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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
iii
mixing for binary bcc, fcc, and hcp solid solutions and ternary fcc solid solutions.
Binary hcp and bcc SQS’s for interstitial solid solutions are used to calculate
enthalpy of mixing for α-Hf (hcp) and β-Hf (bcc) phases in the Hf-O system to be
used in the thermodynamic modeling of the Hf-Si-O system.
This thesis shows that first-principles studies of SQS’s can provide insight into
the understanding of mixing behavior for solid solution phases and calculated thermodynamic properties, for example enthalpy of mixing, can be readily used in
thermodynamic modeling to overcome scarce and uncertain experimental data.
iv
Table of Contents
List of Figures
x
List of Tables
xv
Acknowledgments
xviii
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 . . . . . . . . . . . . . . . . . . . . . . . . . .
29
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binary solid
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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
5.2
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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parameters
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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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227
the ternary fcc solution phase 234
. . . . . . . . . . . . . . . . . . . 234
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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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235
235
235
236
236
237
237
237
237
238
238
239
240
List of Figures
2.1 Ternary phase diagram showing three-phase equilibrium[2]. . . . . .
2.2 Isothermal and isoplethal phase diagrams of the Hf-Si-O system at
1 atm[3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Heat capacity of aluminum from the SGTE pure element database[11].
2.4 Gibbs energies of the individual phases of pure aluminum. The
reference state is given as fcc phase at all temperatures. . . . . . . .
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. . . . . . . . . . . . . . . . . . . . . . . .
2.6 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. . . . . . . . . . . . . . . . . . . . . . . .
2.7 Illustration describing the interaction of the different end-members
within a two-sublattice model. Colon separates sublattices and
comma separates interacting species. . . . . . . . . . . . . . . . . .
2.8 The entire procedure of the CALPHAD approach from Kumar and
Wollants [24] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
3.2
3.3
3.4
3.5
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Radial distribution analysis of Cd50 Mg50 SQS’s. The dotted lines
under the smoothed and fitted curves are the error between the two
curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
9
11
13
16
17
18
21
22
36
37
40
48
49
3.6
3.7
3.8
3.9
3.10
3.11
3.12
4.1
4.2
4.3
4.4
4.5
Radial distribution analysis of Mg50 Zr50 SQS’s. The dotted lines
under the smoothed and fitted curves are the error between the two
curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calculated and experimental results of mixing enthalpy and lattice
parameters for the Cd-Mg system . . . . . . . . . . . . . . . . . . .
Calculated enthalpy of mixing in the Mg-Zr system compared with
a previous thermodynamic assessment[18]. Both reference states are
the hcp structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calculated and experimental results of mixing enthalpy and lattice
parameters for the Al-Mg system . . . . . . . . . . . . . . . . . . .
Enthalpy of formation of the Mo-Ru system with both first principles and CALPHAD lattice stabilities. Reference states are bcc for
Mo and hcp for Ru. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Enthalpy of mixing for the Hf-Ti, Hf-Zr and Ti-Zr binary hcp solutions calculated from first-principles calculations and CALPHAD
thermodynamic models. All the reference states are hcp structures.
Calculated DOS of Ti1−x Zrx hcp solid solutions from (a) SQS and
(b) CPA[38] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Liquidus lines at various temperatures in the Al-Mg-Si system from
the COST507 database[7]. Ternary interaction parameters for the
liquid phase are 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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. . . . . . . . . .
xi
50
55
56
58
61
62
64
72
73
78
80
82
4.6
4.7
5.1
5.2
5.3
5.4
5.5
5.6
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. . . . . . . . . . . . . . . . .
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. . . . . . . . . . . . . . . . . . . . . . . . .
84
85
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
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. . . . . . . . . . . . . . . 98
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. . . . . . . . . . . . . . . 99
The electronegativity vs the metallic radius for a coordination number of 12 (Darken-Gurry) map. . . . . . . . . . . . . . . . . . . . . 101
xii
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
6.3
6.4
7.1
7.2
7.3
7.4
7.5
103
105
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
Calculated lattice parameters of α-Hf with experimental data[8, 9,
24, 40]. Scale for a-axis is left and for c is right. . . . . . . . . . . . 143
xiii
7.6
7.7
7.8
7.9
7.10
7.11
8.1
8.2
Calculated Hf-rich side of the Hf-O phase diagram with experimental
data from Domagala and Ruh[9]. . . . . . . . . . . . . . . . . . . .
Calculated partial enthalpy of mixing of oxygen in the α-Hf with
experimental data[22] at 1323K. . . . . . . . . . . . . . . . . . . . .
Calculated Hf-O phase diagram. . . . . . . . . . . . . . . . . . . . .
Calculated HfO2 -SiO2 pseudo-binary phase diagram. . . . . . . . .
Calculated isothermal section of Hf-Si-O at (a) 500K and (b) 1000K
at 1 atm. Tie lines are drawn inside the two phase regions. The
vertical cross section between HfO2 and Si is the isopleth in 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 Å. Errors indicate the difference in the number
of atoms calculated through the sum of peak areas and those expected in each coordination shell. . . . . . . . . . . . . . . . . . . .
First nearest-neighbors average bond lengths for the fully relaxed
hcp SQS of the seven binaries studied in this work. Uncertainty
corresponds to the standard deviation of the bond length distributions.
Pair and multi-site correlation functions of ternary fcc SQS-N structures when 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. . . . . . . . . . . .
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
41
42
44
51
52
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). . . . . . . . . . . . . . . . . . 81
Cohesive energies of selected bivalent metals, Ca, Sr, and Yb, from
Ref. [18]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
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. 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. .
First-principles calculations results of α-Hf special quasirandom structures. F R and SP represent ’Fully Relaxed’ and ’Symmetry Preserved’, respectively. Oxygen atoms are excluded for the symmetry
check. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
First-principles calculations results of β-Hf special quasirandom structures. F R and SP represent ’Fully Relaxed’ and ’Symmetry Preserved’, respectively. Oxygen atoms are excluded for the symmetry
check. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Thermodynamic parameters of the Hf-Si-O ternary system (in S.I.
units). Gibbs energies for pure elements and gas phases are respectively from the SGTE pure elements database[44] and the SSUB
database[36]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xvii
135
136
137
150
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. Isothermal and isoplethal phase diagrams of the Hf-Si-O system at 1 atm[3].
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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A
Figure 2.8. The entire procedure of the CALPHAD approach from Kumar and Wollants
[24]
which can be used in the thermodynamic modeling falls into two categories:
thermodynamic data and phase diagram data. Afterwards, it is necessary to
critically evaluate the compiled data. If one finds two conflicting datasets for
the same property, then either one or the other is correct or both are wrong.
Both datasets cannot be correct at the same 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 Schrödinger equation is too complex to solve with modern
computing resources.5
Ĥ(R1 , R2 · · · , RN )Ψ(r1 , r2 , · · · , rn ) = ET Ψ(r1 , r2 , · · · , rn )
(2.24)
where Rj ’s are the ionic coordinates, ri ’s are the electronic coordinates, Ĥ is the
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 Schrödinger’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 Schrödinger 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
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32
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B. Joensson, B. Sundman, and J. Aagren. A new method of describing lattice
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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 ( 13 , 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
number of atoms within bins of 10−3 Å, up to the fifth coordination shell. In
order to eliminate high frequency noise, the raw data was scaled and smoothed
through Gaussian smearing with a characteristic distance of 0.01 Å. 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 3.14Å, 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
coordination shells merge at 5.40Å, enclosing 18 atoms. It can be expected that if
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 Å. 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 Hämäläinen 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 Hämäläinen 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
Al-Mg 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
(M o1−x Rux ) − (1 − x)[E bcc (M o) − E hcp (M o)]
= Hmix
(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
-2
0
2
4
Energy (eV)
6
Zr
8
-6
-4
-2
0
2
4
Energy (eV)
6
8
Figure 3.12. Calculated DOS of Ti1−x Zrx hcp solid solutions from (a) SQS and (b)
CPA[38]
3.6
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
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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.
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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
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[33] K. C. H. Kumar, P. Wollants, and L. Delaey. Thermodynamic assessment of
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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
G
bin
=
c−1 X
c
X
i=1 j>i
xs
G
tern
=
c−2 X
c−1 X
c
X
xi 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
G
tern
=
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
-1.0
-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, 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
G
fcc
=
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
G
fcc
=
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
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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
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[6] Z. Fang and Q. Zhang. A new model for predicting thermodynamic properties
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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. 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.
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[12] G. Kresse and D. Joubert. From ultrasoft pseudopotentials to the projector
augmented-wave method. Phys. Rev. B., 59(3):1758–1775, 1999.
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[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.
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[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
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[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
5
0
-2
Enthalpy of Mixing, kJ/mol
Enthalpy of Mixing, kJ/mol
0
-4
-6
-8
-10
-12
-14
-5
-10
-15
-20
-16
-25
-18
0
0.2
0.4
0.6
Mole Fraction, Cu
0.8
0
1.0
(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
1.5
1.0
0.5
0
-2.5
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
(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
-4.5
-4.0
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) 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
-4
-6
-8
-10
Si
-12
-12
Al
Al
Cu
Cu
-14
-14
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
(a) Enthalpy of mixing for the fcc phase in the AlCu-Si system when xCu /xSi =1
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
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[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
Group
parameter (Å)
(eV/atom)
(kJ/mol-atom)
I 4̄3d
9.726
-4.0749
-5.7542
fcc-Cu
F d3̄m
3.632
-3.6376
-
diamond-Si
F m3̄m
5.468
-5.4315
-
-Cu15 Si4
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
k
− xSi )
(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
15
0
-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
-15
-18
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) 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
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
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Light Metal Alloys, volume 2 of COST 507: Definition of Thermochemical and
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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 R3̄ and P 3̄1c 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
Lattice parameters (Å)
Total energy
∆Hf
a
b
c
(eV/atom)
(kJ/mol-atom)
HCP A3 (Hf)
P 63 /mmc
3.198
3.198
5.053
-9.8320
-
Diamond A4 (Si)
F d3̄m
5.468
5.468
5.468
-5.4315
-
-4.7936
-
Gas (O2 )
α-Hf (HfO0.5 )
P 3̄m1
3.225
3.225
5.150
-9.9718
-175.511
Im3̄m
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
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
Space
Total Energy
∆H mix
Group
(eV/atom)
(kJ/mol-atom)
%
Hf
O
Va
25
8
6
18
FR
Pm
-9.9280
-227.299
8
6
18
SP
Im3̄m
-9.0165
-139.345
4
6
6
FR
C2/m
-10.0088
-315.520
4
6
6
SP
Im3̄m
-8.5718
-176.866
8
12
12
FR
Pm
-9.9636
-311.160
8
12
12
SP
Im3̄m
-8.6081
-180.370
8
18
6
FR
Pm
-9.4727
-307.100
8
18
6
SP
Im3̄m
-8.1752
-181.908
50
75
(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)
v i y Ci
(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 ) − 12 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
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[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
179
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)<0:1E-5
LABEL ADEL
TABLE_VALUES
983
900
800
700
600
500
400
300
200
TABLE_END
$====================================================================$
$********************************************************************$
$
GAMMA
$
$********************************************************************$
$====================================================================$
$********************************************************************$
$ Gamma should NOT appear above 1004K
$********************************************************************$
TABLE_HEAD 9050
CREATE_NEW_EQUILIBRIUM @@,1
180
CHANGE_STATUS PHASE HCP=ENTERED 1
CHANGE_STATUS PHASE DELTA=ENTERED 1
CHANGE_STATUS PHASE GAMMA=DORMANT
SET_CONDITION P=P0, X(SI)=0.1641, T=@1, N=1
EXPERIMENT DGM(GAMMA)<0:1E-5
LABEL AGAM
TABLE_VALUES
1003
1013
1023
1033
1043
1059
TABLE_END
$====================================================================$
$********************************************************************$
$
EPSILON
$
$********************************************************************$
$====================================================================$
$********************************************************************$
$ Epsilon should NOT appear above 1073K
$********************************************************************$
TABLE_HEAD 9100
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE DELTA=ENTERED 1
CHANGE_STATUS PHASE EPSILON=ENTERED 1
CHANGE_STATUS PHASE GAMMA=ENTERED 1
SET_CONDITION P=P0, X(SI)=0.2, T=@1, N=1
EXPERIMENT DGM(GAMMA)<0:1E-5
LABEL AEPS
TABLE_VALUES
1074
1076
1078
1080
1082
1084
1086
1088
1090
1092
TABLE_END
TABLE_HEAD 9125
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE LIQUID=ENTERED 1
CHANGE_STATUS PHASE EPSILON=DORMANT
SET_CONDITION P=P0, X(SI)=0.2, T=@1, N=1
EXPERIMENT DGM(GAMMA)<0:1E-5
LABEL AEPS
TABLE_VALUES
1157.89
1133.52
1136.84
1142.38
1150.14
TABLE_END
$====================================================================$
$********************************************************************$
$
F.C.C.
$
181
$********************************************************************$
$====================================================================$
$********************************************************************$
$ To make fcc correct with gamma
$********************************************************************$
TABLE_HEAD 9150
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE FCC=ENTERED 1
CHANGE_STATUS PHASE GAMMA=DORMANT
SET_CONDITION P=P0, X(SI)=@1, T=@2, N=1
EXPERIMENT DGM(GAMMA)<0:1E-5
LABEL AFCC
TABLE_VALUES
$x(Si)
T
0.0773
719.18
0.0784
747.31
0.0795
771.61
0.0817
798.46
0.0839
825.32
0.0850
854.73
0.0871
884.14
0.0904
912.27
0.0915
940.40
0.0970
969.82
0.0991
999.23
TABLE_END
$********************************************************************$
$ To prevent fcc goes to further to the liquid phase region
$********************************************************************$
TABLE_HEAD 9200
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE LIQUID=ENTERED 1
CHANGE_STATUS PHASE FCC=DORMANT
SET_CONDITION P=P0, X(SI)=@1, T=@2, N=1
EXPERIMENT DGM(FCC)<0:1E-5
LABEL AFCC
TABLE_VALUES
$x(Si)
T
0.0110
1542.94
0.0165
1374.52
0.0193
1414.40
0.0220
1463.16
0.0385
1529.64
0.0385
1356.79
0.0523
1432.13
0.0550
1339.06
0.0606
1392.24
0.0606
1516.34
0.0743
1312.47
0.0771
1480.89
0.0853
1401.11
0.0881
1285.87
0.0936
1352.35
0.1018
1449.86
0.1046
1259.28
0.1101
1370.08
0.1128
1432.13
0.1156
1330.19
0.1211
1237.12
0.1349
1347.92
0.1349
1409.97
0.1376
1206.09
182
0.1376
1290.30
0.1431
1396.68
0.1541
1161.77
0.1541
1259.28
0.1651
1228.25
0.1651
1361.22
0.1651
1285.87
0.1734
1135.18
0.1761
1197.23
0.1789
1241.55
0.1817
1325.76
0.1872
1223.82
0.1899
1170.64
0.1954
1294.74
0.1982
1192.80
0.2092
1170.64
0.2147
1250.42
0.2284
1219.39
0.2312
1161.77
0.2395
1201.66
0.2615
1161.77
0.2670
1183.93
0.2972
1144.04
0.3138
1157.34
TABLE_END
$********************************************************************$
$ To prevent fcc goes to further to the hcp phase region
$********************************************************************$
TABLE_HEAD 9250
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE HCP=ENTERED 1
CHANGE_STATUS PHASE FCC=DORMANT
SET_CONDITION P=P0, X(SI)=@1, T=@2, N=1
EXPERIMENT DGM(FCC)<0:1E-5
LABEL AFCC
TABLE_VALUES
$x(Si)
T
0.1176
856.49
0.1203
896.36
0.1242
965.03
0.1258
1001.58
0.1269
934.02
0.1286
1055.85
0.1302
1082.44
0.1302
959.49
0.1308
1089.08
0.1313
1104.59
0.1313
1099.05
0.1313
1094.62
0.1324
1100.16
0.1357
1086.87
0.1379
999.37
0.1401
1037.03
0.1412
1060.28
TABLE_END
$********************************************************************$
$ To prevent fcc goes to further to the hcp + gamma phase region
$********************************************************************$
TABLE_HEAD 9300
CREATE_NEW_EQUILIBRIUM @@,1
183
CHANGE_STATUS PHASE HCP=ENTERED 1
CHANGE_STATUS PHASE GAMMA=ENTERED 1
CHANGE_STATUS PHASE FCC=DORMANT
SET_CONDITION P=P0, X(SI)=@1, T=@2, N=1
EXPERIMENT DGM(FCC)<0:1E-5
LABEL AFCC
TABLE_VALUES
$x(Si)
T
0.1170
827.69
0.1231
881.96
0.1275
920.73
0.1324
953.96
0.1396
997.15
TABLE_END
$********************************************************************$
$ To prevent fcc goes to further to the hcp + bcc phase region
$********************************************************************$
TABLE_HEAD 9350
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE HCP=ENTERED 1
CHANGE_STATUS PHASE BCC=ENTERED 1
CHANGE_STATUS PHASE FCC=DORMANT
SET_CONDITION P=P0, X(SI)=@1, T=@2, N=1
EXPERIMENT DGM(FCC)<0:1E-5
LABEL AFCC
TABLE_VALUES
$x(Si)
T
0.1330
1113.45
0.1363
1096.84
0.1390
1082.44
0.1429
1063.61
TABLE_END
$====================================================================$
$********************************************************************$
$
H.C.P.
