Determination of the stretch tensor for structural transformations
DDetermination of the stretch tensor for structural transformations
Xian Chen ( 陈 弦 ),
1, 2, ∗ Yintao Song ( 宋寅 韬 ), Nobumichi Tamura, and Richard D. James Aerospace Engineering and Mechanics,University of Minnesota, Minneapolis, MN 55455 USA Advanced Light Source, Lawrence Berkeley National Lab, CA 94702 USA (Dated: October 20, 2018)
Abstract
The transformation stretch tensor plays an essential role in the evaluation of conditions of compatibilitybetween phases and the use of the Cauchy-Born rule. This tensor is difficult to measure directly fromexperiment. We give an algorithm for the determination of the transformation stretch tensor from x-raymeasurements of structure and lattice parameters. When evaluated on some traditional and emerging phasetransformations the algorithm gives unexpected results.
PACS numbers: 61.50.Ks a r X i v : . [ c ond - m a t . m t r l - s c i ] J a n he structural transformations commonly occur in application of functional materials. Typicalexamples of phase transformation driven phenomena include shape memory alloys, ferroelectric-ity, piezoelectricity, colossal magnetoresistance and superconductivity. It has been demonstratedthat material reliability depends, essentially, on the reversibility of the transformation. It is there-fore important to understand how reversibility can be achieved and how the transformation occursat the lattice and atomic level. The transformation stretch tensor , U , is the stretch part of thelinear transformation that maps the crystal structure from its initial phase to the final phase [1–4].Recently, the reversibility, the thermal hysteresis, and the resistance to cyclic degradation of func-tional materials have been linked to properties of the transformation stretch tensor. For example,when the middle eigenvalue λ of U is tuned to the value 1 by compositional changes, the mea-sured width of the thermal hysteresis loop drops precipitously to near 0 in diverse alloy systems[5–7]. Assuming the Cauchy-Born rule for martensitic materials [3, 8, 9], the condition λ = cofactor conditions ( λ = | U − e | = | Ue | =
1, where e is unit vector on a 2-fold symmetry axis of austenite), leadto even lower hysteresis and significantly enhanced reversibility during cyclic transformation [10]. U also plays an important role in determining the elastically favored orientations of precipitatesfor diffusional transformations [4, 11]. FIG. 1. Non-uniqueness of Cauchy-Born deformation gradient from (a) square lattice to (b) oblique lattice.Red, blue and green balls represent different atomic species. Gray dots are lattice points
2n principle, the determination of the stretch tensor U for a structural transformation is straight-forward. Suppose the primitive lattice vectors of initial and final phases are, respectively, linearlyindependent vectors a i and b i for i = , , ... d where d is the dimension of the lattice. A nonsingularlinear transformation F can be defined uniquely by Fa i = b i , i = , , ... d , (1)and the polar decomposition of F is written F = QU , where Q is orthogonal and U is positive-definite and symmetric, called the transformation stretch tensor. The notation a i → b i denotesthe lattice correspondence . In the case of transformation in Fig. 1, one choice of the latticecorrespondence can be a → b , a → b where a = [ , ] , a = [ , ] and b = [ a , ] and b =[ b cos β , b sin β ] .As is well-known [12, 13], F and U are not uniquely determined by the two lattices. This fol-lows from the fact that there are infinitely many choices of lattice correspondence. From Fig. 1, thealternative set of vectors a and a + a describes the same lattice (a), which results in a differentcorrespondence from (a) to (b). This obviously changes the F and thus the transformation stretchtensor U . More generally, any two sets of primitive lattice vectors for a