Continuum model for dislocation structures of semicoherent interfaces
CContinuum model for dislocation structures of semicoherentinterfaces
Luchan Zhang, Xiaoxue Qin, Yang Xiang ∗ Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay,Kowloon, Hong Kong
Abstract
In order to relieve the misfitting elastic energy, the hetero-interfaces become semicoherentby forming networks of dislocations. These microscopic structures strongly influence thematerials properties associated with the development of advanced materials. We developa continuum model for the dislocation structures of semicoherent interfaces. The classicalFrank-Bilby equation that governs the dislocation structures on semicoherent interfaces isnot able to determine a unique solution. The available methods in the literature either usefurther information from atomistic simulations or consider only special cases (dislocationswith no more than two Burgers vectors) where the Frank-Bilby equation has a unique so-lution. In our continuum model, the dislocation structure of a semicoherent interface isobtained by minimizing the energy of the equilibrium dislocation network with respect toall the possible Burgers vectors, subject to the constraint of the Frank-Bilby equation. Thecontinuum model is validated by comparisons with atomistic simulation results.
Keywords:
Semicoherent interfaces, Dislocations, Frank-Bilby equation, Energyminimization
1. Introduction
The interfaces between different materials or different phases commonly form semicoherentstructures that consist of discrete dislocation networks to accommodate the lattice misfit ∗ Corresponding author
Email address: [email protected] (Yang Xiang)
Preprint submitted to Elsevier December 18, 2020 a r X i v : . [ c ond - m a t . m t r l - s c i ] D ec etween the two materials [1, 2, 3]. Such semicoherent interfaces play essential roles in themechanical, electronic and plasticity properties that are associated with the developmentof novel composite materials and alloys [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15].These properties strongly depend on the characteristics of the dislocation networks of thesemicoherent interfaces.A semicoherent interface is composed of a network of misfitting dislocations and coherentregions separated by these dislocations. The equilibrium dislocation structure on a semico-herent interface is governed by the Frank-Bilby equation [16, 17], which determines the netBurgers vector content B ( p ) crossing a probe vector p on the interface. The Frank-Bilbyequation strongly depends on the reference state and the possible Burgers vectors definedassociated with it, on which there have been some in-depth discussions in the literature,e.g., [6, 7, 8, 9, 10, 11, 12, 13].However, even though the reference state and all the associated possible Burgers vectorson a semicoherent interface are determined, the Frank-Bilby equation is still not able togive a unique dislocation structure. See the example discussed at the end of Sec. 2. Wang et al. [7, 9] have developed an atomically informed Frank-Bilby theory by combining theclassical Frank-Bilby theory and atomistic simulations to determine the reference lattice andinterfacial dislocation structure of a heterogeneous interface. The dislocation line directionsand their Burgers vectors in the dislocation structure in the Frank-Bilby equation are in-formed by atomistic simulation results. Vattre and Demkowicz, Abdolrahim and Demkowicz[8, 10, 11] have formulated approaches to determine the reference state for the interfacialmisfitting dislocation arrays, linking the Frank–Bilby equation and anisotropic elasticitytheory under the condition of vanishing far-field stresses. They considered two sets of mis-fitting dislocations (i.e., dislocations with two possible Burgers vectors) in their theory, forwhich the Frank–Bilby equation is able to give a unique solution. Generalization of thisapproach has been proposed by Vattre [12] to incorporate hexagonal misfitting dislocationnetworks with new dislocation segments with the third Burgers vector formed by dislocationreaction, from the lozenge dislocation network of two sets of dislocations solved using