... Science and Engineering. âSignatures are on file in the Graduate School. ...... [33] R. Arróyave, L. Kaufman, and T. W. Eagar. Thermodynamic modeling of.
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
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.
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
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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.
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Table of Contents
List of Figures
x
List of Tables
xv
Acknowledgments
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Chapter 1 Introduction 1.1 Phase diagram calculations . . . . . . . . . . . . . . . . . . . . . . . 1.2 Atomistic simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 Computational methodology 2.1 Introduction . . . . . . . . . . . . . . . . . 2.2 CALPHAD approach . . . . . . . . . . . . 2.2.1 Theoretical background . . . . . . . 2.2.1.1 Gibbs energy formalism . 2.2.1.2 Unary . . . . . . . . . . . 2.2.1.3 Binary . . . . . . . . . . . 2.2.1.4 Multicomponent . . . . . 2.2.2 Procedure of CALPHAD modeling 2.2.3 Automation of CALPHAD . . . . . 2.3 First-principles calculations . . . . . . . . 2.3.1 Density functional theory . . . . . 2.3.2 Ordered phase . . . . . . . . . . . . 2.3.3 Disordered phase . . . . . . . . . .
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2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 3 Special quasirandom structures for substitutional solutions 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 3.2 Correlation function . . . . . . . . . . . . . . . . . 3.3 First-Principles methodology . . . . . . . . . . . . . 3.4 Generation of special quasirandom structures . . . . 3.5 Results and discussions . . . . . . . . . . . . . . . . 3.5.1 Analysis of relaxed structures . . . . . . . . 3.5.2 Radial distribution analysis . . . . . . . . . 3.5.3 Bond length analysis . . . . . . . . . . . . . 3.5.4 Enthalpy of mixing . . . . . . . . . . . . . . 3.5.5 Cd-Mg . . . . . . . . . . . . . . . . . . . . . 3.5.6 Mg-Zr . . . . . . . . . . . . . . . . . . . . . 3.5.7 Al-Mg . . . . . . . . . . . . . . . . . . . . . 3.5.8 Mo-Ru . . . . . . . . . . . . . . . . . . . . . 3.5.9 IVA transition metal alloys . . . . . . . . . . 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 4 Special quasirandom structures for ternary fcc solid solutions 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Ternary interaction parameters . . . . . . . . . . . . . . . . . . . 4.3 Ternary fcc special quasirandom structures . . . . . . . . . . . . . 4.3.1 Correlation functions . . . . . . . . . . . . . . . . . . . . . 4.3.2 Generation of ternary SQS . . . . . . . . . . . . . . . . . . 4.4 First-principles methodology . . . . . . . . . . . . . . . . . . . . . 4.5 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Binary SQS’s for the Ca-Sr-Yb system . . . . . . . . . . . 4.5.2 Ternary SQS’s for the Ca-Sr-Yb system . . . . . . . . . . . 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 5 Solid solution phases in the Al-Cu-Mg-Si system 90 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 vi
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Enthalpy of mixing for binary solid solutions . 5.2.1 Miedema’s model . . . . . . . . . . . . 5.2.2 Binary special quasirandom structures 5.3 Ternary fcc solid solutions: Al-Cu-Mg, Al-Cu-Si, and Al-Mg-Si . . . . . . 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . Chapter 6 Thermodynamic modeling of the 6.1 Introduction . . . . . . . . . . . 6.2 Review of previous work . . . . 6.3 First-principles calculations . . 6.3.1 Intermetallic compounds 6.3.2 Solid solution phases . . 6.3.3 Methodology . . . . . . 6.4 Thermodynamic modeling . . . 6.4.1 Solution phases . . . . . 6.4.2 Ordered phases . . . . . 6.5 Results and discussions . . . . . 6.6 Conclusion . . . . . . . . . . . . References . . . . . . . . . . . . . . .
Cu-Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 7 Thermodynamic modeling of the Hf-Si-O system 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 7.2 Experimental data . . . . . . . . . . . . . . . . . . 7.2.1 Phase diagram data . . . . . . . . . . . . . . 7.2.1.1 Hf-O . . . . . . . . . . . . . . . . . 7.2.1.2 Hf-Si-O . . . . . . . . . . . . . . . 7.2.2 Thermochemical data . . . . . . . . . . . . . 7.3 First-principles calculations . . . . . . . . . . . . . 7.3.1 Methodology . . . . . . . . . . . . . . . . . 7.3.2 Ordered phases . . . . . . . . . . . . . . . . 7.3.3 Oxygen gas calculation . . . . . . . . . . . . 7.3.4 Interstitial solid solution phases: from SQS . 7.4 Thermodynamic modeling . . . . . . . . . . . . . . 7.4.1 Hf-O . . . . . . . . . . . . . . . . . . . . . . 7.4.1.1 HCP and BCC . . . . . . . . . . . 7.4.1.2 Ionic liquid . . . . . . . . . . . . . vii
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7.4.1.3 Gas . . . . 7.4.1.4 Polymorphs 7.4.2 Si-O . . . . . . . . . 7.4.3 Hf-Si . . . . . . . . . 7.4.4 Hf-Si-O . . . . . . . 7.5 Results and discussion . . . 7.6 Conclusion . . . . . . . . . . References . . . . . . . . . . . . . Chapter 8 Conclusion and future work 8.1 Conclusion . . . . . . . . . . 8.2 Future works . . . . . . . . 8.2.1 Statistical analysis . 8.2.2 Sensitivity analysis of References . . . . . . . . . . . . .
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Appendix A The input files used in Thermo-Calc 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 B.1 A1 B1 C1 . . . . . B.1.1 SQS-3 . . B.1.2 SQS-6 . . B.1.3 SQS-9 . .
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the ternary fcc solution phase234 . . . . . . . . . . . . . . . . . . . 234 . . . . . . . . . . . . . . . . . . . 234 . . . . . . . . . . . . . . . . . . . 234 . . . . . . . . . . . . . . . . . . . 234
B.1.4 SQS-15 . B.1.5 SQS-18 . B.1.6 SQS-24 . B.1.7 SQS-36 . B.1.8 SQS-48 . B.2 A2 B1 C1 . . . . B.2.1 SQS-4 . B.2.2 SQS-8 . B.2.3 SQS-16 . B.2.4 SQS-24 . B.2.5 SQS-32 . B.2.6 SQS-48 . B.2.7 SQS-64 .
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List of Figures 2.1 2.2 2.3 2.4 2.5
2.6
2.7
2.8 3.1 3.2 3.3
3.4
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]. 13 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, -50000 J/mol, have been given to all k L parameters. . . . . . . . . . . . . . . . . . . . . . . Contribution to the total Gibbs energy (G) from mechanical mixing xs (Gom ), ideal mixing (∆Gideal m ), and excess energy of mixing (∆Gm ) 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 Ax B1−x disordered phase with/without local relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crystal structures of the A1−x Bx 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, P1. . . . . Radial distribution analysis of Hf50 Zr50 SQS’s. The dotted lines under the smoothed and fitted curves are the error between the two curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Radial distribution analysis of Cd50 Mg50 SQS’s. The dotted lines under the smoothed and fitted curves are the error between the two curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Radial distribution analysis of Mg50 Zr50 SQS’s. The dotted lines under the smoothed and fitted curves are the error between the two curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Calculated and experimental results of mixing enthalpy and lattice parameters for the Cd-Mg system . . . . . . . . . . . . . . . . . . . 3.8 Calculated enthalpy of mixing in the Mg-Zr system compared with a previous thermodynamic assessment[18]. Both reference states are the hcp structure. . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Calculated and experimental results of mixing enthalpy and lattice parameters for the Al-Mg system . . . . . . . . . . . . . . . . . . . 3.10 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. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 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. 3.12 Calculated DOS of Ti1−x Zrx hcp solid solutions from (a) SQS and (b) CPA[38] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1
4.2
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Liquidus lines at various temperatures in the Al-Mg-Si system from the COST507 database[7]. Ternary interaction parameters for the liquid phase are LAl = +4125.86 − 0.51573T , LM g = −47961.64 + 5.9952T , and LSi = +25813.8 − 3.22672T . 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=0. . . . . . . . . . . . . . . . . . . . . . . . . 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, P1. . . . . . . . . . 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 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Enthalpy of mixing for the fcc phases in the binaries of the Ca-SrYb system. Open and closed symbols represent symmetry preserved and fully relaxed calculations of SQS’s, respectively. . . . . . . . . . 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 (LCaSrYb = 46652 J/mol) and three independent ternary interaction parameters (LCa = 10636, LSr = 98254, and LYb = 31062 J/mol), respectively. . . . . . . . . . . . . . . . . Radial distribution analysis of Ca1 Sr1 Yb1 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. . . . . . . . . . . . . .
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The electronegativity vs the metallic radius for a coordination number of 12 (Darken-Gurry) map. . . . . . . . . . . . . . . . . . . . 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 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]. . . . 5.9 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]. . . . 5.10 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]. . . . . . . . . . . . . . . . . . . . . . . . . 6.1 6.2
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. 106
. 107
Enthalpies of formation for the Cu-Si system from previous modelings[1, 2]. Reference states for Cu and Si are fcc and diamond, respectively. 115 Calculated enthalpy of formation of the Cu-Si system with firstprinciples calculation of -Cu15 Si4 . Reference states are fcc-Cu and diamond-Si. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 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]. 120 Calculated phase diagram of the Cu-Si with experimental data[20– 24] in the present work. . . . . . . . . . . . . . . . . . . . . . . . . . 121 Proposed phase diagram of the Hf-O system from Massalski[18]. . . 127 Calculated Si-O phase diagram from Hallstedt[37]. . . . . . . . . . . 140 Calculated Hf-Si phase diagram from Zhao et al. [38]. . . . . . . . . 141 First-principles calculations results of hypothetical compounds (HfO0.5 and HfO3 ) 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. . . . . . . . 142 xiii
7.5
Calculated lattice parameters of α-Hf with experimental data[8, 9, 24, 40]. Scale for a-axis is left and for c is right. . . . . . . . . . . 7.6 Calculated Hf-rich side of the Hf-O phase diagram with experimental data from Domagala and Ruh[9]. . . . . . . . . . . . . . . . . 7.7 Calculated partial enthalpy of mixing of oxygen in the α-Hf with experimental data[22] at 1323K. . . . . . . . . . . . . . . . . . . . 7.8 Calculated Hf-O phase diagram. . . . . . . . . . . . . . . . . . . . 7.9 Calculated HfO2 -SiO2 pseudo-binary phase diagram. . . . . . . . 7.10 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 HfO2 and Si is the isopleth in Figure 7.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Calculated isopleth of HfO2 -Si at 1 atm. Hafnium dioxide is left and silicon is right. Polymorphs of HfO2 , monoclinic, tetragonal, and cubic, are given in parentheses. The phases in the bracket are zero amount. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 8.2
. 143 . 144 . 145 . 146 . 147
. 148
. 149
Enthalpy of mixing for the liquid phase in the Mg-Si system from two different modeling[1, 2] with experimental data[3, 4]. . . . . . . 158 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. . . . . . . . . . . . . . . . . . 159
xiv
List of Tables 3.1
3.2
3.3
3.4
3.5
4.1
4.2
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 Πk,m 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 Cd0.25 Mg0.75 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 ˚ A. 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
41
.
42
.
44
.
51
.
52
Pair and multi-site correlation functions of ternary fcc SQS-N structures when xA = xB = xC = 13 . The number in the square bracket next to Πk,m 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 xA = 21 , xB = xC = 14 . The number in the square bracket next to Πk,m is the number of equivalent figures at the same distance in the structure, the so-called degeneracy factor. . . .
xv
76
77
4.3 4.4 4.5
5.1 5.2 6.1
6.2
7.1
7.2
7.3
7.4
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81 81
83
Selected binary solid solution phases in the Al-Cu-Mg-Si system. Sublattice models are taken from previous thermodynamic modelings. 93 Coordination numbers of selected structures. . . . . . . . . . . . . . 102 First-principles results of -Cu15 Si4 and its Standard Element Reference (SER), fcc-Cu and diamond-Si. By definition, ∆Hf of pure elements are zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 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]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 First-principles calculation results of pure elements, hypothetical compounds (α, β-Hf), and stable compounds (HfO2 , SiO2 , and HfSiO4 ). By definition, ∆Hf 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 ( 23 1 1 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 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 Πk,m is the number of equivalent figures at the same distance in the structure. . . . . . . . . . . . . . . . . . . . . . . .
xvi
. 130
. 133
. 134
. 135
7.5
7.6
7.7
7.8
Pair and multi-site correlation functions of SQS-N structures for β solid solution. The number in the square bracket next to Πk,m is the number of equivalent figures at the same distance in the structure. 135 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 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]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
xvii
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. This thesis could not be finished without her support and love.
xviii
To my father...
xix
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
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.
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
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
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. f f We can assume that ∆H0K ' ∆H298.15K since there is almost no entropy effect at room temperature. 3 A. van de Walle, Ph.D. thesis, M.I.T., 2000 2
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
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.
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
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.
Liquidus surfaces L+β L+α Solidus surface
Solidus surface
L L+α+β
β Solvus surface
Solvus surface
B
C α+β α
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
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 CALPHAD 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.
2.2.1
Theoretical background
2.2.1.1
Gibbs energy formalism
By definition, Gibbs energy consists of enthalpy and entropy terms as G = H − TS
(2.1)
and the polynomial of Gibbs energy as a function of temperature is usually given
11
1.0
Hf2Si+hcp
0.9
Mo
le F
rac
tio
n,
Hf
0.8
0 0
Hf2Si+Hf3Si2+hcp Hf3Si2 +hcp
0.6
0.5
0.3 Gas +HfSiO4 +HfO2
HfO2+Hf3Si2+Hf5Si4 HfO2+Hf5Si4+HfSi
HfO2+hcp +Hf3Si2
HfO2 +HfSi+HfSi2
0.4
0.2 0.1
0.7
HfSiO4+HfO2 +HfSi2 HfSiO4 +HfSi2+diamond
Gas+ HfSiO4 HfSiO4+Quartz +Quartz+diamond
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
(a) Isothermal section of Hf-Si-O at 500K
5000 Gas
4500 4000
Gas+L1 Gas +L1+L2
Temperature, K
3500 3000
Gas+L2 L1+L2 +HfO2(t)
Gas+L1+HfO2(c)
2500
Gas+L1+HfO2(t)
2000
L1+L2+HfO2(m)
L1+HfO2(m)+HfSiO4
1500
L1+L2 L1+HfSiO4
HfO2(m)+diamond[+L2]
1000
L1+L2 +HfSiO4
HfO2(m)+diamond[+HfSiO4] 543.53
500
HfSiO4+HfO2(m) +HfSi2
0 0
HfSiO4+diamond +HfSi2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
(b) Isoplethal section of HfO2 -Si from Hf-Si-O
Figure 2.2. atm[3].
Isothermal and isoplethal phase diagrams of the Hf-Si-O system at 1
12 as: G − H SER = a + bT + cT ln T + dT 2 + eT 3 + f T −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 298.15K/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 ∂T
= −b − c(1 + ln T ) − 2dT − 3eT 2 + f T −2
(2.3)
In the same manner, the enthalpy can be derived as: H = G + T S = a − cT − dT 2 − 2eT 3 + 2f T −1
(2.4)
Heat capacity at constant pressure, Cp , is the ratio of the heat added to increase temperature: Cp =
∂H ∂T
(2.5)
Therefore, the heat capacity derived from the Gibbs energy in Eqn. 2.2 is now: Cp =
∂H ∂T
=T
∂S ∂T
= −c − 2dT − 6eT 2 − 2f T −2
(2.6)
From Eqn. 2.6, the empirical heat capacity can be rewritten as the well-known Meyer-Kelly expression: Cp = a0 + b0 T + c0 T 2 + d0 T −2
(2.7)
where, a0 , b0 , c0 , and d0 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.
13
34
Heat Capacity, J/mol-K
32
FCC_A1
Liquid
30
28
26
24 0
500
1000 1500 Temperature, K
2000
Figure 2.3. Heat capacity of aluminum from the SGTE pure element database[11].
2.2.1.2
Unary
Unary systems1 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.
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 T7 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.
Cps
=
Cpl (T )
+
[Cps (Tm )
−
Cpl (Tm )]
T Tm
−10 (T > Tm )
" −9 # T T − Tm Tm − + 1− · 10 Tm 90
( ∆(Gsm − Glm ) = [Cps (Tm ) − Cpl (Tm )] ·
Cpl
=
Cps (T )
+
[Cpl (Tm )
−
Cps (Tm )] (
∆(Gsm
−
Glm )
=
[Cps (Tm )
−
Cpl (Tm )]
·
T Tm
(2.8) )
6 (T < Tm ) "
Tm − T − + 1− 6
T Tm
7 # ·
Tm 42
(2.9) )
where Cps and Cpl are heat capacities of solid and liquid phases, and Gsm and Glm 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
15 database[11]. Above the melting temperature, 933.47K, 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. 2.2.1.3
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: xs Gm = Gom + ∆Gideal mix + ∆Gmix
(2.10)
16
10 BCC_A2, BCC_A12
Gibbs Energy, kJ/mol
5
CUB_A13 HCP_A3
0
FCC_A1
-5
Liquid
-10
-15 0
500
1000 1500 Temperature, K
2000
Figure 2.4. Gibbs energies of the individual phases of pure aluminum. The reference state is given as fcc phase at all temperatures.
Gom is the contribution of mechanical mixing from the pure elements A and B, denoted by: Gom = xA GoA + xB GoB
(2.11)
∆Gideal mix is the contribution of the interaction between components. Assuming random mixing and discounting short-range order, the Bragg-Williams approximation[18] can be used: ∆Gideal mix = RT (xA ln xA + xB ln xB )
(2.12)
The excess term ∆Gxs mix , is used to characterize the deviation of the compound from ideal solution behavior. This expression is generally defined using a RedlichKister polynomial[19]:
17
5
Gibbs Energy, kJ/mol
0
-5
-10 1st order interaction parameters 2 nd order interaction parameters 3rd order interaction parameters Total excess Gibbs energy
-15 0
0.2
0.4 0.6 Mole Fraction, B
0.8
1.0
Figure 2.5. Geometry of the Redlich-Kister type polynomial interaction parameters in the A-B binary. Arbitrary values, -50000 J/mol, have been given to all k L parameters.
∆Gxs mix
= xA xB
n X
k
LA,B (xA − xB )k
(2.13)
k=0
where k LA,B is the k-th order interaction parameter and normally described as: k
LA,B = 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:
18
5
G
Contribution to G, kJ/mol
0
G oB
o m
G oA
-5
∆G ideal m
-10
∆G xsm
-15
G -20 0
0.2
0.4 0.6 Mole Fraction, B
0.8
1.0
(a) Negative Gxs m
10
∆G xsm
Contribution to G, kJ/mol
8 6 4
G oB
2
G
0
Gom
G oA
-2
Miscibility Gap
-4
∆G ideal m
-6 -8 0
0.2
0.4 0.6 Mole Fraction, B
0.8
1.0
(b) Positive Gxs m
Figure 2.6. Contribution to the total Gibbs energy (G) from mechanical mixing (Gom ), xs ideal mixing (∆Gideal m ), and excess energy of mixing (∆Gm ) in the A-B binary system.
19
o
a Bb − xA HASER − xB HBSER = a + bT + cT ln T + GA m
X
di T i
(2.15)
where the coefficients a, b, c, and di are model parameters. The coefficients c and di are related to the specific heat, Cp , and are often not used as model parameters for compounds with no specific heat data. Assuming the Neumann-Kopp rule2 holds, the Gibbs energy can be expressed as: o
+ xB o GSER + a + bT GAa Bb = xA o GSER A B
(2.16)
is the molar Gibbs energy of a pure element i for SER, and a and b where o GSER i are the enthalpy and entropy of formation with respect to pure elements A and B for compound Aa Bb . For compounds with appreciable homogeneity ranges, multiple sublattice models are used to describe such phases[20–23]. For example, assume a compound Aa Bb 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. Gom is defined much the same way as Eqn. 2.11: Gom = yAI yAII GoA:A + yAI yBII GoA:B + yBI yAII GoB:A + yBI yBII GoB:B
(2.17)
describing the contribution from each end-member of the sublattice model, where yiI and yiII are the site fractions of each element i, respectively. o Gi:j is the Gibbs energy of the compound where each sublattice is occupied by i and j, respectively. The ideal mixing term, ∆Gideal mix , is described as:
I I I I II II II II ∆Gideal mix = aRT (yA ln yA + yB ln yB ) + bRT (yA ln yA + yB ln yB ) 2
(2.18)
The heat capacity of a compound is calculated as the simple sum of the Cp ’s of the constituent elements at the same temperature.
20 Lastly, the excess Gibbs energy term describes the interaction within each sublattice3 :
! I I ∆Gxs mix = yA yB
yAII
X
k
LA,B:A (yAI − yBI )k + yBII
X
k
LA,B:B (yAI − yBI )k
(2.19)
k=0
k=0
! +yAII yBII
yAI
X
k
LA:A,B (yAII − yBII )k + yBI
X
k
LB:A,B (yAII − yBII )k
k=0
k=0
The interaction between the different end-members is shown schematically in Figure 2.7. 2.2.1.4
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: Go =
X
xi Goi
(2.20)
X
(2.21)
The ternary ideal mixing is: ∆Gideal mix = RT
xi ln xi
The excess Gibbs energy consists of binary and ternary interactions ∆Gxs mix =
XX i
j>i
xxI + | i {zj ij}
from binary
XXX i
j>i k>j
x x x I +··· | i j{zk ijk}
(2.22)
from ternary
where Iijk = x0i Li + x1j Lj + x2k Lk 3
(2.23)
For the notation used, a colon separates the different sublattices, while a comma separates species within a specific sublattice.
21
o
o
GBaA
G
aBb
B:A
B:B k
y IB
k
k
LA,B:A k
A:A
y IIB
LB:A,B
LA,B:A,B
k
LA,B:B
LA:A,B
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.
2.2.2
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
22
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The entire procedure of the CALPHAD approach from Kumar and
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 time4 . 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
23 to accommodate potential extrapolation to the higher-order system. iii. Weighting experimental data In thermodynamic modeling, thermodynamic data are preferred since they are typically for single phases. However, the uncertainty of thermodynamic data is usually large because most experimental measurements are indirect methods. Phase diagram data give the relative phase stabilities involving more than one phase, but the accuracy of such measurements are higher. Thus, different weights need to be given to the data according to the relative importance and accuracy of the experimental measurements or theoretical calculations. However, this process is rather time consuming and subjective to the modeler. iv. Model parameters evaluation The model parameters then need to be evaluated to reproduce all the accepted thermochemical and phase diagram data. A model can only be considered valid if it can reproduce all of the accepted data accurately. Otherwise a modeler has to come up with a better model. Thereafter, the new model parameters have to be evaluated with the same experimental datasets.
2.2.3
Automation of CALPHAD
The procedure of the CALPHAD approach has some space for further improvement. CALPHAD modeling has been done exclusively by experienced computational thermodynamic experts. Even for them, updating an existing database is a daunting task since it is time consuming and repetitive work. The current issues of the CALPHAD approach are summarized as follows: i) Understanding the phase stability of a system is not only the aim of the CALPHAD community but also an initial task for experimentalists or those in industry. However, it is not always possible to have a complete thermodynamic modeling of a designated alloy system. Since CALPHAD modeling requires a certain dexterity which takes some time to obtain, if a system has not been thermodynamically modeled, then those without experience have two choices: wait until the thermodynamic description of a system becomes available or
24 start a series of experiments to investigate the phase stability. Neither can be done within a reasonable amount of time. ii) A thermodynamic database has to be updated whenever critical experimental data which cannot be reproduced with the existing model become available. A model could be modified based on the new data, however, the parameter evaluation procedure would be similar to what has been done before. Since all the previously accepted data should still be used, there will be only a slight change in the entire dataset. If the dataset alters a lower-order system, then all higher-order systems containing this lower-order system have to be updated as well. iii) All the experimental data used in the parameter evaluation process will perish after a system has been modeled. The evaluated parameters of individual phases are published in a journal article, but the data used in the thermodynamic modeling are shown to be compared with the calculated result. Therefore, those who want to update the existing database has to collect all the data used to develop the database again. In the present thesis, CALPHAD modeling has been automated in order to resolve the problems addressed above. All the experimental and theoretical information of a phase and its thermodynamic model and corresponding model parameters are stored in an XML (eXtensible Markup Language) database. A user can add new data to the existing data and interactively select datasets to be used in the parameter evaluation process. If there are enough data for thermodynamic modeling, then unary, binary, and multicomponent systems can be modeled with minimal user interruption. Repetitive parameter evaluation process can be done automatically so that even experts in the CALPHAD community can benefit from using this tool. Furthermore, all the experts’ strategies of CALPHAD modeling have been implemented in the automation tool, so that those who are less experienced with computational thermodynamics can develop a thermodynamic database of their own. Therefore, this automation tool is useful for saving time developing a robust thermodynamic model and improving the quality of the model easily.
25
2.3
First-principles calculations
More often than not, available data are insufficient for robust thermodynamic modeling of a system. As discussed earlier in this chapter, thermodynamic data is preferred in a thermodynamic modeling and data of a liquid phase is relatively easy to get while data of a solid phase is not. Thus, thermodynamic properties for the liquid phase, such as enthalpy of mixing, partial Gibbs energy, and activity, can be obtained and modeled to reproduce those thermodynamic properties quite satisfactorily. For a solid phase, however, measurement of the thermodynamic properties are usually not accurate, even for binary intermediate phases. Consequently, thermodynamic parameters for solid phases are sometimes evaluated with insufficient thermodynamic data. However, phase diagram information can be reproduced even with completely wrong enthalpy and entropy of formation values as long as the relation among the Gibbs energies of the phases in equilibrium is correct. Thus, correct thermodynamic properties of solid phases are valuable to constrain the thermodynamic model. Furthermore, it is required to have energies for metastable or even unstable phases for phase diagram construction within the CALPHAD method; however, such values cannot be accurately measured experimentally. Recent progress in high performance computing has made the atomistic simulations of crystalline solids based on quantum mechanics possible [25]. With only the input of atomic number and corresponding structural data, various properties, including thermodynamic properties can be calculated within the atomistic simulations. In principle, a phase diagram can be constructed completely from first-principles calculations; however, this method has not reached the level of industrial applications yet. Regardless, first-principles electronic structure calculations can be used to support CALPHAD modeling, e.g., enthalpy of formation of intermetallic compounds as well as enthalpy of mixing of solid solutions[26–28]. Although first-principles results cannot be used as they are in the CALPHAD approach to construct an accurate phase diagram, they can be good starting points for optimization. Also thermodynamic characteristics of a solid phase can be analyzed from first-principles study. In the following, the Density Functional Theory (DFT), upon which first-principles calculations are based, is briefly introduced.
26
2.3.1
Density functional theory
Solid phases with an ideal crystal structure are defined by indefinitely repeating unit cells via translational symmetry. Even with these periodic boundary conditions, the multi-electron Schr¨odinger equation is too complex to solve with modern computing resources.5 ˆ 1 , R2 · · · , RN )Ψ(r1 , r2 , · · · , rn ) = ET Ψ(r1 , r2 , · · · , rn ) H(R
(2.24)
ˆ is the where Rj ’s are the ionic coordinates, ri ’s are the electronic coordinates, H Hamiltonian operator, ET is the total energy of the electrons, and Ψ is the wavefunction. Hohenberg and Kohn[29, 30] have shown that the total energy of the electrons in an external potential (ions) is a functional of the charge density, ρ(~r). ET = E[ρ(~r)]
(2.25)
First-principles calculations are based on the Density Functional Theory (DFT) [29, 30]. According to DFT, the total energy of a system can be uniquely defined by the electron charge density, ρ(~r). The original many-electron Schr¨odinger’s equation is then converted into a set of Kohn-Sham equations, one for each electron in the system:
"
# Z N e2 ~2 2 e2 X ZI ρ(~r0 ) 3 0 + − ∇ − d ~r + VXC [ρ(~r)] ψ(~r) = εi ψi (~r) ~ I | 4πε0 2me i 4πεo I=1 |~r − R |~r − ~r0 | (2.26) The exchange correlation potential VXC [ρ(~r)] is given by the functional deriva-
tive: VXC [ρ(~r)] =
δ EXC [ρ(~r)] δρ(~r)
(2.27)
The exact form of the exchange correlation energy EXC [ρ(~r)] is unknown. The most widely used approximation is the Local Density Approximation (LDA)[31], which assumes that the exchange correlation is only a functional of the local density 5 An exact solution to the Schr¨odinger equation is known only for the one electron problem: hydrogen.
27 in the form: Z EXC [ρ(~r)] =
ρ(~r)εXC [ρ(~r)]d3~r
(2.28)
where εXC [ρ(~r)]d3~r is the exchange correlation energy of a homogeneous electron gas of the same charge density. One significant limitation of LDA is its overbinding of solids: lattice parameters are usually underpredicted[25] while cohesive energies are usually overpredicted. Another widely used approximation is the Generalized Gradient Approximation (GGA)[32]. GGA is an improvement on LDA by considering not only the local charge density but also its gradient. Although it is subjective, the calculation results using GGA generally agree better with experiments than those with LDA. The total energy of any compound can be readily calculated from first-principles. The Vienna Ab initio Simulation Package (VASP)[33] processes the E[ρ](R1 , R2 , · · · , RN ) to obtain the charge density that minimizes the total energy. This minimized value is the ground state total energy, E0 [{Ri }], where Ri represents the equilibrium lattice constant and atomic positions. This total energy can be further minimized with respect to the atomic positions to obtain the stable structures of the system. The total energies obtained from the first-principles calculations can be converted into formation enthalpies. For an Aa Bb compound, for example, the ∆H(Aa Bb ) is obtained by using the following equation: ∆H(Aa Bb ) = E(Aa Bb ) − xA E(A) − xB E(B)
(2.29)
where E’s are the first-principles calculated total energies of structure Aa Bb and pure elements A and B each fully relaxed to reference structures.
2.3.2
Ordered phase
First-principles calculations of ordered phases are relatively straightforward. In principle, DFT codes can calculate its total energy as long as its crystal structure can be defined. In order to minimize the computing expense, the smallest structure which imposes its original symmetry6 is used. Even structures with several dozens 6
Usually the primitive cell of the structure
28 of atoms can be readily calculated with the current computing power although the efficiency of a calculation is highly dependent upon the symmetry and the number of atoms of a structure.
2.3.3
Disordered phase
As discussed in the previous sections, the first-principles electronic structure calculations of perfectly ordered periodic structures are relatively straightforward. Problems arise, however, when attempting to use these methods to study the thermochemical properties of random solid solutions since an approximation must be made in order to simulate a random atomic configuration through a periodic structure. The most widely used approaches in the literature can be summarized as follows: i) The most direct approach is the supercell method. In this case, the sites of the supercell can be randomly occupied by either A or B atoms to yield the desired A1−x Bx composition. In order to reproduce the statistics corresponding to a random alloy, such supercells needs to be very large. This approach is, therefore, computationally prohibitive when the size of the supercell is on the order of hundreds of atoms. ii) Another technique, the Coherent Potential Approximation (CPA)[34] method, is a single-site approximation that models the random alloy as an ordered lattice of effective atoms. These are constructed from the criterion that the average scattering of electrons off the alloy components should vanish[35]. In this method, local relaxations are not considered explicitly and the effects of alloying on the distribution of local environments cannot be taken into account. Local relaxations have been shown to significantly affect the properties of random solutions [36], especially when the constituent atoms vary greatly in size and, therefore, their omission constitutes a major drawback. Although the local relaxation energy can be taken into account[35], these corrections rely on cluster expansions of the relaxation energy of ordered structures and the distribution of local environments is not explicitly considered. Additionally, such corrections are system specific.
29 iii) A third option is to apply the Cluster Expansion approach[37]. In this case, a generalized Ising model is used and the spin variables can be related to the occupation of either atom A or B in the parent lattice. In order to obtain an expression for the configurational energy of the solid phase, the energies of multiple configurations (typically in the order of a few dozens) based on the parent lattice must be calculated to obtain the parameters that describe the energy of any given A1−x Bx composition. This approach typically relies on the calculation of the energies of a few dozen ordered structures. In the techniques outlined above, there are serious limitations in terms of either the computing power required (supercells, cluster expansion) or the ability to accurately represent the local environments of random solutions (CPA). Ideally, one would like to be able to accurately calculate the thermodynamic and physical properties of a random solution with as small a supercell as possible so that accurate first-principles methods can be applied. This has become possible thanks to the development of Special Quasirandom Structures (SQS) to be discussed in two chapters.
2.4
Conclusion
Reliable phase diagrams can be constructed from the Gibbs energy functions of individual phases of a system and thermodynamic descriptions of the individual phases can be modeled/evaluated from the available experimental data. Two types of data, thermochemical data and phase diagram data, can be used in a thermodynamic modeling. Thermochemical data are preferred in a thermodynamic modeling since they can be directly derived from the Gibbs energy of a phase, while phase diagram data only give the relation among the Gibbs energies of the phases in equilibrium. However, it is not always possible to have enough experimental data for a robust thermodynamic modeling. For a liquid phase, thermochemical data can be accurately measured from experiments while that of solid phases cannot be. First-principles calculations can provide valuable information, especially for solid phases, in the CALPHAD modeling. First-principles calculations of ordered
30 phases are rather straightforward since their structural data can be easily defined. The enthalpy of formation from the calculated total energy of ordered phases can be readily used in the CALPHAD approach. Solid solution phases also have great importance in thermodynamic modeling. However, it is difficult to determine their thermodynamic properties experimentally. There are some limitations in current theoretical calculation methods of solid solution phases as well. In next two chapters, Special quasirandom structures (SQS), specially designed ordered structures of binaries (bcc, fcc, and hcp) and ternary (fcc) are introduced to obtain thermodynamic properties of a solid solution phase efficiently.
