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Intermetallics 16 (2008) 42e51www.elsevier.com/locate/intermet
Prediction of the ordering behaviours of the orthorhombic phasebased on Ti2AlNb alloys by combining thermodynamic model
with ab initio calculation
Bo Wu a,b,*, Matvei Zinkevich b, Fritz Aldinger b, Maoyou Chu c, Jianyun Shen c
a College of Materials Science and Engineering, Fuzhou University, Shangjie, Minhou, Fuzhou 350108, PR Chinab Max-Planck-Institut fur Metallforschung und Institut fur Nichtmetallische Anorganische Materialien der Universitat Stuttgart,
Heisenbergstrasse 3, 70569 Stuttgart, Germanyc General Research Institute for Non-ferrous Metals, 301#, Xinjiekou Waidajie 2, 100088 Beijing, PR China
Received 17 December 2006; received in revised form 6 July 2007; accepted 14 July 2007
Available online 4 September 2007
Abstract
The orderedisorder transformation of intermetallics is of fundamental and technical importance. The ordering behaviours of the O phasebased on Ti2AlNb alloys are predicted by combining thermodynamic model with ab initio calculation. The site occupying tendencies of theconstituent elements are studied for the first time theoretically without referring experimental data as input. The predicted results show thatAl atoms always tend to occupy the g (4c1) sublattice, Ti atoms tend to occupy the a (8g) sublattice and Nb atoms the b (4c2) sublattice.The ordering tendencies of Ti and Nb atoms decrease with the increase of temperature, while the site occupation of Al atoms is weakly depen-dent on the temperature. The orderedisorder transformation belongs to a second-order transition with a continuous character. It is also predictedthat for the nonstoichiometric O phase with Al contents higher than 25 at.%, the site occupancies of the excess Al atoms prefer the b sublattice.The predicted site occupancy fractions and order parameters agree well with the reliable experimental data. The prediction has been improvedcompared with the GorskyeBraggeWilliams model, as well as our early LMTO-ASA calculations.� 2007 Elsevier Ltd. All rights reserved.
Keywords: A. Ternary alloy systems; B. Order/disorder transformations; D. Site occupancy; E. Ab initio calculations; E. Phase stability, prediction
1. Introduction
Ti2AlNb-based alloys are of great technological interest inhigh temperature applications since they have a higherstrength-to-density ratio and better room temperature ductilityand fracture toughness than conventional titanium aluminides,with no sacrifice in elevated temperature properties. They haveattracted considerable attention during the past two decades
* Corresponding author. College of Materials Science and Engineering, Fuz-
hou University, Shangjie, Minhou, Fuzhou 350108, PR China. Tel.: þ86 591
88071272; fax: þ86 591 22866537.
E-mail addresses: [emailprotected] (B. Wu), [emailprotected]
(M. Zinkevich), [emailprotected] (F. Aldinger), chumaoyou@hotmail.
com (M. Chu), [emailprotected] (J. Shen).
0966-9795/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.intermet.2007.07.007
[1e4] since Banerjee et al. [5,6] from India identified an or-dered orthorhombic phase in Ti3AleNb system and designatedit as the O phase in 1988. Ti2AlNb-based alloys may consist ofO, B2 and/or a2 phases depending on the alloy compositionand heat treatment condition [1,2,4]. And the O phase is thedominating phase, so its orderedisorder transformation is offundamental and technical importance [7]. So far, the investi-gations concerning the orderedisorder transformation of the Ophase are very limited. Mozer et al. [8] studied the site occu-pations of O phase based on the Ti2AlNb stoichiometry byneutron diffraction (ND). Muraleedharan et al. [9] studiedthe site occupations of alloying elements in the orthorhombicphase with composition as Tie27.5AlezNb (where z¼ 25, 20or 17.5) by atom location channelling enhanced microanalysis(ALCHEMI) and simulated the orderedisorder transformation
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43B. Wu et al. / Intermetallics 16 (2008) 42e51
of the O phase with aluminum content fixed at 25 at.% by us-ing a GorskyeBraggeWilliams (GeBeW) model. Severalyears later, new experimental data were reported from thesame laboratory in India [11]. These two sets of data do notagree with each other and no full disorder phase was reportedin the new experiment at 1173 K. Wu et al. [10] and Singhet al. [11] predicted the orderedisorder transformation and re-curred the ‘‘weak’’ first-order transition character in an ex-tended composition range with GeBeW model. Sarosi et al.