Thermodynamic modeling of the Re–Si–B system Ying Yang a, * , Shuanglin Chen a , Y. Austin Chang b a CompuTherm LLC, Madison, WI, USA b Department of Materials Science and Engineering, University of Wisconsin-Madison, Madison, WI, USA article info Article history: Received 6 May 2009 Received in revised form 26 May 2009 Accepted 5 June 2009 Available online 3 July 2009 Keywords: A. Intermetallics, miscellaneous A. Silicides, various B. Phase diagrams B. Thermodynamic and thermochemical properties E. Phase diagram prediction abstract The Re–Si and Re–B systems were thermodynamically modeled using the Calphad method. The obtained thermodynamic descriptions for the Re–Si and Re–B systems reproduce not only the phase equilibrium but also thermodynamic property data. The Re–Si–B thermodynamic description was then established through extrapolation of binary ones of Re–Si and Re–B developed in this work and B–Si from literature. The calculated Re–Si–B isothermal section at 1200 C agrees well with experimental data. The calculated ternary liquidus projection predicts the liquid–solid phase equilibria in the Re–Si–B system for the first time, which can be used as a guidance for experimental design of future study. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Refractory Metal Intermetallic Composites (RMICs) based on the Mo–Si–B system have been studied as candidate structural mate- rials to be used in oxidizing environments at temperatures higher than the upper limit for the Ni-based superalloys [1,2]. Extensive efforts on alloying strategies have been put to improve the room- temperature fracture toughness of the Mo–Si–B alloys [1,3,4]. Re has been found to be an effective alloying addition for this purpose [5]. While Re additions are beneficial for the low-temperature fracture toughness, the oxidation test suggested a worse oxidation resistance of the Mo–11Si–11B–12Re alloy as compared to the corresponding alloy without Re addition [5]. Microstructural analysis showed that the Mo–11Si–11B–12Re alloy contains a large volume fraction of a third phase with the composition of Mo 5 Re 2 Si in addition to the (Mo) and T2 phases [5]. The ‘‘Mo 5 Re 2 Si’’ phase has a lower Si concentration (12.5 at%) than the Mo 3 Si phase in the Mo– 11Si–11B alloy, therefore, it is expected to be less oxidation resistant than Mo 3 Si. Whether the ‘‘Mo 5 Re 2 Si’’ phase is a new phase or extension of a binary phase and there are other effects on the phase equilibria in the Mo–Si–B alloys due to Re additions needs a systematic study on the phase relationship in the Mo–Si–B–Re system, from which the optimal amount of Re addition can be estimated for the Mo–Si–B alloys. In this study, we use the Calphad method [6] to develop thermodynamic models of the Re–Si–B system, which are the essential building blocks for constructing thermodynamic description of the Mo–Si–B–Re system. The ther- modynamic description of the Re–Si–B system is built upon those of its three constituent binary systems, i.e., Re–Si, Re–B, and B–Si. The B–Si and Re–Si systems have already been thermodynami- cally modelled. While thermodynamic description of B–Si devel- oped by Fries and Lukas [7] was directly used in this work, that of Re–Si by Shao[8] cannot be used for the reasons to be discussed in Section 2.1 . The calculated B–Si phase diagram based on Fries and Lukas’s work [7] is shown in Fig. 1 . This work consists of the remodeling of the Re–Si binary, the modeling of the Re–B binary, and the development of the thermodynamic description of Re–Si–B through extrapolation of thermodynamic models of the B–Si, Re–Si, and Re–B binaries. The Re–Si–B isothermal section at 1200 C and liquidus projection will be calculated from this ternary thermody- namic description and compared with available experimental data. 2. Literature review 2.1. Re–Si System Shao [8] has performed the first thermodynamic modeling for this system in which the phase equilibria and thermodynamic data in literature have been discussed in detail. Therefore, these literature data were briefly reviewed here. The literature data of the Re–Si * Corresponding author. Fax: þ1 608 262 8353. E-mail address: [email protected](Y. Yang). Contents lists available at ScienceDirect Intermetallics journal homepage: www.elsevier.com/locate/intermet 0966-9795/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.intermet.2009.06.003 Intermetallics 18 (2010) 51–56
0966-9795/$ – see front matter � 2009 Elsevier Ltd.doi:10.1016/j.intermet.2009.06.003
a b s t r a c t
The Re–Si and Re–B systems were thermodynamically modeled using the Calphad method. The obtainedthermodynamic descriptions for the Re–Si and Re–B systems reproduce not only the phase equilibriumbut also thermodynamic property data. The Re–Si–B thermodynamic description was then establishedthrough extrapolation of binary ones of Re–Si and Re–B developed in this work and B–Si from literature.The calculated Re–Si–B isothermal section at 1200 �C agrees well with experimental data. The calculatedternary liquidus projection predicts the liquid–solid phase equilibria in the Re–Si–B system for the firsttime, which can be used as a guidance for experimental design of future study.
