Table Of ContentDetermination of the stretch tensor for structural transformations
Xian Chen (陈弦),1,2,∗ Yintao Song (宋寅韬),1 Nobumichi Tamura,2 and Richard D. James1
1Aerospace Engineering and Mechanics,
University of Minnesota, Minneapolis, MN 55455 USA
2Advanced Light Source, Lawrence Berkeley National Lab, CA 94702 USA
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(Dated: January22,2015)
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Abstract
n
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J Thetransformationstretchtensorplaysanessentialroleintheevaluationofconditionsofcompatibility
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between phases and the use of the Cauchy-Born rule. This tensor is difficult to measure directly from
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] experiment. We give an algorithm for the determination of the transformation stretch tensor from x-ray
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s measurementsofstructureandlatticeparameters. Whenevaluatedonsometraditionalandemergingphase
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r transformationsthealgorithmgivesunexpectedresults.
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The structural transformations commonly occur in application of functional materials. Typical
examples of phase transformation driven phenomena include shape memory alloys, ferroelectric-
ity, piezoelectricity, colossal magnetoresistance and superconductivity. It has been demonstrated
that material reliability depends, essentially, on the reversibility of the transformation. It is there-
foreimportanttounderstandhowreversibilitycanbeachievedandhowthetransformationoccurs
at the lattice and atomic level. The transformation stretch tensor, U, is the stretch part of the
linear transformation that maps the crystal structure from its initial phase to the final phase [1–4].
Recently,thereversibility,thethermalhysteresis,andtheresistancetocyclicdegradationoffunc-
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-
2
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 = 1
2
implies a special condition of compatibility between phases by which the undistorted austenite
phaseandasingleundistortedvariantofthemartensitephasefitperfectlytogetherataninterface.
Even stronger conditions of compatibility known as the cofactor conditions (λ =1 together with
2
either |U−1e|=1 or |Ue|=1, where e is unit vector on a 2-fold symmetry axis of austenite), lead
toevenlowerhysteresisandsignificantlyenhancedreversibilityduringcyclictransformation[10].
U also plays an important role in determining the elastically favored orientations of precipitates
fordiffusionaltransformations[4,11].
FIG.1. Non-uniquenessofCauchy-Borndeformationgradientfrom(a)squarelatticeto(b)obliquelattice.
Red,blueandgreenballsrepresentdifferentatomicspecies. Graydotsarelatticepoints
2
Inprinciple,thedeterminationofthestretchtensorUforastructuraltransformationisstraight-
forward. Suppose the primitive lattice vectors of initial and final phases are, respectively, linearly
independentvectorsa andb fori=1,2,...dwheredisthedimensionofthelattice. Anonsingular
i i
lineartransformationFcanbedefineduniquelyby
Fa =b, i=1,2,...d, (1)
i i
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 → b denotes
i i
the lattice correspondence. In the case of transformation in Fig. 1, one choice of the lattice
correspondence can be a →b , a →b where a =[1,0], a =[0,1] and b =[a,0] and b =
1 1 2 2 1 2 1 2
[bcosβ,bsinβ].
As is well-known [12, 13], F and U are not uniquely determined by the two lattices. This fol-
lowsfromthefactthatthereareinfinitelymanychoicesoflatticecorrespondence. FromFig.1,the
alternative set of vectors a and a +a describes the same lattice (a), which results in a different
1 1 2
correspondence from (a) to (b). This obviously changes the F and thus the transformation stretch
tensorU. Moregenerally,anytwosetsofprimitivelatticevectorsforagivenlatticearerelatedby
alatticeinvarianttransformation[2]i.e.,aunimodularmatrixofintegers. Ifweallowaninvariant
transformation for both initial and final phases, the ambiguity of F is F → Λ FΛ−1 where Λ
(f) (i) (i)
andΛ denotethelatticeinvarianttransformationforinitialandfinallattices,respectively.
