Kinetic Pathways of Phase Transformations in Two-Phase Ti ...

Kinetic Pathways of Phase Transformations in Two-Phase
Ti Alloys

TAE WOOK HEO, DONALD S. SHIH, and LONG-QING CHEN

Possible phase transformation kinetic pathways from the high temperature b phase to the low
temperature (a + b) two-phase Ti alloys were analyzed using the graphical thermodynamic
method and the assumption that diffusionless and displacive transformations take place much
faster than phase separation which requires long-range diffusion. It is shown that depending on
the composition of a b-stabilizing element, many transformation mechanisms are possible,
involving competing continuous and discontinuous displacive/diffusional transformations. We
discuss the proposed phase transformation sequences employing existing experimental micro-
structures.

DOI: 10.1007/s11661-014-2269-2
Ó The Minerals, Metals & Materials Society and ASM International 2014

I. INTRODUCTION associated with complicated competitions between
nucleation-and-growth and spinodal decomposition, or
TITANIUM (Ti) alloys have been extensively utilized between continuous and discontinuous displacive trans-
formations. It is a significant challenge to distinguish the
in aerospace applications, biomedical devices, and different mechanisms experimentally. For example, the
chemical processing equipment owing to their excellent spinodal decomposition process of an intermediate a¢ or
strength to weight ratio and corrosion resistance.[1] Pure a¢¢ phase with high solute content or b phase during the
Ti has two allotropic forms, hexagonal-close packed phase transformation is not easily detectable in exper-
(hcp) a and body-centered cubic (bcc) b, and it under- iments, since the decomposed solute-rich and solute-
goes b to a allotropic transformation upon cooling. poor phases exhibit a very small difference in lattice
Most commercial Ti alloys in structural applications parameters.
display (a + b) two-phase microstructures for high
strength.[2] Incorporating 3d transition metals such as Systematic theoretical analyses on phase transforma-
V, Mn, Fe, and Mo as alloying components which tion mechanisms have previously been employed to
stabilize the b phase[3] makes it possible for both phases understand coupled kinetic processes. For example,
to coexist. The mechanical properties of Ti alloys are Khachaturyan, Lindsey, and Morris theoretically inves-
very sensitive to the spatial configurations of the two tigated the equilibrium between a disordered solid
phases in the microstructure. Therefore, the prediction solution and an L12 ordered phase as well as the phase
of microstructural evolution of the phases plays a key transformation paths from a quenched disordered phase
role in the optimization of the mechanical properties of to the two-phase field in Al-Li alloys.[17] Soffa and
Ti alloys. Laughlin[18] applied a graphical thermodynamic method
to theoretically analyze concurrent clustering or order-
Different thermo-mechanical processing routes pro- ing processes. Fan and Chen discussed the possibility of
duce a wide spectrum of complex (a + b) two-phase spinodal decomposition during the structural phase
microstructures such as fully lamellar structure (or transformation in a ZrO2-Y2O3 system,[19] and Ni
basket-weave and Widmansta¨ tten structures) and bimo- et al.[20] discussed the transformation sequences associ-
dal (duplex) structure containing lamellae with primary ated with a cubic to tetragonal structural change
a phases displaying globular morphology.[2,4] There accompanied by a decomposition process. There have
have been a number of experimental efforts to under- also been attempts to model the formation of (a + b)
stand the phase transformations and microstructural two-phase microstructures of Ti alloys.[21,22] However,
evolution of binary[3,5–12] or multicomponent[13–16] Ti the systematic approach to analyzing the phase trans-
alloys. The phase transformations in Ti alloys are formation mechanisms in Ti alloys has not been well
established despite a number of existing thermodynamic
TAE WOOK HEO, formerly Postdoctoral Scholar with the analyses.[7,23]
Department of Materials Science and Engineering, The Pennsylvania
State University, University Park, PA 16802, is now Postdoctoral The main objective of this paper is to analyze the
Research Staff Member with the Condensed Matter and Materials possible phase transformation mechanisms in (a + b)
Division, Lawrence Livermore National Laboratory, Livermore, CA two-phase Ti alloys using graphical thermodynamics.[18]
94550. Contact e-mail: [email protected], [email protected] DONALD The phase transformation sequences that may undergo
S. SHIH, Technical Fellow, is with the Boeing Research & Technol- either nucleation-and-growth or a continuous transfor-
ogy, St. Louis, MO 63166. LONG-QING CHEN, Distinguished mation such as spinodal decomposition and continuous
Professor, is with the Department of Materials Science and Engineer- displacive transformation are analyzed following Refer-
ing, The Pennsylvania State University. ences 19 and 20. Microstructural features associated

