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Monovacancy in copper: Trapping efficiency for hydrogen and oxygen impurities
Computational Materials Science 84 (2014) 122–128
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Monovacancy in copper: Trapping efficiency for hydrogen and oxygen impurities P.A. Korzhavyi 1, R. Sandström ⇑ Department of Materials Science and Engineering, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden
a r t i c l e
i n f o
Article history: Received 18 September 2013 Received in revised form 22 November 2013 Accepted 27 November 2013
Keywords: Hydrogen trapping Point defects Ab initio calculations
a b s t r a c t The structure and binding energy of vacancy–impurity complexes in copper are studied using firstprinciples calculations based on density functional theory. A single vacancy is found to be able to trap up to six hydrogen atoms which tend to be situated inside the vacancy at off-center positions (related to the octahedral interstitial positions of the ideal fcc lattice). The binding energy of an H atom dissolved in the Cu lattice (octahedral interstitial position) to a vacancy is calculated to be about 0.24 eV, practically independent of the number of H atoms already trapped by the vacancy, up to the saturation with 6 hydrogens. For an oxygen impurity in Cu, a monovacancy is shown to be a deep trap (with a binding energy of 0.95 eV). The position of a trapped O atom inside a vacancy is off-center, almost a half-way from the nearest octahedral interstitial to the vacancy center. Such a vacancy–O cluster is shown to be a deep trap for dissolved hydrogen (the calculated binding energy is 1.23 eV). The trapping results in the formation of an OH-group, where the H atom is situated near the vacancy center, and the O atom is displaced from the center along a h1 0 0i direction towards a nearby octahedral interstitial position. Further hydrogenation of the monovacancy–OH cluster is calculated to be energetically unfavourable. McNabb–Forster’s equations are generalised to describe the competition between a deep hydrogen trap and a shallow one. It is demonstrated that the deep trap is almost fully filled, which explains why some of hydrogen is strongly bound and cannot be removed without vacuum treatment at elevated temperatures. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Interactions of dissolved hydrogen with lattice defects in metals have been studied quite thoroughly in the past [1–5], presently they are again of high interest in connection with the applications of metals as catalysts, parts of batteries and fuel cells, materials for hydrogen production and storage [6–8], as well as in connection with the recent advances in the understanding of hydrogen-related effects on the mechanical properties of metallic materials [9–14]. The solubility, mobility, and trapping of hydrogen in copper metal, as well as the stable and metastable atomic configurations in the ternary Cu–O–H system [15], are of particular interest also in connection with the Swedish plan of spent nuclear fuel disposal in a deep geological repository using copper canisters [16–18]. First-principles calculations and atomistic simulations provide important information about the atomic-scale structures and atomistic mechanisms involved in hydrogen-related effects in metallic materials [19–22]. In the present work, the first-principles approach is used in order to study possible interactions of ⇑ Corresponding author. Tel.: +46 8 790 91 93; fax: +46 8 20 76 81. 1
E-mail addresses: [email protected] (P.A. Korzhavyi), [email protected] (R. Sandström). Principal corresponding author.
0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.11.065
hydrogen and oxygen impurities with a single vacancy (monovacancy) in copper. 2. Calculation The electronic structure and total energy calculations were based on density functional theory [23,24] in the generalised gradient approximation (GGA) using the formulation by Perdew et al. [25]. Ultrasoft pseudopotentials due to Vanderbilt [26] and a plane-wave basis set with the energy cutoff 50 Ry for the wave function and 340 Ry for the density were used in the calculations that employed PWscf code of the Quantum ESPRESSO distribution [27]. Copper metal was modelled using a 108-site supercell obtained by a 3 3 3 multiplication of a face-centered cubic (fcc) unit cell containing 4 atoms. The cases considered in our calculations included a structure with the perfect site occupancy, a supercell with one vacancy, supercell containing H or O impurities, and supercells containing one vacancy and several impurities in various configurations. The calculations were performed at a fixed lattice parameter of the supercell corresponding to three times the calculated equilibrium lattice parameter a0 = 3.63 Å of pure fcc Cu. The shape of the supercell was also kept fixed, while the internal degrees of freedom (atomic positions) were allowed to
P.A. Korzhavyi, R. Sandström / Computational Materials Science 84 (2014) 122–128
relax. The local symmetry of the defect configurations was intentionally broken before relaxing the atomic positions. The Brillouin zone integration was carried out using a 2 2 2 Monkhorst– Pack mesh of k-points. The so calculated total energy, as a function of the atomic coordinates, represents the potential energy surface for atomic motion. A next step should be to solve the quantum-mechanical equations of atomic motion, which is essential for light impurities like hydrogen even at zero Kelvin. In this study, however, we limit ourselves to just computing the potential energy surface and comparing the depths of its deepest energy minima, in order to locate the trapping sites for impurities and to estimate the corresponding binding energies. One should thus keep in mind that the calculated binding energies do not contain the zero-point energy contribution, which mostly cancels out. The omitted zero-point energy contribution does not exceed 10% of the computed value, which is negligible in all cases except one where a hydrogen impurity enters an oxygen-contaminated vacancy to form a hydroxyl group. In the latter case the binding energy should be lowered by about 0.1 eV. 3. Results and discussion 3.1. Single point defects The formation energy of a monovacancy in copper metal can be derived using the calculated total energies, EðCuN1 Þ of a supercell containing N 1 atoms with one vacancy and EðCuN Þ of a defectfree supercell containing N atoms, as follows:
EF1V ¼ EðCuN1 Þ
N1 EðCuN Þ: N
ð1Þ
The calculated value EF1V ¼ 1:07 eV is in agreement with previously calculated values which scatter in the range from 1.0 to 1.4 eV [28–33] (and closer to lower bound of that range, together with other theoretical values obtained using relaxed atomic configurations and GGA [31–33]) as well with experimental values, 0.92–1.31 eV, of the vacancy formation energy in copper [34–36]. Estimates of the solution energy for X = {H,O} impurities in copper may be obtained from the energies EðCuN X1 Þ of supercells each containing one of these impurities in the most stable (calculated to be octahedral) interstitial position, the energy of a defect-free Cu supercell EðCuN Þ, the energies EðX1 Þ of free H and O atoms, and experimental atomization energies DEðX2 Þ ¼ 2EðX1 Þ EðX2 Þ of H2 and O2 molecules:
1 DEsol ¼ EðCuN X1 EðCuN Þ EðX1 Þ þ DEðX2 Þ: 2
ð2Þ
The necessity to use experimental atomization energies comes from the fact that GGA is known [37] to describe the stability of H2 and O2 molecules rather inaccurately (the errors are too large even for crude estimates) while treating the bulk copper metal relatively accurately. For the sake of simplicity, let us exclude the energy of ionic zeropoint motion from all the terms in Eq. (2). Then the last term in Eq. (2) becomes equal to 2.374 eV for H2 and 2.613 eV for O2 [37], and one arrives at the following estimates: DEsol ðH2 in CuÞ ¼ þ0:41 eV and DEsol ðO2 in CuÞ ¼ 0:34 eV. These values are in good qualitative agreement with the solution energies that can be extracted from the experimental solubility of hydrogen (+0.44 eV, Ref. [38]) and oxygen (0.53 eV, Refs. [39,40]) in the copper metal. Let us now analyse the site preference and mobility of a single H or O impurity in the otherwise defect-free Cu. As follows from the calculated potential energy maps presented in Fig. 1(a and b), both H and O impurities prefer octahedral interstitial positions in the fcc lattice. However, the potential exhibits additional local minima at the tetrahedral interstitial positions. From Fig. 1(a and
123
b) one can also deduce that the minimum-energy migration path for these interstitial impurities through the fcc lattice of copper goes from an octahedral position via a neighbouring tetrahedral position to a next octahedral position. The calculated potential profiles for H and O impurities along such a path are shown in Fig. 1(c and d). The barrier heights are in qualitative agreement with the experimental activation energies of impurity diffusion for hydrogen and deuterium (D), DQ ðH in CuÞ ¼ 0:40 eV and DQ ðD in CuÞ ¼ 0:38 eV [41], and for oxygen in copper, DQ ðO in CuÞ ¼ 0:69 eV [42]. Overall, the potential profile is much steeper for oxygen than for hydrogen, which correlates well with the strength of chemical interaction of these elements with copper. 3.2. Vacancy–impurity pairs Our calculations show that, if a monovacancy is present in the crystal lattice of copper, a single H or O impurity will be absorbed by the defect from an interstitial position in the bulk. Figs. 2 and 3 show the calculated potential energy maps for H and O impurities inside a monovacancy. The origin of potential energy is chosen to correspond to the preferred octahedral interstitial position by each impurity in bulk Cu, as in the maps shown in Fig. 1(a and b). The figures show that each of the impurities inside a Cu vacancy tends to occupy an off-center position, displaced from the vacancy center along one of the h1 0 0i family directions toward an adjacent octa hedral interstitial position 0; 0; 12 . There are also eight metastable minima, located some distance away from the vacancy center in the h1 1 1i directions, which can be related to the tetrahedral interstitial positions 14 ; 14 ; 14 adjacent to the vacant site. The binding energy EB1V ðXÞ of an impurity X to a vacancy (the energy needed in order to move the impurity from the lowest energy minimum inside the vacancy to a regular octahedral interstitial position in the bulk) is calculated to be EB1V ðHÞ ¼ 0:24 eV and EB1V ðHÞ ¼ 0:95 eV. Quite similar to the case of a perfect crystal lattice, the potential energy profile inside a vacancy is found to be much steeper (and deeper) for an oxygen than for a hydrogen impurity. The oxygenvacancy binding energy is calculated here to be almost as large as the vacancy formation energy, suggesting that such a defect complex must be very stable. 3.3. Hydrogen binding to atomic defects in copper Experimental positron annihilation [43] and ion channelling [44] data, as well as theoretical data calculated using effective medium theory [45], are available on the hydrogen-vacancy binding energy in copper, consistently suggesting a somewhat larger value of about 0.4 eV. The origin of the difference with the present well-converged result (0.24 eV) is not very clear and calls for attention. Here we address the question briefly, since it lies beyond the scope of the present work, and leave a more detailed consideration for future investigations. First of all, we note that the vacancy–hydrogen binding energy value obtained in this work practically coincides with the experimental value 0.22 eV that is usually assigned to hydrogen binding to a self-interstitial in copper metal [43]. Our calculations for a hydrogen impurity at a self-interstitial defect in copper (whose optimised geometry is obtained to be a [1 0 0] dumbbell) reveal two sets of shallow potential energy minima, 0.04 and 0.07 eV, in which the H is located near octahedral interstitial positions at distances, respectively, 0:502a0 and 0:874a0 away from the lattice site occupied by the dumbbell. The latter configuration, corresponding to the lower energy minimum, is depicted in Fig. 4(a). Fig. 4(b) shows the lowest-energy configuration obtained in this work for a hydrogen atom at a di-vacancy in copper. The
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Fig. 1. Potential energy map for hydrogen (a) and oxygen (b) impurities in the (1 1 0) cross-section of Cu crystal lattice. The fcc lattice positions of Cu atoms are shown by filled circles. The crosses and triangles indicate, respectively, the octahedral and tetrahedral interstitial positions. Migration barriers along the minimum energy path for interstitial hydrogen (c) and oxygen (d) impurities in copper. The initial ð0; 0; 12Þ and the final ð12 ; 12 ; 12Þ coordinates in the path correspond to an octahedral interstitial position of the impurity; the midpoint corresponds to a tetrahedral position, e.g., ð14 ; 14 ; 14Þ. The calculated points in (c and d) are indicated by open circles; the lines are guide to the eye.
Fig. 2. Potential energy map for a single hydrogen impurity inside a vacancy in copper. The vertical dot-dashed line separates the (1 0 0) and (1 1 0) cross-sections cutting through the vacant lattice site (indicated by an open circle in the origin). The notations for the lattice sites and interstitial positions are the same as in Fig. 1. The blue coloured regions correspond to the potential energy minima for a hydrogen impurity.
Fig. 3. Potential energy map for a single oxygen impurity inside a vacancy in copper. The vertical dot-dashed line separates the (1 0 0) and (1 1 0) cross-sections cutting through the vacant lattice site (indicated by the open circle in the origin). The notations for the lattice sites and interstitial positions are the same as in Fig. 1. The deep blue coloured regions correspond to the potential energy minima for an oxygen impurity.
