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Thermodynamic modeling of the Cu–Si system
Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 520–526
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Computer Coupling of Phase Diagrams and Thermochemistry journal homepage: www.elsevier.com/locate/calphad
Thermodynamic modeling of the Cu–Si system Dongwon Shin 1 , James E. Saal, Zi-Kui Liu ∗ Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA
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Article history: Received 6 February 2008 Received in revised form 20 May 2008 Accepted 26 May 2008 Available online 20 June 2008 Keywords: CALPHAD Cu–Si First-principles calculations Solid solutions Special quasirandom structures
a b s t r a c t The thermodynamic description of the Cu–Si system has been updated with first-principles calculations of the -Cu15 Si4 phase and solid solution phases. Calculated enthalpy of formation for the -phase indicates that enthalpies of formation for intermetallic compounds in Cu–Si must be negative, which were evaluated as positive in the previous thermodynamic modelings. Enthalpies of mixing for the solid solution phases, bcc, fcc, and hcp, are also calculated from first-principles study of Special Quasirandom Structures (SQS) and all the solution phases exhibit similar mixing behavior. It is concluded that firstprinciples calculations of solid phases including solution phases can supplement scarce experimental data and be readily used in thermodynamic modelings. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction The CALPHAD approach aims to model the Gibbs energy functions of individual phases in a system as a function of temperature and, if needed, pressure. It starts with compiling all available data related to a system, then thermodynamic models of each phase are established and the collected data are scrutinized. Afterwards, model parameters of each phase are evaluated to reproduce the accepted data. There are two types of data which are used in a thermodynamic modeling: thermochemical data and phase equilibrium data. Thermochemical data, such as heat capacity, enthalpy of mixing and activity, are valuable in thermodynamic modeling since they can be directly derived from the Gibbs energy functions of individual phases. In principle, if one can measure enough thermochemical data of individual phases in a system with high precision, then a state-of-the-art phase diagram can be readily calculated from the Gibbs energies evaluated from those measured data. Unfortunately, thermochemical data cannot be measured accurately enough to be exclusively used in a thermodynamic modeling. Furthermore, the number of phases in industrial alloys is fairly large so that even the least amount of needed thermochemical measurements for a thermodynamic modeling are immense. Therefore, it is impractical to obtain reliable Gibbs
∗
Corresponding author. E-mail address: [email protected] (Z.-K. Liu).
1 Present address: Department of Materials Science and Engineering, Northwestern University, IL 60208, USA. 0364-5916/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2008.05.003
energy functions purely from thermochemical data due to the lack of quality and quantity of the data. In contrast, phase equilibrium data can be measured from experiments with high precision, specifically thermal analysis, but such data only provide the relationship of the Gibbs energies of the phases which are in equilibrium. Nevertheless, phase equilibrium data are still valuable since they constrain the Gibbs energies of the phases to reproduce themselves. On the whole, these two types of experimental data are complementary to each other in the CALPHAD approach. 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. More often, solid phases, including solid solution phases, have very few thermochemical data – or sometimes no data at all – since obtaining a single solid phase for reliable thermodynamic measurement itself is quite demanding. Furthermore, a comprehensive thermodynamic understanding of solid phases at low temperatures also becomes impractical due to sluggish kinetics. Thus when there is not enough thermochemical data of a phase in a thermodynamic modeling, its thermochemical properties have to be guessed based on phase equilibrium data, if available. In this instance, evaluating thermodynamic parameters of a phase only from phase diagram data may pose a problem since there are an infinite number of plausible sets of parameters to satisfy phase equilibria with incorrect Gibbs energies of the relevant phases. Previous thermodynamic modelings of the Cu–Si system [1,2] exactly fall under this circumstance. As important alloying elements in Al and Mg alloys, the Cu–Si system has been studied extensively. However, most reported data are limited to thermochemical data of the liquid phase and phase equilibrium data. As a result, both thermodynamic modelings could reproduce the Cu–Si
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unavailability of thermochemical data for any solid phases. 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. Fig. 1 shows the calculated enthalpy of formation from the previous two thermodynamic modelings of the Cu–Si system [1,2]. Fries and Jantzen [3] pointed out the problem of positive enthalpies of formation and suggested a calorimetric study of intermetallic compounds in the Cu–Si system. These thermodynamic descriptions are valid within the system since the relative values of the Gibbs energies for individual phases are satisfied. However, when extrapolated to a higher order system, they may predict incorrect phase stabilities. Therefore, the thermodynamic description of the Cu–Si system needs to be updated to have better Gibbs energy functions for the intermetallic compounds. Fig. 1. Calculated enthalpies of formation for the Cu–Si system at 298.15 K from previous thermodynamic modelings [1,2]. Dotted vertical lines correspond to the stoichiometry of four intermetallic compounds. Reference states for Cu and Si are fcc and diamond, respectively.
phase diagram correctly but evaluated the enthalpies of formation for intermetallic compounds as positive. For these unusually evaluated enthalpies of formations for intermetallic compounds in the Cu–Si system, Fries and Jantzen [3] suggested that further experimental verification is needed. Unfortunately, thermochemical measurements of the solid phases are still unavailable. In this regard, theoretical calculations that can provide thermochemical data of a solid phase would be extremely valuable in the CALPHAD modeling. For the last decade, it has been steadily demonstrated that thermochemical properties of solid phases can be reliably determined from first-principles calculations [4,5]. The total energy of a phase can be calculated from first-principles as long as the crystal structure is known and then its reference states’ energies are subtracted to obtain the enthalpy of formation. Combined with phase equilibrium data, first-principles energetics 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, of which the crystal structures are unknown, are inferred correspondingly from that of -Cu15 Si4 . Furthermore, three solid solution phases in the Cu–Si system, i.e. bcc, fcc, and hcp, are also calculated from first-principles via Special Quasirandom Structures (SQS)[6]. All the first-principles results are combined with phase equilibrium data and a better thermodynamic description of the Cu–Si system is obtained. 2. Review of previous work The Cu–Si system was comprehensively assessed by Olesinski and Abbaschian [7] with 11 phases: liquid, fcc-Cu and diamond–Si, bcc (β ) and hcp (κ ) solid solution phases, cubic phases ( and γ ); tetragonal phase (δ ); rhombohedral phase (η) and its low temperature forms (η0 and η00 ). The first thermodynamic description of the Cu–Si system can be found in the COST 507 database [1] containing four stoichiometric intermetallic compounds: η-Cu19 Si6 , γ -Cu56 Si11 , -Cu4 Si, and δ -Cu33 Si7 . This database has been updated by Yan and Chang [2]. The stoichiometry of the -phase has been changed from Cu4 Si to Cu15 Si4 and better agreement with experimentally determined phase equilibrium data was obtained. However, both thermodynamic modelings were based on thermochemical data for the liquid phase and phase equilibrium data due to the
