The Pennsylvania State University The Graduate School THERMODYNAMIC PROPERTIES OF SOLID SOLUTIONS FROM SPECIAL QUASIRANDOM STRUCTURES

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1 The Pennsylvania State University The Graduate School THERMODYNAMIC PROPERTIES OF SOLID SOLUTIONS FROM SPECIAL QUASIRANDOM STRUCTURES AND CALPHAD MODELING: APPLICATION TO AL CU MG SI AND HF SI O A Thesis in Materials Science and Engineering by Dongwon Shin c 2007 Dongwon Shin Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2007

2 The thesis of Dongwon Shin was reviewed and approved by the following: Zi-Kui Liu Professor of Materials Science and Engineering Thesis Co-Advisor Co-Chair of Committee Long-Qing Chen Professor of Materials Science and Engineering Thesis Co-Advisor Co-Chair of Committee Jorge O. Sofo Associate Professor of Physics Vincent H. Crespi Professor of Physics Gary L. Messing Distinguished Professor of Ceramic Science and Engineering Head of the Department of Materials Science and Engineering Signatures are on file in the Graduate School.

3 Abstract This thesis focuses on calculating thermodynamic properties of solid solution phases from first-principles studies for the CALPHAD thermodynamic modeling. Since thermodynamic properties of solid solutions cannot be determined accurately through experimental measurements various efforts have been made to estimate them from theoretical calculations. First-principles studies of Special Quasirandom Structures (SQS) deserve special attention among the available approaches. SQS s are structural templates whose correlation functions are very close to those of completely random solid solutions thus can be applied to any relevant system by switching the atomic numbers in first-principles calculations. Moreover the effect of local relaxation can be considered by fully relaxing the structure. In this thesis SQS s for both substitutional and interstitial solid solutions are considered. For substitutional solid solutions binary hcp SQS s and ternary fcc SQS s are generated. First-principles results of those SQS s are compared with experimental data and/or thermodynamic modelings where available and verified that they are capable of reproducing thermodynamic properties of substitutional binary hcp and ternary fcc solid solutions respectively. For interstitial solid solution binary hcp and bcc SQS s are generated by considering the mixing of vacancy and interstitial atoms while the atoms in the parental structures are considered as frozen. SQS s for substitutional solid solutions are applied to the Al-Cu-Mg-Si system with previously developed binary fcc and bcc SQS s to investigate the enthalpy of iii

4 mixing for binary bcc fcc and hcp solid solutions and ternary fcc solid solutions. Binary hcp and bcc SQS s for interstitial solid solutions are used to calculate enthalpy of mixing for α-hf (hcp) and β-hf (bcc) phases in the Hf-O system to be used in the thermodynamic modeling of the Hf-Si-O system. This thesis shows that first-principles studies of SQS s can provide insight into the understanding of mixing behavior for solid solution phases and calculated thermodynamic properties for example enthalpy of mixing can be readily used in thermodynamic modeling to overcome scarce and uncertain experimental data. iv

5 Table of Contents List of Figures List of Tables Acknowledgments x xv xviii Chapter 1 Introduction Phase diagram calculations Atomistic simulation Overview Chapter 2 Computational methodology Introduction CALPHAD approach Theoretical background Gibbs energy formalism Unary Binary Multicomponent Procedure of CALPHAD modeling Automation of CALPHAD First-principles calculations Density functional theory Ordered phase Disordered phase v

6 2.4 Conclusion References Chapter 3 Special quasirandom structures for substitutional binary solid solutions Introduction Correlation function First-Principles methodology Generation of special quasirandom structures Results and discussions Analysis of relaxed structures Radial distribution analysis Bond length analysis Enthalpy of mixing Cd-Mg Mg-Zr Al-Mg Mo-Ru IVA transition metal alloys Conclusion References Chapter 4 Special quasirandom structures for ternary fcc solid solutions Introduction Ternary interaction parameters Ternary fcc special quasirandom structures Correlation functions Generation of ternary SQS First-principles methodology Results and discussions Binary SQS s for the Ca-Sr-Yb system Ternary SQS s for the Ca-Sr-Yb system Conclusion References Chapter 5 Solid solution phases in the Al-Cu-Mg-Si system Introduction vi

7 5.2 Enthalpy of mixing for binary solid solutions Miedema s model Binary special quasirandom structures Ternary fcc solid solutions: Al-Cu-Mg Al-Cu-Si and Al-Mg-Si Conclusion References Chapter 6 Thermodynamic modeling of the Cu-Si system Introduction Review of previous work First-principles calculations Intermetallic compounds Solid solution phases Methodology Thermodynamic modeling Solution phases Ordered phases Results and discussions Conclusion References Chapter 7 Thermodynamic modeling of the Hf-Si-O system Introduction Experimental data Phase diagram data Hf-O Hf-Si-O Thermochemical data First-principles calculations Methodology Ordered phases Oxygen gas calculation Interstitial solid solution phases: from SQS Thermodynamic modeling Hf-O HCP and BCC Ionic liquid vii

8 Gas Polymorphs of HfO Si-O Hf-Si Hf-Si-O Results and discussion Conclusion References Chapter 8 Conclusion and future work Conclusion Future works Statistical analysis Sensitivity analysis of model parameters References Appendix A The input files used in Thermo-Calc 161 A.1 The Cu-Si system A.1.1 Setup file A.1.2 POP file A.1.3 EXP file A.1.4 TDB file A.2 The Hf-O system A.2.1 Setup file A.2.2 POP file A.2.3 EXP file A.2.4 TDB file A.3 The Hf-Si-O system A.3.1 Setup file A.3.2 POP file A.3.3 TDB file Appendix B Special quasirandom structures for the ternary fcc solution phase 234 B.1 A 1 B 1 C B.1.1 SQS B.1.2 SQS B.1.3 SQS viii

9 B.1.4 SQS B.1.5 SQS B.1.6 SQS B.1.7 SQS B.1.8 SQS B.2 A 2 B 1 C B.2.1 SQS B.2.2 SQS B.2.3 SQS B.2.4 SQS B.2.5 SQS B.2.6 SQS B.2.7 SQS ix

