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Atomistic simulations of grain boundary transformation under high pressures in MgO
Physica B xxx (xxxx) xxx–xxx
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Atomistic simulations of grain boundary transformation under high pressures in MgO ⁎
T. Yokoia,b, , M. Yoshiyab,c a b c
Department of Materials Science and Engineering, Nagoya University, Aichi, Japan Department of Adaptive Machine Systems, Osaka University, Osaka, Japan Nanostructures Research Laboratory, Japan Fine Ceramics Center, Aichi, Japan
A R T I C L E I N F O
A BS T RAC T
Keywords: Grain boundary Atomistic simulations MgO High pressure
This study focuses on transformation of grain boundary (GB) structures under high pressures up to 60 GPa by using a simulated annealing technique with molecular dynamics and lattice statics calculations for various symmetric tilt GBs (STGBs) of MgO. It is found that except for the Σ3(111)/[110] that is a rather stable GB, all the STGBs studied transform into a metastable structure more than once at threshold pressures. In addition, the GBs with an open-core structure and small tilt angle are found to be more “flexible” to transform into different structures than the GBs with a dense structure. For polycrystalline MgO, therefore, GBs may also exhibit GB transformation under high pressures and flexible GBs may govern overall transformation and deformation. These findings also suggest that polycrystals sintered at high pressures consist of more pressure-resistant GBs than those at normal pressures.
1. Introduction Grain boundary (GB) frequently has impacts on the material properties of polycrystalline oxides. For example, recent experimental studies with bicrystals and theoretical calculations have indicated that GB segregation (GBS) behavior [1–3] and GB diffusivity [4,5] vary with GBs, suggesting that microscopic or atomic-level GB characters, such as GB plane and atomic arrangement, play important roles in these GB phenomena. A better understanding of GB structure in the atomic level is a key issue for material design of polycrystalline oxides. Nevertheless, it is still difficult to determine precisely atomic arrangements at oxide GBs, partly because intrinsic and extrinsic factors, which is either intentional or unintentional, can modify GB structures. Previous studies showed that “pure” GB structures obtained by calculations frequently contradict observations with transmission electron microscopy (TEM) and scanning TEM (STEM) even for welldefined symmetrical tilt GBs (STGBs) and such “pure” GBs are speculated to transform into different structures by various factors, e. g. the formation of Schottky pairs at GBs in NiO [6–8], GBS of unintentional impurities in MgO [9–11], and surface conditions of samples in Al2O3 [12–14]. Therefore, it is essential to understand structural deviation from pure GB structures under given conditions and environments, first of all.
⁎
Pressure is also a crucial factor that affects both the macroscopic and microscopic states of materials, and recent studies have explored new bulk phases under high pressure for a wide variety of oxides [15– 18]. GB structure under high pressure has been also studied as an interesting issue for theoretical study and basic materials science. In this case, it is difficult to experimentally observe the pressure dependence of GB structure, and thus previous studies employed theoretical calculations to STGBs for MgO as a prototype oxide system and indicated that the STGBs are transformed into different structures under high pressures up to 100 GPa [19–22]. In addition, these studies showed that the formation energy of vacancies at GBs decreases as pressure increases, indicating that pressure can promotes the creation of point defects at GBs. However, the previous studies focused only on simple well-defined STGBs, namely the Σ5(210), Σ5(310) and Σ17(410) GBs with the [001] tilt axis, all of which have the open-channel structures identical to each other. At present, it is still unclear whether GB-structural transformation reported for the simple well-defined GBs is the general phenomenon for GBs of MgO or not. In order to tackle this issue, we investigate high-pressure transformation for thirteen STGBs of MgO with the [001] and [110] tilt axis, which cover various types of GB structures at zero pressure, by using simulated annealing (SA) with molecular dynamics (MD) and lattice statics calculations. The remaining part of this work consists of the following sections: the next section describes details of construction of
Corresponding author at: Department of Materials Science and Engineering, Nagoya University, Aichi, Japan. E-mail address: [email protected] (T. Yokoi).
http://dx.doi.org/10.1016/j.physb.2017.03.014 Received 31 December 2016; Received in revised form 1 March 2017; Accepted 4 March 2017 0921-4526/ © 2017 Elsevier B.V. All rights reserved.
Please cite this article as: Yokoi, T., Physica B (2017), http://dx.doi.org/10.1016/j.physb.2017.03.014
Physica B xxx (xxxx) xxx–xxx
T. Yokoi, M. Yoshiya
Fig. 1. (a) Schematic illustration of a starting GB model with a disordered structure sandwiched by two grains with a misorientation angle 2Θ. First, a MD simulation is run with a single-crystal model a 8000 K to disorder the atomic arrangement completely. Then, the structure is inserted between two grains that are individually tilted by a misorientation angle of θ°. (b) Conditions of the SA technique with MD simulations in terms of temperature and holding time, and snapshots of atomic arrangement at each of the temperature during MD simulations.
