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Coherency dislocations in grain boundaries
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Coherency dislocations in grain boundaries J.B. Yang , K.Q. Li , J.X. Yan , L.L. Li , Z.J. Zhang , Z.G. Yang , Z.F. Zhang PII: DOI: Reference:
S2589-1529(19)30278-9 https://doi.org/10.1016/j.mtla.2019.100482 MTLA 100482
To appear in:
Materialia
Received date: Accepted date:
14 September 2019 18 September 2019
Please cite this article as: J.B. Yang , K.Q. Li , J.X. Yan , L.L. Li , Z.J. Zhang , Z.G. Yang , Z.F. Zhang , Coherency dislocations in grain boundaries, Materialia (2019), doi: https://doi.org/10.1016/j.mtla.2019.100482
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd on behalf of Acta Materialia Inc.
Coherency dislocations in grain boundaries J.B. Yang1,2,*, K.Q. Li1,2, J.X. Yan1,2, L.L. Li1,2, Z.J. Zhang1,2, Z.G. Yang3, Z.F. Zhang1,2,† 1
Laboratory of Fatigue and Fracture for Materials, Institute of Metal Research, Chinese Academy of Sciences,
72 Wenhua Road, Shenyang 110016, China 2
School of Materials Science and Engineering, University of Science and Technology of China, Hefei 230026,
China 3
Key Laboratory of Advanced Materials of Ministry of Education, School of Materials Science and
Engineering, Tsinghua University, Beijing 100084, China
Abstract Although coherency dislocation has been proposed for several decades, it is not yet accepted or even well known in the previous studies about grain boundaries. Here direct atomistic simulations are specially designed to prove the existence of coherency dislocations in grain boundaries. Our data show that coherency dislocations possess their own displacement and stress fields as conventional dislocations. In contrast to the prevailing model of grain boundaries with anti-coherency dislocations only, we support that coherency dislocations must be considered together with anti-coherency dislocations to provide a correct description of boundary structures and related phenomena. Keywords: Dislocations; Grain boundary defects; Interface structure; Atomistic simulation; Analytical methods * [email protected] †
[email protected] 1
Discrete dislocations exist widely in grain boundaries [1-4]. Each of them possesses the elementary properties of conventional dislocations, and influences our quantitative understanding of a number of properties of grain boundaries and phenomena in polycrystalline materials [5-15]. Correct knowledge about their characteristics and properties is therefore crucial for designing materials properties by using grain boundaries. From a theoretical perspective, Olson and Cohen proposed that two distinct types of discrete dislocations are needed, namely, coherency and anti-coherency dislocations with Burgers vectors of opposite sign, to describe the motion and the elastic fields of general boundaries in crystalline materials [16]. Interphase boundaries are frequently described with both coherency and anticoherency dislocations [2], especially in martensitic nucleation and growth [1, 17-19]. In contrast, grain boundaries have been long modeled with anti-coherency dislocations only [1-3]. Taking as example the earliest classical tilt wall model proposed 85 years ago [20], a small-angle symmetrical tilt grain boundary (SASTGB) was thought to be a set of parallel edge dislocations, all of which belong to anti-coherency dislocations in the Olson-Cohen nomenclature. This tilt wall model with anti-coherency dislocations only is the prototype of other grain boundaries with the most widespread application [1-3]. The existence of anti-coherency dislocations in various types of grain boundaries has been confirmed by overwhelming experimental and atomistic simulation evidence [1-4, 8, 9, 21-26]. However, to our knowledge, there is no such evidence to support or refute the concept of coherency dislocation. In other words, it still remains a great unknown whether coherency dislocations exist in grain boundaries. In this work, we provide the first supporting evidence for the existence of coherency dislocations in grain boundaries by atomistic simulation in which the displacement and stress fields of coherency dislocations are confirmed in a quantitative manner. In particular, no 2
presumption is made to determine these two fields in simulations. This study has general significance in understanding boundary properties and related subjects because it decidedly shows that coherency dislocations take the same important role as anti-coherency dislocations in grain boundaries. The coherency/anti-coherency dislocation approach is first of all applied to grain boundaries. Suppose that two grains α and β meet along a planar grain boundary with a normal vector n. The Burgers vector content Ba of anti-coherency dislocations across a probe vector p in the boundary n is specified as follows:
Ba Tp .
(1)
Here T (S -1α Sβ-1 ) , where Sα and Sβ, respectively, represent the lattice transformation matrices that distort a reference lattice r into the grains α and β. Eq. (1) is known as the Frank-Bilby equation (FBE) [1-3, 27, 28]. In this work, the superscripts „-1‟ and „T‟ denote the inverse and transpose of a matrix. The Burgers vector content Bc of coherency dislocations is given accordingly: Bc = -Ba .
