THERMODYNAMIC ASSESSMENT OF THE TI AL NB, TI AL CR, AND TI AL MO SYSTEMS

Transcription

1 THERMODYNAMIC ASSESSMENT OF THE TI AL NB, TI AL CR, AND TI AL MO SYSTEMS By DAMIAN M. CUPID A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA

2 c 2009 Damian M. Cupid 2

3 This dissertation is dedicated to Samuel Cox, who unfortunately could not be there till the end. 3

4 ACKNOWLEDGMENTS I would first like to acknowledge my advisors, Prof. Hans-Jürgen Seifert and Prof. Fereshteh Ebrahimi. Without their help, advice, and encouragement, this work would not be possible. A special thanks is extended to Dr. Olga Fabrichnaya, who has taught me everything I know about thermodynamic optimization. I thank also my past and present committee members: Prof. Phillpot, Prof. Sigmund, Prof. Sinnott, and Prof. Lear. I acknowledge all members of the research group in Florida: Orlando Rios, Mike Kessler, and Sonalika Goyel. Their experimental work was tremendously helpful. I also acknowledge the undergraduate students: Jonah Klemm-Toole, Daniel Heinz, and Tabea Wilk, who participated at various stages in this project. Members of the research group at Freiberg University of Mining and Technology also require special acknowledgement: Frau Galina Savinykh and Mario Kriegel and Dmytro Pavlyuchkov. Mario Kriegel deserves special attention as he worked intensively on the optimization of the Ti Al Cr system. I am grateful to my friends in Florida and in Germany. They have made my stay on both sides of the Atlantic memorable. Last, I would like to deeply acknowledge my parents. Without their support along every step of the way, this would be impossible. This work was supported by the University of Florida College of Engineering Alumni Fellowship and by the NSF/AFOSR under grant number DMR

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTER 1 INTRODUCTION BACKGROUND AND THEORY Thermodynamic Modeling Modeling of the Pure Elements Modeling of Stoichiometric Phases Modeling of Substitution Solutions Extrapolation methods Modeling of Ordered Phases The Compound Energy Formalism Phases with Order Disorder Transformations Thermodynamic Optimization Experimental Data Theoretical Data Gibbs Free Energy Minimization Optimization Method: The Least Squares Method REVIEW OF THE Ti Al Nb SYSTEM Phases in the Ti Al Nb System Experimentally Determined Phase Equilibria in the Ti Al Nb System The Liquidus Surface Isothermal Sections Vertical Sections Thermodynamic Descriptions of the Ti Al Nb System Thermodynamic Description of Kattner and Boettinger Thermodynamic Description of Servant and Ansara Reasons for re-optimization Thermodynamic Description of Witusiewicz et al Thermodynamic description of the Ti Al system Thermodynamic description of the Al Nb system Thermodynamic Description of the Ti Al Nb system

6 4 KEY EXPERIMENTAL DATA FOR OPTIMIZATION Tie Line Data Alloy DTA Peak Analysis for Alloy Alloy DTA Peak Analysis for Alloy RE-OPTIMIZATION OF THE Ti Al Nb SYSTEM Unary Data and Binary Sub-Sections Thermodynamic Optimization Optimization Strategy Selection of Thermodynamic Parameters The Sigma phase The Delta Phase The Beta Phase The Gamma Phase The Ordered Beta Phase The Disordered Alpha and Ordered Alpha 2 Phases The Liquid Phase RESULTS OF THE RE-OPTIMIZATION Liquidus and Solidus Projections Isothermal Sections Vertical Sections Phase Fraction Diagrams THE Ti-Cr SYSTEM Phase Equilibria in the Ti Cr System Thermodynamic Descriptions of the Ti Cr System Lattice Stability of Pure Cr Number of Laves Phases Modeled Modeling of the Laves Phases The Homogeneity Ranges of the Laves Phases The Activity of Cr in the Beta Phase Optimization of the Parameters for the Binary Ti Cr System Selection of Thermodynamic Parameters for the Laves Phases Optimization Strategy Optimized Results THE Ti-Al-Cr SYSTEM Phases in the Ti Al Cr System Experimentally Determined Phase Equilibria Isothermal Sections

7 8.2.2 The Liquidus Surface Critical Assessments of the Ti Al Cr System Thermodynamic Descriptions of the Ti Al Cr System Thermodynamic Description of Saunders Reasons for re-optimization Re-optimization of the Ti Al Cr System Optimization Strategy The alpha2 phase The ternary Ti(Al,Cr)2 laves phase The ternary tau phase The beta phase The ordered beta phase The TiCr2 laves phases Results of the Re-optimization REVIEW OF THE TI-AL-MO SYSTEM Binary Subsystems Phases in the Ti Al Mo System Review of Critical Assessments in the Region from 0 to 20 at.%ti Thermodynamic Descriptions of the Ti Al Mo System RE-OPTIMIZATION OF THE AL-MO AND TI-AL-MO SYSTEMS Review of the Al Mo System Thermodynamic Descriptions for the Al Mo System Re-optimization of the Al Mo System Optimization Strategy Selection of Adjustable Parameters The beta phase The liquid phase The Al-rich intermetallic phases The AlMo3 phase The (Al) phase The Results Re-optimization of the Ti Al Mo System Optimization Strategy Selection of Parameters The eta phase The beta phase The delta phase The Results CONCLUSIONS AND SUGGESTED FUTURE WORK Conclusions Future Work

8 A THERMODYNAMIC PARAMETERS FOR THE TI-AL-NB SYSTEM B THERMODYNAMIC PARAMETERS FOR THE TI-AL-CR SYSTEM C THERMODYNAMIC PARAMETERS FOR THE TI-AL-MO SYSTEM REFERENCES BIOGRAPHICAL SKETCH

9 Table LIST OF TABLES page 3-1 Phases in the Ti Al Nb system Thermodynamic modeling of the phases used in the description of Kattner and Boettinger [9] Thermodynamic modeling of the phases in the Ti Al Nb system of Servant and Ansara [10] Thermodynamic modeling of the phases in the Ti Al Nb system of Witusiewicz et al. [17] Key experimental works in the Ti Al Nb system Tie line and tie triangle data for alloys A2, A3, and A133 that were used for the optimization Transition temperatures for alloy 11 and alloy Crystal structure of the σ phase [135] Phases in the Ti Cr system Homogeneity range of the Laves phases determined by Chen et al. [148] Crystal structure of the α TiCr 2 Laves phase Crystal structure of the β TiCr 2 Laves phase Crystal structure of the γ TiCr 2 Laves phase Data on the Laves phases that were used for the optimization Stable solid phases in the Ti Al Cr system Stable solid phases in the Ti Al Mo system Invariant reactions in the Al Mo system which were accepted for the optimization based on a critical evaluation of the available literature A-1 Thermodynamic Description for the Ti Al Nb System B-1 Thermodynamic Description for the Ti Al Cr System C-1 Thermodynamic Description for the Ti Al Mo System

10 Figure LIST OF FIGURES page 2-1 Schematic of the CALPHAD method [62] The contribution to the excess Gibbs free energy of mixing from the first four terms in the Redlich-Kister polynomial Graphical representation of the A) Kohler, B) Colinet, and C) Muggianu and D) Toop ternary extrapolation methods. In all diagrams, the open circle represents a point of ternary composition (x i, x j, x k ). The filled squares show the points on the i j binary which will make a contribution to the Gibbs free energy of the ternary composition. The Kohler and Muggianu extrapolation methods use only one point along the i j binary whereas the Colinet extrapolation method uses two points along the i j binary. One point gives the mole fraction of i and the other point gives the mole fraction of j that will be used. Therefore, in the Colinet method, x bin i + x bin j 1. In Toop extrapolation, the points along the i k and j k binaries are chosen at constant x k, but the point along the i j binary is chosen using in the same way as the Kohler method The surface of reference for a hypothetical compound (A, B) p (C, D) q plotted above the composition square The calculated A) Ti Al, B) Nb Al, and C) Ti Nb constituent binary systems used in the thermodynamic dataset of Kattner and Boettinger. Figures taken from [9] Calculated liquidus using the thermodynamic description of Kattner and Boettinger. Figure taken from [9] The calculated A) Ti Al, B) Nb Al, and C) Ti Nb constituent binary systems used in the thermodynamic dataset of Servant and Ansara [10] Isothermal section at 1273 K calculated using the thermodynamic description of Servant and Ansara [10]. The experimental data identified as [1992Men] and [1998Hel] refer to the works of Menon et al. [109] and Hellwig et al. [101] respectively Isothermal section at 1373 K calculated using the thermodynamic description of Servant and Ansara [10]. The experimental data identified as [1999Eck] and [2002Leo] refer to the works of Eckert et al. [108] and Leonard et al. [107] respectively Isothermal section at 1473 K calculated using the thermodynamic description of Servant and Ansara [10]. The experimental data identified as [1992Men], [1995Zdz], and [1998Hel] refer to the works of Menon et al. [109], Zdziobek et al. [13], and Hellwig et al. [101] respectively

11 3-7 Isothermal section at 1673 K calculated using the thermodynamic description of Servant and Ansara [10]. The experimental data identified as [1998Wan] refer to the work of Wang et al. [99]. Although there seems to be quite a large disagreement between the calculated isothermal section and the superimposed experimental data, it is possible that Wang et al. did not heat treat their alloys for sufficiently long periods of time at 1673 K to achieve the equilibrium microstructures Isothermal section at 1813 K calculated using the thermodynamic description of Servant and Ansara [10]. The experimental data identified as [2000Leo] refers to data obtained from Leonard and Vasudevan [14]. In this work, an alloy of composition Ti 25 at.%al 60 at.%nb which was heat treated at 1813 K and quenched was shown to be in the three phase region β+δ+σ However, calculations with this dataset indicate that an alloy of this composition, shown by the blue triangle, is in the δ + σ two-phase region Isothermal section at 1923 K calculated using the thermodynamic description of Servant and Ansara [10]. The experimental data identified as [1992Men] refers to the work of Menon et al. [109] Liquidus surface calculated using the thermodynamic description of Servant and Ansara [10]. The experimental data identified as [1992Fen], [1996Ara], and [2000Leo] refer to the works of Feng et al. [98], Leonard et al. [15], and d Aragão and Ebrahimi [4] respectively. The positions of alloy 11 with nominal composition Ti 45 at.%al 18 at.%nb and alloy 12 with nominal composition Ti 45 at.%al 27 at.%nb are also indicated. These alloys have also been shown to solidify first as β phase from the liquid. Although these alloys will be discussed later in this work, they are indicated here for convenience The calculated Ti Al phase diagram using the descriptions of A) Witusiewicz et al. [17]. The calculated Ti Al phase diagram of Saunders [18] (B) is repeated here for comparison Comparison between calculated and experimentally determined standard enthalpies of formation of alloys in the Nb Al system at 298 K. The reference state for Al is the fcc phase and the reference state for Nb is the bcc phase. The dashed line shows the results of the calculation using the dataset of Servant and Ansara [104] and the solid line shows the results of the calculation using the dataset of Witusiewicz et al. [17]. Experimental points were taken from George et al. [117] Comparison between calculated and experimentally determined enthalpies of formation of alloys in the Nb Al system. The reference state for Al is the liquid and the reference state for Nb is the bcc phase. The dashed line shows the results of the calculation using the dataset of Servant and Ansara [104] and the solid line shows the results of the calculation using the dataset of Witusiewicz et al. [17]. Both calculations were performed at 1669 K. Experimental points were taken from Mahdouk et al. [116]

12 3-14 The calculated Nb Al phase diagrams using the thermodynamic descriptions of Witusiewicz et al. [17] and Servant and Ansara [104]. The main differences between the phase diagrams are the congruent melting temperature of the η phase and the eutectic reaction temperature between the liquid, σ, and η phases. Servant and Ansara accepted the levitation thermal analysis measurements of Jorda et al. [118], while Witusiewicz et al. determined these temperatures using optical pyrometry and differential thermal analysis Isothermal section at 1813 K calculated using the thermodynamic description of Witusiewicz et al. [17]. The experimental data identified as [2000Leo] refers to data obtained from Leonard and Vasudevan [14]. In this work, an alloy of composition Ti 25 at.%al 60 at.%nb which was heat treated at 1813 K and quenched was shown to be in the three phase region β+δ+σ However, calculations with this dataset indicate that an alloy of this composition, shown by the blue triangle, is in the δ + σ two-phase region Liquidus surface in the Ti Al Nb system calculated using the thermodynamic description of Witusiewicz et al. [17]. All alloys indicated on the diagram should be in the primary crystallization field of the β phase. This suggests that the primary crystallization fields of the δ and γ phases extend too much into the ternary Solidus surface in the Ti Al Nb system calculated using the thermodynamic description of Witusiewicz et al. [17]. For an alloy to solidify as single phase β, not only should it be located in the primary crystallization field of β, but it should also be located to the right of the β solidus DTA curve for alloy 11 measured at heating and cooling rates of 10 o C/min Gibbs triangle showing the position of Alloy 11 as well as the position of the three phase triangle σ + β + γ measured by Hellwig et al. [101]. This three phase triangle was determined by heat treating an alloy of composition Ti 40.5 at.%al 24.8 at.%nb for 48 hours at 1200 o C and then water quenching. The alloy was shown to contain the β, σ, and γ phases, the compositions of which were measured using EPMA and are shown as open triangles. The alloy is indicated as the filled red circle. According to these results, Alloy 11 should already be in the three phase region σ + β + γ at 1200 o C Peak separation of the two overlapped peaks using Voigt functions. The peak representing the σ + γ σ + β + γ transition is shown in red and the peak representing the σ + β + γ β + γ transition is shown in green Peak analysis for the σ + γ σ + β + γ transformation. The first derivative is shown in blue. The transition temperature was selected based on the point at which the value of the first derivative deviated from zero

13 4-5 Peak analysis for the σ + β + γ β + γ transformations. The first derivative is shown in blue. The transition temperature was selected based on the point at which the value of the first derivative deviated from zero DTA curve for alloy 12 measured at heating and cooling rates of 10 o C/min DTA curve for alloy 12 showing the overlapped peaks on heating. Because of the presence of the shallow peak, shown by the blue rectangle, many peak separation methods must be evaluated The results of peak separation using three peaks. In this method, all peak parameters were refined. The results show that although two phase transformation peaks are fairly well modeled, a third peak, shown in red, appears at an improbable position between the two phase transformation peaks The results of peak separation using three peaks. In this approach, the centroid of one peak was fixed at 1273 o C in an attempt to fix the position of the first peak. Although the cumulative peak in good agreement with the original DTA signal, the phase transition temperatures associated with the first two peaks are approximately equal. This is also a highly improbable situation The results of peak separation using only two peaks. The cumulative peak is in good agreement with the original DTA signal. However, in an attempt to take into account the shallow peak, the parameters of the first peak are refined to produce a peak that is much wider than expected. This introduces quite a large error in the analysis of the phase transition temperature using the method of first deviation from the baseline using the first derivative The overlapped peaks on heating for alloy 12 indicating the choice of temperatures for the corresponding phase transformations o C is selected as the start temperature of the σ+γ β+σ+γ transition and 1323 o C is selected as the start temperature of the β + σ + γ β + γ transition Calculated binary Ti Nb diagram using the description of Zhang et al. [129] Unit cell of the σ phase. The Al1 and Al2 atoms are located at the 2a and 8i 2 Wyckoff positions respectively. Since the 2a and 8i 2 positions each have the same coordination number, the atoms in these positions occupy the same sublattice. The Nb2 and Nb3 atoms, which are located at the 8i 2 and 8j positions, occupy the same sublattice as they each have a coordination number of 14 and the Nb1 atom in the 4f position occupies its own sublattice. This results in the model (Nb) 16 (Al) 10 (Nb) 4 when there is no mixing on either of the sublattices The end members of the σ phase. The dashed line from Nb:Al:Nb to Ti:Al:Nb schematically illustrates the influence of the 0 L σ Nb,Ti:Al:Nb mixing parameter The end members of the δ phase. The dashed line from Nb:Al to Ti:Al schematically illustrates the influence of the 0 L δ Nb,Ti:Al parameter

14 5-5 The starting values of the A) 0 L β Al,Nb,Ti, B)1 L β Al,Nb,Ti, and C)2 L β Al,Nb,Ti parameters as a function of temperature. There was no possibility to fit these parameters as linear functions of temperature. Therefore, the 1 L β Al,Nb,Ti and 2 L β Al,Nb,Ti parameters were modeled as cubic functions and the 2 L β Al,Nb,Ti parameter was modeled as a quadratic function The end members of the γ phase. The dashed lines show the influence of the six mixing parameters of type 0 L γ i,j:k, which represent the condition of mixing of two different components on one sublattice with a second sublattice singly occupied by a third component The values of the 0 L γ Al:Nb,Ti at 1923 K, 1683 K, and 1473 K. Expressing this parameter as a linear function of temperature only will not result in a good fit to the experimental phase equilibria. Therefore, a quadratic variation with temperature was chosen Calculated liquidus surface for the Ti Al Nb system. The experimental data identified as [1992Fen], [1996Ara], and [2000Leo] refer to the works of Feng et al. [98], Leonard et al. [15], and d Aragão and Ebrahimi [4] respectively. The calculated liquidus surface is in good agreement with the literature showing that all alloys indicated solidify as β phase Calculated solidus surface in the Ti Al Nb system. The position of the β solidus shows that all alloys solidify as single phase β except for the alloy of composition Ti 48 at.%al 25 at.%nb from Feng et al. [98] and alloy 12. After solidification, these alloys exist as two phase β + γ Scheil reaction scheme for equilibria with the liquid in the Ti Al Nb system. There are five invariant reactions: three transition reactions, one eutectic reaction, and one peritectic reaction Calculated isothermal section at 1923 K. The experimental data identified as [1992Men] refers to the work of Menon et al. [109] Calculated isothermal section at 1813 K. The experimental data identified as [2000Leo] refers to data obtained from Leonard and Vasudevan [14]. In this work, an alloy of composition Ti 25 at.%al 60 at.%nb which was heat treated at 1813 K and quenched was shown to be in the three phase region β + δ + σ Calculated isothermal section at 1783 K. Tie line and tie triangle data for alloys A2, A3, and A133 are included. The tie lines shown in blue were determined after the description was re-optimized. Therefore, the current description is also able to predict the phase equilibria at 1783 K Calculated isothermal section at 1683 K. Tie line and tie triangle data for alloys A2, A3, and A133 are included

15 6-8 Calculated isothermal section at 1673 K. The experimental data of Wang et al. [99] is superimposed. While the dataset is able to calculate phase equilibria between β and δ, δ and σ, σ and γ, σ and η, and γ and η phases, other phase equilibria cannot be calculated. However, Wang et al. [99] may have insufficiently heat treated their alloys at 1673 K. Therefore, they may not have measured the equilibrium tie lines Calculated isothermal section at 1613 K. There is good agreement with the β γ tie line for alloy Calculated isothermal section at 1513 K. There is good agreemtn with the β γ tie line for alloy Calculated isothermal section at 1473 K. The experimental data identified as [1992Men], [1995Zdz], and [1998Hel] refer to the works of Menon et al. [109], Zdziobek et al. [13], and Hellwig et al. [101] respectively Calculated isothermal section at 1373 K. The experimental data identified as [1999Eck] and [2002Leo] refer to the works of Eckert et al. [108] and Leonard et al. [107] respectively Calculated isothermal section at 1273 K. The experimental data identified as [1992Men] and [1998Hel] refer to the works of Menon et al. [109] and Hellwig et al. [101] respectively Calculated isopleth through the nominal compositions of alloy 11 and alloy 12. There is good agreement between the calculations and the solid state transformations of alloys 11 and 12 measured using DTA. Alloy 11 solidifies as single phase β. However, there is no single phase β field for alloy 12. Instead, directly after solidification, β and γ should appear Calculated isopleth at 40 at.%al. Transformation temperatures measured using DTA from Leonard et al. [15] are included Calculated phase fraction diagram for alloy Calculated phase fraction diagram for alloy Unit cell of the α TiCr 2 Laves phase. A two sublattice model with one Ti-rich and one Cr-rich sublattice is used Unit cell of the β TiCr 2 Laves phase. The Ti1 and Ti2 atoms, which occupy the 4e and 4f positions respectively, occupy the same sublattice because they have the same coordination numbers, and the Cr1, Cr2, and Cr3 atoms, which occupy the 4f, 6g, and 6h positions, occupy the same sublattice as they have the same coordination numbers

16 7-3 Unit cell of the γ TiCr 2 Laves phase. The Ti atoms occupy the 4f position, and the Cr1 and Cr2 atoms occupy the 2a and 6h positions respectively. Since the Cr1 and Cr2 atoms have the same coordination numbers, they are placed on the same sublattice The Calculated partial Ti Cr phase diagram from 60 at.%cr to 70 at.%cr using the description of Saunders [18]. Only the low temperature α TiCr 2 and a high temperature γ TiCr 2 Laves phases are modeled. There homogeneity range of the α TiCr 2 phase is in good agreement with the experimental work of Chen et al. [148, 149]. However, the agreement of the homogeneity range of the γ TiCr 2 phase with the work of Chen et al. [148, 149] is not so good Graph showing the dependence of the G γ TiCr 2 Cr:Ti parameter for the γ TiCr 2 phase as the solid line. The linear dependence of the G β TiCr 2 Cr:Ti parameter is also included as the dashed line Calculated Ti Cr phase diagram using the new description. The experimental data from the literature is superimposed Calculated partial Ti Cr phase diagram from 60 at.%cr to 70 at.%cr. There is good agreement of the homogeneity range of the α TiCr 2, β TiCr 2 and γ TiCr 2 Laves phases with the experimental work of Chen et al. [148, 149] Calculated activity of Cr in the β phase compared to the experimental data of Pool et al. [152]. Since the new dataset uses the same thermodynamic parameters for the β phase as in Saunders [18], there is no difference in the activity curves for Cr between the new dataset and the dataset of Saunders [18] Calculated binary Al Cr phase diagram using the description of Saunders [18] Calculated binary Al Cr phase diagram from the description of Liang et al. [193] Comparison between the A) isothermal section at 1073 K calculated using the dataset of Saunders [18] and the B) assessed 1073 K isothermal section of Bochvar et al. [19] Comparison between the A) isothermal section at 1273 K calculated using the dataset of Saunders [18] and the B) assessed 1273 K isothermal section of Bochvar et al. [19] Comparison between the A) calculated [18]and B) assessed [19] liquidus surface The end members of the ternary γ Ti(Al,Cr) 2 phase. The dashed line from Al:Ti to Cr:Ti schematically illustrates the influence of the 0 L γ Al,Cr:Ti parameter K isothermal section calculated using the new description for the Ti Al Cr system. Experimental data of Jewett et al. [97, 177, 179] are superimposed

17 K isothermal section calculated using the new description for the Ti Al Cr system. Experimental data of Jewett et al. [97, 177, 179] are superimposed Liquidus surface calculated using the new description of the Ti Al Cr system Partial liquidus surface calculated using the new description of the Ti Al Cr system Scheil reaction scheme for the Ti Al Cr system Assessed Ti Mo phase diagram according to Massalski [201] Ti Mo phase diagram calculated using the dataset of Saunders [18] Assessed Al Mo phase diagram according to the Schuster [210] Calculated Al Mo phase diagram from the description of Saunders [18] Assessed 1873 K isothermal section from the work of Tretyachenko et al. [226]. The alloy of composition Ti 52 at.%al 45 at.%mo, which has been shown to solidify as single phase β [27], is in the AlMo + liquid two phase field. Therefore, this alloy solidifies as the AlMo phase Assessed partial solidus in the Ti Al Mo system from the work of Tretyachenko et al. [226]. The alloy of composition Ti 52 at.%al 45 at.%mo, which should solidify as single phase β [27], is instead in the β + AlMo + Al 63 Mo 37 three phase field. The group 2 and group 3 alloys of Nino et al. [27] were shown to solidify as single phase β Calculated 1773 K isothermal section using the description of Saunders [18]. The alloy of composition Ti 52 at.%al 45 at.%mo, which should be single phase β according to the work of Nino et al. [27], is in the liquid + β + Al 63 Mo 37 three phase field Phase equilibria in the region from 0 at.%ti to 20 at.%ti from Nino et al. [27]. The alloy of composition Ti 52 at.%al 45 at.%mo is indicated Calculated 1540 K isothermal sections using the description of Saunders [18]. Since the β phase does not extend to high enough Al compositions, the invariant reaction β+ Al 8 Mo 3 δ + η cannot be calculated Phase equilibria in the region from 0 at.%ti to 20 at.%ti from Nino et al. [27]. The invariant reaction β+ Al 8 Mo 3 δ + η, as well as the position of the β phase boundary at 1540 K, is indicated

18 9-11 Liquidus surface calculated using the description for the Ti Al Mo system of Saunders [18]. The alloy of composition Ti 52 at.%al 45 at.%mo is in the primary crystallization field of the δ phase. However, Nino et al. [27] showed that this alloy should solidify as single phase β. Additionally, the group 3 alloys investigated by Nino et al. [27] also solidify as single phase β. However, the calculated β solidus does not extend to high enough Al compositions for this to be reproduced Partial Al Mo phase diagram from 70 at.%al to 100 at.%al constructed by Eumann et al. [251] Al Mo phase diagram calculated using the new description. The experimental data from the literature are superimposed Al Mo partial phase diagram calculated using the new description. The experimental data from the literature are superimposed Calculated and experimentally determined enthalpies of formation of the phases in the Al Mo system. [1982Shi] refers to the work of Shilo and Franzen [249] and [1993Mes] refers to the work of Meschel and Kleppa [252] End members for the η phase which is described using the compound energy formalism as (Al*, Mo, Ti) 0.75 (Al, Mo*, Ti) 0.25 where the asterisk identifies the major species on each sublattice. The end members for the δ phase with model (Al, Mo*, Ti) 0.75 (Al*, Mo, Ti) 0.25 are the same. The interaction parameter 0 L δ Mo,Ti:Al, shown as the dashed line, influenced the extension of the δ phase into the ternary Liquidus surface calculated with the new description for the Ti Al Mo system. All alloys indicated were shown to solidify as single phase β in the work of Nino et al. [27], with which the new liquidus surface is in very good agreement Isothermal section at 1773 K calculated using the new description for the Ti Al Mo system. [2003Nin] refers to the work of Nino et al. [27] Isothermal section at 1673 K calculated using the new description for the Ti Al Mo system. [2003Nin] refers to the work of Nino et al. [27] Isothermal section at 1540 K calculated using the new description for the Ti Al Mo system. [2003Nin] refers to the work of Nino et al. [27] Isopleth calculated through 50 at.%al. The solid state transformation temperatures measured using thermal analysis from Nino et al. [27] are indicated. Alloys identified by filled squares were single phase β at the respective temperatures

19 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THERMODYNAMIC ASSESSMENT OF THE TI AL NB, TI AL CR, AND TI AL MO SYSTEMS Chair: Fereshteh Ebrahimi Major: Materials Science and Engineering By Damian M. Cupid December 2009 Two-phase alloys based on a microstructure of disconnected sigma Nb2Al precipitates within a gamma TiAl matrix are promising materials for gas turbine blades because of their expected creep resistance and fracture toughness properties. To optimize alloy microstructures, respective alloys should solidify as single phase beta, and, on aging transform to the two phase microstructures. To extend the high temperature single phase beta field to optimal compositions, beta stabilizers such as Cr and Mo may be used. The CALculation of PHAse Diagrams (CALPHAD) method is a powerful tool that can be used to guide materials design through the application of computational thermodynamics to the calculation of phase diagrams in multi-component systems. Existing thermodynamic descriptions for the Ti Al Nb, Ti Al Cr, and Ti Al Mo systems could not reproduce experimentally determined phase diagrams; therefore, CALPHAD based re-optimizations of the thermodynamic parameters of the phases in the multi-component system descriptions were required. The re-optimized description for the Ti Al Nb system calculates the experimentally observed extension of the primary crystallization field of the beta phase, the existence of the single phase beta field at sub-solidus temperatures, and the solid state phase transformations and phase transformation temperatures of two experimentally investigated alloys. 19