$
$********************************************************************$
$====================================================================$
$********************************************************************$
$ To prevent hcp goes to further to the fcc phase region
$********************************************************************$
TABLE_HEAD 9400
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE FCC=ENTERED 1
CHANGE_STATUS PHASE HCP=DORMANT
SET_CONDITION P=P0, X(SI)=@1, T=@2, N=1
EXPERIMENT DGM(HCP)<0:1E-5
LABEL AHCP
TABLE_VALUES
$x(Si)
T
0.1088
1108.03
0.1080
1074.79
0.1071
1130.19
0.1055
1048.20
0.1038
1011.63
0.1022
957.34
0.1022
981.72
0.1005
924.10
0.0973
878.67
0.0964
1157.89
0.0948
826.59
184
0.0923
804.43
0.0874
1175.62
0.0775
1197.78
TABLE_END
$********************************************************************$
$ To prevent hcp goes to further to the bcc phase region
$********************************************************************$
TABLE_HEAD 9450
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE BCC=ENTERED 1
CHANGE_STATUS PHASE HCP=DORMANT
SET_CONDITION P=P0, X(SI)=@1, T=@2, N=1
EXPERIMENT DGM(HCP)<0:1E-5
LABEL AHCP
TABLE_VALUES
$x(Si)
T
0.1574
1070.36
0.1558
1075.90
0.1541
1082.55
0.1533
1090.30
0.1516
1098.06
0.1508
1105.82
0.1492
1111.36
0.1475
1122.44
TABLE_END
$********************************************************************$
$ To prevent hcp goes to further to the liquid + fcc phase region
$********************************************************************$
TABLE_HEAD 9500
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE LIQUID=ENTERED 1
CHANGE_STATUS PHASE BCC=ENTERED 1
CHANGE_STATUS PHASE HCP=DORMANT
SET_CONDITION P=P0, X(SI)=@1, T=@2, N=1
EXPERIMENT DGM(HCP)<0:1E-5
LABEL AHCP
TABLE_VALUES
$x(Si)
T
0.1516
1130.19
0.1360
1167.87
0.1319
1129.09
0.1203
1196.68
0.1129
1130.19
0.1055
1147.92
0.0981
1166.76
0.0940
1178.95
0.0849
1198.89
TABLE_END
$********************************************************************$
$ To prevent hcp goes to further to the liquid phase region
$********************************************************************$
TABLE_HEAD 9550
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE LIQUID=ENTERED 1
CHANGE_STATUS PHASE HCP=DORMANT
SET_CONDITION P=P0, X(SI)=@1, T=@2, N=1
EXPERIMENT DGM(HCP)<0:1E-5
LABEL AHCP
185
TABLE_VALUES
$x(Si)
T
0.2942
1099.17
0.2852
1113.57
0.2654
1125.76
0.2497
1134.63
0.2357
1134.63
0.2102
1122.44
0.1912
1100.28
0.1805
1109.14
0.1607
1130.19
0.1549
1141.27
0.1475
1157.89
0.1409
1173.41
0.1343
1185.60
0.1277
1197.78
TABLE_END
$====================================================================$
$********************************************************************$
$
B.C.C.
$
$********************************************************************$
$====================================================================$
$********************************************************************$
$ To prevent bcc goes to further to the liquid phase region
$********************************************************************$
TABLE_HEAD 9600
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE LIQUID=ENTERED 1
CHANGE_STATUS PHASE BCC=DORMANT
SET_CONDITION P=P0, X(SI)=@1, T=@2, N=1
EXPERIMENT DGM(BCC)<0:1E-5
LABEL ABCC
TABLE_VALUES
$x(Si)
T
0.2069
1115.79
0.2011
1111.36
0.1953
1103.60
0.1920
1096.95
0.1904
1099.17
0.1821
1104.71
0.1780
1110.25
0.1714
1115.79
0.1640
1123.55
0.1582
1127.98
0.1541
1140.17
0.1500
1147.92
0.1442
1161.22
0.1376
1173.41
0.1310
1187.81
TABLE_END
$********************************************************************$
$ To prevent bcc goes to further to the hcp phase region
$********************************************************************$
TABLE_HEAD 9650
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE HCP=ENTERED 1
CHANGE_STATUS PHASE BCC=DORMANT
SET_CONDITION P=P0, X(SI)=@1, T=@2, N=1
EXPERIMENT DGM(BCC)<0:1E-5
LABEL ABCC
TABLE_VALUES
186
$x(Si)
T
0.1409
1057.06
0.1401
1064.82
0.1401
1045.98
0.1393
1034.90
0.1376
1072.58
0.1352
1084.76
0.1327
1095.84
0.1310
1105.82
TABLE_END
$********************************************************************$
$ To prevent bcc goes to further to the liquid + fcc phase region
$********************************************************************$
TABLE_HEAD 9700
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE LIQUID=ENTERED 1
CHANGE_STATUS PHASE FCC=ENTERED 1
CHANGE_STATUS PHASE BCC=DORMANT
SET_CONDITION P=P0, X(SI)=@1, T=@2, N=1
EXPERIMENT DGM(BCC)<0:1E-5
LABEL ABCC
TABLE_VALUES
$x(Si)
T
0.1541
1129.09
0.1467
1145.71
0.1442
1129.09
0.1360
1170.08
0.1302
1129.09
0.1253
1191.14
0.1137
1130.19
0.1038
1154.57
0.0940
1176.73
0.0865
1195.57
TABLE_END
SAVE_WORKSPACES
187
A.1.3
EXP file
$===================================================================$
$
Map for this Cu-Si exp file
$
$===================================================================$
$*******************************************************************$
$ PROLOG 1 : Partial Enthalpy of Mixing in the liquid ˜1900K
$
$ PROLOG 2 : Enthalpy of Mixing in the liquid
$
$ PROLOG 3 : Activity of liquid at 1753K
$*******************************************************************$
$ DATASET 1 : Partial Enthalpy of Mixing in the liquid ˜1900K
$
$ DATASET 2 : Enthalpy of Mixing in the liquid
$
$ DATASET 3 : Activity of liquid at 1753K
$
$ DATASET 4 : Phase diagram data
$
$*******************************************************************$
PROLOG 1
XSCALE
0.00000
1.00000
YSCALE
-120.000
20.0000
XTYPE LINEAR
YTYPE LINEAR
XLENGTH
11.5000
YLENGTH
11.5000
TITLE Partial Enthalpy of Mixing
XTEXT Mole Fraction, Si
YTEXT Partial Enthalpy of Mixing, kJ/mol
PROLOG 2
XSCALE
0.000000
YSCALE
-18000
XTYPE LINEAR
YTYPE LINEAR
XLENGTH
11.5000
YLENGTH
11.5000
TITLE Enthalpy of Mixing
XTEXT Mole Fraction, Si
YTEXT Enthalpy of Mixing
PROLOG 3
XSCALE
0.00000
YSCALE
0.00000
XTYPE LINEAR
YTYPE LINEAR
XLENGTH
11.5000
YLENGTH
11.5000
TITLE
XTEXT Mole Fraction, Si
YTEXT Activity
1.00000
-0.157588E-02
1.00000
1.00000
$*******************************************************************$
$===================================================================$
$
Partial Enthalpy of Mixing for liquid
$
$===================================================================$
$*******************************************************************$
DATASET 1
CHARSIZE 0.2
SYM 0.2
BLOCK X=C1; Y=C2;
GOC=C3,WAS;
$ Partial Enthaly of Mixing Si of liquid phase at 1370K
$ R. Castanet, J. Chem. Thermodynamics, 1979, 11, 787-791
$ X(Si) Partial Enthalpy (kJ/mol)
$
188
0.00
-86.4 S100
0.05
-85.2
0.10
-82.0
0.15
-71.6
0.20
-41.2
0.25
-16.6
0.30
-0.6
0.35
3.0
0.40
4.4
0.45
5.0
BLOCKEND
BLOCK X=C1; Y=C2;
GOC=C3,WAS;
$ Partial Enthaly of Mixing, Cu in liquid phase
$ 1997 Victor Witusiewicz et al., Z. Metallkd. 88 (1997)
$ X(Si) Partial Enthalpy (kJ/mol)
0.0000
0.0
S201
0.0530
1.0
0.0540
0.3
0.1010
-0.4
0.1450
-5.9
0.2010
-10.8
0.2590
-14.3
0.2970
-16.9
0.3545
-19.8
0.3580
-19.9
0.3980
-19.1
0.4530
-15.6
0.4780
-20.1
0.4820
-19.6
0.5433
-24.4
0.5572
-23.9
0.5728
-24.2
0.5829
-23.3
0.5937
-18.0
0.6061
-23.5
0.6195
-23.8
0.6320
-20.8
0.6425
-24.7
0.6546
-27.1
0.6674
-24.2
0.6791
-25.2
0.6898
-20.9
0.6991
-25.3
0.7106
-27.4
0.7230
-22.4
0.7358
-27.3
0.7470
-23.1
0.7608
-25.1
0.7735
-24.1
0.7863
-24.2
0.8009
-23.7
0.8164
-24.3
0.8280
-17.3
0.8412
-18.7
0.8537
-16.8
0.8657
-22.6
0.8801
-18.4
0.8912
-13.2
0.9066
-14.9
0.9224
-16.2
0.9381
-13.3
0.9555
-9.7
0.9705
-10.6
189
0.9859
-14.3
BLOCKEND
BLOCK X=C1; Y=C2;
GOC=C3,WAS;
$ Partial Enthaly of Mixing, Si in liquid phase
$ 1997 Victor Witusiewicz et al., Z. Metallkd. 88 (1997)
$ X(Si) Partial Enthalpy (kJ/mol)
0.0140
-84.4 S201
0.0260
-86.8
0.0370
-78.5
0.0540
-84.4
0.0690
-84.3
0.0860
-73.3
0.1020
-74.8
0.1170
-72.4
0.1330
-60.0
0.1470
-61.9
0.1600
-51.7
0.1750
-36.3
0.1910
-43.3
0.2040
-25.6
0.2180
-26.3
0.2350
-18.9
0.2480
-18.8
0.2620
-8.8
0.2740
5.6
0.2870
3.6
0.3000
-1.5
0.3090
3.1
0.3230
2.8
0.3360
-1.5
0.3480
-3.2
0.3620
-2.2
0.3660
-2.0
0.3780
-5.9
0.3900
-2.2
0.4010
-0.9
0.4090
1.7
0.4210
0.7
0.4340
2.2
0.4460
1.7
0.4580
-4.4
0.4640
-4.1
0.4750
-1.7
0.4860
1.9
0.5461
2.8
0.5484
0.7
0.5959
2.5
0.6446
0.4
0.7011
1.0
0.7488
3.1
0.7878
2.6
0.8424
-0.4
0.8922
0.8
0.9386
-0.2
1.0000
0.0
BLOCKEND
BLOCK X=C1; Y=C2;
GOC=C3,WAS;
$ Partial Enthaly of Mixing, Cu in liquid phase
$ 2000 Victor Witusiewicz et al., Z. Metallkd. 91 (2000)
$ X(Si) Partial Enthalpy (kJ/mol)
0.1181
-0.3 S103
0.1419
-7.1
190
0.1665
0.1870
0.2328
0.2684
0.3110
BLOCKEND
-5.5
-9.6
-14.3
-18.3
-19.8
BLOCK X=C1; Y=C2;
GOC=C3,WAS;
$ Partial Enthaly of Mixing, Si in liquid phase
$ 2000 Victor Witusiewicz et al., Z. Metallkd. 91 (2000)
$ X(Si) Partial Enthalpy (kJ/mol)
0.1086
-76.8 S103
0.1302
-59.7
0.1540
-61.6
0.1768
-49.7
0.1984
-37.3
0.2232
-24.6
0.2387
-15.2
0.2586
-4.6
0.2761
-6.4
0.2996
-2.8
0.3195
-2.4
0.3398
-2.7
BLOCKEND
$*******************************************************************$
$===================================================================$
$
Enthalpy of Mixing for liquid
$
$===================================================================$
$*******************************************************************$
DATASET 2
CHARSIZE 0.2
SYM 0.15
$0.6 0.125 NS101’ Batalin and Sudavtosova
$0.6 0.080 NS102’ Witusiewicz et al.
$0.6 0.040 NS103’ Iguchi et al.
BLOCK X=C1; Y=C2;
GOC=C3,WAS;
$ Enthaly of Mixing of liquid phase at 1370K
$ R. Castanet, J. Chem. Thermodynamics, 1979, 11, 787-791
$ X(Si) Enthalpy (J/mol)
0.0135
-1350.77 S200
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
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
$
191
0.2539
0.2592
0.2709
0.2879
0.2897
0.2959
0.3076
0.3255
0.3318
0.3702
0.3971
0.4132
0.4526
0.4562
BLOCKEND
-14303.50
-14125.50
-14178.40
-13893.40
-13573.30
-13626.40
-13394.80
-12754.10
-13056.20
-11668.00
-11347.00
-10528.60
-9442.62
-9780.30
BLOCK X=C1; Y=C2;
GOC=C3,WAS;
$ Enthaly of Mixing of liquid phase at 1600K
$ 1981 Arpshofen et al., Z. Metallkunde, 72 (1981) 842-846
$ X(Si) Enthalpy (J/mol)
0.973
-290
S101
0.73
-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
BLOCKEND
BLOCK X=C1; Y=C2;
GOC=C3,WAS;
$ Enthaly of Mixing of liquid phase at 1773K
$ 1982 G.I. Batalin and V.S. Sudavtsova
$ X(Si) Enthalpy (J/mol)
0.973
-290 S100
0.948
-380
0.916
-1005
0.897
-1460
0.872
-1880
0.848
-2300
0.828
-2720
0.81
-3055
192
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
0.143
-10670
0.128
-9450
0.108
-8450
0.08
-6230
0.05
-3975
BLOCKEND
BLOCK X=C1; Y=C2;
GOC=C3,WAS;
$ Enthaly of Mixing of liquid phase around 1900K
$ 1997 Victor Witusiewicz et al., Z. Metallkd. 88 (1997)
$ X(Si)
Enthalpy (J/mol)
0.014
-1000 S102
0.026
-2100
0.037
-2700
0.053
-3600
0.054
-4400
0.054
-4400
0.069
-5700
0.086
-6500
0.101
-8000
0.102
-8300
0.117
-9700
0.133
-10100
0.145
-13300
0.147
-12000
0.16
-12000
0.175
-11400
0.191
-14700
0.201
-14600
0.204
-12600
0.218
-14200
193
0.235
0.248
0.259
0.262
0.274
0.287
0.297
0.3
0.309
0.323
0.336
0.348
0.3545
0.358
0.362
0.366
0.378
0.39
0.398
0.401
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
-14100
-15200
-13000