and metastable structures within limited time. On the other hand, the disadvantage is that SA simulations are sometimes trapped into localminimum GB structures at which atomic arrangements are significantly disordered even after geometrical optimizations. In this work, such highly disordered GBs are not considered as a metastable GB structure and excluded. In order to understand systematically the pressure dependence of GB structure, we study several STGBs with the [001] and [110] tilt axis varying their misorientation angles and thereby atomic arrangements at the GBs. Table 1 lists the STGBs studied in this work and their GB energy and excess volume per GB area at 0 GPa. The structural features and energetics of these STGBs will be discussed in detail elsewhere. The GB energy at pressure P, ΔE GB(P ), is evaluated by the following equation:
a GB model and computational conditions for SA techniques. Section 3 discusses results of GB structures obtained at high pressures and changes in GB energy and excess volume. In addition, the effect of temperature on the formation of metastable GB structures is also discussed. The final section is devoted to conclusions of this work. 2. Computational details Fig. 1 shows a schematic picture of construction of a GB model and computational conditions of MD simulations. We employ SA techniques with MD to obtain energetically stable and metastable GB structures under pressures up to 60 GPa. An initial GB model under zero pressure is constructed by inserting a disordered bulk structure between two grains tilted at a given misorientation angle. The disordered structure is created by holding a single-crystal model at 8000 K for 20 ps. The first SA starts at 4000 K with the NVT condition to suppress the expansion of the disordered structure and to solidify the structure. Then the NVT condition is switched to the NPT condition at T < 4000 K to allow a simulation box to change its lattice constants. At this stage, the temperature is decreased in a stepwise manner with a temperature decrement ΔT of 250 K at T ≥ 2500 K or 500 K at T < 2500 K . A holding time at the temperature is set to 100 ps at T ≥ 3000 K , 50 ps at 1000 K ≤T < 3000 K , or 25 ps at T < 1000 K . After a MD runs for 25 ps at 100 K, the GB structure is numerically optimized with a conjugate gradient algorithm of a static lattice energy minimization. This study uses an empirical potential set of the Buckingham type reported in a previous study [23]. The cut-off radius of the empirical potential is set to 20 Å. The formal charges of a Mg and O ion are assumed to be +2 and −2, respectively. All calculations in this work are performed with the LAMMPS code [24]. The second SA starts at 3000 K using the zero-pressure GB structures, which obtained from the first SA, at a pressure from 2 GPa to 60 GPa with an interval of 2 GPa. The temperature is decreased stepwise with ΔT = 250 K at T ≥ 2000 K or ΔT = 500 K at T < 2000 K , and a holding time is set to 50 ps at T ≥ 1000 K and 25 ps at 500 K and 100 K. Finally a numerical optimization is performed at the same pressure as that of the SA. In addition, the effect of pressure release on the GB structure is studied. A GB structure obtained under a high pressure is again numerically optimized at zero pressure, and its atomic arrangement and GB energy are evaluated. The advantage of the SA techniques used in this work is that atomic diffusion is effectively promoted by thermal fluctuation whose magnitude depends on a given temperature. As a result, it becomes easier to overcome energy barrier for GB transformation between different GB structures, which allows us to search a wide range of possible stable
ΔE GB(P ) =
E GB(P ) − nE BULK (P ) 2A(P )
GB
(1)
BULK
(P ) are the lattice energy at pressure P of a GB where E (P ) and E model and an unit cell of MgO, respectively, n is the number of unit cells contained in the GB model, and A(P) is the GB area at P. It should be noted that ΔE GB(P ) corresponds to the increase of internal energy per GB area, not the Gibbs free energy, since ΔE GB(P ) does not consider the contribution of entropy and PV term. Nevertheless, it is possible to understand the trend of the pressure dependence of GB structure by Table 1 STGBs studied in this work and their GB energy and excess volume per GB area for the most stable GB structures. These values are obtained under zero pressure.
2
Grain boundary
Misorientation angle 2Θ (°)
GB energy (J/m2)
Excess volume per GB area (Å)
Σ65(810)/[001] Σ13(510)/[001] Σ5(310)/[001] Σ5(210)/[001] Σ13(320)/[001] Σ61(650)/[001] Σ65(551)/[110]
14.3 22.6 36.7 53.1 67.4 79.6 16.1
1.78 1.95 1.95 1.76 2.06 1.41 2.28
0.90 1.13 1.23 0.99 0.39 0.25 0.30
Σ33(441)/[110]
20.0
2.49
0.35
Σ9(221)/[110]
38.9
2.74
0.66
Σ11(332)/[110]
50.5
2.65
0.50
Σ3(111)/[110]
70.5
0.86
0.01
Σ17(223)/[110]
93.4
2.68
0.44
Σ3(112)/[110]
109.5
2.52
0.91
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Fig. 2. GB structures obtained by SA technique at zero pressure and high pressures up to 60 GPa for the [001] and (b) [110] STGBs. Yellow and red balls represent Mg and O ions, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
evaluating ΔE GB(P ). The excess volume per GB area under pressure, ΔV GB(P ), are also evaluated by using the following equation:
ΔV GB(P ) = GB
V GB(P ) − nV BULK (P ) 2A(P )
At present it is technically difficult to compare our results with experimental results since TEM/STEM observations are usually performed at normal pressures and further GB segregation of unintentional impurities (e. g. Ca, Al, and Ti) is also believed to induce GB transformation [9–11]. Nevertheless, experimentally observed GBs [9– 11] are similar to the obtained metastable GBs with a dense structure at high pressures. This similarity may indicate that external factors, such as pressure and impurity segregation, induce a same kind of GB transformation at least for STGBs. Interestingly, the Σ65(810) GB exhibits various types of GB structures: at least four different stable and metastable GBs are predicted under high pressures (Fig. 2(a)). Moreover, additional two metastable GBs are predicted after pressure release as shown in Fig. 3. The two structures are clearly different from those in Fig. 2(a). The similar trend, that is, the formation of various metastable GBs depending on pressure and its release, is also observed for the Σ13(510) GB. The similarities between the two GBs are the characteristic open-core structure at zero pressure and small misorientation angles compared with the other STGBs. Therefore, GBs with the two features may potentially be transformed into various metastable GB structures. Fig. 4 shows ΔE GB(P ) as a function of pressure. These values are obtained based on GB structures after SA simulations and subsequent static energy minimizations. This result also indicates that the pressure dependence of the GBs is very different with each other in terms of
(2)
BULK
(P ) are the volume of the GB model and unit where V (P ) and V cell, respectively. It turned out that the PV term is negligibly small compared with the dominant internal energy term. 2.1. Results and discussion This section firstly shows GB structures obtained by SA techniques at high pressures for the individual GBs comparing their zeropressure structures, and discusses their energetics based on ΔE GB(P ) and ΔV GB(P ). Fig. 2 shows results of GB structures at each pressure for several STGBs. It is found that most of the STGBs studied exhibit transformation into metastable structures more than once except for the Σ3(111) STGB that has a significantly lower values of ΔE GB(P = 0) and ΔV GB(P = 0) than the other GBs, as listed in Table 1. For these metastable structures the open-core atomic arrangements collapse and more denser structures are formed. These results indicate that the existence of metastable GB structures for MgO is a major phenomenon. If this phenomenon also occurs for general GBs, it is expected that GB structures in polycrystalline MgO are transformed by applying high pressure. 3
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Fig. 3. Metastable GB structures after pressure release from an initial pressure of (a) 40 GPa and (b) 46 GPa for the Σ65(810)/[001] GB.
