Theoretical Research on Phase Transformations in Metastableb-Titanium Alloys

CHENG LIN, ZHILIN LIU, and YONGQING ZHAO

The theoretical study of phase transformation in metastable b-titanium alloys is of primaryimportance due to its effect on the properties and applications of the alloys. On the basis of theempirical electron theory (EET) of solids and molecules, phase transformations such as thestability of the b phase, the separation of the b phase, and the eutectoid reaction of the b-phaseenrichment in b-stabilizing elements were investigated with the use of the electron structureparameters of phases in this article. The results provided a theoretical basis for the compositiondesign of the metastable b-titanium alloys.

DOI: 10.1007/s11661-009-9798-0� The Minerals, Metals & Materials Society and ASM International 2009

I. INTRODUCTION

THE metastable b-titanium alloys have been widelyapplied to aviation industry due to their high strength-to-density ratio, good hardenability, excellent fatigue/crack-propagation behavior, and corrosion resistance.In recent years, many investigators have carried outprofound experimental researches on metastableb-titanium alloys.[1–4] With the rapid development ofcomputers, many numerical methods, such as ab-initiosimulation, molecular dynamics (MD) simulation, andMonte Carlo simulation, have become increasinglypopular in the calculation prediction of the propertiesof materials. Although ab-initio simulation is convincingbecause of its safe basis in quantum theory, it is onlysuitable for systems with small clusters.[5,6] With the MDmethod, calculations involving millions of atoms arenow feasible with the use of massively parallel comput-ers.[7] However, these methods still cannot treat theelectron structure of materials actually used in industrywith huge numbers of atoms. In this regard, empiricaltheory has special superiority.

In 1978, Yu established the empirical electron theory(EET) of solids and molecules[8] on the basis ofPauling’s electron theory of metal and the quantumtheory; the content and calculation methods of the EEThave been published in detail in References 9 through11. Using the bond length difference (BLD) method inthe theory, the covalent electron pair numbers of allbonds in a crystal with a known lattice constant caneasily be calculated, so that the bond energy can also becalculated. Benefiting from the EET, investigatorssuccessively studied solid solution, compound, electrical

conductivity, magnetic property, phase diagrams, phasetransition, microstructures, phase interfaces, mechanicalproperty, etc.[12–18]

In order to retain the entire b phase upon quenchingfrom above the b transus, the molybdenum equiva-lence should be greater than 10.[19] While metastableb-titanium alloys contain more b stabilizers, such as Mo,V, Cr, Fe, etc., metastable b-titanium alloys are primar-ily strengthened through the precipitation of the aphase, including elements such as Al, Zr, or Sn. Whenthe content of the alloy elements is higher, the metasta-ble b phase is decomposed into a b phase that is bothpoor in b-stabilizing elements and rich in b-stabilizingelements at 500 �C and 550 �C, respectively. This isknown as the separation of the b phase. Furthermore,the metastable b-titanium alloys containing aluminumcan effect even more quickly the transition of b¢ fi aafter the decomposition reaction of b fi b + b¢. How-ever, the mechanisms of phase transitions are not clearat the electron structure level.The goal of this article is to present some new

understandings about the stability of the b phase, theseparation of the b phase, and the eutectoid reaction ofthe b-phase enriching in b-stabilizing elements, throughthe use of the electron structure parameters of metasta-ble b-titanium alloys.

II. CALCULATION OF VALENCE ELECTRONSTRUCTURE

The metastable b-titanium alloys after solution treat-ment above the b-transformation temperature hadentirely b phase with equiaxed grains in the quenchedmicrostructure. The quenched microstructure containsthe body-centered b-Ti structure unit or cell, the b-Ti-Mstructure unit (M atom into b-Ti cell), and the b-Ti-Mx-My structure unit (Mx and My atoms into the b-Ti cell);the properties of the metastable b-titanium alloys aretogether decided by the valence electron structures ofthese structure units.

CHENG LIN, Doctoral Student, and ZHILIN LIU, Professor, arewith the College of Materials and Chemical Engineering, LiaoningUniversity of Technology, Jinzhou 121001, Liaoning, People’sRepublic of China. YONGQING ZHAO, Professor, is with theNorthwest Institute for Nonferrous Metal Research, Xi’an 710016,Shanxi, People’s Republic of China. Contact e-mail: lgveslc@tom.com

Manuscript submitted July 22, 2008.Article published online February 25, 2009

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 40A, MAY 2009—1049

A. Calculations of Valence Electron Structure and BondEnergy of b-Ti

1. Head and tail states and hybrid table of titaniumIn EET, while atoms form molecules or solids, the

valence electrons are described according to theirdistribution and their effect as covalent electrons nc,lattice electrons nl, magnetic electrons nm, and dumbpair electrons nd, denoted by d, / (or ), ›, and ||,respectively, where / denotes one-lattice electron anddenotes two-lattice electrons.

In EET, the head and tail states of titanium (A type)are given as

d s p

h state: s2p1d1

t state: s1p1d2

l = 2, m = 1, n = 1, s = 0; l¢ = 1, m¢ = 1, n¢ = 2,s¢ = 1

For titanium, there are Rh(1) = 0.13530 nm andRt(1) = 0.10530 nm.