3. First-principles calculations First-principles calculations, based on density functional theory (DFT), can provide helpful insight into the characteristics of thermodynamic behavior of solid phases, especially enthalpy of formation [4,5]. In fact, first-principles calculations determine the total energy of a phase at 0 K. Nevertheless the enthalpy of formation derived from first-principles at 0 K can be treated as that of at 298.15 K since enthalpies of formation are often independent of temperature. Although there is a systematic error from the implemented approximations to simplify the complex electron calculations in the first-principles calculations, it is compensated by subtracting its reference states to calculate the enthalpy of formation. Entropy of formation, then, can be evaluated with the phase equilibrium data, such as phase transformation temperatures, in a thermodynamic modeling. Therefore, the enthalpy of formation from first-principles calculations has great importance in the CALPHAD approach as a good starting point to evaluate reliable Gibbs energy functions for intermetallic phases. In the following section, first-principles calculations of the solid phases in the Cu–Si system have been used as supplementary experimental data within the CALPHAD modeling. 3.1. Intermetallic compounds The enthalpies of formation for the ordered phases in the Cu–Si system can be determined from: Cux Siy
1Hf
' E (Cux Siy ) −
x x+y
E fcc (Cu) −
y x+y
E dia (Si)
(1)
where E’s are total energies from first-principles calculations, and the reference states for compounds are set as fcc for Cu and diamond for Si, respectively. The existence of several intermetallic compounds are reported in Olesinski and Abbaschian [7], but only the crystal structure of -Cu15 Si4 has been reported [8]. Therefore, the enthalpy of formation for the -phase can be determined from first-principles calculations from Eq. (1). For the first-principles calculations of binary solid phases in the Cu–Si system with VASP, one has four different choices for the pseudopotential of Cu. For exchange correlations, Local Density Approximation (LDA) or Generalized Gradient Approximation (GGA) can be used, and each with two different pseudopotentials for Cu: Cu and Cu_pv. PV pseudopotential treats p-orbital as a valence, and includes more electrons in calculations. It is generally known that LDA overbinds and thus underestimates lattice parameters, while GGA slightly overcorrects them [9]. Even though it is not always the case, experimental lattice parameter measurements are closer to those calculated with GGA than LDA in
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Table 1 First-principles results of the -Cu15 Si4 phase and its Standard Element Reference (SER), fcc-Cu and diamond–Si Phases
fcc-Cu
Space group
¯ Fm3m
Total energy
1Hf
Calc. (Å)
Exp. (Å)
Error (%)
(eV/atom)
(kJ/mol atom)
Cu Cu_pva Cu Cu_pv
3.6362 3.6263 3.5233 3.5202
3.6152
0.58 0.31 −2.54 −2.63
−3.7274 −3.6374 −4.6978 −4.6161
– – – –
GGA LDA
Sia Si
5.4676 5.4029
5.4309
0.68
−0.52
−5.4315 −5.9606
– –
GGA
Cu Cu_pva Cu Cu_pv
9.7599 9.7307 9.4931 9.4762
9.6940
0.68 0.38 −2.07 −2.25
−4.1433 −4.0751 −5.0769 −5.0144
−5.509 −5.790 −40.800 −11.120
Pseudopotential
GGA LDA
Diamond–Si
-Cu15 Si4
¯ Fd3m ¯ Fm3m
LDA
Lattice parameter
By definition, 1Hf of pure elements are zero. a
Used in the present work.
many cases. As Table 1 shows both LDA and GGA could reproduce the experimental lattice parameter of Cu15 Si4 quite satisfactorily. However, for the formation enthalpy calculations, there is a notable difference with LDA. With the Cu potential, ∆Hf (Cu15 Si4 ) is calculated to be ∼−11 kJ/mol, while that of Cu_pv is ∼−41 kJ/mol. For GGA, ∆Hf (Cu15 Si4 ) is consistently obtained to be ∼−6 kJ/mol, regardless of pseudopotentials for Cu. As shortly discussed above, LDA tends to overbind and it occasionally predicts enthalpies of formation for intermetallic compounds too negative. Hence GGA pseudopotentials have been chosen in the present work to avoid such overbindings. First-principles results of -Cu15 Si4 and its reference states are summarized in Table 1. As discussed earlier, 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 -Cu15 Si4 phase in Table 1 clearly indicates that it is negative and thus the enthalpies of formation for the other intermetallic compounds should also be negative. It is worth noting here that the enthalpies of formation for the three compounds other than -Cu15 Si4 should be very close to that of the -phase since the stoichiometry of the four intermetallic compounds are so close to each other. 3.2. Solid solution phases Besides the four intermetallic compounds, the Cu–Si system also 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 range including metastable regions since the extrapolation to a higher order system has to be considered. Unfortunately, thermochemical measurements for solid solution phases, such as enthalpy of mixing, are difficult to determine experimentally due to the limited solubility, while many efforts have been made to determine the phase equilibrium regarding the solid solution phases [10–14]. In order to overcome the scarce experimentally determined thermochemical data of solid solution phases from experiments, various efforts have been made to estimate them from theoretical calculations. First-principles study of Special Quasirandom Structures (SQS) deserves special attention among the currently available approaches. SQSs are structural templates whose correlation functions are very close to those of completely random solid solutions, thus they can be applied to any relevant system by changing the atomic numbers in first-principles calculations. Moreover, the effect of local relaxation can be considered by fully relaxing the structure. Such SQSs have been successfully applied to calculate the mixing enthalpies of binary solid solutions for bcc, fcc,
and hcp phases [15–17]. Only SQSs at three compositions (x = 0.25, 0.5, and 0.75) can be obtained in the A1−x Bx substitutional solutions with adequately sized supercells,2 but they can provide sufficient information on the mixing behavior of a solid solution in a binary system. In the present work, enthalpy of mixing for bcc, fcc, and hcp solid solution phases are calculated from their binary SQSs and used in the thermodynamic modeling of the Cu–Si system. The supercell structures of binary SQSs are from previous works [15–17]. 3.3. Methodology The Vienna Ab initio Simulation Package (VASP) [18] was used to perform the electronic structure calculations based on DFT. The projector augmented wave (PAW) method [19] was chosen and the generalized gradient approximation (GGA) [20] was used to take into account exchange and correlation contributions to the Hamiltonian of the ion–electron system. A constant energy cutoff of 461 eV was used to calculate the electronic structures of all the compounds. 5000 k-points per reciprocal atom based on the Monkhorst–Pack scheme for the Brillouin-zone sampling was used. It should be noted that different geometrical optimization schemes have been applied to ordered and disordered phases (SQSs) in the first-principles calculations. While the ordered structures in the Cu–Si system shown in Table 1 are completely relaxed, the SQSs for solid solutions are relaxed in two different ways: (i) fully relaxed to consider the local relaxation effect as in the calculations for the ordered phases, and (ii) constrained to preserve the original symmetry. In principle, all first-principles calculations must be fully relaxed to find their lowest energy configurations. However, the symmetries of SQSs are as low as P1 although all the atom positions are those of their higher order phases, i.e. bcc, fcc, and hcp. Hence the atoms may largely deviate from the original perfect lattice sites when fully relaxed during structural optimization due to the local relaxation caused by the different interatomic bondings, i.e. A–A, B–B, and A–B. In the CALPHAD approach, a solution phase is described throughout
2 In principle, increasing the size of SQSs increases the probability of finding SQSs with better correlation functions. However, the size of SQSs has to be quite large to satisfy correlation functions at a composition other than x = 0.25, 0.5, and 0.75, since the correlation function of different shapes, k, is expressed as (2x − 1)k in substitutional binary solid solutions. In order to reliably calculate thermodynamic properties of solid solution phases from first-principles calculations of SQSs, the number of atoms in an SQS has to be at least 8 per atom in the primitive cell of the target structure. For cubic phases, such as bcc and fcc, SQSs with 8 atoms are more than enough. However, hcp phase with 2 atoms in the primitive cell needs 16 atom SQSs for convergence. See Refs. [15–17] for further discussion.
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(a) Enthalpy of formation for the Cu–Si system with first-principles calculation of -Cu15 Si4 at 0 K.
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(b) Entropy of formation for the Cu–Si system.