themethod in Ref. [8, 10]. Further generalizations have been made by Vattre and Pan [13] for2nteraction and movements of various dislocations in anisotropic bicrystals with semicoher-ent interfaces. There are also simulations for the dislocation/disconnection structures onsemicoherent interfaces of precipitates with prescribed Burgers vectors [4, 5].In this paper, we present a continuum model to obtain the dislocation structure of asemicoherent interface, given the reference state and all possible Burgers vectors. In thecontinuum model, the energy of the equilibrium misfitting dislocation network is minimizedwith respect to all possible Burgers vectors subject to the constraint of the Frank-Bilbyequation. The continuum model is based on the orientation-dependent dislocation densitiesof the dislocation structure. Since the Frank-Bilby equation holds, the long-range elasticenergy vanishes, and the energy of the heterogeneous interface consists of only the localenergy of the equilibrium dislocation network, for which the continuum formulation for thelocal energy of dislocation arrays [18] is used. When the dislocation network consists ofstraight dislocations, our continuum model gives the exact solution (i.e., exact dislocationline directions and inter-dislocation distances) of the dislocation network. We also developan identification method based on dislocation reactions to recover the exact dislocationnetwork (e.g., the hexagonal network) from the orientation-dependent dislocation densitiesobtained in the continuum model. This model is a generalization of the method proposedin Ref. [19] for finding dislocation structures of low angle grain boundaries.Numerically, the constrained minimization problem in our continuum model is solvedby the penalty method. We use our continuum model to study the fcc/bcc semicoherentinterfaces with the Nishiyama-Wassermann (NW) and Kurdjumov-Sachs (KS) orientationrelations. Comparisons with atomistic simulations show that our continuum model canprovide excellent predictions of the dislocation structures of semicoherent interfaces.This paper is organized as follows. In Sec. 2, we review the semicoherent interfaces andthe Frank-Bilby equation. In Sec. 3, we present our continuum model for the dislocationstructure. The reference lattices and possible Burgers vectors of some semicoherent interfacesare reviewed in Sec. 4. In Sec. 5, we apply our continuum model to obtain the dislocationstructures of Cu/Nb semicoherent interfaces, and compare the results with those of atomisticsimulations in Ref. [7, 9]. Conclusion and discussion are made in Sec. 6.3 . Semicoherent interfaces and Frank-Bilby equation We first review the semicoherent interfaces and the Frank-Bilby equation. The geometry ofa bicrystal hetero-interface is illustrated schematically in Fig. 1(a). Two materials (or twophases) α and β with different lattice structures are joined, and a hetero-interface is formedbetween them. The interface plane is set as the xy plane. When the lattice structures ofthe adjacent crystals are similar, and the lattice spacing difference between the unstrainedadjacent crystals are relatively small, the interface usually becomes semicoherent by forminga network of misfit dislocations on the interface, and atoms in two adjacent lattices areadjusted by additional strains or rotations. Figure 1(b) demonstrates a natural dichromaticpattern of the semicoherent interface between fcc(110)/bcc(001). βα y xz (a) x k [001] fcc k [010] bcc y k [ ¯ ] f cc k [ ] b cc (b) Figure 1: (a) Illustration of the bicrystal hetero-interface. (b) The natural dichromatic pattern of the interface betweenfcc(110)/bcc(001).