Fig. 5. Correlation between zero-pressure GB energy ΔE GB(P = 0) and high-pressure GB energy ΔE GB(P ) .
ΔE GB(P ) and its gradient. For example, the GBs with the open structure at 0 GPa, e.g. the Σ5(310), Σ5(210), and Σ3(112) GBs, show abrupt changes in ΔE GB(P ). In contrast, the Σ13(320) and Σ61(650) GBs, which have the dense structure, show almost a linear change of ΔE GB(P ) in all range of pressure. The trend suggests that the relative stability between GBs depends on pressure, that is, stable GBs at 0 GPa are not necessarily preferred at high pressures. Therefore, it is speculated that the distribution of GBs at 0 GPa in polycrystalline MgO is modified by applying a high pressure and consequently the population of individual GBs is changed depending on the pressure. The Σ3(111) GB is clearly distinct from the other GBs in the sense that the increase of ΔE GB(P ) remains within the range of 0– 0.13 J/m2 , indicating that the energy increase at a pressure is almost as small as in a single-crystal model. This result is not
surprising since this particular GB corresponds to a twin boundary having no deficiency in coordination number. Even under high pressures up to 60 GPa, ΔE GB(P ) of this GB is smaller than next lowest ΔE GB(P = 0). Consequently, the difference in ΔE GB(P ) between the Σ3(111) and the other GBs becomes more and more large as an applied pressure increases. This result probably implies that the relative stability of the Σ3(111) to other GBs is enhanced with the increase of pressure, and consequently the population of the Σ3(111) becomes much larger than other GBs. At present, this speculation is just based on our theoretical results of the internal energy, and thus further study is required. Fig. 5 shows the correlation between ΔE GB(P = 0) and ΔE GB(P ) at pressures between 10 GPa and 40 GPa. There is a positive correlation between two GB energies at 10 GPa (the grey points) while the
Fig. 4. GB energy as a function of pressure for (a) the [001] and (b) [110] STGBs.
4
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Fig. 6. Decrease of excess volume per GB area from 0 GPa as a function of pressure for (a) the [001] and (b) [110] STGBs.
from not only excess volume but also other dominant factors that are related to atomic details at GBs. Fig. 8 shows ΔE GB(P → 0) relative to ΔE GB(0) as function of initial pressure from which an applied pressure is released at 0 K. The description ΔE GB(P → 0) means that an initial pressure P is discretely changed to 0 GPa. Some of the GBs sustain a positive value of ΔE GB(P → 0), indicating that metastable structures are maintained even after the pressure release. An irreversible GB transformation was reported in a previous theoretical study [19]. As in Fig. 3, such metastable structures differ from those under high pressures, and thus high pressure and its release may result in metastable GB structures that are usually inaccessible with high pressure alone. Finally, a path of GB transformation is discussed based on results of SA simulations and static energy minimizations, as shown in Fig. 9. Results of the Σ5(210) GB are shown as an example since the GB shows a clear change in the GB energy with the increase of pressure. Both two approaches apply high pressures to the GB structure, but SA simulations are performed at finite temperatures (the black points) while static energy minimizations at 0 K (the grey points). The difference leads to a change in the threshold pressure of the formation of metastable/stable structures. For SA simulations, the metastable structures are formed at the lower pressures than static energy minimizations: the lattice energy discretely increases at 8 GPa and 46 GPa. The quadratic curves fitted to the black or grey data points show that the metastable structures obtained from SA simulations correspond to the most stable structures obtained from static energy minimizations. Presumably, GB structures obtained from SA simulations are energetically stable at high temperatures and high pressures, while the structures change from stable to metastable ones with the decrease of temperature. The metastable structures are maintained if there is an energy barrier for transformation between the stable and metastable structures. It is therefore expected that metastable GB structures are formed by high-pressure sintering, whose atomic arrangements differ from those by high-pressure compression at low temperature. In addition, such metastable GBs may be sustained after pressure release depending on GB characters.
correlation becomes weak with the increase of pressure. This reason is that some of the GBs transform into significantly different structures at higher pressures, which involves a sharp increase in ΔE GB. Fig. 6 shows the decrease of ΔV GB(P ) from 0 GPa as a function of pressure. The trend also depends on the individual GBs, although all the GBs show the decrease of ΔV GB(0) with the increase of pressure. The GB transformation is clearly detected as a sharp change in ΔV GB(0) in the cases of the GBs with the open structure, while the GBs with the dense structure show relatively a small volume change at the threshold pressure of the GB transformation. The correlation between the rate change of ΔE GB(P ) and ΔV GB(P ) is shown in Fig. 7. There is a clear correlation between the two quantities for one GB, but a comparison between GBs indicates a significant variation in the data points. The result suggests that a difference in relative stability between GBs result
Fig. 7. Correlation between the rate change of ΔE GB(P ) and ΔV GB(P ) for all the STGBs.