(f)
The linear transformation F represents the change of periodicity of the two phases. The indi-
vidualatomsdenotedbythered,blueandgreenballsinFig.1mayshuffleinvariousways,giving
risetodifferentspacegroupsymmetries,butitisthelineartransformationFthatrelatestomacro-
scopic deformation and therefore to conditions of compatibility [1, 7–9, 12, 14–18]. This idea is
formalized by the weak Cauchy-Born rule [8, 19]. This rule is used to define the dependence on
deformation of the free energy at continuum scale from the free energy density at atomistic scale
for complex lattices with multiple atoms per unit cell and inhomogeneous deformations. Inho-
mogeneous deformations y(x) locally satisfy the same rule as above: b = ∇ya, where a and
i i i
b represent the local periodicity. Note that we use a geometrically exact description here. A
i
geometrically linear description (i.e., as in linear elasticity) would not be sufficiently accurate to
describetransformationshereforthepurposesofimposingtheconditionsofcompatibility(see[7]
forcalculationsoftheerrorinvariouscases).
Basedonanaturalintuitionthat“amodeofatomicshiftrequiresminimummotion”[20],Bain
3
proposed a famous lattice correspondence in 1924 for the formation of bcc αFe from fcc γFe .
Thecorrespondencehasbeenwell-acceptedandappliedtostudynumerousphasetransformations
[2, 3, 21–25]. To illustrate how easy the Bain correspondence misses the smallest strain, we
constructanexampleoftransformationfromabcclatticewitha =1toamonocliniclatticewith
0
a=0.961,b=1.363,c=1.541,andβ =97.78◦. Fig.2(a)showsthebcclatticewithtwosublattice
unit cells (red and blue). Conventional wisdom would say that the Bain correspondence (red →
grayinFig.2(b),bottom)isappropriateforthistransformation. However,ouralgorithmproposed
later 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 similar
to the primitive cell of monoclinic lattice. Fig. 2(b) shows the comparison of distortions for both
transformation mechanisms. Notice that both mechanisms give exactly the same final monoclinic
lattice. However, by quantitative calculation, the principle strains for the new correspondence are
infactsmallerthanthosefortheBaincorrespondence.
FIG.2. Theleastatomicmovementsduringthestructuraltransformation. (a)Thebcclatticeandtwoofits
sublattices(redandblue)ofsize4. (b)Comparisonbetweenthesebccsublatticeunitcellsandtheprimitive
cellofthefinalphase(gray;forclarityatomsintheunitcellarenotshown).
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 of
4
U Cr , he examined theoretically (by hand) 1,600 possible transformation mechanisms, and
98.6 1.4
reduced this to three correspondences having the smallest principle strains, which he considered
thelikelycandidates.
Direct experimental measurement of the macroscopic finite strain of transformation, together
withaccuratestructuralcharacterizationbyX-raydiffractionprovidesapossiblewaytodetermine
the 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 fine
microstructures that form during transformation due to constraints of compatibility, and (iii) the
need for an accurate measure of full finite strain tensor along known crystallographic directions.
Wealsonoticedthatusingastate-of-arthighresolutionTEMonapre-orientedsinglecrystalsam-
ple can not definitively remove the ambiguities among many lattice correspondences due to some
inevitable obstacles: tracking the evolution of diffraction spots in a fast structural transformation
process, simultaneously indexing both phases, and most significantly, finding a special zone that
canunambiguouslyrevealthedifferencesamongvariouslatticecorrespondences.
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 between
initial and final lattices. The input to the algorithm is the underlying periodicities (the remaining
space group information is not needed) and the lattice constants of the two phases. The output
from 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. The
results 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 calculations
togivestartingpointsforthedeterminationofenergybarriersorinterfacialdistortionprofiles.
Consider a Bravais lattice L = {∑niei : n1,...nd ∈ Zd} determined by linearly independent
lattice vectors e ,...,e ∈ Rd, i = 1,...,d, and assemble the lattice vectors as the columns of a
1 d
d×d matrixE=(e ,...,e ). L canequivalentlybedenoted
1 d
L =L(E)=(cid:8)r∈Rd :r=Eξ,ξ ∈Zd(cid:9).
Withoutlossofgenerality,byswitchingthesignofe ifnecessary,weassumethatdetE>0. This
1
determinantisthe(d-dimensional)volumeofaunitcellofL(E).