20 Calculations using the new description of the Ti Al Cr system are able to reproduce the ternary extension of the Laves phases based on TiCr2, the stoichiometric Tau phase at composition Ti 67 atomic %Al 8 atomic %Cr, the critically evaluated liquidus surface, and isothermal sections at 1073 K and 1273 K. The continuity of the beta phase to the Al Mo binary in the Ti Al Mo system could only be reproduced through re-optimization of the thermodynamic parameters for the Al Mo binary sub-system. The new Al Mo binary description calculates the congruent melting of the beta phase at 50 atomic %Al, and the new Ti Al Mo description is in excellent agreement with the extension of the single phase beta field to the Al Mo binary and the invariant reaction between beta, delta, Al8Mo3, and eta phases at 1540 K. 20

21 CHAPTER 1 INTRODUCTION There have been continuous efforts in developing high temperature materials for turbine blades in aircraft turbines. This application requires materials with high oxidation resistance, low creep deformation, long fatigue life, good thermal conductivity, and low production cost. Two-phase alloys based on γ TiAl and α 2 Ti 3 Al are promising as these alloys are lighter than the currently used Ni-based superalloys. For example, γ TiAl alloys have a low material density of approximately 4 g/cm 3 [1] whereas the CMSX-10 Ni-based superalloy has a density of 9.05 g/cm 3 [2]. Based on previous work on Ti Al Nb alloys [3, 4], two-phase alloys with a microstructure consisting of a γ TiAl(Nb) matrix reinforced by submicron-sized, disconnected σ Nb 2 Al precipitates are also promising γ TiAl based alloys because of their anticipated high temperature creep properties. Furthermore, additions of refractory elements such as Nb have been shown to improve the creep resistance as well as room temperature ductility of γ TiAl [5 8]. Several basic requirements must be met for design of the γ TiAl + σ Nb 2 Al alloys. First, the γ TiAl + σ-nb 2 Al microstructures should be stable at operating temperatures i.e., there should be no phase transformations upon cooling. Second, the alloys should contain at least 40 at.%al to provide oxidation resistance. Third and most important for microstructural development, the β bcc single phase solid solution should exist as a single-phase at high temperatures and that can be quenched and maintained at room temperature so that the γ TiAl + σ-nb 2 Al microstructures can be produced on aging. Knowledge of the phase equilibria in the Ti Al Nb system can significantly reduce alloy development time. As an example, the liquidus and solidus surfaces indicate which alloys should solidify as single phase β, and isothermal sections show the equilibrium phases in an alloy at a given temperature. Phase diagrams can either be determined experimentally or calculated using thermodynamic descriptions for the multi-component systems. Although there is much experimental information available in the literature 21

22 on the primary solidification fields on the liquidus surface in the Ti Al Nb system, the calculated liquidus projections from the thermodynamic descriptions of Kattner and Boettinger [9] and Servant and Ansara [10] are inconsistent with this data. For example, even though the single phase β field in the Ti Al Nb system has been shown to extend to approximately 40 at.%al at 30 at.%ti in the experimental works of Kaltenbach et al. [11], Perepezko et al. [12], and Zkziobek et al. [13] and to 40 at.%al at 20 at.%ti in the more recent works of Leonard et al. [14, 15], neither of the thermodynamic descriptions reproduce accurately the extension of the single phase β field. Additionally, the descriptions do not reproduce the presence of the single phase β field for an alloy of nominal composition Ti 44.5 at.%al 18.5 at.%nb as determined by Rios et al. [16]. Another inconsistency is that although an alloy of nominal composition Ti 25 at.%al 60 at.%nb, which was heat treated at 1813 K and quenched, has been shown to contain β + δ + σ in the microstructure [14], this alloy is located in the δ + σ two-phase field at 1813 K using the dataset of Servant and Ansara [10]. Since there is much interest in determining phase equilibria in the Ti Al Nb system for alloy design, recently and parallel to this work, a new description for the Ti Al Nb system was developed by Witusiewicz et al. [17], which was published in Although this description reproduces the single phase β field for an alloy of composition Ti 44.5 at.%al 18.5 at.%nb [16], this dataset still cannot calculate the extension of the single phase β field to 20 at.%ti at 40 at.%al or the β + δ + σ three phase field for an alloy of composition Ti 25 at.%al 60 at.%nb at 1813 K [14]. From the basic requirements for alloy design, the optimal two phase γ TiAl + σ Nb 2 Al microstructures should exist at compositions with more than 40 at.%al to provide oxidation resistance. This means that even if the extension of the β phase from Leonard et al. [14, 15] is taken into account, the β phase may, on the first hand, still not be stable enough to be retained on quenching, and, on the second hand, still may not exist at high enough Al compositions to ensure a low volume fraction of σ Nb 2 Al phase on 22

23 aging. Therefore, to extend the single phase β field, known β stabilizers such as Cr and Mo may be used. To understand the effect of Cr and Mo β stabilizers on phase transformations and reactions, thermodynamic descriptions for the quaternary Ti Al Nb Cr and Ti Al Nb Mo systems respectively are required. The key ternary systems required to generate the Ti Al Nb Cr and Ti Al Nb Mo quaternary descriptions are the Ti Al Cr and Ti Al Mo systems respectively. Although thermodynamic datasets for the Ti Al Cr and Ti Al Mo systems are available from Saunders [18], these datasets were created by combining the constituent binaries and extrapolating to the ternary; CALPHAD based assessments for these systems were not performed. Calculations in the Ti Al Cr system using the dataset of Saunders [18] could not reproduce either the liquidus surface from the critical evaluation of Bochvar et al. [19] or the extension of the γ Ti(Al,Cr) 2 Laves phase into the ternary. In this dataset, there is no description for the ternary τ phase, which was found in numerous works in the literature [20 26], and the ternary extension of the γ Ti(Al,Cr) 2 Laves phase along approximately 33 at.%ti is not modeled. Similarly, calculations in the Ti Al Mo system using the dataset of Saunders [18] could not reproduce the phase equilibria in the region from 0 to 20 at.%ti from the experimental work of Nino et al. [27]. For example, although Nino et al. [27] showed the continuity of the β phase to the Al Mo binary at 1773 K and the invariant reaction β + Al 8 Mo 3 δ + η at 1540 K, these features could not be reproduced using the dataset of Saunders [18] as the calculated single β phase field does not extend to high enough Al compositions. Based on the above information, the objective of this dissertation is to use the CALPHAD method to develop thermodynamic descriptions for the Ti Al Nb, Ti Al Cr, and Ti Al Mo systems that are able to reproduce experimental results. Using the CALPHAD approach, thermodynamic models for the phases in the multi-component 23

24 systems were selected based on the crystallography of the phases, the thermodynamic parameters of the models for each phase were optimized to fit the available experimental and theoretical data, and various kinds of phase diagrams were calculated to assess the agreement between the calculated phase diagrams and the experimental data. The PARROT [28] module of THERMO-CALC [29] was used to optimize the thermodynamic end-member and interaction parameters of the phase descriptions, and both THERMO-CALC [29] and PANDAT [30] software packages were used to calculate the various kinds of phase diagrams. The structure of this dissertation is as follows. In Chapter 2, the CALPHAD method is introduced, the thermodynamic modeling of stoichiometric phases, solution phases, and ordered phases using the compound energy formalism is developed, and details of Gibbs free energy minimization and thermodynamic optimization using the least squares method are given. A review of experimental work and thermodynamic calculations in the Ti Al Nb system is given in Chapter 3. Calculated liquidus projections and isothermal sections from the datasets Servant and Ansara [10] and Witusiewicz et al. [17] are compared to the available experimental data, discrepancies between the calculations and the experimental data are highlighted, and reasons for re-optimization are given. Chapter 4 gives a more detailed look at the key experimental data used for the optimization with a strong emphasis placed on the evaluation of thermal analysis data for Alloy 11 and Alloy 12. The specifics of the thermodynamic optimization of the parameters for the Ti Al Nb system are given in Chapter 5, and the results of the optimization are given in Chapter 6. The assessment of the ternary Ti Al Cr system starts in Chapter 7 with a literature review of the binary Ti Cr system, reasons why the Ti Cr dataset of Saunders [18] should be modified to include a description for the high temperature γ TiCr 2 Laves phase, the re-optimization of this system, and the final results. The introduction of the new description for the Ti Cr binary, the re-optimization of the ternary parameters of some of 24

25 the phases in the Ti Al Cr system, and the results of the re-optimization of the Ti Al Cr system are addressed in Chapter 8. Chapters 9 and 10 focus on the work in the Ti Al Mo system. Calculations using the Ti Al Mo dataset of Saunders [18] are compared to the experimental results in the region from 0 to 20 at.%ti from Nino et al. [27]. Chapter 9 concludes with the statement that the only way to reproduce the experimental data in the Ti Al Mo system is with a complete re-optimization of the binary Al Mo system. In Chapter 10, the details of the re-optimization of the Al Mo binary system and of some of the ternary parameters of some of the phases in the Ti Al Mo system are given and the results of both re-optimizations are compared to the experimental data. Conclusions and suggestions for future work are given in Chapter

26 CHAPTER 2 BACKGROUND AND THEORY A phase diagram is a graphical construction in two or three dimensions which shows the equilibrium state of a system as a function of parameters or quantities such as temperature, pressure, chemical potential, or composition. Phase diagrams are important as they act as maps that aid in the processing and development of materials. The classical representation of the binary system is the temperature composition diagram. Ternary systems can be graphically represented as isothermal sections, temperature composition cuts through the ternary system (isopleths), or liquidus and solidus projetions. The work of Pelton and Schmalzried [31] summarizes the various methods for the presentation of phase diagrams in multi-component systems. Phase diagrams can be determined experimentally, but this is a time consuming process with a complexity that drastically increases when additional elements are included. For example, although the phase diagram in a binary system can be determined using combined metallography, thermal analysis, and diffraction methods, the addition of a third element in the ternary system introduces an extra degree of freedom in composition, which phenomenally increases the amount of samples that must be prepared to fully understand and characterize the phase equilibria in the system. This is where the CALculation of PHAse Diagrams (CALPHAD) method plays a great role. The philosophy behind the CALPHAD method is that phase diagrams can not only be experimentally determined but also computed using thermodynamic datasets. The advantage of the CALPHAD method is that, on the one hand, one could predict phase equilibria in regions of the phase diagram that have not yet experimentally been determined. For example, one could predict the isothermal section at a given temperature in a ternary system based on experimentally determined phase equilibria at other temperatures. On the other hand, one could predict the phase equilibria in a multi-component system based on an extrapolation of the thermodynamic description of 26

27 the constituent subsystems. An additional benefit of the CALPHAD method is that one could calculate not only the conventional binary and ternary phase diagrams, but also phase fraction diagrams, which show the change in amount of the equilibrium phases as some condition in the system is changed, and volatility diagrams, which show the partial pressure of gas species that are in equilibrium with condensed phases. In principle, using the CALPHAD method, the investigator is now free to visualize the system from multiple perspectives. It is thus a powerful tool for predicting multi-component phase diagrams and guiding new material development. The CALPHAD method incorporates: 1. The selection of thermodynamic models to represent the various phases within a given system. 2. The expression of the thermodynamic models as analytic functions of temperature, pressure, and composition. 3. The optimization of the thermodynamic parameters using all available experimental and theoretical data. 4. The storage of the optimized parameters in computer-readable thermodynamic datasets. 5. The calculation of phase diagrams and various phase equilibria using the thermodynamic datasets. Points 1 and 2 above will be discussed in Section 2.1 and points 3, 4, and 5 will be discussed in Section 2.2. A schematic of the CALPHAD method is shown in Figure Thermodynamic Modeling Modeling is the selection of specific assumptions, based on physically sound principles, which are used to calculate the properties of a given system [32]. In the CALPHAD method, the Gibbs energy of the phases in a multi-component system is mathematically formulated using a specific model which takes into account some critical feature of the phase. For example, in the ideal gas, there is no interaction between atoms; therefore, the ideal gas model expresses the Gibbs free energy of the ideal gas only as a function of 27

28 temperature and pressure. As another example, to model the regular solution, the entropy of mixing is the same as that for the ideal solution and the enthalpy of mixing is non-zero [33]. Much interest has been directed to the modeling of ordered phases. Since the work of Tammann in 1919 [34] which postulated that atoms occupy specific lattice sites within a structure, theoretical treatments for the formation of ordered structures have been proposed by Borelius et al. [35], Gorsky [36] and Bragg and Williams [37]. Bond-energy models taking into account the interaction and exchange between like and unlike atoms within a lattice were presented by Bethe [38] and Bragg and Williams [39]. Taking these early bond-energy formulations into account, stoichiometric phases were regarded as being composed of multiple sublattices with each sublattice singly occupied by a specific element. In the work of Hillert and Staffansson [40], this formulation was extended to model a class of solution phases consisting of two sublattices with each sublattice occupied by two different elements. In this work, the term surface of reference and the idea of sublattice site fractions were introduced. Additionally, usage was made of the model of Temkin [41], which proposed that the ideal entropy of mixing could be calculated assuming random mixing on each sublattice. This mathematical formalism was generalized to cases with more than two elements on each sublattice by Harvig [42] and Sundman and Årgen [43]. Although the work of Hillert and Staffansson [40] can already be considered as the beginning of the compound energy formalism, the term was only coined in 1986 by Andersson et al. [44] Modeling of the Pure Elements The Gibbs energy of the pure elements depends only on temperature. An expression for the Gibbs free energy can be derived directly from an analytic expression of the measured heat capacity of the element from K, given as [45]: C p = c 2dT 6eT 2 2fT 2 (2 1) 28

29 Using the relation C p = T (ds/dt ) P, the differential entropy ds can be integrated using: S(T ) S(T =T 0 ) ds = and results in an expression of the entropy as: T C P T =T 0 T dt (2 2) S = S(T ) S(T = T 0 ) = b c ln T 2dT 3eT 2 + ft 2 (2 3) In Equation 2 3, b results from the substitution of the lower integration limit T 0 after integration. The enthalpy can also be calculated using the relation C P = (dh/dt ) P which gives the differential equation: yielding the solution: H(T ) H(T =T 0 ) T dh = C p dt T =T 0 (2 4) H = H(T ) H(T = T 0 ) = a ct dt 2 2eT 3 + 2fT (2 5) where a results from substitution of the lower integration limit T 0 after integration. The Gibbs energy is then calculated using the equation G = H T S (2 6) This yields the solution G = G(T ) G(T = T 0 ) = a + bt + ct ln T + dt 2 + et 3 + ft (2 7) where b = b c. In Equations 2 3, 2 7 and 2 5, attention must be given to the terms S(T = T 0 ) and G(T = T 0 ) and H(T = T 0 ) respectively. In all cases, since the analytic description of C P given in Equation 2 1 is valid only at temperatures above K, the lower temperature limit T 0 is customarily chosen as K. Using the third law of thermodynamics, S(T = T 0 ), which is now the entropy of the pure element at K, 29

30 can be calculated as: S(T = T 0 ) = S(298.15K) = C P T dt (2 8) where the expression for C P is valid at temperatures below K and can be determined using low-temperature adiabatic calorimetry. Unlike entropy, however, the Gibbs energy cannot be defined in absolute terms, and some reference for the Gibbs free energy is required. Conventionally, the reference state that is chosen is the enthalpy of the pure element in the same state at K. As an addendum, it is important to note that for convenience in tabulation, the enthalpy of the pure elements in their stable states at K is taken as zero. This means that Equation 2 7 can be re-written as: GHSER i = 0 G φ i (T ) 0 H φ i (298.15K) = a + bt + ct ln T + dt 2 + et 3 + ft (2 9) where GHSER i is the Gibbs energy of the pure element i referred to the enthalpy of its stable state φ at K, denoted as 0 H φ i (298.15K), and Equation 2 5 can be written as: HSER i = 0 H φ i (T ) 0 H φ i (298.15K) = a ct dt 2 2eT 3 + 2fT (2 10) The Gibbs energy of metastable states ϕ can also be referred to the enthalpy of the stable state using the equation: 0 G ϕ i (T ) 0 H φ i (298.15K) = 0 G ϕ i (T ) 0 G φ i (T ) + 0 G φ i (T ) 0 H φ i (298.15K) (2 11) Since the last two terms in Equation 2 11 is exactly equal to GHSER i, the equation can be simplified to the form: 0 G ϕ i (T ) 0 H φ i (298.15K) = 0 G ϕ i (T ) 0 G φ i (T ) + GHSER i (2 12) where the term 0 G ϕ i (T ) 0 G φ i (T ) represents the lattice stability of the element i in the metastable state ϕ, which is defined as the difference in Gibbs energy between the stable ( φ ) and metastable ( ϕ ) states of the element i as a function of temperature. 30

31 2.1.2 Modeling of Stoichiometric Phases A stoichiometric phase is defined as a phase in which no composition variation occurs. Therefore, just as in the case for the pure elements, the Gibbs energy of a stoichiometric phase can be expressed as an analytic function of temperature. In the case of the hypothetical stoichiometric compound A a B b, the Gibbs energy is: G AaB b (T ) a 0H φ A b 0H φ B = a + bt + ct ln T + dt 2 + et 3 + ft (2 13) Using Equation 2 9, the right hand terms of Equation 2 13 can be expressed as: G Aa B b (T ) a 0G φ A b 0H φ B + a GHSER A + b GHSER B = f G Aa B b (T ) + a GHSER A + b GHSER B (2 14) where G AaB b (T ) is the Gibbs energy of formation of the compound A a B b at temperature T. The modeling of stoichiometric phases can be extended to compounds Φ with more than two elements using the expression: G Φ (T ) n x i H φ i (298.15K) = a + bt + ct ln T + dt 2 + et 3 + ft (2 15) i=1 where n is the number of elements in the compound, i represents the stoichiometric coefficients of the elements of the compound according to the formula unit chosen, and φ represents the stable states of the constituent elements Modeling of Substitution Solutions A solution phase is defined as a phase in which composition variation occurs. The model for the substitutional solution, denoted as φ, expresses the molar Gibbs free energy of such a solution phase using the equation: G φ m = G φ,srf m + G φ,id m + G φ,e m (2 16) G φ,srf m is the surface of reference, which is the Gibbs energy of a mechanical mixture of the pure components at the same temperature and pressure. The surface of reference is thus 31

32 defined as: G φ,srf m = where x i is the mole fraction of component i in solution and 0 G φ i energy of the pure component i at the given temperature. n x 0 i G φ i (2 17) i is the molar Gibbs free The third term in Equation 2 16, G φ,id m, is the contribution to the molar Gibbs free energy resulting from a random, ideal mixing of the components. The physical basis for random mixing is the condition where there is no energy of interaction between the components. The contribution to the molar Gibbs energy resulting from random mixing, therefore, comes from the configurational entropy of mixing, and is defined as: G φ,id m = T S φ,id m = RT n x i ln x i (2 18) i The last term in Equation 2 16, G φ,e m, is the excess Gibbs free energy of mixing. It is the difference between the Gibbs free energy of the ideal solution and the Gibbs free energy of the actual solution, and is a measure of the deviation of the actual solution from ideal solution behavior. There are many ways to model the deviation of the actual solution from the ideal solution. One such model was advanced in 1929 by Hildebrand [33] and called the regular solution model. The regular solution is defined as one involving no entropy change when a small amount of one of its components is transferred to it from an ideal solution of the same composition, the total volume remaining unchanged [33]. In this case, the enthalpy of mixing of the solution is not zero (as is the case for the ideal solution), but equal to the excess Gibbs free energy of mixing, which depends only on the composition of the solution. For a binary solution, the simplest regular solution model is obtained when G φ,e m = ωx 1 x 2 where ω is a constant and x 1 and x 2 are the mole fractions of components 1 and 2 in solution respectively. 32

33 Another way to model the excess Gibbs free energy of mixing is to use the Redlich-Kister polynomials [46]. These are expressed as: G φ,e m n = x i x ν j L φ ij (x i x j ) ν (2 19) ν=0 where ν = 0, 1, 2,... and the ν L φ ij interaction parameter can display a linear temperature dependence expressed as: ν L φ ij = ν a φ ij + ν b φ ij T (2 20) The solution model becomes regular when only a temperature independent 0 L φ ij term is used and subregular when only the 1 L φ ij is used. The contributions of the various terms in the Redlich-Kister polynomial to the excess Gibbs energy of mixing is shown in Figure Extrapolation methods To extrapolate to higher order systems, a point must be selected on the binary side whose value of Gibss free energy of mixing will make some weighted contribution to the Gibbs free energy of the multi-component alloy. The Kohler [47], Colinet [48], Muggianu [49], and Toop [50] methods are various numerical methods which choose the composition on the binary side which contributes to the ternary. These numerical methods are graphically illustrated in Figure 2-3. The extrapolation methods may be expressed in the general form: n 1 G φ,bin.e m = n i=1 j=i+1 ω G φ,e m {x bin i ; x bin j } (2 21) where ω is the weighing factor and G φ,e m is the excess Gibbs free energy of the composition x bin i and x bin j chosen on the binary side. The Kohler method uses the numerical expression: n 1 G φ,bin.e m = i=1 j=i+1 n (x i + x j ) 2 G φ,e m { x i x i + x j ; x j x i + x j } (2 22) 33

34 the Colinet method uses the numerical expression: n 1 G φ,bin.e m = n i=1 j=i+1 x j /2 1 x i G φ,e m {x i ; 1 x i } + x i/2 1 x j G φ,e m {1 x j ; x j } (2 23) and the Muggianu method uses the numerical expression: n 1 G φ,bin.e m = n i=1 j=i+1 4x i x j (1 + x i x j )(1 + x j x i ) Gφ,E m { 1 + x i x j 2 ; 1 + x j x i } (2 24) 2 The weighing factors are chosen so that regular solution behavior is predicted by the binaries. When the Kohler numerical method (Equation 2 22) is evaluated, one obtains the expression: G φ,bin.e m = x i x j [ x i x j x i + x j ] k kl ij (2 25) which reproduces the regular solution behavior only when k is zero. When the Colinet numerical method (Equation 2 23) is evaluated, one obtains the expression: G φ,bin.e m = x ix j 2 [(2x i 1) k + (1 2x j ) k ] kl ij (2 26) which can reproduce both the regular solution parameter 0 L ij and sub-regular solution parameter 1 L ij when k equals zero and one respectively. The Muggianu numerical method, however, reproduces all terms in the Redlich-Kister polynomial (Equation (2 19)) and therefore is the recommended method for the extrapolation of ternary solution behavior based only on an analytic expression for the binaries. Unlike the other methods, the Toop extrapolation method [50] is a non-symmetric method. The numerical expression is given as: G φ,bin.e x i m = ( ) G φ,e m {1 x k ; x k } + ( ) G φ,e m {1 x k ; x k } 1 x k 1 x k x j +(1 x k ) 2 G φ,e m { x i x i + x j ; x j x i + x j } (2 27) From this expression, it is clear that the points along the i k and j k binaries are chosen in a similar fashion, but the point along the i j binary is chosen differently. Therefore, 34

35 the Toop method is customarily used in cases where one of the binaries behaves very differently from the other two binaries. If the behavior of a multi-component solution still cannot be predicted using the binary extrapolation methods, then an additional term is introduced into the expression for the Gibbs free energy of mixing which takes into account interactions of a ternary nature. The excess Gibbs free energy of mixing is then modeled as: G φ,e m = G φ,bin.e m + G φ,tern.e m = n 1 n n 2 x i x ν j L φ ij + i=1 j=i+1 n 1 n i=1 j=i+1 k=j+1 x i x j x k ν L φ ijk (2 28) where the first term represents all binary interactions determined using an applied extrapolation method and the second term represents the ternary interactions. The composition dependence of the ternary ν L φ ijk parameter is expressed as [51]: ν L φ ijk = υ i il φ ijk + υ j jl φ ijk + υ k kl φ ijk (2 29) where υ i = x i + (1 x i x j x k )/3 υ j = x j + (1 x i x j x k )/3 υ k = x k + (1 x i x j x k )/3 (2 30) The expressions for υ i, υ j, and υ k are equal to x i, x j, and x k respectively in the ternary system, with a sum that is always unity in higher order systems Modeling of Ordered Phases The Compound Energy Formalism The compound energy formalism [52] was constructed to thermodynamically model the behavior of solution phases with two or more sublattices in which the composition of at least one of the constituent sublattices is allowed to vary. An example of such a solution can be given as: (A, B) p (C, D) q (2 31) 35

36 where components A and B mix on the first sublattice and components C and D mix on the second sublattice. The components A, B, C, and D may be elements, molecular species, ions, or vacancies, and any component can simultaneously occupy multiple sublattices. The stoichiometric coefficients for the first and second sublattices are p and q respectively, so that one mole of formula units contains p + q moles of atoms. The constitution of the phase is described by the site fraction yi s of a species i on sublattice s as: y s i = ns i j ns j where j indicates summation over all species that occupy sublattice s. Using this (2 32) formalism, i ys i = 1 and s ns = n where n is the total number of sites summed over all sublattices and n s is the total number of sites in sublattice s. n s is thus equal to p for the first sublattice in the hypothetical phase modeled using Equation Since any component can occupy any sublattice, and, depending on the particular crystallographic information, the same component can occupy multiple sublattices, the mole fraction of component i is defined as: x i = s ns y s i s ns (1 y s V a ) (2 33) where y s V a is the site fraction of vacancies on sublattice s. In the limits where only one species occupies each sublattice, stoichiometric compounds are produced, which are defined as the end-member compounds of the solution phase. In the hypothetical phase represented by Equation 2 31, the stoichiometric compounds are A p C q, A p D q, B p C q, and B p D q. Using these precepts, the model of the compound energy formalism is expressed as: G φ m = G φ,srf m + G φ,conf m + G φ,e m (2 34) The key points of the formalism are: The surface of reference G φ,srf m end members. is the summation of the weighted contribution of all 36

37 The Gibbs free energy term related to the configurational entropy of mixing, G φ,conf m, is based on the assumption of random mixing within each sublattice. The excess Gibbs free energy of mixing G φ,e m be modeled using only G φ,srf m and G φ,conf m. is: contains all excess terms which cannot For the hypothetical compound given in Equation 2 31, the surface of reference term G φ,srf m = y 1 Ay 2 C 0G A:C + y 1 Ay 2 D 0G A:D + y 1 By 2 C 0G B:C + y 1 By 2 D 0G B:D (2 35) The surface of reference for the hypothetical compound (A, B) p (C, D) q plotted above the composition square is schematically illustrated in Figure 2-4. The Gibbs free energy term related to random mixing on each sublattice is: G φ,conf m = RT [p(y 1 A ln y 1 A + y 1 B ln y 1 B) + q(y 2 C ln y 2 C + y 2 C ln y 2 D)] (2 36) and the excess Gibbs free energy of mixing is given by: G φ,e m = y 1 Ay 1 B[y 2 CL A,B:A + y 2 DL A,B:D ] + y 2 Cy 2 D[y 1 AL A:D,C + y 1 BL B:C,D ] (2 37) The terms of type L i,j:k represent the case of mixing of i and j on the 1st sublattice with the second sublattice singly occupied by component k whereas terms of type L k:i,j represent the condition of mixing of i and j on the second sublattice with the first sublattice singly occupied by component k. These terms are usually modeled using the Redlich-Kister polynomials given in Equation 2 19 but modified to incorporate the sublattice modeling as: L i,j:k = ν (y 1 i y 1 j ) ν νl i,j:k (2 38) Phases with Order Disorder Transformations There are several systems in which a disordered phase transforms to the ordered structure as temperature is decreased. For example, the fcc phase in the Al Ni system orders to the L1 2 structure. This order disorder transformation was modeled in the binary 37