(2)
The vector Bc is reminiscent of the displacement by coherency in elastic continuum [16]. As is generally admitted [1, 2, 16, 17], coherency dislocations maintain a specific registry between two lattices, while anti-coherency dislocations disrupt locally the lattice registry and separate coherent regions. In order to restrict our attention to coherency dislocations, here we choose small-angle grain boundaries in which it is well established that Sα and Sβ are well-defined misorientation rotation matrices, the reference lattice r is the median lattice and anti-coherency dislocations are easily 3
characterized and visualized [4, 8]. Meanwhile, considering the generality of the FBE [1-4], we need only one representative boundary to demonstrate the existence of coherency dislocation. A SASTGB is undoubtedly an ideal test case because its simple structure facilitates the present quantitative study of coherency dislocation. Nonetheless, the characterization of individual coherency dislocations still remains a challenge because of their extremely small Burgers vector and separation as shown below. A horizontal symmetrical tilt grain boundary is produced in body-centered-cubic Fe as previously described [29, 30]. The interaction of atoms is described with an empirical manybody potential of Fe [31], which has been extensively used to simulate various types of dislocations in a single crystal and grain boundaries [8, 32-34]. The lattice parameter a=2.855Å at the temperature T=0 K. An Fe single crystal (which is selected as the reference lattice r) is created in the box with x|| 1 1 2 , y|| 1 10 and z||[111], where x, y and z are unit vectors. Its top and bottom halves „α‟ and „β‟ are rotated around x at the box centre „O‟ by /2 counterclockwise and clockwise, respectively. The misorientation angle is generated by an integer m by
θ 2atan 6 4 m . As a result, both component crystals have 1 1 2 along x and <-m+0.5
m+0.5 0.5> along y so that the periodic conditions can be applied along x and y to achieve an infinite boundary plane. The six top surface layers are free along z, while the six bottom layers are fixed unless otherwise noted. In this work, m=45 and =1.559º. The box dimensions are Lx=7.348a, Ly=381.873a and Lz=300.0a. The large Ly and Lz are set to avoid the mid-boundary dislocations moving out of the box in association with boundary motion and to exclude the influence from the free and fixed surfaces. After full relaxation of the system, twelve anti-coherency dislocations might be most easily visualized with high-energy atoms in cores, two of which are presented in Fig. 1 (a). They are 4
equally spaced with a Burgers vector [111]/2 (by the Burgers circuit mapping). Now the six top layers are fixed and displaced rigidly as a whole along y with a displacement s and a shear strain increment 2.5×10-4. Then, the SASTGB starts moving upward (its migration distance h) when the shear stress increases linearly with the applied strain to about 80 MPa, which is almost equal to the Peierls stress of an edge dislocation [111]/2 in Fe. Finally, the boundary moves smoothly in glissile fashion and the shear stress oscillates slightly around 80 MPa. The coupling factor s/h measured from the slope in Fig. 1 (b) is 2.7×10-2 and it is in agreement with the relation s/h=2tan/2 for symmetrical tilt grain boundaries in both simulations and experiments [29, 30, 35]. At finite T, the same structure and motion of the SASTGB are observed in simulations. Due to the lattice discreteness Eqs. (1) and (2) have to be quantized to predict the structure of the above simulated SASTGB in the x-y-z coordinate system according to the previous procedure [17, 30]. Their predictions show that the SASTGB possesses a set of anti-coherency dislocations with the Burger vector, line direction and separation: ba=[0 0 ba]T, ξ=[-1 0 0]T and da=31.820a. And it also has a single set of coherency dislocations with a Burgers vector bc=[0 0 -εba]T, a line direction ξ=[-1 0 0] T and a separation dc=0.707a. Here ba=0.866a and ε=1/45. The calculated dislocations are sketched in Fig. 1 (a): the anti-coherency ones are consistent with those characterized in simulation, and the coherency ones are uniformly distributed between 1 1 0 planes. Those dense coherency dislocations could be never detected straightforwardly as anticoherency dislocations even when each atom position is known. The problem then arises of how to demonstrate the existence of coherency dislocations in simulation. The answer to this requires a correct knowledge of the displacement and stress fields generated from dislocations in the simulated SASTGB, which could be deduced from the above predicted dislocation structure. The SASTGB normal vector n=[0 0 1]T in Fig. 1 (a), and the 5
probe vector p=(x, y, 0)T in Eqs. (1) and (2), where x and y are two arbitrary numbers. The displacement across p due to the predicted anti-coherency dislocations is [10, 30, 36]:
ua n ξ da p ba =[0 0 round(y/da)ba ]T , T
(3)
where the rounding function, round(y/da), denotes the number of anti-coherency dislocations across p. The displacement across p arising from the calculated coherency dislocations is
uc n ξ dc p bc =[0 0 -round(y/dc)εba]T . T
(4)
The resultant displacement in the SASTGB n of all the dislocations above can be written as: ut = ua + uc .
(5)
It is obvious that the variations of ua, uc and ut can be represented by their non-zero zcomponents uz changing with y, as presented in Figs. 2 (a) to (c). If the SASTGB n possesses coherency dislocations, when it sweeps through a cluster of atoms along y parallel to the boundary plane, the displacement of these atoms should vary as the sawtooth-shaped ut, rather than the staircase ua with anti-coherency dislocations only. Suppose that a number of parallel edge dislocations are distributed along y with a Burgers vector b (its length b) along z, a line direction along -x, and a separation d. Their stress field can be obtained by adding up the stresses of individual dislocations [1-3, 13]. Its three non-zero stress components yy, zz and yz have the similar spatial distribution and thus the shear component yz is selected as an example:
σ yz b, d , N 2σ 0
z y y N
i N 1
z z 2 y yi
2
2 2
2
.