Given two lattices L(E) and L(E(cid:48)), the d×d nonsingular matrix L satisfying E(cid:48) = EL is
called the correspondence matrix from L(E) to L(E(cid:48)). As noted above, the two lattices L(E)
5
and L(E(cid:48)) are the same if and only if the correspondence matrix L is a unimodular matrix of
integers, or, briefly, L ∈ GL(d,Z). If a correspondence matrix L is a matrix of integers with
|detL|>1, then L(E(cid:48)) is a sublattice of L(E). The quantity |detL| is the volume ratio of the
unitcellofL(E(cid:48))tothatofL(E).
Correspondencematricesareoftenreportedforconventionalratherthanprimitivedescriptions,
particularly for 7 of the 14 types of Bravais lattices in 3D. For example, the conventional descrip-
tionforanfcclatticewithlatticeparametera isanorthogonalbasis,soE =a I=Eχ,where,
0 conv 0
forexample,
1 0 1 1 1 −1
a
E= 0 1 1 0, χ=−1 1 1 .
2
0 1 1 1 −1 1
Here, detχ = 4 so the volume of the conventional unit cell is 4 times that of the primitive cell.
Fromnowon,thesymbolχisreservedforacorrespondencematrixfromtheprimitivetoconven-
tionalunitcellofaBravaislattice: E =Eχ.
conv
WeseekasublatticeofL(E )thatismappedtotheprimitivelatticeofL(E ). (Thealgorithm
A B
can easily handle the case in which we take sublattices of both lattices.) As above, let E =
A
(a ,...,a ) and E = (b ,...,b ). Let (cid:96) ∈ Zd×d, det(cid:96) > 0, be the correspondence matrix giving
1 d B 1 d
the sublattice L(E (cid:96)) that is mapped to the final lattice L(E ) during the transformation. The
A B
basicequation(1)inthiscasebecomesFE (cid:96)=E ,andthetransformationstretchtensorUisthe
A B
uniquepositive-definitesquarerootofFTF.
WeintroducethefollowingfunctionasameasureofthedistancefromUtoI:
dist((cid:96),EA,EB)=(cid:13)(cid:13)(FTF)−1−I(cid:13)(cid:13)2
(2)
=(cid:13)(cid:13)EA(cid:96)E−B1E−BT(cid:96)TETA−I(cid:13)(cid:13)2.
√
(cid:107)·(cid:107) denotes the Euclidean norm, (cid:107)A(cid:107)= trATA. The distance (2) is independent of rigid rota-
tionsofbothlattices,andisparticularlyattractivefromthepointofviewofsymmetry. Physically,
it represents the Lagrangian strain of the structural transformation. The use of inverse of FTF
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 and R are, respectively, in the point groups of L(E ) and L(E ),
A B A B
i.e., L(E ) = L(R E ) and L(E ) = L(R E ), which, by the above implies that there ex-
A A A B B B
ist associated matrices µ and µ such that R E =E µ and R E =E µ then the distance
A B A A A A B B B B
6
transformsas
dist(µ (cid:96)µ ,E ,E )=dist((cid:96),E ,E ). (3)
A B A B A B
Note that µ are integral matrices of determinant ±1, so det(cid:96)=detµ (cid:96)µ . Thus, immediately
A,B A B
oneminimizerofthedistancewithassigneddeterminantgivestheexpectedsymmetry-relatedmin-
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 of
martensite.
As noted above it is typical to report the correspondence matrix in terms of the conventional
basisinsteadoftheprimitiveone. If(cid:96)∗ isaminimizerofdist((cid:96),E ,E )theconversionisdoneby
A B
L∗ =χ−1(cid:96)∗χ . NotethatL∗ isnotnecessarilyamatrixofintegers.
A B
A significant property of the distance function (2) will be used to justify our algorithm below.