31
Bibliography [1] T. B. Massalski. Binary alloy phase diagrams. ASM International, Materials Park, Ohio, 2nd edition, 1990. [2] F. N. Rhines. Phase diagrams in metallurgy: Their development and application. McGraw Hill, New York, 1956. [3] D. Shin, R. Arroyave, and Z.-K. Liu. Thermodynamic modeling of the Hf-Si-O system. CALPHAD, 30(4):375–386, 2006. [4] B. Sundman and F. Aldinger. The Ringberg workshop 1995 on unary data for elements and other end-members of solutions. CALPHAD, 19(4):433–6, 1996. [5] M. W. Chase, I. Ansara, A. Dinsdale, G. Eriksson, G. Grimvall, L. Hoeglund, and H. Yokokawa. Thermodynamic models and data for pure elements and other endmembers of solutions. Group 1: Heat capacity models for crystalline phases from 0K to 6000K. CALPHAD, 19(4):437–47, 1996. [6] J. ˚ Agren, B. Cheynet, M. T. Clavaguera-Mora, K. Hack, J. Hertz, F. Sommer, and U. Kattner. Thermodynamic models and data for pure elements and other endmembers of solutions. Group 2: Extrapolation of the heat capacity in liquid and amorphous phases. CALPHAD, 19(4):449–80, 1996. [7] A. Chang, C. Colinet, M. Hillert, Z. Moser, J. M. Sanchez, N. Saunders, R. E. Watson, and A. Kussmaul. Thermodynamic models and data for pure elements and other endmembers of solutions. Group 3: Estimation of enthalpies for stable and metastable states. CALPHAD, 19(4):481–98, 1996. [8] D. de Fontaine, S. G. Fries, G. Inden, P. Miodownik, R. Schmid-Fetzer, and S.-L. Chen. Thermodynamic models and data for pure elements and other endmembers of solutions. Group 4: λ-transitions. CALPHAD, 19(4):499–536, 1996. [9] B. Burton, T. G. Chart, H. L. Lukas, A. D. Pelton, H. Seifert, and P. Spencer. Thermodynamic models and data for pure elements and other endmembers of solutions. Group 5: Estimation of enthalpies and entropies of transition. CALPHAD, 19(4):537–53, 1996. [10] F. Aldinger, A. F. Guillermet, V. S. Iorich, L. Kaufman, W. A. Oates, H. Ohtani, M. Rand, and M. Schalin. Thermodynamic models and data for pure elements and other endmembers of solutions. Group 6: Periodic system effects. CALPHAD, 19(4):555–71, 1996.
32 [11] A. T. Dinsdale. SGTE data for pure elements. CALPHAD, 15(4):317–425, 1991. [12] J. O. Andersson, A. F. Guillermet, P. Gustafson, M. Hillert, B. Jansson, B. Joensson, B. Sundman, and J. Aagren. A new method of describing lattice stabilities. CALPHAD, 11(1):93–8, 1987. [13] N. Saunders and A. P. Miodownik. CALPHAD (Calculation of Phase Diagrams) : A Comprehensive Guide. Pergamon, Oxford ; New York, 1998. [14] L. Kaufman and H. Bernstein. Computer Calculation of Phase Diagram. Academic Press, New York, 1970. [15] J. O. Andersson, A. F. Guillermet, and P. Gustafson. On the lattice stabilities for chromium, molybdenum and tungsten. CALPHAD, 11(4):361–4, 1987. [16] A. F. Guillermet and W. Huang. Calphad estimates of the lattice stabilities for high-melting bcc metals: vanadium, niobium, and tantalum. Z. Metallkd., 79(2):88–95, 1988. [17] A. F. Guillermet and M. Hillert. A thermodynamic analysis of the CALPHAD approach to phase stability of the transition metals. CALPHAD, 12(4):337– 49, 1988. [18] W. L. Bragg and E. J. Williams. Effect of thermal agitation on atomic arrangement in alloys. II. Proc. Roy. Soc. (London), A151:540–66, 1935. [19] O. Redlich and A. T. Kister. Algebraic representations of thermodynamic properties and the classification of solutions. Ind. Eng. Chem., 40(2):345– 348, 1948. [20] M. Hillert and L. I. Staffansson. Regular solution model for stoichiometric phases and ionic melts. Acta Chem. Scand., 24(10):3618–26, 1970. [21] H. Harvig. Extended version of the regular solution model for stoichiometric phases and ionic melts. Acta Chem. Scand., 25(9):3199–204, 1971. [22] B. Sundman and J. Aagren. A regular solution model for phases with several components and sublattices, suitable for computer applications. J. Phys. Chem. Solids, 42(4):297–301, 1981. [23] J. O. Andersson, A. Fernandez Guillermet, M. Hillert, B. Jansson, and B. Sundman. A compound-energy model of ordering in a phase with sites of different coordination numbers. Acta Metall., 34(3):437–45, 1986. [24] K. C. H. Kumar and P. Wollants. Some guidelines for thermodynamic optimisation of phase diagrams. J. Alloys Compd., 320:189–198, 2001.
33 [25] J. Hafner. Atomic-scale computational materials science. Acta Mater., 48(1): 71–92, 2000. [26] B. P. Burton, N. Dupin, S. G. Fries, G. Grimvall, A. F. Guillermet, P. Miodownik, W. A. Oates, and V. Vinograd. Using ab initio calculations in the CALPHAD environment. Z. Metallkd., 92(6):514–525, 2001. [27] C. Wolverton, X. Y. Yan, R. Vijayaraghavan, and V. Ozolins. Incorporating first-principles energetics in computational thermodynamics approaches. Acta Mater., 50(9):2187–2197, 2002. [28] C. Colinet. Ab-initio calculation of enthalpies of formation of intermetallic compounds and enthalpies of mixing of solid solutions. Intermetallics, 11 (11-12):1095–1102, 2003. [29] P. Hohenberg and W. Kohn. Inhomogeneous electron gas. Phys. Rev., 136 (3B):864–871, 1964. [30] W. Kohn and L. J. Sham. Self-consistent equations including exchange and correlation effects. Phys. Rev., 140:A1133–A1138, 1965. [31] D. M. Ceperley and B. J. Alder. Ground state of the electron gas by a stochastic method. Phys. Rev. Lett., 45(7):566–9, 1980. [32] J. P. Perdew, K. Burke, and Y. Wang. Generalized gradient approximation for the exchange-correlation hole of a many-electron system. Phys. Rev. B., 54(23):16533–16539, 1996. [33] G. Kresse and J. Furthmuller. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci., 6(1):15–50, 1996. [34] P. Soven. Coherent-potential model of substitutional disordered alloys. Phys. Rev., 156(3):809–13, 1967. [35] A. E. Kissavos, S. Shallcross, V. Meded, L. Kaufman, and I. A. Abrikosov. A critical test of ab initio and CALPHAD methods: The structural energy difference between bcc and hcp molybdenum. CALPHAD, 29(1):17–23, 2005. [36] Z. W. Lu, S. H. Wei, and A. Zunger. Large lattice-relaxation-induced electronic level shits in random CuPd alloys. Phys. Rev. B., 44(7):R3387–R3390, 1991. [37] J. M. Sanchez. Cluster expansion and the configurational energy of alloys. Phys. Rev. B., 48(18):R14013–R14015, 1993.
Chapter
3
Special quasirandom structures for substitutional binary solid solutions 3.1
Introduction
The concept of Special Quasirandom Structure (SQS) was first developed by Zunger et al. [1] to mimic random solutions without generating a large supercell or using many configurations. The basic idea consists of creating a small —4∼32 atoms— periodic structure with the target composition that best satisfies the pair and multi-site correlation functions corresponding to a random alloy, up to a certain coordination shell. Upon relaxation, the atoms in the structure are displaced away from their equilibrium positions, creating a distribution of local environments that can be considered to be representative of a random solution, at least up to the first few coordination shells. Provided the interatomic electronic interactions in a given system are relatively short-ranged, the first-principles calculations of the properties of these designed supercells can be expected to yield sensible results, especially when calculating properties that are mostly dependent on the local atomic arrangements, such as enthalpy of mixing, charge transfer, local relaxations, and so forth. It is important to stress that the approach fails whenever a property heavily depends on long-range interactions. The SQS’s for fcc-based alloys and bcc alloys have been generated by Wei
35 et al. [2] and Jiang et al. [3], respectively. In this chapter, two SQS’s capable of mimicking hcp random alloys at 25, 50 and 75 at.% are presented. The proposed SQS’s are characterized in terms of their ability to reproduce the pair and multi-site correlation functions of a random hcp solution. Subsequently, the structures are tested in terms of their ability to reproduce, via first-principles calculations, the properties of certain selected stable or metastable binary hcp solutions, namely Cd-Mg, Mg-Zr, Al-Mg, Mo-Ru, Hf-Ti, Hf-Zr and Ti-Zr. To further analyze the relaxation behavior of the structures, the distribution of first nearest bond lengths as well as the radial distribution for the first few coordination shells is presented. Finally, for each of the selected binaries, the calculated and available experimental lattice parameters and enthalpy of mixing are compared. Results from other techniques for hcp solutions are also presented where available in order to further corroborate the present calculations.
3.2
Correlation function
In order to characterize the statistics of a given atomic arrangement, one can use its correlation functions[4]. Within the context of lattice algebra, a ”spin value”, σ = ±1, can be assigned to each of the sites of the configuration, depending on whether the site is occupied by A or B atoms. Furthermore, all the sites can be grouped in figures, f (k, m), of k vertices, where k=1,2,3,· · · corresponds to a shape, point, pair, and triplet,. . . respectively, spanning a maximum distance of m, where m = 1, 2, 3, · · · is the first, second, and third-nearest neighbors, and so forth. The correlation functions, Πk,m , are the averages of the products of site occupations (±1 for binary alloys and ±1, 0 for ternary alloys) of figure k at a distance m and are useful in describing the atomic distribution. The optimum SQS for a given composition is the one that best satisfies the condition: Πk,m
SQS
∼ = hΠk,m iR
(3.1)
where hΠk,m iR is the correlation function of a random alloy, which is (2x − 1)k in the A1−x Bx substitutional binary alloy, where x is the composition. Two different compositions, i.e. x = 0.5 and 0.75 of SQS’s are considered.
36
3.3
First-Principles methodology
The selected hcp SQS-16 structures were used as geometrical input for the firstprinciples calculations. The Vienna Ab initio Simulation Package (VASP)[5] was used to perform the Density Functional Theory (DFT) electronic structure calculations. The projector augmented wave (PAW) method [6] was chosen and the general gradient approximation (GGA) [7] was used to take into account exchange and correlation contributions to the Hamiltonian of the ion-electron system. A constant energy cutoff of 350 eV was used for all the structures, with 5,000 k -points per reciprocal atom based on the Monkhorst-Pack scheme for the Brillouin-zone integrations. The k -point meshes were centered at the Γ point. The convergence criterion for the calculations was 10 meV with respect to the 16 atoms. Spinpolarization was not taken into account. The generated SQS’s were either fully relaxed, or relaxed without allowing local ion relaxations, i.e., only volume and c/a ratio were optimized. As will be seen below, the full relaxation caused some of the SQS’s to lose the original hcp symmetry. To further illustrate this problem, the same structure with the two different elements, A and B, are given in Figure 3.1. Both are in the square symmetry with lattice parameters aA and aB , respectively.
aB
aA (a) A in the perfect square symmetry
aA (b) B in the perfect square symmetry
Figure 3.1. Two dimensional structures of A and B in their perfect square symmetry.
37 Suppose those two elements are being mixed together to make the square solution phase. If so, then the atoms should be deviated from the original perfect square lattice sites as shown in Figure 3.2(a) due to the local relaxation caused by the different interatomic reactions, i.e. A-A, B-B, and A-B bondings. However, a problem arises when the degree of such local distortions is so big as to cause the collapse of its original square symmetry. The final structure has the configuration for the lowest energy at the given composition, but it does not have its initial symmetry. As discussed already, for the effective extrapolation to the higher-order system, a solution phase should be described throughout the entire composition range in the CALPHAD approach although it cannot be measured experimentally. This is why an SQS’s calculation results are so valuable to CALPHAD modeling. However, the relaxed structure should have the original symmetry in order to stay in the designated phase. Other than that it is just another metastable phase at the composition.
(a) Ax B1−x disordered phase in the com- (b) Ax B1−x disordered phase in the perfect pletely relaxed structure. square symmetry with effective lattice parameter.
Figure 3.2. Two dimensional structures of Ax B1−x disordered phase with/without local relaxation.
In principle, all the electronic structure optimization calculations should be performed with respect to all the degrees of freedom to find the lowest energy configuration. Most DFT software keep the initial symmetry of a structure even
38 though it is metastable or sometimes unstable. However, relaxation of SQS’s yield a symmetry issue since the generated SQS’s have lower symmetry, not its higher original symmetry. All the atoms are at the exact lattice sites of the original structure with higher symmetry, however, some—perhaps most— of the symmetry operations are no longer valid because the atomic distribution within the structure is now close to that of a random solution. Thus, when the SQS’s are relaxed within DFT codes, such software will try to keep the lower symmetry of the SQS’s, not its original symmetry. If a system shows a wide solubility range and the concentration of the SQS falls within the solubility range, then the fully relaxed structure should have the original symmetry when the two different types of atoms in the SQS’s are substituted as one single type of atom. In this regard, it is better to have the calculation result forced to keep the original symmetry to represent the structure before it collapse. Such a symmetry preserved calculation will have the effective lattice parameter which will lead to the same bond lengths regardless of the bonding types as shown in Figure 3.2(b). Of course the total energy of the fully relaxed state with a different structure should be lower than that of symmetry preserved calculation. However, such SQS calculations can give insight to the mixing behavior of a phase.
3.4
Generation of special quasirandom structures
Unlike cubic structures, the order of a given configuration in the hcp lattices relative to a given lattice site may be altered with the variation of c/a ratio. However, these new arrangements will not cause any change in the correlation functions since one can thus use any c/a ratio to generate the hcp SQS’s. As a matter of simplicity, the ideal c/a ratio was considered in generating SQS’s. The major drawback of the SQS method is that the concentrations which can be calculated is typically limited to 25, 50, and 75 at.% since the correlation functions for completely random structures at other compositions are almost impossible to satisfy with the small number of atoms. In principle, one can find a bigger supercell which has a better correlation function than smaller ones. However, such a calculation requires expensive computing. On the other hand, three data points from SQS’s calculations can clearly indicate the mixing behavior of solution phases.
39 Another disadvantage of SQS is that it cannot consider the long range interaction since the size of the structure itself is limited. It is reported that SQS works well with a system where short range interactions are dominant[3, 8]. In the present work, the Alloy Theoretic Automation Toolkit (ATAT)[9] has been used to generate special quasirandom structures for the hcp structure of 8 and 16 sites. The schematic diagrams of the created special quasirandom structure with 16 atoms are shown in Figure 3.3 and the corresponding lattice vectors and atomic positions are listed in Table 3.1. The correlation functions of the generated 8 and 16-atom SQS’s were investigated to verify that they satisfied at least the short-range statistics of an hcp random solution. As is shown in Table 3.2, the 16-atom structures satisfy the pair correlation functions of random alloys up to the fifth and third nearest neighbor for the 50 at.% and the 75 at.% compositions, respectively. On the other hand, Table 3.2 shows that the SQS-8 for 75 at.% could not satisfy the random correlation function even for the first-nearest neighbor pair. Thus, SQS’s with 16 atoms are capable of mimicking a random hcp configuration beyond the first coordination shell. It is important to note that in Table 3.2, and contrary to what is observed in the SQS for cubic structures, some figures have more than one crystallographically inequivalent figure at the same distance. For example, in the case of hcp lattices with the ideal c/a ratio, two pairs may have the same interatomic distance and yet be crystallographically inequivalent. In this case, despite the fact that the two pairs, (0,0,0) and (a,0,0); (0,0,0) and ( 31 , 23 , 12 ), have the same inter-atomic distance, a, they do not share the same symmetry operations. This degeneracy is broken when the c/a ratio deviates from its ideal value. For the sake of efficiency, the initial lattice parameters of the SQS’s were determined from Vegard’s law. By doing so, the c/a ratio was no longer ideal. Afterwards, the correlation functions of the new structures remained the same as long as the corresponding figures were identical. The maximum range over which the correlation function of an SQS mimics that of a random alloy can be increased by increasing the supercell size. As the size of the SQS increases, the probability of finding configurations that mimic random alloys over a wider coordination range increases accordingly. The search algorithm
40
A
B (a) SQS-16 for x=0.5
(b) SQS-16 for x=0.75
Figure 3.3. Crystal structures of the A1−x Bx 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, P1.
41
Table 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
SQS-16
SQS-8
x = 0.5 Lattice vectors 0 −1 −1 −2 −2 0 −2 1 −1 Atomic positions −2 13 −1 23 −1 12 A −1 −1 −1 A −2 0 −1 A 2 1 −1 13 −1 A 3 2 −3 −2 −1 A 2 −2 13 −1 12 A 3 −4 −2 −2 A −3 13 −1 23 −1 12 A −2 −2 −1 B 1 −1 13 −1 23 B 2 −3 −1 −1 B −2 −1 −1 B 2 1 B −1 13 3 2 1 2 1 B 3 3 2 1 1 2 −2 3 −1 3 B 2 −3 −1 −2 B
x = 0.75 Lattice vectors 1 1 1 −1 0 1 0 −4 0 Atomic positions − 13 −2 23 1 12 A − 13 −1 23 1 12 A 0 −3 2 A 0 −3 1 A 0 −2 2 B 0 −1 2 B 0 0 2 B − 13 − 23 1 12 B 1 − 13 1 12 B 3 1 1 2 − 3 −3 3 B 2 0 −2 1 B 1 − 13 −2 23 B 2 0 −1 1 B 1 − 13 −1 23 B 2 0 0 1 B 1 − 13 − 23 B 2
Lattice vectors −1 1 1 1 −1 1 1 1 0 Atomic positions 2 1 1 A 3 3 2 1 2 1 1 A 3 3 2 1 0 1 A 1 1 2 A 0 1 1 B 1 1 1 B 1 1 13 23 B 2 1 13 23 1 12 B
Lattice vectors 1 1 −1 0 −1 −1 −2 2 0 Atomic positions −1 1 −1 A 2 1 − 23 − A 3 2 2 2 13 − 12 B −1 3 −1 1 −2 B 2 1 − 23 −1 B 3 2 0 0 −1 B 0 0 −2 B 1 1 1 − −1 B 3 3 2
42
Table 3.2. 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 Πk,m is the number of equivalent figures at the same distance in the structure, the so-called degeneracy factor.
Π2,1 [6] Π2,1 [6] Π2,2 [6] Π2,3 [2] Π2,4 [12] Π2,4 [6] Π2,5 [12] Π2,6 [6] Π2,7 [12] Π2,8 [12] Π3,1 [12] Π3,1 [2] Π3,1 [2] Π3,2 [24] Π3,3 [6] Π3,3 [6] Π4,1 [4] Π4,2 [12] Π4,2 [12] Π4,3 [6]
Random 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
x=0.5 SQS-16 0 0 0 0 0 0 0 -0.33333 0 0 0 0 0 0 0 0 0 0 0 0.33333
SQS-8 0 0 0 0 0 -0.33333 -0.33333 0.33333 0 0 0.33333 0 0 0 0 0 0 -0.33333 0 0.33333
Random 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.125 0.125 0.125 0.125 0.125 0.125 0.0625 0.0625 0.0625 0.0625
x=0.75 SQS-16 0.25 0.25 0.25 0.25 0.25 0.45833 0.33333 0.16667 0.25000 0.1667 -0.08333 0.25 0.25 -0.04167 -0.08333 -0.08333 0 -0.16667 0 -0.16667
SQS-8 0.16667 0.33333 0.33333 0 0.16667 0 0.33333 0.33333 0.5 0.33333 0.16667 0.5 0.5 0 0.16667 -0.16667 0.5 -0.16667 0 0
used in this work consists of enumerating every possible supercell of a given volume and for each supercell, enumerating every possible atomic configuration. For each configuration, the correlation functions of different figures, i.e. points, pairs, and triplets, are calculated. To save time, the calculation of the correlations is stopped as soon as one of them does not match the random state value. This algorithm becomes prohibitively expensive very rapidly. The generation of a larger SQS could be accomplished by using a Monte-Carlo-like scheme[10], but this is beyond the scope of present work. In fact, 32-atom SQS’s was generated, however, the average total energy difference between 16-atom SQS’s and 32-atom SQS’s in the Cd-Mg system was around only 2 meV per atom. It is concluded that 16-atom SQS is good enough because this size represents a good compromise between accuracy and the computational requirements associated with the necessary first-principles calculations.
43 It is also important to note that finding a good hcp SQS is more difficult than finding an SQS of cubic structures with the same range of matching correlations due to the fact that, for a given range of correlations, there are more symmetrically distinct correlations to match. Additionally, the lower symmetry of the hcp structure implies that there are also many more candidate configurations to search through in order to find a satisfactory SQS. Thus, the number of distinct supercells is larger and the number of symmetrically distinct atomic configurations is larger, in comparison to fcc or bcc lattices. In order to verify the proposed 16-atom SQS’s are adequate for the simulation of hcp random solutions, other SQS’s at 75 at.%, which have random-like pair correlations up to the third nearest-neighbor but that have slightly different correlations for the fourth nearest-neighbor, are calculated. The pair correlation function at 75 at.% of a truly random solution would be (2 × 0.75 − 1)2 = 0.25 and therefore the four SQS’s in Table 3.3 are worse than the one used in the present work. These structures were applied to the Cd 25 at.%-Mg 75 at.% system and, as can be seen in Table 3.3, the associated energy differences are negligible. This is due to the fact that the energetics of this system are dominated by short-range interactions. Thus, as long as the most important pair correlations (up to the third nearest-neighbors in hcp structure with ideal c/a ratio) are satisfied, the SQS’s can successfully be applied to acquire properties of random solutions in which short-range interactions dominate.
3.5 3.5.1
Results and discussions Analysis of relaxed structures
The symmetry of the resulting SQS was checked using the PLATON[11] code before and after the relaxations. Both SQS’s have the lowest symmetry of P1, although all the atoms are sitting on the lattice sites of hcp. The procedure was verified by checking the symmetries of the generated unrelaxed SQS. Once all the sites in the SQS were substituted with the same type of atoms, PLATON identified SQS’s as perfect hcp structures. All the atoms of the initial structures are on their exact hcp lattice sites. However, upon relaxation the atoms may be displaced from these
44
Table 3.3. Pair correlation functions up to the fifth shell and the calculated total energies of other 16 atoms sqs’s for Cd0.25 Mg0.75 are enumerated to be compared with the one used in this work (SQS-16). The total energies are given in the unit, eV /atom. Π2,1 [6] Π2,1 [6] Π2,2 [6] Π2,3 [2] Π2,4 [12] Π2,4 [6] Π2,5 [12] Symmetry Preserved Fully Relaxed
a 0.25 0.25 0.25 0.25 0.20833 0.5 0.5
b 0.25 0.25 0.25 0.25 0.16667 0.5 0.16667
c 0.25 0.25 0.25 0.25 0.16667 0.5 0.33333
d 0.25 0.25 0.25 0.25 0.08333 0.16667 0.33333
SQS-16 0.25 0.25 0.25 0.25 0.25 0.45833 0.33333
-1.3864
-1.3882
-1.3886
-1.3886
-1.3869
-1.3874
-1.3887
-1.3889
-1.3893
-1.3883
ideal positions. According to the definition of an hcp random solution, all the atoms, in this case two different types of atoms, should be at the hcp lattice points —within a certain tolerance— even after the structure has been fully relaxed. The default tolerance of detecting the symmetry of the relaxed structures allowed the atoms to deviate from their original lattice sites by up to 20%. In principle, relaxations should be performed with respect to the degrees of freedom consistent with the initial symmetry of any given configuration. In the particular case of the hcp SQS’s, local relaxations may in some cases be so large that the character of the underlying parent lattice is lost. However, within the CALPHAD methodology, one has to define the Gibbs energy of a phase throughout the entire composition range, regardless of whether the structure is stable or not. In these cases, it is necessary to constrain the relaxations so that they are consistent with the lattice vectors and atom positions of an hcp lattice. Obviously, the energetic contributions due to local relaxations are not considered in this case. The results of these constrained relaxations can therefore be directly compared to those calculations using the CPA. In most cases, local relaxations were not significant. However, in a few instances, it was found that the structure was too distorted to be considered as hcp after the full relaxation. However, this symmetry check was not sufficient to characterize the relaxation behavior of the relaxed SQS.
45 Furthermore, in some of the cases it may be possible for the structure to fail the symmetry test and still retain an hcp-like environment within the first couple of coordination shells, implying that the energetics and other properties calculated from these structures could be characterized as reasonable, although not optimal, approximations of random configurations.
3.5.2
Radial distribution analysis
In order to investigate the local relaxation of the fully relaxed SQS, their radial distribution (RD) was analyzed. In this analysis, the bond distribution and coordination shells were studied to determine whether the relaxed structures maintained the local hcp-like environment. Additionally, this analysis permitted us to quantify the degree of local relaxations up to the fifth coordination shells in the 16-atom SQS. The RD of each of the fully relaxed structures was obtained by counting the ˚, up to the fifth coordination shell. In number of atoms within bins of 10−3 A order to eliminate high frequency noise, the raw data was scaled and smoothed through Gaussian smearing with a characteristic distance of 0.01 ˚ A. Pseudo-Voigt functions were then used to fit each of the smoothed peaks and the goodness of fit was in part determined through the summation of the total areas of the peaks and comparing them to the total number of atoms that were expected within the analyzed coordination shells. The relaxation of the atoms at each coordination shell is quantified by the width of the corresponding peak in the fitted RD. The RD results of selected SQS’s are given in Figure 3.4, 3.5, and 3.6. The unrelaxed, fully relaxed, and non-locally relaxed structures are compared in each case as well as the smoothed bond distributions and their fitted curves. These results are representative of the RD’s obtained for the seven binary systems at the three compositions studied. Figure 3.4(a) shows the RDs for the Hf-Zr SQS at the 50 at.% composition. As can be seen in the figure, the RDs for the unrelaxed and non-locally relaxed SQS are almost identical, implying that in this system Vegard’s Law is closely followed. Furthermore, the RD for the fully relaxed SQS in Figure 3.4(b) shows a rather narrow distribution around each of the the bond-lengths corresponding
46 to the ideal or unrelaxed structure. The system therefore needs to undergo very negligible local relaxations in order to minimize its energy. In the case of the Cd-Mg solution at 50 at.% (Figure 3.5(a)), the RDs of the unrelaxed and non-locally relaxed SQS are more dissimilar. Even in the nonlocally relaxed calculation, the original first coordination shell (corresponding to the six first-nearest neighbors) has split into two different shells (of 4 and 2 atoms) and the position of the peak is noticeably shifted. Upon full relaxation the first two well defined coordination shells of the unrelaxed structure have merged into a single, broad peak at 3.14˚ A, as shown in Figure 3.5(b). This peak now encloses 12 first nearest neighbors. As shown in Table 3.2, Π2,1 and Π2,4 have two different types of pairs. However, since they have the same correlation functions, they cannot be distinguished. In Figure 3.5(b) it is also shown how the fourth and fifth ˚, enclosing 18 atoms. It can be expected that if coordination shells merge at 5.40A the c/a ratio of a relaxed structure is close to ideal and the broadening of nearby shells are wide enough that they merge, then the structure has almost the same radial distribution of an ideal hcp structure, albeit with a larger peak width. Figure 3.6(a) shows the RD for the Mg50 Zr50 composition. Among the three RD’s presented, this one is clearly the one that undergoes the greatest distortion upon full relaxation. Even in the symmetry preserved structures there is a broad bond-length distribution around the peaks of the unrelaxed SQS. With respect to the fully relaxed SQS, it can be seen how the peaks for the fifth and sixth coordination shells have practically merged. In this case, the local environment of each atom within the SQS stops being hcp-like within the first couple of coordination shells. Although the two end members of this binary alloy have an hcp as the stable structure, it is evident from this figure that the SQS arrangement is unstable. In this system, there is a miscibility gap in the hcp phase up to ∼ 900K and the RD reflects the tendency for the system to phase-separate. The results from the peak fitting for all the fully relaxed SQS’s are summarized in Table 3.4. It should be noted that regardless of the system and compositions, the sum of the areas under each peak should converge to a single value, proportional to 50 atoms. For each peak, the error was quantified as the absolute and normalized difference between the expected and actual areas. The error reported in the table is the averaged value for all the peaks in the RD. The broadness of the peaks in the
47 RD is quantified through the full width at half maximum, FWHM. In the table, the reported FWHM corresponds to the average FWHM observed for the coordination shells enclosing a total of 50 atoms. Note that the alloys with the smallest FWHM are Hf-Zr and Cd-Mg. As will be seen later, Hf-Zr behaves almost ideally and Cd-Mg is a system with rather strong attractive interactions between unlike atoms that forms ordered hexagonal structures at the 25 and 75 at.% compositions.
3.5.3
Bond length analysis
In addition to the RD analysis, the bond length analysis was performed (A-A, B-B, and A-B) for all the relaxed SQS’s. In Table 3.5 the bond lengths corresponding to the first nearest neighbors for all the 21 SQS’s are presented. As expected, in the majority of the cases the sequence dii < dij < djj is observed throughout the composition range, where dij corresponds to the bond distance between two different atom types. The two notable exceptions to this trend correspond to the Cd-Mg and Mg-Zr alloys. As will be mentioned below, the Cd-Mg system tends to form rather stable intermetallic compounds at the 25, 50 and 75 at.% compositions, including two hexagonal intermetallic compounds. The calculated enthalpy of mixing in this case—shown in Figure 3.7(a)—is the most negative among seven binaries studied and the fact that the Cd-Mg bonds are shorter than Cd-Cd and Mg-Mg seems to reflect the tendency of this system to order. In the case of the Mg-Zr alloys, the Mg-Zr bonds are longer than Mg-Mg and Zr-Zr, suggesting that this system has a great tendency to phase separate, as indicated by the presence of a large hcp miscibility gap in the Mg-Zr phase diagram [12].
3.5.4
Enthalpy of mixing
It is obvious that if an hcp SQS alloy is not stable with respect to local relaxations, its properties are not accessible through experimental measurements. However, approximate effective properties could still be estimated through CALPHAD modeling. In order to compare the energetics and properties of the calculated SQS’s with the available experiments or previous thermodynamic models, only the non-locally relaxed structures were considered whenever the SQS was identified as unstable. This effectively assumes that the structures in question are constrained to maintain
0
2
4
6
8
10
12
2.5
3
4
4.5 Distance (Å)
5
5.5
(a) RD of Hf50 Zr50 (∆Hmix ∼ 0)
3.5
Original Symmetry preserved Fully relaxed
6
6.5
3
3.5
4
4.5
5 Distance (Å)
5.5
Smoothed Fitted
6
(b) Smoothed and fitted RD’s of fully relaxed Hf50 Zr50
0
0.5
1
1.5
2
2.5
3
3.5
4
6.5
Figure 3.4. Radial distribution analysis of Hf50 Zr50 SQS’s. The dotted lines under the smoothed and fitted curves are the error between the two curves.
Number of bonds
14
Number of bonds
4.5
48
0
2
4
6
8
10
12
14
2.5
3
4
4.5 Distance (Å)
5
5.5
(a) RD of Cd50 Mg50 (∆Hmix < 0)
3.5
Original Symmetry preserved Fully relaxed
6
6.5
Number of bonds
3
3.5
4
4.5
5 Distance (Å)
5.5
6
(b) Smoothed and fitted RD’s of fully relaxed Cd50 Mg50
0
0.5
1
1.5
2
Smoothed Fitted
6.5
Figure 3.5. Radial distribution analysis of Cd50 Mg50 SQS’s. The dotted lines under the smoothed and fitted curves are the error between the two curves.
Number of bonds
2.5
49
0
2
4
6
8
10
12
14
2.5
3
4
4.5 Distance (Å)
5
5.5
(a) RD of Mg50 Zr50 (∆Hmix > 0)
3.5
Original Symmetry preserved Fully relaxed
6
6.5
Number of bonds
3
3.5
4
4.5
5 Distance (Å)
5.5
6
(b) Smoothed and fitted RD’s of fully relaxed Mg50 Zr50
0
0.5
1
Smoothed Fitted
6.5
Figure 3.6. Radial distribution analysis of Mg50 Zr50 SQS’s. The dotted lines under the smoothed and fitted curves are the error between the two curves.
Number of bonds
1.5
50
a
FWHM Error, % Symmetry FWHM Error, % Symmetry FWHM Error, % Symmetry
Cd-Mg 0.06 ± 0.01 0.72 PASS 0.07 ± 0.02 0.30 PASS 0.04 ± 0.01 2.05 PASS
Mg-Zr 0.09 ± 0.03 0.39 PASS 0.15 ± 0.02 1.42 FAIL 0.09 ± 0.03 1.22 PASS
Al-Mg 0.08 ± 0.02 0.47 PASS 0.15 ± 0.07 1.28 FAIL 0.10 ± 0.02 0.26 PASS
Mo-Ru N/Aa N/A FAIL 0.13 ± 0.01 1.90 PASS 0.07 ± 0.02 1.93 PASS
Hf-Ti 0.11 ± 0.03 1.07 PASS 0.16 ± 0.02 0.35 PASS 0.11 ± 0.06 0.26 PASS
Hf-Zr 0.02 ± 0.00 1.84 PASS 0.03 ± 0.01 1.84 PASS 0.03 ± 0.00 1.01 PASS
Ti-Zr 0.16 ± 0.05 1.27 FAIL 0.19 ± 0.06 2.39 PASS 0.13 ± 0.07 0.96 PASS
The radial distribution analysis of Mo 75 at.%-Ru 25 at.% was not possible since it completely lost its symmetry as hcp.