5
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Fig. 8. GB energy relative to ΔE GB(0) after pressure release to zero as a function of applied pressure for (a) the [001] and (b) [110] STGBs. After the pressure release, a static structural optimization is performed.
Acknowledgements This work was supported by Council for Science, Technology and Innovation (CSTI), Cross-ministerial Strategic Innovation Promotion Program (SIP), “Structural Materials for Innovation” (Funding agency: JST), and Grant-in-Aid for Scientific Research on Innovative Areas “Exploration of nanostructure-property relationships for materials innovation” from MEXT, Japan. References [1] T. Gemming, S. Nufer, W. Kurtz, M. Rühle, Structure and Chemistry of Symmetrical Tilt Grain Boundaries in α-Al2O3: II, Bicrystals with Y at the Interface, J. Am. Ceram. Soc. 86 (4) (2003) 590–594. [2] N. Shibata, F. Oba, T. Yamamoto, Y. Ikuhara, Structure, energy and solute segregation behaviour of [110] symmetric tilt grain boundaries in yttria-stabilized cubic zirconia, Philos. Mag. 84 (2004) 2381–2415. [3] B. Feng, T. Yokoi, A. Kumamoto, M. Yoshiya, Y. Ikuhara, N. Shibata, Atomically ordered solute segregation behaviour in an oxide grain boundary, Nat. Commun. 7 (2016) 11079. [4] T. Nakagawa, H. Nishimura, I. Sakaguchi, N. Shibata, K. Matsunaga, T. Yamamoto, Y. Ikuhara, Grain boundary character dependence of oxygen grain boundary diffusion in α-Al2O3, Scr. Mater. 65 (2011) 544–547. [5] F. Ye, C.Y. Yin, D.R. Ou, T. Mori, Relationship between lattice mismatch and ionic conduction of grain boundary in YSZ, Prog. Mat. Sci. Mater. Int. 24 (2013) 83–86. [6] K.L. Merkle, Atomic resolution electron microscopy of NiO grain boundaries, Ultramicroscopy 22 (1987) 57–70. [7] K.L. Merkle, High-resolution electron microscopy of interfaces in fcc materials, Ultramicroscopy 37 (1991) 130–152. [8] D.J. Harris, J.H. Harding, G.W. Watson, Computer simulation of the reactive element effect in NiO grain boundaries, Acta Mater. 48 (2000) 3039–3048. [9] Y. Yan, M.F. Chisholm, G. Duscher, A. Maiti, S.J. Pennycook, S.T. Pantelides, Impurity-induced structural transformation of a MgO grain boundary, Phys. Rev. Lett. 81 (1998) 3675–3678. [10] Z. Wang, M. Saito, K.P. McKenna, L. Gu, S. Tsukimoto, A.L. Shluger, Y. Ikuhara, Atom-resolved imaging of ordered defect superstructures at individual grain boundaries, Nature 479 (2011) 380–383. [11] M. Saito, Z. Wang, Ikuhara, Selective impurity segregation at a near-Σ5 grain boundary in MgO, J. Mater. Sci. 49 (2014) 3956–3961. [12] T. Höche, P.R. Kenway, H.-J. Kleebe, M.W. Finnis, M. Rühle, The structure of special grain boundaries in α-Al2O3, J. Phys. Chem. Solids 55 (1994) 1067–1082. [13] S. Fabris, C. Elsässer, Σ13(1014) twin in α-Al2O3: a model for a general grain boundary, Phys. Rev. B 64 (2001) 245117. [14] S. Azuma, N. Shibata, T. Mizoguchi, S.D. Findlay, K. Nakamura, Y. Ikuhara, Atomic structure, energetics, and chemical bonding of Y doped Σ13 grain boundaries in αAl2O3, Philos. Mag. 93 (2013) 1158–1171. [15] S. Ono, K. Funakoshi, A. Nozawa, T. Kikegawa, High-pressure phase transitions in SnO2, J. Appl. Phys. 97 (2005) 073523. [16] F.X. Zhang, J.W. Wang, J. Lian, M.K. Lang, U. Becker, R.C. Ewing, Phase Stability
Fig. 9. Increase of lattice energy per unit cell vs pressure for the Σ5(210)/[001] GB. The black points represents results of SA, while the grey points are obtained from geometrical optimizations at 0 K without SA. The red, blue and green curved lines are obtained by fitting quadratic curves to the data points, which corresponds to three different GB structures. The dotted vertical lines indicate the intersection points of these curved lines. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3. Conclusions This work studied stable and metastable structures under high pressures up to 60 GPa for various STGBs of MgO by using SA techniques with MD and lattice statics. Most of the GBs show transformation to a metastable structure, while the threshold pressure of transformation is very different from GB to GB. In particular, the GBs with the open structure at 0 GPa and a small misorientation angle exhibit various metastable structure under high pressures and its release. Therefore, it is expected that GB transformation under high pressure also occur in real polycrystalline MgO and specific GBs are dominantly transformed into different structures. 6
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[17]
[18]
[19]
[20]
and Pressure Dependence of Defect Formation in Gd2Ti2O7 and Gd2Zr2O7 Pyrochlores, Phys. Rev. Lett. 100 (2008) 045503. A.B. Belonoshko, S. Arapan, R. Martonak, A. Rosengren, MgO phase diagram from first principles in a wide pressure-temperature range, Phys. Rev. B 81 (2010) 054110. D. Zagorac, J.C. Schön, J. Zagorac, M. Jansen, Prediction of structure candidates for zinc oxide as a function of pressure and investigation of their electronic properties, Phys. Rev. B 89 (2014) 075201. D.J. Harris, G.W. Watson, S.C. Parker, Computer simulation of pressure-induced structural transitions in MgO [001] tilt grain boundaries, Am. Mineral. 84 (1999) 138–143. D.J. Harris, G.W. Watson, S.C. Parker, Atomistic simulation studies on the effect of
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[22]
[23]
[24]