10 List of Figures 2.1 Ternary phase diagram showing three-phase equilibrium[2] Isothermal and isoplethal phase diagrams of the Hf-Si-O system at 1 atm[3] Heat capacity of aluminum from the SGTE pure element database[11] Gibbs energies of the individual phases of pure aluminum. The reference state is given as fcc phase at all temperatures Geometry of the Redlich-Kister type polynomial interaction parameters in the A-B binary. Arbitrary values J/mol have been given to all k L parameters Contribution to the total Gibbs energy (G) from mechanical mixing (G o m) ideal mixing ( G ideal m ) and excess energy of mixing ( G xs m ) in the A-B binary system Illustration describing the interaction of the different end-members within a two-sublattice model. Colon separates sublattices and comma separates interacting species The entire procedure of the CALPHAD approach from Kumar and Wollants [24] Two dimensional structures of A and B in their perfect square symmetry Two dimensional structures of A x B 1 x disordered phase with/without local relaxation Crystal structures of the A 1 x B x binary hcp SQS-16 structures in their ideal unrelaxed forms. All the atoms are at the ideal hcp sites even though both structures have the space group P Radial distribution analysis of Hf 50 Zr 50 SQS s. The dotted lines under the smoothed and fitted curves are the error between the two curves Radial distribution analysis of Cd 50 Mg 50 SQS s. The dotted lines under the smoothed and fitted curves are the error between the two curves x

11 3.6 Radial distribution analysis of Mg 50 Zr 50 SQS s. The dotted lines under the smoothed and fitted curves are the error between the two curves Calculated and experimental results of mixing enthalpy and lattice parameters for the Cd-Mg system Calculated enthalpy of mixing in the Mg-Zr system compared with a previous thermodynamic assessment[18]. Both reference states are the hcp structure Calculated and experimental results of mixing enthalpy and lattice parameters for the Al-Mg system Enthalpy of formation of the Mo-Ru system with both first principles and CALPHAD lattice stabilities. Reference states are bcc for Mo and hcp for Ru Enthalpy of mixing for the Hf-Ti Hf-Zr and Ti-Zr binary hcp solutions calculated from first-principles calculations and CALPHAD thermodynamic models. All the reference states are hcp structures Calculated DOS of Ti 1 x Zr x hcp solid solutions from (a) SQS and (b) CPA[38] Liquidus lines at various temperatures in the Al-Mg-Si system from the COST507 database[7]. Ternary interaction parameters for the liquid phase are L Al = T L Mg = T and L Si = T. Dotted lines represent the liquidus lines without ternary interaction parameters Arbitrary ternary interaction parameters are given in the fcc phase of the Al-Mg-Si system from the COST507 database[7] to see the impact of ternary parameters. Pure extrapolation from the binaries is the curve when L= Crystal structures of the ternary fcc SQS-N structures in their ideal unrelaxed forms. All the atoms are at the ideal fcc sites even though both structures have the space group P Calculated phase diagrams of three binaries in the Ca-Sr-Yb system. The interaction parameters for the bcc and fcc phases are evaluated identically. The evaluated thermodynamic parameters are listed in Table Enthalpy of mixing for the fcc phases in the binaries of the Ca-Sr- Yb system. Open and closed symbols represent symmetry preserved and fully relaxed calculations of SQS s respectively xi

12 4.6 Calculated enthalpy of mixing for the fcc phase in the Ca-Sr-Yb system with first-principles results of ternary SQS s. Solid lines are extrapolated result from the combined binaries from binary SQS s. Open and closed symbols represent symmetry preserved and fully relaxed calculations of SQS s respectively. Dashed and dotted lines represent the evaluated enthalpy of mixing with an identical ternary interaction parameter (L CaSrYb = J/mol) and three independent ternary interaction parameters (L Ca = L Sr = and L Yb = J/mol) respectively Radial distribution analysis of Ca 1 Sr 1 Yb 1 ternary fcc SQS s. The dotted lines under the smoothed and fitted curves are the error between the two curves Enthalpy of mixing for the solution phases in the Al-Cu system with first-principles calculations of binary SQS s (symbols) and previous thermodynamic modeling (solid lines)[4]. Open and closed symbols represent symmetry preserved and fully relaxed calculations of SQS s respectively Enthalpy of mixing for the solution phases in the Al-Mg system with first-principles calculations of binary SQS s (symbols) and previous thermodynamic modeling (solid lines)[5]. Open and closed symbols represent symmetry preserved and fully relaxed calculations of SQS s respectively Enthalpy of mixing for the solution phases in the Al-Si system with first-principles calculations of binary SQS s (symbols) and previous thermodynamic modeling (solid lines)[17]. Open and closed symbols represent symmetry preserved and fully relaxed calculations of SQS s respectively Enthalpy of mixing for the solution phases in the Cu-Mg system with first-principles calculations of binary SQS s (symbols) and previous thermodynamic modeling (solid lines)[3]. Open symbols represent symmetry preserved calculations of SQS s Enthalpy of mixing for the solution phases in the Mg-Si system with first-principles calculations of binary SQS s (symbols) and previous thermodynamic modeling (solid lines)[7]. Open symbols represent symmetry preserved calculations of SQS s The electronegativity vs the metallic radius for a coordination number of 12 (Darken-Gurry) map xii