Manuscript submitted February 3, 2014. METALLURGICAL AND MATERIALS TRANSACTIONS A
Article published online April 2, 2014

3438—VOLUME 45A, JULY 2014

with the possible kinetic pathways are demonstrated Figure 2. Since the intermediate a¢ phase or a¢¢ phase
using the existing experimental microstructures. with a high content of b-stabilizer can be decomposed
into solute-rich and solute-poor phases, i.e., there is a
II. PHASE STABILITIES AND KINETIC miscibility gap for the a phase,[1,24] we assume a double-
PATHWAYS well type energy curve for fa(X). However, the single-
well type energy curve is used for fb(X) for simplicity. As
We consider a Ti-M binary alloy system for simplicity a matter of fact, there may exist a miscibility gap in the b
where M is a b (isomorphous) stabilizing element such phase as the presence of oxygen might open up a
as V, Mo, or Nb. The incorporation of M decreases the miscibility gap[8] although the b miscibility gap is
b to a transition temperature, and the schematic phase believed to be at lower temperature.[7,15] In this case,
diagram of a binary alloy system is shown in Figure 1.[1] the free energy function of the b phase should display
We confine our attention to compositions and temper- the double-well type curve (see the dashed line in
ature changes at which an alloy undergoes the phase Figures 2 and 3) as suggested in Reference 23.
transformation from the high temperature b phase to
the low temperature (a + b) dual phases as indicated The thermodynamic stability of a solid solution can
with a blue solid arrow in Figure 1. Since a recent be determined by analyzing the topological properties of
research reported that no transient ordering was found the free energy curves as a function of composition and
in the system,[3] it is not addressed in our analysis. order parameter.[18] In particular, we can specify the
instability regimes by taking into consideration the
We explore the possible kinetic mechanisms involving second derivatives of the free energy with respect to g
the formation of intermediate phases or the existence of and X. We also assume that a displacive structural
metastable states by analyzing the thermodynamic phase transformation, which only requires shear and atomic
stabilities. The phase transformations in Ti alloys shuffles, takes place much faster than phase separation
involve both solute diffusion and displacive structural which requires long-range diffusion. The combination of
transformation from a bcc (b) structure during cooling. thermodynamic stability of a and b phases at a given
With regard to the displacive structural transformation, composition and temperature coupled with the assump-
the product phase can be a¢ (hcp, low solute content) or tion with respect to the relative rate of structural
a¢¢ (orthorhombic or distorted hexagonal, high solute transformations and phase separation would allow us
content) phase depending on the solute content. The to determine the kinetic pathways for a given Ti alloy.
criterion to determine the occurrence of a¢ or a¢¢ phase in
terms of solute content, i.e., the a¢/a¢¢ boundary as a An undercooled b phase can transform to the a phase
function of solute composition, for some selected binary at a fixed composition if it leads to a free energy
Ti alloy systems with transition metals was discussed in
Reference 1. We represent the state of an alloy using Fig. 2—Schematic diagram of free energy curves of a and b phases.
composition (X) of solute contents and order parameter
(g) for the structural identification, i.e., g = 0 represents
bcc (b) structure, and g = 1 represents displacively
transformed structures from the b phase, i.e., hcp (a or
a¢) or orthorhombic (a¢¢) structure. The local specific free
energy is expressed as a function of those two variables,
f(X, g). The two phases have separate free energy curves
fa(X) and fb(X), and the free energy function f(X, g)
becomes fa(X) or fb(X) depending on the structural state
of the system, i.e., the order parameter value. The
schematic diagram of the free energy curves is shown in

Fig. 1—Schematic phase diagram of a Ti-M system.[1] Fig. 3—Energy pathways of the variation of the structural order
METALLURGICAL AND MATERIALS TRANSACTIONS A parameter in a f-g-X space.