A25 B75
A50 B50
A75 B25
Compositions
Table 3.4. 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 ˚ A. Errors indicate the difference in the number of atoms calculated through the sum of peak areas and those expected in each coordination shell.
51
A0 B100
A25 B75
A50 B50
A75 B25
Compositions A100 B0
Bonds A-A A-A A-B B-B A-A A-B B-B A-A A-B B-B B-B
Cd-Mg 3.07 3.17 ± 0.10 3.16 ± 0.11 3.18 ± 0.10 3.16 ± 0.04 3.12 ± 0.04 3.15 ± 0.03 3.16 ± 0.01 3.14 ± 0.02 3.15 ± 0.01 3.18
Mg-Zr 3.18 3.18 ± 0.03 3.18 ± 0.05 3.12 ± 0.10 3.16 ± 0.04 3.20 ± 0.06 3.14 ± 0.08 3.15 ± 0.04 3.19 ± 0.04 3.18 ± 0.04 3.19
Al-Mg 2.87 2.92 ± 0.03 2.95 ± 0.03 2.96 ± 0.03 2.98 ± 0.06 3.02 ± 0.06 3.07 ± 0.08 3.06 ± 0.04 3.08 ± 0.04 3.11 ± 0.03 3.18 2.81 ± 0.08 2.75 ± 0.04 2.75 ± 0.04 2.73 ± 0.04 2.73 ± 0.04 2.71 ± 0.04 2.68
N/A
Mo-Ru 2.75
Hf-Ti 3.13 3.14 ± 0.05 3.10 ± 0.05 3.09 ± 0.06 3.09 ± 0.06 3.05 ± 0.07 3.00 ± 0.06 3.02 ± 0.05 3.00 ± 0.06 2.95 ± 0.05 2.87
Hf-Zr 3.13 3.18 ± 0.03 3.18 ± 0.03 3.18 ± 0.04 3.18 ± 0.03 3.19 ± 0.03 3.20 ± 0.03 3.19 ± 0.03 3.19 ± 0.03 3.20 ± 0.04 3.19
Ti-Zr 2.87 2.96 ± 0.07 3.02 ± 0.07 3.04 ± 0.06 3.00 ± 0.09 3.06 ± 0.08 3.12 ± 0.08 3.09 ± 0.08 3.11 ± 0.06 3.17 ± 0.06 3.19
Table 3.5. 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.
52
53 their symmetry. The total energies of the structures under symmetry-preserving relaxations are obviously higher since the relaxation is not considered. However, one can consider these calculated thermochemical properties as an upper bound which can still be of great use when attempting to generate thermodynamically consistent models based on the combined first-principles/CALPHAD approach. The enthalpies of mixing for these alloys were calculated at the 25, 50, and 75 at.% concentrations through the expression: ∆H(A1−x Bx ) = E(A1−x Bx ) − (1 − x)E(A) − xE(B)
(3.2)
where E(A) and E(B) are the reference energies of the pure components in their hcp ground state. In the following sections, the generated SQS’s are tested by calculating the crystallographic, thermodynamic and electronic properties of hcp random solutions in seven binary systems, Cd-Mg, Mg-Zr, Al-Mg, Mo-Ru, Hf-Ti, Hf-Zr, and TiZr. The results of the calculations are then compared with existing experimental information as well as previous calculations.
3.5.5
Cd-Mg
In the Cd-Mg system, both elements have the same valence and almost the same atomic volumes. Consequently, there is a wide hcp solid solution range as well as order/disorder transitions in the central, low temperature region of the phase diagram. In fact, at the 25 and 75 at.% compositions there are ordered intermetallic phases with hexagonal symmetries. Figure 3.7(a) compares the enthalpy of mixing calculated from the fully relaxed and symmetry preserved SQS with the results from cluster expansion[13]. The results by Asta et al. [13] at 900K are presented for comparison since it is to be expected that these values would be rather close to the calculated enthalpy of completely disordered structures. The previous and current calculations are also compared with the experimental measurements as reported in Hultgren[14] at 543K. The first thing to note from Figure 3.7(a) is that the fully relaxed and symmetry preserved calculations are very close in energy, implying negligible local relaxation. Additionally, the present calculations are remarkably close (∼1 kJ/mol)
54 to the experimental measurements. By comparing the SQS enthalpy of mixing with the results from the cluster expansion calculations [13], it is obvious that the former is more capable of reproducing the experimental measurements. Formation enthalpies of the three ordered phases in the Cd-Mg system, Cd3 Mg, CdMg, and CdMg3 are also presented. The measurements from Hultgren[14] deviate from the calculated results from Asta et al. [13] and this work. Cd and Mg are known as very active elements and it is likely that reaction with oxygen present during the measurements may have introduced some systematic errors. Furthermore, the measurements were conducted at relatively low temperatures, making it difficult for the systems to equilibrate. Nevertheless, experiments and calculations agree that these three compounds constitute the ground state of the Cd-Mg system. Figure 3.7(b) also shows that the present calculations are able to reproduce the available measurements on the variation of the lattice parameters of hcp Cd-Mg alloys with composition, as well as the deviation of these parameters from Vegard’s Law. This deviation is mainly related to the rather large difference in c/a ratio between Cd and Mg. The c/a ratio of Cd is one of the largest ones of all the stable hcp structures in the periodic table.
3.5.6
Mg-Zr
The Mg-Zr system is important due to the grain refining effects of Zr in magnesium alloys. According to the assessment of the available experimental data by NayebHashemi and Clark [12], the Mg-Zr system shows very little solubility in the three solution phases, bcc, hcp and liquid. In fact, the low temperature hcp phase exhibits a broad miscibility gap up to 923K, corresponding to the peritectic reaction hcp + liquid → hcp [12]. Our calculations yielded a positive enthalpy of mixing, confirming the trends derived from the thermodynamic model developed by H¨am¨al¨ainen et al. [18]. In the case of the full relaxation, however, it was observed that the Mg50 Zr50 SQS was unstable with respect to local relaxations. The instability at this composition and the large, positive enthalpy of mixing indicate that the system has a strong tendency to phase-separate. By comparing the fully relaxed and the symmetry
55 2 SQS:Fully Relaxed SQS:Symmetry Preserved Disordered phase(1963 Hultgren et al.) Disordered phase(1993 Asta et al.) Ordered phases(1952 Buck et al.) Ordered phases(This work) Orderd phases(1993 Asta et al.)
Enthalpy of formation (kJ/mol)
0
-2
-4
-6
-8
-10
-12 0
0.25
0.5
0.75
1
Mole Fraction, Mg (a) Calculated enthalpy of mixing for the disordered hcp phase in the Cd-Mg system with SQS at T=0K, Cluster Variation Method (CVM)[13] at T=900K, and experiment[14] at T=543K 6
c
5.5
Lattice Parameters (Å)
5 4.5
SQS:Fully Relaxed SQS:Symmetry Preserved 1940 Hume-Rothery et al. 1957 von Batchelder et al. 1959 Hardie et al.
4 3.5 3
a
2.5 2
c/a
1.5 0
0.25
0.5
0.75
1
Mole Fraction, Mg (b) Calculated lattice parameters of the Cd-Mg system compared with experimental data[15–17]
Figure 3.7. Calculated and experimental results of mixing enthalpy and lattice parameters for the Cd-Mg system
56
14 SQS:Fully Relaxed SQS:Symmetry Preserved 1999 Hämäläinen et al.(assessment) 2005 Arroyave et al.(assessment)
Enthalpy of mixing (kJ/mol)
12 10 8 6 4 2 0 0
0.25
0.5
0.75
1
Mole Fraction, Zr Figure 3.8. Calculated enthalpy of mixing in the Mg-Zr system compared with a previous thermodynamic assessment[18]. Both reference states are the hcp structure.
preserved structures, it has been estimated that the local relaxation energy lowers the mixing enthalpy of the random hcp SQS by about 2 kJ/mol in this system. Figure 3.8 shows the calculated mixing enthalpy for the Mg-Zr hcp SQS with the symmetry preserved calculations, as well as the mixing enthalpy calculated from the thermodynamic model by H¨am¨al¨ainen et al. [18], which was fitted only through phase diagram data. It is remarkable that the maximum difference between the CALPHAD model and the present hcp SQS calculations is ∼3 kJ/mol. The CALPHAD model, however, does not correctly describe the asymmetry of the mixing enthalpy indicated by the first-principles calculations. The results of the hcp SQS calculations for the Mg-Zr system have recently been used to obtain a better thermodynamic description of the Mg-Zr system [19] and, as can be seen in the figure, this description is better at describing the trends in the calculated enthalpy of mixing.
57
3.5.7
Al-Mg
As one of the most important industrial alloys, the Al-Mg system has been studied extensively recently[20–22]. This system has two eutectic reactions and shows solubility within both the fcc and hcp phases. However, the solubility ranges are not wide enough so there is only limited experimental information for the properties of the hcp phase. The maximum equilibrium solubility of Al in the Mg-rich hcp phase is around 12 at.%. In Figure 3.9(a) the calculated enthalpy of mixing is slightly positive. The fully relaxed calculations show that the SQS with the 50 at.% composition was unstable with respect to local relaxations. This can be explained by the strong interaction between Al and Mg, as evident from the tendency of this system to form intermetallic compounds at the middle of the phase diagram, such as β-Al140 Mg89 , γ-Al12 Mg17 , and ε-Al30 Mg23 . At the 25 and 75 at.% compositions the SQS’s were stable with respect to local relaxations because both elements have a close-packed structure. Figure 3.9(a) shows that the present fully relaxed calculations are in excellent agreement with the most recent CALPHAD assessments[20, 21]. Note also that in this case, and contrary to what is observed in the Cd-Mg binary, the energy change associated with local relaxation is not negligible, although it is still within ∼1 kJ/mol. Additionally, the calculated lattice parameters agree very well with the experimental measurements of Mg-rich hcp alloys, as can be seen in Figure 3.9(b). It is important to note that the lattice parameter measurements of metastable hcp alloys from Luo et al. [23](77.4 and 87.8 Mg at.%) are lying on the extrapolated line between the 75 at.% SQS and the pure Mg calculations. This is another example of how SQS’s can be successfully used in calculating the properties of an hcp solid solution with a narrow solubility range and mixed with non-hcp elements, even in the metastable regions of the phase diagram.
3.5.8
Mo-Ru
The Mo-Ru system shows a wide solubility range in both the bcc and hcp solutions. In the Ru-rich hcp solution, the maximum solubility of Mo in the hcp-Ru matrix is up to 50 at.%. For the hcp SQS, the calculations at Mo25 Ru75 and Mo50 Ru50
58 4 SQS:Fully Relaxed SQS:Symmetry Preserved 1997 Liang et al.(assessment) COST 507 Zhong et al.
Enthalpy of mixing (kJ/mol)
3.5 3 2.5 2 1.5 1 0.5 0 0
0.25
0.5
0.75
1
Mole Fraction, Mg (a) Calculated enthalpy of mixing for the hcp phase in the Al-Mg system compared with assessed data[20–22]. Reference states are hcp for both elements. 5.5
Lattice Parameters (Å)
5
4.5
SQS:Fully Relaxed SQS:Symmetry Preserved 1941 Hume-Rothery et al. 1942 Raynor et al. 1950 Busk et al. 1957 von Batchelder et al. 1959 Hardie et al. 1964 Luo et al.
4
3.5
3
2.5 0
0.25
0.5
0.75
1
Mole Fraction, Mg (b) Calculated lattice parameters of the hcp phase in the AlMg system compared with experimental data[16, 17, 23– 26]
Figure 3.9. Calculated and experimental results of mixing enthalpy and lattice parameters for the Al-Mg system
59 retained the original hcp symmetry but Mo75 Ru25 did not. The instability of the Mo-rich hcp SQS is not surprising since the Mo-rich bcc region is stable over a wide region of the phase diagram. As shown by Wang et al. [27], elements whose ground state is bcc are not stable in an hcp lattice and viceversa (bcc Ti, Zr and Hf are only stabilized at high temperature due to anharmonic effects). Thus hcp compositions close to the bcc-side would be dynamically unstable and would have a very large driving force to decrease their energy by transforming to bcc. Recently, Kissavos et al. [28] calculated the enthalpy of mixing for disordered hcp Mo-Ru alloys through the CPA in which relaxation energies were estimated by locally relaxing selected multi-site atomic arrangements. Enthalpy of formation for hcp solutions were calculated from Eqn. 3.3 shown below. The enthalpy of mixing of the disordered hcp phase can be evaluated accordingly based on the so-called lattice stability[29], E bcc (M o) − E hcp (M o).
∆f H(M o1−x Rux ) = E hcp (M o1−x Rux ) − (1 − x)E bcc (M o) − xE hcp (Ru) = E hcp (M o1−x Rux ) − (1 − x)E hcp (M o) − xE hcp (Ru) − (1 − x)E bcc (M o) + (1 − x)E hcp (M o) hcp = Hmix (M o1−x Rux ) − (1 − x)[E bcc (M o) − E hcp (M o)]
(3.3)
For some transition elements, the disagreement between the two approaches is quite significant[30]. Mo is one such case, with the structural energy difference between bcc and hcp from first-principles calculations and the CALPHAD approach differing by over 30 kJ/mol. After a rather extensive analysis, Kissavos et al. [28] arrived at the conclusion that in order to reproduce enthalpy values close enough to the available experimental data[31] the CALPHAD lattice stability (11.55 kJ/mol) needed to be used for the value of the bcc → hcp promotion energy. The SQS and CPA calculations are compared with the experimental measurements in Figure 3.10. On the assumption that the experimental measurements by Kleykamp [31] are correct, the derived enthalpy of formation of the hcp Mo-Ru system from the first-principles calculated lattice stability with the SQS and CPA approach in Figure 3.10(a) did not agree with the experimental observation at all since the first-principles bcc → hcp lattice stability for Mo is 42 kJ/mol. Given
60 this lattice stability, the only way in which the first-principles calculations within both the SQS and CPA approaches would match the experimental results would be for the calculated enthalpy of mixing to be very negative, which is not the case. In fact, as can be seen in Figure 3.10(a), the SQS and CPA calculations are very close to each other. On the other hand, the enthalpy of formation derived from the CALPHAD lattice stability in Figure 3.10(b) shows a better agreement than that from the first-principles lattice stability. It is important to note that the CALPHAD lattice stability was obtained through the extrapolation of phase boundaries in phase diagrams with Mo and stable hcp elements and, therefore, are empirical. The reason why such an empirical approach would yield a much better agreement with experimental data is still the source of intense debate within the CALPHAD community and has not been resolved as of now. The main conclusion of this section, however, is that the SQS’s were able to reproduce the thermodynamic properties of hcp alloys as good as or better than the CPA method while at the same time allowing for the ion positions to locally relax around their equilibrium positions.
3.5.9
IVA transition metal alloys
The group IVA transition metals, Ti, Zr, and Hf have hcp structure at low temperatures and transform to bcc at higher temperatures due to the effects of anharmonic vibrations. When they form a binary system with each other, they show complete solubility for both the hcp and bcc solutions without forming any intermetallic compound phases in the middle. The Hf-Ti binary is reported to have a low temperature miscibility gap and was modeled with a positive enthalpy of mixing by Bittermann and Rogl [32]. Figure 3.11(a) shows remarkable agreement between the fully relaxed first-principles calculations and the thermodynamic model, which was obtained by fitting the experimental phase boundary data. Despite the fact that the local relaxation energies are rather large (∼ 4 kJ/mol), the lattice parameters in both cases agree with each other and with the experimental results[34–36]. In the case of the Ti-Zr binary, although no low-temperature miscibility gap has been reported, Kumar et al. [33] found that the enthalpy of mixing for the hcp
61 50
Enthalpy of formation (kJ/mol)
40 SQS:Fully relaxed SQS:Symmetry preserved CPA 1988 Kleykamp
30
20 10
0
-10
-20 0
0.25
0.5
0.75
1
Mole Fraction, Ru (a) Enthalpy of formation of hcp phase in the Mo-Ru system from SQS’s (this work) and CPA[28]. Total energy of hcp Mo is obtained from first-principles calculations in both cases.
Enthalpy of formation (kJ/mol)
15
10
SQS:Fully relaxed SQS:Symmetry preserved CPA 1988 Kleykamp
5
0
-5
-10
0
0.25
0.5
0.75
1
Mole Fraction, Ru (b) Enthalpy of formation of hcp phase in the Mo-Ru system from SQS’s and CPA. Total energy of hcp Mo is derived from the SGTE (Scientific Group Thermodata Europe) lattice stability
Figure 3.10. 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.
62
7
10
SQS:Fully Relaxed SQS:Symmetry Preserved 1994 Kumar et al.(assessment)
Enthalpy of mixing (kJ/mol)
Mixing Enthalpy (kJ/mol)
6 5
4 3
2 SQS:Fully Relaxed SQS:Symmetry Preserved 1997 Bittermann et al.(assessment)
1
8
6
4
2
0
0 0
0.25
0.5
0.75
1
0
0.25
Mole Fraction, Ti
0.5
0.75
(a) Calculated enthalpy of mixing for the hcp phase in the Hf-Ti system compared with a previous assessment[32]
(b) Calculated enthalpy of mixing for the hcp phase in the Ti-Zr system compared with a previous assessment[33]
1
Enthalpy of mixing (kJ/mol)
SQS:Fully Relaxed SQS:Symmetry Preserved 0.8
0.6
0.4
0.2
0 0
1
Mole Fraction, Zr
0.25
0.5
0.75
1
Mole Fraction, Zr
(c) Calculated enthalpy of mixing for the hcp phase in the Hf-Zr system. ∆Hmix ' 0
Figure 3.11. 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.
63 solutions in this binary was positive through fitting of phase diagram data. Our results confirm this finding, although with even more positive enthalpy. They are in fact similar in value to those calculated in the Hf-Ti alloys, suggesting that a low temperature miscibility gap may also be present in this binary. In the Hf-Zr system no miscibility gap has been reported. The hcp phase was modeled as an ideal solution (∆Hmix = 0) in the CALPHAD assessment[37]. The present calculations suggest that the enthalpy of mixing of this system is positive, although rather small. In this case, it is expected that any miscibility gap would only occur at very low temperatures. The three systems described in this section are chemically very similar, having the same number of electrons in the d bands. Electronic effects due to changes in the widths and shapes of the DOS of the d bands are not expected to be significant in determining the alloying energetics. Charge transfer effects are also expected to be negligible. The enthalpy observed can then be explained by only considering the atomic size mismatch between the different elements. As was shown in Table 3.5, the Hf-Zr hcp alloys have the smallest difference in their lattice parameter, thus explaining their very small positive enthalpy of mixing. As a final analysis of the ability of the generated SQS to reproduce the properties of random hcp alloys, Figure 3.12 shows the alloying effects on the electronic DOS in Ti-Zr hcp alloys. The figure also presents the results obtained through the CPA approach by Kudrnovsky et al. [38]. As can be seen in the figure, both calculations predict that the DOS corresponding to the occupied d states are virtually insensitive to alloying. The overall shape of the d -DOS remains relatively invariant. Since Ti and Zr have the same number of valence electrons, the fermi level remains essentially unchanged as the concentration varies from pure Zr to pure Ti. On the other hand, alloying effects are more pronounced in the d -DOS corresponding to the unoccupied states. Figure 3.12 shows how the broad peak at ∼ 4.5 eV of the d -DOS for Zr is gradually transformed into a narrow peak at ∼ 3.0 eV as the Ti content in the alloy is increased. The results from the CPA and the first-principles SQS calculations thus agree with each other, confirming the present results.
64
(b) Ti
Ti
Ti0.75Zr0.25
Ti0.75Zr0.25
DOS (arbitrary units)
DOS (arbitrary units)
(a)
Ti0.5 Zr0.5
Ti0.25Zr0.75
Ti0.5 Zr0.5
Ti0.25Zr0.75
Zr
-6
-4
Figure 3.12. CPA[38]
3.6
-2
0 2 4 Energy (eV)
6
Zr
8
-6
-4
-2
0 2 4 Energy (eV)
6
8
Calculated DOS of Ti1−x Zrx hcp solid solutions from (a) SQS and (b)
Conclusion
Periodic special quasirandom structures with 16 atoms for binary hcp substitutional alloys at three different compositions, 25, 50, and 75 at.%, have been created to mimic the pair and multi-site correlations of random solutions. The generated SQS’s were tested in seven different binaries and showed fairly good agreement with existing experimental lattice parameters either enthalpy of mixing and/or CALPHAD assessments. Analysis of the radial distribution and bond lengths in the 21 calculated SQS’s, yielded a detailed account of the local relaxations in the hcp solutions and has been shown the useful way of characterizing the degree relaxation over several coordination shells. It should also be noted that when using enthalpy of mixing to derive formation enthalpy to compare with experimental measurements, there can be a severe discrepancy between theoretical calculations and experimental data when the lattice stability, or structural energy difference, from first-principles calculation is problematic such as the Mo-Ru system in this work. This problem remains as an
65 unsolved issue. These SQS’s can be applied directly to any substitutional binary alloys to investigate the mixing behavior of random hcp solutions via first-principles calculations without creating new potentials, as in the coherent potential approximation (CPA) or calculating other structures in the cluster expansion. Although the size of the current SQS’s is not large enough to generate a supercell which can satisfy its correlation function at more than just three compositions (x = 0.25, 0.5, and 0.75 in A1−x Bx binary), calculations for these compositions can yield valuable information about the overall behavior of the alloys.
66
Bibliography [1] A. Zunger, S. H. Wei, L. G. Ferreira, and J. E. Bernard. Special Quasirandom Structures. Phys. Rev. Lett., 65(3):353–6, 1990. [2] S. H. Wei, L. G. Ferreira, J. E. Bernard, and A. Zunger. Electronic properties of random alloys: Special Quasirandom Structures. Phys. Rev. B., 42(15): 9622–49, 1990. [3] C. Jiang, C. Wolverton, J. Sofo, L.-Q. Chen, and Z.-K. Liu. First-principles study of binary bcc alloys using special quasirandom structures. Phys. Rev. B., 69(21):214202/1–214202/10, 2004. [4] G. Inden and W. Pitsch. Atomic ordering. In Phase transformations in materials, volume 5 of Materials science and technology: A comprehensive treatment, pages 497–552. VCH, Weinheim ; New York, 1991. [5] G. Kresse and J. Furthmuller. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci., 6(1):15–50, 1996. [6] G. Kresse and D. Joubert. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B., 59(3):1758–1775, 1999. [7] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais. Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation. Phys. Rev. B., 46(11):6671–87, 1992. [8] D. Shin, R. Arr´oyave, Z.-K. Liu, and A. van de Walle. Thermodynamic properties of binary HCP solution phases from special quasirandom structures. Phys. Rev. B., 74(2):024204/1–024204/13, 2006. [9] A. van de Walle, M. Asta, and G. Ceder. The alloy theoretic automated toolkit: A user guide. CALPHAD, 26(4):539–553, 2002. [10] I. A. Abrikosov, S. I. Simak, B. Johansson, A. V. Ruban, and H. L. Skriver. Locally self-consistent green function approach to the electronic structure problem. Phys. Rev. B., 56(15):9319–9334, 1997. [11] A. L. Spek. Single-crystal structure validation with the program PLATON. J. Appl. Crystallogr., 36(1):7–13, 2003. [12] A. A. Nayeb-Hashemi and J. B. Clark. The Mg–Zr (Magnesium–Zirconium) system. Bulletin of Alloy Phase Diagrams, 6(3):246–250, 1985.
67 [13] M. Asta, R. McCormack, and D. de Fontaine. Theoretical study of alloy phase stability in the cadmium-magnesium system. Phys. Rev. B., 48(2): 948–66, 1993. [14] R. R. Hultgren. Selected values of thermodynamic properties of metals and alloys. Wiley series on the science and technology of materials. Wiley, New York, 1963. [15] W. Hume-Rothery and G. V. Raynor. Equilibrium and lattice-spacing relations in the system magnesium-cadmium. Proc. Roy. Soc. (London), A174: 471–86, 1940. [16] F. W. Von Batchelder and R. F. Raeuchle. Lattice constants and brillouin zone overlap in dilute magnesium alloys. Phys. Rev., 105:59–61, 1957. [17] D. Hardie and R. N. Parkins. Lattice spacing relations in magnesium solid solutions. Philosophical Magazine, 4(8):815–25, 1959. [18] M. H¨am¨al¨ainen, N. Bochvar, L. L. Rokhlin, and K. Zeng. Thermodynamic evaluation of the Cu-Mg-Zr system. J. Alloys Compd., 285(1-2):162–166, 1999. [19] R. Arr´oyave, D. Shin, and Z.-K. Liu. Modification of the thermodynamic model for the Mg-Zr system. CALPHAD, 25:230–238, 2005. [20] H. Liang, S. L. Chen, and Y. A. Chang. A thermodynamic description of the Al-Mg-Zn system. Metall. Trans. A, 28A(9):1725–1734, 1997. [21] I. Ansara, A. T. Dinsdale, and M. H. Rand. Thermochemical Database for Light Metal Alloys, volume 2 of COST 507: Definition of Thermochemical and Thermophysical Properties to Provide a Database for the Development of New Light Alloys. European Cooperation in the Field of Scientific and Technical Research, 1998. [22] Y. Zhong, M. Yang, and Z.-K. Liu. Contribution of first-principles energetics to Al-Mg thermodynamic modeling. CALPHAD, 29(4):303–311, 2005. [23] H. L. Luo, C. C. Chao, and P. Duwez. Metastable solid solutions in aluminummagnesium alloys. Trans. AIME, 230(6):1488–90, 1964. [24] W. Hume-Rothery and G. V. Raynor. The apparent sizes of atoms in metallic crystals with special reference to aluminum and indium, and the electronic state of magnesium. Proc. Roy. Soc. (London), A177:27–37, 1941. [25] G. V. Raynor. The lattice spacings of the primary solid solutions in magnesium of the metals of group III B and of tin and lead. Proc. Roy. Soc. (London), A180:107–21, 1942.
68 [26] R. S. Busk. Lattice parameters of magnesium alloys. Trans. AIME, 188 (Trans.):1460–4, 1950. [27] L. G. Wang, M. Sob, and Z. Zhang. Instability of higher-energy phases in simple transition metals. J. Phys. Chem. Solids, 64:863–872, 2003. [28] A. E. Kissavos, S. Shallcross, V. Meded, L. Kaufman, and I. A. Abrikosov. A critical test of ab initio and CALPHAD methods: The structural energy difference between bcc and hcp molybdenum. CALPHAD, 29(1):17–23, 2005. [29] N. Saunders and A. P. Miodownik. CALPHAD (Calculation of Phase Diagrams) : A Comprehensive Guide. Pergamon, Oxford ; New York, 1998. [30] Y. Wang, S. Curtarolo, C. Jiang, R. Arroyave, T. Wang, G. Ceder, L.-Q. Chen, and Z.-K. Liu. Ab initio lattice stability in comparison with CALPHAD lattice stability. CALPHAD, 28(1):79–90, 2004. [31] H. Kleykamp. Thermodynamics of the molybdenum-ruthenium system. J. Less-Common Met., 144(1):79–86, 1988. [32] H. Bittermann and P. Rogl. Critical assessment and thermodynamic calculation of the ternary system boron-hafnium-titanium (B-Hf-Ti). J. Phase Equlib., 18(1):24–47, 1997. [33] K. C. H. Kumar, P. Wollants, and L. Delaey. Thermodynamic assessment of the Ti-Zr system and calculation of the Nb-Ti-Zr phase diagram. J. Alloys Compd., 206(1):121–27, 1994. [34] M. A. Tylkina, A. I. Pekarev, and E. M. Savitskii. Phase diagram of the titanium-hafnium system. Zh. Neorg. Khim., 4:2320–22, 1959. [35] Y. A. Chang. Ternary phase equilibria in Transition Metal-Boron-CarbonSilicon systems. Technical Report AFML-TR-65-2, Part II, Vol. V, Air Force Materials Laboratory, 1966. [36] E. Rudy. Compilation of phase diagram data. Technical Report AFML-TR65-2, Part V, Air Force Materials Laboratory, 1969. [37] H. Bittermann and P. Rogl. Critical assessment and thermodynamic calculation of the ternary system C-Hf-Zr (carbon-zirconium-hafnium). J. Phase Equlib., 23(3):218–235, 2002. [38] J. Kudrnovsky, V. Drchal, M. Sob, and O. Jepsen. Electronic structure in random hexagonal-close-packed transition metal alloys by the tight-binding linear-muffin-tin-orbital coherent-potential method. Phys. Rev. B., 43(6): 4622–8, 1991.
Chapter
4
Special quasirandom structures for ternary fcc solid solutions 4.1
Introduction
Thermodynamic properties of a solution phase in a ternary or higher order system are usually extrapolated from the binary components plus ternary parameters. Since the most dominant interatomic reaction in a multicomponent system is that of the binaries, accurate thermodynamic descriptions which are capable of reproducing the characteristics of binary solution phases are prerequisites to a successful multicomponent thermodynamic modeling. In this regard, considerable efforts have been made to combine thermodynamic descriptions for binary solution phases to be used in higher order systems [1–6]. Obtaining accurate thermochemical data for solid solution phases is difficult even for a binary since it is hard to reach a complete thermodynamic equilibrium at low temperatures where solid phases are stable and formation of compounds. As the number of elements increases in a multicomponent system, the complexity of acquiring reliable data also increases. Consequently, interaction parameters for the excess Gibbs energy of binary solid solution phases are usually evaluated only from phase diagram data and that of ternary are usually omitted due to the lack of data. In this chapter, two different fcc SQS’s in an A-B-C ternary system, with the
70 compositions being at xA = xB = xC =
1 3
and xA = 12 , xB = xC = 14 , are devel-
oped to investigate the enthalpy of mixing for ternary fcc solid solutions. In this chapter, the impact of ternary interaction parameters on a ternary solution phase is briefly reviewed first. Then the generated ternary fcc SQS’s are characterized in terms of their atomic arrangement to reproduce the pair and multi-site correlation functions of completely random fcc solid solutions. Finally, the generated SQS’s are applied to the Ca-Sr-Yb system which supposedly has fcc solid solution phases throughout the entire composition range in all three binaries and ternary without order/disorder transitions.
4.2
Ternary interaction parameters
The Gibbs energy of a solution phase, φ, with c elements are expressed as φ
G =
c X
xi Go,φ i
+ RT
i=1
c X
xi ln xi +xs Gbin,φ +xs Gtern,φ + · · ·
(4.1)
i=1
where xs Gbin,φ and xs Gtern,φ are the excess Gibbs energies of the subordinate binary and ternary systems, respectively. The excess Gibbs energies for binary and ternary systems can be further described as: xs
bin
G
=
c−1 X c X i=1 j>i
xs
tern
G
=
c−2 X c−1 X c X
x i xj
n X
υ
Lφij (xi − xj )υ
(4.2)
υ=0
xi xj xk (Lφi xi + Lφj xj + Lφk xk )
(4.3)
i=1 j>i k>j
If all three L-parameters are identical, as in a regular solution[4], Lφi = Lφj = Lφk = Lφijk
(4.4)
then the ternary excess Gibbs energy shown in Eqn. 4.3 can be further simplified to: xs
tern
G
=
c−2 X c−1 X c X i=1 j>i k>j
xi xj xk Lφijk
(4.5)
71 since xi + xj + xk = 1 in a ternary. The Gibbs energy of solution phases described above can be equally applied to a liquid phase and solid solution phases in binary and ternary systems. For a liquid phase, ternary interaction parameters are often used to fit the experimentally measured phase diagram data and, if possible, thermochemical data. The effect of ternary interaction parameters on the liquidus lines at various temperatures in the Al-Mg-Si system from the COST507 database[7] is shown in Figure 4.1. The three independent ternary interaction parameters for the liquid phase, LAl , LMg , and LSi , did not significantly change the liquidus projection of the Al-Mg-Si system. Figure 4.2 shows the effect of ternary interaction parameters on a thermodynamic property, activity in this case, of the fcc phase in the Al-Mg-Si system. Contrary to the effect of ternary interaction parameters on the phase diagram, the activity of a ternary system can vary quite significantly with ternary parameters. For a solid solution phase, ternary interaction parameters are usually set to zero since experimental data for ternary solid solutions are scarce. Even though it is difficult to get any data for ternary solid solutions and the multicomponent interatomic reactions tend to be weak, to model a ternary solid solution phase as ideal —by using zero value parameter for ternary interactions— may cause a problem when ternary interactions in a system is not negligible. For example, when extrapolated Gibbs energies from constituent binaries are not accurate enough to describe the thermodynamic characteristics of a ternary solution phase, then the Gibbs energy of a new phase in the ternary, such as a ternary compound, is forced to have incorrect values in order to satisfy ternary phase equilibria from experimental observations. However, such incorrect Gibbs energy for the ternary solid solution phase will be inherited to the higher order system without being noticed since the phase diagram data can be reproduced with the erroneous Gibbs energy. Therefore, thermochemical data for ternary solid solution phases are indeed necessary to evaluate ternary interaction parameters accurately for both phase diagram data and thermodynamic properties. Ternary SQS’s, as shown in binary SQS’s, can provide thermodynamic properties of solid solution phases, such as enthalpy of mixing, which can be readily used in a thermodynamic modeling.
72
1.0
fS i in
liq u
id
0.9
0.6
no rac tio
0.7
0.3
1300K
0.5
1200K
0.4
le F Mo
0.8
1100
K
K 1000
1300K 1200K
0.2 1100K
0.1
1000K
0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction of Mg in liquid
Figure 4.1. Liquidus lines at various temperatures in the Al-Mg-Si system from the COST507 database[7]. Ternary interaction parameters for the liquid phase are LAl = +4125.86 − 0.51573T , LM g = −47961.64 + 5.9952T , and LSi = +25813.8 − 3.22672T . Dotted lines represent the liquidus lines without ternary interaction parameters.