13 5.7 Enthalpy of mixing for the fcc phase in the Cu-Mg-Si system from the COST507 database[3]. Reference states for all elements are the fcc phase Enthalpy of mixing for the fcc phase in the Al-Cu-Mg system from first-principles calculations of ternary SQS s. Solid lines are extrapolated result from the combined Al-Cu[4] Cu-Mg[24] and Mg-Al[5] databases. Dashed lines are from the COST507 database[3] Enthalpy of mixing for the fcc phase in the Al-Cu-Si system from first-principles calculations of ternary SQS s. Solid lines are extrapolated result from the combined Al-Cu[4] Cu-Si[6] and Si-Al[17] databases. Dashed lines are from the COST507 database[3] Enthalpy of mixing for the fcc phase in the Al-Mg-Si system from first-principles calculations of ternary SQS s. Solid lines are extrapolated result from the combined Al-Mg[5] Mg-Si (from binary SQS s) and Si-Al[17] databases. Dashed lines are from the COST507 database[3] Enthalpies of formation for the Cu-Si system from previous modelings[1 2]. Reference states for Cu and Si are fcc and diamond respectively Calculated enthalpy of formation of the Cu-Si system with firstprinciples calculation of ɛ-cu 15 Si 4. Reference states are fcc-cu and diamond-si Calculated enthalpies of mixing of the solution phases in the Cu-Si system with first-principles results. Open and closed symbols are symmetry preserved and fully relaxed calculations of SQS s respectively. Dashed lines are from previous thermodynamic modeling[2] Calculated phase diagram of the Cu-Si with experimental data[20 24] in the present work Proposed phase diagram of the Hf-O system from Massalski[18] Calculated Si-O phase diagram from Hallstedt[37] Calculated Hf-Si phase diagram from Zhao et al. [38] First-principles calculations results of hypothetical compounds (HfO 0.5 and HfO 3 ) and special quasirandom structures for α and β solid solutions with the evaluated results. Reference states for Hf of α and β solid solutions are given as hcp. Fully relaxed calculations of β solid solution have been excluded from this comparison since the calculation results completely lost their bcc symmetry Calculated lattice parameters of α-hf with experimental data[ ]. Scale for a-axis is left and for c is right xiii

14 7.6 Calculated Hf-rich side of the Hf-O phase diagram with experimental data from Domagala and Ruh[9] Calculated partial enthalpy of mixing of oxygen in the α-hf with experimental data[22] at 1323K Calculated Hf-O phase diagram Calculated HfO 2 -SiO 2 pseudo-binary phase diagram Calculated isothermal section of Hf-Si-O at (a) 500K and (b) 1000K at 1 atm. Tie lines are drawn inside the two phase regions. The vertical cross section between HfO 2 and Si is the isopleth in Figure Calculated isopleth of HfO 2 -Si at 1 atm. Hafnium dioxide is left and silicon is right. Polymorphs of HfO 2 monoclinic tetragonal and cubic are given in parentheses. The phases in the bracket are zero amount Enthalpy of mixing for the liquid phase in the Mg-Si system from two different modeling[1 2] with experimental data[3 4] Two different version of calculated phase diagrams for the Mg-Si system from different databases with experimental measurements[3 5 7]. The interaction parameters for the liquid phase in each database are listed inside the phase diagrams xiv

15 List of Tables 3.1 Structural descriptions of the SQS-N structures for the binary hcp solid solution. Lattice vectors and atomic positions are given in fractional coordinates of the hcp lattice. Atomic positions are given for the ideal unrelaxed hcp sites Pair and multi-site correlation functions of SQS-N structures when the c/a ratio is ideal. The number in the square bracket next to Π km is the number of equivalent figures at the same distance in the structure the so-called degeneracy factor Pair correlation functions up to the fifth shell and the calculated total energies of other 16 atoms sqs s for Cd 0.25 Mg 0.75 are enumerated to be compared with the one used in this work (SQS-16). The total energies are given in the unit ev/atom Results of radial distribution analysis for the seven binaries studied in this work. FWHM shows the averaged full width at half maximum and is given in Å. Errors indicate the difference in the number of atoms calculated through the sum of peak areas and those expected in each coordination shell First nearest-neighbors average bond lengths for the fully relaxed hcp SQS of the seven binaries studied in this work. Uncertainty corresponds to the standard deviation of the bond length distributions Pair and multi-site correlation functions of ternary fcc SQS-N structures when x A = x B = x C = 1. The number in the square bracket 3 next to Π km is the number of equivalent figures at the same distance in the structure the so-called degeneracy factor Pair and multi-site correlation functions of ternary fcc SQS-N structures when x A = 1 x 2 B = x C = 1. The number in the square bracket 4 next to Π km is the number of equivalent figures at the same distance in the structure the so-called degeneracy factor xv

16 4.3 Thermodynamic parameters of the binaries in the Ca-Sr-Yb system evaluated in this work (in S.I. units) Cohesive energies of selected bivalent metals Ca Sr and Yb from Ref. [18] First nearest-neighbor average bond lengths for the fully relaxed fcc SQS-8 of the three binaries in the Ca-Sr-Yb system. Uncertainty corresponds to the standard deviation of the bond length distributions Selected binary solid solution phases in the Al-Cu-Mg-Si system. Sublattice models are taken from previous thermodynamic modelings Coordination numbers of selected structures First-principles results of ɛ-cu 15 Si 4 and its Standard Element Reference (SER) fcc-cu and diamond-si. By definition H f of pure elements are zero Thermodynamic parameters for the Cu-Si system (all in S.I. units). Gibbs energies for pure elements are from the SGTE pure element database[25] First-principles calculation results of pure elements hypothetical compounds (α β-hf) and stable compounds (HfO 2 SiO 2 and HfSiO 4 ). By definition H f of pure elements are zero. Reference states for all the compounds are SER Structural descriptions of the SQS-N structures for the α solid solution. Lattice vectors and atom/vacancy positions are given in fractional coordinates of the supercell. Atomic positions are given for the ideal unrelaxed hcp sites. Translated Hf positions are not listed. Original Hf positions in the primitive cell are (0 0 0) and ( ) Structural descriptions of the SQS-N structures for the β solid solution. Lattice vectors and atom/vacancy positions are given in fractional coordinates of the supercell. Atomic positions are given for the ideal unrelaxed bcc sites. Translated Hf positions are not listed. The original Hf position in the primitive cell is (0 0 0) Pair and multi-site correlation functions of SQS-N structures for α solid solution when the c/a ratio is ideal. The number in the square bracket next to Π km is the number of equivalent figures at the same distance in the structure xvi

