the present study, effect of dislocation on static and dynamic features of hydrogen in a-vanadium hydride was analyzed by classical MD simulation. 2. Modeling and simulation of dislocation The most fundamental property of dislocation geometry is the Burgers vector b. Depending on the angle between b and dislocation line vector d, dislocation is classified into three types: edge (b is perpendicular to d), screw (b is parallel to d), and mix (others). All types of dislocation can be modeled within a rectangular MD cell with three-dimensional periodic boundary conditions by introducing pairs of positive and negative dislocations having the Burgers vectors +b and b, respectively. Java applet DLstudio [7] was designed to construct such periodic models for arbitrary dislocation in arbitrary crystal. In this study, we deal with two representative dislocations of b.c.c, metal: edge dislocation with d//
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H. Ogawa / Journal of Alloys and Compounds xxx (2015) xxx–xxx
1 1 and b = [1 1 1]a/2 on the slip plane (0 1 1), and screw disloca½2 tion with d//b = [1 1 1]a/2. Detailed settings of the models are given in Table 1. In both models, dislocation line extends toward z infinitely and the dislocation cores initially locate on a rectangular grid on xy plane with about 3 nm spacing. Assumed dislocation density in these models is about 105 lm2. Number of dislocation cores and their arrangement in periodic cell are different between edge and screw models. The edge dislocation model has two cores in a periodic cell, and their replica images of the same sign align toward y direction. This arrangement corresponds to dislocation array on very small angle, tilt grain boundary. In the case of screw dislocation model, two positive and two negative cores locate alternatively on a rectangular grid in periodic cell. This arrangement is equivalent to the periodic, two core model with slant y-axis which is frequently used for screw dislocation calculation [8]. Cell thickness toward z-axis is related to the degree of static and kinetic freedom of atoms along dislocation line. A very thin models having very few atomic layers, which is inevitably adopted in firstprinciples calculation, will lead artificial results on dynamics of hydrogen atoms and dislocation motion by closely neighbored replica images. Thickness of present dislocation models is about 3 nm which is fairly large to reduce the effects from neighboring replica images. A dislocation-free, single crystal model was also calculated for comparison. Appropriate numbers of hydrogen atoms were initially added to the models randomly and homogeneously to the interstitial T-sites. Assumed hydride composition is a-VH0.0625. Many-body and pairwise potentials were assumed for V–V, V–H and H–H interactions. Pair potential of V–H was calculated by fitting the potential energy surface of H atom at more than 100 interstitial points in b.c.c. vanadium [5] by using the VASP code [9,10], and the other terms were assumed the literature values [11,12]. By these potential functions, experimental data [13–17] of a, b and c phases of vanadium hydrides were successfully reproduced as shown in Table 2. At the beginning of simulation, all atoms were freely relaxed from the initial configuration. MD simulation was carried out as constant-NPT ensemble at 0.1 MPa for up to 2 ns at each target temperatures from 150 K to 600 K. 3. Results and discussion 3.1. Simulated core structure and hydrogen atom distribution near dislocation As in real metals, dislocation in MD simulation is movable and possibly causes pair annihilation [5]. In the present models Table 1 Settings of the edge and screw dislocation models. Vectors a and b denote the cubic lattice vector and Burgers vector, respectively, Lx, Ly and Lz are the edge vectors of the periodic cell, and core(+) and core() denote the positive and negative dislocation, respectively. Dislocation line vector d is parallel to Lz in both models. Property
Unit
Edge
Screw
b Lx Ly Lz |Lx| |Ly| |Lz| NV NH Ncore
1/2 [1 1 1] 12 [1 1 1] 8 [0 1 1] 1 1] 4 [2
(x, y)core(+)
a a a a nm nm nm /cell /cell /cell 105 lm2 (Lx, Ly)
6.31 3.45 2.98 4608 288 2 0.92 (1/4, 1/2)
(x, y)core()
(Lx, Ly)
(3/4, 1/2)
1/2 [1 1 1] 1] 8 [2 1 14 [0 1 1] 6 [1 1 1] 5.96 6.02 3.16 8064 504 4 1.11 (1/4, 1/4), (3/4, 3/4) (3/4, 1/4), (1/4, 3/4)
qdisl
Table 2 Comparison of simulated and experimental value at 300 K on lattice constants a and c, surface tension c, and hydrogen diffusion coefficient DH of vanadium and its hydride.