(6)
i
6
Here the total dislocation number is 2(N+1), the dislocation position yi=(i+0.5)d, the parameter
σ 0 μb 4π1 ν , the shear modulus =89GPa and the Poisson ratio =0.285. When r z 2 y yi
Now let us apply Eq. (6) to the predicted coherency dislocations. The shear stress of an infinite coherency dislocation array (N is infinitely large) can be expressed by σ yz εba , d c , , which decreases exponentially to zero with increasing |z| as |z|>dc/2 [3]. Owing to the extremely small separation dc, the shear stress of coherency dislocations cannot be detected out of the boundary plane, as revealed in our atomistic simulations. A different situation exists for a finite coherency dislocation array. When the SASTGB n is designed to be a boundary segment with four anti-coherency dislocations (Ly/3), it has 180 coherency dislocations as predicted. The shear stresses of coherency and anti-coherency dislocations σ yz εba , dc , 2 ε 1 and σ yz ba , d a ,1 at the middle of the SASTGB segment are plotted in Figs. 3 (a) and (b). In the top and bottom grains, coherency dislocations have quite high stress regions (|yz|>440MPa) when |z|>da, opposite in sign to those of anti-coherency dislocations. If this SASTGB segment has no coherency dislocations, its yz distribution would be identical to Fig. 3 (b) generated in terms of anti-coherency dislocations only. According to the above analysis, two special simulations of the SASTGB n are performed to verify the predicted displacement and stress fields of coherency dislocations. During one simulation about the motion of the infinite SASTGB, an array of marker atoms parallel to y (|y|<2da) is selected in advance at a distance of da to the boundary plane in the top grain. After these atoms are swept by the SASTGB n their displacements coincide with ut, as plotted in Fig. 2 (c). This indicates that the displacement of coherency dislocations uc exist together with ua of 7
anti-coherency dislocations in the SASTGB n. It should be noted that the displacements uc and ua in Eqs. (3) and (4) are derived on the basis of the lattice registry defined in Eqs. (1) and (2), whereas the displacement field in simulation is obtained with no aid from these formula. In another simulation, the periodic boundary conditions along y are canceled, and additionally, two slots along the boundary are created by removing the atoms in the shaded areas in Fig. 4 (a). The slots and the free surfaces have no stress effect in the region |y|
dislocations could vanish around boundary planes as we analyzed above. Consequently, although coherency dislocations were not known/accepted, some boundary properties could be correctly predicted, e.g. the classical Read-Shockley equation about boundary energy [3, 14]. In contrast, this prevailing model could probably give wrong interpretation on grain growth and rotation in nanograined materials [7, 12, 21-23], because coherency dislocations and their effects cannot be ignored in reduced grains. Furthermore, coherency rather than anti-coherency dislocations are objects capable of generating grain growth and rotation (or more generally, homogeneous lattice transformation) [16]. Coherency dislocations are distinct from secondary dislocations [39] although both of them could be partial dislocations in grain crystal lattices. Secondary dislocations belong to anticoherency dislocations as primary dislocations, separated by coherent regions, different from coherency dislocations uniformly distributed in boundaries, as sketched in Figs. 2.5 and 2.9 in Ref. [2]. Without considering dislocation dissociation, in the selected displacement-shiftcomplete (DSC) reference lattice, secondary dislocations are usually “perfect” anti-coherency dislocations (since their Burgers vectors are the translation vectors of the DSC lattice), while coherency dislocations are commonly “partial” dislocations. In addition, coherency dislocations are different from disconnections of SASTGBs [40, 41] because their Burgers vectors are perpendicular to each other. Besides, The SASTGB in Fig. 1 (b) in Ref. [40] has only one set of boundary dislocations, with a Burgers vector almost identical to that of anti-coherency dislocations in Fig. 1 (a). Here is given a theoretical consideration of the energy contribution E to the simulation system due to the boundary segment in Fig. 4 (a) (as a planar defect in the bi-crystal) to prove the existence necessity of coherency dislocations in grain boundaries. It is assumed that the 9
boundary segment in Fig. 4 (a) contains a set of anti-coherency dislocations ba with a predetermined number na (na=4 in the above simulation). On the one hand, E can be estimated as E=nadaLx, independent to Ly and Lz. Here is the energy of the boundary per unit area, possessing a finite value (here ≈0.015J/m2) due to the thermodynamic limit [1-3]. On the other hand, if the boundary segment in Fig. 4 (a) is thought to be composed of na anti-coherency dislocations, without coherency dislocations, E can be regarded as the elastic energy arising from these anti-coherency dislocations which can be approximated as a whole to be a superdislocation with a Burgers vector naba. In this case, it is apparent that E increases with Ly and/or Lz, because of the long-range stresses of these anti-coherency dislocations as shown in Fig. 3 (b). Thus, E will easily become beyond nadaLx, and thus it will break through the thermodynamic limit in the light of the energy of a straight dislocation naba per unit length along x (here we simply neglect the energy loss from the slots in Fig. 4 (a) because of their small volume fraction). However, if these anti-coherency dislocations are canceled out by coherency dislocations, as seen from the elastic field of the neutralized boundary segment in Fig. 4 (c), due to the rapid decay of stresses with the distance to the boundary plane, E is restricted within several atomic layers near the boundary (the cohesive energy change of atoms in simulation), in accord with the thermodynamic limit. And most importantly, during the above derivation, the boundary segment can be replaced by a grain (phase) boundary of arbitrary type provided that a finite number (na) of anti-coherency dislocations are predetermined to be parallel to x in Fig. 4 (a). All the experimentally observed SASTGBs are finite boundary segments. Their elastic distortion fields have been measured and analyzed at atomic scale [21, 25, 42, 43], and they disappear rapidly away from the boundary plane, in agreement with that in Fig. 4 (c). Without coherency dislocations, by comparison of Fig. 4 (c) with the above mentioned high stress regions 10
when |z|>da in Fig. 2 (b), it is obvious that the local stresses will become larger by several orders of magnitude. In future experiments, the quantitative effects of free surfaces [44-46] and boundary size [21] on boundary elastic fields need to be thoroughly analyzed. An alternative theory suggests that the continuous displacement Bc in Eq. (2) can be directly used, instead of coherency dislocations and their displacement field uc in Eq. (4) with fine stairs (which have a riser height εba and a terrace length dc) [6, 47]. As was previously discussed [16], such continuum representation is a convenient approximation, but coherency (and anti-coherency) dislocations reflect the discreteness of the true boundary structure since the crystal structure is discrete. The coherency/anti-coherency dislocation approach, in which coherency dislocations can advantageously move by individuals, has made considerable strides in the description of the motion mechanisms of general boundaries and the associated deformations and energetics [1719]. In summary, coherency dislocations are proved to exist in grain boundaries, with the same displacement and stress fields as conventional dislocations. They must be considered when modeling and predicting grain boundary properties with dislocations. It is worth re-investigating grain boundary properties which were modeled and evaluated by only using anti-coherency dislocations in previous studies. And the combination effect of coherency and anti-coherency dislocations warrants further exploration.