VOLUME 45A, JULY 2014—3439

Fig. 4—(a) Phase instability of initial b phase, (b) phase instability of intermediate a¢ or a¢¢ phase, and (c) 4 different subdivisions of a composition
range.

reduction. The b phase can be either metastable or Figure 4(c) is rapidly quenched, it undergoes bcc to hcp
displacive structural transformation continuously without
unstable depending on the content of b-stabilizer in the composition change (congruently), resulting in an inter-
mediate a¢ phase. The a¢ phase within this composition
alloy, i.e., b phase becomes more stable as the alloy range is metastable with respect to the decomposition. The
next step is the nucleation-and-growth of b phase from the
contains more b-stabilizer. It is supported by the fact supersaturated a¢ solid solution, leading to an equilibrium
(a + b) two-phase mixture. The kinetic pathway I is
that the b phase retention upon cooling occurs when the illustrated in Figure 5(a) and expressed as the following:

alloy includes high content of b-stabilizer as well as the

phase diagram of a binary Ti alloy in Figure 1 where the

b transus decreases as the b-stabilizer composition

ninuccrleeaasteios.n-Tahned-tgrraonwsftohrmatatwiohnichma@@g2y2fgta¼k0>e p0la(hceighthrsooulugthe Pathway I: β α ' α + β [1]

caonnistmentw, imtheta@@sg22ftagb¼l0e<) o0r spinodal (or continuous) mech- We use a solid arrow to represent the continuous
(low solute content, unstable). reaction, e.g., continuous displacive transformation or
spinodal decomposition, and a dashed arrow to denote
Figure 3 illustrates the schematic possible energy path- the nucleation-and-growth in the figures as well as the
expressions of kinetic pathways.
ways in f-X-g space for the variations of g with fixed
An undercooled b phase with a composition within
compositions. The state U on the figure is unstable with regime II undergoes more complicated transformations.
The b phase transforms continuously and congruently to
respect to the displacive structural transformation, and the supersaturated a¢ phase with the hcp structure or a¢¢
phase with the distorted hexagonal or orthorhombic
thus, the transformation occurs spontaneously and structure depending on solute content. The a¢ or a¢¢ phase
within this composition regime is, however, unstable with
continuously without the need to overcome a nucleation respect to the decomposition. Therefore, the decompo-
sition process occurs through the spinodal mechanism
barrier. On the other hand, the state M is metastable which produces solute-rich (a1) and solute-poor (a2)
phases. Figure 6(a) shows the experimental evidence of
with respect to the displacive structural transformation, spinodal decomposition within the a¢¢ phase which leads
to the modulated structure consisting of solute-rich and
and it undergoes the transformation through the nucle- solute-poor phases.[24] The composition of the solute-
poor a phase gradually reaches the equilibrium a phase
ation-and-growth mechanism due to an energy barrier. composition keeping the hcp structure, whereas the
solute-rich a phase experiences the hcp (a1) to bcc (b)
The whole composition range is then divided into two structural change when the composition exceeds the
critical composition where fa and fb intersect each other.
regimes according to the transformation mechanism as Eventually, the composition of the b phase reaches the
equilibrium composition of b phase. Therefore, the
illustrated in Figure 4(a). kinetic pathway can be summarized as the following:

The transformed a phase can subsequently undergo

phase separation due to a compositional metastability or

instability of a phase with respect to its decomposition

through either nucleation-and-growth or spinodal mech-

anism, respectively, depending on the curvature of the

local free energy, .@ 2 fa The composition range where the

@X2

a phase is unstable with respect to the decomposition is

specified by @ 2 fa <0 (Figure 4(b)).[25] Phase separation
@X2

occurs through the nucleation-and-growth process out-

side the instability regime.