Fixing E and E , the distance function can be trivially extended to a function over real matri-
A B
ces, f(L) = dist(L,E ,E ). Denoting X = E LE−1E−TLTET and using X ·I ≤ (cid:107)X (cid:107)(cid:107)I(cid:107) =
A B L A B B A L L
√
3(cid:107)X (cid:107),wehave
L
f(L)=(cid:107)X (cid:107)2−2X ·I+3
L L
√ √ (4)
(cid:62)(cid:107)X (cid:107)2−2 3(cid:107)X (cid:107)+3=((cid:107)X (cid:107)− 3)2,
L L L
Chooseanyintegralmatrix(cid:96) anddefineC = f((cid:96) ). By(4)theminimizer(s)of f(L)necessarily
1 1 1
√ √ √
lieintheboundedset(cid:107)X (cid:107)≤ 3+ C ,thatis,(cid:107)X (cid:107)2≤3+C +2 3C .Letα betheminimum
L 1 L 1 1
of(cid:107)X (cid:107)2 undertheconstraint(cid:107)L(cid:107)=1,thenwehave
L
(cid:112)
α(cid:107)L(cid:107)4 (cid:54)(cid:107)X (cid:107)2 <3+C +2 3C . (5)
L 1 1
√
Thatis,alltheL’ssuchthat f(L)<Cliveinthespherewiththeradiusof((3+C +2 3C )/α)1/4
1 1
inR9.
HereisabriefoutlineofthealgorithmforthedeterminationoftheN besttransformationstretch
tensorsandtheirassociatedlatticecorrespondences:
1. Calculatetheprimitivebasesandthetransformationmatricesfortheconventionalcellsfrom
the input lattice parameters: E and χ . Calculate α by minimizing the term X with
A,B A,B L
respecttoLforall(cid:107)L(cid:107)=1.
2. Choose N integral matrices (cid:96), i=1,...,N as the initial guess of the solution list such that
i
det(cid:96) isclosetodetE /detE anddist((cid:96),E ,E )issmall.
i B A i A B
7
(a) (b)
Bain corr.
New corr.
m=4
n ]
Bai u.
.
c a
1] [
0 e
0
[ c
w n
N e a
t
s
0-layer di
1-layer
2
[010] [100]
c c
0 2 4 6 8 10 12 14 16
number of modulation
FIG.3. TwopossiblelatticecorrespondencesinanFCCtomonoclinictransformation. (a)(010)projection
of the FCC lattice: the dark (resp. light) atoms are in the y=0 (resp. y=1/2) planes. The solid blue
and 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=1,2,3,4. (b) shows the
dependence of the values of the distance function on the modulation of the monoclinic c-axis for the two
latticecorrespondences.
3. LetC bethemaximum f((cid:96))for(cid:96)’sinthesolutionlist.
1 i i
4. Calculate the distance for all integral matrices in the sphere of radius of ((3+C +
1
√
2 3C )/α)1/4. Update the solution list as necessary. If the solution list is changed, re-
1
peatfromstep3.
5. Foreachsolution(cid:96),calculatetheCauchy-BorndeformationgradientF =E (E (cid:96))−1 and
i i B A i
the transformation stretch tensor U = (FTF)1/2. Finally, rewrite all the solutions in the
i i i
conventionalbases: L∗ =χ−1(cid:96)χ .
i A i 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 integers
satisfyingtherigorousbounds(5).
InFig.3wegiveanexamplecomputedbythealgorithmthatrevealsaswitchfromBaincorre-
spondencetoanewcorrespondencewithincreasinglatticecomplexity. Consideratransformation
from an fcc lattice with lattice parameter a = 2 to a monoclinic lattice with lattice parameters
0
a=1.41, b=1.99, c=1.42m, β =86◦, where the integer m>0 denotes the modulation along
monoclinic c-axis. Fig. 3(a) shows the undeformed fcc lattice projected onto (010) plane. The
8
two correspondences given by the algorithm are depicted for the m=4 case. Fig. 3(b) shows the
change in distance function for the two correspondences with m varying from 1 to 16. Initially
Bain correspondence is much smaller than the new one, however it loses its privilege after the
7th modulation. The results suggest that both kinds of lattice correspondence can be feasible in a
structural transformation for some special lattice parameters, and in this case m=7 has this spe-
cial status. As mentioned above, these long stacking period structures are common in martensitic
phasetransformations.
