Acta Materialia 106 (2016) 313e321
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The electric charge and climb of edge dislocations in perovskite oxides: The case of high-pressure MgSiO3 bridgmanite P. Hirel a, *, P. Carrez a, E. Clouet b, P. Cordier a a b
^t. C6, Univ. Lille 1, 59655 Villeneuve d'Ascq, France Unit e Mat eriaux Et Transformations, Ba CEA, DEN, Service de Recherches de M etallurgie Physique, UPSay, F-91191 Gif-sur-Yvette, France
a r t i c l e i n f o
a b s t r a c t
Article history: Received 10 November 2015 Received in revised form 4 January 2016 Accepted 8 January 2016 Available online xxx
Dislocation climb is expected to play a major role in high-temperature creep of complex ionic oxides, however the fundamental mechanisms for climb are poorly understood in this class of materials. In this work we investigate the interaction of vacancies with an edge dislocation by means of atomic-scale simulations in high-pressure MgSiO3 bridgmanite, a mineral whose natural conditions of deformation are favorable to climb. The evaluation of the interaction with oxygen vacancies reveals that the dislocation favors a non-stoichiometric, oxygen-poor configuration, associated with a positive electric charge of maximum value þ 9.17 1011 C m1. This result is in qualitative agreement with experimental observations in related perovskite oxides. Subsequently, the interactions between dislocations and vacancies are dominated by electrostatics, with binding enthalpies of several eV in the vicinity of the dislocation core, superseding elastic effects. These results shed a new light on dislocation-vacancy interactions and dislocation climb in this class of materials. © 2016 Acta Materialia Inc. Published by Elsevier Ltd. This is an open access article under the CC BY-NCND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords: Computer simulation Perovskites Dislocations Vacancies Climb
1. Introduction Dislocation climb is a non-conservative process that occurs when a dislocation absorbs or emits vacancies. This mechanism allows for dislocations with edge components to move out of their glide plane. Favoring unpinning, climb plays an essential role in high-temperature creep. Although theoretical models for climb were formulated early in the development of the theory of dislocations [1e4], actual experimental observation of climb has been very scarce and remains very challenging even today. Edelin and Poirier were among the first to obtain a climb deformation rate by deforming single crystals of hexagonal close-packed magnesium and beryllium-copper alloys at high temperature, along the c axis to inhibit dislocation glide [5e7]. Recently climb has been demonstrated to be the prevalent deformation mechanism in icosahedral quasicrystals in response to the high lattice friction of highly corrugated glide plane in those structures [8,9]. Climb thus appears as an alternative to dislocation glide when this mechanism is inhibited. Since the seminal work of Eshelby, Newey, Pratt and Lidiard [10],
* Corresponding author. E-mail address:
[email protected] (P. Hirel).
it is admitted that dislocations in ionic materials should carry an electric charge, because of the difference in formation energy of anion and cation vacancies. Charged dislocations are then surrounded by a cloud of charge-compensating defects, called DebyeeHückel cloud. Direct evidence of charged dislocations were obtained in some materials with the rocksalt structure, e.g. LiF [11], NaCl [12], or doped KCl [13,14]. In perovskite-type materials, recent electron energy loss spectrum (EELS) measurements as well as atomic-scale simulations have revealed that dislocations tend to be oxygen-deficient, e.g. in SrTiO3 [15e19] or in BaTiO3 [20], however their electric charge was not determined. Yet one can expect electrically charged dislocations to interact strongly with charged point defects, therefore the determination of the charge of dislocations is of primary importance to the understanding of the processes related to climb. Because climb is a slow process that depends on the mobility of vacancies, molecular dynamics simulations are usually not suited to investigate it because of the short time-scales accessible to this method. However static calculations can be utilized to quantify the interactions between vacancies and dislocations, and bring understanding of the elementary processes of climb [21e23]. In this work we investigate dislocation-vacancy interactions, the elementary mechanism of dislocation climb, in bridgmanite, the main constituent of the Earth's lower mantle. The natural
http://dx.doi.org/10.1016/j.actamat.2016.01.019 1359-6454/© 2016 Acta Materialia Inc. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-ncnd/4.0/).
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conditions of deformation of this mineral are favorable to dislocation climb: grain size larger than 1 mm [24], high temperature (T ¼ 15002500 K) and extremely slow strain rates (_ε ¼ 1012 1016 s1). The study focuses on the [100](010) edge dislocation, which belongs to one of the most favorable slip systems in bridgmanite and whose glide properties were already investigated [25e28]. It is shown that at finite temperature this dislocation favors an oxygen-deficient configuration, and carries an intrinsic positive electric charge. As a result the vacancies, which are also electrically charged, interact strongly with the dislocation through the Coulomb interaction. Based on these results we propose a scenario for dislocation climb in bridgmanite.