With the combined structural instability of the b phase

and composition instability of the a phase, the whole

composition range within the two-phase regime can be Pathway II: b ! ða0 or a00Þ ! a1ðsolute-richÞ ½2Š
þ a2ðsolute-poorÞ ! a þ b;
divided into four different sub-regimes for the kinetic

pathways as shown in Figure 4(c). The hatched portion of

free energy curve of b phase (fb) represents the unstable The corresponding graphical representation is shown
state of b phase with respect to the structural transforma- in Figure 5(b).

tion. If a solid solution of composition within regime I in

3440—VOLUME 45A, JULY 2014 METALLURGICAL AND MATERIALS TRANSACTIONS A

Fig. 5—Phase transformation pathway in (a) the regime I, (b) the regime II, (c) the regime III, and (d) the regime IV.

Fig. 6—Experimental micrographs of intermediate phases: (a) modulated structure within a martensite plate (orthorhombic a¢¢) of a Ti-Mo alloy
(decomposition of a¢¢ phase),[24] (b) morphology of an x phase,[15] and (c) modulated structure within a b phase of a Ti-6Al-4V alloy (decompo-
sition of b phase).[15]

The phase transformation sequence in regime III is under even rapid quenching hardly happens.[24] How-
similar to those in regime II. The only difference is the ever, existing structural defects, e.g., grain boundaries
first stage within which the initial quenched b phase is and dislocations, would promote the nucleation of
structurally metastable with respect to the displacive martensitic phase. The product phase would be more
structural transformation, and thus, the transformation likely a¢¢ phase with the high solute content rather than
takes place through a nucleation-and-growth mecha- a¢ phase. The subsequent sequence is the same as that of
nism. In fact, the structural transformation from a b the regime II, i.e., the kinetic sequences in pathway III
phase with such a high content of b-stabilizing element (shown in Figure 5(c)) is

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 45A, JULY 2014—3441

Pathway III: β α" α1(solute-rich) + α2(solute-poor) α + β [3]