-13600
-10200
-11100
-12700
-12900
-11600
-11600
-13000
-13400
-12600
-12600
-12800
-12700
-13900
-12200
-11600
-11300
-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
194
0.8537
0.8657
0.8801
0.8912
0.8922
0.9066
0.9224
0.9381
0.9386
0.9555
0.9705
0.9859
BLOCKEND
-1810
-2510
-1840
-1190
-980
-1270
-1250
-900
-910
-530
-410
-260
BLOCK X=C1; Y=C2;
GOC=C3,WAS;
$ Enthaly of Mixing of liquid phase around 1393K
$ 1997 Iguchi et al., J. Iron Steel Inst. Jpn., 63, 275-284
$ X(Si) Enthalpy (J/mol)
0.1995 -12260 S201
0.1732 -11860
0.1492 -10181
0.1263 -8352
0.0988 -7576
0.0818 -5496
0.0541 -3591
0.0277 -2038
BLOCKEND
BLOCK X=C1; Y=C2;
GOC=C3,WAS;
$ Enthaly of Mixing of liquid phase around 1900K
$ 1997 Victor Witusiewicz et al., Z. Metallkd. 88 (1997)
$ X(Si)
Enthalpy (J/mol)
0.0977
-9500 S202
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
BLOCKEND
$*******************************************************************$
$===================================================================$
$
Activity of liquid
$
$===================================================================$
$*******************************************************************$
DATASET 3
CHARSIZE 0.2
SYM 0.2
195
BLOCK X=C1; Y=C2;
GOC=C3,WAS;
$ Activity of the liquid phase at 1753K
$ Bergman et al., J. Chem. Thermo. 1986, 18, 835-845
$ x(Si) ACR(Si)
0.097
0.0005 S100
0.180
0.007
0.248
0.054
0.272
0.077
0.324
0.079
0.392
0.187
0.475
0.37
0.775
0.82
0.885
0.9
BLOCKEND
BLOCK X=C1; Y=C2;
GOC=C3,WAS;
$ Activity of the liquid phase at 1753K
$ Bergman et al., J. Chem. Thermo. 1986, 18, 835-845
$ x(Si) ACR(Cu)
0.097
0.76 S100
0.180
0.64
0.248
0.36
0.272
0.33
0.324
0.2
0.392
0.14
0.475
0.13
0.775
0.045
0.885
0.025
BLOCKEND
BLOCK X=C1; Y=C2;
GOC=C3,WAD
$ Activity of Liquid at 1773K
$ Miki et al., ISIJ International, 42 (2002), 1071-1076
$ x(Si) ACR(Si)
LINETYPE 3
1.0000 1.0000 M
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
0.0000 0.0000
BLOCKEND
BLOCK X=C1; Y=C2;
GOC=C3,WAD;
$ Activity of Liquid at 1773K
$ Miki et al., ISIJ International, 42 (2002), 1071-1076
196
$ x(Si) ACR(Cu)
LINETYPE 3
0.0000 1.0000 M
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
1.0000 0.0000
BLOCKEND
BLOCK X=C1; Y=C2;
GOC=C3,WAS;
$ Activity of Liquid at 1783K
$ K. Sano et al., Mem. Fac. Eng. Nagoya Univ., 8 (1956) 127
$ x(Si) ACR(Si)
0.8729 0.8837 S201
0.8181 0.9003
0.7054 0.6981
0.5783 0.4820
0.5639 0.4848
0.5264 0.3961
0.4773 0.2576
0.4426 0.2493
0.3675 0.1524
0.3040 0.1108
0.2231 0.1025
0.2173 0.0720
BLOCKEND
BLOCK X=C1; Y=C2;
GOC=C3,WAS;
$ Activity of Liquid at 1743K
$ K. Sano et al., Mem. Fac. Eng. Nagoya Univ., 8 (1956) 127
$ x(Si) ACR(Si)
0.5783 0.5429 S101
0.5292 0.3850
0.4513 0.1911
0.4397 0.2188
0.1451 0.0443
0.1105 0.0277
0.0903 0.0194
0.0758 0.0166
BLOCKEND
$BLOCK X=C1; Y=C2;
GOC=C3,WAS;
$$ Activity of Liquid at 1700K
$$ G. Riekert et al.,
$$ x(Si)
acr(Cu)
$0.0
1.00 S200
$0.1
0.88
$0.2
0.75
197
$0.3
0.66
$0.4
0.64
$0.5
0.63
$0.6
0.62
$0.7
0.57
$0.8
0.46
$0.9
0.27
$1.0
0.00
$BLOCKEND
$
$BLOCK X=C1; Y=C2;
GOC=C3,WAS;
$$ Activity of Liquid at 1700K
$$ G. Riekert et al.,
$$ x(Si)
acr(Si)
$0.0
0.00 S200
$0.1
0.17
$0.2
0.43
$0.3
0.65
$0.4
0.69
$0.5
0.71
$0.6
0.73
$0.7
0.77
$0.8
0.89
$0.9
0.91
$1.0
1.00
$BLOCKEND
$*******************************************************************$
$===================================================================$
$
Phase diagram data
$
$===================================================================$
$*******************************************************************$
DATASET 4
CHARSIZE 0.2
SYM 0.15
BLOCK X=C1; Y=C2;
$ Phase diagram data
$ 1907 E. Rudolfi
$ x(Si) Temperature
0.0007
1356.97
0.0207
1330.53
0.0208
1344.01
0.0409
1327.52
0.0418
1274.13
0.0625
1297.70
0.0635
1304.48
0.0635
1224.73
0.0835
1280.34
0.0836
1172.44
0.1000
1237.59
0.1025
1248.01
0.1033
1116.62
0.1036
981.74
0.1250
1196.79
0.1273
1116.93
0.1277
1198.43
0.1284
982.01
0.1292
1087.54
0.1417
1121.09
GOC=C3,WAS;
198
0.1424
0.1426
0.1432
0.1513
0.1519
0.1520
0.1526
0.1556
0.1563
0.1570
0.1607
0.1611
0.1626
0.1676
0.1677
0.1688
0.1748
0.1753
0.1820
0.1824
0.1962
0.1972
0.2092
0.2099
0.2212
0.2213
0.2221
0.2308
0.2310
0.2392
0.2529
0.2531
0.3338
0.3592
0.3610
0.4273
0.4920
0.5987
0.6902
0.7686
0.8357
0.8945
0.9464
0.9920
BLOCKEND
986.19
1081.18
1173.23
1145.36
1077.32
1127.75
981.71
1135.82
1125.82
1066.57
1120.97
997.22
1057.56
997.55
1049.74
1115.38
1111.63
994.33
985.85
1105.47
1091.44
988.77
1109.09
1098.51
1125.03
1098.05
981.31
1129.90
982.12
1133.66
1126.04
1073.26
1077.66
1105.26
1071.02
1262.95
1328.63
1423.70
1482.40
1536.81
1582.69
1624.29
1653.06
1669.01
BLOCK X=C1; Y=C2;
$ Phase diagram data
$ 1928 Smith - I
$ x(Si) Temperature
0.1098
1130.85
0.1071
1138.54
0.0971
1167.11
0.0837
1195.59
0.0769
1216.48
0.0655
1241.70
0.0545
1259.16
0.0441
1283.29
0.0248
1317.19
BLOCKEND
GOC=C3,WAS;
BLOCK X=C1; Y=C2;
$ Phase diagram data
$ 1928 Smith - II
GOC=C3,WAS;
199
$ x(Si)
0.1586
0.1551
0.1542
0.1540
0.1488
0.1462
0.1424
0.1347
0.1248
0.1182
0.1085
0.0981
0.0852
0.0779
0.0663
0.0550
0.0455
0.0257
BLOCKEND
Temperature
1126.70
1138.80
1139.89
1146.54
1156.37
1159.62
1157.29
1183.69
1204.49
1223.16
1242.86
1253.67
1275.50
1289.73
1296.07
1313.53
1319.94
1338.29
BLOCK X=C1; Y=C2;
$ Phase diagram data
$ 1929 Smith - I
$ x(Si) Temperature
0.1293
1206.09
0.1384
1182.70
0.1467
1155.97
0.1513
1157.64
0.1533
1155.97
0.1581
1145.94
0.1594
1139.26
0.1608
1135.92
0.2127
1105.85
0.2224
1117.54
0.2285
1124.22
0.2315
1125.90
0.2357
1129.24
0.2393
1127.57
0.2458
1127.57
0.2482
1129.24
0.2500
1127.57
0.2547
1127.57
0.2994
1092.48
0.3229
1092.48
0.3355
1120.88
0.3386
1129.24
0.3475
1137.59
0.3581
1177.69
BLOCKEND
GOC=C3,WAS;
BLOCK X=C1; Y=C2;
$ Phase diagram data
$ 1929 Smith - II
$ x(Si) Temperature
0.1687
999.33
0.1749
999.51
0.1755
973.33
0.1756
992.77
0.1757
953.05
0.1788
998.78
0.1788
992.87
0.1813
973.50
GOC=C3,WAS;
200
0.1819
0.1847
0.1901
0.1913
0.1943
BLOCKEND
953.24
993.88
993.20
1004.22
1022.90
BLOCK X=C1; Y=C2;
$ Phase diagram data
$ 1940 Smith
$ x(Si) Temperature
0.0814
666.36
0.0871
716.44
0.0923
766.50
0.0980
816.58
0.0985
822.14
0.0989
827.70
0.0994
834.37
0.0998
845.47
0.1012
865.48
0.1039
915.47
0.1064
965.46
0.1065
1140.74
0.1077
989.91
0.1089
1014.34
0.1100
1039.89
0.1111
1065.44
0.1112
833.58
0.1116
1125.35
0.1116
845.80
0.1125
1113.18
0.1127
865.80
0.1136
845.85
0.1150
915.78
0.1167
865.91
0.1176
965.78
0.1188
988.00
0.1199
1015.76
0.1212
1040.20
0.1223
1065.75
0.1234
1085.75
0.1236
1097.96
0.1241
1109.07
0.1244
917.16
0.1265
1108.02
0.1291
1098.11
0.1321
1087.11
0.1337
966.23
0.1394
1064.02
0.1402
989.72
0.1432
1039.72
0.1435
1116.28
0.1442
1015.35
0.1454
1107.46
0.1456
1118.56
BLOCKEND
GOC=C3,WAS;
BLOCK X=C1; Y=C2;
$ Phase diagram data
$ 1940 Andersen
$ x(Si) Temperature
0.0808
658.59
0.0890
732.88
0.0936
773.02
GOC=C3,WAS;
201
0.0945
0.0956
0.0964
0.0966
0.0970
0.0975
0.0977
0.0980
0.1003
0.1012
0.1040
0.1129
0.1144
0.1260
0.1190
0.1138
0.1118
0.1138
0.1164
0.1179
0.1235
0.1284
0.1396
BLOCKEND
785.81
797.75
811.38
816.49
822.45
832.68
840.33
852.25
903.34
923.77
973.18
1083.19
1096.85
1025.95
904.98
865.66
852.83
853.77
864.92
873.49
902.62
927.47
983.19
202
A.1.4
TDB file
$
ELEMENT
ELEMENT
ELEMENT
ELEMENT
/VA
CU
SI
ELECTRON_GAS
VACUUM
FCC_A1
DIAMOND_FCC_A4
0.0000E+00
0.0000E+00
6.3546E+01
2.8085E+01
0.0000E+00
0.0000E+00
5.0041E+03
3.2175E+03
0.0000E+00!
0.0000E+00!
3.3150E+01!
1.8820E+01!
FUNCTION GHSERCU
2.98140E+02 -7770.458+130.485235*T-24.112392*T*LN(T)
-.00265684*T**2+1.29223E-07*T**3+52478*T**(-1); 1.35777E+03 Y
-13542.026+183.803828*T-31.38*T*LN(T)+3.64167E+29*T**(-9);
3.20000E+03 N !
FUNCTION GHSERSI
2.98140E+02 -8162.609+137.236859*T-22.8317533*T*LN(T)
-.001912904*T**2-3.552E-09*T**3+176667*T**(-1); 1.68700E+03 Y
-9457.642+167.281367*T-27.196*T*LN(T)-4.20369E+30*T**(-9);
3.60000E+03 N !
FUNCTION UN_ASS 298.15 0; 300 N !
TYPE_DEFINITION % SEQ *!
DEFINE_SYSTEM_DEFAULT SPECIE 2 !
DEFAULT_COMMAND DEF_SYS_ELEMENT VA !
PHASE LIQUID:L % 1 1.0 !
CONSTITUENT LIQUID:L :CU,SI :
!
PARAMETER G(LIQUID,CU;0) 2.98140E+02 +12964.735-9.511904*T+GHSERCU#
-5.8489E-21*T**7; 1.35777E+03 Y
-46.545+173.881484*T-31.38*T*LN(T); 3.20000E+03 N REF:0 !
PARAMETER G(LIQUID,SI;0) 2.98140E+02 +50696.36-30.099439*T+GHSERSI#
+2.09307E-21*T**7; 1.68700E+03 Y
+40370.523+137.722298*T-27.196*T*LN(T); 3.60000E+03 N REF:0 !
PARAMETER G(LIQUID,CU,SI;0) 2.98150E+02 -38763+5.653*T;
6.00000E+03
N REF:0 !
PARAMETER G(LIQUID,CU,SI;1) 2.98150E+02 -52442+25.307*T;
6.00000E+03
N REF:0 !
PARAMETER G(LIQUID,CU,SI;2) 2.98150E+02 -29485+14.742*T;
6.00000E+03
N REF:0 !
TYPE_DEFINITION & GES A_P_D BCC_A2 MAGNETIC
PHASE BCC_A2 %& 2 1
3 !
CONSTITUENT BCC_A2 :CU,SI : VA : !
-1.0
4.00000E-01 !
PARAMETER G(BCC_A2,CU:VA;0) 2.98140E+02 +4017-1.255*T+GHSERCU#;
3.20000E+03 N REF:0 !
PARAMETER G(BCC_A2,SI:VA;0) 2.98140E+02 +47000-22.5*T+GHSERSI#;
3.60000E+03 N REF:0 !
PARAMETER G(BCC_A2,CU,SI:VA;0) 2.98150E+02 -8742-12.281*T;
6.00000E+03
N REF:0 !
PARAMETER G(BCC_A2,CU,SI:VA;1) 2.98150E+02 -68822+8.906*T;
6.00000E+03
N REF:0 !
TYPE_DEFINITION ’ GES A_P_D CBCC_A12 MAGNETIC
PHASE CBCC_A12 %’ 2 1
1 !
CONSTITUENT CBCC_A12 :SI : VA : !
PARAMETER G(CBCC_A12,SI:VA;0)
3.60000E+03 N REF:0 !
PHASE CUB_A13
%
2 1
1 !
2.98140E+02
-3.0
2.80000E-01 !
+50208-20.377*T+GHSERSI#;
203
CONSTITUENT CUB_A13
:SI : VA :
PARAMETER G(CUB_A13,SI:VA;0)
3.60000E+03 N REF:0 !
2.98140E+02
PHASE DELTA % 2 .825
.175 !
CONSTITUENT DELTA :CU : SI :
PARAMETER G(DELTA,CU:SI;0)
-5215-1.6*T;
6.00000E+03
!
!
2.98150E+02
N REF:0 !
PHASE DIAMOND_A4 % 1 1.0 !
CONSTITUENT DIAMOND_A4 :SI :
PARAMETER G(DIAMOND_A4,SI;0)
REF:0 !
+47279-20.377*T+GHSERSI#;
+.825*GHSERCU#+.175*GHSERSI#
!
2.98140E+02
+GHSERSI#;
3.60000E+03
N
PHASE EPSILON % 2 .789474
.210526 !
CONSTITUENT EPSILON :CU : SI : !
PARAMETER G(EPSILON,CU:SI;0) 2.98150E+02 +.789474*GHSERCU#
+.210526*GHSERSI#-6136-1.386*T;
6.00000E+03
N REF:0 !
PHASE ETA % 2 .76
.24 !
CONSTITUENT ETA :CU : SI :
!
PARAMETER G(ETA,CU:SI;0) 2.98150E+02
-1.801*T;
6.00000E+03
N REF:0 !
+.76*GHSERCU#+.24*GHSERSI#-6255
TYPE_DEFINITION ( GES A_P_D FCC_A1 MAGNETIC
PHASE FCC_A1 %( 2 1
1 !
CONSTITUENT FCC_A1 :CU,SI : VA : !
-3.0
2.80000E-01 !
PARAMETER G(FCC_A1,CU:VA;0) 2.98140E+02 +GHSERCU#; 3.20000E+03 N
REF:0 !
PARAMETER G(FCC_A1,SI:VA;0) 2.98140E+02 +51000-21.8*T+GHSERSI#;
3.60000E+03 N REF:0 !
PARAMETER G(FCC_A1,CU,SI:VA;0) 2.98150E+02 -34176+7.017*T;
6.00000E+03
N REF:0 !
PARAMETER G(FCC_A1,CU,SI:VA;1) 2.98150E+02 -26169-7.43*T;
6.00000E+03
N REF:0 !
PHASE GAMMA % 2 .835821
.164179 !
CONSTITUENT GAMMA :CU : SI : !
PARAMETER G(GAMMA,CU:SI;0) 2.98150E+02 +.835821*GHSERCU#
+.164179*GHSERSI#-6005-.5*T;
6.00000E+03
N REF:0 !
TYPE_DEFINITION ) GES A_P_D HCP_A3 MAGNETIC
PHASE HCP_A3 %) 2 1
.5 !
CONSTITUENT HCP_A3 :CU,SI : VA : !
-3.0
2.80000E-01 !
PARAMETER G(HCP_A3,CU:VA;0) 2.98140E+02 +600+.2*T+GHSERCU#;
3.20000E+03 N REF:0 !
PARAMETER G(HCP_A3,SI:VA;0) 2.98140E+02 +49200-20.8*T+GHSERSI#;
3.60000E+03 N REF:0 !
PARAMETER G(HCP_A3,CU,SI:VA;0) 2.98150E+02 -23124-2.221*T;
204
6.00000E+03
N REF:0 !
PARAMETER G(HCP_A3,CU,SI:VA;1)
6.00000E+03
N REF:0 !
LIST_OF_REFERENCES
NUMBER SOURCE
!