Fig. 2. Calculated enthalpy and entropy of formation for the Cu–Si system at 298.15 K. Reference states for Cu and Si are fcc and diamond phases, respectively.
the entire composition range for the effective extrapolation to higher-order systems. Thus the relaxed structure in first-principles calculations should have the same symmetry as the original phase. However, if the degree of such a local distortion is too large, it may result in the loss of their original symmetries when all lattice sites are substituted by one single type of atom. In such instances, symmetry preserved calculations can be performed to estimate the mixing energy solid solution by freezing the degree of freedom for local relaxation. For such symmetry preserved calculations, only the volume is relaxed for the cubic phases (bcc and fcc), while both the volume and shape are relaxed for hexagonal phases (hcp) to optimize the c /a ratio. It should be emphasized here that preserving the symmetry during the structural optimization of SQSs is a biased approach to fit the first-principles energetics of solid solution phases into the thermodynamic model. There is no fully satisfactory way to calculate mechanically or dynamically unstable solid solution phases from first-principles calculations. Nevertheless, energies from symmetry preserved SQS calculations should not be used to evaluate thermodynamic parameters of solid solution phases in the CALPHAD approach, since they cannot reflect the energetics of truly random solid solutions within the CALPHAD formalism.
4.2. Solution phases The liquid and fcc phases are described with a one sublattice model which is equivalent to a substitutional solution. Since the bcc phase of Cu–Si forms continuous solid solutions with the β phase in the ternary Al–Cu–Si system, and the hcp phase of Cu–Si extends into the ternary Cu–Mg–Si system, these two intermediate phases are also modeled as disordered solution phases. The molar Gibbs energies of the four solution phases are described as: φ
φ
Gφm = xCu o GCu + xSi o GSi + RT (xCu ln xCu + xSi ln xSi ) + xs Gφ
(3)
where xi represents the mole fraction of component i. The first two terms on the right-hand side represent the Gibbs energy of the mechanical mixing of the two elements, the third term corresponds to the ideal mixing and the fourth term is the excess Gibbs energy. The Redlich–Kister polynomial [21] is used to represent the excess Gibbs energy of these phases: xs
Gφ = xCu xSi
X
k
φ
LCu,Si (xCu − xSi )k
(4)
k=0
with k
φ
LCu,Si = k a + k bT
(5)
4. Thermodynamic modeling
where k a and k b are model parameters evaluated from experimental data and first-principles calculations.
4.1. Ordered phases
5. Results and discussions
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 Gibbs energies of the four intermetallic compound phases are described as:
The calculated enthalpy of formation for the Cu–Si system is shown in Fig. 2(a) with first-principles result of the -Cu15 Si4 phase superimposed. It is assumed that formation enthalpies of four intermetallic compounds in the Cu–Si system should be close to each other on the convex hull since their stoichiometry is very similar. Thus formation enthalpies of intermetallic compounds other than -Cu15 Si4 are initially estimated to be laid on the conceptual convex hull, which is determined from firstprinciples formation enthalpy of -phase. Formation enthalpies and entropies of four intermetallic compounds are then evaluated to satisfy experimentally observed phase equilibrium data from the literature thereafter (see Yan and Chang [2] and references therein).
φ
o dia Gφm = xCu o Gfcc Cu + xSi GSi + 1Gf
φ
φ
φ
(2)
where 1Gf = 1Hf − T 1Sf , represents the Gibbs energy of formation of the stoichiometric phases. 1Hf and 1Sf are enthalpy and entropy of formation, respectively. 1Hf of -Cu15 Si4 is obtained from first-principles and the others are inferred from that of -Cu15 Si4 .
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(a) Calculated enthalpy of mixing of the liquid phase in the Cu–Si system at 1773 K with experimental data [28–33]. Reference states for both elements are the liquid phase.
(b) Calculated enthalpy of mixing of the bcc phase at 298.15 K. Reference states for both elements are bcc phase.
(c) Calculated enthalpy of mixing of the fcc phase at 298.15 K. Reference states for both elements are fcc phase.
(d) Calculated enthalpy of mixing of the hcp phase at 298.15 K. Reference states for both elements are hcp phase.
Fig. 3. Calculated enthalpies of mixing of the solution phases in the Cu–Si system are given as solid lines with first-principles results in the present work. Open and closed symbols in (b), (c), and (d) are symmetry preserved and fully relaxed first-principles calculations of SQSs at 0 K, respectively. Dotted lines are from COST507 [1] and dashed lines are from Yan and Chang [2].
Fig. 2(b) shows the calculated entropy of formation for the Cu–Si system. Although any experimental measurements or theoretical calculations have never been reported to validate evaluated entropies of formation, the same trend could be found in a similar system, Ni–Si, which has been assessed by Du and Schuster [22]. In their work, the intermetallic compounds are predicted to have positive entropies of formation, much as they are in the present work. It is mentioned in their work that enthalpies of formation for the intermetallic compounds in the Ni–Si system were evaluated from quite reliable experimental measurements. Fig. 3 shows the calculated enthalpy of mixing for the solid solution phases, i.e. bcc, fcc, and hcp, and the liquid phase in the Cu–Si system. First-principles calculations of the SQSs in the present work and the calculated results from previous modelings [1,2] are shown together for comparison. It should be noted that none of the previous thermodynamic modelings used thermochemical data for solid solution phases to evaluate their parameters. However, both results agree quite satisfactorily with first-principles results of SQSs in the present work. All the fully relaxed calculations have been used in the thermodynamic parameters evaluation and symmetry preserved calculations are
shown in Fig. 3 only for comparison. In Fig. 3(b), the fully relaxed bcc SQS at 50% Si completely lost its bcc symmetry. Therefore, its energy was not considered during the optimization. However, as discussed in Section 3.3, the symmetry preserved calculations reproduce the overall mixing behavior of all solid solution phases quite satisfactorily. Another intriguing observation in Fig. 3 is that the tendency, and the absolute values, for the enthalpies of mixing obtained from SQSs in the Cu–Si system are close to each other in all phases except bcc, which exhibits a rather positive enthalpy of mixing in the Si-rich side. Conventional alloy theories, such as Hume-Rothery rules [23–25] and Darken–Gurry methods [26], can explain the similar mixing behavior of two elements in different structures. Both methods are not for estimating the enthalpy of mixing for binaries but for predicting the solubility limit empirically without considering the Gibbs energies of individual phases in a binary system. Furthermore, the existence of intermetallic compounds strongly affects a solubility limit. 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.