[12] studied the order behaviours for Tie23Ale20Nb andTie27.5Ale23Nb orthorhombic alloy by ALCHEMI andND methods and characterized them with a series of orderingtie triangles. The crystallographic information [5,8,13] of thecompletely ordered O phase based on stoichiometric Ti2AlNbis given in Table 1. For the convenience of statement, generally,in Ti2AlNb-based O phase, a denotes the 8g sublattice, b de-notes the 4c2 sublattice and g denotes the 4c1 sublattice. Ofthe total number of sites, 50% belongs to the a sublattice,25% belongs to the b sublattice, and the rest belongs to the gsublattice. The available experimental results of site occupancyfractions and order parameters are summarized in Table 2. FromTable 2, it is seen that it is not easy to measure the site occu-pancy fractions, especially in the case of nonstoichiometric Ophase. In this paper, the experimental data available at 973 Kfrom Mozer et al. [8] by ND technique for the stoichiometriccompound Ti2AlNb (Tie25Ale25Nb) are accepted as reliableexperimental results. And considering the big difference of theexperimental data for some compositions in Refs. [9,11], for thenonstoichiometric alloy, the experimental data of Tie27.5Ale25Nb from Ref. [11] at 973 K as well as the experimental data ofTie27.5Ale20Nb from Ref. [9] at 1173 K are not accepted.Furthermore, when the GeBeW model is employed to predictthe orderedisorder transformation of O phase based onTi2AlNb alloys, the inherent limitations very often hamper usto get a reliable result. Firstly, one has to refer the experimentalresults in the GeBeW model, while it is not easy to obtain thereliable experimental data in most case as mentioned above.Secondly, only the first nearest neighbor interactions are takeninto account in the GeBeW model. In fact, the distance be-tween the second nearest neighbors is just about two times ofthe distance between the first nearest neighbors (the distance be-tween the first nearest neighbors is 2.9� 0.15 A and the dis-tance between the second nearest neighbors is 4.8� 0.5 A),so one can image that the error of the interaction energy is
Table 1
The crystallographic information of the completely ordered O phase based on
Ti2AlNb stoichiometry [5,8,13]
Element Multiplicity,
Wyckoff
notation
Point
symmetry
Internal parameter of atomic coordinates
x y z
Ti a (8g) m 0.2310 0.9041 0.25
Al g (4c1) m2m 0 0.1633 0.25
Nb b (4c2) m2m 0 0.6357 0.25
Orthorhombic, space group: Cmcm, no. 63; prototype: NaHg; Pearson symbol:
oC16; lattice parameters: a¼ 6.09 A, b¼ 9.57 A, c¼4.67 A, a¼ b¼ g¼ 90�,V¼ 271.93 A3.
unacceptable in the real many-body system. Servant and Ansara[14] described the ordering behaviours of the O phase with ther-modynamic models, and the experimental results from Mura-leedharan et al. [9] were adopted. Due to the complication ofthe phase relationship in the TieAleNb ternary system anda large number of thermodynamics’ parameters to optimize,the exact site occupancy fractions of the alloying elementswere still unavailable.
With the availability of powerful computers, reliablemodels, and efficient algorithms, nowadays, the combinationof ab initio electronic structure calculation and computationalthermodynamics has attracted more and more attention[15e17]. In our early work [18,19], the order behaviours ofthe O phase based on Ti2AlNb alloys were studied by combin-ing the general three-sublattice model with ab initio calcula-tion [16,17], where the method formulated within the linearmuffin-tin-orbital method in the atomic-sphere approximation(LMTO-ASA) [20] was employed to execute the ab initio totalenergy calculations. However, the LMTO-ASA method doesnot seem very convenient in systems presenting large size mis-match and in structures which are not closely packed, the pre-dicted results are not satisfied although it has been improvedcompared with the GeBeW model. The most accurate andtherefore the most computationally intensive techniques arethe full-potential methods [15], nevertheless, the projectoraugmented wave (PAW) potential method [21,22] is thestate-of-the-art in electronic calculations because it is almostas fast as the usual ultrasoft pseudopotential (US-PP) method[23,24] and gives energies very close to the best full-potentiallinearized augmented-plane-wave (FLAPW) calculations [25].To conclude the currently used methods, the Vienna ab initiosimulation package (VASP) [26e28] can be used with successin metallic system. VASP is based on the density functionaltheory (DFT) [29] within the local density approximations(LDAs) [30] or the generalized gradient approximations(GGAs) [31e33]. It is a plane-wave code. In the present paper,the ordering behaviours of the O phase based on Ti2AlNb alloyare revisited and VASP is employed to fulfill the ab initio totalenergy calculation. The paper is presented in five sections. InSection 2, the general three-sublattice model is described. InSection 3, the details of the ab initio calculation on the stablepure elements and the end-member compounds are presented.In Section 4, the site occupancy fractions are calculated andthe ordering behaviours are analyzed. The conclusions are pre-sented in Section 5.