38 Al Ni system by Ansara et al. [53, 54] and extended to the ternary Al Cr Ni system in the work of Dupin et al [55]. The Gibbs energy of the ordered and disordered phases can be described using the same Gibbs energy function G m which is expressed as: G m = G dis m + G ord m (2 39) where G dis m is the Gibbs energy of the disordered phase, modeled as a substitutional solution, and G ord m is the additional contribution to the Gibbs free energy of the phase resulting from ordering. This additional contribution is described as: G ord m = G ord m (y s i ) G ord m (y s i = x i ) (2 40) where G ord m (y s i ) is the Gibbs energy of the ordered phase, expressed using the compound energy formalism, with the site fractions y s i of component i on sublattice s, and G ord m (y s i = x i ) is the Gibbs free energy contribution of the disordered phase to the ordered phase, calculated when the site fractions of each component y s i on each sublattice is the same and equal to the overall composition of the phase. In order to ensure the stability of the disordered phase, G m must always have an extremum when y i = x i. When the disordered phase is stable, this extremum must be a minimum. To take into account this additional stipulation of the thermodynamic model, the condition dg m used. dy s i = 0 for all temperatures is 2.2 Thermodynamic Optimization Thermodynamic optimization is the process where the adjustable parameters of the analytic expressions of the thermodynamic models are optimized so that the thermodynamic dataset is able to reproduce the available experimental and theoretical data. 38

39 2.2.1 Experimental Data Several kinds of experimental data can be used for thermodynamic optimization. One example is equilibrium tie-line data, i.e., the composition of the phases in equilibrium with each other at a given temperature preferentially determined using microprobe analysis. Another example is data from thermal analysis (both differential thermal analysis (DTA) and differential scanning calorimetry (DSC)), which give information on the temperature at which phase transformations or reactions take place. A suitably calibrated DSC can also be used to measure the heat capacity of a a given phase as well as the enthalpy of a given reaction. Chemical potential data is also quite important, and can be determined using galvanic cells or vapor pressure measurements. Both the Knudsen effusion method and the Langmuir vaporization method enables vapor pressure determination by measuring the rate of evaporation of a given species. These methods are often combined with the mass spectrometer to separate the various gas species. Calorimetric data determined from direct-reaction calorimetry or solution calorimetry can give information on the enthalpy of formation of condensed phases and the enthalpy of mixing of, for example, the liquid phase. Low temperature adiabatic calorimetry is used to measure the c p of a substance in the temperature range from close to 0 K to room temperature. Integration using Equation 2 2 will give the entropy of that substance. Drop calorimetry can also be used to measure the standard enthalpy of formation H 298K and enthalpy increment of a substance Theoretical Data During the last few years, much research effort has been focused on using data obtained from ab initio methods for CALPHAD based assessments. Ab initio methods can provide fundamental information that is not experimentally available [56], such as the enthalpies of formation and enthalpies of tranformation for real or hypothetical compounds. Many of these works use density functional theory (DFT) in the generalized gradient approximation (GGA) to calculate the ground state energies of the elements and 39

40 compounds. Wang et al. [57] calculated the lattice stabilities of 78 of the most important elements and compared the results to the SGTE lattice stabilities of Dinsdale [58]. The ab initio results were found comparable to the SGTE lattice stability data except for the transition metals. As an example of a combined CALPHAD + ab initio approach, Abe et al. [59] used ab initio calculations of the enthalpies of formation of solid phases, cluster variation method calculations to estimate the free energies at finite temperatures, and available experimental data in a CALPHAD type assessment of the Ir Ni system. A good review of the application of ab initio electronic structure calculations in the construction of phase diagrams of metallic systems with complex phases is given by Šob et al. [60] Gibbs Free Energy Minimization In Section 2.1, the analytic expressions for the Gibbs free energies of the pure elements, stoichiometric phases, solution phases, and phases modeled using the compound energy formalism were developed. The total Gibbs free energy of a system, summed over all phases φ, is given as: G = φ m φ G φ m (2 41) where m φ and G φ m are the number of moles and the Gibbs free energy per mole of phase φ respectively. The condition of equilibrium takes place when there is a minimum in the Gibbs free energy of the system. Therefore, to determine the equilibrium state of a system, Equation 2 41 should be minimized under the conditions that and i m φ s φ y φ,s i = 1 (2 42) n s y φ,s i = N i (2 43) where Equation 2 42 means that the sum of the site fractions y i of component i on sublattice s of phase φ is one and Equation 2 43 calculates the total number of moles N of component i within the system. This problem can be solved though the introduction of 40

41 Lagrangian multipliers µ i for each element and λ φ for each phase, where µ i is directly the chemical potential of component i. The Lagrangian for the system is then expressed as: L = φ m φ G φ m + i µ i [ φ m φ s n s y φ,s i N i ] + φ λ φ [ i y φ,s i 1] (2 44) The minimum of the Gibbs free energy can now be determined by the minimum of the Lagrangian function. Differentiating the Lagrangian function with respect to y φ,s i and m φ and setting the derivatives equal to zero yields respectively L y φ,s i = m φ Gφ m + µ y φ,s i m φ n s + λ φ = 0 (2 45) i for each component i and L m φ = Gφ m i µ i s n s y φ,s i = 0 (2 46) Equations 2 42, 2 43, 2 45, and 2 46 are then four non-linear equations with a set of unknowns m φ, y φ,s i, µ i, and λ φ which must be solved iteratively to yield the solution for the Gibbs free energy minimum at a given temperature and overall composition. In the first step, an initial guess is made of the site fractions of the components on the sublattices of the phases. Then, the chemical potentials of all components i are calculated using this initial guess of the site fractions for the phases and substituted into Equation The Newton-Raphson iterative method is then used to calculate refined compositions of the phases using Equation These are the compositions of the j + 1 iterative step, which are then used to calculate the chemical potentials of the elements in the different phases, and substituted into Equation 2 45 again. This is continued until the whole calculation converges sufficiently [61] Optimization Method: The Least Squares Method In the CALPHAD method, the Gibbs free energy of the phases in a given system are expressed as analytic functions with thermodynamic parameters from a particular model. Using the analytic functions, it is possible to generate a calculated value F i of a 41

42 property which is compared to the measured value L i of the same property under the same conditions. This measured value can be either experimental or theoretical. The difference between the measured value F i and the calculated value L i is multiplied by some weighing factor, w i, which gives the error v i : (F i L i ) w i = v i (2 47) The aim of thermodynamic optimization is to reduce the error to a minimum value. The condition of best fit is obtained when the sum of the squares of the errors is a minimum. That means that: n vi 2 = Min (2 48) i=1 where n experimental data values are chosen. The solution is obtained when the derivative of the sum of the squares with respect to the m adjustable coefficients C j is equal to zero. This condition is expressed as a series of m equations: C j n vi 2 = 2 i=1 n i=1 v i v i C j = 0 (j = 1,..., m) (2 49) To make the m equations linear, the error term v i is expanded as a Taylor series of the corrections C l to the adjustable coefficients and truncated after the linear term: v i = v 0 i + m l=1 v i C j C l (2 50) When Equation 2 50 is substituted into Equation 2 49, a series of m linear equations is generated: or l=1 n {vi 0 + i=1 m l=1 m n v i ( v i ) C l = C j C l i=1 v i C l C l } v i C j = 0 (j = 1,..., m) (2 51) n i=1 v 0 i v i C j (j = 1,..., m) (2 52) This system of m linear equations is solved iteratively, starting with an initial value of the coefficients C l. As a measure of the fit of the resulting calculated values of the properties 42

43 of the system to the experimental values of the properties of the system, the mean square error, given as: is used. mean square error = n i=1 v 2 i n m (2 53) 43

44 Figure 2-1. Schematic of the CALPHAD method [62]. 44

45 L G m [kj/mol] L 2 L L Mole Fraction B Figure 2-2. The contribution to the excess Gibbs free energy of mixing from the first four terms in the Redlich-Kister polynomial. 45

46 A Kohler B Colinet C Muggianu D Toop Figure 2-3. Graphical representation of the A) Kohler, B) Colinet, and C) Muggianu and D) Toop ternary extrapolation methods. In all diagrams, the open circle represents a point of ternary composition (x i, x j, x k ). The filled squares show the points on the i j binary which will make a contribution to the Gibbs free energy of the ternary composition. The Kohler and Muggianu extrapolation methods use only one point along the i j binary whereas the Colinet extrapolation method uses two points along the i j binary. One point gives the mole fraction of i and the other point gives the mole fraction of j that will be used. Therefore, in the Colinet method, x bin i + x bin j 1. In Toop extrapolation, the points along the i k and j k binaries are chosen at constant x k, but the point along the i j binary is chosen using in the same way as the Kohler method. 46

47 Figure 2-4. The surface of reference for a hypothetical compound (A, B) p (C, D) q plotted above the composition square. 47

48 CHAPTER 3 REVIEW OF THE TI AL NB SYSTEM 3.1 Phases in the Ti Al Nb System The crystallographic data of the solid phases in the Ti-Al-Nb system are given in Table 3-1. The σ and δ phases, which originate in the Nb-Al binary, show large solubilities for Ti and the η phase is a continuous solid solution between NbAl 3 and TiAl 3. The β and α and solid solution phases exhibit disorder-order transformations to the β 0 and α 2 phases respectively. Lattice site occupations of the β 0, α 2, and γ phases have been studied in the literature. Banerjee et al. [63] investigated a two-phase β 0 + α 2 alloy of composition Ti 25.6 at.%al 10.1 at.%nb and showed that Ti atoms occupy one sublattice and Al atoms with some Nb and Ti atoms occupy the second sublattice in the β 0 phase. Ordering separation of Ti and Al has also been observed in the β 0 phase by Hou and Fraser [64], Chaumat et al. [65], and Leonard and Vijay [66], but for the high Nb containing β 0 alloy of composition Ti 15 at.%al 68 at.%nb, ordering separation to form Nb and Al rich sublattices is also possible [66]. In the α 2 and γ phases, Nb atoms substitute on the Ti sites [67]. There are additionally two accepted ternary phases. The orthorhombic O phase [68 83] with Cmcm symmetry, based on the composition Ti 2 AlNb, is formed from ternary ordering of the α 2 Ti 3 Al lattice sites. Ti with some Nb occupies the 8g site, Nb with some Ti occupies the 4c2 site, and Al occupies the 4c1 site [70]. Transformation to the O phase can be either from the higher temperature β phase through intermediate hexagonal (α and α 2 ) phases or from the higher temperature β 0 phase through a martensitic type transformation [76, 77]. The reaction kinetics of the β 0 to O phase transformation were studied using differential thermal analysis [80]. The O phase undergoes a continuous ordering transformation [78, 82, 84] from the disordered state, where Nb and Ti randomly occupy the 8g and 4c2 lattice sites, to the ordered state. The O phase forms generally at temperatures lower than approximately 1273 K [72, 78, 79, 81, 81]. 48

49 The hexagonal ω phase with stoichiometry Ti 4 Al 3 Nb [85 87] forms from the higher temperature B2 phase by displacive transitions and chemical ordering through the metastable ω phase to the stable ω phase [87, 88]. Sadi et al. [81] investigated the long term heat-treatment and continuous cooling behavior of Ti 29.7 at.%al 21.8 at.%nb and Ti 23.4 at.%al 31.7 at.%nb alloys. After 1500 hours at 923 K, ω phase was detected in the Ti 29.7 at.%al 21.8 at.%nb alloy. Phase transformations to the metastable ω and ω and stable ω phases were observed in both alloys using dilatometry and using differential thermal analysis for alloy Ti 29.7 at.%al 21.8 at.%nb [81]. Other ternary phases have been found but have either later been disproved or their existence shown to be uncertain. An isolated T2 phase with bcc-type structure was presented in an isothermal section at 1473 K [12], but the formation of the T2 phase was later investigated by Jackson and Lee [89] and shown not to exist. The ordered γ 1 phase has been presented in various works [90 93]. This phase is suggested to form through continuous ordering of Nb atoms on Ti sites in the γ phase until at sufficiently large Nb concentrations (>18 at.%), Nb atoms occupy a specific lattice site in the γ 1 phase [90]. The stoichiometric formula of γ 1, Ti 4 Nb 3 Al 9, as well as the space group and Wyckoff positions of the structure are given in the work of Chen et al. [94]. Although Liu et al. [95] produced the structure from annealing an alloy of composition Ti 48 at.%al 10at.%Nb at at 1073 K for 34 hours and have used invariant line theory to predict the crystallographic features and morphology of the γ 1 precipitate [96], the existence or non-existence of the γ 1 phase is still not clear as, in an independent work, Jewett [97] could not find the γ 1 phase in an alloy of composition Ti 51 at.%al 23 at.%nb heat treated at 1473 K for 180 hours. 49

50 3.2 Experimentally Determined Phase Equilibria in the Ti Al Nb System The Liquidus Surface Several liquidus surfaces constructed using the analysis of as-cast microstructures exist in the literature. In 1989, Kaltenbach et al. [11] presented one of the first liquidus projections. The key features of this liquidus surface presented include: 1. A relatively large extension of the primary crystallization field for the β phase. 2. A transition reaction liquid + δ σ + β at 2073 K. 3. A transition reaction liquid + β σ + γ at 1623 K. 4. A transition reaction liquid + ζ β + γ at 1573 K. 5. An eutectic reaction between liquid, σ, η, and γ at 1523 K. In this work, however, no indication was made of a primary crystallization field for the α phase. Kaltenbach et al. [11] also presented a Scheil reaction scheme for the Ti Al Nb system. Also in 1989, Perepezko et al. presented a liquidus projection for the Ti Al Nb system [12]. In addition to microstructural observation of as cast alloys, Perepezko et al. also used information from differential thermal analysis of selected alloys. In this work: 1. A transition reaction liquid + η σ + γ was proposed between the liquid, σ, γ,and η phases instead of the eutectic reaction suggested by Kaltenbach et al. [11]. 2. A maximum was indicated in the liquid + η + γ univariant line. 3. The character of the transition reaction between the liquid, σ, γ, and β was changed to liquid + σ β + γ. 4. A primary crystallization field for the α phase was presented. No significant changes were made to the extension of the primary crystallization field of the β phase, although the invariant reaction between liquid, β, δ, and σ phases was shown to take place at lower Ti contents. In 1995, Zdziobek et al. constructed a new liquidus surface which was also based on the microstructural analysis of as cast alloys [13]. In this contribution, the character of 50

51 all invariant reactions was the same as that shown in the work of Perepezko et al. [12]. However, a much smaller extension of the primary crystallization of the β phase was given. There are some indications in the literature that the primary crystallization of the β field should extend to even higher Al contents than is shown in the works of Kaltenbach et al. [11], Perepezko et al. [12], and Zdziobek et al. [13]. For example, d Aragão and Ebrahimi reported that the β transus for an alloy of composition Ti 40 at.%al 27 at.%nb is 1723 K [4]. While this alloy is in the β primary crystallization field in the liquidus projections of Kaltenbach et al. [11] and Perepezko et al. [12], this alloy is close to the liquid + β + δ and liquid + β + σ univariant lines in the liquidus projection of Zdziobek et al [13]. Additionally, an alloy of composition Ti 48 at.%al 25 at.%nb was claimed to have been heat treated in the single phase β region [98]. However, this alloy lies in the γ crystallization field in the liquidus projections of Perepzko et al. [12] and Zdziobek et al. [13] and in the σ primary crystallization field in the liquidus surface constructed by Kaltenbach et al. [11]. As a result of these inconsistencies, Leonard et al. published two works to clarify the extension of the primary crystallization of the β phase [14, 15]. Based on the as cast microstructures of three alloys of nominal composition Ti 40 at.%al 30 at.%nb, Ti 40 at.%al 35 at.%nb, and Ti 40 at.%al 40 at.%nb, they were able to confirm that the primary crystallization of the β phase did indeed extend to much higher Al compositions since all of these alloys solidified as single phase β. They suggested a revision to the liquidus surface of Zdziobek et al. [13] with a much larger extension of the β phase primary crystallization region Isothermal Sections Complete isothermal sections at 1673 K [91, 99], 1473 K [11 13, 100, 101], 1423 K [91, 93], and 1273 K [101] based on experimental data are available in the literature. The 1673 K, 1473 K, and 1423 K isothermal sections of Chen et al. [91] and Ding et al. [93] 51

52 show the γ 1 phase, which was not accepted in the present research work as the existence or non-existence of the γ 1 phase is still unclear. The 1673 K isotherms [91, 99] indicate large solubilities of Ti in the δ and σ phases and Nb in the α phase. For example, the solubility of Ti in the δ phase is 28 at.% Ti at 17 at.% Al, the solubility of Ti in the σ phase is 37 at.%ti at 22 at.% Al, and the solubility of Nb in the α phase is 23 at.% Nb at 54 at.% Ti. These high solubilities would correspond to quite large extensions of the δ, σ, α and γ phases in the liquidus surface, which would contradict the works of Leonard et al. [14, 15]. One explanation for these reported large solubilities may come from the heat treatment history of the alloys under investigation. The samples were heat treated at 1473 K for 160 hours followed by furnace cooling, and were then annealed at 1673 K for 4.5 hours. It is quite possible that the annealing time at 1673 K was insufficient for equilibrium to occur, so that the extensions of the δ, σ, α, and γ phases may result from compositions of the phases taken from insufficiently equilibriated microstructures. The microstructures may in fact represent more closely the possible equilibira at 1473 K than the equilibria at 1673 K Vertical Sections Vertical sections from experimental data include the Ti 27.5 at.% Al to Nb [78], Ti 22 at.%al to Nb [79, 102], and TiAl to TiNb sections [72, 79]. These sections were constructed to show the development of phase equilibria regarding formation of the O phase. The vertical section from Al 3 Ti to Nb 3 Ti is also reported [12]. Witusiewicz et al. [17] presented differential thermal analysis data on vertical sections at 45 at.% Al, 47 at.% Al, 8 at.% Nb, through Ti 87.2 Nb 12.8 and Nb 2.2 Al 97.8, through Ti 72.8 Nb 27.2 and Ti 31.6 Al 68.4, and through Ti 70 Al 30 and Ti 51 Nb 49 and also showed the calculated vertical sections Ti 27.5 at.% Al to Nb and Al 3 Ti to Nb 3 Ti. 3.3 Thermodynamic Descriptions of the Ti Al Nb System The three thermodynamic descriptions for the Ti Al Nb system available in the literature were developed by Kattner and Boettinger in 1992 [9], Servant and Ansara in 1998 [10], and Witusiewicz et al. in 2009 [17]. In all three works, an overview of the 52

53 available experimental data is given, the thermodynamic models for the phases are presented, and an optimization procedure is described. All works include a calculated liquidus projection as well as selected calculated isothermal sections and isopleths Thermodynamic Description of Kattner and Boettinger In the thermodynamic assessment of Kattner and Boettinger [9], new thermodynamic descriptions for the Ti Al and Al Nb constituent binaries were developed but the description of the Ti Nb system was taken from Kaufman [103]. Although this work was published after the availability of the SGTE data for the pure elements [58], there is no indication of if this data were used to represent the Gibbs energies and lattice stabilities of the pure elements. The calculated constituent binaries are shown in Figure 3-1. In the ternary system, the liquid, (β Ti,Nb), (α Ti), and (Al) phases were modeled as substitutional solutions, the Ti 2 Al 5, TiAl 2, and Ti 4 Al 3 Nb phases were modeled as stoichiometric compounds, and the Nb 3 Al, Nb 2 Al, (Ti,Nb)Al 3, TiAl, Ti 3 Al, and Ti 2 AlNb phases were modeled using the compound energy formalism. The modeling of the phases is given in Table 3-2 where the nomenclature for the phases is taken from Kattner and Boettinger [9]. Although this dataset is able to predict key experimental phase equilibria at 1473 K and 1373 K that were available at that time, the calculated liquidus surface, shown in Figure 3-2, indicates too large extensions of primary crystallization of the Nb 3 Al and Nb 2 Al phases, and, consequently, an insufficient extension of the primary crystallization of the (β Ti,Nb) phase. However, the calculated liquidus surface is in good qualitative agreement with the liquidus surface of Kaltenbach et al. [11] i.e., the calculated liquidus surface predicts correctly the phases which occur at the invariant reactions. In some cases, however, the character of the invariant reactions are different. Of note is that this dataset calculates a eutectic reaction between the liquid, Nb 2 Al, (Ti,Nb)Al 3, and TiAl phases. 53

54 This work did not treat the formation of the β 0 phase, i.e., only the disordered β phase is modeled Thermodynamic Description of Servant and Ansara In this description, the SGTE data for the pure elements Al, Nb, and Ti were used [58] and the parameters for the Ti Al, Al Nb, and Ti Nb constituent binary sub-systems were taken from Saunders [18], Servant and Ansara [104], and Hari Kumar [105] respectively. Although Saunders did not originally model the α α 2 order disorder transformation in the Ti Al system, a description for this transformation was included in the dataset for the constituent Ti Al binary which was used in the Ti Al Nb ternary of Servant and Ansara without changing the phase diagram [10]. The calculated constituent binaries are shown in Figure 3-3. In the ternary system, the liquid, β, α, and (Al) phases were modeled as substitutional solutions, the Ti 5 Al 11, TiAl 2, and Ti 4 Al 3 Nb phases were modeled as stoichiometric compounds, and the δ Nb 3 Al, σ Nb 2 Al, β 0, η (Ti,Nb)Al 3, γ TiAl, α 2 Ti 3 Al, O 1 Ti 2 AlNb and O 2 Ti 2 AlNb phases were modeled using the compound energy formalism. Phases with order disorder transformations (disordered β to β 0 and disordered α to α 2 Ti 3 Al) were modeled using the same Gibbs energy function. Although there is an order disorder transformation from the disordered O 1 phase to the ordered O 2 phase, this order disorder transformation was not modeled, and both phases were treated as separate phases in the dataset. In a later work, Servant and Ansara modeled the order disorder transition of the O phase using the same Gibbs energy description for both phases [106]. The thermodynamic modeling of the phases used in this description is given in Table Reasons for re-optimization Calculations were performed with the dataset of Servant and Ansara [10] to assess its suitability for use in predicting phase equilibria in the Ti Al Nb system. The calculated isothermal sections at 1273 K (Figure 3-4), 1373 K (Figure 3-5), 1473 K (Figure 3-6), 1673 K (Figure 3-7), 1813 K (Figure 3-8), and 1923 K (Figure 3-9) are presented. The 54

55 calculations indicate that the description is quite good at predicting phase equilibria at temperatures of approximately 1473 K and lower. For example, although the β δ, δ σ, σ γ, and β + δ + σ phase equilibria determined by Leonard et al. [107] and the σ + γ + η phase equilibria from Eckert et al. [108] at 1373 K became available only after the dataset was published, the calculated isothermal section at 1373 K (Figure 3-5) shows that the dataset predicts such phase equilibria. However, the database could not correctly predict all experimentally observed phase equilibria at higher temperatures. For example, although the β + δ tie-line at 1923 K from Menon et al. [109] could be calculated (Figure 3-9), the β + δ + σ three phase equilibria at 1813 K observed in an alloy of composition Ti 25 at.%al 60 at.%nb, which was heat treated at this temperature and then quenched by Leonard et al. [14], could not. Instead, an alloy of this composition is in the δ + σ two phase field as is indicated in the calculated isothermal section at 1813 K (Figure 3-8). A comparison of the calculated isothermal section at 1673 K with the phase equilibria data determined by Wang et al. [99] indicates remarkable inconsistencies. One possible reason for the inconsistencies could be that Wang et al. [99] did not heat treat the alloys for a long enough time at 1673 K to yield the equilibrium microstructures. Therefore, the calculated isothermal section with experimental data at 1673 K in Figure 3-7 is included only to illustrate the difference between the calculations and the experimental data. When one compares the calculated liquidus surface (Figure 3-10) with experimental data, more inconsistencies appear. For example, the alloys of composition Ti 40 at.%al 30 at.%nb, Ti 40 at.%al 35 at.%nb, and Ti 40 at.%al 40 at.%nb, which have been shown to solidify as single phase β by Leonard et al. [14, 15], are in the primary crystallization field of the σ phase. Additionally, the alloy of composition Ti 48 at.%al 25 at.%nb, which was heat treated by Feng et al. in the single phase β field [98], is in the primary crystallization field of the γ phase. As well, the alloy of composition Ti 40 at.%al 27 at.%nb, which has been shown to have a β transus temperature of 1723 K 55

56 [4], is in the primary crystallization field of the σ phase. All inconsistencies between the calculated liquidus surface and the experimental data indicate that the calculated primary crystallization field of the β phase does not extend to high enough Al compositions, and that the primary crystallization fields of the δ, σ, and γ phases extend too much into the ternary. Taking into account the inconsistencies between the calculated liquidus surface and experimental work, as well as the inconsistency of the calculated isothermal section at 1813 K, the thermodynamic description of the Ti Al Nb system developed by Servant et al. [10] should be re-optimized. The primary crystallization field of the β phase should be extended to be in agreement with the literature, but the new dataset should still be able to predict, within good agreement, the experimental phase equilibria at lower temperatures Thermodynamic Description of Witusiewicz et al. Parallel to this work, another thermodynamic description for the Ti Al Nb system was developed by Witusiewicz et al. [17] for use as the constituent ternary in the thermodynamic description of the quaternary Al B Nb-Ti system. In this work, the SGTE data for the pure elements Al, Nb, and Ti were taken from Dinsdale [58] and the description of the Ti Nb system was taken from Hari Kumar [105]. However, new descriptions for both the Ti Al [110] and Al Nb [17] binaries were developed Thermodynamic description of the Ti Al system In 2006, Schuster and Palm published a critical assessment of the Ti Al system [111], highlighting inconsistencies between various representations of the Ti Al phase diagram and the available experimental data. The key points of interest are: 1. The presence of a melting point maximum for the β phase at 8.5 at.% Al based on the work of Ogden et al. [112], which was accepted in the evaluation of Schuster and Palm [111]. The melting point maximum for the β phase is calculated at 20 at. % Al in the description of Saunders [18]. 56

57 2. The presence of β β 0 ordering in the Ti rich region. Ordering of the β phase was investigated in the work of Ohnuma et al. [113]. This was not accepted in the critical evaluation of Schuster and Palm [111] and not modeled in the thermodynamic assessment of Saunders [18]. 3. The peritectoid formation of the α 2 phase according to the reaction β + α α 2 or the congruent formation of the α 2 phase from the α phase. The peritectoid formation of α 2 was accepted in the evaluation of Schuster and Palm [111] but not reproduced using the thermodynamic description of Saunders [18]. 4. The existence and stability of the Ti 3 Al 5 phase. This phase was considered to be metastable in the work of Schuster and Palm [111] and was not included in the description of Saunders [18]. 5. The presence of two phases Ti 1 x Al 1+x and Ti 2+x Al 5 x [114, 115] or one one-dimensional anti-phase structure (1d-APS) [111] in the region between the Al rich boundary of the γ TiAl phase and the Ti rich boundary of the η TiAl 3 phase at temperatures between 1450 K and 1720 K. The thermodynamic description of Saunders includes only the stoichiometric Ti 5 Al 11 phase in this region [18]. In 2008, Witusiewicz et al. [110] optimized a thermodynamic description for the Ti Al system using information from the critical assessment of Schuster and Palm. The new description: 1. Includes a melting point maximum of the β phase at 9.6 at. %Al and 1963 K. 2. Models the second order order disorder β β 0 transformation using the same Gibbs energy function for the disordered and ordered phases. 3. Calculates the peritectoid formation of α 2 according to the reaction β + α α 2 at 1432 K. 4. Models the Ti 3 Al 5 phase as a stoichiometric phase which forms at 1083 K. 5. Models the 1d-APS as ζ Ti 2+x Al 5 x. Witusiewicz et al. did not model the order disorder transformation of the disordered α phase to the ordered α 2 phase using the same Gibbs free energy function. The calculated binary Ti Al phase diagram using this new description is shown in Figure 3-11 along with the calculated Ti Al binary from Saunders for comparison. 57