Frank and Bilby proposed a theory that provides an geometry constraint of the equilib-rium dislocation structure on a semicoherent interface [16, 17]. Given the reference lattice,the Frank-Bilby theory determines the net Burgers vector B ( p ) crossing an interfacial probevector p as B ( p ) = ( S − β − S − α ) p , where S − α , S − β respectively are inverse matrices of thedistortion transformation matrices S α , S β that map the lattice vectors from the naturalunstrained lattices α , β to the reference lattice.4nce the reference lattice is determined, there will be finite number of possible Burg-ers vectors associated with it, which are the lattice vectors in the reference lattice. Wedenote these Burgers vectors by b j , j = 1 , , · · · , J . The net Burgers vector B ( p ) can beexpressed in the reference lattice as B ( p ) = (cid:80) Jj =1 ( N j · p ) b j . Recall that in the classicaldislocation model of grain boundaries and interfaces [3, 20], the dislocation structure on aboundary/interface is described by the reciprocal vector N lying in the boundary/interfaceplane that is perpendicular to the dislocation and has magnitude N = 1 /D , where D is theinter-dislocation distance. The local dislocation line direction is ξ = ( N /N ) × n , where n isthe unit normal vector of the interface (which is in the + z direction here). The dislocationdensity is N . For multiple arrays of dislocations on the interface, the dislocation array withBurgers vectors b j are represented by N j , j = 1 , , · · · , J . Accordingly, the Frank-Bilbyequation can be written as J (cid:88) j =1 ( N j · p ) b j = ( S − β − S − α ) p , (1)for any interfacial probe vector p .The Frank-Bilby equation in general is not able to uniquely determine the disloca-tion structure. For example, on the fcc(110)/bcc(001) interface with [001] fcc (cid:107) [010] bcc and[1¯10] fcc (cid:107) [100] bcc , there are four possible Burgers vectors [7], leading to 8 unknowns of thefour vectors N j = ( N jx , N jy ) for the orientations and inter-dislocation distances of the foursets of Burgers vectors; whereas there are only 4 equations in Frank-Bilby theory, whichare not enough to determine the 8 unknown quantities. Only when there are no more thantwo possible Burgers vectors in the semicoherent interface, the Frank-Bilby equation canuniquely determine a dislocation structure.Our continuum model will be based on the given reference state. In this paper, we simplyadopt the median lattice (average of the two lattices) with isotropic elasticity. In practice,the median lattice or one of the adjacent lattices have often been used as the reference lattice[16, 21, 3]. Especially, the median lattice [16] is an excellent approximation of the referencelattice for symmetric and isotropic interfaces, leading to equal partition of the elastic fieldsof the two crystals. Recently, methods of determining the reference lattice in general cases5uch as anisotropic or unsymmetrical interfaces have also been developed [6, 7, 8, 9, 10, 11].The reference lattices and possible Burgers vectors of some hetero-interfaces will bereviewed in Sec. 4.
3. Continuum model
Now we present a continuum model to obtain the dislocation structure of the semicoherenthetero-interface with the given reference state (or equivalently, the distortion transformationmatrices S α and S β ) and all possible Burgers vectors ( b j , j = 1 , , · · · , J ). Since Frank-Bilbyequation is not sufficient to uniquely determined the dislocation structure on the interface,we identify the equilibrium dislocation structure by minimizing the local energy associatedwith the constituent dislocations of the interface subject to the constraint of the Frank-Bilbyequation. That is, we solve the following constrained energy minimization problem for the dislocation structure:minimize E = (cid:90) S γdS, (2)with γ = J (cid:88) j =1 µb j π (1 − ν ) (cid:18) − ν ( N j × n · b j ) b j N j (cid:19) N j log 1 r g (cid:113) N j + (cid:15) , (3)subject to h = J (cid:88) j =1 ( N j · p ) b j − ( S − β − S − α ) p = . (4)Here S is a periodic cell on the interface plane, γ is the interface energy density, µ is theshear modulus, ν is the Poisson ratio, b j is the length of the j -th Burgers vector, r g is aparameter associated with the dislocation core size, (cid:15) is some small positive regularizationparameter to avoid the numerical singularity when N j = 0. The constraint in Eq. (4) is theFrank-Bilby equation, which holds for any vector p on the interface.The interface energy γ in Eq. (3) is based on the local energy of dislocation arrays interms of dislocation densities [18]. Since Frank-Bilby equation is equivalent to cancellationof the long-range elastic