Table I shows the results calculated by the algorithm for six materials. The types of transfor-
mation are diverse and the principle stretches are consistent with the references. Among these
examples, we list two solutions for Zn Au Cu . The material has been recently found to sat-
45 30 25
isfy the cofactor conditions (the 2 constraints on U explained in paragraph 1) [7], which have
been shown [10] to promote unusually low thermal hysteresis (≈ 2◦C) and enhanced reversibil-
ity, owing to a fluid-like flexible martensite microstructure. It was believed [10] to transform
by 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 the
cofactor conditions. To investigate this further, the same sample of Zn Au Cu used in [10]
45 30 25
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 collected
continuously as heating/cooling through the transformation temperature. These patterns were an-
alyzed and indexed using the XMAS software [28]. The orientation relationships are determined
as the closest parallelisms of the crystallographic planes and zone axes between the indexed Laue
patterns of austenite and martensite respectively. They are (206) ||(203¯4) , (204) ||(102¯6) ,
a m a m
[211¯] ||[269¯1] ,[010] ||[010] and[11¯0] ||[89¯1] (seesupplementaryforindexeddiffractionpat-
a m a m a m
terns). However,thisdeterminationwithacceptederrorbarsdoesnotdefinitivelydistinguishthese
two mechanisms, since these relationships are so close that one could imagine that both mecha-
nismsoccursimultaneouslyinthematerial.
In addition to the reversible martensitic transformation, the algorithm is applicable to a wide
range of phase transformations even if the initial and final crystal structures do not have a
group/sub-group relation. Examples are Ti Mn and Sb Te /PbTe (Table I). The algorithm can
95 5 2 3
be also applied to organic materials when the molecular chains have sufficient periodicity. One
extreme example is the polymorphic transformation between two triclinic lattices in terephthalic
9
TABLE I. Transformation principle stretches (p. s.), the associated lattice correspondences (lat. cor.) and
derivedorientationrelationships(o. r.) forvariousphase-transformingmaterials
materials p. s. lat. cor. derivedo. r.
0.9363 [101] →[100] (204) ||(1¯026)
2 2 L21 M L21 M
1.0017 [010] →[010] [11¯0] ||[89¯1]
L21 M L21 M
Zn Au Cu [10] 1.0589 [4¯05] →[001] [211¯] ||[269¯1]
45 30 25 L21 M L21 M
L2 →M18R 0.9363 [1¯01¯] →[100] (204) ||(1¯027)
1 2 2 L21 M L21 M
1.0006 [010] →[010] [11¯0] ||[99¯1]
L21 M L21 M
1.0600 [909¯] →[001] [211¯] ||[279¯1]
2 2 L21 M L21 M
0.9178 [12012]A→[100]B (110)β1||(121)γ(cid:48)
CuAl Ni [29]
30 4
β →γ(cid:48) 1.0231 [010]A→[010]B [11¯1¯]β1||[21¯0]γ(cid:48)
1 1.0619 [1¯01] →[001]
2 2 A B
0.9052 [010] →[100] (11¯1¯) ||(214)
c h c h
Ti Mn [16]
95 5
1.0164 [1¯ 11] →[010] [1¯2¯1] ||[201¯]
222 c h c h
bcc→hexagonal
1.1086 [101] →[001]
c h
0.9791 [112¯]β(cid:48) →[100]β(cid:48)(cid:48) (100)β(cid:48)||(111)β(cid:48)(cid:48)
Ru Nb [30]
50 50
1.0024 [11¯0]β(cid:48) →[010]β(cid:48)(cid:48) [011]β(cid:48)||[11¯0]β(cid:48)(cid:48)
β(cid:48)→β(cid:48)
1.0169 [111¯]β(cid:48) →[001]β(cid:48)(cid:48)
0.9384 [1¯ 10] →[100] (1¯10) ||(010)
22 c h c h
Sb Te /PbTe[4]
2 3
0.9384 [011] →[010] [001] ||[4¯81¯]
22 c h c h
fcc→hexagonal
1.0779 [222¯] →[001]
c h
0.8244 [01¯2¯] →[100] [100] ||[112]
I II I II
Terephthalicacid[31]
0.9373 [110] →[010] [010] ||[102]
I II I II
triclinicI→triclinicII
1.3424 [001] →[001] [110] ||[010]
I II I II
acid (see Table I). In this case the calculated principle stretches agree well with the measured
macroscopicdeformationofthepolymorphictransformationofthismaterial.
We thank Liping Liu, Robert Kohn and Kaushik Bhattacharya for helpful discussions during
the preparation of this work. XC, YS, and RDJ acknowledge the support of the MURI project
Managing the Mosaic of Microstructure (FA9550-12-1-0458, administered by AFOSR), NSF-
10