a
Property
Phase
MD
Exp.
a (nm) a (nm) c (nm) a (nm) c{1 0 0} (J m2) c{1 10} (J m2) DH (109 m2 s1)
a
.304 .579 .682 .418 3.23 2.55 6.1
.303 [13] .600 [14] .662 [14] .428 [13] 3.18 [15]a 2.55 [16] 6 [17]
b b
c pure V pure V
a
Calculation.
including small amount of hydrogen, both edge and screw dislocations remained during the simulation. Firstly, core structure of each dislocation type was examined. The relaxed positions of vanadium atoms on the slip plane of edge dislocation show small displacements from the straight line along the Burgers vector as shown in Fig. 1. The trace of dislocated atoms resembles to the extension of edge dislocation suggested by Cohen et al. [18] for b.c.c. metals by combination of three partials as b = {[0 1 1] + 2[2 1 1] + [0 1 1]}a/8. In the case of screw dislocation, on the other hand, two types are known for the candidate core structure for b.c.c. metals as hard and easy [19]. Simulated core structure in this study was easy-type. Formation of easy-type core is consistent with experimental [20] and simulation [21] results on b.c.c. transition metals. During the initial relaxation, hydrogen atoms aggregated in the vicinity of dislocation lines. Fig. 2 shows the number density profiles of hydrogen atoms after relaxation as functions of distance from dislocation core. Both edge and screw dislocations attract hydrogen atoms to their core regions by about 4 times of that at far from dislocation. The average profiles (solid line) in both dislocation models show similar Gaussian-like curves with half-height radius of about 0.5 nm from core centers. Hydrogen trapping at dislocation was observed by small angle neutron scattering experiment [22,23] and the radius of segregated region was estimated to be about 1 nm [24]. Anisotropic distribution of hydrogen atoms toward x and y direction (dashed lines) was found in edge dislocation model: aggregated region extends up to about 1 nm toward x (along the slip plane) but about a half toward y (perpendicular to the slip plane). This anisotropy is probably cause by the dislocation partialization mentioned above. Similar anisotropic distribution in site energies was calculated for edge dislocation in b.c.c. iron [25]. In
x
[111]
z [211]
[211]
[011]
Fig. 1. Positions of vanadium atoms (small red circles) on a slip plane relative to the lattice under the plane (large green circles) (For interpretation of the (0 1 1) references to color in this figure legend, the reader is referred to the web version of this article.).
Please cite this article in press as: H. Ogawa, J. Alloys Comp. (2015), http://dx.doi.org/10.1016/j.jallcom.2015.01.060
3
(a)
edge average
toward x
toward y
distance from dislocation core / nm
Number density of H (normalized)
Number density of H (normalized)
H. Ogawa / Journal of Alloys and Compounds xxx (2015) xxx–xxx
(b)
screw
average toward x toward y
distance from dislocation core / nm
Fig. 2. Distribution of hydrogen atoms in edge (a) and screw (b) dislocation models as functions of direction and distance from dislocation core after relaxation at 300 K.
the case of screw dislocation, on the other hand, hydrogen atoms distribute cylindrically along dislocation line. 3.2. Temperature dependence of hydrogen diffusivity in dislocation models
Diffusion coefficient / (m 2 /s)
Fig. 3 shows the temperature variation of simulated hydrogen diffusivity in dislocation-free and edge/screw dislocation models. Self-diffusion coefficients were calculated by ordinary meansquared displacement method [26]. The experimental data in the literatures [17] scatters especially in the lower temperature region as in the figure. Kiuchi and McLellan [4] suggested that the data scattering was caused by incompleteness of experimental samples such as surface effect. The simulated values of dislocation-free model trace the upper bound of scattered experimental data. This result is reasonable since the MD model is surface-free. In comparison with dislocation-free model, dislocation models gave a half- to one-order lower hydrogen diffusivity. This result is rather contradictory to the discussion on palladium [27,28]. There are several reports on hydrogen diffusion at dislocation in vanadium or b.c.c metals. Schaumann et al. [29] discussed pipe diffusion of hydrogen or deuterium in deformed vanadium and suggested it is unlikely within possible dislocation density range. Experimental results on deformed iron ([30], [4] and references therein) suggested that dislocation provides trapping sites for hydrogen and generally lower the diffusivity. Recent simulation studies also reveal that fast diffusion may occur in f.c.c. metals [31] but does not at grain boundary [32] or dislocation [33] in b.c.c. iron. These results suggests that hydrogen pipe-diffusion along dislocation is not evident for b.c.c. metals.