17 7.5 Pair and multi-site correlation functions of SQS-N structures for β solid solution. The number in the square bracket next to Π km is the number of equivalent figures at the same distance in the structure First-principles calculations results of α-hf special quasirandom structures. F R and SP represent Fully Relaxed and Symmetry Preserved respectively. Oxygen atoms are excluded for the symmetry check First-principles calculations results of β-hf special quasirandom structures. F R and SP represent Fully Relaxed and Symmetry Preserved respectively. Oxygen atoms are excluded for the symmetry check Thermodynamic parameters of the Hf-Si-O ternary system (in S.I. units). Gibbs energies for pure elements and gas phases are respectively from the SGTE pure elements database[44] and the SSUB database[36] xvii

18 Acknowledgments I would like to thank: My advisors Zi-Kui Liu and Long-Qing Chen for their advice and support throughout my days at Penn State. Special thanks go to Zi-Kui who showed me the way to be a good materials scientist via his famous TKC theory and ten pan cakes story. The committee members of my thesis Vincent Crespi and Jorge Sofo for their careful reading of my thesis. Phases Research Laboratory members especially Ray Bill Sara and Yu for their stimulating discussions about thermodynamics and basically all the other subjects. James Saal should be acknowledged for his patient proofreading of my thesis. All my good friends in MATSE department who showed me that State College is a great town to have lots of fun. My father who always encouraged me to be a scientist and patiently waited for my long journey. My wife Sanghee for everything else. without her support and love. This thesis could not be finished xviii

19 To my father... xix

20 Chapter 1 Introduction 1.1 Phase diagram calculations Phase diagrams depict the phase stability of an alloy with respect to various conditions e.g. temperature composition and sometimes pressure and are often considered as an initial roadmap in materials science to locate a condition to be or not to be based on the phases of interest. Metallic silicides for example are detrimental in growing a metal oxide on a silicon substrate as a gate oxide material due to their metallic conductivity which deteriorate its dielectric property as a thin film capacitor. Hence finding the optimum conditions such as temperature compositions of metals and silicon and oxygen partial pressure to fabricate a stable metal oxide/silicon interface is the highest priority in complementary metal-oxide semiconductor (CMOS) integrated circuit production. In principle an empirical phase diagram can be constructed by compiling the experimental phase equilibrium data measured in the dimensions such as temperature-composition and temperature-pressure. Unfortunately it is almost impossible to draw a reliable phase diagram solely from experiments since the range is too wide to be investigated. Exceptions can be made when a system is rather simple or has been studied extensively so that the accumulated data are sufficient for manual illustration. However most industrial alloys are multicomponent systems with a large number of phases and consequently there are many degrees of freedom in the phase diagram space. By conducting trial-anderror-scheme experiments of such multicomponent systems only a partial phase

21 2 diagram can be obtained far from a comprehensive understanding of the system. Furthermore conducting a series of experiments to synthesize phase stabilities of a system within a reasonable period of time is also very doubtful. Alternatively phase diagrams can be calculated from the Gibbs energies of individual phases in a system. The Gibbs energy is minimized when the conditions are fixed such as temperature and pressure. Then the area where a phase or phases have the lowest Gibbs energy can be obtained with respect to the given conditions. For example temperature-composition phase diagram can be obtained for a binary system. At any given temperature 1 the Gibbs energy is minimized with respect to the composition and the regions for homogeneous phase(s) can be calculated from the Gibbs energies at different temperatures. However minimizing the Gibbs energy in order to visualize the phase stability of a system become a daunting task as the number of elements in a system increases since the number of phases increases correspondingly. Thus it is inevitable to take advantage of computational thermodynamics for efficient and robust phase diagram calculations in multicomponent systems. Thermodynamic modeling using the CALPHAD (CALculation of PHAse Diagrams) method attempts to describe the Gibbs energies of individual phases of a system through empirical models whose parameters are evaluated using experimental information based on the crystal structures so-called sublattice model. From these thermodynamic descriptions phase diagrams other than compositiontemperature can be readily calculated. Furthermore the Gibbs energies of a higher-order system can be extrapolated from the lower-order systems and any new phases of the higher-order system can be introduced. The CALPHAD approach however is as good as the experimental data used to evaluate them and is therefore limited by the availability of accurate experimental data. There are two types of experimental data that can be used in CALPHAD modeling in order to evaluate Gibbs energies. One is thermochemical data and the other is phase diagram data. Thermochemical data such as enthalpy of formation enthalpy of mixing and activity are extremely useful in the parameter evaluation process since they can be directly derived from the Gibbs energy functions while phase diagram data such as liquidus solidus and invariant reactions gives only 1 Pressure is usually fixed as 1 atm.

22 3 indirect relationships between the phases are in that equilibrium. For example heat capacity one of the representative thermochemical data can be derived from the second derivative of the Gibbs energy with respect to temperature so that parameters for the Gibbs energy of a phase can be directly evaluated from heat capacity data. While a melting point the temperature where the liquid and solid phases are in equilibrium can be reproduced with any Gibbs energy curves for the liquid and solid phases as long as they are crossing each other at the temperature. In principle if one can measure enough thermochemical data of individual phases in a system for thermodynamic modeling and the measured data are absolutely precise then a state-of-the-art phase diagram can be readily calculated from the Gibbs energies evaluated from those measured data. Unfortunately thermochemical measurements cannot be measured accurately enough to be exclusively used in thermodynamic modeling without phase diagram data. Since most measurement methods such as calorimetry and Electromotive Force (EMF) are indirect the uncertainties those measurements are fairly large. Furthermore a number of phases in industrial alloys are quite significant so that even the least amount of needed thermochemical measurements for a thermodynamic modeling are enormous. Therefore it is almost unachievable to calculate a reliable phase diagram purely from thermochemical data due to the lack of quality and quantity of the data. On the other hand phase diagram data can be easily and accurately measured from experiments. For example once a composition is fixed then temperatures for phase transformations such as melting or solidification can be measured via thermal analysis equipment with high precision. However there are an infinite number of plausible solutions in the Gibbs energy functions which satisfy the relationship between the corresponding phases in the equilibrium. Therefore the calculated phase diagram of the system with the Gibbs energy functions evaluated only from the phase equilibria is superficially fine but there may be a substantial problem when it is extrapolated to its higher-order system. For example an incorrect thermodynamic description of an intermediate phase in a binary system will propagate an error to a ternary quaternary and higher-order system which uses the Gibbs energy functions of the binary system. When the problematic binary description is combined with other systems the extrapolated phase diagram