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pressure on diffusion at the MgO 410/[001] tilt grain boundary, Phys. Rev. B 64 (2001) 134101. A.K. Verma, B.B. Karki, First-principles simulations of MgO tilt grain boundary: structure and vacancy formation at high pressure, Am. Mineral. 95 (2010) 1035–1041. B.B. Karki, D.B. Ghosh, A.K. Verma, First-principles prediction of pressureenhanced defect segregation and migration at MgO grain boundaries, Am. Mineral. 100 (2015) 1053–1058. F. Landuzzi, L. Pasquini, S. Giusepponi, M. Celino, A. Montone, P.L. Palla, F. Cleri, Molecular dynamics of ionic self-diffusion at an MgO grain boundary, J. Mater. Sci. 50 (2015) 2502–2509. 〈http://lammps.sandia.gov/index.html〉.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Atomistic simulations of grain boundary transformation under high pressures in MgO ⁎
T. Yokoia,b, , M. Yoshiyab,c a b c
Department of Materials Science and Engineering, Nagoya University, Aichi, Japan Department of Adaptive Machine Systems, Osaka University, Osaka, Japan Nanostructures Research Laboratory, Japan Fine Ceramics Center, Aichi, Japan
A R T I C L E I N F O
A BS T RAC T
Keywords: Grain boundary Atomistic simulations MgO High pressure
This study focuses on transformation of grain boundary (GB) structures under high pressures up to 60 GPa by using a simulated annealing technique with molecular dynamics and lattice statics calculations for various symmetric tilt GBs (STGBs) of MgO. It is found that except for the Σ3(111)/[110] that is a rather stable GB, all the STGBs studied transform into a metastable structure more than once at threshold pressures. In addition, the GBs with an open-core structure and small tilt angle are found to be more “flexible” to transform into different structures than the GBs with a dense structure. For polycrystalline MgO, therefore, GBs may also exhibit GB transformation under high pressures and flexible GBs may govern overall transformation and deformation. These findings also suggest that polycrystals sintered at high pressures consist of more pressure-resistant GBs than those at normal pressures.
1. Introduction Grain boundary (GB) frequently has impacts on the material properties of polycrystalline oxides. For example, recent experimental studies with bicrystals and theoretical calculations have indicated that GB segregation (GBS) behavior [1–3] and GB diffusivity [4,5] vary with GBs, suggesting that microscopic or atomic-level GB characters, such as GB plane and atomic arrangement, play important roles in these GB phenomena. A better understanding of GB structure in the atomic level is a key issue for material design of polycrystalline oxides. Nevertheless, it is still difficult to determine precisely atomic arrangements at oxide GBs, partly because intrinsic and extrinsic factors, which is either intentional or unintentional, can modify GB structures. Previous studies showed that “pure” GB structures obtained by calculations frequently contradict observations with transmission electron microscopy (TEM) and scanning TEM (STEM) even for welldefined symmetrical tilt GBs (STGBs) and such “pure” GBs are speculated to transform into different structures by various factors, e. g. the formation of Schottky pairs at GBs in NiO [6–8], GBS of unintentional impurities in MgO [9–11], and surface conditions of samples in Al2O3 [12–14]. Therefore, it is essential to understand structural deviation from pure GB structures under given conditions and environments, first of all.
⁎
Pressure is also a crucial factor that affects both the macroscopic and microscopic states of materials, and recent studies have explored new bulk phases under high pressure for a wide variety of oxides [15– 18]. GB structure under high pressure has been also studied as an interesting issue for theoretical study and basic materials science. In this case, it is difficult to experimentally observe the pressure dependence of GB structure, and thus previous studies employed theoretical calculations to STGBs for MgO as a prototype oxide system and indicated that the STGBs are transformed into different structures under high pressures up to 100 GPa [19–22]. In addition, these studies showed that the formation energy of vacancies at GBs decreases as pressure increases, indicating that pressure can promotes the creation of point defects at GBs. However, the previous studies focused only on simple well-defined STGBs, namely the Σ5(210), Σ5(310) and Σ17(410) GBs with the [001] tilt axis, all of which have the open-channel structures identical to each other. At present, it is still unclear whether GB-structural transformation reported for the simple well-defined GBs is the general phenomenon for GBs of MgO or not. In order to tackle this issue, we investigate high-pressure transformation for thirteen STGBs of MgO with the [001] and [110] tilt axis, which cover various types of GB structures at zero pressure, by using simulated annealing (SA) with molecular dynamics (MD) and lattice statics calculations. The remaining part of this work consists of the following sections: the next section describes details of construction of
Corresponding author at: Department of Materials Science and Engineering, Nagoya University, Aichi, Japan. E-mail address: [email protected] (T. Yokoi).
http://dx.doi.org/10.1016/j.physb.2017.03.014 Received 31 December 2016; Received in revised form 1 March 2017; Accepted 4 March 2017 0921-4526/ © 2017 Elsevier B.V. All rights reserved.
