Calculation of the P-T phase diagram and tendency toward decomposition in equiatomic TiZr alloy

0.29 MB·English

by  V. Yu. Trubitsin

Calculation of the P-T phase diagram and tendency toward decomposition in
equiatomic TiZr alloy
∗
V.Yu. Trubitsin, E.B. Dolgusheva
Physical-Technical Institute, Ural Branch of Russian Academy of Sciences, 132 Kirov Str.,426001 Izhevsk, Russia
(Dated: January 11, 2010)
Electronic, structural and thermodynamic properties of theequiatomic alloy TiZr are calculated
within the electron density functional theory and the Debye-Gru¨neisen model. The calculated
valuesofthelatticeparametersaandc/aagree wellwiththeexperimentaldatafortheα,ω andβ
0 phases. Theωphaseisshowntobestableatatmosphericpressureandlowtemperatures;itremains
1 energeticallypreferableuptoT =600K.TheαphaseoftheTiZralloybecomesstableintherange
0 600K<T <900K,andtheβ phaseat temperaturesabove900K.Theconstructedphasediagram
2 qualitatively agrees with the experimental data available. The tendency toward decomposition in
n the equiatomic alloy ω−TiZr is studied. It is shown that in the ground state the ω phase of the
a ordered equiatomic alloy TiZr has a tendency toward ordering, rather than decomposition.
J
1 PACSnumbers: 63.20.Ry,05.10.Gg,63.20.Kr,71.15.Nc
Keywords: Phasediagram,decomposition,alloy,zirconium,titanium
1

]
i Introduction changes in the electron structure of the alloy. They sup-
c
posed that the large difference in the atomic volume be-
s
- tween the two phases points to the existence of an s-d
rl It has been experimentally found that the TiZr sys- electronic transition in ω−TiZr. Later, phase separation
t tem is characterizedby full solubility of its constituents.
m ofahexagonalTiZrωphasewasexperimentallydetected
As in pure titanium and zirconium, three phases (α, β
. and ω) are observedin the TiZr alloy1–3. The structural in Ref.5. The ω → ω1+ω2 decomposition was revealed
at α → β transformations of the equiatomic alloy were ex- after a prolonged heat treatment at P = 5.5±0.6 GPa
m tensivelystudiedinRef.1bythedifferentialthermalanal- and T = 440±300 C. It was supposed that in a wide
concentrationrangeatpressuresabovethetripleequilib-
- ysis (DTA) at temperatures up to 1023K, andpressures
d upto7GPa. Itwasfoundthattheβ →αtransitiontem- riumpoint, the ω phase mayexistinthe TixZr1−x alloy
n only as a metastable one that persists due to low diffu-
perature, being equal to 852 K at atmospheric pressure,
o sive mobility of its constituents. The decomposition of
decreases with pressure down to the triple equilibrium
[c point of the α, β and ω phases (Ptr = 4.9±0.3 GPa, the ω−TiZr solid solution into two ω phases of different
1 Ttr = 733±30 K). At pressures above the triple point setxrpuecrtiumreenwtaalsruesseudltsasobatnaianletedrninatRiveef.4e.xplanation for the
the β phase transforms immediately to the ω phase with
v Upto nowthe electronstructure andstructuraltrans-
a light positive slope of the equilibrium line. If a sample
0
formations of the equiatomic alloy TiZr have never been
0 is cooled to room temperature at a pressure of 6 GPa,
calculated. Below we present the results of our theoret-
7 and then unloaded, one can obtain at atmospheric pres-
1 sure a metastable ω phase which on heating transforms ical calculations of electronic, structural and thermody-
1. into an α phase in the temperature interval from 698 K namic properties ofthe equiatomic alloy TiZr performed
within the framework of the electron density functional
0 to743K.Coolingoftheβ phaseinthepressurerange2.8
theory and the Debye-Gru¨neisen model. The tendency
0 -4.8GParesultsintheformationofatwo-phasemixture
1 of a stable α and a metastable ω phase. The structural of the ordered equiatomic alloy ω−TiZr to decompose is
v: α→ω transformations in the TiZr alloy were studied in also investigated.