23 4 maybe completely incorrect however it cannot be noticed unless there are enough data in the higher-order system to prove that the extrapolated result is not trustworthy. It is also likely to happen that the Gibbs energy of any new phase in the higher-order system has to be evaluated improperly to satisfy the phase stability with the intermediate phase in the binary system. The characteristics of the two different kinds of experimental data thermochemical data and phase diagram data are complementary to each other in the CALPHAD approach. Thermochemical data are needed to investigate the thermodynamic characteristics of a phase for modeling. Phase diagram data are also needed to adjust the Gibbs energies of the phases in a system since the accuracy of thermochemical data are usually far from good enough to evaluate precise Gibbs energy functions for reliable phase diagram calculations. However it is not always feasible to have enough real experimental data for the thermodynamic modeling of a system. Alternatively data for a thermodynamic modeling can be obtained from theoretical calculations as well when experimental data are scarce. 1.2 Atomistic simulation In order to have a complete thermodynamic description of a phase throughout the entire composition range a model and whose parameters which precisely reproduces the thermodynamic characteristics of the phase is required. A thermodynamic model of a phase can be established based on the experimental observation and the parameters used in the model can be evaluated to minimize the error between the calculated values from the model and the raw experimental data. Thus the reliability of the thermodynamic model of a phase is highly sensitive to experimental information regarding the phase. Unfortunately it is not always possible to compile enough experimental results to have a reliable thermodynamic model for all the phases in a system. This limitation however can be overcome by using theoretical calculations such as ab initio calculations (also known as firstprinciples calculations) which are capable of predicting the physical properties of phases with no experimental input. Over the last couple of decades atomistic level simulations have become a reality thanks to the drastic development of computing technology. Such small scale

24 5 computer simulations for material science has made it possible to conduct virtual experiments of candidate materials for almost any solid state properties. Based on the periodic nature of solid phases only geometric information and the corresponding atom types of the structure are needed as inputs and such atomistic calculations are able to compute various properties for example formation energy interfacial energy activation energy and many more. Thermodynamic properties especially enthalpy of formation derived from the total energy calculation are valuable to CALPHAD modeling since they can provide phase stabilities at room temperature. 2 Despite the powerful ability of atomistic calculations to obtain thermodynamic properties of a phase these methods are not yet able to calculate the thermochemistry of materials especially multicomponent multiphase systems with the precision required in industry. In this regard it is interesting to notice the complementarity between virtual and real experiments: What is difficult to measure is easy to compute and vice-versa. 3 For example a phase boundary between two phases in the binary system can be easily and precisely measured via thermal analysis like DTA. However the uncertainty of the calculated phase boundary from the individually evaluated Gibbs energies of two phases is quite high. On the contrary the calculation of thermochemical properties such as the formation energy of a solid phase is straightforward within atomistic level calculations even though the phase is binary ternary or higher-order. Measuring reliable thermodynamic properties of a single solid phase from experiments is usually difficult. First obtaining a satisfactory purity for the single phase is demanding. Also such low temperatures where the solid phases are stable it is hard to reach thermodynamic equilibrium so that the measured values might be that of a non-equilibrium state. Furthermore it is sometimes necessary to have the thermodynamic description for the metastable even unstable phases within the CALPHAD approach; however this is completely beyond the ability of experiments. Thus experimental measurement and theoretical calculations are complementary to each other within the CALPHAD approach. 2 We can assume that H f 0K Hf K since there is almost no entropy effect at room temperature. 3 A. van de Walle Ph.D. thesis M.I.T. 2000

25 6 From calculated thermodynamic properties a good approximation of individual phases can be made when there is not enough experimental data available. Subsequently the parameters used in the model can be adjusted to satisfy the phase relationship based on the experimentally measured phase diagram data while still satisfying the thermochemical data of each phase. With this hybrid CALPHAD/first-principles calculation approach it is possible to construct a robust thermodynamic description of a system much more efficiently than with the conventional CALPHAD/experiment approach. 1.3 Overview In the present thesis a comprehensive discussion of the CALPHAD approach and supplementary first-principles calculations mainly focused for the thermodynamic modeling is presented. The organization of this thesis is as follows: In Chapter 2 computational methodology for the CALPHAD approach and first-principles calculations are discussed in detail. The theoretical background and current status of the CALPHAD approach are addressed in the chapter. Automation of the CALPHAD approach for those interested in developing a thermodynamic database is also presented. The latter half of this chapter is spent explaining how first-principles results can be correlated with the CALPHAD approach. The current limitations of first-principles calculations for the thermodynamic modeling are also considered. Chapter 3 mainly deals with the calculations of thermodynamic properties for binary solid solution phases from first-principles. Special quasirandom structures (SQS) specially designed ordered structures which mimic the atomic configuration of the completely random solid solution are introduced in this chapter. Generation of hcp SQS s calculation of generated structures within the first-principles methodology and the validation of calculated results with existing experimental data or previous calculations are given in the chapter. Special quasirandom structures for the ternary fcc phase have also been created and are critically evaluated in Chapter 4. The developed computational methodologies are applied to a conventional metallic Al-Cu-Mg-Si quaternary system in Chapters 5 and 6. In Chapter 7 the CALPHAD/first-principles approach has been applied

26 7 to the Hf-Si-O system which is important in CMOS (Ceramic Metal Oxide System) and Special Quasirandom Structures have been expanded to interstitial solid solution phases as well.

