CommunicationModeling of Incommensurate xStructure in the Zr-Nb Alloys

BIN TANG, Y.-W. CUI, HUI CHANG,HONGCHAO KOU, JINSHAN LI, and LIAN ZHOU

Kanzaki force induced by substitutional solutes wasintroduced as a composition-dependent oscillatingdefect field in the Ginzburg–Landau model to simulatethe pre-transition structure preceding the b-to-x phasetransformation. A three-dimensional (3-D) modulatedincommensurate structure of the x phase composed ofalternating x and b phases was predicted to be presentin the moderately concentrated Zr-Nb alloys. Themodeling work suggests that the Kanzaki force acts as aresistance that generates and retains the incommensu-rate structure of the x phase.

DOI: 10.1007/s11661-012-1214-5� The Minerals, Metals & Materials Society and ASMInternational 2012

The metastable x phase present in the bcc phase ofGroup IVb (Zr, Ti, Hf) transition metals and their alloyshas been a subject of intense interest since thex phasewasdetected.[1–6] Over the years, a substantial amount ofexperimental and theoretical work has been carried out inorder to understand the various aspects of the x phasetransformation, e.g., microstructure, atomic mechanism,and pre-transition effect.[7–9] It has been well known for along time that the b to x phase transformation exhibitspronounced pre-transition effects, e.g., diffuse intensityand offset of peak position from crystallinex positions inthe electron diffraction pattern and phonon softeningalong the wave vector kx = 2/3 111h i.[10,11] It is alsoknown that the formation of the x phase with sharpreflections is typically limited to the Ti and Zr alloys with<10 at. pct solute content, and the diffuse intensity andoffset increase with the solute content.[12] These featureshave prompted experimental work[13,14] and structuralmodeling in the field.[15,16] Cook[16] proposed a hetero-phase fluctuation mechanism that explained the offsetfeature by constructing an incommensurate x structure.Realizing the central role of the sizemismatch between theatomics species in the formation of diffuse x, Sinkler andLuzzi[17] modified the Cook’s model and devised aphenomenological structural model that reproduced the

variation of diffuse x peaks with solute composition andsolute group number. On the other hand, the x-typefluctuation was understood quantitatively asmediated bythe local lattice distortions, i.e., the Kanzaki force, for thesubstitutional Ti-V[12] and Zr alloys[18] and interstitialalloys Nb(O/N)x.

[19] The Kanzaki force is defined as avirtual force that a point defect (for example, solute atom,impurity, and vacancy) exerts on the neighboring hostatoms and, in turn, causes a local displacement of the hostatoms within the defect core.[20,21] Many of the predic-tions of atomic displacement based on the Kanzaki forcemodel have been confirmed experimentally,[22,23] and theKanzaki force concept has beenwidely used to analyze theimpurity-induced strain field[24,25] and elastic interac-tions[26] in metals.Recently, a new approach has been developed to

introduce an oscillating stress field into the Ginzburg–Landau (GL) model for the description of first-orderphase transformation,[27] for instance, the Kanzaki forcesurrounding structural defects or the Peierls barrier.This encouraged us to use a similar method, i.e., linkingthe incommensurate microstructure to the Kanzakiforce, to model the formation of the incommensuratestructure that consists of a decomposed dual-phasestructure, as illustrated by the lattice study for the Cu-Zn alloys,[28,29] and to understand the variation ofdiffuse x peaks and microstructure formation withsolute content and temperature.The GL energy describing the athermal x transfor-

mation can be written as

G ¼ZV

1

2

Xi;j;p

kðpÞij

@gp@ri

� �@gp@rj

� �þX4p¼1ðAg2p þ Bg3pÞ

"