i detail in Ref.3. Investigating TiZr samples under shear-
X
strain conditions at pressures up to 9 GPa at tempera-
r tures300and77K,theauthorsarrivedattheconclusion I. CALCULATION TECHNIQUE
a
thatinequiatomicTiZrtheequilibriumα→ωboundary
is situated on the P-T diagram at 6.6 GPa. In the same The electron structure and total energy were calcu-
paper the phase diagram of TiZr was constructed in the lated by the scalar relativistic full-potential linearized
regular-solution approximation, and the triple point pa- augmented-plane-wave (FPLAPW) method, using the
rameters were calculated (P =8.5 GPa, T =693 K). As WIEN2K package7. To ensure the desired accuracy of
may be seen, these values differ substantially from those the total energy calculation, the number of plane waves
of Ref.1. wasdefinedbytheconditionRK =7,the totalnum-
max
Detailed studies performed in Ref.4 have shown that ber of k-points in the Brillouin zone was equal to 3000,
in the region of high pressures and temperatures there 3000, 600 for the β, α and ω phase, respectively. The
exist two ω phases (ω and ω1) that differ in atomic vol- total and partial densities of states were obtained by a
ume by about14%. The authorssuggestedthe existence modified tetrahedron method6. The atomic radii were
ofa isostructuralphasetransitionω−ω1 connectedwith the same for all phases and pressures: 2.42 a.u. for Zr,
2
and 2.26 a.u. for Ti.
0.13
b
In Fig.1 are shown the crystal structures of the β, α,
0.12 TiZr a
and ω phases of the equiatomic alloy TiZr used in the w
calculation. It is seen that the β phase was represented 0.11
y)
byastructureoftheCsCltypewithTiatomsatthecube R
sites and a Zr atom at the center. The α phase had a y ( 0.10
g
hexagonalclose-packedlatticeinwhichoneatomwasZr, er 0.09
n
the other Ti. Finally, to describe the ω phase we used a E 0.08
hexagonal lattice with 6 atoms per cell (an ω structure
0.07
doubled along the z axis). The atomic arrangement and
species in this case were chosen as follows: (0,0,0) - Zr, 0.06
1 2 1 2 1 1 1 1 2 3 16 18 20 22 24
( , , ) - Ti, ( , , ) - Zr, (0,0, ) - Ti, ( , , ) - Zr,
(32,31,43)-Ti. T3hu3s,t4heβ andαph2aseswere3re3pr4esented Volume (Å3)
3 3 4
by layers of Ti and Zr alternating along the z axis, and
in the ω phase layers of Ti and Zr were separated by a
FIG.2: Volumedependenceofthetotalenergyofequiatomic
mixed Ti-Zr layer. TiZr alloy for theβ, α and ω phases
anharmonic effects. The curves obtained for the volume
dependence of the electron subsystem total energy are
shown in Fig. 2. The energy zero in figure corresponds
to -8906.0 Ry.
As seen from the figure, the energy minimum in the
ground state falls on the ω phase. And only at the
FIG. 1: Crystal structure of the β, α and ω phases of
equiatomicTiZr. The Zr atoms are grey, the Ti atoms are relative volume change V/V0 = 0.75 the β phase be-
dark. comes energetically preferable. A similar situation was
observed in pure Ti and Zr as well. However the dif-
ference in energy between the α and ω phases in TiZr
Inthehexagonalstructurestheratioc/awasoptimized amounts to ∆Eα−ω =6 mRy, while in pure Ti and Zr it
for the experimental volume values. In the following, is0.8mRyand1mRy,respectively. Henceitfollowsthat
when calculating the volume dependence of the total en- in the equiatomic alloyTiZr the stability regionofthe ω
ergy, the c/a ratio was considered to be constant. In phaseshouldbemuchlargerintemperaturethaninpure
Table I are listed the calculated and experimental equi- metals Ti,Zr. The equilibrium values of the volume are
librium values of the TiZr lattice parameters. Veq =20.47˚A3, 20.52˚A3, and 19.97˚A3 for the α, ω and
β phases, respectively.
Figure 3 presents the volume dependence of the free
TABLE I: Equilibrium values of the TiZr lattice parameters
energy at different temperatures. The free energy of the
in atomic units
TiZr alloy was calculated in the Debye-Gru¨neisen model
with allowancemade for the contributionsfromthe elec-
acalc. (c/a)calc. aexp. [1] (c/a)exp. [1] aexp. [3] (c/a)exp. [3] tronentropy. Thetechniqueofcalculatingthislatterhas
beendescribedindetailinRef.9. Asseenfromthefigure,
β 6.457 1 - 1 - - the relationship between the energies of different struc-
α 5.860 1.583 5.866 1.583 - - tures changes with temperature. So, at 300 K and zero
ω 9.122 0.617 9.152 0.617 9.131 0.616 pressuretheenergyminimumfalls,asinthegroundstate,
ontheω phasewhichremainsenergeticallypreferableup
toT =600K.Inthetemperaturerange600K<T<900K
It is seen from the table that the lattice parameters a itistheαphaseofTiZralloywhichbecomesstable,while
and c/a obtained in our calculation agree well with the above 900 K it is the β phase.