2.98150E+02
-48482+4.615*T;
205
A.2
A.2.1
The Hf-O system
Setup file
GOTO_MODULE DATABASE
SWITCH USER HFO_SSUB
DEFINE_SYSTEM HF O HF+4 O-2 HF1O1 HF1O2 O O2 O3
GET
GOTO_MODULE GIBBS_ENERGY_SYSTEM
ENTER_PHASE IONIC_LIQ Y HF+4; O-2,VA; NO NO
AMEND_PHASE_DESCRIPTION HCP_A3 NEW_CONSTITUENT 2 O
AMEND_PHASE_DESCRIPTION BCC_A2 NEW_CONSTITUENT 2 O
GOTO_MODULE PARROT
ENTER_PARAMETER G(IONIC_LIQ,HF+4:O-2;0) 298.15 2*F11197+2.52e5-86.7975992*T;
6000 N ENTER_PARAMETER G(IONIC_LIQ,HF+4:VA;0) 298.15 GHFLIQ; 6000 N
ENTER_PARAMETER G(IONIC_LIQ,HF+4:O-2,VA;0) 298.15 V1+V2*T; 6000 N
ENTER_PARAMETER G(IONIC_LIQ,HF+4:O-2,VA;1) 298.15 V3+V4*T; 6000 N
ENTER_PARAMETER G(IONIC_LIQ,HF+4:O-2,VA;2) 298.15 V5+V6*T; 6000 N
ENTER_PARAMETER G(HCP_A3,HF:O;0) 298.15 GHSERHF+0.5*GHSEROO+V11+V12*T; 6000 N
ENTER_PARAMETER G(HCP_A3,HF:O,VA;0) 298.15 V13+V14*T; 6000 N ENTER_PARAMETER
G(HCP_A3,HF:O,VA;1) 298.15 V15+V16*T; 6000 N ENTER_PARAMETER
G(HCP_A3,HF:O,VA;2) 298.15 V17+V18*T; 6000 N
ENTER_PARAMETER G(BCC_A2,HF:O;0) 298.15 GHSERHF+3*GHSEROO+V21+V22*T; 6000 N
ENTER_PARAMETER G(BCC_A2,HF:O,VA;0) 298.15 V23+V24*T; 6000 N ENTER_PARAMETER
G(BCC_A2,HF:O,VA;1) 298.15 V25+V26*T; 6000 N ENTER_PARAMETER
G(BCC_A2,HF:O,VA;2) 298.15 V27+V28*T; 6000 N
CREATE_NEW_STORE_FILE HFO
SAVE_PARROT_WORKSPACES
SET_INTERACTIVE
206
A.2.2
POP file
ENTER_SYMBOL CONSTANT P0=101325 R0=8.314 DX=0.05 DH=200 DT=3
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$
PHASE DIAGRAM DATA
$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$$$$$$$$$$$$$$
$$ TWO PHASE $$
$$$$$$$$$$$$$$$
$$$$$$$$$$$$$$$$$$$$$$$
$$ ALPHA PHASE
$$
$$ CONGRUENT MELTING $$
$$$$$$$$$$$$$$$$$$$$$$$
CREATE_NEW_EQUILIBRIUM 100,1
CHANGE_STATUS PHASE IONIC_LIQ,HCP_A3=FIXED 1
SET_CONDITION P=P0 X(IONIC_LIQ,O)-X(HCP_A3,O)=0
EXPERIMENT X(O)=0.15:0.01
EXPERIMENT T=2773.15:1
SET_ALL_START_VALUES 2773.15 Y
LABEL AM01
$$$$$$$$$$$$$$$
$$ HCP + BCC $$
$$$$$$$$$$$$$$$
TABLE_HEAD 1000
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE BCC_A2=FIXED 1
CHANGE_STATUS PHASE HCP_A3=FIXED 0
SET_CONDITION X(BCC_A2,O)=@1,P=P0
EXPERIMENT T=@2:DT
LABEL AP01
SET_ALL_START 2051.07 Y
TABLE_VALUES
0.00211049 2051.07
0.00446927 2086.46
0.0072005
2121.34
0.00931099 2155.75
0.0112973
2190.16
0.0134078
2224.56
0.0155183
2258.97
0.0177529
2293.37
0.0196151
2327.78
0.0212291
2362.19
0.0234637
2396.6
0.0253259
2431.01
0.0269398
2467.88
0.0302917
2499.31
TABLE_END
TABLE_HEAD 1100
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE HCP_A3=FIXED 1
CHANGE_STATUS PHASE BCC_A2=FIXED 0
SET_CONDITION X(HCP_A3,O)=@1,P=P0
EXPERIMENT T=@2:DT
LABEL AP02
TABLE_VALUES
0.00558659 2035.76
0.0122905
2060.25
207
0.0187461
0.0244569
0.0299193
0.0351335
0.0400993
0.0448169
0.0495345
0.0543762
0.0588454
0.0623215
0.0653011
0.0687772
0.0723774
0.0754811
TABLE_END
2085.23
2111.71
2139.66
2168.6
2198.53
2228.96
2258.89
2288.33
2317.29
2348.23
2381.14
2414.54
2446.95
2480.36
$$$$$$$$$$$$$$$$$$$$$$$$
$$ IONIC LIQUID + BCC $$
$$$$$$$$$$$$$$$$$$$$$$$$
TABLE_HEAD 2000
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE IONIC_LIQ=FIXED 1
CHANGE_STATUS PHASE BCC_A2=FIXED 0
SET_CONDITION X(IONIC_LIQ,O)=@1,P=P0
EXPERIMENT T=@2:DT
LABEL AP05
TABLE_VALUES
0.00223464 2510.6
0.00558659 2512.02
0.00806952 2518.38
TABLE_END
TABLE_HEAD 2100
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE BCC_A2=FIXED 1
CHANGE_STATUS PHASE IONIC_LIQ=FIXED 0
SET_CONDITION T=@2,P=P0
EXPERIMENT X(BCC_A2,O)=@1:DX
LABEL AP06
TABLE_VALUES
0.00670391 2512
0.0122905
2516.34
0.0167598
2516.26
0.0212291
2520.62
0.027933
2524.93
TABLE_END
$$$$$$$$$$$$$$$$$$$$$
$$ IONIC_LIQ + HCP $$
$$$$$$$$$$$$$$$$$$$$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$ LEFT from CONGRUENT MELTING $$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
TABLE_HEAD 3000
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE IONIC_LIQ=FIXED 1
CHANGE_STATUS PHASE HCP_A3=FIXED 0
SET_CONDITION X(IONIC_LIQ,O)=@1,P=P0
EXPERIMENT T=@2:DT
208
LABEL AP07
TABLE_VALUES
0.0301 2578.18
0.0480 2622.17
0.0692 2674.95
0.0893 2714.47
0.1128 2749.5
0.1262 2762.57
TABLE_END
TABLE_HEAD 3100
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE HCP_A3=FIXED 1
CHANGE_STATUS PHASE IONIC_LIQ=FIXED 0
SET_CONDITION X(HCP_A3,O)=@1,P=P0
EXPERIMENT T=@2:DT
LABEL AP08
TABLE_VALUES
0.0815 2541.91
0.0893 2603.77
0.1027 2674.4
0.1162 2714.02
0.1284 2740.39
0.1374 2757.95
TABLE_END
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$ RIGHT from CONGRUENT MELTING $$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
TABLE_HEAD 3200
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE IONIC_LIQ=FIXED 0
CHANGE_STATUS PHASE HCP_A3=FIXED 1
SET_CONDITION X(IONIC_LIQ,O)=@1,P=P0
EXPERIMENT T=@2:DT
LABEL AP09
TABLE_VALUES
0.1720 2739.67
0.1798 2717.4
0.1865 2686.29
0.1955 2646.29
0.2022 2597.47
0.2100 2544.2
0.2145 2513.13
TABLE_END
TABLE_HEAD 3300
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE HCP_A3=FIXED 0
CHANGE_STATUS PHASE IONIC_LIQ=FIXED 1
SET_CONDITION X(HCP_A3,O)=@1,P=P0
EXPERIMENT T=@2:DT
LABEL AP10
TABLE_VALUES
0.1731 2766.22
0.1832 2757.19
0.1932 2743.74
0.2055 2725.82
0.2201 2699.02
0.2312 2672.26
0.2435 2641.06
0.2569 2605.41
0.2692 2565.36
209
0.2782 2534.21
TABLE_END
$$$$$$$$$$$$$$$$$$$$$$$$$$
$$ IONIC_LIQ + HF1O2_S2 $$
$$$$$$$$$$$$$$$$$$$$$$$$$$
TABLE_HEAD 4000
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE IONIC_LIQ=FIXED 1
CHANGE_STATUS PHASE HF1O2_S2=FIXED 0
SET_CONDITION X(IONIC_LIQ,O)=@1,P=P0
EXPERIMENT T=@2:1%
LABEL AP11
TABLE_VALUES
0.3292 2568.27
0.3575 2654.23
0.3836 2727.27
TABLE_END
$$$$$$$$$$$$$$$$$$$$$$$$$$
$$ IONIC_LIQ + HF1O2_S3 $$
$$$$$$$$$$$$$$$$$$$$$$$$$$
TABLE_HEAD 4100
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE IONIC_LIQ=FIXED 1
CHANGE_STATUS PHASE HF1O2_S3=FIXED 0
SET_CONDITION X(IONIC_LIQ,O)=@1,P=P0
EXPERIMENT T=@2:1%
LABEL AP12
TABLE_VALUES
0.4118 2795.97
0.4434 2864.63
0.4783 2924.61
0.5075 2967.4
0.5367 3001.55
0.5692 3031.34
0.5983 3048.23
0.6307 3065.08
0.6552 3069.08
TABLE_END
$$$$$$$$$$$$$$$$$$
$$ THREE PHASES $$
$$$$$$$$$$$$$$$$$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$ PERITECTIC REACTION AT HF RICH SIDE $$
$$ LIQUID + HCP ->> BCC
$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
CREATE_NEW_EQUILIBRIUM 2,1
CHANGE_STATUS PHASE IONIC_LIQ,BCC_A2,HCP_A3=FIXED 1
SET_CONDITION P=P0
EXPERIMENT T=2525.15:1
EXPERIMENT X(IONIC_LIQ,O)=0.01:3%
EXPERIMENT X(BCC_A2,O)=0.03:3%
EXPERIMENT X(HCP_A3,O)=0.08:3%
LABEL APR1
SET_ALL_START_VALUES 2525.15 Y
210
$ PERITECTIC REACTION AT HF RICH SIDE
ENTER_SYMBOL FUNCTION DLQB=X(IONIC_LIQ,O)-X(BCC_A2,O);
CREATE_NEW_EQUILIBRIUM 3,1
CHANGE_STATUS PHASE IONIC_LIQ,BCC_A2,HCP_A3=FIXED 1
SET_CONDITION P=P0
EXPERIMENT T=2525.15:1
EXPERIMENT DLQB>0:1E-2
LABEL APR1
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$ EUTECTIC REACTION AT HFO2 SIDE $$
$$ IONIC_LIQ -> HCP + HF1O2_S2
$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
CREATE_NEW_EQUILIBRIUM 4,1
CHANGE_STATUS PHASE IONIC_LIQ,HCP_A3,HF1O2_S2=FIXED 1
SET_CONDITION P=P0
EXPERIMENT T=2473.15:DT
EXPERIMENT X(IONIC_LIQ,O)=0.3:1%
EXPERIMENT X(HCP_A3,O)=0.22:3%
SET_ALL_START_VALUES 2473.15 Y
LABEL AER1
$$$$$$$$$$$$$$$$$$$$$$$$$
$$ THERMOCHEMICAL DATA $$
$$$$$$$$$$$$$$$$$$$$$$$$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$ PARTIAL MOLAR ENTHALPY OF MIXING $$
$$ HCP_ALPHA PHASE
$$
$$ 1984 G. BOUREAU
$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
ENTER_SYMBOL FUNCTION HPO=2*(MUR(O)-T*MUR(O).T);
TABLE_HEAD 5000
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE HCP_A3=FIXED 1
SET_REFERENCE_STATE HF HCP_A3 * 1E5
SET_REFERENCE_STATE O GAS * 1E5
SET_CONDITION T=1323,P=P0,X(HCP_A3,O)=@1
EXPERIMENT HPO=@2:5%
SET_ALL_START_VALUES Y
LABEL AT01
TABLE_VALUES
0.020238253 -1191140
0.02603629 -1191610
0.032953184 -1200060
0.040366319 -1205220
0.045372855 -1186030
0.052318645 -1184870
0.058582438 -1195420
0.064490122 -1201290
0.069809582 -1176480
0.076120382 -1185150
0.082079109 -1191960
0.086657148 -1175810
0.092729406 -1195020
0.098742656 -1196670
0.104933962 -1196920
0.109537565 -1184290
0.114573326 -1190390
0.12030142 -1192040
0.125242527 -1175660
211
0.131073255
0.135404328
0.140626208
0.146031223
0.149988737
0.15595432
0.160257331
0.165189748
0.170065938
0.174468626
0.179260234
0.18353417
0.188192421
0.192801077
0.197167607
0.20167393
0.204918059
0.209715442
0.218040127
0.222506267
0.226178685
0.230751479
0.23825078
0.242846413
0.246670283
0.254223889
TABLE_END
-1178950
-1173570
-1179200
-1172890
-1162360
-1162610
-1167070
-1167080
-1165920
-1145560
-1121450
-1133640
-1139270
-1141860
-1127350
-1130410
-1134870
-1163220
-1136540
-1143820
-1124860
-1120890
-1117870
-1136620
-1145060
-1137120
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$ ENTHALPY OF FORMATION: HCP $$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
CREATE_NEW_EQUILIBRIUM 5,1
CHANGE_STATUS PHASE HCP_A3=FIXED 1
SET_REFERENCE_STATE HF HCP_A3 * 1E5
SET_REFERENCE_STATE O GAS * 1E5
SET_CONDITION T=298.15,P=P0,X(HCP_A3,O)=0.333333
EXPERIMENT HMR=-175511:5%
LABEL AT02
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$ ENTHALPY OF FORMATION: BCC $$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
CREATE_NEW_EQUILIBRIUM 6,1
CHANGE_STATUS PHASE BCC_A2=FIXED 1
SET_REFERENCE_STATE HF HCP_A3 * 1E5
SET_REFERENCE_STATE O GAS * 1E5
SET_CONDITION T=298.15,P=P0,X(BCC_A2,O)=0.7499999
EXPERIMENT HMR=-161308:25%
LABEL AT03
$$$$$$$$$$$$$$$$
$$ SQS OF HCP $$
$$$$$$$$$$$$$$$$
TABLE_HEAD 6000
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE HCP_A3=FIXED 1
SET_REFERENCE_STATE HF HCP_A3 * 1E5
SET_REFERENCE_STATE O GAS * 1E5
SET_CONDITION T=1323,P=P0,X(HCP_A3,O)=@1
EXPERIMENT HMR=@2:5%
SET_ALL_START_VALUES Y
LABEL AT04
212
TABLE_VALUES
0.2
-109480.7993
0.2
-108075.4063
0.2
-109186.9879
0.2
-107818.3877
0.2
-109230.1085
0.2
-107768.465
TABLE_END
$$$$$$$$$$$$$$$$$$$$$$$$$$
$$ STABILITY CONDITIONS $$
$$ FOR GAS PHASE AT H.T $$
$$$$$$$$$$$$$$$$$$$$$$$$$$
TABLE_HEAD 7000
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE GAS=F 1
CHANGE-STATUS PHASE IONIC_LIQ=D
SET-CONDITION P=P0, X(GAS,O)=@1,T=@2
EXPERIMENT DGM(IONIC_LIQ)<0:1E-5
LABEL AT05
TABLE_VALUES
0.1 5000
0.2 5000
0.3 5000
0.4 5000
0.4555 5000
0.4555 5500
0.4555 6000
TABLE_END
$$$$$$$$$$$$$$$$$$$$$$$$$$
$$ STABILITY CONDITIONS $$
$$ FOR IONIC LIQUID
$$
$$$$$$$$$$$$$$$$$$$$$$$$$$
TABLE_HEAD 7100
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE IONIC_LIQ=F 1
CHANGE-STATUS PHASE GAS,HCP_A3,BCC_A2,HF1O2_S,HF1O2_S2,HF1O2_S3=D
SET-CONDITION P=P0, X(IONIC_LIQ,O)=@1,T=@2
EXPERIMENT DGM(GAS)<0:1E-5
EXPERIMENT DGM(HCP_A3)<0:1E-5
EXPERIMENT DGM(BCC_A2)<0:1E-5
EXPERIMENT DGM(HF1O2_S)<0:1E-5
EXPERIMENT DGM(HF1O2_S2)<0:1E-5
EXPERIMENT DGM(HF1O2_S3)<0:1E-5
LABEL AT06
TABLE_VALUES
0.026 3361.5
0.028 2724.38
0.028 3188.37
0.028 3036.01
0.030 2869.81
0.065 2765.93
0.065 2966.76
0.068 3112.19
0.070 3326.87
0.116 2876.73
0.116 3008.31
0.116 3153.74
0.116 2807.48
0.120 3361.5
0.156 2980.61
213
0.156 3132.96
0.160 2821.33
0.162 3361.5
0.2
2862.88
0.204 3015.24
0.208 3146.81
0.208 2793.63
0.210 3361.5
0.257 2925.21
0.261 3188.37
0.263 3036.01
0.265 2759
0.268 3382.27
0.305 2869.81
0.307 2759
0.309 3139.89
0.309 3008.31
0.314 2655.12
0.314 3347.65
0.358 2565.1
0.358 2973.68
0.358 2835.18
0.358 3119.11
0.358 2696.68
0.358 3257.62
0.358 3396.12
0.4
3396.12
0.4
2842.11
0.4
3257.62
0.4
2703.6
0.4
2973.68
0.4
3119.11
0.441 2835.18
0.443 2973.68
0.443 2752.08
0.443 3250.69
0.443 3112.19
0.446 3396.12
0.487 3250.69
0.487 2973.68
0.487 2849.03
0.487 3112.19
0.487 3396.12
0.531 3403.05
0.531 2980.61
0.531 3250.69
0.531 2883.66
0.531 3112.19
0.573 3257.62
0.575 2987.53
0.575 3119.11
0.575 3403.05
0.615 3119.11
0.615 3022.16
0.617 3257.62
0.617 3409.97
TABLE_END
TABLE_HEAD 7500
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE HCP_A3=F 1
CHANGE_STATUS PHASE BCC_A2=D
CHANGE-STATUS PHASE HF1O2_S=D
CHANGE-STATUS PHASE HF1O2_S2=D
214
SET-CONDITION P=P0, X(HCP_A3,O)=@1,T=@2
EXPERIMENT DGM(BCC_A2)<0:1E-5
EXPERIMENT DGM(HF1O2_S)<0:1E-5
EXPERIMENT DGM(HF1O2_S2)<0:1E-5
LABEL AT06
TABLE_VALUES
0.0087 1200.83
0.0131 1941.83
0.0131 1408.59
0.0153 1803.32
0.0153 1616.34
0.0219 1096.95
0.0329 2018.01
0.0351 2114.96
0.0417 1803.32
0.0417 1574.79
0.0461 1387.81
0.0505 1131.58
0.0571 2246.54
0.0593 2101.11
0.0659 1907.2
0.0703 1782.55
0.0813 1567.87
0.0835 2440.44
0.0835 2322.71
0.0901 1443.21
0.0923 2156.51
0.0923 1235.46
0.1010 2585.87
0.1032 1193.91
0.1054 1969.53
0.1142 1713.3
0.1186 1547.09
0.1230 1124.65
0.1252 1367.04
0.1252 2668.98
0.1362 2232.69
0.1384 2468.14