þCX4p¼1

g2p

!2

þFcpl

35dVþ Felastic ½1�

where gp is the order parameter with p standing for thefour different crystallographic orientation variants ofthe x phase and kðpÞij is the gradient energy coefficientthat reflects the crystallographic anisotropy of the inter-facial boundary. The second and third terms of Eq. [1]are the Landau-type chemical-free energy; the fourthterm, Fcpl, is the coupling term between the orderparameter and the Kanzaki force around Nb solute;and the last term, Felastic, is the coherent strain energyproduced by the lattice misfit between the x and bphases, respectively. Note that the calculation of thechemical-free energy and the elastic energy Felastic hasbeen addressed already in our previous articles.[30,31]

The coupling term, Fcpl, was explicitly added to thefree energy expansion as –W(k)u(k), where W(k) is theFourier transform of the Kanzaki force acting on hostatoms. It is apparent that the coupling is introduced as aspatially oscillating field that generates an athermalresistance to interface propagation for athermal b to xtransformation.[32] The other approach is to add a termto the elastic interactions.[33] Figure 1(a) illustrates theKanzaki forces acting on the host Zr atoms aroundthe interstitial defect at the octahedral site and aroundthe substitutional Nb atom that is smaller than the host

BIN TANG, Doctoral Candidate, HUI CHANG andHONGCHAO KOU, Associate Professors, and JINSHAN LI andLIAN ZHOU, Professors, are with the State Key Laboratory ofSolidification Processing, Northwestern Polytechnical University,Xi’an 710072, P.R. China. Y.-W. CUI, Staff Researcher, is with theComputational Alloys Design Group, IMDEA Materials, 28040Madrid, Spain. Contact e-mail: yuwen.cui@imdea.orgorg

Manuscript submitted September 22, 2011.Article published online June 7, 2012

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 43A, AUGUST 2012—2581

Zr atom in the center. Note that the forces around theinterstitial defect are shown up to the third-nearestneighbors, i.e., fin1 , f

in2 , and fin3 , while the forces around

the Nb solutes are only shown to the nearest neighbors,fsub1 . Figure 1(b) is a schematic representation of theathermal x structure that forms out of the b parentphase and the 111h i stacking sequence of the bccstructure showing the effect of the Kanzaki forcesaround the two types of structural defects on theformation of x and anti-x structures. It can be seenthat the Kanzaki force induced by the Nb solute has acomponent in the direction that favors energetically theunfavorable anti-x structure; in contrast, the interstitialdefect tends to stabilize the x structure.

According to the Kanzaki model shown inFigures 1(a) and (b), the static displacement aroundthe Nb solute acting on the nearest collapse (111) planescan be expressed via a coupling term between W(k) andthe lattice Green’s function GðkÞ:Fcpl ¼ n �

XWþðkÞGðkÞWðkÞ � ujul

¼ n0 �X

WðkÞuðkÞ � g2 ½2�

in which uj and ul are the planar displacements,j and l denote the plane numbers shown in

Figure 1(c), n and n¢ are the adjustable phenomeno-logical coefficients, and u(k) is the Fourier transformof the atom displacement. Assuming that the Nb sol-utes are uniformly distributed in the parent phase,the magnitude of W(k) can be described byFsub ¼ Asub � sin2ðp � r=ksubÞ. Note that the wavelengthksub and the amplitude Asub, both as a function ofthe Nb content, can be calculated from the macro-scopic material parameters cij (elastic constants), ab

(lattice parameter) and dab/dxNb.[12] In this work, a

3-D Landau model was developed for the incommen-surate x phase; therefore, the Kanzaki force wasdepicted as

Fsub ¼ Asub � sin2pxki

� �� sin2 py

kj

� �� sin2 pz

kk

� �½3�

where ki, kj and kk are the wavelengths of the defectfiled along the X, Y, and Z directions in a Cartesiancoordinate system, respectively. For the sake of simpli-fication, we assumed that ki ¼ kj ¼ kk ¼ k and thewavelength of the defect field is k ¼ 3ab � 10abXNb.The spatio-temporal evolution of the order parametercan be described by the time-dependent Ginzburg–Landau equation.[34]

bcc > < 111

2 sub f 1

sub f 2

in f 1

1

1 2 3 4

0

bcc > < 111

2 sub f 1

sub f 2

in f 1

1

1 2 3 4

0

(a)

(b)

Fig. 1—(a) The Kanzaki model: the Kanzaki forces acting on the nearest neighbor atoms around an interstitial impurity at the octahedral siteand a substitutional Nb solute at the center. The x-like displacement along the 111h ib direction is shown by a purple arrow. (b) Formation ofthe x phase from a bcc-Zr lattice along the 111h ib direction. h1 and h2 represent the angles between fin1 and 111h ib and between fsub1 and 111h ib,respectively.