10–8
10–9
10–10
MD (edge disl.) MD (screw disl.) MD (dislocation-free) experiments
10 3 K / Temperature Fig. 3. Arrhenius plots of simulated hydrogen self-diffusion coefficients of edge/ screw dislocation and dislocation-free models in comparison with experimental values [17]. Simulated values are the average of whole hydrogen atoms in the model.
On the other hand, most of experimental or simulation studies on pipe diffusion suggest lower activation energy at dislocation core compared with crystal region [27,28,31]. It may be worth mentioning that the Arrhenius plots in Fig. 3 show slightly different slopes in higher and lower temperature regions. The fitted activation energies at 600–300 K and 300–150 K in the screw dislocation model were 68 and 51 meV, respectively. Similar inflection in Arrhenius plot was also observed in deformed polycrystalline Pd [28]. 3.3. Detailed analyses of hydrogen diffusion at dislocation line Fig. 4 shows directional anisotropy of hydrogen diffusivity toward perpendicular (x, y) or parallel (z) to dislocation lines at 300 K. Calculation was made by dividing hydrogen atoms in each distance range according to their initial positions. In the case of edge dislocation, diffusivity along dislocation line is dominant in whole region especially at the core center. In other word, so-called pipe diffusion possibly occurs at edge dislocation. It should be noted, however, absolute value of diffusion coefficient along edge dislocation line is still lower than the single crystal values shown in Fig. 3. Comparatively higher diffusivity toward x (parallel to the slip plane) than that of y (perpendicular to the slip plane) is possibly related to the anisotropic distribution of hydrogen atoms shown in Fig. 2(a). In the case of screw dislocation, on the other hand, diffusivity along dislocation line is not dominant compared with other directions, hence pipe diffusion does not occur in the present model as pointed out for b.c.c. iron [33]. This result can be understood by the core structure of easy-type [21] having reversely winded atomic bonds at the core center which seems to be not efficient for hopping motion of hydrogen atoms compared with hard-type. The other important factor of hydrogen diffusivity in defective structure is dependency on hydrogen concentration [22,32,35]. Orimo et al. [34] discussed hydrogen diffusivity in nanostructured vanadium by NMR measurement. Their 10 nm grain sample includes disordered inter-granular region by about 30% which is close to the ratio of dislocated region in the present model. The H/V ratio of their sample, however, was 0.67 where the bulk diffusion is dominant [32]. Considering the hydrogen saturation at dislocation region, analysis on lower hydrogen concentration is necessary [35]. Heller and Wipf [36] measured hydrogen diffusivity in vanadium at H/V 6 0.013 and showed hydrogen diffusivity decreases with increasing hydrogen concentration. Unfortunately, dislocation simulation at very low H/V ratio is not stable because of high dislocation mobility [5], and only the screw dislocation model was successful at H/V = 0.01. The resulted diffusivity was about 1.5 times of H/V = 0.06 which is consistent with the experimental [36] and simulation [32] results.
Please cite this article in press as: H. Ogawa, J. Alloys Comp. (2015), http://dx.doi.org/10.1016/j.jallcom.2015.01.060
(a)
x y z
edge
distance from dislocation core / nm
directional component of H diffusivity
H. Ogawa / Journal of Alloys and Compounds xxx (2015) xxx–xxx
directional component of H diffusivity
4
(b)
x y z
screw
distance from dislocation core / nm
Fig. 4. Anisotropic diffusivity of hydrogen atoms along edge (a) and screw (b) dislocation lines at 300 K as functions of distance from dislocation core. The values were normalized by the average diffusivity of three directions to be unity.