field [16, 17, 20, 3], there is only local energy of the constituentdislocations on the interface when Frank-Bilby equation in Eq. (4) holds, as that for the grain6oundaries in homogeneous materials [20, 3]. The elastic constants in γ in Eq. (3) can bechosen as the averages of those of the two materials: e.g., µ = ( µ α + µ β ) / ν = ( ν α + ν β ) / γ in Eq. (3) is based on densities of dislocations. When all theconstituent dislocations on the interface are straight, the obtained vectors N j ’s give the exactdislocation structure. When the dislocation network consists of disconnected dislocationsegments, e.g., the hexagonal network, our continuum model gives the line directions anddensities of these dislocations (line direction N j /N j and density N j ); and in this case, wewill present a method to recover the exact hexagonal network from the obtained dislocationdensities and line directions; see the end of this section.The constraint of the Frank-Bilby formula in Eq. (4) holds for any probe vector p if andonly if it holds for the two basis vectors of the xy plane: p = p = (1 ,
0) and p = p = (0 , h = ( h , h ) T = with h and h being the Frank-Bilby formula in Eq. (4) when theprobe vector p is set to be p and p , respectively.In addition to the misfit, the Frank-Bilby equation in Eq. (4) may also include furthertwist and/or tilt of the two crystals α and β through the transformation matrices S α and S β [6, 7, 8, 9, 10, 11, 12, 13]. When there are rotations around an axis perpendicular to theinterface (i.e., twist), assuming that the rotation angles of the natural lattices of the twomaterials and the reference lattice are θ α , θ β and θ , respectively, the Burgers vectors in therotated reference lattice can be calculated using the rotation matrix R θ as b R i = R θ b i , andthe distortion transformation matrices are S R α = R θ S α R T α and S R β = R θ S β R T β , where b i , S α and S β are those in the un-rotated state. Remarks :1. The interface energy formula in Eq. (3) can also be generalized to include elasticanisotropy based on the energy of straight dislocations with appropriate pre-logarithmicenergy coefficients [20].2. We adopt the local energy of the constituent dislocations of the interface for a simpleform and efficient calculation of the continuum model. In principle, the local energy inEq. (3) can be replaced by the full elastic energy with elastic anisotropy and/or different7lastic constants in the two materials [22, 8, 10, 11, 12, 13] for more accurate results.Numerically, the constrained minimization problem can be solved by the penalty method[23], in which it is approximated by the following unconstrained minimization problem :minimize Q = (cid:90) S (cid:16) γ + α p (cid:107) ˜ h (cid:107) (cid:17) dS, (5)where α p > α p → + ∞ , the solution of this unconstrained minimization problem convergesto the solution of the constrained minimization problem [23]. (Other method such as theaugmented Lagrangian method can also be used to solve this constrained minimizationproblem [23].)This unconstrained problem is still very challenging to solve due to the nonconvexity ofthe interface energy. We make a further simplification by considering uniform distributionsof straight dislocations on the interface. In this case, each N j = ( N jx , N jy ) is a constantvector, and the problem is reduced to minimize q = γ + α p (cid:107) ˜ h (cid:107) / N jx , N jy , j = 1 , , · · · , J , which leads to the following evolution equations with anartificial time:( N jx ) t = − (cid:18) ∂γ∂N jx + α p ∂c∂N jx (cid:19) , ( N jy ) t = − (cid:18) ∂γ∂N jy + α p ∂c∂N jy (cid:19) , (6)for j = 1 , , · · · , J , where c = (cid:107) ˜ h (cid:107) /
2, and ˜ h = ( h , h , h , h ) T with h = (cid:80) Jj =1 b jx N jx − ( S − β [1 , − S − α [1 , h = (cid:80) Jj =1 b jy N jx − ( S − β [2 , − S − α [2 , h = (cid:80) Jj =1 b jx N jy − ( S − β [1 , − S − α [1 , h = (cid:80) Jj =1 b jy N jy − ( S − β [2 , − S − α [2 , α and β is also considered, in the Frank-Bilby equation in Eq. (4), S α and S β will be 3 × b j , p and h will be vectors in three dimensions, and ˜ h will be a6 × Identification of dislocation structure from dislocation densities.