58 Thermodynamic description of the Al Nb system A new thermodynamic description for the Al Nb system was developed by Witusiewicz et al. in 2009 [17]. The optimization of this system took into account data on the enthalpies of formation of the δ Nb 3 Al, σ Nb 2 Al and η NbAl 3 phases from Mahdouk et al. [116] and George et al. [117] and of the β phase at 3 and 6 at.%al which was also measured by George et al. These data were not available to Servant and Ansara at the time of their Nb Al assessment [104]. The comparison between the calculated [17, 104] and experimentally determined enthalpies of formation at 298 K measured by George et al. [117] using the emf method are shown in Figure 3-12 and the comparison between the calculated and experimentally determined enthalpies of formation of the alloys at 1699 K and 1533 K compared to the results of Mahdouk et al. [116] measured using solution calorimetery are shown in Figure The calculations show that the dataset of Witusiewicz et al. [17] is in better agreement with the enthalpy of formation data of the phases in the Al Nb system than the dataset of Servant and Ansara [104]. The calculated Al Nb phase diagrams from the assessments of Witusiewicz et al. [17] and Servant and Ansara [104] are shown in Figure 3-14 for comparison. The main differences between the calculated phase diagrams are the temperature of congruent melting of the η phase and the eutectic reaction temperature between the liquid, σ, and η phases. Servant et al. [104] took into account the work of Jorda et al. [118], who, using levitation thermal analysis, measured the congruent melting point of the η phase as 1953±5 K and the eutectic reaction temperature between the liquid, σ, and η phases as 1863±5 K. Therefore, the calculation of the eutectic reaction temperature between the liquid, σ, and η phases is within the experimental error of Jorda et al. [118], and the congruent melting of the η phase is in better agreement with the work of Jorda et al. [118] than is the congruent melting of the η phase calculated using the dataset of Witusiewicz et al. [17]. Witusiewicz et al. [17] used high-temperature pyrometry to determine the congruent melting of the η phase and differential thermal analysis to measure the eutectic 58

59 reaction temperature between the liquid, σ, and η phases. Their measurements gave a congruent melting point of 1987±6 K and an eutectic reaction temperature of 1844 K. Therefore, their calculation is in excellent agreement with their experimental work Thermodynamic Description of the Ti Al Nb system The models for the phases used in the dataset of Witusiewicz et al. [17] are given in Table 3-4. In this work, the liquid, β, and α phases were modeled as substitutional solutions, the Ti 4 Al 3 Nb and Ti 3 Al 5 phases were modeled as stoichiometric compounds, and the δ Nb 3 Al, σ Nb 2 Al, η (Ti 1 x Nb x )Al 3, γ TiAl, β 0, α 2, ζ Ti 2+x Al 5 x, TiAl 2, O 1 Ti 2 AlNb, and O 2 Ti 2 AlNb phases were treated using the compound energy formalism. Although there is an order disorder transformation for the α (α α 2 ), β ( β β 0 ), and O (O 1 O 2 ) phases, only the β and β 0 phases were modeled using the same Gibbs energy function. Calculations were also performed with this dataset to evaluate its ability to calculate correctly the high temperature phase equilibria in the Ti Al Nb system. The calculations indicate that there are still some inconsistencies with the literature. For example, the calculated 1813 K isotherm (Figure 3-15) shows that this dataset still can not predict the β + δ + σ three-phase field for an alloy of composition Ti 25 at.%al 60 at.%nb which was determined using heat treatment and quench experiments by Leonard and Vasudevan [14]. The calculated liquidus surface (Figure 3-16) also indicates that the primary crystallization field of the β phase does not extend to high enough Al compositions since some of the alloys which have been shown to solidify as β phase are in the primary crystallization fields of γ and σ. The solidus surface (Figure 3-17) shows that not only does the primary crystallization field of the β phase not extend to high enough Al compositions, but also the β solidus, shown as the red line in Figure 3-17, does not extend to high enough Al compositions. All alloys to the left of the β solidus will solidify as single phase β from the liquid, and alloys to the immediate right will, on cooling, pass through a three phase field liquid + β + x where x can be either the δ, the σ, the γ, or 59

60 the α phase, depending on alloy composition. Therefore, as an additional constraint, not only should all β solidifying alloys be located in the primary crystallization field of the β phase, but the β solidus should also extend to high enough Al compositions so that these alloys solidify as single phase β. 60

61 Table 3-1. Phases in the Ti Al Nb system. Phase Prototype Pearson symbol Space group Structure report σ Nb 2 Al σcrfe tp30 P 4 2 /mnm D8 b δ Nb 3 Al Cr 3 Si cp8 P m 3n A15 ζ Ti 5 Al 11 Al 3 Zr ti16 I4/mmm D0 23 β W ci2 Im 3m A2 β 0 CsCl cp2 P m 3m B2 α Mg hp2 P 6 3 /mmc A3 α 2 Ti 3 Al Ni 3 Sn hp8 P 6 3 /mmc D0 19 γ TiAl AuCu tp2 P 4/mmm L1 0 ε TiAl 2 Ga 2 Hf ti24 I4 1 /amd... η (Ti,Nb)Al 3 Al 3 Ti ti8 I4/mmm D0 22 (Al) Cu cf4 F m 3m Al O Ti 2 NbAl NaHg oc16 Cmcm... ω Ti 4 NbAl 3 InNi 2 hp6 P 6 3 /mmc B8 2 61

62 A Ti Al B Al Nb C Ti Nb Figure 3-1. The calculated A) Ti Al, B) Nb Al, and C) Ti Nb constituent binary systems used in the thermodynamic dataset of Kattner and Boettinger. Figures taken from [9]. 62

63 Table 3-2. Thermodynamic modeling of the phases used in the description of Kattner and Boettinger [9]. Phase Thermodynamic Model Liquid (Al,Nb,Ti) (β Ti,Nb) (Al,Nb,Ti) (α Ti) (Al,Nb,Ti) (Al) (Al,Nb,Ti) Ti 2 Al 5 (Ti) 2 (Al) 5 TiAl 2 (Ti)(Al) 2 Ti 4 Al 3 Nb (Ti) 4 (Al) 3 Nb Nb 3 Al (Al,Nb,Ti) 0.75 (Al,Nb,Ti) 0.25 Nb 2 Al (Al,Nb,Ti) (Al,Nb,Ti) (Nb) (Ti,Nb)Al 3 (Al,Nb,Ti) 0.25 (Al,Nb,Ti) 0.75 TiAl (Al,Nb,Ti) 0.5 (Al,Nb,Ti) 0.5 Ti 3 Al (Al,Nb,Ti) 0.75 (Al,Nb,Ti) 0.25 Ti 2 AlNb (Nb,Ti) 0.5 (Nb,Ti) 0.25 (Al)

64 Figure 3-2. Calculated liquidus using the thermodynamic description of Kattner and Boettinger. Figure taken from [9]. 64

65 A Ti Al B Al Nb C Ti Nb Figure 3-3. The calculated A) Ti Al, B) Nb Al, and C) Ti Nb constituent binary systems used in the thermodynamic dataset of Servant and Ansara [10]. 65

66 Table 3-3. Thermodynamic modeling of the phases in the Ti Al Nb system of Servant and Ansara [10]. Phase Thermodynamic Model Liquid (Al,Nb,Ti) (β) (Al,Nb,Ti) (α) (Al,Nb,Ti) (Al) (Al,Nb,Ti) Ti 2 Al 5 (Ti) 2 (Al) 5 TiAl 2 (Ti)(Al) 2 Ti 4 Al 3 Nb (Ti) 4 (Al) 3 Nb δ Nb 3 Al (Al,Nb,Ti) 0.75 (Al,Nb,Ti) 0.25 σ Nb 2 Al (Al,Nb,Ti) (Al,Nb,Ti) (Nb) η (Ti,Nb)Al 3 (Al,Nb,Ti) 0.25 (Al,Nb,Ti) 0.75 TiAl (Al,Nb,Ti) 0.5 (Al,Nb,Ti) 0.5 β 0 (Al,Nb,Ti) 0.5 (Al,Nb,Ti) 0.5 α 2 Ti 3 Al (Al,Nb,Ti) 0.75 (Al,Nb,Ti) 0.25 O 1 Ti 2 AlNb (Al,Nb,Ti) 0.75 (Al,Nb,Ti) 0.25 O 2 Ti 2 AlNb (Al,Nb,Ti) 0.50 (Al,Nb,Ti) 0.25 (Al,Nb,Ti) 0.25 Figure 3-4. Isothermal section at 1273 K calculated using the thermodynamic description of Servant and Ansara [10]. The experimental data identified as [1992Men] and [1998Hel] refer to the works of Menon et al. [109] and Hellwig et al. [101] respectively. 66

67 Figure 3-5. Isothermal section at 1373 K calculated using the thermodynamic description of Servant and Ansara [10]. The experimental data identified as [1999Eck] and [2002Leo] refer to the works of Eckert et al. [108] and Leonard et al. [107] respectively. Figure 3-6. Isothermal section at 1473 K calculated using the thermodynamic description of Servant and Ansara [10]. The experimental data identified as [1992Men], [1995Zdz], and [1998Hel] refer to the works of Menon et al. [109], Zdziobek et al. [13], and Hellwig et al. [101] respectively. 67

68 Figure 3-7. Isothermal section at 1673 K calculated using the thermodynamic description of Servant and Ansara [10]. The experimental data identified as [1998Wan] refer to the work of Wang et al. [99]. Although there seems to be quite a large disagreement between the calculated isothermal section and the superimposed experimental data, it is possible that Wang et al. did not heat treat their alloys for sufficiently long periods of time at 1673 K to achieve the equilibrium microstructures. Figure 3-8. Isothermal section at 1813 K calculated using the thermodynamic description of Servant and Ansara [10]. The experimental data identified as [2000Leo] refers to data obtained from Leonard and Vasudevan [14]. In this work, an alloy of composition Ti 25 at.%al 60 at.%nb which was heat treated at 1813 K and quenched was shown to be in the three phase region β + δ + σ However, calculations with this dataset indicate that an alloy of this composition, shown by the blue triangle, is in the δ + σ two-phase region. 68

69 Figure 3-9. Isothermal section at 1923 K calculated using the thermodynamic description of Servant and Ansara [10]. The experimental data identified as [1992Men] refers to the work of Menon et al. [109]. Figure Liquidus surface calculated using the thermodynamic description of Servant and Ansara [10]. The experimental data identified as [1992Fen], [1996Ara], and [2000Leo] refer to the works of Feng et al. [98], Leonard et al. [15], and d Aragão and Ebrahimi [4] respectively. The positions of alloy 11 with nominal composition Ti 45 at.%al 18 at.%nb and alloy 12 with nominal composition Ti 45 at.%al 27 at.%nb are also indicated. These alloys have also been shown to solidify first as β phase from the liquid. Although these alloys will be discussed later in this work, they are indicated here for convenience. 69

70 A B Figure The calculated Ti Al phase diagram using the descriptions of A) Witusiewicz et al. [17]. The calculated Ti Al phase diagram of Saunders [18] (B) is repeated here for comparison. 70

71 Figure Comparison between calculated and experimentally determined standard enthalpies of formation of alloys in the Nb Al system at 298 K. The reference state for Al is the fcc phase and the reference state for Nb is the bcc phase. The dashed line shows the results of the calculation using the dataset of Servant and Ansara [104] and the solid line shows the results of the calculation using the dataset of Witusiewicz et al. [17]. Experimental points were taken from George et al. [117]. 71

72 Figure Comparison between calculated and experimentally determined enthalpies of formation of alloys in the Nb Al system. The reference state for Al is the liquid and the reference state for Nb is the bcc phase. The dashed line shows the results of the calculation using the dataset of Servant and Ansara [104] and the solid line shows the results of the calculation using the dataset of Witusiewicz et al. [17]. Both calculations were performed at 1669 K. Experimental points were taken from Mahdouk et al. [116]. 72

73 Figure The calculated Nb Al phase diagrams using the thermodynamic descriptions of Witusiewicz et al. [17] and Servant and Ansara [104]. The main differences between the phase diagrams are the congruent melting temperature of the η phase and the eutectic reaction temperature between the liquid, σ, and η phases. Servant and Ansara accepted the levitation thermal analysis measurements of Jorda et al. [118], while Witusiewicz et al. determined these temperatures using optical pyrometry and differential thermal analysis. 73

74 Table 3-4. Thermodynamic modeling of the phases in the Ti Al Nb system of Witusiewicz et al. [17]. Phase Thermodynamic Model Liquid (Al,Nb,Ti) (β) (Al,Nb,Ti) 1 (Va) 3 (α) (Al,Nb,Ti) 1 (Va) 0.5 (Al) (Al,Ti) 1 (Va) 1 Ti 4 Al 3 Nb (Ti) 4 (Al) 3 Nb Ti 3 Al 5 (Ti) 3 Al 5 δ Nb 3 Al (Al,Nb,Ti) 0.75 (Al,Nb,Ti) 0.25 σ Nb 2 Al (Al,Nb,Ti) (Al,Nb,Ti) (Nb,Ti) η (Ti 1 x Nb x )Al 3 (Al,Nb,Ti) 1 (Al,Nb,Ti) 3 γ TiAl (Al,Nb,Ti) 1 (Al,Nb,Ti) 1 β 0 (Al,Nb,Ti) 0.5 (Al,Nb,Ti) 0.5 (Va) 3 α 2 Ti 3 Al (Al,Nb,Ti) 3 (Al,Nb,Ti) 1 ζ Ti 2+x Al 5 x (Al,Nb,Ti) 2 (Al,Nb,Ti) 5 TiAl 2 (Al,Nb,Ti) 1 (Al,Nb,Ti) 2 O 1 Ti 2 AlNb (Al,Nb,Ti) 0.75 (Al,Nb,Ti) 0.25 O 2 Ti 2 AlNb (Al,Nb,Ti) 0.50 (Al,Nb,Ti) 0.25 (Al,Nb,Ti) 0.25 Figure Isothermal section at 1813 K calculated using the thermodynamic description of Witusiewicz et al. [17]. The experimental data identified as [2000Leo] refers to data obtained from Leonard and Vasudevan [14]. In this work, an alloy of composition Ti 25 at.%al 60 at.%nb which was heat treated at 1813 K and quenched was shown to be in the three phase region β + δ + σ However, calculations with this dataset indicate that an alloy of this composition, shown by the blue triangle, is in the δ + σ two-phase region. 74

75 Figure Liquidus surface in the Ti Al Nb system calculated using the thermodynamic description of Witusiewicz et al. [17]. All alloys indicated on the diagram should be in the primary crystallization field of the β phase. This suggests that the primary crystallization fields of the δ and γ phases extend too much into the ternary. Figure Solidus surface in the Ti Al Nb system calculated using the thermodynamic description of Witusiewicz et al. [17]. For an alloy to solidify as single phase β, not only should it be located in the primary crystallization field of β, but it should also be located to the right of the β solidus. 75

76 CHAPTER 4 KEY EXPERIMENTAL DATA FOR OPTIMIZATION The key experimental data from the literature on the Ti Al Nb system are given in Table 4-1. These works clarify the crystallography of stable and metastable phases, present equilibrium tie line and tie triangle data, and suggest the topography of the liquidus surface. Therefore, although it is not possible that all works are directly taken into account in the optimization, all works are important for a full understanding of the phase equilibria in the Ti Al Nb system. The works which were used in the current optimization are identified with an asterisk. Several alloys were investigated experimentally using the techniques of XRD and HTXRD for phase identification, SEM and TEM for microstructure analysis, EPMA for phase composition determination, and DTA for phase transformation temperatures. All experimental work was performed at the University of Florida by Orlando Rios, Michael S. Kesler, and Sonalika Goyel under the supervision and advice of Prof. Fereshteh Ebrahimi. The specific details on alloy preparation and analysis are given in the Ph.D. thesis of Rios [119]. Only the results of this work which were directly used in the optimization of the parameters for the Ti Al Nb system will be discussed here. As this section focuses on the analysis of original experimental data, which were given in temperature units of o C, all temperature units in this section will also be given in o C for convenience. 4.1 Tie Line Data Tie line data from heat treatment and quench experiments at 1510 o C and 1410 o C for alloys A2 and A3 and at 1520 o C and 1420 o C for alloy A133 [119] were taken into account in the optimization. The bulk compositions of the alloys and the composition of the equilibrium phases are given in Table 4-2. Alloys A2 and A3 were three phase σ +η +γ at 1510 o C and 1410 o C whereas alloy A133 was two phase σ + γ at 1520 o C and 1420 o C. 76

77 4.2 Alloy 11 5 g buttons of alloy 11 were prepared by arc-melting in a non-consumable tungsten electrode arc melter from high purity pure elements. The nominal composition of alloy 11 is Ti 45 at.%al 18 at.%nb and the actual composition of alloy 11 measured using EPMA is Ti at.%al 18.1 at.%nb. The DTA of alloy 11 heated at 10 o C/min to 1600 o C gave a solidus temperature of 1532 o C. Based on this result, additional DTA measurements were performed to determine the solid state reaction temperatures for alloy 11. For these measurements, alloy 11 was heated in the DTA to 1500 o C, held at this temperature for 30 min, and then cycled between 1100 o C and 1500 o C at various heating rates of 10, 5, and 1 o C/min. The DTA result for alloy 11 at a heating rate of 10 o C/min is shown in Figure 4-1. The DTA curve shows two exothermic peaks on cooling and two overlapped peaks on heating followed by an extended peak and a return to the baseline. Through HTXRD, the details of which can be found in [119], combined with the analysis of the as-cast microstructure and the analysis of a sample which was heated at 1500 o C for 1 hour, cooled to 1343 o C at 12 o C/min, held at 1343 o C for 2 hours and then quenched, the phases before and after each peak on heating were identified. At 1100 o C, only the σ and γ phases are present. As the alloy is heated, β phase forms according to the reaction σ + γ σ + β + γ to produce the three phase field marked as σ + β + γ. As temperature increases, the σ phase disappears according to the reaction σ + β + γ β + γ leaving only the β and γ phases. As temperature is further increased, the amount of γ phase decreases and the amount of β phase increases until the β transus is reached. At this temperature, the reaction β + γ β takes place so that at higher temperatures, the alloy is in the single phase β region. Since the transition temperatures of alloy 11 measured by DTA will be used for optimization, an assessment of the literature must be made to evaluate the agreement between the literature and the experimental data. Hellwig et al. [101] heat treated an 77

78 alloy of actual composition Ti 40.5 at.%al 24.8 at.%nb for 48 hours followed by water quenching. This alloy was shown to contain the β, γ, and σ phases in the microstructure after the water quench. The compositions of the phases as determined by EPMA, as well as the composition of the heat treated alloy, are shown in Figure 4-2. Also included in Figure 4-2 is the composition of alloy 11. Alloy 11 is shown to exist in the σ + β + γ three phase region at 1200 o C measured by Hellwig et al. [101]. This figure indicates, then, that a correct interpretation of the DTA curve for alloy 11 should yield a result that the transformation σ + γ σ + β + γ takes place at a temperature lower than 1200 o C. Also, since alloy 11 is located close to the β + γ boundary of the σ + β + γ three phase triangle, the transition σ + β + γ β + γ should take place at a temperature minimally higher than 1200 o C. If these are not the cases, then either the measured transformation temperatures of alloy 11 or the experimental work of Hellwig et al. [101] would have to be rejected in the assessment DTA Peak Analysis for Alloy 11 Since the DTA curve on heating for alloy 11 shows two overlapped peaks for the phase transformations σ + γ σ + β + γ and σ + β + γ β + γ, the overlapped peaks should be completely separated to accurately determine the phase transformation temperatures required for optimization. One method of peak separation is to express each peak as an analytic function with adjustable parameters. The sum of the two analytic functions should reproduce, within some error, the original overlapped peaks. This method was successfully applied in the DTA peak analysis for phase transformations in the Fe Mn Ni system [120]. One peak function which can be used is the Voigt function, which is a combination of the Lorentzian and Gaussian peak functions. It is expressed as: y = y 0 + A 2 ln 2 w L π 3/2 wg 2 e t2 ( ln 2 w L w G ) 2 + ( 4 ln 2 x x c w G t) 2 dt (4 1) 78

79 where y 0 is the offset from the y-axis, x c is the center of the peak, A is the area of the peak, w G is the Gaussian width, and w L is the Lorentzian width. These are the five adjustable parameters which may be refined during curve fitting. The result of the fit to two Voigt functions is shown in Figure 4-3. The calculation was performed using the program Microcal Origin. The separated peak representing the σ + γ σ + β + γ transition is shown in red and the separated peak representing the σ + β + γ β + γ transition is shown in green. The cumulative peak is shown in blue. The peak separation results indicate that there is a good agreement between the analytically determined overlapped DTA peak using the two Voigt functions and the measured overlapped peak. Each peak was separately analyzed to determine the onset temperatures for the respective phase transformations (Figure 4-4 and 4-5). The phase transformation temperature was chosen as the point at which the first derivative curve deviated from zero. Using this analysis, the transformation temperature for the reaction σ + γ σ + β + γ was selected as 1196 o C and the transformation temperature for the reaction σ + β + γ β + γ was selected as 1241 o C. These values are in good agreement with the experimental data of Hellwig et al. [101], since the σ + γ σ + β + γ transformation occurs at a temperature lower than 1200 o C and the σ + γ σ + β + γ transformation takes place at 1241 o C, which is not much higher than 1200 o C, indicating good agreement with the position of alloy 11 being close to the β + γ boundary of the σ + β + γ three phase triangle of Hellwig et al. [101]. One clear disadvantage of this method is that only symmetric peaks can be simulated. In reality, however, the peak for a phase transformation measured using DTA is typically asymmetric. Therefore, some error is introduced into the evaluation of the onset temperatures. The other possibility which could be used to separate the overlapped peaks into two distinct peaks is to use a lower heating rate. However, this method is time 79

80 consuming, leads to less DTA signal sensitivity, and may not necessarily result in peak separation [120]. 4.3 Alloy 12 5 g buttons of alloy 12 were also arc melted from high purity pure elements. The nominal composition of alloy 12 is Ti 45 at.%al 27 at.%nb and the actual composition of alloy 12 used for the analysis, measured using EPMA, is Ti 44.9 at.%al 28.5 at.%nb. DTA was also performed on a sample of alloy 12. The alloy was heated to 1550 o C, held at this temperature for 30 min, and then cycled between 1100 o C and 1550 o C at heating rates of 10, 5, and 1 o C/min. The DTA curve for alloy 12 heated at 10 o C/min is given in Figure 4-6. The DTA curve shows two exothermic peaks on cooling and two or three overlapped endothermic peaks on heating. Through analysis of the as-cast structure for alloy 12, as well as the analysis of microstructures from heat treating and quenching experiments at 1343 o C, the phase equilibria indicated on Figure 4-6 was confirmed DTA Peak Analysis for Alloy 12 Peak separation using Voigt functions was also performed in an attempt to determine phase transformation temperatures on heating from σ + γ to β + σ + γ and from β + σ + γ to β + γ. However, because of the shallow peak on the shoulder of the first heating peak, indicated by the blue rectangle in Figure 4-7, multiple methods peak separation methods were evaluated. In the first approach, the overlapped peaks were fit to three Voigt peak functions and all adjustable parameters for the Voigt functions (five parameters per peak, therefore fifteen parameters in all) were refined. This first approach was performed to take into account the possibility of the shallow peak shoulder also being a valid DTA heat flow response to an actual phase transformation. The result of this fit is shown in Figure 4-8. The result indicates that two phase transformations, one shown by the blue peak and the other shown by the green peak, are fairly well modeled. However, the shallow peak 80

81 shoulder was not reproduced. Instead, the third peak, shown in red, appeared at a highly improbable position between the first and second Voigt function peaks. This means that although the cumulative peak is in good agreement with the original DTA heat flow signal, this analysis could not be used. In the second approach, the centroid of the third peak was fixed at 1273 o C and all other parameters were refined. The result of this fit is shown in Figure 4-9. This fit shows that there is also a reasonably good agreement between the refined cumulative signal and the original DTA signal. Using this fit procedure, however, the first and second peaks (shown as the red and blue peaks respectively) are reproduced with the same transition temperature when the transition temperature is determined using the method of first deviation from the baseline calculated using the first derivative of the curves. This is also a highly improbable solution as it would suggest that the alloy is in the β + σ + γ three phase region for only a maximum of 5 o C. In the last approach, only two Voigt peaks were modeled and all parameters were refined. The result of this fit is shown in Figure This fit shows that there is very good agreement between the refined cumulative signal and the original DTA signal. Using this fit procedure, the shallow peak is not reproduced. However, the first peak, shown in green, is quite extended i.e., the full width at half maximum is more than can be expected. The reason for this extension is that the parameters for the first peak are refined to take into account the shallow peak. This resulting peak extension introduces quite a large error in the determination of the transition temperature for this peak. Therefore, although this solution is not improbable, the error in the transition temperature is simply too large for this temperature to be used in any optimization. Since none of the above peak fitting methods can be used with accuracy, the transition temperatures were chosen as 1269 o C for the σ + γ β + σ + γ transformation and 1323 o C for the β + σ + γ β + γ phase transformation. The temperature of the σ + γ β + σ + γ transformation corresponds to the onset point of the first 81

82 shallow peak and the temperature of the maximum of the overlapped peaks is chosen as the temperature of the σ + γ β + σ + γ phase transformation. The choice of temperatures are shown in Figure Some error is necessarily introduced with these transition temperature choices. For example, although the maximum is chosen as the start temperature for the σ + γ β + σ + γ reaction, it is clear that an extrapolation of the second peak to lower temperatures, in the absence of the first peak, would give the start temperature for the reaction at a lower temperature. The phase transformation temperatures for alloy 11 and alloy 12 which were used for optimization are summarized in Table

83 Table 4-1. Key experimental works in the Ti Al Nb system Reference Experimental Methods Phase Equilibria Temperature [K] [63] Metallography (TEM with EDX) Site occupations in β 0 Temperature not indicated [64]* TEM Sublattice occupation of β 0 for selected alloys [65] Neutron diffraction Sublattice occupation of β 0 for selected alloys. 1873, 1573, 1273 β β 0 transformation temperatures for selected alloys [66]* Metallography (TEM with EDS) XRD Site occupations in the γ and α 0 phases Samples heat treated at 1273 K ALCHEMI [68]* Metallography (OM, Identification of the O 973 K TEM) phase [69] Metallography (OM, α 2 β , 1173, 1073 SEM, TEM) XRD [70] Metallography (OM, TEM) Neutron diffraction [71] Metallography (SEM, TEM) XRD Structure refinement of the O phase 973 Temperature of formation of the O phase in a Ti 24.4 at.% 15.5 at.%nb alloy β 0 O 973 [72] Metallography (OM, SEM, TEM) XRD [73] Metallography (TEM) ω O (no tie-line data) 1173, 1123, 1073, 1023, 973 [74] Metallography (TEM) Site occupations of the O phase formed in the alloy Ti 24 at.%al 27 at.%nb [75] Metallography (OM, SEM, TEM) Effect of oxygen contamination on phase equilibria in alloys Ti 22 at.%al 23 at.%nb and Ti 22 at.%al 27 at.%nb [77] Metallography (TEM) Transformation paths to the O phase [78] Metallography (SEM, TEM) α 2 β 0, β 0 O Peritectoid formation of the O phase: β 0 + α 2 O (1273 K) 1173, , 1373, , 1423, 1398, 1373, 1323, 1273, 1223, 1173, 1073,