It is natural to consider that low hydrogen diffusivity near dislocation cores affects structural restoration during repeated hydrogen sorption. The previous study [5] showed that dislocation migration speed to pair-annihilation is lowered by hydrogen adsorption, and remaining dislocations will cause dead storage of hydrogen to reduce the storage capacity. This mechanism supplies a good explanation for the results of Kim et al. [6] on the reduction of hydrogen storage capacity and structure variation. Atomistic simulation will be useful for investigating such variation in hydrogen storage property by dislocation. 4. Summary Effects of edge and screw dislocation on dynamics of hydrogen atoms in vanadium hydrides were studied by classical MD simulation. Simulated core structure was found to be slightly extended for edge dislocation, and easy-type for screw dislocation. Distribution of hydrogen atoms near dislocation core has planner shape for edge dislocation, and cylindrical for screw dislocation. Hydrogen diffusivity in both dislocation models was found to be lower by a half- to one-order than that of dislocation-free model. Directional diffusivity of hydrogen toward dislocation line, so called pipe diffusion, was higher for edge but lower for screw compared with those toward other directions. The lowered hydrogen diffusivity near dislocation is considered to be an origin of structure variation and degradation of vanadium hydride during repeated hydrogen sorption. References [1] [2] [3] [4]
A. Pundt, R. Kirchheim, Ann. Rev. Mater. Res. 36 (2006) 555–608. E.W. Hart, Acta Metall. 5 (1957) 597. G.R. Love, Acta Metall. 12 (1964) 731–737. K. Kiuchi, R.B. McLellan, Acta Metall. 31 (1983) 961–984.
[5] H. Ogawa, J. Alloys Comp. 580 (2013) S131–S134. [6] H. Kim, K. Sakaki, H. Ogawa, Y. Nakamura, J. Nakamura, E. Akiba, A. Machida, T. Watanuki, T. Proffen, J. Phys. Chem. C 117 (2013) 26543–26550. [7] H. Ogawa, DLstudio, Java applet on the web, 2012. [8] for example F. Shimizu, S. Ogata, H. Kimizuka, T. Kano, J. Li, H. Kaburaki, J. Earth Simulator 7 (2007) 17–21. [9] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558–561. [10] G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169–11186. [11] M.W. Finnis, J.E. Sinclair, Phil. Mag. A 50 (1984) 45–55. [12] M.S. Daw, M.I. Baskes, Phys. Rev. B 29 (1984) 6443–6453. [13] F.D. Manchester (Ed.), Phase Diagrams of Binary Hydrogen Alloys, ASM International, 2000. [14] Y. Noda, K. Masumoto, S. Koike, T. Suzuki, S. Sato, Acta Cryst. B42 (1986) 529– 533. [15] L. Vitos, A.V. Ruban, H.L. Skriver, J. Kolla´r, Surf. Sci. 411 (1998) 186–202. [16] F.R. de Boer, R. Boom, W.C.M. Mattens, A.R. Miedema, A.K. Niessen, Cohesion in Metals, North-Holland, Amsterdam, 1988. [17] J. Völkl, G. Alefeld, Hydrogen in Metals I, Springer-Verlag, 1978. pp. 321–348. [18] J.B. Cohen, R. Hinton, K. Lay, S. Sass, Acta Metall. 10 (1962) 894–895. [19] V. Vítek, R.C. Perrin, D.K. Bowen, Phil. Mag. 21 (1970) 1049–1073. [20] V. Vitek, Crystal Lattice Defects 5 (1974) 1–34. [21] W. Xu, J.A. Moriarty, Phys. Rev. B 54 (1996) 6941–6951. [22] R. Kirchheim, Prog. Mater. Sci. 32 (1988) 261–325. [23] D.K. Ross, K.L. Stefanopoulos, K.S. Forcey, I. Iordanova, Z. Phys. Chem. 183 (1994) S29–S37. [24] M. Maxelon, A. Pundt, W. Pyckhout-Hintzen, J. Barker, R. Kirchheim, Acta Mater. 49 (2001) 2625–2634. [25] S. Taketomi, R. Matsumoto, N. Miyazaki, J. Mater. Sci. 43 (2008) 1166–1169. [26] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Oxford, 1987. [27] R. Kirchheim, Acta Metall. 29 (1981) 835–843. [28] B.J. Heuser, D.R. Trinkle, N. Jalarvo, J. Serio, E.J. Schivone, E. Manontov, M. Tyagi, Phys. Rev. Let. 113 (2014) 025504. [29] G. Schaumann, J. Völkl, G. Alefeld, Phys. Stat. Sol. 42 (1970) 401–413. [30] A.J. Kumnick, H.H. Johnson, Acta Mater. 28 (1980) 33–39. [31] Y. Tang, J.A. El-Awady, Phys. Rev. B 86 (2012) 174102. [32] Y.A. Du, J. Rogal, R. Drautz, Phys. Rev. B 86 (2012) 74110. [33] H. Kimizuka, S. Ogata, Phys. Rev. B 84 (2011) 024116. [34] S. Orimo, F. Kinmerle, G. Majer, Phys. Rev. B 63 (2001) 094307. [35] S.M. Myers, M.I. Baskes, H.K. Birnbaum, J.W. Corbett, G.G. DeLeo, S.K. Estreicher, E.E. Haller, P. Jena, N.M. Johnson, R. Kirchheim, S.J. Pearton, M.J. Stavola, Rev. Modern Phys. 64 (1992) 559–617. [36] R. Heller, H. Wipf, Phys. Stat. Sol. 33 (1976) 525–529.