This work was supported by the Program of “One Hundred Talented People” of the Chinese Academy of Sciences (JBY) and the National Natural Science Foundation of China (NSFC) under Grant Nos. 51571198, 51771206, 51790482 and 51871223.
11
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[21] C.Y. Wang, K. Du, K.P. Song, X.L. Ye, L. Qi, S.Y. He, D.M. Tang, N. Lu, H.J. Jin, F. Li, and H.Q. Ye, Phys. Rev. Lett. 120, 186102 (2018). [22] L.H. Wang, J. Teng, P. Liu, A. Hirata, E. Ma, Z. Zhang, M.W. Chen and X.D. Han, Nature Comm. 5, 4402 (2014). [23] X.P. Zhang, J. Han, J.J. Plombon, A.P. Sutton, D.J. Srolovitz, and J.J. Boland, Science 357, 397 (2017). [24] N.N. Kumar, E. Martinez, B.K. Dutta, G.K. Dey, and A. Caro, Phys. Rev. B 87, 054106 (2013). [25] Y. Nohara, E. Tochigi, N. Shibata, T. Yamamoto, and Y. Ikuhara, J. Electron Microscopy 59, S117 (2010). [26] R.A. Marks, S.T. Taylor, E. Mammana, R. Gronsky, and A.M. Glaeser, Nature Mater. 3, 682(2004). [27] J.W. Christian, Metall. Trans. A 13A, 509 (1982). [28] W.T. Read, Dislocations in crystals, (McGraw-Hill, New York, 1953). [29] J.W. Cahn, Y. Mishin, and A. Suzuki, Acta Mater. 54, 4953 (2006). [30] J.B. Yang, Z.G. Yang, Y. Nagai, and M. Hasegawa, Philos. Mag. Lett. 89, 605 (2009). [31] G.J. Ackland, M.I. Mendelev, D.J. Srolovitz, and A.V. Barashev, J. Phys. 16, S2629 (2004). [32] Y. Fan, Y.N. Osetsky, S. Yip, and B. Yildiz, Phys. Rev. Lett. 109, 135503 (2012) [33] D. Terentyev, P. Grammatikopoulos, D.J. Bacon, Y.N. Osetsky, Acta Mater. 56, 5034(2008) [34] J. Chaussidon, M. Fivel, and D. Rodney, Acta Mater. 54, 3407 (2006). [35] D.A. Molodov, V.A. Ivanov, and G. Gottstein, Acta Mater. 55, 1843 (2007). [36] K.M. Knowles, Philos. Mag. A 46, 951 (1982).
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[37] The influence of free surfaces and slots is estimated quantitatively by simulation of a single Fe crystal with the same box dimensions and boundary conditions. [38] K.S. Cheung and S. Yip, J. Appl. Phys. 70, 5688 (1991). [39] W. Bollmann, Crystal lattices, interfaces, matrices. Geneva: W. Bollmann; 1982. [40] J.P. Hirth, R.C. Pond, and J. Lothe. Acta Mater. 54, 4237 (2006) [41]J.P. Hirth, R.C. Pond, R.G. Hoagland, X.Y. Liu, and J. Wang, Prog. Mater. Sci. 58, 749(2013). [42] R. Bonnet, M. Couillard, S. Dhouibi, and S. Neily, Phys. Status Solidi B 253, 1545 (2016). [43] C.L. Johnson, M.J. Hÿtch, and P.R. Buseck, PNAS 101, 17936 (2004). [44] B.G. Mendis, Y. Mishin, C.S. Hartley, and K.J. Hemker, Philos. Mag. 86, 4607(2006). [45] R. Bonnet, Philos. Mag. A 73, 1193(1996). [46] P. Humble, Philos. Mag. A 51, 355(1985). [47] R. Bonnet, Philos. Mag. A 43, 1165 (1981).