4.3
Ternary fcc special quasirandom structures
Much like binary SQS’s, first-principles calculations of ternary SQS’s should be able to reproduce thermodynamic properties of ternary solid solutions since their atomic configurations, which are represented as correlation functions, are very close to that of ternary solid solutions. Therefore, understanding the correlation functions of a ternary solid solution is essential. Correlation functions of disordered structures are well derived in Inden and Pitsch [8]. In the following section, the correlation functions for binary and ternary systems are briefly summarized.
4.3.1
Correlation functions
The normalized correlation functions, Πk , in crystalline structures are defined as
73
1.0
Si
0.9 0.8
Al
Mg
Activity
0.7 0.6 0.5 Ternary parameter L in kJ/mol +20 0 -20 -50
0.4 0.3 0.2 0.1 0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Al
Figure 4.2. 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=0.
1 c2 ...ck Πk = Πc12...k =
1 X c1 −1 c2 −1 σ σ2 · · · σkck −1 N k site 1
(4.6)
where the sum is over all the distinctive k-site clusters, which are geometrically equivalent, in the N lattice sites structure. When k=1, 2, 3, . . . , then k-site clusters are point, pair, triplets, and so forth. The superscript ck takes values 2, 3, · · · , C, with C as the number of constituents, which represents a constituent ck at a given lattice site. Site operators are denoted as σk , where the subscript k indicates that the k-th constituent is located in the corresponding site of different clusters. For binary systems when C = 2, conventional values of the site operators σk are ±1 depending on whether a lattice site is occupied by A or B atoms. According to Eqn. 4.6, the normalized point correlation function for constituent site 2 (atom B P 2−1 site) is given as Π21 = N11 σ1 with σ1 = 1 or -1. It is noted here that the atom
74 sites do not need to be distinguished in a binary system since they are switchable. Assuming the fractions of A and B are xA and xB , respectively (xA + xB = 1), then Π21 = xA − xB . For a k-site cluster, the normalized correlation functions for the binary solid solution is formulated as Πk = (xA − xB )k
(4.7)
For ternary systems when C = 3, the values of the site operators σk take 1, 0, or -1 if a lattice site is occupied by A, B, or C atoms, respectively. The normalized point correlation function for B atom sites (the second constituent site) can be P 2−1 given as Π21 = N11 σ1 with σ1 = +1, 0, or, -1. For C atom sites (the third P 3−1 constituent site) the function can be given as Π31 = N11 σ1 with σ1 = +1, 0, or, -1. Assuming the fractions of A, B, and C are xA , xB , and xC , respectively (xA + xB + xC = 1), then Π21 = xA − xC and Π31 = xA + xC . The vanishing of xB is due to when the site operator is 0. For a k-site cluster with nB B atom sites and nC C atom sites (nB + nC = k), the normalized correlation functions for the ternary solid solution is denoted as Πk = (xA − xC )nB (xA + xC )nC with nB + nC = k
4.3.2
(4.8)
Generation of ternary SQS
In the present work, two different ternary fcc SQS’s are generated. The first SQS is at the equimolar composition where xA = xB = xC = xA =
1 , 2
xB = xC =
1 . 4
1 3
and the second is at
By switching the occupation of the A atoms in the second
SQS with either B or C atoms, two other SQS’s can be obtained where xB = 12 , xC = xA =
1 4
and xC = 12 , xA = xB = 41 . Therefore, enthalpy of mixing at four
different compositions in a ternary system can be determined from first-principles total energy calculations of ternary fcc SQS’s by ∆H(Aa Bb Cc ) ≈ E(Aa Bb Cc ) − xA E(A) − xB E(B) − xC E(C)
(4.9)
where E represents the total energy of each structure and the reference states for all pure elements are given as fcc. When the number of atoms in the SQS is less than 24, the Alloy Theoretic
75 Automation Toolkit (ATAT)[9] has been used to generate ternary fcc SQS’s. Since the ATAT enumerates all the atomic configurations within each supercell and then checks its correlation functions, the time needed to find SQS’s increases exponentially as the size of a supercell increases. For the sake of efficiency, to find SQS’s bigger than 24-lattice sites, a Monte-Carlo-like scheme[10] has been used. In each supercell with different lattice vectors, atom positions are randomly exchanged between the atoms and correlation functions of a supercell are calculated after every alternation. If correlation functions of the new state are getting closer to that of random solutions, then the new configuration is accepted. Otherwise the new state is discarded and another configuration will be generated from the previous one. This process continues until the atomic arrangement of a supercell converges to its closest correlation functions representing the completely random solution. In both methods, direct search via ATAT and Monte-Carlo-like scheme, a supercell whose correlation functions matches best with that of a completely random structure is chosen as the SQS at a given number of lattice sites. The selected SQS’s at two different compositions, SQS-24 when xA = xB = xC =
1 3
and SQS-36 when xA =
1 2
and xB = xC = 41 , are shown in Figure 4.3.
These two SQS’s are selected for later calculations because they are adequate with respect to the size and correlation functions in each concentration. Also the uncertainty from different sizes of SQS’s are converged within 1 meV /atom with these SQS’s. The space group of both structures are P1 with all the atoms at their ideal fcc sites. The correlation functions of the generated two SQS’s are given in Tables 4.1 and 4.2, respectively.
4.4
First-principles methodology
The Vienna Ab initio Simulation Package (VASP)[11] was used to perform the Density Functional Theory (DFT) electronic structure calculations. The projector augmented wave (PAW) method[12] was chosen and the generalized gradient approximation (GGA)[13] was used to take into account exchange and correlation contributions to the Hamiltonian of the ion-electron system. An energy cutoff of 364 eV was used to calculate the electronic structures of all the SQS’s. 5,000 kpoints per reciprocal atom based on the Monkhorst-Pack scheme for the Brillouin-
Table 4.1. Pair and multi-site correlation functions of ternary fcc SQS-N structures when xA = xB = xC = 31 . The number in the square bracket next to Πk,m is the number of equivalent figures at the same distance in the structure, the so-called degeneracy factor. SQS-N Random 3 6 9 15 18 24 36 48 Π2,1 [6] 0 0 0 0 0 0 0 0 0 Π2,1 [12] 0 0 0 0 0 0 0 0 0 Π2,1 [6] 0 0 0 0 0 0 0 0 0 Π2,2 [3] 0 0.25 -0.125 0 0 0 0 0 0 Π2,2 [6] 0 0 0 0 0 0 0 0 0 Π2,2 [3] 0 0.25 -0.125 0 0 0 0 0 0 Π2,3 [12] 0 -0.25 -0.0625 -0.0625 0 -0.01042 0 0 0 Π2,3 [24] 0 0 0 0 0 0 0 0 0 Π2,3 [12] 0 -0.25 -0.0625 -0.0625 -0.06667 -0.01042 0 0 0 Π2,4 [6] 0 0 0 0 -0.075 0 0 -0.04167 0 Π2,4 [12] 0 0 0 0 -0.01443 0.0842 0 0 -0.02255 Π2,4 [6] 0 0 0 0 -0.05833 0.09722 0.04167 -0.04167 0.09896 Π3,1 [8] 0 0.125 -0.01563 0.03125 0.04063 0.03125 0.01953 -0.00391 0.01953 Π3,1 [24] 0 0 0 0 -0.03789 0 0.01353 0.00226 0.00338 Π3,1 [24] 0 -0.125 0.01563 -0.03125 0.00938 -0.03125 -0.00391 -0.02734 0 Π3,1 [8] 0 0 0 0 -0.00541 0 0.01353 -0.00226 0.01691 Π3,2 [12] 0 -0.125 0.0625 0 -0.025 0 -0.01562 0 -0.00391 Π3,2 [24] 0 0 0 0 0.2165 0.01804 0 0.00902 0 Π3,2 [12] 0 0 0 0 0 -0.03608 0.02706 -0.01804 0.00677 Π3,2 [24] 0 0.125 -0.0625 0 -0.0125 0.01042 0.01563 0.01563 -0.00781 Π3,2 [12] 0 0.125 -0.0625 0 -0.025 -0.02083 -0.01562 -0.01042 -0.02734 Π3,2 [12] 0 0 0 0 0 0 0.00902 0 0.00226
76
Table 4.2. Pair and multi-site correlation functions of ternary fcc SQS-N structures when xA = 12 , xB = xC = 41 . The number in the square bracket next to Πk,m is the number of equivalent figures at the same distance in the structure, the so-called degeneracy factor. SQS-N Random 4 8 16 24 32 48 64 Π2,1 [6] 0.0625 0.0625 0.0625 0.0625 0.0625 0.0625 0.0625 0.0625 Π2,1 [12] 0 0 0 0 0 0 0 0 Π2,1 [6] 0 -0.0625 0 0 0 0 0 0 Π2,2 [3] 0.0625 -0.125 0.0625 0.0625 -0.0625 0.0625 0.0625 0.0625 Π2,2 [6] 0 0 0 0 0.09021 0 0 0 Π2,2 [3] 0 0.125 0.0625 -0.0625 0.08333 0 0 0 Π2,3 [12] 0.0625 0.0625 -0.00781 0.074219 0.0625 0.0625 0.0625 0.05957 Π2,3 [24] 0 0 0 0.006766 0.02255 0 -0.002255 0.003383 Π2,3 [12] 0 -0.0625 -0.10156 0.011719 0.02604 0 -0.002604 -0.006836 Π2,4 [6] 0.0625 -0.125 0.015625 0.085938 0.15625 0.0625 0.0625 0.361328 Π2,4 [12] 0 0 0 -0.040595 0 -0.05413 0.009021 -0.010149 Π2,4 [6] 0 0.125 -0.046875 -0.023437 0.07292 0 -0.03125 0.193359 Π3,1 [8] -0.015625 -0.015625 0.089844 -0.068359 -0.015625 -0.05518 0.001953 -0.015625 Π3,1 [24] 0 0 0 0.010149 -0.013532 0.00254 -0.003383 0.010149 Π3,1 [24] 0 0.015625 -0.058594 0.005859 -0.015625 0.01025 -0.005859 0.008789 Π3,1 [8] 0 0 0 -0.010149 0.013532 0.00761 0.02368 0.005074 Π3,2 [12] -0.015625 0.03125 -0.015625 0.019531 0.015625 -0.05078 -0.033203 -0.050781 Π3,2 [24] 0 0 0 0.010149 -0.036084 -0.01015 0 -0.025372 Π3,2 [12] 0 0 0 -0.040595 -0.027063 -0.02030 -0.003383 0.030446 Π3,2 [24] 0 -0.03125 0.046875 0.005859 0.007813 0 0.003906 -0.008789 Π3,2 [12] 0 -0.03125 0.078125 0.007813 -0.036458 -0.01172 0.013672 -0.017578 Π3,2 [12] 0 0 0 0.006766 0 0 0.001128 -0.020297
77
78
A
B
C
(a) SQS-24 when xA = xB = xC =
1 3
(b) SQS-36 when xA = 12 , xB = xC =
1 4
Figure 4.3. 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, P1.
zone sampling was used. In all first-principles calculations of ternary fcc SQS’s, structures are relaxed in two ways as in Chapter 3: full relaxation and symmetry preserved relaxation.
79
4.5
Results and discussions
In this work, the Ca-Sr-Yb system has been selected to apply the generated ternary fcc SQS’s which supposedly has complete solubility in the fcc phase for all binaries and ternary without any reported order/disorder transition. Both the Ca-Sr and Ca-Yb systems show complete solubility for both fcc and bcc phases at low and high temperatures respectively without intermetallic compounds[14, 15]. There is no reported phase diagram for the Sr-Yb system, however, from the similarity of the two binary systems, Ca-Sr and Ca-Yb, it can be postulated that Sr-Yb also has complete solubility for both fcc and bcc phases. Consequently, it can be cautiously expected that the combined ternary, the Ca-Sr-Yb system, would have the fcc solid solution phase throughout the entire composition range at low temperatures.
4.5.1
Binary SQS’s for the Ca-Sr-Yb system
Prior to applying the ternary SQS’s to the Ca-Sr-Yb system, the mixing behavior of the fcc phase in binaries was investigated through 8-atom binary fcc SQS’s at three different compositions, namely x=0.25, 0.5, and 0.75 in A1−x Bx alloys. Calculated enthalpies of mixing from binary fcc SQS’s are combined with experimental data from the literature to evaluate parameters for each binary. For the sake of simplicity, parameters for bcc have been modeled as identical to fcc. The congruent melting of bcc is observed in the Ca-Sr and Yb-Ca systems, thus the Sr-Yb system has been evaluated to have it as well on the assumption that the Sr-Yb system would have the same trend. The evaluated parameters are listed in Table 4.3 and the calculated phase diagrams of three binaries are shown in Figure 4.4. Calculated enthalpies of mixing for the fcc phase of the three binaries are shown in Figure 4.5 with first-principles calculations of binary fcc SQS’s. All nine SQS calculations have retained the fcc symmetry after the full relaxation and the difference between fully relaxed and symmetry preserved structures are at most ∼1 kJ/mol. It is intriguing to see that only the Sr-Yb system has an enthalpy of mixing close to zero among the three binaries in Figure 4.5, which implies that Sr-Yb is likely to have ideal mixing in the fcc phase. The other two systems, Ca-Sr and Yb-Ca, have rather negative (∼3 kJ/mol) enthalpies of mixing at the Ca-rich
80
1150
1100
Liquid
1958Sch
1100
Liquid
1050
1000
Temperature, K
Temperature, K
1050
950
bcc
900 850
bcc
1000 950 900
fcc
800 850 750
fcc
700
800 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Sr
(a) Calculated Ca-Sr phase diagram with experimental data[16]
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Yb
(b) Calculated Sr-Yb phase diagram
1150
Liquid
1100
Temperature, K
1050 1000
bcc 950 900 850 800
fcc
750 1968Sod
700 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Ca
(c) Calculated Yb-Ca phase diagram with experimental data[17]
Figure 4.4. 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 4.3.
81
Table 4.3. Thermodynamic parameters of the binaries in the Ca-Sr-Yb system evaluated in this work (in S.I. units). System Ca-Sr Sr-Yb Yb-Ca
Phase Liquid
Evaluated parameters 1 Liquid LLiquid Ca,Sr = 1609 − 14.825T , LCa,Sr = 263
0
bcc, fcc
0
Lbcc,fcc Ca,Sr = −11465
Liquid
0
LLiquid Sr,Yb = 2000
bcc, fcc
0
Lbcc,fcc Sr,Yb = 2672
Liquid
0
1 Liquid LLiquid Yb,Ca = −8229, LYb,Ca = 280
bcc, fcc
0
Lbcc,fcc Yb,Ca = −7668
Table 4.4. Cohesive energies of selected bivalent metals, Ca, Sr, and Yb, from Ref. [18]. Elements Ca Sr Yb
Cohesive energy (eV/atom) 1.84 1.72 1.60
side indicating that Ca tends to be ordered with Sr and Yb when mixed to form fcc solid solutions. The stronger ordering tendency of Ca over Sr and Yb in the Ca-Sr-Yb system can be simply explained since Ca has the largest cohesive energy of the three elements as shown in Table 4.4. It is widely believed that elements with larger cohesive energy have a more negative enthalpy of mixing and Figure 4.5 supports the relation. Sr also has an ordering tendency with Ca and Yb but not as strong as Ca. Yb has the smallest cohesive energy and intriguingly tends to weaken the ordering effects from Ca and Sr at the Yb-rich sides. The bond length analysis for the fully relaxed SQS’s in Table 4.5 shows that first nearest-neighbor average bond lengths follow Vegard’s law closely in all calculations. This observation means that the lattice parameter of the fcc solid solution varies linearly with the composition change and there is no significant distortion due to the ordering in all three binaries.1 1 Binary SQS calculations for the hcp solid solution in the Cd-Mg system, which has three ordered phases, Cd3 Mg, CdMg, and CdMg3 , show that the change of lattice parameters with composition do not follow Vegard’s law and agreed well with experimental measurements. See Figure 3.7(b).
0
2.0
-0.5
1.5 Enthalpy of Mixing, kJ/mol
Enthalpy of Mixing, kJ/mol
82
-1.0 -1.5 -2.0 -2.5
1.0 0.5 0 -0.5
-3.0 -3.5
-1.0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Sr
(a) Enthalpy of mixing for the fcc phase in the CaSr system
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Yb
(b) Enthalpy of mixing for the fcc phase in the SrYb system
0.5
Enthalpy of Mixing, kJ/mol
0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 -3.5 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Ca
(c) Enthalpy of mixing for the fcc phase in the YbCa system
Figure 4.5. 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.
83
Table 4.5. 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. Compositions A100 B0 A75 B25
A50 B50
A25 B75 A0 B100
4.5.2
Bonds A-A A-A A-B B-B A-A A-B B-B A-A A-B B-B B-B
Ca-Sr 3.887 3.956±0.045 4.018±0.036 4.003±0.012 4.054±0.040 4.057±0.032 4.126±0.058 4.156±0.017 4.140±0.032 4.204±0.048 4.253
Sr-Yb 4.253 4.180±0.039 4.120±0.030 4.147±0.007 4.086±0.039 4.055±0.047 3.995±0.092 3.983±0.005 3.989±0.035 3.924±0.054 3.819
Yb-Ca 3.819 3.842±0.002 3.843±0.002 3.842±0.001 3.884±0.005 3.883±0.006 3.880±0.005 3.888±0.003 3.889±0.002 3.887±0.003 3.887
Ternary SQS’s for the Ca-Sr-Yb system
It is shown from the binary fcc SQS’s calculations that there is an ordering due to Ca and Sr with other elements, while Yb tends to weaken the ordering effects from Ca and Sr. In this regard, it will be interesting to see the ternary interaction. First-principles calculations of ternary fcc SQS’s at four different compositions in the Ca-Sr-Yb system, namely xCa = xSr = xYb = 13 ; xCa = 12 , xSr = xYb = 14 ; xSr =
1 , 2
xCa = xYb =
1 ; 4
and xYb =
1 , 2
xCa = xSr =
1 , 4
have been considered
to investigate the ternary interactions. Three isoplethal sections, connecting the equimolar composition and three other compositions when xi = 1/2, xj /xk = 1, are selected to see the enthalpy of mixing for the Ca-Sr-Yb ternary system. Calculated enthalpies of mixing from ternary fcc SQS’s are shown in Figure 4.6 including extrapolated results from the three binaries and improved enthalpies of mixing to reproduce ternary fcc SQS’s results by introducing ternary interaction parameters. All the fully relaxed ternary SQS’s have preserved the fcc symmetry and enthalpy of mixing from first-principles calculations are close to zero within 1 kJ/mol. Figure 4.7 shows the radial distribution analysis of the fully relaxed SQS at the equimolar composition. The narrow distribution along each of the the bond-lengths corresponding to the ideal structure indicates that the effect of local relaxation is very small since enthalpy of mixing at the equimolar composition is almost zero. This is because the ordering of Ca and Sr are suppressed by the
1.5
1.5
1.0
1.0
0.5
0.5
Enthalpy of Mixing, kJ/mol
Enthalpy of Mixing, kJ/mol
84
0 -0.5 -1.0 -1.5 -2.0
Yb
-2.5
0 -0.5 -1.0 -1.5 -2.0
Yb
-2.5 Ca
Sr
Ca
-3.0
Sr
-3.0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Ca
(a) Enthalpy of mixing for the fcc phase in the CaSr-Yb system when xSr /xYb =1
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Sr
(b) Enthalpy of mixing for the fcc phase in the CaSr-Yb system when xCa /xYb =1
1.5
Enthalpy of Mixing, kJ/mol
1.0 0.5 0 -0.5 -1.0 -1.5 -2.0
Yb
-2.5 Ca
Sr
-3.0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Yb
(c) Enthalpy of mixing for the fcc phase in the CaSr-Yb system when xCa /xSr =1
Figure 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 (LCaSrYb = 46652 J/mol) and three independent ternary interaction parameters (LCa = 10636, LSr = 98254, and LYb = 31062 J/mol), respectively.
0
5
10
15
20
25
3
5
7
Distance, (Å)
6
8
9
(a) Radial distribution of Ca1 Sr1 Yb1 (∆Hmix ∼ 0)
4
Symmetry preserved Fully relaxed
10
Number of bonds 3
4
5
7 Distance, (Å)
6
8
Smoothed Fitted
9 (b) Smoothed and fitted RD’s of fully relaxed Ca1 Sr1 Yb1
0
1
2
3
4
5
10
Figure 4.7. Radial distribution analysis of Ca1 Sr1 Yb1 ternary fcc SQS’s. The dotted lines under the smoothed and fitted curves are the error between the two curves.
Number of bonds
30
85
86 presence of Yb. The tendencies of ordering and phase separation are balanced as zero at this composition. The Ca-rich ternary SQS, Ca2 SrYb, shows a negative enthalpy of mixing about -1 kJ/mol in Figure 4.6(a). This is not as negative as in the Ca-rich sides in binary calculations since the amount of Yb has been decreased with respect to the SQS at the equimolar composition. The CaSr2 Yb and CaSrYb2 calculations are even slightly positive and the extrapolations from binaries have the same tendency as well. As can be seen in Figure 4.6, the binary extrapolation cannot reproduce the enthalpy of mixing from ternary SQS’s in the Ca-Sr-Yb fcc solid solution. Thus, ternary interaction parameters are introduced to improve the ternary enthalpy of mixing. According to Eqn.4.3, the contribution from the ternary excess Gibbs energy for the fcc phase in the Ca-Sr-Yb system can be denoted as xs
fcc
G
=
c−2 X c−1 X c X
fcc fcc xCa xSr xYb (Lfcc Ca xCa + LSr xSr + LYb xYb )
(4.10)
i=1 j>i k>j
or xs
fcc
G
=
c−2 X c−1 X c X
xCa xSr xYb Lfcc CaSrYb
(4.11)
i=1 j>i k>j
as simplified in Eqn.4.5 when the ternary fcc is considered as a regular solution. When three independent ternary interaction parameters (LCa = 10636, LSr = 98254, and LYb = 31062 J/mol) are used, slightly better agreement with ternary SQS’s has been made than the identical interaction parameter (LCaSrYb = 46652 J/mol). The calculated enthalpies of mixing at the equimolar composition are evaluated as the same value regardless of the interaction parameters since all the data are equally weighted.2 It should be emphasized here that interaction parameters of a phase have to be evaluated with all the relevant data, such as phase diagram data, to reproduce overall properties, otherwise the thermodynamic description will be biased to the only data used (enthalpy of mixing in this case). Thus, the evaluation strategy of interaction parameters for the ternary fcc solid solution in the Ca-Sr-Yb system has to be considered carefully with other data. 2
LCaSrYb ' (LCa + LSr + LYb )/3
87
4.6
Conclusion
In the present work, two ternary fcc SQS’s at different compositions, xA = xB = xC =
1 3
and xA = 12 , xB = xC = 14 , are generated and their correlation functions
are satisfactorily close to that of random fcc solid solutions. Since there are no experimental data for ternary fcc solid solutions to compare with ternary SQS’s, the generated SQS’s are applied to the Ca-Sr-Yb system which supposedly has a complete solubility range without order/disorder transitions in ternary fcc solid solutions. Binary fcc SQS’s are applied to three binaries in the Ca-Sr-Yb system and show that mixing of binary fcc solid solutions are not ideal, and there are ordering due to Ca and Sr with other elements, while Yb tends to weaken those ordering effects. First-principles results of four ternary SQS’s at xCa = xSr = xYb =
1 ; 3
xCa =
xCa = xSr =
1 4
1 , 2
xSr = xYb =
1 ; 4
xSr =
1 , 2
xCa = xYb =
1 ; 4
and xYb =
1 , 2
preserved the fcc symmetry after the full relaxation. Enthalpy of
mixing from ternary SQS’s in the Ca-Sr-Yb is close to zero, and confirmed by the radial distribution analysis. It can be explained that the ordering of Ca and Sr with other elements are weakened by Yb as expected from binary interactions, and ternary fcc SQS’s could successfully reproduce ternary interactions. Therefore, it can be concluded that the generated ternary fcc SQS’s are able to reproduce thermodynamic properties of ternary fcc solid solutions and readily can be applied to other systems.
88
Bibliography [1] F. Kohler. Estimation of the thermodynamic data for a ternary system from the corresponding binary systems. Monatsh. Chem., 91:738–40, 1960. [2] G. W. Toop. Predicting ternary activities using binary data. Trans. AIME, 233(5):850–5, 1965. [3] Y. M. Muggianu, M. Gambino, and J. P. Bros. Enthalpies of formation of liquid alloys bismuth-gallium-tin at 723K. Choice of an analytical representation of integral and partial excess functions of mixing. J. Chim. Phys. Phys.-Chim. Biol., 72(1):83–8, 1975. [4] M. Hillert. Empirical methods of predicting and representing thermodynamic properties of ternary solution phases. CALPHAD, 4(1):1–12, 1980. [5] K. C. Chou. A general solution model for predicting ternary thermodynamic properties. CALPHAD, 19(3):315–325, 1995. [6] Z. Fang and Q. Zhang. A new model for predicting thermodynamic properties of ternary metallic solution from binary components. J. Chem. Thermodyn., 38(8):1079–1083, 2006. [7] I. Ansara, A. T. Dinsdale, and M. H. Rand. Thermochemical Database for Light Metal Alloys, volume 2 of COST 507: Definition of Thermochemical and Thermophysical Properties to Provide a Database for the Development of New Light Alloys. European Cooperation in the Field of Scientific and Technical Research, 1998. [8] G. Inden and W. Pitsch. Atomic ordering. In Phase transformations in materials, volume 5 of Materials science and technology: A comprehensive treatment, pages 497–552. VCH, Weinheim ; New York, 1991. [9] A. van de Walle, M. Asta, and G. Ceder. The alloy theoretic automated toolkit: A user guide. CALPHAD, 26(4):539–553, 2002. [10] I. A. Abrikosov, S. I. Simak, B. Johansson, A. V. Ruban, and H. L. Skriver. Locally self-consistent green function approach to the electronic structure problem. Phys. Rev. B., 56(15):9319–9334, 1997. [11] G. Kresse and J. Furthmuller. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci., 6(1):15–50, 1996. [12] G. Kresse and D. Joubert. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B., 59(3):1758–1775, 1999.
89 [13] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais. Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation. Phys. Rev. B., 46(11):6671–87, 1992. [14] C. Alcock and V. Itkin. The Ca–Sr(Calcium–Strontium) system. Bulletin of Alloy Phase Diagrams, 7(5):455–457, 1986. [15] Jr. Gschneidner, K. A. The Ca–Yb(Calcium–Ytterbium) system. Bulletin of Alloy Phase Diagrams, 8(6):521–522, 1987. [16] J. C. Schottmiller, A. J. King, and F. A. Kanda. The calcium-strontium metal phase system. J. Phys. Chem., 62:1446–9, 1958. [17] S. D. Soderquist and F. X. Kayser. Calcium-ytterbium system. J. LessCommon Met., 16(4):361–5, 1968. [18] C. Kittel. Introduction to solid state physics. Wiley, Hoboken, NJ, 8th edition, 2005.
Chapter
5
Solid solution phases in the Al-Cu-Mg-Si system 5.1
Introduction
Al-Cu-Mg-Si alloys possess excellent mechanical properties as light metal alloys[1] and have received renewed attention for automotive applications. The principal strengthening mechanism of the Al-Cu-Mg-Si alloys is the growth of precipitates within the aluminum matrix, mainly controlled by the heat treatment process. In this situation, optimization of the heat treatment conditions, such as temperature and composition, to promote precipitate strengthening are very important. Therefore, understanding the phase stability of the Al-Cu-Mg-Si quaternary system as a function of composition and temperature is essential. However, the Al-Cu-Mg-Si quaternary system is too complex to be studied only from experimental exploration of the phase stability. Theoretical calculations, such as formation energies of intermetallic compounds from first-principles studies[2], have been elaborated to calculate thermodynamic properties of individual phases to be used in the thermodynamic modeling. With the support from first-principles calculations[3–5] to determine thermodynamic properties of solid phases and newly available experimental data[6, 7], thermodynamic databases for the Al-Cu-Mg-Si system has been constantly updated. However, thermodynamic descriptions for solid solution phases —usually interaction parameters for excess
91 Gibbs energy— are mostly evaluated only from phase diagram data (binary, and sometimes ternary) or left as zero to be considered as ideal solutions (ternary or higher) since thermochemical data for solid solution phases is difficult to obtain from experiments. As discussed in the previous chapters, enthalpy of mixing for solid solution phases can be obtained from first-principles calculations of SQS’s. The calculated enthalpy of mixing can be combined with the experimentally determined phase diagram data for solid solutions and better thermodynamic descriptions for solid solutions can be obtained than ones evaluated solely from phase diagram data. In this chapter, all the previously developed SQS’s for substitutional binary solid solutions (bcc, fcc, and hcp) and ternary solid solution (fcc) in the previous chapters are applied to study the enthalpy of mixing for the solid solution phases.
5.2
Enthalpy of mixing for binary solid solutions
Determining reliable enthalpy of mixing data for a binary solution has great importance in the thermodynamic modeling of a binary system since the thermodynamic descriptions of a solution phase in a ternary or higher order system can be efficiently extrapolated from its constituent binaries. In principle, enthalpy of mixing for binary solid solutions can be obtained experimentally from various techniques, such as calorimetry and EMF (Electromotive Force) when there are no intermetallic compounds. However, the reactions at low temperatures are too slow to reach the equilibrium states within a reasonable amount of time. Therefore uncertainties of measurements are usually quite large for solid solutions. In order to supplement scarce and uncertain experimental enthalpy of mixing data for solid solutions, a considerable amount of effort has been made to estimate enthalpy of mixing for solid solutions from theoretical calculations. However, most representative methods to determine the enthalpy of mixing for solid solutions require calculating many supercells (cluster expansion) or developing a new potential which cannot consider the local relaxation properly (CPA). In the following sections, two approaches to calculate enthalpy of mixing for binary solid solution phases, simple Miedema’s model and SQS, are discussed.
92
5.2.1
Miedema’s model
Miedema’s model[8] simply treats metallic atoms as macroscopic pieces of atoms and applied to calculate enthalpy of mixing for bcc, fcc, and hcp solid solutions in 4-d transition metals satisfactorily[9]. The formalism of Miedema’s model is given as
2/3
∆HM ie = fAB
1/3
1/3
2xVA {−P (φ∗A − φ∗B )2 + Q[(nW S )A − (nW S )B ]2 − R} −1/3
−1/3
(5.1)
(nW S )A + (nW S )B
where fAB indicates the degree to which an A atom is surrounded by dissimilar neighboring B atoms and P, Q, and R are empirical proportionality constants. Miedema’s model requires three basic parameters: the molar volume V , the electron density at the Wigner-Seitz boundary nW S , and the electronegativity φ∗ . The molar volume is easily accessible via either experiments or calculations; the density nW S is proposed to be proportional to the ratio of (B/V )1/2 where B is the bulk modulus, which is also well defined. Therefore, V and nW S are easily determinable, while the third parameter, the electronegativity φ∗ , is rather difficult to obtain. As can be inferred from Eqn. 5.1, Miedema’s model is highly sensitive to the parameters used within. As a result, when the parameters of an element are not reliable, the result is unacceptable as shown by Chen and Podloucky [10] for Zr. Miedema’s model is indeed useful to approximate the mixing behavior of binary solid solutions. However, it is not accurate enough to be used in the thermodynamic modeling.
5.2.2
Binary special quasirandom structures
Unlike the simple Miedema’s model, more reliable enthalpy of mixing for solid solutions can be obtained from first-principles calculations of SQS’s because SQS’s are designed to reproduce the atomic arrangement of completely random solid solutions as shown in previous two chapters. Another advantage of SQS calculations is that it is more efficient than conventional cluster expansion and CPA methods. Since SQS’s are structural templates, they can be readily applied to any system by switching the atomic numbers and local relaxation can be considered by fully
93 relaxing the SQS’s within first-principles calculations. Therefore, first-principles study of binary SQS’s is the most efficient way to calculate the enthalpy of mixing for binary solid solutions in Al-Cu-Mg-Si. There are six binaries in the Al-Cu-Mg-Si quaternary system: Al-Cu, Al-Mg, Al-Si, Cu-Mg, Cu-Si, and Mg-Si. As shown in Table 5.1, most binary systems in Al-Cu-Mg-Si exhibit homogeneity ranges of solid solution phases though limited. Therefore, interaction parameters for the excess Gibbs energy of binary solution phases are required to reproduce those homogeneity ranges in phase diagram calculations. Table 5.1. Selected binary solid solution phases in the Al-Cu-Mg-Si system. Sublattice models are taken from previous thermodynamic modelings. System
Phase
Al-Cu
(Al) β (Cu) (Al) γ (Mg) (Al) (Si) (Cu) Cu2 Mg (Cu) κ β
Al-Mg
Al-Si Cu-Mg Cu-Si
Pearson symbol cF 4 cI2 cF 4 cF 4 cI58 hP 2 cF 4 cF 8 cF 4 cF 24 cF 4 hP 2 cI2
Prototype Cu W Cu Cu ∼(αMn) Mg Cu C(diamond) Cu Cu2 Mg Cu Mg W
Sublattice model (Al,Cu) (Al,Cu) (Al,Cu) (Al,Mg) (Mg)5 (Al,Mg)12 (Al,Mg)12 (Al,Mg) (Al,Si) (Al,Si) (Cu,Mg) (Cu,Mg)2 (Cu,Mg)1 (Cu,Si) (Cu,Si) (Cu,Si)
Composition range 0∼2.5 at.% Cu 71∼80 81.6∼100 0∼18.6 at.% Mg 45∼60.5 69∼100 0∼1.5 at.% Si 99.9984∼100 0∼7 at.% Mg 31∼35.3 0∼11.25 at.% Si 11.05∼14.5 14.2∼17.2
Reference [11]
[12]
[13] [14] [15]
The enthalpies of mixing for practical solid solution phases, i.e. fcc, hcp and bcc, of five binaries1 in the Al-Cu-Mg-Si systems are calculated from binary SQS’s at three different compositions, where x=0.25, 0.5, and 0.75 in A1−x Bx binary. For the Al-Si system, the diamond phase has also been considered. VASP[16] was used for first-principles calculations, and cutoff energy was set to be 25% larger than the default value to include more wavefunctions in each calculation. SQS’s were relaxed in two different ways: fully relaxed to consider the local relaxation effect and constrained to preserve the original symmetry in order to stay in the 1
The Cu-Si system will be separately discussed in Chapter 6.