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Table 2 Thermodynamic parameters for the Cu–Si system (all in S.I. units) Phases
Sublattice model
Evaluated descriptions
Liquid
(Cu, Si)
0 1
Lfcc Cu,Si Lfcc Cu,Si 0 fcc LCu,Si 0 fcc LCu,Si 1 fcc LCu,Si
= −41 305 + 26.591T = −37 829 − 33.805T = −34 106 − 1.908T = −42 204 + 13.891T = −1102 − 18.178T
0
= −26485 + 15.151T = −85 212 − 11.583T = −21 740 + 3.9T = −40 000 − 4.39T = −100 = −26 447 + 10.216T = −47 275 − 8.517T
1 2 0 1 2
(Cu, Si)
Liq
= −35 111 + 8.311T = −39 688 + 14.275T = −49 937 + 29.790T = −31 810 + 18.008T = −38 764 + 12T = −52 431 + 27.457T = −29 427 + 14.775T
0
0
1
bcc
(Cu, Si)
Lbcc Cu,Si Lbcc Cu,Si 0 bcc LCu,Si 1 bcc LCu,Si 2 bcc LCu,Si 0 bcc LCu,Si 1 bcc LCu,Si 1
hcp
(Cu, Si)
0 1 0 1 2
hcp
LCu,Si = −30 194 + 7.381T
η-Cu19 Si6
(Cu)19 (Si)6
[1]
[2]
This work [1] [2] This work [1]
[2] This work
hcp
LCu,Si = −62 319 − 6.517T hcp
LCu,Si = −19 948 − 2.356T
[1]
hcp
LCu,Si = −23 800 − 1.97T hcp
LCu,Si = −20
hcp LCu,Si 1 hcp LCu,Si 0
Reference This work
LCu,Si = −49 747 + 15.454T
Liq LCu,Si Liq LCu,Si Liq LCu,Si Liq LCu,Si Liq LCu,Si Liq LCu,Si Liq LCu,Si
2
fcc
Liq
LCu,Si = −39729 + 4.278T
19 o 25 19 o 25 19 o 25
= −26 799 − 0.732T = −28 065 − 0.029T
Gfcc Cu + Gfcc Cu + Gfcc Cu
+
6 o 25 6 o 25 6 o 25
[2]
Gdia Si − 5065 − 4.508T
This work
Gdia Si + 3345 − 7.324T
[1]
+ 1368 − 7.198T
[2]
Gdia Si
-Cu15 Si4 -Cu4 Si
(Cu)15 (Si)4 (Cu)4 (Si)1
4 o dia Gfcc Cu + 19 GSi − 5142 − 3.909T −7984.87 + 171.70094T − 30.93528T ln T
This work [1]
-Cu15 Si4
(Cu)15 (Si)4
+0.002149728T 2 + 1.0267 × 10−7 T 3 + 77 316T −1 15 o fcc 4 o dia GCu + 19 GSi + 1241 − 6.720T 19
[2]
γ -Cu56 Si11
(Cu)56 (Si)11
δ -Cu33 Si7
(Cu)33 (Si)7
15 o 19
56 o 67 56 o 67 56 o 67
Gfcc Cu
33 o 40 33 o 40 33 o 40
Gfcc Cu
+
Gfcc Cu + Gfcc Cu +
+
Gfcc Cu + Gfcc Cu +
11 o 67 11 o 67 11 o 67
Gdia Si
7 o 40 7 o 40 7 o 40
Gdia Si
− 4014 − 3.978T
This work
Gdia Si + 1030 − 1.608T
[1]
Gdia Si + 1046 − 5.875T
[2]
− 2660 − 5.594T
This work
Gdia Si + 2830 − 7.025T
[1]
Gdia Si + 1972 − 6.966T
[2]
Gibbs energies for pure elements are from the SGTE pure element database[34].
The well-known first rule of Hume-Rothery states that the size difference of two elements must be less than 15%, and the second rule requires them to have similar electronegativities in order to exhibit extensive solid solubilities. Darken and Gurry [26] showed that Hume-Rothery’s first and second rules can be applied simultaneously to predict the formation of solid solutions. From the Darken–Gurry map shown in Fig. 4, large solubility can be expected if two elements are close to each other. The distances between Cu and Si is fairly short in the Darken–Gurry map when the coordination number is 12, and, therefore, it is expected that they are very likely to form solid solution phases. As a matter of fact, the Cu–Si system is very unusual to have all three representative solid solution phases, bcc, fcc, and hcp. However, the solubility ranges of the solid solution phases are not significantly wide due to the existence of intermetallic compounds in the Cu–Si system. The slightly different mixing behavior of bcc from those of fcc, hcp, and liquid in the Cu–Si system is attributed to different
coordination numbers. Recently, Gschneidner and Verkade [27] presented the semi-empirical 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 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 their coordination numbers are bigger than those of group IVB elements’ stable structure, diamond. Similarly, fcc and hcp are less directional than bcc since their coordination number is larger by 50%. Therefore, one would expect the enthalpy of mixing for bcc to behave differently from that of fcc and hcp, as is the case shown in Fig. 3. The calculated Cu–Si phase diagram is compared with experimental phase equilibrium data [10–14] in Fig. 5 and all evaluated parameters for the Cu–Si system are listed in Table 2.
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results can work as constraints in the thermodynamic modeling to prevent having incorrect parameters of a phase. Acknowledgments This work is funded by the National Science Foundation (NSF) through Grant No. DMR-0205232. First-principles calculations were carried out on the LION clusters at the Pennsylvania State University supported in part by the NSF grants (DMR-9983532, DMR-0122638, and DMR-0205232) and in part by the Materials Simulation Center and the Graduate Education and Research Services at the Pennsylvania State University. References
Fig. 4. The electronegativity v s. the atomic radius for a coordination number of 12 (Darken–Gurry map)[27].
Fig. 5. Calculated phase diagram of the Cu–Si system with experimental data [10– 14] in the present work.
6. Conclusion The complete self-consistent thermodynamic description of the Cu–Si system has been obtained with first-principles calculations of solid phases and phase equilibrium data. The enthalpy of formation for -Cu15 Si4 from first-principles calculations clearly indicates that the formation enthalpies of the intermetallic compounds in the system, must be negative, which were previously evaluated as positive. The enthalpies of mixing for the three solid solution phases, bcc, fcc, and hcp, are also calculated from first-principles studies of Special Quasirandom Structures. It is shown that the challenge of scarce thermochemical data for solid phases, even for solid solution phases, can be overcome by incorporating first-principles energetics. Those first-principles
[1] I. Ansara, A.T. Dinsdale, M.H. Rand, Thermochemical Database for Light Metal Alloys, volume 2 of COST 507: Definition of Thermochemical and Thermophysical Properties to Provide a Database for the Development of New Light Alloys. European Cooperation in the Field of Scientific and Technical Research, 1998. [2] X. Yan, Y.A. Chang, J. Alloys Compounds 308 (1–2) (2000) 221–229. [3] S.G. Fries, T. Jantzen, Thermochim. Acta 314 (1–2) (1998) 23–33. [4] C. Wolverton, X.Y. Yan, R. Vijayaraghavan, V. Ozolinš, ¸ Acta Mater. 50 (9) (2002) 2187–2197. [5] C. Colinet, Intermetallics 11 (11–12) (2003) 1095–1102. [6] A. Zunger, S.H. Wei, L.G. Ferreira, J.E. Bernard, Phys. Rev. Lett. 65 (3) (1990) 353–356. [7] R. Olesinski, G. Abbaschian, Bull. Alloy Phase Diagrams 7 (2) (1986) 170–178. [8] F.R. Morral, A. Westgren, Arkiv Kemi, Mineral. Geol. 11B (37) (1934) 6. [9] J. Hafner, Acta Mater. 48 (1) (2000) 71–92. [10] E. Rudolfi, Z. Anorg. Chem. 53 (1907) 216–227. [11] C.S. Smith, J. Inst. Met. 476 (1928) 12. [12] C.S. Smith, Trans. AIME 83 (1929) 414. [13] C.S. Smith, Trans. AIME 137 (1940) 313. [14] A.G.H. Andersen, Trans. AIME 137 (1940) 334. [15] S.H. Wei, L.G. Ferreira, J.E. Bernard, A. Zunger, Phys. Rev. B. 42 (15) (1990) 9622–9649. [16] C. Jiang, C. Wolverton, J. Sofo, L.-Q. Chen, Z.-K. Liu, Phys. Rev. B. 69 (21) (2004) 214202. [17] D. Shin, R. Arróyave, Z.