0.1450 2128.81
0.1472 2703.6
0.1494 2537.4
0.1516 1997.23
0.1582 1810.25
0.1582 1637.12
0.1604 1408.59
0.1626 1221.61
0.1648 2198.06
0.1670 1006.93
0.1736 2578.95
0.1780 2481.99
0.1780 2371.19
0.1824 1069.25
0.1846 1831.02
0.1846 2031.86
0.1890 1277.01
0.1912 1554.02
TABLE_END
TABLE_HEAD 8000
CREATE_NEW_EQUILIBRIUM @@,1
CHANGE_STATUS PHASE HCP_A3=F 1
CHANGE-STATUS PHASE HF1O2_S=D
SET-CONDITION P=P0, X(HCP_A3,O)=@1,T=@2
EXPERIMENT DGM(HF1O2_S)>0:1E-5
215
LABEL AT06
TABLE_VALUES
0.263014 1450.28
0.257534 1281.77
0.257534 1494.48
0.254795 1165.75
0.252055 1022.1
0.246575 596.685
0.243836 803.867
0.243836 701.657
0.243836 914.365
TABLE_END
SAVE
216
A.2.3
EXP file
DATASET
1 HFO2 Liquidus
ATTRIBUTE CENTER
CHAR 0.25
SYM 0.25
BLOCK X=C1; Y=C2; GOC=C3,MAWS1
3.69857147979123e-001
2.69290053114932e+003 S1
3.67967795916904e-001
2.68732619622013e+003
3.65983760396705e-001
2.68175178080267e+003
3.63999724876505e-001
2.67617736538522e+003
3.62110445985429e-001
2.67050857748714e+003
3.60221167094352e-001
2.66483978958906e+003
3.58331888203275e-001
2.65917100169099e+003
3.56442609312199e-001
2.65350221379291e+003
3.54648013879102e-001
2.64783350638308e+003
3.52806076717016e-001
2.64216475872913e+003
3.50964212726071e-001
2.63640155810630e+003
3.49074933834995e-001
2.63073277020822e+003
3.47138166872643e-001
2.62525284800378e+003
3.45296229710557e-001
2.61958410034983e+003
3.43501707448603e-001
2.61382093997113e+003
3.41659770286516e-001
2.60815219231718e+003
3.39770491395440e-001
2.60248340441910e+003
3.37975969133485e-001
2.59672024404039e+003
3.36228861771664e-001
2.59086267093693e+003
3.34529169309974e-001
2.58491068510872e+003
3.32829476848285e-001
2.57895869928051e+003
3.31177126115586e-001
2.57300675369642e+003
3.29477433653896e-001
2.56705476786821e+003
3.27635569662952e-001
2.56129156724538e+003
3.25746217600733e-001
2.55571723231618e+003
3.23951695338779e-001
2.54995407193748e+003
3.22157246247967e-001
2.54409645858989e+003
3.20410065715003e-001
2.53833333845531e+003
3.18757641811161e-001
2.53247584584011e+003
3.17105364249604e-001
2.52642944728714e+003
3.15453013516905e-001
2.52047750170306e+003
3.13800662784206e-001
2.51452555611897e+003
3.12148238880364e-001
2.50866806350377e+003
3.10495814976523e-001
2.50281057088857e+003
3.08843537414966e-001
2.49676417233560e+003
3.07191186682267e-001
2.49081222675152e+003
3.05586397191984e-001
2.48457696250491e+003
3.03839655685873e-001
2.47824712455704e+003
5.20216015845943e-001
2.99660038970950e+003
5.23101080810941e-001
3.00019205741886e+003
5.25986145775940e-001
3.00378372512821e+003
5.17378512123361e-001
2.99272540333763e+003
5.14588423300911e-001
2.98875600424100e+003
5.46089404891784e-001
3.02571391766344e+003
5.49021884756914e-001
3.02921117264804e+003
5.52191804978418e-001
3.03176409916446e+003
5.43251754826917e-001
3.02202783722933e+003
5.40224225648095e-001
3.01900276660089e+003
5.71068935264759e-001
3.04755380237499e+003
5.74143879343713e-001
3.05048446027868e+003
5.77313799565217e-001
3.05303738679510e+003
5.67756111802578e-001
3.04613419075278e+003
5.64776217037315e-001
3.04273134849293e+003
5.94348700004902e-001
3.06353615422721e+003
5.97518620226406e-001
3.06608908074362e+003
6.00831224175160e-001
3.06779205127248e+003
5.91131145369840e-001
3.06136099934219e+003
217
5.87866102663503e-001
BLOCKEND
3.05937471015081e+003
DATASET
2 HCP Liquidus
ATTRIBUTE CENTER
CHAR 0.25
SYM 0.25
BLOCK X=C1; Y=C2; GOC=C3,MAWS1
2.4399700E-02 2.5886500E+03 S2
2.4163100E-02 2.5703300E+03
6.4243100E-02 2.6435800E+03
1.0129600E-01 2.7009900E+03
1.0133100E-01 2.7183200E+03
1.0135000E-01 2.7277300E+03
1.0176800E-01 2.7371400E+03
1.0158900E-01 2.7470500E+03
BLOCKEND
DATASET
3 HCP+BCC Phase Boundary
ATTRIBUTE CENTER
CHAR 0.25
SYM 0.25
BLOCK X=C1; Y=C2; GOC=C3,MAWS1
2.7454100E-02 2.4230000E+03 S3
1.9564900E-02 2.2730000E+03
7.6825100E-03 2.1230000E+03
7.1780100E-02 2.4230000E+03
5.1111500E-02 2.2730000E+03
2.4055800E-02 2.1230000E+03
BLOCKEND
DATASET
4 BCC
ATTRIBUTE CENTER
CHAR 0.25
SYM 0.25
CLIP OFF
0.02 2950 NWS101’ ˜Transformed a
0.02 2910 NWS8’ ˜Melting
0.02 2870 NWS318’ ˜a+HfOˆDO2$
0.02 2830 NWS135’ ˜a+b
0.02 2790 NWS100’ ˜a
BLOCK X=C1; Y=C2; GOC=C3,MAWS1
0.000
2117.12 S101
0.000
2265.4
0.0116704
2265.5
0.000
2416.78
0.0239353
2418.02
0.000128713 2521.5
BLOCKEND
DATASET
5 BCC+HCP
ATTRIBUTE CENTER
CHAR 0.25
SYM 0.25
BLOCK X=C1; Y=C2; GOC=C3,MAWS2
0.0118575
2119.3 S135
0.0222421
2118.35
0.0228804
2268.7
0.0328496
2267.75
0.0424009
2268.87
0.0426239
2419.22
0.0621444
2419.39
BLOCKEND
218
DATASET
6 HCP
ATTRIBUTE CENTER
CHAR 0.25
SYM 0.25
BLOCK X=C1; Y=C2; GOC=C3,MAWS3
0.0313781
2119.47 S100
0.040932
2118.51
0.0513153
2118.6
0.0608679
2118.69
0.0716665
2118.78
0.0808038
2118.86
0.0907718
2118.94
0.0999091
2119.02
0.144351
2118.37
0.184637
2119.75
0.0523702
2267.92
0.0627535
2268.01
0.0714755
2268.09
0.0822754
2267.14
0.090582
2267.21
0.0993026
2268.33
0.144575
2267.68
0.184447
2268.02
0.082911
2419.57
0.100356
2418.68
0.144797
2419.06
0.184668
2419.41
0.100639
2522.37
0.145079
2522.75
0.184536
2523.09
BLOCKEND
DATASET 7 HCP+HFO2
ATTRIBUTE CENTER
CHAR 0.25
SYM 0.25
BLOCK X=C1; Y=C2; GOC=G3,MAWS4
0.22399
2525.51 S318
0.22288
2418.7
0.254027
2421.04
0.223074
2267.32
0.253392
2268.62
0.253997
2120.35
BLOCKEND
DATASET 8 Melting
ATTRIBUTE CENTER
CHAR 0.25
SYM 0.25
BLOCK X=C1;Y=C2;GOC=G3,MAWS5
0.0150447 2526.93 S8
0.0264167 2574.53
0.0264419 2590.91
0.0664802 2648.85
0.104367
2750.96
0.104551
2739.54
0.104325
2723.67
0.104298
2706.3
0.15
2773.15
BLOCKEND
DATASET 10 Add type of data here
ATTRIBUTE CENTER
CHAR 0.25
219
SYM 0.25
BLOCK X=C1; Y=C2/1E3; GOC=C3,MAWS1
1.0000000E-08 -1.1910000E+06 S100
1.4070800E-02 -1.1912600E+06
1.9889800E-02 -1.1915800E+06
2.6186700E-02 -1.2000500E+06
3.3300100E-02 -1.2051400E+06
4.0031500E-02 -1.1860100E+06
4.6815600E-02 -1.1849700E+06
5.2658500E-02 -1.1952300E+06
5.7942600E-02 -1.2010600E+06
6.5047400E-02 -1.1765400E+06
7.0890500E-02 -1.1852200E+06
7.6549500E-02 -1.1916800E+06
8.2269100E-02 -1.1756200E+06
8.7157200E-02 -1.1948700E+06
9.2778800E-02 -1.1965700E+06
9.9351400E-02 -1.1968000E+06
1.0447900E-01 -1.1842300E+06
1.0941400E-01 -1.1903700E+06
1.1512100E-01 -1.1917600E+06
1.2137500E-01 -1.1757000E+06
1.2693200E-01 -1.1790900E+06
1.3148300E-01 -1.1736000E+06
1.3646300E-01 -1.1794300E+06
1.4204800E-01 -1.1725700E+06
1.4668800E-01 -1.1623200E+06
1.5271100E-01 -1.1625500E+06
1.5681000E-01 -1.1668900E+06
1.6159200E-01 -1.1669000E+06
1.6648300E-01 -1.1656400E+06
1.7202600E-01 -1.1456800E+06
1.8186000E-01 -1.1337500E+06
1.8630000E-01 -1.1391500E+06
1.9063100E-01 -1.1418100E+06
1.9588300E-01 -1.1274400E+06
2.0000200E-01 -1.1305200E+06
2.0303700E-01 -1.1344400E+06
2.0685000E-01 -1.1631000E+06
2.1626100E-01 -1.1362700E+06
2.2042500E-01 -1.1436800E+06
2.2488200E-01 -1.1248700E+06
2.2953000E-01 -1.1206600E+06
2.3734300E-01 -1.1178200E+06
2.4124600E-01 -1.1364400E+06
2.4461900E-01 -1.1448000E+06
2.5257700E-01 -1.1367900E+06
BLOCKEND
DATASET 11
ATTRIBUTE CENTER
CHAR 0.25
SYM 0.25
BLOCK X=C1; Y=C2/1E3; GOC=C3,MAWS1
0.00425881 -769299 S100
0.0321019 -925915
0.0514983 -953837
0.0426986 -986544
0.058964 -1.08506e+06
0.0834363 -1.12061e+06
0.113811 -1.21172e+06
0.131485 -1.20939e+06
0.143627 -1.19439e+06
0.195257 -1.20511e+06
220
0.210296
0.224706
0.258401
0.274014
BLOCKEND
-1.06659e+06
-1.10462e+06
-1.1152e+06
-1.11796e+06
221
A.2.4
TDB file
$
ELEMENT
ELEMENT
ELEMENT
ELEMENT
/VA
HF
O
ELECTRON_GAS
VACUUM
HCP_A3
1/2_MOLE_O2(G)
SPECIES
SPECIES
SPECIES
SPECIES
SPECIES
SPECIES
HF+4
HF1O1
HF1O2
O-2
O2
O3
0.0000E+00
0.0000E+00
1.7849E+02
1.5999E+01
0.0000E+00
0.0000E+00
0.0000E+00
4.3410E+03
0.0000E+00!
0.0000E+00!
0.0000E+00!
1.0252E+02!
HF1/+4!
HF1O1!
HF1O2!
O1/-2!
O2!
O3!
FUNCTION GHSERHF
2.98140E+02 -6987.297+110.744026*T-22.7075*T*LN(T)
-.004146145*T**2-4.77E-10*T**3-22590*T**(-1); 2.50600E+03 Y
-1446776.33+6193.60999*T-787.536383*T*LN(T)+.1735215*T**2
-7.575759E-06*T**3+5.01742495E+08*T**(-1); 3.00000E+03 N !
FUNCTION GHSEROO
2.98140E+02 -3480.87-25.503038*T-11.1355*T*LN(T)
-.005098875*T**2+6.61845833E-07*T**3-38365*T**(-1); 1.00000E+03 Y
-6568.763+12.659879*T-16.8138*T*LN(T)-5.957975E-04*T**2+6.781E-09*T**3
+262905*T**(-1); 3.30000E+03 Y
-13986.728+31.259624*T-18.9536*T*LN(T)-4.25243E-04*T**2
+1.0721E-08*T**3+4383200*T**(-1); 6.00000E+03 N !
FUNCTION F11148T
2.98140E+02 +617056.219-63.2709603*T-18.39562*T*LN(T)
-.0012918935*T**2-8.82478833E-07*T**3-52100.45*T**(-1); 7.00000E+02 Y
+620483.245-117.717981*T-9.924584*T*LN(T)-.010280685*T**2
+8.59787167E-07*T**3-299415.5*T**(-1); 1.70000E+03 Y
+579429.545+154.489923*T-46.59529*T*LN(T)+.0043662005*T**2
-2.43450833E-07*T**3+8372780*T**(-1); 4.20000E+03 Y
+774112.735-449.953417*T+26.29293*T*LN(T)-.007829545*T**2
+1.4093445E-07*T**3-89758000*T**(-1); 8.80000E+03 Y
+240587.568+415.824682*T-69.61305*T*LN(T)-1.3868485E-04*T**2
+2.49491833E-08*T**3+4.6926785E+08*T**(-1); 1.00000E+04 N !
FUNCTION F11183T
2.98140E+02 +62940.8071-40.9156148*T-28.3658*T*LN(T)
-.007276495*T**2+1.01958767E-06*T**3+58458.65*T**(-1); 7.00000E+02 Y
+59559.5027+19.378148*T-37.89078*T*LN(T)+.0036336835*T**2
-1.1975895E-06*T**3+247046.45*T**(-1); 1.40000E+03 Y
+103760.564-342.615352*T+12.49522*T*LN(T)-.021788325*T**2
+1.198578E-06*T**3-6989295*T**(-1); 2.60000E+03 Y
-28663.0213+312.032338*T-71.76128*T*LN(T)+.00184396*T**2
-3.57374E-08*T**3+31649665*T**(-1); 5.70000E+03 Y
-41768.4312+310.198914*T-71.03452*T*LN(T)+.0012058915*T**2
-8.191305E-09*T**3+50801050*T**(-1); 1.00000E+04 N !
FUNCTION F11203T
2.98140E+02 -229480.942+8.72983769*T-40.25422*T*LN(T)
-.01666109*T**2+2.97927E-06*T**3+237328.6*T**(-1); 7.00000E+02 Y
-238953.708+135.078913*T-59.39669*T*LN(T)+8.697345E-04*T**2
-9.73477667E-08*T**3+1130732*T**(-1); 2.70000E+03 Y
-156234.064-146.3845*T-25.24511*T*LN(T)-.004732982*T**2
+3.490855E-08*T**3-34168770*T**(-1); 5.00000E+03 Y
+60613.7736-782.454499*T+50.79205*T*LN(T)-.01648782*T**2
+3.73003333E-07*T**3-1.49192E+08*T**(-1); 6.00000E+03 N !
FUNCTION F13349T
2.98140E+02 +243206.494-20.8612582*T-21.01555*T*LN(T)
+1.2687055E-04*T**2-1.23131283E-08*T**3-42897.09*T**(-1); 2.95000E+03
Y
+252301.423-52.0847281*T-17.21188*T*LN(T)-5.413565E-04*T**2
+7.64520667E-09*T**3-3973170.5*T**(-1); 6.00000E+03 N !
FUNCTION F13704T
2.98140E+02 -6960.6927-51.1831467*T-22.25862*T*LN(T)
-.01023867*T**2+1.339947E-06*T**3-76749.55*T**(-1); 9.00000E+02 Y
-13136.0174+24.7432966*T-33.55726*T*LN(T)-.0012348985*T**2
+1.66943333E-08*T**3+539886*T**(-1); 3.70000E+03 Y
+14154.6459-51.485458*T-24.47978*T*LN(T)-.002634759*T**2
+6.01544333E-08*T**3-15120935*T**(-1); 9.60000E+03 Y
222
-314316.629+515.068037*T-87.56143*T*LN(T)+.0025787245*T**2
-1.878765E-08*T**3+2.9052515E+08*T**(-1); 1.85000E+04 Y
-108797.175+288.483019*T-63.737*T*LN(T)+.0014375*T**2-9E-09*T**3
+.25153895*T**(-1); 2.00000E+04 N !
FUNCTION F14021T
2.98140E+02 +130696.944-37.9096643*T-27.58118*T*LN(T)
-.02763076*T**2+4.60539333E-06*T**3+99530.45*T**(-1); 7.00000E+02 Y
+114760.623+176.626737*T-60.10286*T*LN(T)+.00206456*T**2
-5.17486667E-07*T**3+1572175*T**(-1); 1.30000E+03 Y
+49468.3956+710.09482*T-134.3696*T*LN(T)+.039707355*T**2
-4.10457667E-06*T**3+12362250*T**(-1); 2.10000E+03 Y
+866367.075-3566.80563*T+421.2001*T*LN(T)-.1284109*T**2
+5.44768833E-06*T**3-2.1304835E+08*T**(-1); 2.80000E+03 Y
+409416.383-1950.70834*T+223.4437*T*LN(T)-.0922361*T**2
+4.306855E-06*T**3-21589870*T**(-1); 3.50000E+03 Y
-1866338.6+6101.13383*T-764.8435*T*LN(T)+.09852775*T**2
-2.59784667E-06*T**3+9.610855E+08*T**(-1); 4.90000E+03 Y
+97590.043+890.798361*T-149.9608*T*LN(T)+.01283575*T**2
-3.555105E-07*T**3-2.1699975E+08*T**(-1); 6.00000E+03 N !
FUNCTION F11197T
2.98140E+02 -1140482.96+414.221582*T-69.26978*T*LN(T)
-.00578349*T**2+1.18556783E-10*T**3+553972*T**(-1); 2.10000E+03 Y
-1175625.83+641.774324*T-98.4*T*LN(T)+3.284007E-15*T**2
-1.716525E-19*T**3+5.40406E-06*T**(-1); 2.79300E+03 Y
-1180094.63+656.070118*T-100*T*LN(T)+1.175826E-14*T**2
-4.54488667E-19*T**3+4.7303325E-05*T**(-1); 3.07300E+03 Y
-1195459.63+701.222166*T-105*T*LN(T)+6.60493E-17*T**2-1.98101E-21*T**3
+3.081755E-07*T**(-1); 6.00000E+03 N !
FUNCTION GHFLIQ
2.98140E+02 +27402.256-10.953093*T+GHSERHF#;
1.00000E+03 Y
+49731.499-149.91739*T+12.116812*T*LN(T)-.021262021*T**2
+1.376466E-06*T**3-4449699*T**(-1); 2.50600E+03 Y
-4247.217+265.470523*T-44*T*LN(T); 3.00000E+03 N !
FUNCTION UN_ASS
2.98140E+02 0.0 ; 3.00000E+02 N !
TYPE_DEFINITION % SEQ *!
DEFINE_SYSTEM_DEFAULT SPECIE 2 !
DEFAULT_COMMAND DEF_SYS_ELEMENT VA !
PHASE GAS:G % 1 1.0 !
CONSTITUENT GAS:G :HF,HF1O1,HF1O2,O,O2,O3 :
!