� 2009 Elsevier Ltd. All rights reserved.
1. Introduction
Refractory Metal Intermetallic Composites (RMICs) based on theMo–Si–B system have been studied as candidate structural mate-rials to be used in oxidizing environments at temperatures higherthan the upper limit for the Ni-based superalloys [1,2]. Extensiveefforts on alloying strategies have been put to improve the room-temperature fracture toughness of the Mo–Si–B alloys [1,3,4]. Rehas been found to be an effective alloying addition for this purpose[5]. While Re additions are beneficial for the low-temperaturefracture toughness, the oxidation test suggested a worse oxidationresistance of the Mo–11Si–11B–12Re alloy as compared to thecorresponding alloy without Re addition [5]. Microstructuralanalysis showed that the Mo–11Si–11B–12Re alloy contains a largevolume fraction of a third phase with the composition of Mo5Re2Siin addition to the (Mo) and T2 phases [5]. The ‘‘Mo5Re2Si’’ phase hasa lower Si concentration (12.5 at%) than the Mo3Si phase in the Mo–11Si–11B alloy, therefore, it is expected to be less oxidation resistantthan Mo3Si. Whether the ‘‘Mo5Re2Si’’ phase is a new phase orextension of a binary phase and there are other effects on the phaseequilibria in the Mo–Si–B alloys due to Re additions needsa systematic study on the phase relationship in the Mo–Si–B–Resystem, from which the optimal amount of Re addition can be
ang).
All rights reserved.
estimated for the Mo–Si–B alloys. In this study, we use the Calphadmethod [6] to develop thermodynamic models of the Re–Si–Bsystem, which are the essential building blocks for constructingthermodynamic description of the Mo–Si–B–Re system. The ther-modynamic description of the Re–Si–B system is built upon thoseof its three constituent binary systems, i.e., Re–Si, Re–B, and B–Si.
The B–Si and Re–Si systems have already been thermodynami-cally modelled. While thermodynamic description of B–Si devel-oped by Fries and Lukas [7] was directly used in this work, that ofRe–Si by Shao[8] cannot be used for the reasons to be discussed inSection 2.1. The calculated B–Si phase diagram based on Fries andLukas’s work [7] is shown in Fig. 1. This work consists of theremodeling of the Re–Si binary, the modeling of the Re–B binary,and the development of the thermodynamic description of Re–Si–Bthrough extrapolation of thermodynamic models of the B–Si, Re–Si,and Re–B binaries. The Re–Si–B isothermal section at 1200 �C andliquidus projection will be calculated from this ternary thermody-namic description and compared with available experimental data.
2. Literature review
2.1. Re–Si System
Shao [8] has performed the first thermodynamic modeling forthis system in which the phase equilibria and thermodynamic datain literature have been discussed in detail. Therefore, these literaturedata were briefly reviewed here. The literature data of the Re–Si
Fig. 1. Calculated B–Si phase diagram based on thermodynamic description developedby Fries and Lukas [7].