Now wepresent a method that recovers the exact dislocation structure based on the densities andorientations of the constituent dislocations obtained in our continuum model. As mentioned8bove, when all the constituent dislocations on the interface are straight, the obtained vec-tors N j ’s give the exact dislocation structure directly. In a general dislocation structure,due to dislocation reactions, the dislocations may not necessarily be continuous straightlines, and they may form hexagons (not necessarily regular) with disconnected dislocationsegments; see Fig. 5(b) for an example. In the identification method, we calculate the exactlength and orientation of each dislocation segment in the hexagonal network based on thedislocation densities and orientations obtained by the continuum model. A a a b b b Figure 2: A hexagonal dislocation structure that consists of dislocations with Burgers vectors b , b and b , whose linedirections are ξξξ , ξξξ , and ξξξ , respectively. Vectors a and a are the two sides of the periodic parallelogram unit cell. The areaof a unit cell is A = (cid:107) a × a (cid:107) . Consider a hexagonal network with dislocations of three Burgers vectors b , b , and b ,in which dislocations may have reactions, e.g, b = b + b and b < b + b ; see Fig. 2.In this hexagonal network, as we discussed before, the direction of dislocation generated by b j is ξξξ j = ( N j /N j ) × n , and the density of these dislocations is N j . Consider a periodicparallelogram cell as shown in Fig. 2. It can be calculated that the b -, b -, and b -dislocation segments in the parallelogram, written in the vector form, are l = A N × n , l = A N × n , l = A N × n , respectively, where n is the normal vector of the interface,and A is the area of the periodic parallelogram cell. Using these results, it can be solvedthat in a periodic parallelogram cell, the length of each dislocation segment is l j = N j (cid:107) N × N (cid:107) + (cid:107) N × N (cid:107) + (cid:107) N × N (cid:107) , j = 1 , , , (7)9sing this formula of length of each dislocation segment l j and its direction ξξξ j = ( N j /N j ) × n ,we can draw the exact hexagonal network structure based on the dislocation densities andorientations represented by { N j } in the continuum model.
4. Reference lattice and possible Burgers vectors
In this section, we briefly review the reference lattice and possible Burgers vectors for thedislocation network on a semicoherent interface. The distortion transformation matrices S α and S β are defined as the matrices mapping from the dichromatic patterns of the two lattices α and β to the reference lattice, and the possible Burgers vectors ar the lattice vectors inthe reference lattice [3]. In practice, the median lattice (average of the two lattices) or oneof the adjacent lattices have often been used as the reference lattice [16, 21, 3]. Especially,the median lattice [16] is an excellent approximation of the reference lattice for symmetricand isotropic interfaces, leading to equal partition of the elastic fields of the two crystals.Recently, methods of determining the reference lattice in general cases such as anisotropicor unsymmetrical interfaces have also been developed [6, 7, 8, 9, 10, 11].Figure 3 shows the reference lattices and possible Burgers vectors of several semicoherentinterfaces between fcc and bcc lattices that have been studied recently in the literature[6, 7, 8, 9, 10, 11, 12, 13, 14]. In Fig. 3(a), the interface orientation relationships are(110) fcc (cid:107) (001) bcc , [001] fcc (cid:107) [010] bcc (in x -direction) and [1¯10] fcc (cid:107) [100] bcc (in y -direction). Thenatural dichromatic patterns of the fcc and bcc lattices are rectangles with parallel sides, asshown by the red and blue dashed lines, respectively, in Fig. 3(a). The coherent dichromaticpattern of the reference lattice is also a rectangle pattern in between the fcc and bcc naturaldichromatic patterns, as shown by the black dashed lines in Fig. 3(a). When the medianlattice (the average of the fcc and bcc natural dichromatic patterns) is adopted as thereference lattice, the possible Burgers vectors of dislocations on the semicoherent interface10 k [001] fcc k [010] bcc y k [ ¯ ] f cc k [ ] b cc b b b b (a) xy [001] fcc [ ¯ ] f cc [010] bcc [ ] b cc b b b b (b) x k [11 ¯ fcc k [1 ¯ bcc y k [ ¯ ] f cc k [ ] b cc b b b (c) x k [11 ¯ fcc k [1 ¯ bcc y k [ ¯ ] f cc k [ ¯ ] b cc b b b (d) Figure 3: Geometry of the dichromatic patterns, reference states and possible Burgers vectors of some classical semicoherentinterfaces between fcc and bcc lattices. The natural dichromatic patterns of the fcc and bcc lattices and the reference lattice (themedian lattice) are polygons with red, blue, and black dashed lines, respectively. The possible Burgers vectors defined in thereference lattice are shown by black arrows. (a) Interface with the orientation relationship (110) fcc (cid:107) (001) bcc , [001] fcc (cid:107) [010] bcc ,and [1¯10] fcc (cid:107) [100] bcc . The natural dichromatic patterns are rectangles. (b) Interface with the orientation relationship of thatin (a) with further rotations around an axis perpendicular to the interface. The reference lattice and possible Burgers vectorscan all be obtained by rotations from those in (a). (c) Interface with the NW orientation relationship: (111) fcc (cid:107) (110) bcc ,[11¯2] fcc (cid:107) [1¯10] bcc , and [1¯10] fcc (cid:107) [001] bcc . The natural dichromatic patterns are hexagons. (d) Interface with the KS orientationrelationship: (111) fcc (cid:107) (110) bcc , [11¯2] fcc (cid:107) [1¯12] bcc , and [¯110] fcc (cid:107) [1¯11] bcc . The fcc and bcc lattices, reference lattice patterns andBurgers vectors can be obtained by rotating the corresponding ones in the NW orientation relationship in (c) around an axisperpendicular to the interface. are the lattice vectors of the reference lattice: b = (cid:0) a fcc + a bcc , (cid:1) , b = (cid:16) , √ a fcc + a bcc (cid:17) , b = (cid:16) a fcc + a bcc , √ a fcc + a bcc (cid:17) , b = (cid:16) a fcc + a bcc , − (cid:16) √ a fcc + a bcc (cid:17)(cid:17) , (8)11nd the distortion transformation matrices mapping from the fcc and bcc lattices to thereference lattice are S α = a fcc + a bcc a fcc √ a fcc + a bcc √ a fcc , S β = a fcc + a bcc a bcc √ a fcc + a bcc a bcc , (9)where a fcc and a bcc are the lattice constants of the fcc and bcc lattices.Figure 3(b) shows the orientation relationship of the fcc(110)/bcc(001) interface inFig. 3(a) with further rotations around an axis perpendicular to the interface. The rotationangles of the natural fcc, bcc lattices and the reference lattice are θ α , θ β and θ , respectively.The Burgers vectors in the rotated reference lattice can be calculated by multiplying theBurgers vectors in the un-rotated reference lattice by the rotation matrix R θ , i.e., b R i = R θ b i , with R θ = cos θ − sin θ sin θ cos θ . (10)The distortion transformation matrices are S R α = R θ S α R T α , S R β = R θ S β R T β . (11)Figure 3(c) demonstrates the classical Nishiyama-Wassermann (NW) orientation re-lationship: (111) fcc (cid:107) (110) bcc , [11¯2] fcc (cid:107) [1¯10] bcc (in x -direction), and [1¯10] fcc (cid:107) [001] bcc (in y -direction). The natural dichromatic patterns of fcc and bcc lattices are hexagonal patterns,and the coherent dichromatic pattern of the reference lattice, which is the median latticebetween the fcc and bcc lattices, is also a hexagonal pattern. The possible Burgers vectorsdefined in the reference lattice are b = (cid:16) − (cid:16) √ a fcc + √ a bcc (cid:17) , − (cid:16) √ a fcc + a bcc (cid:17)(cid:17) , b = (cid:16) − (cid:16) √ a fcc + √ a bcc (cid:17) , √ a fcc + a bcc (cid:17) , b = (cid:16) , √ a fcc + a bcc (cid:17) , (12)and the distortion transformation matrices are S α = √ a fcc + √ a bcc √ a fcc √ a fcc + a bcc √ a fcc , S β = √ a fcc + √ a bcc √ a bcc √ a fcc + a bcc a fcc . (13)12nother classical orientation relationship of the interface jointing (111) fcc (cid:107) (110) bcc is theKurdjumov-Sachs (KS) orientation relationship with [11¯2] fcc (cid:107) [1¯12] bcc (in x -direction) and[¯110] fcc (cid:107) [1¯11] bcc (in y -direction); see Fig. 3(d). It can be obtained by rotating the fcc andbcc lattices in NW orientation relationship around an axis perpendicular to the interface.The possible Burgers vectors and the distortion transformation matrices can also be obtainedfrom those in the NW orientation relationship by rotations using Eqs. (10) and (11).