experimentaldata. The greatestdiscrepancy is observed
forthelatticeconstantoftheωphase. Itshouldbenoted
thatinRef.1 the lattice parameterswerecalculatedfor a II. PHASE DIAGRAM
metastable ω structure at atmospheric pressure, and in
Ref.3 for pressure-strainedsamples. The phase diagram of TiZr based on the analysis of
Thetotalenergyofeachstructurewascalculatedfor7 Gibbs potentials for different structures is presented in
valuesofthecellvolumeV. Thedataobtainedwerethen Fig.4. The results of calculation are shown by the solid
interpolated using the technique proposed by Moruzzi8. line. The dotted line denotes the experimental equilib-
Such an interpolation scheme, together with the Debye- rium boundaries for the α, β and ω phases of TiZr ob-
Gru¨neisen model, makes it possible to include implicitly tained in Ref.1. The experimental values of the α−ω
3
rium point of the α and ω phases was determined under
T=300K T=900K
shear-strain conditions at pressures up to 9 GPa. The
−0.02 −0.02
b b shearing strain is known to lower the pressure at which
−0.025 wa −0.03 wa the phase transition begins. Presumably for this reason
the authors of Ref.3 failed to obtain the α−ω transition
−0.04
y) −0.03 atroomtemperatureunderquasi-hydrostaticconditions.
y (R −0.035 −0.05 In Ref.2 it was shown by X-ray diffraction method that
g the α phase of TiZr remains the sole stable phase under
er −0.06
en −0.04 quasi-hydrostaticpressureupto12.2GPa. Onlyfrom5.5
e −0.07 GPaon,becomesdominatingtheω phasewhichremains
e
Fr −0.045 −0.08 stableupto56.9GPa. Atpressuresabove56.9GPathere
forms a high-pressure phase with a bcc lattice.
−0.05 −0.09
−0.055 −0.1 1000
16 18 20 22 24 16 18 20 22 24
900 TiZr
T=600K T=1200K 800 b
−−00.0.0225 ab −−00..0032 ab URE (K) 700 a
−0.03 w −0.04 w RAT 600
y (Ry) −0.035 −0.05 TEMPE 500 w
g −0.04 −0.06
er 400
n −0.045 −0.07
e
e −0.05 −0.08 300
e
Fr −0.055 −0.09 0 10 20 30 40 50
PRESSURE(GPa)
−0.06 −0.1
−0.065 −0.11 FIG.4: TheP-TphasediagramofTiZr. Thesolidlineshows
16 18 20 22 24 16 18 20 22 24
thecalculationresults. Thedottedlinesareconstructedfrom
Volume (Å3) Volume (Å3) the experimental data [1]. The experimental values for the
•α−ω transition at room temperature are taken from Refs.:
- [3], (cid:4) - [2].
FIG. 3: Free energy of the α, ω and β phases of TiZr at
different temperatures
As seen from Fig.4, in our calculation at atmospheric
pressure and low temperatures the ω phase is stable,
there occurs no α−ω transition at room temperature.
transition at room temperature are taken from Refs.2,3. Note that in our calculations of pure Ti10 and Zr11, in
complete agreement with the experimental data, the α
Onthewhole,agoodagreementofthecalculatedtriple
phase is stable at atmospheric pressure and room tem-
point(P =4.2GPa,T =720K)withtheexper-
theor theor perature, and the ω phase is stable only under pressure.
imentalvalues P =4.9±0.3GPa, T =733±30K1
exp exp ThattheωphaseinTiZratnormalconditionsisenerget-
is observed. At zeropressurethe calculatedtemperature
ically preferable,immediately follows from a comparison
of the β −α transition is Ttheor = 943 K. This value is
β−α of the calculated free energies (see Fig.3). Recall that
higherthantheexperimentalone,Texp =852K,defined
β−α the difference in energy between the α and ω structures
in Ref.1 as the average of the temperatures of the tran- in the equiatomic TiZr alloy is almost five times greater
sition onset on heating and cooling. It should be noted than in pure titanium and zirconium.