Y. Yang et al. / Intermetallics 18 (2010) 51–5652
system up to 1996 were reviewed by Gokhale and Abbaschian [9].The evaluated phase diagram is shown in Fig. 2(a), which is mainlybased on the experimental work done by Jorda et al. [10] There aresix phases in the Re–Si system: liquid, (Re), Re2Si, ReSi, ReSi1.8, and(Si). The assessed crystal structure data of phases in this system arelisted in Table 1. ReSi1.8 was named as ReSi2 phase to be moreconsistent with its prototype MoSi2 crystal structure. The thermo-dynamic data have been summarized in Shao’s work [8] when heperformed the thermodynamic modeling of the Re–Si system,which are listed in Table 2. A later measurement after Shao’s workon the enthalpy of formation of Re2Si by Meschel and Kleppa [11]using high temperature direct synthesis calorimetry is also listed inTable 2. The invariant reactions in this system are listed in Table 3.Even though Shao’s thermodynamic modeling can generallyreproduce the most of experimental data, it cannot be used in thiswork for the following two reasons. Firstly, the lattice stability of Rein Shao’s work is not consistent with the currently used one in SGTE(an acronym for Scientific Group for Thermodata Europe) database[12]. Secondly, since ReSi2 phase has the same crystal structure asMoSi2 [9], they should be described by the same thermodynamic
a
Fig. 2. (a) Assessed experimental phase diagram [9], and (b) calculated Re–Si phase di
model. However, the ReSi2 phase was modeled as the stoichiometricphase ReSi1.8 in Shao’s work, which makes it difficult to model theMoSi2 and ReSi1.8 as the same phase in the Mo–Si–B–Re system. Inthis work, we will use the current accepted lattice stability of Re, anda two-sublattice model for the ReSi2 phase for developing the Re–Sithermodynamic description.
2.2. Re–B System
Portnoi and Romashov [13,14] investigated the phase diagram ofthe Re–B system. They prepared the Re–B alloys from 99.4 at% B and99.9 at% Re. The melting and thermal analysis were performedunder the inert gas with a Ti getter to reduce nitrogen and oxygen.The temperature was measured using an optical pyrometer whosereadings were calibrated against the melting points of the pureelements. The errors in the initial melting temperatures and finalmelting temperatures were estimated to be 1.5% and 2–2.5%respectively. The annealed alloys were subjected to X-Ray structuralanalysis, metallographic examination, electron probe microanalysisand hardness measurement. The borides Re3B, Re7B3 and ReB2 werediscovered in their experiments. They also found that the compo-sition of ReB2 deviated from stoichiometric ratio to higher B contentas the temperature increases. ReB2 and liquid form two eutecticreactions with Re7B3 and boron, respectively. The experimentallydetermined Re–B phase diagram is shown in Fig. 3(a). The phasenames and crystal structures are listed in Table 1. The invariantreactions in the Re–B system are listed in Table 4.
There are limited experimental data available on thermody-namic property for the Re–B system. Meschel and Kleppa [15]measured the standard enthalpy of formation of ReB2.5 to be�21.5� 0.5 kJ/g$atom at 1473� 2 K using high-temperature directsynthesis calorimetry. They chose ReB2.5 instead of ReB2 becausethey could not get a single phase at the composition of ReB2, whichconfirmed the work of Portnoi and Romashov [13] on that thecomposition of ReB2 deviates from the stoichiometric ratio. Baeh-ren and Vollath [16] detected the standard enthalpy of formation ofRe3B to be �32� 2.5 kJ/g$atom from the reaction between BN andmetallic powders: xMþ yBN¼MxByþ 1/2yN2, under different N2
partial pressures. The beginning of the reaction was detected withthe aid of a thermal balance. The reaction products were identifiedby X-ray diffraction. Other than these limited measurements, DeBoer et al. [17] estimated the enthalpies of formation of Re3B and
x(Si)Re Si
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11000
1400
1800
2200
2600
3000
3400
Liquid
(Re)
Re2SiReSi ReSi2
(Si)
b
Tem
peratu
re d
eg
ree C
agram. Squares denote the critical data of the invariant reactions listed in Table 3.