given lattice are related bya lattice invariant transformation [2] i.e., a unimodular matrix of integers. If we allow an invarianttransformation for both initial and final phases, the ambiguity of F is F → Λ (f) F Λ − where Λ (i) and Λ (f) denote the lattice invariant transformation for initial and final lattices, respectively.The linear transformation F represents the change of periodicity of the two phases. The indi-vidual atoms denoted by the red, blue and green balls in Fig. 1 may shuffle in various ways, givingrise to different space group symmetries, but it is the linear transformation F that relates to macro-scopic deformation and therefore to conditions of compatibility [1, 7–9, 12, 14–18]. This idea isformalized by the weak Cauchy-Born rule [8, 19]. This rule is used to define the dependence ondeformation of the free energy at continuum scale from the free energy density at atomistic scalefor complex lattices with multiple atoms per unit cell and inhomogeneous deformations. Inho-mogeneous deformations y ( x ) locally satisfy the same rule as above: b i = ∇ y a i , where a i and b i represent the local periodicity. Note that we use a geometrically exact description here. Ageometrically linear description (i.e., as in linear elasticity) would not be sufficiently accurate todescribe transformations here for the purposes of imposing the conditions of compatibility (see [7]for calculations of the error in various cases).Based on a natural intuition that “a mode of atomic shift requires minimum motion” [20], Bain3roposed a famous lattice correspondence in 1924 for the formation of bcc α Fe from fcc γ Fe .The correspondence has been well-accepted and applied to study numerous phase transformations[2, 3, 21–25]. To illustrate how easy the Bain correspondence misses the smallest strain, weconstruct an example of transformation from a bcc lattice with a = a = . b = . c = . β = . ◦ . Fig. 2(a) shows the bcc lattice with two sublatticeunit cells (red and blue). Conventional wisdom would say that the Bain correspondence (red → gray in Fig. 2(b), bottom) is appropriate for this transformation. However, our algorithm proposedlater in this letter reveals an unexpected alternative correspondence (blue → gray, Fig. 2 (b), top).Both contain 4 lattice points ( n =
4) in the unit cells, and the shape and size of them are similarto the primitive cell of monoclinic lattice. Fig. 2(b) shows the comparison of distortions for bothtransformation mechanisms. Notice that both mechanisms give exactly the same final monocliniclattice. However, by quantitative calculation, the principle strains for the new correspondence arein fact smaller than those for the Bain correspondence.
FIG. 2. The least atomic movements during the structural transformation. (a) The bcc lattice and two of itssublattices (red and blue) of size 4. (b) Comparison between these bcc sublattice unit cells and the primitivecell of the final phase (gray; for clarity atoms in the unit cell are not shown).
The significance of finding the correct lattice correspondence for structural phase transforma-tions is emphasized in the literature [12, 13]. The problem was well-appreciated by Lomer [26]as early as the mid-1950s. In his study of the mechanism of the β → α phase transformation of4 . Cr . , he examined theoretically (by hand) 1,600 possible transformation mechanisms, andreduced this to three correspondences having the smallest principle strains, which he consideredthe likely candidates.Direct experimental measurement of the macroscopic finite strain of transformation, togetherwith accurate structural characterization by X-ray diffraction provides a possible way to determinethe lattice correspondence and thus the transformation stretch tensor. But this is technically dif-ficult due to (i) the need for an oriented single crystal, (ii) the need to remove the inevitable finemicrostructures that form during