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Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jalcom
Atomistic simulation of hydrogen dynamics near dislocations in vanadium hydrides Hiroshi Ogawa ⇑ National Institute of Advanced Industrial Science and Technology (NRI-AIST), 1-1-1, Umezono, Tsukuba, Ibaraki 305-8568, Japan
a r t i c l e
i n f o
Article history: Available online xxxx Keywords: Dislocation Hydrogen Vanadium hydride Diffusion Molecular dynamics
a b s t r a c t Kinetics of interstitial hydrogen atoms near dislocation cores were analyzed by atomistic simulation. Classical molecular dynamics method was applied to model structures of edge and screw dislocations in a-phase vanadium hydride. Simulation showed that hydrogen atoms aggregate near dislocation cores. The spatial distribution of hydrogen has a planner shape at edge dislocation due to dislocation partialization, and a cylindrical shape at screw dislocation. Simulated self-diffusion coefficients of hydrogen atoms in dislocation models were a half- to one-order lower than that of dislocation-free model. Arrhenius plot of self-diffusivity showed slightly different activation energies for edge and screw dislocations. Directional dependency of hydrogen diffusion near dislocation showed high and low diffusivity along edge and screw dislocation lines, respectively, hence so called ‘pipe diffusion’ possibly occur at edge dislocation but does not at screw dislocation. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction Lattice defects are known to play important roles on materials property of metals. They are inevitably generated during hydrogen adsorption and desorption processes, and possibly affect hydrogen storage property. Hydrogen atoms firstly adsorb on metallic surface (planner defect) and migrate via subsurface layer into bulk region by accompanying dislocation (line defect) or point defects at the phase boundary. Some of these defects works as a hydrogen trap and possibly affect hydrogen diffusivity. Dynamical feature of hydrogen atoms in lattice defects is one of the important issues to understand hydrogen storage processes. Relation between lattice defects and hydrogen storage property has been extensively studied. Pundt and Kirchheim [1] reviewed such studies and showed that hydrogen solubility and other properties of thin films or nanoparticles are strongly affected by lattice defects. However, effects of lattice defects, especially dislocation, in bulk region to hydrogen storage property are not fully understood yet. Relation between hydrogen and dislocation has also been studied in aspect of hydrogen embrittlement. It is known that segregation of hydrogen atoms at defects causes degradation of structural materials. One of the interesting topics in this field is so called ‘pipe diffusion’ [2,3] which denotes high diffusivity of atoms along dislocation line. Experimental data of hydrogen diffusivity, however, exhibit large scattering due to incompleteness of the samples such ⇑ Tel.: +81 29 861 9212. E-mail address: [email protected]
as surface [4]. Hence it is difficult to clarify the pure effect of dislocation among several types of lattice defects. Atomistic simulation is useful to understand the individual effect of each defect structure. In the previous study [5], the present author simulated point, line, and planner defects in vanadium hydride by using density functional theory and classical molecular dynamics (MD). Kim et al. [6] recently observed the structure variation of vanadium hydride during hydrogen sorption cycles by X-ray total scattering. They suggested by comparing the observed and simulated values that damping of pair correlation in middle and long ranges after repeated hydrogen sorption could be explained by dislocation pileup. In the present study, effect of dislocation on static and dynamic features of hydrogen in a-vanadium hydride was analyzed by classical MD simulation. 2. Modeling and simulation of dislocation The most fundamental property of dislocation geometry is the Burgers vector b. Depending on the angle between b and dislocation line vector d, dislocation is classified into three types: edge (b is perpendicular to d), screw (b is parallel to d), and mix (others). All types of dislocation can be modeled within a rectangular MD cell with three-dimensional periodic boundary conditions by introducing pairs of positive and negative dislocations having the Burgers vectors +b and b, respectively. Java applet DLstudio [7] was designed to construct such periodic models for arbitrary dislocation in arbitrary crystal. In this study, we deal with two representative dislocations of b.c.c, metal: edge dislocation with d//