Table 1The phases in the Re–Si–B system. The bold font denotes the dominant species.
Phasesymbol
Pearsonsymbol
Thermodynamicmodels
Phase description
L N/A (B,Re,Si) Liquid phase(Re) hP2 (B,Re,Si) hcp_A3 solid solution
in the Re-rich corner(Si) cF8 (B,Si) Diamond solid solution
in the Si-rich corner(B) hR423 (B)93(B,Si)12 beta_rhomb_B solid solution
Note: the values in the first row for each reaction are calculated from this work.
Y. Yang et al. / Intermetallics 18 (2010) 51–56 53
ReB2 using a semi-empirical method. Until recently, first-principle(FP) calculations for the enthalpies of formation of Re3B, ReB2 andRe7B3 became available [18]. All these data are listed in Table 2. Thistable clearly shows that the data obtained from different sourcesdiffer significantly. Therefore, it is necessary to identify the optimalset of thermodynamic property data for these compounds throughthe self-consistent Calphad method by taking both the thermody-namic property and phase equilibrium data into accountsimultaneously.
2.3. Re–Si–B System
The only experimental work available for the Re–Si–B ternarysystem is the isothermal section measured by Chaban et al. [19]
Table 2Comparison of the assessed enthalpies of formation of rhenium silicides in Re–Sisystem and rhenium borides in Re–B systems with literature data.
Systems Compounds Enthalpy of formation(kJ/g$atom)
Methods
Re–Si Re2Si �20 Elemental effusion [24,25]�19.7 Assessment [23]�22 Estimation [17]�12.7� 2.5 Calorimetry [11]�22.7 Assessment [8]�19.367 This work
ReSi �25.4 Elemental effusion [24,25]�26.4 Assessment [23]�28 Estimation [17]�24.8 Assessment [8]�26.356 This work
ReSi2 �28.8 Elemental effusion [24,25]�30.1 Estimation [23]�10 Assessment [17]�31.2 Assessment [8]�33.445 This work
Re–B Re3B �19.2975 FP calculation [18]�20 Estimation [17]�32.22� 2.5 Chemical reaction [16]�21 This work
Re7B3 �22.29 FP calculation [18]�24 This work
ReB2 �43.1 FP calculation [18]�28 Estimation [17]
ReB2.5 �21.5� 0.5 Calorimetry [15]ReB2 �40 This work
They found no ternary compounds in the Re–Si–B system. Thesolubilities of the third component in the binary compounds of theRe–B and Re–Si systems were found to be negligible.
3. Thermodynamic models
The Gibbs energy of pure element i, G�
i , was taken from the latestversion of SGTE database, which is referred to the enthalpy for itsstable state at 298.15 K. There are a total of 13 phases in the Re–Si–Bsystem. They are liquid, (Re), (Si), (B), Re2Si, ReSi, ReSi2, ReB2, Re7B3,Re3B, BnSi, B6Si, and B3Si. Their crystal structures and thermody-namic models are listed in Table 1. (Re), (Si) and (B) are essentiallythe solid solutions based on their elemental crystal structures, i.e.,hcp_A3, diamond_A4, and beta_rhomb_B, respectively. Togetherwith the liquid phase, they were described by the substitutionalsolution model. The Gibbs energy of a substitutional solution phaseis described by
G ¼X
i
xiG�
i þ RTX
i
xiln xi þ Gex (1)
where xi is the mole fraction of element i, R the gas constant, and Ttemperature. For a ternary i–j–k system, the excess Gibbs energy ofmixing Gex is described by summation of binary and ternary excessGibbs energies based on Muggianu method [20]:
Gex ¼ xixj
Xn
Lni�j
�xi � xj
�nþxixk
Xn
Lni�kðxi � xkÞ
nþxjxk
�X
n
Lnj�k
�xj � xk
�nþxixjxkLi�j�k (2)
where Lni�j, Ln
i�k, Lnj�k, and Li�j�k are binary and ternary interaction
parameters, respectively. They will be optimized based on experi-mental data. In this work, no ternary interaction parameters wereneeded to reproduce the ternary experimental data.