After substituting these parameters into the K for-mula,[8–11] the hybridization table of titanium (A type)can be calculated as shown in Table I.

2. Covalent bond distribution of b-Ti crystal cellThe b-Ti is of bcc lattice; the lattice constant

a = 0.33065 nm at 900 �C. Figure 1 shows the atomarrangement and bond distribution of a b-Ti crystal cell.In the crystal cell, only two covalent bonds (representedby Ba (a = A, B)) should not be neglected. The covalentbond length Da formed by two atoms in the cell is alsocalled the experiment bond length; its value can becalculated with the lattice constant a. Therefore, theexperiment bond lengths in ab-Ti cell are as the following:

DA ¼ffiffiffi

3p

2a ¼ 0:28635 nm ½1�

DB ¼ a ¼ 0:33065 nm ½2�

In EET, the number of covalent bonds that have theequivalent bond length and uniform environment is

called the equivalent bond number Ia. Its formula isgiven as the following:[8–11]

Ia ¼ IM�IS�IK ½3�

where IM represents the reference atom number in thestructure or the molecule; IS represents the equivalentbond number for a reference atom to form Ba; and IKis a parameter that equals 1, when the two atoms thatform the bond are the same kind, or 2, when theatoms are of different kinds. Therefore, there are

IA ¼ 1� 8� 1 ¼ 8 IB ¼ 1� 6� 1 ¼ 6

3. Establishment of experiment bond length equationsand calculation of ra ¼ na

nAAccording to the bond length formula in EET, the

bond length of Ba formed by atoms u and v should bedenoted as

Du�va ¼ Ruð1Þ þ Rvð1Þ � b lg na ½4�

where Ru(1) and Rv(1) represent the single-bond radiusof atoms u and v, respectively, na is the shared electron

Table I. Hybridization Table of Titanium (A Type)

r 1 2 3 4 5 6 7 8 9

Chr 1 0.9987 0.9927 0.9882 0.9797 0.9602 0.9038 0.8094 0.8040Ctr 0 0.0013 0.0073 0.0118 0.0203 0.0398 0.0962 0.1906 0.1960nTr 4 4 4 4 4 4 4 4 4nlr 2 1.9974 1.9853 1.9764 1.9595 1.9204 1.8076 1.6188 1.6079ncr 2 2.0026 2.0147 2.0236 2.0405 2.0796 2.1924 2.3812 2.3921Rr(1)/nm 0.13530 0.13526 0.13508 0.13495 0.13469 0.13411 0.13241 0.12958 0.12942r 10 11 12 13 14 15 16 17 18Chr 0.7240 0.6232 0.6025 0.5054 0.4192 0.3712 0.3084 0.2049 0Ctr 0.2760 0.3768 0.3975 0.4946 0.5808 0.6288 0.6916 0.7951 1nTr 4 4 4 4 4 4 4 4 4nlr 1.4480 1.2463 1.2050 1.0107 0.8385 0.7423 0.6168 0.4097 0ncr 2.5520 2.7537 2.7950 2.9893 3.1615 3.2577 3.3832 3.5903 4Rr(1) /nm 0.12702 0.12400 0.12338 0.12046 0.11788 0.11644 0.11455 0.11145 0.10530

Fig. 1—BLD analysis model of b-Ti.

1050—VOLUME 40A, MAY 2009 METALLURGICAL AND MATERIALS TRANSACTIONS A

pair number of the Ba covalent bond, and b is a constantand has the following rule: while nMa <0:25 ornMa >0:75; b ¼ 0:0710 nm; while 0:300 � nMa � 0:700;b ¼ 0:0600 nm; while nMa ¼ 0:250þ e or nMa ¼ 0:750� e;b ¼ 0:0710� 0:22e: Here, nMa is the largest value of all thena in a given structure unit, and 0 £ e < 0.050.

For b-Ti, there are

DA ¼ 2Rð1Þ � b lg nA ½5�

DB ¼ 2Rð1Þ � b lg nB ½6�

Let DB be subtracted from DA. Therefore,

DA�DB ¼ 2R 1ð Þ�b lgnA½ � � 2R 1ð Þ�b lgnB½ � ¼ b lgnBnA

½7�

Let cA = 1 and cB ¼ nBnA: Then

lg cB ¼ lgnBnA¼ DA �DBð Þ=b ½8�

By substituting the experiment bond lengths DA andDB into Eq. [8], the ratio cB ¼ nB

nAcan be calculated.