2. Sublattice model
The thermodynamics properties of the O phase based onTi2AlNb can be considered by the following general three-sublattice model according to the crystallographic information
�Tiya
TiAlya
AlNbya
Nb
�0:5
�Tiyg
TiAlyg
AlNbyg
Nb
�0:25
�Tiyb
TiAlyb
AlNbyb
Nb
�0:25
ð1Þ
Table 2
Results of ALCHEMI and ND
Temp. (K) Alloy ( yAlezNb) Calculated site occupancy fraction (%) from Expt. data Order parameter sa Reference
a (8g) g (4c1) b (4c2)
Ti Al Nb Al Ti Al Nb
1223 27.5e25 64 3 33 100 62 3 35 0.02b [9]
27.5e20 69 3 28 100 71 3 26 0.02b [9]
27.5e17.5 76 3 21 100 70 3 27 0.06b [9]
1173 27.5e25 82 3 15 100 26 3 71 0.56 [9]
27.5e25 0.44 [11]
27.5e20 71 3 26 100 66 3 31 0.05b [9]
27.5e20 0.21 [11]
1073 27.5-25 73 3 24 100 44 3 53 0.29 [9]
27.5e20 76 3 21 100 57 3 40 0.19 [9]
27.5e17.5 84 3 13 100 54 3 43 0.30 [9]
973 27.5e25 77 3 20 100 36 3 61 0.41 [9]
27.5e25 0.78 [11]
27.5e20 79 3 18 100 50 3 47 0.29 [9]
27.5e20 0.46 [11]
25e25 82.3 0 17.7 100 35.4 0 64.6 0.469 [8]
23e20 86 0 14 92 50 0 50 0.36 [12]
a Order parameter s is the site occupancy fraction difference of Ti atoms on the a and b sublattices.b Disorder, according to CBED.
44 B. Wu et al. / Intermetallics 16 (2008) 42e51
where ynm means the site occupancy fraction of the species m
(Ti, Al or Nb) in the sublattice n (a, b or g). Giving an alloycomposition, once the site occupancy fractions in differenttemperature are calculated out, the orderedisorder transforma-tion can be investigated systematically.
For a given alloy composition TixTiAlxAl
NbxNb, where xi rep-
resents the mole fraction of the constituent element in the Ophase alloy. Based on the composition normalization andmass balance, Eqs. (2)e(8) can be obtained
xTiþ xAl þ xNb ¼ 1 ð2Þ
yaTiþ ya
Al þ yaNb ¼ 1 ð3Þ
ybTiþ yb
Al þ ybNb ¼ 1 ð4Þ
ygTiþ yg
Al þ ygNb ¼ 1 ð5Þ
0:5yaTiþ 0:25yb
Ti þ 0:25ygTi ¼ xTi ð6Þ
0:5yaAlþ 0:25yb
Alþ 0:25ygAl ¼ xAl ð7Þ
and
0:5yaNbþ 0:25yb
Nbþ 0:25ygNb ¼ xNb ð8Þ
The Gibbs energy of formation of the O phase from theroom temperature stable pure elements can be calculated by
DG¼ DH� TDS ð9Þ
where DH is the enthalpy of formation of the O phase from theroom temperature stable pure elements and DS is the corre-sponding entropy of formation.
DH can be calculated by
DH ¼X
i¼Ti;Al;Nb
Xj¼Ti;Al;Nb
Xk¼Ti;Al;Nb
yai yg
j ybkDHði:j:kÞ þDEH ð10Þ
where DH(i:j:k) represents the enthalpy of formation of the end-member compound from the room temperature stable pureelements Ti, Al and Nb in ground state (0 K), while the heatcontent difference of the enthalpy of formation between 0 Kand definite temperature is reasonably ignored in the presentwork, which will be discussed later based on the differentheat capacities of the compounds involved in the study. DEHrepresents the excess enthalpy of formation, which is alsoignored in the present work.
For DH(i:j:k), except DH(Ti:Al:Nb), which is accessible byexperiments, the rest end-member compounds are fictional, asmentioned in Section 1, in the early computational thermody-namics, the enthalpies of formation belong to thermodynamicsparameters to optimize [14]. Due to too many parametersinvolved in the optimizing procedure, obvious deviation wasfound between the calculated results and experimental data con-cerning the site occupancies. In the following section, the abinitio total energy method based on DFT is employed to calcu-late the total energy and thereafter to get the enthalpies of forma-tion of the end-member compounds in ground state (0 K).