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Fig. 4. Geometry of the most stable configurations for a H impurity bound to a self-interstitial dumbbell (a) and to a di-vacancy (b) in copper. The atoms are shown as orange (Cu) and blue (H) balls; the crosses indicate empty lattice sites. For clarity, the two Cu atoms in the dumbbell and the two vacancies in the di-vacancy are connected with full lines. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
corresponding binding energy is calculated to be 0.38 eV, which is very close to the experimental value of about 0.4 eV usually ascribed to a hydrogen complex with a monovacancy in copper. Thus, the present calculations yield a spectrum of binding energies of hydrogen to atomic defects in copper. Some of the computed binding energy values are quite similar to those observed experimentally, but the present interpretation is different from the previous one. Furthermore, in the previous interpretation it has been implicitly assumed that one vacancy can bind only one hydrogen atom. As will be discussed below, our calculations show that this assumption does not hold. 3.3.1. Hydrogen absorption capacity of a monovacancy To investigate the absorption capacity for hydrogen of a monovacancy in copper, we performed supercell calculations with several hydrogen atoms occupying a single vacancy. To make up starting configurations for these calculations, we took into account the two sorts of local minima for a single H atom inside a vacancy, which are situated near the octahedral and tetrahedral interstitial positions (see Fig. 2). Other starting configurations were also tried (for example, H atom or H2 molecule at the vacancy center position), but found to be unstable. All the considered structures were relaxed keeping fixed the volume and the shape of each supercell. From the energy of an N-site supercell containing n hydrogen atoms inside one vacancy, EðCuN1 Hvac n Þ, the absorption energy Ea for the nth hydrogen (entering the vacancy from a bulk octahedral interstitial position) can be obtained as the energy of the following oct quasichemical reaction, Hvac ! Hvac n1 þ H n þ Ea , as: vac oct Ea ¼ EðCuN1 Hvac n Þ Emin ðCuN1 Hn1 Þ lH :
ð3Þ
Emin ðCuN1 Hvac n1 Þ
Here is the energy of a supercell representing the most stable configuration of n 1 hydrogen atoms inside the vacancy, and loct H is the chemical potential of hydrogen in the interstitial solid solution in copper. The chemical potential can easily be evaluated from the calculated supercell energies. For example, the following two definitions (respectively, without and with a vacancy present far from the position of the H atom), oct loct H ¼ EðCuN H Þ EðCuN Þ
ð4Þ
oct loct H ¼ EðCuN1 H Þ EðCuN1 Þ;
ð5Þ
and
have been checked to yield practically identical results differing by 0.02 eV. The absorption energies calculated according to Eq. (3) are plotted in Fig. 5. It should be noted that the absorption energy for the most stable configurations (indicated in Fig. 5 using filled
Fig. 5. Absorption energy for Hn impurity clusters inside a vacancy in copper, calculated according to Eqs. (3)–(5). Filled symbols connected by a solid line correspond to the most stable configurations for the given cluster size n. In the first six of them (depicted in Fig. 6) the H atoms are located near the octahedral interstitial positions. Green filled circles show configurations in which the H atoms are located near the tetrahedral interstitial positions surrounding the vacant site. Black open circles indicate the energies of various metastable configurations in which the H atoms may be located near the octahedral positions or at both (octahedral and tetrahedral) positions. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
symbols connected with a solid line) is related to the binding energy as Ea ¼ Eb . The most stable Hn configurations inside a copper vacancy are shown in Fig. 6. Configurations with 1 6 n 6 6, exhibiting the lowest absorption energies (highest binding energies), correspond to H atoms gradually filling up the near-octahedral interstitial positions around the vacant lattice site. It is noteworthy that the absorption energy for these configurations is calculated to be Ea 0:24 eV, almost independently of the number ðn 1Þ of H atoms contained in the vacancy prior to the absorption event. 3.3.2. Hydrogen binding to a vacancy–O complex Quite different absorption capacity for hydrogen is obtained for a monovacancy that already contains an absorbed O atom. Our calculations show that absorption of the first H atom by such a vacancy is strongly exothermic, Ea ¼ 1:23 eV, and results in the formation of a hydroxyl (OH) group inside the vacancy. Six stable configurations of an OH group inside a vacancy correspond to the H atom located near the vacancy center and the O atom displaced from the center about halfway towards a neighbouring octahedral
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calculations show that such a reaction is endothermic, Ea ¼ þ0:48 eV, so that hydrogen absorption capacity of a single vacancy contaminated by oxygen is limited to one H atom (to form a hydroxyl group). Fig. 7(b) shows the most stable OH2 configuration inside a vacancy in copper. Although the total binding energy of such a cluster (relative to all the defects in the bulk solid solution) is quite large, Eb ¼ 1:70 eV, (which is qualitatively similar to the case of an oxygen-contaminated vacancy in tungsten [46]), the cluster inside a vacancy in copper is unstable with respect to a dissociation reaction:
ðOH2 Þvac ! ðOHÞvac þ Hoct : 3.4. Trapping of hydrogen Two major types of traps have been identified for hydrogen above, one for binding to monovacancies (0.24 eV) and the other to vacancy–O clusters (1.23 eV). To compute the amount of H that will actually be trapped, the McNabb and Foster equation will be used. Its original form is [47,48].
dn ¼ kC L ð1 nÞ pn: dt
ð6Þ
Here n is the fraction of the trap that is filled with hydrogen, and C L the hydrogen concentration available for trap formation. Parameter k ¼ m0 exp Emigr =kB T describes the rate at which hydrogen is transferred to the trap and p ¼ m0 exp Emigr þ Eb =kB T the rate at which hydrogen is removed from the trap. Eb is the binding energy of the trap. C L is assumed to be the solubility of hydrogen. Its value can be found from [49]:
C L ¼ C 0 exp ðQ sol =kB T Þ;
ð7Þ
4
m0
where C 0 ¼ 1:152 10 at: ppm and Q sol ¼ 54; 850 J=mol. The and Emigr are related to the diffusion constant for hydrogen,
DH ¼ D0 exp Emigr =kB T
Fig. 6. Geometry of the most stable Hn configurations inside a vacancy in copper. Copper (orange) and hydrogen (blue) atoms a shown as balls, the cross in the middle of each structure indicates the vacancy center. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
with
interstitial position. The geometry of such a vacancy–OH complex in copper is shown in Fig. 7(a). An interesting question is whether a water molecule can be formed by absorbing another H atom from the bulk and adjoining it to the OH group already occupying a vacancy in copper. Our
ð8Þ
where r is the nearest-neighbour distance between octahedral interstitial positions in Cu. Values for D0 ¼ 1:13 106 m2 =s and Emigr ¼ 38; 880 J=mol are given in Ref. [41]. Eq. (6) is valid for a single trap, whereas we would like to analyse the situation with two competing traps. For this purpose Eq. (6) has to be generalised. Instead of the fraction of the trap n that is filled, the amount of hydrogen in the trap N is considered. This is achieved by multiplying Eq. (6) by the maximum amount N 0 in the trap
dN1 ¼ kC L ðN 01 N1 Þ p1 N1 dt dN2 ¼ kC L ðN 02 N1 N2 Þ p2 N 2 : dt
Fig. 7. Geometry of the most stable OH (a) and OH2 (b) atomic configurations inside a vacancy in copper. The atoms are shown as orange (Cu), red (O), and blue (H) balls; the cross in the middle of each structure denotes the vacancy center. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
m0 ¼ D0 =r2 ;
ð9Þ ð10Þ
N 1 ¼ N 01 n and N 2 ¼ N 02 n are the concentration of hydrogen in the two traps. N01 and N02 are the maximum amounts in the traps. Trap 1 is assumed to be the deep trap and trap 2 the more shallow one. In addition to the binding energy, there is another important difference between the two traps. The deep trap is based on the presence of oxygen, which means that a limited amount of hydrogen can be bound to this trap. Since one oxygen atom can bind one hydrogen atom, N 01 is taken as oxygen content in the system. However, for the second trap there is no practical limitation for how much hydrogen that be bound there, because vacancies are abundant point defects in copper. N 02 is chosen as the total amount of hydrogen. Since the hydrogen content must be conserved in the system the amount in trap 1 must be subtracted in the brackets in Eq. (10). These simplifying assumptions allow us to relate model parameters N 01 and N 02 to the concentrations of chemical species (oxygen and hydrogen, respectively) in the material.