2. Methods and models Bridgmanite is a high-pressure magnesium silicate, stable only in the pressure range 26e130 GPa, that crystallizes in a distorted perovskite structure with an orthorhombic lattice (Fig. 1a). The chemical composition of natural bridgmanite is (Mg,Al,Fe)(Si,Al)O3, and in this work we focus on the pure magnesium-rich compound, MgSiO3. We describe it within the Pbnm space group, the occupied Wyckoff positions being (4c) for Mg ions, (4b) for Si ions, and (4c) and (8d) for O ions. All calculations presented in this work are performed using a hydrostatic pressure 30 GPa corresponding to the uppermost lower mantle (about 670e700 km below the Earth's surface). Atomic systems are constructed with our home-made software Atomsk [29], and classical molecular statics simulations are performed with the LAMMPS package [30]. Interatomic interactions are described by a rigid-ion pair potential function where each ion has a fixed electric charge, and ions interact through the Coulomb force and a short-range Buckingham potential. The parameterization by Alfredsson and coworkers is used, which produces accurate lattice constants, elastic constants, and surface energies, in the pressure range 30e140 GPa [31]. We choose this potential because the ion charges it uses follow the relationship qMg ¼ qO ¼ 0.5qSi ¼ 1.672e (where e is the elementary electric charge), which ensures that the charge of dislocations does not arise from an incomplete compensation of the charges of ions. We verified that this potential reproduces the generalized stacking
fault (GSF) energy with accuracy, and yields the same dislocation core structures as the potential used in a previous work [27]. Atomic systems are optimized by means of molecular statics simulations employing a conjugate-gradient algorithm until reaching a residual force on each atom smaller than 1011 eV/Å. Dislocations are constructed following a procedure similar to the one proposed by Marrocchelli et al. in SrTiO3 [19], as illustrated in Fig. 1. The system is 3-D periodic and has a square shape, of size about 150 150 6.7 Å3. Two dislocations of opposite Burgers vectors b¼±[100] are introduced so that the line joining them forms an angle of about 45+ with their glide planes to minimize their elastic interaction [32]. Because the two atomic planes across a (010) cut plane do not have the same stoichiometry, the termination of the extra half-plane is not the same for the [100] and for the ½100 dislocation. In the example shown in Fig. 1, the dislocation of positive Burgers vector has a termination MgSiO, i.e. it is oxygendeficient compared to the bulk, and in the following will be referred to as the P variant. The dislocation of negative Burgers vector has a termination O2, i.e. enriched in oxygen compared to the bulk, and will be referred to as the N variant. The core structures of these dislocations are described in more details in the Results section. In the framework of the chosen rigid-ion potential, a vacancy can only be created by removing an ion with its full charge, e.g. removing an oxygen ion will create an oxygen vacancy of charge þ1.672e. This approach is justified because ab initio calculations have determined that the most energetically favorable state for vacancies is to carry an uncompensated electric charge in other perovskite materials, e.g. SrTiO3 [33], PbZrO3 [34], BaZrO3 [35], CaTiO3 [36]. By analogy we assume that the charged state is the most favorable in bridgmanite as well. The Coulomb interaction is computed with the Ewald summation method, which includes a self-interaction correction compensating for the interactions between periodic replicas of a point charge [37e39], allowing to perform calculations with supercells that are not charge-neutral. In order to evaluate the effective charge of dislocations, we use an atomic-to-continuum approach inspired by the Ewald summation method. Atoms are projected in a plane normal to the dislocation line, and replaced by 2-D Gaussian functions over a finiteelement grid. The magnitude of each Gaussian function is equal to the electric charge of the ion it replaces, and their variance s2 ¼ 9
Fig. 1. (a) The unit cell of MgSiO3 bridgmanite. Mg ions are displayed as large orange spheres, Si ions as medium blue spheres, and oxygen as small red spheres. SiO6 octahedra are shown in transparent blue. In the [001] direction normal to the figure, two layers of Mg and SiO6 octahedra are superimposed. A (010) plane (dashed line) separates an atomic plane of stoichiometry MgSiO, and another one of stoichiometry O2. (b) Sketch of the atomic system constructed, containing a dipole of ±[100](010) edge dislocations. The two dislocations have different core structures, noted P and N. (c) Close-up of the atomic configuration of the P dislocation, which is oxygen-deficient compared to the bulk. The anti-phase boundary (APB) inside the dislocation core is overlaid with a thick dashed line. On each side of the APB, incomplete, truncated SiO5 octahedra are recognizable. The electric charge density map around the P dislocation, as calculated with our atomic-to-continuum method (see text), is reported in the right-hand side. The charge density is in units of e/S where S is the surface of an element of the grid. Areas where the charge density vanishes appear in green, those where it is positive in red, and negative in blue. Two zones of positive charge density appear in red, due to the missing oxygen ions in the incomplete octahedra. (d) Close-up view of the atomic configuration of the N dislocation, which is enriched in oxygen compared to the bulk. This dislocation contains four oxygen ions in excess (arrows) compared to the P dislocation. The electric charge density map (same color conventions as c) shows two zones of negative charge density in blue, due to the excess oxygen ions. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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is chosen so that the Gaussian functions overlap and yield a continuous zero charge density in the bulk material. Where Gaussian functions only partially overlap, the resulting net charge density is non-zero. When the supercell has a non-vanishing total electric charge, a continuous neutralizing background is introduced by shifting all values of the density, similarly to what is done in the Ewald summation method. Then, the integration of the charge density around a defect provides an estimation of its charge. The final result does not depend on small variations of the variance or of the size of elements of the grid. Moreover we verified that applying this approach to a single vacancy produces a net charge opposite to that of the removed ion, e.g. removing a Mg ion results in a net charge of 1.672e, which validates this approach. The energy of interaction of a vacancy with the dislocations is defined as:
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two maxima of negative charge localized on the SiO6 polyhedra, obviously due to the extra oxygen ions. The integration yields a net charge of 0.381 e/L, or 9.10 1011 C.m1, i.e. almost exactly opposite to the charge of the P core, as one would expect from the terminations of the inserted half-planes. 3.2. Construction of charge-neutral dislocations
where Edv is the total energy of the system containing the dislocations and the vacancy, Ed the energy of the system containing only the dislocations, Ev the energy of the system containing only the vacancy, and E0 the energy of the ideal bulk system with no defect. We verified that the system is large enough so that, when the vacancy is close to a dislocation, the effect of the second dislocation is negligible. Note that DiE can also be interpreted as the variation of the formation energy of the vacancy with respect to the bulk (defect-free) material. In order to account for the effects of pressure, the enthalpy of interaction is defined as:
The N variant of the dislocation has four oxygen ions in excess with respect to the P variant. Therefore, in order to construct two equivalent, charge-neutral dislocation cores, two oxygen ions have to be displaced from the N to the P dislocation. Among the possible resulting configurations, the one with the lowest energy was selected. The displacement of the two oxygen ions lowers the total energy of the system by about 17.75 eV, confirming that it is much more favorable. The two dislocations have then identical atomic configurations, represented in Fig. 2a and referred to as the O variant. Inside this dislocation, on each side of the APB there is one complete SiO6 and one incomplete SiO5 octahedra (arrows in Fig. 2a). The density of electric charge in the O dislocation is represented in Fig. 2b. It reveals a small distribution of negative charge at the center, surrounded by three regions of positive charge. These regions of non-vanishing electric charge are due to the displacement field of the dislocation, which prevent the ions from overlapping like in the bulk material. However the integration of the charge density is zero, and overall this dislocation is indeed chargeneutral.