4. Establishment of nA equation and solution of na

In general, a structure unit should be electron neutral;therefore, the covalent electron on all the j atoms in theunit should be distributed on all the a (=A, B, C……)covalent bonds in it. In other words, the total covalentelectron number of all the j atoms

P

j

ncj should equal the

sum of the electron number on all the covalent bonds inthe unit

P

aIana; i.e.,

X

j

ncj ¼X

a

Iana ¼X

a

nAIara ¼ nAX

a

Iara ½9�

nA ¼

P

j

ncj

P

aIara¼ nr

c

8rA þ 6rB½10�

nB ¼ nA�rB ½11�

When the titanium atom is at certain hybrid level r,the covalent electron number nr

c and single-bond radiuscan be found in the hybridization table of titanium(Table I); the covalent electron distribution on bonds,i.e., nA and nB, can be calculated. Because there are 18hybrid levels for the titanium atom under the A type ofhybridization state, calculation of the electron structureof b-Ti needs to be carried out 18 times. Among the 18groups of calculation results, the criterion of EET, i.e.,DDa ¼ Da �Da

�

�

�

�<0:0050 nm; can make clear whichelectron distributions may actually exist.

5. Calculation of the theoretical bond length �Da andascertainment of all the atom states that may exist in thecellIn EET, the covalent bond lengths calculated with the

calculated na are called theoretical bond lengths, whichare denoted by �Da. According to the bond lengthequation of EET, there is

�Da ¼ Ru 1ð Þ þ Rv 1ð Þ � b lg na ¼ 2Rr 1ð Þ � b lg na ½12�

In EET, the BLD denoted by DDa is the absolutevalue of the difference between the experimental bondlength and the theoretical one; i.e.,

DDa ¼ Da � �Daj j ½13�

If DDa < 0.0050 nm, then it is considered that the givenatom state and the corresponding electron distributionare in accordance with reality. In EET, the total groupnumber required for satisfying DDa ¼ Da � �Daj j<0:0050 nm is denoted as rN. The calculation resultsshow that there are only two groups of solutions thatsatisfy DDa ¼ Da � �Daj j<0:0050 nm: Table II shows thevalence electron structure of b-Ti.

6. Calculation of bond energyIn Reference 11, the bond energy of the covalent bond

formed by two identical atoms in crystal is representedby Ea, with the calculation formula as follows:

Ea ¼ b�f� na

�Da½14�

where a represents the bond name (bond order), na

represents the covalent electron pair number on the a

Table II. Valence Electron Structure of b-Ti

Ti:A10 R(1) = 0.12702/nm, nc = 2.55196, nl = 1.44804P

Ir = 9.26726, b = 0.0656 (nm)Bond name Ia Da (nm) �Da (nm) na Ea (kJ/mol) DDa (nm)

DA 8 0.28635 0.29078 0.27537 43.35680 0.00443DB 6 0.33065 0.33508 0.05816 7.94659 0.00443

Ti:A11 R(1) = 0.12400/nm, nc = 2.75370, nl = 1.24630P

Ir = 9.26726, b = 0.0656 (nm)Bond name Ia Da (nm) �Da (nm) na Ea (kJ/mol) DDa (nm)

DA 8 0.28635 0.28256 0.29714 50.02124 0.00379DB 6 0.33065 0.32686 0.06276 9.13316 0.00379

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 40A, MAY 2009—1051

bond, and �Da represents the theoretical bond length ofthe a bond. The term f represents the bond-formingability of the atom hybrid orbital, which can be calcu-lated as

f ¼ffiffiffi

apþ

ffiffiffiffiffiffi

3bp

þ gffiffiffiffiffi

5cp

½15�

where g reflects the contributions of the coupling effectof spin orbit to the bonding ability of the d electron;for elements in the 4th, 5th, and 6th periods of theperiodic table, and g = 1, 1.35, and 1.70, respectively.The terms a ¼ ns

nT; b ¼ np

nT; c ¼ nd

nT; ns; np; and nd; repre-

sent the valence electrons of s, p, and d, respectively;nT represents the total valence electron number. Theterm b is a parameter that represents the shielding fac-tor of the electron to nuclear charge; its value can becalculated as[11]

b ¼ 31:395

n� 0:36d½16�

where n and d are the total effects on the bond energy,which are generated by the shielding effect of the innerelectron in the atom, the coulomb interaction ofelectrons, and the exchange and correlation effect. Fortitanium, n = 2 and d = 1; substituting these intoEq. [16], one has bTi = 19.14329 kJÆnm/mol.

Table I shows the components Chr and Ctr of thehead state (h) and the tail state (t), the total valenceelectron number nT, and the lattice electron nl. There-fore, when the titanium atom is at the 10th hybrid levelof the A-type hybridization, the components a, b, and cfor the s, p, and d electrons, respectively, can becalculated as follows:

a ¼ nsnT¼ lsChr þ l0s0Ctrð Þ=nT

¼ 2� 0� 0:7240þ 1� 1� 0:2760ð Þ=4 ¼ 0:069

b ¼ npnT¼ mChr þm0Ctrð Þ=nT

¼ 1� 0:7240þ 1� 0:2760ð Þ=4 ¼ 0:25

c ¼ ndnT¼ nChr þ n0Ctrð Þ=nT

¼ 1� 0:7240þ 2� 0:2760ð Þ=4 ¼ 0:319

For titanium, g = 1. Substituting a, b, c, and g intoEq. [15], one has

fTi10 ¼ffiffiffiffiffiffiffiffiffiffiffi

0:069p

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3� 0:25p

þ 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

5� 0:319p

¼ 2:3916

Substituting na; �Da; fTi10 ; and bTi into Eq. [14], one has

EA ¼ b�f� nA�DA

¼ bTi�fTi10 �nA�DA

¼ 19:14329� 2:3916� 0:27537

0:29078

¼ 43:35680 kJ=mol

EB ¼ b�f� nB�DB

¼ bTi�fTi10 �nB�DB

¼ 19:14329� 2:3916� 0:05816

0:33508

¼ 7:94659 kJ=mol

Using the same calculation steps, the values of EA andEB can be calculated when titanium is at the 11th hybridlevel of the A-type hybridization; the calculation resultsare listed in Table II.