transformation due to constraints of compatibility, and (iii) theneed for an accurate measure of full finite strain tensor along known crystallographic directions.We also noticed that using a state-of-art high resolution TEM on a pre-oriented single crystal sam-ple can not definitively remove the ambiguities among many lattice correspondences due to someinevitable obstacles: tracking the evolution of diffraction spots in a fast structural transformationprocess, simultaneously indexing both phases, and most significantly, finding a special zone thatcan unambiguously reveal the differences among various lattice correspondences.In this letter we propose an algorithmic approach to search the N best choices of lattice cor-respondence for a structural transformation, by minimizing a particular strain measure betweeninitial and final lattices. The input to the algorithm is the underlying periodicities (the remainingspace group information is not needed) and the lattice constants of the two phases. The outputfrom the algorithm is the N best choices of lattice correspondence and the associated transforma-tion stretch tensors. Users can customize how many solutions they like by manipulating N . Theresults can be used as a reference by the advanced structural characterization facilities for the de-termination of orientation relationships, and it can be integrated with first principles calculationsto give starting points for the determination of energy barriers or interfacial distortion profiles.Consider a Bravais lattice L = { ∑ n i e i : n , . . . n d ∈ Z d } determined by linearly independent lattice vectors e , . . . , e d ∈ R d , i = , . . . , d , and assemble the lattice vectors as the columns of a d × d matrix E = ( e , . . . , e d ) . L can equivalently be denoted L = L ( E ) = (cid:8) r ∈ R d : r = E ξ , ξ ∈ Z d (cid:9) . Without loss of generality, by switching the sign of e if necessary, we assume that det E >
0. Thisdeterminant is the ( d -dimensional) volume of a unit cell of L ( E ) .Given two lattices L ( E ) and L ( E (cid:48) ) , the d × d nonsingular matrix L satisfying E (cid:48) = EL iscalled the correspondence matrix from L ( E ) to L ( E (cid:48) ) . As noted above, the two lattices L ( E ) L ( E (cid:48) ) are the same if and only if the correspondence matrix L is a unimodular matrix ofintegers, or, briefly, L ∈ GL ( d , Z ) . If a correspondence matrix L is a matrix of integers with | det L | >
1, then L ( E (cid:48) ) is a sublattice of L ( E ) . The quantity | det L | is the volume ratio of theunit cell of L ( E (cid:48) ) to that of L ( E ) .Correspondence matrices are often reported for conventional rather than primitive descriptions,particularly for 7 of the 14 types of Bravais lattices in 3D. For example, the conventional descrip-tion for an fcc lattice with lattice parameter a is an orthogonal basis, so E conv = a I = E χ , where,for example, E = a , χ = − − − . Here, det χ = χ is reserved for a correspondence matrix from the primitive to conven-tional unit cell of a Bravais lattice: E conv = E χ .We seek a sublattice of L ( E A ) that is mapped to the primitive lattice of L ( E B ) . (The algorithmcan easily handle the case in which we take sublattices of both lattices.) As above, let E A =( a , ..., a d ) and E B = ( b , ..., b d ) . Let (cid:96) ∈ Z d × d , det (cid:96) >