PARAMETER G(GAS,HF;0) 2.98150E+02 +F11148T#+R#*T*LN(1E-05*P);
6.00000E+03
N REF:6314 !
PARAMETER G(GAS,HF1O1;0) 2.98150E+02 +F11183T#+R#*T*LN(1E-05*P);
6.00000E+03
N REF:6339 !
PARAMETER G(GAS,HF1O2;0) 2.98150E+02 +F11203T#+R#*T*LN(1E-05*P);
6.00000E+03
N REF:6343 !
PARAMETER G(GAS,O;0) 2.98150E+02 +F13349T#+R#*T*LN(1E-05*P);
6.00000E+03
N REF:7612 !
PARAMETER G(GAS,O2;0) 2.98150E+02 +F13704T#+R#*T*LN(1E-05*P);
6.00000E+03
N REF:7737 !
PARAMETER G(GAS,O3;0) 2.98150E+02 +F14021T#+R#*T*LN(1E-05*P);
6.00000E+03
N REF:7889 !
PHASE IONIC_LIQ:Y % 2 1
1 !
CONSTITUENT IONIC_LIQ:Y :HF+4 : O-2,VA :
!
PARAMETER G(IONIC_LIQ,HF+4:O-2;0) 2.98150E+02 +2*F11197T#+252000
-86.7975992*T;
6.00000E+03
N REF:0 !
PARAMETER G(IONIC_LIQ,HF+4:VA;0) 2.98150E+02 +GHFLIQ#;
6.00000E+03
N REF:0 !
PARAMETER G(IONIC_LIQ,HF+4:O-2,VA;0) 2.98150E+02 +50821+4.203*T;
6.00000E+03
N REF:0 !
223
PARAMETER G(IONIC_LIQ,HF+4:O-2,VA;1)
6.00000E+03
N REF:0 !
PARAMETER G(IONIC_LIQ,HF+4:O-2,VA;2)
N REF:0 !
2.98150E+02
+420485-133.3*T;
2.98150E+02
30537;
TYPE_DEFINITION & GES A_P_D BCC_A2 MAGNETIC
PHASE BCC_A2 %& 2 1
3 !
CONSTITUENT BCC_A2 :HF : O,VA : !
-1.0
6.00000E+03
4.00000E-01 !
PARAMETER G(BCC_A2,HF:O;0) 2.98150E+02 +GHSERHF#+3*GHSEROO#-737857
+268.54*T;
6.00000E+03
N REF:0 !
PARAMETER G(BCC_A2,HF:VA;0) 2.98140E+02 +5370.703+103.836026*T
-22.8995*T*LN(T)-.004206605*T**2+8.71923E-07*T**3-22590*T**(-1)
-1.446E-10*T**4; 2.50600E+03 Y
+1912456.77-8624.20573*T+1087.61412*T*LN(T)-.286857065*T**2
+1.3427829E-05*T**3-6.10085091E+08*T**(-1); 3.00000E+03 N REF:0 !
PARAMETER G(BCC_A2,HF:O,VA;0) 2.98150E+02 -981440+20.349*T;
6.00000E+03
N REF:0 !
TYPE_DEFINITION ’ GES A_P_D HCP_A3 MAGNETIC
PHASE HCP_A3 %’ 2 1
.5 !
CONSTITUENT HCP_A3 :HF : O,VA : !
PARAMETER
+41.56*T;
PARAMETER
REF:0 !
PARAMETER
REF:0 !
PARAMETER
REF:0 !
2.80000E-01 !
G(HCP_A3,HF:O;0) 2.98150E+02 +GHSERHF#+.5*GHSEROO#-271214
6.00000E+03
N REF:0 !
G(HCP_A3,HF:VA;0) 2.98140E+02 +GHSERHF#; 3.00000E+03 N
G(HCP_A3,HF:O,VA;0)
2.98150E+02
-31345;
G(HCP_A3,HF:O,VA;1)
2.98150E+02
-6272;
PHASE HF1O2_S % 1 1.0 !
CONSTITUENT HF1O2_S :HF1O2 :
PARAMETER G(HF1O2_S,HF1O2;0)
REF:6341 !
2.98150E+02
PARAMETER G(HF1O2_S2,HF1O2;0)
6.00000E+03
N REF:6341 !
PHASE HF1O2_S3 % 1 1.0 !
CONSTITUENT HF1O2_S3 :HF1O2 :
PARAMETER G(HF1O2_S3,HF1O2;0)
6.00000E+03
N REF:6341 !
LIST_OF_REFERENCES
NUMBER SOURCE
6314
6.00000E+03
6.00000E+03
N
N
!
PHASE HF1O2_S2 % 1 1.0 !
CONSTITUENT HF1O2_S2 :HF1O2 :
HO1
S.G.T.E.
**
6339
T.C.R.A.
-3.0
+F11197T#;
6.00000E+03
N
!
2.98150E+02
+F11197T#+12000-5.71428571*T;
!
2.98150E+02
+F11197T#+30000-12.1589689*T;
224
Class:
6
!
225
A.3
The Hf-Si-O system
A.3.1
Setup file
go d
sw user hfsio
d-sy *
rej const ion 2 va,,,
rej ph *
res pha hf1o2_s hf1o2_s2 hf1o2_s3 ion qua cris trid
get
go g
e-ph HFSIO4
3
1
1
4
HF;
SI;
O;
NO
NO
go pa
e-pa g(hfsio4)
298.15
GHFO2+GSIO2+V1+V2*T;
6000
N
create hfsio
save
226
A.3.2
POP file
DEF-COM HFO2 SIO2 O
$ MELTING OF HAFNON PHASE
CREATE_NEW_EQUILIBRIUM 1,1
CHANGE_STATUS PHASE IONIC_LIQ, HFSIO4, HF1O2_S=FIXED 1
SET_CONDITION P=101325, AC(O)=1
EXPERIMENT T=2023.15:15
LABEL ACGM
$ Total energy from first-principles is -109.22927eV
$ HfSiO4 = -9.102432 eV/atom
$ Hf = -9.831985 eV/atom
$ Si = -5.431504 eV/atom
$ O = -4.89516195 eV/atom
$ delta H(HfSiO4)= -324455 J/mol-atom
CREATE_NEW_EQUILIBRIUM 2,1
CHANGE_STATUS PHASE HFSIO4=FIXED 1
SET_CONDITION P=101325, T=298.15, AC(O)=1 X(HFO2)-X(SIO2)=0
SET_REFERENCE_STATE HFO2 HF1O2_S,,,
SET_REFERENCE_STATE SIO2 QUARTZ,,,
EXPERIMENT HR=-10614.6:5%
LABEL AHMR
SAVE
227
A.3.3
TDB file
ELEMENT
ELEMENT
ELEMENT
ELEMENT
ELEMENT
/VA
HF
O
SI
ELECTRON_GAS
VACUUM
HCP_A3
1/2_MOLE_O2(G)
DIAMOND_FCC_A4
SPECIES
SPECIES
SPECIES
SPECIES
SPECIES
SPECIES
SPECIES
SPECIES
SPECIES
SPECIES
SPECIES
HF+4
HF1O1
HF1O2
O-2
O2
O3
SI+4
SI2
SI3
SIO
SIO2
0.0000E+00
0.0000E+00
1.7849E+02
1.5999E+01
2.8085E+01
0.0000E+00
0.0000E+00
0.0000E+00
4.3410E+03
3.2175E+03
0.0000E+00!
0.0000E+00!
0.0000E+00!
1.0252E+02!
1.8820E+01!
HF1/+4!
HF1O1!
HF1O2!
O1/-2!
O2!
O3!
SI1/+4!
SI2!
SI3!
O1SI1!
O2SI1!
FUNCTION GHSERHF
2.98140E+02 -6987.297+110.744026*T-22.7075*T*LN(T)
-.004146145*T**2-4.77E-10*T**3-22590*T**(-1); 2.50600E+03 Y
-1446776.33+6193.60999*T-787.536383*T*LN(T)+.1735215*T**2
-7.575759E-06*T**3+5.01742495E+08*T**(-1); 3.00000E+03 N !
FUNCTION GHSEROO
2.98140E+02 -3480.87-25.503038*T-11.1355*T*LN(T)
-.005098875*T**2+6.61845833E-07*T**3-38365*T**(-1); 1.00000E+03 Y
-6568.763+12.659879*T-16.8138*T*LN(T)-5.957975E-04*T**2+6.781E-09*T**3
+262905*T**(-1); 3.30000E+03 Y
-13986.728+31.259624*T-18.9536*T*LN(T)-4.25243E-04*T**2
+1.0721E-08*T**3+4383200*T**(-1); 6.00000E+03 N !
FUNCTION GHFGAS
2.98140E+02 +617056.219-63.2709603*T-18.39562*T*LN(T)
-.0012918935*T**2-8.82478833E-07*T**3-52100.45*T**(-1); 7.00000E+02 Y
+620483.245-117.717981*T-9.924584*T*LN(T)-.010280685*T**2
+8.59787167E-07*T**3-299415.5*T**(-1); 1.70000E+03 Y
+579429.545+154.489923*T-46.59529*T*LN(T)+.0043662005*T**2
-2.43450833E-07*T**3+8372780*T**(-1); 4.20000E+03 Y
+774112.735-449.953417*T+26.29293*T*LN(T)-.007829545*T**2
+1.4093445E-07*T**3-89758000*T**(-1); 8.80000E+03 Y
+240587.568+415.824682*T-69.61305*T*LN(T)-1.3868485E-04*T**2
+2.49491833E-08*T**3+4.6926785E+08*T**(-1); 1.00000E+04 N !
FUNCTION GHFOGAS
2.98140E+02 +62940.8071-40.9156148*T-28.3658*T*LN(T)
-.007276495*T**2+1.01958767E-06*T**3+58458.65*T**(-1); 7.00000E+02 Y
+59559.5027+19.378148*T-37.89078*T*LN(T)+.0036336835*T**2
-1.1975895E-06*T**3+247046.45*T**(-1); 1.40000E+03 Y
+103760.564-342.615352*T+12.49522*T*LN(T)-.021788325*T**2
+1.198578E-06*T**3-6989295*T**(-1); 2.60000E+03 Y
-28663.0213+312.032338*T-71.76128*T*LN(T)+.00184396*T**2
-3.57374E-08*T**3+31649665*T**(-1); 5.70000E+03 Y
-41768.4312+310.198914*T-71.03452*T*LN(T)+.0012058915*T**2
-8.191305E-09*T**3+50801050*T**(-1); 1.00000E+04 N !
FUNCTION GHFO2GAS
2.98140E+02 -229480.942+8.72983769*T-40.25422*T*LN(T)
-.01666109*T**2+2.97927E-06*T**3+237328.6*T**(-1); 7.00000E+02 Y
-238953.708+135.078913*T-59.39669*T*LN(T)+8.697345E-04*T**2
-9.73477667E-08*T**3+1130732*T**(-1); 2.70000E+03 Y
-156234.064-146.3845*T-25.24511*T*LN(T)-.004732982*T**2
+3.490855E-08*T**3-34168770*T**(-1); 5.00000E+03 Y
+60613.7736-782.454499*T+50.79205*T*LN(T)-.01648782*T**2
+3.73003333E-07*T**3-1.49192E+08*T**(-1); 6.00000E+03 N !
FUNCTION GOGAS
2.98140E+02 +243206.494-20.8612582*T-21.01555*T*LN(T)
+1.2687055E-04*T**2-1.23131283E-08*T**3-42897.09*T**(-1); 2.95000E+03
Y
+252301.423-52.0847281*T-17.21188*T*LN(T)-5.413565E-04*T**2
+7.64520667E-09*T**3-3973170.5*T**(-1); 6.00000E+03 N !
FUNCTION GO2GAS
2.98140E+02 -6960.6927-51.1831467*T-22.25862*T*LN(T)
228
-.01023867*T**2+1.339947E-06*T**3-76749.55*T**(-1); 9.00000E+02 Y
-13136.0174+24.7432966*T-33.55726*T*LN(T)-.0012348985*T**2
+1.66943333E-08*T**3+539886*T**(-1); 3.70000E+03 Y
+14154.6459-51.485458*T-24.47978*T*LN(T)-.002634759*T**2
+6.01544333E-08*T**3-15120935*T**(-1); 9.60000E+03 Y
-314316.629+515.068037*T-87.56143*T*LN(T)+.0025787245*T**2
-1.878765E-08*T**3+2.9052515E+08*T**(-1); 1.85000E+04 Y
-108797.175+288.483019*T-63.737*T*LN(T)+.0014375*T**2-9E-09*T**3
+.25153895*T**(-1); 2.00000E+04 N !
FUNCTION GO3GAS
2.98140E+02 +130696.944-37.9096643*T-27.58118*T*LN(T)
-.02763076*T**2+4.60539333E-06*T**3+99530.45*T**(-1); 7.00000E+02 Y
+114760.623+176.626737*T-60.10286*T*LN(T)+.00206456*T**2
-5.17486667E-07*T**3+1572175*T**(-1); 1.30000E+03 Y
+49468.3956+710.09482*T-134.3696*T*LN(T)+.039707355*T**2
-4.10457667E-06*T**3+12362250*T**(-1); 2.10000E+03 Y
+866367.075-3566.80563*T+421.2001*T*LN(T)-.1284109*T**2
+5.44768833E-06*T**3-2.1304835E+08*T**(-1); 2.80000E+03 Y
+409416.383-1950.70834*T+223.4437*T*LN(T)-.0922361*T**2
+4.306855E-06*T**3-21589870*T**(-1); 3.50000E+03 Y
-1866338.6+6101.13383*T-764.8435*T*LN(T)+.09852775*T**2
-2.59784667E-06*T**3+9.610855E+08*T**(-1); 4.90000E+03 Y
+97590.043+890.798361*T-149.9608*T*LN(T)+.01283575*T**2
-3.555105E-07*T**3-2.1699975E+08*T**(-1); 6.00000E+03 N !
FUNCTION GHFO2
2.98140E+02 -1140482.96+414.221582*T-69.26978*T*LN(T)
-.00578349*T**2+1.18556783E-10*T**3+553972*T**(-1); 2.10000E+03 Y
-1175625.83+641.774324*T-98.4*T*LN(T)+3.284007E-15*T**2
-1.716525E-19*T**3+5.40406E-06*T**(-1); 2.79300E+03 Y
-1180094.63+656.070118*T-100*T*LN(T)+1.175826E-14*T**2
-4.54488667E-19*T**3+4.7303325E-05*T**(-1); 3.07300E+03 Y
-1195459.63+701.222166*T-105*T*LN(T)+6.60493E-17*T**2-1.98101E-21*T**3
+3.081755E-07*T**(-1); 6.00000E+03 N !
FUNCTION GHFLIQ
2.98140E+02 +27402.256-10.953093*T+GHSERHF#;
1.00000E+03 Y
+49731.499-149.91739*T+12.116812*T*LN(T)-.021262021*T**2
+1.376466E-06*T**3-4449699*T**(-1); 2.50600E+03 Y
-4247.217+265.470523*T-44*T*LN(T); 3.00000E+03 N !
FUNCTION UN_ASS
2.98140E+02 0.0 ; 3.00000E+02 N !
FUNCTION GHSERSI
2.98140E+02 -8162.609+137.236859*T-22.8317533*T*LN(T)
-.001912904*T**2-3.552E-09*T**3+176667*T**(-1); 1.68700E+03 Y
-9457.642+167.281367*T-27.196*T*LN(T)-4.20369E+30*T**(-9);
3.60000E+03 N !
FUNCTION GSILIQ
2.98140E+02 +50696.36-30.099439*T+GHSERSI#
+2.09307E-21*T**7; 1.68700E+03 Y
+40370.523+137.722298*T-27.196*T*LN(T); 3.60000E+03 N !
FUNCTION GSIO2LIQ
2.98140E+02 -923689.98+316.24766*T-52.17*T*LN(T)
-.012002*T**2+6.78E-07*T**3+665550*T**(-1); 2.98000E+03 Y
-957614.21+580.01419*T-87.428*T*LN(T); 4.00000E+03 N !
FUNCTION GCRISTOB
2.98140E+02 -601467.73-8140.2255*T+1399.8908*T*LN(T)
-2.8579085*T**2+.0010408145*T**3-13144016*T**(-1); 3.73000E+02 Y
-1498711.3+13075.913*T-2178.3561*T*LN(T)+3.493609*T**2
-.0010762132*T**3+29100273*T**(-1); 4.53000E+02 Y
-3224538.7+47854.938*T-7860.2125*T*LN(T)+11.817149*T**2
-.0033651832*T**3+1.2750272E+08*T**(-1); 5.43000E+02 Y
-943127.51+493.26056*T-77.5875*T*LN(T)+.003040245*T**2
-4.63118E-07*T**3+2227125*T**(-1); 3.30000E+03 Y
-973891.99+587.05606*T-87.373*T*LN(T); 4.00000E+03 N !
FUNCTION GTRIDYM
2.98140E+02 -918008.73+140.55925*T-25.1574*T*LN(T)
-.0148714*T**2-2.2791833E-05*T**3+66331*T**(-1); 3.88000E+02 Y
-921013.31+224.53808*T-37.8701*T*LN(T)-.02368535*T**2-1.6835E-07*T**3;
4.33000E+02 Y
-919633.42+210.51651*T-35.605*T*LN(T)-.03049985*T**2+4.6255E-06*T**3
-162026*T**(-1); 9.00000E+02 Y
-979377.7+848.3098*T-128.434*T*LN(T)+.03387055*T**2-3.786883E-06*T**3
+7070800*T**(-1); 1.66800E+03 Y
229
-943685.26+493.58035*T-77.5875*T*LN(T)+.003040245*T**2
-4.63118E-07*T**3+2227125*T**(-1); 3.30000E+03 Y
-974449.74+587.37585*T-87.373*T*LN(T); 4.00000E+03 N !