94 same crystal structure2 . The calculated results of binary SQS’s are shown in Figures 5.1 through 5.5 and are compared with enthalpy of mixing from previous thermodynamic modelings, where available. • Al-Cu The Al-Cu system shows quite similar mixing behaviors in all solution phases, including the liquid phase. They all exhibit negative mixing of around -15 kJ/mol, asymmetric to the Cu-rich side. The bcc SQS of Al0.75 Cu0.25 could not retain the symmetry because the bcc phase is only stable around the Cu-rich side. The collapse of fcc Al0.25 Cu0.75 SQS can be explained with the existence of γ0 and γ1 phases. • Al-Mg The solid solution phases in the Al-Mg system commonly tend to have positive enthalpies of mixing around 1.5 kJ/mol in the middle. None of the fully relaxed bcc SQS’s could retain the bcc symmetry but the mixing behavior of symmetry preserved calculations is similar to those in the other solid solutions. • Al-Si The fcc Al0.75 Si0.25 SQS is the only structure to retain its original symmetry, while all the other SQS’s lost their original symmetry since the solubilities of the fcc and diamond phases are very limited. It should be noted that the enthalpy of mixing for the diamond phase is positive. • Cu-Mg All the SQS calculations of the Cu-Mg system have lost the original symmetry and their mixing is found to be positive at around 6∼8 kJ/mol in the middle. It is not surprising that the solubility of Mg in the Cu-rich fcc phase is only about 7 at% and that there is Cu2 Mg C15 laves phase with a homogeneity range and a line compound, CuMg2 . • Mg-Si The Mg-Si system is not listed in Table 5.1 since there is no solubility at all 2
Volume relaxation only for cubic structures, bcc, fcc, and diamond, and both the volume and shape relaxation for hcp to optimize the c/a ratio.
95
0
5
-2 Enthalpy of Mixing, kJ/mol
Enthalpy of Mixing, kJ/mol
0 -4 -6 -8 -10 -12 -14
-5 -10 -15 -20
-16 -18
-25 0
0.2
0.4 0.6 Mole Fraction, Cu
0.8
1.0
0
(b) bcc
0
0
-2
-2 Enthalpy of Mixing, kJ/mol
Enthalpy of Mixing, kJ/mol
(a) Liquid
-4 -6 -8 -10 -12
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Cu
-4 -6 -8 -10 -12 -14
-14
-16 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Cu
(c) fcc
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Cu
(d) hcp
Figure 5.1. Enthalpy of mixing for the solution phases in the Al-Cu system with firstprinciples 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.
0
2.5
-0.5
2.0 Enthalpy of Mixing, kJ/mol
Enthalpy of Mixing, kJ/mol
96
-1.0
-1.5
-2.0
-2.5
1.5
1.0
0.5
0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Mg
0
(b) bcc
3.0
3.0
2.5
2.5 Enthalpy of Mixing, kJ/mol
Enthalpy of Mixing, kJ/mol
(a) Liquid
2.0 1.5 1.0 0.5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Mg
2.0 1.5 1.0 0.5
0 -0.5
0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Mg
(c) fcc
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Mg
(d) hcp
Figure 5.2. Enthalpy of mixing for the solution phases in the Al-Mg system with firstprinciples 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.
0
0
-0.5
-0.5 Enthalpy of Mixing, kJ/mol
Enthalpy of Mixing, kJ/mol
97
-1.0 -1.5 -2.0 -2.5 -3.0 -3.5
-1.0 -1.5 -2.0 -2.5 -3.0 -3.5 -4.0
-4.0
-4.5 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
0
(a) Liquid
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
(b) fcc
10
0
9 -0.5 Enthalpy of Mixing, kJ/mol
Enthalpy of Mixing, kJ/mol
8 7 6 5 4 3 2
-1.0 -1.5 -2.0 -2.5
1 0
-3.0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
(c) diamond
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
(d) hcp
Figure 5.3. Enthalpy of mixing for the solution phases in the Al-Si system with firstprinciples 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.
98
0
6
-1 5 Enthalpy of Mixing, kJ/mol
Enthalpy of Mixing, kJ/mol
-2 -3 -4 -5 -6 -7 -8
4 3 2 1
-9 -10
0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Mg
0
(a) Liquid
(b) bcc
10
6
8
5 Enthalpy of Mixing, kJ/mol
Enthalpy of Mixing, kJ/mol
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Mg
6 4 2 0 -2
4 3 2 1
-4 -6
0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Mg
(c) fcc
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Mg
(d) hcp
Figure 5.4. Enthalpy of mixing for the solution phases in the Cu-Mg system with firstprinciples calculations of binary SQS’s (symbols) and previous thermodynamic modeling (solid lines)[3]. Open symbols represent symmetry preserved calculations of SQS’s.
99
0
0
-3
-0.2 Enthalpy of Mixing, kJ/mol
Enthalpy of Mixing, kJ/mol
-0.4 -6 -9 -12 -15 -18 -21
-0.6 -0.8 -1.0 -1.2 -1.4 -1.6 -1.8
-24
-2.0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
0
(b) bcc
0
0
-1
-0.5 Enthalpy of Mixing, kJ/mol
Enthalpy of Mixing, kJ/mol
(a) Liquid
-2 -3 -4 -5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Mg
-1.0 -1.5 -2.0 -2.5
-6 -7
-3.0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
(c) fcc
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
(d) hcp
Figure 5.5. Enthalpy of mixing for the solution phases in the Mg-Si system with firstprinciples calculations of binary SQS’s (symbols) and previous thermodynamic modeling (solid lines)[7]. Open symbols represent symmetry preserved calculations of SQS’s.
100 at either side. Nevertheless, interaction parameters for solid solution phases still need to be introduced for extrapolation to higher order system. Since all SQS calculations have lost the symmetry due to the structural instability, however, negative enthalpy of mixing curves have commonly obtained from the symmetry preserved calculations. It is intriguing to see that the tendency and even the absolute values for the enthalpies of mixing obtained from SQS’s in the same system are very close to each other regardless of the crystal structure except for the diamond phase in the Al-Si system. Conventional alloy theories, such as Hume-Rothery rules[18–20] and Darken-Gurry methods[21], can explain the similar mixing behavior of two elements in different structures. Both methods are not for estimating the enthalpy of mixing for binaries at all, but for predicting the solubility limit without considering the Gibbs energies of individual phases in a binary system. Furthermore, a solubility limit is strongly affected by intermetallic compounds. Thus these methods cannot be directly applied to estimate the enthalpy of mixing. Nevertheless, such conventional alloy theories are still useful since they can provide insight into how favorably two elements will form a solid solution. The well-known Hume-Rothery’s first rule states that the size difference of two elements must be less than 15% while the second rule requires them to have similar electronegativities in order to form extensive solid solubilities. Darken and Gurry [21] showed that Hume-Rothery’s first and second rule can be applied simultaneously to predict the formation of solid solutions. From the Darken-Gurry map shown in Figure 5.6, large solubility can be expected if the distance between two elements are close to each other. Among the distances between two elements in Al-Cu-Mg-Si, Cu and Si have the shortest in the Darken-Gurry map. Since Cu and Si are very similar to each other in terms of atomic radius and electronegativity, they form bcc, fcc, and hcp solid solution phases in the binary, as shown in Table 5.1. Solid solution phases in the Cu-Si system will be further discussed in Chapter 6. The negative enthalpies of mixing for solid solution phases in Al-Cu and Al-Si are attributed to their fairly close distances in the Darken-Gurry map. The distance between Cu and Mg in the Darken-Gurry map is the furthest among the distances between two elements in Al-Cu-Mg-Si, which is reflected with a positive calculated enthalpy
101
2.4 Mn(7)
2.2
Rh Cr
P
Electronegativity
2
Ni Co
1.8
Cu
Ir
Pd Pt
Tc Mo Ru Os W As Re Sb
Mn(5) Fe
Si
Ge
Au
Ga
V
Zn Ta
Be
Al
1.4
Tl
Bi Np Sn(4) Hg In Pu(5)
Ag
1.6
Pb
Te
U
Po
Sn(2) Po Cd Pu(4.75)
Nb Ti
Ce(4) Am
Hf Zr
Th Sc
Mg
1.2 1
1
1.2
1.4
Lu Er Y,Dy Tm Gd Pm Ho Ce(3) Tb Sm Nd La Pr
1.6
1.8
2
Atomic radius, (Å) Figure 5.6. The electronegativity vs the metallic radius for a coordination number of 12 (Darken-Gurry) map.
of mixing. However, the Darken-Gurry map cannot provide the reason for the positive enthalpy of mixing for the diamond phase in the Al-Si system while the enthalpies of mixing for the other structures are negative. Also, positive enthalpy of mixing for Al-Mg cannot be explained even though they are fairly close to each other in the Darken-Gurry map and vice-versa for Mg-Si. Recently, Gschneidner and Verkade [22] presented a semi-empirical approach, ECS2 method —the Electronic and Crystal Structures, Size method— to better understand the nature of solid solution formation by relating the electronic structure and the crystal structure to solid solution formation. In their work, group
102 IVB elements, C, Si, and Ge, have been categorized as directional elements due to the tetrahedral arrangement of atoms which have the sp3 electronic configuration. In this regard, the common metallic structures, such as bcc, fcc, and hcp, are less directional and there are many bonds since the coordination numbers are bigger than that of group IVB elements, as shown in Table 5.2. This is why the diamond solution phase in the Al-Si system shows different enthalpy of mixing with other phases. However, the reasons behind the positive enthalpy of mixing in the Al-Mg system and the negative enthalpy of mixing in the Mg-Si system remain unsolved from conventional alloy theory even considering the electronic structure.
Table 5.2. Coordination numbers of selected structures. Unit cell
Prototype
Coordination number
diamond
C(diamond)
4
bcc
W
8
fcc
Cu
12
hcp
Mg
12
Laves C15
Cu2 Mg
13.333a
a
Average coordination number from Ref. [23]
From this observation, mixing behavior of other solid solution phases in the same system may be postulated from simple SQS calculations, when the coordination numbers of two different structures are close to each other. For example, mixing of the Cu and Mg atoms of both sublattices in the laves phase Cu2 Mg, whose sublattice model is (Cu,Mg)2 (Cu,Mg), are evaluated to be positive as +13011 and +6599 J/mol respectively in the COST507 database[3] and first-principles results of all the SQS’s for Cu-Mg are also positive, as shown in Figure 5.4. It should be noted that average coordination number of Laves C15 is 13.333 and close to those of fcc and hcp, 12. Although it is a very crude approximation, it could be a good estimation when a designated phase is structurally too complex to be calculated from other theoretical calculation methods since the electron affinity of two elements to create a bond in a similar structure can be considered through the first-principles calculations of the SQS’s. Afterwards, enthalpy of mixing for the phase can be evaluated along with other relevant data, such as phase diagram data, in the thermodynamic modeling.
103
5.3
Ternary fcc solid solutions: Al-Cu-Mg, Al-Cu-Si, and Al-Mg-Si
Figure 5.7 schematically shows the enthalpy of mixing for the fcc solid solution in the Cu-Mg-Si ternary system from the COST507 database[3]. It is extrapolated from the binaries without any ternary interaction parameters.
0
0 -1 -2 -3 -4 -5 -6 -7 -8 -9
Enthalpy of Mixing (kJ/mol)
-3 -6 -9
Cu Mg Si Figure 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.
Due to scarce experimental data, the ternary interaction parameters for solid solution phases are usually assumed to be zero as in Cu-Mg-Si from the COST507 database, but ternary interactions can be important. Cluster expansion can be used to calculate the enthalpy of mixing for a ternary solid solution as in binary. However, the number of needed structures for ternary cluster expansion increases exponentially with one more degree of freedom than binary.3 First-principles cal3
It highly depends on the complexity of the structure. A couple of dozen structures are generally needed for a binary cluster expansion.
104 culations of ternary SQS’s at four different compositions in the A-B-C system, namely xA = xB = xC = 13 ; xA = 21 , xB = xC = 14 ; xB = 12 , xA = xC = 14 ; and xC = 12 , xA = xB = 14 , can comprehensively determine the enthalpy of mixing for ternary solid solution phases as shown in the previous chapter. In this section, the generated ternary fcc SQS’s are applied to three important ternary systems in Al alloys: Al-Cu-Mg, Al-Cu-Si, and Al-Mg-Si. First-principles results of ternary SQS’s in the three ternaries are compared with two thermodynamic modelings: the COST507 database[3] and the newly updated binaries combined. The COST507 database was compiled almost a decade ago and many binary systems in the COST507 project have been updated with newly available experimental data since then; including thermodynamic descriptions of binary fcc solid solutions. If the updated thermodynamic descriptions for binary fcc solid solutions are better than the previous ones, then the extrapolated ternary fcc solid solution must have been improved automatically. First-principles calculations of ternary fcc SQS’s for Al-Cu-Mg, Al-Cu-Si, and Al-Mg-Si systems are shown in Figures 5.8 through 5.10. • Al-Cu-Mg Four first-principles calculations of ternary SQS’s in the Al-Cu-Mg system well agree with the combined Al-Cu[4], Cu-Mg[24], and Mg-Al[5] databases to cc within 2 kJ/mol without any ternary interaction parameters for fcc. 0 LfCu,Mg
has been evaluated as quite positive (∼60 kJ/mol) by Buhler et al. [24] and is, thus, in good agreement with the first-principles data in Figure 5.8(a). For the other two isoplethal sections, both the trend and even absolute values are quite close between first-principles results and the combined binaries. It can be readily concluded that ternary interaction parameters for the fcc solid solution are unnecessary in this ternary system. • Al-Cu-Si The COST507 database and the combined three binaries, Al-Cu[4], Cu-Si[6], and Si-Al[17] databases, are very close to each other in the Al-Cu-Si system with a maximum discrepancy of ∼2 kJ/mol, but both are several kJ lower than the values from the ternary SQS’s. This indicates the use of ternary interaction parameters to make enthalpy of mixing less negative would be
105
15
15 Mg
10 Enthalpy of Mixing, kJ/mol
Enthalpy of Mixing, kJ/mol
10 5 0 -5 Mg
-10
Al
Cu
5 0 -5 -10
Al
Cu
-15
-15 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Al
(a) Enthalpy of mixing for the fcc phase in the AlCu-Mg system when xCu /xMg =1
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Cu
(b) Enthalpy of mixing for the fcc phase in the AlCu-Mg system when xAl /xMg =1
15
Enthalpy of Mixing, kJ/mol
10 5 0 -5 Mg
-10 Al
Cu
-15 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Mg
(c) Enthalpy of mixing for the fcc phase in the AlCu-Mg system when xAl /xCu =1
Figure 5.8. Enthalpy of mixing for the fcc phase in the Al-Cu-Mg system from firstprinciples 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].
0
0
-2
-2 Enthalpy of Mixing, kJ/mol
Enthalpy of Mixing, kJ/mol
106
-4 -6 -8 -10 Si
-12
-4 -6 -8 -10 Si
-12 Al
Cu
-14
Al
Cu
-14 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Al
(a) Enthalpy of mixing for the fcc phase in the AlCu-Si system when xCu /xSi =1
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Cu
(b) Enthalpy of mixing for the fcc phase in the AlCu-Si system when xAl /xSi =1
0
Enthalpy of Mixing, kJ/mol
-2 -4 -6 -8 -10 Si
-12 Al
Cu
-14 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
(c) Enthalpy of mixing for the fcc phase in the AlCu-Si system when xAl /xCu =1
Figure 5.9. Enthalpy of mixing for the fcc phase in the Al-Cu-Si system from firstprinciples 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].
2
2
1
1
0
0
Enthalpy of Mixing, kJ/mol
Enthalpy of Mixing, kJ/mol
107
-1 -2 -3 -4 -5
Si
-6
-1 -2 -3 -4 -5
Si
-6 Al
Mg
-7
Al
Mg
-7 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Al
(a) Enthalpy of mixing for the fcc phase in the AlMg-Si system when xMg /xSi =1
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Mg
(b) Enthalpy of mixing for the fcc phase in the AlMg-Si system when xAl /xSi =1
2
Enthalpy of Mixing, kJ/mol
1 0 -1 -2 -3 -4 -5
Si
-6 Al
Mg
-7 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
(c) Enthalpy of mixing for the fcc phase in the AlMg-Si system when xAl /xMg =1
Figure 5.10. Enthalpy of mixing for the fcc phase in the Al-Mg-Si system from firstprinciples 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].
108 desirable in this case since ternary interactions in this system are rather significant. As shown in Figure 5.6, Cu and Si are very closely located. Also it can be found in Table 5.1 that Cu-Si has three solid solution phases (bcc, fcc, and hcp) as well as four intermetallic compounds[6]. From these observation, it is obvious that there is ordering between Cu and Si. However, it has been weakened by the addition of Al, as shown in Figure 5.9(a). Similarly, addition of Si weakened the ordering between Al and Cu when xAl /xCu = 1. The addition of Cu promoted enthalpy of mixing toward negative direction since Cu tends to be ordered with both Si and Al. From these results, it can be concluded that ternary interaction parameters are necessary to reproduce the ternary interactions even though binary extrapolation results followed the mixing trend correctly. • Al-Mg-Si The discrepancy between COST507 and the combined three binaries, AlMg[5], Mg-Si4 , and Si-Al[17] databases, in the Al-Mg-Si are quite significant. The results from first-principles calculations in this ternary are pretty close to those from the binary combinations, indicating a weak ternary interaction in this system. The discrepancies along xMg /xSi = 1 and xAl /xSi = 1 can be attributed to the difference in the binary sides, while the congest discrepancy between two databases along xAl /xMg = 1 is due to the ternary interactions. It is related to the significant preference of Al-Si and Mg-Si bonds over Al-Mg bonds in the fcc solid solution as can be postulated from enthalpy of mixing for three binaries. As shown from three ternaries, first-principles calculations of ternary SQS’s are extremely valuable to judge whether or not ternary interaction parameters for a solid solution phase in the thermodynamic modeling are needed from only four calculations in a ternary system. 4
0
Interaction parameters for the binary fcc solid solution are evaluated from binary SQS’s as L ' -15 kJ/mol in the present work.
109
5.4
Conclusion
Binary SQS’s (bcc, fcc, and hcp) and ternary fcc SQS’s are applied to the Al-CuMg-Si system to calculate the enthalpy of mixing for solid solution phases. For binary, first-principles calculations of SQS’s at three different compositions have shown that enthalpy of mixing can be reliably obtained. It is also found that within the same system enthalpies of mixing for fcc, bcc, and hcp solid solutions are very close to each other regardless of the phase. The different mixing behavior of the diamond phase is due to the directional bond. Thus, it can be provisionally concluded that the enthalpy of mixing for a solid solution phase with a complicate structure might be estimated from that of simpler solid solutions, such as bcc, fcc, and hcp, as long as the coordination numbers are similar. Ternary fcc SQS’s were also successfully applied in calculating mixing of the ternary solid solutions in three ternary systems. Only four SQS’s, corresponding to four different compositions, are needed for a comprehensive understanding of enthalpy of mixing for ternary solid solutions.
110
Bibliography [1] J. Lee, Y. Han, H. Lee, and M. Kim. Microstructures and mechanical properties of squeeze cast Al-Si-Cu-Mg alloy. J. Korean Inst. Met. Mater., 32(10): 1259–1268, 1994. [2] C. Ravi and C. Wolverton. First-principles study of crystal structure and stability of Al-Mg-Si-(Cu) precipitates. Acta Mater., 52(14):4213–4227, 2004. [3] I. Ansara, A. T. Dinsdale, and M. H. Rand. Thermochemical Database for Light Metal Alloys, volume 2 of COST 507: Definition of Thermochemical and Thermophysical Properties to Provide a Database for the Development of New Light Alloys. European Cooperation in the Field of Scientific and Technical Research, 1998. [4] C. Jiang. Theoretical studies of aluminum and aluminide alloys using calphad and first-principles approach. Ph.d. thesis, The Pennsylvania State Univeristy, 2004. [5] Y. Zhong, M. Yang, and Z.-K. Liu. Contribution of first-principles energetics to Al-Mg thermodynamic modeling. CALPHAD, 29(4):303–311, 2005. [6] X. Yan and Y. A. Chang. A thermodynamic analysis of the Cu-Si system. J. Alloys Compd., 308(1-2):221–229, 2000. [7] X.-Y. Yan, F. Zhang, and Y. A. Chang. A thermodynamic analysis of the Mg-Si system. J. Phase Equlib., 21(4):379–384, 2000. [8] A. R. Miedema, F. R. de Boer, and R. Boom. Model predictions for the enthalpy of formation of transition metal alloys. CALPHAD, 1(4):341–359, 1977. [9] A. R. Miedema and A. K. Niessen. The enthalpy of solution for solid binary alloys of two 4d transition metals. CALPHAD, 7(1):27–36, 1983. [10] X.-Q. Chen and R. Podloucky. Miedema’s model revisited: The parameter φ∗ for Ti, Zr, and Hf. CALPHAD, 30(3):266–269, 2006. [11] J. L. Murray. The Al–Cu (Aluminum–Copper) system. Int. Met. Rev., 30: 211–233, 1985. [12] J. L. Murray. The Al-Mg (Aluminum-Magnesium) system. Bulletin of Alloy Phase Diagrams, 3(1):60–74, 1982. [13] J. L. Murray and A. J. McAlister. The Al–Si (Aluminum–Silicon) system. Bulletin of Alloy Phase Diagrams, 5(1):74–84, 89–90, 1984.
111 [14] A. A. Nayeb-Hashemi and J. B. Clark. The Cu–Mg (Copper–Magnesium) system. Bulletin of Alloy Phase Diagrams, 5(1):36–43, 1984. [15] R. Olesinski and G. Abbaschian. The Cu–Si(Copper–Silicon) system. Bulletin of Alloy Phase Diagrams, 7(2):170–178, 1986. [16] G. Kresse and J. Furthmuller. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci., 6(1):15–50, 1996. [17] T. Wang. Unpublished work. 2006. [18] W. Hume-Rothery, G. W. Mabbott, and K. M. C. Evans. The freezing points, melting points and solid solubility limits of the alloys of silver and copper with the elements of the B subgroups. Philosophical Transactions of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences, 233:1–97, 1934. [19] W. Hume-Rothery. The structure of metals and alloys. Monograph and report series ; no. 1. The Institute of Metals, London,, 1939. [20] W. Hume-Rothery, R. E. Smallman, and C. W. Haworth. The structure of metals and alloys. Monograph and report series ; no. 1. Institute of Metals ; Distributed in North America by the Institute of Metals North American Publications Center, London Brookfield, VT, 5th edition, 1988. [21] L. S. Darken and R. W. Gurry. Physical chemistry of metals. Metallurgy and metallurgical engineering series. McGraw-Hill, New York,, 1953. [22] Jr. Gschneidner, K. A. and M. Verkade. Electronic and crystal structures, size (ECS2 ) model for predicting binary solid solutions. Progress in Materials Science, 49(3-4):411–428, 2004. [23] D. P. Shoemaker and C. B. Shoemaker. Concerning the relative numbers of atomic coordination types in tetrahedrally close packed metal structures. Acta Crystallogr., Sect. B: Struct. Sci., B42(1):3–11, 1986. [24] T. Buhler, S. G. Fries, P. J. Spencer, and H. L. Lukas. A thermodynamic assessment of the Al-Cu-Mg ternary system. J. Phase Equlib., 19(4):317–333, 1998.
Chapter
6
Thermodynamic modeling of the Cu-Si system 6.1
Introduction
The key of the CALPHAD approach is to model the Gibbs energy functions of individual phases in a system as a function of temperature and, if possible, pressure as well. The thermodynamic modeling process starts with collecting all the experimental data or theoretical calculations related to a system. Two different types of data can be used in such thermodynamic modelings: thermochemical data and phase diagram data. Afterwards, the collected data are scrutinized and thermodynamic models of each phase are established. Parameters used in the modeling are evaluated in order to reproduce the accepted data. Thermochemical data, such as enthalpy of mixing and activity, are valuable in thermodynamic modeling since they are directly related to the Gibbs energy functions while phase diagram data can only provide the relationship between phases in equilibrium. However, favorable thermochemical data for solid phases are difficult to determine accurately from experiments or the uncertainty of measured data is too large. Thus, it is not surprising to have not enough thermochemical data for solid phases, such as intermetallic compounds and solid solution phases, to be used in a thermodynamic modeling. However, to include solid phases in a thermodynamic modeling without any relevant thermochemical data, there is no choice but
113 to evaluate parameters only from phase diagram data. It should be stressed here that a thermodynamic description of a phase should be able to reproduce both its thermochemical properties and phase equilibria with other phases. Therefore, evaluating thermodynamic parameters of a phase only from phase diagram data may pose a problem since there are infinite number of plausible sets of parameters to satisfy phase diagram data with incorrect Gibbs energy descriptions of the relevant phases. Previous thermodynamic modelings of the Cu-Si system[1, 2] exactly fall under these circumstances. Due to its importance in Al and Mg alloys, the Cu-Si system has been studied extensively; however, most of reported data are limited to thermodynamic properties of the liquid phase and phase diagram data. As a result, previous modelings[1, 2] which are evaluated from those data can correctly calculate phase diagram of the Cu-Si system from incorrect thermodynamic descriptions of intermetallic compounds. In this regard, first-principles calculations of solid phases are valuable when experimental data for thermodynamic properties are scarce. The total energy of a phase can be calculated from first-principles as long as structural information is available and then used to obtain formation energies for compounds in alloys[3, 4]. Combined with phase diagram data, such first-principles data work as constraints to prevent having incorrect thermodynamic descriptions in the parameter evaluation process. In the present work, the enthalpy of formation for -Cu15 Si4 is calculated from first-principles study. The enthalpies of formation for other intermetallic compounds are evaluated from their relationship with -Cu15 Si4 . The solid solution phases in the Cu-Si system, i.e. fcc, bcc, and hcp, are also calculated from firstprinciples via Special Quasirandom Structures (SQS)[5]. All the first-principles results are combined with phase diagram data and a better thermodynamic description of the Cu-Si system is obtained.
6.2
Review of previous work
The Cu-Si system was comprehensively assessed by Olesinski and Abbaschian [6] with 11 stable phases: liquid, fcc-Cu and diamond-Si, bcc (β) and hcp (κ) solid
114 solution phases, cubic intermediate phases ( and γ); tetragonal intermediate phase (δ); rhombohedral phase (η) and its low temperature forms (η 0 and η 00 ). The first thermodynamic description of the Cu-Si can be found in the COST 507 database[1] and modeled intermetallic compounds, η-Cu19 Si6 , γ-Cu56 Si11 , -Cu4 Si, δCu33 Si7 as stoichiometric compounds due to the lack of data. This database has been updated later by Yan and Chang [2]. The stoichiometry of the -phase has been changed to Cu15 Si4 and better agreement with experimentally determined phase diagram data is obtained. However, there were no published thermodynamic information for the solid phases so that their thermodynamic modelings were purely based on the thermodynamic information for the liquid phase and phase equilibrium data. The evaluated thermodynamic description of the liquid phase could successfully reproduce the accepted data in the modeling, while enthalpies of formation for the intermetallic compounds in their works were evaluated as positive when they should be negative. Figure 6.1 shows the calculated enthalpy of formation from the previous two thermodynamic modelings of the Cu-Si system [1, 2]. Correspondingly, entropies of formation for intermetallic compound phases were also evaluated as positive so as to reproduce the correct phase diagram. Such plausible thermodynamic descriptions are valid within the system since the relativity of Gibbs energies for individual phases are correct. However, when extrapolated to a higher order system, such problematic thermodynamic descriptions of a lower system force to have incorrect Gibbs energies of new phases in the higher order system. In this regard, any thermochemical data of a solid phase can be a pinning point to prevent having incorrect Gibbs energy of the phase.
6.3
First-principles calculations
First-principles calculations, based on density functional theory (DFT), can provide helpful insight into the characteristics of thermodynamic behavior of a solid phase[3, 4]. First-principles calculations determine the total energy of a phase at 0K. The systematic error from the implemented approximations are compensated by subtracting its reference states in order to calculate the enthalpy of formation. Furthermore, the enthalpy of formation derived from first-principles at 0K can be
115
Enthalpy of Formation, J/mol
1500
COST507 2000Yan
1000
500
0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
Figure 6.1. Enthalpies of formation for the Cu-Si system from previous modelings[1, 2]. Reference states for Cu and Si are fcc and diamond, respectively.
treated as that at 298.15K since enthalpies of formation are often independent of temperature. Entropy of formation, then, can be optimized with the phase diagram data. Therefore, the enthalpy of formation from first-principles calculation has great importance in the CALPHAD approach as a good starting point to evaluate accurate Gibbs energy functions. In the following section, first-principles calculations of the solid phases in the Cu-Si system have been discussed to be used as supplementary experimental data within the CALPHAD modeling.
6.3.1
Intermetallic compounds
In principle, the enthalpies of formation for the ordered phases in the Cu-Si system can be determined from: ∆HfCua Sib = E(Cua Sib ) − xCu E f cc (Cu) − xSi E dia (Si)
(6.1)
116 where E’s are total energies and xi is the mole fraction of element, i. The existence of several intermetallic compounds are reported in Olesinski and Abbaschian [6], but only the crystal structure of -Cu15 Si4 has been reported by Morral and Westgren [7]. First-principles results of -Cu15 Si4 and its reference states, fcc-Cu and diamond-Si, are shown in Table 6.1. Table 6.1. First-principles results of -Cu15 Si4 and its Standard Element Reference (SER), fcc-Cu and diamond-Si. By definition, ∆Hf of pure elements are zero.
Phases
Space
Lattice
Total energy
∆Hf
parameter (˚ A)
(eV/atom)
(kJ/mol-atom)
-Cu15 Si4
Group ¯ I 43d
9.726
-4.0749
-5.7542
3.632
-3.6376
-
5.468
-5.4315
-
F d¯3m ¯ diamond-Si F m3m fcc-Cu
As briefly introduced before, enthalpies of formation for the intermetallic compounds in the Cu-Si system should have negative values since the existence of intermediate phases indicates the tendency of ordering between Cu and Si atoms. The calculated enthalpy of formation for the phase clearly verifies that it should be negative and the enthalpies of formation for the other intermetallic compounds should also be negative.
6.3.2
Solid solution phases
The Cu-Si system has three solid solution phases, namely bcc, fcc, and hcp. Although their homogeneous ranges are not notably wide in the Cu-Si system, thermodynamic descriptions for these phases have to be reliable throughout the entire composition, including metastable regions, since the extrapolation to a higher order system has to be considered. However, evaluating the parameters for these solid solution phases only from the phase diagram would be rather ambiguous since there are infinite number of solutions are possible. Special Quasirandom Structures (SQS) proposed by Zunger et al. [5] have been successfully applied to calculate the mixing energies of binary solid solutions for fcc[8], bcc[9], and hcp[10] phases. Only SQS’s at the simple compositions, i.e. x=0.25, 0.5, and 0.75 can be obtained in the A1−x Bx substitutional solutions, but
117 three data points are very useful in investigating the mixing behavior of a solid solution in a binary system. In the present work, enthalpy of mixing for three solid solution phases are calculated from SQS’s and used in the thermodynamic modeling of the Cu-Si system.
6.3.3
Methodology
The Vienna Ab initio Simulation Package (VASP)[11] was used to perform the electronic structure calculations based on Density Functional Theory (DFT). The projector augmented wave (PAW) method[12] was chosen and the generalized gradient approximation (GGA)[13] was used to take into account exchange and correlation contributions to the Hamiltonian of the ion-electron system. An energy cutoff of 461 eV was used to calculate the electronic structures of all the compounds. 5,000 k-points per reciprocal atom based on the Monkhorst-Pack scheme for the Brillouin-zone sampling was used.
6.4 6.4.1
Thermodynamic modeling Solution phases
The liquid and fcc-(Cu) phases are described with a one sublattice model which is equivalent to a substitutional model. Since the β-bcc phase of Cu-Si forms continuous solid solutions with the β phase of Al-Cu in the ternary Al-Cu-Si system, and the κ-hcp phase of Cu-Si extends into the ternary Al-Cu-Si system and Cu-MgSi system, these two intermediate phases are also modeled as disordered solution phases. The molar Gibbs energy of these four phases is described as: Gφ = xCu o GφCu + xSi o GφSi + RT (xCu ln xCu + xSi ln xSi ) + xs Gφ
(6.2)
where xi represents the mole fraction of component i. The first two terms on the right-hand side of the equation represent the Gibbs energy of the mechanical mixture of the components, the third term the ideal Gibbs energy of mixing, and the fourth term the excess Gibbs energy. The Redlich-Kister power series is used to represent excess Gibbs energy of these phases,
118
xs
φ
G = xCu xSi
X
k
LφCu,Si (xCu
− xSi )
k
(6.3)
k=0
with k
LφCu,Si = a + bT
(6.4)
where a and b are model parameters are evaluated from experimental data and first-principles calculations.