84 Table 4-1. Continued Reference Experimental Methods Phase Equilibria Temperature [K] [81] Metallography (OM, SEM, TEM) XRD, DTA, Dilatometry Continuous cooling trasformations of Ti 27.9 at.%al 21.8 at.%nb and Ti 23.4 at.%al , 1533, 1173, 973 [83] Metallography (OM, SEM, TEM) XRD, Dilatometry at.%nb alloys β 0 O Phase transformation temperatures for the β 0, α 2, and O phases [85] Metallography (TEM) Formation of the ω phase in alloys with 25 at.%al and 5-14 at.% Nb [86] Metallography (SEM, EPMA, TEM) XRD Identification of the metastable ω and ω phases β 0 γ α 2 and ω γ α 2 [86] Metallography (OM, TEM) Derivatives of the ω phase [87] XRD Crystallography of the ω phase in an alloy Ti 3 Al 2.25 Nb 0.75 [12]* Metallography (OM, SEM, EPMA, TEM) XRD, DTA Liquidus surface [98]* Metallography (TEM) Determination of a single phase β phase field for an alloy of composition Ti 48 at.%al 26 at.%nb [89]* Metallography (OM, SEM, TEM, EPMA) DTA [90] Metallography (SEM, TEM, HRTEM, EDS) [91]* Metallography (SEM, EPMA) XRD, DSC, DTA [92] Metallography (SEM, EDX, TEM) XRD [93] Metallography (OM, EPMA) XRD 1623, 1173, 1073, , , 1373, , 1373, α γ 1548, 1473 Identification of γ 1 β δ, δ σ, σ γ 1, σ β, α β, α α 2, α γ, γ-η, γ γ 1, η γ 1, γ 1 β, α 2 β, γ TiAl 2 Existence of the γ 1 phase followed by furnace cooling 1673, 1323, 1273 σ γ 1, σ γ, σ γ γ

85 Table 4-1. Continued Reference Experimental Methods Phase Equilibria Temperature [K] [94] Metallography (TEM) Crystallography of the 1673 XRD γ 1 phase [95] Metallography (TEM) CRD Crystallography of γ 1 precipitates in the alloy 1073 [97]* Metallography (SEM, EDX), XRD [11]* Metallography (EPMA) XRD [13]* Metallography (SEM, EPMA, TEM), XRD [99] Metallography (OM, SEM, EDS), XRD, DTA [100]* Metallography (OM, SEM, TEM, EPMA) [101]* Metallography (OM, EPMA, SEM, TEM), XRD [121]* Metallography (SEM, TEM, EPMA), XRD [122] Metallography (OM, SEM, TEM, EPMA), XRD [109]* Metallography (OM, SEM, EPMA, TEM), XRD [123]* Metallography (SEM, EPMA) [102] Metallography (OM, SEM, TEM, EDS in TEM, EPMA), XRD [124]* Metallography (OM, SEM, TEM, EPMA), XRD [108]* Vapor Pressure Measurements (Knudsen effusion spectroscopy) Ti 48 at.%al 10 at.%nb Non existence of the γ 1 phase σ γ Liquidus surface β δ σ and β γ σ Primary crystallization fields on the liquidus surface β σ, σ γ, η γ, σ β Complete isothermal 1673 section Isothermal section 1473 σ β γ, γ η, σ γ η, β 0 γ, β 0 σ γ, β 0 α, β 0 α 2, β 0 γ α, α γ, η TiAl 2, η liquid Single phase β 0, α 2 β Volume fraction of α 2 and β in a Ti 47.2 at.&al 2.14 at.%nb alloy Samples annealed at 1323 β δ 1923, 1473, 1273 β 0 α γ 1473 The Ti 22 at.%al Nb vertical section β δ β β 0 temperature for a Ti at.%al at.%nb alloy σ γ η, β δ σ, β 0 γ σ, β 0 γ, σ γ, β 0 γ α 2, β 0 α 2, σ γ 1311, 1273, 1173, 1088, 1033, , 1473, , 1373,

86 Table 4-1. Continued Reference Experimental Methods Phase Equilibria Temperature [K] [125] EDX, TEM Ti occupation of Nb 1173 sites in γ [126]* Metallpgraphy (EPMA) α γ, α β, β γ, α 2 γ, 1573, 1473, 1273 α β γ [5]* Metallography (OM, SEM, TEM, EPMA) XRD Composition of the α(α 2 ) β(β 0 ) γ tie triangle 1473, 1423, 1373, 1323, 1273 [15]* Metallography (OM, EPMA, SEM, TEM,) microhardness XRD DTA [14]* Metallography (OM, SEM, TEM, EPMA) XRD [107]* Metallography (OM, EPMA, SEM, TEM), XRD [127] Metallography (OM, SEM, EBSD), DSC [128] Metallography (OM, TEM), XRD, DTA [17]* Metallography (OM, SEM) XRD, DTA Primary cystallization fields on the liquidus surface Transformation paths, Phase equilibria (phase compositions not included) β δ, δ σ, σ γ, β σ, β δ σ Temperature of the reactions γ + α α and α 2 + γ α 2 + α + γ for alloy Ti 45 at.%al 5 at.%nb α 2 β 0 O, β 0 O, β 0 α 2, α 2 O Liquidus temperature of selected alloys Several solid state transition and melting temperatures are presented for alloys with up to 17.1 at.%nb 1373, 1173, , 1373, 1273,

87 Table 4-2. Tie line and tie triangle data for alloys A2, A3, and A133 that were used for the optimization. Alloy Bulk [at.%] σ [at.%] η [at.%] γ [at.%] Al Nb Ti Al Nb Ti Al Nb Ti Al Nb Ti A2 at A2 at A3 at A3 at A133 at A133 at

88 Figure 4-1. DTA curve for alloy 11 measured at heating and cooling rates of 10 o C/min. 88

89 Figure 4-2. Gibbs triangle showing the position of Alloy 11 as well as the position of the three phase triangle σ + β + γ measured by Hellwig et al. [101]. This three phase triangle was determined by heat treating an alloy of composition Ti 40.5 at.%al 24.8 at.%nb for 48 hours at 1200 o C and then water quenching. The alloy was shown to contain the β, σ, and γ phases, the compositions of which were measured using EPMA and are shown as open triangles. The alloy is indicated as the filled red circle. According to these results, Alloy 11 should already be in the three phase region σ + β + γ at 1200 o C. 89

90 Figure 4-3. Peak separation of the two overlapped peaks using Voigt functions. The peak representing the σ + γ σ + β + γ transition is shown in red and the peak representing the σ + β + γ β + γ transition is shown in green. Figure 4-4. Peak analysis for the σ + γ σ + β + γ transformation. The first derivative is shown in blue. The transition temperature was selected based on the point at which the value of the first derivative deviated from zero. 90

91 Figure 4-5. Peak analysis for the σ + β + γ β + γ transformations. The first derivative is shown in blue. The transition temperature was selected based on the point at which the value of the first derivative deviated from zero. 91

92 Figure 4-6. DTA curve for alloy 12 measured at heating and cooling rates of 10 o C/min. 92

93 Figure 4-7. DTA curve for alloy 12 showing the overlapped peaks on heating. Because of the presence of the shallow peak, shown by the blue rectangle, many peak separation methods must be evaluated. Figure 4-8. The results of peak separation using three peaks. In this method, all peak parameters were refined. The results show that although two phase transformation peaks are fairly well modeled, a third peak, shown in red, appears at an improbable position between the two phase transformation peaks. 93

94 Figure 4-9. The results of peak separation using three peaks. In this approach, the centroid of one peak was fixed at 1273 o C in an attempt to fix the position of the first peak. Although the cumulative peak in good agreement with the original DTA signal, the phase transition temperatures associated with the first two peaks are approximately equal. This is also a highly improbable situation. 94

95 Figure The results of peak separation using only two peaks. The cumulative peak is in good agreement with the original DTA signal. However, in an attempt to take into account the shallow peak, the parameters of the first peak are refined to produce a peak that is much wider than expected. This introduces quite a large error in the analysis of the phase transition temperature using the method of first deviation from the baseline using the first derivative. Figure The overlapped peaks on heating for alloy 12 indicating the choice of temperatures for the corresponding phase transformations o C is selected as the start temperature of the σ + γ β + σ + γ transition and 1323 o C is selected as the start temperature of the β + σ + γ β + γ transition. 95

96 Table 4-3. Transition temperatures for alloy 11 and alloy 12. Alloy σ + γ β + σ + γ β + σ + γ β + γ β + γ β o C 1241 o C 1478 o C o C 1323 o C 96

97 CHAPTER 5 RE-OPTIMIZATION OF THE TI AL NB SYSTEM 5.1 Unary Data and Binary Sub-Sections The data for the pure elements, including lattice stability data, were taken the SGTE dataset [58]. The description for the Al Ti binary sub-system was taken from Saunders [18] and is the same description used by Servant and Ansara [10]. The Al Ti binary of Saunders [18] was modified to model the α and α 2 phases, which undergo an order disorder transformation, using a single Gibbs energy expression [10]. Although there is a new description for the Al Ti system from Witusiewicz et al. [110], this description was not used as the α and α 2 phases are modeled as separate phases (see Section ). The Al Nb system was taken from the work of Witusiewicz et al. [17] as this dataset reproduces better the enthalpy of formation data for the phases in the Al Nb system (see Section ). The Ti Nb system was taken from Zhang et al. [129]. This dataset is recommended as it calculates the metastable miscibility gap in the β phase based on data from Hickman et al. [130] and Moffat and Larbalestier [131, 132]. Additionally, the calculated α/β T 0 curve is above the measured temperatures for the martensitic transformation from the β phase to the hexagonal α martensite quoted in Brown et al. [133]. The calculated binary Ti Nb system is shown in Figure Thermodynamic Optimization Optimization of the parameters was performed using the PARROT [28] module of THERMO-CALC [29] Optimization Strategy The mixing parameters of the σ and δ phases were re-optimized so that an alloy of composition Ti 25 at.%al 40 at.%nb is in the σ + δ + β three phase field at 1813 K for agreement with the work of Leonard and Vasudevan [14]. Next, the ternary mixing parameters of the β phase were re-optimized. As there is no experimental data on the enthalpy of mixing of the β phase, the starting values of the ternary mixing parameters 97

98 were generated using the trial and error method. This is detailed in Section Once the parameters for the σ, δ, and β phases were re-assessed, the parameters representing binary mixing of two components on one sublattice with a singly occupied second sublattice, as well as the ternary mixing parameters of the γ phase were re-optimized. The details of this method are given in Section After the parameters of the γ phase were re-optimized, ternary mixing parameters of the σ phase were introduced to improve the fit of the homogeneity range of the σ phase to the available experimental data. TEM investigations on the single phase β region have shown that the β β 0 boundary is between Ti 25 at.%al 30 at.%nb and Ti 25 at.%al 20 at.%nb at 1473 K [68, 101], at Al contents larger than 25 at.% at 1373 K [68], and at Al contents larger than 23 at.% at 1273 K [101]. Additionally, an alloy of composition Ti 27.9 at.%al 21.8 at.%nb is shown to be β 0 at 1573 K [81]. Therefore, in these composition ranges, the parameters of the β 0 phase were optimized so that the β 0 phase is more stable than the β phase. The parameters for the β 0 phase were modeled as linear functions of temperature, the details of which are given in Section Next, the parameters for the α and α 2 phases were re-optimized to decrease the primary crystallization field of the α phase on the liquidus surface and to calculate the intersection of the α 2 + β 0 and σ + β 0 phase fields at 1273 K [101]. The parameters for the liquid phase were optimized last. The temperature, composition, and eutectic character of reaction between the liquid, σ, γ, and η phases were accepted from the work of Rios [119] and the temperature of the transition reaction liquid + β δ + σ were accepted from the work of Pavlov and Zakharov [134]. Since there is some uncertainty in the characters and temperatures of all other invariant reactions, the ternary mixing parameters for the liquid phase were optimized so that the four alloys of composition 40 at.%al and 36, 30, 24, and 20 at.%ti respectively are located in the primary crystallization field of the β phase to be in agreement with the work of Leonard et al. [15]. 98

99 5.3 Selection of Thermodynamic Parameters The Sigma phase The σ phase has the tp30 CrFe crystal structure. The atoms, Wyckoff positions, and coordination numbers are given in Table 5-1 and the unit cell is shown in Figure 5-2. From the recommendations of Ferro and Cacciamani [135] and Ansara et al. [54] three sublattices should be used with each sublattice containing atoms having the same coordination number. In the above case, the Nb atoms on the 8i 1 and 8j sites occupy one sublattice with total multiplicity 16, the Al atoms on the 2a and 8i 2 sites occupy a second sublattice with multiplicity 10, and the Nb atom on the 4f site occupies its own sublattice with multiplicity 4. This modeling results in the description (Nb) 16 (Al) 10 (Nb) 4. Allowing mixing of Al and Ti on the first sublattice, mixing of Nb and Ti on the second sublattice, and no mixing on the third sublattice, the model for the σ phase is (Al,Nb*,Ti) 16 (Al*,Nb,Ti) 10 (Nb) 4 which reduces to: (Al,Nb*,Ti) (Al*,Nb,Ti) (Nb) (5 1) when one mole of formula units is used and where the asterisk identifies the major species on each sublattice. The locations of the end members of this description are shown on the Gibbs triangle in Figure 5-3. All thermodynamic descriptions for the end members were taken from Witusiewicz et al. [17] except the end members G σ Ti:Al:Nb and Gσ Ti:Ti:Nb, which were taken from the description of Servant and Ansara [10]. Servant and Ansara also included sixteen more parameters to model the σ phase. These are the 0 L σ Al,Ti:i:Nb, 0 L σ Nb,Ti:i:Nb, 0 L σ i:al,nb:nb, 0 L σ i:al,ti:nb, and 0 L σ i:nb,ti:nb parameters, where i=al, Nb, Ti, as well as the 0 L σ Al,Nb:Ti:Nb parameter [10]. This study revealed that these parameters were shown to have no influence on the ternary phase equilibria with the σ phase. Additionally, the removal of these parameters did not affect the Ti Al or Ti Nb constituent binaries, i.e., the σ phase was not stabilized in either the Ti Al or the Ti Nb constituent binary. In fact, the 0 L σ Nb,Ti:i:Nb and 0 L σ i:nb,ti:nb parameters for i=nb and Ti were added by Servant 99

100 and Ansara to prevent the stabilization of the σ phase to the Ti Nb binary. However, the simpler solution, which was adopted in this approach, is to add a slightly positive term of J/mol to the G σ Ti:Nb:Nb end member. The optimized parameters were the 0 L σ Nb,Ti:Al:Nb parameter and the 1 L σ Al,Nb,Ti:i:Nb parameters where i=al,nb,ti. The 0 L σ Nb,Ti:Al:Nb parameter controls the extension of the σ phase into the ternary at 33 at.%al. The influence of this parameter is illustrated as the dashed line in Figure 5-3. It was used particularly to fit the extension of the σ phase at 1813 K to be in agreement with the β + δ + σ tie triangle of Leonard and Vasudevan [14]. Starting values of the 0 L σ Nb,Ti:Al:Nb parameter were generated by assuming a linear temperature dependence between the value of the parameter at 1813 K and the value of the parameter at 1373 K required to fit the β +δ+σ tie triangle of Leonard and Vasudevan at 1813 K [14] and the β + δ + σ tie triangle of Leonard et al. at 1373 K [107] respectively. The 1 L σ Al,Nb,Ti:i:Nb parameters were used to improve the fit of the homogeneity range of the σ phase to the σ + γ + η tie triangle data for alloys A2 and A3 at 1783 K and 1683 K [119], to the σ + γ tie line data for alloy A133 at 1783 K and 1683 K [119], to the phase diagram data at 1473 K [13, 101, 108], 1373 K [107], and 1273 K [101], and as well to the temperatures of the phase transformations β + γ β + γ + σ and β + σ + γ σ + γ for alloys 11 and 12. The 0 L σ Al,Nb,Ti:i:Nb and 2 L σ Al,Nb,Ti:i:Nb parameters were set to zero as they did not affect the phase boundaries of the σ phase. All optimized parameters were modeled using a linear temperature dependence The Delta Phase The δ phase which has the cp8 Cr 3 Si structure is modeled using the compound energy formalism as: (Al,Nb*,Ti) 0.75 (Al*,Nb,Ti) 0.25 (5 2) where the asterisk identifies the major species on each sublattice. The location of the end members generated using this description are shown on the Gibbs triangle in Figure 5-4. The descriptions for the G δ Al:Al, Gδ Al:Nb, Gδ Nb:Al, and Gδ Nb:Nb end members were taken from 100

101 Witusiewicz et al. [17] whereas the descriptions for all other end members were taken from the dataset of Servant and Ansara [10]. The mixing parameter 0 L δ Nb:Al,Nb was also taken from Witusiewicz et al. [17] and the 0 L δ Ti:Al,Ti and 0 L δ Al:Al,Ti parameters were kept from the original description of Servant and Ansara [10]. Servant and Ansara used eleven more parameters to fit the phase equilibria with the δ phase. These are the 0 L δ i:nb,ti and 0 L δ Al,Nb:i parameters with i=al,nb,ti, the 0 L δ i:al,ti parameters with i=al,ti, the 0 L δ Nb,Ti:i parameters with i=nb,ti, and the 0 L δ Nb:Al,Ti parameter. These parameters had negligible influence on the phase boundaries of the δ phase, and were therefore set to zero. The only parameter which was optimized in this work was the 0 L δ Nb,Ti:Al parameter, which controls the extension of the δ phase into the ternary at 25 at.%al. The influence of this parameter is shown as the dashed line in Figure 5-4. This parameter was modeled with a linear temperature dependence. The starting values of the 0 L δ Nb,Ti:Al parameter were generated by using the value of the parameter at 1813 K and the value of the parameter at 1373 K required to fit the β + δ + σ tie triangle of Leonard and Vasudevan at 1813 K [14] and the β + δ + σ tie triangle of Leonard et al. at 1373 K [107] respectively The Beta Phase The β phase which has the ci2 W structure is modeled as the substitutional solution (Al,Nb,Ti). In this work, only the 0 L β Al,Nb,Ti, 1 L β Al,Nb,Ti, and the 2 L β Al,Nb,Ti parameters were optimized. In the original description from Servant and Ansara, the constraint 0 L β Al,Nb,Ti = 1 L β Al,Nb,Ti = 2 L β Al,Nb,Ti was used to model the β phase [10]. However, during the course of this assessment, it was found that there was no possibility to fit the extension of the β phase to higher Al compositions using this constraint. Therefore, each of the parameters was assessed individually. To choose correct starting values of the 0 L β Al,Nb,Ti, 1 L β Al,Nb,Ti, and 2 L β Al,Nb,Ti parameters, the values of the parameters required to produce a good fit to the phase equilibria at 1473 K [13, 101], and 1783 K [119], to calculate correctly the β + δ + σ tie triangle at 1813 K [14], and to produce reasonable tie lines at 2000 K and 2100 K were selected. The values 101

102 of the 0 L β Al,Nb,Ti, 1 L β Al,Nb,Ti, and 2 L β Al,Nb,Ti parameters at 1473 K, 1783 K, 1813 K, 2000 K, and 2100 K are shown in Figure 5-5. Figure 5-5 shows that there is no possibility to fit the equilibria defining these parameters as linear functions of temperature only. Therefore, the 0 L β Al,Nb,Ti and 1 L β Al,Nb,Ti parameters were modeled as cubic functions of temperature and the 2 L β Al,Nb,Ti parameter was modeled as a quadratic function of temperature. The starting values of the 0 L β Al,Nb,Ti, 1 L β Al,Nb,Ti, and 2 L β Al,Nb,Ti parameters were then further optimized The Gamma Phase The γ phase which has the tp2 AuCu structure is modeled using the compound energy formalism as: (Al,Nb,Ti) 0.5 (Al,Nb,Ti) 0.5 (5 3) The description for all end members are the same as used by Servant and Ansara [10] except for the G σ Nb:Ti and Gσ Ti:Ti parameters. In the original dataset, these parameters were expressed as: G σ Nb:Ti = G σ Nb:Ti = 0.5 GHSER Nb GHSER Ti (5 4) However, in this description, the positive term J/mol was not used. The location of all end members on the Gibbs triangle is shown in Figure 5-6. The parameters which were optimized in this work are the parameters of type 0 L γ i,j:k, which represent mixing of two components on one sublattice with a third, different component on a singly occupied second sublattice, and parameters of type 0 L γ i,j,k:i, 1 L γ i,j,k:i, and 2 L γ i,j,k:i, which represent the mixing of three components on one sublattice with a singly occupied second sublattice. To reduce the number of optimizing variables, the constraint 0 L γ i,j:k = 0 L γ k:i,j was used for the 0 L γ i,j:k parameters, and the constraint Lγ i,j,k:i = L γ i:i,j,k was used for the Lγ i,j,k:i parameters. The binary parameters 0 L γ Al:Al,Nb = 0 L γ Nb:Al,Nb = 0 L γ Al,Nb:Al = 0 L γ Al,Nb:Al were included in the optimization as they were required to shift the γ phase to higher Nb compositions 102

103 to be able to fit the σ + γ + η tie triangles at 1783 K and 1683 K [119]. The γ phase was not stabilized in the Al Nb binary through addition of these parameters. All optimized parameters were modeled as linear functions of temperature except the 0 L γ Al:Nb,Ti = 0 L γ Nb,TI:Al parameter. This parameter is a key parameter used to fit the composition of the γ phase in the γ + σ tie line at 1683 K [119], to fit the composition of the γ phase in the σ + γ + η tie-triangle at 1473 K [101], and to calculate a high Nb containing γ phase at 1923 K. The values of this parameter at 1923 K, 1683 K, and 1473 K required to calculate the aforementioned equilibria are shown in Figure 5-7. Figure 5-7 shows that a linear temperature dependence is inadequate to reproduce the required equilibria. For example, if a linear temperature dependence of the parameter was selected using the values of the parameter at 1923 K and 1473 K, then the alloy A133, which contains only σ + γ at 1683 K [119] (see Table 4-2), is calculated in the β + γ + σ tie triangle. Additionally, if the linear temperature dependence of the parameter is chosen for values of the parameter at 1923 K and 1683 K, then the calculated isothermal section at 1473 K is in disagreement with the tie line and tie triangle data from Hellwig et al. [101]. Therefore, a quadratic temperature dependence for this parameter was selected The Ordered Beta Phase The β 0 phase is the ordered cp2 CsCl B2 structure with Ti-rich and Al-rich sublattices respectively. It is modeled using the compound energy formalism as (Al,Nb,Ti) 0.5 (Al,Nb,Ti) 0.5 (5 5) Since a similar model is used for the γ phase (Equation 5 3), the locations of the end members for the β 0 phase are the same as that for the γ phase, and are therefore already illustrated in Figure 5-6. In the model for the β 0 phase, the restrictions of symmetry impose the constraint 0 L β 0 i,j:k = 0 L β 0 k:i,j. The additional constraint 0 L β 0 i,j:a = 0 L β 0 i,j:b = 0 L β 0 i,j:c was used to reduce the number of optimizing variables. The third simplification 0 G β 0 i:j = -0 L β 0 i,j:k, which was used 103

104 in the dataset of Servant and Ansara [10], can reduce the number of optimizing variables from 12 to 6. To assess the influence of the number of optimizing parameters on the phase equilibria, re-optimization of the parameters of the β 0 phase was performed with and without the third simplification. The results indicated that there was no significant change to the phase equilibria. Additionally, there is insufficient experimental data to justify the use of 12 parameters. Therefore, the simplification 0 G β 0 i:j = -0 L β 0 i,j:k, as used by Servant and Ansara [10], was maintained in the present re-optimization. The optimized parameters for the β phase were all modeled as linear functions of temperature. To generate the starting values of the parameters for optimization, the values of the parameters required to fit the β 0 + γ + σ three phase equilibria at 1473 K of Zdziobek et al. [13] and Hellwig et al. [101] and the values of the parameters required to fit the β + γ tie line data for alloy 12 at 1613 K [119] were used to construct a linear temperature dependence The Disordered Alpha and Ordered Alpha 2 Phases The α phase with hp2 Mg structure is modeled as the substitutional solution (Al,Nb,Ti). In this work, only the 0 L α Al,Nb,Ti, 1 L α Al,Nb,Ti, and 2 L α Al,Nb,Ti parameters were optimized. Just as in the case for the β phase, Servant and Ansara [10] modeled the ternary mixing parameters for the α phase using the constraint 0 L β Al,Nb,Ti = 1 L β Al,Nb,Ti = 2 L β Al,Nb,Ti. However, this modeling leads to quite a large extension of the primary crystallization of the α phase on the liquidus surface which is not experimentally confirmed. Therefore, each of the ternary parameters was assessed individually. The D0 19 structure of the α 2 phase is the hcp analogue to the L1 2 structure from fcc ordering with hp8 Ni 3 Sn structure. Constraints used to model G ord M were developed by Dupin and Ansara [55] based on the mathematical equivalence between the 2 sublattice model (A,B,C) 0.75 (A,B,C) 0.25 and the (A,B,...) 0.25 (A,B,...) 0.25 (A,B,...) 0.25 (A,B,...) 0.25 four sublattice model where all sublattices are equivalent. Only one parameter was optimized. This parameter, related to the formation energy G AlNbTi2 of the hypothetical compound 104

105 (Ti 2/3 Nb 1/3 ) 3 Al, is given as u in the expression G AlNbTi2 = u AlNb + u AlTi + 2 u NbTi + u (5 6) where u AlNb, u AlTi, and u NbTi are the bond energies taken from the binary systems and u is a correction term that can be optimized to fit to the ternary phase equilibria. As the extension of the α phase was significantly reduced, to ensure the intersection of the α 2 + β 0 and σ + β 0 phase fields at 1273 K [101], the parameter u was re-optimized using the trial-and-error method to increase the stability of the α 2 phase The Liquid Phase The liquid phase is modeled as the substitutional solution (Al,Nb,Ti). The ternary mixing parameters 0 L Liq Al,Nb,Ti, 1 L Liq Al,Nb,Ti, and 2 L Liq Al,Nb,Ti were defined as linear functions of temperature. In the dataset of Servant and Ansara, the constraint 0 L Liq Al,Nb,Ti = 1 L Liq Al,Nb,Ti = 2 L Liq Al,Nb,Ti was also used [10]. However, it was found that there was no possibility to fit the extension of the primary crystallization of the β phase to higher Al compositions using this constraint. Therefore, the parameters were independently assessed. 105

106 Figure 5-1. Calculated binary Ti Nb diagram using the description of Zhang et al. [129]. Table 5-1. Crystal structure of the σ phase [135] Space Group P4 2 /mnm Pearson Symbol tp 30 Atoms Wyckoff Position x y z CN Al 2a Nb 4f Nb 8i Al 8i Nb 8j

107 Figure 5-2. Unit cell of the σ phase. The Al1 and Al2 atoms are located at the 2a and 8i 2 Wyckoff positions respectively. Since the 2a and 8i 2 positions each have the same coordination number, the atoms in these positions occupy the same sublattice. The Nb2 and Nb3 atoms, which are located at the 8i 2 and 8j positions, occupy the same sublattice as they each have a coordination number of 14 and the Nb1 atom in the 4f position occupies its own sublattice. This results in the model (Nb) 16 (Al) 10 (Nb) 4 when there is no mixing on either of the sublattices. 107