7. Calculation of statistical values of electron structureparametersAccording to the viewpoint of the authors of Reference

20, the statistical value n0a of the shared electron pairnumber na on the covalent bond can be calculated as

n0a ¼X

rN

i¼1naiCi ½17�

where rN is the number of atomic configurationssatisfying DDa < 0.005 nm; naI is the covalent electronpair number of the a covalent bond when the cell orstructure unit is at the i atom-state configuration; and Ci

is the probability of occurrence for the i atomicconfiguration; i.e., Ci ¼ 1

rN:

Substituting nA and nB (Table II) into Eq. [17],respectively,

n0A ¼1

20:27537þ 0:29714ð Þ ¼ 0:28626

n0B ¼1

20:05816þ 0:06276ð Þ ¼ 0:06046

According to the viewpoint of the authors of Refer-ence 20, the statistical value E0a of the bond energy Ea onthe covalent bond can also be calculated as

E0a ¼X

rN

i¼1EaiCi ½18�

where Eai is the bond energy of covalence bonds whenthe cell or structure unit is at the i atomic configuration;the meanings of rN and Ci are the same as those ofEq. [17].Substituting EA and EB in Table II into Eq. [18],

respectively, one has

E0A ¼1

243:35680þ 50:02124ð Þ ¼ 46:68902 kJ=mol

E0B ¼1

27:94689þ 9:13316ð Þ ¼ 8:54003 kJ=mol

B. Calculation of Valence Electron Structure of b-Ti-MStructure Unit

When the alloy atom M dissolves into the b-Ti cell,the atom M occupies the position of the titanium atom

1052—VOLUME 40A, MAY 2009 METALLURGICAL AND MATERIALS TRANSACTIONS A

to form the substitution solid solution; therefore, thelattice will be changed after the addition of the alloyingelementM. According to the viewpoint of the authors ofReference 15, the BLD analysis can be carried out onthe basis of the known lattice constant of b-Ti; thevariation in the lattice constant can be reflected by thevariation in the atom states. When the alloy atom Mdissolves into b-Ti, it is difficult to ascertain the atomposition. Therefore, it is supposed that 0.5 titaniumatoms in a b-Ti cell are occupied by the alloy atom M;all the lattice positions will be occupied by a kind of Xatom that is neither a titanium atom nor an M atomwhen the M element is added. It is also supposed thatthe X atom is the mixing atom of 0.75 titanium atomsand 0.25 M atoms; the character parameters of mixedatoms, such as the single-bond radius, covalent electronnumber, bond-forming ability, and shielding factor, arethe weighted average of the component elements of thesolid solution; i.e.,

RXð1Þ ¼3

4RTið1Þ þ

1

4RMð1Þ

nXc ¼3

4nTic þ

1

4nMc

fX ¼3

4fTi þ

1

4fM

bX ¼3

4bTi þ

1

4bM

9

>

>

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

>

>

;

½19�

where R(1) represents the single-bond radius, nc thecovalent electron number of the atom, f the bond-forming ability of the atom, and b the shielding factorof the atom (Appendix I). Thus, the BLD analysismodel of b-Ti-M can be established, as shown inFigure 2. In the crystal cell, only two kinds of covalentbonds should be considered, the bond names, bondlengths, and equivalent bond number Ia of which arethe following:

BX�XA ;DA ¼

ffiffiffi

3p

2a ¼ 0:28635 nm; IA ¼ 1� 8� 1 ¼ 8

BX�XB ;DB ¼ a ¼ 0:33065 nm; IB ¼ 1� 6� 1 ¼ 6

Taking the same BLD analysis steps as with the b-Ti,the valence electron structure parameters n0A and E0A forthe b-Ti-M structure units can be calculated. Table IIIshows the valence electron structure parameters n0A andE0A for b-Ti-M structure units.