Di H ¼ Di E þ V0 dP
3.3. Optimal stoichiometry and charge of the dislocation
Di E ¼ Edv Ed Ev E0
(1)
(2)
where V0 is the volume of the supercell and dP the variation of the pressure due to the vacancy.
3. Results 3.1. Atomic structures of the [100](010) dislocation The system containing the dislocation dipole was optimized, and the atomic configurations of the P and N dislocations are reported in Fig. 1c and d, respectively. Both dislocations contain a characteristic anti-phase boundary (APB), recognizable by the pairs of SiO6 octahedra connected by their edges in the glide plane (dashed lines), as opposed to the bulk where octahedra are connected by their corners. Despite their difference in composition both cores are compact and have a similar spreading of about 13 Å. On one hand, the P dislocation is characterized by incomplete, truncated octahedra on each side of the APB (Fig. 1c), due to the deficiency of oxygen ions. On the other hand, the N dislocation is bordered by complete SiO6 octahedra, marked by thick arrows in Fig. 1d. There are two octahedra per unit length of dislocation on each side of the APB, therefore the N dislocation has four extra oxygen ions per unit line length compared to the P core. We used our atomistic-to-continuum approach to estimate the net electric charge of these dislocations. The electric charge densities are reported in Fig. 1c and d. As expected, the surrounding bulk material has a continuous zero charge density. The P dislocation (Fig. 1c) is characterized by two maxima of positive charge localized on the incomplete SiO5 octahedra. This reflects the fact that the positive charge of cations is not compensated because of missing oxygen ions. The integration of this charge density yields a net charge of þ0.384 e/L, where L is the unit length of the dislocation (equal to the lattice parameter c0 ¼ 6.7063 Å), i.e. þ9.17 1011 C.m1. Similarly the N dislocation is characterized by
At any finite temperature, vacancies naturally form in the material. Typically in ionic materials the anions and cations have very different formation energies and mobilities. In bridgmanite the vacancies with the lowest formation and migration energies are oxygen vacancies [40,41], therefore they are the most mobile ones. When vacancies of all types exist in the material, oxygen vacancies will segregate the most quickly into the dislocations. We wanted to determine how many oxygen vacancies per unit line length the charge-neutral O dislocation can absorb, in order to determine the optimal oxygen vacancy concentration, i.e. the optimal atomic configuration, of the [100](010) edge dislocation. Starting from the O variant, we removed half a column of oxygen ions (i.e. one oxygen ion out of two along the [001] direction), relaxed the system, and computed the corresponding energy of interaction using Eq. (2). By repeating this step for several positions of the vacancy column, we determined if it was more favorable for oxygen vacancies to be in the bulk or in the vicinity of the dislocation. Then the configuration of lowest energy was selected (i.e. the one with the oxygen vacancies at the most favorable site), and the procedure was repeated for a second oxygen vacancy. Note that since the two O dislocations in our setup are identical, the same results are obtained from both of them, but only one is presented below. The results are summarized in Fig. 2. Since the initial O dislocation is charge-neutral (Fig. 2b) one would expect the interaction to be driven by elastic effects, i.e. different interactions above and below the glide plane. Surprinzingly the interaction map in Fig. 2c shows that the dislocation is unequivocally attractive to oxygen vacancies. The interaction is quite short-ranged and vanishes quickly outside the dislocation core. It is more favorable for oxygen vacancies to be inside the dislocation core rather than in the surrounding bulk material, by 5.26 eV. The effect responsible for this peculiar interaction is illustrated in Fig. 3. The (charged) row of vacancies is responsible for a displacement of neighboring ions:
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Fig. 2. (a) Atomic structure of the O dislocation, obtained by removing two oxygen ions from the initial N dislocation (Fig. 1c), and placing them into the P dislocation (Fig. 1d). (b) The electric charge density around the O dislocation is characterized by a central region of negative charge (in blue) and three surrounding regions of positive charge (in red). The integration results in an overall charge-neutral dislocation. (c) Map of the enthalpy of interaction of a single oxygen vacancy with the dislocation of O type. Vanishing values appear in green, positive ones in red, and negative ones in blue. Only oxygen sites are represented (Mg and Si ions are omitted), and some oxygen octahedra are overlaid with black lines. (d) The reduced O dislocation, after it has absorbed one oxygen vacancy per unit line length. Because of the oxygen vacancy, one octahedron is incomplete on the left side of the APB. (e) The electric charge density of the reduced O dislocation. The oxygen vacancy is responsible for a zone of positive charge (in red). (f) Map of the enthalpy of interaction of the reduced O vacancy with a second oxygen vacancy (same color convention as c). Although the dislocation is positively-charged, negative binding enthalpies are still found in the vicinity of the core. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
oxygen ions are attracted to the row of vacancies, while cations are repelled from it. In other words the vacancies are responsible for an ionic polarization of the surrounding material. When the oxygen vacancies approach the dislocation, they induce displacements of ions in the dislocation core that modify the balance of the electric charges. The effect is quite small in magnitude and can barely be seen with the naked eye. To better quantify this effect, we computed a weighted average of the positions of positive and negative charges, using the density of electric charge as weights. Even so, the changes in polarization are small and are best observed when the vacancy is close to the dislocation. The Fig. 3 shows the positions of positive and negative centers of charge of the dislocation for various locations of the row of oxygen vacancies. No matter the position of the vacancies, the dislocation always polarizes with its negative center of charge oriented towards the vacancy. This happens because the dislocation, although chargeneutral, contains regions of positive and negative charges that do not perfectly overlap, hence acting like a dipole. The vacancies are responsible for an induced polarization of the dislocation, which is the reason for the attraction of the vacancy to the dislocation. Fig. 2d shows the atomic configuration of the dislocation after removal of the first oxygen row. This reduced dislocation has gained an incomplete SiO5 octahedron (on the left side of the APB), responsible for a zone of positive electric charge. The total charge of the dislocation becomes positive, about þ3.82 1011 C.m1. In this configuration, zones of positive and negative charges do not coincide, as shown in Fig. 2e (zones in red and blue, respectively), so that the dislocation has the aspect of a line of electric dipoles.