5. Numerical simulation
We apply our continuum simulation model to obtain dislocation structures on the semicoher-ent interfaces of Cu(110)/Nb(001) and Cu(111)/Nb(110), which have been studied recentlyin the literature [6, 7, 8, 9, 10, 11, 12, 13, 14]. The orientation relationships, reference states(the median lattices) and possible Burgers vectors of these interfaces are shown in Fig. 3.The lattice constants of Cu and Nb are a Cu = 0 . a Nb = 0 . N jx = N jy = 0, j = 1 , , · · · , J , when performing the energy gradient mini-mization in Eq. (6). We choose a large value for the penalty parameter α p in Eq. (5) andfurther increases of its value give only negligible changes in the converged dislocation struc-tures. We compare our results with those obtained using atomistic simulations [7, 9]. Thereference lattices of these semicoherent interfaces obtained by atomistic simulations [7, 9]are very close to the median lattices adopted in our simulations. (110) /Nb (001) interface with rectangle pattern We first consider the Cu(110)/Nb(001) interface with the orientation relationship as shownin Fig. 3(a), where the natural Cu (fcc), Nb (bcc), and reference lattices have rectanglepatterns. There are four possible Burgers vectors b j , j = 1 , , ,
4. See Eqs. (8) and (9) forthe formulas of these Burgers vectors and the distortion matrices S α and S β .The dislocation structure obtained using our continuum model is shown in Fig. 4. Thedislocation structure is a rectangular network that consists of two arrays of dislocations with13 b b Figure 4: Dislocation structure of this Cu(110)/Nb(001) interface calculated using our continuum model. It is a rectangularnetwork that consists of two arrays of dislocations with Burgers vectors b and b (red vertical lines and blue horizontal lines,respectively). Unit: nm. Burgers vectors b , b and represented by the reciprocal vectors N = ( − . , / nm, N = (0 , . / nm, respectively. These two arrays of dislocations are both edge dis-locations, and are in the + y ([1¯10] Cu (cid:107) [100] Nb ) and + x ([001] Cu (cid:107) [010] Nb ) directions, withinter-dislocation distances D = 1 /N = 3 . D = 1 /N = 1 . b and b do not appear in the converged dislocationstructure, i.e., N , N converge to in the simulation. These results of rectangular net-work, dislocation line directions and inter-dislocation distances agree excellently with thoseobtained using atomistic simulation in Ref. [7], in which the inter-dislocation distances inthe two dislocation arrays are D = 3 . D = 1 . (111) /Nb (110) interface with NW orientation relationship We then consider the Cu(111)/Nb(110) interface with hexagonal pattern in the classicalNW orientation relationship as shown in Fig. 3(c). The natural dichromatic patterns of Cu(fcc) and Nb (bcc) lattices and the coherent dichromatic pattern of the reference lattice arehexagonal patterns. The three possible Burgers vectors defined in the reference lattice andthe distortion matrices S α and S β are given in Eqs. (12) and (13).In the dislocation network of this semicoherent interface obtained using our contin-uum model, dislocations with all the three Burgers vectors are present; see Fig. 5(a). The14 b b b (a) -8 -4 0 4 8 x (nm) -0.2-0.100.10.2 D i s r e g i s t r y ( n m ) (b) -4 -2 0 2 4 y (nm) -0.5-0.2500.250.5 D i s r e g i s t r y ( n m ) (c) Figure 5: (a) Dislocation structure calculated using our continuum model. The dislocation structure consists of a hexagonalnetwork of three arrays of dislocations with Burgers vectors b , b and b , shown by red, blue and black line segments,respectively. Unit: nm. (b) and (c) Disregistries along some cross-section lines obtained by atomistic simulations in Ref. [9](reproduced from their data): (b) disregistry in the x direction ([11¯2] Cu (cid:107) [1¯10] Nb ) along a cross-section line parallel to the x axis, and (c) disregistry in the y direction ([1¯10] Cu (cid:107) [001] Nb ) along a cross-section line parallel to the y axis. The black andred lines show the actual disregistry and the unrelaxed uniform disregistry, respectively. The ⊥ symbols show the locations ofdislocations. N = ( − . , − . N = ( − . , . N = (0 , . ξξξ = ( − . , . ξξξ = (0 . , . ξξξ = (1 ,