Please cite this article as: Yokoi, T., Physica B (2017), http://dx.doi.org/10.1016/j.physb.2017.03.014
Physica B xxx (xxxx) xxx–xxx
T. Yokoi, M. Yoshiya
Fig. 1. (a) Schematic illustration of a starting GB model with a disordered structure sandwiched by two grains with a misorientation angle 2Θ. First, a MD simulation is run with a single-crystal model a 8000 K to disorder the atomic arrangement completely. Then, the structure is inserted between two grains that are individually tilted by a misorientation angle of θ°. (b) Conditions of the SA technique with MD simulations in terms of temperature and holding time, and snapshots of atomic arrangement at each of the temperature during MD simulations.
and metastable structures within limited time. On the other hand, the disadvantage is that SA simulations are sometimes trapped into localminimum GB structures at which atomic arrangements are significantly disordered even after geometrical optimizations. In this work, such highly disordered GBs are not considered as a metastable GB structure and excluded. In order to understand systematically the pressure dependence of GB structure, we study several STGBs with the [001] and [110] tilt axis varying their misorientation angles and thereby atomic arrangements at the GBs. Table 1 lists the STGBs studied in this work and their GB energy and excess volume per GB area at 0 GPa. The structural features and energetics of these STGBs will be discussed in detail elsewhere. The GB energy at pressure P, ΔE GB(P ), is evaluated by the following equation:
a GB model and computational conditions for SA techniques. Section 3 discusses results of GB structures obtained at high pressures and changes in GB energy and excess volume. In addition, the effect of temperature on the formation of metastable GB structures is also discussed. The final section is devoted to conclusions of this work. 2. Computational details Fig. 1 shows a schematic picture of construction of a GB model and computational conditions of MD simulations. We employ SA techniques with MD to obtain energetically stable and metastable GB structures under pressures up to 60 GPa. An initial GB model under zero pressure is constructed by inserting a disordered bulk structure between two grains tilted at a given misorientation angle. The disordered structure is created by holding a single-crystal model at 8000 K for 20 ps. The first SA starts at 4000 K with the NVT condition to suppress the expansion of the disordered structure and to solidify the structure. Then the NVT condition is switched to the NPT condition at T < 4000 K to allow a simulation box to change its lattice constants. At this stage, the temperature is decreased in a stepwise manner with a temperature decrement ΔT of 250 K at T ≥ 2500 K or 500 K at T < 2500 K . A holding time at the temperature is set to 100 ps at T ≥ 3000 K , 50 ps at 1000 K ≤T < 3000 K , or 25 ps at T < 1000 K . After a MD runs for 25 ps at 100 K, the GB structure is numerically optimized with a conjugate gradient algorithm of a static lattice energy minimization. This study uses an empirical potential set of the Buckingham type reported in a previous study [23]. The cut-off radius of the empirical potential is set to 20 Å. The formal charges of a Mg and O ion are assumed to be +2 and −2, respectively. All calculations in this work are performed with the LAMMPS code [24]. The second SA starts at 3000 K using the zero-pressure GB structures, which obtained from the first SA, at a pressure from 2 GPa to 60 GPa with an interval of 2 GPa. The temperature is decreased stepwise with ΔT = 250 K at T ≥ 2000 K or ΔT = 500 K at T < 2000 K , and a holding time is set to 50 ps at T ≥ 1000 K and 25 ps at 500 K and 100 K. Finally a numerical optimization is performed at the same pressure as that of the SA. In addition, the effect of pressure release on the GB structure is studied. A GB structure obtained under a high pressure is again numerically optimized at zero pressure, and its atomic arrangement and GB energy are evaluated. The advantage of the SA techniques used in this work is that atomic diffusion is effectively promoted by thermal fluctuation whose magnitude depends on a given temperature. As a result, it becomes easier to overcome energy barrier for GB transformation between different GB structures, which allows us to search a wide range of possible stable
ΔE GB(P ) =
E GB(P ) − nE BULK (P ) 2A(P )
GB
(1)
BULK
(P ) are the lattice energy at pressure P of a GB where E (P ) and E model and an unit cell of MgO, respectively, n is the number of unit cells contained in the GB model, and A(P) is the GB area at P. It should be noted that ΔE GB(P ) corresponds to the increase of internal energy per GB area, not the Gibbs free energy, since ΔE GB(P ) does not consider the contribution of entropy and PV term. Nevertheless, it is possible to understand the trend of the pressure dependence of GB structure by Table 1 STGBs studied in this work and their GB energy and excess volume per GB area for the most stable GB structures. These values are obtained under zero pressure.
2
Grain boundary
Misorientation angle 2Θ (°)
GB energy (J/m2)
Excess volume per GB area (Å)
Σ65(810)/[001] Σ13(510)/[001] Σ5(310)/[001] Σ5(210)/[001] Σ13(320)/[001] Σ61(650)/[001] Σ65(551)/[110]
14.3 22.6 36.7 53.1 67.4 79.6 16.1
1.78 1.95 1.95 1.76 2.06 1.41 2.28
0.90 1.13 1.23 0.99 0.39 0.25 0.30
Σ33(441)/[110]
20.0
2.49
0.35
Σ9(221)/[110]
38.9
2.74
0.66
Σ11(332)/[110]
50.5
2.65
0.50
Σ3(111)/[110]
70.5
0.86
0.01
Σ17(223)/[110]
93.4
2.68
0.44
Σ3(112)/[110]
109.5
2.52
0.91
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Fig. 2. GB structures obtained by SA technique at zero pressure and high pressures up to 60 GPa for the [001] and (b) [110] STGBs. Yellow and red balls represent Mg and O ions, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
evaluating ΔE GB(P ). The excess volume per GB area under pressure, ΔV GB(P ), are also evaluated by using the following equation:
ΔV GB(P ) = GB
V GB(P ) − nV BULK (P ) 2A(P )
At present it is technically difficult to compare our results with experimental results since TEM/STEM observations are usually performed at normal pressures and further GB segregation of unintentional impurities (e. g. Ca, Al, and Ti) is also believed to induce GB transformation [9–11]. Nevertheless, experimentally observed GBs [9– 11] are similar to the obtained metastable GBs with a dense structure at high pressures. This similarity may indicate that external factors, such as pressure and impurity segregation, induce a same kind of GB transformation at least for STGBs. Interestingly, the Σ65(810) GB exhibits various types of GB structures: at least four different stable and metastable GBs are predicted under high pressures (Fig. 2(a)). Moreover, additional two metastable GBs are predicted after pressure release as shown in Fig. 3. The two structures are clearly different from those in Fig. 2(a). The similar trend, that is, the formation of various metastable GBs depending on pressure and its release, is also observed for the Σ13(510) GB. The similarities between the two GBs are the characteristic open-core structure at zero pressure and small misorientation angles compared with the other STGBs. Therefore, GBs with the two features may potentially be transformed into various metastable GB structures. Fig. 4 shows ΔE GB(P ) as a function of pressure. These values are obtained based on GB structures after SA simulations and subsequent static energy minimizations. This result also indicates that the pressure dependence of the GBs is very different with each other in terms of