6.4.2
Ordered phases
For the sake of simplicity, the -Cu15 Si4 , η-Cu19 Si6 , γ-Cu56 Si11 and δ-Cu33 Si7 phases are treated as stoichiometric compounds. The unimportant low solubility of Cu in diamond-Si is negligible, and it is better to treat as an solute phase. The Gibbs energy of these five phases are described as: cc Gφ = xCu o GfCu + xSi o Gdia Si + ∆Gf
(6.5)
where ∆Gf = ∆Hf − T ∆Sf , represents the Gibbs energy of formation of the stoichiometric phase. ∆Hf and ∆Sf are enthalpy and entropy of formation, respectively. ∆Hf of -Cu15 Si4 is obtained from first-principles and the others are estimated from that of -Cu15 Si4 .
6.5
Results and discussions
The calculated enthalpy of formation for the Cu-Si system is shown in Figure 6.2 with first-principles calculations of -Cu15 Si4 showing satisfactory agreement. Enthalpies of formation for the other intermetallic compounds are evaluated correspondingly and the obtained convex hull for the Cu-Si seems reasonable. Enthalpy of mixing for the solid solution phases, i.e. bcc, fcc, and hcp, in the Cu-Si system are calculated and compared with the previous modeling[2] and first-principles results in Figure 6.3. Calculated results are satisfactorily close to first-principles results. An interesting observation in these calculations is that the mixing behavior of three different solid solutions are quite similar to each other as
119
0
Enthalpy of Formation, kJ/mol
-1 -2 -3 -4 -5 -6 -7 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
Figure 6.2. Calculated enthalpy of formation of the Cu-Si system with first-principles calculation of -Cu15 Si4 . Reference states are fcc-Cu and diamond-Si.
discussed in Chapter 5. Cu shows strong ordering tendency with Si in all three solid solutions while Si-rich sides have even positive enthalpy of mixing in fcc phase. The Cu-Si phase diagram is calculated as shown with experimental phase diagram data[20–24] in Figure 6.4 and all evaluated parameters for the Cu-Si system are listed in Table 6.2.
6.6
Conclusion
The complete self-consistent thermodynamic description of the Cu-Si system has been obtained. Enthalpies of formation for the intermetallic compounds are evaluated from first-principles calculations of -Cu15 Si4 , which was previously evaluated to have positive values. The enthalpy of mixing for the three solid solution phases, bcc, fcc, and hcp, are also obtained from first-principles calculations via Special Quasirandom Structures.
120
0
15
-2
-6 -8 -10 1977Igu (1393K) 1979Cas (1370K) 1981Arp (1600K) 1982Bat (1773K) 1997Wit (1900K) 2000Wit (1281K)
-12 -14 -16
Enthalpy of Mixing, kJ/mol
Enthalpy of mixing, kJ/mol
10 -4
0 -5 -10
-18
-15 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
(a) Calculated enthalpy of mixing of the liquid phase in the Cu-Si system with experimental data[14–19]. Reference states for both elements are the liquid phase.
0
0
2
-2
0
-4 -6 -8 -10
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
(b) Calculated enthalpy of mixing of the bcc phase. Reference states for both elements are fcc phase.
Enthalpy of Mixing, kJ/mol
Enthalpy of Mixing, kJ/mol
5
-2 -4 -6 -8 -10
-12
-12 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
(c) Calculated enthalpy of mixing of the fcc phase. Reference states for both elements are fcc phase.
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
(d) Calculated enthalpy of mixing of the hcp phase. Reference states for both elements are hcp phase.
Figure 6.3. 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].
121
1800 1600
Temperature, K
1400 1200 1000 800
1907Rud 1928Smi 1929Smi 1940Smi 1940And
600 400 200 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
Figure 6.4. Calculated phase diagram of the Cu-Si with experimental data[20–24] in the present work.
Table 6.2. 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]. Phases Liquid
Sublattice model (Cu,Si)
fcc
(Cu,Si)
bcc
(Cu,Si)
hcp
(Cu,Si)
η-Cu19 Si6 -Cu15 Si4 γ-Cu56 Si11 δ-Cu33 Si7
(Cu)19 (Si)6 (Cu)15 (Si)4 (Cu)56 (Si)11 (Cu)33 (Si)7
Evaluated descriptions 0 Liq LCu,Si = -38763 + 5.653T 1 Liq LCu,Si = -52442 + 25.307T 2 Liq LCu,Si = -29485 + 14.742T 0 fcc LCu,Si = -34176 + 7.017T 1 fcc LCu,Si = -26169 - 7.430T 0 bcc LCu,Si = -8742 - 12.281T 1 bcc LCu,Si = -68822 + 8.906T 0 hcp LCu,Si = -23124 -2.221T 1 hcp LCu,Si = -48482 + 4.615T o dia 0.76o Gfcc Cu +0.24 GSi -6255-1.801T o dia 0.789474o Gfcc Cu +0.210526 GSi -6136-1.386T o fcc o dia 0.835821 GCu +0.164179 GSi -6005-0.500T o dia 0.825o Gfcc Cu +0.175 GSi -5215-1.600T
122 It is shown that the challenge of scarce thermodynamic data for solid phases, even for solid solution phases, can be overcome by implementing first-principles results. Those first-principles results can work as constraints in the thermodynamic modeling to prevent having incorrect parameters of a phase.
123
Bibliography [1] I. Ansara, A. T. Dinsdale, and M. H. Rand. Thermochemical Database for Light Metal Alloys, volume 2 of COST 507: Definition of Thermochemical and Thermophysical Properties to Provide a Database for the Development of New Light Alloys. European Cooperation in the Field of Scientific and Technical Research, 1998. [2] X. Yan and Y. A. Chang. A thermodynamic analysis of the Cu-Si system. J. Alloys Compd., 308(1-2):221–229, 2000. [3] C. Wolverton, X. Y. Yan, R. Vijayaraghavan, and V. Ozolins. Incorporating first-principles energetics in computational thermodynamics approaches. Acta Mater., 50(9):2187–2197, 2002. [4] C. Colinet. Ab-initio calculation of enthalpies of formation of intermetallic compounds and enthalpies of mixing of solid solutions. Intermetallics, 11 (11-12):1095–1102, 2003. [5] A. Zunger, S. H. Wei, L. G. Ferreira, and J. E. Bernard. Special Quasirandom Structures. Phys. Rev. Lett., 65(3):353–6, 1990. [6] R. Olesinski and G. Abbaschian. The Cu–Si(Copper–Silicon) system. Bulletin of Alloy Phase Diagrams, 7(2):170–178, 1986. [7] F. R. Morral and A. Westgren. The crystal structure of a complex coppersilicon compound. Arkiv Kemi, Mineral. Geol., 11B(No. 37):6 pp., 1934. [8] S. H. Wei, L. G. Ferreira, J. E. Bernard, and A. Zunger. Electronic properties of random alloys: Special Quasirandom Structures. Phys. Rev. B., 42(15): 9622–49, 1990. [9] C. Jiang, C. Wolverton, J. Sofo, L.-Q. Chen, and Z.-K. Liu. First-principles study of binary bcc alloys using special quasirandom structures. Phys. Rev. B., 69(21):214202/1–214202/10, 2004. [10] D. Shin, R. Arr´oyave, Z.-K. Liu, and A. van de Walle. Thermodynamic properties of binary HCP solution phases from special quasirandom structures. Phys. Rev. B., 74(2):024204/1–024204/13, 2006. [11] G. Kresse and J. Furthmuller. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci., 6(1):15–50, 1996. [12] G. Kresse and D. Joubert. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B., 59(3):1758–1775, 1999.
124 [13] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais. Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation. Phys. Rev. B., 46(11):6671–87, 1992. [14] Y. Iguchi, H. Shimoji, S. Banya, and T. Fuwa. Calorimetric study the heat of mixing of copper alloys at 1120C. J. Iron Steel Inst. Jpn., 63(2):275–84, 1977. [15] R. Castanet. Thermodynamic investigation of copper + silicon melts. J. Chem. Thermodyn., 11(8):787–9, 1979. [16] I. Arpshofen, M. J. Pool, U. Gerling, F. Sommer, E. Schultheiss, and B. Predel. Experimental determination of the integral mixing enthalpies in the leadcopper and silicon-copper binary systems at 1600 k. Z. Metallkd., 72(12): 842–6, 1981. [17] G. I. Batalin and V. S. Sudavtsova. Thermodynamic properties of coppersilicon molten alloys. Izvest. Akad. Nauk SSSR, Neorg. Mater., 18(1):155–7, 1982. [18] V. Witusiewicz, I. Arpshofen, and F. Sommer. Thermodynamics of liquid Cu-Si and Cu-Zr alloys. Z. Metallkd., 88(11):866–872, 1997. [19] V. Witusiewicz, I. Arpshofen, H.-J. Seifert, F. Sommer, and F. Aldinger. Enthalpy of mixing of liquid Cu-Ni-Si alloys. Z. Metallkd., 91(2):128–142, 2000. [20] E. Rudolfi. The silicides of copper. Z. Anorg. Chem., 53:216–27, 1907. [21] C. S. Smith. The alpha-phase boundary of the copper-silicon system. J. Inst. Met., 476(advance copy):12, 1928. [22] C. S. Smith. The constitution of the copper-silicon system. Trans. AIME, 83: 414, 1929. [23] C. S. Smith. Constitution and microstructure of copper-rich silicon-copper alloys. Trans. AIME, 137:313, 1940. [24] A. G. H. Andersen. The alpha solubility limit and the first intermediary phase in the copper-silicon system. Trans. AIME, 137:334, 1940. [25] A. T. Dinsdale. SGTE data for pure elements. CALPHAD, 15(4):317–425, 1991.
Chapter
7
Thermodynamic modeling of the Hf-Si-O system 7.1
Introduction
The Hf-O system has been considered as one of the most important systems in various industrial fields, such as nuclear materials and high temperature/pressure materials. Hafnium dioxide is known as the least volatile of all the oxides and its high melting point, extreme chemical inertness, and high thermal neutron capture cross section make it suitable for use as control rods or neutron shielding. All of these reasons can make HfO2 a promising refractory material for future nuclear applications[1]. Recently, the Hf-O system has also attracted considerable attention for semiconductor materials. The thickness of current gate oxide material (SiO2 in general) in advanced complementary metal-oxide semiconductor (CMOS) integrated circuits has continuously decreased and reached the current process limit[2]. One solution to further improve their performance is to use alternative materials with higher dielectric constants (k ), such as ZrO2 and HfO2 [3, 4]. In this regard, thermodynamic stability calculations results showed that the interface between HfO2 and Si is found to be stable with respect to the formation of silicides whereas the ZrO2 /Si interface is not[5]. The other important group IVA transition metals, such as Ti and Zr, which show very similar behavior as Hf, with oxygen are modeled recently by Waldner
126 and Eriksson [6] and Wang et al. [7] respectively. All these systems commonly have a wide oxygen solubility ranges in the hcp phase, up to 33 at.%(Ti-O), 29 at.%(ZrO), and 20 at.%(Hf-O) at room temperature, as derived from higher temperature measurements. In the present work, the Hf-O system has been modeled with the existing experimental data and first-principles calculations results. Afterwards, combining with the thermodynamic parameters of the Hf-Si and the Si-O binary systems, the thermodynamic description of the Hf-Si-O ternary system is obtained, and the stability diagrams pertinent to thin film processing, such as the HfO2 -SiO2 pseudo-binary, the isopleth of HfO2 -Si, and isothermal sections are calculated.
7.2
Experimental data
7.2.1
Phase diagram data
7.2.1.1
Hf-O
Many investigations have been conducted to clarify the phase diagram of the Hf-O system[8–11]. The main features of the phase diagram of the Hf-O system include: the wide extent of the α-Hf solid solution, the congruent melting of HfO2 , and the allotropic transformations of HfO2 . The phase diagrams suggested by Rudy and Stecher [8] and Domagala and Ruh [9] are quite similar to each other, except the formation of the β-Hf phase and the eutectic reaction involving liquid, α and β-Hf phases. For the hafnium rich-side, Rudy and Stecher [8] proposed a eutectic reaction at 2273K, while Domagala and Ruh [9] suggested a peritectic reaction around 2523K. Similar to other group IVA transition metals, such as Ti, and Zr, it is strongly believed that Hf also has a peritectic reaction at the Hf-rich side. Another minor disagreement between these two phase diagram determinations is the eutectic reaction, Liquid → α + HfO2 . Rudy and Stecher [8] suggested composition of oxygen in the liquid as 40 at.% O and 2453±40K, while Domagala and Ruh [9] proposed 37 at.% O and ∼2473K. For the α-Hf solid solution, these two works[8, 9] are in quite good agreement with each other. Rudy and Stecher [8] found that α-Hf dissolves up to 20.5 at.% oxygen at 1623K and that the solubility range is almost independent of tempera-
127 ture; this shows consistency with observations by Domagala and Ruh [9] and they quoted a solubility of oxygen in α-Hf of 18.6 at.% at 1273K. Ruh and Patel [11] proposed a tentative phase diagram for the HfO2 -rich portion of the Hf-HfO2 system on the basis of metallographic data. They suggested the existence of solid solution regions for both cubic and tetragonal phases deviated from the stoichiometric composition of HfO2 . Since the important phases in the Hf-O system are the polymorphs of the HfO2 phases, i.e. monoclinic, tetragonal, and cubic, they have been studied extensively[12–16]. Allotropic transformations of the HfO2 phase have been well summarized by Wang et al. [17]. The suggested phase diagram of the Hf-O system by Massalski[18] is shown in Figure 7.1 .
HfO2
3200
o
2800
L 2231oC
Temperature, C
2810 C
2400
o
2500 C
cubic
1 3
2200oC
8
22
(βHf)
2000
tetragonal o
1670 C
o
1743 C
1600
(αHf) 1200
monoclinic
800 0
10
20
30
40
50
60
70
80
Atomic Percent Oxygen Figure 7.1. Proposed phase diagram of the Hf-O system from Massalski[18].
7.2.1.2
Hf-Si-O
Not many studies have been conducted regarding the phase stabilities of the Hf-SiO ternary system. Speer and Cooper [19] reported a ternary compound, Hafnon,
128 with the chemical formula of HfSiO4 and the crystal structure as I41 /amd. One of the most important phase diagrams of the Hf-Si-O system is the HfO2 -SiO2 pseudobinary that includes HfSiO4 since the phase stabilities of this pseudo-binary are pertinent to the processing of the dielectric thin film. Parfenenkov et al. [20] determined the melting of HfSiO4 at 2023±15K.
7.2.2
Thermochemical data
As discussed in the introduction, the Hf-O system has a wide range of oxygen solubility in the hcp phase. Hirabayashi et al. [21] studied the order/disorder transformation of interstitial oxygen in hafnium around 10 ∼ 20at.% by electron microscopy and neutron and X-ray diffractions. This work revealed that two types of interstitial superstructures are formed in the hypo- and hyper-stoichiometric compositions near HfO1/6 below 700K which have R¯3 and P ¯31c symmetries, respectively. For the completely disordered hcp phase at high temperature, Boureau and Gerdanian[22] measured the partial molar enthalpy of solution of oxygen in α-Hf solid solution at 1323K as a function of oxygen content using a Tian-Calvet-type microcalorimeter. Previously, it was almost impossible to measure the extremely low oxygen pressure in equilibrium with hafnium-oxygen solutions. Therefore derivation from the second law of thermodynamics was the only way to acquire thermodynamic information of solid solution phases[23, 24]. The major difficulty of direct measurement is making sure that all the hafnium surface is accessible at the same time to oxygen. Boureau and Gerdanian improved the accuracy of measurement by solving the geometrical effect of specimen and oxygen contact. The observed phase boundary of α-Hf in this work is consistent with that of previous phase diagram studies[8, 9] as O/Hf=0.255.
7.3 7.3.1
First-principles calculations Methodology
The Vienna Ab initio Simulation Package (VASP)[25] was used to perform the electronic structure calculations based on Density Functional Theory (DFT). The
129 projector augmented wave (PAW) method[26] was chosen and the general gradient approximation (GGA)[27] was used to take into account exchange and correlation contributions to the Hamiltonian of the ion-electron system. An energy cutoff of 500 eV was used to calculate the electronic structures of all the compounds. 5,000 k-points per reciprocal atom based on the Monkhorst-Pack scheme for the Brillouin-zone sampling was used. The k-point meshes were centered at the Γ point for the hcp calculations.
7.3.2
Ordered phases
The ordered structures of the Hf-Si-O system calculated in this work can be categorized into three groups. First, pure elements, i.e. hcp hafnium and diamond silicon are calculated for the reference states. Second, hypothetical compounds, the end-members of the α, β solid solutions (HfO0.5 and HfO3 ), are also calculated. The stable compounds, monoclinic HfO2 , quartz SiO2 , and the ternary compounds, HfSiO4 are calculated as well. The calculated results of the ordered structures are listed in Table 7.1. The enthalpy of those compounds are calculated from Eqn. 7.1:
Hfx Siy Oz
∆Hf
= H(Hfx Siy Oz −
x H(Hf) x+y+z
−
y H(Si) x+y+z
−
z H(O) x+y+z
(7.1)
where H corresponds to the enthalpies of the compound and reference structures. The reference states for Hf and Si were the hcp and diamond structures, respectively. In the case of condensed phases, the effects of lattice vibrations and other degrees of freedom (i.e. electronic, magnetic) can be neglected at low temperatures. Moreover, due to their rather small molar volumes, the P V contributions to their enthalpies (H ≡ U + P V ) can also be neglected. Thus their enthalpies ,H, can be replaced by the calculated first-principles total energies at 0K in Eqn. 7.1. For the oxygen gas, the selected reference state was diatomic oxygen, O2 . In this case, the contributions due to vibrational and translational degrees of freedom, as well as the P V work term, the molar volume of O2 is much larger than that of the condensed phases and cannot be neglected. In the following section, the determination of the correct reference state for O2 will be briefly discussed.
130
Table 7.1. First-principles calculation results of pure elements, hypothetical compounds (α, β-Hf), and stable compounds (HfO2 , SiO2 , and HfSiO4 ). By definition, ∆Hf of pure elements are zero. Reference states for all the compounds are SER. Phases
Space Group
HCP A3 (Hf)
P 63 /mmc ¯ F d3m
Diamond A4 (Si)
Lattice parameters (˚ A)
Total energy
∆Hf
a
b
c
(eV/atom)
(kJ/mol-atom)
3.198
3.198
5.053
-9.8320
-
5.468
5.468
5.468
-5.4315
-
-4.7936
-
Gas (O2 ) P¯ 3m1 Im¯ 3m
3.225
3.225
5.150
-9.9718
-175.511
4.364
4.364
4.364
-7.7253
-161.308
Monoclinic(HfO2 )
P 21 /c
5.135
5.194
5.314
-10.2101
-360.563
Quartz (SiO2 )
P 32 21
5.007
5.007
5.496
-7.9581
-284.809
HfSiO4
I41 /amd
6.616
6.616
6.004
-9.1024
-324.453
α-Hf (HfO0.5 ) β-Hf (HfO3 ) a
-1.769b a b
β=99.56◦ Reference states are monoclinic (HfO2 ) and quartz (SiO2 ).
7.3.3
Oxygen gas calculation
The selected reference state for oxygen corresponds to the diatomic molecule at 298K. Two oxygen atoms were placed in a ’big box’ to model the oxygen molecule gas and completely relaxed to find the lowest energy configuration. In order to properly account for the net magnetic moment of this molecule, spin polarization was considered. It was necessary to take into account the contributions of vibrational, rotational and translational degrees of freedom at finite temperature. Under the harmonic oscillator-rigid rotor approximation at temperatures greater than the characteristic rotational temperature— 2.07K for O2 —the internal energy of the O2 molecule is given by McQuarrie [28]: E(T ) = kB T
5 Θν Θν /T + + Θν /T 2 2 e −1
+ E0
(7.2)
where Θν is the characteristic vibrational temperature. The first term in Eqn. 7.2 corresponds to the contributions due to translational and rotational degrees of freedom and the second and third terms correspond to vibrational contributions.
131 The last term, E0 , corresponds to the energy of the ground electronic state at 0K. By including the P V = kB T term of an ideal, non-interacting gas in Eqn. 7.2, the enthalpy of the diatomic O2 molecule can be obtained. The difference between the ground state electronic energy and the enthalpy of O2 , per atom, is given by: O2
H(T )
−
E0O2
kB T = 2
7 Θν Θν /T + + Θν /T 2 2 e −1
(7.3)
In the case of O2 , the characteristic vibrational temperature, Θν is 2256K[28]. At 298K, the value of H(T )O2 − E0O2 is +0.0936 eV/atom or +9.03 kJ/mol-atom.
7.3.4
Interstitial solid solution phases: from SQS
In order to calculate the enthalpies of mixing of oxygen and vacancy for both hcp and bcc phases in the Hf-O system, special quasirandom structures (SQS)[29] are used which are supercells with correlation functions as close as possible to those of a completely random solution phase. In order to introduce the special quasirandom structure, it is convenient to understand the concept of correlation functions. Correlation functions, Πk,m , are defined as the products of site occupation numbers of different figures, k, such as point, pair, triplet (when k = 1, 2, 3, . . . ) and so forth. The correlation functions of each figure can be grouped together based on the distance from a lattice site as mth nearest-neighbors. In any atomic arrangements, the geometrical correlation between atoms in the structure can be defined as to whether it is ordered or disordered. For a completely disordered structure, the surrounding environment of an atom at any given sites should be the same as all the other lattice sites. For a binary system, a spin variable, σ = ±1, can be assigned to different types of atomic occupations and their products represent the correlation functions of binary alloys. The correlation function of a random alloy is simply described as (2x − 1)k in the A1−x Bx substitutional binary alloy, where x is the composition. Once such a supercell satisfies the correlation function of a target structure, it can be easily transferred to other systems by simply switching the types of atoms in the structure. The major drawback of SQS is that the concentration which can be calculated is typically limited to 25, 50, and 75 at.% since the correlation functions
132 for completely random structures other than those three compositions are almost impossible to satisfy with a small number of atoms. In principle, one can find a bigger supercell which has better correlation functions than smaller ones; however, such a calculation requires expensive computing. On the other hand, three data points from SQS calculations can give good indication of the mixing behavior of solution phases. Another disadvantage of SQS is that it cannot consider the long range interaction since the size of the supercell itself is limited. It is reported that SQS works well with a system where short range interactions are dominant[30, 31]. In order to consider the interstitial oxygen atoms in the solution phases, the sublattice models for the hcp and bcc phases are (Hf)1 (O,Va)0.5 and (Hf)1 (O,Va)3 , respectively. Three different compositions, i.e. yO = 0.25, 0.5, and 0.75 with yO representing the mole fraction of oxygen in the hcp and bcc interstitial sites, were considered and only two structures were generated in both phases since the structures of yO = 0.25 and 0.75 are switchable to each other. For the hcp phase, α-Hf solid solution, the total number of lattice sites considered were 24, 36, and 48. Since only the mixing between oxygen and vacancies are considered to generate a SQS for the hcp and bcc phases, the hafnium ions are excluded from the correlation function calculations. Therefore, the total number of oxygen and vacancies are 8, 12, and 16, respectively. For the bcc phase, β-Hf solid solution, the total number of lattice sites that were considered for the mixing of oxygen and vacancy were 12 and 24 with total number of sites being 16 and 32, respectively. The complete descriptions of the SQS’s for α and β solid solutions are listed in Tables 7.2 and 7.3, and their correlation functions are given in Tables 7.4 and 7.5. Finding bigger cells than these were prohibited by the limited computing resources. The generated SQS’s were fully relaxed, and relaxed without allowing local ion relaxations, i.e. only volume for bcc and volume as well as c/a ratio for hcp were optimized. Theoretically, all the first-principles calculations should be fully relaxed to find the lowest energy configurations. However, the structure should lie on the energy curve vs. geometrical degree of freedom of the same phase. If the fully relaxed final structure does not have the same crystal structure as initial input, it is not the phase of interest any longer. Thus it is necessary to force the structure to keep its parent symmetry. The calculated results of α and β solid solution phases are listed in Table 7.6
− 43 − 43 − 43
− 13 − 13
− 23 −2 23 −1 13
− 23 −2 31
−1 43
−1 23
−2 23 −4 31
− 43
− 13 −1 43
−1 23 −2 31
− 43
− 23 −3 31
−1 23 −3 31
− 23 −1 31
−1 23 −4 31
− 23 −3 31
−1 23 −3 31
− 23 −4 31
−1 23 −4 31
− 23 −2 31
−1 23
1 4 1 4 1 14 1 4 1 14 1 14
1 4 1 14 1 4 1 14 1 4 1 4
1 −1 1 −2 −2 1 −1 −3 −1
− 43
Va
1 3 −1 13 −1 13 −1 13
1 3 2 −1 3 − 23 −2 23
O
Lattice vectors
1 0 −1 −1 −2 0 −3 0 −1
36
24
SQS-N
50
Oxygen % 48
1 3 1 13 1 13 2 13 3 13 1 3 1 13 1 3
1 3 1 23 3 13 1 13 2 13 3 13 2 13 3 13
− 34
− 34
− 34
2 3 2 3 − 13 − 13 −1 13
−1 13
− 13
− 13
2 3 2 3 − 13 − 13
−1 43
−1 43
−1 43
− 34
− 34
− 34
− 34
− 34
−1 43
−1 43
−1 43
−1 43
−1 13 −1 43
− 13
− 13
−1 13
0 −2 0 0 0 −2 4 2 0
2
1 −1
48
1 3 1 3 2 −3 1 13 − 23 1 3 1 13 − 23 1 3 − 23 1 3 1 3 1 3 1 13 −1 23
2 3 2 3 1 32 − 31 − 31 2 3 2 3 2 3 − 31 − 31 −1 31 − 31 − 31 −1 31 2 3 − 31
−2 34
− 34
−1 34
−1 34
− 34
−1 34
−2 34
−2 34
−1 34
−1 34
−1 34
−1 34
−1 34
−2 34
−2 34
−2 34
0 −2 −1 −2 1 −2
− 23
75
Table 7.2. 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 ( 23 31 12 ).
133
134
Table 7.3. 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). Oxygen % SQS-N
50
Lattice vectors O
Va
0.5 −0.5 −0.5 0 −0.5 −0.5 0 0 0
16 0.5 1.5 −1.5 0 1 0 −0.5 1 0.5
1.5 0.5 −0.5 0.5 1 1 1 1.5 1.5
−0.5 1 0.5 −0.5 −0.5 0 −0.5 −1 0 −0.5 0 0.5 0 0.5 1 −0.5 0.5 1
−1 0 0 −1 −1 −0.5 −1 −0.5 −1 −1 −1 −0.5 −1 −0.5 −0.5
32 0 1 −2 0 0.5 0 0.5 0.5 −1 −0.5 −0.5 0 −0.5 −0.5 −1
0 −1 −2 −0.5 −1.5 −1.5 −1 −1 −1.5 −2.5 −1 −2 −2 −2 −2
−0.5 −1 −1 −0.5 −1 −0.5 −0.5 −1 −1 −0.5 −0.5 −0.5
−1 −1 −1.5 0 −1.5 −1.5 −1 0 −0.5 0 −1 −0.5
−2.5 −2.5 −2.5 −0.5 −2 −2 −3 −1.5 −1.5 −1 −1.5 −1
75 32 0 0 0 2 −1 0 −0.5 2 −0.5 1 −1 1 −1 0.5 0.5 2 1 2 −1 1.5 −1 1 −1 0.5 −1 1.5 −0.5 1.5 −0.5 1 −1 0.5 −0.5 0.5 −0.5 2 0.5 1.5 −0.5 1 −1 0.5 −1 2 −1 1.5 −0.5 1 −1 1.5 −0.5 0.5 −0.5 2
2 0 0 1.5 1.5 1.5 1.5 0.5 0.5 0.5 0.5 0.5 1 1 1 1 1 1 2 2 2 1.5 1.5 0.5 2 2 2
135
Table 7.4. 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 Πk,m is the number of equivalent figures at the same distance in the structure.
Oxygen %
50
75
SQS-N
Random
24
36
48
Random
48
Π2,1 [3]
0
0
0
0
0.25
0.25
Π2,2 [1]
0
0
0
0
0.25
0.25
Π2,3 [3]
0
0
0
0
0.25
0.25
Π2,4 [6]
0
0
-0.16667
0
0.25
0.20833
Π2,5 [3]
0
0
-0.11111
0
0.25
0.25
Π2,6 [3]
0
-0.16667
0.11111
0
0.25
0.41667
Π3,1 [2]
0
0
-0.33333
-0.25
0.125
0.25
Π3,1 [6]
0
0
0.11111
-0.08333
0.125
0.08333
Π3,2 [2]
0
0
0.33333
0.25
0.125
0.25
Table 7.5. Pair and multi-site correlation functions of SQS-N structures for β solid solution. The number in the square bracket next to Πk,m is the number of equivalent figures at the same distance in the structure.
Oxygen %
50
75
SQS-N
Random
16
32
Random
32
Π2,1 [6]
0
0
0
0.25
0.25
Π2,2 [12]
0
0
0
0.25
0.25
Π2,3 [12]
0
0
0
0.25
0.25
Π2,4 [6]
0
0.16667
0
0.25
0.33333
Π2,4 [3]
0
0
0
0.25
0.33333
Π2,5 [24]
0
-0.29167
0
0.25
0.25
Π2,6 [24]
0
-0.08333
0
0.25
0.25
Π3,1 [12]
0
-0.16667
0
0.125
0.16667
Π3,2 [8]
0
0
0
0.125
0
Π3,3 [48]
0
0.08333
0.08333
0.125
0.125
136 and 7.7. More detailed discussion of calculation results of special quasirandom structures and optimized results are found in a later section. Table 7.6. 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.
Oxygen
Atoms
Space
Total Energy
∆H mix
Group
(eV/atom)
(kJ/mol-atom)
Symmetry
%
Hf
O
Va
25
32
4
12
FR
P 63 /mmc
-9.9140
-61.9280
32
4
12
SP
P 63 /mmc
-9.9072
-61.2715
16
4
4
FR
P 63 /mmc
-9.9590
-109.481
16
4
4
SP
P 63 /mmc
-9.9444
-108.075
24
6
6
FR
P 63 /mmc
-9.9560
-109.187
24
6
6
SP
P 63 /mmc
-9.9418
-107.818
32
8
8
FR
P 63 /mmc
-9.9564
-109.230
32
8
8
SP
P 63 /mmc
-9.9413
-107.768
32
12
4
FR
P 63 /mmc
-9.9755
-146.428
32
12
4
SP
P 63 /mmc
-9.9541
-144.356
50
75
7.4 7.4.1
Thermodynamic modeling Hf-O
Seven phases are modeled in the Hf-O system: hcp, bcc, ionic liquid, gas, and three polymorphs of HfO2 : monoclinic, tetragonal, and cubic. Detailed discussions of individual phases are given below. 7.4.1.1
HCP and BCC
It has been reported that the stable solid phases of group IVA transition metals (Ti, Zr, and Hf), the hcp and bcc phases, dissolve oxygen interstitially into their octahedral sites[32]. The solid solutions of Hf-O are modeled with the two sublattice model, with one sublattice occupied only by hafnium and the other one occupied by both oxygen and vacancies:
137
Table 7.7. 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.
Oxygen
Atoms
Symmetry
%
Hf
O
Va
25
8
6
18
FR
8
6
18
SP
4
6
6
FR
4
6
6
SP
8
12
12
FR
8
12
12
SP
8
18
6
FR
8
18
6
SP
50
75
Space
Total Energy
∆H mix
Group
(eV/atom)
(kJ/mol-atom)
Pm ¯ Im3m
-9.9280
-227.299
-9.0165
-139.345
C2/m Im¯3m
-10.0088
-315.520
-8.5718
-176.866
Pm Im¯3m
-9.9636
-311.160
-8.6081
-180.370
Pm Im¯3m
-9.4727
-307.100
-8.1752
-181.908
(Hf)1 (O, Va)c where c corresponds to the ratio of interstitial sites to lattice sites in each structure. For the hcp phase, the ratio is derived as c =
1 2
from the pure hcp sublattice model,
consistent with the previous thermodynamic modeling of Ti-O[6] and Zr-O[33]. For the bcc phase, the stoichiometric ratio c is equal to 3 as in the conventional bcc phase. The Gibbs energies of the hcp and bcc phase can be described as:
GHCP,BCC = yO 0 GHf Oc + yV a 0 GHf V ac + cRT (yO ln yO + yV a ln yV a ) + Gxs m m Gex m = yO yV a
k X
k
L(Hf :O,V a) (T )(yO − yV a )k
(7.4) (7.5)
k=0
where o GHf Oc is the standard Gibbs energy of the hypothetical oxide HfOc , which is one of the end members that establishes the reference surface for this model; 0
GHf V ac corresponds to the standard Gibbs energy of the pure bcc and hcp phases
and the chemical interaction for oxygen and vacancies in the second sublattice is P given by yO yV a k L(Hf :O,V a) (T )(yO − yV a )k . This is identical to a Redlich-Kister
138 polynomial[34]. 7.4.1.2
Ionic liquid
The liquid phase region goes from pure liquid hafnium to stoichiometric liquid HfO2 . The ionic two-sublattice model is used for the liquid phase[35]: −v
(Ci+vi )P (Aj j , V a, Bk0 )Q where Ci+vi corresponds to the cation, i, with valence, +vi ; Aj to the anion, j, with valence, −vj ; Va are hypothetical vacancies added for electro-neutrality when the liquid is away from stoichiometry, having a valence equal to the average charge of 0 the cation, Q; and BK represents any neutral component dissolved in the liquid.