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FIG. 1. (a) The SASTGB near (111) simulated in Fe. The black atoms possess a higher energy than gray ones. The FBE predicted coherency and anti-coherency dislocations are superimposed and indicated by small and large edge dislocation arrays, respectively. Only one 1 1 2 layer (parallel to the paper plane) is presented here so that half of 1 10 planes are lost. (b) The variation of the migration distance h with the sliding distance s of the SASTGB during motion. The circles are measured from the simulation results. See text for parameters.
15
16
FIG. 2. The displacement fields of (a) the anti-coherency and (b) coherency dislocations predicted from the FBE. Their superposition in (c) is in agreement with the simulation results (circles). See text for parameters.
17
FIG. 3. The shear stress fields (MPa) of (a) the coherency and (b) anti-coherency dislocations predicted from the FBE for the boundary segment with a length Ly/3 along y.
18
FIG. 4. (a) Schematic of the boundary segment with a length Ly/3 along y. Four anti-coherency dislocations are denoted by Ei, where the dislocation number i = -2 to 1, as defined in Eq. (6). The shaded areas are filled with no atoms to create two slots. The shear stress field (MPa) of the boundary segment (b) in simulation and (c) by the sum of shear stresses from the coherency and anti-coherency dislocations predicted from the FBE. See text for parameters. 19
Coherency dislocations in grain boundaries J.B. Yang , K.Q. Li , J.X. Yan , L.L. Li , Z.J. Zhang , Z.G. Yang , Z.F. Zhang PII: DOI: Reference:
S2589-1529(19)30278-9 https://doi.org/10.1016/j.mtla.2019.100482 MTLA 100482
To appear in:
Materialia
Received date: Accepted date:
14 September 2019 18 September 2019
Please cite this article as: J.B. Yang , K.Q. Li , J.X. Yan , L.L. Li , Z.J. Zhang , Z.G. Yang , Z.F. Zhang , Coherency dislocations in grain boundaries, Materialia (2019), doi: https://doi.org/10.1016/j.mtla.2019.100482
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd on behalf of Acta Materialia Inc.
Coherency dislocations in grain boundaries J.B. Yang1,2,*, K.Q. Li1,2, J.X. Yan1,2, L.L. Li1,2, Z.J. Zhang1,2, Z.G. Yang3, Z.F. Zhang1,2,† 1
Laboratory of Fatigue and Fracture for Materials, Institute of Metal Research, Chinese Academy of Sciences,
72 Wenhua Road, Shenyang 110016, China 2
School of Materials Science and Engineering, University of Science and Technology of China, Hefei 230026,
China 3
Key Laboratory of Advanced Materials of Ministry of Education, School of Materials Science and
Engineering, Tsinghua University, Beijing 100084, China
Abstract Although coherency dislocation has been proposed for several decades, it is not yet accepted or even well known in the previous studies about grain boundaries. Here direct atomistic simulations are specially designed to prove the existence of coherency dislocations in grain boundaries. Our data show that coherency dislocations possess their own displacement and stress fields as conventional dislocations. In contrast to the prevailing model of grain boundaries with anti-coherency dislocations only, we support that coherency dislocations must be considered together with anti-coherency dislocations to provide a correct description of boundary structures and related phenomena. Keywords: Dislocations; Grain boundary defects; Interface structure; Atomistic simulation; Analytical methods * [email protected] †
[email protected] 1
Discrete dislocations exist widely in grain boundaries [1-4]. Each of them possesses the elementary properties of conventional dislocations, and influences our quantitative understanding of a number of properties of grain boundaries and phenomena in polycrystalline materials [5-15]. Correct knowledge about their characteristics and properties is therefore crucial for designing materials properties by using grain boundaries. From a theoretical perspective, Olson and Cohen proposed that two distinct types of discrete dislocations are needed, namely, coherency and anti-coherency dislocations with Burgers vectors of opposite sign, to describe the motion and the elastic fields of general boundaries in crystalline materials [16]. Interphase boundaries are frequently described with both coherency and anticoherency dislocations [2], especially in martensitic nucleation and growth [1, 17-19]. In contrast, grain boundaries have been long modeled with anti-coherency dislocations only [1-3]. Taking as example the earliest classical tilt wall model proposed 85 years ago [20], a small-angle symmetrical tilt grain boundary (SASTGB) was thought to be a set of parallel edge dislocations, all of which belong to anti-coherency dislocations in the Olson-Cohen nomenclature. This tilt wall model with anti-coherency dislocations only is the prototype of other grain boundaries with the most widespread application [1-3]. The existence of anti-coherency dislocations in various types of grain boundaries has been confirmed by overwhelming experimental and atomistic simulation evidence [1-4, 8, 9, 21-26]. However, to our knowledge, there is no such evidence to support or refute the concept of coherency dislocation. In other words, it still remains a great unknown whether coherency dislocations exist in grain boundaries. In this work, we provide the first supporting evidence for the existence of coherency dislocations in grain boundaries by atomistic simulation in which the displacement and stress fields of coherency dislocations are confirmed in a quantitative manner. In particular, no 2
presumption is made to determine these two fields in simulations. This study has general significance in understanding boundary properties and related subjects because it decidedly shows that coherency dislocations take the same important role as anti-coherency dislocations in grain boundaries. The coherency/anti-coherency dislocation approach is first of all applied to grain boundaries. Suppose that two grains α and β meet along a planar grain boundary with a normal vector n. The Burgers vector content Ba of anti-coherency dislocations across a probe vector p in the boundary n is specified as follows:
Ba Tp .