Despite its positive charge, this dislocation is still found to be attractive to oxygen vacancies, as illustrated in Fig. 2f. The positively-charged oxygen vacancies are mainly attracted to the negatively-charged zone of the dislocation. Inside the core the binding enthalpy is 3.1 eV, i.e. weaker than in the case of the O dislocation. Finally, after the removal of two columns of oxygen ions the dislocation is in the P configuration, as shown in Fig. 1c, and its electric charge is 9.17 1011 C.m1. Because of this important positive electric charge, the P dislocation is then completely repulsive to oxygen vacancies, as will be presented in the next section. In conclusion, our results indicate that it is favorable for the [100](010) dislocation to be deficient in oxygen vacancies, hence to bear a positive electric charge. In the limit of a super-saturation of oxygen vacancies, the dislocation can absorb up to two oxygen vacancies per unit line length compared to the charge-neutral O core. These results are consistent with studies found in the literature showing that dislocations tend to be oxygen-poor in other perovskite materials [15e18,20,42,19].
3.4. Interaction of the P dislocation with vacancies In equilibrium with a population of oxygen vacancies, the oxygen-deficient configuration of the [100](010) edge dislocation appears to be the most favorable. However this dislocation is electrically charged and, as presented in the introduction, it is expected to be surrounded by a cloud of charge-compensating defects. In bridgmanite the positive charge of the P dislocation can be
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compensated by negatively-charged Mg and Si vacancies, and naturally these charged defects will interact with the dislocation. In order to quantify this interaction, the system is duplicated along the direction of the dislocation lines. A size corresponding to four supercells along the dislocation line (24 Å) is found to give a good convergence of the interaction energies. A vacancy (Mg or Si) is introduced, the atom positions are optimized with a conjugategradient algorithm, and the interaction with the dislocation is evaluated (Eq. (2)). The procedure is repeated for vacancy sites within a radius of 30 Å around the dislocation. The maps of interaction of the P dislocation with vacancies are reported in Fig. 4. As stated earlier this dislocation is entirely repulsive to oxygen vacancies, since it is already saturated in oxygen vacancies. The interaction is radial, and much more long-range than before, because of the electric charge carried by the dislocation. By opposition, the P dislocation is strongly attractive to cation vacancies. Again the interaction is radial and long-range because of its electrostatic nature. A Mg vacancy has a binding enthalpy of 11 eV when it is at the most favorable site inside the dislocation core, while the Si vacancy has binding enthalpy of about 22 eV at its most favorable position. The factor of 2 between the two defects reflects quite well the ratio of their charges q(VSi)/q(VMg)¼2, again due to the electrostatic nature of the interaction. Fig. 4b, e and h show the elastic contribution of pressure, i.e. the term VdP in Eq. (2). When the vacancy is far from the dislocation, its formation enthalpy has a value close to that of the bulk, therefore VdP ¼ 0. When it approaches the dislocation, the vacancy is responsible for small variations of the enthalpy term, of the order of 0.2 eV. It can be seen that the pressure in the system does not depend much on the position of the vacancies, and the contribution of this term is quite small compared to the electrostatic effects. Finally, Fig. 5 shows the atomic structure of the P dislocation after it has absorbed cation vacancies. After absorbing one Mg vacancy the dislocation core becomes narrower, and the octahedra are not connected by their edges anymore. After absorbing a Si vacancy, the atomic configuration of the dislocation changes even more significantly: the stacking fault is spread in the (100) climb plane, as illustrated in Fig. 5b. This stacking fault is an APB, as evidenced by the SiO6 octahedra connected by their edges in the (100) climb plane. This configuration looks very similar to that of another dislocation, the [010](100) edge dislocation that also dissociates by climb when it is submitted to pressures above 80 GPa [27]. These results show that the absorption of cation vacancies allow the dislocation to move out of its glide plane, and initiate the process of climb. They also show that the dislocation adopts different atomic configurations as it absorbs vacancies during its climb. 4. Discussion 4.1. The charge of the edge dislocation According to Eshelby, in ionic materials at thermal equilibrium dislocations are electrically charged [10], because the formation energies of cation and anion vacancies are different. These arguments were mainly based on the atomic structure and mechanical properties of binary ionic materials with rocksalt structure, like
Fig. 3. Electric charge density when an oxygen vacancy is at various positions around the O dislocation. The dotted circle indicates the position of the vacancy row, and the
4 and . signs mark the positions of the centers of charge of the dislocation. The color convention is the same as in Fig. 1. (a) Vacancy at the far top left of the dislocation. (b) Vacancy on the left side of the dislocation. (c) Vacancy at the right of the dislocation. (d) Vacancy below the dislocation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 4. Interaction of the P dislocation with (a,b,c) a single oxygen vacancy, (d,e,f) a Mg vacancy, (g,h,i) a Si vacancy, visualized with Atomeye [59]. The crystal orientation is identical to the one in Fig. 1. The (010) glide plane of the dislocation is shown with a horizontal dashed line, and a plane normal to the glide plane, used in the bottom graphs, is shown with a vertical dotted line. (a,d,g) Maps of the enthalpy of interaction (Eq. (2)). Only the possible lattice sites for the relevant vacancy are represented (other ions are omitted), and ions are colored according to their binding enthalpy DiH. The interaction enthalpy is always positive for the oxygen vacancy, and always negative for the cation vacancies. (b,e,h) Contribution of the work of pressure. Ions are colored according to the variation VdP induced by their removal. (c,f,i) Graphs showing the enthalpy of interaction DiH when the vacancy belongs to the atomic plane above the glide plane (filled triangles), below the glide plane (empty triangles), and to the plane normal to the glide plane (circles). The thick, dashed black lines are the results of the fits of Eq. (3).