A b phase in the regime IV is metastable with respect III. EXPERIMENTAL MICROSTRUCTURES
to bcc to hcp (or orthorhombic) displacive transforma- AND KINETIC PATHWAYS
tion upon cooling, and the free energy of b phase is
lower than that of a phase with a same composition. In We analyzed the possible kinetic pathways during the
fact, multiple pathways are possible depending on the phase transformations only in a binary Ti-M system as an
presence of the intermediate phases in this regime. First example. However, the analyses are generally applicable
of all, the direct formation of stable a phase from b to any multicomponent systems. Commercial Ti alloys
phase through the nucleation-and-growth process, usually contain both a-stabilizer and b-stabilizer, although
which mostly happens heterogeneously, without the we considered alloys only containing b-stabilizers. It
occurrence of intermediate phases is expected. The should be noted that the decrease of b-stabilizer compo-
existing structural defects can be favorable nucleation sition is equivalent to the increase of a-stabilizer compo-
sites for a plates when the alloy is slowly cooled down or sition and vice versa. Moreover, the phase diagram of a
isothermally processed at the relatively high temperature multicomponent system can be reduced to a quasi-binary
as in the previous case. The kinetic pathway within the phase diagram. For example, a quasi-vertical section of a
regime IV can be expressed by phase diagram of a Ti-Al-V ternary system is very similar
to that of the binary system shown in Figure 1 except the
Pathway IV: β α + β [4] formation of other phases such as Ti3Al in the presence of
Al as an a-stabilizer.[4] In other words, the consideration of
and the pathway is represented in Figure 5(d) as a only one b-stabilizer composition would be able to capture
representative pathway in this regime. The pathway the main physical consequences of phase transformations
expressed in Eq. [4] is essentially inclusive. In this involving multiple alloying elements. The main scenario
regime, it is also possible that the formation of precur- for the possible kinetic pathways in multicomponent
sors such as b¢ or x phases can nucleate a plates from alloys would be very similar to those proposed in the
the metastable b phase depending on the heat treatment above section. Therefore, we demonstrate how our
history.[1,7,26,27] For example, x phases which are known proposed kinetic pathways in Section II can be employed
to exhibit the ellipsoidal or cuboidal shape[1] may to understand the underlying phase transformation mech-
precipitate when the alloy is quenched, since the direct anisms leading to experimentally observed typical micro-
bcc to hcp displacive reaction is suppressed due to high structural features by employing some examples of
b-stabilizer content[1,23,28] in this regime. Figure 6(b) existing experimental microstructures of general Ti alloys
shows an example of experimental observations of the x in the following section.
phase.[15] The x precipitates play a role as nucleation
sites of a phase during the aging process at higher A. Basket-Weave Microstructure
temperature.[10,12] In this case, the kinetic pathway in
A basket-weave microstructure is associated with the
Eq. [4] can be extended as β β + ω α + β . structural instability of the b phase. The formation of
Another possible kinetic pathway in this regime is that this type of microstructure can be explained using the
the metastable b phase (with respect to the displacive kinetic pathways I or II. The kinetic pathway I and II,
transformation) containing high b-stabilizer content can upon rapid cooling, involve a continuous displacive
decompose to (b + b¢) through either spinodal or transformation due to the structural instability. Figure
nucleation-and-growth mechanisms, and the solute lean 7(a) shows the corresponding experimental micrograph
phase (b¢) displacively transforms to x phase by the of a microstructure resulting from the displacive trans-
nucleation-and-growth mechanism[15] which acts as a formation.[24] It is known that the features of the
nucleation site of the a phase. Figure 6(c) shows a structure depend on the composition of alloying ele-
modulate microstructure consisting of solute-rich and ments. As the composition of b-stabilizer increases, the
solute-poor phases resulting from the decomposition of martensite plate becomes thinner and more acicular.[1,24]
the b phase.[15] The kinetic pathway in Eq. [4] can be The subsequent phase transformation step is the forma-
tion of the b phase through either nucleation-and-
extend as β β + β ' β + ω α + β in this case. growth (pathway I) or spinodal decomposition (path-
It should be noted that, upon sufficiently slow way II). For the case of pathway I, the nucleation of b
phases can occur homogeneously at the a¢ plate interior
cooling from the b phase field to (a + b) two-phase or heterogeneously at the a¢/a¢ boundaries. As a matter
field, the phase transformation starts from the temper- of fact, the heterogeneous nucleation is a more probable
ature just below the transition temperature at which the mechanism of b phase formation. When the solute
b phase is metastable. Therefore, the phase transfor- composition exceeds the critical point of the composi-
mation takes place through the nucleation-and-growth tional instability, the supersaturated a¢ or a¢¢ phase is
mechanism similar to the kinetic pathway III (Eq. [3]) spinodally decomposed within the a¢ or a¢¢ phase
or IV (Eq. [4]). The a phase particles are also expected
to heterogeneously nucleate at the existing structural METALLURGICAL AND MATERIALS TRANSACTIONS A
defects.

3442—VOLUME 45A, JULY 2014

Fig. 7—Experimental micrographs of (a) a microstructure of a Ti-Mo alloy resulting from a displacive phase transformation[24] and (b) a fully
basket-weave microstructure of a Ti-6Al-4V alloy.[29]

Fig. 8—Experimental microstructures of Ti-6Al-4V under slow cooling at (a) 473 K/h (200 °C/h)[30] and (b) 323 K/h (50 °C/h).[30]