0, be the correspondence matrix givingthe sublattice L ( E A (cid:96) ) that is mapped to the final lattice L ( E B ) during the transformation. Thebasic equation (1) in this case becomes FE A (cid:96) = E B , and the transformation stretch tensor U is theunique positive-definite square root of F T F .We introduce the following function as a measure of the distance from U to I :dist ( (cid:96) , E A , E B ) = (cid:13)(cid:13) ( F T F ) − − I (cid:13)(cid:13) = (cid:13)(cid:13) E A (cid:96) E − B E − TB (cid:96) T E TA − I (cid:13)(cid:13) . (2) (cid:107) · (cid:107) denotes the Euclidean norm, (cid:107) A (cid:107) = √ tr A T A . The distance (2) is independent of rigid rota-tions of both lattices, and is particularly attractive from the point of view of symmetry. Physically,it represents the Lagrangian strain of the structural transformation. The use of inverse of F T F avoids possible noninvertibility of (cid:96) that may arise during the minimization process. In addition,this norm is exactly preserved by point group transformations of both Bravais lattices. That is,if orthogonal tensors R A and R B are, respectively, in the point groups of L ( E A ) and L ( E B ) ,i.e., L ( E A ) = L ( R A E A ) and L ( E B ) = L ( R B E B ) , which, by the above implies that there ex-ist associated matrices µ A and µ B such that R A E A = E A µ A and R B E B = E B µ B then the distance6ransforms as dist ( µ A (cid:96)µ B , E A , E B ) = dist ( (cid:96) , E A , E B ) . (3)Note that µ A , B are integral matrices of determinant ±
1, so det (cid:96) = det µ A (cid:96)µ B . Thus, immediatelyone minimizer of the distance with assigned determinant gives the expected symmetry-related min-imizers. Physically, in the typical case of a symmetry-lowering transformation, e.g. the marten-sitic transformation, the distance function (2) automatically gives the equi-minimizing variants ofmartensite.As noted above it is typical to report the correspondence matrix in terms of the conventionalbasis instead of the primitive one. If (cid:96) ∗ is a minimizer of dist ( (cid:96) , E A , E B ) the conversion is done by L ∗ = χ − A (cid:96) ∗ χ B . Note that L ∗ is not necessarily a matrix of integers.A significant property of the distance function (2) will be used to justify our algorithm below.Fixing E A and E B , the distance function can be trivially extended to a function over real matri-ces, f ( L ) = dist ( L , E A , E B ) . Denoting X L = E A LE − B E − TB L T E TA and using X L · I ≤ (cid:107) X L (cid:107) (cid:107) I (cid:107) = √ (cid:107) X L (cid:107) , we have f ( L ) = (cid:107) X L (cid:107) − X L · I + (cid:62) (cid:107) X L (cid:107) − √ (cid:107) X L (cid:107) + = ( (cid:107) X L (cid:107) − √ ) , (4)Choose any integral matrix (cid:96) and define C = f ( (cid:96) ) . By (4) the minimizer(s) of f ( L ) necessarilylie in the bounded set (cid:107) X L (cid:107) ≤ √ + √ C , that is, (cid:107) X L (cid:107) ≤ + C + √ C . Let α be the minimumof (cid:107) X L (cid:107) under the constraint (cid:107) L (cid:107) =
1, then we have α (cid:107) L (cid:107) (cid:54) (cid:107) X L (cid:107) < + C + (cid:112) C . (5)That is, all the L ’s such that f ( L ) < C live in the sphere with the radius of (( + C + √ C ) / α ) / in R .Here is a brief outline of the algorithm for the determination of the N best transformation stretchtensors and their associated lattice correspondences:1. Calculate the primitive bases and the transformation matrices for the conventional cells fromthe input lattice parameters: E A, B and χ A, B . Calculate α by minimizing the term X L withrespect to L for all (cid:107) L (cid:107) = N integral matrices (cid:96) i , i = , . . . , N as the initial guess of the solution list such thatdet (cid:96) i is close to det E B / det E A and dist ( (cid:96) i , E A , E B ) is small.7 number of modulation d i s t a n c e [ a . u . ] (b) Bain corr.New corr. m =4 (a) [100] c [ ] c [010] c B a i n N e w -layer -layer FIG. 3. Two possible lattice correspondences in an FCC to monoclinic transformation. (a) (010) projectionof the FCC lattice: the dark (resp. light) atoms are in the y = y = /
2) planes. The solid blueand red lines represent the two lattice correspondences respectively for m =
4, where the the Bain corre-spondence is in blue. The dashed blue lines indicate the modulation numbers m = , , ,
4. (b) shows thedependence of the values of the distance function on the modulation of the monoclinic c -axis for the twolattice correspondences.