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Nstat ¼ Nmax = 1 þ expðEb;i =kB TÞ=C L : i i
ð11Þ N max i
The index i = 1 and 2 represents the two traps. is the maximum hydrogen content in trap i. N max is taken as N 01 and N max 1 2 approximately as N 02 N 01 since trap 1 is fully or almost fully filled. If the exponential with the binding energy in Eq. (11) is much smaller than the solubility, the trap content is almost the maximum one. This is the case for the deep trap. On the other hand, if the exponential with the binding energy is much larger than the solubility, the trap is only filled to a small extent. This is characteristic of the shallow trap. If the binding energy is large, the temperature dependence is larger than that of the solubility and the trap content will decrease somewhat with temperature. In the reverse situation with a small binding energy, the trap content will increase with temperature. From the analysis above it is evident that a deep trap will be almost fully filled with hydrogen for temperatures up to the melting point. This has important technical implications. The hydrogen bound to the deep trap will be very difficult to remove. This has also been found both industrially and experimentally. Industrially vacuum treatment of molten copper is used to reduce hydrogen content down to 20 at. ppm [50]. If it was not for the presence of deep traps, the hydrogen could be removed by a simple heat treatment at high temperatures. Scientifically, degassing in ultrahigh vacuum has been applied many times to study gas evolution from copper. For example, Pope and Olson [51] first baked copper specimens in vacuum at 250 °C to remove all loosely bound hydrogen. Then the specimens were degassed in ultrahigh vacuum at 735–945 °C to measure the outflow of hydrogen. Evidently high
Trap content, at. ppm
101
CuOFP
100
873 K
10-1 Trap 1.23 eV 10
Trap 0.24 eV
-2
10-10
time, s
10-5
Fig. 8. Content of hydrogen in two traps with binding energies of 1.23 and 0.24 eV versus time. T = 873 K.
Trap content, at. ppm
23
CuOFP 1.23 eV
20 18
15
13 200
400
600
800
1000
1200
Temperature, C Fig. 9. Content of hydrogen in deep trap with a binding energy of 1.23 eV versus temperature.
CuOFP
Trap content, at. ppm
To illustrate the use of Eqs. (9) and (10), they will be applied to copper material intended for storage of nuclear waste [16]. It is pure copper with about 100 at. ppm phosphorus (Cu-OFP). Its specified maximum hydrogen and oxygen contents are 38 and 20 at. ppm, respectively. In Fig. 8 it is shown how the hydrogen content in the two traps vary with time. The starting values are arbitrarily set to 5 at. ppm. The content in the traps rapidly reaches a stationary condition. The temperature dependence of the stationary contents is illustrated in Figs. 9 and 10. The deep trap in Fig. 9 is fully filled up to temperatures close to the melting point for copper, 1084 °C. The content in the trap decreases somewhat with increasing temperature. The shallow trap is only filled to a small extent and this content is increasing with temperature. Expression for the stationary contents can be derived from Eqs. (9) and (10) by assuming that the time derivatives vanish.
0.24 eV 10−3
10−4
200
400
600
800
1000
1200
Temperature, C Fig. 10. Content of hydrogen in shallow trap with a binding energy of 0.24 eV versus temperature.
temperatures and low pressures are needed to remove the firmly bound hydrogen. 4. Conclusions Defect clusters comprised of one vacancy and several hydrogen and oxygen impurities in copper have been investigated using first-principles calculations and a supercell approach. Single H and O impurities are found to bind to monovacancies with the respective energies of about 0.24 and 0.95 eV. The potential energy profiles for the single impurities inside a vacancy are found to exhibit several minima; the deepest minima are located near the octahedral interstitial positions surrounding the vacant lattice site. The H atoms in the most stable multi-atom configurations inside a vacancy are also found to be located near the positions of potential energy minima calculated for a single hydrogen impurity. Up to the saturation limit of six H atoms per vacancy, the absorption energy is calculated to be almost independent of the number of absorbed atoms. In spite of the large absorption capacity for hydrogen, the formation of H2 molecule(s) inside a vacancy in copper is found to be energetically unfavourable. The presence of an oxygen impurity inside a vacancy is found to limit the absorption capacity of the vacancy to just one hydrogen atom which chemically binds with the oxygen to form a hydroxyl group. Absorption of another hydrogen atom by the
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vacancy–hydroxyl complex to form a water molecule is calculated to be an endothermic reaction. Thus we have found that a single oxygen-contaminated vacancy in copper does not have enough empty space for the formation of a water molecule by the reaction with dissolved hydrogen. The binding of just one hydrogen atom to an oxygen atom explains why a certain amount of hydrogen is firmly bound in the copper, for which high temperatures and low pressures are needed to outflow this hydrogen. Other hydrogen that is not placed in such deep traps can be removed in heat treatments. Other local oxide configurations (such as, for example, oxygencontaining multivacancy clusters or oxygen-segregated grain boundaries) should be investigated in connection with the phenomenon of ’’hydrogen sickness’’ of copper, which is believed to consist in the formation of water steam inside oxygencontaminated copper subject to hydrogen charging [52,53]. Another extension of this work could be a study of the interactions of dissolved hydrogen with dislocation loops, stacking faults, and vacancy clusters [54] in copper, in order to check whether local hydride configurations [55] can spontaneously form there. It has also been proposed that grain boundaries can act as shallow traps [56]. Interactions of hydrogen with impurities and minor alloying elements in oxygen-free copper are also of interest in connection with the observed effects of hydrogen on the mechanical properties of hydrogen-charged oxygen-free copper [57]. Acknowledgements We thank C. Lilja and M. Ganchenkova for comments and suggestions. Financial support by Svensk Kärnbränslehantering AB, the Swedish Nuclear Fuel and Waste Management Company, is gratefully acknowledged. Computational resources for this study, at the National Supercomputer Center, Linköping, Sweden, were provided by the Swedish National Infrastructure for Computing (SNIC). References [1] A. Pisarev, Sov. Atom. Energy 62 (1987) 131–142. [2] S. Linderoth, J. Phys.: Condens. Matter 1 (1989) SA55–SA66. [3] S. Myers, M. Baskes, H. Birnbaum, J. Corbett, G. DeLeo, S. Estreicher, E. Haller, P. Jena, N. Johnson, R. Kirchheim, S. Pearton, M. Stavola, Rev. Mod. Phys. 64 (1992) 559–617. [4] J. Condon, T. Schober, J. Nucl. Mater. 207 (1993) 1–24. [5] R. Kirchheim, Solid State Phys. 59 (2004) 203–291. [6] A. Atkinson, S. Barnett, R. Gorte, J. Irvine, A. McEvoy, M. Mogensen, S. Singhal, J. Vohs, Nat. Mater. 3 (2004) 17–27. [7] N. Ockwig, T. Nenoff, Chem. Rev. 107 (2007) 4078–4110. [8] S. Matar, Prog. Solid State Chem. 38 (2010) 1–37. [9] H. Birnbaum, P. Sofronis, Mater. Sci. Eng. A 176 (1994) 191–202. [10] G. Lu, Q. Zhang, N. Kioussis, E. Kaxiras, Phys. Rev. Lett. 87 (2001) 095501. [11] H. Birnbaum, MRS Bull. 28 (2003) 479–485. [12] Y. Liang, P. Sofronis, N. Aravas, Acta Mater. 51 (2003) 2717–2730.