FUNCTION GSIO2
2.98140E+02 -900936.64-360.892175*T+61.1323*T*LN(T)
-.189203605*T**2+4.9509742E-05*T**3-854401*T**(-1); 5.40000E+02 Y
-1091466.54+2882.67275*T-452.1367*T*LN(T)+.428883845*T**2
-9.0917706E-05*T**3+12476689*T**(-1); 7.70000E+02 Y
-1563481.44+9178.58655*T-1404.5352*T*LN(T)+1.28404426*T**2
-2.35047657E-04*T**3+56402304*T**(-1); 8.48000E+02 Y
-928732.923+356.218325*T-58.4292*T*LN(T)-.00515995*T**2-2.47E-10*T**3
-95113*T**(-1); 1.80000E+03 Y
-924076.574+281.229013*T-47.451*T*LN(T)-.01200315*T**2
+6.78127E-07*T**3+665385*T**(-1); 2.96000E+03 Y
-957997.4+544.992084*T-82.709*T*LN(T); 4.00000E+03 N !
FUNCTION GSIOGAS
2.98140E+02 -106673.891-46.3143573*T-23.81284*T*LN(T)
-.010875415*T**2+1.67414E-06*T**3-21752.77*T**(-1); 8.00000E+02 Y
-114089.896+42.5396478*T-37.02803*T*LN(T)-1.479494E-04*T**2
-2.38484333E-09*T**3+775315*T**(-1); 4.10000E+03 Y
-212235.767+345.258888*T-73.51617*T*LN(T)+.005956135*T**2
-1.96044833E-07*T**3+50917650*T**(-1); 6.00000E+03 N !
FUNCTION GSIO2GAS
2.98140E+02 -335801.165+32.7172035*T-37.09556*T*LN(T)
-.0203048*T**2+3.19765167E-06*T**3+153794.6*T**(-1); 8.00000E+02 Y
-349149.52+195.460769*T-61.37218*T*LN(T)-2.2226765E-04*T**2
+8.75020167E-09*T**3+1559572.5*T**(-1); 4.70000E+03 Y
-352085.975+206.037885*T-62.6768*T*LN(T)+3.952196E-05*T**2
-9.16093333E-10*T**3+2923760*T**(-1); 6.00000E+03 N !
FUNCTION GSIGAS
2.98140E+02 +444169.766-27.9507103*T-21.04097*T*LN(T)
+3.7970425E-04*T**2-9.73407167E-08*T**3-61797.2*T**(-1); 2.00000E+03 Y
+437545.354-12.9504906*T-22.5648*T*LN(T)-2.428964E-04*T**2
+1.1169705E-08*T**3+2760718.5*T**(-1); 5.50000E+03 Y
+405256.153+76.2660207*T-33.14557*T*LN(T)+.0012877205*T**2
-3.00107333E-08*T**3+21167395*T**(-1); 1.00000E+04 N !
FUNCTION GSI2GAS
2.98140E+02 +572963.738+3.71111611*T-35.9832*T*LN(T)
-9.4063E-04*T**2-4.840215E-08*T**3+12476.05*T**(-1); 2.60000E+03 Y
+609343.923-151.775041*T-16.36102*T*LN(T)-.005711295*T**2
+1.72754333E-07*T**3-12777080*T**(-1); 6.00000E+03 N !
FUNCTION GSI3GAS
2.98140E+02 +611231.759+46.355823*T-46.87401*T*LN(T)
-.01734593*T**2+3.536315E-06*T**3+194880.6*T**(-1); 7.00000E+02 Y
+604474.516+159.236355*T-64.54462*T*LN(T)+.0019320005*T**2
-2.38038667E-07*T**3+630559*T**(-1); 2.40000E+03 Y
+613241.601+78.626845*T-53.45528*T*LN(T)-.0027577635*T**2
+9.275105E-08*T**3+606517*T**(-1); 6.00000E+03 N !
TYPE_DEFINITION % SEQ *!
DEFINE_SYSTEM_DEFAULT SPECIE 2 !
DEFAULT_COMMAND DEF_SYS_ELEMENT VA !
PHASE GAS:G % 1 1.0 !
CONSTITUENT GAS:G :HF,HF1O1,HF1O2,O,O2,O3,SI,SI2,SI3,SIO,SIO2 :
!
PARAMETER G(GAS,HF;0) 2.98150E+02 +GHFGAS#+R#*T*LN(1E-05*P);
6.00000E+03
N REF:6314 !
PARAMETER G(GAS,HF1O1;0) 2.98150E+02 +GHFOGAS#+R#*T*LN(1E-05*P);
6.00000E+03
N REF:6339 !
PARAMETER G(GAS,HF1O2;0) 2.98150E+02 +GHFO2GAS#+R#*T*LN(1E-05*P);
6.00000E+03
N REF:6343 !
PARAMETER G(GAS,O;0) 2.98150E+02 +GOGAS#+R#*T*LN(1E-05*P);
6.00000E+03
N REF:7612 !
PARAMETER G(GAS,O2;0) 2.98150E+02 +GO2GAS#+R#*T*LN(1E-05*P);
6.00000E+03
N REF:7737 !
PARAMETER G(GAS,O3;0) 2.98150E+02 +GO3GAS#+R#*T*LN(1E-05*P);
6.00000E+03
N REF:7889 !
PARAMETER G(GAS,SI;0) 2.98150E+02 +GSIGAS#+R#*T*LN(1E-05*P);
230
6.00000E+03
N REF:0 !
PARAMETER G(GAS,SI2;0) 2.98150E+02 +GSI2GAS#+R#*T*LN(1E-05*P);
6.00000E+03
N REF:0 !
PARAMETER G(GAS,SI3;0) 2.98150E+02 +GSI3GAS#+R#*T*LN(1E-05*P);
6.00000E+03
N REF:0 !
PARAMETER G(GAS,SIO;0) 2.98150E+02 +GSIOGAS#+R#*T*LN(1E-05*P);
6.00000E+03
N REF:0 !
PARAMETER G(GAS,SIO2;0) 2.98150E+02 +GSIO2GAS#+R#*T*LN(1E-05*P);
6.00000E+03
N REF:0 !
PHASE IONIC_LIQ:Y % 2 1
1 !
CONSTITUENT IONIC_LIQ:Y :HF+4,SI+4 : O-2,VA,SIO2 :
!
PARAMETER G(IONIC_LIQ,HF+4:O-2;0) 2.98150E+02 +2*GHFO2#+252000
-86.7975992*T;
6.00000E+03
N REF:0 !
PARAMETER G(IONIC_LIQ,SI+4:O-2;0) 2.98150E+02 +2*GSIO2LIQ#+2000000;
6.00000E+03
N REF:0 !
PARAMETER G(IONIC_LIQ,HF+4:VA;0) 2.98150E+02 +GHFLIQ#;
6.00000E+03
N REF:0 !
PARAMETER G(IONIC_LIQ,SI+4:VA;0) 2.98150E+02 +GSILIQ#;
6.00000E+03
N REF:0 !
PARAMETER G(IONIC_LIQ,SIO2;0) 2.98150E+02 +GSIO2LIQ#;
6.00000E+03
N
REF:0 !
PARAMETER G(IONIC_LIQ,HF+4:O-2,VA;0) 2.98150E+02 +50821+4.203*T;
6.00000E+03
N REF:0 !
PARAMETER G(IONIC_LIQ,HF+4:O-2,VA;1) 2.98150E+02 +420485-133.3*T;
6.00000E+03
N REF:0 !
PARAMETER G(IONIC_LIQ,HF+4:O-2,VA;2) 2.98150E+02 30537;
6.00000E+03
N REF:0 !
PARAMETER G(IONIC_LIQ,HF+4,SI+4:VA;0) 2.98150E+02 -177631+6.42546*T;
6.00000E+03
N REF:0 !
PARAMETER G(IONIC_LIQ,HF+4,SI+4:VA;1) 2.98150E+02 -1830;
6.00000E+03
N REF:0 !
PARAMETER G(IONIC_LIQ,SI+4:VA,SIO2;0) 2.98150E+02 +120000+14.4*T;
6.00000E+03
N REF:0 !
TYPE_DEFINITION & GES A_P_D BCC_A2 MAGNETIC
PHASE BCC_A2 %& 2 1
3 !
CONSTITUENT BCC_A2 :HF,SI : O,VA : !
-1.0
4.00000E-01 !
PARAMETER G(BCC_A2,HF:O;0) 2.98150E+02 +GHSERHF#+3*GHSEROO#-737857
+268.54*T;
6.00000E+03
N REF:0 !
PARA G(BCC_A2,SI:O;0) 298.15 0; 6000 N!
PARAMETER G(BCC_A2,HF:VA;0) 2.98140E+02 +5370.703+103.836026*T
-22.8995*T*LN(T)-.004206605*T**2+8.71923E-07*T**3-22590*T**(-1)
-1.446E-10*T**4; 2.50600E+03 Y
+1912456.77-8624.20573*T+1087.61412*T*LN(T)-.286857065*T**2
+1.3427829E-05*T**3-6.10085091E+08*T**(-1); 3.00000E+03 N REF:0 !
PARAMETER G(BCC_A2,SI:VA;0) 2.98140E+02 +47000-22.5*T+GHSERSI#;
3.60000E+03 N REF:0 !
PARAMETER G(BCC_A2,HF:O,VA;0) 2.98150E+02 -981440+20.349*T;
6.00000E+03
N REF:0 !
TYPE_DEFINITION ’ GES A_P_D CBCC_A12 MAGNETIC
PHASE CBCC_A12 %’ 2 1
1 !
CONSTITUENT CBCC_A12 :SI : VA : !
PARAMETER G(CBCC_A12,SI:VA;0)
3.60000E+03 N REF:0 !
2.98140E+02
-3.0
2.80000E-01 !
+50208-20.377*T+GHSERSI#;
231
PHASE CRISTOBALITE % 1 1.0 !
CONSTITUENT CRISTOBALITE :SIO2 :
PARAMETER G(CRISTOBALITE,SIO2;0)
N REF:0 !
PHASE CUB_A13 % 2 1
CONSTITUENT CUB_A13
2.98150E+02
1 !
:SI : VA :
PARAMETER G(CUB_A13,SI:VA;0)
3.60000E+03 N REF:0 !
!
+GCRISTOB#;
6.00000E+03
!
2.98140E+02
PHASE DIAMOND_A4 % 1 1.0 !
CONSTITUENT DIAMOND_A4 :O,SI :
+47279-20.377*T+GHSERSI#;
!
PARAMETER G(DIAMOND_A4,O;0) 2.98150E+02 +.5*GO2GAS#+30000;
6.00000E+03
N REF:0 !
PARAMETER G(DIAMOND_A4,SI;0) 2.98140E+02 +GHSERSI#; 3.60000E+03
REF:0 !
PARAMETER G(DIAMOND_A4,O,SI;0) 2.98150E+02 -340000+85.3*T;
6.00000E+03
N REF:0 !
TYPE_DEFINITION ( GES A_P_D FCC_A1 MAGNETIC
PHASE FCC_A1 %( 2 1
1 !
CONSTITUENT FCC_A1 :HF,SI : VA : !
PARAMETER G(FCC_A1,HF:VA;0)
3.00000E+03 N REF:0 !
PARAMETER G(FCC_A1,SI:VA;0)
3.60000E+03 N REF:0 !
-3.0
2.80000E-01 !
2.98140E+02
+10000-2.2*T+GHSERHF#;
2.98140E+02
+51000-21.8*T+GHSERSI#;
TYPE_DEFINITION ) GES A_P_D HCP_A3 MAGNETIC
PHASE HCP_A3 %) 2 1
.5 !
CONSTITUENT HCP_A3 :HF,SI : O,VA : !
N
-3.0
2.80000E-01 !
PARAMETER G(HCP_A3,HF:O;0) 2.98150E+02 +GHSERHF#+.5*GHSEROO#-271214
+41.56*T;
6.00000E+03
N REF:0 !
PARA G(HCP_A3,SI:O;0) 298.15 0; 6000 N!
PARAMETER G(HCP_A3,HF:VA;0) 2.98140E+02 +GHSERHF#; 3.00000E+03 N
REF:0 !
PARAMETER G(HCP_A3,SI:VA;0) 2.98140E+02 +49200-20.8*T+GHSERSI#;
3.60000E+03 N REF:0 !
PARAMETER G(HCP_A3,HF:O,VA;0) 2.98150E+02 -31345;
6.00000E+03
N
REF:0 !
PARAMETER G(HCP_A3,HF:O,VA;1) 2.98150E+02 -6272;
6.00000E+03
N
REF:0 !
PHASE HF1O2_S % 1 1.0 !
CONSTITUENT HF1O2_S :HF1O2 :
PARAMETER G(HF1O2_S,HF1O2;0)
REF:6341 !
!
2.98150E+02
PHASE HF1O2_S2 % 1 1.0 !
CONSTITUENT HF1O2_S2 :HF1O2 :
PARAMETER G(HF1O2_S2,HF1O2;0)
6.00000E+03
N REF:6341 !
+GHFO2#;
6.00000E+03
N
!
2.98150E+02
+GHFO2#+12000-5.71428571*T;
232
PHASE HF1O2_S3 % 1 1.0 !
CONSTITUENT HF1O2_S3 :HF1O2 :
!
PARAMETER G(HF1O2_S3,HF1O2;0)
6.00000E+03
N REF:6341 !
PHASE HF2SI % 2 2
CONSTITUENT HF2SI
2.98150E+02
1 !
:HF : SI :
!
PARAMETER G(HF2SI,HF:SI;0) 2.98150E+02
+4.6239*T;
6.00000E+03
N REF:0 !
PHASE HF3SI2 % 2 3
CONSTITUENT HF3SI2
2 !
:HF : SI :
3 !
:HF : SI :
4 !
:HF : SI :
1 !
:HF : SI :
2 !
:HF : SI :
+GHSERHF#+GHSERSI#-146026
!
PARAMETER G(HFSI2,HF:SI;0) 2.98150E+02
+31.44*T;
6.00000E+03
N REF:0 !
PHASE HFSIO4 % 3 1
CONSTITUENT HFSIO4
+5*GHSERHF#+4*GHSERSI#-674883
!
PARAMETER G(HFSI,HF:SI;0) 2.98150E+02
+3.9194*T;
6.00000E+03
N REF:0 !
PHASE HFSI2 % 2 1
CONSTITUENT HFSI2
+5*GHSERHF#+3*GHSERSI#-548296
!
PARAMETER G(HF5SI4,HF:SI;0) 2.98150E+02
+18.5004*T;
6.00000E+03
N REF:0 !
PHASE HFSI % 2 1
CONSTITUENT HFSI
+3*GHSERHF#+2*GHSERSI#-383860
!
PARAMETER G(HF5SI3,HF:SI;0) 2.98150E+02
+6.5224*T;
6.00000E+03
N REF:0 !
PHASE HF5SI4 % 2 5
CONSTITUENT HF5SI4
+2*GHSERHF#+GHSERSI#-191355
!
PARAMETER G(HF3SI2,HF:SI;0) 2.98150E+02
+14.148*T;
6.00000E+03
N REF:0 !
PHASE HF5SI3 % 2 5
CONSTITUENT HF5SI3
1
4 !
:HF : SI : O :
+GHSERHF#+2*GHSERSI#-209175
!
PARAMETER G(HFSIO4,HF:SI:O;0) 2.98150E+02
+1.3126925*T;
6.00000E+03
N REF:0 !
PHASE QUARTZ % 1 1.0 !
CONSTITUENT QUARTZ :SIO2 :
PARAMETER G(QUARTZ,SIO2;0)
+GHFO2#+30000-12.1589689*T;
+GSIO2#+GHFO2#-10614.6
!
2.98150E+02
+GSIO2#;
6.00000E+03
N REF:0 !
233
PHASE TRIDYMITE % 1 1.0 !
CONSTITUENT TRIDYMITE :SIO2 :
PARAMETER G(TRIDYMITE,SIO2;0)
REF:0 !
LIST_OF_REFERENCES
NUMBER SOURCE
!
!
2.98150E+02
+GTRIDYM#;
6.00000E+03
N
Appendix
B
Special quasirandom structures for
the ternary fcc solution phase
Special quasirandom structures are N -atom per cell periodic structures designed
to have correlation functions those of completely random alloy. The ternary fcc
SQS’s used in the present work are the following. Lattice vectors are given as a,
b, and c, and atom positions of A, B, and C atoms are given as Ai , Bi , and Ci ,
respectively.