Re3B, Re7B3, Re2Si, and ReSi phases have negligible homogeneityranges at their stoichiometric ratios. Therefore, they were modeledas line compounds. Based on the Kopp–Neumann rule [21], theGibbs energy of formation of the compound ipjq can be described by
Gipjq¼ pG
�
i þ qG�
j þ aþ bT (3)
with constants a and b to be optimized using experimental data.ReB2 and ReSi2 were modeled by two-sublattice models
(metal)1(metalloid)2 with one as the metal sublattice and the other
x(B)Re B
Liquid
(Re)
Re3BRe7B3
ReB2
(B)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11000
1400
1800
2200
2600
3000
3400
Tem
peratu
re d
eg
ree C
ba
Fig. 3. (a) Assessed experimental phase diagram [13] and (b) calculated Re–B phase diagram. Squares denote the critical data of the invariant reactions listed in Table 3.
Table 5
Y. Yang et al. / Intermetallics 18 (2010) 51–5654
as the metalloid sublattice. In order to describe the experimentallyobserved off-stoichiometric phase compositions, anti-site atoms orvacancy were introduced in the model. It has been found that thehomogeneity range of the ReB2 phase is on the B-rich side of thestoichiometric ratio, while that of ReSi2 is on the Re-rich side. Sincethe atom size of B or Si is smaller than that of Re, B or Si may get intothe Re sublattice, not vice versa. But the vacancy (Va) could form inthe small-size metalloid sublattice. Therefore, (Re,B)1(B)2 for ReB2
phase and (Re)1(Si,Va)2 for ReSi2 phase were used in this modeling.The Gibbs energy functions of these two phases are described bycompound energy formalism, as shown in Eqs. (4) and (5),respectively:
GReB2ðRe;BÞ1B2
¼ yIRe�GReB2
Re:B þ yIB�GReB2
Re:B þ13
RT�
yIReln yI
Re þ yIBln yI
B
�
þ yIB yI
Re LReB2Re;B:B (4)
GReSi2ReðSi;VaÞ2
¼ yIISi�GReB2
Re:B þ yIIVa�GReB2
Re:B þ23
RT�
yIISiln yII
Si þ yIIValn yII
Va
�
þ yIISi yII
Va LReSi2Re:Si;Va (5)
where yIRe and yI
B in Eq. (4) are the site fractions of Re and B on thefirst sublattice, respectively.
�G and
�G are the Gibbs energies of the
end-member compounds (B)(B)2 and (Re)(B)2 of the phase ReB2.LReB2
Re;B:B represents the interaction between Re and B on the firstsublattice with only B being present on the second sublattice. Thevariables in Eq. (5) have the similar meanings as those in Eq. (4).
Table 4Invariant reactions in the Re–B system.
Invariant reaction (xB) Temperature (K)
Liquid0:260:269
þ ðReÞ00
/ Re3B0:250:25
2418.592423, Ref. [13]
Liquid0:9510:92
/ ðBÞ10:985
þ ReB20:7610:75
2304.502323, Ref [13]
Liquid0:3570:34
þ Re3B0:250:25
/ Re7B30:30:3
2281.132273, Ref. [13]
Liquid0:7140:742
/ ReB20:7140:742
2671.042673, Ref [13]
Liquid0:4340:418
/ ReB20:6670:652
þ Re7B30:30:3
2092.762103, Ref. [13]
Note: the values in the first row for each reaction are calculated from this work.
The models of BnSi, B6Si and B3Si are adopted from Fries andLukas’s work [7] and are not described here.
After thermodynamic descriptions of three constituent binarieshad been obtained, thermodynamic models of ternary phases weredeveloped through extrapolation of binary ones. Parameters ofthermodynamic models optimized in this study are summarized inTable 5. All the phase diagram calculation and model parameteroptimization were carried out in Pandat and PanOptimizer [22].