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Monovacancy in copper: Trapping efficiency for hydrogen and oxygen impurities P.A. Korzhavyi 1, R. Sandström ⇑ Department of Materials Science and Engineering, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden
a r t i c l e
i n f o
Article history: Received 18 September 2013 Received in revised form 22 November 2013 Accepted 27 November 2013
Keywords: Hydrogen trapping Point defects Ab initio calculations
a b s t r a c t The structure and binding energy of vacancy–impurity complexes in copper are studied using firstprinciples calculations based on density functional theory. A single vacancy is found to be able to trap up to six hydrogen atoms which tend to be situated inside the vacancy at off-center positions (related to the octahedral interstitial positions of the ideal fcc lattice). The binding energy of an H atom dissolved in the Cu lattice (octahedral interstitial position) to a vacancy is calculated to be about 0.24 eV, practically independent of the number of H atoms already trapped by the vacancy, up to the saturation with 6 hydrogens. For an oxygen impurity in Cu, a monovacancy is shown to be a deep trap (with a binding energy of 0.95 eV). The position of a trapped O atom inside a vacancy is off-center, almost a half-way from the nearest octahedral interstitial to the vacancy center. Such a vacancy–O cluster is shown to be a deep trap for dissolved hydrogen (the calculated binding energy is 1.23 eV). The trapping results in the formation of an OH-group, where the H atom is situated near the vacancy center, and the O atom is displaced from the center along a h1 0 0i direction towards a nearby octahedral interstitial position. Further hydrogenation of the monovacancy–OH cluster is calculated to be energetically unfavourable. McNabb–Forster’s equations are generalised to describe the competition between a deep hydrogen trap and a shallow one. It is demonstrated that the deep trap is almost fully filled, which explains why some of hydrogen is strongly bound and cannot be removed without vacuum treatment at elevated temperatures. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Interactions of dissolved hydrogen with lattice defects in metals have been studied quite thoroughly in the past [1–5], presently they are again of high interest in connection with the applications of metals as catalysts, parts of batteries and fuel cells, materials for hydrogen production and storage [6–8], as well as in connection with the recent advances in the understanding of hydrogen-related effects on the mechanical properties of metallic materials [9–14]. The solubility, mobility, and trapping of hydrogen in copper metal, as well as the stable and metastable atomic configurations in the ternary Cu–O–H system [15], are of particular interest also in connection with the Swedish plan of spent nuclear fuel disposal in a deep geological repository using copper canisters [16–18]. First-principles calculations and atomistic simulations provide important information about the atomic-scale structures and atomistic mechanisms involved in hydrogen-related effects in metallic materials [19–22]. In the present work, the first-principles approach is used in order to study possible interactions of ⇑ Corresponding author. Tel.: +46 8 790 91 93; fax: +46 8 20 76 81. 1
E-mail addresses: [email protected] (P.A. Korzhavyi), [email protected] (R. Sandström). Principal corresponding author.
0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.11.065
hydrogen and oxygen impurities with a single vacancy (monovacancy) in copper. 2. Calculation The electronic structure and total energy calculations were based on density functional theory [23,24] in the generalised gradient approximation (GGA) using the formulation by Perdew et al. [25]. Ultrasoft pseudopotentials due to Vanderbilt [26] and a plane-wave basis set with the energy cutoff 50 Ry for the wave function and 340 Ry for the density were used in the calculations that employed PWscf code of the Quantum ESPRESSO distribution [27]. Copper metal was modelled using a 108-site supercell obtained by a 3 3 3 multiplication of a face-centered cubic (fcc) unit cell containing 4 atoms. The cases considered in our calculations included a structure with the perfect site occupancy, a supercell with one vacancy, supercell containing H or O impurities, and supercells containing one vacancy and several impurities in various configurations. The calculations were performed at a fixed lattice parameter of the supercell corresponding to three times the calculated equilibrium lattice parameter a0 = 3.63 Å of pure fcc Cu. The shape of the supercell was also kept fixed, while the internal degrees of freedom (atomic positions) were allowed to
P.A. Korzhavyi, R. Sandström / Computational Materials Science 84 (2014) 122–128
relax. The local symmetry of the defect configurations was intentionally broken before relaxing the atomic positions. The Brillouin zone integration was carried out using a 2 2 2 Monkhorst– Pack mesh of k-points. The so calculated total energy, as a function of the atomic coordinates, represents the potential energy surface for atomic motion. A next step should be to solve the quantum-mechanical equations of atomic motion, which is essential for light impurities like hydrogen even at zero Kelvin. In this study, however, we limit ourselves to just computing the potential energy surface and comparing the depths of its deepest energy minima, in order to locate the trapping sites for impurities and to estimate the corresponding binding energies. One should thus keep in mind that the calculated binding energies do not contain the zero-point energy contribution, which mostly cancels out. The omitted zero-point energy contribution does not exceed 10% of the computed value, which is negligible in all cases except one where a hydrogen impurity enters an oxygen-contaminated vacancy to form a hydroxyl group. In the latter case the binding energy should be lowered by about 0.1 eV. 3. Results and discussion 3.1. Single point defects The formation energy of a monovacancy in copper metal can be derived using the calculated total energies, EðCuN1 Þ of a supercell containing N 1 atoms with one vacancy and EðCuN Þ of a defectfree supercell containing N atoms, as follows:
EF1V ¼ EðCuN1 Þ
N1 EðCuN Þ: N
ð1Þ
The calculated value EF1V ¼ 1:07 eV is in agreement with previously calculated values which scatter in the range from 1.0 to 1.4 eV [28–33] (and closer to lower bound of that range, together with other theoretical values obtained using relaxed atomic configurations and GGA [31–33]) as well with experimental values, 0.92–1.31 eV, of the vacancy formation energy in copper [34–36]. Estimates of the solution energy for X = {H,O} impurities in copper may be obtained from the energies EðCuN X1 Þ of supercells each containing one of these impurities in the most stable (calculated to be octahedral) interstitial position, the energy of a defect-free Cu supercell EðCuN Þ, the energies EðX1 Þ of free H and O atoms, and experimental atomization energies DEðX2 Þ ¼ 2EðX1 Þ EðX2 Þ of H2 and O2 molecules:
1 DEsol ¼ EðCuN X1 EðCuN Þ EðX1 Þ þ DEðX2 Þ: 2
ð2Þ
The necessity to use experimental atomization energies comes from the fact that GGA is known [37] to describe the stability of H2 and O2 molecules rather inaccurately (the errors are too large even for crude estimates) while treating the bulk copper metal relatively accurately. For the sake of simplicity, let us exclude the energy of ionic zeropoint motion from all the terms in Eq. (2). Then the last term in Eq. (2) becomes equal to 2.374 eV for H2 and 2.613 eV for O2 [37], and one arrives at the following estimates: DEsol ðH2 in CuÞ ¼ þ0:41 eV and DEsol ðO2 in CuÞ ¼ 0:34 eV. These values are in good qualitative agreement with the solution energies that can be extracted from the experimental solubility of hydrogen (+0.44 eV, Ref. [38]) and oxygen (0.53 eV, Refs. [39,40]) in the copper metal. Let us now analyse the site preference and mobility of a single H or O impurity in the otherwise defect-free Cu. As follows from the calculated potential energy maps presented in Fig. 1(a and b), both H and O impurities prefer octahedral interstitial positions in the fcc lattice. However, the potential exhibits additional local minima at the tetrahedral interstitial positions. From Fig. 1(a and
123
b) one can also deduce that the minimum-energy migration path for these interstitial impurities through the fcc lattice of copper goes from an octahedral position via a neighbouring tetrahedral position to a next octahedral position. The calculated potential profiles for H and O impurities along such a path are shown in Fig. 1(c and d). The barrier heights are in qualitative agreement with the experimental activation energies of impurity diffusion for hydrogen and deuterium (D), DQ ðH in CuÞ ¼ 0:40 eV and DQ ðD in CuÞ ¼ 0:38 eV [41], and for oxygen in copper, DQ ðO in CuÞ ¼ 0:69 eV [42]. Overall, the potential profile is much steeper for oxygen than for hydrogen, which correlates well with the strength of chemical interaction of these elements with copper. 3.2. Vacancy–impurity pairs Our calculations show that, if a monovacancy is present in the crystal lattice of copper, a single H or O impurity will be absorbed by the defect from an interstitial position in the bulk. Figs. 2 and 3 show the calculated potential energy maps for H and O impurities inside a monovacancy. The origin of potential energy is chosen to correspond to the preferred octahedral interstitial position by each impurity in bulk Cu, as in the maps shown in Fig. 1(a and b). The figures show that each of the impurities inside a Cu vacancy tends to occupy an off-center position, displaced from the vacancy center along one of the h1 0 0i family directions toward an adjacent octa hedral interstitial position 0; 0; 12 . There are also eight metastable minima, located some distance away from the vacancy center in the h1 1 1i directions, which can be related to the tetrahedral interstitial positions 14 ; 14 ; 14 adjacent to the vacant site. The binding energy EB1V ðXÞ of an impurity X to a vacancy (the energy needed in order to move the impurity from the lowest energy minimum inside the vacancy to a regular octahedral interstitial position in the bulk) is calculated to be EB1V ðHÞ ¼ 0:24 eV and EB1V ðHÞ ¼ 0:95 eV. Quite similar to the case of a perfect crystal lattice, the potential energy profile inside a vacancy is found to be much steeper (and deeper) for an oxygen than for a hydrogen impurity. The oxygenvacancy binding energy is calculated here to be almost as large as the vacancy formation energy, suggesting that such a defect complex must be very stable. 3.3. Hydrogen binding to atomic defects in copper Experimental positron annihilation [43] and ion channelling [44] data, as well as theoretical data calculated using effective medium theory [45], are available on the hydrogen-vacancy binding energy in copper, consistently suggesting a somewhat larger value of about 0.4 eV. The origin of the difference with the present well-converged result (0.24 eV) is not very clear and calls for attention. Here we address the question briefly, since it lies beyond the scope of the present work, and leave a more detailed consideration for future investigations. First of all, we note that the vacancy–hydrogen binding energy value obtained in this work practically coincides with the experimental value 0.22 eV that is usually assigned to hydrogen binding to a self-interstitial in copper metal [43]. Our calculations for a hydrogen impurity at a self-interstitial defect in copper (whose optimised geometry is obtained to be a [1 0 0] dumbbell) reveal two sets of shallow potential energy minima, 0.04 and 0.07 eV, in which the H is located near octahedral interstitial positions at distances, respectively, 0:502a0 and 0:874a0 away from the lattice site occupied by the dumbbell. The latter configuration, corresponding to the lower energy minimum, is depicted in Fig. 4(a). Fig. 4(b) shows the lowest-energy configuration obtained in this work for a hydrogen atom at a di-vacancy in copper. The
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Fig. 1. Potential energy map for hydrogen (a) and oxygen (b) impurities in the (1 1 0) cross-section of Cu crystal lattice. The fcc lattice positions of Cu atoms are shown by filled circles. The crosses and triangles indicate, respectively, the octahedral and tetrahedral interstitial positions. Migration barriers along the minimum energy path for interstitial hydrogen (c) and oxygen (d) impurities in copper. The initial ð0; 0; 12Þ and the final ð12 ; 12 ; 12Þ coordinates in the path correspond to an octahedral interstitial position of the impurity; the midpoint corresponds to a tetrahedral position, e.g., ð14 ; 14 ; 14Þ. The calculated points in (c and d) are indicated by open circles; the lines are guide to the eye.
Fig. 2. Potential energy map for a single hydrogen impurity inside a vacancy in copper. The vertical dot-dashed line separates the (1 0 0) and (1 1 0) cross-sections cutting through the vacant lattice site (indicated by an open circle in the origin). The notations for the lattice sites and interstitial positions are the same as in Fig. 1. The blue coloured regions correspond to the potential energy minima for a hydrogen impurity.
Fig. 3. Potential energy map for a single oxygen impurity inside a vacancy in copper. The vertical dot-dashed line separates the (1 0 0) and (1 1 0) cross-sections cutting through the vacant lattice site (indicated by the open circle in the origin). The notations for the lattice sites and interstitial positions are the same as in Fig. 1. The deep blue coloured regions correspond to the potential energy minima for an oxygen impurity.