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Please cite this article in press as: H. Ogawa, J. Alloys Comp. (2015), http://dx.doi.org/10.1016/j.jallcom.2015.01.060
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H. Ogawa / Journal of Alloys and Compounds xxx (2015) xxx–xxx
1 1 and b = [1 1 1]a/2 on the slip plane (0 1 1), and screw disloca½2 tion with d//b = [1 1 1]a/2. Detailed settings of the models are given in Table 1. In both models, dislocation line extends toward z infinitely and the dislocation cores initially locate on a rectangular grid on xy plane with about 3 nm spacing. Assumed dislocation density in these models is about 105 lm2. Number of dislocation cores and their arrangement in periodic cell are different between edge and screw models. The edge dislocation model has two cores in a periodic cell, and their replica images of the same sign align toward y direction. This arrangement corresponds to dislocation array on very small angle, tilt grain boundary. In the case of screw dislocation model, two positive and two negative cores locate alternatively on a rectangular grid in periodic cell. This arrangement is equivalent to the periodic, two core model with slant y-axis which is frequently used for screw dislocation calculation [8]. Cell thickness toward z-axis is related to the degree of static and kinetic freedom of atoms along dislocation line. A very thin models having very few atomic layers, which is inevitably adopted in firstprinciples calculation, will lead artificial results on dynamics of hydrogen atoms and dislocation motion by closely neighbored replica images. Thickness of present dislocation models is about 3 nm which is fairly large to reduce the effects from neighboring replica images. A dislocation-free, single crystal model was also calculated for comparison. Appropriate numbers of hydrogen atoms were initially added to the models randomly and homogeneously to the interstitial T-sites. Assumed hydride composition is a-VH0.0625. Many-body and pairwise potentials were assumed for V–V, V–H and H–H interactions. Pair potential of V–H was calculated by fitting the potential energy surface of H atom at more than 100 interstitial points in b.c.c. vanadium [5] by using the VASP code [9,10], and the other terms were assumed the literature values [11,12]. By these potential functions, experimental data [13–17] of a, b and c phases of vanadium hydrides were successfully reproduced as shown in Table 2. At the beginning of simulation, all atoms were freely relaxed from the initial configuration. MD simulation was carried out as constant-NPT ensemble at 0.1 MPa for up to 2 ns at each target temperatures from 150 K to 600 K. 3. Results and discussion 3.1. Simulated core structure and hydrogen atom distribution near dislocation As in real metals, dislocation in MD simulation is movable and possibly causes pair annihilation [5]. In the present models Table 1 Settings of the edge and screw dislocation models. Vectors a and b denote the cubic lattice vector and Burgers vector, respectively, Lx, Ly and Lz are the edge vectors of the periodic cell, and core(+) and core() denote the positive and negative dislocation, respectively. Dislocation line vector d is parallel to Lz in both models. Property
Unit
Edge
Screw
b Lx Ly Lz |Lx| |Ly| |Lz| NV NH Ncore
1/2 [1 1 1] 12 [1 1 1] 8 [0 1 1] 1 1] 4 [2
(x, y)core(+)
a a a a nm nm nm /cell /cell /cell 105 lm2 (Lx, Ly)
6.31 3.45 2.98 4608 288 2 0.92 (1/4, 1/2)
(x, y)core()
(Lx, Ly)
(3/4, 1/2)
1/2 [1 1 1] 1] 8 [2 1 14 [0 1 1] 6 [1 1 1] 5.96 6.02 3.16 8064 504 4 1.11 (1/4, 1/4), (3/4, 3/4) (3/4, 1/4), (1/4, 3/4)
qdisl
Table 2 Comparison of simulated and experimental value at 300 K on lattice constants a and c, surface tension c, and hydrogen diffusion coefficient DH of vanadium and its hydride.
a
Property
Phase
MD
Exp.
a (nm) a (nm) c (nm) a (nm) c{1 0 0} (J m2) c{1 10} (J m2) DH (109 m2 s1)
a
.304 .579 .682 .418 3.23 2.55 6.1
.303 [13] .600 [14] .662 [14] .428 [13] 3.18 [15]a 2.55 [16] 6 [17]
b b
c pure V pure V
a
Calculation.