Fig. 5. Atomic configuration of the P dislocation after it has absorbed (a) one Mg vacancy; (b) one Si vacancy. The dashed line shows the plane of the stacking fault.
NaCl or MgO. In the easiest 1=2½110ð110Þ slip system of these materials, pure edge dislocations lie along a [001] axis, i.e. the dislocation line is along a row of alternating positive and negative
ions [32]. Therefore by construction, in a charge-neutral dislocation the electric charges compensate perfectly and the density of charge is zero everywhere. Consequently, the Coulomb interaction vanishes and the interaction with vacancies is driven by elastic effects, as it was shown in MgO [43e47], NaCl [48], or NiO [49]. These simulations were performed on dislocations that were constructed to be charge-neutral, without taking temperature into account. In the rocksalt structure the absorption of one vacancy produces a pair of jogs along the dislocation line, and because of the ionic nature of the material the jogs are electrically charged [32]. For instance the removal of one Naþ ion along a 1=2½110ð110Þ dislocation in NaCl produces a pair of jogs of negative sign. In Eshelby's model, at finite temperature the difference in the formation energy of anions and cations results in the dislocation to have more jogs of one sign than the other, naturally causing the dislocation to acquire an electric charge. In a ternary perovskite oxide such as bridgmanite, the atomic structure is much more complex. Even when the dislocation is charge-neutral the electric charges do not compensate perfectly, and zones of positive and negative electric charges still exist as illustrated in Fig. 2b. Electrostatic effects are the dominant term in the interaction with vacancies (Fig. 3). In bridgmanite there seem to be no configuration where the charge-neutral dislocation would interact only elastically with charged vacancies. Because of this interaction, at any finite temperature the dislocation attracts oxygen vacancies and becomes positively charged. This situation is distinct from the one described by Eshelby: in materials with the rocksalt structure the absorption of one vacancy is sufficient to move the dislocation out of its glide plane, i.e. to create a pair of jogs and make it climb. On the contrary, in bridgmanite (and other perovskite materials) the absorption of one or even two oxygen vacancies does not form jogs, and does not move the dislocation out of its glide plane. Indeed the dislocation spreads in the same plane whether it is enriched or depleted in oxygen, i.e. whether it is in the
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N, O, or P configuration, as shown in Figs. 1c,d and 2a. The charge of the dislocation is not due to the formation of jogs like in the rocksalt structure, but only to the concentration of oxygen vacancies inside the dislocation. Our results indicate that the oxygen-deficient, positivelycharged dislocation core is the most favorable one in bridgmanite. If the material is saturated with oxygen vacancies the dislocation charge can reach a maximum of about þ9.17 1011 C.m1. Experimentally, we propose that it may be possible to determine the charge of edge dislocations in bridgmanite (or in an analogue perovskite) by measuring their velocity under an applied electric field, like it was done in other ionic materials [13,14]. 4.2. Interaction with vacancies The positively-charged dislocation is attractive to cation vacancies, which will slowly migrate towards the dislocation core, driven by electrostatic interaction. If we approximate the dislocation as a straight line of lineic charge density l and the vacancy as a point charge q, their interaction is defined as the sum of elastic and electrostatic contributions [10,32]:
Eint ¼ Eelas þ ECoul ¼
mb 1 þ n y2 2ql lnðr Þ þ cst: dV þ 3p 1 n x2 þ y2 ε (3)
where m is the shear modulus of the material, n its Poisson ratio, ε its relative permittivity, b the magnitude of the Burgers vector of the dislocation, x and y the position of the vacancy (i.e. considering pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi that the dislocation is located at x ¼ 0,y ¼ 0), and r ¼ ðx2 þ y2 Þ the dislocation-vacancy distance. In order to confirm the value of the dislocation charge given by our atomic-to-continuum approach, we fitted Eq. (3) to the interaction energy profiles obtained from atomistic calculations for the Mg and Si vacancies. The only fit parameter was the dislocation charge l, and the resulting fitted functions are reported as thick dashed lines in Fig. 4c, f and i. The obtained values are l¼3.37 1010 C.m1 when fitting to the interaction enthalpies of the oxygen vacancy (q(VO)¼þ1.672e), l¼6.68 1010 C.m1 using the data for the Mg vacancy (q(VMg) ¼ 1.672e), and l¼5.01 1010 C.m1 using the data for the Si vacancy (q(VSi) ¼ 3.344e). These values are of the same order of magnitude as our atomic-to-continuum method, the best match being obtained from the fit to data for the oxygen vacancy. The discrepancy can be attributed to the divergence of the model when the dislocationvacancy distance r/0, while in atomistic simulations the energy of the system is always finite. As a result the obtained value of l depends critically on the data points included in the fitting procedure. Including or excluding data points at distances smaller than 5 Å can result in large variations of the estimated dislocation charge. In addition, the Eq. (3) does not account for second-order effects like dipole-point charge interactions, which may play an important role as in the case of the O dislocation (Fig. 3). For these reasons we estimate that the dislocation charge obtained with our atomic-to-continuum approach is more reliable than the one obtained by fitting the simple Eq. (3). Nonetheless the latter enhances the confidence on the order of magnitude of the dislocation charge, both qualitatively and quantitatively. 4.3. Consequences for dislocation climb As introduced by Eshelby, a charged dislocation in an ionic material is surrounded by an atmosphere of vacancies of opposite charge, the DebyeeHückel cloud [10,32]. Our results show that in