-K. Liu, A. van de Walle, Phys. Rev. B. 74 (2) (2006) 024204. [18] G. Kresse, J. Furthmuller, Comput. Mater. Sci. 6 (1) (1996) 15–50. [19] G. Kresse, D. Joubert, Phys. Rev. B. 59 (3) (1999) 1758–1775. [20] J.P. Perdew, Y. Wang, Phys. Rev. B. 45 (23) (1992) 13244–13249. [21] O. Redlich, A.T. Kister, Ind. Eng. Chem. 40 (2) (1948) 345–348. [22] Y. Du, J.C. Schuster, Metall. Mater. Trans. A 30 (9) (1999) 2409–2418. [23] W. Hume-Rothery, G.W. Mabbott, K.M.C. Evans, Philos. Trans. Royal Soc. London, Ser. A: Math. Phys. Eng. Sci. 233 (1934) 1–97. [24] W. Hume-Rothery, The structure of metals and alloys. Monograph and Report Series; no. 1. The Institute of Metals, London, 1939. [25] W. Hume-Rothery, R.E. Smallman, C.W. Haworth, The structure of metals and alloys. Monograph and Report Series; no. 1. Institute of Metals ; Distributed in North America by the Institute of Metals North American Publications Center, London Brookfield, VT, 5th ed., 1988. [26] L.S. Darken, R.W. Gurry, Physical Chemistry of Metals, in: Metallurgy and Metallurgical Engineering Series, McGraw-Hill, New York, 1953. [27] K.A. Gschneidner Jr., M. Verkade, Prog. Mater. Sci. 49 (3–4) (2004) 411–428. [28] Y. Iguchi, H. Shimoji, S. Banya, T. Fuwa, J. Iron Steel Inst. Jpn. 63 (2) (1977) 275–284. [29] R. Castanet, J. Chem. Thermodyn. 11 (8) (1979) 787–789. [30] I. Arpshofen, M.J. Pool, U. Gerling, F. Sommer, E. Schultheiss, B. Predel, Z. Metallkd. 72 (12) (1981) 842–846. [31] G.I. Batalin, V.S. Sudavtsova, Izv. Akad. Nauk SSSR, Neorg. Mater. 18 (1) (1982) 155–157. [32] V. Witusiewicz, I. Arpshofen, F. Sommer, Z. Metallkd. 88 (11) (1997) 866–872. [33] V. Witusiewicz, I. Arpshofen, H.-J. Seifert, F. Sommer, F. Aldinger, Z. Metallkd. 91 (2) (2000) 128–142. [34] A.T. Dinsdale, CALPHAD 15 (4) (1991) 317–425.
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Computer Coupling of Phase Diagrams and Thermochemistry journal homepage: www.elsevier.com/locate/calphad
Thermodynamic modeling of the Cu–Si system Dongwon Shin 1 , James E. Saal, Zi-Kui Liu ∗ Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA
article
info
Article history: Received 6 February 2008 Received in revised form 20 May 2008 Accepted 26 May 2008 Available online 20 June 2008 Keywords: CALPHAD Cu–Si First-principles calculations Solid solutions Special quasirandom structures
a b s t r a c t The thermodynamic description of the Cu–Si system has been updated with first-principles calculations of the -Cu15 Si4 phase and solid solution phases. Calculated enthalpy of formation for the -phase indicates that enthalpies of formation for intermetallic compounds in Cu–Si must be negative, which were evaluated as positive in the previous thermodynamic modelings. Enthalpies of mixing for the solid solution phases, bcc, fcc, and hcp, are also calculated from first-principles study of Special Quasirandom Structures (SQS) and all the solution phases exhibit similar mixing behavior. It is concluded that firstprinciples calculations of solid phases including solution phases can supplement scarce experimental data and be readily used in thermodynamic modelings. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction The CALPHAD approach aims to model the Gibbs energy functions of individual phases in a system as a function of temperature and, if needed, pressure. It starts with compiling all available data related to a system, then thermodynamic models of each phase are established and the collected data are scrutinized. Afterwards, model parameters of each phase are evaluated to reproduce the accepted data. There are two types of data which are used in a thermodynamic modeling: thermochemical data and phase equilibrium data. Thermochemical data, such as heat capacity, enthalpy of mixing and activity, are valuable in thermodynamic modeling since they can be directly derived from the Gibbs energy functions of individual phases. In principle, if one can measure enough thermochemical data of individual phases in a system with high precision, then a state-of-the-art phase diagram can be readily calculated from the Gibbs energies evaluated from those measured data. Unfortunately, thermochemical data cannot be measured accurately enough to be exclusively used in a thermodynamic modeling. Furthermore, the number of phases in industrial alloys is fairly large so that even the least amount of needed thermochemical measurements for a thermodynamic modeling are immense. Therefore, it is impractical to obtain reliable Gibbs
∗
Corresponding author. E-mail address: [email protected] (Z.-K. Liu).
1 Present address: Department of Materials Science and Engineering, Northwestern University, IL 60208, USA. 0364-5916/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2008.05.003
energy functions purely from thermochemical data due to the lack of quality and quantity of the data. In contrast, phase equilibrium data can be measured from experiments with high precision, specifically thermal analysis, but such data only provide the relationship of the Gibbs energies of the phases which are in equilibrium. Nevertheless, phase equilibrium data are still valuable since they constrain the Gibbs energies of the phases to reproduce themselves. On the whole, these two types of experimental data are complementary to each other in the CALPHAD approach. 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. More often, solid phases, including solid solution phases, have very few thermochemical data – or sometimes no data at all – since obtaining a single solid phase for reliable thermodynamic measurement itself is quite demanding. Furthermore, a comprehensive thermodynamic understanding of solid phases at low temperatures also becomes impractical due to sluggish kinetics. Thus when there is not enough thermochemical data of a phase in a thermodynamic modeling, its thermochemical properties have to be guessed based on phase equilibrium data, if available. In this instance, evaluating thermodynamic parameters of a phase only from phase diagram data may pose a problem since there are an infinite number of plausible sets of parameters to satisfy phase equilibria with incorrect Gibbs energies of the relevant phases. Previous thermodynamic modelings of the Cu–Si system [1,2] exactly fall under this circumstance. As important alloying elements in Al and Mg alloys, the Cu–Si system has been studied extensively. However, most reported data are limited to thermochemical data of the liquid phase and phase equilibrium data. As a result, both thermodynamic modelings could reproduce the Cu–Si
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unavailability of thermochemical data for any solid phases. 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. Fig. 1 shows the calculated enthalpy of formation from the previous two thermodynamic modelings of the Cu–Si system [1,2]. Fries and Jantzen [3] pointed out the problem of positive enthalpies of formation and suggested a calorimetric study of intermetallic compounds in the Cu–Si system. These thermodynamic descriptions are valid within the system since the relative values of the Gibbs energies for individual phases are satisfied. However, when extrapolated to a higher order system, they may predict incorrect phase stabilities. Therefore, the thermodynamic description of the Cu–Si system needs to be updated to have better Gibbs energy functions for the intermetallic compounds. Fig. 1. Calculated enthalpies of formation for the Cu–Si system at 298.15 K from previous thermodynamic modelings [1,2]. Dotted vertical lines correspond to the stoichiometry of four intermetallic compounds. Reference states for Cu and Si are fcc and diamond, respectively.