C. Calculation of Valence Electron Structure of b-Ti-Mx-My Structure Unit

Because metastable b-titanium alloys always containvarious alloy elements, two kinds of alloy atoms, Mx

and My, may dissolve simultaneously into a b-Ti cell toform the substitution solid solution b-Ti-Mx-My. WhenMx and My dissolve into the b-Ti cell, it is difficult toascertain the atom positions. Therefore, it is supposedthat the atoms in a b-Ti cell are a kind of X atom. It isalso supposed that the X atom is a mixing atom of 0.25Mx atoms, 0.25 My atoms, and 0.50 titanium atoms; thecharacter parameters of mixed atoms, such as the single-bond radius, covalent electron number, bond-formingability, and shielding factor are the weighted average ofthose of the component elements of the solid solution;i.e.,

RXð1Þ ¼1

2RTið1Þ þ

1

4RMxð1Þ þ

1

4RMyð1Þ

nXc ¼1

2nTic þ

1

4nMxc þ 1

4nMyc

fX ¼1

2fTi þ

1

4fMx þ

1

4fMy

bX ¼1

2bTi þ

1

4bMx þ

1

4bMy

9

>

>

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

>

>

;

½20�

where the meanings of R(1), nc, f, and b are the same asin Eq. [19].Figure 3 shows the BLD analysis model of the b-Ti-

Mx-My solid solution. In the crystal cell, the bond name,bond length, and equivalent bond number Ia are thesame as those in the b-Ti-M structure unit. Using thesame BLD analysis steps as with b-Ti, the valenceelectron structure parameters of the b-Ti-Mx-My solidsolution can be calculated. Table III shows thevalence electron structure parameters n0A and E0A forthe b-Ti-Mx-My solid solution.

III. DISCUSSION AND ANALYSIS

A. Influences of Alloy Elements on Stability of b Phase

In titanium alloys, the phase transition of b to a phasemeets Bragg’s orientation relationship, i.e., (110)b//(0001)a, [111]b//[1120]a. Figure 4 shows the atomarrangement on the (110) plane of the b-Ti cell. Theatoms on the (110) plane of the b-Ti cell can formcovalent bonds. For example, atom 0 can form B03 andB04 covalent bonds with atoms 3 and 4, respectively,and the separation angle between the B03 and B04covalent bonds is 70 deg, 32 minutes. In order toFig. 2—BLD analysis model of b-Ti-M.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 40A, MAY 2009—1053

achieve the transition of b fi a, the separation anglesneed to be changed from 70 deg, 32 minutes to 60 deg;therefore, the atoms on the (110) plane need to bemoved along the 111h i direction.

Taking atom 2 as an example, the movement processis given as follows. Atom 2 can form two kinds ofcovalent bonds with the nearby atoms, i.e., covalentbonds B20 and B21. The former is the B covalent bond(Figure 1); the equivalent bond number of the B20covalent bond equals 6 and the covalent electron pairnumber of the bond is n0B: The latter is the A covalentbond (Figure 1); the equivalent bond number equals 8and the covalent electron pair number is n0A: Becausen0A � n0B; the bonding force between atoms 2 and 1 ismuch larger than the one between atoms 2 and 0;therefore, only when the driving force of the phasetransition breaks up the covalent bond B21 can atom 2move to the position 2 minutes. Because atoms 2, 3, 5,and 6 are equivalent atoms, atoms 3, 5 and 6 can alsomove to the positions of 3, 5, and 6 minutes when atom2 can move to the position 2 minutes. Figure 5 shows

the sketch of the atom movement for the transition ofb fi a. The phase transition of b fi a is then accom-plished. Figure 6 shows the atom arrangement of the(0001) plane of hcp lattice.

Table III. The n0A and E0A for Structure Units in Metastable b-Titanium Alloys

Structure Unit n0A E0A (kJ/mol) rN Structure Unit n0A E0A (kJ/mol) rN

b-Ti-Al 0.26400 38.66381 14 b-Ti-Sn-Mo 0.32071 45.99552 300b-Ti-Al-Fe 0.26655 35.10451 475 b-Ti-Sn-V 0.33109 45.48490 472b-Ti-Al-Cr 0.26905 38.40947 579 b-Ti-Zr 0.30980 54.50137 45b-Ti-Al-Mo 0.27418 41.30620 282 b-Ti- Zr-Cr 0.31197 53.89732 1340b-Ti-Al-V 0.28214 40.60425 463 b-Ti-Zr-Fe 0.31719 50.55063 1434b-Ti -Al-Zr 0.28808 46.04309 388 b-Ti-Zr-Sn 0.33681 51.05280 290b-Ti-Al-Sn 0.28877 36.17637 122 b-Ti-V 0.30213 47.85365 55b-Ti 0.28626 46.68936 2 b-Ti-V-Cr 0.31050 49.80960 1264b-Ti-Cr 0.29088 46.62839 48 b-Ti-V-Mo 0.32312 53.80428 664b-Ti-Cr-Fe 0.29113 43.40459 970 b-Ti-V-Zr 0.32552 56.27058 1240b-Ti-Cr-Sn 0.31754 43.68245 453 b-Ti-Mo 0.28969 47.40079 39b-Ti-Fe 0.29870 43.66662 70 b-Ti-Mo-Cr 0.31438 53.14668 603b-Ti-Fe-V 0.31237 45.87885 933 b-Ti-Mo-Fe 0.31543 49.65754 790b-Ti-Fe-Sn 0.32144 40.45925 499 b-Ti-Mo-Zr 0.31889 57.57535 773b-Ti-Sn 0.31022 43.15808 12 — — — —

Fig. 3—BLD analysis model of b-Ti-Mx-My.

Fig. 4—Atom arrangement on the (110) plane of b-Ti.

Fig. 5—Sketch of atom movement for the transition of b to a.