27 Chapter 2 Computational methodology 2.1 Introduction In this chapter the CALPHAD (CALculation of PHAse Diagram) approach and first-principles calculations used to construct thermodynamic descriptions of a system are introduced. The advantages of both methods as well as their current limitations are discussed in detail. 2.2 CALPHAD approach An equilibrium phase diagram can be treated as an initial road map which visualizes the stable phases of a system as a function of various conditions: temperature pressure and composition. From such phase diagrams one can easily determine an optimized condition to find a phase region with favorable phases or to avoid a phase region with detrimental phases especially for alloy design. Most currently available binary and ternary phase diagrams are manually illustrated from experimental measurements[1]. To measure phase diagram data experimentally DTA (Differential Thermal Analysis) for instance can be used to determine a phase boundary or x-ray diffraction for a phase region. Consequently the uncertainty of the phase diagram is highly dependent upon the amount of accumulated experimental data of a system and the precision of the measurements. Furthermore constructing a phase diagram exclusively from experiments

28 9 is inefficient in terms of cost and time and such experimental phase diagram determination has practical limitations as the number of components in a system increases. Solidus surface Liquidus surfaces L + α L L + α + β L + β β Solidus surface Solvus surface Solvus surface C B α α + β A Figure 2.1. Ternary phase diagram showing three-phase equilibrium[2]. Not only constructing a phase diagram from experiments but also visualizing a phase diagram is vague. Binary phase diagrams can be easily visualized in two dimensions as temperature-composition. In order to depict the compositions of the three elements in ternary systems one has to use a Gibbs triangle in two dimensions. To take temperature into consideration the third dimension has to be introduced in the phase diagram. Understanding a ternary phase diagram in three dimensions is complicated even for a simple system as shown in Figure 2.1. The

29 10 A-B-C ternary system shown in Figure 2.1 has only three phases: α β and liquid. It will be much harder of course to visualize a ternary system with more phases such as compounds from the individual binaries or even ternary compounds. It is usually convenient to plot such three dimensional ternary phase diagrams in two dimensions at a constant temperature or composition. They are respectively called isothermal and isoplethal sections and those of the Hf-Si-O are shown in Figure 2.2. As can be seen in the figures it is convenient to understand the phase stability with respect to the various conditions by slicing the planes in the three dimensional Hf-Si-O ternary phase diagram. However manual illustration of phase diagrams for multicomponent systems are almost impossible when the number of phases in a system is quite significant. Thus it is essential to take advantage of computational aid in depicting complex phase diagrams for multicomponent systems. Computer coupling of phase diagrams and thermochemistry the so-called CAL- PHAD methods makes it possible to easily calculate the equilibrium conditions of a complicated system based on the thermodynamic descriptions of individual phases. The goal of the CALPHAD method is to find mathematical expressions for the Gibbs energy of individual phases as a function of temperature composition and if possible pressure for all phases in a system. From those expressions the phase diagram or any kind of property diagram pertinent to the processing of a relevant system can be readily calculated by minimizing the Gibbs energy. The following is an introduction to basic thermodynamic principles on which the CALPHAD method is based. The Gibbs energy formalism and the characteristics of unary binary and multicomponent systems are presented. Thereafter the procedure and current problems of the CALPHAD approach are also presented Theoretical background Gibbs energy formalism By definition Gibbs energy consists of enthalpy and entropy terms as G = H T S (2.1) and the polynomial of Gibbs energy as a function of temperature is usually given

30 Mole Fraction Hf Gas +HfSiO4 +HfO Gas+ HfSiO4+Quartz HfO2+hcp +Hf3Si2 HfSiO4+HfO2 +HfSi2 Hf3Si2 +hcp HfSiO4 +Quartz+diamond Hf2Si+hcp HfO2 +HfSi+HfSi Mole Fraction Si Hf2Si+Hf3Si2+hcp HfSiO4 +HfSi2+diamond (a) Isothermal section of Hf-Si-O at 500K HfO2+Hf3Si2+Hf5Si4 HfO2+Hf5Si4+HfSi Gas Temperature K Gas+L1 Gas +L1+L2 Gas+L1+HfO2(c) Gas+L1+HfO2(t) L1+HfO2(m)+HfSiO4 HfSiO4+HfO2(m) L1+L2+HfO2(m) HfO2(m)+diamond[+L2] HfO2(m)+diamond[+HfSiO4] +HfSi2 +HfSi Mole Fraction Si Gas+L2 L1+L2 +HfO2(t) L1+HfSiO4 HfSiO4+diamond (b) Isoplethal section of HfO 2 -Si from Hf-Si-O L1+L2 L1+L2 +HfSiO4 Figure 2.2. Isothermal and isoplethal phase diagrams of the Hf-Si-O system at 1 atm[3]

31 12 as: G H SER = a + bt + ct ln T + dt 2 + et 3 + ft 1 (2.2) where a b c d e and f are fitting parameters. In CALPHAD the Gibbs energy of a compound or element is given relative to the stable phase of the elements at K/1atm. This is termed as the stable element reference (SER) by SGTE (Scientific Group Thermodata Europe). Then the entropy derived from the Gibbs energy in Eqn. 2.2 is: S = ( ) G = b c(1 + ln T ) 2dT 3eT 2 + ft 2 (2.3) T In the same manner the enthalpy can be derived as: H = G + T S = a ct dt 2 2eT 3 + 2fT 1 (2.4) Heat capacity at constant pressure C p is the ratio of the heat added to increase temperature: C p = ( ) H T Therefore the heat capacity derived from the Gibbs energy in Eqn. 2.2 is now: C p = ( ) H = T T (2.5) ( ) S = c 2dT 6eT 2 2fT 2 (2.6) T From Eqn. 2.6 the empirical heat capacity can be rewritten as the well-known Meyer-Kelly expression: C p = a + b T + c T 2 + d T 2 (2.7) where a b c and d are fitting parameters which can be evaluated from experimental measurement. In the following sections the principles of the thermodynamic modeling to describe the properties of each phase successfully in the unary binary and multicomponent systems are discussed.