phase diagram correctly but evaluated the enthalpies of formation for intermetallic compounds as positive. For these unusually evaluated enthalpies of formations for intermetallic compounds in the Cu–Si system, Fries and Jantzen [3] suggested that further experimental verification is needed. Unfortunately, thermochemical measurements of the solid phases are still unavailable. In this regard, theoretical calculations that can provide thermochemical data of a solid phase would be extremely valuable in the CALPHAD modeling. For the last decade, it has been steadily demonstrated that thermochemical properties of solid phases can be reliably determined from first-principles calculations [4,5]. The total energy of a phase can be calculated from first-principles as long as the crystal structure is known and then its reference states’ energies are subtracted to obtain the enthalpy of formation. Combined with phase equilibrium data, first-principles energetics 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, of which the crystal structures are unknown, are inferred correspondingly from that of -Cu15 Si4 . Furthermore, three solid solution phases in the Cu–Si system, i.e. bcc, fcc, and hcp, are also calculated from first-principles via Special Quasirandom Structures (SQS)[6]. All the first-principles results are combined with phase equilibrium data and a better thermodynamic description of the Cu–Si system is obtained. 2. Review of previous work The Cu–Si system was comprehensively assessed by Olesinski and Abbaschian [7] with 11 phases: liquid, fcc-Cu and diamond–Si, bcc (β ) and hcp (κ ) solid solution phases, cubic phases ( and γ ); tetragonal phase (δ ); rhombohedral phase (η) and its low temperature forms (η0 and η00 ). The first thermodynamic description of the Cu–Si system can be found in the COST 507 database [1] containing four stoichiometric intermetallic compounds: η-Cu19 Si6 , γ -Cu56 Si11 , -Cu4 Si, and δ -Cu33 Si7 . This database has been updated by Yan and Chang [2]. The stoichiometry of the -phase has been changed from Cu4 Si to Cu15 Si4 and better agreement with experimentally determined phase equilibrium data was obtained. However, both thermodynamic modelings were based on thermochemical data for the liquid phase and phase equilibrium data due to the
3. First-principles calculations First-principles calculations, based on density functional theory (DFT), can provide helpful insight into the characteristics of thermodynamic behavior of solid phases, especially enthalpy of formation [4,5]. In fact, first-principles calculations determine the total energy of a phase at 0 K. Nevertheless the enthalpy of formation derived from first-principles at 0 K can be treated as that of at 298.15 K since enthalpies of formation are often independent of temperature. Although there is a systematic error from the implemented approximations to simplify the complex electron calculations in the first-principles calculations, it is compensated by subtracting its reference states to calculate the enthalpy of formation. Entropy of formation, then, can be evaluated with the phase equilibrium data, such as phase transformation temperatures, in a thermodynamic modeling. Therefore, the enthalpy of formation from first-principles calculations has great importance in the CALPHAD approach as a good starting point to evaluate reliable Gibbs energy functions for intermetallic phases. In the following section, first-principles calculations of the solid phases in the Cu–Si system have been used as supplementary experimental data within the CALPHAD modeling. 3.1. Intermetallic compounds The enthalpies of formation for the ordered phases in the Cu–Si system can be determined from: Cux Siy
1Hf
' E (Cux Siy ) −
x x+y
E fcc (Cu) −
y x+y
E dia (Si)
(1)
where E’s are total energies from first-principles calculations, and the reference states for compounds are set as fcc for Cu and diamond for Si, respectively. The existence of several intermetallic compounds are reported in Olesinski and Abbaschian [7], but only the crystal structure of -Cu15 Si4 has been reported [8]. Therefore, the enthalpy of formation for the -phase can be determined from first-principles calculations from Eq. (1). For the first-principles calculations of binary solid phases in the Cu–Si system with VASP, one has four different choices for the pseudopotential of Cu. For exchange correlations, Local Density Approximation (LDA) or Generalized Gradient Approximation (GGA) can be used, and each with two different pseudopotentials for Cu: Cu and Cu_pv. PV pseudopotential treats p-orbital as a valence, and includes more electrons in calculations. It is generally known that LDA overbinds and thus underestimates lattice parameters, while GGA slightly overcorrects them [9]. Even though it is not always the case, experimental lattice parameter measurements are closer to those calculated with GGA than LDA in
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Table 1 First-principles results of the -Cu15 Si4 phase and its Standard Element Reference (SER), fcc-Cu and diamond–Si Phases
fcc-Cu
Space group
¯ Fm3m
Total energy
1Hf
Calc. (Å)
Exp. (Å)
Error (%)
(eV/atom)
(kJ/mol atom)
Cu Cu_pva Cu Cu_pv
3.6362 3.6263 3.5233 3.5202
3.6152
0.58 0.31 −2.54 −2.63
−3.7274 −3.6374 −4.6978 −4.6161
– – – –
GGA LDA
Sia Si
5.4676 5.4029
5.4309
0.68
−0.52
−5.4315 −5.9606
– –
GGA
Cu Cu_pva Cu Cu_pv
9.7599 9.7307 9.4931 9.4762
9.6940
0.68 0.38 −2.07 −2.25
−4.1433 −4.0751 −5.0769 −5.0144
−5.509 −5.790 −40.800 −11.120
Pseudopotential
GGA LDA
Diamond–Si
-Cu15 Si4
¯ Fd3m ¯ Fm3m
LDA
Lattice parameter
By definition, 1Hf of pure elements are zero. a
Used in the present work.
many cases. As Table 1 shows both LDA and GGA could reproduce the experimental lattice parameter of Cu15 Si4 quite satisfactorily. However, for the formation enthalpy calculations, there is a notable difference with LDA. With the Cu potential, ∆Hf (Cu15 Si4 ) is calculated to be ∼−11 kJ/mol, while that of Cu_pv is ∼−41 kJ/mol. For GGA, ∆Hf (Cu15 Si4 ) is consistently obtained to be ∼−6 kJ/mol, regardless of pseudopotentials for Cu. As shortly discussed above, LDA tends to overbind and it occasionally predicts enthalpies of formation for intermetallic compounds too negative. Hence GGA pseudopotentials have been chosen in the present work to avoid such overbindings. First-principles results of -Cu15 Si4 and its reference states are summarized in Table 1. As discussed earlier, 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 -Cu15 Si4 phase in Table 1 clearly indicates that it is negative and thus the enthalpies of formation for the other intermetallic compounds should also be negative. It is worth noting here that the enthalpies of formation for the three compounds other than -Cu15 Si4 should be very close to that of the -phase since the stoichiometry of the four intermetallic compounds are so close to each other. 3.2. Solid solution phases Besides the four intermetallic compounds, the Cu–Si system also 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 range including metastable regions since the extrapolation to a higher order system has to be considered. Unfortunately, thermochemical measurements for solid solution phases, such as enthalpy of mixing, are difficult to determine experimentally due to the limited solubility, while many efforts have been made to determine the phase equilibrium regarding the solid solution phases [10–14]. In order to overcome the scarce experimentally determined thermochemical data of solid solution phases from experiments, various efforts have been made to estimate them from theoretical calculations. First-principles study of Special Quasirandom Structures (SQS) deserves special attention among the currently available approaches. SQSs are structural templates whose correlation functions are very close to those of completely random solid solutions, thus they can be applied to any relevant system by changing the atomic numbers in first-principles calculations. Moreover, the effect of local relaxation can be considered by fully relaxing the structure. Such SQSs have been successfully applied to calculate the mixing enthalpies of binary solid solutions for bcc, fcc,
and hcp phases [15–17]. Only SQSs at three compositions (x = 0.25, 0.5, and 0.75) can be obtained in the A1−x Bx substitutional solutions with adequately sized supercells,2 but they can provide sufficient information on the mixing behavior of a solid solution in a binary system. In the present work, enthalpy of mixing for bcc, fcc, and hcp solid solution phases are calculated from their binary SQSs and used in the thermodynamic modeling of the Cu–Si system. The supercell structures of binary SQSs are from previous works [15–17]. 3.3. Methodology The Vienna Ab initio Simulation Package (VASP) [18] was used to perform the electronic structure calculations based on DFT. The projector augmented wave (PAW) method [19] was chosen and the generalized gradient approximation (GGA) [20] was used to take into account exchange and correlation contributions to the Hamiltonian of the ion–electron system. A constant energy cutoff of 461 eV was used to calculate the electronic structures of all the compounds. 5000 k-points per reciprocal atom based on the Monkhorst–Pack scheme for the Brillouin-zone sampling was used. It should be noted that different geometrical optimization schemes have been applied to ordered and disordered phases (SQSs) in the first-principles calculations. While the ordered structures in the Cu–Si system shown in Table 1 are completely relaxed, the SQSs for solid solutions are relaxed in two different ways: (i) fully relaxed to consider the local relaxation effect as in the calculations for the ordered phases, and (ii) constrained to preserve the original symmetry. In principle, all first-principles calculations must be fully relaxed to find their lowest energy configurations. However, the symmetries of SQSs are as low as P1 although all the atom positions are those of their higher order phases, i.e. bcc, fcc, and hcp. Hence the atoms may largely deviate from the original perfect lattice sites when fully relaxed during structural optimization due to the local relaxation caused by the different interatomic bondings, i.e. A–A, B–B, and A–B. In the CALPHAD approach, a solution phase is described throughout
2 In principle, increasing the size of SQSs increases the probability of finding SQSs with better correlation functions. However, the size of SQSs has to be quite large to satisfy correlation functions at a composition other than x = 0.25, 0.5, and 0.75, since the correlation function of different shapes, k, is expressed as (2x − 1)k in substitutional binary solid solutions. In order to reliably calculate thermodynamic properties of solid solution phases from first-principles calculations of SQSs, the number of atoms in an SQS has to be at least 8 per atom in the primitive cell of the target structure. For cubic phases, such as bcc and fcc, SQSs with 8 atoms are more than enough. However, hcp phase with 2 atoms in the primitive cell needs 16 atom SQSs for convergence. See Refs. [15–17] for further discussion.