The numbers of sites in the sublattices, P and Q, are varied in such a way that electro-neutrality for all compositions is ensured with y being the site fractions:
P =
X
vj yAj + QyV a
(7.6)
vi yCi
(7.7)
j
Q=
X i
For the particular case of the Hf-O system, this two-sublattice model can be further simplified: (Hf +4 )4−2yO−2 (O−2 , Va−4 )4 The Gibbs energy expression for this system is:
GIonic m
liquid
= yHf +4 yO−2 0 GL(Hf +4 )2 (V a−4 )4 + 4yHf +4 yV a−4 0 G(Hf +4 )(V a−4 ) + RT (4 − 2yO−2 )(yHf +4 ln(yHf +4 )) + RT (4)(yO−2 ln(yO−2 ) + yV a−4 ln(yV a−4 )) + Gex m
Gxs m
= yHf +4 yO−2 yV a−4
(7.8) k X k=0
k
L(Hf +4 :O−2 ,V a−4 ) (T )(yO−2 − yV a−4 )k
(7.9)
139 where 0 GL(Hf +4 )2 (V a−4 )4 corresponds to the standard Gibbs energy for two moles of liquid Hf O2 ; 0 G(Hf +4 )(V a−4 ) is the standard Gibbs energy for pure hafnium liquid and k L(Hf +4 :O−2 :V a−4 ) corresponds to the excess chemical interaction parameters between oxygen and vacancies in the second sublattice. 7.4.1.3
Gas
To describe the oxygen-rich side of the Hf-O system, the gas phase was included in the calculation. The ideal gas model was used and the following six species were considered: (O, O2 , O3 , Hf, HfO, HfO2 ) The Gibbs energy of the gas phase can be described as: 0 0 GGas m = yO ( GO + RT ln(P )) + yO2 ( GO2 + RT ln(P ))
+ yO3 (0 GO3 + RT ln(P )) + yHf (0 GHf + RT ln(P )) + yHf O (0 GHf O + RT ln(P )) + yHf O2 (0 GHf O2 + RT ln(P )) + RT (yO ln(yO ) + yO2 ln(yO2 ) + yO3 ln(yO3 )) + RT (yHf ln(yHf ) + yHf O ln(yHf O ) + yHf O2 ln(yHf O2 ))
(7.10)
with y being the mole fraction of species in the gas phase. All data are taken from the SSUB database[36] 7.4.1.4
Polymorphs of HfO2
Thermodynamic descriptions of three polymorphs of HfO2 have been obtained from the SSUB database[36]. For simplicity, all three phases are modeled as line compounds and the transformation temperatures for monoclinic → tetragonal → cubic → liquid are 2100, 2793, and 3073K, respectively.
7.4.2
Si-O
The Si-O system has been modeled by Hallstedt[37] with an ionic liquid model. Three different polymorphs of silicon dioxides: quartz, tridymite, and cristobalite
140 are included in the system. The Si-O phase diagram is given in Figure 7.2.
4000 Gas
Temperature, K
3500 3000
Gas +L2
L1+Gas
2500
L2+Gas
L1+L2
2000
L1+Crystobalite
Crystobalite+Gas L1+Tridymite
1500
(Si)+Tridymite Tridymite+Gas (Si)+Quartz
1000 0
0.2
0.4 0.6 Mole Fraction, O
Quartz+Gas
0.8
1.0
Figure 7.2. Calculated Si-O phase diagram from Hallstedt[37].
7.4.3
Hf-Si
The Hf-Si system has been extensively studied and modeled by Zhao et al. [38]. Six intermetallic compounds, Hf2 Si, Hf5 Si3 , Hf3 Si2 , Hf5 Si4 , HfSi, and HfSi2 are present. The calculated phase diagram of Hf-Si system is given in Figure 7.3.
7.4.4
Hf-Si-O
In order to be combined with the Hf-O and Si-O systems, the liquid phase of the Hf-Si system was converted to an ionic liquid in the present work. Hf+4 and Si+4 are in the first sublattice of the ionic liquid phase and vacancies have been introduced into the second sublattice for the electro-neutrality. The interaction parameters for the liquid phase from Zhao et al. [38] were used for the mixing of Hf+4 and Si+4 in the first ionic liquid sublattice.
141
3000 2800 Hf5Si3
2600
Liquid
2200
Hf2Si
1600 1400 1200
HfSi2
1800
HfSi
2000
Hf5Si4
Hf3Si2
Temperature, K
2400
1000 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
Figure 7.3. Calculated Hf-Si phase diagram from Zhao et al. [38].
The ternary compound HfSiO4 has been introduced. Due to the lack of experimental data, HfSiO4 has been modeled as a stoichiometric compound.
7.5
Results and discussion
Based on the existing experimental data and results from the first-principles calculations, the model parameters for the Hf-O and Hf-Si-O systems are evaluated. The PARROT module in the Thermo-Calc software has been used[39]. The enthalpy of mixing of the hcp and bcc phases calculated from the present thermodynamic modeling are shown together with the result of first-principles calculations in Figure 7.4. For the α-Hf, they agree well with each other. As shown in Table 7.6, all first-principles calculation results retained their original symmetry as hcp and the calculation results are quite well converged with respect to the size of SQS. On the other hand, results for the β-Hf are not as good as those for the α phase. The difference between fully relaxed and symmetry preserved
142
Enthalpy of Mixing, kJ/mol-atom
50
HCP BCC Hypothetical compounds: α,β-Hf HCP SQS: Fully Relaxed HCP SQS: Symmetry Preserved BCC SQS: Symmetry Preserved
0
-50
-100
-150
-200
-250 0
0.2
0.4
0.6
0.8
Mole Fraction, O Figure 7.4. First-principles calculations results of hypothetical compounds (HfO0.5 and HfO3 ) 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.
calculations is 140 kJ/mol-atom at most and all the fully relaxed calculations could not maintain the bcc symmetry. Such first-principles calculations results can be validated by comparing their calculated lattice parameters with experimental measurements. In Figure 7.5, the calculated lattice parameters of α-Hf, both a and c, are compared with experimental data and quite satisfactorily agree with each other. In Figure 7.6, the Hf-rich side of the Hf-O phase diagram is shown. The congruent melting of α-Hf and the peritectic reaction are reproduced correctly. The calculated phase region shows quite good agreement with the X-ray phase identification results of Domagala and Ruh[9]. The calculated partial enthalpy of mixing of oxygen in the α-Hf is shown in Figure 7.7 and compared with experimental data[22]. As mentioned before, the ac-
143
3.45
3.40
Lattice Parameters, (Å)
5.3
1961 Dagerhamn 1963 Rudy and Stecher 1963 Silver et al. 1965 Domagala and Ruh First-principles
5.2
3.35
5.1
c-axis 3.30
5.0
3.25
4.9
3.20
3.15
4.8
a-axis
0
0.1
0.2
0.3
4.7
Mole Fraction, O Figure 7.5. Calculated lattice parameters of α-Hf with experimental data[8, 9, 24, 40]. Scale for a-axis is left and for c is right.
curacy of measurement has been improved compared to the previous experiments. However, it is still quite difficult to measure the low oxygen pressure. The calculated phase diagram of the entire Hf-O system is shown in Figure 7.8 with the gas phase included. In the present work, the ternary liquid phase is extrapolated from the binaries. The enthalpy of formation of HfSiO4 is obtained from first-principles calculations and the entropy of formation is evaluated from its preitectic reaction, Liquid + HfO2 (Monoclinic) → HfSiO4 , at 2023K as reported by Parfenenkov et al. [20]. The pseudo-binary phase diagram of HfO2 -SiO2 is calculated and shown in Figure 7.9. As discussed in the introduction, HfO2 is a promising candidate to replace SiO2 as the gate dielectric in CMOS transistors due to its high dielectric constant and compatibility with Si in comparison with ZrO2 [5]. In general, during the fabrication of such devices, the films are subjected to temperatures around 1273K for a short period of time[4]. Thus, it is quite essential to understand the thermodynamic
144
3000 Melting α+HfO2 α+β α β
2900 2800
Liquid
Temperature, K
2700 2600 2500 2400 2300 2200 2100 2000 0
0.05
0.10 0.15 0.20 Mole Fraction, O
0.25
0.30
Figure 7.6. Calculated Hf-rich side of the Hf-O phase diagram with experimental data from Domagala and Ruh[9].
stability of HfO2 /SiO2 /Si. From the evaluated thermodynamic database of the Hf-Si-O, the isothermal sections of the Hf-Si-O system can be readily calculated to study the stability of the HfO2 /Si interface. Two different temperatures are selected for the calculations and they are 500K for low temperature processing, such as mist deposition method and rapid thermal processing[41] and 1000K, typical temperature for the epitaxial growth of oxides deposition. Calculated isothermal sections of the Hf-Si-O system at 500K and 1000K are shown in Figure 7.10(a) and 7.10(b), respectively. The two three-phase regions, HfSiO4 +HfO2 +Hf2 Si and HfSiO4 +diamond+Hf2 Si, in the 500K isothermal section should be noticed with respect to the stability of HfO2 /Si interface. Since those regions are intersected by the line connecting HfO2 and Si, HfSi2 can be found in the fabrication of polySi/HfO2 gate stack Metal Oxide Semiconductor Field Effect Transistor (MOSFET) on bulk Si at 500K, while the 1000K calculation result shows that HfO2 is stable with the Si substrate.
145
Partial Enthalpy of Mixing of Oxygen, kJ/mol
-1000
-1050
α+HfO2 -1100
α
-1150
-1200
-1250 0
0.05
0.10 0.15 0.20 Mole Fraction, O
0.25
0.30
Figure 7.7. Calculated partial enthalpy of mixing of oxygen in the α-Hf with experimental data[22] at 1323K.
Isothermal sections, the isopleth between HfO2 -Si is calculated in order to investigate the stability range of HfSi2 in the HfO2 /Si interface and is given in Figure 7.11. It should be noted that HfSiO4 at the low temperature range is zero amount. The calculated result shows that HfSi2 becomes stable below 543K. This result is in agreement with the experimental observation from Gutowski et al. [5]. Their HfO2 film was deposited at 823K and then annealed at 1023K without the formation of any silicides. It should be emphasized that the thermodynamic stability of HfSi2 in the HfO2 /Si interface depends on the formation energy of HfSiO4 . The enthalpy of formation for HfSiO4 is calculated from first-principles calculations since there is no experimental measurement. To further illustrate this, the reference states of the enthalpy of formation for HfSiO4 are defined from the two binary metal oxides (See Eqn. 7.11 and Table 7.1). ∆HfHfSiO4 = H(HfSiO4 ) − 21 H(HfO2 ) − 21 H(SiO2 )
(7.11)
146
5500 Gas
5000
Temperature, K
4500 4000 3500
Liquid
3000
Cubic
2500
β-Hf
2000
Tetragonal
α-Hf Monoclinic
1500 1000 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, O
Figure 7.8. Calculated Hf-O phase diagram.
where H’s are the enthalpies of formation for each structure calculated from firstprinciples given in the unit of eV /atom. The current result of the HfSiO4 calculation predicts that HfSi2 is stable up to 543K. However, the uncertainty of the formation enthalpy of HfSiO4 , which originates from the density functional theory itself, is about ±1 kJ/mol-atom[42]. Thus, the associated decomposition temperature of HfSi2 in the HfO2 /Si interface varies from 382K to 670K within the calculated uncertainty of ∆f HfSiO4 . The recent work from Miyata et al. [43] found the formation of nanometer-scale HfSi2 dots on the newly opened void surface produced by the decomposition of HfO2 /SiO2 films at the oxide/void boundary in vacuum. However, their result cannot be directly compared with the current thermodynamic calculations due to the unknown oxygen partial pressure. All parameters for the Hf-Si-O system are listed in Table 7.8.
147
3500 Liquid
3000
Temperature, K
Cubic+L
2500
Tetragonal+L L+Cristobalite
2000
HfSiO4+L HfSiO4+Cristobalite
1500 Monoclinic + HfSiO4
1000
HfSiO4+Tridymite
HfSiO4+Quartz
500 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, SiO2
Figure 7.9. Calculated HfO2 -SiO2 pseudo-binary phase diagram.
7.6
Conclusion
The complete thermodynamic description of the Hf-Si-O ternary system is developed via the hybrid approach of first-principles calculations and CALPHAD modeling in the present work. In the Hf-O system, special quasirandom structures have been generated to calculate the enthalpies of mixing of oxygen and vacancies in the α and β solid solutions. Calculated enthalpies of mixing of α-Hf are almost identical to the model-calculated value whereas those of β-Hf show significant discrepancy. In the β phase, first-principles calculations could not retain its original symmetry as bcc due to the strong interaction between the atoms in the structure. The calculated enthalpies of mixing from SQS’s results are combined with the enthalpies of formations of those hypothetical compounds calculated from the electronic structure calculations to derive the Gibbs energy of solid solutions in the Hf-O system. In the total energy calculation of oxygen gas, vibrational, rotational and trans-
148
1.0
Hf2Si+hcp
0.9
tio
n,
Hf
0.8 Hf2Si+Hf3Si2+hcp Hf3Si2 +hcp
0.6
HfO2 +HfSi+HfSi2
0.4
0.3
0.2 0.1
HfO2+Hf3Si2+Hf5Si4 HfO2+Hf5Si4+HfSi
HfO2+hcp +Hf3Si2
0.5
rac le F Mo
0.7
HfSiO4+HfO2 +HfSi2
Gas +HfSiO4 +HfO2
HfSiO4 +HfSi2+diamond
Gas+ HfSiO4 HfSiO4+Quartz +Quartz+diamond
0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si (a) 500K
1.0
Hf2Si+hcp
0.9 0.8
Mo le
Fra
cti on
,H
f
0.7
0 0
0.5
Hf3Si2 +hcp
HfO2+Hf3Si2+Hf5Si4 HfO2+Hf5Si4+HfSi HfO2 +HfSi+HfSi2
0.4 0.3
0.2 0.1
0.6
Hf2Si+Hf3Si2+hcp HfO2 +Hf3Si2 +hcp
Gas +HfSiO4 +HfO2
HfSiO4+HfO2 +diamond
Gas+ HfSiO4+Quartz
HfO2+HfSi2+diamond
HfSiO4 +Quartz+diamond
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si (b) 1000K
Figure 7.10. 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 HfO2 and Si is the isopleth in Figure 7.11.
149
5000 Gas
4500 4000
Gas+L1 Gas +L1+L2
Temperature, K
3500 3000
Gas+L2 L1+L2 +HfO2(t)
Gas+L1+HfO2(c)
2500
Gas+L1+HfO2(t)
2000
L1+L2+HfO2(m)
L1+HfO2(m)+HfSiO4
1500
L1+L2 L1+HfSiO4
HfO2(m)+diamond[+L2]
1000
L1+L2 +HfSiO4
HfO2(m)+diamond[+HfSiO4] 543.53
500
HfSiO4+HfO2(m) +HfSi2
0 0
HfSiO4+diamond +HfSi2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
Figure 7.11. Calculated isopleth of HfO2 -Si at 1 atm. Hafnium dioxide is left and silicon is right. Polymorphs of HfO2 , monoclinic, tetragonal, and cubic, are given in parentheses. The phases in the bracket are zero amount.
lational degrees of freedom are considered. With the adjusted total energy of oxygen molecule, the enthalpies of formation from first-principles calculations for both ordered and disordered phases showed good agreement with evaluated values. The Hf-O system has been combined with the Hf-Si and the Si-O systems to calculate the Hf-Si-O ternary system with the ternary compound HfSiO4 introduced from the first-principles calculations. From the Hf-Si-O thermodynamic database, phase stabilities pertinent to thin film processing such as HfO2 -SiO2 pseudo-binary, isothermal sections, and isopleth have been calculated. The thermodynamic calculation results show that the HfO2 /Si interface is stable above 543K, which agrees with previous experimental results. However, due to the uncertainty of HfSiO4 formation energy from first-principles, the stability of HfSi2 in the HfO2 /Si interface is still in question and further experimental investigation is required. It can be concluded that the thermodynamic properties of solid phases can be obtained from first-principles calculations not only for the ordered structures
150
Table 7.8. 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].
Phase Ionic
Sublattice model (Hf
+4
−2
)p (O , Va)q
liquid
hcp
(Hf)1 (O, Va)0.5
bcc
(Hf)1 (O, Va)3
HfSiO4
(Hf)1 (Si)1 (O)4
Evaluated description 0
liquid monoclinic + 252000 − 86.798T GIonic Hf +4 :O−2 = 2GHfO2
0
liquid = 0 GLiquid GIonic Hf Hf +4 :Va
0 Ionic liquid LHf +4 :O−2 ,Va = 50821 + 4.203T 1 Ionic liquid LHf +4 :O−2 ,Va = 420485 − 133.300T 2 Ionic liquid LHf +4 :O−2 ,Va = 30537 0 hcp GHf:Va = 0 Ghcp Hf 0 hcp 0 hcp GHf:O = GHf + 0.50 GGas O − 271214 + 41.560T 0 hcp LHf:O,Va = −31345 1 hcp LHf:O,Va = −6272 0 bcc GHf:Va = 0 Gbcc Hf 0 bcc 0 hcp GHf:O = GHf + 30 GGas O − 737857 + 268.540T 0 hcp LHf:O,Va = −981440 + 20.349T 0 HfSiO4 GHf:Si:O = Gmonoclinic + Gquartz HfO2 SiO2 − 10615 + 1.313T
but also for the solution phases as long as one can find appropriate geometrical input for phases of interest. Special quasirandom structures for a substitutional solution phase is one example. However, one should notice that such a supercell only mimics short-ranged interaction as in metallic alloy systems. As shown in the present work, SQS’s can successfully describe the mixing behavior between oxygen and vacancies in the α solid solution where oxygen concentrations are relatively low, but for the oxygen-rich β phase such interactions between the electrons at the longer distance become important and lead to the collapse of its original structure as bcc when the structure has been fully relaxed.
151
Bibliography [1] K. L. Komarek, P. J. Spencer, and International Atomic Energy Agency. Hafnium : physico-chemical properties of its compounds and alloys. Atomic Energy Review Special issue ; no 8. International Atomic Energy Agency, Vienna, 1981. [2] S. Sayan, E. Garfunkel, T. Nishimura, W. H. Schulte, T. Gustafsson, and G. D. Wilk. Thermal decomposition behavior of the HfO2 /SiO2 /Si system. J. Appl. Phys., 94(2):928–934, 2003. [3] K. J. Hubbard and D. G. Schlom. Thermodynamic stability of binary oxides in contact with silicon. J. Mater. Res., 11(11):2757–2776, 1996. [4] S. Ramanathan, P. C. McIntyre, J. Luning, P. S. Lysaght, Y. Yang, Z. Chen, and S. Stemmer. Phase separation in hafnium silicates for alternative gate dielectrics. J. Electrochem. Soc., 150(10):F173–F177, 2003. [5] M. Gutowski, J. E. Jaffe, C.-L. Liu, M. Stoker, R. I. Hegde, R. S. Rai, and P. J. Tobin. Thermodynamic stability of high-k dielectric metal oxides ZrO2 and HfO2 in contact with Si and SiO2 . Appl. Phys. Lett., 80(11):1897–1899, 2002. [6] P. Waldner and G. Eriksson. Thermodynamic modelling of the system titanium-oxygen. CALPHAD, 23(2):189–218, 1999. [7] C. Wang, M. Zinkevich, and F. Aldinger. On the thermodynamic modeling of the Zr-O system. CALPHAD, 28(3):281–292, 2005. [8] E. Rudy and P. Stecher. The constitution diagram of the hafnium-oxygen system. J. Less-Common Met., 5(No. 1):78–89, 1963. [9] R. F. Domagala and R. Ruh. The hafnium-oxygen system. Am. Soc. Metals, Trans. Quart., 58(2):164–75, 1965. [10] G. I. Ruda, V. V. Vavilova, I. I. Kornilov, L. E. Fykin, and L. D. Panteleev. Nature of phase equilibrium and transformations in the solid state in the hafnium-oxygen system. Izvestiya Akademii Nauk SSSR, Neorganicheskie Materialy, 12(3):461–5, 1976. [11] R. Ruh and V. A. Patel. Proposed phase relations in the hafnium oxide-rich portion of the system hafnium-hafnium oxide. J. Am. Ceram. Soc., 56(11): 606–7, 1973. [12] S. Geller and E. Corenzwit. Crystallographic data. Hafnium oxide, HfO2 (monoclinic). Anal. Chem., 25:1774, 1953.
152 [13] C. E. Curtis, L. M. Doney, and J. R. Johnson. Properties of hafnium oxide, hafnium silicate, calcium hafnate, and hafnium carbide. J. Am. Ceram. Soc., 37:458–65, 1954. [14] J. Adam and M. D. Rogers. The crystal structure of ZrO2 and HfO2 . Acta Cryst., 12:951, 1959. [15] A. G. Boganov, V. S. Rudenko, and L. P. Makarov. X-ray study of Zr and Hf dioxides at temperatures up to 2750 deg. Doklady Akademii Nauk SSSR, 160 (5):1065–8, 1965. [16] D. W. Stacy and D. R. Wilder. Yttria-hafnia system. J. Am. Ceram. Soc., 58(7-8):285–8, 1975. [17] J. Wang, H. P. Li, and R. Stevens. Hafnia and hafnia-toughened ceramics. J. Mater. Sci., 27(20):5397–430, 1992. [18] T. B. Massalski. Binary alloy phase diagrams. ASM International, Materials Park, Ohio, 2nd edition, 1990. [19] J. A. Speer and B. J. Cooper. Crystal structure of synthetic hafnon, HfSiO4 , comparison with zircon and the actinide orthosilicates. Am. Mineral., 67(7-8): 804–8, 1982. [20] V. N. Parfenenkov, R. G. Grebenshchikov, and N. A. Toropov. Phase equilibriums in the hafnium dioxide-silicon dioxide system. Doklady Akademii Nauk SSSR, 185(4):840–2, 1969. [21] M. Hirabayashi, S. Yamaguchi, and T. Arai. Superstructure and orderdisorder transformation of interstitial oxygen in hafnium. J. Phys. Soc. Jpn., 35(2):473–81, 1973. [22] G. Boureau and P. Gerdanian. High temperature thermodynamics of solutions of oxygen in zirconium and hafnium. J. Phys. Chem. Solids, 45(2):141–5, 1984. [23] K. L. Komarek and M. Silver. Thermodynamic properties of zirconiumoxygen, titanium-oxygen, and hafnium-oxygen alloys. Thermodynamics of Nuclear Materials, Proceedings of the Symposium on Thermodynamics of Nuclear Materials, 1962:749–73, 1963. [24] M. D. Silver, P. A. Farrar, and K. L. Komarek. Thermodynamic properties and lattice parameters of hafnium-oxygen alloys. Trans. AIME, 227(4):876– 84, 1963. [25] G. Kresse and J. Furthmuller. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci., 6(1):15–50, 1996.
153 [26] G. Kresse and D. Joubert. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B., 59(3):1758–1775, 1999. [27] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais. Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation. Phys. Rev. B., 46(11):6671–87, 1992. [28] D. A. McQuarrie. Statistical mechanics. University Science Books, Sausalito, Calif., 2000. [29] A. Zunger, S. H. Wei, L. G. Ferreira, and J. E. Bernard. Special Quasirandom Structures. Phys. Rev. Lett., 65(3):353–6, 1990. [30] C. Jiang, C. Wolverton, J. Sofo, L.-Q. Chen, and Z.-K. Liu. First-principles study of binary bcc alloys using special quasirandom structures. Phys. Rev. B., 69(21):214202/1–214202/10, 2004. [31] D. Shin, R. Arr´oyave, Z.-K. Liu, and A. van de Walle. Thermodynamic properties of binary HCP solution phases from special quasirandom structures. Phys. Rev. B., 74(2):024204/1–024204/13, 2006. [32] T. Tsuji. Thermochemistry of IVA transition metal-oxygen solid solutions. J. Nucl. Mater., 247:63–71, 1997. [33] R. Arr´oyave, L. Kaufman, and T. W. Eagar. Thermodynamic modeling of the Zr-O system. CALPHAD, 26(1):95–118, 2002. [34] O. Redlich and A. T. Kister. Algebraic representations of thermodynamic properties and the classification of solutions. Ind. Eng. Chem., 40(2):345– 348, 1948. [35] M. Hillert, B. Jansson, B. Sundman, and J. Aagren. A two-sublattice model for molten solutions with different tendency for ionization. Metall. Trans. A, 16A(2):261–6, 1985. [36] Scientific Group Thermodata Europe (SGTE). Thermodynamic Properties of Inorganic Materials, volume 19 of Landolt-Boernstein New Series, Group IV. Springer, Verlag Berlin Heidelberg, 1999. [37] B. Hallstedt. Thermodynamic assessment of the silicon-oxygen system. CALPHAD, 16(1):53–61, 1992. [38] J. C. Zhao, B. P. Bewlay, M. R. Jackson, and Q. Chen. Hf-Si binary phase diagram determination and thermodynamic modeling. J. Phase Equlib., 21 (1):40–45, 2000.
154 [39] J. O. Andersson, T. Helander, L. Hoglund, P. Shi, and B. Sundman. ThermoCalc & DICTRA, computational tools for materials science. CALPHAD, 26 (2):273–312, 2002. [40] T. Dagerhamn. X-ray study on solid solutions of oxygen in hafnium. Acta Chem. Scand., 15:214–15, 1961. [41] K. Chang, K. Shanmugasundaram, D. O. Lee, P. Roman, C. T. Wu, J. Wang, J. Shallenberger, P. Mumbauer, R. Grant, R. Ridley, G. Dolny, and J. Ruzyllo. Silicon surface treatments in advanced MOS gate processing. Microelectron. Eng., 72(1-4):130–135, 2004. [42] C. Wolverton, X. Y. Yan, R. Vijayaraghavan, and V. Ozolins. Incorporating first-principles energetics in computational thermodynamics approaches. Acta Mater., 50(9):2187–2197, 2002. [43] N. Miyata, Y. Morita, T. Horikawa, T. Nabatame, M. Ichikawa, and A. Toriumi. Two-dimensional void growth during thermal decomposition of thin HfO2 films on Si. Phys. Rev. B., 71(23):233302/1–233302/4, 2005. [44] A. T. Dinsdale. SGTE data for pure elements. CALPHAD, 15(4):317–425, 1991.
Chapter
8
Conclusion and future work 8.1
Conclusion
In this thesis, thermodynamic properties of both substitutional and interstitial solid solutions have been studied from first-principles calculations of Special Quasirandom Structures (SQS) and CALPHAD thermodynamic modeling. The main contributions of the present thesis include: I. Binary hcp SQS’s for the substitutional hcp binary solid solutions are presented. These structures are able to mimic the most important pair and multi-site correlation functions corresponding to perfectly random hcp solutions at three compositions, x=0.25, 0.5 and 0.5 in A1−x Bx binary alloys. Due to the relatively small size of the generated structures, they can be used to calculate the properties of random hcp alloys via first-principles methods. The structures are relaxed in order to find their lowest energy configurations at each composition. The generated SQS’s are applied to seven binary systems (Cd-Mg, Mg-Zr, Al-Mg, Mo-Ru, Hf-Ti, Hf-Zr, and Ti-Zr) and show good agreement with both enthalpy of mixing and lattice parameters measurements from experiments. II. Enthalpies of mixing for six binaries in the Al-Cu-Mg-Si system have been studied via first-principles calculations of binary SQS’s of different structures: bcc, fcc, hcp, and diamond. It is found that the enthalpies of mixing are very similar to each other within the same system except the enthalpy of mixing
156 for the diamond phase in the Al-Si system. This is due to the more directional bonding of group IVB elements in the diamond structure other than bcc, fcc, and hcp phases. It is concluded that the enthalpy of mixing in the same system will be very close to each other regardless of the crystal structure as far as the coordination numbers of two phases are close to each other. III. In the previous thermodynamic modelings of the Cu-Si system, experimental phase diagram data could be reproduced from the incorrect Gibbs energy functions of the intermetallic compounds. The thermodynamic description of the Cu-Si system has been updated with first-principles results in the present thesis. Based on the total energy calculation of -Cu15 Si4 , better enthalpies of formation for the intermetallic compounds are obtained. Enthalpies of mixing for the solid solution phases, bcc, fcc, and hcp, are also calculated from binary SQS’s to be used in the thermodynamic modeling. IV. Ternary SQS’s for the fcc solid solution phase are generated at different compositions, xA = xB = xC =
1 3
and xA = 21 , xB = xC = 14 , with correlation
functions satisfactorily close to that of fcc random solutions. The generated SQS’s are applied to the Ca-Sr-Yb system which supposedly has complete solubility range without order/disorder transitions in ternary fcc solid solutions to calculate the enthalpy of mixing for the fcc solution phase. It is found that ternary SQS’s can provide valuable information about the mixing behavior of ternary solid solution phases and calculated enthalpy of mixing for ternary solid solutions could be used the evaluate ternary interaction parameters for the fcc solid solution phase to improve its thermodynamic description. V. Generated ternary SQS’s are applied to three ternary systems: Al-Cu-Mg, Al-Cu-Si, and Al-Mg-Si. SQS’s at four different compositions are enough to provide comprehensive understanding of enthalpy of mixing for ternary solid solutions. Ternary SQS’s can be used to decide whether or not the thermodynamic descriptions of constituent binaries are correct as well as whether or not ternary interaction parameters for ternary solid solutions are necessary. VI. SQS’s for binary interstitial solid solution phases, i.e. hcp and bcc, are gen-
157 erated to calculate the mixing behavior of oxygen and vacancies in the Hf-O system. The Hf-O system has been thermodynamically modeled by combining existing experimental data and first-principles calculations results through the CALPHAD approach. The Hf-O system was combined with previously modeled Hf-Si and Si-O systems, and the ternary compound in the Hf-SiO system, HfSiO4 , has been introduced to calculate the stability diagrams pertinent to the thin film processing.
8.2 8.2.1
Future works Statistical analysis
The Gibbs energy functions can be evaluated through the existing experimental data. However, there is no standardized way to judge how good the evaluated parameters are in the CALPHAD modeling. Figure 8.1 shows the evaluated result of enthalpy of mixing of the liquid phase in the Mg-Si system from two references [1, 2]. The quantified error, such as standard deviation, between the calculated results and the experimental data used for the parameter evaluation process can be used to check the degree of optimization. Furthermore, since two different types of data, thermodynamic data and phase diagram data, are used in the thermodynamic modeling, it is also necessary to fairly weight the data according to the significance.
8.2.2
Sensitivity analysis of model parameters
The other issue of the CALPHAD approach is determining the minimal number of parameters should be used to model a phase. Figure 8.2 shows two different parameters sets for the liquid phase of the Mg-Si system. For the sake of simplicity, it is desirable to have the minimal set of parameters for a phase. As can be seen in Figure 8.2, the calculated phase diagrams from two different thermodynamic databases are almost identical. However, the number of parameters have used in the two different modelings is quite different. From Figure 8.2(a), the contributions from interaction parameters 3 L and 4 L are almost zero according to the relationship shown in Eqn. 2.13. Among the parameters used to reproduce experimental data, it is necessary to find the pa-
158
0
Enthalpy of Mixing, kJ/mol
-3
1967Eld 1968Gef 1996Feu 2000Yan
-6 -9 -12 -15 -18 -21 -24 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
Figure 8.1. Enthalpy of mixing for the liquid phase in the Mg-Si system from two different modeling[1, 2] with experimental data[3, 4].
rameters with significant contribution to the Gibbs energy. By conducting the sensitivity analysis of the model parameters, it should be able to find more important parameter than others to have the smallest number of parameters for a phase.
159
1800 1909Vog 1940Ray 1968Gef 1977Sch
1600
Temperature, K
1400 1200 1000
L0 = -83864 + 32.444T L1 = 18027 - 19.612T L2 = 2486 - 0.311T L3 = 18541 - 2.318T L4 = -12338 + 1.542T
800 600 400 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
(a) Calculated Mg-Si phase diagram from Feufel et al. [1]
1800 1909Vog 1940Ray 1968Gef 1977Sch
1600
Temperature, K
1400 1200 1000
L0 = -73623.6 + 21.321T L1 = -30000 + 21.438T L2 = 44417.4 - 28.375T
800 600 400 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fraction, Si
(b) Calculated Mg-Si phase diagram from Yan et al. [2]
Figure 8.2. 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.
160
Bibliography [1] H. Feufel, T. Goedecke, H. L. Lukas, and F. Sommer. Investigation of the Al-Mg-Si system by experiments and thermodynamic calculations. J. Alloys Compd., 247(1-2):31–42, 1997. [2] X.-Y. Yan, F. Zhang, and Y. A. Chang. A thermodynamic analysis of the Mg-Si system. J. Phase Equlib., 21(4):379–384, 2000. [3] R. Geffken and E. Miller. Phase diagrams and thermodynamic properties of the magnesium-silicon and magnesium-germanium systems. TMS-AIME, 242 (11):2323–8, 1968. [4] J. M. Eldridge, E. Miller, and K. L. Komarek. Thermodynamic properties of liquid magnesium-silicon alloys. Discussion of the Mg-group IVB systems. TMS-AIME, 239(6):775–81, 1967. [5] R. Vogel. Magnesium-silicon alloys. Z. Anorg. Chem., 61:46, 1909. [6] G. V. Raynor. The constitution of the magnesium-rich alloys in the systems magnesium-lead, magnesium-tin, magnesium-germanium and magnesiumsilicon. J. Inst. Met., 66(Pt. 12):403–26(Paper No. 888), 1940. [7] E. Schuermann and A. Fischer. Melting equilibriums in the ternary system of aluminum-magnesium-silicon. Part 2. Binary system of magnesium-silicon. Giessereiforschung, 29(3):111–13, 1977.