108 Figure 5-3. The end members of the σ phase. The dashed line from Nb:Al:Nb to Ti:Al:Nb schematically illustrates the influence of the 0 L σ Nb,Ti:Al:Nb mixing parameter. Figure 5-4. The end members of the δ phase. The dashed line from Nb:Al to Ti:Al schematically illustrates the influence of the 0 L δ Nb,Ti:Al parameter. 108

109 A The starting values of the 0 L β Al,Nb,Ti parameter as a function of temperature B The starting values of the 1 L β Al,Nb,Ti parameter as a function of temperature C The starting values of the 2 L β Al,Nb,Ti parameter as a function of temperature Figure 5-5. The starting values of the A) 0 L β Al,Nb,Ti, B)1 L β Al,Nb,Ti, and C)2 L β Al,Nb,Ti parameters as a function of temperature. There was no possibility to fit these parameters as linear functions of temperature. Therefore, the 1 L β Al,Nb,Ti and 2 L β Al,Nb,Ti parameters were modeled as cubic functions and the 2 L β Al,Nb,Ti parameter was modeled as a quadratic function. 109

110 Figure 5-6. The end members of the γ phase. The dashed lines show the influence of the six mixing parameters of type 0 L γ i,j:k, which represent the condition of mixing of two different components on one sublattice with a second sublattice singly occupied by a third component. 110

111 Figure 5-7. The values of the 0 L γ Al:Nb,Ti at 1923 K, 1683 K, and 1473 K. Expressing this parameter as a linear function of temperature only will not result in a good fit to the experimental phase equilibria. Therefore, a quadratic variation with temperature was chosen. 111

112 CHAPTER 6 RESULTS OF THE RE-OPTIMIZATION 6.1 Liquidus and Solidus Projections The liquidus surface calculated using the new description is shown in Figure 6-1. There are five invariant reactions with the liquid phase. An eutectic reaction was calculated for the invariant reaction between the liquid, σ, η, and γ phases. This is in agreement with other thermodynamic assessments [9, 10, 17] as well as with the experimental work of Rios [119]. The calculated liquidus surface is in very good agreement with experimental works indicating an extension of the primary crystallization field of the β phase [4, 14, 15, 98, 119] i.e., all alloys in the literature which have been shown to solidify as β phase are in the β primary crystallization field. The calculated soildus surface (Figure 6-2) shows that all alloys except the Ti 48 at.%al 25 at.%nb alloy of Feng et al. [98] and alloy 12 solidify as single phase β. In this case, it was not possible to extend the β solidus to higher Al compositions. Alloy 12 is located directly on the β solidus line but the Ti 48 at.%al 25 at.%nb alloy of Feng et al. [98], which has a higher Al content, solidifies as two phase β + γ. The Scheil reaction scheme for equilibria with the liquid phase is shown in Figure Isothermal Sections Calculated isothermal sections at 1923 K (Figure 6-4), 1813 K (Figure 6-5), 1783 K (Figure 6-6), 1683 K (Figure 6-7), 1673 K (Figure 6-8), 1613 K (Figure 6-9), 1513 K (Figure 6-10), 1473 K (Figure 6-11), 1373 K (Figure 6-12), and 1273 K (Figure 6-13) with superimposed key experimental data are given. The calculated isothermal sections show good agreement with the experimental data. Although the tie line data of Wang et al. [99] were not taken into account in the optimization, they are still superimposed onto the isothermal section at 1673 K. As mentioned before, the data of Wang et al. [99] would suggest a much smaller primary crystallization of the β phase, which is not in agreement 112

113 with other data. The discrepancy between the present calculation and the experimental tie line data at 1673 K [99] is particularly evident in the calculated singe phase β field, where β δ, β γ, and β α tie lines are reported. As the alloys were heat treated at 1473 K for 160 hours, furnace cooled, and then annealed at 1673 K for 4.5 hours, it is possible that there was insufficient time at 1673 K for the β phase to completely dissolve the δ, γ, and α phases that are present in the microstructure prior to annealing at 1673 K. 6.3 Vertical Sections Although there are many vertical sections reported in the literature, only the vertical sections through the nominal compositions of alloy 11 and alloy 12 (Figure 6-14) and the vertical section at 40 at.%al (Figure 6-15) are included here. The calculated vertical sections also show quite a good agreement with the literature [15, 119]. Alloy 11 is in the single phase β field until the β transus at 1750 K is reached, at which point γ forms. However, a similar single phase β field for alloy 12 is not calculated. Instead, alloy 12 passes through a very small three phase field liquid +β + γ (calculated at only 0.2 K), before entering into the β + γ two phase field. Also of note, the calculated solid state transformation of Leonard et al. s alloy 1 [15] is from β to β +γ (Figure 6-15). However, in the original work, this transformation is cited as from β to β + σ. In the original work, only thermal analysis was performed on the alloy to determine solid state transformation temperatures, but no heat treatment and quench experiments were performed at temperatures before and after DTA peaks on this alloy to confirm the resulting microstructures. Therefore, it is equally probable that our phase transformation path calculated here is correct. The vertical sections Ti 27.5 at.%al to Nb [78], Ti 22 at.%al to Nb [79, 102], and TiAl to TiNb [72, 79] and at 45 at.%al, 47 at.%al, 8 at.%nb, through Ti 87.2 Nb 12.8 and Nb 2.2 Al 97.8, through Ti 72.8 Nb 27.2 and Ti 31.6 Al 68.4, and through Ti 70 Al 30 and Ti 51 Nb 49 [17] were not addressed in the present re-optimization as they focus on phase transformations with the α, α 2, β, and O phases which do not greatly impact the phase equilibria in the 113

114 two phase γ + σ region for alloy development, which is our region of interest. The vertical section from Al 3 Ti to Nb 3 Ti [12] can only be reproduced if the η phase is modeled as (Ti,Nb) 1 x Al 3+x, which would allow either an excess or deficiency of Al in the η phase with respect to the line compound (Ti,Nb)Al 3. However, the thermodynamic description for the η phase was not changed from that reported in the work of Servant and Ansara [10]. 6.4 Phase Fraction Diagrams The calculated phase fraction diagrams for alloy 11 and alloy 12 are shown in Figure 6-16 and Figure 6-17 respectively. For alloy 11, after solidification, a single phase β field is observed. With continued cooling, γ phase forms at the β transus at 1750 K. The amount of β phase decreases and the amount of γ phase increases. At 1485 K, σ phase forms, and the amount of γ phase continues to increase and the amount of β phase continues to decrease until at 1463 K, only σ and γ are present. Alloy 12, however, does not solidify as single phase β, but passes through a very small region of liquid +β +γ (calculated at only 0.2 K), before completely solidifying as β +γ. In fact, the amount of β phase directly after solidification is calculated at 99.3 mol % β. This can be directly seen on the calculated solidus surface (Figure 6-2), as alloy 12 is almost directly on the β solidus line. With continued cooling, the amount of β phase decreases and the amount of γ phase increases until at 1596 K, σ phase forms. After formation of σ phase, the amount of γ continues to increase and the amount of β continues to decrease until at 1542 K, only γ and σ phases are present. 114

115 Figure 6-1. Calculated liquidus surface for the Ti Al Nb system. The experimental data identified as [1992Fen], [1996Ara], and [2000Leo] refer to the works of Feng et al. [98], Leonard et al. [15], and d Aragão and Ebrahimi [4] respectively. The calculated liquidus surface is in good agreement with the literature showing that all alloys indicated solidify as β phase. Figure 6-2. Calculated solidus surface in the Ti Al Nb system. The position of the β solidus shows that all alloys solidify as single phase β except for the alloy of composition Ti 48 at.%al 25 at.%nb from Feng et al. [98] and alloy 12. After solidification, these alloys exist as two phase β + γ. 115

116 Figure 6-3. Scheil reaction scheme for equilibria with the liquid in the Ti Al Nb system. There are five invariant reactions: three transition reactions, one eutectic reaction, and one peritectic reaction. 116

117 Figure 6-4. Calculated isothermal section at 1923 K. The experimental data identified as [1992Men] refers to the work of Menon et al. [109]. Figure 6-5. Calculated isothermal section at 1813 K. The experimental data identified as [2000Leo] refers to data obtained from Leonard and Vasudevan [14]. In this work, an alloy of composition Ti 25 at.%al 60 at.%nb which was heat treated at 1813 K and quenched was shown to be in the three phase region β + δ + σ. 117

118 Figure 6-6. Calculated isothermal section at 1783 K. Tie line and tie triangle data for alloys A2, A3, and A133 are included. The tie lines shown in blue were determined after the description was re-optimized. Therefore, the current description is also able to predict the phase equilibria at 1783 K. Figure 6-7. Calculated isothermal section at 1683 K. Tie line and tie triangle data for alloys A2, A3, and A133 are included. 118

119 Figure 6-8. Calculated isothermal section at 1673 K. The experimental data of Wang et al. [99] is superimposed. While the dataset is able to calculate phase equilibria between β and δ, δ and σ, σ and γ, σ and η, and γ and η phases, other phase equilibria cannot be calculated. However, Wang et al. [99] may have insufficiently heat treated their alloys at 1673 K. Therefore, they may not have measured the equilibrium tie lines. Figure 6-9. Calculated isothermal section at 1613 K. There is good agreement with the β γ tie line for alloy

120 Figure Calculated isothermal section at 1513 K. There is good agreemtn with the β γ tie line for alloy 11. Figure Calculated isothermal section at 1473 K. The experimental data identified as [1992Men], [1995Zdz], and [1998Hel] refer to the works of Menon et al. [109], Zdziobek et al. [13], and Hellwig et al. [101] respectively. 120

121 Figure Calculated isothermal section at 1373 K. The experimental data identified as [1999Eck] and [2002Leo] refer to the works of Eckert et al. [108] and Leonard et al. [107] respectively. Figure Calculated isothermal section at 1273 K. The experimental data identified as [1992Men] and [1998Hel] refer to the works of Menon et al. [109] and Hellwig et al. [101] respectively. 121

122 Figure Calculated isopleth through the nominal compositions of alloy 11 and alloy 12. There is good agreement between the calculations and the solid state transformations of alloys 11 and 12 measured using DTA. Alloy 11 solidifies as single phase β. However, there is no single phase β field for alloy 12. Instead, directly after solidification, β and γ should appear. 122

123 Figure Calculated isopleth at 40 at.%al. Transformation temperatures measured using DTA from Leonard et al. [15] are included. 123

124 Figure Calculated phase fraction diagram for alloy 11. Figure Calculated phase fraction diagram for alloy

125 CHAPTER 7 THE TI-CR SYSTEM 7.1 Phase Equilibria in the Ti Cr System The crystallographic data of the solid phases in the Ti Cr system are given in Table 7-1. Since the early works by Vogel and Wenderott [136], McPherson and Fontana [137] and McQuillan [138] measured the Cr content of the Laves phase as 60 at.%, the Laves phase stoichiometry was misidentified as Ti 2 Cr 3. However, Duwez and Fontana [139] and Cuff et al. [140], using X-ray diffraction, corrected the stoichiometry to TiCr 2. There are three modifications of the Laves phase: the low temperature α TiCr 2 phase, the intermediate temperature β TiCr 2 phase, and the high temperature γ TiCr 2 phase. In the early works [ ], the polymorphism of the Laves phase was not taken into account. However, in the experimentally determined phase diagrams for the Ti Cr system shown by Levinger [144], Ageev and Model [145], and Farrar and Margolin [146], both the low temperature α TiCr 2 and the high temperature γ TiCr 2 modifications of the Laves phase were indicated. Svechnikov [147] was the first to include all three modifications of the Laves phase. In this work, the phase transformation temperatures of the Laves phases were measured using DTA [147]. Although the presence of three modifications of the Laves phase was confirmed by Chen et al. [148, 149], later experimental works of Hao and Zeng [150] and Schuster and Du [151] still could not find the β TiCr 2 modification. The works of Farrar and Margolin [146] and Chen et al. [148, 149] address the homogeneity range of the Laves phases. Although Farrar and Margolin [146] measured the homogeneity range of the α TiCr 2 and β TiCr 2 Laves phases from 63 at.%cr to 66 at.%cr, the most comprehensive work done to clarify the homogeneity ranges of the Laves phases was performed by Chen et al. [148, 149]. In this work, several alloys of nominal compositions between Ti 62 at.%cr and Ti 69 at.%cr were heat treated and quenched from 1573 K, 1473 K, and 1273 K for times varying from 4 hours at 1573 K to 500 hours at 1273 K. X-ray diffraction and metallography were used to identify the Laves phases 125

126 in the microstructures. The homogeneity ranges of the Laves phases from Chen et al., measured using EPMA, [148, 149] are given in Table 7-2. One key feature of the Ti Cr phase diagram is the congruent transformation of the γ TiCr 2 Laves phase to the β phase. McQuillan measured the transformation temperature at 1633 K [138], Cuff et al. [140] and VanThyne et al. [142] measured the transformation temperature at 1623 K, and Farrar and Margolin measured the transformation temperature at 1638 K [146]. In a small temperature range between the congruent transformation of the γ TiCr 2 phase and the minimum of the liquidus, the β phase forms a continuous solid solution from Ti to Cr. The activity of chromium in the β phase at compositions between 10 and 90 at.%cr in the temperature range 1523 K, 1593 K, 1633K and 1653 K was measured by Pool et al. using the Knudsen effusion method [152]. An additional feature of the Ti Cr phase diagram is the eutectoid reaction between the β, α, and α TiCr 2 phases, the temperature of which was measured to be between 933 K and 958 K [139, 140, 142, 153, 154], determined using either metallography [139, 142, 153, 154] or dilatometry [140] techniques. Resistivity measurements were used by Mikheev and Chernova [143] but the eutectoid reaction temperature of 993 K which they obtained is much higher than that given in the other literature and is therefore generally not accepted. Various authors investigated the melting of Ti Cr alloys of various compositions. The liquidus curve was measured by VanThyne et al. [142], the liquidus and solidus curves were measured by Mikheyev and Alekasashin [155], Minaeva et al. [156], and Svechnikov and Kocherzhinskii [157], and the solidus curves were measured by McQuillan [138] and Rudy [158]. The results all indicate the presence of a melting point minimum at approximately 1883 K at a composition of 45 at.%cr. The Ti Cr system was critically assessed by Murray [159, 160]. 126

127 7.2 Thermodynamic Descriptions of the Ti Cr System Thermodynamic descriptions for the binary Ti Cr system were developed by Saunders [18], Lee et al. [161], Zhuang et al. [162], and Ghosh [163] respectively. Although all datasets model the liquid, β, and α phases as substitutional solutions, differences in the datasets exist based on: 1. The lattice stability description of pure Cr. 2. The number of Laves phases modeled. 3. The modeling of the Laves phases. 4. The homogeneity range of the Laves phases. 5. The calculated activity of the β phase compared to the experimental work of Pool et al. [152] Lattice Stability of Pure Cr The SGTE recommended thermodynamic description for pure Cr from Dinsdale [58] assigns a temperature of 2810 K to the melting point of pure Cr, which is based on the assessment of Gurvich et al. [164]. The Ti Cr datasets of Saunders [18] and Ghosh [163] use the thermodynamic description of pure Cr from Dinsdale [58]. However, the works of Hultgren et al. [165], Murray [160], Knacke et al. [166], Young [167], Stankus [168] and Dubrovinskaia et al. [169] report the melting point of pure Cr between 2130 K and 2136 K. The thermodynamic description for pure Cr from Smith et al. [170] calculates a melting temperature of 2130 K, which is in agreement with the above works. The dataset of Lee et al. [161] therefore uses the description of pure Cr from Smith et al. [170] and Zhuang et al. [162] modified the lattice stability of pure Cr to calculate a melting point of 2136 K [162]. In the work of Zhuang et al. [162], however, it is not clear if the authors modified only the thermodynamic description of liquid Cr or if they modified the thermodynamic description of liquid Cr as well as the description of β Cr. Because Lee et al. [161] and Zhuang et al. [162] use different descriptions for the lattice stability of 127

128 pure Cr, it is difficult to include these descriptions into datasets for higher order systems including Cr Number of Laves Phases Modeled Although the critical assessment of the Ti Cr phase diagram by Murray [159, 160] clearly indicated three modifications of the Laves phase, Saunders [18] only considered two modifications: the low temperature α TiCr 2 modification and a hexagonal high temperature modification reported as γ TiCr 2. The description of Saunders [18] is in agreement with the experimental works of Hao and Zeng [150] and Schuster and Du [151], in which the β TiCr 2 modification of the Laves phase could not be found. The works of Lee et al. [161], Zhuang et al. [162], and Ghosh [163], however, modeled all three modifications of the Laves phase in agreement with the experimental works of Svechnikov [147] and Chen et al. [148, 149] Modeling of the Laves Phases The α TiCr 2 Laves phase has the cf24 MgCu 2 structure. The atoms, Wyckoff positions, and coordination numbers are given in Table 7-3 and the unit cell is shown in Figure 7-1. The MgCu 2 structure contains only two crystallographic sites; therefore, in accordance with the recommendation of Ansara et al. [171] and Ferro and Cacciamani [135], the two sublattice model (Cr*,Ti) 2 (Ti*,Cr) 1 (7 1) should be used to model the homogeneity range of the α TiCr 2 phase. In the above description, the asterisk identifies the major species in each sublattice. The β TiCr 2 Laves has the hp24 MgNi 2 structure. The atoms, Wyckoff positions, and coordination numbers are given in Table 7-4 and the unit cell is shown in Figure 7-2. The β TiCr 2 phase has five different crystallographic sites. However, using five sublattices to model the β TiCr 2 Laves phase would result in too many end member and interaction parameters which would have to be assessed. Therefore, simplifications taking into account the crystallography of the structure are used to reduce the number of sublattices to be 128

129 modeled. In the first simplification, the 4e and 4f Cr sites may be combined and the 6g and 6h Ti sites may be combined. However, the 4e and 4f sites each have a coordination number of 16 and the 4f, 6g, and 6h Ti sites each have a coordination number of 12. Therefore, as an additional simplification, the 4f site may be combined with the 6g and 6h sites. This results in the model (Cr*,Ti) (Ti*,Cr) 4+4 = (Cr*,Ti) 2 (Ti*,Cr) 1 (7 2) which is also recommended by Ansara et al. [171]. The γ TiCr 2 Laves phase has the hp12 MgZn 2 structure with three crystallographic sites. The atoms, Wyckoff positions, and coordination numbers are given in Table 7-5 and the unit cell is shown in Figure 7-3. Just as in the case for the β TiCr 2 Laves phase, the 2a and 6h Cr sites, which have the same coordination number, can be combined. This simplification results also in the model given in Equation 7 1 [171]. The recommended two-sublattice model for the Laves phases was used by Saunders [18]. Lee et al. [161] and Ghosh [163] also described all three modifications of the Laves phase using the two sublattice model. However, Zhuang et al. [162] used a three sublattice model to describe the β TiCr 2 and γ TiCr 2 Laves phases since they did not use the additional simplification which could be imposed by combining crystallographic sites with the same coordination number into a single sublattice. Therefore, they modeled the β TiCr 2 and γ TiCr 2 phases as: (Cr) 2 (Cr*,Ti) 4 (Ti*,Cr) 6 (7 3) This is also a simplification as there is no evidence in the literature that only the Cr atom occupies the first sublattice The Homogeneity Ranges of the Laves Phases The partial Ti Cr phase diagram calculated using the dataset of Saunders [18] is shown in Figure 7-4. The experimental data on the homogeneity range of the Laves phases 129

130 of Chen et al. [148, 149] are superimposed. Although the calculated homogeneity range of the α TiCr 2 phase is in good agreement with the work of Chen et al. [148, 149], the Ti-rich side of the γ TiCr 2 phase does not extend to high enough Ti contents and the Cr-rich side of the γ TiCr 2 phase does not extend to high enough Cr compositions. As Saunders did not take into account the β TiCr 2 phase, calculations performed using the dataset of Saunders [18] cannot reproduce the single phase β TiCr 2 field of Chen et al. [148, 149]. The homogeneity range of the α TiCr 2 Laves phase calculated using the datasets of Zhuang et al. [162] and Ghosh [163] are not in good agreement with the experimental data of Chen et al. [148, 149]. Although Chen s Ph.D. thesis [148] gives the composition of the α TiCr 2 phase in equilibrium with the Ti-rich β phase for an alloy of composition Ti 62 at.%cr at 1273 K, this data was probably not taken into account in the assessments of Zhuang et al. [162] and Ghosh [163]. Therefore, in these works, the Ti-rich boundary of the α TiCr 2 Laves phase extends to higher Ti compositions than those measured by Chen [149]. However, in comparison to the dataset of Saunders [18], where the homogeneity range of the γ TiCr 2 Laves phase is not in agreement with the work of Chen et al. [148, 149], the homogeneity range of the γ TiCr 2 Laves phase in the works of Zhuang et al. [162] and Ghosh [163] is in good agreement with the measured homogeneity range of Chen et al. [148, 149]. The published paper of Lee et al. [161] does not present an enlarged part of the Ti Cr phase diagram from 60 at.%cr to 70 at.%cr showing details of the homogeneity ranges of the Laves phase modifications. Therefore, the agreement of the homogeneity ranges of the three Laves phases with the work of Chen et al. [148, 149] is unclear The Activity of Cr in the Beta Phase The activity of Cr in the β phase at compositions from 10 at.%cr to 90 at.%cr at 1523 K, 1593 K, 1633 K, and 1653 K respectively were determined by Pool et al. [152] using vapor pressure measurements. The measured activities show quite a large 130

131 variation with temperature. The calculation of Saunders [18] reproduces the activity of Cr in the β phase at 1523 K and 1593 K, whereas the calculations of Zhuang et al. [162] and Ghosh [163] reproduce the activity data for Cr at 1633 K and 1653 K. Zhuang et al. [162] attempted to reproduce the large temperature variation of Cr activity in the β phase in accordance with the measurements of Pool et al. [152] by optimizing the 0 L β Cr,Ti parameter. However, this resulted in a relatively large temperature dependent term for the parameter, which, according to Okamoto [172], may be unrealistic in such a binary system. Ghosh [163] additionally reported that a better fit to the experimental data on the β phase boundary could be achieved if only the higher temperature activity data of Pool et al. [152] is taken into account. However, when the activity of Cr in the β phase is calculated using the dataset of Ghosh [163], there is a larger deviation of the activity of Cr from the ideal at higher temperatures than the deviation of the activity of Cr from the ideal at lower temperatures. This indicates that Ghosh [163] modeled a physically unrealistic behavior of the activity of Cr in the β phase. The publication of Lee et al. [161] gives no indication on the fit of the activity of Cr in the β phase to the experimental results of Pool et al. [152]. 7.3 Optimization of the Parameters for the Binary Ti Cr System Taking into account the above discrepancies between the datasets of Saunders [18], Lee et al. [161], Zhuang et al. [162], and Ghosh [163] and the experimental data, the only clear strategy would be to re-optimize the thermodynamic parameters for the description of the Ti Cr system using one of the existing datasets as a base from which further parameter re-optimization could be performed. The datasets of Zhuang et al. [162] and Lee et al. [161] are not recommended since they used different descriptions for pure Cr than that recommended by SGTE [58]. Additionally, Zhuang et al. [162] modeled the β TiCr 2 and γ TiCr 2 Laves phases using three sublattices which is not recommended by Ansara et al. [171]. It would have been possible to use the dataset of Ghosh [163], except 131

132 that Ghosh inadvertently modeled a physically unrealistic behavior of the activity of Cr in the β phase. The description of Saunders [18] used the lattice stability of pure Cr recommended by SGTE [58], modeled the Laves phases using the two sublattice model recommended by Ansara et al. [171], and modeled a physically realistic behavior of the activity of Cr in the β phase. Therefore, the dataset of Saunders is a good candidate dataset from which further optimization could proceed although Saunders did not take into account all three modifications of the Laves phases. However, an additional justification for the choice of the Saunders dataset is the application of the description of the Ti Cr system to the higher order Ti Al Cr system, which Saunders also modeled using his description for the Ti Cr binary [18] as the constituent binary description. Therefore, if only the descriptions for the two Laves phases in the Ti Cr dataset of Saunders are re-optimized and an additional description for the β TiCr 2 Laves phase is included, then it would still be possible to calculate all other phase equilibria in the Ti Al Cr system that include the β, α, and liquid phases as the descriptions for these phases would not be changed. Since the dataset of Saunders [18] was used, all thermodynamic descriptions for the pure elements were taken from the SGTE recommendations [58]. All re-optimization was performed using the PARROT module [28] of THERMO-CALC [29] Selection of Thermodynamic Parameters for the Laves Phases The thermodynamic model for the α TiCr 2, β TiCr 2, and γ TiCr 2 Laves phases is given in Equation 7 1. Such a description results in the four end-members G φ Cr:Cr, Gφ Cr:Ti, G φ Ti:Cr, and Gφ Ti:Ti where φ represents either of the three Laves phases. The only physically realistic end member is the G φ Cr:Ti term, which corresponds to full occupation of the Cr lattice sites and Ti lattice sites by Cr and Ti atoms respectively. This end member term is modeled as: G φ Cr:Ti = 2 GHSER Cr + GHSER Ti + a + b T (7 4) 132

133 where a and b are coefficients to be optimized. The G φ Cr:Ti parameter affects the temperatures at which the invariant reactions with the Laves phases occur as well as the composition of the Laves phases at the invariant reactions. The G φ Cr:Cr and Gφ Ti:Ti end member parameters are related to the occupation of all lattice sites in the Laves phase structure by only Cr and Ti atoms respectively. As these are physically unrealistic conditions, large positive values were added to description of these end member terms, which are now given as: G φ Cr:Cr = GHSER Cr (7 5) and G φ Ti:Ti = GHSER Ti (7 6) The last end member, G φ Ti:Cr, also represents a physically unrealistic condition where all Cr lattice sites are occupied by Ti atoms and all Ti lattice sites are occupied by Cr atoms. The G φ Ti:Cr end member can be related to all other end members using the reciprocal relation: G φ Ti:Cr = Gφ Ti:Ti + Gφ Cr:Cr Gφ Cr:Ti + Grec (7 7) where G rec is the Gibbs energy of the reciprocal reaction. In the present re-optimization, G rec was assigned a value of 0. The substitution of Equation 7 4, Equation 7 5, and Equation 7 6 into Equation 7 7 results in the expression: G φ Ti:Cr = 2 GHSER Cr + GHSER Ti a b T (7 8) for the G φ Ti:Cr end member term. Therefore, for any of the three Laves phases, the values of the end members G φ Cr:Cr, Gφ Cr:Ti, Gφ Ti:Cr, and Gφ Ti:Ti are optimized by optimizing only the a and b terms in Equation 7 4. The two sublattice model for the Laves phases results in the four binary mixing parameters 0 L φ Cr,Ti:Ti, 0 L φ Cr,Ti:Cr, 0 L φ Cr:Cr,Ti, and 0 L φ Ti:Cr,Ti. To reduce the number of 133