DS can be calculated by
DS¼ SmixþDESþX
i¼Ti;Al;Nb
Xj¼Ti;Al;Nb
Xk¼Ti;Al;Nb
yai yg
j ybkDSTði;j;kÞ
ð11Þ
45B. Wu et al. / Intermetallics 16 (2008) 42e51
where Smix is the ideal mixing entropy of the system based onthe sublattice model of ideal solutions, which can be expressedas
Smix ¼ R
(� 0:5
Xi¼Ti;Al;Nb
yai ln�ya
i
�� 0:25
Xi¼Ti;Al;Nb
ybi ln�yb
i
�
� 0:25X
i¼Ti;Al;Nb
ygi ln½yg
i �)
ð12Þ
DES is the excess entropy of formation, which is ignored inthis paper, and DSTði;j;kÞ represents the entropy of formationof the end-member compound from the room temperature sta-ble pure elements Ti, Al and Nb, which can be calculated by
DSTði;j;kÞ ¼ STði:j:kÞ � 0:5STðiÞ � 0:25STðjÞ � 0:25STðkÞ ð13Þ
The ST in Eq. (13) is the entropy of a material at T (K) andgiven by Eq. (14) based on the third law of thermodynamics
ST ¼ZT
Cp
TdT ð14Þ
where T is the absolute temperature. So
DSTði;j;kÞ ¼ZT
Cpði:j:kÞ
TdT� 0:5
ZT
CpðiÞ
TdT� 0:25
ZT
CpðjÞ
TdT
� 0:25
ZT
CpðkÞ
TdT: ð15Þ
In the present work, the heat capacity, which is mainly con-tributed from the lattice vibration and thermal excitations ofthe electrons, is not studied yet, although it is definitely neces-sary for further investigation by combining ab initio calcula-tion and calorimetric experiments [34e36]. However, it isexpected that DH, the heat content difference of the enthalpyof formation of end-member compound between 0 K and def-inite temperature, as well as DSTði;j;kÞ, the entropy of formationof the end-member compound from the room temperature sta-ble pure elements are considerably low. In a similar case ofoxide ceramics, for example, the heat content difference forthe synthesis reaction of the LaGaO3 perovskite from its con-stituent binary oxides Ga2O3 and La2O3 between 0 and298.15 K is smaller than 1.0 kJ/mol (less than 1% of the quan-tity of the enthalpy of formation of LaGaO3 from Ga2O3 andLa2O3) based on the polynomial expressions of the heat capac-ity (Cp) of the involved compounds [37,38], where the heatcontent difference is calculated by
DH ¼Z298:15
CpðLaGaO3Þ dT� 0:5
Z298:15
CpðGa2O3Þ dT
� 0:5
Z298:15
CpðLa2O3Þ dT ð16Þ
So DS can be reasonably simplified as
DS¼ R
(� 0:5
Xi¼Ti;Al;Nb
yai ln�ya
i
�� 0:25
Xi¼Ti;Al;Nb
ybi ln�yb
i
�
� 0:25X
i¼Ti;Al;Nb
ygi ln½yg
i �)
ð17Þ
According to Eq. (9), the Gibbs energy of formation of theO phase from the stable pure elements can be expressed as
DG¼X
i¼Ti;Al;Nb
Xj¼Ti;Al;Nb
Xk¼Ti;Al;Nb
yai yg
j ybkDHði:j:kÞ
þRT
(0:5
Xi¼Ti;Al;Nb
yai ln�ya
i
�þ 0:25
Xi¼Ti;Al;Nb
ybi ln�yb
i
�
þ 0:25X
i¼Ti;Al;Nb
ygi ln½yg
i �): ð18Þ
When the orderedisorder transformation is in equilibrium, DGmust present a minimum, so the following system of equationsof constraint can be obtained
vDG
vyaTi
¼ vDG
vyaAl
¼ vDG
vyaNb
¼ vDG
vybTi
¼ vDG
vybAl
¼ vDG
vybNb
¼ vDG
vygTi
¼ vDG
vygAl
¼ vDG
vygNb
¼ 0: ð19Þ
In the above three composition variables and nine site occu-pancy variables, only two composition variables and four siteoccupancy fraction variables are independent. The other sixvariables can be derived from the six independent variables.Once the values of DH(i:j:k) are determined, the relationshipof site occupancy fractions with composition and temperaturecan be established, i.e.
ynm ¼ yn
mðxi;TÞ ð20ÞIn order to solve the complex system of partial differential
equations, the calculations of the site occupancy fractions arecarried out using the software package ‘‘Thermo-Calc’’ [39].
3. Ab initio total energy calculations
All calculations presented here are performed with VASP.Calculations are performed at 0 K without pressure andzero-point motion. For the GGA exchange-correlation energy,PerdeweBurkeeErnzerhof parametrization (PAW_PBE)[32,33,40] is used. The eigenstates are expanded in plane-wave basis functions, the ion cores are represented using thePAW potentials [21,22] and the Ti 3d and 4s, Al 3s and 3p,
46 B. Wu et al. / Intermetallics 16 (2008) 42e51
Nb 4p, 5s and 4d states are treated as fully relaxed energybands. The present calculations are performed at practicallyconverged basis sets. In particular, a 4� 4� 4 MonkhorstePack net [41] is used to sample the Brillouin zone of all theend-member compounds, a 15� 15� 15 MonkhorstePacknet is used to sample the Brillouin zone of the fcc Al andbcc Nb pure elements. While for the hcp Ti, the 8� 8� 6gamma centered grids’ method [40] is employed. The kineticenergy cutoff is set at 520 eV. Convergence tests concerning Kpoints and kinetic energy cutoff show that the total energiesconvergence is less than 2 meV/atom. The atomic geometryis optimized using HellmaneFeynman forces and conjugategradients [42]. The total energy, Etot, is minimized with re-spect to the volume (volume relaxation), the shape of the unitecell (cell external relaxation) and the position of the atomswithin the cell (cell internal relaxation) fully. The processwas terminated for an atomic force less than 0.05 eVA�1.The total energies of stable elements and the end-membercompounds of the O phase based on Ti2AlNb are summarizedin Table 3.