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Fig. 4. Geometry of the most stable configurations for a H impurity bound to a self-interstitial dumbbell (a) and to a di-vacancy (b) in copper. The atoms are shown as orange (Cu) and blue (H) balls; the crosses indicate empty lattice sites. For clarity, the two Cu atoms in the dumbbell and the two vacancies in the di-vacancy are connected with full lines. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
corresponding binding energy is calculated to be 0.38 eV, which is very close to the experimental value of about 0.4 eV usually ascribed to a hydrogen complex with a monovacancy in copper. Thus, the present calculations yield a spectrum of binding energies of hydrogen to atomic defects in copper. Some of the computed binding energy values are quite similar to those observed experimentally, but the present interpretation is different from the previous one. Furthermore, in the previous interpretation it has been implicitly assumed that one vacancy can bind only one hydrogen atom. As will be discussed below, our calculations show that this assumption does not hold. 3.3.1. Hydrogen absorption capacity of a monovacancy To investigate the absorption capacity for hydrogen of a monovacancy in copper, we performed supercell calculations with several hydrogen atoms occupying a single vacancy. To make up starting configurations for these calculations, we took into account the two sorts of local minima for a single H atom inside a vacancy, which are situated near the octahedral and tetrahedral interstitial positions (see Fig. 2). Other starting configurations were also tried (for example, H atom or H2 molecule at the vacancy center position), but found to be unstable. All the considered structures were relaxed keeping fixed the volume and the shape of each supercell. From the energy of an N-site supercell containing n hydrogen atoms inside one vacancy, EðCuN1 Hvac n Þ, the absorption energy Ea for the nth hydrogen (entering the vacancy from a bulk octahedral interstitial position) can be obtained as the energy of the following oct quasichemical reaction, Hvac ! Hvac n1 þ H n þ Ea , as: vac oct Ea ¼ EðCuN1 Hvac n Þ Emin ðCuN1 Hn1 Þ lH :
ð3Þ
Emin ðCuN1 Hvac n1 Þ
Here is the energy of a supercell representing the most stable configuration of n 1 hydrogen atoms inside the vacancy, and loct H is the chemical potential of hydrogen in the interstitial solid solution in copper. The chemical potential can easily be evaluated from the calculated supercell energies. For example, the following two definitions (respectively, without and with a vacancy present far from the position of the H atom), oct loct H ¼ EðCuN H Þ EðCuN Þ
ð4Þ
oct loct H ¼ EðCuN1 H Þ EðCuN1 Þ;
ð5Þ
and
have been checked to yield practically identical results differing by 0.02 eV. The absorption energies calculated according to Eq. (3) are plotted in Fig. 5. It should be noted that the absorption energy for the most stable configurations (indicated in Fig. 5 using filled
Fig. 5. Absorption energy for Hn impurity clusters inside a vacancy in copper, calculated according to Eqs. (3)–(5). Filled symbols connected by a solid line correspond to the most stable configurations for the given cluster size n. In the first six of them (depicted in Fig. 6) the H atoms are located near the octahedral interstitial positions. Green filled circles show configurations in which the H atoms are located near the tetrahedral interstitial positions surrounding the vacant site. Black open circles indicate the energies of various metastable configurations in which the H atoms may be located near the octahedral positions or at both (octahedral and tetrahedral) positions. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
symbols connected with a solid line) is related to the binding energy as Ea ¼ Eb . The most stable Hn configurations inside a copper vacancy are shown in Fig. 6. Configurations with 1 6 n 6 6, exhibiting the lowest absorption energies (highest binding energies), correspond to H atoms gradually filling up the near-octahedral interstitial positions around the vacant lattice site. It is noteworthy that the absorption energy for these configurations is calculated to be Ea 0:24 eV, almost independently of the number ðn 1Þ of H atoms contained in the vacancy prior to the absorption event. 3.3.2. Hydrogen binding to a vacancy–O complex Quite different absorption capacity for hydrogen is obtained for a monovacancy that already contains an absorbed O atom. Our calculations show that absorption of the first H atom by such a vacancy is strongly exothermic, Ea ¼ 1:23 eV, and results in the formation of a hydroxyl (OH) group inside the vacancy. Six stable configurations of an OH group inside a vacancy correspond to the H atom located near the vacancy center and the O atom displaced from the center about halfway towards a neighbouring octahedral
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calculations show that such a reaction is endothermic, Ea ¼ þ0:48 eV, so that hydrogen absorption capacity of a single vacancy contaminated by oxygen is limited to one H atom (to form a hydroxyl group). Fig. 7(b) shows the most stable OH2 configuration inside a vacancy in copper. Although the total binding energy of such a cluster (relative to all the defects in the bulk solid solution) is quite large, Eb ¼ 1:70 eV, (which is qualitatively similar to the case of an oxygen-contaminated vacancy in tungsten [46]), the cluster inside a vacancy in copper is unstable with respect to a dissociation reaction:
ðOH2 Þvac ! ðOHÞvac þ Hoct : 3.4. Trapping of hydrogen Two major types of traps have been identified for hydrogen above, one for binding to monovacancies (0.24 eV) and the other to vacancy–O clusters (1.23 eV). To compute the amount of H that will actually be trapped, the McNabb and Foster equation will be used. Its original form is [47,48].
dn ¼ kC L ð1 nÞ pn: dt
ð6Þ
Here n is the fraction of the trap that is filled with hydrogen, and C L the hydrogen concentration available for trap formation. Parameter k ¼ m0 exp Emigr =kB T describes the rate at which hydrogen is transferred to the trap and p ¼ m0 exp Emigr þ Eb =kB T the rate at which hydrogen is removed from the trap. Eb is the binding energy of the trap. C L is assumed to be the solubility of hydrogen. Its value can be found from [49]:
C L ¼ C 0 exp ðQ sol =kB T Þ;
ð7Þ
4
m0
where C 0 ¼ 1:152 10 at: ppm and Q sol ¼ 54; 850 J=mol. The and Emigr are related to the diffusion constant for hydrogen,
DH ¼ D0 exp Emigr =kB T
Fig. 6. Geometry of the most stable Hn configurations inside a vacancy in copper. Copper (orange) and hydrogen (blue) atoms a shown as balls, the cross in the middle of each structure indicates the vacancy center. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
with
interstitial position. The geometry of such a vacancy–OH complex in copper is shown in Fig. 7(a). An interesting question is whether a water molecule can be formed by absorbing another H atom from the bulk and adjoining it to the OH group already occupying a vacancy in copper. Our
ð8Þ
where r is the nearest-neighbour distance between octahedral interstitial positions in Cu. Values for D0 ¼ 1:13 106 m2 =s and Emigr ¼ 38; 880 J=mol are given in Ref. [41]. Eq. (6) is valid for a single trap, whereas we would like to analyse the situation with two competing traps. For this purpose Eq. (6) has to be generalised. Instead of the fraction of the trap n that is filled, the amount of hydrogen in the trap N is considered. This is achieved by multiplying Eq. (6) by the maximum amount N 0 in the trap
dN1 ¼ kC L ðN 01 N1 Þ p1 N1 dt dN2 ¼ kC L ðN 02 N1 N2 Þ p2 N 2 : dt
Fig. 7. Geometry of the most stable OH (a) and OH2 (b) atomic configurations inside a vacancy in copper. The atoms are shown as orange (Cu), red (O), and blue (H) balls; the cross in the middle of each structure denotes the vacancy center. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
m0 ¼ D0 =r2 ;
ð9Þ ð10Þ
N 1 ¼ N 01 n and N 2 ¼ N 02 n are the concentration of hydrogen in the two traps. N01 and N02 are the maximum amounts in the traps. Trap 1 is assumed to be the deep trap and trap 2 the more shallow one. In addition to the binding energy, there is another important difference between the two traps. The deep trap is based on the presence of oxygen, which means that a limited amount of hydrogen can be bound to this trap. Since one oxygen atom can bind one hydrogen atom, N 01 is taken as oxygen content in the system. However, for the second trap there is no practical limitation for how much hydrogen that be bound there, because vacancies are abundant point defects in copper. N 02 is chosen as the total amount of hydrogen. Since the hydrogen content must be conserved in the system the amount in trap 1 must be subtracted in the brackets in Eq. (10). These simplifying assumptions allow us to relate model parameters N 01 and N 02 to the concentrations of chemical species (oxygen and hydrogen, respectively) in the material.