including small amount of hydrogen, both edge and screw dislocations remained during the simulation. Firstly, core structure of each dislocation type was examined. The relaxed positions of vanadium atoms on the slip plane of edge dislocation show small displacements from the straight line along the Burgers vector as shown in Fig. 1. The trace of dislocated atoms resembles to the extension of edge dislocation suggested by Cohen et al. [18] for b.c.c. metals by combination of three partials as b = {[0 1 1] + 2[2 1 1] + [0 1 1]}a/8. In the case of screw dislocation, on the other hand, two types are known for the candidate core structure for b.c.c. metals as hard and easy [19]. Simulated core structure in this study was easy-type. Formation of easy-type core is consistent with experimental [20] and simulation [21] results on b.c.c. transition metals. During the initial relaxation, hydrogen atoms aggregated in the vicinity of dislocation lines. Fig. 2 shows the number density profiles of hydrogen atoms after relaxation as functions of distance from dislocation core. Both edge and screw dislocations attract hydrogen atoms to their core regions by about 4 times of that at far from dislocation. The average profiles (solid line) in both dislocation models show similar Gaussian-like curves with half-height radius of about 0.5 nm from core centers. Hydrogen trapping at dislocation was observed by small angle neutron scattering experiment [22,23] and the radius of segregated region was estimated to be about 1 nm [24]. Anisotropic distribution of hydrogen atoms toward x and y direction (dashed lines) was found in edge dislocation model: aggregated region extends up to about 1 nm toward x (along the slip plane) but about a half toward y (perpendicular to the slip plane). This anisotropy is probably cause by the dislocation partialization mentioned above. Similar anisotropic distribution in site energies was calculated for edge dislocation in b.c.c. iron [25]. In
x
[111]
z [211]
[211]
[011]
Fig. 1. Positions of vanadium atoms (small red circles) on a slip plane relative to the lattice under the plane (large green circles) (For interpretation of the (0 1 1) references to color in this figure legend, the reader is referred to the web version of this article.).
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3
(a)
edge average
toward x
toward y
distance from dislocation core / nm
Number density of H (normalized)
Number density of H (normalized)
H. Ogawa / Journal of Alloys and Compounds xxx (2015) xxx–xxx
(b)
screw
average toward x toward y
distance from dislocation core / nm
Fig. 2. Distribution of hydrogen atoms in edge (a) and screw (b) dislocation models as functions of direction and distance from dislocation core after relaxation at 300 K.
the case of screw dislocation, on the other hand, hydrogen atoms distribute cylindrically along dislocation line. 3.2. Temperature dependence of hydrogen diffusivity in dislocation models
Diffusion coefficient / (m 2 /s)
Fig. 3 shows the temperature variation of simulated hydrogen diffusivity in dislocation-free and edge/screw dislocation models. Self-diffusion coefficients were calculated by ordinary meansquared displacement method [26]. The experimental data in the literatures [17] scatters especially in the lower temperature region as in the figure. Kiuchi and McLellan [4] suggested that the data scattering was caused by incompleteness of experimental samples such as surface effect. The simulated values of dislocation-free model trace the upper bound of scattered experimental data. This result is reasonable since the MD model is surface-free. In comparison with dislocation-free model, dislocation models gave a half- to one-order lower hydrogen diffusivity. This result is rather contradictory to the discussion on palladium [27,28]. There are several reports on hydrogen diffusion at dislocation in vanadium or b.c.c metals. Schaumann et al. [29] discussed pipe diffusion of hydrogen or deuterium in deformed vanadium and suggested it is unlikely within possible dislocation density range. Experimental results on deformed iron ([30], [4] and references therein) suggested that dislocation provides trapping sites for hydrogen and generally lower the diffusivity. Recent simulation studies also reveal that fast diffusion may occur in f.c.c. metals [31] but does not at grain boundary [32] or dislocation [33] in b.c.c. iron. These results suggests that hydrogen pipe-diffusion along dislocation is not evident for b.c.c. metals.
10–8
10–9
10–10
MD (edge disl.) MD (screw disl.) MD (dislocation-free) experiments
10 3 K / Temperature Fig. 3. Arrhenius plots of simulated hydrogen self-diffusion coefficients of edge/ screw dislocation and dislocation-free models in comparison with experimental values [17]. Simulated values are the average of whole hydrogen atoms in the model.