(1)
Here T (S -1α Sβ-1 ) , where Sα and Sβ, respectively, represent the lattice transformation matrices that distort a reference lattice r into the grains α and β. Eq. (1) is known as the Frank-Bilby equation (FBE) [1-3, 27, 28]. In this work, the superscripts „-1‟ and „T‟ denote the inverse and transpose of a matrix. The Burgers vector content Bc of coherency dislocations is given accordingly: Bc = -Ba .
(2)
The vector Bc is reminiscent of the displacement by coherency in elastic continuum [16]. As is generally admitted [1, 2, 16, 17], coherency dislocations maintain a specific registry between two lattices, while anti-coherency dislocations disrupt locally the lattice registry and separate coherent regions. In order to restrict our attention to coherency dislocations, here we choose small-angle grain boundaries in which it is well established that Sα and Sβ are well-defined misorientation rotation matrices, the reference lattice r is the median lattice and anti-coherency dislocations are easily 3
characterized and visualized [4, 8]. Meanwhile, considering the generality of the FBE [1-4], we need only one representative boundary to demonstrate the existence of coherency dislocation. A SASTGB is undoubtedly an ideal test case because its simple structure facilitates the present quantitative study of coherency dislocation. Nonetheless, the characterization of individual coherency dislocations still remains a challenge because of their extremely small Burgers vector and separation as shown below. A horizontal symmetrical tilt grain boundary is produced in body-centered-cubic Fe as previously described [29, 30]. The interaction of atoms is described with an empirical manybody potential of Fe [31], which has been extensively used to simulate various types of dislocations in a single crystal and grain boundaries [8, 32-34]. The lattice parameter a=2.855Å at the temperature T=0 K. An Fe single crystal (which is selected as the reference lattice r) is created in the box with x|| 1 1 2 , y|| 1 10 and z||[111], where x, y and z are unit vectors. Its top and bottom halves „α‟ and „β‟ are rotated around x at the box centre „O‟ by /2 counterclockwise and clockwise, respectively. The misorientation angle is generated by an integer m by
θ 2atan 6 4 m . As a result, both component crystals have 1 1 2 along x and <-m+0.5
m+0.5 0.5> along y so that the periodic conditions can be applied along x and y to achieve an infinite boundary plane. The six top surface layers are free along z, while the six bottom layers are fixed unless otherwise noted. In this work, m=45 and =1.559º. The box dimensions are Lx=7.348a, Ly=381.873a and Lz=300.0a. The large Ly and Lz are set to avoid the mid-boundary dislocations moving out of the box in association with boundary motion and to exclude the influence from the free and fixed surfaces. After full relaxation of the system, twelve anti-coherency dislocations might be most easily visualized with high-energy atoms in cores, two of which are presented in Fig. 1 (a). They are 4
equally spaced with a Burgers vector [111]/2 (by the Burgers circuit mapping). Now the six top layers are fixed and displaced rigidly as a whole along y with a displacement s and a shear strain increment 2.5×10-4. Then, the SASTGB starts moving upward (its migration distance h) when the shear stress increases linearly with the applied strain to about 80 MPa, which is almost equal to the Peierls stress of an edge dislocation [111]/2 in Fe. Finally, the boundary moves smoothly in glissile fashion and the shear stress oscillates slightly around 80 MPa. The coupling factor s/h measured from the slope in Fig. 1 (b) is 2.7×10-2 and it is in agreement with the relation s/h=2tan/2 for symmetrical tilt grain boundaries in both simulations and experiments [29, 30, 35]. At finite T, the same structure and motion of the SASTGB are observed in simulations. Due to the lattice discreteness Eqs. (1) and (2) have to be quantized to predict the structure of the above simulated SASTGB in the x-y-z coordinate system according to the previous procedure [17, 30]. Their predictions show that the SASTGB possesses a set of anti-coherency dislocations with the Burger vector, line direction and separation: ba=[0 0 ba]T, ξ=[-1 0 0]T and da=31.820a. And it also has a single set of coherency dislocations with a Burgers vector bc=[0 0 -εba]T, a line direction ξ=[-1 0 0] T and a separation dc=0.707a. Here ba=0.866a and ε=1/45. The calculated dislocations are sketched in Fig. 1 (a): the anti-coherency ones are consistent with those characterized in simulation, and the coherency ones are uniformly distributed between 1 1 0 planes. Those dense coherency dislocations could be never detected straightforwardly as anticoherency dislocations even when each atom position is known. The problem then arises of how to demonstrate the existence of coherency dislocations in simulation. The answer to this requires a correct knowledge of the displacement and stress fields generated from dislocations in the simulated SASTGB, which could be deduced from the above predicted dislocation structure. The SASTGB normal vector n=[0 0 1]T in Fig. 1 (a), and the 5
probe vector p=(x, y, 0)T in Eqs. (1) and (2), where x and y are two arbitrary numbers. The displacement across p due to the predicted anti-coherency dislocations is [10, 30, 36]:
ua n ξ da p ba =[0 0 round(y/da)ba ]T , T
(3)
where the rounding function, round(y/da), denotes the number of anti-coherency dislocations across p. The displacement across p arising from the calculated coherency dislocations is
uc n ξ dc p bc =[0 0 -round(y/dc)εba]T . T
(4)
The resultant displacement in the SASTGB n of all the dislocations above can be written as: ut = ua + uc .