The comparisons between the experimental and computa-tional results from different sources for pure elements andsome compounds are compiled in Table 4 in order to validatethe present ab initio calculations. It is seen that the predictedlattice constants for the elements and compounds under inves-tigation are generally deviated within 1% of experimentalvalues, which is a typical result of the DFTeGGA, and thepredicted formation energies, DH, of Ti3Al_D019 and Ti2AlNb_Orth. are within the range of experimental values and con-sistent with the earlier first-principle treatments of thisproperty.
The enthalpies of formation of the end-member compound,DH(i:j:k), are calculated according to Eq. (21) as
DHði:j:kÞ ¼ Etotði:j:kÞ � 0:5EtotðiÞ � 0:25EtotðjÞ � 0:25EtotðkÞ: ð21Þ
The calculated enthalpies of formation of the end-membercompounds are summarized in Table 5.
Table 3
The total energies of the stable elements and the end-member compounds of the O
text, eV/atom
Element Etot Element
(a) Pure elements
Ti_HCP �7.7628 Al_FCC
(b) End-member compounds
End-member Etot End-member
Ti:Ti:Ti �7.7500 Al:Ti:Ti
Ti:Ti:Al �7.0407 Al:Ti:Al
Ti:Ti:Nb �8.3327 Al:Ti:Nb
Ti:Al:Ti �7.0407 Al:Al:Ti
Ti:Al:Al �6.1203 Al:Al:Al
Ti:Al:Nb �7.6458 Al:Al:Nb
Ti:Nb:Ti �8.3301 Al:Nb:Ti
Ti:Nb:Al �7.6457 Al:Nb:Al
Ti:Nb:Nb �8.9113 Al:Nb:Nb
4. Results and discussion
The temperature dependences of site occupancy fractionsof the different alloy compositions are plotted in Figs. 1e9.The available experimental data are superimposed with thecorresponding solid symbols. The selected alloy compositionscover four types.
(a) The stoichiometric O phase alloy Tie25Ale25Nb, seeFig. 1. The experimental results from Mozer et al. [8]by ND are superimposed.
(b) The nonstoichiometric O phase alloy with a fixed Al con-stituent on its stoichiometric content 25 at.%, while Ti andNb are of off-stoichiometric content. The representativealloys are Tie25Ale27Nb and Tie25Ale23Nb, seeFigs. 2 and 3, respectively.
(c) The nonstoichiometric O phase alloy with Al contentshigher than 25 at.%. The representative alloys areTie27.5Ale25Nb, Tie27.5Ale20Nb and Tie27.5Ale17.5Nb, see Figs. 4e7, respectively. And Fig. 6 is anenlargement of Fig. 5 in part in order to show the siteoccupying preference of the excess Al atoms clearly.The experimental data from Muraleedharan et al. [9] byALCHEMI are superimposed.
(d) The nonstoichiometric O phase alloy with Al contents lessthan 25 at.%. The representative alloys are Tie24Ale27Nb and Tie23Ale20Nb, see Figs. 8 and 9, respectively.The experimental data from Sarosi et al. [12] by AL-CHEMI are superimposed in Fig. 9.
The above plots of site occupancy fraction under differentcomposition and temperature show that Al atoms alwaystend to occupy the g (4c1) sublattice, Ti atoms tend to occupythe a (8g) sublattice and Nb atoms the b (4c2) sublattice.When the Al content is fixed at 25 at.%, the site occupancyfraction of Al on the g sublattice is nearly 1.0 and the site oc-cupancy fractions of Al on the a and b sublattices are less than0.005, which can be seen in Figs. 1e3. The ordering tendencyof the site occupations of Ti and Nb atoms decreases
phase based on Ti2AlNb with the sublattice model described in Formula (1) in
Etot Element Etot
�3.7413 Nb_BCC �10.2267
Etot End-member Etot
�6.1198 Nb:Ti:Ti �8.9082
�5.0644 Nb:Ti:Al �8.1300
�6.6834 Nb:Ti:Nb �9.5817
�5.0644 Nb:Al:Ti �8.1584
�3.7257 Nb:Al:Al �7.2157
�5.6044 Nb:Al:Nb �8.8117
�6.6800 Nb:Nb:Ti �9.5823
�5.6044 Nb:Nb:Al �8.7968
�7.2155 Nb:Nb:Nb �10.2258
Table 4
The comparison between the experimental and computational results from different sources for pure elements and some compounds, in order to validate the present
ab initio calculations
Element/compound Method Lattice parameter (A) Formation energy
(kJ/mol atom)
Reference
a b c
Ti_HCP Calc. 2.921 4.637
Calc. 2.929 4.628 [43]
Expt. 2.9508 4.6855(3) [44]
Al_FCC Calc. 4.030
Calc. 4.048 [43]
Expt. 4.0493 [45]
Nb_BCC Calc. 3.310
Calc. 3.322 [43]