134 optimizing variables, the simplifications 0 L φ Cr,Ti:Ti = 0 L φ Cr,Ti:Cr = 0 L φ Cr,Ti:* (7 9) and 0 L φ Cr:Cr,Ti = 0 L φ Ti:Cr,Ti = 0 L φ *:Cr,Ti (7 10) were used, corresponding to the assumption that mixing of Cr and Ti atoms on one sublattice is independent of the species occupying the second sublattice. The 0 L φ Cr,Ti:Ti binary interaction parameter influences the position of the Ti-rich Laves phase boundary and the 0 L φ Cr:Cr,Ti parameter influences the position of the Cr-rich Laves phase boundary Optimization Strategy The composition and temperature of the congruent transformation from the γ TiCr 2 phase to the β phase, as well as the temperatures and phase compositions at the invariant reactions with the three Laves phases used for the optimization are given in Table 7-6. These values were chosen based on a critical and careful review of all available experimental data. The C P behavior of the Laves phases was chosen to obey the Neumann-Kopp rule. In the first step, the end member parameter G α TiCr 2 Cr:Ti and the 0 L α TiCr 2 Cr,Ti:* and 0 L α TiCr 2 *:Cr,Ti mixing parameters for the α TiCr 2 phase were re-optimized to fit the homogeneity range of the α TiCr 2 phase and to calculate the eutectoid reaction temperature between the α TiCr 2, β, and α Ti phases at 959 K. Next, the G β TiCr 2 Cr:Ti end member parameter and the 0 L β TiCr 2 Cr,Ti:* and 0 L β TiCr 2 *:Cr,Ti mixing parmaters for the β TiCr 2 phase were re-optimized simultaneously with the parameters for the α TiCr 2 phase to fit the temperatures and phase compositions for the eutectoid decomposition of the β TiCr 2 phase at 1077 K, the peritectoid formation of the α TiCr 2 phase at 1496 K, and the eutectoid reaction between the α TiCr 2, β, and α Ti phases at 959 K. 134

135 In the following step, the γ TiCr 2 phase should be stabilized at temperatures above 1544 K using the G γ TiCr 2 Cr:Ti parameter. Since the G γ TiCr 2 Cr:Ti linear function of temperature (Equation 7 4), values of the G γ TiCr 2 Cr:Ti parameter is expressed as a parameter at two temperatures are required to estimate starting values of a and b in Equation 7 4. The first value of the G γ TiCr 2 Cr:Ti parameter is G γ TiCr 2,1640 K Cr:Ti, which is the value of the G γ TiCr 2 Cr:Ti parameter at 1640 K required to calculate the congruent decomposition of the γ TiCr 2 phase at approximately 1640 K. To estimate a value for this parameter, a slightly negative term of approximately 60 J was added to the value of the G β TiCr 2 Cr:Ti at 1640 K, according to the equation: end member parameter G γ TiCr 2,1640 K Cr:Ti = G β TiCr 2,1640 K Cr:Ti 60 (7 11) Next, G γ TiCr 2,1543 K Cr:Ti, which is the value of the G γ TiCr 2 Cr:Ti parameter at 1543 K, was estimated to calculate the invariant reactions between the β TiCr 2, γ TiCr 2, and Cr-rich β phase and the β TiCr 2, γ TiCr 2 and Ti-rich β phases respectively. The values G γ TiCr 2,1640 K Cr:Ti and G γ TiCr 2,1543 K Cr:Ti were used to build the starting values for the linear dependence of the G γ TiCr 2 Cr:Ti parameter. The linear temperature dependence of the G γ TiCr 2 Cr:Ti and G β TiCr 2 Cr:Ti parameters are shown in Figure 7-5. In the next step, the G γ TiCr 2 Cr:Ti, 0 L γ TiCr 2 Cr,Ti:*, and 0 L γ TiCr 2 *:Cr,Ti parameters were optimized separately to fit the congruent decomposition of the γ TiCr 2 at 1640 K and the homogeneity range of the γ TiCr 2 phase to be in agreement with the work of Chen et al. [148, 149]. In the last step, all end member and interaction parameters of all three modifications of the Laves phases were optimized simultaneously Optimized Results The Ti Cr binary phase diagram calculated with the new descriptions for the Laves phases is shown in Figure 7-6. The experimental data from the literature superimposed. Figure 7-6 shows that there is a very good agreement between the calculated phase 135

136 diagram and the experimental data. The partial Ti Cr phase diagram from 60 at.%cr to 70 at.%cr is shown in Figure 7-7. The experimental data on the homogeneity ranges of the Laves phases from Chen et al. [148, 149], Minaeva et al. [156] and Farrar and Margolin [146] are superimposed. The calculations also show good agreement with the experimental works [146, 148, 149, 156]. Figure 7-8 shows the calculated activity of Cr in the β phase from 10 at.%cr to 90 at.%cr at 1523 K, 1593 K, 1633 K, and 1653 K along with the experimental data points of Pool et al. [152]. Since the parameters for the β phase were not optimized in this work, this diagram is exactly the same as that which would be calculated using the original Ti Cr description of Saunders [18]. 136

137 Table 7-1. Phases in the Ti Cr system. Phase Prototype Pearson symbol Space group Structure report β W ci2 Im 3m A2 (αti) Mg hp2 P 6 3 /mmc A3 α TiCr 2 MgCu 2 cf24 F d 3m C15 β TiCr 2 MgNi 2 hp24 P 6 3 /mmc C36 γ TiCr 2 MgZn 2 hp12 P 6 3 /mmc C14 Table 7-2. Homogeneity range of the Laves phases determined by Chen et al. [148]. Temperature Laves Phase Ti rich boundary Cr rich boundary 1573 K γ TiCr at.%cr 66.3 at.%cr 1473 K β TiCr at.%cr 66.3 at.%cr 1273 K α TiCr at.%cr 66.0 at.%cr 137

138 Figure 7-1. Unit cell of the α TiCr 2 Laves phase. A two sublattice model with one Ti-rich and one Cr-rich sublattice is used. Table 7-3. Crystal structure of the α TiCr 2 Laves phase. Space Group F d 3m Pearson Symbol cf 24 Atoms Wyckoff Position x y z CN Ti 8a Cr 16d

139 Figure 7-2. Unit cell of the β TiCr 2 Laves phase. The Ti1 and Ti2 atoms, which occupy the 4e and 4f positions respectively, occupy the same sublattice because they have the same coordination numbers, and the Cr1, Cr2, and Cr3 atoms, which occupy the 4f, 6g, and 6h positions, occupy the same sublattice as they have the same coordination numbers. Table 7-4. Crystal structure of the β TiCr 2 Laves phase. Space Group P 6 3 /mmc Pearson Symbol hp 24 Atoms Wyckoff Position x y z CN Ti 4e Ti 4f Cr 4f Cr 6g Cr 6h

140 Figure 7-3. Unit cell of the γ TiCr 2 Laves phase. The Ti atoms occupy the 4f position, and the Cr1 and Cr2 atoms occupy the 2a and 6h positions respectively. Since the Cr1 and Cr2 atoms have the same coordination numbers, they are placed on the same sublattice. Table 7-5. Crystal structure of the γ TiCr 2 Laves phase. Space Group P 6 3 /mmc Pearson Symbol hp 12 Atoms Wyckoff Position x y z CN Cr 2a Ti 4f Cr 6h

141 Figure 7-4. The Calculated partial Ti Cr phase diagram from 60 at.%cr to 70 at.%cr using the description of Saunders [18]. Only the low temperature α TiCr 2 and a high temperature γ TiCr 2 Laves phases are modeled. There homogeneity range of the α TiCr 2 phase is in good agreement with the experimental work of Chen et al. [148, 149]. However, the agreement of the homogeneity range of the γ TiCr 2 phase with the work of Chen et al. [148, 149] is not so good. Table 7-6. Data on the Laves phases that were used for the optimization. Reaction φ 1 + φ 2 + φ 3 Temperature [K] Composition [at.%cr] φ 1 φ 2 φ 3 β γ TiCr γ TiCr 2 + β β TiCr γ TiCr 2 β + β TiCr β TiCr 2 + β α TiCr β TiCr 2 β + α TiCr β α TiCr 2 + β(ti)

142 Figure 7-5. Graph showing the dependence of the G γ TiCr 2 Cr:Ti parameter for the γ TiCr 2 phase as the solid line. The linear dependence of the G β TiCr 2 Cr:Ti parameter is also included as the dashed line. Figure 7-6. Calculated Ti Cr phase diagram using the new description. The experimental data from the literature is superimposed. 142

143 Figure 7-7. Calculated partial Ti Cr phase diagram from 60 at.%cr to 70 at.%cr. There is good agreement of the homogeneity range of the α TiCr 2, β TiCr 2 and γ TiCr 2 Laves phases with the experimental work of Chen et al. [148, 149]. 143

144 Figure 7-8. Calculated activity of Cr in the β phase compared to the experimental data of Pool et al. [152]. Since the new dataset uses the same thermodynamic parameters for the β phase as in Saunders [18], there is no difference in the activity curves for Cr between the new dataset and the dataset of Saunders [18]. 144

145 CHAPTER 8 THE TI-AL-CR SYSTEM 8.1 Phases in the Ti Al Cr System The crystallographic data of the solid phases in the Ti Al Cr system are given in Table 8-1. The α Al 8 Cr 5 and Al 7 Cr phases from the Al Cr binary have been shown to exhibit some solubility for Ti. For example, the α Al 8 Cr 5 dissolves up to 10 at.%ti at room temperature [22, 24, 173] and Zoller [174] reported the solubility of Ti in Al 7 Cr as 1.03 at.% whereas Sokolovskaya et al. [175] reported this same solubility as up to 2 at.%ti. However, there is no information on the solubility of Ti in the other binary Al Cr phases available in the literature. The γ TiAl, α 2 Ti 3 Al, and TiAl 2 phases extend into the ternary Ti Al Cr system through dissolution of Cr. The solubility of Cr in γ TiAl is 2 at.%cr at 47 at.%al and 8 at.%cr at 58 at.%al at 1273 K [176]. At 1073 K, the maximum solubility of Cr in γ TiAl is 4.5 at.%cr. The solubility of Cr in α 2 Ti 3 Al is about 2.5 at.%cr in the temperature range 1073 K to 1273 K, and the solubility of Cr in TiAl 2 is 3.5 at.%cr and 6 at.%cr at 1273 K and 1073 K respectively [176]. The solubility of Al in the α TiCr 2 and β TiCr 2 phases from the Ti Cr binary has been shown to be 1 at.%al and 4 at.%al respectively [177, 178]. Three ternary phases are formed in the Ti Al Cr system. The τ phase forms through stabilization of the binary metastable TiAl 3 (m) phase by addition of Cr [20 26]. This phase exhibits a homogeneity range from 7 at.%cr to 15 at.%cr and 23 at.%ti to 27 at.%ti at 1473 K [22], from 6 at.%cr to 13 at.%cr and 60 at.%al to 68 at.%al at 1273 K, and from 6 at.%cr to 10 at.%cr and 64 at.%al to 67 at.%al at 1073 K [179]. According to the works of Mabuchi et al. [173], Ichimaru et al. [180], and Barabash et al. [181], it is possible that the τ phase forms peritectically from the liquid. Ichimaru et al. [180] gave the τ formation temperature as 1623 K although Barabash et al. [181] gave a formation temperature of 1523 K. 145

146 The γ Ti(Al,Cr) 2 phase stabilizes into the ternary at temperatures below the stability of the γ TiCr 2 Ti Cr binary Laves phase by substitution of Al atoms for Cr atoms. The homogeneity range of the γ Ti(Al,Cr) 2 phase at 33 at.%ti extends from 3 at.%al to 20 at.%al at 1323 K [178], from 5 at.%al to 42 at.%al at 1273 K, and from 5 at.%al to 40 at.%al at 1073 K [177]. The deviation of Ti content from the stoichiometric 33.3 at.%ti is measured as 1 at.% at 5 at.%al and 3 at.% at 35 at.%al at 1273 K and 3 at.% at 40 at.%al at 1073 K [176, 177]. Fujita et al. [182] did not find the γ Ti(Al,Cr) 2 phase at 1473 K although it was found at 1323 K in the work of Xu et al. [178]; therefore, the formation of the γ Ti(Al,Cr) 2 phase is assumed to take place at a temperature between 1323 K and 1473 K. Various investigators [22, 24 26, 173, 183] have found the γ TiCr 2 phase to be stable down to room temperature, i.e., there is no evidence of a solid state decomposition of the γ Ti(Al,Cr 2 ) phase at lower temperatures. Ordering of the disordered β phase to the β 0 phase in the Ti-rich region of the Ti Al Cr system has been observed in several works [6, 126, 176, ]. The homogeneity range of the Ti-rich β phase at 1273 K has been measured by Jewett et al. [176, 177]. The formation of the Ti-rich β 0 phase is assumed to take place between 1323 K and 1273 K as Xu et al. [178] did not find the β 0 phase in Ti-rich alloys which were heat treated and quenched from 1323 K, but the phase was found by Jewett et al. [176] and Kainuma et al. [126, 186] at 1273 K. 8.2 Experimentally Determined Phase Equilibria Isothermal Sections Complete Isothermal sections at 1273 K and 1073 K are presented by Jewett et al. [176] based on measured tie line data from heat treatment and quench experiments for various alloys at 1273 K and 1073 K [176, 177, 179]. According to Jewett et al. [176], however, the τ + γ TiAl + TiAl 2 and τ + TiAl 2 + Ti 5 Al 11 three phase fields at 1273 K are not fully characterized. Additionally, phase equilibria at Al-rich compositions above the τ + η (Ti,Cr)Al 3 + α Al 8 Cr 5 three phase field is unclear. The β + γ TiAl 146

147 + α 2 Ti 3 Al tie triangle of Jewett et al. [176] was confirmed in the experimental work of Kainuma et al. [126]. Kainuma et al. [186] also presented α α 2 and α 2 β tie line data at 1273 K which is in good agreement with the 1273 K isothermal section of Jewett et al. [176]. In the isothermal section at 1073 K of Jewett et al. [176], phase equilibria in the Al-rich region at Al compositions higher than the η (Ti,Cr)Al 3 + α Al 9 Cr 4 two phase tie line area remain unclear The Liquidus Surface Very limited information is available in the literature on the liquidus projection in the Ti Al Cr system. Liquidus isotherms in the Ti-rich corner from 50 mass %Ti to 100 mass %Ti were presented in the work of Kornilov et al. [187] based on DTA results, and Ichimaru et al. [180] presented liquidus isothermal lines for the τ phase in the temperature range from 1643 K to 1523 K. Shao and Tsakiropoulos [185] investigated the solidification of a Ti 40at.%Al 10at.%Cr alloy, a Ti 50at.%Al 10at.%Cr alloy, and a Ti 52at.%Al 20at.%Cr alloy, and showed that the first alloy solidified as β phase, the second alloy solidified as α phase and the third alloy solidified as the ternary τ phase. 8.3 Critical Assessments of the Ti Al Cr System The phase equilibria in the Ti Al Cr system was critically assessed by Bochvar et al. [19], Hayes [188] and Raghavan [189] with the most recent assessment performed by Bochvar et al. [19]. In the reviews of Raghavan [189] and Bochvar et al. [19], liquidus surfaces, various complete and partial isothermal sections at temperatures between 1473 K and 770 K, and Scheil reaction schemes are presented. The key differences between the liquidus surfaces of Bochvar et al. and Raghavan are: 1. Raghavan [189] includes the primary crystallization field for the Ti 1 x Al 1+x phase as well as for the Ti 5 Al 11 phase whereas Bochvar et al. [19] show only a primary crystallization field for the Ti 2 Al 5 phase. The reason for this is that Bochvar et al. [19] use the Ti Al phase diagram calculated from the dataset of Witusiewicz et al. [110] as the constituent binary for their assessment whereas Raghavan [189] uses his own assessed Ti Al binary [190] as the constituent binary. The Ti Al binary of 147

148 Witusiewicz et al [17] combines the Ti 1 x Al 1+x and Ti 5 Al 11 phases as one Ti 2 Al 5 phase with a homogeneity range from 62 at.%al to 68 at.%al. Therefore, only the Ti 2 Al 5 phase can be indicated in the liquidus surface of Bovhvar et al. [19]. 2. The assessment of Raghavan [189] shows a much larger primary crystallization field for the Ti 5 Al 11 phase (shown as the Ti 2 Al 5 phase in the assessment of Bochvar et al. [19]). As a result, Raghavan includes the univariant line liquid + β + Ti 5 Al 11 and invariant reactions between the liquid, Ti 5 Al 11, τ, and β phases and between the liquid, Ti 5 Al 11, γ TiAl, and β phases. The assessment of Bochvar et al. [19] includes no such equilibria i.e., the Ti 2 Al 5 phase does not extend to sufficiently high Cr compositions so as to exhibit a univariant reaction with the liquid and β phases on the liquidus surface. 3. The τ phase forms peritectically according to the reaction liquid + Ti 5 Al 11 τ in the assessment of Raghavan [189], but according to the transition reaction liquid + γ TiAl τ + β in the assessment of Bochvar [19]. 8.4 Thermodynamic Descriptions of the Ti Al Cr System Phase diagram calculations in the Ti Al Cr system using thermodynamic descriptions have been published by Gros et al. [191], Shao and Tsakiropoulos [185], and Kaufman [192]. However, neither details of the thermodynamic modeling and optimization strategy nor the final thermodynamic parameters are given. Saunders [18] also developed a thermodynamic dataset for the Ti Al Cr system as part of the COST507 European action. This thermodynamic description was readily available, and was therefore used to perform initial calculations in the Ti Al Cr system to assess its ability to reproduce key phase equilibria. The calculations were compared to the phase diagrams published in the assessment of Bochvar et al. [19] as this assessment is the most recent and most comprehensive assessment available Thermodynamic Description of Saunders The description for the unary elements were taken from the SGTE recommended dataset of Dinsdale [58]. The binary Ti Al description of Saunders, which is used as the constituent binary in the ternary Ti Al Cr system, was discussed in Section Saunders also used his description of the Ti Cr binary as the constituent binary in the description for the Ti Al Cr system. However, as was discussed in Chapter 7, only two 148

149 modifications of the TiCr 2 Laves phases were modeled, and therefore the parameters for the Laves phases were re-assessed to take into account the experimental evidence indicating three modifications of the Laves phase. A description for the Al Cr constituent binary was also developed by Saunders [18] and is the SGTE recommended dataset although another, more recent dataset by Liang et al. [193] exists in the literature. The calculated binary Al Cr system of Saunders [18] is shown in Figure 8-1. Some discrepancies exist between the calculated phase diagram of Saunders [18], the calculated phase diagram of Liang et al. [193] (Figure 8-2), and the numerous experimental work performed on the system. Some of the major discrepancies are: 1. Grushko et al. [194] showed that in the composition range from 30 at.%cr to 42 at.%cr, only one high temperature γ phase with a perfect γ-brass structure and one low temperature γ phase with a rhomboherally distorted γ brass structure exist. The homogeneity range of the γ phase was reproduced in the more recent thermodynamic description of Liang et al. [193]. However, Saunders models two stoichiometric phases, the Al 8 Cr 5 and Al 9 Cr 4 phases, each with a polymorphic transformation from the low temperature modification to the high temperature modification, within this composition range. 2. Grushko et al. [195] found a new phase of composition Al 11 Cr 4 which forms by a peritectoid reaction between the Al 4 Cr and γ phases. This new phase was not included in the dataset of Liang et al. [193] or in the dataset of Saunders [18]. 3. Mahdouk and Gachon [196], using heat treating and quenching experiments, found that the Al 11 Cr 2 phase undergoes eutectoid decomposition to the Al 7 Cr 2 and Al 4 Cr phases at a temperature between 1123 K and 923 K. However, this was not confirmed in the works of Grushko et al. [194, 195, 197]. The eutectoid decomposition of the Al 11 Cr 2 phase is reproduced in the description of Liang et al. [193] but not in the description of Saunders [18]. 4. Helander and Tolochko [198] showed that alloys containing between 58.4 at.%al and 64.8 at.%al heat treated and quenched from 1158 K and 1178 K contained the ordered B2 structure. However, the order disorder transformation of the β phase was not modeled in either the dataset of Liang et al. [193] or the dataset of Saunders [18]. 149

150 Reasons for re-optimization Preliminary calculations were performed using the thermodynamic description of the Ti Al Cr system optimized by Saunders [18] to assess the ability of this dataset to reproduce the phase diagrams given in the assessment of Bochvar et al. [19]. The calculated and assessed isothermal sections at 1073 K and 1273 K are shown in Figure 8-3 and Figure 8-4 respectively. The major discrepancies between the calculated and assessed phase diagrams are: 1. The extension of the ternary γ Ti(Al,Cr) 2 Laves phase, which is indicated in the 1073 K and 1273 K assessed isothermal sections of Bochvar et al. [19], is not reproduced in the description of Saunders [18]. 2. Saunders [18] did not include a description for the ternary τ phase. 3. The isolated single phase β 0 field at approximately 18 at.%cr and 30 at.%al shown in the 1073 K isothermal section assessed by Bochvar et al. [19] is not reproduced using the dataset of Saunders [18]. 4. The α 2 Ti 3 Al phase appears to be stabilized in the Ti-rich corner of the calculated 1073 K isothermal section in the description of Saunders [18]. 5. The calculated solubility of Al in the α TiCr 2 and γ TiCr 2 Laves phases is much lower than that observed experimentally. 6. There is no description for the solubility of Ti in the Al 11 Cr 2, Al 4 Cr, α Al 9 Cr 4, α Al 8 Cr 5, and AlCr 2 phases. The calculated and assessed liquidus surfaces are shown in Figure 8-5. The main difference between the liquidus projections is the presence of a primary crystallization field for the τ phase in the liquidus surface of Bochvar et al. [19]. Since the τ phase is not modeled in the dataset of Saunders [18], the liquidus surface of Saunders shows larger primary crystallization areas for the γ TiAl, η (Ti,Cr)Al 3, and β Al 8 Cr 5 phases. To improve the agreement between the calculated phase diagrams and the phase diagrams from the assessment of Bochvar et al. [19], descriptions for the τ and γ Ti(Al,Cr) 2 Laves phases should be introduced. Additionally, the Al solubility of the α TiCr 2, β TiCr 2, and γ TiCr 2 Laves phases should be modeled for better agreement with the 150

151 experimental data. The description for the α 2 Ti 3 Al should be changed so that the α 2 Ti 3 Al phase is not stabilized in the Ti-rich corner of the calculated 1073 K isothermal section, and last, the parameters of the β and β 0 phases should be re-assessed to reproduce the isolated β 0 field in the 1073 K isothermal section at approximately 18 at.%cr and 30 at.%al. 8.5 Re-optimization of the Ti Al Cr System Optimization Strategy First, the α 2 Ti 3 Al phase was made less stable in the Ti Cr binary by addition of a positive temperature independent term to one of its end members. Second, the parameters of the γ Ti(Al,Cr) 2 phase were optimized to calculate the experimentally observed extension of the γ Ti(Al,Cr) 2 phase into the ternary Ti Al Cr system at 33.3 at.%ti. Third, the τ phase was introduced as a stoichiometric phase with a linear temperature dependence of the Gibbs energy of formation from the elements. The temperature dependent and independent terms were assessed so that three phase equilibria with the τ phase at 1273 K reported in the works of Jewett et al. [176, 177, 179], as well as the presence of a primary crystallization field for the τ phase on the liquidus projection could be calculated. Next, the parameters of the β phase were reoptimized to calculate the β/β + γ Ti(Al,Cr) 2 phase boundaries at 1073 K and 1273 K from Jewett et al. [176, 177, 179]. Once the parameters for the disordered β phase were re-optimized, the parameters of the ordered β 0 phase were re-assessed to reproduce the isolated β 0 field at 1073 K which is given in the works of Jewett et al. [176, 177]. Last, the solubility of Al in the α TiCr 2 and β TiCr 2 Laves phases were modeled to be in agreement with the 1073 K and 1273 K isothermal sections assessed by Bochvar et al. [19] The alpha2 phase Calculations performed using the dataset of Saunders [18] indicated a stabilization of the α 2 Ti 3 Al phase in the Ti Cr system and Ti-rich corner of the Ti Al Cr system. Since this stabilization occurred in Ti-rich corner, a positive term of 4000 J was added to 151

152 the G α 2 T i 3 Al Ti:Ti end member description. This term was sufficient to remove the stabilized α 2 Ti 3 Al phase in the calculated 1073 K isothermal section The ternary Ti(Al,Cr)2 laves phase The ternary γ Ti(Al,Cr) 2 Laves phase has the hp12 MgZn 2 structure. The atoms, Wyckoff positions, and coordination numbers for such a structure were already given in Table 7-5. The two sublattice model given in Equation 7 1, which was used to model the binary Ti Cr Laves phases, was modified to: (Cr*,Ti,Al) 2 (Ti*,Cr,Al) 1 (8 1) where the asterisk identifies the major species in each sublattice, to reproduce the solubility of Al in the ternary γ Ti(Al,Cr) 2 Laves phase. Since there is no evidence in the literature of a separation of the ternary γ Ti(Al,Cr) 2 and binary γ TiCr 2 phases, and since both phases have the same MgZn 2 crystal structure, they were modeled as a single phase. The locations of the end members that result from the description given in Equation 8 1 are indicated on the Gibbs triangle in Figure 8-6. The description for the G γ Cr:Ti end member was taken from the binary Ti Cr dataset which was assessed in the present work (details given in Chapter 7) whereas the descriptions for all other end members were kept from the original Ti Al Cr dataset of Saunders [18]. The 0 L γ *:Cr,Ti, 0 L γ Cr,Ti:*, and 0 L γ Al,Cr:Ti mixing parameters were optimized in this work. The 0 L γ *:Cr,Ti and 0 L γ Cr,Ti:* mixing parameters were accepted from the binary description of the Ti Cr system performed in the present work (Chapter 7). These parameters control the width of the γ Ti(Al,Cr) 2 phase. The 0 L γ Al,Cr:Ti parameter, which was modeled using a linear temperature dependence, influences the extension of the γ Ti(Al,Cr) 2 phase into the ternary. To generate starting values for the temperature independent and temperature dependent terms of the 0 L γ Al,Cr:Ti parameter, two values of this parameter were approximated. The first value, 0 L γ,1273k Al,Cr:Ti, is the value of the 0 L γ Al,Cr:Ti parameter required to calculate an extension of the γ Ti(Al,Cr) 2 phase to approximately 42 at.%al 152

153 at 1273 K to be in agreement with the experimental work of Jewett et al. [177]. The second value, 0 L γ,1073k Al,Cr:Ti, is the value of the 0 L γ Al,Cr:Ti parameter required to calculate the extension of the γ Ti(Al,Cr) 2 phase to approximately 40 at.%al at 1073 K, which was also measured by Jewett et al. [177] The ternary tau phase The τ phase has the cp4 AuCu structure. Although the τ phase exhibits a slight homogeneity range, expressed as (Al 3-y Cr x )Ti 1-x+y, this homogeneity range is quite small. Most authors [24, 173, 199, 200] report the stoichiometry of the τ phase as Ti 25 (Al 67 Cr 8 ), which is obtained when x=y=0.32. Other authors [21, 179] give a homogeneity range of the τ phase which includes this stoichiometry. Therefore, the τ phase is modeled as a stoichiometric phase using the equation: G τ Ti 25 (Al 67 Cr 8 ) = 25 GHSER Ti + 67 GHSER Al + 8 GHSER Cr + a + b T (8 2) where a and b are the parameters to be optimized. Since the τ phase is modeled with a linear temperature dependence, values of G τ Ti 25 (Al 67 Cr 8 ) at two different temperature are required to generate initial, starting values of the a and b parameters in Equation 8 2. The first value of G τ Ti 25 (Al 67 Cr 8 ) was chosen as the value of the parameter required to calculate the β + γ Ti(Al,Cr) 2 + τ, β + α Al 8 Cr 5 + τ, η (Ti,Cr)Al 3 + ζ Ti 5 Al 11 + τ, and γ TiAl + γ Ti(Al,Cr) 2 + τ series of three phase equilibria at 1273 K to be in agreement with the experimental work of Jewett et al. [176, 177, 179]. The second value of G τ Ti 25 (Al 67 Cr 8 ) was chosen to fit the experimental data on the formation of the τ phase at higher temperatures. According to the works of Mabuchi et al. [173], Ichimaru et al. [180], and Barabash et al. [181], it is possible that the τ phase forms at temperatures between 1623 K [180] and 1523 K [181]. Taking into account the 100 K difference in τ phase formation temperatures, the value of the G τ Ti 25 (Al 67 Cr 8 ) parameter was chosen to fit the formation of the τ phase at a temperature between 1623 K and 1523 K, as well as to reproduce primary crystallization fields of 153