2582—VOLUME 43A, AUGUST 2012 METALLURGICAL AND MATERIALS TRANSACTIONS A

In all modeling, we took the 50 9 50 9 50 systemsinto account with the length scale Dx = 0.1 nm, and setn¢ = 3.0 and usub = 0.02 n m. All the parameters of thestrain energy were detailed in a previous article.[30] Inthis work, the modeling was performed for four alloys:Zr-20 at. pct Nb, Zr-15 at. pct Nb, Zr-10 at. pct Nb,and Zr-5 at. pct Nb. We first considered the Zr-20 at.pct Nb alloy and let it be quenched from the b-phasefield to 400 K (127 �C). It yields a 3-D oscillating defectfield with the wavelength

ffiffiffi3p

ab along [111]b(Figure 2(a)). The modeling started with one small xseed setting at the center. Figure 2(b) shows the x particleprecipitates in a shape of ellipsoid and aligns with itsmajor axis along the 111h i direction. This satisfactorilyrepresents the widely observed shape of the athermal xprecipitate in the Zr-Nb alloys;[35] however, the xellipsoid exhibits a rough surface as compared to anideal particle (Figure 2(g)). The slice plot along theð1�10Þb plane in Figure 2(c) unambiguously reveals thatthe precipitate is actually an ellipsoidal domain com-posed of alternating dual phases of x ‘‘frame’’ and b‘‘gap.’’ The obtained ellipsoidal x+ b domains resem-ble the predicted dual-phase incommensurate structureof the x phase for the Cu-Zn[36] and Zr-Nb alloys.[37]

However, since a 3-D oscillating defect was consideredin this work, the predicted structure shows a 3-Dalternating ‘‘x frame+b gap’’ feature vs one-dimensional(1-D) laminated ellipse in the lattice study.[28,36] Wenoticed that the use of the 1-D field in the GL modelwould lead to a 1-D laminated incommensurate structure.The same happens to three other variants (Figures 2(d)through (f)).

The formation of the decomposed dual-phase struc-ture of x phase can be understood in terms of the

coupling term that imposes a periodic resistance to the xtransformation. When quenching the alloy to a temper-ature above the x start temperature T0(x), the couplingterm remains small as the order parameter fluctuatesaround zero, therefore making a negligible contributionto the energy of the system. As a result, the b matrix, asa whole, retains its bcc structure. When reducing thetemperature to T<T0(x), the order parameter departsfrom the zero value; then the Kanzaki force starts toplay its role by fluctuating the GL energy.[14] In the casewhen and where the local GL energy is fluctuated up tothe L0 curve or above, as shown in Figure 3(a), thetransformed x phase, if transformed, will transformback to the b structure, while in the region with a net GLenergy below L0, the x phase will form. This, in turn,develops a decomposed dual-phase structure with alter-nating b and x phases. As the quenching temperaturedrops to 350 K (77 �C), 300 K (27 �C), and 250 K(–23 �C) (Figures 3(b) through (d)) the alternating bgaps in the dual-phase incommensurate structure keepshrinking and the surface becomes smoother. This isassociated with the fact that the same coupling termrelatively contributes less vs the chemical driving force ata lower quenching temperature.As revealed by the experimental observation[38] that the

Zr-Nb alloys with less than 8 at. pct Nb occur, anegligible shift of the intensitymaxima of thex reflectionsaway from the hexagonal positions and the shift becomesignificant as the Nb content increases while the x peaksbroaden. This feature can also be grasped in the snapshotsfor the various Zr-Nb alloys in Figure 4. Figure 4(a)shows the morphology of the incommensurate x phasepredicted for the Zr-5 at. pct Nb alloy with the slice plotin the inset, and Figure 4(d) is the Kanzaki force field. As