32 FCC_A1 Heat Capacity J/mol-K Liquid Temperature K Figure 2.3. Heat capacity of aluminum from the SGTE pure element database[11] Unary Unary systems 1 are the basis for the modeling of binary and higher-order systems. If critical experimental data of a unary system become available and it cannot be reproduced with the existing model then the unary description has to be updated to include the new data. However due to the hierarchical and interconnected characteristics of thermodynamic modeling all model parameters in the thermodynamic descriptions that used the original set of unary parameters must be remodeled. Thus the Gibbs energy of a unary has to be very accurate to reproduce its physical properties correctly and at the same time it should be as simple as possible to be efficiently extrapolated to a higher-order system. There have been a lot of effort to model the unary system effectively and it had been discussed extensively at the Ringberg meeting in 1995[4 10]. The Gibbs energies of the stable and metastable phases for pure elements as a function of temperature and if possible pressure are compiled in the SGTE pure 1 Unary is not only confined to pure elements but also compounds.

33 14 element database[11]. In the SGTE pure element modeling the heat capacity of metastable phases are also defined to describe the Gibbs energies of all the phases throughout the entire temperature region. For the liquid phase below the melting temperature the heat capacity cannot be simply extrapolated linearly because for certain temperatures the liquid phase may have lower entropy than that of the solid phase. Extrapolation of the solid phase could have a similar problem where the solid phase might be stable again[5]. Thus the heat capacity of extrapolated phases are made to approach that of the stable phase forcing the Gibbs energy function to avoid such problems in the SGTE pure element modeling. As a result the heat capacity of the stable solid phase above the melting temperature is modeled to approach that of the liquid phase and vice-versa in SGTE pure elements modeling[12]. In order to achieve this purpose the SGTE model incorporated T 9 and T 7 terms in the solid and liquid phases in the Gibbs energy respectively. Also the heat capacity of the liquid phase has been modeled as a constant based on the heat capacity difference between the solid and liquid phase at the melting point. The mathematical expressions for the heat capacities and Gibbs energies for the solid and liquid phases within the SGTE method are given in following equations. ( T Cp s = Cp(T l ) + [Cp(T s m ) Cp(T l m )] { (G s m G l m) = [C s p(t m ) C l p(t m )] T m T T m 10 ) 10 (T > T m ) (2.8) [ ( ) ] 9 T + 1 T m Tm 90 } ( ) 6 T Cp l = Cp(T s ) + [Cp(T l m ) Cp(T s m )] (T < T m ) (2.9) T { m [ (G s m G l m) = [Cp(T s m ) Cp(T l m )] T ( ) ] } 7 m T T + 1 Tm 6 42 where C s p and C l p are heat capacities of solid and liquid phases and G s m and G l m are molar Gibbs energies of solid and liquid phases. Figure 2.3 shows the heat capacity for the fcc and liquid phases of the pure aluminum from the SGTE pure elements T m

34 15 database[11]. Above the melting temperature K the heat capacity of fcc goes to that of liquid and vice-versa. It should be noted that this SGTE method is not based on any physical observation. In order to yield reasonable Gibbs energy differences between stable and metastable phases the extrapolation of metastable phases is forced to obey certain rules. Therefore SGTE pure element modeling has to be revised as soon as a good physical model which is able to perform more realistic extrapolations becomes available. As discussed earlier the CALPHAD approach aims to describe the Gibbs energy throughout the entire composition range. This involves the extrapolation of the Gibbs energy of stable phases into regions where they are not stable. Consequently the relative Gibbs energies of the allotropic phases phases other than the stable one for the pure elements have to be included in the pure element data[13]. The structural difference in the molar Gibbs energy between the two phase is called lattice stability and is usually assumed to vary linearly with temperature[14]. Previously such structural energy differences have been systematically evaluated with relevant systems phase boundary data since the properties of non-equilibrium states cannot be measured experimentally[15 17]. The Gibbs energies of the individual phases of aluminum from the SGTE pure element database[11] are shown in Figure 2.4. Metastable phases such as bcc cub and hcp are also included with respect to the stable fcc phase Binary Binary is the most critical among the hierarchy of thermodynamic systems because binary interactions are dominant in a multicomponent system. There are three major types of condensed phases in the binary system: solution phases stoichiometric (line) compounds and compounds with a homogenous range. In the following the Gibbs energy formalisms of those phases are presented. For solution phases with one sublattice the substitutional solution model is normally used. The Gibbs energy formalism is expressed as: G m = G o m + G ideal mix + G xs mix (2.10)

35 16 10 BCC_A2 BCC_A12 5 CUB_A13 Gibbs Energy kj/mol 0-5 FCC_A1 HCP_A3 Liquid Temperature K Figure 2.4. Gibbs energies of the individual phases of pure aluminum. The reference state is given as fcc phase at all temperatures. G o m is the contribution of mechanical mixing from the pure elements A and B denoted by: G ideal mix G o m = x A G o A + x B G o B (2.11) is the contribution of the interaction between components. Assuming random mixing and discounting short-range order the Bragg-Williams approximation[18] can be used: G ideal mix = RT (x A ln x A + x B ln x B ) (2.12) The excess term G xs mix is used to characterize the deviation of the compound from ideal solution behavior. This expression is generally defined using a Redlich- Kister polynomial[19]:

36 Gibbs Energy kj/mol st order interaction parameters 2nd order interaction parameters 3rd order interaction parameters Total excess Gibbs energy Mole Fraction B Figure 2.5. Geometry of the Redlich-Kister type polynomial interaction parameters in the A-B binary. Arbitrary values J/mol have been given to all k L parameters. G xs mix = x A x B n k L AB (x A x B ) k (2.13) k=0 where k L AB is the k-th order interaction parameter and normally described as: k L AB = k a + k bt (2.14) where k a and k b are model parameters to be evaluated from experimental information. Contribution to the total Gibbs energy (G) from mechanical mixing ideal mixing and excess energy of mixing in the A-B binary system is shown in Figure 2.6. For stoichiometric compounds without homogeneity ranges the Gibbs energy can be expressed using the SER of the pure elements as follows:

37 o Gm o G B Contribution to G kj/mol o G A ideal G m xs G m G Mole Fraction B (a) Negative G xs m 10 8 xs G m 6 Contribution to G kj/mol o G A -2-4 G o Gm Miscibility Gap o G B -6 ideal G m Mole Fraction B (b) Positive G xs m Figure 2.6. Contribution to the total Gibbs energy (G) from mechanical mixing (G o m) ideal mixing ( G ideal m ) and excess energy of mixing ( G xs m ) in the A-B binary system.