B.1
B.1.1
A1B1C1
SQS-3
a = −1 12 , 12 , 0 ,
b = −1 21 , − 12 , 0 ,
c = 1 21 , 0, 21 ,
A1 = − 12 , 0, 12
B1 = 21 , 0, 12 ,
C1 = −1 21 , 0, 21
B.1.2
SQS-6
a = −1 12 , 0, − 21 ,
b = −1 12 , − 12 , 0 ,
c = (0, 1, 1),
A1 = −1, 21 , 12 ,
A2 = −2, 21 , 12 ,
B1 = (−3, 0, 0),
B2 = (−2, 0, 0),
C1 = −3, 21 , 12 ,
C2 = (−1, 0, 0)
B.1.3
SQS-9
a = 1, 12 , −1 12 ,
b = −1, − 21 , −1 12 ,
c = 12 , 1, 1 12 ,
A1 = − 12 , 0, −1 12 ,
A2 = 0, 21 , −1 12 ,
A3 = (1, 1, −1),
B1 = 12 , 1, − 12 ,
B2 = 0, 12 , − 12 ,
B3 = 12 , 1, −1 12 ,
C1 = 12 , 1, 12 ,
235
C2 =
C3 =
B.1.4
1 1
, , −1 ,
2 2
1 1
, , −2
2 2
SQS-15
a = 21 , − 21 , 1 ,
b = 0, −1 21 , − 12 ,
c = 2, − 21 , − 12 ,
A1 = 2, −1 12 , − 12 ,
A2 = (2, −1, 0),
A3 = 21 , −1, 21 ,
A4 = (1, −2, 0),
A5 = 2 12 , −2 12 , 0 ,
B1 = 1 21 , −1, − 21 ,
B2 = (1, −1, 0),
B3 = 1, −1 12 , 12 ,
B4 = 1 12 , −1, 21 ,
B5 = 21 , − 12 , 0 ,
C1 = (2, −2, 0),
C2 = 1, −1 21 , − 12 ,
C3 = 1 12 , −1 12 , 0 ,
C4 = 21 , −1 12 , 0 ,
C5 = 1 12 , −2, − 12
B.1.5
SQS-18
a = −1, 2 21 , − 12 ,
b = − 12 , 2 21 , −1 ,
c = −1 21 − 1, −1 21 ,
A1 = −1 21 , 3, −1 21 ,
A2 = (−2, 1, −2),
A3 = −1 21 , 0, −1 21 ,
A4 = (−1, 3, −1),
A5 = −1 21 , 4, −1 21 ,
A6 = −1 21 , 1, −1 21 ,
B1 = −2 21 , 4, −2 12 ,
B2 = (−1, 2, −1),
B3 = −1 12 , 2, −1 12 ,
B4 = (−1, 1, −1),
B5 = (−2, 3, −2),
B6 = −2 12 , 3, −2 12 ,
C1 = (−3, 4, −3),
C2 = (−1, 0, −1),
C3 = − 12 , 0, − 12 ,
C4 = − 12 , 1, − 12 ,
C5 = (−2, 4, −2),
C6 = (−2, 2, −2)
B.1.6
SQS-24
a = (3, 1, −1),
b = (−3, 1, −1, ),
c = 0, 12 , 21 ,
A1 = 1, 1 21 , − 21 ,
A2 = 21 , 2, −1 12 ,
A3 = 0, 1 21 , − 21 ,
A4 = 12 , 12 , 0 ,
A5 = (−1, 2, −1),
A6 = −1, 1 12 , − 12 ,
A7 = − 12 , 1, − 12 ,
A8 = −1 12 , 1, − 12 ,
B1 = (1, 2, −1),
B2 = 1 12 , 1 12 , −1 ,
B3 = 1 12 , 1, − 21 ,
B4 = (0, 2, −1),
B5 = 12 , 1, − 12 ,
B6 = − 12 , 2, −1 12 ,
B7 = (−1, 1, 0),
B8 = −1 21 , 1 12 , −1 ,
236
C1 = 2, 1 12 , − 12 ,
C2 = (1, 1, 0),
C3 = 21 , 1 12 , −1 ,
C4 = (0, 1, 0),
C5 = 0, 2 21 , −1 21 ,
C6 = − 12 , 1 21 , −1 ,
C7 = − 12 , 12 , 0 ,
C8 = −2, 1 21 , − 12 ,
B.1.7
SQS-36
a = 0, −1 21 , −1 21 ,
b = −1 21 , 0, −1 21 ,
c = 1 12 1 12 , −1 ,
A1 = 0, − 21 , −2 21 ,
A2 = 0, 12 , −2 21 ,
A3 = 12 , 21 , −3 ,
A4 = − 12 , − 21 , −2 ,
A5 = 1, 21 , −1 21 ,
A6 = (−1, 0, −2),
A7 = − 21 , 12 , −2 ,
A8 = 21 , 1, −2 21 ,
A9 = (0, 0, −1),
A10 = (0, 1, −3),
A11 = 21 , 0, −3 21 ,
A12 = (1, 0, −3),
B1 = − 21 , −1, −2 21 ,
B2 = (0, −1, −2),
B3 = (1, 0, −2),
B4 = −1, − 12 , −2 21 ,
B5 = − 12 , 0, −2 12 ,
B6 = 12 , 21 , −2 ,
B7 = 0, − 21 , −1 12 ,
B8 = 12 , 0, −1 21 ,
B9 = 0, 12 , −3 21 ,
B10 = − 12 , 0, −1 12 ,
B11 = 0, 21 , −1 12 ,
B12 = 12 , 12 , −1 ,
C1 = (−1, −1, −3),
C2 = − 21 , − 12 , −3 ,
C3 = (0, 0, −3),
C4 = 12 , 0, −2 12 ,
C5 = 21 , − 21 , −2 ,
C6 = (0, 0, −2),
C7 = 1, 12 , −2 12 ,
C8 = (0, 1, −2),
C9 = 21 , 1, −1 12 ,
C10 = (1, 1, −2),
C11 = (1, 1, −1),
C12 = (0, 0, −4)
B.1.8
SQS-48
a = 0, 2 21 , −1 12 ,
b = 0, 1 12 , −2 12 ,
c = (−3, −1, 1),
A1 = − 12 , 1, −1 12 ,
A2 = −3, 12 , − 12 ,
A3 = −1, 3 12 , −3 12 ,
A4 = − 12 , 3, −2 12 ,
A5 = (−1, 2, −2),
A6 = − 12 , 2 12 , −2 ,
A7 = −1 12 , 1, −1 12 ,
A8 = −1 12 , 12 , −1 ,
A9 = −1, 12 , − 12 ,
A10 = −2, 2 12 , −2 12 ,
A11 = (−2, 2, −2),
A12 = −1 12 , 2 21 , −2 ,
237
A13 = −2, 1 21 , −1 12 ,
A14 = −1 12 , 2, −1 21 ,
A15 = −2 12 , 0, − 21 ,
A16 = −2, 12 , − 12 ,
B1 = −3, 2 21 , −2 12 ,
B2 = (−3, 2, −2),
B3 = − 21 , 1 12 , −2 ,
B4 = −3, − 12 , 12 ,
B5 = (−3, 3, −3),
B6 = −1 21 , 2, −2 12 ,
B7 = −1 21 , 1 12 , −2 ,
B8 = −1, 1 21 , −1 12 ,
B9 = − 21 , 1 12 , −1 ,
B10 = −2 12 , 1 21 , −2 ,
B11 = −2 21 , 12 , −1 ,
B12 = (−2, 1, −1),
B13 = −2, − 21 , 12 ,
B14 = −2 12 , 2, −1 12 ,
B15 = −2 21 , 1, − 12 ,
B16 = −2 12 , 12 , 0 ,
C1 = − 12 , 2 21 , −3 ,
C2 = − 12 , 2, −2 21 ,
C3 = −3, 1 12 , −1 12 ,
C4 = (−3, 1, −1),
C5 = (−3, 0, 0),
C6 = (−1, 3, −3),
C7 = −1, 2 21 , −2 12 ,
C8 = − 12 , 2, −1 21 ,
C9 = (−1, 1, −1),
C10 = (−1, 0, 0),
C11 = (−2, 3, −3),
C12 = −2 21 , 1, −1 21 ,
C13 = −1 12 , 1 21 , −1 ,
C14 = −1 21 , 1, − 21 ,
C15 = (−2, 0, 0),
C16 = −2 12 , 1 12 , −1
B.2
B.2.1
A2B1C1
SQS-4
a = 12 , 12 , 0 ,
b = (0, 0, −1),
c = (−1, 1, 0),
A1 = − 12 , 1, − 12 ,
A2 = (0, 1, −1),
B1 = − 12 , 1 12 , −1 ,
C1 = 0, 21 , − 12
B.2.2
SQS-8
a = 1 12 , − 21 , 0 ,
b = 1 21 , 0, − 21 ,
c = −1, 1 12 , 1 21 ,
A1 = (2, 1, 1),
A2 = 0, 12 , 12 ,
A3 = (1, 1, 1),
A4 = (0, 1, 1),
B1 = (1, 0, 0),
B2 = 1, 21 , 12 ,
C1 = (2, 0, 0),
C2 = 2, 21 , 12
B.2.3
SQS-16
a = (−2, −1, −1),
b = (−1, −1, −2),
c = 12 − 1, 12 ,
A1 = (−1, −2, −1),
238
A2 = −2, −1 21 , −1 21 ,
A3 = (−2, −2, −2),
A4 = −1 12 , −1 12 , −2 ,
A5 = −1, −1 21 , −1 21 ,
A6 = −2 12 , −3, −2 21 ,
A7 = −1 12 , −2, −1 21 ,
A8 = (−1, −1, −1),
B1 = −1 12 , −1 21 , −1 ,
B2 = −1, −1 21 , − 12 ,
B3 = −1 12 , −2 21 , −2 ,
B4 = −2 21 , −2, −2 21 ,
C1 = − 12 , −1, − 21 ,
C2 = −2, −2 21 , −1 21 ,
C3 = − 12 , −1 12 , −1 ,
C4 = (0, −1, 0)
B.2.4
SQS-24
a = 2 21 , −2, 1 21 ,
b = −1 12 , 2, −2 21 ,
c = 12 , 1, 12 ,
A1 = 1, 21 , − 12 ,
A2 = (0, 2, −1),
A3 = 21 , 1 12 , −1 ,
A4 = 21 , 12 , 0 ,
A5 = (−1, 2, −2),
A6 = 2, − 21 , 12 ,
A7 = 1 12 , − 12 , 0 ,
A8 = (2, 0, 1),
A9 = 1 21 , 12 , 0 ,
A10 = (2, −1, 1),
A11 = 2 21 , −1, 1 21 ,
A12 = 0, 1 21 , −1 12 ,
B1 = 1 12 , 0, 12 ,
B2 = (0, 1, −1),
B3 = 0, 1 21 , − 21 ,
B4 = 0, 12 , − 12 ,
B5 = 1, − 12 , 12 ,
B6 = 12 , 12 , −1 ,
C1 = 21 , 1, − 12 ,
C2 = 1 12 , 1, − 12 ,
C3 = (1, 1, 0),
C4 = 1, 12 , 21 ,
C5 = − 12 , 2, −1 21 ,
C6 = (1, 0, 0)
B.2.5
SQS-32
a = (1, 1, 2),
b = (1, 1, −2),
c = (−1, 1, 0),
A1 = 1 12 , 2, 12 ,
A2 = (0, 2, −1),
A3 = 21 , 1 12 , −1 ,
A4 = 1, 1 12 , 1 12 ,
A5 = 12 , 1, − 21 ,
A6 = 1, 1 12 , − 21 ,
A7 = (1, 3, 0),
A8 = − 12 , 1 12 , 0 ,
A9 = (0, 2, 0),
A10 = 0, 1 12 , − 21 ,
A11 = 12 , 2, − 12 ,
A12 = 12 , 2 12 , −1 ,
A13 = (0, 2, 1),
A14 = 12 , 2, −1 12 ,
A15 = 12 , 2, 12 ,
A16 = (1, 2, −1),
B1 = 12 , 2 12 , 1 ,
239
B2 = 1, 2 21 , 12 ,
B3 = 21 , 2, 1 12 ,
B4 = 1 21 , 2 12 , 0 ,
B5 = (1, 2, 0),
B6 = 1, 1 21 , −1 12 ,
B7 = 21 , 1 12 , 1 ,
B8 = 1, 2 21 , − 12 ,
C1 = (1, 2, 1),
C2 = 12 , 2 21 , 0 ,
C3 = (0, 1, 0),
C4 = 12 , 1 21 , 0 ,
C5 = 12 , 1, 21 ,
C6 = 1, 1 21 , 21 ,
C7 = 0, 1 12 , 21 ,
C8 = 1 12 , 2, − 12
B.2.6
SQS-48
a = 1, 12 , 1 21 ,
b = 1, −1 12 , − 12 ,
c = 1 12 , 2, −2 21 ,
A1 = 1 12 , 12 , 0 ,
A2 = 3, 1 21 , −1 21 ,
A3 = 2 12 , 1 21 , −2 ,
A4 = 2 21 , 1, − 21 ,
A5 = 2, 12 , −1 21 ,
A6 = 2 12 , 12 , −1 ,
A7 = 3 21 , 1, −1 21 ,
A8 = 1, − 21 , − 21 ,
A9 = (2, 0, 0),
A10 = 1 21 , − 21 , 0 ,
A11 = 2, 21 , − 21 ,
A12 = 2, 1 12 , − 21 ,
A13 = 1 12 , 1 21 , −2 ,
A14 = (2, 2, −1),
A15 = 2 12 , 0, − 21 ,
A16 = (2, 0, −2),
A17 = (2, 1, −2),
A18 = 1 12 , −1, − 12 ,
A19 = 2 12 , 12 , −2 ,
A20 = 1 12 , 12 , −1 ,
A21 = (1, 0, −1),
A22 = 1 12 , 0, −1 12 ,
A23 = 2 12 , 1, −1 12 ,
A24 = 2 12 , 1 12 , −1 ,
B1 = 2 12 , 1, −2 12 ,
B2 = 2, − 21 , − 12 ,
B3 = 1 12 , 1, − 21 ,
B4 = 1, 12 , 12 ,
B5 = 1 21 , − 12 , −1 ,
B6 = 1 21 , 0, − 21 ,
B7 = 2, − 12 , 12 ,
B8 = (2, 1, 0),
B9 = (3, 1, −2),
B10 = (2, 1, −1),
B11 = 1 12 , 1 12 , −1 ,
B12 = (1, 1, −1),
C1 = (2, 0, −1),
C2 = (3, 1, −1),
C3 = 2 21 , 2, −1 12 ,
C4 = 1 12 , 1, −1 12 ,
C5 = (1, 0, 0),
C6 = 12 , 0, − 12 ,
C7 = (2, 2, −2),
C8 = 2, 1 12 , −2 12 ,
C9 = 1 12 , 1, 21 ,
C10 = 2, 1 12 , −1 12 ,
C11 = 1 12 , 0, 12 ,
240
C12 = 1, 12 , − 21
B.2.7
SQS-64
a = (1, 0, 1),
b = (0, 1, 1),
c = −5 21 , −5 12 , 5 ,
A1 = −3 21 , −3 12 , 4 ,
A2 = −3 21 , −4, 4 21 ,
A3 = −4, −3 21 , 5 21 ,
A4 = −3, −2 21 , 4 21 ,
A5 = −2 21 , −2 12 , 5 ,
A6 = −2 21 , −2 12 , 3 ,
A7 = −4 21 , −4 12 , 7 ,
A8 = (−5, −5, 5),
A9 = −3 21 , −3 12 , 6 ,
A10 = −2, −1 21 , 2 12 ,
A11 = −5, −4 21 , 6 12 ,
A12 = (−3, −3, 5),
A13 = −4 12 , −4 21 , 6 ,
A14 = −2 21 , −2 21 , 4 ,
A15 = −1 21 , −1 21 , 2 ,
A16 = −3 21 , −3 21 , 5 ,
A17 = −1 21 , −2, 2 21 ,
A18 = (−1, −1, 1),
A19 = 0, − 21 , 1 21 ,
A20 = −3 12 , −3, 4 21 ,
A21 = −3, −3 21 , 4 12 ,
A22 = −2, −1 21 , 3 12 ,
A23 = − 12 , − 21 , 3 ,
A24 = −1, − 21 , 2 21 ,
A25 = −4, −3 12 , 4 12 ,
A26 = −3, −2 21 , 3 12 ,
A27 = − 12 , −1, 1 21 ,
A28 = −2 12 , −2, 3 12 ,
A29 = − 12 , − 12 , 1 ,
A30 = −3 12 , −4, 5 12 ,
A31 = − 12 , 0, 1 21 ,
A32 = − 12 , −1, 2 12 ,
B1 = −1, − 12 , 1 12 ,
B2 = 0, 21 , 1 12 ,
B3 = (−1, −1, 3),
B4 = −4 21 , −4, 5 12 ,
B5 = −4, −4 21 , 5 12 ,
B6 = 12 , 21 , 2 ,
B7 = −5, −4 21 , 5 12 ,
B8 = (−2, −2, 3),
B9 = − 12 , − 12 , 2 ,
B10 = −4 12 , −5, 5 12 ,
B11 = −2, −2 12 , 3 21 ,
B12 = −2 12 , −3, 3 12 ,
B13 = (−4, −4, 6),
B14 = (−1, −1, 2),
B15 = −2 12 , −3, 4 12 ,
B16 = (0, 0, 2),
C1 = −1 12 , −1 12 , 4 ,
C2 = −1, −1 21 , 2 12 ,
C3 = (−2, −2, 4),
C4 = −1 12 , −1 12 , 3 ,
C5 = −4 12 , −5, 6 12 ,
C6 = (0, 0, 1),
C7 = (−3, −3, 3),
C8 = 12 , 0, 1 21 ,
C9 = (−5, −5, 6),
C10 = (−3, −3, 4),
C11 = −1 12 , −1, 2 21 ,
C12 = (−2, −2, 2),
C13 = −4 12 , −4 21 , 5 ,
241
C14 = −1 12 , −2, 3 21 ,
C15 = (−4, −4, 4),
C16 = (−4, −4, 5)
Vita
Dongwon Shin
Dongwon Shin was born on January 11, 1975. He graduated from Pusan National University, Pusan, Korea in 2000 with a B.S. degree in Materials Science and
Engineering. He continued his graduate study at the same university and got his
M.S. degree in Materials Science and Engineering in 2002. Afterwards, Dongwon
enrolled at The Pennsylvania State University for his Ph.D. degree.
Listed below are his publications during his Ph.D. study:
1. D. Shin and Z.-K. Liu. Phase stability of hafnium oxide and zirconium oxide
on silicon substrate. Submitted, 2006.
2. D. Shin, R. Arróyave, and Z.-K. Liu. Thermodynamic modeling of the Hf-Si-O
system. CALPHAD, 30(4):375.386, 2006.
3. D. Shin, R. Arróyave, 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.
4. W. J. Golumbfskie, R. Arróyave, D. Shin, and Z. K. Liu. Finite-temperature
thermodynamic and vibrational properties of Al-Ni-Y compounds via firstprinciples calculations. Acta Mater., 54(8):2291.2304, 2006.
5. R. Arróyave, D. Shin, and Z. K. Liu. Ab initio thermodynamic properties of
stoichiometric phases in the Ni-Al system. Acta Mater., 53(6):1809.1819, 2005.
6. R. Arróyave, D. Shin, and Z. K. Liu. Modification of the thermodynamic model
for the Mg-Zr system. CALPHAD, 29(3):230.238, 2005.
7. S. Zhang, D. Shin, and Z.-K. Liu. Thermodynamic modeling of the Ca-Li-Na
system. CALPHAD, 27(2):235.241, 2003.