Expt. 3.3279 [46]
Ti3Al_D019 Calc. 5.734 4.639 �27.4001
Calc. 5.750 4.650 �26.8227 [47]
Calc. 5.747 4.599 �26.0510 [48]
Calc. 5.613 4.664 �27.0158 [49]
Calc. 5.662 4.586 �27.0158 [50]
Calc. 5.650 4.571 �28.700 [51]
Calc. �26.0510 [52]
Calc. �27.9807 [52]
Expt. 5.768 4.614 �23.1564 to �27.0158 [53]
Expt. 5.773 4.623 �25.0861 to �27.9807 [52]
Ti2AlNb_Orth. Calc. 6.050 9.501 4.670 �26.2830
Calc. 6.091 9.573 4.673 �22.1926 [48]
Calc. 5.8781 9.2895 4.5470 �28.9455 [54]
Expt. 6.09 9.57 4.67 [5,8]
47B. Wu et al. / Intermetallics 16 (2008) 42e51
continuously with the increase of the temperature, while thesite occupation of Al atoms is weakly dependent on the tem-perature, which agrees well with the available experimentaldata. For the nonstoichiometric O phase with Al contentshigher than 25 at.%, the excess Al atoms preferentially occupythe b sublattice and the site occupancy fraction of Al atoms onthe a sublattice is less than 0.005, which is not the case as wasobserved by Muraleedharan et al. [9] that the excess Al atomsoccupy the a sublattice and b sublattice with equal probability.Possible reasons for the disagreement with the experimentaldata are due to the difficulty of the measurement of the site oc-cupancy fraction for the nonstoichiometric compounds.
In practice, the final heat treatment temperature range ofTi2AlNb-based alloy is between 873 K and 1173 K, becausethe phase equilibrium cannot be easily reached due to the lowmobility of atom diffusion when the temperature is lower than
Table 5
The enthalpies of formation of the end-member compound of the O phase bas
J/(mol atom)
End-member DH(i:j:k) End-member
Ti:Ti:Ti 1235.01 Al:Ti:Ti
Ti:Ti:Al �27 330.58 Al:Ti:Al
Ti:Ti:Nb 4446.75 Al:Ti:Nb
Ti:Al:Ti �27 330.58 Al:Al:Ti
Ti:Al:Al �35 530.68 Al:Al:Al
Ti:Al:Nb �26 283.72 Al:Al:Nb
Ti:Nb:Ti 4694.00 Al:Nb:Ti
Ti:Nb:Al �26 271.66 Al:Nb:Al
Ti:Nb:Nb 8050.47 Al:Nb:Nb
873 K, and the O phase will evolve into B2 and/or a2 phaseswhen the temperature is over 1273 K. Considering that the pres-ent calculations agree well with the experimental results in973 K both for the stoichiometric O phase alloy and nonstoi-chiometric O phase alloys (see Figs. 1, 4, 5, 7, 9), we can expectthat the orderedisorder transformation of the O phase to be eas-ier to reach its equilibrium above 973 K. So it is valuable to em-phasize that the kinetic issues do not affect the orderedisordertransformation of the O phase to reach its equilibrium, althoughin an early report [11] it was mentioned that the kinetic effectsincreased the experimental uncertainties.
The site occupancy preference can be understood from theview of the Gibbs energy of the compounds. Taking the stable el-ement as the reference state, the more negative the Gibbs energy,the more stable the structure. Here the ordering behaviours of thestoichiometric Ti2AlNb O phase at 973 K are considered. In the
ed on Ti2AlNb with the sublattice model described in Formula (1) in text,
DH(i:j:k) End-member DH(i:j:k)
�35 482.36 Nb:Ti:Ti 8351.98
�30 655.70 Nb:Ti:Al �13 568.20
�30 426.54 Nb:Ti:Nb 2757.06
�30 655.70 Nb:Al:Ti �16 305.97
1505.17 Nb:Al:Al �22 354.37
�23 325.25 Nb:Al:Nb �19 908.47
�30 100.91 Nb:Nb:Ti 2741.38
�23 325.49 Nb:Nb:Al �18 473.26
�22 336.28 Nb:Nb:Nb 85.63
200 400 600 800 1000 1200 1400 1600 1800 2000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Solid : Expt.
Ti-25Al-25NbyAl,8g
yNb,8g
yTi,8g
yAl,4c1
yNb,4c1
yTi,4c1
yAl,4c2
yNb,4c2
yTi,4c2
Site
occ
upan
cy f
ract
ion
Temperature, K0
Fig. 1. The site occupancy fractions of elements in sublattices vs. temperature
of O phase based on Tie25Ale25Nb with experimental data [8]
superimposed.
200 400 600 800 1000 1200 1400 1600 1800 2000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0Ti-25Al-23Nb
yAl,8g
yNb,8g
yTi,8g
yAl,4c1
yNb,4c1
yTi,4c1
yAl,4c2
yNb,4c2
yTi,4c2
Site
occ
upan
cy f
ract
ion
Temperature, K0
Fig. 3. The site occupancy fractions of elements in sublattices vs. temperature
of O phase based on Tie25Ale23Nb.