1054—VOLUME 40A, MAY 2009 METALLURGICAL AND MATERIALS TRANSACTIONS A

From these analyses, it can be seen that the sharedelectron pair number n0A on the strongest covalent bondof the b phase plays a critical role in the phase transitionof b fi a, i.e., the larger the n0A; the more difficult thephase transition of b fi a. Table III shows the valenceelectron structure parameter n0Aof the b-Ti, b-Ti-M, andb-Ti-Mx-My structure units. It can be seen that thevalues of n0A vary from 0.26001 to 0.28877 for theb-Ti-Al and b-Ti-Al-M structure units, which aresmaller than (or close to) that of the b-Ti cellðn0b�TiA ¼ 0:28626Þ: Therefore, the addition of aluminumhas a beneficial effect on the decrease in the stability ofthe b phase and promotes the phase transition of b fi a.Therefore, aluminum belongs to an a stabilizing ele-ment. However, the values of n0A for the other b-Ti-Mand b-Ti-Mx-My structure units are larger than that ofb-Ti; as a result, the other elements (e.g., Mo, V, Zr, Sn,Cr, and Fe) increase the stability of the b phase andrestrain the phase transition of b fi a, even causing b tobe stable at room temperature.

B. Mechanism of the Separation of the b Phase

In view of the fact that a smaller bond energy E0Acauses a larger diffusion ability,[20] the separation of theb phase that results from the rapid diffusion of alloyingelements in the crystal lattice of the b phase can becharacterized by the bond energy E0A: In the structureunits with a small E0A; diffusion of the atoms occurseasily, permitting the separation of the b phase. Table IIIshows that the values of the bond energy for the b-Ti,b-Ti-M, and b-Ti-Mx-My structure units in metastableb-titanium alloys are different. Therefore, the structureunits with a smaller E0A will be broken up first duringaging at 500 �C or 550 �C; these atoms within thebroken structure units can diffuse rapidly. After a periodof time, the diffusing atoms are accumulated again toform the more stable b¢ phase. The separation of b andb¢ is thereby accomplished.

For b-Ti-Al and b-Ti-Al-M structure units, the valuesof E0A are much smaller; therefore, these structure unitswill be broken up first and the atoms within them caneasily accumulate again to form b phase. In addition,the values of n0A for the b-Ti-Al and b-Ti-Al-M structureunits are also quite small; the stability of the b phase willbe decreased when the aluminum is added. Therefore,

the metastable b-titanium alloys containing an alumi-num element can easily facilitate the transition of b¢ fi afollowing the transition of b fi b +b¢.

C. Mechanism of Controlling the Decompositionof b-Phase Enriching in b- Stabilizing Elements

After its separation, b¢ phase can be transformed intoa phase during a long period of aging. If the b-phaseenriching in b-stabilizing elements contains eutectoidelements, the eutectoid reaction of b fi a+TixMy mayoccur during aging or applications, resulting in thebrittleness of the alloys. Therefore, the eutectoid reac-tion should be controlled.During the eutectoid reaction, the b phase should be

broken up first; the alloy atoms will then diffuse andbecome the new redistribution. When the alloy elementreaches the component of the eutectoid point, theintermetallic TixMy forms. At the same time, the aphase without (or with few) b-stabilizing elements alsoforms. Therefore, the bond energy E0A related todiffusion can be used to characterize the tendency ofthe eutectoid reaction. The larger the E0A of the b phase,the more difficult the diffusion of the alloy atoms andthe smaller the tendency of the eutectoid reaction.Table III shows that the structure units formed by Mo,V, and Zr elements have a larger E0A; the structure unitsformed by Fe, Cr, and Sn elements have a smaller E0A:Table IV shows the values of E0A for the structure unitsformed by Mn, Si, and Cu elements. It can be seen thatthe values of the structure units formed by Si and Cuelements are smaller in comparison to the values of theE0A in Table III. Therefore, the addition of Si and Cuelements, known as the fast eutectoid elements, canaccelerate the eutectoid reaction; the addition of Fe, Cr,Mn, and Sn elements, known as the long eutectoidelements, can slow the speed of the eutectoid reaction.However, b stabilizers such as V, Mo, and Zr cannotproduce a eutectoid reaction, because their structureunits have a much larger n0A: As a consequence, additionof the faster eutectoid elements such as Si and Cu shouldbe avoided, if possible; addition of the long eutectoidelements such as Fe, Cr, Mn, and Sn should also belimited.

IV. CONCLUSIONS

This study focuses on analyzing phase transforma-tions such as the stability of the b phase, the separationof the b phase, and the eutectoid reaction of the b-phaseenriching in b-stabilizing elements in metastableb-titanium alloys with the electron structure parameters

Fig. 6—Atom arrangement on the (0001) plane of a-Ti.