(2)
BULK
(P ) are the volume of the GB model and unit where V (P ) and V cell, respectively. It turned out that the PV term is negligibly small compared with the dominant internal energy term. 2.1. Results and discussion This section firstly shows GB structures obtained by SA techniques at high pressures for the individual GBs comparing their zeropressure structures, and discusses their energetics based on ΔE GB(P ) and ΔV GB(P ). Fig. 2 shows results of GB structures at each pressure for several STGBs. It is found that most of the STGBs studied exhibit transformation into metastable structures more than once except for the Σ3(111) STGB that has a significantly lower values of ΔE GB(P = 0) and ΔV GB(P = 0) than the other GBs, as listed in Table 1. For these metastable structures the open-core atomic arrangements collapse and more denser structures are formed. These results indicate that the existence of metastable GB structures for MgO is a major phenomenon. If this phenomenon also occurs for general GBs, it is expected that GB structures in polycrystalline MgO are transformed by applying high pressure. 3
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Fig. 3. Metastable GB structures after pressure release from an initial pressure of (a) 40 GPa and (b) 46 GPa for the Σ65(810)/[001] GB.
Fig. 5. Correlation between zero-pressure GB energy ΔE GB(P = 0) and high-pressure GB energy ΔE GB(P ) .
ΔE GB(P ) and its gradient. For example, the GBs with the open structure at 0 GPa, e.g. the Σ5(310), Σ5(210), and Σ3(112) GBs, show abrupt changes in ΔE GB(P ). In contrast, the Σ13(320) and Σ61(650) GBs, which have the dense structure, show almost a linear change of ΔE GB(P ) in all range of pressure. The trend suggests that the relative stability between GBs depends on pressure, that is, stable GBs at 0 GPa are not necessarily preferred at high pressures. Therefore, it is speculated that the distribution of GBs at 0 GPa in polycrystalline MgO is modified by applying a high pressure and consequently the population of individual GBs is changed depending on the pressure. The Σ3(111) GB is clearly distinct from the other GBs in the sense that the increase of ΔE GB(P ) remains within the range of 0– 0.13 J/m2 , indicating that the energy increase at a pressure is almost as small as in a single-crystal model. This result is not
surprising since this particular GB corresponds to a twin boundary having no deficiency in coordination number. Even under high pressures up to 60 GPa, ΔE GB(P ) of this GB is smaller than next lowest ΔE GB(P = 0). Consequently, the difference in ΔE GB(P ) between the Σ3(111) and the other GBs becomes more and more large as an applied pressure increases. This result probably implies that the relative stability of the Σ3(111) to other GBs is enhanced with the increase of pressure, and consequently the population of the Σ3(111) becomes much larger than other GBs. At present, this speculation is just based on our theoretical results of the internal energy, and thus further study is required. Fig. 5 shows the correlation between ΔE GB(P = 0) and ΔE GB(P ) at pressures between 10 GPa and 40 GPa. There is a positive correlation between two GB energies at 10 GPa (the grey points) while the
Fig. 4. GB energy as a function of pressure for (a) the [001] and (b) [110] STGBs.
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Fig. 6. Decrease of excess volume per GB area from 0 GPa as a function of pressure for (a) the [001] and (b) [110] STGBs.
from not only excess volume but also other dominant factors that are related to atomic details at GBs. Fig. 8 shows ΔE GB(P → 0) relative to ΔE GB(0) as function of initial pressure from which an applied pressure is released at 0 K. The description ΔE GB(P → 0) means that an initial pressure P is discretely changed to 0 GPa. Some of the GBs sustain a positive value of ΔE GB(P → 0), indicating that metastable structures are maintained even after the pressure release. An irreversible GB transformation was reported in a previous theoretical study [19]. As in Fig. 3, such metastable structures differ from those under high pressures, and thus high pressure and its release may result in metastable GB structures that are usually inaccessible with high pressure alone. Finally, a path of GB transformation is discussed based on results of SA simulations and static energy minimizations, as shown in Fig. 9. Results of the Σ5(210) GB are shown as an example since the GB shows a clear change in the GB energy with the increase of pressure. Both two approaches apply high pressures to the GB structure, but SA simulations are performed at finite temperatures (the black points) while static energy minimizations at 0 K (the grey points). The difference leads to a change in the threshold pressure of the formation of metastable/stable structures. For SA simulations, the metastable structures are formed at the lower pressures than static energy minimizations: the lattice energy discretely increases at 8 GPa and 46 GPa. The quadratic curves fitted to the black or grey data points show that the metastable structures obtained from SA simulations correspond to the most stable structures obtained from static energy minimizations. Presumably, GB structures obtained from SA simulations are energetically stable at high temperatures and high pressures, while the structures change from stable to metastable ones with the decrease of temperature. The metastable structures are maintained if there is an energy barrier for transformation between the stable and metastable structures. It is therefore expected that metastable GB structures are formed by high-pressure sintering, whose atomic arrangements differ from those by high-pressure compression at low temperature. In addition, such metastable GBs may be sustained after pressure release depending on GB characters.
correlation becomes weak with the increase of pressure. This reason is that some of the GBs transform into significantly different structures at higher pressures, which involves a sharp increase in ΔE GB. Fig. 6 shows the decrease of ΔV GB(P ) from 0 GPa as a function of pressure. The trend also depends on the individual GBs, although all the GBs show the decrease of ΔV GB(0) with the increase of pressure. The GB transformation is clearly detected as a sharp change in ΔV GB(0) in the cases of the GBs with the open structure, while the GBs with the dense structure show relatively a small volume change at the threshold pressure of the GB transformation. The correlation between the rate change of ΔE GB(P ) and ΔV GB(P ) is shown in Fig. 7. There is a clear correlation between the two quantities for one GB, but a comparison between GBs indicates a significant variation in the data points. The result suggests that a difference in relative stability between GBs result
Fig. 7. Correlation between the rate change of ΔE GB(P ) and ΔV GB(P ) for all the STGBs.