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bridgmanite the dislocation is positively charged, therefore the DebyeeHückel cloud is composed of negatively-charged Mg and Si vacancies. These cation vacancies are attracted to the charged dislocation, and eventually they also segregate into the dislocation core. This has two effects. First, the absorption of negativelycharged vacancies modifies the electric charge of the dislocation, making it charge-neutral or negatively charged. Our results show that in both cases the dislocation becomes attractive to oxygen vacancies again. By sequentially absorbing oxygen and cation vacancies the dislocation alternates between positive and negative charge. The second effect of the absorption of cation vacancies is to modify the plane in which the dislocation spreads. In other words it allows the dislocation to climb. After absorbing a complete formula unit MgSiO3, a segment of the dislocation can climb by one lattice vector. To summarize, we propose that dislocation climb in bridgmanite, and possibly in other perovskite materials, occurs by the sequential absorption of cation and oxygen vacancies, as illustrated in Fig. 6. The absorption of oxygen vacancies has the role of maintaining the positive charge of the dislocation: indeed if the dislocation is charge-neutral or negatively charged it attracts oxygen vacancies, however their absorption does not change the plane where the dislocation spreads. Subsequently, the positivelycharged dislocation attracts cation vacancies, effectively moving the dislocation out of its glide plane and changing its electric charge. The dislocation then becomes attractive to oxygen vacancies again. The cyclic repetition of this sequence allows the dislocation to climb. Furthermore, the predominance of the Coulomb interaction has a direct consequence on the magnitude of the dislocation-vacancy interactions. In materials where this interaction is governed by elastic effects, the interaction energies are of the order of 0.1e0.5 eV [50,51]. By contrast, in bridgmanite we find energies of the order of several eV to more than 10 eV, i.e. up to two orders of magnitude larger. As a result, the kinetics of dislocation climb in bridgmanite may also be significantly faster than it is in metallic systems. Previous atomic-scale calculations have shown that bridgmanite is a high lattice friction material under high-pressure, making dislocation glide difficult [27,28]. Moreover the DebyeeHückel cloud is responsible for a back-stress acting on the dislocation, so that the minimum stress required to move the dislocation is larger than the Peierls stress predicted by considering an isolated dislocation [10]. We expect this to be true also in bridgmanite, where the minimum stress required to move a dislocation (in Ref. [27]) was probably underestimated because it did not take into account the back stress due to a cloud. In the natural conditions of the Earth's lower mantle the deformation occurs under extremely slow strain rates (_ε ¼ 1012 1016 s1), i.e. the applied stresses are extremely small. Also, the Earth's lower mantle is expected to be an oxygenreducing environment [52], thus favoring the existence of oxygen-poor dislocations rather than oxygen-enriched ones. In such conditions one can expect dislocation glide to be a rare event, and dislocations to stay immobile for very long times. This gives time for the cation vacancies to migrate towards the dislocation cores, despite their low diffusivity [41]. In such extreme hightemperature creep conditions dislocation climb may play an important role. It remains difficult to give an estimate of the climb velocity with the available data. Other studies however, seem to corroborate the fact that dislocation climb can occur easily in perovskite materials. High-temperature creep experiments in BaTiO3 [53] and CaTiO3 [54,55] revealed the formation of features named “scallops” along dislocation lines, which were interpreted as climb-dissociated dislocation loops. The activation of climb at so short time scales suggests an efficient mechanism for the absorption of vacancies by
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Fig. 6. Scenario for dislocation climb. (a) A charge-neutral dislocation surrounded by vacancies of positive and negative charges. The positively-charged oxygen vacancies migrate faster than cation vacancies towards the dislocation. (b) After absorbing neighboring oxygen vacancies the dislocation is positively charged (red) and surrounded by a space-charge region (dashed line), where only cation vacancies remain and form the DebyeeHückel cloud. Cation vacancies migrate slowly towards the dislocation core. (c) After absorbing cation vacancies the dislocation climbs, and becomes attractive to oxygen vacancies again. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
the dislocations, which could be explained by electrostatic interactions. Another supporting experimental fact is the ductile to brittle transition around 1050 K in bulk strontium titanate SrTiO3, attributed to a change in the core structure of the 〈110〉 dislocations [56,57], which are the structural equivalent to the [100] dislocations in bridgmanite. The sudden loss of this slip system at 1050 K contrasts with the fact that these dislocations dissociate in their glide plane and are responsible for the ductility at lower temperature [58]. Since the increase of temperature is accompanied with an increase in the concentration and mobility of vacancies, both enhancing dislocation climb, it is possible that this dislocation is charged and absorbs vacancies. Indeed, recent atomistic simulations have pointed to the predominance of electrostatic effects in dislocation-vacancy interactions in SrTiO3 [19], supporting the conclusions of the present work.
[7] [8]
[9] [10] [11] [12] [13] [14]
[15]
5. Conclusion Atomic-scale calculations were utilized to determine the electric charge of the [100](010) edge dislocation in MgSiO3 perovskite at 30 GPa, and quantify its interaction with intrinsic vacancies. We show that [100](010) edge dislocations in bridgmanite favor an oxygen-deficient composition and carry a positive electric charge. As a result the interaction with vacancies is dominated by the Coulomb interaction rather than by elastic effects. Because of its electric charge, the dislocation alternatively attracts defects of negative and positive electric charges, allowing it to climb. These findings modify significantly the understanding of dislocation climb in bridgmanite, and in perovskite materials in general. The climb velocity of dislocations in this class of materials may be much faster than in non-ionic systems. It appears necessary to pursue this work in order to obtain estimates of climb velocities.