154 the τ, γ TiAl, and ζ Ti 5 Al 11 phases which are in agreement with the assessed liquidus surface of Bochvar et al. [19]. The calculated liquidus surface was found to be in good qualitative agreement with the assessed liquidus projection of Bochvar et al. [19] when the formation temperature of the τ phase was calculated at 1603 K. Therefore, the values of the G τ Ti 25 (Al 67 Cr 8 ) parameter at 1273 K and 1603 K were used to generate starting values of the a and b parameters in Equation The beta phase The β phase, which has the ci2 W structure, is modeled as the substitutional solution (Al,Nb,Ti). In this work, only the 0 L β Al,Nb,Ti, 1 L β Al,Nb,Ti, and the 2 L β Al,Nb,Ti parameters were optimized using experimental data on the β/β + γ Ti(Al,Cr) 2 phase boundaries from Jewett et al. [176, 177, 179] The ordered beta phase The β 0 phase is the ordered cp2 CsCl B2 structure with Ti-rich and Al-rich sublattices respectively. It is modeled using the compound energy formalism as (Al,Nb,Ti) 0.5 (Al,Nb,Ti) 0.5 (8 3) The locations of the end members for the β 0 phase are the same as that for the γ phase in the Ti Al Nb system (Section 5.3.4), and are therefore already illustrated in Figure 5-6. The descriptions for the end members of the β 0 phase were kept from the original description of Saunders [18]. In the model for the β 0 phase, the restrictions of symmetry impose the constraint 0 L β 0 i,j:k = 0 L β 0 k:i,j. Only the 0 L β 0 Al,Cr:Ti = 0 L β 0 Ti:Al,Cr parameters affected the formation of the isolated β 0 field. All other parameters of type 0 L β 0 i,j:k were therefore given a value of 0. The 0 L β 0 Al,Cr:Ti = 0 L β 0 Ti:Al,Cr parameters were modeled with a linear temperature dependence, the starting values for which were selected to fit the phase equilibria with the β 0 phase at 1273 K and 1073 K given in the works of Jewett et al. [176, 177]. 154

155 The TiCr2 laves phases The α TiCr 2 and β TiCr 2 Laves phases were modeled using the compound energy formalism as given in Equation 8 1. The only parameters which influence the solubility of Al in the α TiCr 2 and β TiCr 2 Laves phases are the 0 L α T icr 2 Al,Cr:Ti and 0 L β T icr 2 Al,Cr:Ti parameters respectively. Therefore, these parameters were optimized to be in agreement with the recommendations of Bochvar et al. [19]. 8.6 Results of the Re-optimization The 1273 K and 1073 K isothermal sections calculated using the re-optimized dataset are shown in Figures 8-7 and 8-8 respectively. The experimental data of Jewett et al. [176, 177, 179] are superimposed. The calculated isothermal sections show a good agreement with the experimental data. However, the α 2 Ti 3 Al + β 0 + α TiCr 2 and α 2 + β + α TiCr 2 three phase fields suggested by Bochvar et al. [19] could not be reproduced. Instead, the α 2 Ti 3 Al and β 0 and the α 2 Ti 3 Al and β phases form three phase equilibria with the ternary γ Ti(Al,Cr) 2 phase. The calculated liquidus surface and partial liquidus surface are shown in Figures 8-9 and 8-10 respectively. The Scheil reaction scheme for equilibria with the liquid phase is shown in Figure

156 Table 8-1. Stable solid phases in the Ti Al Cr system. Phase Prototype Pearson Symbol Space group Al 7 Cr V 7 Al 45 mc104 C2/m Al 11 Cr 2 oc584 Cmcm Al 4 Cr MnAl 4 hp574 P 6 3 /mmc α Al 9 Cr 4 Cr 4 Al 9 hr156 R 3m β Al 9 Cr 4 Cu 4 Al 9 ci5 I 43m α Al 8 Cr 5 Cr 5 Al 8 hr78 R 3m β Al 8 Cr 5 Cu 5 Zn 8 ci52 I 43m AlCr 2 MoSi 2 ti6 I4/mmm α TiCr 2 MgCu 2 cf24 F d 3m β TiCr 2 MgNi 2 hp24 P 6 3 /mmc γ TiCr 2 MgZn 2 hp12 P 6 3 /mmc β W ci2 Im 3m β 0 CsCl cp2 P m 3m α Mg hp2 P 6 3 /mmc α 2 Ti 3 Al Ni 3 Sn hp8 P 6 3 /mmc γ TiAl AuCu tp2 P 4/mmm ε TiAl 2 Ga 2 Hf ti24 I4 1 /amd η (Ti,Cr)Al 3 Al 3 Ti ti8 I4/mmm ζ Ti 5 Al 11 Al 3 Zr ti16 I4/mmm (Al) Cu cf4 F m 3m τ (Ti 1 x+y Cr x )Al 3 y AuCu 3 cp4 P m 3m γ Ti(Al,Cr) 2 MgZn 2 hp12 P 6 3 /mmc 156

157 Figure 8-1. Calculated binary Al Cr phase diagram using the description of Saunders [18]. Figure 8-2. Calculated binary Al Cr phase diagram from the description of Liang et al. [193]. 157

158 A B Figure 8-3. Comparison between the A) isothermal section at 1073 K calculated using the dataset of Saunders [18] and the B) assessed 1073 K isothermal section of Bochvar et al. [19]. A B Figure 8-4. Comparison between the A) isothermal section at 1273 K calculated using the dataset of Saunders [18] and the B) assessed 1273 K isothermal section of Bochvar et al. [19]. 158

159 A Figure 8-5. Comparison between the A) calculated [18]and B) assessed [19] liquidus surface. B 159

160 Figure 8-6. The end members of the ternary γ Ti(Al,Cr) 2 phase. The dashed line from Al:Ti to Cr:Ti schematically illustrates the influence of the 0 L γ Al,Cr:Ti parameter. 160

161 Figure K isothermal section calculated using the new description for the Ti Al Cr system. Experimental data of Jewett et al. [97, 177, 179] are superimposed. Figure K isothermal section calculated using the new description for the Ti Al Cr system. Experimental data of Jewett et al. [97, 177, 179] are superimposed. 161

162 Figure 8-9. Liquidus surface calculated using the new description of the Ti Al Cr system. Figure Partial liquidus surface calculated using the new description of the Ti Al Cr system. 162

163 Figure Scheil reaction scheme for the Ti Al Cr system. 163

164 CHAPTER 9 REVIEW OF THE TI-AL-MO SYSTEM 9.1 Binary Subsystems The Ti Al Mo system contains the Ti Al, Ti Mo, and Al Mo binary subsystems. The most recent Ti Al binary subsystems available in the literature were already discussed in Section Critical evaluations of the phase equilibria in the Ti Mo binary subsystem were performed by Murray [160] and Massalski [201]. These critical evaluations accept the presence of a miscibility gap in the bcc phase with a monotectoid reaction at 1123 K based on the metallographic investigation and electrical resistivity measurements of several Ti Mo alloys performed by Terauchi et al. [202]. The critically evaluated Ti Mo phase diagram of Massalski is shown in Figure 9-1. Thermodynamic descriptions for the Ti Mo system were originally developed by Saunders [18] and Shim et al. [203]. The description of Shim et al. [203] was updated by Chung et al. [204] to be able to calculate experimentally observed invariant equilibria in the Ti Mo N system. The assessment of Chung et al. [204] reproduces the miscibility gap in the bcc phase whereas the assessment of Saunders [18] does not take this miscibility gap into account (Figure 9-2). Although Terauchi et al. [202] observed a miscibility gap in the Ti Mo system, which was accepted in the evaluations of Murray [160], Massalski [201], and the assessment of Chung et al. [204], this miscibility gap could not be confirmed in the works of Morniroli and Gantonis [205] and Dupouy and Averbach [206]. Both works used X-ray diffraction to show the tendency for short range ordering, not phase separation, in the bcc solid solution. The results of the first principles calculations of Rubin and Fine [207] also question the presence of the miscibility gap in the bcc phase. Therefore, the Ti Mo description of Saunders [18] is recommended by SGTE. The Al Mo system was critically assessed by Brewer et al. [208], Saunders [209], and Schuster [210]. After the work of Saunders [209], Schuster and Ipser [211] published the 164

165 results of their experimental work on the Al Mo 3 Al 8 region, which were taken into account in the critical evaluation of the Al Mo system of Schuster [210]. The assessed Al Mo diagram from Schuster [210] is given in Figure 9-3 and the calculated Al Mo diagram from Saunders [18] is given in Figure 9-4. There are obvious differences between the diagrams. First, Saunders [18] models the δ AlMo 3 phase as a line compound whereas Schuster [210] shows that the δ AlMo 3 phase exists over a homogeneity range. Second, Schuster [210] shows an eutectic reaction between the liquid, δ AlMo 3, and AlMo phases whereas the dataset of Saunders calculates a peritectic reaction [18]. Last, Schuster [210] includes more phases in the Al Mo 3 Al 8 region than Saunders [18]. 9.2 Phases in the Ti Al Mo System The crystallographic data of the solid phases in the Ti Al Mo system are listed in Table 9-1. There is no information on the solubility of Ti in the the Al 12 Mo, Al 5 Mo, Al 22 Mo 5, Al 17 Mo 4, Al 4 Mo, and Al 63 Mo 37 phases of the Al Mo binary system, but there is some evidence in the literature that the Al 8 Mo 3 and δ AlMo 3 phases dissolve Ti. For example, the equilibrium β + Al 8 Mo 3 + δ AlMo 3 tie-triangle data for an alloy of composition Ti 52 at.%al 45 at.%mo, which was heat treated at 1673 K and quenched, showed that the Al 8 Mo 3 phase dissolved 1 at.%ti and the δ AlMo 3 phase dissolved 2.7 at.%ti [27]. Additionally, an alloy of composition Ti 24 at.%al 58 at.%mo, which was heat treated at 1198 K and quenched, contained only the δ AlMo 3 phase [212], meaning that the δ AlMo 3 phase extends to at least 18 at.%ti at 1198 K. The η (Ti,Mo)Al 3 phase, which is based on the binary TiAl 3 phase, exists over a large homogeneity range in the ternary Ti Al Mo system. Hansen and Raman [212] showed that alloys of composition Ti 64 at.%al 10 at.%mo, Ti 68 at.%al 16 at.%mo, and Ti 75 at.%al 12.5 at.%mo were all single phase η (Ti,Mo)Al 3 at 1198 K. The η (Ti,Mo)Al 3 phase was found to extend to Mo compositions as high as 20 at.%mo and Al compositions as low as 62 at.%al at 1198 K [212]. Eremenko et al. [213] confirmed the large homogeneity range of the η (Ti,Mo)Al 3 phase at 1573 K, and Abdel-Hamid [214] 165

166 found the η (Ti,Mo)Al 3 phase with compositions up to 9.04 at.%mo in slow-cooled Al-rich dilute melts. The large extension of the η (Ti,Mo)Al 3 phase occurs through substitution of Mo on both Al and Ti sites of the TiAl 3 structure [212]. The γ TiAl phase has been shown to dissolve some Mo. Das et al. [215] showed that at 1448 K, the γ TiAl phase contains 1 at.%mo at 52 at.%ti and 3 at.%mo at 48 at.%ti whereas Morris et al. [216] measured the Mo content of the γ TiAl phase in equilibrium with the α 2 Ti 3 Al and β 0 phases at 1173 K and 1473 K as 1 at.%mo at 51 at.%ti and 1.5 at.%mo at 53 at.%ti respectively. Kimura and Hashimoto [217] measured the Mo content of the γ TiAl phase in equilibrium with the α 2 Ti 3 Al and β 0 phases at 1473 K as 0.5 at.%mo at 50at.%Ti, which is much lower than that measured in the work of Das et al. [215]. However, Das et al. [215] heat treated their alloys for 120 hours whereas Kimura and Hashimoto [217] heat treated their alloys for only 4 hours. Therefore, the work of Das et al. [215] may more closely give the equilibrium composition of the γ TiAl phase at 1473 K. Although the maximum solubility of Mo in the α phase has not been measured, Singh and Banerjee [218] showed that an alloy of composition Ti 48 at.%al 2at.%Mo is in the single phase α field at 1673 K. The same authors suggested that the α phase field extends to an alloy of composition Ti 50 at.%al 6 at.%mo at 1673 K although this was not explicitly proved. The Ti-rich α single phase was also observed in an alloy Ti 2.5 at.%al 2.5 at.%mo which was heat treated at 1073 K for 222 hours and quenched [219]. The solubility of Mo in the α 2 Ti 3 Al phase is unclear. Although Banerjee et al. [220] published partial isothermal sections at 1673 K, 1573 K, and 1473 K showing the α 2 Ti 3 Al phase, the microstructures of the investigated alloys were interpreted taking into account a description of the binary Ti Al phase diagram from Margolin [221] which showed the peritectic formation of α 2 Ti 3 Al from the liquid and β phases at 1745 K. However, the critically assessed Ti Al phase diagram of Schuster and Palm [111] shows the peritectoid formation of the α 2 Ti 3 Al phase from the β and α phases at 1473 K. Therefore, it is 166

167 possible that Banerjee et al. [220] misidentified the α phase as the α 2 Ti 3 Al phase. In fact, the partial isothermal sections at 1448 K of Das et al. [100] and at 1573 K and 1473 K of Kimura and Hashimoto [217] show only the α phase. However, Morris et al. [216] claimed to also have the α 2 Ti 3 Al phase at 1473 K. The ordered β 0 phase was first found by Böhm and Löhberg [222] in alloys within the composition range from 60 wt.%mo to 30 wt.%mo at 50 at.%ti which were heat treated at 1073 K and then quenched. Since then, the β 0 phase has been found by many authors. Particularly, Chen et al. [223] found the β 0 phase in an alloy of composition Ti 50 at.%al 15 at.%mo at 1673 K, 1623 K, 1473 K, 1273 K, and 1073 K. Singh et al. [224] performed Rietveld refinement of X-ray and neutron diffraction data for an alloy of composition Ti 25 at.%al 25 at.%mo which was homogenized at 1273 K and furnace cooled. The alloy was found to contain the β 0 phase, and Rietveld refinement of the data showed that Ti atoms occupy A sites and Al and Mo atoms occupy the B sites in the CsCl structure. Excess Mo atoms can also occupy A sites with Ti atoms [224]. There is one ternary phase in the Ti Al Mo system. Hansen and Raman [212] found the ternary σ phase in the single phase alloy of composition Ti 41 at.%al 33 at.%mo at 1198 K. The σ phase was found to have a very small homogeneity range and the stoichiometry Ti 1.5 Al 2 Mo 1.5 was assigned. The presence of the ternary σ phase was confirmed in the work of Eremenko et al. [219]. The σ phase forms through a peritectoid reaction at 1523 K. 9.3 Review of Critical Assessments in the Region from 0 to 20 at.%ti Critical assessments of the phase equilibria in the Ti Al Mo system were performed by Budberg and Schmid-Fetzer [225] based on results published up to 1990, Tretyachenko [226] based on results published up to 2003, and Raghavan [227] based on results published up to The assessment of Tretyachenko [226] is an update to the earlier assessment of Budberg and Schmid-Fetzer [225] to take into account newer experimental work since Tretyachenko used the assessed Al Mo diagram from Schuster [210] 167

168 and the assessed Al Ti diagram from Schmid-Fetzer [228] as the constituent binaries for the assessment whereas Raghvan [227] used the Al Mo from Saunders [209], updated to include the experimental work of Schuster and Ipser [211] on the Al Mo 3 Al 8 region, and the Al Ti system from Raghavan [190]. Both Tretyachenko and Raghavan use the Mo Ti binary from Massalski [201]. Nino et al. [27] published the most comprehensive work on the phase equilibria in the Ti Al Mo system from 0 to 20 at.%ti. Several samples with Al contents ranging from 45 at.%al to 55 at.%al and Ti contents ranging from 3 at.%ti to 17 at.%ti were heat treated at 1773 K, 1723 K, and 1673 K and quenched. DTA was also performed on the alloys to measure solid state transformation temperatures. Based on the results of metallographic analysis and DTA, partial isothermal sections from 0 at.%ti to 20 at.%ti at 1673 K, 1573 K, 1540 K, 1473 K, and 1373 K, as well as an isopleth through 50 at.%al from 0 at.%ti to 20 at.%ti, were constructed. Since the assessment of Tretyachenko [226] was presumably performed before the publication of Nino et al. [27], Tretyachenko could not have taken this work into account in the assessment. Raghavan [190], however, presents isothermal sections at 1573 K, 1473 K, and 1373 K that take into account the work of Nino et al. [27]. One key conclusion of Nino et al. [27] is the continuity of the β phase at 1773 K from the Ti Mo binary to the Al Mo binary. Nino et al. confirmed the β phase continuity since an alloy of composition Ti 52 at.%al 45 at.%mo (which contains only 3 at.%ti and is therefore quite close to the Al Mo binary), showed a single phase β microstructure following heat treatment at 1773 K and quenching [27]. Since Tretyachenko does not include an isothermal section at 1773 K, no direct comparison can be made between the experimental results of Nino et al. [27] and the critical assessment of Tretyachenko [226]. However, Tretyachenko does include an isothermal section at 1873 K, which is shown in Figure 9-5. When the alloy of composition Ti 52 at.%al 45 at.%mo is superimposed on the 1873 K isothermal section, however, it 168

169 is located in the AlMo + liquid two phase field close to the AlMo single phase boundary. More apparent, however, is the fact that the AlMo and β phases do not form a continuous solid solution. Instead, the assessed 1873 K isothermal section shows a two phase AlMo + β field with a maximum solubility of Ti in AlMo at 3 at.%ti. One possible explanation for the discontinuity of the AlMo and β phases is that the AlMo phase is assigned the CsCl B2 crystal structure in the Al Mo assessment of Schuster [210], which was used by Tretyachenko [226]. However, in the original work of Rexer [229], in which the AlMo phase was discovered, the phase was assigned the W-A2 structure after analysis of the x-ray diffraction pattern. In fact, Schuster [210] gives no indication why the AlMo phase was assigned the CsCl B2 structure while the other Al Mo assessments of Saunders [209] and Brewer et al. [208] assign the W-A2 structure to the AlMo phase. Tretyachenko, however, does indicate the phase equilibria at 1873 K close to the Al Mo binary edge using dashed lines, which means that the phase equilibria in this region is unclear. Tretyachenko also included an assessed solidus surface which is shown in Figure 9-6. When the alloy of composition Ti 52 at.%al 45 at.%mo is superimposed on the solidus surface, it is located in the β + AlMo + Al 63 Mo 37 three phase field which is formed from the invariant reaction liquid + AlMo β + Al 63 Mo 37. According to the partial Scheil reaction scheme constructed by Tretyachenko [226], this invariant reaction should take place at 1823 K. Thus, combining the information available from the assessed 1873 K isothermal section, the information available from the assessed solidus surface, and the partial Scheil reaction scheme, the Ti 52 at.%al 45 at.%mo alloy would solidify as liquid liquid + AlMo liquid + AlMo + β β + AlMo + Al 63 Mo 37 and could never exist as single phase β. Therefore, the evaluation of Tretyachenko [226] is not in agreement with the experimental work of Nino et al. [27] for this alloy composition. Nino et al. [27] heat treated four alloys with Ti compositions ranging from 15 at.%ti to 17 at.%ti and Al compositions ranging from 47 at.%al to 50 at.%al at 1723 K and quenched the alloys to analyze the resulting microstructures. Metallographic investigation 169

170 of the alloys showed only single phase β in the microstructures. Since these alloys are inside the area of the β solidus in Figure 9-6 according to Tretyachenko [226], they would solidify as single phase β, which is in good agreement with this aspect of the work of Nino et al. [27]. As stated earlier, Ragahavan [227] took into account the experimental work of Nino et al. [27] in his assessment. Therefore, as only partial isothermal sections from 0 at.%ti to 30 at.%ti are presented at 1573 K, 1473 K, and 1373 K, there is no obvious contradiction with the work of Nino et al. [27]. Although Raghavan does not present any critically assessed liquidus surface, solidus surface, or higher temperature isothermal sections, he does mention that Nino et al. [27] have ruled out the possibility of the existence of two separate b.c.c phases in the composition range that was investigated. Based on the DTA of the several alloys investigated, Nino et al. [27] concluded that the invariant reaction β+ Al 8 Mo 3 δ + η should take place at 1540 K. According to the Scheil reaction scheme of Tretyachenko [226], this reaction occurs at 1598 K and Raghavan [227] accepted the invariant reaction temperature of Nino et al. [27] for his critical assessment. 9.4 Thermodynamic Descriptions of the Ti Al Mo System A thermodynamic description for the Ti Al Mo system is available from Saunders [18]. Calculations were performed with this dataset to assess its ability to predict the phase equilibria in the region from 0 at.%ti to 20 at.%ti that was investigated by Nino et al. [27]. In this dataset, the lattice stabilities for the pure elements were taken from SGTE [58] and the descriptions for all three constituent binaries were developed by Saunders. The calculated isothermal section at 1773 K is shown in Figure 9-7. The alloy of composition Ti 52 at.%al 45 at.%mo, which should be single phase β at 1773 K according to the work of Nino et al. [27], is in the liquid + β + Al 63 Mo 37 three phase field because Saunders [18] did not model the ternary β phase and the AlMo phase as a continuous solid solution. This would indicate that the description for the binary Al Mo 170

171 system should be re-optimized so that the β phase is continuous to AlMo compositions. For comparison, the partial isothermal sections from 0 at.%ti to 20 at.%ti from Nino et al. [27] at 1673 K and 1573 K are given in Figure 9-8, in which the β phase is shown to extend to 6 at.%ti at 1673 K and to 10 at.%ti at 1573 K. Additionally, the invariant reaction β+ Al 8 Mo 3 δ+η could not be calculated at any temperature. For such a transition reaction to take place, two three phase equilibria, β + Al 8 Mo 3 + δ and β + Al 8 Mo 3 + η must exist at temperatures above the invariant reaction temperature. The calculated isothermal section at 1540 K and the partial isothermal section at 1540 K from 0 at.%ti to 20 at.%ti from Nino et al. [27] are given in Figures 9-9 and 9-10 respectively. The partial isothermal section of Nino et al. [27] shows a β phase that extends to 14 at.%ti. The calculated isothermal section at 1540 K shows that while the β + Al 8 Mo 3 + δ three phase field is present, the β phase does not extend to low enough Ti compositions to form the β + η + Al 8 Mo 3 three phase equilibrium at any temperature. The liquidus surface along with the β solidus line calculated using the description of the Ti Al Mo system of Saunders is shown in Figure According to the calculations, the alloy of composition Ti 52 at.%al 45 at.%mo is in the primary crystallization field of the δ phase, which is in disagreement with the work of Nino et al. [27]. Additionally, although the four alloys with Ti compositions ranging from 15 at.%ti to 17 at.%ti and Al compositions ranging from 47 at.%al to 50 at.%al, which have been shown to solidify as single phase β at 1723 K by Nino et al. [27], are located in the primary crystallization area of the β phase, according to the calculations, these alloys do not solidify as single phase β because the β solidus does not extend to high enough Al compositions. The results of these preliminary calculations suggest the following points for re-optimization. These are: 1. Since there are large differences between the assessed Al Mo phase diagram of Schuster [210] and the calculated phase diagram of Saunders [18], the thermodynamic 171

172 parameters of the binary Al Mo system should be re-optimized after a critical assessment of the available literature. 2. The re-optimized Al Mo system should model the Mo-rich β phase and the AlMo phase as the same phase so that the ternary β phase is continuous from the Ti Mo boundary to the Al Mo boundary at 1773 K to be in accordance with the work of Nino et al. [27]. 3. The ternary interaction parameters for the β phase and the parameters for the η phase should be re-optimized so that the invariant equilibrium β+ Al 8 Mo 3 δ + η could be calculated as a stable one. 172

173 Figure 9-1. Assessed Ti Mo phase diagram according to Massalski [201]. Figure 9-2. Ti Mo phase diagram calculated using the dataset of Saunders [18]. 173

174 Figure 9-3. Assessed Al Mo phase diagram according to the Schuster [210]. Figure 9-4. Calculated Al Mo phase diagram from the description of Saunders [18]. 174

175 Table 9-1. Stable solid phases in the Ti Al Mo system. Phase Prototype Pearson Symbol Space group Al 12 Mo WAl 12 ci26 Im 3 Al 5 Mo(r) hp36 R 3c Al 5 Mo(h 1 ) hp60 P 3 Al 5 Mo(h 2 ) WAl 5 hp12 P 6 3 Al 22 Mo 5 of216 F dd2 Al 17 Mo 4 mc84 C2 Al 4 Mo WAl 4 mc30 Cm Al 3+x Mo 1-x WO 3 cp8 P m3n Al 3 Mo(h) Cr 3 Si mc32 Cm Al 8 Mo 3 mc22 c2/m Al 63 Mo 37 AlMo W ci2 Im 3m δ AlMo 3 Cr 3 Si cp8 P m 3n β W ci2 Im 3m β 0 CsCl cp2 P m 3m α Mg hp2 P 6 3 /mmc α 2 Ti 3 Al Ni 3 Sn hp8 P 6 3 /mmc γ TiAl AuCu tp2 P 4/mmc ε TiAl 2 Ga 2 Hf ti24 I4 1 /amd η (Ti,Mo)Al 3 Al 3 Ti ti8 I4/mmm ζ Ti 5 Al 11 Al 3 Zr ti16 I4/mmm (Al) Cu cf4 F m 3m σ Ti 1.5 Al 2 Mo 1.5 σ-crfe tp30 P 4 2 mnm 175

176 Figure 9-5. Assessed 1873 K isothermal section from the work of Tretyachenko et al. [226]. The alloy of composition Ti 52 at.%al 45 at.%mo, which has been shown to solidify as single phase β [27], is in the AlMo + liquid two phase field. Therefore, this alloy solidifies as the AlMo phase. 176

177 Figure 9-6. Assessed partial solidus in the Ti Al Mo system from the work of Tretyachenko et al. [226]. The alloy of composition Ti 52 at.%al 45 at.%mo, which should solidify as single phase β [27], is instead in the β + AlMo + Al 63 Mo 37 three phase field. The group 2 and group 3 alloys of Nino et al. [27] were shown to solidify as single phase β. 177

178 Figure 9-7. Calculated 1773 K isothermal section using the description of Saunders [18]. The alloy of composition Ti 52 at.%al 45 at.%mo, which should be single phase β according to the work of Nino et al. [27], is in the liquid + β + Al 63 Mo 37 three phase field. Figure 9-8. Phase equilibria in the region from 0 at.%ti to 20 at.%ti from Nino et al. [27]. The alloy of composition Ti 52 at.%al 45 at.%mo is indicated. 178

179 Figure 9-9. Calculated 1540 K isothermal sections using the description of Saunders [18]. Since the β phase does not extend to high enough Al compositions, the invariant reaction β+ Al 8 Mo 3 δ + η cannot be calculated. Figure Phase equilibria in the region from 0 at.%ti to 20 at.%ti from Nino et al. [27]. The invariant reaction β+ Al 8 Mo 3 δ + η, as well as the position of the β phase boundary at 1540 K, is indicated. 179