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Fig. 8. GB energy relative to ΔE GB(0) after pressure release to zero as a function of applied pressure for (a) the [001] and (b) [110] STGBs. After the pressure release, a static structural optimization is performed.
Acknowledgements This work was supported by Council for Science, Technology and Innovation (CSTI), Cross-ministerial Strategic Innovation Promotion Program (SIP), “Structural Materials for Innovation” (Funding agency: JST), and Grant-in-Aid for Scientific Research on Innovative Areas “Exploration of nanostructure-property relationships for materials innovation” from MEXT, Japan. References [1] T. Gemming, S. Nufer, W. Kurtz, M. Rühle, Structure and Chemistry of Symmetrical Tilt Grain Boundaries in α-Al2O3: II, Bicrystals with Y at the Interface, J. Am. Ceram. Soc. 86 (4) (2003) 590–594. [2] N. Shibata, F. Oba, T. Yamamoto, Y. Ikuhara, Structure, energy and solute segregation behaviour of [110] symmetric tilt grain boundaries in yttria-stabilized cubic zirconia, Philos. Mag. 84 (2004) 2381–2415. [3] B. Feng, T. Yokoi, A. Kumamoto, M. Yoshiya, Y. Ikuhara, N. Shibata, Atomically ordered solute segregation behaviour in an oxide grain boundary, Nat. Commun. 7 (2016) 11079. [4] T. Nakagawa, H. Nishimura, I. Sakaguchi, N. Shibata, K. Matsunaga, T. Yamamoto, Y. Ikuhara, Grain boundary character dependence of oxygen grain boundary diffusion in α-Al2O3, Scr. Mater. 65 (2011) 544–547. [5] F. Ye, C.Y. Yin, D.R. Ou, T. Mori, Relationship between lattice mismatch and ionic conduction of grain boundary in YSZ, Prog. Mat. Sci. Mater. Int. 24 (2013) 83–86. [6] K.L. Merkle, Atomic resolution electron microscopy of NiO grain boundaries, Ultramicroscopy 22 (1987) 57–70. [7] K.L. Merkle, High-resolution electron microscopy of interfaces in fcc materials, Ultramicroscopy 37 (1991) 130–152. [8] D.J. Harris, J.H. Harding, G.W. Watson, Computer simulation of the reactive element effect in NiO grain boundaries, Acta Mater. 48 (2000) 3039–3048. [9] Y. Yan, M.F. Chisholm, G. Duscher, A. Maiti, S.J. Pennycook, S.T. Pantelides, Impurity-induced structural transformation of a MgO grain boundary, Phys. Rev. Lett. 81 (1998) 3675–3678. [10] Z. Wang, M. Saito, K.P. McKenna, L. Gu, S. Tsukimoto, A.L. Shluger, Y. Ikuhara, Atom-resolved imaging of ordered defect superstructures at individual grain boundaries, Nature 479 (2011) 380–383. [11] M. Saito, Z. Wang, Ikuhara, Selective impurity segregation at a near-Σ5 grain boundary in MgO, J. Mater. Sci. 49 (2014) 3956–3961. [12] T. Höche, P.R. Kenway, H.-J. Kleebe, M.W. Finnis, M. Rühle, The structure of special grain boundaries in α-Al2O3, J. Phys. Chem. Solids 55 (1994) 1067–1082. [13] S. Fabris, C. Elsässer, Σ13(1014) twin in α-Al2O3: a model for a general grain boundary, Phys. Rev. B 64 (2001) 245117. [14] S. Azuma, N. Shibata, T. Mizoguchi, S.D. Findlay, K. Nakamura, Y. Ikuhara, Atomic structure, energetics, and chemical bonding of Y doped Σ13 grain boundaries in αAl2O3, Philos. Mag. 93 (2013) 1158–1171. [15] S. Ono, K. Funakoshi, A. Nozawa, T. Kikegawa, High-pressure phase transitions in SnO2, J. Appl. Phys. 97 (2005) 073523. [16] F.X. Zhang, J.W. Wang, J. Lian, M.K. Lang, U. Becker, R.C. Ewing, Phase Stability
Fig. 9. Increase of lattice energy per unit cell vs pressure for the Σ5(210)/[001] GB. The black points represents results of SA, while the grey points are obtained from geometrical optimizations at 0 K without SA. The red, blue and green curved lines are obtained by fitting quadratic curves to the data points, which corresponds to three different GB structures. The dotted vertical lines indicate the intersection points of these curved lines. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3. Conclusions This work studied stable and metastable structures under high pressures up to 60 GPa for various STGBs of MgO by using SA techniques with MD and lattice statics. Most of the GBs show transformation to a metastable structure, while the threshold pressure of transformation is very different from GB to GB. In particular, the GBs with the open structure at 0 GPa and a small misorientation angle exhibit various metastable structure under high pressures and its release. Therefore, it is expected that GB transformation under high pressure also occur in real polycrystalline MgO and specific GBs are dominantly transformed into different structures. 6
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