(pathway II). The solute-rich regimes become equilib- remaining b grain boundaries, and the nucleated a phase
rium b phases, and the solute-poor regimes become grows into the b grain interior (through pathway III or
equilibrium a phases. The final experimental micro- IV also). Thus, it displays the colony-type microstruc-
structure as a result of above pathways would display ture or fully lamellar structure containing relatively
the basket-weave type, and the morphological features thicker grain boundary a lamellae. The a colonies near
of the microstructure are shown in Figure 7(b).[29] grain boundaries are shown in Figure 8(a)[30] as an
example. The continuous a layers along b grain bound-
B. Grain Boundary Nucleated Microstructure aries are clearly shown in an experimental micrograph in
Figure 8(b).[30]
Upon slow cooling and/or with a high content of
b-stabilizer, the a phase nucleates preferentially at the b C. Bimodal (Duplex) Microstructure
grain boundaries to minimize the strain energy as well as
interfacial energy, which results in the continuous a The proposed kinetic pathways can also be applied to
layer along the grain boundaries[1] as mentioned above. explain the microstructural evolution in bimodal micro-
Depending on the solute composition, the a phase can structures (Figure 9(a)[2]). During the cooling process
nucleate directly or through the intermediate phases from the temperature at which the alloy is homogenized
such as x and b¢ phases, similar to the kinetic pathway in the b phase field to the (a + b) two-phase field, the
IV (or its extensions), or the intermediate a¢¢ phase can microstructural evolution would follow the proposed
first nucleate, which is followed by a phase separation, kinetic pathways depending on the solute composition.
similar to the kinetic pathway III. Further cooling leads The only difference in processing conditions leading to
to additional nucleation of a phase (directly or through the bimodal microstructure from the above cases is the
the intermediate phases such as x and b¢ phases) at the recrystallization temperature. In this case, since the alloy
interface between continuous a layer and b phase or is recrystallized at the (a + b) two-phase field, the

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 45A, JULY 2014—3443

Fig. 9—Experimental micrographs of (a) bimodal (duplex) structure of IMI 834[2] and (b) a fully equiaxed structure of Ti-6.5Al-3.5Mo-1.5Zr-
0.3Si.[31]

equiaxed primary a phase incoherently forms at b grain ACKNOWLEDGMENTS
boundaries or mainly at triple junctions. In many
cases, the recrystallization temperature is close to the b This work is funded by the Center for Computational
transus. Therefore, the b phases, which will be trans- Materials Design (CCMD), a joint National Science
formed to primary a phases, are essentially metastable, Foundation (NSF) Industry/University Cooperative
i.e., the primary a phase formation would follow the Research Center at Penn State (IIP-1034965) and Geor-
kinetic pathway III or IV (or its extensions) depending gia Tech (IIP-1034968).
on the solute content. In addition, the phase transfor-
mation sequences for the remaining b phase to a REFERENCES
lamellae upon cooling after the recrystallization process
would follow the proposed pathways corresponding to 1. G. Lu¨ tjering and J.C. Williams: Titanium, 2nd ed., Springer,
the solute composition of the remaining phase. How- Berlin, 2007.
ever, it should be noted that, if the cooling rate is
sufficiently slow, the primary a phases continue to 2. G. Lu¨ tjering: Mater. Sci. Eng. A, 1998, vol. 243, pp. 32–45.
grow, leading to a fully equiaxed microstructure as 3. I.B. Ramsteiner, A. Scho¨ ps, F. Phillipp, M. Kelsch, H. Reichert,
shown in Figure 9(b)[31] rather than formation of a
lamellae.[1,2] The equiaxed microstructure can be also and H. Dosch: Phys. Rev. B, 2006, vol. 73, p. 024204.
found when the recrystallization temperature is suffi- 4. H.M. Flower: Mater. Sci. Technol., 1990, vol. 6, pp. 1082–92.
ciently low such that the equilibrium volume fraction 5. H.I. Aaronson, W.B. Triplett, and G.M. Andes: Trans. AIME,
of a phase is high.[1,2]
1957, vol. 209, pp. 1227–35.
IV. SUMMARY 6. M.J. Blackburn and J.C. Williams: Trans. Metall. Soc. AIME,