0) in the xy plane, respectively, and their densities are ρ = N = 0 . ρ = N = 0 . ρ = N = 0 . ξ j = ( N j /N j ) × n , where n isthe unit normal vector of the interface. The b -dislocations are edge dislocations, and the b - and b -dislocations are very close to edge, with an angle of 93 ◦ between the dislocationsand their Burgers vectors.These three arrays of dislocations form a hexagonal network, because we have b =( − b ) + b with b < b + b and dislocation reaction from a b -dislocation and a b -dislocation into a b -dislocation is energetically favorable. In this case, dislocations aredisconnected segments instead of connected straight lines. Using the identification methodpresented at the end of Sec. 3 (Eq. (7)), we can calculate the lengths of the three typesof dislocation segments, which are 0 . . . b -, b -,and b -dislocation segments, respectively. Based on these orientations and lengths of thesedislocation segments, we can recover the hexagonal network as shown in Fig. 5(a).We compare the dislocation structure obtained by our continuum model with the atom-istic simulation result in Ref. [9]. Disregistries along some cross-section lines obtained byatomistic simulations in Ref. [9] are shown in Figs. 5(b) and (c). Figure 5(b) shows the dis-registry in the x direction ([11¯2] Cu (cid:107) [1¯10] Nb ) along a cross-section line parallel to the x axis,and this cross-section line intersects with dislocations periodically with average distance of4 . y direction ([1¯10] Cu (cid:107) [001] Nb ) along a cross-section line parallel to the y axis, and thiscross-section line intersects with dislocations periodically with average distance betweentwo neighboring intersecting points 1 . b - and b -dislocations for − ≤ x ≤ . -dislocation segments with inter-dislocation distance of 1 . . b -dislocation segments is 1 . (111) /Nb (110) interface with KS orientation relationship We also consider the Cu(111)/Nb(110) interface with hexagonal pattern in the KS orien-tation relationship as shown in Fig. 3(d), in which there is a relative rotation between thenatural Cu fcc and Nb bcc lattices. The rotation angle of the natural Cu fcc lattice is θ α = 60 ◦ , and that of the natural Nb bcc lattice is θ β = 54 . ◦ . The rotation angle of thereference lattice is approximately chosen as the average of θ α and θ β , i.e., θ = ( θ α + θ β ) / b and b represented by the17
10 -5 0 5 10-505 b b Figure 6: Dislocation structure of the Cu(111)/Nb(110) interface in KS OR calculated using our continuum model. Thedislocation structure consists of a parallelogram network of dislocation arrays with Burgers vectors b and b , shown by redand black lines, respectively. reciprocal vectors N = (0 . , − . / nm and N = ( − . , − . / nm, respec-tively; see Fig. 6 for the obtained dislocation network. These results of the continuum modelmean that the two arrays of dislocations have line directions ξξξ = ( − . , − . ξξξ = ( − . , . D = 2 . D = 1 . x axisare − . ◦ and 90 . ◦ , respectively. We compare this dislocation structure with that ob-tained by atomistic simulation in Ref. [9] (Figs. 8 and 9 in Ref. [9]), and observe excellentagreement between the two results. In the atomistic simulation result in Ref. [9], the inter-dislocation distances in the two arrays of dislocations are D = 2 . D = 1 . ◦ and 90 ◦ with respect to the x -axis,respectively (see Table 2 in Ref. [9]. Notice that the + z direction is pointing downwardthere).
6. Conclusions
In summary, we have developed a continuum model for the dislocation structures of semico-herent interfaces based on constrained energy minimization. In our model, the dislocationstructure of a semicoherent interface is obtained by minimizing the energy of the equilibriumdislocation network with respect to all the possible Burgers vectors, subject to the constraintof the Frank-Bilby equation. Comparisons with atomistic simulation results and results of18ther available models show that our continuum model is able to give excellent predictionsof dislocation structures on semicoherent interfaces.
Acknowledgements
This work was supported by the Hong Kong Research Grants Council General ResearchFund through grant 16302818.
Data availability
The datasets generated in study are available upon request.
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