D. Shin et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 520–526
(a) Enthalpy of formation for the Cu–Si system with first-principles calculation of -Cu15 Si4 at 0 K.
523
(b) Entropy of formation for the Cu–Si system.
Fig. 2. Calculated enthalpy and entropy of formation for the Cu–Si system at 298.15 K. Reference states for Cu and Si are fcc and diamond phases, respectively.
the entire composition range for the effective extrapolation to higher-order systems. Thus the relaxed structure in first-principles calculations should have the same symmetry as the original phase. However, if the degree of such a local distortion is too large, it may result in the loss of their original symmetries when all lattice sites are substituted by one single type of atom. In such instances, symmetry preserved calculations can be performed to estimate the mixing energy solid solution by freezing the degree of freedom for local relaxation. For such symmetry preserved calculations, only the volume is relaxed for the cubic phases (bcc and fcc), while both the volume and shape are relaxed for hexagonal phases (hcp) to optimize the c /a ratio. It should be emphasized here that preserving the symmetry during the structural optimization of SQSs is a biased approach to fit the first-principles energetics of solid solution phases into the thermodynamic model. There is no fully satisfactory way to calculate mechanically or dynamically unstable solid solution phases from first-principles calculations. Nevertheless, energies from symmetry preserved SQS calculations should not be used to evaluate thermodynamic parameters of solid solution phases in the CALPHAD approach, since they cannot reflect the energetics of truly random solid solutions within the CALPHAD formalism.
4.2. Solution phases The liquid and fcc phases are described with a one sublattice model which is equivalent to a substitutional solution. Since the bcc phase of Cu–Si forms continuous solid solutions with the β phase in the ternary Al–Cu–Si system, and the hcp phase of Cu–Si extends into the ternary Cu–Mg–Si system, these two intermediate phases are also modeled as disordered solution phases. The molar Gibbs energies of the four solution phases are described as: φ
φ
Gφm = xCu o GCu + xSi o GSi + RT (xCu ln xCu + xSi ln xSi ) + xs Gφ
(3)
where xi represents the mole fraction of component i. The first two terms on the right-hand side represent the Gibbs energy of the mechanical mixing of the two elements, the third term corresponds to the ideal mixing and the fourth term is the excess Gibbs energy. The Redlich–Kister polynomial [21] is used to represent the excess Gibbs energy of these phases: xs
Gφ = xCu xSi
X
k
φ
LCu,Si (xCu − xSi )k
(4)
k=0
with k
φ
LCu,Si = k a + k bT
(5)
4. Thermodynamic modeling
where k a and k b are model parameters evaluated from experimental data and first-principles calculations.
4.1. Ordered phases
5. Results and discussions
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 Gibbs energies of the four intermetallic compound phases are described as:
The calculated enthalpy of formation for the Cu–Si system is shown in Fig. 2(a) with first-principles result of the -Cu15 Si4 phase superimposed. It is assumed that formation enthalpies of four intermetallic compounds in the Cu–Si system should be close to each other on the convex hull since their stoichiometry is very similar. Thus formation enthalpies of intermetallic compounds other than -Cu15 Si4 are initially estimated to be laid on the conceptual convex hull, which is determined from firstprinciples formation enthalpy of -phase. Formation enthalpies and entropies of four intermetallic compounds are then evaluated to satisfy experimentally observed phase equilibrium data from the literature thereafter (see Yan and Chang [2] and references therein).
φ
o dia Gφm = xCu o Gfcc Cu + xSi GSi + 1Gf
φ
φ
φ
(2)
where 1Gf = 1Hf − T 1Sf , represents the Gibbs energy of formation of the stoichiometric phases. 1Hf and 1Sf are enthalpy and entropy of formation, respectively. 1Hf of -Cu15 Si4 is obtained from first-principles and the others are inferred from that of -Cu15 Si4 .
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D. Shin et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 520–526
(a) Calculated enthalpy of mixing of the liquid phase in the Cu–Si system at 1773 K with experimental data [28–33]. Reference states for both elements are the liquid phase.
(b) Calculated enthalpy of mixing of the bcc phase at 298.15 K. Reference states for both elements are bcc phase.
(c) Calculated enthalpy of mixing of the fcc phase at 298.15 K. Reference states for both elements are fcc phase.
(d) Calculated enthalpy of mixing of the hcp phase at 298.15 K. Reference states for both elements are hcp phase.
Fig. 3. Calculated enthalpies of mixing of the solution phases in the Cu–Si system are given as solid lines with first-principles results in the present work. Open and closed symbols in (b), (c), and (d) are symmetry preserved and fully relaxed first-principles calculations of SQSs at 0 K, respectively. Dotted lines are from COST507 [1] and dashed lines are from Yan and Chang [2].