3. Let C be the maximum f ( (cid:96) i ) for (cid:96) i ’s in the solution list.4. Calculate the distance for all integral matrices in the sphere of radius of (( + C + √ C ) / α ) / . Update the solution list as necessary. If the solution list is changed, re-peat from step 3.5. For each solution (cid:96) i , calculate the Cauchy-Born deformation gradient F i = E B ( E A (cid:96) i ) − andthe transformation stretch tensor U i = ( F Ti F i ) / . Finally, rewrite all the solutions in theconventional bases: L ∗ i = χ − (cid:96) i χ B .Note that the algorithm converges in a finite number of steps and gets all matrices with the N lowest distances (up to the degeneracy in (3)) because it searches through all matrices of integerssatisfying the rigorous bounds (5).In Fig. 3 we give an example computed by the algorithm that reveals a switch from Bain corre-spondence to a new correspondence with increasing lattice complexity. Consider a transformationfrom an fcc lattice with lattice parameter a = a = . , b = . , c = . m , β = ◦ , where the integer m > c -axis. Fig. 3(a) shows the undeformed fcc lattice projected onto ( ) plane. The8wo correspondences given by the algorithm are depicted for the m = m varying from 1 to 16. InitiallyBain correspondence is much smaller than the new one, however it loses its privilege after the7th modulation. The results suggest that both kinds of lattice correspondence can be feasible in astructural transformation for some special lattice parameters, and in this case m = Au Cu . The material has been recently found to sat-isfy the cofactor conditions (the 2 constraints on U explained in paragraph 1) [7], which havebeen shown [10] to promote unusually low thermal hysteresis ( ≈ ◦ C) and enhanced reversibil-ity, owing to a fluid-like flexible martensite microstructure. It was believed [10] to transformby the second solution, Table I. However, the first solution is the one having the smallest trans-formation strain. Coincidentally, the new transformation stretch tensor also satisfies closely thecofactor conditions. To investigate this further, the same sample of Zn Au Cu used in [10]was characterized by synchrotron X-ray Laue microdiffraction. The experiment has been con-ducted on beamline 12.3.2 of the Advanced Light Source, Lawrence Berkeley National Labora-tory. Details on the experimental setup can be found in [27]. The Laue patterns were collectedcontinuously as heating/cooling through the transformation temperature. These patterns were an-alyzed and indexed using the XMAS software [28]. The orientation relationships are determinedas the closest parallelisms of the crystallographic planes and zone axes between the indexed Lauepatterns of austenite and martensite respectively. They are ( ) a || (
20 ¯34 ) m , ( ) a || (
10 ¯26 ) m , [ ] a || [
26 ¯9 1 ] m , [ ] a || [ ] m and [ ] a || [ ] m (see supplementary for indexed diffraction pat-terns). However, this determination with accepted error bars does not definitively distinguish thesetwo mechanisms, since these relationships are so close that one could imagine that both mecha-nisms occur simultaneously in the material.In addition to the reversible martensitic transformation, the algorithm is applicable to a widerange of phase transformations even if the initial and final crystal structures do not have agroup/sub-group relation. Examples are Ti Mn and Sb Te /PbTe (Table I). The algorithm canbe also applied to organic materials when the molecular chains have sufficient periodicity. Oneextreme example is the polymorphic transformation between two triclinic lattices in terephthalic9 ABLE I. Transformation principle stretches (p. s.), the associated lattice correspondences (lat. cor.) andderived orientation relationships (o. r.) for various phase-transforming materials materials p. s. lat. cor. derived o. r.Zn Au Cu [10]L2 → M18R 0 . [ ] L2 → [ ] M ( ) L2 || ( ¯1 0 26 ) M . [ ] L2 → [ ] M [ ] L2 || [ ] M . [ ¯405 ] L2 → [ ] M [ ] L2 || [
26 ¯9 1 ] M . [ ¯12 ¯12 ] L2 → [ ] M ( ) L2 || ( ¯1 0 27 ) M . [ ] L2 → [ ] M [ ] L2 || [ ] M . [ ¯92 ] L2 → [ ] M [ ] L2 || [
27 ¯9 1 ] M CuAl Ni [29] β → γ (cid:48) . [ ] A → [ ] B ( ) β || ( ) γ (cid:48) . [ ] A → [ ] B [ ] β || [ ] γ (cid:48) . [ ¯ ] A → [ ] B Ti Mn [16]bcc → hexagonal 0 . [ ] c → [ ] h ( ) c || ( ) h . [ ¯12 12 12 ] c → [ ] h [ ¯1¯21 ] c || [ ] h . [ ] c → [ ] h Ru Nb [30] β (cid:48) → β (cid:48) . [ ] β (cid:48) → [ ] β (cid:48)(cid:48) ( ) β (cid:48) || ( ) β (cid:48)(cid:48) . [ ] β (cid:48) → [ ] β (cid:48)(cid:48) [ ] β (cid:48) || [ ] β (cid:48)(cid:48) . [ ] β (cid:48) → [ ] β (cid:48)(cid:48) Sb Te / PbTe [4]fcc → hexagonal 0 . [ ¯12 12 ] c → [ ] h ( ¯110 ) c || ( ) h . [
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