38 19 o G AaB b m x A H SER A x B H SER B = a + bt + ct ln T + d i T i (2.15) where the coefficients a b c and d i are model parameters. The coefficients c and d i are related to the specific heat C p and are often not used as model parameters for compounds with no specific heat data. Assuming the Neumann-Kopp rule 2 holds the Gibbs energy can be expressed as: o G AaB b = x A o G SER A + x B o G SER B + a + bt (2.16) where o G SER i is the molar Gibbs energy of a pure element i for SER and a and b are the enthalpy and entropy of formation with respect to pure elements A and B for compound A a B b. For compounds with appreciable homogeneity ranges multiple sublattice models are used to describe such phases[20 23]. For example assume a compound A a B b exhibits a solubility range extending in both directions from the stoichiometric value. Assuming mixing of A in B sites and B in A sites this compound will be modeled using the two-sublattice model (A B) a (A B) b where the subscripts denote the number of sites in each sublattice. The same equation used to previously describe stoichiometric solution phases is used but will be expanded in terms of multiple sublattices. G o m is defined much the same way as Eqn. 2.11: G o m = y I Ay II A G o A:A + y I Ay II B G o A:B + y I By II A G o B:A + y I By II B G o B:B (2.17) describing the contribution from each end-member of the sublattice model where y I i and yi II are the site fractions of each element i respectively. o G i:j is the Gibbs energy of the compound where each sublattice is occupied by i and j respectively. The ideal mixing term G ideal mix is described as: G ideal mix = art (y I A ln y I A + y I B ln y I B) + brt (y II A ln y II A + y II B ln y II B ) (2.18) 2 The heat capacity of a compound is calculated as the simple sum of the C p s of the constituent elements at the same temperature.

39 20 Lastly the excess Gibbs energy term describes the interaction within each sublattice 3 : G xs mix = y I Ay I B +ya II yb II ( ( y II A k=0 ya I k=0 k L AB:A (y I A y I B) k + y II B k L A:AB (y II A y II B ) k + y I B ) k L AB:B (ya I yb) I k (2.19) k=0 k=0 ) k L B:AB (ya II yb II ) k The interaction between the different end-members is shown schematically in Figure Multicomponent The Gibbs energy formalism of ternaries quaternaries and other higher-order multicomponent systems are almost the same as that of binary but with more parameters due to the increased number of elements. For a multicomponent substitutional solution phase the mechanical mixing is denoted by: The ternary ideal mixing is: G o = x i G o i (2.20) G ideal mix = RT x i ln x i (2.21) The excess Gibbs energy consists of binary and ternary interactions where G xs mix = i x i x j I ij + }{{} j>i i from binary x i x j x k I ijk + (2.22) }{{} j>i k>j from ternary I ijk = x 0 i L i + x 1 jl j + x 2 kl k (2.23) 3 For the notation used a colon separates the different sublattices while a comma separates species within a specific sublattice.

40 21 o GBaA o G abb B:A k LB:AB B:B y I B k LAB:A k LAB:AB k LAB:B A:A y II B k LA:AB A:B Figure 2.7. Illustration describing the interaction of the different end-members within a two-sublattice model. Colon separates sublattices and comma separates interacting species. The higher order interactions are usually weak and thus omitted Procedure of CALPHAD modeling A schematic diagram of the CALPHAD procedure is shown in Figure 2.8 with the key steps summarized as follows: i. Acquiring data From a literature search or experimental results all relevant experimental and theoretical information of a designated system must be compiled. The data

41 A s N s 22 C o l l e c t i o n o f e x p e r i m e n t a l t h e r m o c h e m i c a l a n d p h a s e d i a g r a m d a t a D a t a f r o m o w n e x p e r i m e n t s D a t a e s t i m a t i o n u s i n g a b i n i t i o a n d s e m i e m p i r i c a l m e t h o d s a n d t r e n d a n a l y s i s C r i t i c a l e v a l u a t i o n o f t h e e x p e r i m e n t a l d a t a T h e r m o d y n a m i c m o d e l i n g o f p h a s e s A s s e m b l i n g c o m p a t i b l e t h e r m o d y n a m i c d e s c r i p t i o n s o f t h e l o w e r o r d e r s y s t e m s A s s e s s m e n t o r r e a s s e s s m e n t o f t h e l o w e r o r d e r s y s t e m E x p e r i m e n t a l d a t a f r o m h i g h e r o r d e r s y s t e m s S e l e c t i o n o f i n p u t d a t a a n d t h e i r a c c u r a c y W e i g h t i n g o f i n p u t d a t a O p t i m i z a t i o n o f t h e m o d e l p a r a m e t e r s b y w e i g h t e d n o n l i n e a r l e a s t s q u a r e r e g r e s s i o n M o d e l c a l c u l a t i o n a n d c o m p a r i s o n w i t h e x p e r i m e n t a l d a t a E x t r a p o l a t i o n t o h i g h e r o r d e r s y s t e m C o n s t r u c t i o n o f m o d e l p a r a m e t e r d a t a b a m k e w e x p e r i m e n t a l d a t a p p l i c a t i o n s Figure 2.8. The entire procedure of the CALPHAD approach from Kumar and Wollants [24] which can be used in the thermodynamic modeling falls into two categories: thermodynamic data and phase diagram data. Afterwards it is necessary to critically evaluate the compiled data. If one finds two conflicting datasets for the same property then either one or the other is correct or both are wrong. Both datasets cannot be correct at the same time 4. ii. Modeling of individual phases The models for individual phases of the system are based upon the characteristics of each phase by analyzing the collected data. A model should be able 4 Bo Jansson: from Thermo-Calc manual