that a large hysteresis is observedupon the α−β trans- The discrepancy between experiment and theory may
formation in TiZr. At atmospheric pressure the maxima bedue tothefactthatthe calculationwasperformedfor
of thermal peaks in the DTA curves fall on T ∼ 912 K idealcrystallinestructures(seeFig.1),whereastheexper-
on heating and T ∼ 810 K on cooling, the typical peak imental samples were imperfect crystals with lattice de-
width being ∆T ∼ 40 K. With this in mind, one can fects. In particular, it was shown3 that in a TiZr alloy
consider the results of calculation of the α−β equilib- shear-strainedunder pressure the ω phase is represented
rium boundary in the Debye-Gru¨neisen model as quite by aggregations of oblong particles with characteristic
satisfactory. sizeof3−5nm,and15−30nmlong. Ifω−phaseparticles
The greatest discrepancy between the theoretical cal- aresituatedinacoarsegrainofαphase,theyaremainly
culation and the experimental evidence available is ob- locatedatitsboundaries. Itwasalsonoted1 thatvarious
served for the α− ω transition. In Ref.3 the pressure imperfect structures in samples pre-treated in different
at whichthis transitionoccurs atroomtemperature was ways have a noticeable effect on the course of structural
estimated to be Pexp = 6.6 GPa. Note thet equilib- transformations. Evidently, we could not model a real
α−ω
4
structure in first-principles calculations. IItwolayersofpuretitaniumalternatedalongthe z axis
Thecorrectnessofourresultsmaybesupportedbythe with two layers of pure zirconium with no intermediate
following experimental evidence1: firstly, the metastable layer. ForstructureIIIwerechosensixTilayersalternat-
ω phasewasobtainedatatmosphericpressureasaresult ing with six Zr layers without intermediate layer. And
ofcooling ofthe β phase under a pressureof 6 GPa with lastly, in variant IV five Ti layers were separated by an
subsequent unloading at room temperature. Secondly, intermediate layer from five Zr layers.
between 2.2and4.8 GPa oncoolingofthe β phase there The free energy was calculated by the scalar rel-
formsatwo-phasemixtureofastableαandametastable ativistic full-potential linearized augmented-plane-wave
ω phase. And lastly, it was found that at atmospheric (FPLAPW) method, using the WIEN2K package7. In
pressuretheωphaseintheTiZralloy,whenheatedabove the first two variants the number of atoms per unit cell
698 K, transforms into an α phase1. In Ref.3 the tem- was six (3 Ti atoms and 3 Zr atoms). In variants III
perature of this transformation was defined as T = 623 and IV the number of atoms in the cell amounted to 18
K for P = 0.0001 GPa. This value differs by only 13 K (9 atoms of each species). For variants I and II struc-
fromthe temperature Tω→α =610Kwe havecalculated tural optimization of the c/a ratio was performed, and
P=0
for the ω →α transition. theequilibriumatompositionsweredefinedwiththepro-
cedure of minimizing the forces acting on atoms.
1
w −TiZr I
0 II
−1
y) −2
R
m −3
E (
−4
D
−5
−6
FIG. 5: The structure types used in modeling the decompo-
sitioninω−TiZr. TheTiatomsaredark,theZratomslight. −7 13 14 15 16 17 18 19 20 21 22 23
Volume (Å3)
The above evidence suggests, in our opinion, that in FIG. 6: Energy change upon lattice relaxation with respect
totheequilibriumatompositionsinidealωphaseforvariants
the TiZr alloy with ideal crystal lattice the phase dia-
I and II
gram should look as it is depicted in Fig.4. Of course,
it must be taken into account that the Debye-Gru¨neisen
model, used for calculating the thermodynamic poten-
tials, is a rather rough approximation and, obviously, In Fig.6 the volume dependence of the lattice relax-
cannot ensure good accuracy, especially at high temper- ation energy change ∆E = Erelax −E0 is displayed for
atures when the anharmonic effects become of consider- calculation variants I and II. Here E0 is the total energy
able importance. It should be noted that at room tem- ofthe systemwith atomicarrangementcorrespondingto
peraturethepressurecalculatedfortheω →β transition theidealω lattice;Erelax isthesystemenergyaftermin-
isnearlyhalfaslargeastheexperimentalvalue. Because imization of the forces acting on atoms for a given vol-
of low temperatures, we do not believe this discrepancy ume. Asseenfromthefigure,withdecreasingvolumethe
to be connected with the choice of the Debye model for atoms become displaced from the positions correspond-
describing the thermodynamic properties. It is rather ing to the ideal ω lattice, the displacement magnitude
due to the deviation of real alloys from the ideal peri- depending on the volume and the structure type. For
odic structures used in calculating the total energy in the structure of type I, corresponding to the most uni-
the ground state. formdistributionofTiandZratomsatV <15˚A3,there