[16] [17] [18] [19] [20]
[21]
[22] [23] [24]
[25]
Acknowledgements This work was supported by funding from the European Research Council under the Seventh Framework Programme (FP7), ERC grant N.290424 - RheoMan. References [1] N. Mott, The mechanical properties of metals, Proc. Phys. Soc. B 64 (1951) 729. [2] J. Weertman, Theory of steady-state creep based on dislocation climb, J. Appl. Phys. 26 (1955) 1213e1217. [3] J. Lothe, Theory of dislocation climb in metals, J. Appl. Phys. 31 (1960) 1077e1087. [4] F. Nabarro, Steady-state diffusional creep, Philos, Mag 16 (1967) 231e237. e des dislocations au moyen [5] G. Edelin, J. Poirier, Etude de la monte riences de fluage par diffusion dans le magne sium - I. Me canisme de d'expe formation, Philos. Mag. 28 (1973) 1203e1210. de e des dislocations au moyen [6] G. Edelin, J. Poirier, Etude de la monte
[26]
[27]
[28]
[29] [30] [31]
[32]
riences de fluage par diffusion dans le magne sium - II. Mesure de la d'expe e, Philos. Mag. 28 (1973) 1211e1223. vitesse de monte R. Le Hazif, G. Edelin, J.-M. Dupouy, Diffusion creep by dislocation climb in beyllium and Be-Cu single crystals, Metall. Trans. 4 (1973) 1275e1281. F. Mompiou, D. Caillard, Dislocation-climb plasticity: modelling and comparison with the mechanical properties of icosahedral AlPdMn, Acta Mater 56 (2008) 2262e2271. F. Mompiou, D. Caillard, Dislocations and mechanical properties of icosahedral quasicrystals, C.R. Phys. 15 (2014) 82e89. J. Eshelby, C. Newey, P. Pratt, A. Lidiard, Charged dislocations and the strength of ionic crystals, Philos. Mag. 3 (1958) 75. R. Sproull, Charged dislocations in lithium fluoride, Philos. Mag. 5 (1960) 815e831. R. Davidge, The sign of charged dislocations in NaCl, Philos. Mag. 8 (1963) 1369e1377. L. Colombo, T. Kataoka, J. Li, Movement of edge dislocations in KCl by large electric fields, Philos. Mag. A 46 (1982) 211e215. T. Kataoka, L. Colombo, J. Li, Direct measurements of dislocation charges in Ca2þ-doped KCl by using large electric fields, Philos. Mag. A 49 (1984) 395e407. Z. Zhang, W. Sigle, W. Kurtz, M. Rühle, Electronic and atomic structure of a dissociated dislocation in SrTiO3, Phys. Rev. B 66 (2002) 214112. Z. Zhang, W. Sigle, M. Rühle, Atomic and electronic characterization of the a [100] dislocation core in SrTiO3, Phys. Rev. B 66 (2002) 094108. Z. Zhang, W. Sigle, W. Kurtz, HRTEM and EELS study of screw dislocation cores in SrTiO3, Phys. Rev. B 69 (2004) 144103. C. Jia, A. Thust, K. Urban, Atomic-scale analysis of the oxygen configuration at a SrTiO3 dislocation core, Phys. Rev. Lett. 95 (2005) 225506. D. Marrocchelli, L. Sun, B. Yildiz, Dislocations in SrTiO3: easy to reduce but not so fast for oxygen transport, J. Am. Chem. Soc. 137 (2015) 4735e4748. H. Kurata, S. Isojima, M. Kawai, Y. Shimakawa, S. Isoda, Local analysis of the edge dislocation core in BaTiO3 thin film by STEM-EELS, J. Microsc. 236 (2009) 128e131. T. Lau, X. Lin, S. Yip, K. Van Vliet, Atomistic examination of the unit processes and vacancy-dislocation interaction in dislocation climb, Scr. Mater 60 (2009) 399e402. M. Kabir, T. Lau, D. Rodney, S. Yip, K. Van Vliet, Predicting dislocation climb and creep from explicit atomistic details, Phys. Rev. Lett. 105 (2010) 095501. E. Clouet, Predicting dislocation climb: classical modeling versus atomistic simulations, Phys. Rev. B 84 (2011) 092106. V. Solomatov, C. Reese, Grain size variations in the Earth's mantle and the evolution of primordial chemical heterogeneities, J. Geophys. Res. 113 (2008) B07408. P. Cordier, T. Ung ar, L. Zsoldos, G. Tichy, Dislocation creep in MgSiO3 perovskite at conditions of the Earth's uppermost lower mantle, Nature 428 (2004) 837e840. , P. Carrez, P. Cordier, Predicted glide sysD. Mainprice, A. Tommasi, D. Ferre tems and crystal preferred orientations of polycrystalline silicate MgPerovskite at high pressure: Implications for the seismic anisotropy in the lower mantle, Earth Planet. Sci. Lett. 271 (2008) 135e144. P. Hirel, A. Kraych, P. Carrez, P. Cordier, Atomic core structure and mobility of [100](010) and [010](100) dislocations in MgSiO3 perovskite, Acta Mater 79 (2014) 117e125. A. Kraych, Ph. Carrez, P. Hirel, E. Clouet, and P. Cordier, Phys. Rev. B, https:// journals.aps.org/prb/accepted/26073O06T6818c3ed21800f5ada7b75c4d40f3 5af (accepted). P. Hirel, Atomsk: a code for manipulating and converting atomic data files, Comput. Phys. Comm. 197 (2015) 212e219. S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J. Comp. Phys. 117 (1995) 1e19. URL, http://lammps.sandia.gov. M. Alfredsson, J. Brodholt, D. Dobson, A. Oganov, C. Catlow, S. Parker, G. Price, Crystal morphology and surface structures of orthorhombic MgSiO3 perovskite, Phys. Chem. Miner. 31 (2005) 671e682. J. Hirth, J. Lothe, Theory of Dislocations, second ed., McGraw-Hill Publ. Co.,