48 B. Wu et al. / Intermetallics 16 (2008) 42e51
Ti2AlNb O phase, Al atoms preferentially occupy the g sublatticeand Nb atoms preferentially occupy the b sublattice. The reason isthat the Gibbs energies are �26 283.7 J/(mol atom) and�26 271.1 J/(mol atom) for the end-member compoundsðTiÞaðAlÞgðNbÞb and ðTiÞaðNbÞgðAlÞb, respectively, and theGibbs energy is �28 458.2 J/(mol atom) for the equilibriumconfiguration ðTi0:80922Al0:00012Nb0:19066ÞaðAl0:99652Nb0:00252
Ti0:00095ÞgðNb0:61613Ti0:38061Al0:00323Þb.Likewise, the site occupying characters of atoms Ti, Al and
Nb can be understood in other cases according to the Gibbsenergy of the Ti2AlNb-based O phase with differentconfigurations.
Based on the plots of the site occupancy fractions of theTi2AlNb-based O phase alloys vs. different compositionsand temperatures, one can conclude that the essence of theorderedisorder transformation is that the Ti and Nb atomsvary their site occupancy fractions on the sublattices a andb. So in the composition and temperature ranges where theO phase is stable, we define a general order parameter, s, as
200 400 600 800 1000 1200 1400 1600 1800 2000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0Ti-25Al-27Nb
yAl,8g
yNb,8g
yTi,8g
yAl,4c1
yNb,4c1
yTi,4c1
yAl,4c2
yNb,4c2
yTi,4c2
Site
occ
upan
cy f
ract
ion
Temperature, K0
Fig. 2. The site occupancy fractions of elements in sublattices vs. temperature
of O phase based on Tie25Ale27Nb.
described in Eq. (22), which consists with the definition ofthe first paper studying the order behaviours of O phase inTi2AlNb-based alloy [9].
s¼ yaTi � yb
Ti: ð22Þ
Although the definition is rough for the nonstoichiometricO phase alloy, it is nearly rigorous for the alloy with a fixedAl content at 25 at.%, where the Al atoms nearly occupy allthe g sublattice, and there is a relationship as described inEq. (23)
s¼ yaTi � yb
Ti ¼ ybNb � ya
Nb: ð23Þ
The order parameter vs. temperature of the stoichiometricO phase Ti2AlNb is plotted in Fig. 10. The results from differ-ent models and methods are compared, including the GeBeWmodel with TieNb bond energy involved [9e11], sublatticemodel with LMTO-ASA total energy calculation [18,19],and sublattice model with PAW-GGA total energy calculation.
200 400 600 800 1000 1200 1400 1600 1800 2000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Solid : Expt.
Ti-27.5Al-25NbyAl,8g
yNb,8g
yTi,8g
yAl,4c1
yNb,4c1
yTi,4c1
yAl,4c2
yNb,4c2
yTi,4c2
Site
occ
upan
cy f
ract
ion
Temperature, K0
Fig. 4. The site occupancy fractions of elements in sublattices vs. temperature
of O phase based on Tie27.5Ale25Nb with experimental data [9]
superimposed.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Solid : Expt.
Ti-27.5Al-20Nb
yAl,8g
yNb,8g
yTi,8g
yAl,4c1
yNb,4c1
yTi,4c1
yAl,4c2
yNb,4c2
yTi,4c2
Site
occ
upan
cy f
ract
ion
Temperature, K
Fig. 5. The site occupancy fractions of elements in sublattices vs. temperature
of O phase based on Tie27.5Ale20Nb with experimental data [9]
superimposed.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Solid : Expt.
Ti-27.5Al-17.5NbyAl,8g
yNb,8g
yTi,8g
yAl,4c1
yNb,4c1
yTi,4c1
yAl,4c2
yNb,4c2
yTi,4c2
Site
occ
upan
cy f
ract
ion
Temperature, K
Fig. 7. The site occupancy fractions of elements in sublattices vs. temperature
of O phase based on Tie27.5Ale17.5Nb with experimental data [9]
superimposed.
49B. Wu et al. / Intermetallics 16 (2008) 42e51
The reliable experimental result by Mozer et al. [8] with NDtechnique is superimposed.
The order parameters vs. temperature and compositions ofthe O phase are plotted in Fig. 11, and the solid symbols rep-resent the corresponding experimental results, which havebeen summarized in Table 2. From Fig. 11, the effect of alloycompositions and temperature on the order behaviours can beanalyzed roughly.