Fig. 2(b) shows the calculated entropy of formation for the Cu–Si system. Although any experimental measurements or theoretical calculations have never been reported to validate evaluated entropies of formation, the same trend could be found in a similar system, Ni–Si, which has been assessed by Du and Schuster [22]. In their work, the intermetallic compounds are predicted to have positive entropies of formation, much as they are in the present work. It is mentioned in their work that enthalpies of formation for the intermetallic compounds in the Ni–Si system were evaluated from quite reliable experimental measurements. Fig. 3 shows the calculated enthalpy of mixing for the solid solution phases, i.e. bcc, fcc, and hcp, and the liquid phase in the Cu–Si system. First-principles calculations of the SQSs in the present work and the calculated results from previous modelings [1,2] are shown together for comparison. It should be noted that none of the previous thermodynamic modelings used thermochemical data for solid solution phases to evaluate their parameters. However, both results agree quite satisfactorily with first-principles results of SQSs in the present work. All the fully relaxed calculations have been used in the thermodynamic parameters evaluation and symmetry preserved calculations are
shown in Fig. 3 only for comparison. In Fig. 3(b), the fully relaxed bcc SQS at 50% Si completely lost its bcc symmetry. Therefore, its energy was not considered during the optimization. However, as discussed in Section 3.3, the symmetry preserved calculations reproduce the overall mixing behavior of all solid solution phases quite satisfactorily. Another intriguing observation in Fig. 3 is that the tendency, and the absolute values, for the enthalpies of mixing obtained from SQSs in the Cu–Si system are close to each other in all phases except bcc, which exhibits a rather positive enthalpy of mixing in the Si-rich side. Conventional alloy theories, such as Hume-Rothery rules [23–25] and Darken–Gurry methods [26], can explain the similar mixing behavior of two elements in different structures. Both methods are not for estimating the enthalpy of mixing for binaries but for predicting the solubility limit empirically without considering the Gibbs energies of individual phases in a binary system. Furthermore, the existence of intermetallic compounds strongly affects a solubility limit. 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.
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Table 2 Thermodynamic parameters for the Cu–Si system (all in S.I. units) Phases
Sublattice model
Evaluated descriptions
Liquid
(Cu, Si)
0 1
Lfcc Cu,Si Lfcc Cu,Si 0 fcc LCu,Si 0 fcc LCu,Si 1 fcc LCu,Si
= −41 305 + 26.591T = −37 829 − 33.805T = −34 106 − 1.908T = −42 204 + 13.891T = −1102 − 18.178T
0
= −26485 + 15.151T = −85 212 − 11.583T = −21 740 + 3.9T = −40 000 − 4.39T = −100 = −26 447 + 10.216T = −47 275 − 8.517T
1 2 0 1 2
(Cu, Si)
Liq
= −35 111 + 8.311T = −39 688 + 14.275T = −49 937 + 29.790T = −31 810 + 18.008T = −38 764 + 12T = −52 431 + 27.457T = −29 427 + 14.775T
0
0
1
bcc
(Cu, Si)
Lbcc Cu,Si Lbcc Cu,Si 0 bcc LCu,Si 1 bcc LCu,Si 2 bcc LCu,Si 0 bcc LCu,Si 1 bcc LCu,Si 1
hcp
(Cu, Si)
0 1 0 1 2
hcp
LCu,Si = −30 194 + 7.381T
η-Cu19 Si6
(Cu)19 (Si)6
[1]
[2]
This work [1] [2] This work [1]
[2] This work
hcp
LCu,Si = −62 319 − 6.517T hcp
LCu,Si = −19 948 − 2.356T
[1]
hcp
LCu,Si = −23 800 − 1.97T hcp
LCu,Si = −20
hcp LCu,Si 1 hcp LCu,Si 0
Reference This work
LCu,Si = −49 747 + 15.454T
Liq LCu,Si Liq LCu,Si Liq LCu,Si Liq LCu,Si Liq LCu,Si Liq LCu,Si Liq LCu,Si
2
fcc
Liq
LCu,Si = −39729 + 4.278T
19 o 25 19 o 25 19 o 25
= −26 799 − 0.732T = −28 065 − 0.029T
Gfcc Cu + Gfcc Cu + Gfcc Cu
+
6 o 25 6 o 25 6 o 25
[2]
Gdia Si − 5065 − 4.508T
This work
Gdia Si + 3345 − 7.324T
[1]
+ 1368 − 7.198T
[2]
Gdia Si
-Cu15 Si4 -Cu4 Si
(Cu)15 (Si)4 (Cu)4 (Si)1
4 o dia Gfcc Cu + 19 GSi − 5142 − 3.909T −7984.87 + 171.70094T − 30.93528T ln T
This work [1]
-Cu15 Si4
(Cu)15 (Si)4
+0.002149728T 2 + 1.0267 × 10−7 T 3 + 77 316T −1 15 o fcc 4 o dia GCu + 19 GSi + 1241 − 6.720T 19
[2]
γ -Cu56 Si11
(Cu)56 (Si)11
δ -Cu33 Si7
(Cu)33 (Si)7
15 o 19
56 o 67 56 o 67 56 o 67
Gfcc Cu
33 o 40 33 o 40 33 o 40
Gfcc Cu
+
Gfcc Cu + Gfcc Cu +
+
Gfcc Cu + Gfcc Cu +
11 o 67 11 o 67 11 o 67
Gdia Si
7 o 40 7 o 40 7 o 40
Gdia Si
− 4014 − 3.978T
This work
Gdia Si + 1030 − 1.608T
[1]
Gdia Si + 1046 − 5.875T
[2]
− 2660 − 5.594T
This work
Gdia Si + 2830 − 7.025T
[1]
Gdia Si + 1972 − 6.966T
[2]
Gibbs energies for pure elements are from the SGTE pure element database[34].
The well-known first rule of Hume-Rothery states that the size difference of two elements must be less than 15%, and the second rule requires them to have similar electronegativities in order to exhibit extensive solid solubilities. Darken and Gurry [26] showed that Hume-Rothery’s first and second rules can be applied simultaneously to predict the formation of solid solutions. From the Darken–Gurry map shown in Fig. 4, large solubility can be expected if two elements are close to each other. The distances between Cu and Si is fairly short in the Darken–Gurry map when the coordination number is 12, and, therefore, it is expected that they are very likely to form solid solution phases. As a matter of fact, the Cu–Si system is very unusual to have all three representative solid solution phases, bcc, fcc, and hcp. However, the solubility ranges of the solid solution phases are not significantly wide due to the existence of intermetallic compounds in the Cu–Si system. The slightly different mixing behavior of bcc from those of fcc, hcp, and liquid in the Cu–Si system is attributed to different
coordination numbers. Recently, Gschneidner and Verkade [27] presented the semi-empirical 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 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 their coordination numbers are bigger than those of group IVB elements’ stable structure, diamond. Similarly, fcc and hcp are less directional than bcc since their coordination number is larger by 50%. Therefore, one would expect the enthalpy of mixing for bcc to behave differently from that of fcc and hcp, as is the case shown in Fig. 3. The calculated Cu–Si phase diagram is compared with experimental phase equilibrium data [10–14] in Fig. 5 and all evaluated parameters for the Cu–Si system are listed in Table 2.
526
D. Shin et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 520–526
results can work as constraints in the thermodynamic modeling to prevent having incorrect parameters of a phase. Acknowledgments This work is funded by the National Science Foundation (NSF) through Grant No. DMR-0205232. First-principles calculations were carried out on the LION clusters at the Pennsylvania State University supported in part by the NSF grants (DMR-9983532, DMR-0122638, and DMR-0205232) and in part by the Materials Simulation Center and the Graduate Education and Research Services at the Pennsylvania State University. References
Fig. 4. The electronegativity v s. the atomic radius for a coordination number of 12 (Darken–Gurry map)[27].
Fig. 5. Calculated phase diagram of the Cu–Si system with experimental data [10– 14] in the present work.
6. Conclusion The complete self-consistent thermodynamic description of the Cu–Si system has been obtained with first-principles calculations of solid phases and phase equilibrium data. The enthalpy of formation for -Cu15 Si4 from first-principles calculations clearly indicates that the formation enthalpies of the intermetallic compounds in the system, must be negative, which were previously evaluated as positive. The enthalpies of mixing for the three solid solution phases, bcc, fcc, and hcp, are also calculated from first-principles studies of Special Quasirandom Structures. It is shown that the challenge of scarce thermochemical data for solid phases, even for solid solution phases, can be overcome by incorporating first-principles energetics. Those first-principles
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