4. Results and discussions
The thermodynamic modeling results for the Re–Si, Re–B, andRe–Si–B systems are presented below. These results are alsocompared with available experimental data.
4.1. Re–Si System
The calculated Re–Si phase diagram is shown in Fig. 2(b). Theassessed one by Gokhale and Abaschian [9] is shown in Fig. 2(a) forcomparison. The squares in Fig. 2(b) denote the experimentallyobserved critical data of invariant reactions, as listed in Table 3.These data are compared with the calculated ones in Table 3. Thecalculated results are in good agreement with the experimentalones within experimental uncertainties. The ReSi1.8 phase inFig. 2(a) now is modeled as the ReSi2 phase being consistent with
Thermodynamic model parameters for the Re–Si–B system optimized in this study.
Phase: thermodynamic model Parameters
Liquid: (B,Re,Si) L0B;Re ¼ �120487
L1B;Re ¼ �10813:7
L2B;Re ¼ �38822:4
L0Re;Si ¼ �71277:6
L1Re;Si ¼ �15883:5
L2B;Re ¼ �37331:4
Re2Si: (Re)2(Si)1 DG�
Re:Si ¼ �58; 100� 9:03418TReSi2: (Re)1(Si,Va)2 DG
�
Re:Si ¼ � 100;335þ 0:3088TDG
�
Re:Va ¼ 54961:4ReSi: (Re)0.5(Si)0.5 DG
�
Re:Si ¼ �18475:1� 5:89997T
Re3B:(Re)0.75(B)0.25DG
�
Re:B ¼ �21;000� 1:70597T
Re7B3: (Re)0.7(B)0.3 DG�
Re:B ¼ �24;000� 1:8601TReB2: (Re,B)1(B)2 DG
�
B:B ¼ 111;213DG
�
Re:B ¼ �120;000� 1:32936TL0
B;Re:B ¼ �100;000� 29:9987T
x(Si)
.00
00.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1
x(B)
Re
B
Si
Liquidus projection
Re7B3
Re3B
(Re Re2S ReSi2
ReB2
BBnSi
B6Si
(Si)
Fig. 5. Calculated liquidus projection of Re–Si–B system.
Y. Yang et al. / Intermetallics 18 (2010) 51–56 55
its crystal structure. A small homogenity range (less than 2 at%) ispresent for this phase due to the model selection.
Table 2 lists the calculated enthalpies of formation of the threeRe silicides in comparison with the literature data [8,11,17,23–25].The values obtained from this work agree with those assessed byShao [8]. Both assessments agree with the experimental measure-ments from elemental effusion [24,25]. It should be noted that theReSi phase is not a stable phase and the experimentally measuredenthalpy of formation is that of the Re2Si and ReSi2 mixture. Wealso noted that the measured enthalpy of formation (�12.7� 2.5 kJ)of Re2Si by Meschel and Kleppa [11] is much lower than thereported values by other authors [8,17,23–25] and was consideredless accurate during modeling.
4.2. Re–B System
Fig. 3(b) shows the calculated Re–B phase diagram from thecurrently developed thermodynamic description. The squares inthe Fig. 3(b) denote the critical data of invariant reactions, whichare listed in Table 4. The calculated phase diagram is in goodagreement with the experimental one. The B-rich homogeneityrange of ReB2 was successfully described using the two-sublatticemodel: (B,Re)1(B)2. The comparison between the calculated criticaldata and the experimental ones of the invariant reactions is listed inTable 4 and satisfactory agreement is reached.
The calculated enthalpies of formation of the three borides arecompared with literature data [15–18] in Table 2. As discussed early,data reported from different sources were contradictory with eachother. The calculated results are well lined with the first-principlescalculated data [18]. The measured enthalpy formation for ReB2.5 byMeschel and Kleppa [15] is �21.5� 0.5 kJ/g$atom, which is signif-icantly different from the calculated value �40 kJ/g$atom from thiswork and �43.1 kJ/g$atom from the first-principle calculation [18].A similar trend has also been found on the measured enthalpy offormation of Re2Si by Meschel and Kleppa [11] in the Re–Si system.The enthalpy of formation of Re3B reported by Baehren and Vollath[16] is �32� 2.5 kJ/g$atom which is derived from the reactionbetween BN and metallic powders: xMþ yBN¼MxByþ 1/2yN2.