(5)
It is obvious that the variations of ua, uc and ut can be represented by their non-zero zcomponents uz changing with y, as presented in Figs. 2 (a) to (c). If the SASTGB n possesses coherency dislocations, when it sweeps through a cluster of atoms along y parallel to the boundary plane, the displacement of these atoms should vary as the sawtooth-shaped ut, rather than the staircase ua with anti-coherency dislocations only. Suppose that a number of parallel edge dislocations are distributed along y with a Burgers vector b (its length b) along z, a line direction along -x, and a separation d. Their stress field can be obtained by adding up the stresses of individual dislocations [1-3, 13]. Its three non-zero stress components yy, zz and yz have the similar spatial distribution and thus the shear component yz is selected as an example:
σ yz b, d , N 2σ 0
z y y N
i N 1
z z 2 y yi
2
2 2
2
.
(6)
i
6
Here the total dislocation number is 2(N+1), the dislocation position yi=(i+0.5)d, the parameter
σ 0 μb 4π1 ν , the shear modulus =89GPa and the Poisson ratio =0.285. When r z 2 y yi
Now let us apply Eq. (6) to the predicted coherency dislocations. The shear stress of an infinite coherency dislocation array (N is infinitely large) can be expressed by σ yz εba , d c , , which decreases exponentially to zero with increasing |z| as |z|>dc/2 [3]. Owing to the extremely small separation dc, the shear stress of coherency dislocations cannot be detected out of the boundary plane, as revealed in our atomistic simulations. A different situation exists for a finite coherency dislocation array. When the SASTGB n is designed to be a boundary segment with four anti-coherency dislocations (Ly/3), it has 180 coherency dislocations as predicted. The shear stresses of coherency and anti-coherency dislocations σ yz εba , dc , 2 ε 1 and σ yz ba , d a ,1 at the middle of the SASTGB segment are plotted in Figs. 3 (a) and (b). In the top and bottom grains, coherency dislocations have quite high stress regions (|yz|>440MPa) when |z|>da, opposite in sign to those of anti-coherency dislocations. If this SASTGB segment has no coherency dislocations, its yz distribution would be identical to Fig. 3 (b) generated in terms of anti-coherency dislocations only. According to the above analysis, two special simulations of the SASTGB n are performed to verify the predicted displacement and stress fields of coherency dislocations. During one simulation about the motion of the infinite SASTGB, an array of marker atoms parallel to y (|y|<2da) is selected in advance at a distance of da to the boundary plane in the top grain. After these atoms are swept by the SASTGB n their displacements coincide with ut, as plotted in Fig. 2 (c). This indicates that the displacement of coherency dislocations uc exist together with ua of 7
anti-coherency dislocations in the SASTGB n. It should be noted that the displacements uc and ua in Eqs. (3) and (4) are derived on the basis of the lattice registry defined in Eqs. (1) and (2), whereas the displacement field in simulation is obtained with no aid from these formula. In another simulation, the periodic boundary conditions along y are canceled, and additionally, two slots along the boundary are created by removing the atoms in the shaded areas in Fig. 4 (a). The slots and the free surfaces have no stress effect in the region |y|
dislocations could vanish around boundary planes as we analyzed above. Consequently, although coherency dislocations were not known/accepted, some boundary properties could be correctly predicted, e.g. the classical Read-Shockley equation about boundary energy [3, 14]. In contrast, this prevailing model could probably give wrong interpretation on grain growth and rotation in nanograined materials [7, 12, 21-23], because coherency dislocations and their effects cannot be ignored in reduced grains. Furthermore, coherency rather than anti-coherency dislocations are objects capable of generating grain growth and rotation (or more generally, homogeneous lattice transformation) [16]. Coherency dislocations are distinct from secondary dislocations [39] although both of them could be partial dislocations in grain crystal lattices. Secondary dislocations belong to anticoherency dislocations as primary dislocations, separated by coherent regions, different from coherency dislocations uniformly distributed in boundaries, as sketched in Figs. 2.5 and 2.9 in Ref. [2]. Without considering dislocation dissociation, in the selected displacement-shiftcomplete (DSC) reference lattice, secondary dislocations are usually “perfect” anti-coherency dislocations (since their Burgers vectors are the translation vectors of the DSC lattice), while coherency dislocations are commonly “partial” dislocations. In addition, coherency dislocations are different from disconnections of SASTGBs [40, 41] because their Burgers vectors are perpendicular to each other. Besides, The SASTGB in Fig. 1 (b) in Ref. [40] has only one set of boundary dislocations, with a Burgers vector almost identical to that of anti-coherency dislocations in Fig. 1 (a). Here is given a theoretical consideration of the energy contribution E to the simulation system due to the boundary segment in Fig. 4 (a) (as a planar defect in the bi-crystal) to prove the existence necessity of coherency dislocations in grain boundaries. It is assumed that the 9