On the other hand, most of experimental or simulation studies on pipe diffusion suggest lower activation energy at dislocation core compared with crystal region [27,28,31]. It may be worth mentioning that the Arrhenius plots in Fig. 3 show slightly different slopes in higher and lower temperature regions. The fitted activation energies at 600–300 K and 300–150 K in the screw dislocation model were 68 and 51 meV, respectively. Similar inflection in Arrhenius plot was also observed in deformed polycrystalline Pd [28]. 3.3. Detailed analyses of hydrogen diffusion at dislocation line Fig. 4 shows directional anisotropy of hydrogen diffusivity toward perpendicular (x, y) or parallel (z) to dislocation lines at 300 K. Calculation was made by dividing hydrogen atoms in each distance range according to their initial positions. In the case of edge dislocation, diffusivity along dislocation line is dominant in whole region especially at the core center. In other word, so-called pipe diffusion possibly occurs at edge dislocation. It should be noted, however, absolute value of diffusion coefficient along edge dislocation line is still lower than the single crystal values shown in Fig. 3. Comparatively higher diffusivity toward x (parallel to the slip plane) than that of y (perpendicular to the slip plane) is possibly related to the anisotropic distribution of hydrogen atoms shown in Fig. 2(a). In the case of screw dislocation, on the other hand, diffusivity along dislocation line is not dominant compared with other directions, hence pipe diffusion does not occur in the present model as pointed out for b.c.c. iron [33]. This result can be understood by the core structure of easy-type [21] having reversely winded atomic bonds at the core center which seems to be not efficient for hopping motion of hydrogen atoms compared with hard-type. The other important factor of hydrogen diffusivity in defective structure is dependency on hydrogen concentration [22,32,35]. Orimo et al. [34] discussed hydrogen diffusivity in nanostructured vanadium by NMR measurement. Their 10 nm grain sample includes disordered inter-granular region by about 30% which is close to the ratio of dislocated region in the present model. The H/V ratio of their sample, however, was 0.67 where the bulk diffusion is dominant [32]. Considering the hydrogen saturation at dislocation region, analysis on lower hydrogen concentration is necessary [35]. Heller and Wipf [36] measured hydrogen diffusivity in vanadium at H/V 6 0.013 and showed hydrogen diffusivity decreases with increasing hydrogen concentration. Unfortunately, dislocation simulation at very low H/V ratio is not stable because of high dislocation mobility [5], and only the screw dislocation model was successful at H/V = 0.01. The resulted diffusivity was about 1.5 times of H/V = 0.06 which is consistent with the experimental [36] and simulation [32] results.
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(a)
x y z
edge
distance from dislocation core / nm
directional component of H diffusivity
H. Ogawa / Journal of Alloys and Compounds xxx (2015) xxx–xxx
directional component of H diffusivity
4
(b)
x y z
screw
distance from dislocation core / nm
Fig. 4. Anisotropic diffusivity of hydrogen atoms along edge (a) and screw (b) dislocation lines at 300 K as functions of distance from dislocation core. The values were normalized by the average diffusivity of three directions to be unity.
It is natural to consider that low hydrogen diffusivity near dislocation cores affects structural restoration during repeated hydrogen sorption. The previous study [5] showed that dislocation migration speed to pair-annihilation is lowered by hydrogen adsorption, and remaining dislocations will cause dead storage of hydrogen to reduce the storage capacity. This mechanism supplies a good explanation for the results of Kim et al. [6] on the reduction of hydrogen storage capacity and structure variation. Atomistic simulation will be useful for investigating such variation in hydrogen storage property by dislocation. 4. Summary Effects of edge and screw dislocation on dynamics of hydrogen atoms in vanadium hydrides were studied by classical MD simulation. Simulated core structure was found to be slightly extended for edge dislocation, and easy-type for screw dislocation. Distribution of hydrogen atoms near dislocation core has planner shape for edge dislocation, and cylindrical for screw dislocation. Hydrogen diffusivity in both dislocation models was found to be lower by a half- to one-order than that of dislocation-free model. Directional diffusivity of hydrogen toward dislocation line, so called pipe diffusion, was higher for edge but lower for screw compared with those toward other directions. The lowered hydrogen diffusivity near dislocation is considered to be an origin of structure variation and degradation of vanadium hydride during repeated hydrogen sorption. References [1] [2] [3] [4]
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