occurs a sharp decrease of ∆E due to significant atomic
rearrangement. For the two-layer system (II) such a re-
III. CALCULATION OF THE TENDENCY arrangementisnotobservedintheconsideredintervalof
TOWARD DECOMPOSITION volume change. In Fig.7, for V = 13 ˚A3 are depicted
the (110) planes, the arrows indicating the direction of
To estimate the tendency toward decomposition in atomicdisplacementsonrelaxationforthelatticesoftype
ω−TiZr in the ground state, the total energy was cal- I(a)andII(b)(thezaxisispointingupwards). Itisseen
culated for four structures (see Fig.5). Structure I was that in both cases the atomic displacements are directed
represented by layers of pure titanium and pure zirco- only along the z axis. In the two-layer system (II) the
nium alternating along the z axis and separated by in- atoms of titanium and zirconium are displaced in op-
termediatemixedTi-Zrlayers. Inthetwo-layerstructure posite directions, whereas in system I the atomic chain
5
displacement occurs without strain. The displacements
shown in Fig.7(a) correspond to those characteristic of
60
the ω → β transition. The volume value V ≈ 15 ˚A3 at 55 (a) w −TiZr I II
which begins a sharp decrease in ∆E, agrees well with 50
the results of the total energy calculation for the ω and Ry) 45
β phases of TiZr plotted in Fig.2. Based on the data m 40
presented, we can draw an important conclusion that gy ( 35
the pressure value at which occurs the ω → β transi- ner 30
E
tion depends substantially on the ordering type in the 25
equiatomic TiZr alloy. This also indirectly confirms our 20
statement that the disagreement with the experiment 15
15 16 17 18 19 20 21 22 23
concerning the ω−β equilibrium boundary position on
Volume (Å3)
thephasediagramcalculatedintheDebyemodel(Fig.4)
is caused by the presence of inhomogeneities in actual
60
TiZr alloys used in experiments in Refs.1–3. 55 (b) w −TiZr III I
50 IV
Ry) 45
m 40
gy ( 35
ner 30
E
25
20
15
15 16 17 18 19 20 21 22 23
Volume (Å3)
FIG.8: Totalenergyofω−TiZrfordifferenttypesofdecom-
position ( I - IV) designated in accordance with Fig.5.
FIG. 7: The direction of atomic displacements upon lattice
relaxation for calculation variants I (a) and II (b) To summarize, the calculations performed show that in
the ground state the ω phase of TiZr exhibits a ten-
dencytowardorderingandnottowarddecomposition,as
was suggested in Ref.5. The analysis of the total energy
The volume dependence of the total energy for the re-
curves shows that the allowance for temperature effects
laxed structures of type I and II is plotted in Fig.8(a).
in the Debye-Gru¨neisen model will not change the en-
As may be seen, in the whole range of volume change
ergy relation between different calculation variants, and
the energy is minimum for the structure of type I corre-
cannotexplainthe experimentally observedformationof
sponding to the most uniform distribution of Ti and Zr
two ω structures. Besides, we have not found any pe-
atoms. Recall that in variant I the lattice is represented
culiarities connected with the s-d electron transition in
bypure monolayersoftitaniumandzirconiumseparated
the total-energy curves. Thus, also the suggestion ad-
by a mixed Ti-Zr layer, and in variant II by a system vanced in Ref.4 that there exists an isostructural transi-
of two Ti layers alternating with two Zr layers (with no tion ω → ω1 due to the pressure-induced changes in the
intermediate layer). Thus, as decomposition grows, the
electronstructure,isnotconfirmedbythecalculation. In
system energy increases. This tendency persists on fur-
our opinion, the high-temperature decomposition in the
therdecomposition,whichmaybeseeninFig.8(b),where
ω phase of equiatomic TiZr alloy is connected not with
the total energy is plotted versus volume for the struc-
thechangeinelectronstructureunderpressure,butwith
tures of type II, III and IV. In the structure of type IV
peculiarities of the lattice dynamics, in particular, with
fiveTilayersareseparatedfromfiveZrlayersbyamixed
the presence of strongly anharmonic vibrational modes
intermediatelayer,whileinstructureIIItherearesixlay-
which are of crucial importance in stabilization of the ω
ers of each metal without intermediate layer. It is seen lattice of pure titanium and zirconium12.
fromthe figurethatasthe thicknessofpure metallayers
increases(fromonetofivelayers),thesystemenergysig-
nificantlyrises. As couldbe expected,the presenceofan
Acknowledgments
intermediatelayerreducesthetotalenergyofthesystem.
Thismaybeseenfromacomparisonoftheenergyvalues
at V ≈ 18.5 ˚A3 for the calculation variants III and IV The authorsacknowledgethe partialsupportfromthe
(structureIIIispresentedinthefigurebyasinglepoint). RFBR Grants No. 07-02-00973and No. 07-02-96018.
6
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