boundary segment in Fig. 4 (a) contains a set of anti-coherency dislocations ba with a predetermined number na (na=4 in the above simulation). On the one hand, E can be estimated as E=nadaLx, independent to Ly and Lz. Here is the energy of the boundary per unit area, possessing a finite value (here ≈0.015J/m2) due to the thermodynamic limit [1-3]. On the other hand, if the boundary segment in Fig. 4 (a) is thought to be composed of na anti-coherency dislocations, without coherency dislocations, E can be regarded as the elastic energy arising from these anti-coherency dislocations which can be approximated as a whole to be a superdislocation with a Burgers vector naba. In this case, it is apparent that E increases with Ly and/or Lz, because of the long-range stresses of these anti-coherency dislocations as shown in Fig. 3 (b). Thus, E will easily become beyond nadaLx, and thus it will break through the thermodynamic limit in the light of the energy of a straight dislocation naba per unit length along x (here we simply neglect the energy loss from the slots in Fig. 4 (a) because of their small volume fraction). However, if these anti-coherency dislocations are canceled out by coherency dislocations, as seen from the elastic field of the neutralized boundary segment in Fig. 4 (c), due to the rapid decay of stresses with the distance to the boundary plane, E is restricted within several atomic layers near the boundary (the cohesive energy change of atoms in simulation), in accord with the thermodynamic limit. And most importantly, during the above derivation, the boundary segment can be replaced by a grain (phase) boundary of arbitrary type provided that a finite number (na) of anti-coherency dislocations are predetermined to be parallel to x in Fig. 4 (a). All the experimentally observed SASTGBs are finite boundary segments. Their elastic distortion fields have been measured and analyzed at atomic scale [21, 25, 42, 43], and they disappear rapidly away from the boundary plane, in agreement with that in Fig. 4 (c). Without coherency dislocations, by comparison of Fig. 4 (c) with the above mentioned high stress regions 10
when |z|>da in Fig. 2 (b), it is obvious that the local stresses will become larger by several orders of magnitude. In future experiments, the quantitative effects of free surfaces [44-46] and boundary size [21] on boundary elastic fields need to be thoroughly analyzed. An alternative theory suggests that the continuous displacement Bc in Eq. (2) can be directly used, instead of coherency dislocations and their displacement field uc in Eq. (4) with fine stairs (which have a riser height εba and a terrace length dc) [6, 47]. As was previously discussed [16], such continuum representation is a convenient approximation, but coherency (and anti-coherency) dislocations reflect the discreteness of the true boundary structure since the crystal structure is discrete. The coherency/anti-coherency dislocation approach, in which coherency dislocations can advantageously move by individuals, has made considerable strides in the description of the motion mechanisms of general boundaries and the associated deformations and energetics [1719]. In summary, coherency dislocations are proved to exist in grain boundaries, with the same displacement and stress fields as conventional dislocations. They must be considered when modeling and predicting grain boundary properties with dislocations. It is worth re-investigating grain boundary properties which were modeled and evaluated by only using anti-coherency dislocations in previous studies. And the combination effect of coherency and anti-coherency dislocations warrants further exploration.
This work was supported by the Program of “One Hundred Talented People” of the Chinese Academy of Sciences (JBY) and the National Natural Science Foundation of China (NSFC) under Grant Nos. 51571198, 51771206, 51790482 and 51871223.
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[37] The influence of free surfaces and slots is estimated quantitatively by simulation of a single Fe crystal with the same box dimensions and boundary conditions. [38] K.S. Cheung and S. Yip, J. Appl. Phys. 70, 5688 (1991). [39] W. Bollmann, Crystal lattices, interfaces, matrices. Geneva: W. Bollmann; 1982. [40] J.P. Hirth, R.C. Pond, and J. Lothe. Acta Mater. 54, 4237 (2006) [41]J.P. Hirth, R.C. Pond, R.G. Hoagland, X.Y. Liu, and J. Wang, Prog. Mater. Sci. 58, 749(2013). [42] R. Bonnet, M. Couillard, S. Dhouibi, and S. Neily, Phys. Status Solidi B 253, 1545 (2016). [43] C.L. Johnson, M.J. Hÿtch, and P.R. Buseck, PNAS 101, 17936 (2004). [44] B.G. Mendis, Y. Mishin, C.S. Hartley, and K.J. Hemker, Philos. Mag. 86, 4607(2006). [45] R. Bonnet, Philos. Mag. A 73, 1193(1996). [46] P. Humble, Philos. Mag. A 51, 355(1985). [47] R. Bonnet, Philos. Mag. A 43, 1165 (1981).
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FIG. 1. (a) The SASTGB near (111) simulated in Fe. The black atoms possess a higher energy than gray ones. The FBE predicted coherency and anti-coherency dislocations are superimposed and indicated by small and large edge dislocation arrays, respectively. Only one 1 1 2 layer (parallel to the paper plane) is presented here so that half of 1 10 planes are lost. (b) The variation of the migration distance h with the sliding distance s of the SASTGB during motion. The circles are measured from the simulation results. See text for parameters.
15
16
FIG. 2. The displacement fields of (a) the anti-coherency and (b) coherency dislocations predicted from the FBE. Their superposition in (c) is in agreement with the simulation results (circles). See text for parameters.
17
FIG. 3. The shear stress fields (MPa) of (a) the coherency and (b) anti-coherency dislocations predicted from the FBE for the boundary segment with a length Ly/3 along y.
18
FIG. 4. (a) Schematic of the boundary segment with a length Ly/3 along y. Four anti-coherency dislocations are denoted by Ei, where the dislocation number i = -2 to 1, as defined in Eq. (6). The shaded areas are filled with no atoms to create two slots. The shear stress field (MPa) of the boundary segment (b) in simulation and (c) by the sum of shear stresses from the coherency and anti-coherency dislocations predicted from the FBE. See text for parameters. 19