Fig. 2—(a) 3-D distribution of defect fields near the Nb solute in Zr-20 pct Nb alloy. (b) 3-D decomposed dual-phase structure of x phase(½111�variant) induced by the defect field shown in (a) at 400 K (127 �C). (c) The slice plot at ð1�10Þb from (b). (d) through (f) the other threevariants and (g) the x particle without a defect field.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 43A, AUGUST 2012—2583

theNb content increases, thewavelength of the oscillatingfield decreases (Figures 4(a) through (f)) and the magni-tude of the Kanzaki force increases (Figure 4(g)), whichthen enhances the shift and diffuse features. The presentmodeling suggests that the Kanzaki force, previouslyexplained as a physical origin in diffuse scattering featuresin Zr and Ti alloys,[12,18,19] also acts as a resistance thatgenerates and retains the incommensurate structure of thex phase. It is also apparent from the slice plots that thedirection of x layers in the decomposed structure has anangular deviation of 54.7 deg from the 111h i direction ofthe bcc, which is close to the observed ~60 deg.[37] Fromthe present model, the direction of layers or blocksdepends on the actual local distribution of theNb solutes.

According to experimental and calculationresults,[19,39] the radial symmetric Kanzaki force inducedby the Nb solutes has a component that is opposite tothe collapse of x transformation, thus triggering theformation of the incommensurate x phase. In contrast,the interstitial defect or a substitutional solute thatexpands the host lattice favors the retention of an idealx embryo or phase. Therefore, it needs a sign change tothe quadratic coupling term for these alloys. However,

care should be taken when the model is applied to diluteZr-Nb alloys and pure Zr, because the x phase has toadditionally compete with the martensite phase.To summarize, we introduced the Kanzaki force as an

oscillating stress field via a quadratic coupling term inthe GL model to describe the incommensurate structureof the x phase that occurs prior to the b to x phasetransformation. The present modeling, as illustrated forthe Zr-Nb alloys, shows that the incommensurate xphase exhibits a 3-D modulated microstructure com-posed of decomposed dual phases x and b. While thecharacteristic diffuse and offset features of the x-phasetransformation were explained by the Kanzaki forcearound the solute in the past,[12] our modeling appro-priately links the Kanzaki force with the formation of amodulated incommensurate x structure, and finds thatthe composition and temperature dependence of thepredicted incommensurate structure closely resemble thevariation of diffuse and shift features in the electrondiffraction pattern with solute composition and temper-ature observed for the b-to-x transformation in themoderately concentrated Zr-Nb alloys. The developedmodel is expected to also work for interstitial alloys or

elastic F cpl F

) ( K f Δ

f Δ

sub F

n

n ) ( 0 T T =

elastic F cpl F

) ( K f Δ

f Δ

sub F

n

n ) ( 0 T T =

(a)

(b) (c) (d)

Fig. 3—(a) Schematic drawing of the free energy change (Df) of the incommensurate x phase as a function of the order parameter, g. nx and nb

determine the thickness of the x and b layers, respectively. The decomposed dual-phase structure of the x phase for the Zr-20 at. pct Nb alloyquenched from the b phase field to (b) 350 K (77 �C), (c) 300 K (27 �C), and (d) 250 K (–23 �C).

2584—VOLUME 43A, AUGUST 2012 METALLURGICAL AND MATERIALS TRANSACTIONS A

substitutional alloys having a solute that expands thehost lattice.

This work is supported by the fund of the StateKey Laboratory of Solidification Processing in NWPU(Grant No. SKLSP200906) and 111 Project (No.B08040).

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0 . 0 4 0 . 0 8 0 . 1 2 0 . 1 6 0 . 2 0

- 0 . 0 0 6 4

- 0 . 0 0 6 2

- 0 . 0 0 6 0

- 0 . 0 0 5 8

- 0 . 0 0 5 6

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