According to Figs. 1e11, the order tendencies decreasegradually with the increase of the temperature, and theGeBeW model predicts a ‘‘weak’’ first-order transitionwith a discontinuous character at the critical temperature,while the sublattice models predict a second-order transitionwith a continuous character at all temperatures and the orderparameters go continuously to zero at infinite temperature.The results of present simulation agree well with the experi-mental result at 973 K, while some deviations are shown be-tween the calculated and experimentally determined siteoccupancy fractions and order parameters at 1173 K and
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20Ti-27.5Al-20Nb(in part)
yAl,8g
yNb,8g
yNb,4c1
yTi,4c1
yAl,4c2
yTi,4c2
Site
occ
upan
cy f
ract
ion
Temperature, K
Fig. 6. The site occupancy fractions of elements in sublattices vs. temperature
of O phase based on Tie27.5Ale20Nb with part enlarged.
1223 K. Possibly the uncertainty of experimental data inRef. [9] is a reason. Another possible reason is that the presentmodel is still rough to study the first-order orderedisordertransformation. So effective solutions for the scientific ques-tion are proposed. Firstly, measure the site occupancy fractionsof stoichiometric Ti2AlNb O phases which are obtained by us-ing the following heat treatment process: the B2 supertransusprocessed [2] samples followed by water quenching (WQ) andthen aging at 1223 K, 1173 K, or 1073 K, i.e., 1373 K/1 h/WQþ (1223, 1173, or 1073) K/240 h/WQ. Secondly, experi-mentally check the site occupancy tendencies of the excessAl atoms on the a and b sublattices for the nonstoichiometricO phase. Thereafter, the model employed in the present papershould be modified if there is still large deviation. However,considering the inherent limitation of the GeBeW model, itis believed that the present predictions are improved, althoughmore reliable experimental results at high temperatures (tem-perature range from 1073 K to 1273 K) are highly expectedto verify the predictions.
200 400 600 800 1000 1200 1400 1600 1800 2000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0Ti-24Al-27Nb
yAl,8g
yNb,8g
yTi,8g
yAl,4c1
yNb,4c1
yTi,4c1
yAl,4c2
yNb,4c2
yTi,4c2
Site
occ
upan
cy f
ract
ion
Temperature, K0
Fig. 8. The site occupancy fractions of elements in sublattices vs. temperature
of O phase based on Tie24Ale27Nb.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Solid : Expt.
Ti-23Al-20NbyAl,8g
yNb,8g
yTi,8g
yAl,4c1
yNb,4c1
yTi,4c1
yAl,4c2
yNb,4c2
yTi,4c2
Site
occ
upan
cy f
ract
ion
Temperature, K
Fig. 9. The site occupancy fractions of elements in sublattices vs. temperature
of O phase based on Tie23Ale20Nb with experimental data [12]
superimposed.
200 400 600 800 1000 1200 1400 1600 1800 20000.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0 Ti-xAl-yNb
25-25
25-27
25-23
23-20
24-27
27.5-25
27.5-20
27.5-17.5
Solid : Expt.
Ord
er p
aram
eter
Temperature, K0
Fig. 11. The plots of the order parameter vs. temperature and compositions of
the O phase, the solid symbols represent the corresponding experimental
results. See Table 2.
50 B. Wu et al. / Intermetallics 16 (2008) 42e51
5. Conclusions
The orderedisorder transformations of O phase based onTi2AlNb alloys are predicted by combining the generalthree-sublattice model with ab initio calculation. The tenden-cies of site occupations of the constituent elements are studiedfor the first time theoretically without referring experimentaldata as input. The predicted results show that Al atoms alwayspreferentially occupy the g (4c1) sublattice, Ti atoms tend tooccupy the a (8g) sublattice and Nb atoms the b (4c2) sublat-tice. The ordering tendencies of Ti and Nb atoms decreasewith the increase of the temperature, while the site occupationof Al atoms is only weakly dependent on it. In the temperaturerange where the O phase exists, the orderedisorder transfor-mation belongs to a second-order transition with a continuouscharacter. It is also predicted that for the nonstoichiometricO phase with Al contents higher than 25 at.%, the site occu-pancy of the excess Al atoms prefers the b sublattice. The pre-dicted site occupancy fractions and order parameters agree
0 200 400 600 800 1000 1200 1400
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Ti-25Al-25Nb
PAW-GGALMTO-ASAG-B-W Model Mozer et al. by ND
Ord
er p
aram
eter
Temperature, K
Fig. 10. Plots of the order parameter vs. temperature of the stoichiometric O
phase based on Ti2AlNb, the results of three predictions as well as of the neu-
tron diffraction (ND) experiment are compared.
well with the experimental data at 973 K. The prediction hasbeen improved compared with the GorskyeBraggeWilliamsmodel, as well as results based on LMTO-ASA calculationspublished elsewhere [18,19]. More reliable experimentalresults of site occupancy fractions at high temperatures (tem-perature range from 1073 K to 1273 K) are highly expected toverify the predictions.
Acknowledgements
The authors thank Chong Wang for helpful discussions.Dr. Bo Wu gratefully acknowledges the scholarship providedby the Max-Planck-Gesellschaft and the Science Foundationin Fuzhou University (Project no. 826212).
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