0.80 0.2 0.4 0.6 10
0.2
0.4
0.6
0.8
1
x(Si)
x(B)
Re
B
Si
ReB2B3Si
B6Si
BnSi
(B)
Re7B3
ReB3
ReSi2Re2Si
T=1200°C
Fig. 4. Calculated isothermal section of Re–Si–B at 1200 �C compared with experi-mental data by Chaban et al. [19]. Square denotes the alloy located in a two-phaseregion and triangle a three-phase region.
This value is much more negative than the first-principles calcu-lated value [18] and this assessment. It needs to be further inves-tigated by experimental work.
4.3. Re–Si–B System
Based on thermodynamic descriptions of Re–Si and Re–Bdeveloped in this work and that of B–Si from literature, the Re–Si–Bthermodynamic description was obtained through extrapolation ofthe binary ones. The calculated Re–Si–B isothermal section at1200 �C is shown in Fig. 4, compared with experimental data [16].The squares denote the alloy compostions located in the two-phaseregion and triangles are those in three-phase region. The goodagreement between the calculated results and experimental datawas obtained. Since the thermodynamic models of ternary phasesin the Re–Si–B system were directly extrapolated from binary onesand no ternary interaction parameters were introduced, the goodagreement suggests that the thermodynamic models of binaryphases are reliable. Based on the ternary description, the liquidusprojection of the Re–Si–B was also calculated, as shown in Fig. 5.The dash lines are isothermal contours at different temperatures. Itcan be seen from this projection that the liquid surface is muchsteeper in the Re-rich corner and becomes flatter as the composi-tion moves away from the Re-rich corner. There are a total of nineinvariant reactions on the liquidus surface, as listed in Table 6. Ofthese nine reactions, there are three type-I ternary eutectic reac-tions: liquid / (Re)þ Re2Siþ Re3B, liquid / Re7B3þ ReSi2þ ReB2,and liquid / ReSi2þ ReB2þ (Si). The remaining ones are all type-IIreactions. There is no experimental data available to validate the
Table 6Calculated invariant reactions in the Re–Si–B system.
calculated liquidus projection. The calculated diagram can serve asa guide to smartly design alloy compositions for future experi-mental study.
5. Conclusions
The Re–Si system is thermodynamically remodeled in thiswork, which differs from the previous work in that the latestlattice stability of Re was used and the ReSi2 phase was describedby a two-sublattice model (Re)1(Si,Va)2 for consistency with itscrystal structure. The introduction of vacancy (Va) in the metalloidsublattice is for describing the off-stoichiometric Re-rich compo-sitions of this phase. Thermodynamic description of the Re–Bsystem is developed for the first time. The off-stoichiometriccomposition at the B-rich side of the ReB2 is described by the two-sublattice model of (Re,B)1(B)2. Thermodynamic descriptions forthe Re–Si and Re–B systems obtained in this work reproduce thephase equilibrium and thermodynamic property data available inliterature in a self-consistent manner. The thermodynamic modelsof ternary phases in the Re–Si–B system were then obtainedthrough direct extrapolation of binary ones of Re–Si and Re–Bdeveloped in this work and B–Si from literature. The calculatedisothermal section of the Re–Si–B at 1200 �C agrees well withexperimental data. The calculated ternary liquidus projectionpredicts the liquid–solid phase equilibria in the Re–Si–B system forthe first time, which can be used as a guidance for experimentaldesign of future study.
Acknowledgements
This research was supported under Air Force Office of Scientificand Research Contract No. FA9550-09-C-0048 through STTRprogram with Dr. Joan Fuller as the program manager. The authorswish to thank Dr. Jaimie Tiley for his interest and advice on this work.