P. Hirel et al. / Acta Materialia 106 (2016) 313e321 New York, 1982. €g üt, Structural and electronic properties of oxygen va[33] J. Buban, H. Iddir, S. O cancies in cubic and antiferrodistortive phases of SrTiO3, Phys. Rev. B 69 (2004) 180102. [34] Z. Zhang, P. Wu, L. Lu, C. Shu, Study on vacancy formation in ferroelectric PbTiO3 from ab initio, Appl. Phys. Lett. 88 (2006) 142902. € rketun, G. Wahnstro € m, Thermodynamics of doping and va[35] P. Sundell, M. Bjo cancy formation in BaZrO3 perovskite oxide from density functional calculations, Phys. Rev. B 73 (2006) 104112. [36] H. Lee, T. Mizoguchi, T. Yamamoto, Y. Ikuhara, First principles study on intrinsic vacancies in cubic and orthorhombic CaTiO3, Mater. Trans. 50 (2009) 977e983. [37] P. Ewald, Die Berechnung optischer und elektrostatischer Gitterpotentiale, Ann. Phys. 64 (1921) 253. [38] D. Adams, G. Dubey, Taming the Ewald sum in the computer simulation of charged systems, J. Comp. Phys. 72 (1987) 156e176. [39] A. Toukmaji, J. Board Jr., Ewald summation techniques in perspective: a survey, Comp. Phys. Comm. 95 (1996) 73e92. [40] D. Dobson, R. Dohmen, M. Wiedenbeck, Self-diffusion of oxygen and silicon in MgSiO3 perovskite, Earth Planet. Sci. Lett. 270 (2008) 125e129. [41] M. Ammann, J. Brodholt, D. Dobson, DFT study of migration enthalpies in MgSiO3 perovskite, Phys. Chem. Miner. 36 (2009) 151e158. [42] V. Metlenko, A. Ramadan, F. Gunkel, H. Du, H. Schraknepper, S. Hoffmann Eifert, R. Dittmann, R. Waser, A. De Souza, Do dislocations act as atomic autobahns for oxygen in the perovskite oxide SrTiO3? Nanoscale 6 (2014) 12864e12876. [43] C. Woo, M. Puls, M. Norgett, Energies of formation of point defects in perfect and dislocated ionic crystals, J. Phys. 12 (1976) 557. [44] C. Woo, M. Puls, Atomistic breathing shell model calculations of dislocation core configurations in ionic crystals, Philos. Mag. 35 (1977) 727e756. [45] M. Puls, C. So, The core structure of an edge dislocation in NaCl,, Phys. Stat. Sol. 98 (1980) 87e96. [46] J. Rabier, M. Puls, Atomistic model calculations of pipe-diffusion mechanisms
321
in MgO, Philos. Mag. 52 (1985) 461e473. [47] F. Zhang, A. Walker, K. Wright, J. Gale, Defects and dislocations in MgO: atomic scale models of impurity segregation and fast pipe diffusion, J. Mater. Chem. 20 (2010) 10445e10451. [48] J. Rabier, M. Puls, Atomistic calculations of point-defect interaction and migration energies in the core of an edge dislocation in NaCl, Philos. Mag. A 59 (1989) 533e546. [49] J. Rabier, J. Soullard, M. Puls, Atomistic calculations of point-defect interactions with an edge dislocation in NiO, Philos. Mag. A 61 (1990) 99e108. [50] E. Clouet, The vacancy-edge dislocation interaction in fcc metals: a comparison between atomic simulations and elasticity theory, Acta Mater 54 (2006) 3543e3552. [51] Z. Chen, M. Mrovec, P. Gumbsch, Dislocation-vacancy interactions in tungsten, Modelling Simul, Mater. Sci. Eng. 19 (2011) 074002. [52] D. Frost, C. McCammon, The redox state of Earth's mantle,, Annu. Rev. Earth Planet. Sci. 36 (2008) 389e420. [53] N. Doukhan, J. Doukhan, Dislocations in perovskites BaTiO3 and CaTiO3, Phys. Chem. Miner. 13 (1986) 403e410. [54] K. Wright, G. Price, J.-P. Poirier, High-temperature creep of the perovskites CaTiO3 and NaNbO3, Phys. Earth Planet. Int. 74 (1992) 9e22. [55] P. Besson, J. Poirier, G. Price, Dislocations in CaTiO3 perovskite deformed at high-temperature: a transmission electron microscopy study, Phys. Chem. Miner. 23 (1996) 337e344. [56] P. Gumbsch, S. Taeri Baghbadrani, D. Brunner, W. Sigle, M. Rühle, Plasticity and an inverse brittle-to-ductile transition in strontium titanate, Phys. Rev. Lett. 87 (2001) 085505. [57] W. Sigle, C. Sarbu, D. Brunner, M. Rühle, Dislocations in plastically deformed SrTiO3, Philos. Mag. 86 (2006) 4809e4821. [58] P. Hirel, M. Mrovec, C. Els€ asser, Atomistic study of 〈110〉 dislocations in strontium titanate, Acta Mater 60 (2012) 329e338. [59] J. Li, Atomeye: an efficient atomistic configuration viewer, Modelling Simul. Mater. Sci. Eng. 11 (2003) 173.