Table IV. The n0A and E0A for b-Ti-M Structure Units

Formed by Mn, Si, and Cu Elements

Structure Unit n0A E0A (kJ/mol) rN

b-Ti-Si 0.29232 40.37036 12b-Ti-Cu 0.31789 42.20060 51b-Ti-Mn 0.32738 45.96767 74

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 40A, MAY 2009—1055

n0A and E0A: The main conclusions are as follows. Thestability of the b phase can be characterized by theelectron structure parameter n0A; i.e., the smaller the n0A;the less stable the b phase. The separation of the b phasecan be characterized by the electron structure parameterE0A; the alloy elements that can decrease the E0A; such asaluminum, promote the separation of the b phase.Finally, the eutectoid reaction resulting from the diffu-sion can also be characterized by the E0A; i.e., the smallerthe E0A; the easier the eutectoid reaction.

ACKNOWLEDGMENTS

The authors acknowledge the financial support ofthe National Key Basic Research Program of Chinaunder Contract No. 2007CB613807.

APPENDIX I

APPENDIX II

Hybridization states in EET of the elements

1. A type of Ti, Zr, and Hf

d s p

h state: s2p1d1

t state: s1p1d2

l = 2, m = 1, n = 1, s = 0; l¢ = 1, m¢ = 1, n¢ = 2,s¢ = 1

Ti: R(1)h = 0.1353 nm, R(1)t = 0.1053 nmZr: R(1)h = 0.15223 nm, R(1) = 0.12155 nmHf: R(1)h = 0.1481 nm, R(1)t = 0.1232 nm

2. B type of Ti, Zr, and Hf

d s p

h state: d1s1p2

t state: d2s1p1

l = 1, m = 2, n = 1, s = 0; l¢ = 1, m¢ = 1, n¢ = 2,s¢ = 1

Ti: R(1)h = 0.1353 nm, R(1)t = 0.1053 nmZr: R(1)h = 0.15223 nm, R(1)t = 0.12155 nmHf: R(1)h = 0.1481 nm, R(1)t = 0.1232 nm

3. A type of V and Nbd s p

h state: d1s2p2

t state: d3s1p1

l = 2, m = 2, n = 1, s = 0; l¢ = 1, m¢ = 1, n¢ = 3,s¢ = 1

V: R(1)h = 0.139 nm, R(1)t = 0.095 nmNb: R(1)h = 0.15606 nm, R(1)t = 0.11098 nm

4. A1 type of V and Nbd s p

h state: d1s1p3

t state: d2s1p2

l = 1, m = 3, n = 1, s = 0; l¢ = 1, m¢ = 2, n¢ = 2,s¢ = 1

V: R(1)h = 0.139 nm, R(1)t = 0.117 nmNb: R(1)h = 0.15606 nm, R(1)t = 0.13352 nm

Shielding factors b of elements (kJÆnm/mol)

Element b Element b Element b Element b

H 49.05469 Pm 19.14329 Cr 19.14329 B 15.69750Li 49.05469 Sm 19.14329 Mo 19.14329 Al 19.14329Na 49.05469 Eu 19.14329 W 19.14329 C 13.76972K 49.05469 Gd 19.14329 Mn 11.89205 Si 13.76972Rb 49.05469 Tb 19.14329 Tc 11.89205 Ge 11.89205Cs 49.05469 Dy 19.14329 Re 11.89205 Sn 11.89205Be 19.14329 Ho 19.14329 Fe 15.69750 Pb 15.69750Mg 19.14329 Er 19.14329 Co 10.46500 N 9.57165Ca 19.14329 Tm 19.14329 Ni 7.84875 P 11.89205Sr 19.14329 Yb 19.14329 Pd 6.76617 As 19.14329Ba 19.14329 Lu 19.14329 Pt 6.76617 Sb 19.14329Sc 19.14329 Ti 19.14329 Cu 8.62500 Bi 15.69750Y 19.14329 Zr 19.14329 Ag 6.76617 S 15.69750La 19.14329 Hf 19.14329 Au 6.76617 Se 19.14329Ce 19.14329 V 19.14329 Zn 4.10931 Te 15.69750Pr 19.14329 Nb 19.14329 Cd 4.10931 — —Nd 19.14329 Ta 19.14329 Hg 2.48376 — —

1056—VOLUME 40A, MAY 2009 METALLURGICAL AND MATERIALS TRANSACTIONS A

5. B, B1, and B2 types of V

d s p

h state: 3d24s14p2

B type t state: 3d34s14p1

B1 type t state: 3d24s14p1

B2 type t state: 3d34s14p1

l = 1, m = 2, n = 0, s = 0; l¢ = 1, m¢ = 1, n¢ = 1,s¢ = 1

V: R(1)h = 0.161 nm, R(1)t = 0.12433 nm

6. C type of V

d s p

h state: 3d24s24p1

t state: 3d34s14p1

l = 2, m = 1, n = 2, s = 0; l¢ = 1, m¢ = 1, n¢ = 2,s¢ = 1

V: R(1)h = 0.117 nm, R(1)t = 0.106 nm

7. B type of Nbd s p

h state: d 2s1p2

t state: d 3s1p1

l = 1, m = 2, n = 2, s = 0; l¢ = 1, m¢ = 1, n¢ = 3,s¢ = 1

Nb: R(1)h = 0.13352 nm, R(1)t = 0.11098 nm

8. A and A1 types of Cr

d s p

h state: 3d44s2→3d 34s24p1

A type t state: 3d54s1→3d 44s14p1

A1 type t state:

l = 2, m = 1, n = 3, s = 0; l¢ = 1, m¢ = 1, n¢ = 1,s¢ = 1

Cr: R(1)h = 0.1067 nm, R(1) = 0.12337 nm

9. A type of Mo

d s p

h state: d2s2p1p′1

t state: d5s1

l = 2, m = 2, n = 2, s = 0; l¢ = 1, m¢ = 0, n¢ = 3,s¢ = 1

Mo: R(1)h = 0.14007 nm, R(1)t = 0.09728 nm

10. B type of Mod s p

h state: d3s2p1

t state: d3s1 p2

l = 2, m = 1, n = 3, s = 0; l¢ = 1, m¢ = 2, n¢ = 1,s¢ = 1

Mo: R(1)h = 0.12295 nm, R(1)t = 0.14863 nm

11. C type of Mod s p

h state: d4s1p′1

t state: d3s′1p′2

l = 1, m = 1, n = 4, s = 0; l¢ = 1, m¢ = 2, n¢ = 1,s¢ = 1

Mo: R(1)h = 0.10583 nm, R(1)t = 0.14863 nm

12. A and A1 types of Mn

d s p

A type h : d4s1p′2

A1 type h : 3d44s14p′2

t : d4s1p2

l = 1, m = 2, n = 2, s = 0; l¢ = 1, m¢ = 2, n¢ = 4,s¢ = 1

Mn: R(1)h = 0.1164 nm, R(1)t = 0.10091 nm

13. B and B1 types of Mn

d s p

B type h state: 3d44s14p2

t state: 3d54s14p1

B1 type h state: 3d 44s14p2

t state: 3d54s14p1

l = 1, m = 2, n = 2, s = 0; l¢ = 1, m¢ = 1, n¢ = 1,s¢ = 0

Mn: R(1)h = 0.1164 nm, R(1)t = 0.1224 nm

14. C type of Mnd s p

h state: 3d44s14p′2

t state: 3d54s14p1

l = 1, m = 2, n = 1, s = 0; l¢ = 1, m¢ = 1, n¢ = 3,s¢ = 1

Mn: R(1)h = 0.12990 nm, R(1)t = 0.0984 nm

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 40A, MAY 2009—1057

15. A and A1 types of Fe

d s p

A type h : 3d64s2→3d54s2p′1

A1 type h :

t : 3d8→3d64s′1p′1

l = 2, m = 1, n = 2, s = 0; l¢ = 1, m¢ = 1, n¢ = 4,s¢ = 1

Fe: R(1)h = 0.1161 nm, R(1)t = 0.09477 nm

16. B, B1, and B2 types of Fe

d s p

B type h state: 3d64s2→3d54s2p′1

t state: 3d8→3d54s′1p′2

B1 type h state: 3d64s2→3d54s2p′1

t state: 3d8→3d54s′1p′2

B2 type h state: 3d64s2→3d54s2p′1

t state: 3d8→3d54s′1p′2

l = 2, m = 1, n = 2, s = 0; l¢ = 1, m¢ = 2, n¢ = 3,s¢ = 1

Fe: R(1)h = 0.1161 nm, R(1)t = 0.1081 nm

17. C type of Fed s p

h state: 3d54s2p′1

t state: 3d54s′1p′2

l = 2, m = 1, n = 2, s = 0; l¢ = 1, m¢ = 2, n¢ = 0,s¢ = 1

Fe: R(1)h = 0.1161 nm, R(1)t = 0.1481 nm

18. A type of Al

s p

h state: s2p1

t state: s1p2

l = 2, m = 1, n = 0, s = 0; l¢ = 1, m¢ = 2, n¢ = 0,s¢ = 1

Al: R(1)h = 0.1190 nm, R(1)t = 0.1190 nm

19. A type of Si and Sn

s p

h state: s2p2

t state: s1p3

l = 2, m = 2, n = 0, s = 0; l¢ = 1, m¢ = 3, n¢ = 0,

s¢ = 1

Si: R(1)h = 0.1170 nm, R(1)t = 0.1170 nmSn: R(1)h = 0.13990 nm, R(1)t = 0.13990 nm

20. A types of Cu, Ag, and Aud s p

h state: d10s1→d8s1p′2

t state: d10p1→d7s′′1p′3

l = 1, m = 2, n = 2, s = 0; l¢ = 1, m¢ = 3, n¢ = 3,s¢ = 0

Cu: R(1)h = 0.11520 nm, R(1)t = 0.11380 nmAg: R(1)h = 0.13170 nm, R(1)t = 0.13020 nmAu: R(1)h = 0.13190 nm, R(1)t = 0.13030 nm

21. B type of Cud s p

h state: 3d104s1→3d74s14p′3

t state: 3d104p1→3d74s′′14p′3

l = 1, m = 3, n = 2, s = 0; l¢ = 1, m¢ = 3, n¢ = 3,s¢ = 0

Cu: R(1)h = 0.11853 nm, R(1)t = 0.11380 nm

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