References
[1] Berczik DM. Oxidation-resistant molybdenum alloys and composites withboron and silicon for dispersed phases. Application. WO (United TechnologiesCorporation, USA); 1996 p. 23.
[3] Sakidja R, Perepezko JH. Microstructure development in high-temperatureMo–Si–B alloys. MRS Proc 2005;851:557–62.
[4] Schneibel JH, Ritchie RO, Kruzic JJ, Tortorelli PF. Optimizaton of Mo–Si–Bintermetallic alloys. Metall Trans A 2005;36A:525–31.
[5] Wang P, Schenibel JH, Parthasarathy TA Development of a pest-resistant hightemperature Mo–SI–B-XX alloy, private communication; 2008.
[6] Kaufman L, Uhrenius B, Birnie D, Taylor K. Coupled pair potential, thermo-chemical and phase diagram data for transition metal binary systems-VII.Calphad 1984;8:25–66.
[7] Fries S, Lukas HL. B–Si. Cost 507, vol. 2. Luxembourg: European Communities;1991. p. 77–79.
[8] Shao G. Thermodynamic analysis of the Re–Si system. Intermetallics2001;9:1063–8.
[9] Gokhale AB, Abbaschian R. The Re–Si system (rhenium–silicon). J PhaseEquilib 1996;17:451–4.
[10] Jorda JL, Ishikawa M, Muller J. Phase relations and superconductivity in thebinary rhenium-silicon system. J Less-Common Met 1982;85:27–35.
[11] Meschel SV, Kleppa OJ. Standard enthalpies of formation of some 5d transitionmetal silicides by high temperature direct synthesis calorimetry. J AlloysCompd 1998;280:231–9.
[12] Dinsdale AT. SGTE data for pure elements. Calphad 1991;15:317–25.[13] Portnoi KI, Romashov VM. State diagram of the rhenium–boron system.
Porosh Met 1968;8:41–4.[14] Portnoi KI, Romashov VM. Binary phase diagram of some elements with
boron. Review. Porosh Met 1972;12:48–56.[15] Meschel SV, Kleppa OJ. Standard enthalpies of formation of niobium diboride,
molybdenum monoboride, and rhenium diboride by high-temperature directsynthesis calorimetry. Metall Trans A 1993;24A:947–50.
[16] Baehren FD, Vollath D. Determination of the thermodynamic data of molyb-denum boride (Mo2B), tungsten boride (W2B), and rhenium boride (Re3B).Planseeber Pulvermet 1969;17:180–3.
[17] Boer FD, Room R, Mattens W, Miedema A, Niessen A. Cohesion in metals-transition metal alloys; 1989.
[18] Gou H, Wang Z, Zhang J, Yan S, Gao F. Structural stability and elastic andelectronic properties of rhenium borides: first principle investigations. InorgChem 2009;48:581–7 (Washington, DC, U.S.).
[19] Chaban NF, Kuz’ma YB. X-ray diffraction study of rhenium–aluminum–boronand rhenium–silicon–boron ternary systems. Izv Akad Nauk SSSR, NeorgMater 1972;8:1065–8.
[20] Muggianu YM, Gambino M, Bros JP. Enthalpies of formation of liquid alloysbismuth-gallium-tin at 723 deg K. Choice of an analytical representation ofintegral and partial excess functions of mixing. J Chim Phys Phys Chim Biol1975;72:83–8.
[21] Neumann F. Ann Phys Chem 1865;202:123–42.[22] Chen SL, Daniel S, Zhang F, Chang YA, Yan XY, Xie FY, et al. The PANDAT
software package and its applications. Calphad 2002;26:175–88.[23] Chart TG. Thermochemical data for transition metal-silicon systems. High
Temp High Press 1973;5:241–52.[24] Searcy AW, Finnie LN. Stability of solid phases in the ternary systems of silicon
and carbon with rhenium and the six platinum metals. J Am Ceram Soc.1962;45:268–73.
[25] Searcy AW, McNees Jr RA. The silicides of rhenium. J Am Chem Soc.1953;75:1578–80.