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Nstat ¼ Nmax = 1 þ expðEb;i =kB TÞ=C L : i i
ð11Þ N max i
The index i = 1 and 2 represents the two traps. is the maximum hydrogen content in trap i. N max is taken as N 01 and N max 1 2 approximately as N 02 N 01 since trap 1 is fully or almost fully filled. If the exponential with the binding energy in Eq. (11) is much smaller than the solubility, the trap content is almost the maximum one. This is the case for the deep trap. On the other hand, if the exponential with the binding energy is much larger than the solubility, the trap is only filled to a small extent. This is characteristic of the shallow trap. If the binding energy is large, the temperature dependence is larger than that of the solubility and the trap content will decrease somewhat with temperature. In the reverse situation with a small binding energy, the trap content will increase with temperature. From the analysis above it is evident that a deep trap will be almost fully filled with hydrogen for temperatures up to the melting point. This has important technical implications. The hydrogen bound to the deep trap will be very difficult to remove. This has also been found both industrially and experimentally. Industrially vacuum treatment of molten copper is used to reduce hydrogen content down to 20 at. ppm [50]. If it was not for the presence of deep traps, the hydrogen could be removed by a simple heat treatment at high temperatures. Scientifically, degassing in ultrahigh vacuum has been applied many times to study gas evolution from copper. For example, Pope and Olson [51] first baked copper specimens in vacuum at 250 °C to remove all loosely bound hydrogen. Then the specimens were degassed in ultrahigh vacuum at 735–945 °C to measure the outflow of hydrogen. Evidently high
Trap content, at. ppm
101
CuOFP
100
873 K
10-1 Trap 1.23 eV 10
Trap 0.24 eV
-2
10-10
time, s
10-5
Fig. 8. Content of hydrogen in two traps with binding energies of 1.23 and 0.24 eV versus time. T = 873 K.
Trap content, at. ppm
23
CuOFP 1.23 eV
20 18
15
13 200
400
600
800
1000
1200
Temperature, C Fig. 9. Content of hydrogen in deep trap with a binding energy of 1.23 eV versus temperature.
CuOFP
Trap content, at. ppm
To illustrate the use of Eqs. (9) and (10), they will be applied to copper material intended for storage of nuclear waste [16]. It is pure copper with about 100 at. ppm phosphorus (Cu-OFP). Its specified maximum hydrogen and oxygen contents are 38 and 20 at. ppm, respectively. In Fig. 8 it is shown how the hydrogen content in the two traps vary with time. The starting values are arbitrarily set to 5 at. ppm. The content in the traps rapidly reaches a stationary condition. The temperature dependence of the stationary contents is illustrated in Figs. 9 and 10. The deep trap in Fig. 9 is fully filled up to temperatures close to the melting point for copper, 1084 °C. The content in the trap decreases somewhat with increasing temperature. The shallow trap is only filled to a small extent and this content is increasing with temperature. Expression for the stationary contents can be derived from Eqs. (9) and (10) by assuming that the time derivatives vanish.
0.24 eV 10−3
10−4
200
400
600
800
1000
1200
Temperature, C Fig. 10. Content of hydrogen in shallow trap with a binding energy of 0.24 eV versus temperature.
temperatures and low pressures are needed to remove the firmly bound hydrogen. 4. Conclusions Defect clusters comprised of one vacancy and several hydrogen and oxygen impurities in copper have been investigated using first-principles calculations and a supercell approach. Single H and O impurities are found to bind to monovacancies with the respective energies of about 0.24 and 0.95 eV. The potential energy profiles for the single impurities inside a vacancy are found to exhibit several minima; the deepest minima are located near the octahedral interstitial positions surrounding the vacant lattice site. The H atoms in the most stable multi-atom configurations inside a vacancy are also found to be located near the positions of potential energy minima calculated for a single hydrogen impurity. Up to the saturation limit of six H atoms per vacancy, the absorption energy is calculated to be almost independent of the number of absorbed atoms. In spite of the large absorption capacity for hydrogen, the formation of H2 molecule(s) inside a vacancy in copper is found to be energetically unfavourable. The presence of an oxygen impurity inside a vacancy is found to limit the absorption capacity of the vacancy to just one hydrogen atom which chemically binds with the oxygen to form a hydroxyl group. Absorption of another hydrogen atom by the
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vacancy–hydroxyl complex to form a water molecule is calculated to be an endothermic reaction. Thus we have found that a single oxygen-contaminated vacancy in copper does not have enough empty space for the formation of a water molecule by the reaction with dissolved hydrogen. The binding of just one hydrogen atom to an oxygen atom explains why a certain amount of hydrogen is firmly bound in the copper, for which high temperatures and low pressures are needed to outflow this hydrogen. Other hydrogen that is not placed in such deep traps can be removed in heat treatments. Other local oxide configurations (such as, for example, oxygencontaining multivacancy clusters or oxygen-segregated grain boundaries) should be investigated in connection with the phenomenon of ’’hydrogen sickness’’ of copper, which is believed to consist in the formation of water steam inside oxygencontaminated copper subject to hydrogen charging [52,53]. Another extension of this work could be a study of the interactions of dissolved hydrogen with dislocation loops, stacking faults, and vacancy clusters [54] in copper, in order to check whether local hydride configurations [55] can spontaneously form there. It has also been proposed that grain boundaries can act as shallow traps [56]. Interactions of hydrogen with impurities and minor alloying elements in oxygen-free copper are also of interest in connection with the observed effects of hydrogen on the mechanical properties of hydrogen-charged oxygen-free copper [57]. Acknowledgements We thank C. Lilja and M. Ganchenkova for comments and suggestions. Financial support by Svensk Kärnbränslehantering AB, the Swedish Nuclear Fuel and Waste Management Company, is gratefully acknowledged. Computational resources for this study, at the National Supercomputer Center, Linköping, Sweden, were provided by the Swedish National Infrastructure for Computing (SNIC). References [1] A. Pisarev, Sov. Atom. Energy 62 (1987) 131–142. [2] S. Linderoth, J. Phys.: Condens. Matter 1 (1989) SA55–SA66. [3] S. Myers, M. Baskes, H. Birnbaum, J. Corbett, G. DeLeo, S. Estreicher, E. Haller, P. Jena, N. Johnson, R. Kirchheim, S. Pearton, M. Stavola, Rev. Mod. Phys. 64 (1992) 559–617. [4] J. Condon, T. Schober, J. Nucl. Mater. 207 (1993) 1–24. [5] R. Kirchheim, Solid State Phys. 59 (2004) 203–291. [6] A. Atkinson, S. Barnett, R. Gorte, J. Irvine, A. McEvoy, M. Mogensen, S. Singhal, J. Vohs, Nat. Mater. 3 (2004) 17–27. [7] N. Ockwig, T. Nenoff, Chem. Rev. 107 (2007) 4078–4110. [8] S. Matar, Prog. Solid State Chem. 38 (2010) 1–37. [9] H. Birnbaum, P. Sofronis, Mater. Sci. Eng. A 176 (1994) 191–202. [10] G. Lu, Q. Zhang, N. Kioussis, E. Kaxiras, Phys. Rev. Lett. 87 (2001) 095501. [11] H. Birnbaum, MRS Bull. 28 (2003) 479–485. [12] Y. Liang, P. Sofronis, N. Aravas, Acta Mater. 51 (2003) 2717–2730.
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