Appendix
A
The input files used in Thermo-Calc A.1 A.1.1
The Cu-Si system Setup file
GOTO_MODULE DATABASE_RETRIEVAL SWITCH_DATABASE PURE DEFINE_ELEMENT CU SI REJECT PHASE LAVES_C15 GET_DATA GOTO_MODULE GIBBS_ENERGY_SYSTEM ENTER_PHASE ETA,,2 19 6 CU; SI; NO NO ENTER_PHASE EPSILON,,2 15 4 CU; SI; NO NO ENTER_PHASE GAMMA,,2 56 11 CU; SI ; NO NO ENTER_PHASE DELTA,,2 33 7 CU; SI; NO NO GOTO_MODULE PARROT ENTER_PARAMETER G(LIQUID,CU,SI;0) 298.15 V1+V2*T; 6000 N ENTER_PARAMETER G(LIQUID,CU,SI;1) 298.15 V3+V4*T; 6000 N ENTER_PARAMETER G(LIQUID,CU,SI;2) 298.15 V5+V6*T; 6000 N ENTER_PARAMETER G(FCC,CU,SI;0) 298.15 V11+V12*T; 6000 N ENTER_PARAMETER G(FCC,CU,SI;1) 298.15 V13+V14*T; 6000 N ENTER_PARAMETER G(FCC,CU,SI;2) 298.15 V15+V16*T; 6000 N ENTER_PARAMETER G(BCC,CU,SI;0) 298.15 V21+V22*T; 6000 N ENTER_PARAMETER G(BCC,CU,SI;1) 298.15 V23+V24*T; 6000 N ENTER_PARAMETER G(BCC,CU,SI;2) 298.15 V25+V26*T; 6000 N ENTER_PARAMETER G(HCP,CU,SI;0) 298.15 V31+V32*T; 6000 N ENTER_PARAMETER G(HCP,CU,SI;1) 298.15 V33+V34*T; 6000 N ENTER_PARAMETER G(HCP,CU,SI;2) 298.15 V35+V36*T; 6000 N ENTER_PARAMETER G(ETA) 298.15 19*GHSERCU+6*GHSERSI+V41+V42*T; 6000 N ENTER_PARAMETER G(EPSILON) 298.15 15*GHSERCU+4*GHSERSI+V51+V52*T; 6000 N ENTER_PARAMETER G(GAMMA) 298.15 56*GHSERCU+11*GHSERSI+V61+V62*T; 6000 N ENTER_PARAMETER G(DELTA) 298.15 33*GHSERCU+7*GHSERSI+V71+V72*T; 6000 N CREATE_NEW_STORE_FILE CUSI
162
A.1.2
POP file
ENTER_SYMBOL CONSTANT
DX=0.01,P0=101325,DH=200,DT=3, DDT=1
$====================================================================$ $********************************************************************$ $ POP file for Cu-Si binary system $ $********************************************************************$ $====================================================================$ $ 2006. 7. 7. Dongwon Shin $ $====================================================================$ $********************************************************************$ $********************************************************************$ $ PART. I Thermochemical data $ $********************************************************************$ $********************************************************************$
$====================================================================$ $ Enthalpy of formation of Cu15Si4(EPSILON) from first-principles $ $ -154.84541 eV = -4.074879211 eV/atom $ $ fcc Cu = -3.6375707 eV/atom $ $ dia Si = -10.863008 eV = -5.431504 eV/atom $ $ Delta H_{Cu15Si4} = -0.059638342 eV/atom = -5754.174711 J/mol-atom $ $====================================================================$ CREATE_NEW_EQUILIBRIUM 1,1 CHANGE-STATUS PHASE EPSILON=FIXED 1 SET_CONDITION P=P0, T=298.15, AC(SI)=1 SET_REFERENCE_STATE CU FCC,,,,, SET_REFERENCE_STATE SI DIAMOND,,,,, EXPERIMENT HMR=-5754:500 LABEL AHEP $====================================================================$ $ Enthalpy of formation of eta $ $ Estimated from Cu15Si4(EPSILON) $ $ Should be less than the extrapolated convex hull from Cu15Si4 $ $====================================================================$ CREATE_NEW_EQUILIBRIUM 2,1 CHANGE-STATUS PHASE ETA=FIXED 1 SET_CONDITION P=P0, T=298.15, AC(SI)=1 SET_REFERENCE_STATE CU FCC,,,,, SET_REFERENCE_STATE SI DIAMOND,,,,, EXPERIMENT HMR=-6250:1000 LABEL AHET $====================================================================$ $ Enthalpy of formation of gamma $ $ Estimated from Cu15Si4(EPSILON) $ $====================================================================$ CREATE_NEW_EQUILIBRIUM 3,1 CHANGE-STATUS PHASE GAMMA=FIXED 1 SET_CONDITION P=P0, T=298.15, AC(SI)=1 SET_REFERENCE_STATE CU FCC,,,,, SET_REFERENCE_STATE SI DIAMOND,,,,, EXPERIMENT HMR=-4850:1000 LABEL AHGA $====================================================================$ $ Enthalpy of formation of delta $ $ Estimated from Cu15Si4(EPSILON) $ $====================================================================$ CREATE_NEW_EQUILIBRIUM 4,1
163
CHANGE-STATUS PHASE DELTA=FIXED 1 SET_CONDITION P=P0, T=298.15, AC(SI)=1 SET_REFERENCE_STATE CU FCC,,,,, SET_REFERENCE_STATE SI DIAMOND,,,,, EXPERIMENT HMR=-4783:1000 LABEL AHDE $====================================================================$ $********************************************************************$ $ Enthalpy of Mixing at 1773K (1500C) $ $ Liquid $ $ 1982 G.I. Batalin and V.S. Sudavtsova $ $********************************************************************$ $====================================================================$ TABLE_HEAD 1100 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE LIQUID=FIXED 1 SET_REFERENCE_STATE * LIQUID,,, SET-CONDITION P=P0, T=1773, X(LIQUID,SI)=@1 EXPERIMENT HMR=@2:100 LABEL AHML TABLE_VALUES $x(Si) HMR 0.973 -290 0.948 -380 0.916 -1005 0.897 -1460 0.872 -1880 0.848 -2300 0.828 -2720 0.81 -3055 0.795 -3515 0.777 -3975 0.764 -4310 0.75 -4520 0.738 -4690 0.728 -4940 0.711 -5210 0.698 -5440 0.68 -5815 0.667 -6170 0.654 -6235 0.642 -6485 0.63 -6820 0.615 -6990 0.602 -7070 0.591 -7320 0.582 -7450 0.572 -7530 0.564 -7780 0.553 -8075 0.546 -8240 0.539 -8450 0.533 -8700 0.37 -12260 0.352 -12845 0.322 -13540 0.28 -14140 0.272 -13810 0.237 -13810 0.2 -13390 0.197 -13180 0.184 -12600 0.151 -11090
164
0.143 -10670 0.128 -9450 0.108 -8450 0.08 -6230 0.05 -3975 TABLE_END
$====================================================================$ $********************************************************************$ $ Enthalpy of Mixing around 1900K $ $ Liquid $ $ Victor Witusiewicz et al., Z. Metallkd. 88 (1997) 866-872 $ $********************************************************************$ $====================================================================$ TABLE_HEAD 1200 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE LIQUID=FIXED 1 SET_REFERENCE_STATE * LIQUID,,, SET-CONDITION P=P0, X(LIQUID,SI)=@1, T=@2 EXPERIMENT HMR=@3:100 LABEL AHML TABLE_VALUES $ x(Si) Temp. Enthalpy of Mixing 0.014 1900 -1000 0.026 1900 -2100 0.037 1900 -2700 0.053 1901 -3600 0.054 1901 -4400 0.054 1901 -4400 0.069 1902 -5700 0.086 1902 -6500 0.101 1902 -8000 0.102 1902 -8300 0.117 1903 -9700 0.133 1903 -10100 0.145 1904 -13300 0.147 1903 -12000 0.16 1904 -12000 0.175 1905 -11400 0.191 1904 -14700 0.201 1903 -14600 0.204 1904 -12600 0.218 1903 -14200 0.235 1902 -14100 0.248 1902 -15200 0.259 1902 -13000 0.262 1902 -13600 0.274 1902 -10200 0.287 1902 -11100 0.297 1901 -12700 0.3 1902 -12900 0.309 1901 -11600 0.323 1900 -11600 0.336 1900 -13000 0.348 1899 -13400 0.3545 1898 -12600 0.358 1899 -12600 0.362 1899 -12800 0.366 1898 -12700 0.378 1898 -13900 0.39 1897 -12200 0.398 1897 -11600 0.401 1897 -11300
165
0.409 0.421 0.434 0.446 0.453 0.458 0.464 0.475 0.478 0.482 0.486 0.5433 0.5461 0.5484 0.5572 0.5728 0.5829 0.5937 0.5959 0.6061 0.6195 0.632 0.6425 0.6446 0.6546 0.6674 0.6791 0.6898 0.6991 0.7011 0.7106 0.723 0.7358 0.747 0.7488 0.7608 0.7735 0.7863 0.7878 0.8009 0.8164 0.828 0.8412 0.8424 0.8537 0.8657 0.8801 0.8912 0.8922 0.9066 0.9224 0.9381 0.9386 0.9555 0.9705 0.9859 TABLE_END
1897 1897 1898 1898 1898 1898 1898 1898 1899 1899 1899 1899 1900 1900 1899 1900 1900 1901 1901 1901 1902 1902 1903 1903 1903 1903 1903 1903 1904 1904 1905 1905 1905 1906 1905 1906 1906 1906 1906 1906 1906 1906 1905 1905 1905 1905 1905 1905 1905 1905 1904 1904 1904 1904 1904 1903
-10100 -10100 -9100 -9000 -9400 -11600 -11400 -10300 -10000 -10200 -8700 -9960 -9630 -10400 -9780 -9770 -9130 -6800 -7630 -8710 -8410 -6900 -7950 -8050 -8350 -6920 -6830 -5140 -6210 -6700 -6460 -4700 -5710 -4330 -3900 -4180 -4020 -3810 -2980 -3480 -3370 -2020 -2160 -3560 -1810 -2510 -1840 -1190 -980 -1270 -1250 -900 -910 -530 -410 -260
$====================================================================$ $********************************************************************$ $ Enthalpy of Mixing at 1120C (1393K) $ $ Liquid $ $ 1997 Iguchi et al., J. Iron Steel Inst. Jpn., 63, 275-284 $ $********************************************************************$
166
$====================================================================$ TABLE_HEAD 1300 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE LIQUID=FIXED 1 SET_REFERENCE_STATE * LIQUID,,, SET-CONDITION P=P0, X(LIQUID,SI)=@1 T=1393 EXPERIMENT HMR=@2:100 LABEL AHML TABLE_VALUES $ x(Si) HMR 0.1995 -12260 0.1732 -11860 0.1492 -10181 0.1263 -8352 0.0988 -7576 0.0818 -5496 0.0541 -3591 0.0277 -2038 TABLE_END $====================================================================$ $********************************************************************$ $ Enthalpy of Mixing around 1281K $ $ Liquid $ $ Victor Witusiewicz et al., Z. Metallkd. 91 (2000) 128-142 $ $********************************************************************$ $====================================================================$ TABLE_HEAD 1400 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE LIQUID=FIXED 1 SET_REFERENCE_STATE * LIQUID,,, SET-CONDITION P=P0, X(LIQUID,SI)=@1 T=1281 EXPERIMENT HMR=@2:100 LABEL AHML TABLE_VALUES $ x(Si) HMR 0.0977 -9500 0.1194 -11100 0.1168 -10900 0.1437 -12400 0.1402 -12200 0.1679 -13800 0.1651 -13700 0.1885 -14700 0.1855 -14600 0.2113 -15300 0.2350 -15600 0.2305 -15500 0.2468 -15500 0.2703 -15100 0.2666 -15200 0.2856 -14900 0.3135 -14300 0.3084 -14400 0.3305 -13900 TABLE_END $====================================================================$ $********************************************************************$ $ Enthalpy of Mixing at 1600K $ $ Liquid $ $ Ingo Arpshofen et al., Z. Metallkd. 72 (1981) 842-846 $ $********************************************************************$ $====================================================================$
167
TABLE_HEAD 1500 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE LIQUID=FIXED 1 SET_REFERENCE_STATE * LIQUID,,, SET-CONDITION P=P0, X(LIQUID,SI)=@1 T=1600 EXPERIMENT HMR=@2:100 LABEL AHML TABLE_VALUES $ x(Si) HMR 0.730 -5198 0.693 -6619 0.628 -6870 0.565 -9219 0.541 -10494 0.528 -9054 0.458 -11199 0.453 -11888 0.449 -11589 0.427 -11821 0.377 -13131 0.358 -12314 0.341 -13926 0.334 -12953 0.331 -13311 0.313 -14825 0.282 -15377 0.278 -13926 0.234 -15983 0.212 -13809 0.199 -15687 0.181 -14186 0.180 -11669 0.160 -14393 0.148 -11256 0.135 -12498 0.093 -7563 0.056 -4602 0.025 -2076 TABLE_END $====================================================================$ $********************************************************************$ $ Enthalpy of Mixing at 1370K $ $ Liquid $ $ R. Castanet, J. Chem. Thermodynamics, 1979, 11, 787-791 $ $********************************************************************$ $====================================================================$ TABLE_HEAD 1600 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE LIQUID=FIXED 1 SET_REFERENCE_STATE * LIQUID,,, SET-CONDITION P=P0, X(LIQUID,SI)=@1 T=1370 EXPERIMENT HMR=@2:100 LABEL AHML TABLE_VALUES $ x(Si) HMR(J/mol) 0.0135 -1350.77 0.0198 -1937.27 0.0413 -3732.23 0.0431 -3945.52 0.0602 -5402.84 0.0700 -6167.01 0.0745 -6700.23 0.0817 -7108.91
168
0.1059 -9081.57 0.1347 -11178.50 0.1391 -11498.40 0.1508 -12244.70 0.1598 -12457.80 0.1885 -13505.70 0.1920 -13754.50 0.1956 -13950.00 0.2207 -14091.30 0.2306 -14251.00 0.2503 -14410.30 0.2539 -14303.50 0.2592 -14125.50 0.2709 -14178.40 0.2879 -13893.40 0.2897 -13573.30 0.2959 -13626.40 0.3076 -13394.80 0.3255 -12754.10 0.3318 -13056.20 0.3702 -11668.00 0.3971 -11347.00 0.4132 -10528.60 0.4526 -9442.62 0.4562 -9780.30 TABLE_END
$====================================================================$ $********************************************************************$ $ Activity of Liquid at 1773K $ $ Miki et al., ISIJ International, 42 (2002), 1071-1076 $ $********************************************************************$ $====================================================================$ TABLE_HEAD 2000 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE LIQUID=FIXED 1 SET_REFERENCE_STATE * LIQUID,,, SET-CONDITION P=P0, X(LIQUID,SI)=@1 T=1773 EXPERIMENT ACR(SI)=@2:100 LABEL ALAC TABLE_VALUES $ x(Si) ACR(Si) 0.9551 0.9501 0.9496 0.9501 0.9034 0.9000 0.8713 0.8501 0.8254 0.7999 0.7879 0.7501 0.7558 0.6999 0.7503 0.6999 0.7157 0.6500 0.6781 0.5999 0.6435 0.5500 0.6114 0.4998 0.5767 0.4500 0.5392 0.4001 0.5016 0.3500 0.4558 0.3001 0.4124 0.2499 0.3691 0.2001 0.3200 0.1499 0.2623 0.1000 0.1843 0.0499
169
TABLE_END TABLE_HEAD 2100 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE LIQUID=FIXED 1 SET_REFERENCE_STATE * LIQUID,,, SET-CONDITION P=P0, X(LIQUID,SI)=@1 T=1773 EXPERIMENT ACR(CU)=@2:100 LABEL ALAC TABLE_VALUES $ x(Si) ACR(Cu) 0.0341 0.9501 0.0688 0.9000 0.1005 0.8501 0.1323 0.7999 0.1666 0.7501 0.1955 0.6999 0.2244 0.6500 0.2507 0.5999 0.2796 0.5500 0.3114 0.4998 0.3489 0.4500 0.3836 0.4001 0.4240 0.3500 0.4757 0.3001 0.5247 0.2499 0.5825 0.2001 0.6489 0.1499 0.7298 0.1000 0.8367 0.0499 TABLE_END
$====================================================================$ $********************************************************************$ $ Special quasirandom structure calculations $ $ F.C.C. $ $********************************************************************$ $====================================================================$ TABLE_HEAD 2500 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE FCC_A1=FIXED 1 SET_REFERENCE_STATE CU FCC_A1,,, SET_REFERENCE_STATE SI FCC_A1,,, SET-CONDITION P=P0, T=298.15, X(SI)=@1 EXPERIMENT HMR=@2:500 LABEL ASQF TABLE_VALUES $x(Si) Enthalpy (J/mol) 0.25 -9851.8 0.5 -7169.6 0.75 -1062.4 TABLE_END $====================================================================$ $********************************************************************$ $ Special quasirandom structure calculations $ $ B.C.C. $ $********************************************************************$ $====================================================================$ TABLE_HEAD 2510 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE BCC_A2=FIXED 1
170
SET_REFERENCE_STATE CU BCC_A2,,, SET_REFERENCE_STATE SI BCC_A2,,, SET-CONDITION P=P0, T=298.15, X(SI)=@1 EXPERIMENT HMR=@2:500 LABEL ASQB TABLE_VALUES $x(Si) Enthalpy (J/mol) 0.25 -6039.2 0.5 713.7 0.75 9883.3 TABLE_END $====================================================================$ $********************************************************************$ $ Special quasirandom structure calculations $ $ H.C.P. $ $********************************************************************$ $====================================================================$ TABLE_HEAD 2520 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE HCP_A3=FIXED 1 SET_REFERENCE_STATE CU HCP_A3,,, SET_REFERENCE_STATE SI HCP_A3,,, SET-CONDITION P=P0, T=298.15, X(SI)=@1 EXPERIMENT HMR=@2:500 LABEL ASQH TABLE_VALUES $x(Si) Enthalpy (J/mol) 0.25 -9288.q 0.5 -3007.4 0.75 626.2 TABLE_END $********************************************************************$ $********************************************************************$ $====================================================================$ $ PART. II Phase diagram data $ $====================================================================$ $********************************************************************$ $********************************************************************$ $********************************************************************$ $====================================================================$ $ Two phase eqilibria $ $====================================================================$ $********************************************************************$ $====================================================================$ $********************************************************************$ $ Liquidus data $ $********************************************************************$ $====================================================================$ $====================================================================$ $********************************************************************$ $ Si-rich liquidus data $ $ Liquid(1) + Diamond(0) $ $********************************************************************$ $====================================================================$ TABLE_HEAD 3000 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE LIQUID=FIXED 1 CHANGE-STATUS PHASE DIAMOND=FIXED 0 SET-CONDITION P=P0, X(LIQUID,SI)=@1
171
EXPERIMENT T=@2:DT LABEL ALDI TABLE_VALUES $x(Si) T 0.3170 1099.41 0.3775 1219.85 0.4381 1299.24 0.4986 1360.31 0.5592 1409.16 0.6197 1451.91 0.6803 1491.60 0.7408 1531.30 0.8014 1570.99 0.8619 1607.63 0.9225 1644.27 0.9830 1677.86 TABLE_END $====================================================================$ $********************************************************************$ $ Cu-rich liquidus $ $ LIQUID(1) + FCC(0) $ $********************************************************************$ $====================================================================$ TABLE_HEAD 3100 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE LIQUID=FIXED 1 CHANGE-STATUS PHASE FCC=FIXED 0 SET-CONDITION P=P0, X(LIQUID,SI)=@1 EXPERIMENT T=@2:DT LABEL ALFC TABLE_VALUES $x(Si) T 0.0042 1354.20 0.0192 1343.51 0.0342 1329.77 0.0492 1312.98 0.0642 1294.66 0.0793 1273.28 0.0943 1251.91 0.1093 1225.95 0.1243 1196.95 0.1393 1167.94 0.1542 1134.35 TABLE_END $====================================================================$ $********************************************************************$ $ Liquidus with eta $ $ LIQUID(1) + ETA(0) $ $********************************************************************$ $====================================================================$ TABLE_HEAD 3200 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE LIQUID=FIXED 1 CHANGE-STATUS PHASE ETA=FIXED 0 SET-CONDITION P=P0, X(LIQUID,SI)=@1 EXPERIMENT T=@2:1 LABEL ALET TABLE_VALUES $x(Si) T 0.1950 1097.71 0.2000 1105.00 0.2050 1111.45
172
0.2100 1116.79 0.2150 1121.37 0.2200 1125.61 0.2250 1128.24 0.2300 1130.20 0.2350 1131.30 0.2450 1130.96 0.2500 1130.20 0.2550 1128.67 0.2600 1126.38 0.2650 1123.32 0.2700 1119.85 0.2750 1115.27 0.2800 1109.92 0.2850 1106.11 0.2900 1100.00 0.2950 1093.13 0.3000 1085.92 0.3043 1080.92 TABLE_END $====================================================================$ $********************************************************************$ $ Liquidus with bcc $ $ LIQUID(1) + BCC(0) $ $********************************************************************$ $====================================================================$ TABLE_HEAD 3300 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE LIQUID=FIXED 1 CHANGE-STATUS PHASE BCC=FIXED 0 SET-CONDITION P=P0, X(LIQUID,SI)=@1 EXPERIMENT T=@2:DT LABEL ALBC TABLE_VALUES $x(Si) T 0.1600 1124.10 0.1650 1119.97 0.1700 1115.17 0.1750 1109.67 0.1800 1104.50 0.1850 1099.01 TABLE_END $====================================================================$ $********************************************************************$ $ Phase boundaries for solid solutions $ $********************************************************************$ $====================================================================$ $********************************************************************$ $ FCC and others $ $********************************************************************$ $====================================================================$ $********************************************************************$ $ FCC and Liquid $ $ FCC(1) + LIQUID(0) $ $********************************************************************$ $====================================================================$ TABLE_HEAD 3400 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE FCC=FIXED 1 CHANGE-STATUS PHASE LIQUID=FIXED 0
173
SET-CONDITION P=P0, X(FCC,SI)=@1 EXPERIMENT T=@2:DT LABEL AFLI TABLE_VALUES $x(Si) T 0.0062 1349.62 0.0161 1334.35 0.0261 1316.03 0.0361 1299.24 0.0462 1279.39 0.0562 1261.07 0.0662 1238.17 0.0762 1215.27 0.0862 1192.37 0.0962 1166.41 0.1062 1141.98 TABLE_END $====================================================================$ $********************************************************************$ $ FCC and Gamma $ $ FCC(1) + GAMMA(0) $ $********************************************************************$ $====================================================================$ TABLE_HEAD 3500 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE FCC=FIXED 1 CHANGE-STATUS PHASE GAMMA=FIXED 0 SET-CONDITION P=P0, X(FCC,SI)=@1 EXPERIMENT T=@2:DT LABEL AFGA TABLE_VALUES $x(Si) T 0.0431 305.17 0.0472 339.95 0.0514 375.15 0.0555 409.92 0.0592 445.12 0.0628 479.90 0.0664 515.10 0.0696 549.87 0.0732 585.07 0.0768 619.85 0.0799 655.05 0.0841 689.82 0.0877 725.02 0.0913 759.80 0.0945 795.00 TABLE_END $====================================================================$ $********************************************************************$ $ FCC and HCP $ $ FCC(1) + HCP(0) $ $********************************************************************$ $====================================================================$ TABLE_HEAD 3600 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE FCC=FIXED 1 CHANGE-STATUS PHASE HCP=FIXED 0 SET-CONDITION P=P0, X(FCC,SI)=@1 EXPERIMENT T=@2:DT LABEL AFHC TABLE_VALUES
174
$x(Si) T 0.0986 850.13 0.1007 875.15 0.1017 900.17 0.1033 925.19 0.1043 949.79 0.1059 974.81 0.1074 999.83 0.1085 1024.85 0.1100 1049.87 0.1111 1074.89 0.1121 1099.92 TABLE_END $********************************************************************$ $ HCP and others $ $********************************************************************$ $====================================================================$ $********************************************************************$ $ HCP and FCC $ $ HCP(1) + FCC(0) $ $********************************************************************$ $====================================================================$ TABLE_HEAD 3700 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE HCP=FIXED 1 CHANGE-STATUS PHASE FCC=FIXED 0 SET-CONDITION P=P0, X(HCP,SI)=@1 EXPERIMENT T=@2:DT LABEL AHFC TABLE_VALUES $x(Si) T 0.1173 850.13 0.1188 875.15 0.1199 900.17 0.1215 925.19 0.1225 949.79 0.1240 974.81 0.1251 999.83 0.1261 1024.85 0.1277 1049.87 0.1287 1074.89 0.1298 1099.92 TABLE_END $====================================================================$ $********************************************************************$ $ HCP and Gamma $ $ HCP(1) + GAMMA(0) $ $********************************************************************$ $====================================================================$ TABLE_HEAD 3800 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE HCP=FIXED 1 CHANGE-STATUS PHASE GAMMA=FIXED 0 SET-CONDITION P=P0, X(HCP,SI)=@1 EXPERIMENT T=@2:DT LABEL AHGA TABLE_VALUES $x(Si) T 0.1180 851.82 0.1210 877.48 0.1240 904.20
175
0.1270 925.19 0.1300 946.18 0.1330 966.33 0.1360 984.35 TABLE_END $====================================================================$ $********************************************************************$ $ HCP and Delta $ $ HCP(1) + DELTA(0) $ $********************************************************************$ $====================================================================$ TABLE_HEAD 3900 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE HCP=FIXED 1 CHANGE-STATUS PHASE DELTA=FIXED 0 SET-CONDITION P=P0, X(HCP,SI)=@1 EXPERIMENT T=@2:DT LABEL AHDE TABLE_VALUES $x(Si) T 0.1394 1011.92 0.1403 1025.06 0.1410 1038.00 0.1419 1050.93 TABLE_END $====================================================================$ $********************************************************************$ $ HCP and BCC $ $ HCP(1) + BCC(0) $ $********************************************************************$ $====================================================================$ TABLE_HEAD 3950 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE HCP=FIXED 1 CHANGE-STATUS PHASE BCC=FIXED 0 SET-CONDITION P=P0, X(HCP,SI)=@1 EXPERIMENT T=@2:DT LABEL AHBC TABLE_VALUES $x(Si) T 0.1320 1108.40 0.1330 1102.67 0.1340 1098.85 0.1350 1093.13 0.1360 1089.31 0.1370 1083.59 0.1380 1079.77 0.1390 1075.95 0.1400 1070.23 0.1410 1064.50 TABLE_END
$********************************************************************$ $ BCC and others $ $********************************************************************$ $====================================================================$ $********************************************************************$ $ BCC and liquid $ $ BCC(1) + LIQUID(0) $ $********************************************************************$
176
$====================================================================$ TABLE_HEAD 4000 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE BCC=FIXED 1 CHANGE-STATUS PHASE LIQUID=FIXED 0 SET-CONDITION P=P0, X(BCC,SI)=@1 EXPERIMENT T=@2:DT LABEL ABLI TABLE_VALUES $x(Si) T 0.1480 1124.10 0.1510 1121.12 0.1540 1117.00 0.1570 1112.42 0.1600 1108.17 0.1630 1103.49 0.1660 1098.68 TABLE_END $====================================================================$ $********************************************************************$ $ BCC and HCP $ $ BCC(1) + HCP(0) $ $********************************************************************$ $====================================================================$ TABLE_HEAD 4100 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE BCC=FIXED 1 CHANGE-STATUS PHASE HCP=FIXED 0 SET-CONDITION P=P0, X(BCC,SI)=@1 EXPERIMENT T=@2:DT LABEL ABHC TABLE_VALUES $x(Si) T 0.1561 1064.00 0.1554 1067.01 0.1550 1069.98 0.1543 1072.99 0.1540 1076.01 0.1533 1079.02 0.1529 1081.98 0.1522 1085.00 0.1516 1088.01 0.1512 1091.02 0.1505 1093.99 0.1502 1097.00 0.1495 1100.01 0.1491 1103.02 0.1484 1105.99 0.1481 1109.00 0.1474 1112.01 0.1471 1114.98 0.1467 1117.99 0.1464 1121.00 0.1464 1124.01 TABLE_END $====================================================================$ $********************************************************************$ $ BCC and Delta $ $ BCC(1) + DELTA(0) $ $********************************************************************$ $====================================================================$ TABLE_HEAD 4200
177
CREATE_NEW_EQUILIBRIUM @@,1 CHANGE-STATUS PHASE BCC=FIXED 1 CHANGE-STATUS PHASE DELTA=FIXED 0 SET-CONDITION P=P0, X(BCC,SI)=@1 EXPERIMENT T=@2:DT LABEL ABDE TABLE_VALUES $x(Si) T 0.1574 1062.01 0.1581 1064.98 0.1588 1067.99 0.1599 1071.00 0.1606 1074.01 0.1616 1076.98 0.1623 1079.99 0.1633 1083.00 0.1640 1086.01 0.1647 1088.98 0.1657 1091.99 0.1664 1095.00 TABLE_END $********************************************************************$ $====================================================================$ $ Three phase eqilibria $ $====================================================================$ $********************************************************************$
$********************************************************************$ $ Eutectoid: hcp -> fcc + gamma at 825K $ $********************************************************************$ CREATE_NEW_EQUILIBRIUM 5001,1 CHANGE-STATUS PHASE HCP FCC GAMMA=FIXED 1 SET_CONDITION P=P0 EXPERIMENT T=825:1 EXPERIMENT X(HCP,SI)=0.115:DX EXPERIMENT X(FCC,SI)=0.097:DX LABEL AINV $********************************************************************$ $ Eutectoid: delta -> epsilon + gamma at 984K $ $********************************************************************$ CREATE_NEW_EQUILIBRIUM 5002,1 CHANGE-STATUS PHASE DELTA EPSILON GAMMA=FIXED 1 SET_CONDITION P=P0 EXPERIMENT T=984:1 LABEL AINV $********************************************************************$ $ Peritectoid: delta + hcp -> gamma at 1002K $ $********************************************************************$ CREATE_NEW_EQUILIBRIUM 5003,1 CHANGE-STATUS PHASE DELTA HCP GAMMA=FIXED 1 SET_CONDITION P=P0 EXPERIMENT T=1002:1 EXPERIMENT X(HCP,SI)=0.139:DX LABEL AINV $********************************************************************$ $ Eutectoid: bcc -> delta + hcp at 1060K $ $********************************************************************$ CREATE_NEW_EQUILIBRIUM 5004,1 CHANGE-STATUS PHASE BCC DELTA HCP=FIXED 1
178
SET_CONDITION P=P0 EXPERIMENT T=1002:1 EXPERIMENT X(HCP,SI)=0.142:DX EXPERIMENT X(BCC,SI)=0.157:DX LABEL AINV $********************************************************************$ $ Peritectoid: eta + delta -> epsilon at 1073K $ $********************************************************************$ CREATE_NEW_EQUILIBRIUM 5005,1 CHANGE-STATUS PHASE ETA DELTA EPSILON=FIXED 1 SET_CONDITION P=P0 EXPERIMENT T=1073:1 LABEL AINV $********************************************************************$ $ Eutectic: Liquid -> eta + diamond at 1075K $ $********************************************************************$ CREATE_NEW_EQUILIBRIUM 5006,1 CHANGE-STATUS PHASE LIQUID ETA DIAMOND=FIXED 1 SET_CONDITION P=P0 EXPERIMENT T=1075:1 EXPERIMENT X(LIQUID,SI)=0.307:DX LABEL AINV $********************************************************************$ $ Eutectic: Liquid -> eta + delta at 1093K $ $********************************************************************$ CREATE_NEW_EQUILIBRIUM 5007,1 CHANGE-STATUS PHASE LIQUID ETA DELTA=FIXED 1 SET_CONDITION P=P0 EXPERIMENT T=1093:1 LABEL AINV $********************************************************************$ $ Peritectic: Liquid + bcc -> delta at 1097K $ $********************************************************************$ CREATE_NEW_EQUILIBRIUM 5008,1 CHANGE-STATUS PHASE LIQUID BCC DELTA=FIXED 1 SET_CONDITION P=P0 EXPERIMENT T=1093:1 EXPERIMENT X(BCC,SI)=0.167:DX LABEL AINV $********************************************************************$ $ Peritectoid: bcc + fcc -> hcp at 1115K $ $********************************************************************$ CREATE_NEW_EQUILIBRIUM 5009,1 CHANGE-STATUS PHASE BCC FCC HCP=FIXED 1 SET_CONDITION P=P0 EXPERIMENT T=1115:1 EXPERIMENT X(FCC,SI)=0.112:DX EXPERIMENT X(HCP,SI)=0.130:DX EXPERIMENT X(BCC,SI)=0.147:DX LABEL AINV $********************************************************************$ $ Peritectic: Liquid + fcc -> bcc at 1126K $ $********************************************************************$ CREATE_NEW_EQUILIBRIUM 5010,1 CHANGE-STATUS PHASE LIQUID FCC BCC=FIXED 1 SET_CONDITION P=P0 EXPERIMENT T=1126:1 EXPERIMENT X(LIQUID,SI)=0.157:DX
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EXPERIMENT X(FCC,SI)=0.111:DX EXPERIMENT X(BCC,SI)=0.146:DX LABEL AINV $====================================================================$ $********************************************************************$ $ Congruent melting of eta phase $ $********************************************************************$ $====================================================================$ CREATE_NEW_EQUILIBRIUM 5011,1 CHANGE-STATUS PHASE LIQUID ETA=FIXED 1 SET_CONDITION P=P0, X(LIQUID,SI)-X(ETA,SI)=0 EXPERIMENT T=1132:1 LABEL ACME
$********************************************************************$ $********************************************************************$ $ PART. III Stability conditions $ $********************************************************************$ $********************************************************************$ $====================================================================$ $********************************************************************$ $ DELTA $ $********************************************************************$ $====================================================================$ $********************************************************************$ $ Delta should NOT appear below 984K $********************************************************************$ TABLE_HEAD 9000 CREATE_NEW_EQUILIBRIUM @@,1 CHANGE_STATUS PHASE EPSILON=ENTERED 1 CHANGE_STATUS PHASE GAMMA=ENTERED 1 CHANGE_STATUS PHASE DELTA=DORMANT SET_CONDITION P=P0, X(SI)=0.2, T=@1, N=1 EXPERIMENT DGM(DELTA)