We analyzed and determined the possible kinetic 1968, vol. 242, pp. 2461–69.
pathways during the phase transformations in (a + b) 7. M.K. Koul and J.F. Breedis: Acta Metall., 1970, vol. 18, pp. 579–88.
two-phase Ti alloy. It was shown that the whole 8. W. Fuming and H.M. Flower: Mater. Sci. Technol., 1989, vol. 5,
composition range of b-stabilizer within the two-phase
regime can be divided into four different sub-regimes. pp. 1172–77.
Each composition regime displays a different kinetic 9. D.L. Moffat and D.C. Larbalestier: Metall. Trans. A, 1988,
pathway by competing continuous and discontinuous
transformation processes. We demonstrated that all the vol. 19A, pp. 1677–86.
major types of existing experimental microstructures can 10. D.L. Moffat and D.C. Larbalestier: Metall. Trans. A, 1988,
be rationalized using the proposed pathways for
describing the interplay between the structural transfor- vol. 19A, pp. 1687–94.
mation and diffusional process in Ti alloys. A full 11. E.S.K. Menon and R. Krishnan: J. Mater. Sci., 1983, vol. 18,
extension of thermodynamic analysis on the kinetic
mechanisms of phase transformations to multicompo- pp. 365–74.
nent Ti alloys is underway. 12. E.S.K. Menon and R. Krishnan: J. Mater. Sci., 1983, vol. 18,

pp. 375–82.
13. M.A. Dyakova, E.A. Lvova, I.N. Kaganovich, Z.F. Zvereva, and

L.S. Meshchaninova: Russian Metall., 1977, pp. 122–26.
14. C.G. Rhodes and J.C. Williams: Metall. Trans. A, 1975, vol. 6A,

pp. 2103–14.
15. Z. Fan and A.P. Miodownik: J. Mater. Sci., 1994, vol. 29,

pp. 6403–12.
16. S.M.C. van Bohemen, J. Sietsma, and S.v.d. Zwaag: Phys. Rev. B,

2006, vol. 74, pp. 134114.
17. A.G. Khachaturyan, T.F. Lindsey, and J.W. Morris: Metall.

Trans. A, 1988, vol. 19A, pp. 249–58.
18. W.A. Soffa and D.E. Laughlin: Acta Metall., 1989, vol. 37,

pp. 3019–28.
19. D. Fan and L.-Q. Chen: J. Am. Ceram. Soc., 1995, vol. 78,

pp. 1680–86.

3444—VOLUME 45A, JULY 2014 METALLURGICAL AND MATERIALS TRANSACTIONS A

20. Y. Ni, Y.M. Jin, and A.G. Khachaturyan: Acta Mater., 2007, 26. P.D. Frost, W.M. Parris, L.L. Hirsch, J.R. Doig, and C.M.
vol. 55, pp. 4903–14. Schwartz: Trans. ASM, 1954, vol. 46, pp. 231–56.

21. Y. Wang, N. Ma, Q. Chen, F. Zhang, S.-L. Chen, and Y.A. 27. G. Aurelio, A.F. Guillermet, G.J. Cuello, and J. Campo: Metall.
Chang: JOM, 2005, vol. 57, pp. 32–39. Mater. Trans. A, 2002, vol. 33A, pp. 1307–17.

22. Q. Chen, N. Ma, K. Wu, and Y. Wang: Scripta Mater., 2004, 28. F.R. Brotzen, E.L. Harmon, and A.R. Troiano: Trans. AIME,
vol. 50, pp. 471–76. 1955, vol. 203, pp. 413–19.

23. R.B. Gullberg, R. Taggart, and D.H. Polonis: J. Mater. Sci., 1971, 29. J.H. Zuo, Z.G. Wang, and E.H. Han: Mater. Sci. Eng., 2008,
vol. 6, pp. 384–89. vol. 473, pp. 147–52.

24. R. Davis, H.M. Flower, and D.R.F. West: J. Mater. Sci., 1979, 30. N. Stanford and P.S. Bate: Acta Mater., 2004, vol. 52, pp. 5215–24.
vol. 14, pp. 712–22. 31. L.J. Huang, L. Geng, A.B. Li, G.S. Wang, and X.P. Cui: Mater.

25. J.W. Cahn: Acta Metall., 1961, vol. 9, pp. 795–801. Sci. Eng. A, 2008, vol. 489A, pp. 330–36.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 45A, JULY 2014—3445